The Economic Order Quantity (EOQ) zone is beneficial for giving some latitude in picking the lot sizes in a continuous time inventory problem, but it is not suitable for a discrete time inventory problem, the discrete lot sizing (DLS) problem. In this paper, a novel enumeration method is proposed and coded as a user-friendly computerized scheduling system to “visualize” the complex DLS problems by projecting the entire feasible solutions on a two dimension space, where setup frequency and total cost are placed on the horizontal and the vertical axis respectively. First, the zone around the optimal solution in the DLS problem is demonstrated always smooth and this phenomenon is defined as DLS zone in which giving a small penalty cost from the optimal solution brings several alternative solutions for picking the lot sizes. Second, even if the penalty costs are changed, the computerized scheduling system is able to determine the adaptive decision zone and find the included alternative solutions. The flexibility in picking the lot sizes for discrete time inventory problems is significantly enhanced since decision makers are enabled to choose a preferable solution from the DLS zone.
From the perspective of the continuous time scale, constant demand rate, and infinite planning horizon, Harris (1913) introduced the Economic Order Quantity (EOQ) model. The total cost curve of EOQ model is available to reflect the total costs corresponding to various order quantities and it is usually a shape of bowl (Solomon, 1959). Most importantly, a region around the lowest point (Q∗) on the total cost curve is relative flat and this phenomenon is the well-known “EOQ zone”, displayed in Fig. 1 ( Stevenson, 2002). From practical standpoint, the EOQ zone gives some latitude in picking the lot sizes because the change in total cost among Q∗ and a number of order quantities included in the “zone” is not too much. Therefore, the EOQ zone is an adaptive decision zone in which there are several alternative solutions for decision making. From academic standpoint, Solomon (1959) was motivated by the EOQ zone to introduce the mathematical approach for defining the economic lot size range. Decision makers could set an acceptable penalty cost in advance and they would produce anywhere within a range of quantities relatively. However, the EOQ zone is only beneficial for giving some latitude in picking the lot sizes in a continuous time inventory problem but not suitable for a discrete time inventory problem, the material requirement planning (MRP) problem.
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Fig. 1.
An adaptive decision zone (EOQ zone) in a continuous time inventory problem.
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MRP is an approach used in production scheduling to determine the required parts and materials for end items (Fakhrzad & Khademi Zare, 2009). MRP system was introduced in the 1950s in US and it had received widespread acceptance in enterprises (Sum, Png, & Yang, 1993). Newman and Sridharan (1992) undertook a comprehensive survey of US companies including machine tools, defense electronics, medical equipment, automobile, plastics, computers, components, and furniture. Their survey results indicated that MRP was the most widely used system for production planning and control (56% of the companies reported using a MRP system).
From the perspective of discrete time scale, dynamic demand, and finite planning horizon, Wagner and Whitin (1958) firstly introduced a standard forward dynamic programming formulation to conduct the discrete lot sizing (DLS) problem in MRP system. Wagner and Whitin’s model could be adopted to get the optimal production plan (PP) in a single-stage environment. Then, in a multi-stage environment, Zangwill (1969) proposed a backward recursive algorithm to find the optimal PP set (PPS) for the DLS problem with a serial production structure (each stage has at most one direct predecessor and one immediate successor). By virtue of the optimal PP or PPS (in terms of a single- or multi-stage version), decision makers are enabled to determine how many quantities have to be produced in which periods at the operation stage.
Since solving the DLS problems is particularly formidable (Bahl, Ritzman, & Gupta, 1987), developing the quantitative approaches for finding an optimal or suboptimal solution efficiently was always a main stream in the past literature. As only one solution can be provided to decision makers, it is the “stationary strategy” for decision making. In practice, however, decision makers desire to know more than an optimal or suboptimal solution, but the flexibility in picking the lot sizes for the DLS problems was given relatively little attention.
The major purposes of this work are to: (1) explore whether an adaptive decision zone as well as a phenomenon of EOQ zone exists in the DLS problems; (2) find the included alternative solutions when the total cost of the optimal solution is changed. The research results are available to fill the gap regarding the flexibility in decision making for the DLS problems and to bring about some interesting research directions in OR/MS.
An adaptive decision zone as well as a phenomenon of the EOQ zone is demonstrated always existing in the DLS problems with single- and multi-stage versions and it is defined as DLS zone. Within the DLS zone, the change in total cost among the optimal solution and a number of alternative solutions is not too much. In other words, giving a small penalty cost from the optimal solution brings several alternative solutions for decision making. Since the developed computerized scheduling system is able to determine the adaptive decision zone and find the included alternative solutions completely, decision makers are enabled to select a preferable solution from the DLS zone even if various penalty costs are taken into account. Therefore, the flexibility in picking the lot sizes in a discrete time inventory problem is significantly enhanced. The further advanced research will extend the proposed method to handle the multi-stage DLS problem with a more complex production structure and to explore whether an adaptive decision zone also exists in the problem.
