Research Article Multiobjective Optimization of Allocated...

Research ArticleMultiobjective Optimization of Allocated Exchange PortfolioModel and SolutionmdashA Case Study in Iran

Mostafa Ekhtiari

Department of Industrial Management Management and Accounting Shahid Beheshti University Tehran Iran

Correspondence should be addressed to Mostafa Ekhtiari m ektiariyahoocom

Received 21 September 2013 Accepted 13 November 2013 Published 30 January 2014

Academic Editors X-L Luo and Y Shi

Copyright copy 2014 Mostafa Ekhtiari This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a triobjective model for optimization of allocated exchange portfolio The objectives of this model areminimizing risk and investment initial cost (by adopting two synchronic policies of buying and selling assets) and maximizingreturn to optimize allocated portfolios (APs) In an AP an investor by considering previous investment experiences and marketconditions selects the within portfolio assets Then considering proposed model the assets proportion of AP is optimized for alimited time horizon In optimizing a multiobjective problem of an AP risk and return objectives are measured on the basis ofstandard deviation of assets dairy return and dairy return mean within AP assets respectively We present a set of interobjectivestrade-offs along with an analysis of IranMelli bank investment in an exchange AP usingWeightedGlobal Criterion (WGC)methodwith assumption 119901 = 1 2 andinfin to optimize the proposed model Results of WGCmodel (in all p = 1 2 andinfin) represent that USdollar exchange in comparison with other exchanges was rather the fewest exchange proportion in Iran Melli bank exchange APwhich this is consistent with Iran exchange investment policy of more concentration on other exchanges

1 Introduction

Markowitz [1] was the first one who quantified risk infinancial management He presented that finance decisionsshould be made on the basis of risk and return and finallyhe presented efficient frontier in finance decisions All pointson this line were optimal That is investor obtains minimumrisk and maximum return on a certain level of return andon a certain level of risk respectively [1] According to theMarkowitz approach risk and future returns of a portfolio arerandom variables which are controlled by the following twoparameters

(i) Portfolio return ismeasured by dairy return expectedvalue of assets

(ii) Portfolio risk is measured by standard deviation ofassets dairy return (see eg [2ndash4])

Markowitz mean-variance model was a quadratic plan-ning model by some lateral constraints [2] Roy [5] presentedan approach to determine the optimum level of risk andreturn He considered the implications of minimizing theupper bound of the chance of a dread event when the

information available about the joint probability distributionof future occurrences is confined to the first and secondmoments Sharpe [6] established portfolio scientific man-agement by his researches He introduced 120573 sensitivitycoefficient which indicates changes of stock return rate ascompared with changes of market return rate that is calledrelative risk Some of the latest works concerning portfolioselection problems are Vafaei Jahan and Akbarzadeh-T [7]Amiri et al [8] Zhang et al [9] and Li et al [10]

Importance of finance risk management significantlydeveloped from the 1970s As in that decade fall of exchangerate constancy system and oil birate crisis were seen While anumber of finance bankrupts such as the fall of internationalstock market in 1987 Mexican crisis in 1995 Asian crisis in1997 andOrange-County and Barings bank collapses in 1994the attitude to this tendency becomes strong These cases allshowed inability of available risk management tools financesystem elegance and consequences of possible finance crisis[11]

First let us have some economic definition of riskInvestment dictionary defines risk as investment potentialloss which is computable [12] Besides introducing risk as

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014 Article ID 708387 16 pageshttpdxdoiorg1011552014708387

2 Chinese Journal of Mathematics

a numerical index Markowitz also defines it as multicycli-cal standard deviation of a variable For example risk ofexchange rate during the years 2000 to 2008 is standarddeviation of exchange rate in these years [1]

Generally economic investors consider two differentkinds of portfolio investment decisions a number of themusing multiobjective economic models and analyzing themtry to select and allocate the best composition of assets forinvestment in a portfolio (see eg [13 14]) In this caseinvestor may have an optimal composition of allocated assetson which investment may not be possible because of marketconditions For example now foreign exchange investmentpolicy of Iran offers less concentration on exchanges likeAmerican dollar Another group of investors select portfolioassets using previous experiences and market conditionsand then by the use of multiobjective economic modelsand heuristic methods optimize allocated portfolio (AP)for a limited time horizon in the future Furthermore inthis kind of investment portfolio management decisions willbe often repeated at the end of the specified time horizon(see eg [15 16]) For instance banks are of economicagencies which considering past exchange trade-off extentuse AP optimization in a finance term for future investmentin an exchange portfolio From AP optimizing advantagesare making opportunity for increasing assets (buy policy) ordecreasing existent assets (sell policy)

Our purpose in this paper will be the latter kind ofinvestmentThepointwhich should be considered by investorin selecting the assets within an AP is the existence of balancein assets two by two overlapping So that if all of AP assets twoby two overlapping are positive in a moment of time it willmean that there is a high probability for obtaining investmentmore profit andmore return aswell But if all of AP assets twoby two overlapping are negative in a moment of time it willmean that there is a low probability for obtaining investmentmore profit and more return as well So in order to obtainproper tradeoffs of risk and return in the balance case wehave to have both negative and positive overlapping for twoby two of assets in an AP

Generally the process of an AP optimization is as followsfirst the investor introduces a certain level of hisher assetsThen using past information the analyst draws out requiredstatistic indices and performs optimization process by offer-ing amultiobjectivemodel and will provide a set of objectivesof different optimal levels for the investor Finally consideringlimited time horizon in future the investor will make thelast decision about increase or decrease of within AP assetslevel

Having various objectives is helpful to make betterdecisions in the future Most of investors are interested inhaving information like investment cost risk and returnto make better decisions According to economic theorieseveryone can have different compositions of risk and returnlevels on the basis of which risk and return trade-offsmake no difference for us Pareto frontier will be drawn byconnecting these points together Of course one does notpractically draw these graphs to make decision and performsthe decision making process rather by an ocular viewpointbased on recognition of ones own specifics

When investors accept higher level of risk they willobtain more return which is called risk premium Because ofcondition changes various people accept different levels ofrisk which is more on the basis of their received informationfrom market as their psychological and behavioral specificsat the time of decision making On the other hand all riskscould not be eliminated due to the fact of considering thestraight relation between risk and return the opportunitieswill be eliminated too So we have to pay attention to risk andreturn simultaneously Obviously acceptance of higher riskwill be accompanied by more return

On the other hand one of themost important goals whichinvestors always want to decrease is investment initial costPeople always want an investment in an economic field byhigher capital return and lower risk and investment initialcost Of course it is an idealistic approach it should beadded that investment in an APmay not be always associatedwith buying new assets and investment may even begin byselling present assets Anyway it is considerable for investorto inform about assets buying or selling extent by regard ofreturn and risk objectives So in this paper considering bankinvestment on an exchange AP we will model investmentinitial cost objective on the basis of an APrsquos specifics as wellas risk and return objectives

The risk we consider in this paper is exchange rate riskwhich is caused by change in exchange rate All companieswhich are out of political borders are dealing with countrieshaving different currencies that are exposed to exchangerate risk Exchange rate risk influences the organizationability to payback the foreign loans Also may cause theorganization not to be able to perform its commitmentsfor forward purchasing of goods from foreign markets Inthe other word changes of exchange rate influence goodsand capital market and may even have destructive effectsFinance institute working in exchange market should consultfor sufficient coverage against future fluctuations of exchangerate It is clear that banks and finance institutes will sustaingreat losses if they do not consider the optimal compositionof an exchange portfolio and each exchangersquos state in theinternational markets So computation of exchange rate riskby banks is effective in decreasing the loss caused by exchangerate fluctuations

Iran has experienced vast changes of exchange rate andits destructive effects during 20 years Most of industrialplans in Iran which were profitable on the basis of the timeexchange rate at the time of startup and economic studybecame bankrupt after decrease of Rials (formal currency ofIran) value because of dependence on imported raw materialand companies went bankrupt as well Of these industrieswe can name matting industry which became bankruptafter exchange rate changed from 1750 Rials to 8000 RialsAlso recently most of investors importers and banks havesustained losses because of extreme increase of Euro priceand fluctuations of other exchanges Thus government triesto compensate this loss by giving loan to investors throughwithdrawal of country exchange reserve account So it seemsnecessary for Iranian investors to know risk level for foreigninvestment so that knowing will prevent disadvantagescaused by irrelevant discount of country income Second

Chinese Journal of Mathematics 3

it seems that Iranian investors beside the return objectivedo not consider risk objective so much or they do notpay enough attention to it as an important objective forinvestment Whereas return and risk objectives should beassessed together And the third consumption of irrelevantcosts when the investor knows nothing about expected riskand return level of hisher investment this influences futuredecisions So it is always favored for investors to find asolution to decrease investment initial cost beside decreaseof risk and increase of future return

Our main motivations for presenting this paper is lackof a monorate exchange regime in Iran before 2002 and thetraditional viewpoint of Iranian investors which has oftenfailed In this paper using Markowitz mean-variance modeland adding a objective function of investment cost for anexchange AP which include five major exchanges presentin foreign investment portfolio of Iran Melli bank we opti-mize triobjective problem by the Weighted Global Criterion(WGC) method and consider interobjectives tradeoffs ofinvestment risk return and initial cost by making considerinter-objectives trade-offs of investment risk return andinitial cost by making changes in preference weights of theobjectives After evaluation of results based on the norms119901 = 1 2infin is presented a proper procedure to investor(bank) for making decision about investment in a one yeartime horizon

