A hybrid Tabu-SA algorithm for location-inventory model with considering capacity levels and uncertain demands

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1 ISSN , Eglad, UK Joural of Iformatio ad Computig Sciece Vol. 3, No. 4, 8, pp A hybrid Tabu-SA algorithm for locatio-ivetory model with cosiderig capacity levels ad ucertai demads Nader Azad 1, Hamid Davoudpour Departmet of Idustrial Egieerig, Amirabir Uiversity of Techology, P.O. Box , Tehra, Ira Received Jauary 8, 8, accepted July 18, 8) Abstract. I this paper, we preset a complex distributio etwor desig problem i supply chai system which icludes locatio ad ivetory decisios. Customers demad is geerated radomly ad each distributio ceter maitais a certai amout of safety stoc i order to achieve a certai service level for the customers it services. Ulie most of past research, our model allows for multiple levels of capacities available to the distributio ceters. This cosideratio helps to achieve the capacity utilizatio to a high level as our computatioal results show this fact. We show that this problem ca be formulated as a o-liear iteger programmig model. A hybrid heuristic combiig Tabu search with Simulated Aealig SA) sharig the same tabu list is developed for solvig the problem. We comprise the hybrid algorithm with the optimal solutio, Simulated Aealig algorithm ad Tabu search algorithm. The results idicate that the method is efficiet for a wide variety of problem sizes. Keywords: facility locatio, ivetory, itegrated supply chai desig, capacity level, simulated aealig, tabu search. 1. Itroductio All compaies that aim to be competitive o the maret have to pay attetio to their orgaizatios related to the etire supply chai. I particular, compaies have to aalyze the supply chai i order to improve the customer service level without a ucotrolled growth of cost. I few words, compaies have to icrease the efficiecy of their logistics operatios. Traditioally, supply chai models have focused o either the strategic aspects of supply chai desig or the tactical aspects, but ot both simultaeously. Oze [11] shows that cost savig ca be obtaied by cosiderig locatio ad ivetory decisios simultaeously istead of the sequetial approach, where locatio decisios are made before the ivetory decisios. For a thorough review of facility locatio problem see Hamacher ad Drezer [6], Barmel ad Simchi- Levi [], ad Dasi [3]. Recetly some authors have icorporated ivetory cotrol decisios ito the facility locatio problem, Simchi-Levi [16] cosiders a hierarchical plaig model for stochastic distributio system, i which the locatios ad demad of customers are determied accordig to some probability distributio. Differet decisios are grouped ito three classes: strategic plaig, tactical plaig, ad operatioal cotrol. Jayarama [7] icorporates the ivetory costs ito a facility locatio problem, assumig fixed lot sizes ad determiistic demads. Nozic ad Turquist [1] icorporate ivetory costs assumig the demads to arrive i a Poisso maer ad a base stoc ivetory policy. Barahoal ad Jese [1] solve a locatio problem with a fixed cost for stocig a give product at a distributio ceter. She [1], studied the oit locatio-ivetory model i which locatio, shipmet ad o-liear safety stoc ivetory costs are icluded i the same model. Teo et.al [18] preset a approximatio algorithm for the problem of choosig distributio ceters to miimize locatio ad ivetory costs, igorig trasportatio cost. Vidal ad Geotschalcx [19] study a global supply chai desig model that maximizes 1 Correspodig Author., Tel : Address: dr_azad@yahoo.com Nader Azad) Published by World Academic Press, World Academic Uio

