A Multi-Product Capacitated Inventory-Location Model with Risk Pooling
¹Noura A. Al Dhaheri, ²Ali Diabat
¹Department of Engineering Systems and Management, Masdar Institute, UAE 2 MASDAR Institute of Science and Technology, United Arab Emirates naldhaheri@masdar.ac.ae; adiabat@masdar.ac.ae
Abstract - In this paper, a novel formulation for the capacitated warehouse inventory-location model with risk pooling for multiple products is proposed. A single plant ships different types of products to retailers via a network of warehouses. The locations and inventory policies of the warehouses are chosen so as to minimize the sum of fixed facility location, transportation, and inventory carrying costs. The warehouses retain safety stock so as to maintain appropriate service levels in the face of uncertain demand at the retailers for multiple products. Keywords - integer programming, location-inventory, multiple products, supply chain optimization

I. INTRODUCTION Supply Chain Management spans all movement and storage of raw materials, work-in-process inventory, and finished goods from point of origin to point of consumption [1]. It involves decisions on facility location, technology selection, inventory management, and distribution. These decisions can be categorized into three different levels: strategic, tactical, and operational. Particularly in today’s competitive business environment, the importance of integrating these decisions so as to minimize costs and maximize customer satisfaction cannot be underestimated. Much of the research literature treats the different decision levels separately; few papers deal with optimizing jointly over both the tactical and operational levels, and even fewer involve multiple products. In this paper, we study a multi-product capacitated inventory-location model with risk Pooling (MPILMRP), which considers the impact of tactical and operational level decisions (e.g., inventory management and distribution) on strategic decisions (e.g., facility location). Our MPILMRP model considers a three-tiered supply chain network consisting of suppliers; distribution centers (DCs) and customers. This model integrates the traditional location problem with the inventory problem for multiple products, taking into consideration capacity constraints and the risk pooling effect. The remainder of the paper is organized as follows: Section 2 reviews the literature on inventory models, location theory and multi-commodity location problems; Section 3 introduces our multi-product capacitated inventory-location model with risk pooling (MPILMRP); Section 4 provides numerical results and a sensitivity analysis; and Section 5 concludes with directions for future research. II. LITERATURE REVIEW

Traditionally, the supply chain management literature has considered strategic, tactical and operational level decisions independently. Strategic level decisions are treated by the location theory literature, whereas operational supply chain decisions, which affect product movement and distribution, are treated by the inventory theory literature. Consequently, our work is very much related to location theory, inventory theory, integrated inventory-location models, and multiple-product supply chain design. Next we provide a short review of the literature in each of these areas. Location theory centers around optimizing the number of the DCs needed to satisfy the demand, identifying their locations and assigning retailers to them while minimizing the fixed facility location and transportation costs. Most location models that have appeared in the literature have disregarded inventory-related costs. Daskin [2] and Drezner [3] provide an overview of the location theory literature. Our model is linked to the capacitated facility location problem (CFLP) in that both models impose a capacity constraint. The classical CFLP problem, as studied by, for example, Sridharan [4], Tragantalerngsak et al. [5], Cortinhal and Captivo [6], Klose and Drexl [7], Klose and Gortz [8], and Liu et al. [9] does not take account of inventory costs and is modeled as an integer linear programming problem. However, in this paper, we take account of inventory costs and non-linearity appears in both the objective function and the capacity constraint. Inventory theory has a rich literature. One of the earliest inventory theory papers to quantify the benefits of risk pooling, assuming uniform or negligible transportation costs, is Eppen’s 1979 paper [10]. Eppen and Shrage later extended this work to nonzero lead times and nonperishable products [11]. Other extensions followed: Federgruen [12] provided approximations for a more general version of the Eppen and Schrage model [11] More recently, much attention has been focused on the single-product location-inventory problem. Baumol and Wolfe [13] were among the first to consider incorporating inventory costs into location models. Their model extends the uncapacitated facility location problem (UFLP) by adding a square root term to the objective function, resulting in an NP-Hard problem [14]. An extensive review of single-commodity location-inventory models is provided in Shen et al paper[15] and Kaijun et al paper[9]. The majority of these models either ignore the nonlinearity of the inventory costs, or approximate these costs, except for those of Erlebacher and Meller et al [16] and Daskin et al. [17] We now turn to the multi-commodity location problem. The pioneering work of Warszawski [18] and

