Research Article Integrated Location-Production-Distribution Planning in a Multiproducts Supply Chain Network Design Model

Transcription

1 Mathematical Poblems in Engineeing Volume 2015, Aticle ID , 13 pages Reseach Aticle Integated Location-Poduction-Distibution Planning in a Multipoducts Supply Chain Netwok Design Model Vincent F. Yu, 1 Nu Mayke Eka Nomasai, 1 and Huynh Tung Luong 2 1 DepatmentofIndustialManagement,NationalTaiwanUnivesityofScienceandTechnology,No.43,Section4, Keelung Road, Taipei 10607, Taiwan 2 Industial and Manufactuing Engineeing, Asian Institute of Technology, 58 Moo 9, Paholyothin Highway Klong Luang, Pathumthani 12120, Thailand Coespondence should be addessed to Nu Mayke Eka Nomasai; mayke sai@yahoo.com Received 31 Octobe 2014; Revised 7 Febuay 2015; Accepted 23 Febuay 2015 Academic Edito: Xuefeng Chen Copyight 2015 Vincent F. Yu et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. This pape poposes integated location, poduction, and distibution planning fo the supply chain netwok design which focuses on selecting the appopiate locations to build a new plant and distibution cente while deciding the poduction and distibution of the poduct. We examine a multiechelon supply chain that includes supplies, plants, and distibution centes and develop a mathematical model that aims at minimizing the total cost of the supply chain. In paticula, the mathematical model consides the decision of how many plants and distibution centes to open and whee to open them, as well as the allocation in each echelon. The LINGO softwae is used to solve the model fo some poblem cases. The study conducts vaious numeical expeiments to illustate the applicability of the developed model. Results show that, in small and medium size of poblem, the optimal solution can be found using this solve. Sensitivity analysis is also conducted and shows that custome demand paamete has the geatest impact on the optimal solution. 1. Intoduction A supply chain is a netwok that consists of a set of geogaphical facilities (supplies, plants, and waehouses o distibution cente). Though those facilities, thee is mateial flow fom supplie, plant, waehouse, and end in the custome. It aims at binging the ight amount of the ight poduct to the ight place at the ight time [1]. Moeove, a supply chain netwok design is a stategic decision that has high isk and long-tem impact in the supply chain system. The impact of efficiency supply chain has become moe impotant on the business competitiveness [2]. The topic has tiggeed both eseaches and pactitiones to pay moe attention to the supply chain netwok design. Many studies have been conducted to help the pactitione in making the best decision on a supply chain netwok. Indeed, detemining the best supply chain netwok is a challenge, stating with poblem identification, poblem fomulation, and its final solution and decision. Today s competition among companies and maket s globalizationhaveesultedinfimsdevelopingasupplychain that can espond quickly to customes need. In the cuent business envionment, a company has to educe costs while impoving its custome sevice level to emain competitive [3], which also helps maintain pofit magins. In ode to achieve these goals, a company should appopiately select the location of the factoy and the distibution cente [4 6]. Accoding to Altipamak et al. [7], an optimal, efficient, and effective supply chain platfom is povided by supply chain netwok (SCN) design, which also helps to impove supply chain pefomance. Moeove, Ballou [8] noted that the SCN design goal is to maximize the financial atio, which is elevant to the objective of gaining the maximum etun of investment at the minimum cost. Supply chain management is divided into two levels: stategic and opeational. The stategic level pimaily is about the cost-effective location of facilities (plants and distibution centes), the flow of poducts thoughout the entie supply chain system, and the assignment in each echelon [9 12]. The opeational level is about the safety stock of each poduct in each facility, the eplenishment size, fequency,

2 2 Mathematical Poblems in Engineeing tanspotation, and lead time, and the custome sevice level. Accoding to Beamon [13], detemining an effective supply chain is an impotant component in supply chain design. In addition, the decisions egading in which facilities the poduct should be made and how to seve customes ae vey citical [14]. This pape povides a system optimization pespective in stategic planning fo a supply chain netwok design that allows simultaneously detemining the best location of facilities, aw mateial flow, and poduct flow on vaious echelons. Pevious eseach on stategic planning fo supply chain stats by consideing the basic poblems that have seveal chaacteistic, namely, single-peiod, single-poduct, single-echelon, and deteministic [15 25]. Howeve, this is not sufficient to cope with the ealistic poblem. Theefoe, many extensions to the basic poblem ae needed to make the poblem moe ealistic. In this case, ou pape consides multipeiod, multipoduct, and multiechelon which ae still in