The paper continues as follows in Section 2 a triobjectivemodel by objectives of investment risk return and initial coston the basis of Markowitz mean-variance model is offeredWGCmethod along with a review of its literature for solutionof our proposal model is presented in Section 2 Also inorder to determine maximum expected loss of an AP ina future time horizon the Value-at-Risk (VaR) method isintroduced in this sectionNext in Section 3 we illustrate ourproposalmodel on an exchange APwhich includes fivemajorexchanges present in Iran Melli bank and analyze obtainedresults based on the norms 119901 = 1 2 and infin and finallySection 4 presents conclusions and final remarks

2 Problem Modelin

Markowitz [1] mean-variance model obtains optimal riskvalue for an explicit level of return by minimizing varianceof total within portfolio assets

Here we model our proposal triobjective model bymaking a change inMarkowitzmean-variancemodel where 119894(for 119894 = 1 2 119898) is number of existent assets in AP 119909

119894(for

119894 = 1 2 119898) is the decision variable of asset proportion119894th (for 119894 = 1 2 119898) in optimal AP and 119877

119894is daily

return random variable of asset 119894th (for 119894 = 1 2 119898) withnormally distributed that is computed as follows

119877119894119905= ln(

119901119894119905

119901119894(119905minus1)

) 119894 = 1 2 119898 (1)

where 119901119894119905is price of asset 119894th (for 119894 = 1 2 119898) in day 119905th of

understudy term and119877119894119905is logarithmic return of asset 119894th (for

119894 = 1 2 119898) in day 119905th of understudy term 119864(119877119894) is daily

return mean of asset 119894th (for 119894 = 1 2 119898) and Var(119877119894) is

daily return variance of asset 119894th (for 119894 = 1 2 119898) Also1205902

AP = sum119898

119894=1Var(119877

119894119909119894) and 119877AP = sum

119898

119894=1119864(119877119894119909119894) are return

variance and mean of all AP assets respectivelyIn an AP if119872

119894(existent)119873119894(existent) and 119901119894 are present valueof existent asset 119894th (for 119894 = 1 2 119898) number of existentasset 119894th (for 119894 = 1 2 119898) and price of asset 119894th (for 119894 =1 2 119898) in the last day of understudy term respectivelythen

119873119894(existent) =

119872119894(existent)

119901119894

119894 = 1 2 119898 (2)

Also if 119909119894(existent) is existent proportion of asset 119894th (for 119894 =

1 2 119898) of an AP in the last day of understudy term then

119909119894(existent) =

119873119894(existent)

sum119898

119894=1119873119894(existent)

119894 = 1 2 119898 (3)

where finally 119864(119877119894) times 119909119894(existent) is minimum aspiration level

of return belonging to asset 119894th (for 119894 = 1 2 119898) in an APin the last day of understudy term and sum5

119894=1119864(119877119894) times 119909119894(existent)

is minimum aspiration level of same AP in the last day ofunderstudy term for all assets as well

In this section we propose a new objective for Markowitzmodel which is called investment initial cost Purpose ofthis objective is to minimize investment cost This minimuminvestment initial cost means minimizing new assets buyingcost but includes AP assets selling subject as well Ourpurpose is decrease of new assets buying cost that maysometimes cause income earning from selling existent assetsThus in this paper we consider final results obtained fromwithin AP investment initial cost objective by two variables119862minus

AP (income variable caused by selling AP existent assets)and 119862+AP (cost variable of new assets buying)

Investor initiates the investment on the basis of last pricepresent for each asset sole So as the investor wants to knowrisk and return level of hisher investment before investmenton an AP in a new finance term heshe could be presenteda set of tradeoffs between investment risk return and initialcost objectives Because each asset price in understudy termchanges a little so we use least period demand method topredict future price for performing assets sell or buy policyso that assets future price in beginning days of time horizon ispredicted to be equal to their present price So in general wecan show AP cost objective function as a linear combinationof the number of assets which should be bought (119910

119894) and the

price of each asset in the last understudy day (119901119894) Results can

be seen as

119862AP =119898

sum

119894=1

119901119894119910119894 (4)

Considering situation of existent assets our purpose isbuying new assets for futureThus 119910

1198941is total number of asset

119894th (for 119894 = 1 2 119898) which we have at present 1199101198942is total

number of asset 119894th (for 119894 = 1 2 119898) which we will have atthe future and 119910

119894is number of asset 119894th (for 119894 = 1 2 119898)


which we must buy regardless of situation of existent assetsSo (4) can be also written as

119862AP =119898

sum

119894=1

119901119894(1199101198942minus 1199101198941) (5)

If 119910 = sum119898119894=1119873119894(existent) is the total number of existent assets

in AP then

1199101198942= 119910119909119894 119894 = 1 2 119898 (6)

Considering (5) and (6) we have

119862AP =119898

sum

119894=1

119901119894(119910119909119894minus 1199101198941) (7)

The following should be noted about (7)

(i) If sum119898119894=1119901119894119910119909119894gt sum119898

119894=11199011198941199101198941 then 119862+AP = 119862AP gt 0 that

is buy policy is offered for investment and optimalvalue of objective119862AP is considered as minimum costof new assets buying

(ii) If sum119898119894=1119901119894119910119909119894lt sum119898

119894=11199011198941199101198941 then 119862minusAP = 119862AP lt 0 that

is sell policy is offered for investment and |119862AP| isconsidered as maximum income obtained of sellingexistent assets

(iii) If 119910119909119894= 1199101198941 then 119862AP = 0 (for 119894 = 1 2 119898) that is

from cost point of view investment would be properby existent assets

It should be added that if we consider assets buy policy119862AP is buy initial cost objective and it is important thatwe minimize it Also if we consider assets sell policy 119862APis income objective obtained of selling the assets and it isimportant that we maximize it So as presupposition weconsider 119862AP as investment initial cost objective by policiesof selling or buying the assets So our proposed triobjectivemodel is problem (P1)

(P1) Opt (1205902

ΑP 119877AP 119862AP) (8)

st119898

sum

119894=1

119909119894= 1 (9)

119909119894ge 0 119894 = 1 2 119898 (10)

Problem (P1) is a constrained triobjective decision modelthat incorporates tradeoffs between competing objectives ofrisk return and cost for investment Equation (8) is theobjective vector to be optimized with respect to the factthat investment risk of existent assets in AP is wished to beminimized return obtained from investment onAP is wishedto bemaximized and initial cost of investment onAP assets iswished to be minimized Equation (9) is the same constraintof variance-covariance primary model which is presentedhere This constraint implies that sum of total proportionsof existent assets in AP will always be equal to one Also(10) guarantees which each asset proportion in optimal APbe non-negative

In (P1) minimizing AP daily return variance is used tominimize risk objective Besides because a portfoliorsquos returnis measured by assets daily return expected value so in (P1)return objective will be maximized by linear combination ofAP assets daily return mean [2]

It should be considered that solving (P1) does not yieldonly one optimal solution and yields a set of optimal non-dominated solutions which are on Pareto frontier instead Todescribe the concept of optimality in which we are interestedwe will introduce next a few definitions

Definition 1 Given two vectors 119909 119910 isin 119877119896 one may say that

119909 ge 119910 if 119909119894ge 119910119894for 119894 = 1 2 119896 and that 119909 dominates 119910

(denoted by 119909 ≻ 119910) if 119909 ge 119910 and 119909 = 119910Consider a biobjective optimization problem with three

different solutions 1 2 and 3 where solutions 1 and 2 are dis-played with vectors 119909 and 119910 respectively The ideal solutionis displayed with 4 Function 119865

1needs to be maximized and

1198652needs to be minimized (see Figure 1)

Comparing solutions 1 and 2 solution 1 is better thansolution 2 in terms of both objective functions So it can besaid that 119909 dominates 119910 and we display this with 119909 ≻ 119910

Definition 2 One may say that a vector of decision variables119909 isin 119878 sub 119877

119899 (119878 is the feasible space) is nondominated withrespect to 119878 if there does not exist another 1199091015840 isin 119878 such that119891(119909) ≻ 119891(119909

1015840)

In Figure 1 if solutions 1 and 3 are displayedwith vectors119909and 119911 respectively then comparing 1 and 3 we see 3 is betterthan 1 in terms of 119865

1 whereas 1 is better than 3 in terms of

1198652 where 119909 ≻ 119911 and 119911 ≻ 119909 So in here vectors 119909 and 119911 are

nondominated with respect to each other

Definition 3 One may say that a vector of decision variable119909lowastisin 119878 sub 119877

119899 is Pareto optimal if it is nondominated withrespect to 119878

Let suppose that 119909lowast notin 119878 be a solution such as 4 Inthis state the above assumption is violated because 119909lowast is adominated solution which dominates all other solutions So119909lowast can be a solution such as 1 or 3 which are nondominated

Definition 4 The Pareto optimal set 119875lowast is defined by

119875lowast= 119909 isin 119865 | 119909 is Pareto optimal (11)

Definition 5 The Pareto Frontier PFlowast is defined by

PFlowast = 119891 (119909) isin 119877119896 | 119909 isin 119875lowast (12)

21 The WGC Method Of the proper assessment methodswhen investor information are unavailable are methodsrelated to 119897

119901-norm family so that by change of objectives

importance weight there is no need for investorrsquos primaryinformation In such methods investor will not be disturbedbut analyst should be able to consider assumptions aboutinvestorrsquos preferences For incorporating weights in GC weuse approach (13) (for more details see [17])


3

1

4

2

F

F2

1

S

Figure 1 Illustration of feasible space and ideal solution for abiobjective problem with objectives maximize and minimize

