Optimal Model and Algorithm for Multi-Commodity Logistics Network Design Considering Stochastic Demand and Inventory Control

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1 Systems Engineering Theory & Practice Voume 29, Issue 4, Apri 2009 Onine Engish edition of the Chinese anguage journa Cite this artice as: SETP, 2009, 29(4): Optima Mode and Agorithm for Muti-Commodity Logistics Network Design Considering Stochastic Demand and Inventory Contro QIN Jin 12,, SHI Feng 1, MIAO Li-xin 2, TAN Gui-jun 1 1. Schoo of Traffic and Transportation Engineering, Centra South University, Changsha , China 2. Research Center for Logistics, Graduate Schoo at Shenzhen, Tsinghua University, Shenzhen , China Abstract: A simutaneous approach that incorporates inventory contro decision into faciity ocation mode is proposed, which is used to sove the muti-commodity ogistics network design probem. Based on the assumption that the stochastic demands of the retaiers are norma distributed, a non-inear mixed integer programming mode that simutaneousy described the inventory decision and the faciity ocation decision is presented, in which the objective is to minimize the tota cost that incuding ocation costs, inventory costs, and transportation costs under the certain service eve. The combined simuated anneaing (CSA) agorithm is deveoped to sove the probem. The mode and effectiveness of the agorithm are carified by the computationa experiments. Key words: muti-commodity; ogistics network design; stochastic demand; optimization mode; simuated anneaing agorithm 1 Introduction In a high competitive environment, the manufacturing companies must pay cose attention to their inventory management. To optimize their inventory system, the companies shoud sove two critica probems. First, they must seect the proper paces that the commodities saving, namey, the sites and the number of stocking ocations or ogistics nodes (LNs). Second, they must determine the amount of commodities to maintain in each LN. So in the ogistics network design probem, the faciity ocation probem and inventory decision probem are two key subprobems and both of them are highy reated. But in many iteratures, the above two probems aways were studied as the faciity ocation probem [1 3] and the inventory contro probem [4 5] separatey. The decision-making resuts in incompatibiity and inconformity at different eves, which coud affect the rationaity of the fina strategy decisions. In addition, the demands of the retaiers for the commodities are aways uncertain in the rea word, but in the research on the ogistics network design probem, the demands were aways considered as a deterministic variabes in order to simpify the anaysis and modeing. Furthermore, the companies shoud maintain a certain stock to satisfy the stochastic demands as far as possibe. They are required to contro their inventory cost because the inventory cost is increasing foowing the inventory amount, so the companies must seect the scientific inventory poicies. Based on the assumption that the stochastic demand of the each retaier is norma distributed, the probem that integrated the faciity ocation probem and inventory contro probem is studied in this artice, which coud increase the rationaity and scientificity of the decisions. For the singe commodity ogistics network design probem considering the inventory cost, the iteratures [6-8] ignored many factors which have infuence on the inventory cost, and ony added the cost as the non-inear function of the commodity quantity to the objective function; the iterature [9] studied the joint ocation-inventory probem under two specia cases: the variance of demand was proportiona to the mean and the demand had zero variance, and restructured the mode into a set-covering integer programming mode; the iterature [10] deveoped a more efficiency agorithm for the specia cases in the iterature [9]; the iterature [11] anayzed the transportation cost considering the vehice routing in the ogistics network, but the order number was considered as a continuous variabe in the formuation derivation; the iterature [12] investigated the trade-offs probem between the service eve and service cost making use of the existed mode in the iteratures [9-10], and proposed a weighting method and a heuristic soution approach based on genetic agorithms to sove the probem. The iteratures that studied the ogistics network design probem with muti-commodity are few for the present at home and abroad. The iterature [13] simpified the inventory cost of the commodities as in the iteratures [6-8] and proposed the Lagrange agorithm to sove the probem; the iteratures [14 15] regarded the inventory cost as the inear function of commodity quantity; the iterature [16] deveoped the mode framework of muti-commodity dynamic capacitated faciity ocation and reported on their computa- Received date: November 28, 2007 Corresponding author: Te: ; E-mai: qinjin@mai.csu.edu.cn Foundation item: Supported by the China Postdoctora Science Foundation funded project (No ); China Nationa Nature Science Foundation funded project (No ) Copyright c 2009, Systems Engineering Society of China. Pubished by Esevier BV. A rights reserved.

