University of Nigeria Research Publications

Aut

hor UGWUANYI, Ukamaka Cynthia

PG/M.Sc/04/35894

Title

A Goal Programming Approach for MultiObjective Function in a Production Company.

Facu

lty Physical Sciences

Dep

artm

ent Statistics

Dat

e August, 2007

Sign

atur

e

A GOAL PROGRAMMING APPROACH FOR MULTI

OBJECTIVE FUNCTION IN A PRODUCTION COMPANY.

BY

UGWUANYI, UKAMAKA CYNTHIA

(PGIM.Sc1 041 35594)

BEING A PROJECT SUBMITTED IN PARTIAL

FULFILLMENT OF THE REQUIREMENT FOR THE

AWARD OF Me Sc. DEGREE IN STATISTICS,

UNIVERSITY OF NIGERIA, NSUKKA.

AUGUST, 2007.

CERTIFICATION

This work embodied in this project is original and has not been in

substance for any other degrees in this or any other university.

,.-+A- Supervisor +-- ----- --- Head of department 1Mr. C h u k w W.1.E Dr. F. 1 Ugwuowo Department of statistics, Department of statistics,

University of Nigeria,Nsukka University of Nigeria Nsukka

...................................

External examiner

DEDTCATTON

This project is dedicated to my heloved husband Mr. Orumie S.T

ACKNOWLEDGMENT

My profound gratitude goes to God Almighty for his divine protection

through out the programme, 1 am indebted to practically everyone who has

touched this field, and 1 here bow 70 all statisticians and operations

researchers , in particular.

I am so thartkfi~l to my supervisor Mr. Chukwu W.1.E for his

supervisory role and fatherIy advice in this work

I appreciate in no sxnaU measure the contribution of my lecturers

towards creating enabling environment for research. Worth acknowledging

are my colleagues - Arnazigo, Ugo, Mbanefo, Ukekwe and my bosom friend

Francis for their encouragement.

FinalIy, I am grateful to the management and staff of Chidera bread,

Shiroro Niger state, in particular Okey - the baker for providing data and

wseful information for. this work.

ABSTRACT

Production planning pIays a centra! role in the successful management of any

production-oriented company, It is typical!y rntdti ob-iective in nature and the

management thereof generally consists of resolving rnany conflicting ob-jectives.

This research presents a goal programming (GP) rncthodo~ogy for solving nrultiple

conflicting objectives in a production company so that growth, devcIopment and

security will he achieved. The model developed in this work used costs of

production, saIes, profit incurred, resources utilized and machine utilized. Data

collected from a bread industry in Shiroro Niger States was used in testing the

adequacy of the model developed. The problem is formulated as preanptive goal

programming model and solved using Tora(2000-2003). The key contribution to

modeling and saving the conflicting aspects of the relationship between the muIti

ob-jectives in the manufacturing company.

TABLE OF CONTENT

Certi fication -- - -. -- -- -- --

Dedication -- -- -- -- -- --

Acknowledgement -- -- -- -- --

Abslract -- -- -- -- -- --

CHAPTER ONE: GENERAL OVERVlEW

1 .I Introduction - - -- -- -- --

1.2 ~Motivation -- -- -- -- -- --

I .3 Significance and justification -- -- --

1.4 Objectives -- -- -- -- --

1.5 Scope and Limitation -- -- -- --

CHAPTER TWO: LTTERATURE, REVIEW

2.1 Literature Review -- -- -- - -

CHAPTER THREE: lMETHODOLOGY

3.1 Introduction -- -- -- -- --

3.2 ?'enninoIogies -- -- -- -- --

3.2.1 -Modeling -- -- -- -- --

3.2.2 G o d Programming -- - - -- --

3.2.2 Cost -- -- -- -- -- -- --

3.2.4 Revenue -- -- -- -- --

3.3 h4ulti objective function in a production company's

3.3.1 Parameters and Vnriable notations -- -- --

3.32 The fonnulnte multiple objective equation -- --

3.4 Mode1 FormuIation -- -- -- - - - -

3.4.1 Minimize cost of p r o c h i o n -- - - -- - - I & 3.4.2 Minimize resource utilization -- -- - - - -TR 3.4.3 Maximize mach utilization -- -- -- -- -- -- -- 19 3.4.4 Maximize sales revenue -- -- -- -- -- -- -- -- 19 3.4.5 Maximize profit -a -- -- -- -- -- -- -20

3.5 Goal priority structure -- -- -- -- - - -- -- -20

CHAPTER FOUR: MODELING THE COMPANY DATA

USTNG THE FORMULATED PGPP

4.1 Introduction -- -- -- -- -- -- -- -- -22

4.2 The data -- -- -- -- -- -- -- -- -- -22 4.3 The analysis -- -- -- -- -- -- -- - - --23

4.4 Mode1 results and discussion -- -- -- -- -- -- --27

CHAPTER W E : SUMMARY, RECOmTENDATJONS, AND

CQNCLUSTON

5.1 Summary -- -- -- - -

5.2 Conclusion -- -- -- --

5.3 Recommendation -- -- - -

References -- -- -- --

Appendices -- -- -- --

CHAPTER ONE

f .l INTRODl.JCT[OY

Any organizaticm whether profif oriented or not usudly has more than one

objective goal: to attail1 mainly bemuse of its diverse nature of activities. A Company

has a lot of resources which when propcrly allocated will lead to effectiveness and

efficiency in terms or physical outfit and high quality inn terms of thc value of the

output. For instance. in any business organization, profitability is important as an

oQjeclive, as well as cost minimization, sales maximization, resource utilization and

machine utilization. By this, the achievement of one might lead to the nonachievcment

of thc other objective.

Varied actions are taken in the course of achieving the organizational

objectives. These actions are numerous that ordered arrangement will become vet),

imperative especially where one action will determine the next action to be takcn.

Because of the complex nature of managing organimtiun, it has become very

imperative that dcep analyses have to be done to have informed decision, which will

make the achievement of the set of predetermined objedives pssiblc. Inherent in this

is the fact that no manager of m y arganizalion has full infortnation of factors or

~ariables that will affc'ct the aclivities of the organi~ation. In essence, therefore the

organization will have to come up with many well defined objcctives and strategies. to

employ, which wilI necessitate reaching a con~prornise of objectives so as not to end

up i~nderachieving the goals already set.

Goal progranming (GP) is a technique, which is used in harmonizing

obiectives within an organization fat. achieving ob,jectives set by the organization.

This tmlmique draws extcnsively f b m the properties of linear programming. It is

another form or means of reaching a compromise among goals or activities. This is

because of the fact that a company has more rhan one ob-jectives ICY be achieved and

more than one strategy in place to achieve the objectives. Goal programming is

tlmetbre developed to indicate mathematically, the quantities of each product

(product mix) to be produced with the aim of achieving the set p a l s or objectives.

