یک روش تعیین اندازه دسته تولید اقتصادی معکوس برای استخراج پارامترهای هزینه عرضه کننده کالا

Recent literature on supply chain coordination offers a wide range of game theoretic and optimization approaches that ensure efficient planning in the supply chain, but assume that the involved parties have complete information about each other. However, in reality, complete information is rarely available, and those models alone do not present any incentive for the parties to reveal their private information, e.g., the cost parameters that they use when solving their planning problems. This paper proposes an inverse lot-sizing model for eliciting the cost parameters of a supplier from historic demand vs. optimal delivery lot-size pairs, gathered during repeated earlier encounters. It is assumed that the supplier solves a single-item, multi-period, uncapacitated lot-sizing problem with backlogs to optimality to calculate its lot-sizes, and the buyer is aware of this fact. The inverse lot-sizing problem is reformulated to an inverse shortest path problem, which is, in turn, solved as a linear program. This model is used to compute the ratios of the supplier's cost parameters, i.e., the setup, the holding, and the backlog cost parameters consistent with all the historic samples. The elicited cost parameters can be used as input for various game theoretic or bilevel optimization models for supply chain coordination. Computational experiments on randomly generated problem instances indicate that the approach is very efficient in predicting future supplier actions from the historic records.

Planning inventories in a supply chain necessarily calls for the interaction of autonomous partners operating with distinct, potentially conflicting objectives, different decision mechanism and asymmetric information. Satisfying external demand requires the interaction of these partners in the supply chain. The literature offers a wide spectrum of coordination mechanisms (Albrecht, 2010 and Váncza et al., 2011) based on game theoretic and optimization approaches, which make different assumptions on the information available to the different partners. Nevertheless, there is a considerable gap between the incomplete information models, which usually assume a single encounter of the buyer and the supplier with some well-defined asymmetric information situation, and the complete information models, which consider that the companies are mutually aware of their partners’ decision situation. Namely, in case of repeated encounters, a significant amount of information is hidden in historic records of earlier interactions. These records can contain earlier orders, delivery lot-sizes, or delivery lead times. Furthermore, by the widespread application of tracking and tracing systems (Holmström et al., 2010 and Ilie-Zudor et al., 2011), the buyer can observe even the production lot-sizes and the manufacturing parameters applied by the supplier. Exploiting this information enables a company to use well-informed, e.g., Stackelberg or bilevel optimization approaches for planning its production and logistics, providing a considerable competitive advantage compared to using models with restricted information. In this paper, we tackle the issue of how the historic records of earlier encounters between a buyer and a supplier can be utilized in decision making. We take the stance of the buyer and aim at eliciting the cost parameters of the rational supplier's decision problem. It is assumed that the buyer possesses a historic record of demand vs. delivery lot-size pairs. It is noted that the same approach could be used for eliciting the supplier's cost parameters in its production lot-sizing problem, given that the production lot-sizes are observed. We introduce an inverse combinatorial approach to eliciting the cost parameters of a supplier who determines its delivery periods and quantities by solving a single-item, multi-period, uncapacitated lot-sizing problem with backlogs (ULSB). Specifically, the proposed model computes the ratios of the supplier's setup, holding, and backlog cost parameters. It is noted that eliciting the absolute values of the cost parameters from the above input data is impossible, since the optimal delivery lot-sizes according to the ULSB model are invariant to the multiplication of the cost parameters by a common constant. To the best of our knowledge, this is the first inverse lot-sizing model investigated in the literature. The elicited cost parameters can be useful in various scenarios involving a buyer–supplier relationship. A specific application is the utilization of the elicited cost parameters as inputs to one of the recent Stackelberg or bilevel approaches to lot-sizing in supply chains (Kovács et al., 2013). Such models require the knowledge of the supplier's cost parameters, but the coordination mechanisms themselves do not present any incentive for the supplier to reveal their true values. It is emphasized that in the above applications, eliciting the ratios of the cost parameters is sufficient, since, likewise the ULSB problem, the rational actions of the parties are insensible to the multiplication of the cost parameters by a common constant. Hence, our method can be a precious complement of those supply chain coordination approaches. On the other hand, a shortcoming of the approach is that it cannot compute the absolute values of the cost parameters, which can be an important limitation in other applications, e.g., in price negotiations. In what follows, the related literature is surveyed first (Section 2). Then, the problem is defined formally and the inverse optimization solution method is introduced (Section 3). Next, the results of computational experiments are presented (Section 4), and finally, the paper is concluded with a discussion of the application opportunities and the directions for future research (Section 5).

This paper proposed a novel technique for eliciting a supplier's cost parameters from earlier records of supply request vs. delivery lot-size pairs by using an inverse optimization approach. An inverse ULSB lot-sizing model, and its reformulation to a linear program was introduced. The approach was evaluated in computational experiments, where the cost parameters could be elicited with an accuracy of 5–16%, whereas future supplier actions could be predicted with a success rate of up to 98% from 50 samples, with a planning horizon of 10 time units. We consider that the achieved precision is sufficient in most supply chain applications. The above model can be useful in various scenarios involving a buyer–supplier relationship. Beyond the obvious and general benefit of knowing the partner's cost parameters, the approach enables the use of complete information models to supply chain coordination even when a part of the required information is not explicitly available, but it is encoded in historic records about earlier encounters of the parties. A specific application derives from the numerous recent cooperation mechanism for lot-sizing in supply chains based on Stackelberg games. Such models assume that the buyer knows the cost parameters of the supplier, but the coordination mechanisms themselves do not present any incentive for the supplier to reveal the true values of its cost parameters. Hence, the proposed approach can be regarded as a precious complement of recent game theoretic and bilevel approaches to coordinating supply chains. On the other hand, the approach elicits only the ratios of the different cost parameters, not their absolute values, which can be an important limitation in some applications, e.g., for price negotiations. The most important direction for future work is extending the approach to handle noisy samples, i.e., allowing the supplier to slightly deviate from its optimal ULSB solutions. Another interesting direction is the extension of the model towards richer lot-sizing models. Some extensions, e.g., costs varying over time, can be easily added to the mathematical model, but the resulting high number of free decision variables, and in turn, different feasible solutions, make it complicated to achieve useful results. Other extensions, such as finite capacities, require changing the core mathematical model as well. Our long-term objective is the composition of a portfolio of inverse lot-sizing models, which is applicable to eliciting also the lot-sizing model applied by the supplier. The key idea is performing the elicitation using each model in the portfolio, on samples regarded as noisy. The model that fits with the least noise is accepted as the model applied by the supplier.