An integrated supply chain design model with random disruptions consideration

Afican Jounal of Business Management Vol. 4(1), pp. 393-401, 18 Septembe, 010 Available online at http://www.academicounals.og/ajbm ISSN 1993-833 010 Academic Jounals Full Length Reseach Pape An integated supply chain design model with andom disuptions consideation Mi-Bahado Ayanezhad, Seyed Gholameza Jalali and Amin Jabbazadeh* Depatment of Industial Engineeing, Ian Univesity of Science and Technology, Tehan, Ian. Accepted 6 August, 010 This pape investigates a supply chain design poblem whee distibution centes ae subect to andom disuptions. As a esult of disuptions, one o moe of the distibution centes may fail to seve the customes. It is assumed that customes have andom demands; thus, each distibution cente maintains some amount of safety stocks in ode to povide suitable sevice level fo the customes it seves. The poposed model fo this study is fomulated as a nonlinea intege pogamming to minimize the expected total cost which includes costs of location, inventoy, tanspotation and lost sales. The model simultaneously detemines the location of distibution centes and the allocation of customes to distibution centes. In ode to solve the esulted mathematical model, an efficient solution appoach based on genetic algoithm is developed. Finally, computational esults fo seveal instances of the poblem ae pesented to demonstate the effectiveness of the poposed algoithm. Key wods: Supply chain management, inventoy management, disuptions, location model. INTRODUCTION Taditional supply chain design models typically assume that facilities will neve fail. Howeve, in the eal wold cases, facilities ae always vulneable to disuptions of vaious sots due to natual disastes, stikes, changes of owneship and othe factos (Snyde and Daskin, 005). Thee ae many evidences that facility disuptions can be costly. Fo instance, a eseach by Hendicks and Singhal (003) examines stock maket eactions when fims publicly announce that they ae expeiencing supply chain disuptions. Results of the study of 519 supply chain poblem announcements eveal that stock maket eactions educed shaeholde value by 10.8%. Also, delivey failue of two citical pats esulted in ove $.6 billion loss fo the Boeing (Radou, 00). Both Huicane Katina and Rita caused shutdowns of numeous facilities and consequently significant economic losses (Baionuevo and Deutsch, 005). These examples and othe events demonstate cucial need to plan fo facility disuptions in designing supply chain systems so that they pefom well even afte disuptions. *Coesponding autho. E-mail: amin@iust.ac.i. This pape pesents an integated supply chain design model, which consides impacts of the facility disuptions on both the stategic facility location and tactical inventoy decisions. Specifically, we study a supply chain system which compised a single supplie, distibution centes (DCs) and customes. It is assumed that customes have uncoelated pobabilistic demands with nomal pobability distibution. In this supply chain system, a supplie ships one type of poduct to customes in ode to satisfy thei demands. DCs function as the diect intemediay between the supplie and customes fo the shipment of the poduct, that is, DCs combine the odes fom diffeent customes and then ode fom the supplie. Each DC etains safety stocks in ode to ensue pe-specific level of sevice. A key poblem is that DCs ae always subect to disuptions. As a esult, each DC, at any time, may become unavailable and fail to seve the customes. In ode to ovecome this poblem, we adopt the stategy that each custome is assigned to multiple DCs: a pimay DC and a numbe of backup DCs. A pimay DC, assigned to a custome, is esponsible fo satisfying all the demands of the custome in nomal cicumstances. As soon as the pimay DC becomes unavailable, the fist

394 Af. J. Bus. Manage. backup DC which is allocated to the custome povides its demands. If the fist backup DC is disupted, then the second backup DC will seve the custome's demands, and so on. If all of the DCs assigned to the custome ae disupted, o when the total cost of satisfying the custome's demands becomes highe than lost sales cost, the custome is not seved. In this case, the system incus lost sales cost. The model detemines the location of DCs and assignment of customes to DCs in ode to minimize the expected total cost which includes: (1) the fixed cost to locate DCs, () the woking inventoy cost at the DCs, (3) the safety stock cost at the located DCs, (4) the shipment