119897119901-norm =

119870

sum

119896=1

119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119901

1119901

(13)

where 119909 = (1199091 1199092 119909

119898) The formulation in (13) is called

standard weighted global criterion formulation Minimizing(13) is sufficient for Pareto optimality as long as 119908

119896gt 0 (for

119896 = 1 2 119870) [17]For each Pareto optimal point 119909

119901 there exists a vector

119908 = (1199081 1199082 119908

119870) and a scalar 119901 such that 119909

119901is a solution

to (13) The value of 119901 determines to what extent a method isable to capture all of the Pareto optimal points (with changein vector119908) even when the feasible spacemay be nonconvexWith (13) using higher values for119901 increases the effectivenessof the method in providing the complete Pareto optimal set[18] However using a higher value for 119901 enables one to bettercapture all Pareto optimal points (with change in 119908) Theweighted min-max formulation which is a special case ofthe WGC approach with 119901 = infin has the following format([19 20] and [21])

(P2) min 119910

st 119910 ge 119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119896 = 1 2 119870

119892119897 (119909) le 119887119897 119897 = 1 2 119871

119910 ge 0

(14)

Using (P2) can provide the complete Pareto optimal setso that it provides a necessary condition for Pareto optimality[19]

In set of WGC methods the goal is to minimize theexistence objective functions deviation from amultiobjectivemodel related to an ideal solution Yu [20] called the idealpoint 119909lowast as a utopia point We optimize each objectivefunction separately to reach utopia point and for 119909 isin 119878It means that in this state ideal solution is obtained fromsolving 119870monoobjective problems as follows

(P3) optimize 119891119896 (119909) 119896 = 1 2 119870

st 119892119897 (119909) le 0 119897 = 1 2 119871

(15)

where utopia point coordinates are 1198911(119909lowast1) 1198912(119909lowast2)

119891119870(119909lowast119870) and 119909lowast119870 optimizes 119896th objective Meanwhile

119909119896

lowastis vector of nadir solution So we canminimize119870 problem

for each objective function in solution space (if objectivesmaximizing is supposed) to reach nadir solution

Considering approach (13) if all 119891119896(119909) are of maximizing

type then 119908119896shows weight of objective 119896th (for 119896 =

1 2 119870) with 0 lt 119908119896lt 1 Also 1 le 119901 le infin shows

indicating parameter of 119897119901-norm family Value 119901 indicates

emphasis degree on present deviations so that the biggerthis value is the more emphasis on biggest deviation willbe If 119901 = infin it means that the biggest present deviation isconsidered for optimizing Usually values 119901 = 1 2 and infinare used in computations Anyway value 119901 may depend oninvestors mental criteria Given values 119908

119896 solution obtained

from minimizing the approach (13) is known as a consistentsolution

So far WGC approach has been widely applied inengineering sciences (see eg [22]) There is no significantstudy performed about application WGC method to solveoptimization portfolio problems On the other hand consid-ering WGC method ability to represent Pareto optimal setit seems that there are no researches performed about usingthis method for optimizing the APs so far So another partof our motivations to present this paper is WGC methodrsquoseffectiveness in representing a complete set of Pareto optimalpoints in optimizing portfolio problems

Using approach (13) we formulate (P1) in the formof (P4)based on the WGC method

(P4) min 1199081(119885lowast1+ 1198651

119885lowast1 minus 1198851lowast

)

119901

+1199082(119885lowast2minus 1198652


)

119901

+ 1199083(119885lowast3+ 1198653


)

119901

1119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(16)

where 119908119896is relative importance weight of objective 119896th (for

119896 = 1 2 3) where 0 lt 119908119896lt 1 and 119908

1+ 1199082+ 1199083=

1 Also 119885lowast119896 (for 119896 = 1 2 3) is utopia value of objectivefunction 119896th (here maximum value of objective function 119896this in solution space) 119885119896

lowast(for 119896 = 1 2 3) is nadir value of

objective function 119896th (here minimum value of objectivefunction 119896th is in solution space) and 119865

119896(for 119896 = 1 2 3) is

the 119896th objective function 1198651is the risk objective function

1198652is return objective function and 119865

3is the initial cost

objective function The tradeoffs between objectives is doneby changing119908

119896values 119901 is parameter of final utility function

for which values 1 2 and infin are supposed in this paperConsidering (P4) interobjectives trade-offs general processis as follows first we suppose that investor wants importance


weight of risk objective be 09 So using WGC methodanalyst obtains a set of tradeoffs between return and costobjectives by assuming importance weight of risk objectiveto be constant heshe decreases importance weight of riskobjective by a descending manner and tradeoffs betweeninvestment return and cost objectives will be reregistered

22 The VaR Method One of the most popular techniquesto determine maximum expected loss of an asset or portfolioin a future time horizon and with a given explicit confidencelevel (VaR definition) is the VaRmethod Dowd et al [23] forcomputation of the VaR associated with normally distributedlog-returns in a long-term applied the following

VaRAP (119879) = 119872 minus119872cl

= 119872 minus exp (119877AP119879 + 120572cl120590APradic119879 + ln (119872))

(17)

Generally considering (17) VaRAP(119879) is VaR of total APfor time horizon understudy in the future 119879 days and 119872 istotal present value of AP assets So in here we have

119872 =

119898

sum

119894=1

119872119894(existent) (18)

Also 120590AP is standard deviation of AP and the VaRconfidence level is cl and we consider VaR over a horizon of119879 days119872cl is the (1 minus cl) percentile (or critical percentile) ofthe terminal value of the portfolio after a holding period of119879 days and 120572cl is the standard normal variate associated withour chosen confidence level (eg so 120572cl = minus1645 if we havea 95 confidence level see eg [24])

3 Case Study

In order to perform tradeoffs or future risk coverage ordiversify exchange reserves Iranian banks perform exchangebuying and selling One of these banks is Bank Melli Iran

which officially started its banking operation in 1928 Theinitial capital of this Iranian bank was about 20000000 RialsNowadays enjoying 85 years of experience and about 3200branches this bank as an important Iran economic andfinance agency has an important role in proving coun-tryrsquos enormous economic goals by absorbing communityrsquoswandering capitals and using them for production Alsofrom international viewpoint Bank Melli Iran with 16 activebranches enjoys distinguished position in rendering bankingservices The most important actions of Bank Melli Iran ininternational field include opening various deposit accountsperforming currency drafts affairs issuing currency under-writing opening confirming covering and conformingdocumentary credits and so forth

Here we consider an exchange AP including five mainexchanges in Iran Melli bank exchange investment portfolioThese five exchanges include US dollar England poundSwitzerland frank Euro and Japan 100 yen The point whichinvestor Melli bank considers after yielding the results is theproportion of US dollars Right now Iran foreign exchangeinvestment policy necessitates less concentration on thisexchange Understudy data include these five exchanges dailyrate from 25 March 2002 to 19 March 2012 This studied termis short because of lack of exchange monorate regime in Iranexchange policy in years before 2002

Here 1199091 1199092 1199093 1199094and 119909

5are exchanges proportion of

USdollar England pound Switzerland frank Euro and Japan100 yen of Melli bank total exchange AP respectively Table 1illustrates statistic indices obtained from these five exchangesdaily rates during the study term

Also variance-covariancematrix obtained from these fiveexchanges daily return during the study term is according toTable 2

Present value of Iran Melli bank exchange AP andminimum aspiration level of AP return in the last day ofstudy term (19 March 2012) along with other information arepresented in Table 3

By considering information of Table 3 we can rewrite(P4) in the form

(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901

+1199082(119885lowast2minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)


After simplifying and normalizing constraint coefficientsrelated to cost objective (by dividing above constraint

coefficients in the biggest mentioned constraint coefficient)we can rewrite (P5) in the form

(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)

Table 1 Illustration of statistic indices drawnout of daily rates of fiveexchanges dollar pound frank Euro and 100 yen from 25 March2002 to 19 March 2012

Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)

USA dollar 0000084973 0005978413England pound 0000272112 0000050975Switzerland frank 0000314821 0006777075Euro 0000349021 0000028221Japan 100 yen 0000170238 0000034704

where the utopia and nadir values of each objective functionare according to Table 4

Considering Table 4 in the best condition third objectivefunction is of 119862minusAP variable kind and offers assets sellingpolicy where normalized income is equal to 02948852 unitAlso and in the worst conditions it is of 119862+AP variablekind and offers assets buying policy where normalizedcost value (disregard to its mark) is equal to 02123636unit So considering (P6) and information of Table 4 wehave

(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)

To optimize (P7) we first suppose 119901 = 1 and optimizethe (P7) by changing objectives importance weight To solve(P7) objectives are given various weights in 99 iterations(9 11-fold set of iterations) All obtained results from solving(P7) are presented in Table 8 (in the Appendix (optimal value

of third objective function is considered in the form of minus1198653

in all figures and tables of the appendix for this paper Thepositive values and values which are specified by negativemark (disregard to their mark) are considered as incomesand buying costs resp)) by software Lingo 110 In Table 8


Table 2 Daily return variance-covariance matrix of five exchanges dollar pound frank Euro and 100 yen from 25 March 2002 to 19 March2012

Exchange USA dollar England pound Switzerland Frank Euro Japan 100 yenUSA dollar 000597841295 minus000000275182 000000333301 000000233057 000000336630England pound 000005097503 000001564795 000001375091 000000578037Switzerland frank 000677707505 000002699918 000001576213Euro 000002822086 000000969165Japan 100 yen 000003470445

Table 3 Present value price number existent proportion daily return mean and minimum aspiration level of return specifics of fiveexchanges USA dollar England pound Switzerland frank Euro and Japan 100 yen in Melli bank exchange AP in 19 March 2012

