A Mixed Integer Programming Model for Production Inventory Network Design Xue-Ming Yuan and Soumen Saha Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore 638075 E-mail:
[email protected] Abstract-The paper considers a production inventory network whose multiple plants and warehouses are located in different geographic regions with customer demands for multiple products, and discusses how to design such kind of network so that the total cost is minimized. We formulate the problem as a mixed integer programming model, and conduct the numerical experiments to provide the managerial insights into production inventory network design.
I.
INTRODUCTION
Production inventory network design is an increasing important research topic in both academia and industry, particularly, with the increasing trend of globalized economy and offshore sourcing. There have been more and more enterprises which are attempting to set up their business entities beyond their own countries or regions. Thus, it is beneficial to the enterprise if there is some decision support or guidelines on how to design its production inventory network across countries or regions with minimal total cost. To address production inventory network design problems, there have been some conventional approaches in the literature, including optimizers, heuristics, simulators, and hybrid procedures. Reference [1] discusses some heuristics approaches for capacitated location problems. Those approaches are ADD, DROP, SHIFT, ALA (alternate location allocation), and VSM (vertex substitution method). Reference [2] discusses the model formulations and solution approaches for facility location problems, and reviews some of the contributions. Reference [3] offers a literature review on the problem of integrating and coordinating parts of the supply chain. References [4] and [5] present a survey and synthesis on plant location problems. In single-echelon facility location problems, the deliveries are made directly from the first echelon facilities (such as plants) to customers, where as in two-echelon facility location problems, the deliveries are made from the first-echelon facilities to customers via the second-echelon facilities (such as warehouses). Reference [6] formulates an incapacitated facility location problem for a two-echelon distribution system, and uses a Branch and Bound approach to solve the problem. One constraint that the model enforces is that a warehouse must be located wherever a plant is located. Reference [7] modifies the facility location problem, and solves it using a dual-based Branch and Bound algorithm. Reference [8] offers an integrated analysis of two-echelon production-distribution problem, where a single plant manufactures multiple commodities, and supplies to the customers via warehouses.
A special scenario, in which each customer receives supply from exactly one facility, is the single-source facility location problem. References [9] and [10] address the issue of singlesource. When the allocation variable to distribute products to customers from open facilities is restricted to zero-one variables, the problem becomes much harder to solve. Reference [11] presents a two-echelon, capacitated, singlesource, multi-commodity facility location problem. It uses a Lagrangean-based heuristics to solve the problem, which provides efficient results in comparison to linear programming relaxation. Reference [12] formulates a two-echelon, singlesource, capacitated facility location problem, and presents six Lagrangean-based heuristics to solve. The results show that the Lagrangean-based heuristics provide lower bounds that are much better than those obtained from a linear programming relaxation. This approach indicates that a Branch and Bound algorithm based on Lagrangean relaxation could generate more effective results efficiently. Reference [13] describes a twoechelon, single-source, capacitated facility location problem, and uses the most efficient Lagrangean-based heuristics to solve the problem. In three-echelon facility location problem, raw materials are shipped from vendors to manufacturing plants, and finished goods are distributed from the plants to the customers through warehouses. Reference [14] considers a three-echelon, capacitated, single-source, multi-commodity facility location problem. It provides a mixed integer programming formulation for the integrated model, and presents an efficient Lagrangeanbased heuristics solution procedure. Once the planning horizon is extended to more than a period, the location problem becomes more complex and dynamic. Reference [15] considers a multi-period, multi-commodity production-distribution problem. It decomposes the problem into two components: a generalized network and a small linear non-network components, and solves the network component with efficient pure network solution technique. Reference [16] presents a multi-period, two-echelon, multi-commodity, capacitated facility location problem. It proposes an algorithm that consists of two parts: in the first part Branch and Bound is used to generate a list of candidate solutions for each period, and then dynamic programming is used to find the optimal sequence of configurations over the multi-period planning horizons. Reference [17] discusses a multi-period, two-echelon, multi-commodity capacitated facility location problem. It formulates a mixed integer programming problem, and