cost fom DCs to customes, and (5) the lost sales cost. The emainde of this pape is oganized as follows. Some elevant models in the liteatue ae discussed, then, the integated supply chain design model fo the poblem is poposed. Also, an efficient solution appoach fo the mathematical model is developed. Finally, the elated computational esults along with conclusions ae povided. Liteatue eview In the ecent yeas, eseaches have focused on the integated models in which location and nonlinea inventoy costs ae included in the same model. Fo instance, Elebache and Melle (000) povide a oint location inventoy model with complicated nonlinea obective function. They applied a continuous appoximation along with some heuistics techniques to solve the model. Shen (000), Shen et al. (003) and Daskin et al. (00) intoduce a location model with isk pooling (LMRP) that incopoates inventoy decisions into the location model. Shen (000) and Shen et al. (003) use column geneation, while Daskin et al. (00) pesents Lagangian elaxation to solve LMRP. Anothe efficient appoach to solve the LMRP is pesented by Shu et al. (005). Shen and Daskin (005) extend the LMRP to include a custome sevice element and popose useful techniques fo evaluation of cost/sevice tade-offs. A pofit-maximizing supply chain netwok design model is studied by Shen (006), whee DCs can chage diffeent pices. Ozsen et al. (008) develop LMRP when each DC has limited capacity. Shen and Qi (007) study an integated supply chain design model that contains location, inventoy and outing decisions; in fact, they add outing decisions to the LMRP famewok. Ozsen et al. (009) analyzed the effect of multi-soucing by intoducing a capacitated location-inventoy model that minimizes the sum of the fixed location costs, the tanspotation costs and the inventoy costs. Snyde et al. (007) poposed a stochastic vesion of LMRP (called SLMRP) that handles uncetainty by descibing discete scenaios. The goal of SLMRP is to minimize the expected system cost acoss identified scenaios. Howeve, the authos ague on how to use SLMRP to solve multi-commodity and multi-peiod poblems. Souiaan et al. (008) examined a two-stage supply chain with a poduction facility in which the eplenishment lead time at a DC depends on the volume of flow though the DC. They pesented a genetic algoithm to solve the model and imply that the poposed algoithm outpefoms the Lagangian elaxation appoach. The eade efeed to Shen (007) and Melo et al. (009) fo a thoough eview of the integated supply chain design models. Anothe body of liteatue which is closely elated to the pesent pape is the liteatue on facility location with disuptions. Snyde and Daskin (005) examine facility location poblems in which facilities may fail with a given pobability. Thei models minimize the weighted sum of two obectives: the fist obective epesents the cost of the system when no disuptions occu, while the second obective indicates the expected tanspotation cost afte accounting fo disuptions. They assume that facilities have equal pobability of failue. Beman et al. (007) and Lim et al. (009) developed models that ae simila to Snyde and Daskin's (005) models, which pemitted diffeent facilities to have diffeent failue pobabilities. Cui et al. (010) also elax the unifom failue pobability assumption in the Snyde and Daskin's (005) models using a continuum appoximation model. Howeve, they ignoe inventoy costs in thei model. Qi et al. (010) poposed an integated location-inventoy model, whee the supplie and etailes ae disupted andomly. Thei model assumes that the demands ae deteministic and the lead time fo ode pocessing is zeo. Related models ae studied by Chuch and Scapaa (007) and Scapaa and Chuch (008). They fomulated poblem in places whee existed facilities can be potected against disuptions by fotification esouces. Since fotification esouces ae limited in thei poblem, potecting all the facilities is not possible. They fomulate models that detemine what facilities must be potected so that the impact of intediction on the emaining system opeation is minimized. Snyde et al. (006) pesent a tutoial eviewing a boad ange of models fo designing supply chains esilient to disuptions. Snyde and Daskin (007) investigate models fo the design of eliable facility location systems unde a vaiety of isk measuements. The pesent pape diffes fom the ealie liteatue on facility location models with disuptions. Fist, the poposed model in this study does not ignoe nonlinea inventoy costs. Moeove, demands fo customes ae consideed pobabilistic instead of deteministic. This eseach, also, is diffeent fom the ealie liteatue on the oint location-inventoy models. Namely, in the pape, we dismiss the common estictive assumption in LMRP that the vaiance and mean of the daily demand ae equal fo each custome. Besides, thee is no assumption

