Production, Manufacturing and LogisticsAn inventory model with generalized type demand, deterioration and backorder rates
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.
Section snippets
Assumptions and notation
We generalize the inventory model of Mandal and Pal, 1998, Wu and Ouyang, 2000, Deng et al., 2007, Skouri et al., 2009 with the following notation and assumptions for the deterministic inventory replenishment policy with a general-type demand and backlog rate.
- (1)
T is the finite time horizon under consideration.
- (2)
t1 is the time when the inventory level reaches zero.
- (3)
is the optimal solution for t1.
- (4)
c1 is the cost of each deteriorated item.
- (5)
c2 is the inventory holding cost per unit of time.
- (6)
c3 is the
Review of previous results
Let us directly quote the results from Skouri et al. (2009). Those interested readers may consult Skouri et al. (2009) for the detailed derivation.
Eq. (14) of Skouri et al. (2009) showed that the total cost, TC(t1), takes the formEqs. (15), (16), (21) of Skouri et al. (2009) imply thatwhere D(t1) is the demand rate andTo find the optimal solution,
Our proposed inventory model
We consider the inventory model that starts with stock. This model was first proposed by Hill (1995), and then further investigated by Mandal and Pal, 1998, Wu and Ouyang, 2000, Deng et al., 2007, Skouri et al., 2009. Replenishment occurs at time t = 0 when the inventory level attains its maximum, S. From t = 0 to t1, the inventory level reduces due to both demand and deterioration, with demand rate D(t), and deterioration rate θ(t). At t1, the inventory level reaches zero, then shortage is allowed
Discussion
Now, let us review the solution procedure in Deng et al., 2007, Skouri et al., 2009. They divided the inventory model according to the expression of the demand function into several problems. For example, in Skouri et al. (2009), they assumed two cases: (1) t1 ⩽ μ and (2) t1 ⩾ μ according to the different behaviors of the demand function D(t). However, our findings point out that it is redundant to divide the entire inventory cycle into different cases. Our approach and result will significantly
An alternative approach for our findings
We provide a different approach from managerial point of view to discuss our findings. Suppose there is an item with demand quantity Q that happens at time t(Q). Then there are two replenishment policies: (a) fulfill demand from the stock, or (b) satisfy demand from the backorder. If we decide to fulfill demand from the stock then we need to save at time t = 0. Note that the solution of is . The beginning stock is . After
Conclusion
We extend several previous results with respect to ramp type demand to show the existence and uniqueness of the optimal solution. We prove that the optimal solution is independent of the demand function. It provides an explanation for a previously unexplained phenomenon that occurred in previous studies: why inventory models with different ramp type demand rates end up with the same optimal solution.
Acknowledgments
The author thanks the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 99-2410-H-606-005.
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