proposes a Lagrangean relaxation and heuristics procedure to solve the problem. In the aforementioned papers, customer demands are assumed to be deterministic. Reference [18] studies the case of stochastic demand, and presents a multi-stage productiondistribution problem. At each stage, sub-problems are defined, and control policies are used. A heuristics optimization procedure is introduced. Reference [19] also considers stochastic demand, and presents a joint location-inventory problem involving a single supplier, and multiple retailers with variable demand. Due to the variability some amount of safety stock is maintained to achieve suitable service level. The riskpooling benefits are achieved by allowing some retailers to act as distribution centers. It formulates this problem as a nonlinear integer programming model, and restructures it into a setcovering integer programming model, and then solves the pricing problem associated with column generation algorithm efficiently for two practical cases. The existing literature generally models the plants as multifunctional plants, i.e., all plants in the network produce the entire product range in the product portfolio. In a mass production scenario, it makes sense to assume the plants multifunctional. However, nowadays enterprises are increasingly drifting away from the strategy of mass production, and localizing the products based on regional needs and tastes. The demands of the localized versions of the products are not high enough to offset the higher costs of establishing multifunctional plants. Instead of making the plants multi-functional, it is economically viable to dedicate the plants towards manufacturing one or two but not all product types from the product portfolio. In this paper, we consider a two-echelon, multi-commodity, capacitated facility location problem where the plants are nonmultifunctional (see Fig. 1). The plants supply multiple product types to the customers via warehouses. This problem is different from the facility location problem presented in [11] in
1
1
1
2
2
2
K
J
I
Plants
Inventory
Fig. 1. The production inventory network
Customers
the following aspects. In their model all the plants are multifunctional, while all the plants are non-multifunctional in our model. Their model constrains the aggregate capacity of each plant. In this model, each plant has a capacity for producing each product type. In addition to the fixed cost of establishing a plant in a site, there is additional fixed cost of making the plant capable of manufacturing each product type in our model. This paper formulates the production inventory network design problem as a mixed integer programming model. The formulation requires a set of key decisions such as how to choose warehouses from a set of potential warehouse location sites, how to choose plants from a set of potential plant location sites, how to assign customers to the opened warehouses and plants, and how to decide which plants will manufacture what product types. The model has been developed from the strategic perspective in which an enterprise desires to set-up the whole production inventory network. The model can also be used at the operational level, where the network exists, and decisions pertaining to allocation of customer demand among existing facilities have to be made. The organization of the paper is as follows. In Section II, we introduce some mathematical notations, and formulate the problem as a mixed integer programming model. In Section III, we conduct the numerical experiments by using LINGO optimization suite, and provide the managerial insights gained from the numerical results. Section IV summarizes the findings, and concludes the paper.
II. PROBLEM FORMULATION We discuss the production inventory network in a multiple product environment. The customers place the demand for multiple product types, which are manufactured by the plants, and supplied to them through the warehouses. Each plant has a fixed capacity for producing a certain product type, thus capacities of the plants are not aggregate rather they are product specific. The capacities of the warehouses are aggregate. The main objective is to minimize the total fixed and variable costs associated with the production inventory network. There are two types of costs. One type of cost is the production costs which include the fixed costs of establishing the plants, additional fixed costs of making the plants capable of producing various product types, and variable costs of manufacturing the products. Another type of cost is the distribution costs which incorporate both the fixed costs of establishing the warehouses, and variable costs of distributing the products. The problem we address in this paper is to locate required set of warehouses and plants among the set of potential sites, assign customers to the warehouses and plants, and decide which plants will manufacture what product types, so as to minimize the total cost incurred. We introduce the following notations to describe our model. I = {1,…,I}, the set of customers,
J = {1, 2,…,J}, the set of potential warehouse sites; K = {1, 2,…,K}, the set of potential plant sites; L = {1, 2,…,L}, the set of product types.
Min Z = ∑ e j u j + j il ijkl
j
k
∑∑ f
+
k k
k
∑∑∑∑ a x i
∑d v
g ijkl +
l
k
∑∑∑∑ a x
We define the decision variables as follows.
1 uj = 0 1 vk = 0 1 wkl = 0 xijkl
if warehouse is opened at site j , ∀ j ∈ J;
otherwise,
if plant is opened at site k , ∀ k ∈ K ; otherwise, if plant opened at site k produces product type l, ∀ k ∈ K , l ∈ L; otherwise,
= the fraction of customer i's demand for product type l supplied by plant k via warehouse j , ∀ i ∈ I , j ∈ J , k ∈ K , l ∈ L.