Exchange 119872119894 (existent) 119901

119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)

USA dollar 17000000000 8904 1909254 0005589454 0000084973 0000000475England pound 662000000000 17934 36913126 0108065349 0000272112 0000029406Switzerland frank 167000000000 8837 18897816 0055324469 0000314821 0000017417Euro 3930000000000 14111 278506130 0815343090 0000349021 0000284572Japan 100 yen 49000000000 9150 5355191 0015677638 0000170238 0000002669

Total 4825000000000 341581517 1 0000334539

two first columns show numbers of iterations in 9 11-foldset of iterations Three second columns indicate changes ofobjectives importance weight In five third columns the valueof optimal proportion of each exchange in exchange APconsidering the changes of objectives weights is shown andfinally in last three columns optimal values of each objectiveare shown in each iteration

31 Evaluating Pareto Optimal Points Specifics In order toanalyze Pareto optimal points in this section consideringoptimal results of each objective we examine Pareto optimalpoint set for obtained results and indicate that all obtainedresults are considered as Pareto optimal point set First letsintroduce some vector variables 119883119895lowast is optimal vector ofmodel variables in iteration 119895th (for 119895 = 1 2 119899) ofsolution (ie vector of optimal solution in iteration 119895th ofsolution) and 119865

119895lowast is vector of objectives optimal value initeration 119895th (for 119895 = 1 2 119899) of solution Also 119882119895lowast isvector of objectives importance weight in iteration 119895th (for119895 = 1 2 119899) of solution Table 8 presents a set of obtainedoptimal points based on WGC method It also should bementioned that all optimal values of third column are ofvariable 119862minusAP and finally sell policy of AP existent assets isoffered for future investment So the purpose is to maximizethe positive values of minus119865

3column For better understanding

Figure 2 shows Pareto optimal set obtained from solving (P7)along with utopia and nadir points

One of the most important specifics of Pareto optimalset is that all optimal points are nondominated Let usdefine being dominated to make clear the concept of beingnondominated

Definition 6 A solution 119909119894lowast is said to dominate the othersolution119883119895lowast if the following conditions are satisfied

(i) the solution 119909119894lowast is not worse than119883119895lowast in all objectivesor 119891119896(119909119894lowast) ⋫ 119891119896(119883119895lowast) for all 119896 = 1 2 119870

(ii) the solution 119909119894lowast is strictly better than 119883119895lowast in at leastone objective or 119891

119896(119909119894lowast) ⊲ 119891

119896(119883119895lowast) for at least one

119896 = 1 2 119870

We can say about the obtained results in Table 8 thatall solutions in each set of iterations is nondominated Forexample consider iterations 119895 = 7 and 119895 = 8 The results ofthese two iterations will be

1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)

Considering results of the two above iterations at risk09 importance weight and by increasing importance weightof return objective and decreasing investment importanceweight of cost objective by considering vectors1198837lowast and1198838lowastthere is any proportion for dollar pound and Japan 100 yen


exchanges in optimal AP and proportion of frank (Euro)exchange is decreasing (increasing) in each set of iterations

What is implied from values of vectors 1198657lowast and 1198658lowast is thatrisk objective has improved 00000003518 unit and the thirdobjective offers assets selling policy to decrease investmentinitial cost objective so that this normalized income in eachtwo iterations will be 00033819825 unit and 00022219703unit respectively In other words the extent of incomeresulting of selling the assets has become worse Also theresults indicate that return value in these two iterations hasimproved 00000001349 unit In this case it is said that riskobjective decreases by decrease of selling the assets in eachset of iterations and vice versa So considering Definition 6solutions of these two iterations are nondominated

Arrangement manner of Pareto optimal set relative toutopia and nadir points is shown in Figure 2 Pareto optimalset is established between two mentioned points so that itis more inclined toward utopia point and has the maximum

distance from nadir point Actually external points of solu-tion space which are close to utopia point and far from nadirpoint are introduced as Pareto optimal pointsThis somehowindicates that interobjectives tradeoffs are in a manner thatdistance between Pareto optimal space and utopia point willbeminimized and distance between Pareto optimal space andnadir point will be maximized

32 Making Changes in Value of Norm 119901 Because 119901 valuechanges are by investorrsquos discretion now we suppose thatinvestor considers value of norm 119901 = 2 andinfin We optimized(P7) by software Lingo 110 under condition 119901 = 2 andTable 9 (in The Appendix) shows all results in 99 iterationsAccording towhatwas said about119901 = 1 under this conditionPareto optimal space is between utopia and nadir points tooand tends to become closer to utopia point (see Figure 3)

Finally we optimize (P7) under condition 119901 = infin In thiscondition considering (P2) approach (P7) is aweightedmin-max model So (P7) can be rewritten in the form of (P8) asfollows

(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)

Table 4 Utopia and nadir values related to each one of theobjectives

Function Utopia Nadirminus1198651

minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600

Results of solving (P8) by software Lingo 110 in 99iterations are shown in Table 10 (in the Appendix) AlsoFigure 4 shows Pareto optimal set obtained from solving thismodel

Considering results obtained from values changes ofnorm 119901 it can be added that except nadir point and iterations119895 = 11 22 33 44 and 55 (from results of norm 119901 = infin) asset

Table 5 Results obtained for each objective considering norms of119901 = 1 2 andinfin and by assumption 119908

1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262

sell policy will be offered in other results obtained from threeexamined states

33 Results Evaluation The most important criterion forexamining obtained results is results conformity level withinvestorrsquos proposed goals As mentioned before considering


Table 6 Information obtained of WGC method results with assumption 119901 = 1 2 andinfin

Objective function 119865lowast

1119865lowast

2minus119865lowast

3

Objective name Risk Rate of return Income Cost119901 = 1

Mean 0000385422 0000275693 0141152048 mdashMin 0000028221 0000170333 0000806933 mdashMax 0006777075 0000349021 0294885220 mdash

119901 = 2


119901 = infin

Mean 0000789979 0000281186 0179850336 minus0002756123lowast

Min 0000027132 0000172041 0000094981 minus0000255917lowastlowast

Max 0003797903 0000348978 0286319160 minus0007842553lowastlowastlowast

Notes lowastMean of cost value obtained (disregard its negative mark)lowastlowastMinimum of cost value obtained (disregard its negative mark)lowastlowastlowastMaximum of cost value obtained (disregard its negative mark)

Table 7 Summary of Table 6 information

Objective 119901 = 1 119901 = 2 119901 = infin

Min Risk 0000028221 0000028248 0000027132Max Rate ofReturn 0000349021 0000348946 0000348978

Max Income 0294885220 0281424452 0286319160Min Cost mdash mdash 0000255917

Iran foreign exchange investment policy investor considersless concentration on US dollar For example the results ofTable 8 indicate that in each 11-fold set of iterations by having1199081constant and increasing 119908

2and decreasing 119908

3 we see

decrease of dollar and Japan 100 yen exchanges proportionand increase of Euro exchange proportion in each set ofiterations so that proportion of these exchanges is often zeroAlso there is no guarantee for investment on pound exchangeIt can be said about frank exchange that there is the firstincrease and then decrease trends in each set of iterations

Finally Tables 8 9 and 10 indicate that the average ofthe most exchange proportion in AP belongs to the Euroexchange followed by the Japan 100 yen frank dollar andpound exchanges respectively So considering all resultsobtained with assumption 119901 = 1 2 andinfin investor obtainshisher first goal

Figures 5 6 and 7 show arrangement of Pareto optimal ofall results of 119901 = 1 2 andinfin norms between two utopia andnadir points in three different bidimensional graphs Figure 5shows tradeoffs between two first and third objectives As itis seen in this graph increase of investment risk objectiveresults in increase of income objective obtained from assetssell and vice versa decrease of obtained income value is alongwith decrease of investment risk value Also Tables 8 9 and10 show these changes in each 11-fold set of two 119865

1and minus119865

3

columns results

02

4

00204

0

2

4

6

8

Utopiapoint Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3

Figure 2 Pareto optimal set obtained from solving (P7) withassumption 119901 = 1

Figure 6 shows tradeoffs between two second and thirdobjectives The objective is increase of investment returnvalue and increase of income value obtained from assets sellResults correctness can be seen in Figure 6 too

Also tradeoffs between two first and second objectivescan be examined in Figure 7 Because the purpose is decreaseof first objective and increase of second objective so thisgraph indicates that we will expect increase (or decrease)of investment return value by increase (or decrease) ofinvestment risk value

Now suppose that investor makes no difference betweenobjectives and wants analyst to reexamine the results fordifferent norms of 119901 = 1 2 andinfin considering the equalityof objectives importance So by assumption 119908

1= 1199082= 1199083

and1199081+1199082+1199083= 1 the objectives results will be according

to Table 5Complete specifications related toTable 5 information are

inserted in iteration 119895 = 100 of Tables 8 9 and 10 As itis clear in Table 5 third objective offers assets sell policy byassumption 119908

1= 1199082= 1199083 On the other hand under


02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3

Figure 4 Pareto optimal set obtained from solving (P7) withassumption 119901 = infin

this objectiveminimumrisk value andmaximum investmentreturn value are related to solution obtained in norm 119901 =

1 and maximum income value obtained from assets sell isrelated to solution obtained in norm 119901 = infin These valueshave been bolded in Table 5

In here we consider assessment objectives on the basisof mean minimum and maximum value indices in order toassess results of WGC method in three different approacheswith assumption 119901 = 1 2 andinfin

Third objective function optimal values in Table 6 are ofboth income and cost kind so that the objective is to increaseincome and to decrease cost Generally results of Table 6 canpresent different options of case selection to investor for finaldecision making