2 QIN Jin, et a./systems Engineering Theory & Practice, 2009, 29(4): tiona experience with standard mathematica programming software, but the inventory cost in the mode was a inear function of demand quantity too. 2 Assumptions and notations To formuate the mode, the foowing notations are used: i denotes the candidate LN site, i = 1, 2,, N ; j denotes the retaier, j = 1, 2,, M; denotes the commodity, = 1, 2,, L. F i is the fixed cost of ocating at candidate LN site i; V i denotes the storage capacity of candidate LN i; RPi is the reorder point of the inventory poicy in LN i for commodity, and when the inventory amount of commodity at LN i decreases to RPi, then an order is triggered; Q i is the order quantity per order; λ is the occupation of storage capacity per unit commodity ; d j and u j are the mean and the standard deviation of the demand at retaier j for commodity ; Dj and U j are the mean and the standard deviation of the demand at candidate LN i for commodity ; LTi is the ead time (in days) from suppier to LN i for commodity, namey, the suppier takes the required ead time to fufi an incoming order from LN i for commodity ; Hi is the inventory hoding cost per unit commodity at LN i; Oi is the fixed cost of paces an order for commodity at LN i; Ri is the cost to ship per unit commodity from the suppier to candidate LN i; Ti is the eapsed time between two consecutive orders for commodity at LN i; C is the distribution cost of per unit commodity between the LN i and the retaier j; θ 1 and θ 2 are the weighted factors associated with transportation cost and inventory cost, respectivey; α is the united service eve in the system, 0 < α < 1, that is to say, the fi rate of a demands for a commodities must not ess than α in a LNs; K is the panning horizon; γ is bank interest rate and β is discount rate (cacuated by γ ); P is the maximum number for the LNs that aowed to ocate. And set the binary variabes as: { 1, if a LN setup on site i X i = 0, otherwise Y = 1, if LN i services retaier j for commodity 0, otherwise Note that when X i = 0, there is no commodities shoud pass LN i and vice versa. Some rationa assumptions are proposed as foowing: (1) Each factory in the network produces ony one commodity; (2) The demand of each retaier for each commodity is uncertain and satisfies a norma distribution. Moreover, a the demands are independent. (3) Each demand is serviced by ony one LN, namey, the demand coud not be partitioned. (4) LN i performs an inventory contro poicy (RPi, Q i ), that is to say, a fixed quantity RP i is ordered to the suppier, once the inventory quantity fas to or beow a reorder point Q i, and the factory coud satisfy the demand of retaier i for commodity after ead time LTi. (5) A LNs in the network shoud have the same service eve, namey, the fi rates for the demands in a LNs are identica. Based on the above assumptions, we get: D i = U i = d jy, i = 1, 2,, N. (1) u jy, i = 1, 2,, N. (2) The fixed ocation costs of LNs are disposabe and the other costs are invested per day, so in order to keep the consistency of a cost in unit time, the ocation cost shoud be shared in day in the panning horizon. The absorption rate coud be computed as foowing: β = K h=1 3 System cost anaysis γ (1 + γ) h 1. (3) According to the above assumptions, the LN i performs an inventory poicy (RPi, Q i ) to meet the stochastic demand pattern. But even the order is triggered, the commodities shoud receive after LTi days. So once an order is submitted, the inventory commodities shoud cover the demand produced in ead time LTi with a certain probabiity α. The probabiity α is known as the given service eve of the system. So the eve-of-service constrains at LN i for commodity can be expressed as foows: P (D(LT i ) RP i ) = α; i,. (4) where D(LTi ) is the random demand quantity during the ead time at LN i for commodity. Based on the inventory poicy (RPi, Q i ) and the assumption of normay distributed demand, RPi can be determined as foows: RP i = D ilt i + Z α LT i U i. (5) where Z α is the vaue of the standard norma distribution, which denotes the uniform service eve and is identica in the network. For simpicity, we et Z = Z α in foowing anaysis. So the average hoding cost rate in each LN i for commodity, based on the expression (5), coud be written as: HiZ LTi Ui + H iq i/2. (6) The first term Hi Z LTi U i in (6), is the average expenditure associated with safety stock kept at LN i. The second term Hi Q i /2 is the average expenditure incurred due to the hoding the order quantity Q i, which is the inventory used to cover the demand arisen between two successive orders. Thus, the operation cost during this period at LN i for commodity is given by: RiQ i + Oi + (HiZ LTi Ui + H iq i/2)ti. (7) Then we divide the expression (7) by Ti (Ti = Q i /D i ), the operation cost rate incurred at LN i for commodity is given by the foowing expression: ( ) Ri + O i Q Di + HiZ LTi Ui + H iq i/2. (8) i

