Capacitated lot sizing with linked lots for general product structures in job shops

In this paper, we propose a mixed integer programming (MIP) model for a multi-level multi resource capacitated lot sizing and scheduling problem with a set of constraints to track dependent demand balances, that is, the amount left over after allocating the available inventory to the dependent demands. A part of this leftover amount may be kept as a reservation quantity to meet dependent demands of the following period under capacity restrictions. These constraints are necessary because we assume independent demands as well as dependent demands for all items in the product structure. They also are used to tighten the domain of on hand and backorder inventory levels. Although we allow backorders for independent demands only, this is not possible for dependent demands as backorders will disturb the whole demand balance of the product structure. Determination of setup costs is a crucial task when developing lot sizing and scheduling models, especially in a capacitated manufacturing environment with backorders. In this respect, the capacitated lot sizing with linked lot sizes (CLSPL) model we formulate needs not to consider setup costs to avoid unnecessary setups thanks to the new set of constraints, and to obtain feasible lot sizes and schedules. Finally, a numerical example and computational results in a job shop environment are also given, and future research directions are provided. 2009 Elsevier Ltd. All rights reserved. 1. Background and motivation Material Requirements Planning (MRP), developed by Orlicky (1976), is the most popular production planning and scheduling system in practice. MRP provides the right part at the right time for the right customer, i.e., it aims to plan the end item requirements of the master production schedule. First, MRP systems are characterized by their rapid adaptability to dynamic changes in a production/inventory system, and ability to determine the production and inventory requirement several periods in advance. Limitations, for example, are its inability to perform comprehensive capacity planning, using constant and inflated lead times, and its lack of a fluent shop floor extension due to myopic solution methodology (see Pochet & Woolsey, 2006). To address these limitations, it may be necessary to develop an optimization approach to reach the desired goal of simultaneously improving the productivity and flexibility of an MRP system. A comprehensive survey of the different optimization approaches, that were developed in the lot sizing and scheduling literature can be found in Drexl and Kimms (1997). Lot sizing and scheduling literature generally is categorized into two groups, which are the small and big time buckets. Grouping is generally based on specifications of production and the length of planning period. While at most two different items can be produced on a resource in small time bucket models, number of different lots that can be produced is not restricted in big-bucket models. Since at most one setup is allowed in small time bucket models, the sequence of different lots is already known. In other words, small time bucket models solve the lot sizing and scheduling problem together. On the other hand, a primitive form of big time bucket models, which is known as capacitated lot sizing problem (CLSP), cannot answer the scheduling question. However, recent studies on big time buckets try to determine at least the first and the last lots by linking adjacent periods. While periods in small bucket models usually correspond to small time slots such as hours or shifts, big-bucket models deal with planning horizons usually less than 6 months i.e., days, weeks or months (Drexl & Kimms, 1997). The main characteristic of bigbucket models is that they do not restrict the number of items produced in any period. The capacitated lot sizing problem (CLSP) is typical of big-bucket models. It is an extension of the Wagner–Whitin model to integrate capacity limitations. Since the size of a bucket is larger than the setup times, the error due to nonpreservation of the setup state between adjacent periods is negligible. Different forms of the mixed integer programming model for solving the CLSP are surveyed in Karimi, Ghomi, and Wilson (2003). In a manufacturing environment, there are instances where developing a feasible schedule is only possible when setup states 0360-8352/$ see front matter 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.10.002 q This manuscript was processed by Area Editor Mohamad Y. Jaber. * Corresponding author. Tel.: +90 232 4888258; fax: +90 232 4888536. E-mail addresses: [email protected] (C. Oztürk), arslan.ornek@ieu. edu.tr (A.M. Ornek). Computers & Industrial Engineering 58 (2010) 151–164