2 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp the after-tax profits of a multi-atioal corporatio. Their model simultaeously cosiderig trasferrig prices, trasportatio cost allocatio, ivetory costs ad their impact o the selectio of iteratioal trasportatio modes. Erlebacher ad Meller [5] formulate a o-liear iteger locatio-ivetory model. They use a cotiuous approximatio for solvig the problem. Dasi et.al [4] apply lagragia relaxatio to solve the locatio-ivetory model. She et.al [14] preset a locatio-ivetory model that is similar to the model of She [1] ad use colum geeratio for solvig the problem. Mirada ad Garrido [9] preset a locatio-ivetory model that is similar to the model of Dasi et.al [4], ad they apply lagragia relaxatio method for solvig the model. Shu et.al [15] study a more geeral locatio ivetory model ad use colum geeratio for solvig the problem. She [13] propose multi-commodity locatio-ivetory model ad use colum geeratio for solvig the problem. I this paper we preset a itegrated stochastic supply chai desig model which optimizes locatio ad ivetory decisios, simultaeously. Oe maor drawbac i most of past research is that they assume the capacity of distributio ceters are ow ad assigmet of customers to the distributio ceters are limited by the capacity of distributio ceters. I our model, we use differet capacity levels for distributio ceters that mae the problem more realistic ad assigmet of customers to the distributio ceters more flexible. We assume each customer has ucertai demad that follows a give ormal distributio. The goal of our model is to choose a set of distributio ceters to serve the customers ad allocate customers to the opeed distributio ceters, ad to determie the ivetory policy based o the iformatio of customers demad i order to miimize the total expected cost of locatig distributio ceters, shipmet, trasportatio ad ivetory costs, while esurig a pre-specified level of service. Oe of the possible obectives i supply chai desig models is to maximize the capacity utilizatio of distributio ceters. Our results show that use of capacity levels for distributio ceters icrease the capacity utilizatio to a high level. The remider of this paper is orgaized as follows. I sectio, mathematical formulatio of the problem is preseted. I sectio 3, the hybrid Tabu-SA algorithm is developed for solvig the problem. Sectio 4, discusses some computatioal results. Fially, sectio 5 cotais some coclusios ad future research developmet.. Model formulatio I our model we cosider the followig assumptios: We assume that each customer has ucertai demad that follows a ormal probability distributio, ad customers demads are idepedet. We assume to ow capacity levels for distributio ceters, ad the compay pays a fixed locatio cost for opeig a distributio ceter with a capacity level. We assume that the compay pays a fixed cost for each order placed at a distributio ceter ad a holdig cost for ivetory at each distributio ceter. The distributio ceters hold worig ivetory ad safety stoc ivetory iteded to buffer the system agaist stoc out durig orderig lead times. The distributio ceters are assumed to follow a Q, R) ivetory policy. Before presetig the model, let us itroduce the otatios that will be used throughout the paper: Idex sets K : Set of customers. J : Set of potetial distributio ceters. N : Set of capacity levels available to the potetial distributio ceters. Parameters ad otatios µ : Mea daily demad at customer, K ) σ : Variace of daily demad at customer, K ) JIC for subscriptio: publishig@wau.org.u

3 9 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model F : Fixed cost for opeig ad operatig distributio ceter with capacity level, J, N ) J b : Capacity with level for the potetial distributio ceter, J, N) h : Ivetory holdig cost per uit of product per year at distributio ceter, J ) p : Fixed cost per order placed to the supplier by distributio ceter, J ) L : Distributio ceter lead time i days, J ) g : Fixed cost per shipmet from the supplier to distributio ceter, J ) a : Cost per uit of a shipmet from the supplier to potetial distributio ceter, J ) c : Cost per uit of shipmet from distributio ceter to customer J, K ) α : Desired percetage of customer orders satisfied fill rate). z α : Stadard ormal deviate such that χ : Days per year. p z z ) α. α = Decisio variables U 1 = 1 = if customer is assiged to distributio ceter. K, J ) otherwise if distributio ceter is opeed with capacity level J, N) otherwise Obective fuctio The obective fuctio miimizes the followig costs: The fixed cost of locatig distributio ceters with capacity level, give by the term: F U J N The aual shipmet cost from distributio ceters to the customers, give by the term: χ c µ J K The expected aual ivetory cost: is the sum of worig ivetory ad safety stoc costs. Worig ivetory represetig product that has bee ordered from the supplier but ot yet requested by the customers ad safety stoc cost ivetory iteded to buffer the system agaist stoc outs durig orderig lead times. We assume that distributio ceters orders ivetory from the supplier usig a Q, R) policy with service level costraits. Worig ivetory cost icludes the fixed costs of placig orders as well as the shipmet costs from the supplier to the distributio ceters as well a the holdig costs of worig ivetory. Let D deote the total aual demad goig through distributio ceter D = µ ) ad w umber of shipmets per year from the supplier. The the total aual cost of orderig ivetory from the supplier to distributio ceter is give by: D D P w+ g + a w+ h w 1) w We assume that cost of shippig a order of size M form the supplier to distributio ceter is give by the term of: g + a M ). JIC for cotributio: editor@ic.org.u

