Uncertain Supply Chain Management

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1 Uncertan Supply Chan Management 5 (2017) Contents lsts avalable at GrowngScence Uncertan Supply Chan Management homepage: Developng a locaton-nventory-routng model usng METRIC approach n nventory polcy Farhad Habb, Ehsan Asad and Seyed Jafar Sadjad * Department of Industral Engneerng, Iran Unversty of Scence and Technology, Tehran, Iran C H R O N I C L E A B S T R A C T Artcle hstory: Receved December 2, 2016 Receved n revsed format December 10, 2016 Accepted Aprl Avalable onlne Aprl Keywords: LIRP model Integrated supply chan Metrc approach Metaheurstc algorthm One-for-one replenshment polcy Locatng, routng and nventory control n producton and dstrbuton centers are the most mportant decsons n supply chan management. Because of the dependence of these decsons to each other, consderng these three aspects smultaneously can have a huge mpact on cost reducton. In ths study, frst, a locaton-nventory model s developed by utlzng METRIC approach and then, METRIC approach s appled to the locaton-nventory-routng model. The ntended supply chan ncludes a suppler, dstrbutors and retalers, and the nventory control polcy s mplemented for both the dstrbutors and retalers. Retalers' demand follows Posson dstrbuton and the lead-tme s also consdered probablstc and s affected by the shortage n dstrbuton centers. Gven that the presented model belongs to the class of NP-hard problems, a hybrd metaheurstc soluton method s also presented to solve the resulted problem. The proposed hybrd metaheurstc algorthm contans a Smulated Annealng algorthm, to optmze the locaton-routng problem, and a Genetc Algorthm, to optmze the nventory problem. Also, to evaluate the performance of hybrd algorthm, a comparson between the results of the proposed hybrd algorthm and the exact solutons obtaned from Lngo software s provded and, fnally, the results are analyzed Growng Scence Ltd. All rghts reserved. 1. Introducton Today, the rapd economc changes and ntense compettve pressures n global marets has led companes and organzatons to pay more attenton to the supply chan management. The organzatons and companes whch are nvolved wth chan structure wll attan greater benefts n the competton. Based on the mpact of tme horzon, Researchers often dvde supply chan decsons nto three types of strategc decsons, tactcal decsons and operatonal decsons. Strategc decsons deal wth locatng the facltes that has sgnfcantly longer term nfluence on the organzatons. Tactcal decsons deal wth nventory management and dstrbuton decsons are consdered as the types of the operatonal decsons wth the mnmum tme perod nfluence. In the past, these decsons were taen separately and ths was caused sub-optmalty and rsng costs. Recently, the researchers found that consderng the decsons lsted smultaneously n one model, wll result n enormous savngs (Hassat & Dabat, 2011). The man purpose of ths study s to optmze the decsons related to dstrbutors' locaton and allocatng retalers to them, determne optmal nventory polcy for dstrbutors and * Correspondng author Tel.: E-mal address: sjsadjad@ust.ac.r (S.J. Sadjad) 2017 Growng Scence Ltd. All rghts reserved. do: /j.uscm

2 338 retalers and determne the optmal route of servng to retalers, smultaneously; so that the total cost related to locaton, nventory and transportaton s to be mnmzed. 2. Lterature revew In the 1980s, organzatons were loong for technques and strateges that they can use to cut producton costs and compete n dfferent marets. Hence, the supply chan ssue was rased so that costs are reduced through the created potental opportuntes. Dstrbuton networ as one of the major causes of supply chan costs, affects the chan performance. Therefore, choosng a sutable and ntegrated dstrbuton networ can facltate the achevement of varous targets of supply chan. The members of the supply chan nclude supplers, warehouses related to raw materals, producton centers, dstrbutors, retalers and fnal customers and the desgn of a dstrbuton networ contans three major problems of assgnng locaton, vehcle routng and nventory control. In the past two decades, much research has been done to develop an ntegrated dstrbuton networ, manly focused on ntegratng two of the above ssues. Hence, the locaton-routng, locaton-nventory and nventory-routng problems were ntroduced. Locaton-Routng Problems (LRP) are one of the oldest ntegraton ssues that have been studed snce the md-1970s. Ths nd of problems was dscussed to fnd the locaton and sutable number of facltes as well as dstrbuton routs for the vehcles. (See, e.g., Mn et al., 1998; Nagy & Salh, 2007; Prodhon & Prns, 2014). Inventory-Routng Problems (IRP), whch were consdered n the early 1980s, are related to storage and dstrbuton of products among a set of customers durng a specfc plannng horzon. (See, e.g., Campbell et al., 1998; Bertazz et al., 2008; Cordeau et al., 2007). Locaton-Inventory Problems (LIP) nclude those of the decsons whch are related to nventory reserves n faclty locaton problem. These problems have been consdered by many researchers durng the past decade (Owen & Dasn, 1998). (See also, e.g., Melo et al., 2009; Farahan et al., 2015). Locaton-Inventory-Routng Problems (LIRP) mae ntegrated decsons on faclty locaton, nventory control and materals orderng, and transport routes plannng ssues. So, we can say that n ths category of problems, strategc, tactcal and operatonal decsons are consdered smultaneously. Research Lu and Lee (2003) s one of the basc research on LIRP ssues where a heurstc method was proposed for solvng these types of problems. Then, Lu and Ln (2005) provded a new method to solve the model presented by Lu and Lee (2003) n whch, the LIRP problem was dvded nto the warehouses locatonallocaton and routng-nventory problems, and then, the fnal soluton was provded by two phases of generatng the ntal soluton and usng hybrd soluton algorthm of TS and SA. Javd and Azad (2010) have modeled the LIRP problem n the sngle-product and sngle-perod mode wth consderng an nfnte capacty for supplers and probablstc demand for customers that follows the normal dstrbuton. Provdng mult-level capacty for each of dstrbuton centers and selectng among them maes ths artcle dstnct from the other related artcles. Wang et al. (2008) entered reverse logstcs ssues nto the LIRP problems. The closed loop logstcs system, consdered by them, was a three-level supply chan and ncludes a plant, several logstcs centers and several dstrbuton or collecton ponts. The model provded by them, was a sngle-product model and has been developed n mult-perod mode. Jang and Ma (2009) have consdered the LIRP problem n closed loop logstcs systems to collect and recycle the products whose lfe has expred. Ther model assumptons were consstent wth the hypotheses of research Wang et al. (2008), and only a two-stage heurstc algorthm based on Genetc Algorthm (GA) has been developed. Neooghadrl et al. (2014) presented a b-objectve and mult-product model for the LIRP problem where the total cost and maxmum average tme of delverng products to customers s mnmzed, smultaneously. Ambrosno and Scutella (2005) have consdered the LIRP problem wth four layers ncludng the plant, central warehouse, regonal warehouses and customers. The am of ther problem was to determne the locaton and number of dfferent facltes n a new networ or to mprove an exstng networ. Seyedhossen et al. (2014) ntroduced an ntegrated model of LIRP to consder the possblty of falure n dstrbuton centers. The model ntroduced by them, smultaneously determnes the dstrbutors' locatons that should be open,