Ayanezhad et al. 395 that the demands of all customes must be satisfied necessaily. Finally, unlike the liteatue on oint locationinventoy models, facility disuptions ae consideed in the pesented model fo making it moe ealistic. Model fomulation Hee, a model fo the poblem is fomulated. The obective is to minimize the expected total cost including: (1) the fixed cost to locate DCs, () the woking inventoy cost at the located DCs (containing ode costs, shipment costs fom supplie to DCs and holding costs), (3) the safety stock cost at the located DCs, (4) the shipment cost fom located DCs to customes, and (5) the lost sale cost of not seving customes. The following notations will be used thoughout the pape: I : set of customes indexed by i ; J : set of candidate DC locations indexed by ; i : mean of daily demand at custome i ; : vaiance of daily demand at customei ; i f : fixed cost of locating a DC at ; F : fixed cost of placing an ode at ; g : fixed cost pe shipment fom the plant to DC ; a : pe-unit shipment cost fom the plant to DC ; h : inventoy holding cost pe unit of poduct pe yea; d i : pe-unit cost to ship fom DC to customei ; : desied pecentage of customes odes that ae satisfied; : weight facto associated with the shipment cost; θ : weight facto associated with the inventoy cost; z : standad nomal deviate, such that P ( z z ) = ; L : lead time fom supplie to DCs, in days; : numbe of days in a yea u i : penalty cost of not seving the costume i, pe unit of demand (it can be intepeted as lost sales cost o the cost of seving custome i by puchasing poduct fom a competito); q : pobability of disuptions fo each DC; P : numbe of DCs that must be located; Without loss of geneality, it is assumed that sets I and J ae the same (Daskin et al., 00). Shipment cost Tanspoting the poduct fom each DC to each custome i that has linea shipment cost. Let Di denote the expected (annual) demand of custome i which is assigned to DC. Then, the shipment cost fom DC to the customes can be obtained by: i i I d D (1) i added to the setj ; it is assumed that the dummy DCu neve faces disuptions. Assigning the customei to this dummy DC, epesents not seving the custome i. Regading DCu, we assume that it has the shipment cost diu = u i to custome i I and thee is no othe cost. Woking inventoy cost Hee, details of the inventoy policy used in the DCs ae given. Each DC odes fom the supplie using an appoximation to the (Q, ) model with Type I sevice (Nahimas, 1997). This appoximation consists of two steps. The fist step detemines the quantity of ode at each DC using economic ode quantity model (EOQ). In the next step, the eode point at each DC is detemined in the way that the pobability of a stockout does not exceed the specified value. The eason why we utilize this two-step appoach is that it povides a good appoximation to the optimal ode quantity and ode point values (Zheng, 199; Axsate, 1996). Following the two-step appoach, fist, the ode quantity is detemined. Let D denote the expected total (annual) demand that = is assigned to the DC (it is obvious that D D ) and n be the unknown numbe of odes pe yea. Then, the expected shipment D size pe shipment fom supplie to DC is equal to n. Futhemoe, the annual woking inventoy cost at DC (including ode, shipment fom supplie to DC and holding cost) is obtained by: a D hd Fn + ( g + ) n + n ( n ) The fist tem of equation () is the fixed cost of placing n odes, while the second tem indicates the cost of shipping n odes of size D n i I. The last tem epesents the cost of holding aveage D units of inventoy pe yea. To detemine the optimal numbe ( n ) of odes pe yea, we take the deivative of () in espect to n and set the deivative to zeo: hd F + g = 0 ( n ) Solving Equation (3) fo n, we obtain n = i () (3) hd. ( F + g ) Plugging this into (), an annual woking inventoy cost can be calculated as follows: hd ( F + g ) + a D (4) Lost sales cost In ode to model the lost sales cost, dummy DC with index u is Safety stock cost Each DC etains a cetain amount of safety stocks to deal with