deteministic situation to make the basic stategic planning poblem moe easonable. A supply chain netwok design model helps manages conduct stategic planning fo thei company by selecting the best facility location that minimizes the total cost of the supply chain. The poposed multipoduct supply chain netwok design model heein helps in choosing the appopiate location of a new plant and distibution cente as well as the distibution of the poduct and aw mateials when the demand vaies duing the diffeent time peiod. Moeove, multiechelon which epesents the multitype of facility is the cucial aspect to be consideed in stategic planning. The pape is oganized as follows. Section 2 pesents pevious eseach to find the gap between this study and ealie elated eseach. Section 3 descibes the poblem definition and the poposed mathematical model. Section 4 contains the numeical expeiments fo the small and medium cases. Section 5 offes a sensitivity analysis esult of the poposed model. Finally, Section6 consists of the conclusion and suggestions fo futue eseach. 2. Liteatue Review Seveal eseach integated supply chain netwok designs have been developed to help pactitiones solve thei supply chainplanning.syaifetal.[26] studiedamultiechelon, single-poduct logistic chain netwok model and poposed a novel technique as the solution method, called the spanning tee-based genetic algoithm (st-ga). The model is fomulated by using a mixed intege line pogamming (MILP) model. Thei model only consides a single-poduct. To demonstate the effectiveness and efficiency of thei poposed method, it is compaed to the taditional matix-based genetic algoithm (m-ga). The expeiment esult shows that the poposed method pesents a bette solution almost all time and also pefoms bette in computational time and memoy fo computation. Jakeman et al. [27] consideed the stategic and opeational planning level decision in thei eseach by developing a static model fo a multiechelon, multipoduct supply chain netwok design. They examined the single souce distibution system. Fo thei solution, they used Lagangian elaxation and a heuistic algoithm that utilizes the Lagangian solution. The esult of thei computation shows that the solution method is both efficient and effective. Shen [20] poposed a supply chain netwok design model with pofit maximization as the objective function, but it consides only a single-poduct. In addition, the company may lose the custome if the poduct s pice is highe than the custome eseve pice. Altipamak et al. [28] studied a single-poduct, multiechelon, and multiobjective SCN design. They set up a solution pocedue based on the genetic algoithm (GA) to find the optimal solution to thei poblem. The multiobjective optimization poblem consists of many optimal solutions, called Paeto-optimal solutions. The poblem is fomulated as a multiobjective mixed intege nonlinea pogamming model. The objectives ae to minimize total cost, maximize custome sevice, and maximize utilization of the distibution centes (DCs). Altipamak et al. [7] pesented a solution pocedue fo a multipoduct supply chain netwok (SCN) design based on the steady-state genetic algoithm (ssga) with a new encoding stuctue. They consideed a single souce, multipoduct, andmultiechelonsupplychainnetwokdesigninwhichthe numbe of customes and thei demands ae assumed to be known. The poblem, which is the NP-had poblem, is povided in mixed intege pogamming fomulation. In ode to investigate the effectiveness of the ssga, thee othe heuistic appoaches ae also used: Lagangian heuistic (LH), hybid genetic algoithm (hga), and simulated annealing. The expeiment s esults show that ssga has a bette solution than the othe heuistic appoaches used. Ying-Hua [29] adopts the model developed by Altipamak et al. [7], which consides a single souce, multipoduct, and multiechelon supply chain netwok design, but the model only has multisouces instead of a single souce. Additionally, the plants and DCs that ae open ae known. To veify the efficiency of his poposed method, he compaed it to othe algoithms, such as mathematical pogamming, the simple genetic algoithm, the coevolutionay genetic algoithm, and the constaint-satisfaction genetic algoithm. The expeimental esult in Taiwan s textile industy shows that the poposed method of Ying-Hua [29] pefoms bette than othe eseaches methods. Bhutta et al. [30] developed an integated location, poduction, distibution, and investment mixed intege linea pogamming (MILP) model in a two-echelon, multipoduct, multipeiod, and flexible facility capacitated with maximum pofit as the objective function. Cóccola et al. [31] set up an integated poduction and distibution MILP model in a multiechelon, multipoduct, and single-peiod setting with minimum total cost as the objective function. They conducted an empiical numeical expeiment on six Euopean counties. Fahimnia et al. [32] pesented an integated poduction and distibution planning MILP model fo a two-echelon SC that consides seveal eal wold vaiables and constaints. They used GA to optimize the model and solved the medium-size case poblem in thei numeical expeiment. In addition, Bashii et al. [33] and Badi et al. [34] developed a multiple-echelon, multiple-commodity