3 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 339 how to allocate customers to dstrbutors and to allocate customers to the actve routes, the order of servng products to them n each route, and nventory polcy n each dstrbuton center. Mult-Echelon Technque for Recoverable Item Control (METRIC) approach was frstly ntroduced by Sherbrooe (1968) to optmze the nventory level of ar force spare parts warehouses, where, the tme between two demands for reparng or replacng the parts s probablstc and follows the exponental dstrbuton. He calculated some performance measures usng queung theory theorems to determne the optmal nventory level. After that, Mucstadt (1973) generalzed Sherbrooe's METRIC approach, and consdered a herarchcal structure for the products. He assumed that when a part needs to be repared, exactly one of ts sub-components (modules) needs reparng. Slay (1984) developed a METRIC model, nown as VARI-METRIC, n whch the average number of tems under repar s equal to ts varance and he used the negatve bnomal dstrbuton for ths assumpton. Graves (1985) ndependently used ths dstrbuton for proposng exact and approxmate methods n a mult-level problem. Afterwards, Sherbrooe (1986) consdered a model smlar to Mucstadt (1973); however, he used a dfferent approxmate approach and wth the numercal examples, showed that a sgnfcant mprovement can be provded n the accuracy of the METRIC method by usng hs proposed approach. The Sherbrooe's model was later developed as a standard model by researchers and dfferent soluton methods were used to solve t. For example, Wang et al. (2000) developed a two-level reparable nventory system where, the lead tme for replenshment was consdered probablstc. Rustenburg et al. (2003) developed an exact model for a mult-level problem and then, Wong et al. (2005) developed an analytcal model to determne the optmal nventory levels of spare parts n a reparable tem nventory system wth consderng several plants and dstrbutors and assumng the mult-product mode. Andersson and Melehors (2001) presented a two-level model wth a central warehouse and several retalers that customers' demand for the retalers consdered probablstc and follows the Posson dstrbuton. In ther model, all the lead tmes were fxed and used nventory polcy was ( S 1, S) wth the contnuous revew. They used METRIC method n ther modelng and evaluated the proposed model by usng 13 test problems. In ths paper, a LIRP model s presented wth consderng the METRIC approach to modelng the nventory polcy that has the followng aspects of nnovaton compared to the other smlar researches: Through revewng the locaton nventory-routng problems lterature, t can be seen that none of the studes and researches conducted n ths feld has consdered the ( S 1, S) nventory polcy, although t s nown as an mportant nventory polcy. Therefore, n ths study, ths polcy s beng used to model the problem. Also, n the most studes, the uncertanty n transport tme that leads to uncertanty n lead tme has not been consdered. Whle n ths paper, lead tme s consdered as a functon of shortage. In most papers, because of the normal dstrbuton nature and ease of deployment, customers' demand s consdered as the functon of normal dstrbuton; whle the demand can be better approxmated usng the Posson dstrbuton. In ths paper, retalers' demand follows the Posson dstrbuton. METRIC approach s one of the new approaches to determne the amount of nventory level n the ( S 1, S) nventory polcy. In ths paper, the METRIC approach s used to model the locaton-nventory and locaton-nventory-routng problems. 3. The locaton-nventory problem wth METRIC approach 3.1. Problem defnton The supply chan studed n ths paper s conssted of three levels ncludng a suppler, potental dstrbutors and retalers. Also, t s assumed that the locatons of suppler and retalers are specfed and we am to determne the number and locatons of dstrbutors to meet the retalers' demands. The