Table 7 which summarizes important information ofTable 6 indicates that minimum risk value of exchange APinvestment is in 119901 = infin and iteration 119895 = 11 Also maximumrate of expected return of investment is in norm 119901 = 1 andin one of two of the last iteration of each set of solution Andmaximum value of income obtained from assets sell can beexpected in norm 119901 = 1 and in iterations 119895 = 93 or 119895 = 94And finally minimum cost value of new assets buy for AP canbe considered in norm 119901 = infin and in iteration = 55 Thesevalues have been bolded in Table 7

Considering Table 6 and mean values of objectivesresults we can have another assessment so that in general if

0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3

Figure 5 Pareto optimal set arrangement considering two first andthird objectives

0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3

Figure 6 Pareto optimal set arrangement considering two secondand third objectives

investor gives priority to risk objective heshe will be offeredto use obtained results in norm 119901 = 1 If objective of returnis given priority using norm 119901 = 2 is offered and if the firstpriority is given to investment cost objective (by adopting twosynchronic policies of buying and selling assets) the results ofnorm 119901 = infin will be offered

Considering the results shown in Figures 2 3 and 4 it canbe seen that that increase of p-norm increases effectivenessdegree of the WGC model for finding more number ofsolutions in Pareto frontier of the triobjective problem Sothat increasing p-norm we can better track down the optimalsolutions in Pareto frontier and diversity of the obtainedresults can help investor to make a better decision

34 Risk Acceptance Levels and Computing VaR of InvestmentConsidering obtained results we now suppose that investorselects risk as hisher main objective So in this case investordecides which level of obtained values will be chosen for finalselection By considering importance weights of objectiveswe say the following

(i) In interval 01 le 1199081le 03 risk acceptance level is low

and investor in case of selecting is not a risky person


Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof

columnminus119865lowast 3areincom

e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith

assumption119901=infin

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum

nminus119865lowast 3arec

ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1

Figure 7 Pareto optimal set arrangement considering two first andsecond objectives

(ii) In interval 04 le 1199081le 06 risk acceptance level

is mean and investor in case of selecting is a rathercautious person

(iii) In interval 07 le 1199081le 09 risk acceptance level is high

and investor in case of selecting is a risky person

Now let us suppose that considering Table 10 resultsinvestor selects iteration 119895 = 94 as hisher optimal solutionBecause of thinking important of risk objective heshe asksanalyst about VaR of exchange AP for one-year future termby 250 working days For this vectors 11988394lowast and 11986594lowast are asfollows

11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)

By considering Table 3 and (17) and (18) and vector 11986594lowastwe have VaRAP(119879 = 250 days) = 377377351262862 Rials sothat maximum loss during future 250 days and in confidencelevel of 95 will not exceed 377377351262862 Rials

4 Conclusions

In the beginning of this paper by defining risk we indicatedspecifics of an AP As mentioned before an AP is a portfoliothat investor considering market conditions and examiningexperiences and making balance in assets two by two over-lapping selects assets for investment It is also said that APoptimization would usually be repeated for a finance termwith limited time horizon Then we proposed a triobjectivemodel for synchronic optimization of investment risk andreturn and initial cost on the basis of Markowitz mean-variance model and indicated our motivations for offeringthis model Then results obtained from these interobjectivestradeoffs were analyzed for an AP including five majorexchanges present in Iran Melli bank investment portfolioThe WGC method (with assumption 119901 = 1 2 and infin) wasused to present results of interobjectives tradeoffs

Then trade-off manner of obtained results under norms119901 = 1 2 and infin were examined on the basis of three bidi-mensional graphs It was also seen that third objective offeredincome obtained from asset sell policy to investor on thebasis of objectives importance weight equality assumptionAnd finally the best value of each objective was identified andexamined on the basis of results of all three different normsof 119901

Then risk acceptance different levels were indicated forobtained results and for one of the levels considering theconcept of the VaR method we computed VaR of investmentduring a limited time horizon 250 days After examiningobtained results it was seen that except nadir point anditerations 119895 = 11 22 33 44 and 55 of Table 10 which offernew asset buying policy all obtained results offered AP assetselling policy in three examined norms Also it is importantabout the case study results that in three examined normsin comparison with other exchanges US dollar exchangeproportion was rather the fewest exchange proportion inIran Melli bank exchange AP in all norms 119901 = 1 2 andinfin So these results could be considered as a proper trendfor decision making of Iran exchange investment policybased on more concentration on other exchanges and lessconcentration on exchanges like US dollar

In this paper a proposal model was introduced consid-ering investment initial cost objective (by two asset sellingor buying policy) Of our proposal model advantages wereto adopt synchronic policy of AP assets selling or buyingand consistency of the proposal model with investor goalsAlso because there were a lot of numbers of existent assets inunderstudy AP proposal model adopted asset selling policyin all iterations but some iteration Adopting asset selling orbuying policy by the model would be sensitive to the numberof AP existent assets

Appendix

See Tables 8 9 and 10

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H M Markowitz ldquoPortfolio selectionrdquo The Journal of Financevol 7 no 1 pp 77ndash91 1952

[2] S Benati and R Rizzi ldquoA mixed integer linear programmingformulation of the optimal meanvalue-at-risk portfolio prob-lemrdquo European Journal of Operational Research vol 176 no 1pp 423ndash434 2007

[3] A Prekopa Stochastic Programming Kluwer Academic Lon-don UK 1995

[4] R E Steuer Y Qi and M Hirschberger ldquoMultiple objectives inportfolio selectionrdquo Journal of Financial DecisionMaking vol 1no 1 pp 5ndash20 2005

[5] A D Roy ldquoSafety first and the holding of assetsrdquo Econometricavol 20 no 3 pp 431ndash449 1952


[6] W Sharpe ldquoCapital asset prices a theory of market equilibriumunder conditions of riskrdquo The Journal of Finance vol 19 no 3pp 425ndash442 1964

[7] M Vafaei Jahan and M-R Akbarzadeh-T ldquoExternal optimiza-tion vs learning automata strategies for spin selection inportfolio selection problemsrdquo Applied Soft Computing vol 12no 10 pp 3276ndash3284 2012

[8] M Amiri M Ekhtiari and M Yazdani ldquoNadir compromiseprogramming a model for optimization of multi-objectiveportfolio problemrdquoExpert SystemswithApplications vol 38 no6 pp 7222ndash7226 2011

[9] W Zhang Y Zhang X Yang and W Xu ldquoA class of on-lineportfolio selection algorithms based on linear learningrdquoAppliedMathematics and Computation vol 218 no 24 pp 11832ndash118412012

[10] X Li B Shou and Z Qin ldquoAn expected regret minimizationportfolio selection modelrdquo European Journal of OperationalResearch vol 218 no 2 pp 484ndash492 2012

[11] K Tolikas A Koulakiotis and R A Brown ldquoExtreme riskand value-at-risk in the German stock marketrdquo The EuropeanJournal of Finance vol 13 no 4 pp 373ndash395 2007

[12] S S Hildreth The Dictionary of Investment Terms DearbonFinancial Chicago Ill USA 1989

[13] S Arnone A Loraschi and A Tettamanzi ldquoA genetic approachto portfolio selectionrdquo Neural Network World vol 3 no 6 pp597ndash604 1993

[14] GVedarajan L C Chan andD EGoldberg ldquoInvestment port-folio optimization using genetic algorithmsrdquo in Late BreakingPapers at the Genetic Programming Conference J R Koza Edpp 255ndash263 Stanford University Stanford Calif USA 1997

[15] W Briec K Kerstens andO Jokung ldquoMean-variance-skewnessportfolio performance gauging a general shortage function anddual approachrdquoManagement Science vol 53 no 1 pp 135ndash1492007

[16] H P Sharma D Ghosh and D K Sharma ldquoCredit unionportfoliomanagement an application of goal interval program-mingrdquo Academy of Banking Studies Journal vol 6 no 1-2 pp39ndash60 2007

[17] V Chankong and Y Y HaimesMultiobjectiva Decision MakingTheory andMethodology Elsevier Science New York NY USA1983

[18] A Messac and A Ismail-Yahaya ldquoRequired relationshipbetween objective function and pareto frontier orders practicalimplicationsrdquo The American Institute of Aeronautics and Astro-nautics Journal vol 39 no 11 pp 2168ndash2174 2001

[19] K Miettinen Nonlinear Multi-Objective Optimization KluwerAcademic Boston Mass USA 1999

[20] P L Yu ldquoA class of solutions for group decision problemsrdquoManagement Science vol 19 no 8 pp 936ndash946 1973

[21] M Zeleny Multiple Criteria Decision Making Mc Graw HillNew York NY USA 1982

[22] Y L Chen ldquoWeighted-norm approach for multiobjective VArplanningrdquo IEE Proceedings vol 145 no 4 pp 369ndash374 1998

[23] K DowdD Blake andA Cairns ldquoLong-term value at riskrdquoTheJournal of Risk Finance vol 5 no 2 pp 52ndash57 2004

[24] KDowdMeasuringMarket Risk JohnWileyampSonsNewYorkNY USA 2nd edition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a numerical index Markowitz also defines it as multicycli-cal standard deviation of a variable For example risk ofexchange rate during the years 2000 to 2008 is standarddeviation of exchange rate in these years [1]

