Using a genetic algorithm to optimize the total cost for a location-routing-inventory problem in a supply chain with risk pooling

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1 Joural of Applied Operatioal Research (2012) 4(1), Tadbir Operatioal Research Group Ltd. All rights reserved. ISSN (Prit), ISSN (Olie) Usig a geetic algorithm to optimize the total cost for a locatio-routig-ivetory problem i a supply chai with risk poolig Fatemeh Forouzafar 1 ad Reza Tavakkoli-Moghaddam 2, * 1 South Tehra Brach, Islamic Azad Uiversity, Tehra, Ira 2 College of Egieerig, Uiversity of Tehra, Tehra, Ira Abstract. This paper addresses a problem of desigig a multi-echelo supply chai with the sigle sourcig type ad the related ivetory systems. We also presets a ovel mathematical model cosiderig the risk-poolig, lead time, multi-echelo ivetory uder the demad ucertaity, routig of vehicles from distributio ceters (DCs) to customers i order to give services i a stochastic supply chai system, simultaeously. This problem is formulated as a mix iteger o-liear programmig model. The aim of this model is to determie, the umber of the located distributio ceters, their locatios ad capacity levels, to allocate customers to distributio ceters ad distributio ceters to suppliers optimally. I additio, it is also to determie the et lead time of distributio ceters ad the ivetory cotrol decisios. I additio, it is to determie the service time of every distributio ceters ad routig decisios. All these are doe i a way that the total system cost is miimized. The GAMS software is used to solve the preseted model for small-size problems. The give problem belogs to the class of NP-hard oes. Hece, to solve the large-sized istaces, a geetic algorithm is used. The sesitivity aalysis has bee performed to ivestigate the impact of effective parameters o the fial solutios. Keywords: supply chai; ivetory cotrol; risk-poolig; ucertaity; capacity levels * Received October Accepted Jauary 2012 Itroductio Nowadays, the competitio i busiess eviromet is icreasig cosiderably. The life cycle of products is becomig shorter, customer demads are gettig more ucertai, ad the lead time o their service is gettig very effective. The demad s variety ca be recogized as oe of the importat sources of ucertaity i a supply chai (Gupta et al. 2000; Park et al. 2010). Risk poolig is a strategy to redesig the supply chai, the productio process or the product to either reduce the ucertaity the firm faces or hedge ucertaity so that the firm is i a better positio to mitigate the cosequece of ucertaity. The proposed risk-poolig strategy ad cetralizig the ivetory at distributio ceters are cosidered as oe of the effective ways to maage such a demad ucertaity for achievig appropriate service levels to customers. Due to the icreasig pressure for remaiig competitio i the global market place, optimizig ivetories across the supply chai has become a maor challege for the process of idustries to reduce the costs ad improve the customer service. This challege requires itegratig ivetory maagemet i a supply chai etwork desig (You ad Grossma, 2009). The lead time is oe of the effective * Correspodece: Reza Tavakkoli-Moghaddam, Departmet of Idustrial Egieerig, College of Egieerig, Uiversity of Tehra, P.O. Box Tehra, Ira. tavakoli@ut.ac.ir