4 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp The first term of 1) represets the total fixed cost of placig w orders per year. The secod term represets the delivery cost from the supplier to distributio ceter per year, ad the third term is the cost of average worig ivetory. By taig the derivative of this expressio with respect to w, we obtai the optimal value of w is give by: h D w= ) P + g ) By substitutig ) ito the total cost 1), we obtai the optimal value of total aual worig ivetory associated with distributio ceter is give by: h D P + g ) + a D = h P + g ) χ µ + a χ µ 3) σ The yearly safety stoc cost at distributio ceter is give by: θ h zα σ ad = L σ The holdig cost for the safety stoc at distributio ceter is: The expected aual ivetory cost is give by the term: h z α L σ h P + g ) χ µ + a χ µ + h zα L σ J The problem formulatig is as follows: MIN : Subect to: F U J N + J K χ c µ + h P + g ) µ + a χ µ + h z L J χ α σ 4) K U J N U = 1 K 5) 1, J 6) χ µ b U J 7) {,1} {,1} N J, K J, N 8) The model miimizes the total expected costs made of: the fixed cost for opeig distributio ceters, the aual shipmet cost from distributio ceters to the customers, ad the expected aual ivetory cost. Costraits 5) esure that each customer is assiged to exactly oe distributio ceter. Costraits 6) esure that each distributio ceter ca be assiged at most oe capacity level. Costraits 7) are the capacity costraits associated with the distributio ceters. Costraits 8) eforce the itegrality restrictios o the biary variables. 3. Solutio approach A hybrid heuristic combiig Simulated Aealig with Tabu Search sharig the same tabu list is used for solvig the problem. I sectio 3.1, 3. we describe the SA algorithm ad Tabu search algorithm, JIC for subscriptio: publishig@wau.org.u

5 94 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model respectively, ad i sectio 3.3, we describe the hybrid Tabu-SA algorithm which we use for solvig the problem Simulated aealig algorithm Simulated aealig SA) is oe of the ovel algorithms which was iitially preseted by Kirpatric et.al [8]. The SA methodology draws its aalogy from the aealig process of solids. I the aealig process, a solid is heated to a high temperature ad gradually cooled to a low temperature to be crystallized. As the heatig process allows the atoms to move radomly, if the coolig is doe too rapidly, it gives the atoms eough time to alig themselves i order to reach a miimum eergy state that amed stability or equipmet. This aalogy ca be used i combiatorial optimizatio i which the state of solid correspods to the feasible solutio, the eergy at each state correspods to the improvemet i the obective fuctio ad the miimum eergy state will be the optimal solutio. The Steps of SA algorithm are show i Fig.1. =φ Select a iitial solutio, =, = While T < ST ) Do S = While S < L ) Do Geerate solutio C = C ) - C ) i the eighborhood of, If C the = S = S +1 If ) C < ) = Ed If C the Else Geerate y U,1) Radomly Set If z = e C T y < z the = S = S +1 Ed If Ed If Ed While T = C T Ed While Fig.1. SA algorithm JIC for cotributio: editor@ic.org.u

6 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp The SA parameters are as follows: T : Iitial temperature, C : Rate of the curret temperature decreases coolig schedule), ST : Freezig temperature the temperature at which the desired eergy level is reached), L : Number of accepted solutio at each temperature, S : Couter for the umber of accepted solutio at each temperature, : A feasible solutio C ) : The value of obective fuctio for, SA uses a stochastic approach to direct the search. It allows the search to proceed to eighborig state eve if the move causes the value of the obective fuctio become worse. This importat feature, ca allow it to prevet fallig i the local optimum trap. SA guides the origial local search method i the followig way. The algorithm starts with a iitial solutio for the problem. I the ier cycle of the SA, repeated while S < L, a eighborig solutio of the curret solutio is geerated. If this move decreases the obective fuctio, or leaves it uchaged, the the move is always accepted. Moves, which icrease the T e c obective fuctio value, are accepted with a probability to allow the search to escape a local optimum. The value of the temperature decreases i each iteratio of the outer cycle of the algorithm. Obviously the probability of acceptig worst solutio decreases as the temperature decreases i each outer cycle. Two importat issues that eed to be defied whe adoptig this geeral algorithm to a specific problem are the procedures to geerate both iitial solutio ad eighborig solutios. 3.. Tabu Search algorithm The overall approach i Tabu search algorithm is to avoid etraimet i cycles by forbiddig or pealizig moves which tae the solutio, i the ext iteratio, to poits i the solutio space previously visited. Tabu search uses a local or eighborhood search procedure to iteratively move from a solutio to a solutio i the eighborhood of, util some stoppig criterio has bee satisfied. To explore regios of the search space that would be left uexplored by the local search procedure, tabu search modifies * the eighborhood structure of each solutio as the search progresses. The solutios admitted to N ), the ew eighborhood, are determied through the use of special memory structures. The search the progresses * by iteratively movig from a solutio to a solutio i N ). * Perhaps the most importat type of short-term memory to determie the solutios i N ) ; also, the oe that gives its ame to tabu search, is the use of a tabu list. Tabu list cotais the solutios that have bee visited i the recet past less tha m moves ago, where m is the tabu teure). Solutios i the tabu list are * excluded from N ). The steps of tabu search are show i Fig.. JIC for subscriptio: publishig@wau.org.u