3 Mathematical Poblems in Engineeing 3 mathematical model fo stategic and tactical planning. The model is developed as a MILP model in fou echelons, but they did not conside satisfying the demand constaint. Many papes have developed a supply chain netwok design though a mixed intege pogamming (MIP) model [35, 36]. Howeve, in fact, the quantity of the commodity is usually an intege. Ou pape consides location, poduction, and distibution planning in the supply chain netwok design poblem with multiechelon, multipoduct, and multipeiod chaacteistics in which the poposed model is pue intege linea pogamming (PILP) model, having fou echelons, multipoduct, and multipeiod demand and satisfying a demand constaint. Consideation of using PILP is intended fo poviding quality guaantees of optimality [37]. Moeove, its application can be used fo low volume discete manufactuing company of lage equipment. In tems of multiplicity, ou pape consides the most complex model in the aea of integated poduction and distibution planning. 3. Poblem Definition and Model Fomulation Development of an efficient and effective supply chain is vey citical to achieving good pefomance. Theefoe, indepth analysis is needed when opening a new plant and new distibution cente in the appopiate location. Aside fom that, multiple poducts instead of a single-poduct need to be consideed in the poblem of supply chain netwok design and taking into account that the intege quantity in the supply chain netwok design is moe applicable. To deal with this poblem, this pape develops a pue intege linea pogamming (PILP) model that focuses on detemining the locations of the plants and distibution centes, as well as the numbe of those facilities, so that custome needs ae satisfied at a minimum total cost duing the planning hoizon. This eseach focuses on the supply chain design poblem with the following chaacteistics. (1) The distibution netwok unde consideation is a multiechelon and multipoduct supply chain netwok. (2) Demand in each time peiod (yealy) is deteministic and known. (3) The plant o DC does not need to be opened at the beginning of the planning hoizon, and when one is opened, it will not be closed. (4) Customes can eceive the poduct fom multiple DCs. This eseach develops a mathematical model that helps to detemine the numbe and locations of plants and distibution centes in a supply netwok and the assignment-elated demand allocation in each echelon. Figue 1 depicts the system consideed in this eseach. Accoding to Jayaaman and Pikul [38], the key components of supply chain modeling that should be consideed by the model builde ae supply chain dives, supply chain constaints, and supply chain decision vaiables of the model. Supply chain dives epesent the goal setting of the model, supply chain constaints epesentthelimitationsontheangeofdecisionaltenatives, and supply chain decision vaiables ae the components that set limits on the ange of decision outcomes. The objective function of this model is to minimize the total cost of the system. Accoding to Fahimnia et al. [39], the total cost in the poduction and distibution netwok natually consists of the poduction cost and distibution cost.poductioncostisthesumofthefixedopeningcost and the vaiable poduction cost, while the distibution cost is the sum of the fixed cost of opening the distibution cente cost, the vaiable inventoy cost, and tanspotation cost. Theefoe, the total cost in this model consists of cost to open theplant,costtopuchaseandtanspotawmateialfom supplie to plant, cost to manufactue the poducts, cost to open the DCs, cost to tanspot the poduct fom plant to DCs, inventoy holding cost of each poduct in DCs, and cost totanspotthepoductfomdcstocustome. The decision vaiables in poduction and distibution planning consist of the supplie stage and distibution stage. In the supplie stage, the decision vaiables consist of how many supplies should be thee, how many quantity and fequency of shipment fom each supplie, what is the configuation of the supplie-plant distibution netwok, and wheeaetheselectedlocationsofthesuppliesandplants. The decision vaiables in the distibution stage consist of how many distibution centes to opeate, whee should they be located, and inventoy in the distibution cente [40, 41]. Theefoe, we decide that the decision vaiables in this eseach encompass detemining whee the plant and distibution cente will be opened, thei distibution, and the poduction of the plant when it is opened. The mathematical model of this eseach is developed basedonthemodelofaltipamaketal.