2 F Forouzafar ad R Tavakkoli-Moghaddam 3 factors i the safety stock levels due to the customer demad ucertaity (Park et al. 2010). Surely, the lower level for product, is cosidered as a additioal value that ca gai a log term or short term competitive beefit i the market. I the recet decades, the topic of multi-depot heterogeeous vehicle routig problem (MDHVRP) is proposed to icrease the productivity ad efficiecy of trasportatio systems, i which this model leads to the least cost fuctio by miimizig the umber of vehicles (Bettielli et al. 2011). Oe of the importat factors of the total productivity ad profitability of a supply chai is to cosider its distributio etwork, which ca be used to achieve the various supply chai obectives. Desigig a distributio etwork cosists of three sub-problems; amely, locatio-allocatio, vehicle routig, ad ivetory cotrol. I the literature, there are some research studies amalgamatig two of the above sub-problems, such as locatio-routig, ivetory-routig, ad locatio-ivetory problems (Ahmadi Javid ad Azad, 2010). These three sub-problems of a distributio etwork desig are cosidered i few papers simultaeously. However, i this paper for the first time, we preset a model to cocurretly optimize the locatio, allocatio, capacity, lead time, ivetory, ad routig decisios with risk-poolig i a stochastic supply chai system. Locatio-routig-ivetory problems are laid i the odetermiistic polyomial-hard category, due to the itrisic complexity of calculatios. Therefore, solvig large-sized problems is ot possible by liear programmig usig ordiary operatioal research software i a reasoable time. A geetic algorithm (GA) has uique characteristics compared to other meta-heuristic methods. The followig advatages have bee added i the revised paper (Goldberge, 1989). A geetic algorithm (GA) works with the codig of the parameter set, ot the parameters themselves. It is a populatio-based solutio, ot a sigle poit. It uses probabilistic trasitio rules, ot determiistic rules. It trades-off betwee exploratio ad exploitatio. It works o the umber of variables at the same time. It is capable of workig with ay kids of the obective fuctios ad costraits i liear ad/or o-liear forms withi ay solutio space (discrete or cotiuous). Cosequetly, it is applied i this paper with respect to the model complexity. It is a bio-ispired algorithm take from the ature ad it is also oe of the most popular meta-heuristics, which is applied i may optimizatio problems with differet fuctios. The previous iformatio is derived ad used i searchig for promisig solutios withi the solutio space. Furthermore, it has bee utilized for solvig supply chai problems extesively. For further study, readers may refer to Wag et al. 2011; Kaa et al. 2010; Yu et al. 2009; Zegordi et al ad Chag et al Problem formulatio The mathematical model of the cosidered problem, miimizes the fixed cost of locatig the opeed distributio ceters, the safety stock costs of distributio ceter by cosiderig ucertaity i customer s demad, ivetory orderig ad holdig costs ad also vehicles routig begiig from a distributio ceter (DC) with the aim of replyig ad coverig to the devoted customer s demads to that DC by cosiderig the risk-poolig. The importat assumptios i this paper are as follows. 1) Oe kid of product is ivolved (Paksoy ad Chag, 2010). 2) Each distributio ceter is assumed to follow a (Q i, R ) ivetory policy (Ahmadi Javid ad Azad, 2010). 3) A sigle sourcig strategy is cosidered i the whole supply chai (Park et al., 2010). 4) It is cosidered that the customers demads after reachig to retailer are idepedet ad follows a ormal distributio (Park et al., 2010; Ahmadi Javid ad Azad, 2010). 5) Each plat ca give ay kids of services i ay amout of demads to the related devoted distributio ceters. 6) We cosider differet capacity levels for each distributio ceter, ad fially oe capacity for each of them is selected. 7) Each DC with the limited capacity carries o-had ivetory to satisfy demads from customer demad zoes as well as safety stock to deal with the mutability of the customer demads at customer demad zoes to attai risk-poolig profits (Park et al., 2010).

3 4 Joural of Applied Operatioal Research Vol. 4, No. 1 8) All customers must be served. 9) The umber of available vehicles for each type ad the umber of allowed routes for each DC are limited (Bettielli et al., 2011) 10) To determie all the feasible routes, the followig factors are take ito accout: - Each customer must be visited by oly oe vehicle. - Each route begis at a DC ad eds at the same DC. - The sum of the demads of the customers served i each route must ot exceed the capacity of the associated vehicle. - Each of the distributio ceters ad the vehicles has the various, limited ad determied capacity (Bettielli et al., 2011; Mariakis ad Mariaki, 2010). Model formulatio Followig are the otatios itroduced for the mathematical descriptio of the proposed model. Idices I Set of plats idexed by i J Set of cadidate DC locatios idexed by K Set of customer demad zoes idexed by k N Set of capacity levels available to DC ( J) V Set of vehicles Set of all feasible routes usig a vehicle of type v (v V ) from DC ( J) v Parameters F Yearly fixed cost for opeig ad operatig distributio ceter with capacity level ( N, J) Safety stock factor of DC ( J) h k Mea demad at customer demad zoe k 2 k Variace of demad at customer demad zoe k A b ti Uit ivetory holdig cost at DC ( J), (aually) Fixed ivetory orderig cost at DC Capacity with level for DC Order processig time of DC if it is served by plat i; icludig material hadlig time of DC, trasportatio time from plat i to DC, ad ivetory review period. S Service time of plat i i Cr Cost of each demad uit i route r (these costs iclude the fixed cost of vehicle plus the trasportatio cost of each demad uit i route r. the metioed trasportatio cost for each demad uit is ot related to customer demad zoe ad it is cosidered fixed for all locatios i each route r. Number of available vehicles of each type v g v Number of routes associated with each distributio ceter Biary coefficiets P kr 1 if ad oly if customer k is visited by route r; ad 0, otherwise