3 QIN Jin, et a./systems Engineering Theory & Practice, 2009, 29(4): Then the tota operation cost rate for the entire ogistics network can be written as: =1 [ HiZ LTi U i +H iq i/2+(r i + O i Q i )D i ]. (9) The transportation cost rate of the ogistics network is: =1 C d jy = =1 CD i. (10) Because the fixed ocation cost is one-off, but the other costs are counted in days, so in order to be consistent with other costs in unit time, the fixed ocation cost shoud be converted into day-cost, and it can be written as: β β F i X i. (11) So the tota cost of the ogistics system is: F i X i + θ 2 +θ 1 N =1 N =1 ( ) C + Ri + O i Q Di i (HiZ LTi Ui + H iq i/2). (12) The above tota cost function is the objective function of the optimization mode. But there are too many variabes in this objective function. It shoud be restructured and simpified considering the optima order quantity Q i. We consider that there is no capacity restriction on the order quantity. Thus, differentiating the tota cost function in terms of Q i (i = 1, 2,, N; = 1, 2,, L), and set the derivation to zero (minimizing the system cost in a centraized approach), then we coud obtain: O i Hi θ 1 2 θ 2 (Q D i )2 i = 0. (13) From the expression (13), we coud get the optima order quantity: ˆQ i = 2θ 2 Oi D i θ 1 Hi, i = 1, 2,, N. (14) Note that the expression (14) corresponds to the same outcome of the cassica economic order quantity (EOQ) mode. But the expression (14) differs from static-sequentia EOQ mode, because the former depends on the configuration of network, given by the variabes X i and Y. Thus, if we modify the network, then the optima order quantity ˆQ i wi change, simiar to the best response function. We coud repace equations (1), (2), and (14) into the expression (12), then the tota system cost can be expressed as foowing: β F i X i + θ 2 N =1 (C + R i)d jy +θ 1 N + =1 =1 HiZ 4 Mode formuation LT i M u j Y 2θ 1 θ 2 Hi O M i d j Y. (15) The optimization mode which coud describe the muti-commodity ogistics network design probem that incorporates inventory contro decision into faciity ocation decision is formuated as foowing: min Ψ = β s.t. F i X i + θ 2 +θ 1 N + =1 =1 N HiZ =1 LT i (C + R i)d jy M u j Y, 2θ 1 θ 2 Hi O M i d j Y, (16) =1 Y = 1, j,, (17) λ d jy V i X i, i, (18) Y X i, i, j,, (19) X i P, (20) X i, Y {0, 1}, i, j,, (21) V i 0, i. (22) The mode determines that where to open LNs ess than P sites in N candidate sites and served the stochastic demands of M retaiers with certain service eve. Each retaier shoud be serviced by ony one LN. The mode must ocate LNs and assign retaiers to the opened LNs considering capacity constraints. Constraint (17) stipuates that each retaier must be assigned to exacty one LN. Constraint (18) ensures that the tota demands of retaiers who are assigned to the same LN does not exceed its capacity. Constraint (19) states that the assignments can ony be assigned to open LNs. Constraints (20) is the number restriction in opening the LNs. Constraints (21) and (22) are standard integraity constraints. The configuration of the above mode is simiar to that of the mode in the iterature [9], but they are different in