396 Af. J. Bus. Manage. possible stockouts duing eplenishment lead time. Assuming that lead time demand at the DC is nomally distibuted with expected vaiance of V, the needed safety stock to guaantee that the stockouts occu with a pobability of o less, which is z V. Maintaining this amount of safety stocks incus the holding cost at DC which is calculated by: hz V (5) Integated model To detemine the locations of the DCs and customes-dcs assignment, two sets of decision vaiables ae defined: X = 1, if is selected as a DC location, and 0, if othewise fo each J ; Y i = 1, if custome i at level is assigned to a DC located at, and 0 if othewise fo i I, J, and = 0,1,, P-1. To claify vaiablesy, note that each custome i I is assigned to multiple DCs at multiple levels. In fact, custome i is assigned to its pimay DC at level 0, its fist backup DC at level 1, its second backup DC at level and its th backup DC at level. Recall that in a nomal cicumstance, the custome is seved by its pimay DC. Howeve, when the pimay DC is disupted, the custome is seved by its fist back up DC. If the fist back up DC fails, the custome's demand is povided by its second back up DC and so on. Notewothy, if custome i is assigned to dummy DC u at level, thee is no need to assign it to any othe DC at uppe level s (whee s > ). The eason is that the dummy DC u neve fails and does not equie any backup DC. Now D andd can be witten in tems of decision vaiables: i i = (1 ) χµ i i = 0 1 = = P i (1 ) χµ i i i I i I = 0 D q q Y (6) D D q q Y (7) Also, consideing the fact that custome demands ae uncoelated, the expected vaiance of lead time demand at the DC, V, can be obtained based on the decision vaiables: P 1 i i i I = 0 V = L (1 q ) q σ Y (8) To explain (6) and (8), note that each custome i I is seved by its th back up DC (name it ) if distibution cente does not face disuptions (this occus with pobability 1 q ) and if all the distibution centes, which wee assigned to the custome i at lowe levels (levels 0, 1, -1) ae not disupted (this occus with pobabilityq ). Now the model can be fomulated as follows: Min f X + β ( 1 q ) q d i χµ iyi J J i I = 0 θ h ( F + β g ) ( 1 q ) q χµ iy i J i I = 0 + p 1 + β a ( 1 q ) q χµ iyi ) J i I = 0 + θhz α L ( 1 q ) q σ i Y J i I = 0 subect to: 1 Y + Y = 1 i I, = 0, 1,,..., P -1 (10) i ius J s = 0 Y X i I, J, = 0, 1,,..., P -1 (11) i X Y J (1) JJ 0 X = 1 (13) u J X = P + 1 (14) Y 1 i I, J (15) i = 0 X {0,1} i J (16) Y {0,1} i I, J, = 0, 1,,..., P -1 (17) i The obective function (9) is composed of fou components sepaated by paentheses. The fist component epesents the fixed cost of locating DCs, while the second pat indicates the expected shipment cost fom the DCs to customes. Recall that we added dummy DC u to the set J in ode to take into account lost sales cost in the model. Consideing Equations (4) and (7), it is easy to find that the thid component epesents the woking inventoy cost. Finally, the fouth pat indicates safety stock cost and can be obtained by consideing Equations (5) and (8). Constaints (10) stipulate that fo each custome i and each level, custome i should be assigned to exactly one DC at level unless i was assigned to DC u at any lowe level s (s < ). In othe wods, if custome i at level s is assigned to dummy DC u, it is not assigned to any of the DCs at any highe level (s < ). Note that in case = 0, we take 1 Y ius s = 0 (9) = 0. Constaints (11) state that customes can only be assigned to candidate sites that ae selected as DCs. Constaints (1) equie that if a DC is located at, this DC should seve the custome at as a pimay DC. Constaint (13) equies

Ayanezhad et al. 397 the dummy DC u to be located. Constaint (14) assues that the numbe of located DCs should be exactly P + 1 (this means that P distibution centes must be located in addition to dummy DC u). Constaints (15) state that a custome cannot be assigned to a given DC at moe than one level. Howeve, constaints (16) and (17) ae binay constaints. Solution appoach Solving the model (9) in the simplest condition (when q = θ = 0, f = a = 0 fo each J, β = 1, and ui ae extemely lage fo each i I ) is identical to solving the P-median poblem which is NP-had (Gaey and Johnson, 1979). This shows that solving the study s model in easonable time is extemely had. Theefoe, in ode to solve this complex nonlinea model, we use an efficient meta-heuistic method based on genetic algoithm (GA) simila to Zhou and Liu (003). GA is a stochastic seach and heuistic optimization technique based on the mechanism of natual genetics which has been successfully applied to vaious complex poblems. It stats with an initial set of andom solution called population. Each solution in the