4 F Forouzafar ad R Tavakkoli-Moghaddam 5 Decisio variables U i Z k r L 1 if distributio ceter is opeed with capacity level ; ad 0, otherwise 1 if distributio ceter is served by plat i; ad 0, otherwise 1 if customer k is assiged to distributio ceter ; ad 0, otherwise 1 if ad oly if route r is selected; ad 0, otherwise Net lead time of DC Service time of DC Q Order size at distributio ceter The problem formulatio is as follows. Z A h Q Miimize s.t. J v V k K r v N U 2 k k i F U h L k Z k J N 2 J k K i I J k K Q i i I J 1 P C k kr r r J (1) kk Z k k N b U J (2) ii Q i kk L 2 k Z k N b U J (3) L ( Si ti ) i J (4) ii ii i N U J (5) J ii Z k i 1 1 k K (6) J (7) N U Z k J, k K (8) i i I Z k J, k K (9) P kr J vv r v r 1 k K (10)

5 6 Joural of Applied Operatioal Research Vol. 4, No. 1 J r v vv r v r r v g v V (11) J (12) vv r v kk 0,1 i 0, 1 U Z k 0, 1 r 0, 1 L Q Z P 1 J (13) k kr J J J r J, N J, i I J, k K r v J, vv This model miimizes the total expected cost cosistig of the fixed cost for opeig distributio ceters with a certai capacity level, the expected aual ivetory cost, ad the aual routig cost. Costrait (1) esures that each distributio ceter ca be assiged to oly oe capacity level. Costraits (2) ad (3) are the capacity costraits associated with the distributio ceters. Costrait (4) imposes limits o the miimum amout of et lead time of DC. Costraits (5) states that if the distributio ceter with capacity is opeed, it is serviced by a plat. Costraits (6) ad (7) represet the sigle-sourcig costraits for each customer demad zoe ad each DC, respectively. Costrait (8) esures that if the distributio ceter is allocated to the customer k, that ceter should certaily be established by a determied capacity level. Costrait (9) makes sure that if the distributio ceter gives the service to the customer k, that ceter must get services from a plat. Costrait (10) is stadard set coverig costraits, modelig assumptio 6. Costraits (11) ad (12) impose limits o the maximum umber of available vehicles of each type ad maximum umber of permitted routes for each DC, modelig assumptio 7. Costrait (13) implies that there is at least oe customer i oe selected route. Costrait (14) eforces the itegrality restrictios o the biary variables. Fially, Costrait (15) eforces the o-egativity restrictios o the other decisio variables. (14) (15) Solutio methodology Some differet small-sized problems have bee solved by the covetioal brach-ad-boud imbedded i the GAMS (Geeral Algebraic Modelig System) software i order to cosider the feasibility ad validity of the preseted mathematical model i small-sized problems. To solve large-sized problems, a geetic algorithm (GA) is proposed. Chromosome defiitio It is obvious that the solutio represetatio is the base of ay meta-heuristic approach. Four oe-dimesioal matrices are used to demostrate the solutio. The first matrix is the 1 m oe (m is the umber of the distributio ceters) ad deotes that each distributio ceter is established with its capacity levels. Each array of the matrix is correspodig to a umber betwee 1 ad N, as show i Figure 1. N 1 N 2 N 3. N m Fig. 1. Presetatio of the first matrix