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Warszawski and Peer [19] was later extended in a number of ways [14] [20] [21]. All these models consider fixed location costs and linear transportation costs, and assume that each warehouse stocks at most one commodity. Geoffrion and Graves [22] considered the capacitated version of the multi-commodity location problem which imposes capacity constraints on the plants and the DCs. They also assumed that each customer must be served by a single DC or directly from a plant. Hind and Basta [23] relaxed these assumptions and solved the resulting model using a branch-and-bound method. Lee [24] and Mazzola and Neebe [25] studied a multi-product capacitated facility-location problem with choice of facility type. All the above works assume deterministic demand. We are not aware of any previous research that explores the capacitated version of LMRP problem with stochastic demand for multiple products. However, for single products, Shen [26], Daskin et al. [17] and Shen et al. [15] studied the impact of inventory costs on location decisions in a stochastic demand environment. Shen [27] considered a multi-commodity version of the LMRP without a capacity constraint. Liu et al. [9]developed a capacitated version of the LMRP for a multi-channel supply chain and suggested a Lagrangian relaxation-based algorithm to solve it. The work of Ozsen et al. [28] is more specifically related to our work, since they study a capacitated LMRP that incorporates tactical inventory management decisions into strategic facility location decisions. Our model can be considered as the multi-product version of their capacitated model; it thus inherits the property of their model that a warehouse can order more frequently so as to lower the inventory levels at the warehouse, thus loosening the capacity constraints. In this article, we extend the capacitated warehouse location model with risk pooling of [28] to accommodate multiple products; this is one of the first supply chain design models for capacitated warehouses that considers multiple products (or product categories), economies of scale, and risk pooling. This model considers a cost term at each DC that exhibits initial economies of scale and then, after a given increase in demand, exhibits an exponential increase which helps in evaluating the tradeoff between risk-pooling benefits and ordering costs incurred at a given DC as more retailers are assigned to it. This model also captures the interdependence between inventory levels and capacity limitations at the DCs. We describe the model and its formulation in the next section. III. MODEL FORMULATION In this section we introduce our multi-product capacitated inventory location model with risk pooling (MPILMRP). This model integrates the traditional location problem with the inventory problem for multiple products, taking into consideration capacity constraints and the risk pooling effect. Here a “product” or a commodity can either represent a specific product or a

product category. Also, we will use the terms “customers” and “retailers” interchangeably. The MPILMRP models the storage and the transportation of different products from a supplier to a set of distribution centers which supply the demands of a set of customers. We require that the location of the supplier and the customers are known and that the supplier has the capacity to meet an arbitrarily large demand for each product. We assume that the different products can be stored in the same warehouse within the available space depending on their specific volume. We also assume each customer is supplied by a single DC via direct shipment. In addition, we consider the daily demands to follow a Poisson process[15], [17] and [28] The MPILMRP problem can be stated as: given a set of candidate distribution centers, a set of customers and a set of different products, determine the location of the DCs and their inventory policy – the reorder time, reorder quantity and safety stock – to serve the customers’ demands for different products at minimum cost. In this framework, for each DC to meet the customers’ stochastic demand, it holds two different types of inventory: working inventory and safety stock. The working inventory is determined by the ordering policy at each DC, while the safety stock is determined by a constraint that keeps the probability of a stockout below a specified threshold α. We use the following notation: set of products or product categories I set of customers J set of candidate DC locations fixed annualized cost of locating a DC at j fixed cost of placing an order from DC j to the supplier fixed cost of shipping an order from the supplier to DC j per‐unit cost to ship product p from the supplier to DC j annual holding cost per item of product p volume of product p inventory capacity at DC j (i.e. in cubic feet) expected annual demand for product p at DC j expected annual working inventory cost at DC j lead time in days for deliveries from the plant to the DC j unit transportation cost for product p between DC j and customer i Mean daily demand of customer i for product p weight factor associated with transportation costs weight factor associated with inventory costs x number of days in a year ( = 365) α Threshold on the probability of stockouts standard Normal deviate such that P(Z ≤ zα) = α In addition, we define the following decision variables:

