مشکل تعیین اندازه دسته تولید قدرتمند با خواسته های نا شناخته

The paper deals with a lot sizing problem with ill-known demands modeled by fuzzy intervals whose membership functions are possibility distributions for the values of the uncertain demands. Optimization criteria, in the setting of possibility theory, that lead to choose robust production plans under fuzzy demands are given. Some algorithms for determining optimal robust production plans with respect to the proposed criteria, and for evaluating production plans are provided. Some computational experiments are presented.

Nowadays, companies do not compete as independent entities but as a part of collaborative supply chains. Uncertainty in demands creates a risk in a supply chain as backordering, obsolete inventory due to the bullwhip effect [1]. To reduce this risk two different approaches exist that are considered here. The first approach consists of a collaboration between the customer and the supplier and the second one consists of an integration of uncertainty into a planning process. The collaborative processesmainly aim to reduce a risk in a supply chain [2]. This is done by enforcing a coordination in a supply chain. Two approaches can be applied: vertical and horizontal. The vertical approach is a centralized decision making that synchronizes a supply chain (the most common way to coordinate within companies). The horizontal one refers to the collaborative planning, in which a supply chain can be seen as a chain, where actors are independent entities [3]. The industrial collaborative planning has been standardized for implementing a cooperation between retailers and manufactures. This process is called Collaborative Planning, Forecasting and Replenishment (CPFR) [4]. More precisely, the collaborative processes are usually characterized by a set of point-to-point (customer/supplier) relationships with a partial information sharing [2,5]. In the collaborative supply chain, a procurement plan is built and propagated through a supply chain. Namely, the procurement plan is composed of three horizons: freezing, flexible and free ones [2]. Quantities in the freezing horizon are crisp and cannot be modified, quantities in the flexible horizon are intervals and can be modified under constraints imposed by a previous procurement plan. In the free horizon quantities can be modified without constraints. Another way to reduce a risk in a supply chain is to integrate the uncertainty in a planning process. In the literature, three different sources of uncertainty are distinguished (see [6] for a review): demand, process and supply. These uncertainties are due to difficulties to access to available historical data allowing to determine a probability distribution. In this paper, we focus on the collaborative supply chain (a supply chain, where actors are independent entities) under uncertain demands. In most companies today, especially in aeronautic companies, actors use the Manufacturing Resource Planning (MRPII) to plan their production. MRPII is a planning control process composed of three processes (the production process, the procurement process and the distribution process) and three levels [7]: the strategic level (computing commercial and industrial plans), the tactical level (the Master Production Scheduling (MPS) and the Material Requirement Planning (MRP)) and the operational level (a detailed scheduling and a shop floor control). MRPII have been also extended to take into account: the imprecision on quantities of demands (MPS) [8], the imprecision on quantities of demands and uncertain orders [9] (MRP) and the imprecision on quantities and on dates of demands with uncertain order dates [10] (MRP). In this paper, we wish to investigate the part of the MRPII process. Namely, the procurement process in the tactical level in the collaborative context. Our purpose is to help the decision maker of a procurement service to evaluate a performance of a given procurement plan with ill-known gross requirements and to compute a procurement plan in a collaborative supply chain (with and without supplier capacity sharing due to a procurement contract) with ill-known gross requirements. Several production planning problems have been adapted to the case of fuzzy demands: economic order quantity [11,12], multi-period planning [8–10,13–17], and the problem of supply chain planning (production distribution, centralized supply chain) [18–23]. In the literature, there are two popular families of approaches for coping with fuzzy parameters. In the first family, a defuzzification is first performed and then deterministic optimization methods are used [20,21]. In the second one, the objective is expressed in the setting of possibility theory [24] and credibility theory [25]. We can distinguish: the possibilistic programming (a fuzzy mathematical programming) in which a solution optimizing a criterion based on the possibility measure is built [16,17], the credibility measure based programming in which the credibility measure is used to guaranty a service level (chance constraints on the inventory level) [26] or the goal is to choose a solution that optimizes a criterion based on the credibility measure [13] and a decision support based on the propagation of the uncertainty to the inventory level and backordering level [8–10]. Here, we restrict our attention to uncertainty propagation in MRP (the tactical level) [8–10] and we propose methods both for evaluating a procurement plan in terms of costs under uncertain demands and for computing a procurement plan which minimizes the impact of uncertainty on costs, since the approaches proposed in the literature are not able to do this. Popular setting of problems for hedging against uncertainty of parameters is robust optimization [27]. In the robust optimization setting the uncertainty is modeled by specifying a set of all possible realizations of the parameters called scenarios. No probability distribution in the scenario set is given. The value of each parameter may fall within a given closed interval and the set of scenarios is the Cartesian product of these intervals. Then, in order to choose a solution, two optimization criteria, called the min–max and the min–max regret, can be adopted. Under the min–max criterion, we seek a solution that minimizes the largest cost over all scenarios. Under the min–max regret criterion we wish to find a solution, which minimizes the largest deviation from optimum over all scenarios. In this paper, we are interested in computing a robust procurement plan (with and without delivering capacity of the supplier sharing). The delivering capacity are composed of two bounds: the lower one being the minimal accepted quantity that is sent to the customer and the upper bound which is due to a production capacity of the supplier. Moreover the customer accepts to have backordering but it is more penalized than inventory. This problem is equivalent to the problem of production planning with backordering, more precisely to a certain version of the lot sizing problem (see, e.g. [28,29]), where: the procured quantities are production quantities, a production plan; delivering constraints are production constraints, capacity limits on production plans; and the gross requirements are demands. Thus, the problem consists in finding a production plan that fulfills capacity limits andminimizes the total cost of storage and backordering subject to the conditions of satisfying each demand. It is efficiently solvable when the demands are precisely known (see, e.g. [30–32]). However, the demands are seldom precisely known in advance and the uncertainty must be taken into account. In this paper, we consider the above problem with uncertain demands modeled by fuzzy intervals. The membership function of a fuzzy interval is a possibility distribution describing, for each value of the demand, the extent to which it is a possible value. In other words, it means that the value of this demand belongs to a -cut of the fuzzy interval with the degree of necessity (confidence) 1 − . To evaluate a production plan, we assign to it, degrees of possibility and necessity that its cost does not exceed a given threshold and a degree of necessity that costs of the plan fall within a given fuzzy goal. In order to find “robust solutions” under fuzzy demands, we apply two criteria. The first one consists in choosing a production plan which maximizes the degree of necessity (certainty) that its cost does not exceed a given threshold. The second criterion is weaker than the first one and consists in choosing a plan with the maximum degree of necessity that costs of the plan fall within a given fuzzy goal. A similar criterion has been proposed in [33] for discrete optimization problems with fuzzy costs. We provide some methods for finding a robust production plan with respect to the proposed criteria as well as for evaluating a given production plan under fuzzy-valued demands which heavily rely on methods for finding a robust production plan, called optimal robust production plan, in the problem of production planning under interval-valued demands with the robust min–max criterion. Namely, it turns out that the considered fuzzy problems can be reduced to examining a family of the interval problems with the min–max criterion. Therefore, we generalize in this way the min–max criterion under the interval structure of uncertainty to the fuzzy case. The paper is organized as follows. In Section 2, we recall some notions of possibility theory. In Section 3, we present a lot-size problem with backorders and precise demands. In Section 4, we present our results. Namely, we investigate the interval case, that is the lot-size problem with backorders in which uncertain demands are specified as closed intervals. We construct algorithms for finding an optimal robust production plan (a polynomial algorithm for the case without capacity limits and an iterative algorithm for the case with capacity limits) and for evaluating a given production plan (linear and mixed integer programming methods, a pseudo-polynomial algorithm). An experimental evidence of the efficiency of the proposed algorithms is provided. In Section 5, we extend our results from the previous section to the fuzzy case. We study the lot-size problem with backorders with uncertain demands modeled by fuzzy intervals in a setting of possibility theory. We provide methods for seeking a robust production plan with respect to two proposed criteria as well as for evaluating a given production plan under fuzzy-valued demands (the methods heavily rely on the ones from the interval case). The efficiency of the methods is confirmed experimentally.