We now formulate the objective function as
wkl + h
il ijkl kl
i
j
k
l
Input parameters: ail = demand by customer i for product type l, ∀ i∈I, l∈L; subject to the constraints cj = capacity of warehouse j, ∀ j∈J. It represents the ail xijkl ≤ c j u j ∀ j∈J, number of units of products, which the warehouse can i k l handle in the time horizon for which we are designing the network; a il xijkl ≤ bkl wkl ∀ k∈K, l∈L, i j bkl = capacity of plant k for producing product type l, ∀ k∈K, l∈L. It represents the number of units of the xijkl = 1 ∀ i∈I, l∈L, product type, which the plant can produce in the time j k horizon for which we are designing the network; uj ≤ s , ej = fixed cost of establishing warehouse at site j, ∀ j∈J; j dk = fixed cost of establishing plant at site k, ∀ k∈K; fkl = additional fixed cost of making the plant k capable of vk ≤ r , producing product type l, ∀ k∈K, l∈L; k gijkl = cost of distributing one unit of product type l from plant k to customer i through warehouse j, ∀ i∈I, j∈J, k∈K, xijkl ≤ u j ∀ i∈I, j∈J, k∈K, l∈L, l∈L; hkl = cost of manufacturing one unit of product type l at plant xijkl ≤ wkl ∀ i∈I, j∈J, k∈K, l∈L, k, ∀ k∈K, l∈L; s= maximum number of warehouses that can be opened; wkl ≤ vk ∀ k∈K, l∈L, r= maximum number of plants that can be opened. We assume the demand of a customer for a certain product type can be met by more than one plant. A plant can supply a certain product type to the customers via more than one warehouse.
kl
l
∑∑∑
(1)
∑∑
(2)
∑∑
(3)
∑
(4)
∑
(5)
u j , v k , wkl ∈ {0,1} , xijkl ≥ 0 ∀ i∈I, j∈J, k∈K, l∈L.
(6) (7) (8) (9) (10)
Constraint set (1) ensures that the customer demand serviced by a warehouse does not exceed its capacity. Constraint set (2) ensures that the customer demand for a certain product type produced at a plant does not exceed its capacity. Constraint set (3) ensures that the demand of each customer for each product type is fulfilled. Constraint set (4) limits the maximum number of warehouses that can be opened. Constraint set (5) limits the maximum number of plants that can be opened. Constraint set (6) ensures that assignments are made to the opened warehouses only. Constraint sets (7) and (8) ensure that assignments are made to the opened plants only. Constraint set (9) ensures that the variables are binary. Constraint set (10) ensures that the variable is non-negative.
III. NUMERICAL EXPERIMENTS This section conducts the numerical experiments to solve the optimization problem in Section II and provides the managerial insights into designing the production inventory network, which are gained from the numerical results. Several examples of different sizes are generated randomly so as to represent various realistic logistics situations. A set is represented by four parameters: number of customers (I),
number of potential warehouses (J), number of potential plants (K), and number of product types (L). We use the LINGO optimization suite to solve the mixed integer programming problem. The capacities of the potential warehouses (cj) and plants (bkl), the fixed costs of establishing potential warehouses (ej) and plants (dk), the unit manufacturing costs (hkl), and unit distribution costs (gijkl) are generated from uniform distributions, respectively. The following are the distributions used to generate the values of the coefficients. cj = Uniform (5000, 13000) bkl = Uniform (5000, 15000 ej = Uniform (160000, 320000) dk = Uniform (190000, 350000) fkl = Constant (45000) gijkl = Uniform (4, 14) hkl = Uniform (8, 28) For all instances belonging to the same set (i.e., with the same number of customers, potential warehouses, potential plants, and product types), the values of these coefficients remain unchanged. We perform the experiments for the following six settings: 1) {5 customers, 3 potential warehouses, 2 potential plants, 2 product types} 2) {5 customers, 3 potential warehouses, 2 potential plants, 3 product types} 3) {10 customers, 5 potential warehouses, 3 potential plants, 2 product types} 4) {10 customers, 5 potential warehouses, 3 potential plants, 3 product types} 5) {15 customers, 8 potential warehouses, 4 potential plants, 2 product types} 6) {15 customers, 8 potential