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1 if we locate a DC at candidate site j 0 otherwise 1 if the DC at location j serves Y customer i s demand for product p 0 otherwise Q : the reorder quantity at DC j for product p We begin by analyzing the cost components associated with the total cost objective function as shown in Fig.1. X

parameter ν for each product p which gives the volume that one unit of product p occupies at the DC. Taking all of the above into consideration, the capacity constraint for DC j under a worst-case scenario is: Q ν z L L µ Y C, j

P

I

(i)

I

µ Y

Fig.1. Cost associated with MPILMRP

Consider now the inventory problem at the DC’s. We assume that each DC uses a (Q, r) inventory policy with type I service [29],[30], and [17]. To formulate the inventory model we follow the two step approach used by Axsater[31]. The first step is to determine the order quantity (Q) in the EOQ model, and the second step is to determine the reordering point (r). This approach yields an inventory policy that has a maximum relative error of .125 as compared to the optimal (Q,r) policy[32], well within the acceptable range. Hence, following Axsater’s approach, we use the EOQ model to find the order quantity Q, and then use this result to determine the reorder point r and the corresponding safety stock level that ensures that stock outs occur with a probability of α or less[28]. Assuming that the daily demands at each customer are uncorrelated over time and across customers, and approximating the Poisson demand distribution with a Normal distribution, which is a good approximation when the demands are sufficiently large [33][28], we find that the safety stock of product p needed to ensure that stockouts occur with a probability of α or less at a DC j which supplies a set of retailers S I is: z L ∑
S

The first term represents the sum over products of the total volume ν of the reorder quantity Q at location j. The second term represents the sum over all products of the volume of the available safety stock at DC j. The third term represents the sum over all products of the volume of the expected demand for each product during the lead time. Now that we have decided on the capacity constraint, we can formulate the inventory model accordingly. Hence, the basic multi product inventory capacitated EOQ model that we need to solve to determine the order quantity Q for product p at DC j is: Minimize G Q F ∑
D PQ

β g ∑
Q P

D PQ

(ii)

β ∑ P a D Subject to: (i) 0 Q

θ ∑

(iii)

µ

Given that the annual expected demand at DC j for x ∑ S µ , the product p can be expressed as D above expression for the safety stock can be rewritten as z L D /x. We interpret the capacity constraints to mean that the maximum inventory at a DC should not exceed its capacity at any point in time. It should be noted that only two activities affect the inventory, namely reordering a quantity Q from the supplier, which increases the inventory, and satisfying customers’ demands, which reduces it. Thus, if no customer orders occur at the time the reorder quantity Q arrives, the inventory will reach its maximum capacity, since the amount of space that the warehouse needs is proportional to peak inventory, not annual flow or average inventory, as pointed out by Simchi-Levi et al.[33]. Furthermore, given that the MPILMRP deals with different products of different volumes, the capacity of each DC should be defined in terms of available volume. We therefore introduce a

Recall that G Q , the objective function of the EOQ model above, is the expected annual cost of the working inventory at location j, which is a function of the reorder have the following quantity. The terms of G Q interpretations: the first term represents the fixed cost of ordering quantity Q of product p at location j; the second term represents the cost of transporting goods from the plant to DC j for all orders; the third term represents the total cost of transporting goods from the DCs to the retailers for all products; and the last term represents the total holding costs, on average, for all products. Here we assume that all orders have the same fixed cost for placing the order regardless of the type of order and the fixed cost of shipping the order. However the per-unit shipping costs may vary from one product to another. Also recall that β and θ are weight factors associated with transportation and inventory costs, respectively; they are introduced to facilitate testing of the relative importance of transportation and inventory costs as compared to the fixed facility location costs. Now that we know the optimal annual cost to maintain the working inventory we can integrate the EOQ model into the location model to obtain the MPILMRP as follows:

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Minimize

IV. NUMERICAL ANALYSIS fX J f DC

F f x∑
P

Iµ

Y

Q x∑
P Iµ

g β x
DC I

Y

Q
P

a µ Y (iv)

θ
P

h Q 2

θ
P

h z

L

I

µ Y

In this section, we first explain the design of our experiments and then summarize the numerical results. We tested our formulation using the BARON solver in GAMS to understand the effect of the capacity, the number of products, and the weight factors associated with the transportation and inventory costs on the results. We ran our experiments with three different numbers of products: a single product, two products, and three products. We used a 5-node data set consisting of the five nodes with the highest demand from Daskin’s 49-node data set (2; pp. 480–482), which represents the capitals of the lower 48 United States plus Washington, DC. Each of the nodes represents a retail location and also a candidate DC location. The demands for the first product are those associated with the five selected nodes, while the demands for other products are those associated with nodes that are selected randomly from the 49-node data set. The resulting mean demands for each node and for each product, and the fixed costs for each node are shown in Table 1 below. μ μ μ i/j Fixed Cost 1 297.6002 48.91769 42.19973 115800 2 179.9046 48.77185 40.40587 101800 3 169.8651 48.66692 36.85296 72600 4 129.3793 47.81468 36.65228 72400 5 118.8164 43.75099 34.86703 38400 Table 1: Demands for 3 products and fixed costs for 5-node data set

βx
DC

I

d µ Y
P

Subject to: ∑ Y
JY

1, i

i

I, p I, j

P J, p
DC

(v)
P
DC

X,

(vi)

Q
P

z L C, j

L

ν
I

I

µ Y

µ Y

(i)

J,

Q Y X

0,
Q

j

J, p

P I, j J, p P

(iii) (ix) (x)

0,1 , i 0,1 , j

J

The objective function (iv) includes the fixed cost of locating distribution centers, the cost of transporting goods from the supplier to the DCs, the cost of transporting goods from DCs to retailers, the safety stock cost, and the holding costs and fixed-order costs at the DCs. Constraints (v) require that each customer be assigned to exactly one DC. Constraints (vi) require that customers can be assigned only to open DCs. Constraints (i) and (iii) of the capacitated EOQ problem. Constraints (ix) and (x) require the decision variables X and Y to be binary. This model has nonlinear terms in both the objective function and the constraints, unlike the UFLP or the LMRP. Also, unlike the LMRP, we must explicitly solve for the order quantity variables Q .

We set the unit cost of shipping from candidate DC , to the great circle to customer for product p, distance between these locations times the product number p for each of the 3 products. We also set for each product p to 1. The fixed ordering costs F and shipping costs g were set to 10, while the variable shipping costs a was set to 5 for all DCs and for all products. The holding costs were set to 1, and z was set to 1.96, corresponding to a 97.5% service level. The lead time, , and the days per year were set to 1. Even though is 1, the differences between the daily parameters and yearly parameters are realized through the weights β and θ. We set β and θ to 0.0005 and 0.01, and then to 0.00001 and 0.001, respectively, in effect decreasing the relative ratio of the weighting on transportation costs to the weighting on inventory costs from 0.05 to 0.01, in addition to making the fixed facility costs relatively more important. Also to vary the difficulty of the instances, we ran our experiments using different values of the DC capacities, as illustrated in Table 2 of Appendix A.