In this paper, we have proposed methods to compute a robust procurement plan in the collaborative supply chain, where the customer uses a version of MRP with ill-known demands to plan a production. This problem is a certain version of the lot sizing problem with ill-known demands modeled by fuzzy intervals, whose membership functions are regarded as possibility distributions for the values of the unknown demands.We have introduced, in this setting, the degrees of possibility and necessity that the cost of a plan does not exceed a given threshold and a degree of necessity that costs of a plan fall within a given fuzzy goal, which allows us to evaluate a given production plan. Moreover, we have provided methods for computing these degrees. For finding robust production plans under fuzzy demands, we have proposed two criteria: the first one consists in choosing a production plan which maximizes the degree of necessity that its cost does not exceed a given threshold, the second criterion is softer than the first one and consists in choosing a plan with the maximum degree of necessity that costs of the plan fall within a given fuzzy goal. We have constructed the algorithms for determining optimal robust production plans with respect to the criteria and confirmed their efficiency experimentally. The criteria are a generalization, to the fuzzy case, of the known from the literature the min–max criterion. Consequently, we have shown in the paper that there exists a link between interval uncertainty with the min–max criterion and possibilistic uncertainty with the necessity based criteria. It turns out that the evaluation of a production plan and choosing a plan in the fuzzy-valued problem are not harder than in the interval-valued case. The difficulty of solving the fuzzy problems lies in the interval case, since it is reduced to solving a small number of interval problems. Therefore, we have discussed first the interval-valued case. In this case, we have considered the problem of determining the optimal interval of possible costs of a production plan, which allowed us to evaluate the plan. Determining the optimal bounds of the interval boils down to computing optimistic and worst case scenarios.We have proposed linear programming based method for computing an optimistic scenario and mixed integer programming and dynamic programming methods for computing a worst case scenario. We have also identified a polynomial solvable case. For computing an optimal robust production plan, we have provided a polynomial algorithm and iterative one for the cases: with no capacity limits and with capacity limits, respectively. Then we have extended the methods introduced for the interval-valued problem to the fuzzy-valued one. There is still an open question concerning the complexity status of computing a worst case scenario of a given production plan. The problem is pseudo-polynomially solvable and polynomially solvable under certain assumptions and seems to be a core of most of the problems considered in the paper. These assumptions are nearly realistic and make possible extension of our approach to the case where a procurement plan is given for a family of product. In other words, when the sum of quantities procured has to respect supplier capacity constraints which are computed from a previous procurement plan. This problem is equivalent to the multi-item capacitated lot sizing problem. The fact that the complexity status is still open creates the possibility to find a polynomial algorithm and to extend our approach to the multi-item, multi-level capacitated lot sizing problem without the assumptions. So, it is an interesting topic of further research.