The Production-Distribution Problem in the Supply Chain Network using Genetic Algorithm

Transcription

1 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// The Producion-Disribuion Problem in he Supply Chain Nework using Geneic Algorihm Suk-Jae Jeong College o Business Adminisraion, Kwangwoon Universiy. 20 Kwangwoon-ro, Nowon-gu, Seoul, Souh Korea. Orcid: Absrac This sudy deals wih inding an opimal soluion or minimizing he oal cos o producion and disribuion problems in supply chain nework. Firs, we presened an inegraed mahemaical model ha saisies he minimum cos in he supply chain. To solve he presened mahemaical model, we used a geneic algorihm wih an excellen searching abiliy or complicaed soluion space. To represen he given model eecively, he marix based real-number coding schema is used. The dierence rae o he objecive uncion value or he erminaion condiion is applied. Compuaional experimenal resuls show ha he real size problems we encounered can be solved wihin a reasonable ime. Keywords: Supply chain nework Geneic algorihm Producion and disribuion problem INTRODUCTION As he organizaion o companies becomes more complicaed and diversiied, and as he compeiion among companies is becoming more inense, he ineres or supply chain is increasing. The supply chain is largely divided ino he supply sage where raw maerials are supplied o, he producion sage where producs are produced and assembled, and he disribuion sage where he producs are ranspored o he DCs(Disribuion Ceners) and cusomers according o demand. The previous sudies have mainly invesigaed each sage separaely wihou considering he complex ineracions beween producion and disribuion aciviies in he supply chain nework. Recenly, an inegraed producion and disribuion problem, which concerns each sage in supply chain nework simulaneously, is widely invesigaed. The inegraed producion and disribuion problem is a very complicaed problem due o is many variables and consrains. Thereore, many echniques were applied o ind he bes soluion or his problem wihin a reasonable ime. We shall inroduce some sudies ha deal simulaneously wih producion and disribuion problems. Bylka (1999) presened a disribuion and invenory decision model under single-vendor, muli-buyer, single-produc, and muli-period circumsances. Moreover, Erengüç e al. (1999) reviewed he sudy o he inegraed producion/disribuion planning in supply chains. They proposed he ramework or analyzing supply chains and ideniied he relevan decisions a each sage o he supply chain. Chandra and Fisher (1994) perormed sudies on he producion scheduling and vehicle rouing problems in order o minimize he ixed, invenory and delivery coss in he single producion aciliy, muliproduc circumsances. Flipo and Finke (2001) perormed sudies by modeling he producion-disribuion problems as a muli-aciliy, muli-produc, muli-period nework low problem. Burn e al. (1985) perormed sudies on he shipmen o he compleed producs in order o minimize he producion, mainenance, and delivery coss or he simpliied manuacuring sysem. Williams (1981) developed he heurisic algorihm o he join producion-disribuion scheduling, which decides he bach size in order o minimize he average invenory mainenance cos in a given period. Cohen (1998) sudied on he sochasic demand in he muli sage producion-disribuion sysem. Zuo (1991) sudied on he heurisic model which assigns producs o he producion plan rom he large scale agriculural producion and disribuion sysem. Kim and Lee (2000) presened a mulisage producion and disribuion-planning problem under he capaciy resricion condiions. Sim and Park (2000) used a heurisic mehod o solve similar problems. Also, Syari e al. (2002) used a GA o solve he supply nework problems. This sudy ocuses he invenory, producion and ransporaion problems in he supply chain nework. This problem is well known as an NP-hard problem. We presened mahemaical models o solve he inegraed producion and disribuion problem in he supply chain nework. As he soluion mehod, he marix-based geneic algorihm is proposed. We implemened our proposed model using a commercial geneic algorihm-based opimizer, Evolver or Microso Excel, and showed ha he soluion can be obained wihin a relaively reasonable ime. This paper is organized as ollows: In Secion 2, deails o he problem are described and he mahemaical model or his problem is given. We describe he srucures o he geneic algorihm and he problem solving process in Secion 3. In Secion 4, compuaional experimen resuls are provided and discussed or his problem. Finally, some concluding remarks are presened in Secion