Generally economic investors consider two differentkinds of portfolio investment decisions a number of themusing multiobjective economic models and analyzing themtry to select and allocate the best composition of assets forinvestment in a portfolio (see eg [13 14]) In this caseinvestor may have an optimal composition of allocated assetson which investment may not be possible because of marketconditions For example now foreign exchange investmentpolicy of Iran offers less concentration on exchanges likeAmerican dollar Another group of investors select portfolioassets using previous experiences and market conditionsand then by the use of multiobjective economic modelsand heuristic methods optimize allocated portfolio (AP)for a limited time horizon in the future Furthermore inthis kind of investment portfolio management decisions willbe often repeated at the end of the specified time horizon(see eg [15 16]) For instance banks are of economicagencies which considering past exchange trade-off extentuse AP optimization in a finance term for future investmentin an exchange portfolio From AP optimizing advantagesare making opportunity for increasing assets (buy policy) ordecreasing existent assets (sell policy)

Our purpose in this paper will be the latter kind ofinvestmentThepointwhich should be considered by investorin selecting the assets within an AP is the existence of balancein assets two by two overlapping So that if all of AP assets twoby two overlapping are positive in a moment of time it willmean that there is a high probability for obtaining investmentmore profit andmore return aswell But if all of AP assets twoby two overlapping are negative in a moment of time it willmean that there is a low probability for obtaining investmentmore profit and more return as well So in order to obtainproper tradeoffs of risk and return in the balance case wehave to have both negative and positive overlapping for twoby two of assets in an AP

Generally the process of an AP optimization is as followsfirst the investor introduces a certain level of hisher assetsThen using past information the analyst draws out requiredstatistic indices and performs optimization process by offer-ing amultiobjectivemodel and will provide a set of objectivesof different optimal levels for the investor Finally consideringlimited time horizon in future the investor will make thelast decision about increase or decrease of within AP assetslevel

Having various objectives is helpful to make betterdecisions in the future Most of investors are interested inhaving information like investment cost risk and returnto make better decisions According to economic theorieseveryone can have different compositions of risk and returnlevels on the basis of which risk and return trade-offsmake no difference for us Pareto frontier will be drawn byconnecting these points together Of course one does notpractically draw these graphs to make decision and performsthe decision making process rather by an ocular viewpointbased on recognition of ones own specifics

When investors accept higher level of risk they willobtain more return which is called risk premium Because ofcondition changes various people accept different levels ofrisk which is more on the basis of their received informationfrom market as their psychological and behavioral specificsat the time of decision making On the other hand all riskscould not be eliminated due to the fact of considering thestraight relation between risk and return the opportunitieswill be eliminated too So we have to pay attention to risk andreturn simultaneously Obviously acceptance of higher riskwill be accompanied by more return

On the other hand one of themost important goals whichinvestors always want to decrease is investment initial costPeople always want an investment in an economic field byhigher capital return and lower risk and investment initialcost Of course it is an idealistic approach it should beadded that investment in an APmay not be always associatedwith buying new assets and investment may even begin byselling present assets Anyway it is considerable for investorto inform about assets buying or selling extent by regard ofreturn and risk objectives So in this paper considering bankinvestment on an exchange AP we will model investmentinitial cost objective on the basis of an APrsquos specifics as wellas risk and return objectives

The risk we consider in this paper is exchange rate riskwhich is caused by change in exchange rate All companieswhich are out of political borders are dealing with countrieshaving different currencies that are exposed to exchangerate risk Exchange rate risk influences the organizationability to payback the foreign loans Also may cause theorganization not to be able to perform its commitmentsfor forward purchasing of goods from foreign markets Inthe other word changes of exchange rate influence goodsand capital market and may even have destructive effectsFinance institute working in exchange market should consultfor sufficient coverage against future fluctuations of exchangerate It is clear that banks and finance institutes will sustaingreat losses if they do not consider the optimal compositionof an exchange portfolio and each exchangersquos state in theinternational markets So computation of exchange rate riskby banks is effective in decreasing the loss caused by exchangerate fluctuations

Iran has experienced vast changes of exchange rate andits destructive effects during 20 years Most of industrialplans in Iran which were profitable on the basis of the timeexchange rate at the time of startup and economic studybecame bankrupt after decrease of Rials (formal currency ofIran) value because of dependence on imported raw materialand companies went bankrupt as well Of these industrieswe can name matting industry which became bankruptafter exchange rate changed from 1750 Rials to 8000 RialsAlso recently most of investors importers and banks havesustained losses because of extreme increase of Euro priceand fluctuations of other exchanges Thus government triesto compensate this loss by giving loan to investors throughwithdrawal of country exchange reserve account So it seemsnecessary for Iranian investors to know risk level for foreigninvestment so that knowing will prevent disadvantagescaused by irrelevant discount of country income Second





2 Problem Modelin



119894(for


119894is daily


119877119894119905= ln(

119901119894119905

119901119894(119905minus1)

) 119894 = 1 2 119898 (1)






AP = sum119898

119894=1Var(119877

119894119909119894) and 119877AP = sum

119898

119894=1119864(119877119894119909119894) are return



119873119894(existent) =

119872119894(existent)

119901119894

119894 = 1 2 119898 (2)



119909119894(existent) =

119873119894(existent)

sum119898

119894=1119873119894(existent)

119894 = 1 2 119898 (3)



119894=1119864(119877119894) times 119909119894(existent)





119894) and the


be seen as

119862AP =119898

sum

119894=1

119901119894119910119894 (4)








119862AP =119898

sum

119894=1

119901119894(1199101198942minus 1199101198941) (5)


in AP then

1199101198942= 119910119909119894 119894 = 1 2 119898 (6)


119862AP =119898

sum

119894=1

119901119894(119910119909119894minus 1199101198941) (7)


(i) If sum119898119894=1119901119894119910119909119894gt sum119898

119894=11199011198941199101198941 then 119862+AP = 119862AP gt 0 that


(ii) If sum119898119894=1119901119894119910119909119894lt sum119898






(P1) Opt (1205902

ΑP 119877AP 119862AP) (8)

st119898

sum

119894=1

119909119894= 1 (9)

119909119894ge 0 119894 = 1 2 119898 (10)













1015840)
















3

1

4

2

F

F2

1

S


119897119901-norm =

119870

sum

119896=1

119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119901

1119901

(13)

where 119909 = (1199091 1199092 119909



119896gt 0 (for



119908 = (1199081 1199082 119908


119901is a solution


(P2) min 119910


119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119896 = 1 2 119870

119892119897 (119909) le 119887119897 119897 = 1 2 119871

119910 ge 0

(14)



(P3) optimize 119891119896 (119909) 119896 = 1 2 119870

st 119892119897 (119909) le 0 119897 = 1 2 119871

(15)



119909119896












(P4) min 1199081(119885lowast1+ 1198651


)

119901

+1199082(119885lowast2minus 1198652


)

119901

+ 1199083(119885lowast3+ 1198653


)

119901

1119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(16)


119896 = 1 2 3) where 0 lt 119908119896lt 1 and 119908

1+ 1199082+ 1199083=




119896(for 119896 = 1 2 3) is










VaRAP (119879) = 119872 minus119872cl


(17)


119872 =

119898

sum

119894=1

119872119894(existent) (18)


3 Case Study




Here 1199091 1199092 1199093 1199094and 119909






(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)




(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2 Problem Modelin



119894(for


119894is daily


119877119894119905= ln(

119901119894119905

119901119894(119905minus1)

) 119894 = 1 2 119898 (1)






AP = sum119898

119894=1Var(119877

119894119909119894) and 119877AP = sum

119898

119894=1119864(119877119894119909119894) are return



119873119894(existent) =

119872119894(existent)

119901119894

119894 = 1 2 119898 (2)



119909119894(existent) =

119873119894(existent)

sum119898

119894=1119873119894(existent)

119894 = 1 2 119898 (3)



119894=1119864(119877119894) times 119909119894(existent)





119894) and the


be seen as

119862AP =119898

sum

119894=1

119901119894119910119894 (4)








119862AP =119898

sum

119894=1

119901119894(1199101198942minus 1199101198941) (5)


in AP then

1199101198942= 119910119909119894 119894 = 1 2 119898 (6)


119862AP =119898

sum

119894=1

119901119894(119910119909119894minus 1199101198941) (7)


(i) If sum119898119894=1119901119894119910119909119894gt sum119898

119894=11199011198941199101198941 then 119862+AP = 119862AP gt 0 that


(ii) If sum119898119894=1119901119894119910119909119894lt sum119898






(P1) Opt (1205902

ΑP 119877AP 119862AP) (8)

st119898

sum

119894=1

119909119894= 1 (9)

119909119894ge 0 119894 = 1 2 119898 (10)













1015840)
















3

1

4

2

F

F2

1

S


119897119901-norm =

119870

sum

119896=1

119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119901

1119901

(13)

where 119909 = (1199091 1199092 119909



119896gt 0 (for



119908 = (1199081 1199082 119908


119901is a solution


(P2) min 119910


119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119896 = 1 2 119870

119892119897 (119909) le 119887119897 119897 = 1 2 119871

119910 ge 0

(14)



(P3) optimize 119891119896 (119909) 119896 = 1 2 119870

st 119892119897 (119909) le 0 119897 = 1 2 119871

(15)



119909119896












(P4) min 1199081(119885lowast1+ 1198651


)

119901

+1199082(119885lowast2minus 1198652


)

119901

+ 1199083(119885lowast3+ 1198653


)

119901

1119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(16)


119896 = 1 2 3) where 0 lt 119908119896lt 1 and 119908

1+ 1199082+ 1199083=




119896(for 119896 = 1 2 3) is










VaRAP (119879) = 119872 minus119872cl


(17)


119872 =

119898

sum

119894=1

119872119894(existent) (18)


3 Case Study




Here 1199091 1199092 1199093 1199094and 119909






(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)




