Research Article Delivery Time Reliability Model of ...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 879472 5 pageshttpdxdoiorg1011552013879472

Research ArticleDelivery Time Reliability Model of Logistics Network

Liusan Wu1 Qingmei Tan1 and Yuehui Zhang2

1 School of Economics and Management Nanjing University of Aeronautics and Astronautics Nanjing 211106 China2Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China

Correspondence should be addressed to Liusan Wu wuhusheng563520126com

Received 17 January 2013 Accepted 14 March 2013

Academic Editor Engang Tian

Copyright copy 2013 Liusan Wu et al 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

Natural disasters like earthquake and flood will surely destroy the existing traffic network usually accompanied by delivery delayor even network collapse A logistics-network-related delivery time reliability model defined by a shortest-time entropy is proposedas a means to estimate the actual delivery time reliability The less the entropy is the stronger the delivery time reliability remainsand vice versaThe shortest delivery time is computed separately based on two different assumptions If a path is concerned withoutcapacity restriction the shortest delivery time is positively related to the length of the shortest path and if a path is concerned withcapacity restriction a minimax programming model is built to figure up the shortest delivery time Finally an example is utilizedto confirm the validity and practicality of the proposed approach

1 Introduction

The ldquograph theoryrdquo a mathematical approach born as soon asOla solved ldquoSeven Bridgesrdquo in 1736 has laid a solid foundationfor the development of network theories In several decadeswith a variety of models such as random network small-world network and scale-free network proposedmore atten-tion has been paid to network study [1ndash3] As well knownnetworks exist in every corner of humanrsquos real life for exam-ple traffic network communication network and logisticsnetwork In recent years many emergency events (such asNew Yorkrsquos Power Outage in 2003 Wenchuan Earthquake in2008) occur frequently in the world wide which had causedenormous economic loss and casualties Since materials andinformation are transported and transmitted via networks itis of vital importance to deliver them to the right demandpoint as soon as possible when an unexpected event occursfor any delay may result in loss of or damage to peoplersquos livesand properties In this sense the study or research dealingwith the reliability of networks is significant in theory and inpractice

The network reliability may be defined as an abil-ity or probability that a network system has to com-pletely fulfill customer-tailored communications tasks duringthe stipulated successive operation procedure Currently

the reliability-network-related research mainly concentrateson the invulnerability of complex networks that is theirendurance to attacks [4ndash8] The literature concerning net-work reliability has prospered ranging from the networkconnectivity reliability network capacity reliability to net-work performance reliability and the like [9] Among themthe network connectivity reliabilitymerely considers networktopology and introduces the ldquoprobability of connectivityachieved by networkrdquo as a reliability measurement criterionthe network capacity reliability adopts the ldquoprobability ofexistence of paths that satisfies a certain flow demandrdquo as themeasurement standard by considering the capacity of linksand nodes within the network the network performancereliability refers to the impact of network performancersquosdynamic change on its reliability mainly characterized by theldquoprobability of certain performance parameters not exceed-ing the stipulated thresholdrdquo

Entropy was originally introduced as a thermodynamicconcept and widely used to measure an unordered systemThen entropy had begun to drawmuch attention in the field ofcomplex systems research since it served as a physical devicecapable of describing the structure of a complex systemFrom the macroscopic viewpoint entropy is reckoned as ametric of energy distribution uniformity in a system andit is able to indicate whether the current situation is stable

2 Mathematical Problems in Engineering

as well as the systemrsquos change trend The more uniformthe energy distribution is the greater the entropy achievesOtherwise the smaller the entropy achieves Qiu and herteam has brought entropy in management decision-makingprocedure and then proposed the management entropytheory [10ndash12] In recent years many scholars use generalizedentropy to describe the reliability of a network [13ndash15] Basedon previous research results this paper puts forward theconcept of shortest delivery time entropy involved in logisticsnetworks Moreover the entropy is used to represent deliverytime reliability of a logistics network featured by a negativerelationship with the shortest delivery time reliability