2 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// MATHEMATICAL MODEL In order o saisy he cusomer s demand or various producs wih a minimum cos, we should deermine how many producs are produced and ranspored. Fig. 1 is an example o he inegraed supply chain model. Our sudy represens he problem wih hree sages or an eecive cos analysis. The irs sage is a supplier sage. A he supplier sage, suppliers produce raw maerials. The second sage is a acory sage. A he acory sage, he raw maerials rom he suppliers are ransormed ino he inal producs. The las sage is a disribuion cener sage. In he inegraed supply chain model, he producion plans are deermined and he proper amoun o producion and ransporaion ha can saisy all capaciies and demands wih a minimum cos is assigned o each plan. In his sudy, he model using a mixed ineger linear programming is proposed. The model is represened as ollows. Figure 1: The supply chain nework Parameers S s S S d h is h i h P h Pd CT is acory CT pd CT pdc` c CP is CP p K pd K p K is TD TF TS : ixed cos a supplier s : ixed cos a acory : ixed cos a DC d : uni cos o invenory o source i a supplier s : uni cos o invenory o source i a acory : uni cos o invenory o produc p a acory : uni cos o invenory o produc p a DC d : uni cos o ransporaion rom supplier s o : uni cos o ransporaion rom acory o DC d : uni cos o ransporaion rom DC d o cusomer : uni cos o producing o source i a supplier s : uni cos o producing o produc p a acory : capaciy or produc p a DC d : capaciy or produc p a acory : capaciy or he source i a supplier s : capaciy or all DCs : capaciy or all acories : capaciy or all suppliers Problem assumpions, parameers and variables Assumpion We impose he ollowing assumpions o design he mahemaical model. Firs, muli-produc and muli-period are considered. Second, he cusomer s demand a each period is given. Indices i : number o sources ( i = 1, 2,, i ) p : number o producs ( p = 1, 2,, p ) s : number o suppliers ( s = 1, 2,, s ) : number o acories ( = 1, 2,, ) d : number o disribuion ceners (d = 1, 2,, d ) c : number o cusomers ( c = 1, 2,, c ) : number o periods(=1,2, ) Variables P is P p I is I i I p I pd T is T pd : producion amoun o raw maerial i a supplier s a end o period : producion amoun o produc p a acory a end o period : invenory amoun o raw maerial i a supplier s a end o period : invenory amoun o raw maerial i a acory a end o period : invenory amoun o produc p a acory a end o period : invenory amoun o produc p a DC d a end o period : ransporaion amoun o raw maerial i a supplier s o acory a end o period : ransporaion amoun o produc p a acory o DC d a end o period 13571

