Multi-commodity inventory-location problem with two different review inventory control policies and modular stochastic capacity constraints

Abstract

In this article, we introduce two novel multi-commodity inventory-location models considering continuous and periodic review inventory control policies and modular stochastic capacity constraints. The models address a logistic problem in which a single plant supplies a set of commodities to warehouses where they serve a set of customers or retailers. The problem consists of determining which warehouses should be opened, which commodities are assigned, and which customers should be served by the located warehouses; as well as their reorder points and order sizes in order to minimize costs of the system while satisfying service level requirements. This problem can be formulated as a mixed-integer nonlinear programming model, which is non-convex in terms of modular stochastic capacity constraints and the objective function. A Lagrangian relaxation and the subgradient method solution approach is proposed. We consider the relaxation of three sets of constraints, including customer assignment, warehouse demand, and variance constraints. Thus, we develop a Lagrangian heuristic to determine a feasible integer solution at each iteration of the subgradient method. An experimental study shows that the proposed algorithm provides good quality gaps and near-optimal solutions in a short time. It also evinces significant impacts of the selected inventory control policy into total costs and network design, including risk pooling effects, when it is compared with different review period values and continuous review.

Introduction

Strong economic turbulence and empowered and informed consumers have driven an intense competition between companies and made logistics and the supply chain increasingly crucial in today’s global markets. So, the focus has been on reducing distribution costs and their impact on “everyday low prices” (e.g., Wal-Mart). Due to these reasons, the current trend is the integration of the entire value chain in order to make more efficient all the activities involved in the process of supply, transformation, and distribution of the final products with high standards of quality and service to the client. To reach this high performance, companies need an optimal supply chain network design (SCND), which also involves the integration of strategic, tactical, and operational decisions in order to achieve a sustainable competitive advantage (Shen, 2007).

In the framework of supply chain management, strategic, tactical, and operational decisions have traditionally been made in a hierarchical scheme due to the different nature, scope and time frame of the associated problems (Fahimnia, Parkinson, Rachaniotis, Mohamed, & Goh, 2013). This approach sometimes results in multiple conflicting or non-feasible decisions, which requires making decisions that integrate multiple levels, despite the inherent complexities of modeling. Besides, in a highly dynamic and uncertain business environment, strategic decisions like facility location may need to be revised to improve the efficiency of the supply chain once tactical and operational plans have been developed such as inventory control, that is, a “closed-loop planning approach,” which considers feedback from and to all decision levels (Farahani, Rashidi Bajgan, Fahimnia, & Kaviani, 2015).

To respond to the previous integration challenge, pioneer researchers have studied the inventory-location problem (ILP) over the past decades, assuming a single commodity scenario and stochastic demand without regard to inventory capacity of warehouses (e.g., Erlebacher and Meller, 2000, Daskin et al., 2002). Nevertheless, the quantity of products supplied by plants is usually limited by the capacity of the warehouses. To fill this gap, some authors have incorporated stochastic capacity constraints, both in the single commodity scenario (e.g., Miranda and Garrido, 2006, Miranda and Garrido, 2008, Ozsen et al., 2008, Ozsen et al., 2009) and multi-commodity scenario (e.g., Atamtürk et al., 2012, Dai et al., 2018). All the previous researchers assume that each location operates a continuous review (s, Q) control policy under which an order of size Q is submitted to the plant whenever the inventory level at a given location falls to level s. Such a system induces an entirely different order cycle for each location, generating subsequent coordination challenges at the operational level (Berman, Krass, & Tajbakhsh, 2012). Only a few researchers have studied ILP assuming an (R, S) periodic review inventory control policy in an uncapacitated single commodity scenario (Berman et al., 2012), with deterministic constraints (Vahdani, Soltani, Yazdani, & Meysam Mousavi, 2017), and multi-commodity scenario with deterministic capacity constraints (Yao, Lee, Jaruphongsa, Tan, & Hui, 2010). Also, the papers that consider an (R, s, S) periodic review inventory policy with stochastic capacity constraints are not many, for example, Cabrera et al. (2013), and Araya-Sassi, Miranda, and Paredes-Belmar (2018). These studies consider a single commodity scenario. Accordingly, in the current literature, no ILP model addresses a multi-commodity scenario with an (R, s, S) periodic review inventory policy with stochastic capacity constraints.