Goal Prograinrning asks management of the organization to set some estimated targets

for each goal and assign priority to then1 i.e. lo rank than in order of importance. The

management only has to say which goaI is more important than the other, it need not

say how much.

Preemptive goal programming is developed to help decision nlakers to come

close as satisfactorily as possible to achieving organimtional goals. It is solved by

achieving higher order gods first, to the extent possible, before achieving or

considering lower order goals.

1.2 MOTIVATION

The multi objective models in the context of planning werc formulated and

solved in recent past to provide information on the trade off among multi objectives.

How ever, although it represents a viable approach to production planning, multiple

objectives GoaI programming is not as wide spread among manufacturing companies

as desired. The modeling approach of Goal programming (GP) does not maxinlize or

minimize the objective function directly. but seeks to ~uinirnite the deviations (both

positive and negative) between the desired goals and then results obtained according

to priorities.

So, it is clear that Production Company is an arca where GP can be applied

very efficiently. The reason is that, the Gp technique has the potential to modeling and

solving thc conflicting aspect of the relationship between the multi objectives in the

manufacturing company.

1.3 STGNTFICRWCE AND JCFiTITTCATION

This work serks to study is to deveIop, apply and evahate a rnulti ob-jective goaI

programlning mode1 to a real life manufacruring situations to slmw the trade off

between different, sometimes conflicting goals concerning the company that is, a

nlodcl that aim at s~:mi~ltancously maximizing sale, profit, resource utilization,

machine utilization and at the same time minimizing cost More specifically

p~~emptiue goal programming utilization techniques(PGPP) are empIoyed. For

illustration: goal prioriy structures have been considered which will guide and assist

the decision maker for achieving the organizationa1 goals for optimum utilization of

resources, available time, sale and profit in improving companies' competitiveness.

The common thread to all the goals is the desire to place organization in a better

competitive position such that growth, security and survival of the companies are

guaranteed.

I .4 0R.JECTTVES

1) To use goal programming approach to development a multi objective strategy

in a company.

2) To set and test the priority structure using data obtained from bread industry.

1.5 SCOPE A N D LIhiTTATION OF WORK

The study deals on modeling multi objectives in a company, particularly, how

the bread industry aIlocates its resources. and ability of the labor force to utilize the

available resources with the ultimate aim of maximizing profit, minimizing cost,

maximizing sale, resotlrcc utilization and rninirnizing h e .

Chapter hvo gives brief description of the related literature review. In chapter

three, definition of relevant terms, multi o biective has becn developed with goal

programming forrnularion (GP) formulation. Testing of the goal priority structure and

discussion of results i: given in chapter 4. FinalIy, chapter 5 summarizes the main

conclusions and gives directions for further research.

Some of the findings predictions from the analysis of data may be restricted to the

bread industry.

CHAPTER TWO

2.1 LITERATtJRE lU3WEW

A lot of research has been carried out in the applications of goal programming

in different fields. So we review some f the scholarly work done in this area.

Kenn~th, et 21 (1975) prrsentcd a GP model that allowed for multiple,

conflicting goals in natural resource allocation management's decision problems.

Results were pprovidecl for a management area in mountainous Colorado state forest

located in northern Colorado. The trade offs between goah were demonstrated by

comparison of resuIts from muItiple runs in which the order of goal preferences

varied. GP was shown to be a very flexible decision-aiding tool, wl-lich can handle any

decision problem formulated by linear programming more efficiently.

Sundaran (1978) applied a goal programming technique in mctal cutting for

seIecting Ievels of mgchining parameters in a fine turning operation on AISI 4140

steeI using cemented tungsten carbide tools. The objectives he considered werc

finishing turning deprh in one pass and finishing lurning depth within a stipulated

time. The goaI programming combined the logic of optimization in mathematical

programming with the decision makers desire to satisfy several goals.

Prenlchandra (1 893) developed a goal prognmming nlodcl for solving problem of

making project decisions that involved a large number of interrelated activities-the

planning and scheduling project management. These problems arose in areas such as

product development, production planning and controlling and setting up of

production facilities. He found that the solution obtained from using Linear

Programming (LP) in deciding the optimal crash plan to complete the project within

the desired time period was not effective and showed that a goal programming

approach can bc used efficiently in such decision-making problems.

Claudia et a1 (1994) applied multiple objective goal programming techniques in

management of the !Mark Twain National Forest in Missouri;. Accurate market values

were not available for some forest products (r.g. dispersed) and therefore, instead of

exact coefficients, their approximations (Fuzzy numbers) were dealt with in the

modeling phase. The applicability of f imy rndtiple objective programming

techniques for resource allocation problems in forest planning were demonstrated.

Springer (1995) Presented a review of current literature on the branch of multi-

critcria modeling known as goal programming. The result of the investigations of the

ttvo main goal pryyamming methods, lexicographic and weighted goal progranming

together with their distinct application areas were reporled. Some guidelines to the

scope of goal programming as an appIication tool were given and methods of

determining which pmblem areas were best suited to the different goal programming

approaches were proposed. The correlation behveen the methods of assigning weights

and priorities and the standard of the results were also ascertained.

Lucia et a1 (1999) presented a Goal Programming methodology for solving

maintenance schcduling (MS) of thermal generating units under cconon~ic and

rrl iahility criteria.

The advantages of a muIti-criteria approach was demonstrated by comparing the

effect that cost reliability have on each other in power plants maintenance scheduling.

The problem was formulated as a large-scaIe mixed integer goal programming

problem integrated in the mathematical programming language (GAiM) .It was shown

that diflerent optirnizstion criteria gave different optinlal MS sdutions. the n m c

antagonistic the more difficult,

The mcthodoIogy ~l lowed the system planner to choose explicitly thc compromise

behveen both criteria using a control parameter determining the increase in cost

aIlowed to leveling the reliability of the systcm. Weekly maintenance scheduIing of

the large scale Spanish power system for a year period illustrated the GP

methodolog.

Ertugrul et a1 (2002) presented a combined analytic network process (Am) and a

zero one goal programming (ZOGP) approach in product planning in quality function

depIoyment (QFD) to incorporate customers' needs and the product technical

requirements (PTRs) systematically into the product design phase.

Numerical examples were presented to ilhstrate the application of the decision

approach. It considercd the interdependence between the customers' needs and PTRs.

and inner dependence within then~selvts, along with the resource limitations.

The ZOGP model was constructed to detrrrninc the set of PTRs that would take

into account in the product design phase considering resource limitations and multi-

ob-iective nature of the probIem (important levels of product technical requirement

usine AN?, cost budget, estendibility level and rnanufactnmbility level goals). The

ZOGP model provided feasible and more consistent solution.