population is called chomosome and each component of chomosome is designated by gene. The chomosomes evolve though successive iteations, called geneations. Duing each geneation, the chomosomes ae evaluated, using some measues of fitness. To ceate the next geneation, new chomosomes (called offsping) ae fomed by cossove o mutation opeatos. Cossove opeato combines two chomosomes fom cuent geneation, while mutation opeato modifies a chomosome to fom offsping. A new geneation is ceated by (a) selecting some of the cuent chomosomes (called paents) and offsping based on the fitness values, and (b) eecting othes so as to keep the population size constant. Howeve, fitte chomosomes have highe pobabilities of being selected. Afte seveal geneations, the algoithms convege to the best chomosome, which may epesent the optimum o suboptimal solution to the poblem (Gen and Cheng, 1996). Chomosome epesentation In the study s GA-based appoach, each chomosome is epesented as a single dimensional aay having two kinds of genes: location and assignment genes. If m denotes the numbe of candidate DCs, each abitay chomosome C can be demonstated by, C = ( X, Y i ) = ( X 1, X 1,, X m, Y 10, Y 11,, Y 1( ), Y 0, Y 1,, Y ( ) Y m 0, Y m1,, Y m ( ) ). Whee,, X coespond to the location genes and Y i to the assignment genes. Associated with each candidate DC location, thee is a location gene in the chomosome. Location genes detemine whee the DCs ae located. Moe pecisely, if X = 1, it means that candidate site is selected as a DC location, while if X = 0, candidate location is not chosen as a DC site. Note that the gene Xm + 1 coesponds to dummy DC u; thus, it always takes the value 1. Also, associated with each custome, thee aeassignment genes. Assignment genes detemine the assignment of customes to DCs at diffeent levels. Specifically, Yi = epesents that custome i at level is assigned to DC. If custome i at level is assigned to a dummy DC u, the coesponding assignment gene takes the value of m+1; in othe wods, Yi = m + 1. In this case, the custome is not assigned to any DC at uppe levels, that is, Yis = 0 fo s >. Initialized chomosome population In the study, we initialize pop-size chomosomes fom the feasible egion andomly. In ode to achieve feasible chomosomes, constaints (10) - (15) must be consideed closely. Based on the model fomulation in constaints (10), if custome i at level s is allocated to dummy DC u, it is not assigned to any DC at uppe levels than s. Thus, if Yis = m + 1, then Y = 0 whee s <. i Constaints (11) state that customes can be only allocated to the located DCs, that is, if Yi =, then X = 1. Accoding to constaints (1), if a DC is located at, custome at level 0 is assigned to this DC. Hence, if X = 1, theny 0 =. Based on constaints (13) and (14), excluding dummy DC u, the numbe of located distibution centes must be P. Theefoe, the numbe of vaiables X taking the value of 1 is equal to P. Accoding to constaints (15), if any custome i at level is assigned to a given DC, it cannot be assigned to i at any level of s, whees. Consequently, if Yi =, then Yis fos. Chomosomes fitness The ank-based evaluation function is defined as the obective function (9) fo the chomosomes. In fact, we calculate the obective function (9) fo each of the chomosomes. Obviously, the chomosome which esults in less value of the obective function (9) has the bette ank.

398 Af. J. Bus. Manage. Cossove opeato Cossove opeato geneates offsping chomosomes by meging paent chomosomes. In ode to detemine which of the chomosomesc k, k = 1,,, pop-size ae selected as paents fo cossove opeation, we epeat the following pocedue fom k = 1 to pop-size, that is, by geneating a andom numbe fom the C inteval [0, 1], the chomosome k will be selected as a paent povided that < Pc, whee the paametepc is the pobability of cossove. Then andomly, we goup the selected paentsc 1, C, C 3, to the pais ( C 1, C ), ( C 3, C 4 ),. Without loss of geneality, the cossove opeato on each pai by ( C 1, C ) will be explained. Cossove opeato assigns each custome i at level in offsping chomosome eithe to the DC which is allocated to custome i at level in paent chomosomec 1, o to the DC which is assigned to custome i at level in paent chomosome C. This occus andomly and with a pobability of 0.5. Howeve, the esulted offsping may be infeasible. If a custome is allocated to the dummy DC u at any level, it is not assigned to any DC at uppe levels, but if a custome is allocated to an unselected candidate DC site, this infeasibility is emoved by