In accordance with the above, most of the ILP models present in the literature address a single commodity scenario. However, the real-world supply chain networks must naturally manage multiple product groups with very different features and attributes, e.g., perishable products and electro domestics. In turn, these products can generally be characterized by their mean and variance of demand and service level required to satisfy the customer’s needs. These contrasts of products impact both location and inventory decisions at DCs. Therefore, the optimal SCND must include DCs that capture these discrepancies, incorporating modular capacities for each product according to the respective geographic zone needs, in order to minimize the total operation cost. Another relevant issue in an integrated SCND is the selection of suitable inventory control policy. This policy must mainly address the problem of how much and how often to order the products. To comply with this policy, we must decide how often the inventory level should be determined, i.e., specify the review interval (R), which is the time that elapses between two consecutive moments at which the stock level is known. An extreme case is a continuous review; that is, the inventory level is always known. According to Silver, Pyke, and Thomas (2017), in practice, a continuous review is usually not required. Instead, each transaction, such as shipment or receipt, triggers an instantaneous updating of the stock level, which is called transaction reporting control. Even using a point-of-sale data collecting system, involving electronic scanning, inventory decisions are usually made periodically (e.g., end of the day). Thus, considering the inventory reordering decision, the minimum review period, R, is one day. The periodic review policy differs from most ILP present in the literature, which assumes a continuous review for DCs inventory control and cannot be properly utilized in SCND when a periodic review is considered. Thus, the main motivation of this research is addressing the previous gaps noted in the literature.

In this article, we introduce the multi-commodity inventory-location problem with continuous and periodic review policies, incorporating modular stochastic capacity constraints, and we formulate a mixed-integer nonlinear programming model for solving it. These two novel models are based on Multi-commodity Location Problem (MLP) following Warszawski and Peer, 1973, Karkazis and Boffey, 1981, and Neebe and Khumawala (1981), integrating inventory control decisions jointly. The structure of these MLP models is based on a modular opening decision, which is related to the inventory capacity setting of the DC for each commodity and not an aggregate capacity for all commodities. This is like most of the multi-commodity inventory-location models present in the specialized literature. This new approach to modeling the modular capacities at the DCs is coherent with the modern view of dynamic capacity planning (Alumur et al., 2016, Correia and Melo, 2017, Jena et al., 2015). We consider a three-echelon supply chain system in which a single plant serves a set of DCs, which further serve a set of end retailers, each with stochastic demand in a multiple commodities scenario (see Fig. 1). We assume that only one DC must serve each retailer for each product or commodity, that is, a single-sourcing distribution policy. We compare a continuous review policy (s, Q) with a periodic review policy (s, S) for each DC, where R is the period review, s is the reorder point, S is the objective inventory level, and $Q$ is the order quantity.

In this paper, we assumed that the demands and capacity constraints were random variables, and the distribution functions could be obtained via past return data. However, when there is not enough data or there are situations where the past data cannot reflect the demands and capacity constraints (e.g., new products) we can use experts’ estimations according to their knowledge and judgments rather than historical data (Liu et al., 2017, Chen et al., 2017a).

Thereby, we develop two inventory-location models in which stochastic inventory capacity constraints, expected inventory and ordering costs are defined using both continuous review and periodic review strategies. We formulate these inventory-location models analyzing the expected safety stock, cyclic inventory, order quantities and peak inventory levels for each potential warehouse. These mixed-integer non-linear programming (MINLP) models are NP-Hard because they are an extension of the Capacitated Facility Location Problem (CFLP), which is already NP-hard.

We propose a solution approach based on Lagrangian relaxation and the subgradient method to tackle the high complexity of the analyzed problems. The decomposition strategy is based on the relaxation of three different constraints of the original problem. Later, we decompose this relaxed problem in a subproblem for each warehouse and commodity, which is, in turn, separated into a location and inventory subproblem. Additionally, to achieve a feasible integer solution at each iteration, a Lagrangian heuristic is developed. This Lagrangian heuristic addresses the warehouse selection and a retailer's greedy assignment, followed by 1-OPT and 2-OPT local search improvements. We solve these using instances of 20 potential sites to locate warehouses, 40 retailers, and 5 commodities. The Lagrangian relaxation algorithm that we propose obtains competitive computational times with good quality gaps and near-optimal solutions. Also, the incorporation of periodic review policy in this model is relevant for those firms in which a continuous review policy implementation is not feasible because it involves a high investment, or there is a need to reduce costs for the inventory control system, especially for those highly demanded items. Considering all these attributes, inventory-location models more accurately represent the complexity faced by distribution companies today.

Based on the above descriptions, we summarize the main contributions of this paper that differentiate it from the other relevant research as follows: First, we introduce what we call the multi-commodity inventory-location problem with continuous and periodic review policies, incorporating modular stochastic capacity constraints. Then, we formulate a mixed-integer nonlinear programming model for solving it. Second, we solve the models using a Lagrangian relaxation approach and subgradient method. Third, we demonstrate that the proposed algorithms provide near-optimal solutions in a short time. Fourth, we compare the two models and algorithms, considering the two different review policies.

The paper is organized as follows: In Section 2, we present a literature review of the inventory-location models. In Section 3, we introduce the formulation of the inventory-location models with periodic and continuous review and modular stochastic capacity constraints. Section 4 introduces the proposed solution approach based on Lagrangian relaxation. Section 5 presents and analyzes the numerical results. Finally, Section 6 presents conclusions, managerial insights, and suggestions for future research.