EIizabeth ct a1 (2002) descrikd River wares {RW) optimization capabilities and

its use by Tennessee v a k y Authority operations schedulers. The River ware is a

flexible gencral river basin modeling tools that aIlows water resources engineers to

both stimulate and optimize the management of multipurpose reservoir system for

daily operations. Input data requirements included Physical and economic

characteristics of the system, prioritized policy goals and paranlcters for automatic

linearization. He generated and efficiently solved a multi-objective, preemptive Linear

Goal programming (MOPLGP) formulation of a reservoir system. An advanced

feature of the RW Isas that both the physical model of the river basin and the

clperating policy were defined and easily modified by the model through an interactive

graphical user interhce. Modifications were incorporated into the LPGP. Thc RW7s

combination of detailed system representation policy expression flexibility, and

computationaI speed make it suitable for use in routine daily scheduling of large

complex multi-objective reservoir system.

In all, PGP approach was chosen for three main reasons: deterministic

optimization was accepted given the relativeIy short time horizon for operational

modeIing, GPLP cou!d model the multiple objectives and physical aspects of

rcsenfoir systems in 3 sufficientIy realistic manner. and GPLP was sufficiently

efficient and robust to be used in daily operations.

Rafael (2002) developed lexicographic integer goal progran~ming model for

the efficient assignment of the financial resource of a Spanish University and applied

it to a particuIar case of the University of Malaga. The problem used 3 I21 integer

variables (22 variables by department, having the University of Malaga 142

department), 712' hard constraints and 1420 soft constraints associatcd with goals of

the pmhlem. The goals were allocated into five priority level. And finally, was shown

that genetic algorithm obtained the same optima1 solution with goal programming but

within a savings of computational time of mare than 80%.

Taylor et a1 (2003) deveIoped a rn~dti-objective model to solve the production

planning problctns fix multinational lingerie company in Mong Kong. in which the

profit is maximized but production penalties resulting from the going over / under

quotas and the change in workforce levels were minimized. Different marlagerial

production loading plans were evaluated according to changes in future policy and

situation in order to enhance the practical in~plications of the model. 'The multi-site

production planning problems considered the production loading plans among

manufacturing factorirs subject to certain restrictions, such as production impod /

export quotas imposed by regulatory requirement of different nations, the use of

manufachiring factorirs / Iocations with regard to customers' preferences, as well as

production capacity, workforce level, storage space and rcsource condi~ions of the

factories,

Flowers et a1 (2003) developed a goal prograrnming approach to rlw waste-fixel-

blending process that considers the divcrse ob-jectives of fuel managers (cg. to blend

hazardous waste into fud, maintaining environmental replatory rquin-ments etc.]. A

real-world case study at a cement kiln illustrated the effectiveness of this approach,

whcre the impleme~tation followed principles of team builders and quality

management.

Ade-jobi et a1 (2003) applied a Linear goal programming technique to model the

farm-family crop production enterprise in the Savannah zone of Nigeria and

developed an optimal crop combination that ivouid enablc the small holder Farmers

n m t their most important goals of providing food for the farniIy through out the year.

The goaI programming results revealed that only 4 out of the 18 basic cropping

activities identified in the study area entered the programme. The 4 activities and their

hectare allocations were millet 1 maize/rice (1.20 ha), followed by maizelguinea

cordcowpea (0.94 ha), thcn followed by millet / cowpea (0.16 ha), and lastly by

maize / cowpea / millet (0.04). But one striking featurc of his plant is that there were

no sole cropping cntrrprises included in the model.

Latinopoulas ct al (2005) creatcd, applied and evaluated a GP model that aimed

at simuItancous maximization of fanner's welfare and the nlininlization of the

consequent environmental burden in alIocatiorr or land and water resources in irrigated

Agriculture. Weighted and Lexicographic GP technique were employed and

implemented on a representative area in the Loudias River Basin in Greece to seek for

a compromising solution-in terms of area and water allocation (under different crops)-

resulting in figures that came as close as possible to the decision makers economic

social and environmental goaIs. The information that was incorporated into the

selected goals includes farmers' welfare, characterized by securing income and

employment levels. as well as enviroruncntal benefits, such as water resources

protection from excessive appIication of fertilizers and from unsustainable use of

irrization water severrl weights or priority levels were assigned on the above goals.

according to thc intentions of thc decision nlakcr, that differentiated the final

allocation of resources,

Me hrther exarnincd the difkrent final outcome that arose when the targets of

the various economic and environmental goals were relaxed in order 10 rcduce the

infortnation bias from the decision makers as well as to better perceive the indirect

relationship between some competitive goals.

Douglas et a1 (2006) developed a n~ethodology to estimate enlpirically the

weights for a multiple goal objective function of SenegaIese subsistence farmers. The

mcthodology includes a farmer-oriented goal preference survey and an application of

multidimensional scaling technique to the survey data. A con~parison of model.

performance under the multiple-goal objective filnction with a profit maximization

objective function did not indicate there were distinctive advantages to using either

function.

Nhantumbo et al (2006) presented a Weighted Goal Programming (WGP)

approach for planning inanagen~ent and use of woodlands as wcll as a framework for

policy analysis. The mcthodology was employed to reconcile dcnland of households,

private sector. and government of M i o m b woodland of South Africa. The approach

was based on househoId and private sector, which linked into a Miombo woodIand

model, Miombo God Programming Model (MIOMROGP). The MIOMBOGP

provided a framework for evaIua~ing the impact, on these two sectors and woodlands,

nf some govemrnent macroeconomic policies as well as some forestry and

agricultural ,- policies.

Finally, there were; scarcity of and l' or rinreliability of data to estimate the

coefficients with distort. The household organization and the mode1 results, inability

of the decision makers like farmers and ~nicfdlcrnen to list and state in a consistent

manner priorities or weight's the attach to each targct level, also making an

appropriate choice of the number irf variables and constraints capable of producing

meaningful results as we11 as interpretation that reflects the decision makers' space i.e.

interests, activities and goals.

The multi-objective models in the context of manuraachiring were formulated and

solved in recent past to provide information on the trade off among multi-objectives.

However, although it represents a viable approach to production. plaru~ing, MOGP is

not as widespread among manufacturing companies as desired.

The modeling approach of goal progi-itmming docs not maximize or minimize the

objective hnction directly as in Linear Programn~ing~ but seeks to minimize the

deviations (both positive and negative) between thu desired goals and then results

obtained according to priorities.

A commonly used generalized model for goal programming is (Ravindran 1979).

n

Minimize z= C u:, p, ( d , ' + dl-) 1

Where

Pi is the preemptive. factor / priority leve! nssigned to ,each objective in rank order.

W, means non-negative constant representing the relative weights assigned with

priority level to the deviational variable d: and d( for each j"' corresponding goal

bi.

Xii means the decision variable coeffkients

WhiIe equation;

f I ) Represents the okjec~ive h indon, wl~ich minimizes the weighted sum of the

deviation variable.

(2) Represents the goal constraints relating the dccision variabIe (so) to target (bj)

(3) Represents standa~d non-negativity restrictions on ail variables.