[28]. They set up a mathematical model that consides a single-poduct, fou echelons, a single souce, and static demand. Ou pape s mathematical model has multipoducts, 4 echelons, multisouces, and multipeiod demand chaacteistics. To meet fluctuating custome demand, the end poducts and the infomation exchange ae conducted egulaly though plants and distibution centes within a given poduction and sevice netwok. Indicatos of supply chain pefomance such as fill ate, custome sevice level, associated cost, and capability of esponse can be obtained unde diffeent netwok configuations though an evaluation of the supply chain netwok configuation itself. Diffeent netwok configuations involve diffeent stock levels of aw mateials, subassemblies and end poducts, distibution cente locations, poduction policy (make-to-stock o make-to-ode), poduction capacity (amount and flexibility), allocation ule fo limited supplies, and tanspotation modes. The common multiechelon supply chain netwok (MSCN) poblem seaches fo a netwok configuation at a minimum cost. This is a NP-had (nondeteministic polynomial-time had) poblem that employs a mathematical pogamming fomulation as a natual way to build an NP-had poblem, although it is not an efficient pocedue. In Yeh [42], some paametes ae known in advance, namely, the numbes and capacities (demand) of supplies, plants, distibution centes (DCs) and customes, the unit tanspotation cost between supplies and plants, plants

4 4 Mathematical Poblems in Engineeing D p,o p W j,g j,c ji b pjt m jit d it Q spt s= 1 S supplie (fixed location) p= 1 P j= 1 J plant distibution cente (location to be selected) (location to be selected) i= 1 l custome zone (fixed location) Figue 1: The supply chain netwok unde consideation. and DCs, and DCs and customes, as well as the fixed cost fo opeating plants and DCs. The goal of his eseach is to identify the locations of plants and DCs and the quantities shipped between the vaious points that minimize total cost and tanspotation costs. The poblem in his eseach is fomulated using a pue intege pogamming (PILP) model. The poposed model uses the notations shown in Notations section. The poblem is fomulated as follows: Min subject to t=t Z={E p O p1 + p t=t + t=1 s p + {F j G j1 + j t=t + t=1 t=t + t=1 p j j t=2 E p (O pt O p(t 1) )} t=t B Vsp Q Vspt + V t=t t=2 K pj b pjt t=t h jt Y j + t=1 p F j (G jt G j(t 1) )} t=1 j i A p q pt L ji m jit, (1) m jit d it, i, t, (2) j p b pjt a + h j(t 1) a W j G jt j, t, (3) 1 q PR pt D p O pt p, t, (4) b pjt q pt, p, t, (5) j U V q pt Q Vspt V,p,t, (6) s Q Vspt C sv V,s,t, (7) p h jt = p b pjt +h j(t 1) i m jit j,, t, (8) O pt O p(t 1) t=2,...,t, (9) G jt G j(t 1) t=2,...,t, (10) O pt = {0, 1}, (11) G jt = {0, 1}, (12) Q Vspt 0,and intege, (13) q pt 0,and intege, (14) b pjt 0,and intege, (15) m jit 0,and intege, (16) h jt 0,and intege, (17) h j0 =0 j,. (18) Equation (1) shows the objective function of the model. Equation (2) is the constaint fo satisfying custome demand. Equation (3) is the capacity constaint fo DC j. Equation (4) is the capacity constaint fo plant p. Equation (5) is the limitation of the poduct that is tanspoted fom plant p to all DCs. Equation (6) is the equiement of aw mateial V fo poduction. Equation (7) is the capacity constaint fo the supplie. Equation (8) is the inventoy

5 Mathematical Poblems in Engineeing 5 Table 1: Data fo instances in the numeical expeiment. Single-peiod Multipeiod Multipeiod small-sized poblem small-sized poblem medium-sized poblem Numbe of customes Numbe of locations fo distibution centes Numbe of locations fo plants Numbe of supplies Numbe of poducts Numbe of aw mateials Length of planning hoizon 1 2, 5, 10 4 Table 2: Paamete values fo the numeical expeiment. NumbePaamete Geneated using 1 Costtoopentheplantp(E p ) Intege unifom distibution U(25000, 30000) 2 Maximum capacity of plant p(d p ) Intege U(18, 22) 3 Poduction ate of manufactuing poduct (PR ) Intege U(10, 15) 4 Cost of tanspoting and puchasing aw mateial V fom supplie s to plant p(b Vsp ) U(10, 15) 5 Unit manufactuing cost of poduct at plant p(a p ) U(8, 10) 6 Costoftanspotingpoduct fom plant p to DC j(k pj ) U(4,8) 7 Capacity of supplie s fo aw mateial V(C sv ) Intege U(1250, 1500) 8 Utilization ate of aw mateial v pe unit of finished poduct (U V ) Intege U(1, 5) 9 CosttoopenDCj(F j ) Intege U(20000, 30000) 10 Capacity of DC j(w j ) Intege U(250, 350) 11 Space equiement ate of poduct on a DC(a ) U(1,2) 12 Demand at custome zone i fo poduct in time peiod t(d it ) Intege U(30, 50) 13 Unit inventoy holding cost of poduct in DC j(y j ) U(5, 10) 14 Cost of tanspoting poduct fom DC j to custome i(l ji ) U(8, 12) balance equation of poduct in DC j at time peiod t. Equation (9) ensues that the plant only opens once. Equation (10) ensues that the DC only opens once. Equations (11)-(12) ae binay constaints fo the decision vaiables. Equations (13) (17) give the equiement of nonnegativity. Equation (18) shows the initial inventoy in DC at the beginning of the planning hoizon. 