4 340 nventory polcy ntended to hold the nventory at the retalers and dstrbutors comply wth ( S 1, S) polcy and the purpose of the problem s to determne the optmal nventory level for the retalers and dstrbutors, so that the holdng and shortage costs wll be mnmzed. Demands of the retalers follow the Posson process and therefore, the demand of each dstrbutor wll be equal to sum of the demands of the retalers whch are assgned to that dstrbutor. Lead (Procurement) tme to receve the retalers' orders from dstrbutors s the functon of the shortage n each dstrbutor and s consdered probablstc. Ths functon s obtaned based on queung theory rules whch are descrbed below. In ths case, t s assumed that the suppler has no supply capacty constrant. Fg. 1 shows the overall structure of the ntended supply chan. Fg. 1. Overall structure of consdered supply chan networ Regardng the above LIRP problem, the followng assumptons have been consdered: Consdered supply chan has three levels ncludng a suppler and several dstrbutors and retalers. Suppler and retalers have fxed and predetermned locatons and the objectve s to determne the optmal locatons for the dstrbutors. The nventory have been held only n the retalers' and dstrbutors' warehouses. The Inventory polcy wll be determned only for retalers and dstrbutors. The nventory polcy ntended for the retalers and dstrbutors s ( S 1, S). Demands of the retalers are probablstc and follow the Posson dstrbuton. Lead tme for replenshng retalers by dstrbutors s the functon of shortage. So, ths tme s also probablstc. The most mportant purposes of the problem are as follows: 1. Determne the optmal locaton of dstrbutors

5 2. Allocatng retalers to the localzed dstrbutors F. Habb et al. /Uncertan Supply Chan Management 5 (2017) Determne the optmal amount of nventory for the retalers and dstrbutors So that the total cost of desgnng supply chan ncludng nventory holdng and shortage costs, the cost of establshment and locaton, and the costs of orderng and purchase are mnmzed Problem notatons Notatons used for the locaton-nventory problem s as follows. These symbols are smlar to the locaton-nventory-routng problem whch s descrbed n the followng, and the addtonal symbols related to the locaton-nventory-routng problem wll be descrbed n the same secton. Sets Set of retalers 1, 2,..., I Set of dstrbuters 1, 2,..., K Parameters f The fxed cost for establshng dstrbutor h Holdng cost per unt for dstrbuter Shortage cost per unt for dstrbuter h Holdng cost per unt for retaler Shortage cost per unt for retaler The annual demand rate for retaler C The purchase cost per unt related to dstrbutor for buyng from suppler A Orderng cost for dstrbutor C The purchase cost per unt related to retaler for buyng from dstrbutor A Orderng cost related to retaler for buyng from dstrbutor The carryng tme from suppler to dstrbutor The carryng tme from dstrbutor to retaler Decson Varables I I 0 I Average nventory for dstrbutor Average shortage for dstrbutor Average nventory for retaler

6 342 I Average shortage for retaler Annual demand rate receved from retalers for dstrbutor W 0 Delay random varable for dstrbutor warehouse, due to nventory shortage z y The average lead tme (procurement tme) of retaler to receve the product A bnary varable that equals to 1 f the dstrbutor s establshed and 0 otherwse A bnary varable that equals to 1 f the retaler s allocated to dstrbutor and 0 otherwse 3.3. Model descrpton As mentoned earler, n ths supply chan, dstrbutors' and retalers' nventory polcy comples wth the ( S 1, S) nventory polcy. In ths polcy, upon recept of a demand request and response to t, nventory levels reduced by one unt and by reducng each unt of nventory, an order wll be sent to replace the dropped tem. The demand whch retalers receve s probablstc and follows the Posson dstrbuton wth the rate of. Snce each retaler s only allocated to one dstrbutor, the demand rate for a dstrbutor s equal to sum of the demand rates of retalers that have been assgned to t whch s calculated from the Eq. (1): y I K (1) The tme to transport the products from suppler to dstrbutor s equal to and the tme of transportng from dstrbutor to retaler s equal to. In addton to the transport tme from the dstrbutor to the retaler, a lead tme s consdered to meet the needs of retalers. Ths lead tme has a value when a dstrbutor s faced wth shortage and the retaler s forced to wat untl the dstrbutor was replenshed. To calculate ths tme accordng to 'Lttle's law', we have (Wang et al., 2008): W I. K (2) When the shortage s zero, the lead tme for elmnaton the shortage wll also be zero and f the shortage s postve, retalers wll wat to get the product untl the shortage condton s resolved. So, the retaler's total lead tme for replenshment wll be equal to: W y, K I (3) whch s equal to the sum of the lead tmes for retalers to get the product and the delay tme due to shortage condton. To calculate the average nventory and the average shortage wth the METRIC approach, we conduct as follows. The probablty that the nventory level of dstrbutor s equal to j s: I j S D j D S j Pr Pr ( ) Pr ( ) K (4) Accordng to a Posson dstrbuton, we have:

7 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 343 e Pr D( ) S j S0 j ( ) S j! K (5) So, the average nventory level n the dstrbutor s calculated from the followng expected value: I S 0 S0 j e ( ) j K (6) j1 S j! Also, the average shortage for the dstrbutor can also be determned as follows (Wang et al., 2008): I 1 S0 j e ( ) j. K (7) j S j! Because of ts nfnte lmt, t s dffcult to solve ths equaton. Hence, we use the followng method to get the average shortage: E I I I (8) I I E I (9) On the other hand, we have: E I S (10) So the average shortage s equal to: I I S K (11) In the same way, the average nventory and average shortage can be calculated for the retalers wth ths dfference that lead tme of retalers, whch s calculated above, s equal to. The average nventory and average shortage values for the retaler are equal to the followng equatons: S j S j e ( ) I j I (12) j1 S j j! I I S I (13) 3.4. Mathematcal model Accordng to the above descrpton, mathematcal model for the locaton-nventory problem wll be as follows K I K C A C A y mn z fz h I I h I I K K I (14)

8 344 subject to y 1 I (15) K y z K; I (16) y K (17) I I S S j e ( ) j K (18) j1 S j! I I S K (19) I W0 K (20) W0 y I (21) I K S j S j e ( ) j I (22) j1 S j j! I I S I (23) z 0,1 K (24) y 0,1 K; I (25) S nt I (26) S nt K (27) The frst part of the equaton 14 or the objectve functon s devoted to the dstrbutors' locatng costs. The next part reflects the retalers' nventory holdng and shortage costs, the dstrbutors' nventory holdng and shortage costs, the retalers' purchasng and orderng costs and the dstrbutors' purchasng and orderng costs. Constrant 15 ensures that each retaler s only allocated to one dstrbutor. Constrant 16 states that a retaler cannot be allocated to an nactve dstrbutor. Constrants 17 to 23 are fully descrbed before provdng the mathematcal model. Constrants 24 to 27 are the standard constrants of the mathematcal model. 4. Locaton-Inventory-Routng Problem wth METRIC approach 4.1. Problem defnton In the locaton-inventory problem proposed n the prevous secton, t was sad that several dstrbutors are selected from the potental dstrbutors set and retalers are allocated to these actvated dstrbutors such a way that the total cost s mnmzed. In the locaton-nventory-routng problem, n addton to prevous decsons, the order of servng and the optmum route for transportng products from dstrbutors to retalers are determned Problem notatons To develop the model presented n the prevous secton and proposng the locaton-inventory-routng model n ths secton, we frstly descrbe the symbols. Most of the symbols of ths model are dentcal to those of the prevous model, and the followng symbols should be added to prevous symbols. Sets v Set of vehcles v 1, 2,..., V

9 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 345 Parameters T The transportng cost from node g to node h g h Q Vehcles capacty Decson varables X A bnary varable that equals to 1 f the node h s mmedately served after g hv g by the vehcle v and 0 otherwse; B A bnary varable that equals to 1 f the vehcle v from the dstrbutor ghv pass through the nodes of g and h to serve the retaler and 0 otherwse; N Auxlary varable for sub-tour elmnaton n route of vehcle v whch s g v defned for retaler g 4.3. Mathematcal model z f z h I I h0 I0 0 I0 K I K mn C A C A y T X gh ghv K K I gm hm vv subject to y 1 I (29) K y z K; I (30) y K (31) I I S S j e ( ) j K (32) j1 S j! I I S K (33) W I K (34) T B K; I (35) gh ghv vv hi gm y X B K; I; gm ; v V (36) gv gv Xghv Bhlv B ghv K; I; hi; li; gm ; v V (37) B ghv X ghv K; I; gm ; hm ; v V (38) Bghv z vv K; I; gm ; h M (39) W0 y I (40) I K S j S j e ( ) j I (41) j1 S j j! (28)

10 346 I I S I (42) Xgv Q gm I v V (43) X gv 1 gm vv I (44) X v 1 K I v V (45) Xghv Xhgv 0 gm gm hm ; v V (46) Xhv Xhv y 1 K; I; v V (47) hm hm Nv Ngv I Xgv I 1 g, I; v V (48) 0,1 0,1 0,1 0,1 z K (49) y K; I (50) X gh, M; v V (51) ghv B K; I; hi; gm ; v V (52) S ghv nt I (53) S nt K (54) The frst part of the objectve functon s smlar to the locaton-nventory problem objectve functon. The fnal part of the objectve functon represents the routng cost of vehcles and sees a path that has the mnmum cost for servng the needs of retalers. Constrants 29 to 34 and 40 to 42 are dentcal to the constrants of locaton-nventory problem presented n the prevous secton. The remanng constrants are as follows: Constrant 35 calculates the transportaton tme from the retaler to the dstrbutor. Constrant 36 states that f the retaler s allocated to the dstrbutor ( y 1) then there s a path from node g to ths retaler ( X gv 1), so, the path of node g to s surely n the path of dstrbutor to ( Bgv 1). Constrant 37 s also conceptually smlar to the prevous constrant. Ths constrant states that f there s a path from node g to node h, the path h wll be n the path from the dstrbutor to the retaler, then surely, the path g h wll be also n the path of dstrbutor to the retaler. Constrant 38 states that when a path can be n the path of dstrbutor to the retaler that t was actve; otherwse, t s not possble to pass from the path and reach the retaler from the dstrbutor. Constrant 39 states that we can have a path from a dstrbutor f the dstrbutor s actve. Constrants 36 to 39 are supplements of the constrant 35 so that they cause the correct value for the varable B. Constrant 43 s the vehcles capacty constrant. Constrant 44 states that each retaler only assgned to a vehcle path. Constrant 45 ensures that several dstrbutors cannot serve a path. Constrant 46 states that f a vehcle s entered to a node whch s related to a retaler or a dstrbutor, t wll also ext from that node. Constrant 47 represents that a path cannot pass a retaler unless t s allocated to a dstrbutor. Constrant 48 s the sub-tour elmnaton constrant and ensures that each path has a dstrbutor as the start and end ponts of the route. Also, constrants 49 to 54 are the standard constrants of mathematcal model. 5. Soluton approach The Genetc Algorthm (GA) was proposed n 1975 by John Holland. The basc dea of GA s based on evolutonary n nature. Consder a certan populaton of people n whch the lmtaton of envronmental resources has caused that they try for survval and reproducton based on ther ftness. In the solvng problem procure, consder a functon that should be maxmzed. We can randomly generate a set of