2 Shortest Delivery Time Entropy ofLogistics Networks

21 Model Assumptions and Symbols Description The as-sumptions are made as follows

(1) Vehicles move at a fixed speed when delivering goodsand materials

(2) Vehicles are infinite in quantity of which each deliversonce and departs from the present network immedi-ately to save time

(3) The materials are allowed to be mixed and loadedwith the total weight considered regardless of theirclassification

(4) The time of loading and unloading is ignored

For a directed network 119866 = (119881 119864 119862119863) with 119899 nodesinvolved 119881 = V

119894| 119894 = 1 2 119899 is the vertices set and

119864 = 119890119894119895 is arcs set Additionally 119888

119894119895denotes the capacity

restriction of arc 119890119894119895 119889119894119895denotes the length of arc 119890

119894119895119891denotes

the length of the shortest path for 119866 119891119894119895denotes the length

of the shortest path when 119890119894119895is attacked and fails 119902 denotes

the quantity of required materials V denotes the vehiclersquostraveling speed and 1198661015840 denotes the impaired network when119890119894119895is attacked and fails

22 Shortest Delivery Time Entropy of Logistics NetworkWhen the path of V

119894V119895is attacked and becomes invalid our

research will focus on two timing points as follows

(1) 1199050 the shortest time of delivering materials on condi-

tion that all paths are in good condition


tion that 119890119894119895is attacked and loses effectiveness

Under the circumstances 1199050119905119894119895is utilized to describe the

network reliability in case that the path V119894V119895suffers attack

and thus fails Definitely the longer the length of the arc thehigher the probability of the arc being invalid after an attackThus the probability may be expressed by 119889

119894119895sumV119894V119895isin119864

119889119894119895

As we all know the entropy can be applied to illustrate asystemrsquos reliability Therefore we put forward the conceptof shortest delivery time entropy of logistics network based

on the networkrsquos maximum flow entropy and shortest pathentropy expressed by

119867 = minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln1199050

119905119894119895

(1)

Seen from the formula the shortest delivery time entropy oflogistics network is generalized entropy so the value of 119867may be very large and even can go to infinite The smallerthe value of 119867 the stronger the delivery time reliability ofthe logistics network otherwise the weaker the delivery timereliability of the logistics network

23 Shortest Delivery Time Entropy of Logistics Networkwithout Flow Restriction When the path is unrestricted byflow capacity the shortest delivery time is positively relatedto the shortest path length Referring to its definitions inprevious literature it is available

1199050=119891

V 119905

119894119895=119891119894119895

V (2)

Substituting the above equation into the formula of shortestdelivery time entropy of logistics network we get the follow-ing

119867 = minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln1199050

119905119894119895

= minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln119891V

119891119894119895V

= minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln119891

119891119894119895

(3)

Obviously the smaller the value of 119867 the stronger thedelivery time reliability of logistics network However thelarger the value of119867 the weaker the delivery time reliabilityof logistics network

24 Shortest Delivery Time Entropy of Logistics Network withFlowRestriction When the path is restricted by flow capacitywith assumption of total119898 different paths departing from V

1

to V119899to deliver goods and materials the 119896th path is denoted

by 119901119896 and the flow of 119901

119896from 119890

119894119895is denoted by 119862

119894119895119896 Because

119901119896may not pass 119890

119894119895 we introduce 120575

119894119895as

120575119894119895=

0 119901119896not pass 119890

119894119895

1 119901119896pass 119890

119894119895

(4)

where the flow of 119901119896is denoted by 119862

119896 then

119862119894119895119896= 120575119894119895sdot 119862119896 (5)

The flow of each path may not be unique Example 1 willclarify it

Mathematical Problems in Engineering 3

(2 2

)

(3 3)

(3 2)

(3 2)

