Trade-off between capacity invested and inventory needed

Abstract

A multi-item capacitated make-to-order production system with considerable demand fluctuations is discussed. The relationship between the available capacity and the inventory needed to meet customer requirements with a pre-defined service level is modeled. Furthermore, the total cost for both capacity and inventory is minimized and it is shown that, assuming negligible change-over times, the double of the surplus inventory cost has to be equal to the excess capacity cost to ensure minimum total cost.

Introduction

According to the hierarchical planning approach, see Hax and Meal (1975), strategic and long-term decisions are made by top management whereas operational and short term tasks are performed by lower-level management. In general, capacity investment decisions are considered as strategic and inventory management as operational tasks. Practice in enterprises as well as most models in the literature view capacity investment decisions separately to the inventory needed to meet customer requirements with restricted available capacity.

In this paper, a relationship between the available capacity and the inventory needed to fulfill the customer orders with a required service level is developed. It is shown that there is a considerable impact of the available capacity (investment decision) on the inventory needed. Furthermore, the optimum capital invested in capacity is determined by minimizing the total cost for capacity as well as cost for capital invested in the required inventory. A make-to-order (MTO) production system under dynamic customer demand with considerable fluctuations is considered. Furthermore, the buying behavior of the customers is modeled by the statistical distribution of the customer required delivery lead time. The multi-item capacitated production model is based on a Fixed Order Period lot-sizing policy and production orders are released if the remaining time to due date is shorter than the work ahead window. Two strategies for the management of short term demand peaks are compared. The first strategy involves dedicating some excess capacity to expected future demand peaks while the second requires pre-producing on stock known customer orders with due dates far in the future.

The research objective of the paper is twofold:

• development of a function with respect to available capacity, describing the inventory needed to meet the customer requirement with a pre-defined service level

• development of a model to minimize the total costs for capacity as well as costs for capital invested in inventory.

The related literature focuses mainly on capacity and inventory management. A good survey of capacity expansion and capacity management literature is given in Luss (1982) and more recently in Van Mieghem (2003). For a good review of inventory management see Silver et al. (1998). In the literature references below, models which combine the capacity and the inventory are discussed.

Bish et al. (2005) investigated a multi-period two-stage supply chain, comprising two uncapacitated suppliers and two capacitated plants over an infinite horizon. They studied the performance measured in sales, production variability, variability propagating upstream, component inventory and outbound distribution assuming an MTO environment with lost sales under several capacity allocation policies to manage short-term order variability.

Bradley and Glynn (2002) considered a joint optimization of capacity and inventory decisions in a single-product, single-stage, single server, produce-to-stock and limited capacity manufacturing model with the objective of minimizing the long-run average operating cost due to penalty, holding and capacity costs. This was one of the first research contributions addressing the joint decision of capacity and inventory. Angelus and Porteus (2002) as well as Van Mieghem and Rudi (2002) also addressed the joint (inventory against capacity investment) decision problem.

Raman and Kim (2002) stated that the error induced by ignoring the impact of inventory holding costs can be substantial. In their model with a high gross margin, unpredictable demand and high obsolescence risk, it was demonstrated that the reduction of inventory enabled by a higher flexible capacity invested can lead to a reduction of the sum of capacity and inventory cost.

Van Mieghem (1998) discussed optimal investment decisions on capacity at the strategic level by discussing the trade-off between revenue and capital investment costs. He stated that the greatest increase in revenue can be generated by flexible capacity when demands are negatively correlated.

Rajagopalan and Swaminathan (2001) explored the interaction between production planning and capacity acquisition in a multi-item and multi-period environment with known, varying and long-term growing demand. They discussed the trade-off between early investment (use of excess capacity for smaller lot-sizes to reduce the inventory) and later investment (use of excess capacity for building up inventories to meet the demand growth with postponed machine purchase).

Mincsovics et al. (2009) discuss a production system with a certain permanent capacity and contingent capacity to meet non-stationary stochastic demand in an MTS production system. They develop a model to economically evaluate the balance between inventory, permanent capacity and contingent capacity. They show that the value of flexibility decreases with an increasing capacity acquisition leadtime.

Zhang et al. (2004) considered a discrete-time capacity expansion problem involving multiple product families, multiple machine types and non-stationary stochastic demand with no finished goods inventory and backorders. Their objective was to minimize the sum of capacity investment and the cost of lost sales.

A framework for the modeling and analysis of MTO, MTS and delay product differentiation (DD) is developed by Gupta and Benjaafar (2004). They discuss a model for minimization of inventory costs subject to a service level constraint in a multi-product production system. Based on a certain accepted customer waiting time on which the service level constraint is based, the inventory holding costs are evaluated for an MTS, MTO and DD production systems. No FGI is held in this model based on the assumption that a customer does not require a certain leadtime but is satisfied whenever the real customer waiting time is shorter than the accepted customer waiting time.

The remainder of this article is organized as follows. In the next section the model, which is based on four steps, is introduced. These steps are customer required capacity determination, describing the buying behavior of the customers by the order characteristic, calculating the inventory needed and finally formulating as well as solving the cost minimization problem. In Section 3 numerical illustrations and comparison of four different settings are presented. Section 4 concludes and all proofs are summarized in the Appendix A.