11 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 347 canddate solutons and consder the functon value as the ftness level. The hgher ths value s, the hgher the ndvdual qualty s. Accordng to ths ftness, the better ndvduals are selected as the parents for creatng the next generaton (usng crossover and mutaton operators). Fg. 2 shows a smple pseudocode for the genetc algorthm. BEGIN END INITIALIZE populaton wth random canddate solutons EVALUATE each canddate REPEAT UNTIL termnaton condton s satsfed, DO SELECT parents RECOMBINE pars of parents MUTATE the resultng offsprng EVALUATE new canddates SELECT ndvduals for the next generaton Fg. 2. Genetc Algorthm (GA) pseudocode In 1935, Metropols et al. (1953) presented a smple algorthm to smulate the process of heatng and coolng of metals to reach the thermal equlbrum. Later, they used ths algorthm to solve the optmzaton problems. In ths algorthm, wth startng from an ntal state, the system s placed under the effects of a change. If the change leads to a reducton n the objectve functon (n a mnmzaton problem) t wll be accepted. Otherwse, f the change leads to an ncrease n the objectve functon, t E wll be accepted wth the probablty of exp( ) where E represents the dfference between two T objectve functons and T ndcates the temperature. In fact, the acceptance process, when the objectve functon wll be worse, a number s randomly selected from the range of zero to one and then, f the E number s less than or equal to exp( ), the change wll be accepted, and otherwse, t s rejected. T Ths approach avod gettng stuc on a local optmum soluton durng the search process. Fg. 3 shows a pseudocode for the Smulated Annealng (SA) algorthm. Input: Coolng schedule s s 0 /* Generaton of the ntal soluton */ T /* Startng temperature */ Tmax Repeat Repeat /* At a fxed temperature */ Generate a random neghbor s E f( s) f( s) If E 0 Then s s /* Accept the neghbor soluton */ Else Accept s wth a probablty T e Untl Equlbrum condton /* e.g. a gven number of teratons executed at each temperature T */ T g( T) /* Updatng temperature */ Untl Stoppng crtera satsfed /* e.g. T T */ mn Output: Best founded soluton E Fg. 3. Smulated Annealng (SA) algorthm pseudocode Snce the model presented n ths paper belongs to the class of NP-hard problems, a hybrd metaheurstc algorthm s used to solve the model. In ths algorthm, the nventory decsons have been optmzed by the GA and routng, locaton and allocaton decsons have been optmzed by the SA algorthm. The reason of usng these two algorthms for solvng locaton-routng and nventory problems arses from the conclusons n prevous studes. For example, n solvng the locaton-routng problem presented by Vncent et al. (2010), SA has been used to solve the problem. We can also refer the studes Neooghadrl et al. (2014), Javd and Azad (2010), and Seyedhossen et al. (2014), whch

12 348 have used genetc algorthm to solve ther problems. Here, one-dmensonal matrx s used to show the solutons. For example, consder the followng matrx. In ths one-dmensonal matrx, the negatve values ndcate the dstrbutors and the postve values ndcate the path. The number of zeros s calculated from the equaton 55 (Vncent et al., 2010). n, Q (55) where represents the demand of retal and Q represents the vehcle capacty and * represents the smallest nteger greater than or equal to *. Fg. 4 shows a feasble soluton for the problem Fg. 4. A feasble soluton matrx Above matrx can be shown as Fg. 5: (10) <25> <21> (16) <24> (19) (11) (8) (2) (15) (6) <23> (9) (14) (18) (17) (12) (1) (3) Legend (4) Opened Depot (20) <22> (7) Closed Depot Customer (13) (5) Vehcle route Fg. 5. A schematc representaton for consdered matrx Ths answer matrx s mproved at each teraton by usng SA algorthm and t s transferred to GA. Based on ths answer, GA generates and mproves the nventory problem soluton and then the soluton obtaned at two stages has been combned together and the total soluton wll be obtaned. In other