(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862AP =119898

sum

119894=1

119901119894(1199101198942minus 1199101198941) (5)


in AP then

1199101198942= 119910119909119894 119894 = 1 2 119898 (6)


119862AP =119898

sum

119894=1

119901119894(119910119909119894minus 1199101198941) (7)


(i) If sum119898119894=1119901119894119910119909119894gt sum119898

119894=11199011198941199101198941 then 119862+AP = 119862AP gt 0 that


(ii) If sum119898119894=1119901119894119910119909119894lt sum119898






(P1) Opt (1205902

ΑP 119877AP 119862AP) (8)

st119898

sum

119894=1

119909119894= 1 (9)

119909119894ge 0 119894 = 1 2 119898 (10)













1015840)
















3

1

4

2

F

F2

1

S


119897119901-norm =

119870

sum

119896=1

119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119901

1119901

(13)

where 119909 = (1199091 1199092 119909



119896gt 0 (for



119908 = (1199081 1199082 119908


119901is a solution


(P2) min 119910


119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119896 = 1 2 119870

119892119897 (119909) le 119887119897 119897 = 1 2 119871

119910 ge 0

(14)



(P3) optimize 119891119896 (119909) 119896 = 1 2 119870

st 119892119897 (119909) le 0 119897 = 1 2 119871

(15)



119909119896












(P4) min 1199081(119885lowast1+ 1198651


)

119901

+1199082(119885lowast2minus 1198652


)

119901

+ 1199083(119885lowast3+ 1198653


)

119901

1119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(16)


119896 = 1 2 3) where 0 lt 119908119896lt 1 and 119908

1+ 1199082+ 1199083=




119896(for 119896 = 1 2 3) is










VaRAP (119879) = 119872 minus119872cl


(17)


119872 =

119898

sum

119894=1

119872119894(existent) (18)


3 Case Study




Here 1199091 1199092 1199093 1199094and 119909






(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)




(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3

1

4

2

F

F2

1

S


119897119901-norm =

119870

sum

119896=1

119908119896(119891119896(119909lowast119896) minus 119891119896 (119909)

119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119901

1119901

(13)

where 119909 = (1199091 1199092 119909



119896gt 0 (for



119908 = (1199081 1199082 119908


119901is a solution


(P2) min 119910


119891119896(119909lowast119896) minus 119891

119896(119909119896lowast))

119896 = 1 2 119870

119892119897 (119909) le 119887119897 119897 = 1 2 119871

119910 ge 0

(14)



(P3) optimize 119891119896 (119909) 119896 = 1 2 119870

st 119892119897 (119909) le 0 119897 = 1 2 119871

(15)



119909119896












(P4) min 1199081(119885lowast1+ 1198651


)

119901

+1199082(119885lowast2minus 1198652


)

119901

+ 1199083(119885lowast3+ 1198653


)

119901

1119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(16)


119896 = 1 2 3) where 0 lt 119908119896lt 1 and 119908

1+ 1199082+ 1199083=




119896(for 119896 = 1 2 3) is










VaRAP (119879) = 119872 minus119872cl


(17)


119872 =

119898

sum

119894=1

119872119894(existent) (18)


3 Case Study




Here 1199091 1199092 1199093 1199094and 119909






(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)




(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












VaRAP (119879) = 119872 minus119872cl


(17)


119872 =

119898

sum

119894=1

119872119894(existent) (18)


3 Case Study




Here 1199091 1199092 1199093 1199094and 119909






(P5) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(

119885lowast3+ (8904 (341581517119909

1minus 1909254) + 17934 (341581517119909

2minus 36913126) + 8837 (341581517119909

3minus 18897816)

+14111 (3415815171199094minus 278506130) + 9150 (341581517119909

5minus 5355191))


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(19)




(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(P6) min 1199081

((

(

119885lowast1+ (0005978413119909

2

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4+ 347045119864 minus 05119909

2

5minus 550364119864

minus0611990911199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094+ 673259119864 minus 06119909

11199095+ 312959119864 minus 05119909

21199093+ 275018119864

minus0511990921199094+ 115607119864 minus 05119909

21199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)


))

)

119901


1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)


)

119901

+1199083(119885lowast3+ (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)


)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(20)


Exchange (for 119894 = 1 2 119898) 119864(119877119894) Var(119877

119894)




(P7) min 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119901

+1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119901

+1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119901

st119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

(21)









119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894119873119894 (existent) 119909

119894 (existent) 119864(119877119894) 119864(119877

119894) times 119909119894 (existent)


Total 4825000000000 341581517 1 0000334539










119896(119909119894lowast) ⊲ 119891


119896 = 1 2 119870


1198827lowast= (09 006 004)

1198837lowast= (0 0 0008756436 0991243600 0)

1198657lowast= (00000287172 00003487215 00033819825)

1198828lowast= (09 007 003)

1198838lowast= (0 0 0004811658 0995188300 0)

1198658lowast= (00000283654 00003488564 00022219703)

(22)









(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(P8) min 119910

st 119910 ge 1199081

((((((

(

minus00000182639 + (00059784131199092

1+ 50975119864 minus 05119909

2

2+ 0006777075119909

2

3+ 282209119864 minus 05119909

2

4

+347045119864 minus 051199092

5minus 550364119864 minus 06119909

11199092+ 666602119864 minus 06119909

11199093+ 466114119864 minus 06119909

11199094

+673259119864 minus 0611990911199095+ 312959119864 minus 05119909

21199093+ 275018119864 minus 05119909

21199094+ 115607119864

minus0511990921199095+ 539984119864 minus 05119909

31199094+ 315243119864 minus 05119909

31199095+ 193833119864 minus 05119909

41199095)

minus00000182639 + 0006777075

))))))

)

119910 ge 1199082(0000349021 minus (849735119864 minus 05119909

1+ 0000272112119909

2+ 0000314821119909

3+ 0000349021119909

4+ 0000170238119909

5)

0000349021 minus 00000849735)

119910 ge 1199083(02948852 + (0496487119119909

1+ 1199092+ 0492751199119909

3+ 0786829486119909

4+ 0510204082119909

5minus 0787636419)

02948852 + 02123636)

119898

sum

119894=1

119909119894= 1

119909119894ge 0 119894 = 1 2 119898

119910 ge 0

(23)