locating the DC in that candidate location. If the numbe of located DCs exceeds P + 1, the numbe of selected DCs is educed top + 1 by closing some DCs andomly. The customes which ae allocated to these closed DCs ae allocated andomly to one of the opened DCs. If a custome is assigned to a given DC in seveal levels, the assignments of custome at uppe levels ae modified, that is, the custome at uppe levels is allocated to othe DCs andomly in ode to pevent fom assigning the custome to a given DC at moe than one level. Mutation opeato Mutation opeato may modify chomosomesc k, k = 1, and pop-size to fom offsping chomosomes. In ode to detemine which of the chomosomes Ck undego mutation, the will be a epetition of the following pactice fom k = 1 to pop-size: by geneating a andom numbe fom the inteval [0, 1], the chomosomec k will be selected as a paent povided that < Pm, whee the paamete < Pm is the pobability of mutation. Each selected chomosome is modified by one of the two following types of mutation, seveal times (each type of mutation occued with a pobability of 0.5). The fist type of mutation geneates offsping by modifying the assignment genes of paent chomosome. Namely, in the fist type of mutation, two located DCs ae selected andomly; let s and t denote them. Then, if any custome in paent chomosome is assigned to s, that custome will be assigned to t and if any custome is assigned to t, it will be allocated to s. The second type of mutation modifies location genes of paent chomosome to fom offsping. Indeed, the second type of mutation andomly selects a location in which no DC is located; let t denotes it. Next, a DC is selected andomly fom the located DCs and is named s. This type of mutation closes DC s instead of locating a DC at t. Then, all the customes assigned to DC s, ae allocated to DC t. Note that this is simila to the cossove pocess, in that if the esulted offsping does not belong to a feasible egion, it is epaied to be a feasible chomosome. COMPUTATION RESULTS AND DISCUSSION Hee, the study summaizes the computational expeience with the genetic algoithm outlined in the pevious section. The algoithm was coded in Visual Basic.Net 008 and executed on Pentium 5 compute with 1.00 GB RAM and.00 GHz CPU. Evaluating obustness of the GA The poposed genetic algoithm was tested on the 49-node, 88-node and 150-node data sets descibed in Daskin (1995). The 49-node data set indicates the capitals of the lowe 48 United States plus Washington, DC; while the 88-node data set epesents the 50 lagest cities in the 1990 U.S. census along with the 49-node data set, minus duplicates; and the 150-node data set includes the 150 lagest cities in the 1990 U.S. census. Fo all thee data sets, the mean and vaiance of daily demand wee obtained by dividing the population data given in Daskin (1995) by 1000. The pe-unit cost to ship fom DC to custome i was set to the geat-cicle distance between i and in Daskin (1995). Fixed costs of locating DCs wee gained by dividing the fixed costs in Daskin (1995) by 10 fo the 49-node poblem and by 100 fo 88-node poblem. Fo the 150-node poblem, fixed costs of locating DCs wee set to 10000. Howeve, we set holding cost to be 1, q = 0.05, P = 0, β = 0.00001, θ = 0.001, χ = 1, zα = 1.96, L = 1, F = 10, g = 10, a = 5 fo all J and u i = 1000 fo all i I. Although 1 χ = may seem unealistic, the diffeence between the daily and yealy paametes can be ealized though the weights andθ. Tables 1, and 3 summaize the esults fo the computational study on 49-node, 88-node and 150-node poblems. The numbe of geneations was set to 400 and was consideed as a stopping ule of GA. The column labeled pop-size gives the numbe of initial feasible chomosomes, while Pc gives the pobability of cossove and Pm gives the pobability of mutation. The column maked CPU indicates the total numbe of CPU seconds equied and cost indicates the obective value of the solution. The last column gives the paamete eo which shows the deviations of obective values (costs). This paamete can be obtained by: (obective value - the best obective value) / the best obective value, whee the best obective value is the least value in the column maked cost. It follows fom Tables 1-3 that the eo fo the 49-node, 88-node and 150-node poblems does not exceed 0.000008, 0.000007 and 0.00000, espectively. These small values of eo show that costs diffe little fom each othe when diffeent paametes ae selected. We can also see fom the thee tables, when the paametes ae vaied, that the CPU time changes slightly.