13 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 349 words, accordng to the soluton obtaned by the SA algorthm, GA generates and mproves a soluton for the nventory problem. In ths way, the GA, accordng to the receved answer, extract the number of actve dstrbutors and form the ntal soluton. For example, f the number of actve dstrbutors s equal to 2 and the number of actve retalers s equal to 6, the soluton wll be consdered as an 8-bt vector whose values ndcate the nventory level S for the actve dstrbutors and retalers. In ths case, n the GA, the chromosome can be shown as follows Fg. 6. An example of nventory decsons n the form of chromosome To generate neghborhood n ths algorthm, two operators of crossover and mutaton have been used. To generate neghborhood n crossover operator, there are two modes n whch, two chromosomes are selected as parents and then, the offsprngs are generated by usng one of the followng methods: 1. Gven the followng two chromosomes n Fg. 7, n the frst method, a random number s generated less than the sze of chromosomes. Suppose that n ths example our random number s 5. It means that the chromosomes replace the ffth elements onwards wth each other. Parent chromosome 1 Parent chromosome Chld chromosome 1 Chld chromosome Fg. 7. Neghborhood generaton procedure n the frst method 2. In the second case, the neghborhood generaton s smlar to the frst method wth the dfference that the fracton and dsplacement occurs n two ponts. Parent chromosome 1 Parent chromosome Chld chromosome 1 Chld chromosome Fg. 8. Neghborhood generaton procedure n the second method

14 350 Also, n the mutaton operator a random number s generated and replaced wth one of the parent's chromosomes elements. For example, f the random number s equal to 15, mutaton wll occur as follows: Parent chromosome Chld chromosome Fg. 9. Mutaton operator The hybrd algorthm proposed to solve the problem s shown schematcally n Fg. 10. Smulated Annealng (SA) algorthm Start Generatng ntal soluton for locaton-routng problem Neghborhood generaton Solvng locaton-routng problem by smulated annealng algorthm Genetc Algorthm (GA) Solvng nventory problem by genetc algorthm No Stop condton has been met or not? Summng two solutons result of solvng two algorthms yes Dsplayng problem soluton End Fg. 10. Schematc representaton of the proposed hybrd algorthm Ths hybrd algorthm starts wth the smulated annealng algorthm and contnues wth generatng ntal solutons for locaton-routng problem. The solutons wll be mproved by usng ths algorthm. The soluton of the SA algorthm wll be transferred to genetc algorthm. The GA generates and mproves the nventory problem soluton accordng to the soluton receved from the prevous stage. The solutons obtaned from the two stages sum together and f the stop condton s meet, the algorthm wll end. Otherwse, the algorthm contnues and the answer also mproves.

15 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) Computatonal results In ths secton, n order to valdate the presented model as well as the proposed hybrd metaheurstc algorthm, the computatonal results have been expressed. As a result, the studed model was coded usng the proposed metaheurstc soluton algorthm provded n the software Matlab 2015a and was run on a system wth Wndows 7-64bt OS, core GHz processor, and 8 GB of RAM. In order to produce varous test problems to solve the problem by the proposed algorthm, the randomly generated data whch are followed the unform dstrbuton was used for the values of model parameters. Table 1 shows the values and statstcal dstrbuton of these parameters. Table 1 Statstcal dstrbuton used for problem parameters Parameter A Unform dstrbuton parameters f h h C A C A Tg h 0 5 In order to ncrease the effcency of the proposed soluton algorthms, Taguch method was used to tune ther parameters. The tuned parameters for the two algorthms of SA and GA are shown n Tables 2 and 3, respectvely. Table 2 Tuned parameters of smulated annealng algorthm Tntal Tmn R ISA-man B ISA-nner Table 3 Tuned parameters of genetc algorthm Pop Sze Pcrossover Pmutaton IGA The presented model s solved 3 tmes n 3 szes of small, medum and large wth consderng the values of the Table 1 for the parameters and the results of solved problems n two exact and hybrd metaheurstc methods are presented n Table 4. Also, the error percentage n metaheurstc method compared to the exact method can be seen n the last column of Table 4 as follows,

16 352 Table 4 The results from solvng proposed LIRP model Problem Sze Small Medum Large Number of dstrbuters Number of retalers Exact Metaheurstc Error percentage Cost Tme Cost Tme % % % % % ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** The large sze problems s not solvable by usng Lngo software As can be seen n Fg. 11, the use of metaheurstc soluton algorthm decreases the computaton tme n solvng large problems. By enlargng the sze of problem, the computaton tme of exact method exponentally ncreases Computatonal tme Percent error Problem Problem Exact method Metaheurstc method Real error Estmated error Fg. 11. Comparng exact and metaheurstc methods Fg. 12. The error of metaheurstc method based on computatonal tme compared to the exact method The error percentage values obtaned from metaheurstc algorthm compared to the exact method s dsplayed n Fg. 12. Red lne shows the predcted values for the error determned by the movng average method wth consderng n 5. As can be seen, the error percentage s sgnfcantly nconsderable and accordng to more approprate computatonal tme for medum and large problems, the use of proposed soluton method s more effcent. 7. Senstvty analyss For the senstvty analyss, the ffth problem of Table 4 whch s conssted of 6 dstrbutors and 15 retalers has been studed. To now whether the problem has proper behavor n the face of changng parameters, t s necessary to observe the effects of parameters on three values n the problem. These

