Capacitated Lot Sizing Problem (CLSP)

The capacitated lot sizing problem (CLSP) is one of the most important and most difficult problems in production planning (Karimi et al. 2003). The capacitated lot sizing problem was considered to be NP hard by Florian and Klein (1971) and Bitran and Yanesse (1982). While the CLSP has been considered NP hard several algorithms and other solution methods have been developed to address the production planning problem. For example Chen and Thizy (1990) prove that lagrangian relaxation can be used to solve the problem. They also formulated and solve the multi-item capacitated lot size problem using the shortest path formulation. Since then several algorithms have been developed to solve the problem in less time. Karimi et al. (2003) have performed an extensive literature review of the models and algorithms related to the CLSP and should be referred to for further explanation of the models and algorithms.

The CLSP has been considered with bounded inventories with direction application to the warehouse environment. Love (1972) was the first to examine an inventory bounded problem. In the paper a single product facility dynamic lot size

problem with bounded inventory levels was examined and an algorithm was developed to solve the problem.

Page and Paul (1976) considered the problem of maintaining inventory for

multiple products when there is a restriction on the maximum inventory investment or on the maximum amount of warehouse space. In order to solve the problem they used an equal order interval method and found that this method was significantly better than the well known method of lagrangian multipliers. While Page and Paul (1976) did not use a

37 CLSP, they applied the idea of bounded inventory to the warehouse setting using the EOQ production planning model.

Gutierrez et al. (2002) examined a relevant class of production inventory systems when the inventory levels were bounded. They solved a dynamic lot size problem in which the order quantities were restricted by the warehouse capacity. Demand was known, shortages were not permitted, and inventory levels could exceed storage capacity. The model was previously solved by Love (1973), however they different algorithm to solve the problem, which reduced the amount of required computational time.

Sedeno-Noda et al. (2004) examined a dynamic lot sizing problem in which the inventory levels were restricted by the warehouse capacities. It was assumed that the demand was known and shortages were not allowed. An O (TlogT) algorithm was used to solve the model.

Chu and Chu (2008) examined the single item dynamic lot sizing model with bounded inventory and outsourcing, where production capacity was assumed to be unlimited. The inventory was bounded by the storage capacity of the warehouse. Their model with the addition of the bounded inventory and outsourcing had industrial

application; they used a strongly polynomial algorithm to optimally solve the problem. Lui and Tu (2008) examined a production planning problem where the inventory capacity was the limiting factor. The problem was further complicated by the inventory capacity being constantly bounded, not allowing backlogging, and stating that production and stock out costs were not decreasing. They formulated the model and developed an algorithm that was considered to be O (T2).

38 Minner (2009) analyzed the replenishment of multiple products to satisfy dynamic demand when the warehouse capacity or the available inventory budget was limited. A savings based heuristic was suggested for the warehouse scheduling problem and three simple approaches to the replenishment of multiple products with dynamic demand and limited warehouse capacity. These results were also compared with a mixed integer programming solver.

The capacitated lot sizing problem with bounded inventory has also been extended to include multiple products. Absi and Kedad-Sidhoum (2008) examined the multi-item capacitated lot sizing problem with setup time and shortages. The problem was to minimize demand shortages, the setup, the inventory and the production costs. The formulation was an extension of the CLSP with the addition of parameters and variables to minimize setup times and shortages costs. This model was also capacitated by bounded inventory.

Hariga and Jackson (1996) presented the warehouse scheduling problem (WSP) which was a multi-item warehouse problem that was limited by the floor space within the warehouse. The focus of the paper was to optimize space utilization, through order sizes and delivery scheduling. A cyclic schedule was made that minimized the long run average inventory and order costs per unit time without violating a warehouse space capacity constraint.

While all of these papers have demonstrated that the capacitated lot sizing problem can be applied to a warehouse none of them have combined storage assignment for the reduction of storage and handling costs to the model. A reason for this could be

39 that the problem was considered NP hard. However it has been found in literature that the problem can be formed and solved as a Mixed Integer Program (MIP).

Rizk et al. (2006) examined a multi-item lot sizing problem with dynamic demand and formulated it as a mixed integer program to plan the supply of a family of items under piecewise linear resource costs. The problem was decomposed using lagrangian relaxation and a lagrangian heuristic was used to solve the problem.

Pochet and Wolsey (1991) solved the multi-item lot sizing problem as a mixed integer program. The paper aimed to show some slow but significant progress in solving a variety of multi-item lot sizing problems. They examined the capacitated and

uncapacitated problem and solution approaches for each. They concluded that the problem with both individual and joint production capacity constraints as well as problems with machine start up costs can be solved efficiently as mixed integer programs.

Pochet and Wolsey (2006) later went on to write a book on production planning by mixed integer programming where they explain how to formulate the uncapacitated and capacitated lot sizing problem as a mixed integer program and described

computational software, and algorithms that can solve these models.