6 F Forouzafar ad R Tavakkoli-Moghaddam 7 The secod matrix is the 1 m ad represets the assigmet of the distributio ceters to the plats. Each array of the matrix is associated with a umber betwee 1 ad ( is the umber of plats) as show i Figure 2. I 1 I 2 I 3. I m Fig. 2. Presetatio the secod matrix The third matrix is the 1 k oe (k is the umber of the customers) that shows the customers allocatio to the distributors. The associated arrays are determied by a umber betwee 1 ad m, as show i Figure 3. J 1 J 2 J 3. J k Fig. 3. Presetatio of the third matrix The fourth matrix is the 1 r oe (r is the umber of the routes) that shows the selected routes, as show i Figure 4. Each array of the matrix belogs to the umbers 0 or 1. It ca be uderstood that the give route has bee used if the i-th cell of the matrix is 1; otherwise, it is 0. A 1 A 2 A 3. A r Fig. 4. Presetatio of the fourth matrix Establishig a iitial populatio The first step is to geerate a iitial populatio from the chromosomes oce so that each oe idicates to a specific solutio. The required feasible solutios are geerated radomly i this sectio. Fitess fuctio The fitess fuctio is similar to cosidered obective fuctio. As the chromosomes are formed ad modified, the obective fuctio value is calculated for each oe to ustify it. Samplig mechaism The samplig mechaism pertais how the chromosomes are chose with respect to the samplig space. The bi-touramet approach is applied i this paper, i which the best solutio is selected from the populatio ad the the ext optimal solutio is selected from the rest. Crossover operator I this paper, we use a two-poit crossover operator for all the four matrices. However, it should be oted that the obtaied solutios from the crossover operator may be ifeasible. Thus, they must be trasformed ito feasible solutios by modifyig practices. I this operator, two radom idices are geerated i the iterval of 1 ad the legth of the matrix for each oe. The first ad secod offsprig are also geerated as follows. The first part of the first paret + the secod part of the secod paret + the third part of the first paret. The first part of the secod paret + the secod part of the first paret + the third part of the secod paret. Mutatio operator Variable Neighborhood Search (VNS) is used i this paper for the mutatio structure. The VNS structure applies four Neighborhood Search Structures (NSS). These four structures are used i the framework of VNS ad the etire structure ca be demostrated by Figure 5. The pseudo-code of our VNS is as follows.

7 8 Joural of Applied Operatioal Research Vol. 4, No. 1 {for each iput particle K=1 While the stoppig criterio is met do New particle=apply NSS type k (Iput particle) If ew particle is better tha iput particle the K=1 Iput particle= ew particle; Else K=k+1 If k=5 the K=1 Edif Edif Edwhile } Fig. 5. Shows the VNS algorithm Strategy i dealig with the costraits The mutatio operator is desiged i a way that o ifeasible solutio ca be geerated. Just, the crossover operator may lead to ifeasible solutios. Sice ew solutios are geerated durig the algorithm implemetatio, a specific procedure has bee deployed to check whether the costraits are satisfied by the give solutio or ot. Hece, if it is ecessary, the feasible solutios remai ad ifeasible solutios ca be trasformed ito feasible oes. The metioed procedure tries to trasform the solutio ito a acceptable oe, wheever oe or more costraits are dissatisfied by the obtaied solutio. Stoppig criterio The algorithm is termiated whe it caot fid a ew solutio aymore, or i other words the obective fuctio values do ot chage. Desig of experimets To ivestigate the validity ad feasibility of the proposed mathematical model, differet small-sized problems are solved by the covetioal brach-ad-boud (B&B) solver i the GAMS software. I order to do that, te radom istaces are take ito accout. Afterwards, the results are compared with those of the GA to validate the approach. The obtaied results of the GA are compared with the exact oes of GAMS, as show i Table 1. The parameters are set with respect to the followig itervals. Establishmet cost of distributio ceters by differet capacity levels ~ U[ ] Stock level for each distributio ceter ~ U[0.2, 2.3] Aual ivetory holdig cost ~ U[4, 15] Average demad of each customer ~U[1, 11] Variace of the customers demads ~U[0, 3] Capacity level of the distributio ceters ~U[1, 30] Processig time of each plat ~U[1, 4] Service time of distributio ceters ~U[1, 5] Fixed orderig cost i each distributio ceter ~U[15, 60] Cost of each demad uit i route r ~U[100, 650] Available vehicles type v ~U[2, 3] Number of possible routes related to distributor ~U[1, r]

