رقابت تحت تعیین اندازه دسته تولید پویای توانا شده با اکتساب ظرفیت

Lot-sizing and capacity planning are important supply chain decisions, and competition and cooperation affect the performance of these decisions. In this paper, we look into the dynamic lot-sizing and resource competition problem of an industry consisting of multiple firms. A capacity competition model combining the complexity of time-varying demand with cost functions and economies of scale arising from dynamic lot-sizing costs is developed. Each firm can replenish inventory at the beginning of each period in a finite planning horizon. Fixed as well as variable production costs incur for each production setup, along with inventory carrying costs. The individual production lots of each firm are limited by a constant capacity restriction, which is purchased up front for the planning horizon. The capacity can be purchased from a spot market, and the capacity acquisition cost fluctuates with the total capacity demand of all the competing firms. We solve the competition model and establish the existence of a capacity equilibrium over the firms and the associated optimal dynamic lot-sizing plan for each firm under mild conditions.

One of the fundamental problems in operations management is determining the investment in capacity. A firm's capacity determines its maximal potential production. To acquire capacity is usually cost and time consuming, and once the investment is made, the cost is often partially or completely irreversible, as installed capacity is difficult to adjust in the short term. Moreover, the decision on how much capacity to acquire also strongly influences the action space for future operations planning. To invest in too much capacity wastes resources that could be used for other important operation activities, such as new product development and marketing; to invest in too little capacity means long waiting times, missed sales opportunities and lost revenue. Therefore, it is necessary to find an effective and comprehensive method to determine the proper capacity configuration for operations. Increasing the capacity does not necessarily improve the operational performance, even if the product profit margins are large, because capacity acquisition cost is usually negative correlated to the production cost and often affected by the competitive resource environment. In addition, the competitors' other decisions, such as the timing of production and quantity, also affect capacity acquisition cost and investment performance. Game-theoretic modelling has been an effective method of describing and solving competition problems. In this paper, we solve a game-theoretic model of capacity competition problem over a finite-period planning horizon for a multiple-firm industry that uses a common resource to produce its products. For each firm, its best-response problem is a single-item capacity acquisition and lot-sizing problem. The best-response problem considers a single-production facility that produces a single product item to satisfy a deterministic demand stream. The best-response problem for individual firms simultaneously determines an optimal capacity and a lot-size plan over the planning horizon. The capacity acquisition, production and inventory holding costs are considered. We formulate the problem as a cost minimizing Mixed Integer Non-Linear Programming (MINLP) model. This general problem class is impossible to solve using a polynomial time algorithm. Thus, we discretize the possible capacity choices and solve it for each of those. The major difference between the best-response problem and the classical capacitated lot-sizing problems is that the capacity level is an internal decision in our model. Given the capacity competition model, we discuss the capacity equilibrium and associated optimal dynamic lot-sizing plans by analyzing the resulted best-response problem. We introduce an approximation for a firm's best-response function, showing through a numerical study that its use results in only a minor difference to the actual cost figures but still has desirable properties. We then proceed to analyze the competitive problem and show the existence of an equilibrium under modest assumptions. To the best of our knowledge, this is the first study to address lot-sizing problems considering resource competition. Moreover, since the complexity of the capacity competition problem, the approximated solutions are acceptable in practice. The remainder of this paper is organized as follows. We review the relevant studies in Section 2. Section 3 introduces the relevant notation and the basic competitive model. Section 4 first describes the best-response problem that an individual firm faces when making its purchasing and lot-sizing decisions. In Section 5, we show our suggested solution in a structure of the game which results in an equilibrium following a standard procedure. Finally, a computational study and numerical examples are discussed in Section 6.

This paper considers a multiple-firm lot-sizing problem with resource competition. We model and solve the competition game and discuss the equilibrium behaviors of the firms. As a best-response problem of a firm, a typical capacity acquisition and lot-sizing problem is solved by line search. The algorithm solves the capacity acquisition, production, and inventory decisions simultaneously for multiple firms iteratively. In order to tackle the complexity of dynamic lot-sizing problem and potential discontinuity of its cost function, a close approximation is applied to substitute the dynamic lot-sizing cost. Under the mild conditions, we show the existence and uniqueness of equilibrium, and furthermore, the equilibrium converges within finite iterations of computation. In addition, the extension of multiple products share a common resource can be easily adapted into our method by solving the approximation problem of multiple product lot-sizing problem. In the present study, we only consider a rather simple structure of the resource competition and dynamic lot-sizing problem. First, the analysis is limited to the deterministic supply and demand. This leaves the future research opportunities on the capacity acquisition and competition problem under random supply and demand uncertainty. In addition, we only consider a constant capacity setting over the planning horizon. It would also be interesting to analyze the time-varying capacity situation. If the capacity can be purchased or disposed of in each period, it could lead to a solution for a dynamic competition game and the problem would be much more complicated.