From the literature it is clear that the goal programming approach has been

applied for a variety of applications. It appears that Production planning is an area

where goal programming can be applied very efficiently. The reason is that from t l x

literature, the goal programming tcchnique has the potentiaI to modeling and solving

the conflicting aspects of the relationship bttween the m d t i objectives in

manufacturing company.

CHAPTER THREE

3.1 TNRODUCTIOk:

Quality products increase rcvenue (sales) and profits. The profit in turn also

depends on dlc cost of production, r a u r c e s nsed and machine utilized. Therefore the

model to be deveIoped includes some of the most important issues reIating to

production planning. In this regard, the management is to make a decision that will

achieve these objectives as dose as possiblc .We therefore note that interest is on the

GP modeling of the above important objectives.

Before a fonnal representation of the research method terminoIogy as appIied

here will be stated.

3.2 TERMINOLOGJ'

3.2.1 MOIlELLlNG: A model is a simplified representation of a real system and

phenomenon. It is a fonnal description of a real system. ModeIs are mere abstractions

revcaling the features that are relevant to the reaI system behavior undor shrdy. The

nature of models that are appropriate for management decision and planning is such

that can be used to represent for example production planning problems. The type of

model that can be appropriate far management will include model that can be used to

represent management plans in numeric or algcbraic forms.

Tho model is commonly used with the intention to gain insight into the general nature

of a particular problem in terms of what particular factor is responsible and h o ~ ~ .

However, there are a number of purposes for which a model can be constructed.

Predictive purpose: Pere the model is used to indicate what to do in the system e.g.

policy model

Control purpose: The model wilI be used to monitor behavior, compute variances

bctwcen what ought to be and what is actually obtained. and decide what remedy is

appropriate e.g. Standard Costing model e.t.c.

Heuristic yuqmse: Hcre, the mode1 is used to review new facts about the system,

which may be initially unknown in the system e.g. Quality Control Modcl.

Measumtive model: Here, the model provides standard or yardstick or value

reference for the asstmment of the performance of the system e.g. Budget

So. in this work we consider heuristic model and we represcnt it mathematically.

Matl~ematicaI model are mathematical statements of the facts of a situation in

question. They are in form of symbols which represents the constituents of the

situation specifq.ing the way in which this constituents interrelates e.g. Y=f(x) is a

mathernaticaI statement which states that the system represented by symbol Y is a

function of the system reprcsented by x.

Various types of mathematical statements exist. This means that mathematical

models arc not mere abstractions but represents reality. They enable us to understand

the behavior of the real system in question. For example, it is used in physics to

describe the behavior of physical being; in chemistry to describc the behavior of

compounds and eIcments; in Engineering to guidc in construction and thus, in

Operations Research to hclp in understanding the khavior of an operating system so

as to bc abIe to make decisions on such systerns. Thcl-efore, here the modcl will heIp

in understanding the relationship between the company's objective goal in order to

place organization in a better competitive position such that growth ,security and

survival of the orgnnimtion are guaranteed.

3.2.2GOAL PROGRAMMTNG

Goal programming is one of the oldest multi criteria decision making

techniques aiming at optimizing severaI goals and at the same time minimize the

deviation for each of the objectives from the desired target. The concept of goal

programming evolved as a result of unsolvable linear progan~ming prohIem and the

occurrence of thc conflicting multiple objectives goaI. Multiple objectives arise in

production companies hecause of several departments with different functions, In fact

the basic conccpt of goal programming is "whether goals are attainable or not, an

objective may be statcd in which optimization gives a result which come as close as

possihle to the indicated goals"

Goal programming provides the management with estinlates of achievement or non-

achievement of his defined and ranked gods.

The objective of goaI programming is to nlinimize the achievement of each

actual goal level. If non achievement is minimized to zero, the cxact atlainment of the

goal has k e n accomplished. For a single goal problem, the formulation and solution

is similar to linear programming with one exception. The exception is that if co~npletc

goal attainment is not possible. goal programming wilI provide a solution and

information to the decision makers.

In problem with more than one goal, the manager must rank the goals in

order of importance. The procedure is to minimize the deviational variables of the

highest priority goal. and proceed to the next lower goal. Deviation from this goaI is

then minimized, the other goals are considered in order of priority but lower order

goals are onIy achieved as long as they do not distract from the attainment of the

higher priority god.

In order to minimize either underachievement or overachievement of a particular

goal, a variabk called a" deviational variable" is assigned to the goal. This variablc

represents the magnitude by which the goal IeveI is not achieved. If the value of the

deviational variable is small, thc goal is more nearly achieved than if the value is

relatively large. i.e. oprimality occur when dcviational variables of the different goals

have been minimized to the smallest possible value in order of importance.

In general the principle idea of goal programming is to cofivert original

multiple objective into a single goal. The resulting model yields what is usuaIly called

an efticient solution because it may not be optimum with respect to all the conflicting

objectives of the problem.

There are two aIgorithms for solving p a 1 programming problems. Both

methods convert the multiple goals into a sin& objective fiinction. In the weights

methods, the single objective function is the weighted sum of the frlnctions

representing the goals of the problems, i.c. it considers a11 goals sirnuItaneously within

a composite objective fimction, comprising the sum of all respective deviations of the

goals from their aspiration levels. The deviations are then weighted according to the

relative importance of each goal. To avoid the possible bias effect of the solution to

different measurenlent unit goal, normalization takrs place (i.e. the model minimizes

the sum of the deviations from the target). Tlre preemptive method starts by

prioritizing the goals in ordcr of importance. 1.e. i t is based on thc logic that in some

decision making sprems, some goals seems to prevail. The procedures begin with

comparing a11 the alternatives with respect to the higher priority goals and continue

with the next priorie until only one alternative is left. The mode! is then optimized

using one goal. at a timc such that the optimum vaIue of a higher priority goal is never

dcgmded by a lower priority goal.

The two methods do not generally producc the same solution and ncither is one

method, however. superior to the other because each teclulique is designed to satisfy

certain dccision makers' prefere~~ccs.

Defining the &ovc methods mathematically,

Weighted method: Suppose that the goal programming model has n goals and that

the it" goal given as;

Minimize G,, i=l .. .n Where; Gi is the owecrivc function for the it" goal.

Then the combined objective function used in the weights method is dcfined as;

Minimize z=w,G, +w:Gz+. . .w,,G, The parameter w,, i= I.. . n represents positive weights that reflects the decision

makers prekrences regarding the relative in~portance of each goal. For example w,=l

signifies that a11 goals carry equal weights. The determination of the specific values of

these weights is subjcdve.

