مدل تعیین اندازه دسته تولید تطبیق شده : یک ابزار قدرتمند برای تصمیم گیری تدارکات

Starting from the seminal intuitions that led to the developments of the Economic Order Quantity model and of the formulation of the Dynamic Lot Sizing Problem, inventory models have been widely employed in the academic literature and in corporate practice to solve a wide range of theoretical and real-world problems, as, through simple modifications to the original models, it is possible to accommodate and describe a broad set of situations taking place in complex supply chains and logistics systems. The aim of this paper is to highlight, once more, the powerfulness of these seminal contributions by showing how the mathematical formulation of the Capacitated Lot Sizing Problem can be easily adapted to solve some further practical logistics applications (mainly arising in the field of coordination of transportation services) not strictly related to manufacturing and production environment. Mathematical formulations and computational experiences will be provided to support these statements.

The history of inventory problems can be rooted back to the Economic Order Quantity (EOQ) model presented by Harris (1913), also known as the Wilson Lot Size formula, since it was firstly used in practice by Wilson (1934). The EOQ model assumes the presence of a single item whose demand is continuous (with a constant known rate) and an infinite planning horizon. The solution of the model is easy and provides the optimal quantity to be ordered, balancing the setup and inventory holding costs. However, with the same assumptions, in presence of multiple items and capacity restrictions the model becomes NP-hard (Hsu, 1983). The Dynamic Lot Sizing Problem (in the following, generically referred to as DLSP or Lot Sizing), first proposed by Wagner and Whitin (1958), can be considered as an extension of the EOQ model. In this new version, on a discrete time scale, deterministic dynamic demand and finite time horizon are considered while the objective function is the same basic trade-off between setup and inventory costs. Starting from these seminal papers, further variants of the problem have been introduced. These are mainly concerned with the extension to the multi-item case (Barany et al., 1984), the introduction of several conditions about the costs, limitations on production capacities (leading to the Capacitated Lot Sizing Problem, in the following CLSP) (Bitran and Yanasse, 1982) and possible additional features regarding, for instance, demand uncertainty (Brandimarte, 2006), setup costs and/or times (Trigeiro et al., 1989), linked lot sizes (Suerie and Stadtler, 2003), alternative suppliers (Basnet and Leung, 2005). Combinations of these aspects can provide models with very different complexities. Interesting reviews about models and methods to tackle Lot Sizing problems have been published by Kuik et al. (1994), Drexl and Kimms (1997), Karimi et al. (2003), and Jans and Degraeve (2008), while a rich textbook on the topic has been provided by Pochet and Wolsey (2006). Jans and Degraeve (2008) compiled a very interesting and complete survey devoted to describe actual and potential variants of the problem. The authors highlight how most of them are inspired by specific real life applications and, in particular, they focus on a variety of industrial production planning problems. The application of the Lot Sizing model, and its variants, to real-world problems constitute a very active research strand (see, for instance: Rezaei and Davoodi, 2011, Ferreira et al., 2012 and Liao et al., 2012). In this paper we want to show how, through an appropriate interpretation of the elements of the model, Lot Sizing formulations can also be effectively used to face further practical logistic problems, outside of the classical field of production and manufacturing planning. Therefore, rather than providing original models, the aim of the paper is to show how standard formulations can be used to support decisions in other contexts of applications; in this sense, the more established these models are, the more powerful and insightful will be their adaptation, as existing results in terms of formulations and solution approaches can be easily exploited. The remainder of this paper is arranged as follows. In the next section we introduce the mathematical model of the CLSP, considering the single item and the multi-item versions. Then we illustrate a general framework indicating how these models can be used to describe different kinds of logistic problems. In particular three specific examples are introduced and discussed: the optimization of the departure schedule for a bus terminal; the management of a logistic cross-dock platform; and the optimization of an airport check-in gates configuration. For the above problems, we explain how they can be formulated, through few adaptations, starting from the CLSP model. Furthermore, some case studies (related to real-world situations) are presented, showing how these models can be solved in limited computational times and be used as decision support tools. Finally, some concluding remarks and directions for future research are drawn.

Inventory models have been widely employed in the academic literature and in corporate practice to solve a wide range of theoretical and real-world problems, mainly related to production planning and scheduling. However, if we look at Capacitated Lot-Sizing model as a general model of flow control, it is possible to use it to describe and formulate a wide variety of optimization problems. The aim of this paper has been highlighting opportunities of using, through simple adaptations of the basic version, this model to solve some practical logistics applications not strictly related to the manufacturing and production environment. In particular, we have illustrated how three different applications can be effectively formulated through this approach. The application of the implemented models to real-world case studies has shown the possibility of obtaining optimal solutions in reasonable computational times by utilizing a commercial solver. In addition to this aspect, many other advantages can be derived from the use of CLSP models: the possibility of immediately including operational constraints and conditions able to effectively describe real case situations; the availability of a vast developed literature which can be exploited to derive mathematical conditions to describe real-life constraints, to benefit from theoretical results and to implement effective and well-established solution methods (exact and/or heuristic). Future research could consider a more in-depth analysis of the described applications in order to verify the potential of this model framework to reproduce effectively more complex operational aspects that can occur in real cases; nevertheless, new contexts and fields in which the adaptation of this approach can be effective and useful could also be explored.