4. Numeical Expeiment This pape examines both small-sized and medium-sized poblems. We conducted the expeiment mainly to show the costsavingsadvantageofthepoposedintegatedmodelove pevious elated published models. Tables 1 and 2 pesent the indices and paametes of the model, espectively. Table 1 gives the data of the test instances. Thee ae five test instances: one single-peiod small-sized poblem (Instance1),theemultipeiodsmall-sizepoblems(Instances2,3,and4),andonemultipeiodmedium-sizepoblem (Instance 5). Instance 1 will be used as ou base fo the compaative analysis to show the advantages of integating multiple peiod planning. In Instance 1, the numbes of custome, DC, plant, supplie, poduct, and aw mateials ae all set to be 2, except fo the planning hoizon which is set to be 1 yea. We adopt the same data fo Instances 2, 3, and 4 with diffeent planning hoizons of 2, 5, and 10 yeas, coespondingly. Lastly, in Instance 5, all paametes ae set to be 4. Table 2 gives the coesponding paamete value of the model. The model consists of 14 paametes which ae all geneated using unifom distibution, some in integes and some in eal numbes. Ou poposed model is exactly solved using LINGO. The model fomulation using the LINGO famewok consists of thee sections: (1) sets of vaiables and paametes; (2) coesponding data sets; (3) mathematical model. The LINGO solve used banch and bound method to solve the poblem. This method is an intelligent enumeation pocess seeking a sequence of bette and bette solutions until the best solution is found. In the pocess of finding the best solution, the memoy is updated with the best objective function value found so fa. This pocess continues until no futhe impovement can be found. The esults of location and assignment planning fo the small-size numeical expeiment can be seen in Tables 3 6. The esults of location and assignment planning fo the medium-size one can be seen in Table 7. Table 3 shows the netwok design esults of single-peiod small-sized poblem. This implies that having plant 2 opened mateials 1 and 2 only come fom supplie 2. Plant 2 delives all poducts 1 and 2 to DC 1; then DC 1 delives all poducts

6 6 Mathematical Poblems in Engineeing Table 3: Location and assignment planning fo the single-peiod small-sized poblem. Optimal solution(objective function) $81, Peiod Supplie(Plant aw mateial ) Plant(DC poduct ) DC(Cus poduct ) 1 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) Table 4: Results fo multipeiod small-size poblem, T=2yeas. Optimal solution(objective function) $94, Peiod Supplie(Plant aw mateial ) Plant(DC poduct ) DC(Cus poduct ) 1 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 2 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) Table 5: Netwok design esult fo multipeiod small-size poblem, T=5. Optimal solution(objective function) $137,372.6 Peiod Supplie(Plant aw mateial ) Plant(DC poduct ) DC(Cus poduct ) 1 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 2 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 3 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 4 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 5 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) Table 6: Netwok design esults fo multipeiod small-size poblem, T=10. Optimal solution(objective function) $207,537.2 Peiod Supplie(Plant aw mateial ) Plant(DC poduct ) DC(Cus poduct ) 1 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 2 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 3 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 4 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 5 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 6 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 7 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 8 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 9 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) 10 2(2 1,2 2 ) 2(1 1,1 2 ) 1(1 1,1 2,2 1,2 2 ) to all customes. The same way of explanation uses fo the solution configuations in the othe columns and fo the succeeding tables until Table 7.Plant 2 delives all poducts 1and2toDC2;thenDC2distibutesthesepoductstoall customes. This gives us the minimum total cost of $81, Table 4 shows the optimal netwok design fo Instance 2 (T = 2). The same optimal solution as in Table 3 is obtained fo this instance fo all peiods. Howeve, the optimal total cost is diffeent with the value of $94, Consequently, we obseve a 17% incease in total cost by doubling T to 2 yeas. Tables 5 and 6 show the optimal netwok design fo Instances 3 and 4 with T=5and T=10,espectively.The same solution as in Table 3 is obtained fo all peiods in both instances, except that they have diffeent optimal total costs. We have total cost of $137,372.6 fo Instance 3 (T = 5) and $207,537.2 fo Instance 4 (T = 10). Hee, we obseve a 51% incease in total cost by doubling the planning hoizon to 10 yeas. Table 7 shows the netwok design esults of mediumsizedpoblem.onlyplants1,3,and4aeopened.supplie 1 delives aw mateials 2 and 3 to plant 1, aw mateials 2 and 4 to plant 3, and aw mateial 3 to plant 4. Supplie 2 delives aw mateials 1 and 4 to plant 1. Supplie 3 delives aw mateial 2 to plant 1, aw mateial 3 to plant 3, and aw mateial4toplant4.supplie4delivesawmateial1to plant 2 and aw mateials 1 and 2 to plant 4. The same solution is obtained fo all peiods, except in peiod 3. In peiod 3, supplie 2 only delives poduct to plants 3 and 4. All DCs 1, 2,3,and4aeopenedfoallpeiods.Allpeiodhasdiffeent plant-dc solution mix. Futhemoe, each DC delives thei coesponding poduct to all customes with diffeent DCcustome solution mix in peiod 4.

