مدل برنامه ریزی تولید یکپارچه برای قالب و آیتم های انتهایی

This study develops a mathematical modelling framework for simultaneously generating production plans for molds and the end items that are made with them. The inputs considered are the item demand (assumed constant over an infinite planning horizon), holding costs and shortage costs, together with the molds’ statistical lifetime distribution (in terms of number of uses) and costs pertaining to amortization, preventive replacements and corrective replacements.

A company producing souvenirs and ornaments (hereafter called end items) out of a semi-precious metal wanted to improve its production planning capability, but doing so was hampered by the fact that the rubber molds in which the molten metal is poured are subject to random failure. Numerous reasons account for the variability in the molds’ lifetime, some of which are labor-related (for instance the care with which a mold is opened and an item is released), and others that are due to factors such as quality of raw materials and pre-heat temperature. This paper proposes a nonlinear stochastic optimization framework for jointly establishing, under the assumption of a single item with constant demand over an infinite planning horizon, the production rates of the items and the molds. The objective function to be minimized is comprised of mold amortization costs, preventive replacement costs and failure replacement costs as well as item holding costs and shortage costs. Although mold shortages may also occur, the resulting costs are assumed to be entirely reflected in the ensuing shortages of end items. Note that the decision structure considered in this paper is traditionally split up, both in practice and in the literature, with different objectives in mind. That is, a production plan for end items would be hatched in view of fulfilling customer orders, following which a production plan for molds would be devised considering the mold availability and the end item production plan. By contrast, a mathematical programming model integrating both decisions is developed in Section 2, with concluding remarks given in Section 3. We now give a brief literature review from the combined production and maintenance realm. Van der Dyun Schouten and Vanneste (1995) proposed a preventive maintenance policy based on the age of the machine and the capacity of a buffer stock for a production line consisting of two machines. Meller and Kim (1996) studied the impact of preventive maintenance on a system with two machines and a fixed buffer stock capacity between the machines. Srinivasan and Lee (1996) analyzed the combined effects of preventive maintenance and production/inventory policies on the operating costs of the production unit. For their part, Cheung and Hausmann (1997) proposed the simultaneous optimization of stock and an age-type maintenance policy. Boukas and Haurie (1990), Gharbi and Kenne (2000) and Kenne and Gharbi (2001) consider the ordering of the production flow and preventive maintenance by using a Markov model. Sarker and Haque (2000) carried out a simulation of a production system with a random failure rate in order to analyze jointly spare parts provisioning and maintenance. Iravani and Duenyas (2002) presented an integrated maintenance/repair and production/inventory model using a Markov decision process. Rezg et al. (2004) presented the joint optimization of preventive maintenance and stock control on a production line made up of N machines. Kenne and Gharbi (2004) studied the stochastic optimization of a problem of production control with corrective and preventive maintenance. They propose a method to find the optimal age of preventive maintenance and production rates for a production system composed of identical machines. In Aghezzaf et al. (2007) the objective is to determine an integrated production and maintenance plan that minimizes the expected total production and maintenance costs over a finite planning horizon. Meantime, Panagiotidou and Tagaras (2006) developed a model for the optimization of preventive maintenance procedures in a production process that may operate in one of two different quality states and is also subject to failure. Cormier and Rezg (2007) examined a somewhat similar problem to that studied here, but estimated the cost per period by means of simulation for a set of production rates. Based on this output, a cost function was then formulated via experimental design, which finally allowed an optimum to be determined analytically. This method is less precise than the one presented in the present paper, but has the advantage of yielding results even when the (probability density function) pdf of the mold's lifetime is not known analytically. Let us now proceed with the mathematical formulation for the problem at hand.

The purpose of this paper was first to define a new applied research area, namely, simultaneously finding the optimal production rates for end items and the molds with which they are made. It is assumed that the single item demand is constant over an infinite planning horizon and that the molds are characterized in part by a statistical lifetime distribution. Total costs to be minimized include mold amortization costs, preventive replacement costs and failure replacement costs as well as item holding costs and shortage costs. The resulting nonlinear stochastic programming model draws upon results from the age-based preventive maintenance literature. A simplified version of the above optimization model, as depicted in Fig. 1, where it is assumed that View the MathML sourceL-x¯R(N)<T<L, was implemented in Mathcad (the simplification allows the “min” on the integral bounds in (6) and (7) to be resolved beforehand and (8) to be re-written as D(L−T)/L). However, it is recommended that a global optimization procedure be used for solving (1)–(9). Extending this study to a multi-item, multi-mold, multi-period environment would require the introduction of three variable indexes. The recent study by Aghezzaf et al. (2007), which considered lot sizing in a capacitated production system subject to random failure, hints as to the challenges involved. It appears that heuristics would be required for situations with many items, molds, and/or periods, see e.g., Silver et al. (1998, pp. 341–343).