1.1. Number of Products versus Costs
We analyzed the effect of the number of products stocked on total costs. Based on Problem 1 in Table 2 of Appendix A, we observed that as the number of products

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stocked increases, the overall costs increase as well (Figure 2).
140000 120000

Total Cost

100000 80000 60000 40000 20000 0
Prob.1 Prob.2 Prob.3

Number of Products

Fig. 2. Number of products versus costs

quantity and safety stock – to serve the customers’ demands for different products at minimum cost. This model can be considered as the multi-product version of the capacitated inventory-location model introduced by Ozsen et al..[28]. However; we generalize their capacity constraint by introducing the product volume as the basis for assessing capacity. In the near future, we plan to add certain service constraints to our model. Also, we plan to solve this problem using heuristic evolutionary search algorithms such as Genetic Algorithms. APPENDIX A
In this appendix, we provide the table of data used in the analysis.

1.2. DC Capacity versus Costs
We analyzed the effect of the size of the facility on total costs. Three problems are involved in this analysis, differing only in the capacity of each facility; these are Problems 6, 4, and 2 in Table 2 of Appendix A. We observed that costs decrease as the capacity of the DCs is increased (Fig. 3). Note that very large values of the capacity constraint are equivalent to having no capacity constraint at all.
400000 350000 300000 250000 200000 150000 100000 50000 0
Prob.6

Table 2 problem design

In table 3 below we illustrate summery of the numerical result associated with the problems designed above and that has been tested using the BARON solver on the GAMS.
Overall Cpacity
Prob.4 Prob.2

Total cost

Fig. 3: DC capacity versus costs

1.3. Number Of DCs versus transportation costs
We also analyzed the effect of transportation costs on the number of DCs opened. We observed that a larger value of the weight on transportations costs (β) results in more DCs being opened (Fig.4).
5 Numper of open DCs 4 3 2 1 0
0.00001 0.0005

Problem No. Open DCs Total cost 1 5 54259.854 2 5 67455.433 3 5 119280 4 1,3 244600 5 1,3 244840 6 1,2,3,5 350840 7 1,2,3 380070 Table 3Summary of the numerical results

REFERENCES
[1]. Cooper, M.C., Lambert, D.M., & Pagh, J. s.l. : “Supply Chain Management: More Than a New Name for Logistics”. The International Journal of Logistics Management, Vol 8, Iss 1, pp 1–14, 1997. [2]. Daskin, M.S. “Network and discrete location: models, algorithms, and applications”, Wiley, New York, 1995. [3]. Drezner, Z. “Facility location: A survey of applications and methods”, Springer, New York, 1995. [4]. Sridharan, R. “The capacitated plant location problem", European Journal of Operational Research 87 (2), pp. 203–213, 1995. [5]. Tragantalerngsak, S., Holt, J., Ronnqvist, M. “Lagrangian heuristics for the two-echelon, single-source, capacitated facility location problem”, European Journal of Operational Research 102 (3), pp. 611–625,1997. [6]. Cortinhal, M.J., Captivo, M.E., “Upper and lower bounds for the single source capacitated location problem”, European Journal of Operational Research 151 (2), pp. 333–351, 2003.

weight on transportation costs (β)

Fig 4. Number of DCs opened versus weight on transportation costs

The numerical results for all problems designed are shown in table 3 in Appendix A. V. CONCLUSION In this paper, we have outlined a formulation for a multi-product capacitated inventory-location model with risk pooling. The model determines the location of the DCs and their inventory policy – the reorder time, reorder

