An inventory model with generalized type demand, deterioration and backorder rates

Abstract

This study is motivated by the paper of Skouri et al. [Skouri, Konstantaras, Papachristos, Ganas, European Journal of Operational Research 192 (1) (2009) 79–92]. We extend their inventory model from ramp type demand rate and Weibull deterioration rate to arbitrary demand rate and arbitrary deterioration rate in the consideration of partial backorder. We demonstrate that the optimal solution is actually independent of demand. That is, for a finite time horizon, any attempt at tackling targeted inventory models under ramp type or any other types of the demand becomes redundant. Our analytical approach dramatically simplifies the solution procedure.

Introduction

Demand has been always one of the most influential factors in the decisions relating to inventory and production activities. Although various formations of consumption tendency have been studied, such as constant demand (e.g., Padmanabhan and Vrat, 1990, Chu et al., 2004), price dependent demand (e.g., Abad, 1996, Abad, 2001), time-dependent demand (e.g., Resh et al., 1976, Henery, 1979, Sachan, 1984, Dave, 1989, Teng, 1996, Teng et al., 2002) and time-and-price dependent demand (e.g., Wee, 1995).

The ramp type demand has attracted a great deal of interest from researchers. Since Hill (1995), one of the pioneers, developed an inventory model with ramp type demand that begins with a linear increasing demand until to the turning point, denoted as μ, proposed by previous researchers, then it becomes a constant demand. There has been a movement towards developing this type of inventory system for minimum cost and maximum profit problems. For examples, several articles of Mandal and Pal (1998) focused on decay product. Wu et al. (1999) were concerned with backlog rates relative to the waiting time. Wu and Ouyang (2000) tried to build an inventory system under two replenishment policies: starting with shortage or without shortage. Wu (2001) considered the deteriorated items satisfying Weibull distribution. Giri et al. (2003) dealt with more generalized three-parameter Weibull deterioration distribution. Deng (2005) extended the inventory model of Wu et al. (1999) for the situation where the in-stock period is shorter than μ. Manna and Chaudhuri (2006) set up a model where the deterioration is dependent on time. Panda et al. (2007) constructed an inventory model with a comprehensive ramp type demand. Deng et al. (2007) contributed to the revisions of Mandal and Pal, 1998, Wu and Ouyang, 2000. Panda et al. (2008) examined the cyclic deterioration items. Wu et al. (2008) studied the maximum profit problem with the stock-dependent selling rate. They developed two inventory models all related to the conversion of the ramp type demand, and then examined the optimal solution for each case.

Recently, Skouri et al. (2009) developed an order level inventory model for deteriorating items. The model is fairly general in which the demand rate could be any function of time until reaching its stabilization while the backlog rate is any non-increasing function of the customer’s waiting time up to the next replenishment.

Another factor that has an effect on the development of the inventory and production activities is deterioration. Deterioration is viewed by most managers as a trade-off among the policies of inventory level, replenishment point, and backorder. Some examples include Papachristos and Skouri, 2000, Abad, 2001 assuming a time-dependent deterioration rate, and Dye et al. (2005) adopting a general-type deterioration rate and so on.

Along with all above studies that center around different types of demand and deterioration rates, this paper aims at developing a cost-minimum inventory model in the considerations of the general-type demand and deterioration with partial backorder. This paper shows that under the finite time horizon, an optimal solution is independent of demand. In other words, there is a general property that is not only shared for ramp type but also for any kind of demand behaviors. The analysis and results of this presentation thereupon points out that the lengthy discussion of Skouri et al. (2009) in considering the two-domain segments at the turning point of ramp type demand function is superfluous. This paper also introduces an analytical procedure that is not limited to only certain types, such as Weibull, but works for any type of deterioration rates.

The remainder of this paper is organized as follows: Section 2 defines the relative assumptions and notation. Section 3 reviews the results of Skouri et al. (2009). Section 4 provides our proposed inventory model. Section 5 presents a different approach from managerial viewpoint to discuss our findings. Two examples are employed to illustrate our finding in Section 6. Finally, we draw a conclusion in Section 7.