The Preemptive methad; in the preemptive method. the decision maker must rank

the goals of the problem in order of importance. Given an n-goal situation, the

objectives of the problem are written as;

Minimize GI=pl (highest priority)

Minirnizc Gi~p , ( z " ~ lo the highest priority)

Minimize G,=p, f lowesi: priority)

The variable pi is either d: or di- representing goal i. The solution procedure

considers one g o d at a time, starting with the highest priority,GI and terminating with

the lowest priority G,.

3.2.2 GUST

Cost is considered as the over a11 expenditure involved in p d u c i n g a giving

quantity of commodity. Multiplying the unit cost of prcduction by the total quantity of

a commodity produced derives it.

3.23: REVENUE

Revenue refers to the money derived by producer or firm from business

activities from the sak of his or its products. It consists of all money received by a

business for rendering a service from the sale of con~~ndi t ies . In this research.

revenue is regarded as the total amount of money, which the company derives from

the sale of its product. It is calculated by multiplying the total quantity sold by the

price of the commodity.

3.3 MULTI OR.IF,CTIVE WNCTTONS IN A PROnUCTION COMPANY

Since there are varied interests, companies often do have many objectives, which

are aimed at satisFying the various interest groups. It is imperative however that

organization as entities will' want to grow, survive and be secured within their

operating environmenr.

The company considered in Ithis research involves the production of multiple

(differcnt) product types, using the company's existing facilities .The management

wants to avoid undcr utilization of machines, resources and at the same time wants to

nlinimize costs, as we11 as maximizing sales margin and profit. DetaiIs of variabIes

and the objcctive functions representing the various performance criteria are presented

as fdIows.

3.3.1 PARAMETERS AND VARMARLE NOTATTONS

K = The product type (k = I . . . n) UF = The unit profit h m K ' ~ product

P = TotaI profit (Target Profit)

J = Thc machine type (J --: I ... rn) M, = TotaI available machine

&& = The machine capacity required for K ' ~ product

Ck = Produc t ioncos to f~~~~roduc t

Sk = Sales revenue from unit of Ch product

Yk = Quantity of product K [No. of product K1 produced per period

& = Amount of resource (materials) needed for K ' ~ product

A = Resource (materials) availabIe

S = Total sales (Target)

VARABLES Y = Thlt product type to be produced per period

The foIIowing criteria are incorporated in the model

Required raw materials and type (resource uti limtion)

Product type

Cost of production

Sales revenue

Profit realized

m Machine utilization

These importam criteria are thus;

Minimize produetion cost

hlaximize resource utiIimtion

Maximize machine utilization

Maximiz~ sales rejcnue

Masimize pro fit

Which are formulated thus:

3.3.2 The formulated multiple objective equation

Equation (I)-(5) represent functional relationship b e t w e n the production type K

(decision variable). and the various performancc measures of the manufacturing

processes (the management).

The total cost of production per product is represented as Ck.Yk in equation (1).

CK is the unit cost of production of kth product. The resource utilization and machine

capacity utilization can be obtained through equation (2) and (3)

The objective of sales revenue is formulated through equation (4),which is represented

as SK YK for the K" product and equation (5) is the profit UK.YK.

Sales maximization objectives aims at improving the cash in flow of the

organization (company) whilc profit nlaximization does not place much premium on

cash flo\v but on high rate of return. Sales are maximized when marginal revenue is

zero, where as profit is maximized when marginal revenue is equal to marginal cost

and since marginal cost cannot be equal to zero, sales ~naximization will not guarantee

profit maximization. This implies that the two objectives are important to the

organization in the attempt to establish competitive advantage in thc market.

It is clcar that many of the above objectives are conflicting and the decision

problem of evaluating their trade-off is a carnplica~ed one. In such case, the conditions

of the manufacturing processes (company) shouId be treated appmpriatcfy to reflect

the decision makers tarset on various objectives into the planning process.

The conmon thread to all the goals listed a b v e is the dcsire to plaee

organization in a k t t e r competitive position such that growth. security and survival of

the company are guaranteed.

All other &jec!ives, apart from the profit maximization arc geared towards

achieving a favorable lcvel of rehlm on investment

Goal prwgramming approach has been sought for the above model due to the

conflicting objectives.

3.4MODEL FORMULATION (GP MODEL) FOR THE ABOVE EQUATIONS

To formulate the model, the parameters used for input to the GP model in each

priority structure should be given or else estimated by the company. Therefore the

company personnel are invoIved and also encouraged to take major role in

FormuIation. All mode1 parameters are assumed ro be deterministic and constant.

The goals are formulated thus:

3.41 MINIIWZE COST OF PRODUCTON

The total cos~ of production per product is the sum of the machine cost, cost of

resources, and other costs. The managcment in consultation with the operation

management department decides the production cost parameter.

The goaI of minimizing the production cost for the product type k can be

( I ) Min d;

Where b-l=underach'tevement in prvduction cost goal

I d',=overachiwement in production cost god

Each product is composed of one or more materials. The amount of the

material mix required to produce one unit of each product type is required. The

materials requircd and availabk is estimated based on the quantity needed for

producing one unit of product type. Avcrclge quantity of thc product inix i s taken into

consideration in the fisation of ak. The averagc quantity mix for each product k is

obtained from the process plan. The production manager of the company provides the

amount of resources available in thc planning horizon. The positive deviation from the

goal can be eliminated fionl the objective function .The goal of minimizing the under

utilization OF the resource can be represented as

( 5 ) Mlln((f;

S.t 1

S (1, Y, + d; - d,' = .A ....................... 17 1

Where,

d~=unden~tiIization of resources

d2+=nv~r utilization of resources

The machine capacity required and available is estimated based on the time

needed for producing one unit of product type. Time spent on the product is takcn

into consideration in the fixation of mk. The manufacturing time for each product k on

machine is obtained from the process plan. The production manager OF the company

provides the capacity available in the planning horizon for each machine. The positive

deviation from the goal can be eliminated from the objective function .The goal of

minimizing the u d e r utilization of the machine can be represented as

Where M=available capacity of machine {goal)

d3--under utilization of rnachinc

3.4.4 RlAXIMIZF SALES REVEUE

The sales parameter depends on the company sales target in thc planning

horizon .The marketing department estirnatcs unit sales contribution from each

product by using thc previot~s sales data. In view of past saIes records, the

management fcels that the sales goal should be S naira, which depends on the total

gross margin of the product type. This goal can rbe represented as:

(2 ) Min dl'

S. t n

X.TkYk + d - - d l = S ........................... L .4 (9 1

Where;

d-4=underachievement of the sales revenue goa 1

d''4=uverachievernent of the sales revenue goal.

However the ovcr ad~ievcment of saIes goal is accepted and hence positive

deviation from the goal is eliminated from the objective function.