4 QIN Jin, et a./systems Engineering Theory & Practice, 2009, 29(4): essence. First, the mode in this artice investigates the muticommodity ogistics network but not singe commodity ogistics network. Second, in the mode derivation, the iterature [9] regarded the order number as a continuous variabe and took the derivation of the objective function with respect to the number. But in this artice, the order quantity is considered as the continuous variabe. And the atter accords much more with the EOQ theory. In addition, the fixed costs are aocation to per year in panning horizon by absorption rate in the objective function in this artice, which coud keep the cost uniformity in time unit, and which is ignored in a existed iteratures. 5 Combined simuated anneaing agorithm The above optimization mode for muti-commodity ogistics network design probem is a typica arge-scae noninear mixed integer programming mode. The Lagrangian agorithm and some meta-heuristics agorithms are aways used to sove these probems. But the Lagrangian agorithm is intuitive and compicated and infexibe, so in order to increase the adaptabiity of the soving approach, the simuated anneaing agorithm is seected to sove the mode. Qin [18 19] deveoped the combined simuated anneaing (CSA) agorithm to sove the ogistics network design probem. The agorithm is divided into two ayers: the outer ayer agorithm (OLA) and the inner ayer agorithm (ILA). The OLA optimizes the faciity ocation decision, and the ILA optimizes the demand aocation based on the resuts of OLA. The step-by-step procedures of the OLA are given as beow: Step 1 Design the anneaing schedue tabe. Set the initia temperature t, cooing rate ξ = 0.9, warming rate τ = 1, the set of tabu ists is Ω, the initia soution S is obtained by randomy assignment, and et remembered optima soution S = S, Ω Ω S, λ(s) = n, K = 1; Step 2 Perform the outer ayer neighboring function on S, and get the new soution S, if S Ω, then repeat Step 2; ese, et Ω Ω S, λ(s ) = n, and update the tabu ength of the objects in the tabu ist, namey, for S Ω, perform λ(s) λ(s) 1 and if λ(s) = 0, then Ω Ω/S, go to Step 3 ; Step 3 Perform ILA based on soution S ; Step 4 If C(S ) < C( S), then set S = S ; Step 5 If C(S ) < C(S), set S = S ; or ese generate a random number ρ from (0,1), if ρ < exp[ (C(S ) C(S))/(τ t)], set S = S or ese the new soution is not accepted;; Step 6 K = K + 1. If the iteration number K does not meet the criterion for samping in the same temperature, then return to Step 2; otherwise, perform the cooing temperature operation, t t ξ, and set K = 0, go to Step 7; Step 7 If the increasing temperature conditions are not met, then return to Step 2. Otherwise, perform the increasing temperature action, set τ τ + C/K; Step 8 Convergence check. If the stop criterion is not met, then go to Step 2; otherwise, stop the computation and output the optima soution S. The detaied steps of the ILA are given as foowing: Tabe 1. OFVs of initia soutions and the corresponding optima soutions No. OFV of OFV of Gap initia soution optima soution (%) Step 1 Regard soution S as the initia soution and the current optima soution, then generate the inner ayer neighboring soution S from S ; Step 2 If C(S ) < C(S ), then set S = S ; otherwise, generate a random number ρ from (0, 1), if ρ < exp[ (C(S ) C(S ))/t], then set S = S, or ese the new soution is not accepted; Step 3 If it does not meet the stop criterion, then go to Step 1; otherwise, stop the computation and return to the OLA. Note that in the above agorithms, C(S) denotes the objective function vaue (tota system cost) of soution S. And the standard SA is improved by taking the advantage of tabu principe in tabu search agorithm. The increasing temperature action is performed in the agorithm too, which coud avoid the agorithm trapped at oca minima in the ate and improve the efficiency of searching procedures. 6 Numerica exampe There are 20 candidate LNs, 50 retaiers, and 5 commodities in the numerica exampe. The panning horizon K = 10 year, interest rate γ = 0.05, the weighted factor of inventory cost is θ 1 = 2 and the weighted factor of transportation cost is θ 2 = 1, the service eve for a LNs is α = 0.99, the max number of LNs aowed to aocate is P = 10. Other data are obtained by stochastic generation, and the detaied information coud be met in the iterature [18]. In the stochastic search agorithm, the quaity of the obtained optima soution may be dependent on the initia soution. To test the mode and the agorithm, we sove ten optima soutions based on ten different initia soutions. Tabe 1 shows the resuts of the ten computations (OFV denotes the objective function vaue). From Tabe 1, we coud find that even the initia soutions are different, the objective function vaues of the optima soutions obtained by the CSA agorithm are very cosed, and the gap between the maxima vaue ( ) and the minimum vaue ( ) is ony 0.09%. Compared with the initia soutions, the objective vaues of optima soutions have 16.82%-31.72% cost