17 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 353 three values nclude: objectve functon value, optmal amount of base stoc ( S ) for dstrbuters, and optmal amount of base stoc ( S ) for retalers. Senstvty analyss based on the dstrbutors' orderng cost: In ths analyss, the relatonshp between the objectve functon and the dstrbutors' orderng cost has been studed. As can be seen n Fg. 13, wth ncreasng n the dstrbutors' orderng cost, the objectve functon value ncreases lnearly. In other words, there s a drect relaton between these two values. 7.5e+6 1.1e+7 7.0e+6 1.0e+7 Value of objectve functon 6.5e+6 6.0e+6 5.5e+6 5.0e+6 Value of objectve functon 9.0e+6 8.0e+6 7.0e+6 6.0e+6 5.0e+6 4.5e+6 4.0e+6 4.0e e Dstrbutors' orderng cost Retalers' orderng cost Fg. 13. Senstvty analyss of objectve functon based on dstrbutors' orderng cost Fg. 14. Senstvty analyss of objectve functon based on retalers' orderng cost Also, the relatonshp between objectve functon and the retalers' orderng cost has been nvestgated. In ths analyss, t s also observed that the objectve functon value ncreases wth an ncrease n the retalers' orderng cost. Ths analyss confrms logcal relatons. Because by observng the objectve functon, we can also understand ths ssue. In the objectve functon, t was observed that the value of A has a drect mpact on the objectve functon and the value of objectve functon ncreases by ncreasng n ts value. Senstvty analyss based on transportng tme from the suppler to dstrbutors (n days): Another analyss that should be done s the effect of transportng tme from suppler to dstrbutor on other values. The effect of ths parameter on the three values of the objectve functon, the dstrbutors' optmal base stoc ( S ) and retalers' optmal base stoc ( S ) has been studed Value of objectve functon Dstrbutors' base stoc Dstrbutors' transportng tme Dstrbutors' transportng tme Fg. 15. Senstvty analyss of objectve functon based on dstrbutors' transportng tme Fg. 16. Senstvty analyss of dstrbutors' base stoc based on dstrbutors' transportng tme

18 354 As can be seen n Fg. 15, wth ncreasng n transportng tme from suppler to dstrbutor, the value of objectve functon also ncreases. Ths ncrease can be the result of the ncrease n nventory level that should be hold. Because by ncreasng n the transport tme, the amount of nventory ncreases to prevent the creaton of a massve shortage.also, Fg. 16 and Fg. 17 confrm ths ssue Retalers' base stoc Value of objectve functon Dstrbutors' transportng tme Dstrbutors' purchase cost Fg. 17. Senstvty analyss of retalers' base stoc based on dstrbutors' transportng tme Fg. 18. Senstvty analyss of objectve functon based on dstrbutors' purchase cost Senstvty analyss of objectve functon based on dstrbutors' purchasng cost from the suppler: Fg. 18 shows the effect of the dstrbutor's purchasng cost from the suppler on the objectve functon. It s reasonable that wth ncreasng trend n ths cost, the value of the objectve functon also ncreases. Senstvty analyss of retalers' and dstrbutors' optmal base stoc: By ncreasng n the shortage cost, to avod facng shortage, the value of base stoc ( S ) wll ncrease. Also, f the nventory hoodng cost ncreases, naturally the amount of holdng nventory should be reduced so that the nventory holdng cost s beng adjusted. These ssues can be seen n Fg. 19 and Fg. 20, respectvely Dstrbutors' base stoc Retalers' base stoc Dstrbutors' shortage cost Retalers' shortage cost Fg. 19. Senstvty analyss based on shortage costs

19 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) Dstrbutors' base stoc Retalers' base stoc Dstrbutors' holdng cost Retalers' holdng cost Fg. 20. Senstvty analyss based on holdng costs Senstvty analyss based on demand rate: Another mportant parameter that needs to be analyzed s the demand rate. The effects of ths parameter on three values of objectve functon, retaler's base stoc, and dstrbutors' base stoc s also studed. As can be seen, ncreasng n ths parameter leads to ncreasng n amount of ths three values. Ths ssue s not out of mnd, because by ncreasng n demand, the nventory level should be ncreased to reduce possblty of facng the shortage. 1.1e e Value of objectve functon 9.0e+4 8.0e+4 7.0e+4 6.0e+4 Retalers' base stoc e e Demand rate Demand rate Fg. 21. Senstvty analyss of objectve functon based on demand rate Fg. 22. Senstvty analyss of retalers' base stoc based on demand rate Accordng to Fg. 21 and observng the objectve functon changes trend towards ncreasng demand, we can found that the objectve functon has also ncreased. Ths ncrease s absolutely logcal, because wth ncreasng demand rate, holdng, shortage and orderng costs also ncreases. Fg. 22 represents that by ncreasng n rate of demand, the optmal nventory level of S n retalers wll also ncrease. Ths ssue s occurred due to ncreased demand to avod shortage. Increasng retalers' demand also causes the ncrease n the dstrbutors' demand. So, t s logcal that the amount of S related to dstrbutors s ncreased by ncreasng the retalers' demand. Fg. 23 also confrms ths theme. Senstvty analyss of objectve functon based on dstrbutors' and retalers' holdng costs, smultaneously: For better understandng, the mpact of dstrbutors' and retalers' holdng costs are analyzed smultaneously. Fg. 24 shows the effects of these two parameters. In ths analyss, dstrbutors' holdng cost ncreases from low to hgh value and the retalers' holdng costs has decreased from hgh to low