7 Mathematical Poblems in Engineeing 7 Table 7: Results fo medium-size poblem with T=4. Optimal solution(objective function) $510, Peiod Supplie(Plant aw mateial ) Plant(DC poduct ) DC(Cus poduct ) (1 2,1 3,3 2,3 4,4 3 ); 2(1 1,1 4 ); 3(1 2,3 3,4 4 ); 4(3 1,4 1,4 2 ) 1(1 2,1 3,3 2,3 4,4 3 ); 2(1 1,1 4 ); 3(1 2,3 3,4 4 ); 4(3 1,4 1,4 2 ) 1(1 2,1 3,3 2,3 4,4 3 ); 2(1 1,1 4 ); 3(3 3,4 4 ); 4(3 1,4 1,4 2 ) 1(1 2,1 3,3 2,3 4,4 3 ); 2(1 1,1 4 ); 3(1 2,3 3,4 4 ); 4(3 1,4 1,4 2 ) 1(2 4,3 3,3 4,4 2,4 4 ); 3(1 1,1 3,2 1,3 2 ); 4(1 2,3 2,3 3,4 2,4 3 ) 1(2 4,3 3,3 4,4 2,4 4 ); 3(1 1,1 3,2 1,3 2 ); 4(1 2,3 2,3 3,4 3 ) 1(2 4,3 3,3 4,4 2,4 4 ); 3(1 1,1 3,2 1,3 2 ); 4(1 2,3 2,3 3,4 2,4 3 ) 1(1 2,2 4,3 4,4 2,4 4 ); 3(1 1,1 3,2 1,3 2 ); 4(1 2,3 2,3 3,4 3 ) 1(1 2,2 3,3 1,4 1,4 2,4 3 ); 2(1 1,2 1,3 4,4 1,4 4 ); 3(1 4,2 2,3 2,4 3 ); 4(1 3,2 4,3 3,3 4,4 2 ) 1(1 2,2 3,3 1,4 1,4 2,4 3 ); 2(1 1,2 1,3 4,4 1,4 4 ); 3(1 4,2 2,3 2,4 3 ); 4(1 3,2 4,3 3,3 4,4 2 ) 1(1 2,2 3,3 1,4 1,4 2,4 3 ); 2(1 1,2 1,3 4,4 1,4 4 ); 3(1 4,2 2,3 2,4 3 ); 4(1 3,2 4,3 3,3 4,4 2 ) 1(1 2,2 3,3 1,4 1,4 3 ); 2(1 1,2 1,3 4,4 1,4 4 ); 3(1 4,2 2,3 2,4 3 ); 4(1 3,2 4,3 3,3 4,4 2 ) Table 8: Netwok design esults fo medium-size poblem with T=4. Yea Annual cost ($) Total Single-peiod Multipeiod (T =10) Cost saving Table 8 shows the potential cost savings fo multiplepeiod plan vesus single-peiod plan. Fo example, given the optimal cost, we have fo Instance 1 a total of $ which is equied fo a plan done individually at the beginning of each yea. On the othe hand, given the optimal cost of $207,537.2 foinstance4,wecanoughlyestimateaveageannualcost fo the entie planning hoizon by dividing total cost by 10 yeas. The potential estimated total saving is $603,518. It should be noted that we intend to geneate a faily concentated demand acoss peiods. In this way, thee will be no much effect to the optimal netwok design acoss the planning hoizon. With this, the esults imply that the cost inceases with the inceases in planning hoizon but not linealyasopposedtosingle-yeaplanning.thefixedcostelement of the total cost is distibuted ove the numbe of peiods (yeas) included in the plan. As this numbe inceases, this fixed cost will be stetched ove the yeas. Thus, it ultimately gives us annual savings compaed to single-yea plans. Tables 9, 10, and 11 show the poduction-distibution plan fo Instances 1, 2, and 5, espectively. In these tables, the following optimal values ae indicated, namely, poduction quantities needed fo plants to manufactue to fully fulfill custome demand, aw mateials equiements fom supplies, finished poduct quantities to be tansfeed fom plant to DC, and finished poduct quantities to be deliveed fom DC to customes. Table 9: Poduction-distibution plan fo the single-peiod smallsized poblem. Oigin Destination Peiod Supplie 2 Plant Plant 2 DC DC1 Custome Custome Table 10: Poduction-distibution plan fo the multipeiod smallsized poblem, T=2. Oigin Destination Peiod 1 Peiod Supplie 2 Plant Plant 2 DC DC1 Custome Custome Sensitivity Analysis Jakeman et al. [27] noted that sensitivity analysis is one step in developing a model. Sensitivity analysis can also assist in executing the model [40]. Sensitivity analysis looks

8 8 Mathematical Poblems in Engineeing Table 11: Poduction-distibution plan fo the multipeiod medium-sized poblem. Oigin Destination Peiod 1 Peiod 2 Peiod 3 Peiod Supplie Plant Plant 2 Plant Plant Plant Plant Plant Plant Plant Plant Plant DC1 7 1 DC DC DC DC DC DC DC4 DC DC DC DC Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust Cust at the influence of paametes changes upon the objective function. In addition, sensitivity analysis can chaacteize the uncetainty in the paamete [12]. We theefoe conduct sensitivity analysis to analyze the changes in the decision vaiables and the objective function when the paamete values ae changed. This section investigates the changes in decision vaiables, namely, O pt, Q Vspt, b pjt, G jt,andm jit, the changes in the netwok configuation, and the changes in objective function. We use Instance 2 to conduct the sensitivity analysis. We pefomed the one-at-a-time (OAT) method in this analysis. In this method, we commonly used one paamete to change at a time while leaving all othes at thei baseline values. This method of sensitivity analysis consides the paamete s vaiability and its associated influence in the output model [43]. Table 12 pesents the scenaios fo