1

2

3 4

Figure 1 The original delivery network

Example 1 In order to illustrate the fact that the flow ofeach path may not be unique we design an original deliverynetwork (see Figure 1) where each point pair (119909 119910) of thegiven arc means the length of 119909 and the capacity restrictionof 119910 respectively

1199011means the path V

1rarr V2rarr V3rarr V4and 119901

2means

the path V1rarr V3rarr V4 Therefore

0 le 1198621le 2

0 le 1198622le 2

0 le 1198621+ 1198622le 3

(6)

So 1198621and 119862

2may not be unique

Assuming that 119879119896is the length of 119901

119896and 119877

119896is the deliv-

ery times of the 119896-th path Since the flow of arc 119890119894119895assigned to

all119898 paths is not allowed to exceed the capacity restriction ofarc 119890119894119895 we have the following

119898

sum

119896=1

119862119894119895119896le 119888119894119895 (7)

Besides the total number of thematerials delivered is not lessthan 119902 thus

119898

sum

119896=1

(119862119896times 119877119896) ge 119902 (8)

The total time spent in delivering materials is as follows

max119879119896

Vtimes 119877119896 (9)

Accordingly whenmax(119879119896V)times119877

119896 reachesminimumvalue

the time spent on materials delivery via the network is theshortest thereby we have the following minimax program-ming model

1199050= minmax

119879119896

Vtimes 119877119896

st119898

sum

119896=1

(119862119896times 119877119896) ge 119902

119898

sum

119896=1

119862119894119895119896le 119888119894119895 119877119896isin 119873

(10)

At present there emerged a lot of algorithms solvingminimaxmodels which is unnecessarily repeated here [16ndash20]

(2 2

)

(3 2)

(3 2)(4 2)

(3 5)

1

2

3

4

(3 3)(3 3)

5

Figure 2 The original delivery network 119866(1)

(2 2

)

(3 3)

(3 2)

(3 5)

(5 1)

1

2

34


Note that when arc 119890119894119895is attacked and fails other nodes

and arcs constitute a newnetwork 1198661015840 By usingmodel (10) wecan obtain 119905

119894119895 Similarly by substituting 119905

0and 119905119894119895into the for-

mula of shortest delivery time entropy of logistics networkwe get the time reliability of delivering materialsThe smallerthe value of 119867 the stronger the delivery time reliability ofthe logistics network otherwise the larger the value of119867 theweaker the delivery time reliability of the logistics network

3 Case Study

Example 2 To illustrate the validity and practicality of ourresearch we design two original delivery networks 119866(1) and119866(2) (see Figures 2 and 3) and then we compute the shortest

delivery time entropy of logistics network respectively 119902(1) =17 and 119902

(2)= 16 represent the quantity of materials to be

delivered and V(1) = 4 and V(2) = 3 denote the delivery speedof a vehicle respectively

From model (10) we learn 119905(1)0

= 875 and 119905(2)0

= 8 Theprobability of the arc to be attacked and 119905

119894119895are obtained and

shown in Tables 1 and 2Add 119905

(1)

0 119905(2)0 119905(1)119894119895 119905(2)119894119895 119889(1)119894119895sumV119894V119895isin119864

119889(1)

119894119895and 119889

(2)

119894119895

sumV119894V119895isin119864119889(2)

119894119895into the formula of shortest delivery time entropy

of logistics network we get the following

119867(1)= minus sum

V119894V119895isin119864

119889(1)

119894119895

sumV119894V119895isin119864119889(1)

119894119895

ln119905(1)

0

119905(1)

119894119895

= minus (2

21ln 875

1575+3

21ln 875

18

+3

21ln 875875

+3

21ln 875105

+3

21ln 875875

+4

21ln 875

12+3

21ln 875

1575)

= 03292


Table 1 The probability of the arc being invalid after an attack of 119866(1) and the delivery time

Impaired path 11989012

11989013

11989023

11989024

11989034

11989035

11989045

119905(1)

1198941198951575 18 875 105 875 12 1575

119889(1)

119894119895sumV119894V119895isin119864

119889(1)