20 356 value. The amount of ncreasng and decreasng have occurred smultaneously and have equalled to a same value. However, the objectve functon value s ncreased by these changes, the analyss shows that the mpact of dstrbutors' holdng cost on the objectve functon s more than that of retalers' holdng cost. Ths ssue shows that the objectve functon s more senstve to dstrbutors' holdng cost parameter Retalers' holdng cost Dstrbutors' base stoc Value of objectve functon Demand rate Fg. 23. Senstvty analyss of dstrbutors' base stoc based on demand rate Dstrbutors' hodng cost Fg. 24. Senstvty analyss of objectve functon based on dstrbutors' and retalers' holdng cost So, accordng to ths analyss, what the managers and decson maers need to consder, s a greater focus on dstrbutors' holdng cost than the retalers' holdng cost; because the reducton n dstrbutors' holdng cost has greater postve mpact than the reducton n retalers' holdng cost. The same analyss can be appled to the shortage cost as follows. Senstvty analyss of objectve functon based on dstrbutors' and retalers' shortage costs, smultaneously: In ths analyss, the effects of dstrbutors' and retalers' shortage cost parameters on the objectve functon are analyzed smultaneously. As can be seen n Fg. 25, the shortage cost ncreases for the dstrbutor and smultaneously reduces for the retaler. We can also conclude that the dstrbutors' shortage cost has a greater effect on the objectve functon Retalers' shortage cost Value of objectve functon Dstrbutors' shortage cost Fg. 25. Senstvty analyss of objectve functon based on dstrbutors' and retalers' shortage costs 8. Concluson and future research As t s expressed earler, a supply chan nvolves dfferent levels of decson-mang n whch macro level ncludes strategc and long-term decsons. The next level ncludes medum-term or tactcal

21 F. Habb et al. /Uncertan Supply Chan Management 5 (2017) 357 decsons and the operatonal and short-term decsons are n the next level. Organzatons are requred to contnuously mprove ther performance to survve n ths compettve world and one of the most mportant mprovements can be consdered n the decsons related to the supply chan. Locaton and allocaton are the two nds of strategc decsons; nventory management and control s one of the tactcal decsons and the decsons related to transportaton problem can be mentoned as the operatonal decsons. Therefore, n ths study, an ntegrated supply chan model wth consderng uncertan condtons s proposed, whch optmzes all locaton, nventory, transportaton and routng decsons. In ths study, we frstly presented the locaton-nventory model wth consderng uncertanty condton and usng METRIC approach and then, the model was developed by addng the routng decsons. The polcy whch s consdered for nventory, was the ( S 1, S) polcy. Snce the presented model belongs to the class of NP-hard problems, a hybrd GA-SA algorthm s proposed to solve the problem. Fnally, 12 test problems n small, medum and large szes have been solved and the effect of the parameters on the dstrbutors' and retalers' base stocs ( S ) and the objectve functon were analyzed. It s recommended for the future research that the problem are also taen nto consderaton wth the am of mnmzng transport tme and mnmzng the rs. Also, the use of mult-objectve and twolevel models s also among the ssues that have receved less attenton n locaton nventory-routng problems. Utlzng the green supply chan topc s another development that could ncrease the attractveness of the ssue; snce green supply chan management s one of the most mportant needs of today's world such a way that many governments have encouraged organzatons to fulfll these green goals. Locaton- nventory-routng model has the potental to be wdely used n green supply chan optmzaton. References Ambrosno, D., & Scutella, M. G. (2005). Dstrbuton networ desgn: New problems and related models. European Journal of Operatonal Research, 165(3), Andersson, J., & Melehors, P. (2001). A two-echelon nventory model wth lost sales. Internatonal Journal of Producton Economcs, 69(3), Bertazz, L., Savelsbergh, M., & Speranza, M. G. (2008). Inventory routng. In The vehcle routng problem: latest advances and new challenges (pp ). Sprnger US. Campbell, A., Clare, L., Kleywegt, A., & Savelsbergh, M. (1998). The nventory routng problem. In Fleet management and logstcs (pp ). Sprnger US. Cordeau, J. F., Laporte, G., Savelsbergh, M. W., & Vgo, D. (2007). Vehcle routng. Handboos n operatons research and management scence, 14, Farahan, R. Z., Rashd Bajgan, H., Fahmna, B., & Kavan, M. (2015). Locaton-nventory problem n supply chans: a modellng revew. Internatonal Journal of Producton Research, 53(12), Graves, S. C. (1985). A mult-echelon nventory model for a reparable tem wth one-for-one replenshment. Management Scence, 31(10), Hassat, A., & Dabat, A. (2011). A locaton-nventory-routng problem wth pershable products. In Proceedngs of the 41st Internatonal Conference on Computers and Industral Engneerng. Javd, A. A., & Azad, N. (2010). Incorporatng locaton, routng and nventory decsons n supply chan networ desgn. Transportaton Research Part E: Logstcs and Transportaton Revew, 46(5), Jang, S., & Ma, Z. (2009). A hybrd genetc algorthm for the stochastc dynamc locaton-routngnventory problem n closed-loop logstcs system for reusng end-of-use products. In Internatonal Conference on Transportaton Engneerng 2009 (pp ). Lu, S. C., & Lee, S. B. (2003). A two-phase heurstc method for the mult-depot locaton routng problem tang nventory control decsons nto consderaton. The Internatonal Journal of Advanced Manufacturng Technology, 22(11-12),

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