7 96 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model =φ Select a iitial solutio, =, = r = While r < G ) Do Geerate solutio C = C ) - C ) i the eighborhood of, If the cadidate move is i the tabu list, the If ) C < ) C the =, r = r + 1 =, update the tabu list Else Geerate aother solutio i the eighborhood of Ed If Else If C = the, r = r + 1 Update the tabu list If ) C < ) = Ed If Else C the Geerate aother solutio Ed If Ed If Ed While i the eighborhood of Fig.. Tabu Search algorithm 3.3. The proposed hybrid algorithm A hybrid heuristic combiig Simulated Aealig with Tabu Search sharig the same tabu list is used for solvig the problem. The steps of proposed hybrid Tabu-SA based heuristic are as follows: Step1: = φ, select a iitial solutio, ) =, = Step: Geerate solutio i the eighborhood of. Step3: Is the cadidate move i the tabu list? If yes, go to Step 4. Otherwise go to Step 5. JIC for cotributio: editor@ic.org.u

8 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp Step4: If C ) C ) the =, =, update the tabu list ad go to Step 6, otherwise go to step for choosig aother cadidate move. Step5: Let C = C ) - C ) If C the =, r = r + 1, update the tabu list ad if C ) < C ) the =. C T 5.. If C > the y U,1), z = e. If y < z the =. Step6: Should the procedure stop uder temperature T? If yes, go to Step 7, otherwise go to Step. Whe the umber of accepted solutios uder temperature T reaches to a predefied value, the followig coditio should be checed: AOV c AOV AOV where AOV c is the average obective value of accepted solutios uder the temperature T, AOV b is the average obective value of accepted solutios before the temperature T, ε is a predefied equilibrium value < ε < 1 ). If the above coditio is satisfied, the procedure stops uder temperature T. This coditio was proposed by Siscim & Golde [17]. Step7: T = C T. Step8: Is the stoppig criterio T < ST ) matched? If yes, stop, otherwise, go to Step. The steps of proposed hybrid Tabu-SA Algorithm are show i Fig.3. =φ Select a iitial solutio, b b ε =, = While T < ST ) Do While AOV c AOV AOV b b ε ) Do Geerate solutio C = C ) - C ) i the eighborhood of, If the cadidate move is i the tabu list, the If ) C the C < ) = Else, Geerate aother solutio Ed If Else If C = If ) =, update the tabu list the, update the tabu list C the C < ) i the eighborhood of JIC for subscriptio: publishig@wau.org.u

9 98 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model = Ed If Else Geerate y U,1) Radomly Set If z = e C T y < z the = Ed If Ed If Ed If Ed While T = C, update the tabu list T Ed While Fig.3. Proposed hybrid Tabu-SA Algorithm I the sectio 3.3.1, 3.3., we describe the iitial solutio costructio ad differet types of move for geeratig the cadidate move which we use for hybrid Tabu-SA algorithm Iitial solutio costructio For obtaiig the iitial solutio, first we assig customers to the distributio ceters, radomly. For each of the opeed distributio ceter the capacity level is selected radomly. The procedure for obtaiig the iitial solutio is as follows. Step1: Put customers ito a set K. Step: 1- Select a customer from K radomly. - Delete the customer from K. Step3: Select a distributio ceter radomly. Step4: If we select this distributio ceter for the first time the select a capacity level for this distributio ceter radomly. Step5: If remaiig capacity of the distributio ceter is greater tha the demad of the customer the assig the customer to the distributio ceter ad go to Step 6 otherwise go to Step 3 for selectig aother distributio ceter. Step6: Is K empty? If yes, stop, otherwise go to Step Improvig the iitial solutio I this phase, the mai obective is to improve the iitial solutio. We apply four differet types of move for geeratig a cadidate move: mov1, mov, mov3, mov4. We geerate a cadidate move from to the cadidate solutio ) usig oe of the four moves radomly. Mov1: Radomly, oe of the opeed distributio ceters a ) is closed ad all of the customers are reallocated amog the remaiig opeed distributio ceters. If the remaiig capacities of the opeed distributio ceters are ot eough for the customers of a the we radomly select a ope distributio ceter ad icrease its capacity level to oe higher level. The procedure of mov1 is as follows: Step1: Select a ope distributio ceter radomly a ). Let to a. Step: Select a customer ) from D, radomly. D be the set of customers that assiged Step3: Determie the opeed distributio ceters that have eough remaiig capacity for demad of JIC for cotributio: editor@ic.org.u