Ayanezhad et al. 399 Table 1. Compaison solution of 49-node poblem. Pop-size Pc Pm CPU Cost Eo 1 5 1 0.33 4 979078.3 0.000007 5 0.95 0.3 4 979079.8 0.000008 3 5 0.75 0.1 4 979075.6 0.000004 4 5 0.65 0.1 4 97907.1 0.000001 5 5 0.7 0.1 4 979077.7 0.000006 6 30 0.9 0.3 5 979071.5 0.000000 7 30 0.95 0.5 5 979074.8 0.000003 8 30 0.8 0.15 5 979073.3 0.00000 9 30 0.8 0.5 5 979077.8 0.000006 10 30 0.75 0.5 5 979076.8 0.000005 Table. Compaison solution of 88-node poblem. Pop-size Pc Pm CPU Cost Eo 1 5 1 0.33 117469 0.00000 5 0.9 0.3 1174631 0.000003 3 5 0.8 0. 1174635 0.000007 4 5 0.75 0.1 1174634 0.000006 5 5 0.65 0.05 1174630 0.000003 6 30 1 0.33 3 117467 0.000000 7 30 0.95 0.5 3 117469 0.00000 8 30 0.8 0.1 3 117469 0.00000 9 30 0.75 0.05 3 1174630 0.000003 10 30 0.75 0.09 3 117468 0.000001 Table 3. Compaison solution of 150-node poblem. Pop-size Pc Pm CPU Cost Eo 1 5 1 0.33 11 000073 0.00000 5 0.95 0.3 11 000074 0.00000 3 5 0.8 0. 11 000069 0.000000 4 5 0.7 0.15 11 000071 0.000001 5 5 0.95 0.5 11 00007 0.000001 6 30 0.9 0.3 14 00007 0.000001 7 30 0.8 0.3 14 000069 0.000000 8 30 0.85 0.3 14 000070 0.000000 9 30 0.95 0.3 14 000069 0.000000 10 30 0.85 0.5 14 000070 0.000000 slightly. Thus, the study s genetic algoithm is obust to the paamete setting and effective to solve the model. Measuing pefomance of the GA Hee, we compae solutions fom the study s genetic algoithm with LINGO 8.00 optimization softwae and finally compute the eo. Fo this expeiment, five data sets wee employed: a 10-node, 15-node, 0-node, 5-node and 30-node data set. These five data sets espectively consist of 10, 15, 0, 5 and 30 nodes with the highest demands fom the 49-node data set given in Daskin (1995). Fo each data set, diffeent values of

400 Af. J. Bus. Manage. Table 4. Compaing GA with Lingo 8.00 optimization softwae solution. Nodes P GA time* Lingo time* GA cost Lingo cost Eo 1 10 0.01 0.0004 5 1 1 83513.9 83504.6 0.000033 10 0.01 0.0004 3 1 146.7 1461.1 0.00007 3 10 0.001 0.00001 5 1 11 8331.8 8330.9 0.000003 4 10 0.001 0.00001 3 1 14603.1 14601.5 0.000011 5 15 0.01 0.0004 8 1 7 47317.6 4796. 0.000045 6 15 0.01 0.0004 6 1 6 334534 334507 0.000081 7 15 0.001 0.00001 8 1 9 471.6 4710.1 0.000005 8 15 0.001 0.00001 6 1 373 33434.1 33431.1 0.000009 9 0 0.01 0.0004 1 5 759517.8 759511.6 0.000008 10 0 0.01 0.0004 10 38 614548.9 61454.7 0.000010 11 0 0.001 0.00001 1 3974 7593.5 75931.9 0.000001 1 0 0.001 0.00001 10 171 61435.4 61434.9 0.000001 13 5 0.01 0.0004 1 99 734867.8 734861 0.000009 14 5 0.01 0.0004 10 84 59776.1 59766 0.000017 15 5 0.001 0.00001 1 764 73468.4 73466.9 0.00000 16 5 0.001 0.00001 10 07 5959.1 5957 0.000004 17 30 0.01 0.0004 1 410 689514.6 689498 0.00004 18 30 0.01 0.0004 10 16 5569.9 556905 0.00003 19 30 0.001 0.00001 1 3 3850 6893.7 68931 0.00000 0 30 0.001 0.00001 10 3 30 55673.4 55671 0.000004 *Time is in second. This wok can be extended in some diections; howeve, it would be inteesting to model the poblem when DCs have diffeent o dependent pobabilities of disuptions. Also, the model can be expanded to conside the possibility of disuptions fo the supplie. Anothe development fo this study could be adding the constaints on the maximum capacity of inventoy at DCs o on the maximum demand that can be povided by a supplie. REFERENCES Axsate S (1996). Using the deteministic EOQ fomula in stochastic inventoy contol. Manage. Sci., 4 (6): 830 834. Baionuevo A, Deutsch CH (005). A Distibution System Bought to Its Knees. New Yok Times. p. C1, Sep. 1. Beman O, Kass D, Menezes MBC (007). Facility eliability issues in netwok p-median poblems: stategic centalization and co-location effects. Ope. Res., 55(1): 33 350. Chuch RL, Scapaa MP (007). Potecting citical assets: the -intediction median poblem