5 QIN Jin, et a./systems Engineering Theory & Practice, 2009, 29(4): Totacost Change rate of tota cost Iteration number of OLA Figure 1. Tota cost vs. iteration number Figure 2. Tota cost vs. service eve α saving. So it can prove the proposed optimization mode and agorithm is rationa and avaiabe. Figure 1 iustrates the objective function vaue (tota cost) Ψ varying with the iteration number of OLA in the ten computing processes. We coud find that in a processes, the CSA agorithm coud search the optima soution in short time and the objective function vaues of the different optima soutions are very cose. Figure 2 shows the variation trend of the tota cost Ψ when the service eve varied from 50%-99%. We coud find out that the optima objective function vaue is increasing when the service eve raises. And the greater the service eve is, the faster the objective function vaue - the tota system cost increases. Apparenty, the case accords with the fact perfecty. Change rate of tota cost Change rate of unit cost Transportation cost Inventorycost Demanddeviation Figure 3. Tota cost vs. unit cost and demand deviation Figure 3 iustrates that when the unit transportation cost, unit hoding cost, and demand deviation changing, the reative change rate of the tota cost Ψ. In the figure, we coud find that the variation of the unit transportation cost has most impact on the tota cost. And when the unit transportation cost varied from 50%-50%, the change rate of the tota cost reaches neary 90%, which exceed the impaction of unit hoding cost and demand deviation by far. Based on the above anaysis, we coud concude that the factors, incuding service eve, demand deviation, hoding cost, and transportation cost, coud affect the tota cost of the ogistics network, and that these factors are directy proportions to the tota cost, namey, the tota cost is increased as the vaues of these factors increases. In addition, the service eve and transportation cost have more infuence on the tota cost. So from the companies view point, if they want to save the tota cost of the ogistics network system, they shoud pay more attention to seect the most economic transportation mode and the appropriate service eve. 7 Concusions In muti-commodity ogistics environment, the two decision probems, which are faciity inventory probem and ocation probem, are integrated as a unity probem to be studied considering the demands of the retaiers a are uncertain and satisfy norma distribution. A decision optimization mode that incorporates the inventory contro decision into faciity ocation decision is deveoped, in which the objective is to minimize the tota cost, incuding fixed LN ocation cost, inventory cost, order cost, and transportation cost, under the precondition of satisfying the certain service eve, namey, the given fi rate for the stochastic demands. The mode coud describe the ogistics network design probem that the commodities which have high hoding costs and the demands are uncertain more reasonaby. The CSA agorithm is proposed to sove the arge scae exampe generated randomy. Computationa resuts show the effectiveness of the mode and the agorithm. Furthermore, the infuence of the factors, incuding the service eve, demand deviation, unit hoding cost, and unit transportation cost, on the tota cost of the ogistics network system is investigated. References [1] Kose A, Drex A. Faciity ocation modes for distribution system design. European Journa of Operationa Research, 2005, 162(1): [2] Xu D C, Du D L. The k-eve faciity ocation game. Operations Research Letters, 2006, 34(4): [3] Arostegui M A, Kadipasaogu S N, Khumawaa B M. An empirica comparison of tabu search, simuated anneaing, and genetic agorithms for faciities ocation probems. Internationa Journa of Production Economics, 2006, 103(2): [4] Ben-Daya M, Noman S M, Hariga M. AIntegrated inventory contro and inspection poicies with deterministic demand. Computers and Operations Research, 2006, 33(6): [5] Arda Y, Hennet J C. Inventory contro in a muti-suppier system. Internationa Journa of Production Economics, 2006, 104(2):

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