9 Mathematical Poblems in Engineeing 9 Table 12: Scenaios used in sensitivity analysis. Paamete Distibution to geneate andom vaiable Scenaios Intege unifom E p distibution Intege unifom D p distibution Intege unifom PR distibution B Vsp Unifom distibution A p Unifom distibution K pj Unifom distibution C sv Intege unifom distibution Intege unifom U V distibution Intege unifom F j distibution Intege unifom W j distibution a Unifom distibution d it Intege unifom distibution Y j Unifom distibution L ji Unifom distibution

10 10 Mathematical Poblems in Engineeing Table 13: Result of sensitivity analysis. Numbe Paamete Impact Z O pt G jt Q Vspt b pjt m jit 1 Costtoopentheplantp(E p ) 2 Maximum capacity of plant p(d p ) 3 Poduction ate of manufactuing poduct (PR ) 4 Cost of tanspoting and puchasing aw mateial v fom supplie s to plant p(b Vsp ) 5 Unit manufactuing cost of poduct at plant p(a p ) 6 Costoftanspotingpoduct fom plant p to DC j(k pj ) 7 Capacity of supplie s fo aw mateial V(C sv ) 8 Utilization ate of aw mateial v pe unit of finished poduct (U V ) 9 CosttoopenDCj(F j ) 10 Capacity of DC j(w j ) 11 Space equiement ate of poduct in a DC(a ) 12 Demand at custome zone i fo poduct in time peiod t(d it ) 13 Unit inventoy holding cost of poduct in DC j(y j ) 14 Cost of tanspoting poduct fom DC j to custome i(l ji ) L ji Y j d it a W j F j U C s K pj A p B sp PR D p E p Figue 2: Pecentage impact of each paamete. sensitivity analysis. The anges fo all paametes values ae given using the indicated distibution. Table 13 showstheesultsofthesensitivityanalysis.the impact fo each paamete change is indicated with check maks unde the coesponding columns fo the objective function and decision vaiables. Changes in inventoy holdingcostshownoimpacttoobjectivefunctionandanyof the decision vaiables. Paametes with the lowest impact ae opening plant cost, manufactuing cost, tanspotation cost to DC, opening DC cost, space equiement ate in DC, and tanspotation cost to custome. Paametes with the medium impact ae plant capacity, poduction ate, tanspotation cost to plant, supplie capacity, aw mateial utilization ate, and DC capacity. Finally, custome demand gives the highest impact. In addition, the impact of the vaiation paamete in the objective function and decision vaiable ae calculated as the weight of the influence, wheeby the highe the value, the geate impact to the decision vaiables. It is calculated by dividing the numbe of check maks (impact) in objective function and decision vaiables in each paamete by total impact. The total weight of all paametes is 1. The lowest, medium, and highest impact weights ae 0.03, 0.11, and 0.17, espectively. The impact of each paamete to the decision vaiable is significantly diffeent. It is depicted in Figue 2. Clealy, custome demand gives the geatest impact on the model s solution. Moeove, significant impact to the decision vaiables such as those unde distibution netwok is also evealed. The manageial implication fom this esult is that the netwok configuation that the long-tem plan may povide is an impotant decision-making input. 6. Conclusions and Recommendations We developed an integated model fo the poblem of location, poduction, and distibution of multipoduct, fouechelon, and multipeiod supply chain netwok design. The model is coded using LINGO pogam and implemented it fo the small-sized and medium-sized poblems. The numeical expeiment illustates the applicability of thepoposedmodel.withuptofouechelonsincludedinthe