10 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp customer. Let O be the opeed distributio ceters which have eough remaiig capacity. Step4: If O is empty, we ca ot fid the opeed distributio ceter that have eough remaiig capacity) the go to Step 5, otherwise go to step 8. Step 5: If all of the opeed distributio ceters have the highest capacity level. The stop ad exit from this move ad select a move radomly for geeratig a cadidate move, otherwise go to Step 6. Step6: Radomly select oe of the opeed distributio ceters. Step7: If this distributio ceter has the highest capacity level the go to Step 6 for selectig aother ope distributio ceter, otherwise icrease its capacity level to oe higher level ad assig customer to this distributio ceter ad go to Step 9. Step8: Select a distributio ceter from O radomly ad assig customer to this distributio ceter. Step9: Delete the customer from D. Step1: Is D empty? If yes, close a ad stop, otherwise go to Step. I Step 5 of the mov1 by the time mov1 is ot performed as may umber as max-mov1, we give up mov1 ad we do ot apply mov1 durig the algorithm, ote that max-mov1 is as a iput for the heuristic method. I fact, mov1 is termiated whe a max-mov1 umber of moves are ot performed based o Step 5. Mov: I this move we select two ope distributio ceters radomly, a i, a ad a. I this move capacities of a i ad a are checed for servig the customers. ), ad exchage a i Mov3: Oe of the opeed distributio ceters a i ) is closed radomly, ad a closed distributio ceter a ) is opeed radomly ad oe of the capacity levels is selected for a radomly. The we assig all of the customers correspodig to the elimiated distributio ceter a i ) to the ew opeed distributio ceter a ). I this move the capacity of a is checed for servig the customers. Mov4: Select two ope distributio ceters, radomly, v i, v ). The radomly select a customer c i ) i v i ad a customer c ) i v ad exchage c i ad c. I this move we must chec the capacities of distributio ceters. 4. Computatioal results The computatioal experimets described i this sectio were desiged to evaluate the performace of our overall solutio procedure with respect to a series of test problems. It was coded i visual basic 6 ad ru o a Petium 4 with.8 GB processor. The daily demad of the customers was draw from a uiform distributio betwee 1 ad 5. The variaces of daily demads of customers were draw from a uiform distributio betwee 1 ad 3. Also we use the followig parameter values. h Is uiformly draw from [, 4] p Is uiformly draw from [15, ] L Is uiformly draw from [6, 1] g Is uiformly draw from [15, ] a Is uiformly draw from [, 5] c Is uiformly draw from [, 5] χ = 5, z α = % service level) Four capacity levels are used for the capacities of the potetial distributio ceters. If we let D JIC for subscriptio: publishig@wau.org.u