8 F Forouzafar ad R Tavakkoli-Moghaddam 9 Table 1. Obtaied results of small-sized problems for the GA ad GAMS. Problem GAMS GA Gap (%) OFV CPU time (sec.) OFV CPU time (sec.) The results idicate that the obtaied obective fuctio values for the GA ad GAMS are the same i small-sized istaces. However, the CPU times of the GA are less tha GAMS. Give the assumptios ad parameters, 30 radom istaces are cosidered for medium ad large-sized problems. The results are preseted i Table 2 i terms of the CPU time ad the obective fuctio value (OFV). Each istace is solved for five times by the GA ad the average value is illustrated i the tables. Table 2. Results obtaied by the proposed GA. Medium-sized problems Large-sized problems Problem No. OFV CPU time (sec.) Problem No. OFV CPU time (sec.)

9 10 Joural of Applied Operatioal Research Vol. 4, No. 1 I order to show the proper performace of the GA for the give problems, the followig assumptios should be take ito accout. Rates of the mutatio ad crossover operators are assumed to be 0.1 ad 0.8, respectively. Local-iteratio is assumed equal to 5. Populatio size is assumed to be 100. Iitial populatio is geerated radomly. The parameters are set with respect to the followig itervals. Etablishmet cost of distributio ceters by differet capacity ~ U[1,40] Stock level for each distributio ceter ~ U[0, 3] Aual ivetory holdig cost ~ U[1, 40] Average demad of each customer ~U[1, 100] Variace of the customers demads ~U[1, 4] Capacity level of the distributio ceters ~U[1000, 2500] Processig time of each plat ~U[1, 40] Service time of distributio ceters ~U[1, 40] Fixed orderig cost i each distributio ceter ~U[1, 40] Cost of each demad uit i route r ~U[1, 100] Available vehicles type v ~U[1, 10] Number of possible routes related to distributor ~U[1, r] Sesitivity aalysis The effective rates of mutatio ad crossover ad the efficiet populatio size are give below. The rate of the mutatio is equal to 0.1, ad the value of the crossover rate is cosidered equal to 0.6, 0.7 ad 0.8. The three levels ca be see i Figure 6, so that each level shows the combiatio of the rate of two operators (Naderi et al., 2009). a 1 : rate 0.6 for crossover ad 0.1 for mutatio. a 2 : rate 0.7 for crossover ad 0.1 for mutatio. a 3 : rate 0.8 for crossover ad 0.1 for mutatio. The vertical axis shows the value of a criterio, amely relative percetage deviatio (RPD), which is calculated by: Algsol mi RPD mi sol sol 100 where the Alg sol is the obective fuctio value for each problem by combiig the parameters ad Mi sol is the miimum obective fuctio value i all the combiatios. I fact, each istace is ru by each of the three combiatios ad the RPD criterio is calculated for each oe. The results are preseted i Figure 6. I this figure, it is obvious that the best crossover rate is 0.8. I additio, the crossover rate is equal to 0.8 ad the mutatio rates are set to 0.1 ad 0.2. Two levels are see i this figure, so that each oe is a combiatio of the rate of two operators give below. a 1 : the rates 0.8 for crossover ad 0.1 for mutatio. a 2 : the rates 0.8 for crossover ad 0.2 for mutatio. I fact, each problem is carried out by each of two possible combiatios ad the RPD criterio is calculated for each problem. It is cocluded that the best mutatio rate is 0.1 as depicted i Figure 7. (16)

10 Data Data F Forouzafar ad R Tavakkoli-Moghaddam aalysis for crossover rate 95% CI for the Mea a1 a2 a3 RPD Fig. 6. RPD diagram with respect to combiatios of the fixed mutatio rates ad differet Crossover rates 3.0 aalysis for mutatio rate 95% CI for the Mea RPD a1 a2 Fig. 7. RPD diagram with respect to combiatios of the fixed Crossover rates ad differet mutatio rates Figure 8 pertais to the populatio size. Four differet sizes (i.e., 30, 50, 100 ad 200) are cosidered as show below: a: shows size 30 b: shows size 50 c: shows size 100 d: shows size 200 Likewise, the vertical axis shows the RPD criterio.