S is the saIes revenue g o d fixcd by the management

3.4.5 FIAXIMIZT? PROFIT

The total profit is estimated as the sum of the saks minus the total cost. The

profit parameter depends on the company profit target in the planning horimn .The

marketing department estin~ates unit profit contribution from each product by using

the prcfit data. In vizw of past records. the management feels that the profit g o d

should be P naira. which depend on the sale and the total expenditure. Howzvcr the

ovcr achievement of 9rofit goal is nccepted and hence positive deviation from the goal

is eliminated from the objective

Function. This goal can be represented as

Where

d5+=ovcrachievement on the profit target

d5-~nderachievenlent on the profit target

Equation (6)-(10) reprzsents the manufacturers' goal.

In equation (61, the production cost for the product volumes of product K is

minimized ( i t . over achievements in production cost goal is minimixd). In equation

(7) undcr utilization of resources is minimized In cquation ( 8 ) idle capacity of

machine is minimized. In equation (9, under ackicvement of the sale revenue : p i 1 is minimized. In equation (10) under achievcn~ent of the profits is minin~ized.

.A5 GOAL PR.TORIY STRUCTRE

In practices. some of the above business goals conflicts in the sense that some

are intcrestcd in cash flow position of' the business, while others are interested in

profit. It is bccause of this co~lflict between ob-jxtives that companies resort to listing

thcir objectives according to degree of important. Objective prioritization is means of

setting the conflict, thus stating explicitly which of rhc ob-iectives ere irnportcmt to

t l~em and thus arranged with the most important one corning first,

A good prior;ty condition is nothing but a hierarchical representation of the

goaI priorities. which reflect the decision maker's prefermces. Pm!uction and

marketing personnel are actively involvcd in the selection and the prioritizing OF the

various goals.

In order to test the goal programming model. goal priority stn~cture is to be

forrnuIated based on the preferences that the company" top management erpressed

especially to suit the specific market conditions based on the above goals.

The preemptive fx tor for thc finalizzd goal priority structure is defined Fxlow;

PI ensure that the production cost is minimized

Pz ensure that underutiIimtion of resources is minimizd

P3 cnsure that idle capncity of machine is minimized

P4 ensure that sales target is met

P5 ensure that undcr-achievement of profit is minimized

Thus the objective is the minimization of deviation from various goals imposed

by the company. Thus;

Min.

Z = Pldl'+P2 b?-+P3 d

CHAPTER FOUR

TESTFVG THE PREEMPTIVE GOAL PRROGEWhIMINC FORMULATED..

4.1 INTRODUCTION

In this chapter, we model, !st. 2nd mdyze the company data. First we set priority to the objectives and thereafter, a rnuIti obiective goal programming (PGPY) is

analyzed using tora (2303)

4.2 DATA COLLECTION

The organizaticln selected for the present study is a bread indust~y in Shiroro

Niger state. To mainlain the secrccy of data, we hide the company's name while

explaining the history of the case organization. The company produces two difTerent

sizes of bread with the same quality. They are called big loaf and small loaf. The

company has fifteen wrkers who are working in different sections of the company

namdy: milling, mixing, and baking sections, Their performance till now is

satisfactory

The data were collected from the fiIe in the company manager's ofice for one

day and it includes;

. ..Sizes of bread

. . .All the raw materials used and the quantity used for each size of the bread in kg

...' The costs of productim

. . .Selling price per unit. of each type in Naira

. . .The machine hour spent in producing each type

. . .The total profit realized from each type

... The total saIm per day in Naira

The raw rnaterbisls used for the production includes sugar, salt. saccharine,

tablcts, nutmeg, EDCoiI, yeast, baking powder, milk flavor, flour, water and fire wood

for the baking. The main machine used is constructcd which include the foIlowing

paths

..Basin for mixing of bread

..Roller for milling

..Pan for scaling to determine the size of the bread 4 - C ; -+

..And then ovum for baking % =d T ~ A * X I I *

The table1 in appendices contains the quantity of raw materials in kg, machine

time requirements per product size in hour per day and costs of production in Naira

Let Y1=big loaf and

Y2xSmalI loaf.

The cost of leather is 300 for Y ,and 150 forYz

Thcreforc the total ran, material is 739.8kg and the available raw material is 8OOkg

The total cost of production is #541166 for Y1 and 53966.66 for Y2

The total available tine is 6hours.

The sample i n p ~ ~ t data is summarized in appendices table 2

The machine avaiIable time is at least 6hours

'Thc profit target is at least #30000

Al costs; profit and sales arc in Naria

The company's sales target is at least ff140000

From the priority structure, the formulated god programming model is thus:

Min Z = Pldt++P2 d2-+P3 d

Z

541 I 6.66XI+53966.66X7+d1--dlA 4

489.8X +389. 8X2 +dc-d2+ =500

6,Y +6X? +d3--d3' =G

7 0 0 0 0 ~ ~ +7500ux2 +d +---d4' = 140000

I 58X3.34Xl+21033.34X2 + d i - - d: =30000

A l1 variable non-nega tives

To minimize thc second priority. the value of the first goal becomes a

constraint, thereby increasing the constraint equation to six. That is

Min Z = P2 d2-

s .r 541 1~ .66X~+53966 .66~~+d1ld l+ r=o

1411 variable non-negatives

All variable non-negat:ives

Min Z = Pq d i

s.1

54 I 16.6~~~+53966.66~~+d~--d~+

ISS83.34X1+21 O3?.34,Yz

All variable non-negatives

AN variable non-negatives

Where ki, kii. kiii, kiv represents the value of thc result output obtained for the first to

the fourth priority.

Relow are the result outputs:

4.3 MODEL RESULTS AND DISCUSSION

The proposed h;lOGP mode1 is tested using as impute, the firms data bascd on

the jnfomation from the decision maker on the objectives, targets and priorities. A

sarnpIe input data was given in table (6) in the appendices for thc preemptive goal

priority structure. The priority structure was executed using Tora software package

(2000-2002) with which are finaIixd bascd on the policies of the management of the

case shidy.

Rrnaka p l , p2, p3. p4, p5 arc the MOGP model output for the priority structure

for the objectives. The goals arc completely minimized .The model output results

dispIay the values and Ihe deviational variables in N , kg. hrs basis.

The following results were yield .The Amaka p l was based on cost

minimization. The cost of production is minimized to the fullest since the deviation

variable is minimized to zero. Goal 4 and 5 were Favored. So optimization of P4 and

P5 is not necessary i.e. salcs and profit targets were met. In An~aka F2, the goal 2 is

not violated. The resourccs are filly utilized. This is because of the fact that the

deviation variable is minimiscd to zero. The optimal solution is to avoid product

shortage. It is clear that lsl, 2". 31d, 4'h and 5'h goal are not violated and end up to

nearly the same solution, which satisfies all the goals without any deviation.