11 3 represets total demad requiremets D = Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model K µ ) ad J be the umber of potetial distributio ceters ad c is a radom umber betwee.8 ad 1. for each distributio ceter. The we defie for each D distributio ceter: cap ) = c, [ A] is the iteger part of A), so the differet capacities of the J potetial distributio ceter are computed as follows b = cap ), b = 1.5 cap ), b = cap ), b =.5 cap ) Fixed set up costs of locatig ad operatig distributio ceters are as follows. Let for each distributio ceter were draw from a uiform distributio betwee 45 ad 55. The fixed set up cost for distributio ceter is computed as follows [.65 ], F = [.9 ], F = [ 1.1 ], F = [ 1. ] F = 35 Our goal i this sectio is to fid out 1) performace of the heuristic algorithm, ad ) what the beefit of cosiderig capacity levels is ad how impact to the capacity utilizatio ad how capacity utilizatio varies with the chagig umber of capacity levels Performace of the hybrid algorithm Compariso of optimal solutio ad hybrid algorithm Table 1Compariso of optimal solutio ad hybrid algorithm Optimal Solutio Hybrid Algorithm NO. # Customers # DCs Cost CPU time Cost CPU time Gap%) hours limit hours limit hours limit Gap: Gap from optimal solutio %). For evaluatig the proposed heuristic, sevetee problems are solved by LINGO.8 software table.1.). The two o-liear terms of the obective fuctio 4) the aual worig ivetory cost ad the holdig cost for safety stoc) are the cocave terms, ad the other terms of the obective fuctio are liear, so the obective fuctio 4) is absolutely cocave ad the problem is to miimize the cocave iteger programmig model. So the solutio which is obtaied by brach & boud methods such as LINGO software) is optimal ad we ca compare our heuristic solutio by optimal solutio which is obtaied by LINGO.8. Parameter tuig is a matter of serious cocer for ay optimizatio problem as it iduces good performaces. The parameters affect the worig of the hybrid algorithm, drastically. For each problem, the tuig of the parameters is doe by carryig out radom experimets. It ca be see that the solutios of the proposed hybrid algorithm are optimal or ear optimal) i differet problems table.1.). The average CPU time are less tha or equal to 137 secods for proposed hybrid algorithm CPU times are i the secods). JIC for cotributio: editor@ic.org.u

12 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp However, the maximal average CPU time for obtaiig the optimal solutios is equal to 8493 secods, ad for istace 15 to 17 by a 3 hours time limit, LINGO ca ot fid the optimal solutio, ad the hybrid algorithm i this istace is better tha the solutios that are obtaied by LINGO Compariso of hybrid algorithm with SA algorithm ad Tabu search algorithm I this sectio, we compare our hybrid algorithm with SA method ad Tabu search method. The procedure for obtaiig iitial solutio ad cadidate move, we use i SA method ad tabu search method, are the same to the procedure of obtaiig solutio ad cadidate move i hybrid algorithm. I SA algorithm ad Tabu search algorithm, for each problem, the tuig of the parameters is doe by carryig out radom experimets The compariso of hybrid algorith with SA algorithm ad Tabu search are show i table. It ca be see that the solutio quality i hybrid algorithm is better tha the solutio quality i SA algorithm ad Tabu Search algorithm. Table compariso of hybrid algorithm with SA Algorithm ad Tabu Search Algorithm Hybrid Algorithm SA Algorithm Tabu Algorithm NO. # Customers # DCs Cost CPU Time Cost CPU Time Cost CPU Time Impact of cosiderig capacity levels I this sectio, we study the beefit of cosiderig capacity levels o the capacity utilizatio, ad we show that how capacity utilizatio varies with the chagig umber of capacity levels. For comparig the capacity utilizatio by varyig the umber of capacity levels, we cosider three differet umber of capacity levels. = 4, 5, 6. The capacities ad fixed costs of four capacity levels were described i the sectio 4. For five capacity levels we use the capacities ad fixed costs as follows b = cap ), b = 1.5 cap ), b = 1.5 cap ), b = cap ), b =.5 cap ) F = [.65 ], F = [.78 ], F = [.9 ], F = [ 1.1 ], F = [ ] For six capacity levels we use: b = cap ), b = 1.5 cap ), b = 1.5 cap ), b = 1.75 cap ), b = cap ), 6 b =.5 cap ) F [.65 ], F =.78, F =.9, F = 1, F = = [ ] [ ] [ ] [ ], F = [ ] Table.3. shows the computatioal results. By varyig the umber of capacity levels, we ca see clearly the impact of umber of capacity levels o the average capacity utilizatio of the opeed distributio ceters. JIC for subscriptio: publishig@wau.org.u