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[7]. Klose, A., Drexl, A. “Facility location models for distribution system design”, European Journal of Operational Research 162 (1), pp. 4–29, 2005. [8]. Klose, A., Gortz, S., “A branch-and-price algorithm for the capacitated facility location problem”, European Journal of Operational Research 179 (3), pp. 1109–1125, 2007. [9]. Kaijun Liu, Yonghong Zhou , Zigang Zhang, “Capacitated location model with online demand pooling in a multichannel supply chain”, European Journal of Operational Research 207, pp. 218–231, 2010. [10]. Eppen, G.D., “Effects of centralization on expected costs in a multi-location newsboy problem”, Management Science 25 (5), pp. 498–501,1979. [11]. Eppen, G., Schrage, L, “Centralized ordering policies in a multi-warehouse system with lead times and random demands. In: Schwarz, L., (Ed.), Multi-Level Production/Inventory Control Systems: Theory and Practice”, Management Sciences, vol. 16, pp. 51–68, 1981. [12]. Federgruen, A., Zipkin, P., “Approximations of dynamic, multilocation production and inventory problems”, Management Science 30 (1), pp. 69–84, 1984. [13]. W.J., Wolfe, P. “A warehouse location problem. Baumol,”, Operations Research 6 (2), pp. 252–263,1958. [14]. Karkazis, J. and Boffey, T.B “The multi-commodity facilities location problem”, The Journal of the Operational Research Society, 32, pp. 803–814,1981. [15]. Shen, Z.J.M., Coullard, C., Daskin, M.S., “A joint locationinventory model”, Transportation Science 37 (1), pp. 40– 55, 2003. [16]. Erlebacher, S.J., Meller, R.D. s.l, “The interaction of location and inventory in designing distribution systems”, IIE Transactions 32,155–166, 2000. [17]. M.S. Daskin, C. coullard, and Z.-J.M. Shen, “An inventorylocation model: Formulation, solution algorithm and computational results” Ann Oper Res, Vol. 110, 83-106, 2002. [18]. Warszawski, A. “Multi-dimensional location problems”, Operational Research Quarterly, pp. 165–179, 1973. [19]. Warszawski, A. and Peer, S., “Optimizing the location of facilities on a building site”, Operational Research Quarterly, 24, pp. 35-44, 1973. [20]. Klincewicz, J.G., Luss,H. andRosenberg, E., “Optimal and heuristic algorithms for multiproduct uncapacitated facility location”, European Journal of Operational Research, 26, pp. 251–258, 1986. [21]. Klincewicz, J. G. and Luss, H., “A dual-based algorithm for multiproduct uncapacitated facility location”, Transportation Science, 21, pp. 198–206, 1987. [22]. Geoffrion, A.M. and Graves G.W., “Multicommodity distribution system design by Benders decomposition”, Management Science, 20, pp. 822–844, 1974. [23]. Hindi, K.S. and Basta, T., “Computationally efficient solution of a multiproduct, two-stage distribution-location problem”, The Journal of the Operational Research Society, 45, pp. 1316–1323, 1994. [24]. Lee, C.Y., “An optimal algorithm for the multiproduct capacitated facility location problem with a choice of facility type” Computers and Operations Research, 18, pp. 167–182, 1991. [25]. Mazzola, J.B. and Neebe, A.W., “Lagrangian-relaxationbased solution procedures for a multi-product capacitated facility location problem with choice of facility type”, European Journal of Operational Research, 115, pp. 285– 299,1999.

[26]. Shen, Z.J, “Approximation algorithms for various supply chain problems.” Ph.D thesis. Evanston : Northwestern University, Department of Industrial Engineering and Management Sciences, 2000. [27]. Shen, Z.J.M., 2005. “A multi-commodity supply chain design problem” IIE Transactions 37 (8), 753–762, 2005. [28]. Ozsen, L., Coullard, C., Daskin, M.S. “Capacitated Warehouse Location Model with Risk Pooling”, Naval Research Logistics, pp. 295-312, 2008. [29]. Nahmias, S. , Production and Operations Management, 3rd edn., Irwin, Chicago, IL, 1997. [30]. Hopp, W. and Spearman, M.L. “Factory Physics: Foundations of Manufacturing Management”, Irwin, Chicago : IL, 1996. [31]. Axsater, S., “Using the deterministic EOQ formula in stochastic inventory control”, Management Science, 42, 830–834, 1996. [32]. Zipkin, P.H., “Foundation of Inventory Management”. Irwin Burr Ridge, IL,1997. [33]. D.C. Montgomery, G.C. Runger and N.F. Hubele, “Engineering statistics” New York ,Wiley, 1998. [34]. D. Simchi-Levi, P. Kaminsky, and E. Simchi-Levi. Designing and managing the supply chain: Concepts, strategies, and case studies. Boston, MA: McGrawHill/Irwin, 2003.

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