119894119895221 321 321 321 321 421 321



11989013

11989023

11989024

11989034

119905(2)

119894119895 12 21 283 8 1123

119889(2)

119894119895sumV119894V119895isin119864

119889(2)

119894119895 216 316 316 516 316


V119894V119895isin119864

119889(2)

119894119895

sumV119894V119895isin119864119889(2)

119894119895

ln119905(2)

0

119905(2)

119894119895

= minus (2

16ln 8

12+3

16ln 8

21+3

16ln 8 times 3

28

+5

16ln 88+3

16ln 8 times 3

112)

= 05494

(11)

According to entropy value of 119867 the delivery timereliability of logistics network is estimatedThat is the smallerthe value of 119867 the stronger the delivery time reliability oflogistics network otherwise the larger the value of 119867 theweaker the delivery time reliability of logistics network Since119867(1)

lt 119867(2) the delivery time reliability of 119866(1) is stronger

than 119866(2)

4 Conclusion

The objective of this research is to develop a heuristicapproach that can be used to evaluate the reliability of alogistics network in emergency context A vehicle howeverhas to choose circumvention in case a certain path ceasesto be effective once attacked by an unexpected event whichactually extends delivery time Therefore further in-depthstudy is needed to take into account ineffective path repairingproblems and to develop a more representative methodapplicable for the specific condition where an arc suffersattacks and becomes invalid

Acknowledgments

The authors are grateful to the editor and anonymousreviewers for their valuable suggestions which improved thepaper This paper was partially supported by the NationalNatural Science Foundation of China (no 71073079 no11271257) the National Social Science Foundation of China(no12BGL104) the Funding of Jiangsu Innovation Programfor Graduate Education (no CXLX12 0174) and the Fun-damental Research Funds for the Central Universities (noNC2012009)

References

[1] P Erdos and A Renyi ldquoOn random graphs Irdquo PublicacionsMatematiques vol 6 pp 290ndash297 1959

[2] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[3] A-L Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquoAmericanAssociation for theAdvancement of Sciencevol 286 no 5439 pp 509ndash512 1999

[4] J Wu and Y J Tan ldquoStudy on measure of complex networkinvulnerabilityrdquo Journal of Systems Engineering vol 20 no 2pp 128ndash131 2005

[5] Y J Tan J Wu and H Z Deng ldquoProgress in invulnerability ofcomplex networksrdquo Journal of University of Shanghai For Scienceand Technology vol 33 no 6 pp 653ndash668 2011

[6] H Z Deng J Wu Y Li X Lv and Y J Tan ldquoInfluence ofcomplex network topologic structureon system invulnerabilityrdquoSystems Engineering and Electronics vol 30 no 12 pp 2425ndash2428 2008

[7] R Albert H Jeong and A L Barabasi ldquoError and attacktolerance of complex networksrdquo Nature vol 406 no 6794 pp378ndash382 2000

[8] S Sun Z X Liu Z Q Chen and Z Yuan ldquoError andattack tolerance of evolving networks with local preferentialattachmentrdquo Physica A vol 373 no 2 pp 851ndash860 2007

[9] Y N Jiang R Y Li N Huang et al ldquoSurvey on networkreliability evaluation methodsrdquo Computer Science vol 39 no5 pp 9ndash18 2012

[10] W H Qiu Entropy Theory and Its Application in ManagementDecision China Electric Power Press Beijing China 2011

[11] L YMaWHQiu and YQ Yang ldquoRiskmodeling and entropydecision on large complex projectrdquo Journal of Beijing Universityof Aeronautics and Astronautics vol 36 no 2 pp 184ndash187 2010

[12] R X Zhou S C Liu and W H Qiu ldquoSurvey of applications ofentropy in decision analysisrdquo Control and Decision vol 23 no4 pp 361ndash371 2008

[13] L S Wu and Q M Tan ldquoA study of the stability of emergencylogistics network based on network Entropyrdquo ContemporaryFinance amp Economics no 7 pp 60ndash68 2012