11 Data 12 Joural of Applied Operatioal Research Vol. 4, No. 1 where the Alg sol is the obective fuctio for each level of the populatio size for each problem, which is obtaied by the algorithm ad Mi sol is the miimum calculated value amog all the cosidered size levels for each problem. As it is observed from Figure 8, the most efficiet ad reliable populatio size is aalysis for popsize 95% CI for the Mea a b c d RPD Fig. 8. RPD diagram regardig the populatio size Coclusios I this paper, a ew mathematical model for desigig the multi-echelo supply chai has bee preseted by cosiderig the ivetory uder ucertai demads, risk-poolig, lead time ad vehicles routig. This model has bee formulated for the first time as a locatio-ivetory-routig problem with a risk-poolig strategy i a multi-echelo supply chai. Feasibility of the developed model was checked by presetig small-sized radom istaces ad solvig them by commercial optimizatio software. The, the results obtaied from the GA were compared to the exact oes of GAMS i small-sized istaces i order to validatig the GA. The results showed that the CPU times were less for the GA i compariso with those of GAMS. A umber of medium ad large-sized problems were solved by the proposed GA because of the NP-hardess of the give problems. Some future studies are as follows: cosiderig each parameter as a fuzzy, time widows, multi-period plaig ad solvig the preseted model by the use of heuristic or other meta-heuristic algorithms. Refereces Ahmadi Javid, A., Azad, N., Icorporatig locatio, routig ad ivetory decisios i supply chai etwork desig. Trasportatio Research Part E: Logistics ad Trasportatio Review 46, Bettielli, A., Ceselli, A., Righii, G., A brach-adcut-ad-price algorithm for the multi-depot heterogeeous vehicle routig problem with time widows. Trasportatio Research - Part C: Emergig Techologies 19, Chag, Y.-H., Adoptig co-evolutio ad costraitsatisfactio cocept o geetic algorithms to solve supply chai etwork desig problems., Expert Systems with Applicatios 37, Goldberge, D.E., Geetic algorithms i search, optimizatio ad machie learig, Addiso-Wesley Publishig, Gupta, A., Maraas, C.D., McDoald, C.M Mid-term supply chai plaig uder demad ucertaity:

12 F Forouzafar ad R Tavakkoli-Moghaddam 13 Customer demad satisfactio ad ivetory maagemet. Computers & Chemical Egieerig 24, Kaa, G., Sasikumar, P., Devika, K., A geetic algorithm approach for solvig a closed loop supply chai model: A case of battery recyclig. Applied Mathematical Modelig 34, Mariakis, Y., Mariaki, M., A hybrid geeticparticle swarm optimizatio algorithm for the vehicle routig problem. Expert Systems with Applicatios 37, Naderi, B., Khalili, M., Tavakkoli-Moghaddam, R., A hybrid artificial immue algorithm for a realistic variat of ob shops to miimize the total completio time. Computers & Idustrial Egieerig 56, Paksoy, T., Chag, C.T., Revised multi-choice goal programmig for multi-period, multi-stage ivetory cotrolled supply chai model with popup stores i Guerrilla marketig. Applied Mathematical Modelig 34, Park, S., Lee, T.E., Sug, C.S., A three-level supply chai etwork desig model with risk-poolig ad lead times. Trasportatio Research - Part E: Logistics ad Trasportatio Review 46, Wag, K.-J., Makod, B., Liu, S.-Y., Locatio ad allocatio decisios i a two-echelo supply chai with stochastic demad A geetic-algorithm based solutio. Expert Systems with Applicatios 38, You, F., Grossma, I.E., Optimal desig of largescale supply chai with multi-echelo ivetory ad risk poolig uder demad ucertaity. Computer Aided Chemical Egieerig 26, Yu, Y., Moo, C., Kim, D., Hybrid geetic algorithm with adaptive local search scheme for solvig multistage-based supply chai problems. Computers & Idustrial Egieerig 56, Zegordi, S.H., Kamal Abadi, I.N., Beheshti Nia, M.A., A ovel geetic algorithm for solvig productio ad trasportatio schedulig i a two-stage supply chai. Computers & Idustrial Egieerig 58,