3 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// T pdc B is B pd Z s = : ransporaion amoun o produc p a DC d o cusomer c a end o period : demand o raw maerial i a acory o supplier s a end o period : demand o produc p a DC d o acory a end o period 1, i producion akes place a supplier s 0, oherwise s.. B T i, s, (4) is is T I 1 P I i,, p, (5) p 1 i p is d i pd p I T P I i, s,, (6) i Z {0,1} (7) I, I, B, T, T 0 i, s, d,, p, (8) i p is is pd Z = Z d = 1, i producion akes place a acory 0, oherwise 1, i DC d is opened 0, oherwise A he acory sage, he objecive uncion is o minimize he sum o he ixed, invenory, producion, and ransporaion coss. Consrain (4) means ha he oal amoun o purchasing a acories equals he oal amoun o ransporaion a suppliers. Consrain (5) is concerned wih he invenory o producs a acory during any given period, and Consrain (6) is concerned wih he invenory o raw maerials a acory during any given period. Consrain (7) is he binary variable o he acory. Mahemaical model ormulaion Supplier sage min z s.. 1 is I ishis SS Z S P CPis is i s s i s i s I P T I i, s, (1) is is is Z {0,1} s (2) S I, T 0 is is i, s,, (3) A he supplier sage, he objecive uncion is o minimize he producion, invenory, mainenance and ixed coss o raw maerials. The consrain (1) is concerned wih he raw maerial invenory a suppliers during any given period. Consrain (2) is he binary variable o he supplier. Facory sage min z I h I i hi p p p i P pcpp p p d T pd CT pd S T is Z CT is Disribuion cener sage min z s.. I h pd pd d p d S d Z d + p d T pcdctpcd B pd Tpd p, d, (9) I 1 T I (10) pd pdc pd p, d, c c T K pdc pd d, p, (11) Z {0,1} d (12) d I, B, T, T 0, c,, p, d (13) pd pd pcd pd The objecive uncion a he DC sage is o minimize he sum o he ixed, invenory, and ransporaion coss. Consrain (9) means ha he oal amoun o purchasing a DCs equals he oal amoun o ransporaion a acories. Consrain (10) is concerned wih he invenory o producs a DC during any given period, and consrain (11) ensures ha he ransporaion amoun should no exceed he capaciy o DCs. Consrain (12) is he binary variable o DC. I can be he opimal supply chain nework ha minimizes he invenory, producion and ransporaion coss i each sage model can be inegraed ino one

4 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// GENETIC ALGORITHM Geneic algorihm is widely used as a mehod o solve he complicaed opimizaion problems. The algorihm is easily applied o all opimizaion problems, and i has an excellen searching abiliy or complicaed soluion spaces. I is very suiable or solving large-scale mahemaical problems wih many variables and consrains and has meris o easily adding consrains or alering objecive uncions. A general procedure o a geneic algorihm is as ollows. begin end 0; iniialize P(); evaluae P() ; while (no erminaion condiion) do begin end recombine P() o yield C(); evaluae C(); selec P(+1) rom P() and C(); + 1 ; Represenaion o he chromosome The irs sep in building a geneic algorihm is o presen he poenial soluion as a geneic represenaion. The represenaion may be dependen on he problem o be solved. In his sudy, real encoding is applied o design consrains easily. Chromosomes o real encoding were presened as a marix-based represenaion. This sudy considered 3 suppliers, 3 acory, 3 disribuion ceners and 3 cusomers. Figure 2 shows he represenaion o chromosome a he period. The irs, second and hird sub-srings are he binary variables a suppliers, acories, and DCs. The ourh is a decision variable or he oal amoun o ransporaion o cusomer. The ih and sixh are he decision variables or he oal amoun o ransporaion o DCs and he oal invenory a DCs. The nex 3 sub-srings represen he decision variables or he amoun o ransporaion o acories and he amoun o producion and invenory a acories. The las wo sub-srings represen he decision variables or he amoun o producion o raw maerials and he oal invenory a suppliers. Geneic operaions Crossover The crossover operaion used in his sudy is a uniorm crossover. When selecing a pair o paren chromosomes, he uniorm crossover produces a emplae, exchanges parenal genes on 1 in he emplae, and produces heir osprings. The exchanged genes are deermined by a emplae, and he emplae is randomly produced. In Fig. 3, he crossover operaion in his sudy is shown. Figure 2: The illusraion o represenaion Figure 3: The illusraion o crossover operaion 13573