warehouses, 4 potential plants, 3 product types}. For each setting, we vary the demand placed by each customer for each product type from 600 to 1000 units in a step of 100 units. Table 1 shows the results about the number of opened warehouses, the number of opened plants, the total setup cost and the total optimal cost in each scenario. For the case of 10 customers, 5 potential warehouses, 3 potential plants and 3 product types, when each customer placing a demand of 1000 units for each product type, the optimal solution requires to open four warehouses, that is, warehouse 1, warehouse 3, warehouse 4 and warehouse 5, and to set up two plants, plant 2 and plant 3. Plant 2 manufactures all the three types of products, type 1, type 2 and type 3, and plant 3 manufactures the two types of products, type 1 and type 3. The total setup cost of the production inventory network is 1858612. The total optimal cost is 2345699. For a certain setting, when increasing the demand, the total set-up cost remains the same, but the total optimal cost increases. From Table I, for the scenario of 5 customers, 3 potential warehouses, 2 potential plants and 3 product types, when the demand increases from 600 to 1000, the total set-up cost remains unchanged at 6890940, where as the total optimal
cost increases from 796540 to 866940. However, for the scenario of for the scenario of 15 customers, 8 potential warehouses, 4 potential plants and 2 product types, when the demand increases from 600 to 700, the total set-up cost remains unchanged at 851211, whereas the total optimal cost increases from 1087526 to 1173907. When the demand increases from 800 to 1000, the total setup cost increases from 992918 to 1525126, and the total optimal cost also increases from 1363178 to 1983833. From the above observation, it can be concluded that the total setup cost has the different trend from the total optimal cost. The opened warehouses and plants being under-utilized at lower demands are able to handle the increased demands. Therefore, the setup cost remains the same, where as manufacturing and distribution costs increase proportionately with demand. This leads to the increase of the total optimal cost. We also use uniform distribution to represent the demand placed by each customer for each type of product, and vary it from Uniform (100,1000) to Uniform (500,1400). Since the conclusions derived from fixed demands, and from demands represented by uniform distributions are same, we are omitting the details in the paper. IV. CONCLUDING REMARKS In the paper, we have formulated the production inventory network design problem as a mixed integer programming model, and discussed the issue of non-multifunctional plants in multi-commodity, two-echelon environment. The model has been developed from the strategic perspective in which an enterprise desires to set up the whole production inventory network. The model can also be used at the operational level, where the network exists, and decisions pertaining to allocation of customer demands among existing facilities have to be made. The model presented here can be expanded to multiple time horizons. The products may be held in inventory at the plants and warehouses to satisfy later period demands. The issues pertaining to increase in manufacturing, distribution and inventory holding costs, and demand over multiple time horizons will be addressed in our future research. REFERENCES [1] [2] [3] [4] [5]
S. K. Jocobsen, “Heuristics for the capacitated plant location model”, European Journal of Operational Research, vol. 12, pp. 253-261, 1983. C. H. Aikens, “Facility location models for distribution planning”, European Journal of Operational Research, vol. 22, pp. 263-279, 1985. R. Bhatnagar, P. Chandra, and S. K. Goyal, “Models for multi-plant coordination”, European Journal of Operational Research, vol. 67, pp. 141-160, 1993. J. Krarup, and P. M. Pruzan, “The simple plant location problem: survey and synthesis”, European Journal of Operational Research, vol. 12, pp. 36-81, 1983. C. S. Revelle, and G. Laporte, “The plant location problem: new model and research prospects”, Operations Research, vol. 44, no. 6, pp. 862-874, 1996.