Howver, the ccrnpromising solution seems to be a quite acccptabIe one as the

final deviation from the aspiration level arc confirmed enough. Thc results show that

from the optima1 company plan, the company wil l incur a cost of 936473.82 This

kind of result reflect a typical day to day production where there are several minimum

costs while producing tangible goods. The maxin~r~ln machine hours decreases to 5.94,

the sales revenue generated is 93365.77 and finally the profit generated is NI9839S9

and all the company goals were completely achieved to the hllest extent given the

priority ranking,

From the examination of the output results of the MOGP model, it can be seen

that the mode1 performs well in communicating the trade- offs among conflicting

goals.

CHAPTER FIVE

5.1: SUMhTARY

Development of GP models and their applications to the real life manufacturing

problems have received an increasing attention during the past sevcral years as a

ponwful decision making tool for the problems that involve multiple conflicting

objectives. .Modern manufacturing process is complex owing to increased uncertainty

En competitive markets and rapid tecImologicaI dcveIopments. Production

management under this scenario is challenging and in such, i t is necessaly to

determine the optimum production plan to assist the dccision maker to achieve the

organization goals for optin~um utilization of resources. reaching sales target, profit

target and to minimize cost so that growth, development and security of the

organization is achiewd. The MOGP presents the optimal company plan generated

under the assumption that cost minimization resource utilization, machine ~itilization

profit and saIe maximization are thc underlying criteria for the company gro~vth and

development. The MOGP modd prescntcd in this paper would he useful to

manufach~rcrs cspccially to find out an optinlm level ofpr~duction actitities in terms

of utilization of machitic and resources.

The MOGP results can be usefi~l to other functional areas such as marketing

and finance For routine planning. Some of thc specific decision making in this convex

are:

1.

2,

3.

4.

5.

The production cost under identified product scenarios.

Under utilization of machine.

Under utilization of resources at the same production volume combination.

The achievement of sales revenue god.

The achievement of profit goal.

The results are expected to guide the production manager to estimate the effect

of product mix changes on machine hour so that security of the organization will be

achieved.

In this way, the MOGP output may act as link between the firms' slrategies and

plans that enables the fir111 to achieve its goals. Also the mode1 result nil1 act as an

cRc'ctive comrnunica~ion tools between the levels of managers to enhancc the

productivity of the system and as for improvement of the business.

5.2 CQNCLUSTOY

This paper investigates the MOGP approach when applied to a real life case

situation that took into account cost of production, sales, resource utilization, profit

and machine utilization; with an intension to evaluate the trade-offs between the

dif'ferent goals of production planning environment under multiple conflicting

objectives. To simp@ this illustration, the probIem was limited in scope to an

application of only o ~ e product but different sizes of the case study. Application such

as this could be easily expanded to deal with more complex real world xobIems that

involve different product, customers and the environmental goals. We noticed that the

resuIt shows that the model can be an effective planning tools to aid dccision maker

faced with multiple conflicting gods of production planning art significant

dctc~miners of any firm's success.

In addition the model is computationally feasible. it requires approximately a

one minutes to obtain the results for the selected data.

In this paper, h result obtained shows that the PGPP model developed performs

well in communicating the trade-offs among the various objectives so that growth,

de\dopment and survival of the conlpany is guarantecd.

5.3 RECOMMEh7)ATION

It is important to point out that the production cost goal may not satisfy

(resources and machine goal) but they end up to better number. These are subject to

perception and the intensions of the decision maker. So attention shouId be paid on

setting the goals, priorities and target correctly & according to the characteristics of

thc reference area. The priority structure is recommended to the case orgsnimtion for

further consideration.

Decision makers should be fully informed or confident about the targets, the

priority levels and the ordering of preferences in order to avoid the model from

producing non rational dccisions in cases.

We also encourage researchers to explore the use of Analytical Hierarchical

Process ( A l p ) for determining priorities and also recommend the use of regression

analysis for estimation of various parameters

Further research should be carried on the same topic to include product

volume.

REFERENCES

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Goal programming, JournaI of range management, Vol, 28. No. 6 .

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metal cutting. International J. Prod. Res. 16: 375 - 382.

3* Premchandra, El. C. (1993). Quantity discount decision under conditions of

multipIe itcms, muItiple suppliers and resource limitation. International Journal

of Production Research Vol. 29, No 10, pp 1953 - 196 1.

4. Douglas Barnett, Bran BIakc, and Bruce A. Mc Carl (2006). Goal

programming via multidin~ensional scaling applied to SenegaIese subsistence

farms. Int.J. of Advanced Manufacturing Technology Vol. 20, No. 4, pp 1-10.

5. Claudia Anderle. ,Mario Fedrizzi and Silvia Giove (1994), Fuuy muItiple

objective programming techniques in nlodeling forest planning. Department of

Computer Science, Eotvos Lorand University.

6. Lricia D., Duan G,, Lei N., Wang J. S. (19%) Analylic Hierarchy Process

based decision nmdeling in CAPP dcvebpment tools. International Journal of

advanced manuF~cturing t~hnalogy Vol. 15, No. 1 , PP 26 - 3 T .

7. Elizabeth A. Escl-~enbach, Tima thy Mviagce, and Morgan Gorar~fla (2002),

MuItiobjective operations of reservoir systzms via Goal progran-rn~ing,

International Journal Operations and Production Management.

8. Rafael Caballero. JuIian Molina and Angcl Torrics (2002). A Genetic

algorithm to solve an Integer Goal programming model for the higher

education. Department of Applied Economics (Mathematics). University of

Malaga.

Taylor, F. W. (2003). On the art of cuthg m6aEs. Trans. ASME 28: 31 - 35.

Dale A. Flowers and Kevin Lindermana [ 2 0 3 ) , Hazardous w a r e disposal: A

waste - fuel blending approach. Departrncnt of Operations, weather head

School of Management.

Adeiobi, A. 0.. Kormawa, P. M.. Manyong, W . M. and Olayerni, J. K. (2003),

Optima1 crop combinations ut~cler limited resource conditions: Application of

linear Goal programming model to small hoIdrr farmers in the driver Savannah

zone of Nigeria. Department of Agrkultural Economics, University of Ibadan,

Nigeria.

Latinopoulos, D. and Mylopoulos, Y. (2005). O p h a l allocation of land and

water resources in irrigated AgricuIture by means of Goal programming:

Application in Loudias River Basin. Global Nest Journal Vol. 7, ho. 3, PP 264

- 273.

Douglas Banwtt, Bran Blake, and Bruce A. Mc Carl (2006). Goal

programming Via muItidimensiona1 scaling applied to Senegalese subsistence

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Nhantumbo and G. Kowero. 14 Goal programming n m k l fix planning

management of Miornbo Woodlands. Vd. [ 1) PP 257 - 283.

Opsiku, 0.0. Abosede, A.J., Ka.inlaS.0, Napoleon S, (1994)basics of

Onerations Research.