5 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Muaion The muaion randomly alers genes during he process o copying a chromosome rom one generaion o he nex. Aer producing random variable beween 0 and 1 or each gene, i a variable ges a number ha is less han or equal o he muaion rae, hen ha variable is muaed. The chromosome genes are real numbers. I muaed, he number o swapping exchanged ino randomly occurred values beween he upper and lower limi o consrains. In Fig. 4, he muaion operaion in his sudy is shown. Figure 4: The illusraion o muaion operaion sraegies dealing wih he consrains. The sraegies are classiied ino rejecing sraegy, repairing sraegy, modiying geneic operaor sraegy, penaly sraegy (Michalewicz 1991, Michalewicz 1996, Gen 1997). The sraegy in his sudy is a repairing sraegy. When he produced ospring chromosome violaes he consrains, i is sen o he paren chromosome unil being a easible soluion. Terminaion condiion Generally, he 2 cases are used as he erminaion condiions. Firs, he GA is erminaed when reaching he given generaion. Secondly, When here is no change in soluion improvemen during he given compuaion ime, he generaion is erminaed. This sudy considers he second case as he erminaion condiion. The generaion sops when he dierence rae beween he preceding bes soluion and he curren bes soluion is close enough o be accepable. Then, we assume ha we ound he opimal soluion. Selecion The selecion in his sudy uses a rank-based selecion which is known as he simples and mos eecive way o conrol selecion. This is designed o improve he problems o he early convergence o roulee wheel selecion (Whiely, 1989). In he selecion, he rank is deermined according o he iness size, and an individual is seleced hrough a sochasic universal sampling. Consrains When applying a geneic algorihm or complicaed opimizaion problems, i is imporan o consider how o deal wih he consrains. Because he presened problem in his sudy has many consrains, he geneic operaor can produce an ineasible soluion. Recen sudies have proposed a ew EXPERIMENTS AND RESULTS Compuaional experimens have been provided in order o demonsrae he eeciveness and reasonabiliy o he proposed mahemaical model. Considering he complexiy o solving such a model, we have applied a geneic algorihm. We implemen our proposed model using a commercial geneic algorihm-based opimizer, Evolver or Microso Excel. The proposed model is esed by using he condiions or he es problem given in Table 1. The iniial invenory a each sage or he es problem is given in Table 2. The capaciy, demand and he ixed cos are given in Table 3, and he relaed coss a each sage are given in Tables 4 hrough 10. As or he geneic operaor parameers, he crossover rae is se o 0.5 and he muaion rae is se o 0.1. The populaion size is 30 and he number o he generaion is se o perorm every 1000 imes. Table 1: The condiion or he es problem Number o suppliers Number o acories Number o DCs Number o cusomers Number o producs and raw maerials Table 2: Iniial Invenory Producs(Sources) DCs Facories Suppliers

6 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Table 3: Capaciy, demand and ixed cos Suppliers Facories DCs Source capaciy Produc capaciy Fixed Produc capaciy Fixed cos cos 1 2 ixed cos Table 4: Shipping cos o acory Source1 Facories Source2 Facories Suppliers Suppliers Table 5: Shipping cos o DC Produc1 DCs Produc2 DCs Facories Facories Table 6: Shipping cos o cusomer Produc1 Cusomers Produc2 Cusomers DC DC Table 7: Producion cos a supplier Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source

7 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Table 8: Producion cos a supplier Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source Table 9: Invenory cos a supplier Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source Table 10: Invenory cos a supplier Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source Table 11: Experimen resuls Generaion # Toal cos Dierence rae Generaion # Toal cos Dierence rae ,476, ,821, ,104, ,746, ,426, ,711, ,370, ,689, ,806, ,699, ,638, ,765, ,160, ,897, ,274, ,057, ,196, ,210, ,023, ,371, Table 11 shows he experimen resuls o he oal cos and dierence rae. Fig. 5 shows he change o objecive uncion values. Fig. 6 shows he change o he dierence raes in objecive uncion values. When he dierence rae is less han 0.2% aer generaions, he generaion is erminaed. The opimal soluion or his experimen is 539,746,

8 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// 740,000, ,000, ,000, ,000,000 Objec Funcion 660,000, ,000, ,000, ,000, ,000, ,000, ,000, ,000, ,000, The number o Generaion Figure 5: Bes soluions in each generaion Dierenc rae The number o Generaion Figure 6: Dierence raes o he bes soluions in each generaion Table 12 hrough 19, we show ha he bes producion and disribuion planning or his problem is given by our represened model