with fotification. Geogaphical. Anal., 39(1): 19 146. Cui T, Ouyang Y, Shen ZJM (010). Reliable facility location design unde the isk of disuptions. Fothcoming in Opeations Reseach. Daskin MS (1995). Netwok and discete location: models, algoithms, and applications. Wiley. New Yok. Daskin MS, Coullad C, Shen ZJ (00). An inventoy-location model: fomulation, solution algoithm and computational esults. Anna. Ope. Res., 110(1): 83 106. Elebache S J, Melle RD (000). The inteaction of location and inventoy in designing distibution systems. IIE. 3(1): 155 166. Gaey MR, Johnson DS (1979). Computes and Intactability: A Guide to the Theoy of NP-Completeness WH, Feeman and Co. New Yok. Gen M, Cheng R (1996). Genetic algoithms and engineeing design. Wiley. New Yok Hendicks K, Singhal V (003). The effect of supply chain glitches shaeholde wealth. J. Ope. Manage., 1(1): 501 5. Lim M, Daskin MS, Bassamboo A, Chopa S (009). A facility eliability poblem: Fomulation, popeties and algoithm. Naval Res. Logistics. 57(1): 58-70. Melo M, Nickel S, Saldanha-da-Gama F (009). Facility location and supply chain management A eview. Eu. J. Ope. Res., 196: 401-41. Nahimas S (1997). Poduction and opeations management. Thid Edition, Iwin, Chicago. Ozsen L, Coullad CR, Daskin MS (008). Capacitated waehouse location model with isk pooling. Naval Res. Logistics. 55(4) (1): 95 31. Ozsen L, Daskin MS, Coullad CR (009). Facility Location Modeling and Inventoy Management with Multisoucing. Tansp. Sci. 43: 455-47. Qi L, Shen ZJ, Snyde LV (010). The effect of supply disuptions on supply chain design decisions. Fothcoming in Tanspotation Science. Radou N (00). Adapting to Supply Netwok Change. Foeste Reseach Tech Stategy Repot. Foeste Reseach. Cambidge. MA. Scapaa MP, Chuch RL (008). A bilevel mixed-intege pogam fo citical infastuctue potection planning. Comput. Ope. Res., 35(6): 1905 193. Shen ZJ (000). Efficient algoithms fo vaious supply chain poblems. Ph.D. Dissetation. Nothwesten Univesity. Shen ZJ (006). A Pofit Maximizing Supply Chain Netwok Design Model with Demand Choice Flexibility. Opeations. Res. Lett., 34: 673-68. Shen ZJ (007). Integated supply chain design models: a suvey and futue eseach diection. J. Ind. Manage. Optim., 3(1): 1-7. Shen ZJ, Coullad C, Daskin MS (003). A oint location-inventoy

Ayanezhad et al. 401 model. Tansp. Sci., 37(1): 40 55. Shen ZJ, Daskin M (005). Tade-offs Between Custome Sevice and Cost in Integated Supply Chain Design. M&SOM. 7(1): 188 07. Shen ZJ, Qi L (007). Incopoating inventoy and outing costs in stategic location models. Eu. J. Ope. Res., 179(1): 37 389. Shu J, Teo CP, Shen ZJ (005). Stochastic tanspotation-inventoy netwok design poblem. Ope. Res., 53(1): 48 60. Snyde LV, Daskin MS (005). Reliability models fo facility location: the expected failue cost case. Tansp. Sci., 39(1): 400 416. Snyde LV, Daskin MS (007). Models fo eliable supply chain netwok design, in Citical Infastuctue: Reliability and Vulneability AT, Muay, Gubesic TH (eds)., Spinge-Velag, Belin. 13: 57 89. Snyde LV, Daskin MS, Teo CP (007). The stochastic location model with isk pooling. Eu. J. Ope. Res., 179(1): 11 138. Snyde LV, Scapaa MP, Daskin MS, Chuch RL (006). Planning fo disuptions in supply chain netwoks. in TutORials in Opeations Reseach. M.P. Johnson, B. Noman, and N. Secomandi (eds). INFORMS, Baltimoe, MD. 9: 34 57. Souiaan K, Ozsen L, Uzsoy R (008). A genetic algoithm fo a single poduct netwok design model with lead time and safety stock consideations. Eu. J. Ope. Res., 197: 599 608. Zheng YS (199). On popeties of stochastic inventoy systems. Manage. Sci., 38(1): 87 103. Zhou J, Liu B (003). New stochastic models fo capacitated location allocation poblem. Comput. Ind. Eng., 45(1) pp. 111 15.