13 3 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model Table 3 Impact of cosiderig capacity level A.C.U NO. # Customers # DCs =4 =5 = A.C.U: Average capacity utilizatio for the opeed distributio ceters %) We observe that average capacity utilizatios for the opeed distributio ceters for the differet umber of capacity levels are very high, also we observe that, whe we icrease umber of capacity levels, the average capacity utilizatios for the opeed distributio ceters also icrease. 5. Coclusios I this paper we have outlied a itegrated stochastic supply chai desig model which optimizes locatio, ivetory ad trasportatio decisios. We assumed each customer has ucertai demad that follows a ormal distributio. The goal of our model was to choose a set of distributio ceters to serve the customers ad allocate customers to the opeed distributio ceters, ad to determie the ivetory policy based o the iformatio of customers demad i order to miimize the total expected cost, also, we used differet capacity levels for distributio ceters that mae the problem more realistic ad assigmet of customers to the distributio ceters more flexible. A hybrid heuristic combiig Tabu search with Simulated Aealig sharig the same tabu list was developed for solvig the problem. We comprised the hybrid algorithm with the optimal solutio, SA algorithm ad Tabu search method. The results of extesive computatioal tests idicated that the hybrid algorithm is both effective ad efficiet for a wide variety of problem sizes. Also, we showed that cosideratio of cosiderig capacity levels helps to achieve the capacity utilizatio to a high level. For future wors it is iterestig to cosider the multi-period ad multi product model. 6. Refereces [1] Barahoa, F., Jese, D.. Plat Locatio with Miimum Ivetory. Mathematical Programmig. 1998, 83: [] Bramel, J., Simchi-Levi, D.. The Logic of Logistics. Spriger-Verlag, New or,. [3] Dasi, M.S.. Networ ad Discrete Locatio: Models Algorithms ad Applicatios. Wiley-Itersciece, New or, [4] Dasi, M., Coullard, C., She, Z.J.. A ivetory-locatio model: Formulatio, solutio algorithm ad computatioal results. Aals of Operatios Research., 11: [5] Erlebacher, S.J., Meller, R.D.. The iteractio of locatio ad ivetory i desigig distributio systems. IIE Trasactio., 3: [6] Hamacher, H., Drezer, Z.. Facility Locatio: Applicatios ad Theory. Spriger-Verlag, Berli,. [7] Jayarama, V.. Trasportatio, facility locatio ad ivetory issues i distributio etwor desig. Iteratioal Joural of Physical Distributio ad Logistics Maagemet. 1998, 18 5), [8] Kirpatric, S., Gelatt, C. & Vecchi, M.. Optimizatio by simulated aealig. Sciece. 1983, : [9] Mirada, P.A., Garrido, R.A.. Icorporatig ivetory cotrol decisios ito a strategic distributio etwor JIC for cotributio: editor@ic.org.u

14 Joural of Iformatio ad Computig Sciece, 3 8) 4, pp desig model with stochastic demad. Trasportatio Research, Part E. 4, 4 3): [1] Nozic, L.K., Turquist, M.. Itegratig ivetory impacts ito a fixed-charge model for locatig distributio ceters. Trasportatio Research, Part E, Logistics ad Trasportatios Review. 1998, 34 3), [11] Oze, L.. Locatio-ivetory plaig models: Capacity issues ad solutio algorithms. Dissertatio for the Degree of Doctor of Philosophy, Northwester Uiversity, 4. [1] She, Z.J.. Efficiet algorithms for various supply chai problems. Ph.D. Dissertatio, Northwester Uiversity, Evasto, IL,. [13] She. Z.J.. A multi-commodity supply chai desig problem. IIE Trasactios. 5, 37: [14] She, Z.J., Coullard, C., Dasi, M.. A oit locatio-ivetory model. Trasportatio Sciece. 3, 37: [15] Shu, J., Teo, C.P., She, Z.J.. Stochastic trasportatio-ivetory etwor desig problem. Operatios Research. 5, 53: [16] Simchi-Levi, D.. Hierarchical desig for probabilistic distributio systems i euclidea spaces. Maagemet Sciece. 199, 38: [17] Siscim, C.C., Golde, B.L.. Optimizatio by simulated aealig: a prelimiary computatioal study for the TSP. Witer Simulatio Coferece, [18] Teo, C.-P., Ou, J., Goh, M.. Impact o Ivetory Costs with Cosolidatio of Distributio Ceters. IIE Trasactios. 1, 33: [19] Vidal, C.J., Goetschalcx, M.. A global supply chai model with trasfer pricig ad trasportatio cost allocatio. Europea Joural of Operatioal Research. 1, 19: JIC for subscriptio: publishig@wau.org.u

15 34 Nader Azad, et al: A Hybrid Tabu-SA Algorithm for Locatio-Ivetory Model JIC for cotributio: editor@ic.org.u