[6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19]
L. Kaufman, M. V. Eede, and P. Hansen, “A plant and warehouse location problem”, Operational Research Quarterly, vol. 28, pp. 547-554, 1977. L. Gao, and E. P. Robinson Jr., “A Dual-based optimization procedure for the two-echelon incapacitated facility location problem”, Naval Research Logistics, vol. 39, pp. 191-212, 1992. G. Barbarasoglu, and D. Özgür, “A Lagrangean relaxation approach to an integrated production and two-echelon distribution system”, International Conference on Industrial Engineering and Production Management, 1997. H. Pirkul, “Efficient algorithms for the capacitated concentrator location problem”, Computers and Operations Research, pp. 197-208, 1987. R. Sridhran, “A Lagrangean heuristics for the capacitated plant location problem with single source constraints”, European Journal of Operational Research, vol. 66, pp. 305-312, 1993. H. Pirkul, and V. Jayaraman, “Production, transportation, and distribution planning in a multi-commodity tri-echelon system”, Transportation Science, vol. 30, no. 4, pp. 291-302, 1996. S. Tragantalerngsak, J. Holt, and M. Rönnqvist, “Lagrangean relaxation heuristics for two-echelon single-source capacitated facility location planning problem”, European Journal of Operational Research, vol. 102, pp. 611-625, 1997. S. Tragantalerngsak, J. Holt, and M. Rönnqvist, “An exact method for the two-echelon, single-source, capacitated facility location problem”, European Journal of Operational Research, vol. 123, pp. 473-489, 2000. V. Jayaraman, and H. Pirkul, “Planning and coordination of production and distribution facilities for multiple commodities”, European Journal of Operational Research, vol. 133, pp. 394-408, 2001. D. Klingman, J. Mote, and N. Phillips, “A logistics planning system at W. R. Grace”, Operations Research, vol. 36, pp. 811-822, 1988. C. Canel, B. M. Khumawala, J. Law, and A. Loh, “An algorithm for the capacitated, multi-commodity, multi-period facility location problem”, Computers and Operations Research, vol. 28, pp. 411-427, 2001. Y. Hinojosa, and J. Puerto, F.R. Fernandez, “A multi-period, twoechelon, multi-commodity capacitated plant location problem”, European Journal of Operational Research, vol. 123, pp. 271-291, 2000. M. Cohen, and H. Lee, “Strategic analysis of integrated productiondistribution systems: models and methods”, Operations Research, vol. 36, pp. 216-228, 1988. Z. L. M. Shen, C. Coullard, and M. S. Daskin, “A joint location-inventory model”, Transportation Science, vol. 37, no. 1, pp. 40-55, 2003.
TABLE I RESULTS FOR THE NUMBERS OF OPENED WAREHOUSES AND PLANTS, THE TOTAL SETUP COSTS AND THE TOTAL OPTIMAL COSTS
customer(I)* warehouse(J)* plant(K)
product types (L)
demand
opened warehouses
opened plants (product types they manufacture)
total set-up cost
total optimal cost
5*3*2
2
600 700 800 900 1,000
1 1 1 1 1
2 (1, 2) 2 (1, 2) 2 (1, 2) 2 (1, 2) 2 (1, 2)
690,940 690,940 690,940 690,940 690,940
796,540 814,140 831,740 849,340 866,940
5*3*2
3
600 700 800 900 1,000
1 1 1,3 1,3 1,3
1 (1, 2, 3) 1 (1, 2, 3) 1 (1, 2, 3) 1 (1, 2, 3) 1 (1, 2, 3)
735,901 735,901 1,016,338 1,016,338 1,016,338
903,901 931,901 1,216,338 1,241,338 1,266,338
10*5*3
2
600 700 800 900 1,000
1,5 1,5 1,5 1,3 1,4,5
3 (1, 2) 3 (1, 2) 3 (1, 2) 3 (1, 2) 2 (2); 3 (1, 2)
839,396 839,396 839,396 918,169 1,488,175
1,083,596 1,124,296 1,171,859 1,291,715 1,752,463
10*5*3
3
600 700 800 900 1,000
1,3 1,4,5 1,3,5 1,2,4,5 1,3,4,5
2 (1, 2, 3) 2 (1, 2, 3) 2 (1, 2, 3) 2 (1, 2, 3) 2 (1, 2, 3); 3(1, 3)
1,016,377 1,193,717 1,218,041 1,480,113 1,858,612
1,355,377 1,571,608 1,726,190 2,030,861 2,345,699
15*8*4
2
600 700 800 900 1,000
1,8 1,8 5,6,8 1,6,8 1,3,8
3 (1, 2) 3 (1, 2) 3 (1, 2) 3 (1, 2) 3 (1, 2); 4 (1, 2)
851,211 851,211 992,918 1,044,528 1,525,126
1,087,526 1,173,907 1,363,178 1,448,924 1,983,833
15*8*4
3
600 700 800 900 1,000
1,6,8 1,5,6,8 1,3,6,8 1,4,5,6,8 1,3,4,5,6,8
2 (1, 2, 3) 2 (1, 2, 3) 2 (1, 2, 3); 4 (1) 2 (1, 2, 3); 4 (1) 2 (1, 2, 3); 4(1, 2, 3)
1,142,736 1,344,400 1,771,651 1,948,991 2,319,428
1,608,632 1,876,396 2,337,231 2,575,246 3,014,373
