ترکیب تصمیمات کنترل موجودی به یک شبکه توزیع مدل طراحی استراتژیک با تقاضای تصادفی

In this paper, we propose a simultaneous approach to incorporate inventory control decisions––such as economic order quantity and safety stock decisions––into typical facility location models, which are used to solve the distribution network design problem. A simultaneous model is developed considering a stochastic demand, modeling also the risk pooling phenomenon. We present a non-linear-mixed-integer model and a heuristic solution approach, based on Lagrangian relaxation and the sub-gradient method. In a numerical application, we found that the potential cost reduction, compared to the traditional approach, increases when the holding costs and/or the variability of demand are higher.

The standard literature on supply chain management classifies the problems into three hierarchical levels: strategic (long term), tactical (medium term), and operational (short term), though the limits between each level remain unclear. The usual approach to solve these problems has typically been to tackle them in isolation from one another. In practice, strategic decisions are made by top managers, while the tactical and operational decisions are made by bottom level managers. This situation tends to promote incompatibilities and incoherence between each level. For instance, facility location problems are considered as strategic, imposing a strong simplification regarding tactical and operational aspects directly related to the optimal location. Examples of these tactical/operational aspects are the inventory control policy, the choice of transportation mode/capacity, warehouse design and management, vehicle routing, among others. The aim of this paper is to incorporate tactical/operational decisions into to the facility location problem solution scheme. Specifically, inventory management decisions will be simultaneously modeled with the distribution network design. This inclusion acquires especial relevance in the presence of high holding costs (e.g. frozen food industry) and high-variability demands. An example is provided to illustrate the relevance of this issue. Fig. 1 shows a distribution network where a single plant supplies products to regional warehouses, and these distribute products to retailers or customers. The ownership of the chain is assumed to belong to a single decision maker responsible for the holding cost at each facility, as well as all the transportation costs. In Fig. 1, warehouse 1 sends products to retailers 1 and 2, each of which has a stochastic demand with means d1 and d2, respectively, and variances u1 and u2, respectively. Warehouse 2 supplies products to retailers 3, 4 and 5, and warehouse 3, to retailers 6 and 7. The operation of the warehouses incurs two type of cost: one is proportional to the average supplied demand (made up of holding and handling costs), and the other one is proportional to the standard deviation of supplied demand (due to the safety stock). Under constant lead times and levels of service, the safety stock is proportional to standard deviation of the supplied demand.Thus, safety stock kept in warehouse 1, must be proportional to View the MathML source; safety stock in the warehouse 2 must be proportional to View the MathML source, and the one in warehouse 3 must be proportional to View the MathML source. Clearly the system’s cost depends on the retailers assignment scheme. For instance, if we closed warehouse 1 and its clients were assigned to warehouse 2, significant cost changes occur. The fixed-installation cost of the warehouse 1 would be eliminated, and the transportation costs (from plant to warehouses and from warehouses to retailers) would change; safety stock cost would be reduced, because total safety stock kept on warehouse 2 would be proportional to View the MathML source (clearly lower than View the MathML source). This situation is known in the literature as risk pooling. This paper presents a non-linear-mixed-integer model to find an optimal configuration of network, considering the installation, transportation, ordering and holding, distribution network design with risk pooling effect model (DNDRP), along with a solution approach based on Lagrangian relaxation. In the case analyzed in this paper, the plant location is known and fixed. Therefore, the transportation costs between the plant and the warehouses grouped into a single arc cost between warehouses and retailers. This situation can be easily modified, for a more general setting. The inventory policy at the plant is not modeled explicitly. The model stated on this paper is an extension of the classical capacitated facility location problem, which is already NP-hard. Thus, if we use this model to solve a great instance (which are easy to find in the real world), any commercial package will take a lot of time to solve the model for these instances, or even will not be able to solve it. Then, is necessary to develop a solving approach. The numerical results developed in this paper consider a relative little instance, especially to compare the results obtained to solve optimally the model, with to solve heuristically this model. The next section presents a literature review. In Section 3 the inventory control policy is analyzed and the objective function of the problem is developed. Section 4 presents the DNDRP model, which solve simultaneously facility location and inventory control decisions. Section 5, presents a solution approach to solve the DNDRP model. The solution approach is based on a combination of Lagrangian relaxation and the sub-gradient method. Section 6 reports the numerical results and their interpretation. Finally, Section 7 presents the conclusions and some future research lines.

In the model introduced in our paper, DNDRP, which is used to design the distribution network, we optimize the magnitude for the orders of warehouses to plant. In this optimization, is interesting to note that the isolated optimization of the order quantities, gives the same results of the centralized optimization of them. The difference between the centralized and the isolated optimization, consists in the cost parameters considered (ordering and holding cost). The isolated optimization considers the cost parameters of the warehouses, while the centralized one, considers the cost parameters associated with the entire system. We observe, in the numerical application discussed in Section 6, that the total cost reduction is higher as the holding cost, ordering cost, lead times and/or level of service (measured as the probability of satisfying all the demands) increase. These elements generate the expenditure in inventory management, and can be modified by the decision makers within a supply chain. In addition, when the variability of demand increases, the DNDRP achieves higher cost reductions. Note that the complexity of the analyzed problem precludes the general use of exact algorithms, especially for large size networks. Under these scenarios, the proposed LR heuristic offers an efficient solution method. We conclude that in supply chains where the products are perishable or of high value, as the frozen food among others, the simultaneous approach (considering DNDRP model along with the LR heuristic) appears as a valuable (and easy to work with) tool for assisting the decision makers in the hard task of designing a distribution network. Furthermore, the time required to solve the DNDRP using the LR heuristic can be significantly improved if we considered a more efficient procedure to solve SP3k at each iteration. For example, if we simultaneously relax the constraints associated with capacity at each warehouse (constraints (30) and/or (40)), and we develop a procedure more complex to search a feasible solution on each step, we can improve the heuristic algorithm used to solve DNDRP (DNDRP-LR). In terms of future research, it would be interesting to apply this simultaneous methodology to more complex supply chains with more stages, considering the inventory at the production plants, or even considering the production process and raw materials replenishment. Furthermore it is possible to consider different levels of shared information between plants and warehouses, allowing to model the bullwhip effect and its impact on the distribution network design. Another unexplored extension of this methodology is the consideration of a more complex inventory model for a supply chain with multiple products and multiple periods. Finally, this approach can be incorporated into other strategic logistics and supply chain management problems, such as transportation network design, covering problems associated with facility location models.