9 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Table 12: Resuls o he amoun o ransporaion o acory in each period =1 =2 =3 =4 Source1 Facories Source2 Facories Suppliers Suppliers Table 13: Resuls o he amoun o ransporaion o DC in each period =1 =2 =3 =4 Produc1 Cusomers Produc2 Cusomers DCs DCs

10 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Table 14: Resuls o he amoun o producion a supplier in each period Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source2 = = = = Table 15: Resuls o he amoun o ransporaion o cusomer in each period =1 =2 =3 =4 Produc1 Cusomers Produc2 Cusomers DCs DCs Table 16: Resuls o he amoun o producion a acory in each period Facory 1 Facory 2 Facory 3 Produc1 Produc2 Produc1 Produc2 Produc1 Produc2 = = = =

11 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// Table 17: Resuls o he amoun o invenory a supplier in each period Supplier 1 Supplier 2 Supplier 3 Source1 Source2 Source1 Source2 Source1 Source2 = = = = Table 18: Resuls o he amoun o invenory a acory in each period Facory 1 Facory 2 Facory 3 Produc1 Produc2 Produc1 Produc2 Produc1 Produc2 = = = = Table 19: Resuls o he amoun o invenory a DC in each period DC 1 DC 2 DC 3 Produc1 Produc2 Produc1 Produc2 Produc1 Produc2 = = = = CONCLUDING REMARKS In his sudy, we represened he mahemaical model considering he muli-sage, muli-produc, muli-period in he supply chain nework. The given model has many decision variables and consrains and requires los o compuaion ime. Considering he complexiy o he model, he geneic algorihm has been applied or solving his model. We used a real-number coding as he geneic represenaion o represen he given model eecively. We considered he repair sraegy as he consrains and dierence rae o he objecive uncion value as erminaion condiion. Compuaional experimenal resuls show ha he real size problems we encounered can be solved wihin a reasonable ime. ACKNOWLEDGEMENTS The work repored in his paper was conduced during he sabbaical year o Kwangwoon Universiy in REFERENCES [1] Burns L Hall, W Blumeneld D, Dazango C (1985) Disribuion sraegies ha minimize ransporaion and invenory coss. Operaions Research 33(3) : [2] Bylka S (1999) A dynamic model or he singlevendor, muli-buyer problem. Inernaional Journal o Producion Economics 59: [3] Chandra P, Fisher M (1994) Coordinaion o producion and disribuion planning. European Journal o Operaional Research 72 :

12 Inernaional Journal o Applied Engineering Research ISSN Volume 12, Number 23 (2017) pp Research India Publicaions. hp:// [4] Cohen M, Lee H (1988) Sraegic analysis o inegraed producion-disribuion sysems: models and mehods. Operaions Research 36 : [5] Erengüç S S, Simpson N Vakharia A J (1999) Inegraed producion/disribuion planning in supply chains: An invied review. European Journal O Operaional Research 115: [6] Flipo C D, Finke G (2001) An inegraed model or an indusrial producion-disribuion problem. IIE Transacions 33 : [7] Kim H K, Lee Y H (2000) Producion/disribuion planning in SCM using simulaion and opimizaion model. Proceedings o Korean supply chain Managemen sociey [8] Sim E, Jang Y, Park J (2000) A sudy on he supply chain nework design considering muli-level, muliproduc, capaciaed aciliy. Proceedings o Korean supply Chain Managemen Sociey [9] Williams j(1981) Heurisic echniques or simulaneous scheduling o producion and disribuion in muli-echelon srucures : heory and empirical comparisons. Managemen Science 27 : [10] Zou M, Kuo W, McOrbers K (1991) Applicaion o mahemaical programming o a large-scale agriculural producion and disribuion sysem. Journal o he Operaional Research Sociey 42(8) :