11 Mathematical Poblems in Engineeing 11 supply chain and as much as ten yeas of planning hoizon, we demonstate cost savings advantage fo the poposed model. Lastly, we detemine the impact of all paametes involved. Custome demand gives the geatest impact on the model s solution. The applicability of the poposed model aises, fo example, mainly on manufactuing industies such as automobile, electonic, and funitue industies. We efe to the latest publishedmodelsandbuilduponthosetoaddesssome impotant featues. Howeve, thee ae some limitations of this study that may need futhe attention fo futue diections. The model belongs to the static and deteministic class with known demand. Solving this netwok flow poblem involving fou stages is computationally expensive using exact methods. Thus, we ae limited with the size that the LINGO pogam can handle. This is when heuistic algoithms pove to be useful. Fo example, a tend in natueinspied algoithms such as genetic algoithm (GA) and simulated annealing (SA) to name a few is known to solve NP-had discete combinatoial optimization poblems with high quality at faste computation speeds. Notations Indices I: Set of customes (i I) J: Set of potential locations of distibution centes (j J) P: Set of potential locations of plants (p P) S: Set of supplies (s S) R:Setofpoducts( R) V: Set of aw mateials (V V) T: Length of planning hoizon (T yeas). Notations Coesponding to the Activities in the Plant Paametes E p :Costtoopentheplantp ($) D p : Maximum capacity of plant p (time unit) B Vsp : Unit cost of tanspoting and puchasing aw mateial V fom supplie s to plant p ($/unit of aw mateial) A p : Unit manufactuing cost of poduct at plant p ($/poduct) PR : Poduction ate of manufactuing poduct (poduct/time unit) K pj :Costoftanspotingpoduct fom plant p to DC j ($/poduct) C sv : Capacity of supplie s fo aw mateial V (aw mateial unit) U V : Utilization ate of aw mateial V pe unit of finished poduct (aw mateial unit/poduct). Vaiables O pt :1ifplantpis opened in time peiod t;0othewise Q Vspt :QuantityofawmateialV shipped fom supplie s to plant p in time peiod t (aw mateial unit) q pt :Quantityofpoduct poduced by plant p in time peiod t (poduct) b pjt :Quantityofpoductshipped fom plant p to DC j in time peiod t (poduct). Notations Coesponding to the Activities in DCs Paametes Vaiables F j :CosttoopenDCj($) W j : Capacity of DC j (volume) a : Space equiement ate of poduct in DC (volume/poduct) d it :Demandatcustomezoneifo poduct in time peiod t (poduct) Y j :Unitinventoyholdingcostofpoduct in DC j ($/poduct) L ji :Costoftanspotingpoduct fom DC j to custome i ($/poduct). G jt :1ifDCjis opened at time peiod t;0othewise m jit :Quantityofpoduct shipped fom DC j to custome i in time peiod t (poduct) h jt :Quantityofpoduct in DC j at the end of time peiod t (poduct). Conflict of Inteests The authos declae that thee is no conflict of inteests egading the publication of this pape. Acknowledgments The authos thank the editos and efeees fo thei helpful comments and suggestions. This wok was patially suppoted by the National Science Council of Taiwan unde Gant NSC E MY3. This suppot is gatefully acknowledged. Refeences [1] J. Xu, Y. He, and M. Gen, A class of andom fuzzy pogamming and its application to supply chain design, Computes and Industial Engineeing,vol.56,no.3,pp ,2009. [2] Z. Tang, M. Goetschalckx, and L. McGinnis, Modeling-based design of stategic supply chain netwoks fo aicaft manufactuing, Pocedia Compute Science, vol. 16, pp , [3] D. J. Thomas and P. M. Giffin, Coodinated supply chain management, Euopean Jounal of Opeational Reseach, vol. 94, no. 1, pp. 1 15, 1996.

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