[14] B Wang H W Tang C H Guo and Z Xiu ldquoEntropyoptimization of scale-free networksrsquo robustness to randomfailuresrdquo Physica A vol 363 no 2 pp 591ndash596 2006

[15] Y R ZhangMXian andGYWang ldquoAquantitative evaluationtechnique of attack effect of computer network based onnetwork entropyrdquo Journal of China Institute of Communicationsvol 25 no 11 pp 158ndash165 2004


[16] L Q Yong P M Sun and J K Zhang ldquoMaximum entropysocial cognitive optimization algorithm for a class of nonlinearminimax problemsrdquo Computer Engineering and Applicationsvol 46 no 26 pp 36ndash42 2010

[17] G Y Yang and Y H Zhen ldquoResearch on solving method ofmulti-objective decision-makingrdquoMathematics in Practice andTheory vol 42 no 2 pp 108ndash115 2012

[18] Y L Zheng L H Ma and J X Qian ldquoSGA (simplex-genetic algorimth) a universal algorithm for solving minimaxproblemrdquo System Engineering Theory and Practice vol 22 no12 pp 33ndash87 2002

[19] Z B Zhu X Cai and J B Jian ldquoAn improved SQP algorithm forsolving minimax problemsrdquo Applied Mathematics Letters vol22 no 4 pp 464ndash469 2009

[20] Q-J Hu Y Chen N-P Chen and X-Q Li ldquoA modifiedSQP algorithm forminimax problemsrdquo Journal ofMathematicalAnalysis and Applications vol 360 no 1 pp 211ndash222 2009

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Stochastic AnalysisInternational Journal of


as well as the systemrsquos change trend The more uniformthe energy distribution is the greater the entropy achievesOtherwise the smaller the entropy achieves Qiu and herteam has brought entropy in management decision-makingprocedure and then proposed the management entropytheory [10ndash12] In recent years many scholars use generalizedentropy to describe the reliability of a network [13ndash15] Basedon previous research results this paper puts forward theconcept of shortest delivery time entropy involved in logisticsnetworks Moreover the entropy is used to represent deliverytime reliability of a logistics network featured by a negativerelationship with the shortest delivery time reliability

2 Shortest Delivery Time Entropy ofLogistics Networks

21 Model Assumptions and Symbols Description The as-sumptions are made as follows

(1) Vehicles move at a fixed speed when delivering goodsand materials

(2) Vehicles are infinite in quantity of which each deliversonce and departs from the present network immedi-ately to save time

(3) The materials are allowed to be mixed and loadedwith the total weight considered regardless of theirclassification

(4) The time of loading and unloading is ignored

For a directed network 119866 = (119881 119864 119862119863) with 119899 nodesinvolved 119881 = V

119894| 119894 = 1 2 119899 is the vertices set and

119864 = 119890119894119895 is arcs set Additionally 119888

119894119895denotes the capacity

restriction of arc 119890119894119895 119889119894119895denotes the length of arc 119890

119894119895119891denotes

the length of the shortest path for 119866 119891119894119895denotes the length

of the shortest path when 119890119894119895is attacked and fails 119902 denotes

the quantity of required materials V denotes the vehiclersquostraveling speed and 1198661015840 denotes the impaired network when119890119894119895is attacked and fails

22 Shortest Delivery Time Entropy of Logistics NetworkWhen the path of V

119894V119895is attacked and becomes invalid our

research will focus on two timing points as follows


tion that all paths are in good condition


tion that 119890119894119895is attacked and loses effectiveness

Under the circumstances 1199050119905119894119895is utilized to describe the

network reliability in case that the path V119894V119895suffers attack

and thus fails Definitely the longer the length of the arc thehigher the probability of the arc being invalid after an attackThus the probability may be expressed by 119889