Title AMAKA (PI) Final Iteration No ' 8 Objective Value = O -

VariAde

x l : r2: x3: x4: x5: re: x7: ri5. x9. x10: x l l : x12:

Condraint

Variable

Y 1 : xz X 3 X 4 x 5 : f i: x7: x8: rs: x10: XI 1: x12: . - .. . -

Constraint

Value

0.1532 1.8846 0.0000

1 1 0000.0000 0.0000

1007.6 159 0.0000 2.2272 0.0000 Q 0000 o.oooi1 0.0000

RHS

D.OOW 500.0000

10.0000 l ~ ~ . O O C O 30000.0000

Obj Coeff 3bj Val Contrib

Current Ohj C m f f

Current RHS

Min Obi CoeR -

0.0000 Q.0GUO 0.0000 0.0000~ 0.0000 O.OG# 0.0000 0.0000 0.000(3; o.oooa 0.0oUOi 0.0000

Max Obj Cne'R

0.0006 O.WU0 infinily 0 0000 infinity 0.0000 infinity 0 0000 infinity infmily infinity infinity

Min RHS Max RHS

F!educed Cost

Dual Price

Tflie. AMAKA (7%

Lower Bound 0.00 Uppar Bound infiniiy llnrestr'rl jyln)? n

Lower Bound 0 ..M3 Upper Bound lnfinily Unmslr'd y h j? n

0.00 infinity

n

Constraint

Variable

x l : x2: x3 : xd. k5: xr3 : x7: xR. x3: x la: x l l : r12:

. - Constrsinf

. .. LINEAR PR-RAMMI-NGPUTPUT SUMMARY

Value --

0.00 1). 68 0.00

Rfi473.62 0.00 0. CU 5.94 C .03

93365.77 0.02

1 9839.59 [3 .m

RHS

Current 0hi Coeff

0.00 C. CC 0.00 0 .m 0.00

0.00 0 .!;?I 0.00 f f . c1:1 0.00 0

Obi Coaff -

0.00 C.bT

0 C3 0 .CD 0.00 1 .C5 0.00 0 .C3 0.00 0.co 0 00 0 .Z3

Obi Val Contrib

Min Obj CwH

'3.00 - 735?.i?C

0.00 0 . L O 0 UO p 0.00 Q Cc.3 P.00 ci

Lower Bound Upper Bol~nd Unwsir'd (ylrr)?

LINEAR PROGRAMMING OUTPUT SUMMARY

Tirle- AMAKA (P.7; Final Iteration No.: 8 Obiactive Value = 0 - . . - . - .

Variable Value Obj Coeff Obj Val Conlrib - - . - - . - -- . - - - -. . -

x i : 0.00 0.04) 0.00 x2. 0.W 0.W U.00 x3. 0.00 0.00 0.00 X4.. 35473.02 0.03 0.00 r5: 0.00 0.00 0.00 Iwri. U.CQ 0.C3 C.CO x7. 5.94 0 . w 0.00 ~ 8 . 0 .W t .03 0 i;D x9: 83385.77 0.00 0.00 XIO: 0. cc C.CC C.CO x t l : 1 9839.59 0.00 0.00 x l 2 0.23 0 C3 U .C3

Constraint RHS Slack-lliur$us+ -

1 ('1 0.00 0.00 2 (=I . 5 y ctq 17 3 I=) 10.00 0.00 4 (=) 1-?pr?Gi&iy iinn-yi~C+c

Min Obi CoeR Mr;x Obj Caeff * - -

0.00 infinity -.h bnify i j C ~ I G

0.W Infinity c I . x 2.C3 0.00 infinity

C-;I ~r*irn;ry 0.00 infinity c Q? i .~ftni!y 0.06 infinity (1. i -- j r n h r i l i Y 0. 00 intinily ra r j i + inhrr.1 , b

Min RHS Maw RHS

Reduced Cost

0.00 g I;.? 0 00

Tille AMAKA (P4j

x l Mit~irrrlze G.03 Subject to I 1, cd I f i j ? I 2 ) 739.80

21 5 ix~ ( 4) 65000. DO ( 57 t[>fG?. ;i

Lower Boimd 0.03 Upper Round infinity U F I T P ~ ~ ~ ' ? {yin)? r~

0 .# inf niiy

n

Lower Bound 0.00 Upper Sound infinity Unr-tr'd (vln)7 n

0.00 infinity

I1

Variable . -

x i : L?. x.7, x4: x5: xB. x7 : ra: r9: x10. x7 7 : XI 2

Value - - . . - . . -

0.00 0.68 0.00

35473.82 0.00 0.W 5.94 0.03

93365.77 C.CC

13839.59 0 .XI

Cnj Val Contrib -

0.00 0.00 0.00 0.00 0.00 u CO Q. 00 U .co 0.00 Q.CO 0.00 g.c3

Currenl Obj Coerf

0.W [l c113 0.00 0 !;3 0.00 D i i l 0.00 0 . a 0.00 1 ii;l 0.00 IG

Max Obj CoeA - - inlinily

{! i

Lower Round Upper n o r ~ r r r l Unreqlr'tl j y k ) 7

Min RliS Maw RHS

APPENDIX 2

LISTS OF TABLES

Tablell); Raw materials in kg with cost of each Qpe in Naira

Salt I 4.5 4.5 3 00

Saccharine 0.1 0.1 100 1

I

Nutmeg 0.2

Y cas t 0 . 3 0.3 (200 1 ,

Baking powder 0.3 0.3 1300 1 1

Milk flavor 0.01 0 . 0 1 pTH Butter favor 0. I 0.1 1 200 , I

1

Water 1 192.5 i192.5 1- I I

Flour 250 -1- 250 3 8000 Grand nut oil 5.6 5.6 2000

Workers 2266.66 -- Firewood 1000 -- Label 1000

Raw Materials {kg) y I Cost (#)

Sugar 3 6 36 I

I

Table 3: Numher nf llnits o f prodrlcts prodiirct'l per day and thc apprnvcd

I I Product s i x - -mnrGm(iij-

I produced I

m o o p---

Table 4: Workers and thcir snlarim --

I Workers -

I

'I'olal prridwr sold

Arnount per month 1 Amount pcr day

- - - -- - . - -. - - .- - 4000 each -p3.3?*13=1733?.3

Table 5: Is the cast of prodilction per day, and the prnfit

Tflblt: 6: The sample input data

I Performance I Parameter used / Parameter value

Product size I Cost

measure I I I I Y1

Sales I Profit

1 production I I utilization I I Machine nlk 6

utilization I I Profit incurred uk 10883.34

--

Parameter 7

Cover Page: UGWUANYI_Ukamaka_2007_35894PreliminariesTitle PageCertificationDedicationAcknowledgementAbstractTable of Content

Chapter One: IntroductionChapter Two: Literature ReviewChapter Three: IntroductionChapter Four: Testing the Preemptive Goal Programme FormulatedChapter Five: SummaryReferencesAppendix

2009-05-07T06:48:34-0700Ojionuka ArinzeI have reviewed this document