minus00000182639 minus00067770751198652

000034902100 00000849730minus1198653

029488520000 minus0212363600




1= 1199082= 1199083

Objective 119901 = 1 119901 = 2 119901 = infin

Min (1198651) 0000371116 0001022573 0001388164

Max (1198652) 0000341307 0000297773 0000295503

Max (minus1198653) 0067135173 0172916129 0192074262






1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1119865lowast

2minus119865lowast

3



119901 = 2


119901 = infin






Objective 119901 = 1 119901 = 2 119901 = infin





3 we see




1and minus119865

3

columns results

02

4

00204

0

2

4

6

8


Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3





1= 1199082= 1199083






02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02

4

00204

0

2

4

6

8

Utopiapoint

Nadir point

Pareto optimal set

minus04minus02

times10minus3

times10minus4

F1

F2

minusF3


0 1 2 3 40

050

2

4

6

8

Utopiapoint

Pareto optimal set

Nadir point

minus05

times10minus3

times10minus4

F1

F2minusF3








0 2 4 6 8

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus3

F1

minusF3


0 1 2 3 4

0

02

04 Utopia point

Nadir point

Pareto optimalset

minus04

minus02

times10minus4

F2

minusF3








Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table8Re

sults

ofWGCmetho

dwith

assumption119901=1

Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

00001

00999

0006871283

00004708435

00988420300

00000345306

00001703329

02776087571

209

001

009

0004696536

0000

7527940

00987775500

00000346430

00001709260

02776281561

309

002

008

0002499822

00010375920

00987124300

00000349232

00001715250

02776476956

409

003

007

0000303108

00013223910

00986473000

00000353709

00001721241

02776672811

509

004

006

00

0015664

130

0075280620

0909055300

00000323410

00001859617

02568811641

609

005

005

00

0012701230

0987298800

000000292790

00003485866

000

45420655

709

006

004

00

0008756436

099124360

00

000

00287172

000

03487215

00033819825

809

007

003

00

00048116

580995188300

000000283654

00003488564

00022219703

909

008

002

00

000

0866

845

0999133200

0000

00282239

000

03489914

00010618182

1009

009

001

00

01

0000

00282209

000

03490210

000

08069331

1109

00999

000

010

00

10

000

00282209

000

03490210

000

08069331

212

08

00001

01999

0008974214

00007087938

00983937800

00000347003

00001704976

02776791658

1308

002

018

00040

56237

00013463850

00982479900

00000352702

00001718388

02777229662

1408

004

016

00

0019869360

00980130600

00000366285

0000173110

802777791354

1508

006

014

00

0026263540

00973736500

00000383869

00001740353

02778906914

1608

008

012

00

0032657720

00967342300

000

0040

6986

000

01749598

02780022985

1708

01

01

00

0028351590

097164840

00

00000335784

00003480514

0009144

5280

1808

012

008

00

0019475800

0980524200

000000307341

00003483549

00065343430

1908

014

006

00

00106

00010

0989400000

000000289536

00003486585

00039241580

2008

016

004

00

0001724232

0998275800

0000

00282368

000

03489620

00013139671

2108

018

002

00

01

0000

00282209

000

03490210

000

08069331

2208

01999

000

010

00

10

000

00282209

000

03490210

000

08069331

323

07

00001

02999

0011678020

00010147340

00978174600

00000351099

00001707094

027776964

5124

07

003

027

0003233185

00021095760

00975671100

00000367854

00001730124

0277844

8457

2507

006

024

00

0032066230

00967933800

000

0040

4614

000

01748742

02779919702

86

02

064

016

00

0024873570

097512640

00

00000323371

00003481703

00081217336

8702

072

008

00

01

0000

00282209

000

03490210

000

08069331

8802

07999

000

010

00

10

000

00282209

000

03490210

000

08069331

989

01

000

0108999

0141458800

0015699660

00

0701544

600

000

030804

11000

01808756

02821127656

9001

009

081

00

0387325700

00612674300

00010372128

000

02262387

02841922874

9101

018

072

00

0617516200

00382483800

00025967994

000

02595203

02882097752

9201

027

063

00

0847706

600

0015229340

0000

48749247

000

02928020

02922272613

9301

036

054

00

10

0000

67770750

000

03148210

02948852202

9401

045

045

00

10

0000

67770750

000

03148210

02948852202

9501

054

036

00

0694796100

0305203900

000032856555

000

03252590

02051313800

9601

063

027

00

0375267800

0624732200

0000

09780617

000

03361868

011116

5044

997

01

072

018

00

0055739370

0944260600

000000

490602

0000347114

700171986952

9801

081

009

00

01

0000

00282209

000

03490210

000

08069331

9901

08999

000

010

00

10

000

00282209

000

03490210

000

08069331

Remarkallresultsof


e


Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table9Re

sults

ofWGCmetho

dwith

assumption119901=2

Set

j1199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0018229950

00028518200

00953251800

000

0040

0116

000

01728069

02781801472

209

001

009

00

0119

077500

0086929170

0793993300

000

012306

44000

02029960

02554637815

309

002

008

00

014294560

00208825900

064

8228500

000

016144

93000

02282400

0222160

6012

409

003

007

00

0154138700

0317146

600

0528714800

000

01820135

000

02492243

019239164

465

09

004

006

00

0158642900

0415838300

0425518800

000

01908465

000

02675199

01651696755

609

005

005

00

0158385300

0507443800

033417100

0000

019044

67000

02838602

01398247201

709

006

004

00

015397000

0059399940

0025203060

0000

01818879

000

02986964

0115804

2336

809

007

003

00

0145161700

0677493900

0177344

400

000

01653018

000

03123503

00925538035

909

008

002

00

0130576800

0760494300

0108928900

000

01397009

000

03250806

00693392358

1009

009

001

00

0105515300

0848209300

004

6275360

000

01015781

000

03371391

004

46376634

1109

00999

000

010

00007650378

099234960

00

000

00285973

000

03487593

00030567605

212

08

000

0101999

002991746

00

0039864350

00930218200

000

00475182

000

01734508

02785384568

1308

002

018

00

0156186200

0083826670

0759987100

000

01912475

000

02078066

02569696656

1408

004

016

00

018571540

00200518500

0613766100

000

02559406

000

02329386

02252050954

1508

006

014

00

019959200

00300829100

0496578900

000

029046

45000

02534151

01968689455

1608

008

012

00

0205190500

0398024700

0396784800

000

03053087

000

02710651

01709097596

1708

01

01

00

0204883800

048582340

00309292800

000

03047496

000

02867177

01466170559

1808

012

008

00

0199402200

05964

20100

023117

7700

000

02906059

000

03008708

0123396

4152

1908

014

006

00

0188426800

0651025100

0160548200

000

02630769

000

03138735

01003607957

2008

016

004

00

0170199200

0733624200

0096176590

000

02204524

000

03260054

00774637183

2108

018

002

00

013878740

0082355060

00037661990

000

01566672

000

03375411

00520395650

2208

01999

000

010

00005895560

0994104

400

0000

002844

12000

03488194

00025407208

323

07

000

0102999

0039009630

00049063780

0091192660

0000

00559352

000

01740057

02788237303

2407

003

027

00

0186056900

0081543590

0732399500

000

02596792

000

02117

173

02581225432

2507

006

024

00

0220191800

01940

0960

0058579860

0000

03501300

000

02367596

02276073350

86

02

064

016

00

0398776800

0577263500

0023959700

000110

01341

000

03310992

01247063931

8702

072

008

00

0327132700

0672867300

0000

07499171

000

03378331

00970095572

8802

07999

000

010

00002425452

0997574500

0000

00282547

000

03489380

00015202436

989

01

000

0108999

01160

46900

0013753340

00

07464

19800

000

02319630

000

01802283

02814244516

9001

009

081

00

046560260

00062508280

0471889200

00014860814

000

02487317

02682670259

9101

018

072

00

0536634300

013633960

00327026100

00019662209

000

02722014

02490831559

9201

027

063

00

0568920100

019828140

00232798500

00022076880

000

02879435

02325119

608

9301

036

054

00

058185400

00255105800

0163040

100

00023089775

000

02999728

02170186730

9401

045

045

00

058137840

003106

87800

0107933800

00023061672

000

03098411

02016349282

9501

054

036

00

0569100

400

036828160

00062618070

00022117

786

000

03183627

01854886977

9601

063

027

00

0543837700

0431747800

00244

14500

00020229669

000

03260569

01674914633

9701

072

018

00

0496141900

0503858100

000016888863

000

03320529

01467114

932

9801

081

009

00

0398874200

0601125800

000011013820

000

03353795

01181071746

9901

08999

000

010

00002207111

0997792900

0000

00282484

000

03489455

00014559879

Remarkallresultsof


e


Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table10R

esultsof

WGCmetho

dwith


Set

1198951199081

1199082

1199083

119909lowast 1

119909lowast 2

119909lowast 3

119909lowast 4

119909lowast 5

119865lowast 1

119865lowast 2

minus119865lowast 3

11

09

000

0100999

0028899780

00029577030

00941589200

000

00427289

000

017204

0702783438020

209

001

009

00

0070433610

0062785360

0866781000

00000630236

000019164

6502612935966

309

002

008

00

0101006200

0174237800

0724755900

000

00939332

000

02159925

0230996

6359

409

003

007

00

01164

28100

0278216500

060535540

0000

011400

42000

02368119

02025025916

509

004

006

00

012375140

00377299100

0498949500

000

01245607

000

02555850

017522164

006

09

005

005

00

012511340

00473218300

040

1668300

000

012646

84000

02729307

01487117

233

709

006

004

00

012118

9100

0567500

900

031131000

0000

01203131

000

02892194

01225622706

809

007

003

00

0111

7466

000661831700

02264

21700

000

0106

4638

000

03047189

00963031761

909

008

002

00

0095342160

0758644900

014601300

0000

00850784

000

03196557

00692358501

1009

009

001

00

006

6968140

0863314500

0069717330

000

00560311

000

03342664

00397864172

1109

00999

000

010

004

057525

00959424800

0000

00271318

000

03459004

ndash0007842553

212

08

00001

01999

00547117

300

0058635980

00886652300

000007046

6400001740508

02792061779

1308

002

018

00

0113395300

006

095540

00825649300

000

01152060

000

01975308

026254960

0114

08

004

016

00

0152208900

0166894800

0680896300

000

01807255

000

02220828

02339214799

1508

006

014

00

017237540

00264806700

0562817900

000

02227532

000

02425035

02071885246

1608

008

012

00

0181971200

0357986500

046

0042300

000

0244

7231

000

02605498

01815800991

1708

01

01

00

018364260

0044

864860

00367708800

000

02487024

000

027700

0301565298298

53

05

04

01

00

0272137800

064

1502500

0086359700

000

05250173

000

03242742

01047260381

5405

045

005

00

020730340

0075987400

00032822650

000

03167791

000

03360631

00708499014

5505

04999

000

010

000

4986061

00995013900

0000

00280779

000

03486375

ndash0000255917

656

04

000

0105999

0134142200

00148261500

00717596300

000

02785519

000

01802365

02818599514

5704

006

054

00

0270078900

005428144

0067563960

0000

05175179

000

02189914

02671303981

5804

012

048

00

0343366

600

013948060

00517152800

000

08184342

000

0244

8197

02448411970

87

02

072

008

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

8802

07999

000

010

0001259991

00998740

000

0000

00281845

000

03489241

000

05383473

989

01

000

0108999

0267611800

00298861900

00433526300

00010453894

000

01906306

02863191600

9001

009

081

00

055621340

0004

2093510

040

1693100

00021109316

000

02581826

02754957253

9101

018

072

00

066

834960

00092874300

0238776100

00030382942

000

02834743

02634055738

9201

027

063

00

0720372500

0133459500

0146167900

000352700

49000

02982519

02530866771

9301

036

054

00

0743590300

0171686800

0084722810

000375746

72000

030844

3202429172472

9401

045

045

00

074751660

00211943200

004

0540250

00037979028

000

03162080

02318497581

9501

054

036

00

07344

77100

0258321800

0007201089

00036682709

000

03226144

02187927134

9601

063

027

00

0657337100

0342662900

000029437968

000

03265401

01941155015

9701

072

018

00

0528083700

0471916300

000019096811

000

03309605

0156104

8830

9801

081

009

00

033215000

0066785000

00

000

07722369

000

03376615

00984850362

9901

08999

000

010

000

0561126

00999438900

0000

00282047

000

03489779

000

06872967

Remarknegativ

evalueso

fcolum


ostsandpo

sitivev

aluesa

reincomes


0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 80

1

2

3

4Utopia point

Nadir point

Pareto optimalset

times10minus3

times10minus4

F2

F1







11988394lowast

= (0 0 0747516600 0211943200 0040540250)

11986594lowast

= (00037979028 00003162080 02318497581)

(24)


4 Conclusions





Appendix




References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Multiobjective Optimization of Allocated...

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