119894119895sumV119894V119895isin119864

119889119894119895

As we all know the entropy can be applied to illustrate asystemrsquos reliability Therefore we put forward the conceptof shortest delivery time entropy of logistics network based

on the networkrsquos maximum flow entropy and shortest pathentropy expressed by

119867 = minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln1199050

119905119894119895

(1)

Seen from the formula the shortest delivery time entropy oflogistics network is generalized entropy so the value of 119867may be very large and even can go to infinite The smallerthe value of 119867 the stronger the delivery time reliability ofthe logistics network otherwise the weaker the delivery timereliability of the logistics network

23 Shortest Delivery Time Entropy of Logistics Networkwithout Flow Restriction When the path is unrestricted byflow capacity the shortest delivery time is positively relatedto the shortest path length Referring to its definitions inprevious literature it is available

1199050=119891

V 119905

119894119895=119891119894119895

V (2)

Substituting the above equation into the formula of shortestdelivery time entropy of logistics network we get the follow-ing

119867 = minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln1199050

119905119894119895

= minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln119891V

119891119894119895V

= minus sum

V119894V119895isin119864

119889119894119895

sumV119894V119895isin119864119889119894119895

ln119891

119891119894119895

(3)

Obviously the smaller the value of 119867 the stronger thedelivery time reliability of logistics network However thelarger the value of119867 the weaker the delivery time reliabilityof logistics network

24 Shortest Delivery Time Entropy of Logistics Network withFlowRestriction When the path is restricted by flow capacitywith assumption of total119898 different paths departing from V

1

to V119899to deliver goods and materials the 119896th path is denoted

by 119901119896 and the flow of 119901

119896from 119890

119894119895is denoted by 119862

119894119895119896 Because

119901119896may not pass 119890

119894119895 we introduce 120575

119894119895as

120575119894119895=

0 119901119896not pass 119890

119894119895

1 119901119896pass 119890

119894119895

(4)

where the flow of 119901119896is denoted by 119862

119896 then

119862119894119895119896= 120575119894119895sdot 119862119896 (5)

The flow of each path may not be unique Example 1 willclarify it


(2 2

)

(3 3)

(3 2)

(3 2)

1

2

3 4





2means


0 le 1198621le 2

0 le 1198622le 2

0 le 1198621+ 1198622le 3

(6)

So 1198621and 119862

2may not be unique


119896and 119877

119896is the deliv-



119898

sum

119896=1

119862119894119895119896le 119888119894119895 (7)


119898

sum

119896=1

(119862119896times 119877119896) ge 119902 (8)


max119879119896

Vtimes 119877119896 (9)




1199050= minmax

119879119896

Vtimes 119877119896

st119898

sum

119896=1

(119862119896times 119877119896) ge 119902

119898

sum

119896=1

119862119894119895119896le 119888119894119895 119877119896isin 119873

(10)


(2 2

)

(3 2)

(3 2)(4 2)

(3 5)

1

2

3

4

(3 3)(3 3)

5


(2 2

)

(3 3)

(3 2)

(3 5)

(5 1)

1

2

34





0and 119905119894119895into the for-


3 Case Study






= 875 and 119905(2)0




(1)

0 119905(2)0 119905(1)119894119895 119905(2)119894119895 119889(1)119894119895sumV119894V119895isin119864

119889(1)

119894119895and 119889

(2)

119894119895

sumV119894V119895isin119864119889(2)




V119894V119895isin119864

119889(1)

119894119895

sumV119894V119895isin119864119889(1)

119894119895

ln119905(1)

0

119905(1)

119894119895

= minus (2

21ln 875

1575+3

21ln 875

18

+3

21ln 875875

+3

21ln 875105

+3

21ln 875875

+4

21ln 875

12+3

21ln 875

1575)

= 03292




11989013

11989023

11989024

11989034

11989035

11989045

119905(1)

1198941198951575 18 875 105 875 12 1575

119889(1)

119894119895sumV119894V119895isin119864

119889(1)

119894119895221 321 321 321 321 421 321



11989013

11989023

11989024

11989034

119905(2)

119894119895 12 21 283 8 1123

119889(2)

119894119895sumV119894V119895isin119864

119889(2)

119894119895 216 316 316 516 316


V119894V119895isin119864

119889(2)

119894119895

sumV119894V119895isin119864119889(2)

119894119895

ln119905(2)

0

119905(2)

119894119895

= minus (2

16ln 8

12+3

16ln 8

21+3

16ln 8 times 3

28

+5

16ln 88+3

16ln 8 times 3

112)

= 05494

(11)



than 119866(2)

4 Conclusion


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(2 2

)

(3 3)

(3 2)

(3 2)

1

2

3 4





2means


0 le 1198621le 2

0 le 1198622le 2

0 le 1198621+ 1198622le 3

(6)

So 1198621and 119862

2may not be unique


119896and 119877

119896is the deliv-



119898

sum

119896=1

119862119894119895119896le 119888119894119895 (7)


119898

sum

119896=1

(119862119896times 119877119896) ge 119902 (8)


max119879119896

Vtimes 119877119896 (9)




1199050= minmax

119879119896

Vtimes 119877119896

st119898

sum

119896=1

(119862119896times 119877119896) ge 119902

119898

sum

119896=1

119862119894119895119896le 119888119894119895 119877119896isin 119873

(10)


(2 2

)

(3 2)

(3 2)(4 2)

(3 5)

1

2

3

4

(3 3)(3 3)

5


(2 2

)

(3 3)

(3 2)

(3 5)

(5 1)

1

2

34





0and 119905119894119895into the for-


3 Case Study






= 875 and 119905(2)0




(1)

0 119905(2)0 119905(1)119894119895 119905(2)119894119895 119889(1)119894119895sumV119894V119895isin119864

119889(1)

119894119895and 119889

(2)

119894119895

sumV119894V119895isin119864119889(2)




V119894V119895isin119864

119889(1)

119894119895

sumV119894V119895isin119864119889(1)

119894119895

ln119905(1)

0

119905(1)

119894119895

= minus (2

21ln 875

1575+3

21ln 875

18

+3

21ln 875875

+3

21ln 875105

+3

21ln 875875

+4

21ln 875

12+3

21ln 875

1575)

= 03292




11989013

11989023

11989024

11989034

11989035

11989045

119905(1)

1198941198951575 18 875 105 875 12 1575

119889(1)

119894119895sumV119894V119895isin119864

119889(1)

119894119895221 321 321 321 321 421 321



11989013

11989023

11989024

11989034

119905(2)

119894119895 12 21 283 8 1123

119889(2)

119894119895sumV119894V119895isin119864

119889(2)

119894119895 216 316 316 516 316


V119894V119895isin119864

119889(2)

119894119895

sumV119894V119895isin119864119889(2)

119894119895

ln119905(2)

0

119905(2)

119894119895

= minus (2

16ln 8

12+3

16ln 8

21+3

16ln 8 times 3

28

+5

16ln 88+3

16ln 8 times 3

112)

= 05494

(11)



than 119866(2)

4 Conclusion


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












11989013

11989023

11989024

11989034

11989035

11989045

119905(1)

1198941198951575 18 875 105 875 12 1575

119889(1)

119894119895sumV119894V119895isin119864

119889(1)

119894119895221 321 321 321 321 421 321



11989013

11989023

11989024

11989034

119905(2)

119894119895 12 21 283 8 1123

119889(2)

119894119895sumV119894V119895isin119864

119889(2)

119894119895 216 316 316 516 316


V119894V119895isin119864

119889(2)

119894119895

sumV119894V119895isin119864119889(2)

119894119895

ln119905(2)

0

119905(2)

119894119895

= minus (2

16ln 8

12+3

16ln 8

21+3

16ln 8 times 3

28

+5

16ln 88+3

16ln 8 times 3

112)

= 05494

(11)



than 119866(2)

4 Conclusion


Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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