مدل های موجودی کار انباشته دو انبار با مشتقات جزئی همراه با سه پارامتر وخامت توزیع وایپول تحت تورم

In many inventory systems, the effects of deterioration of physical goods have drawn much attention of various researchers in recent years. The more the deterioration is, the more the order quantity would be. According to such consideration, taking the deterioration rate into account is necessary. Thus, in this paper, I develop the two-warehouse partial backlogging inventory model introduced in Yang (2006) to incorporate three-parameter Weibull deterioration distribution. The objective is to derive the optimal replenishment policy that minimizes the net present value of total relevant cost per unit time. Two alternative models are compared based on the minimum cost approach. The study shows that the optimal solution not only exists but also is unique. Model 2 is still less expensive to operate than Model 1 in the proposed model. Finally, some numerical examples and sensitivity analysis on parameters are made to validate the results of the proposed inventory system.

In general, a certain product deteriorates with time varying during the normal storage period, such as seasonal food, vegetables, fruit and others. Thus, to control and maintain the inventory of deteriorating items to satisfy customer's demand or retailer's order is important. In the past few decades, the analysis of deteriorating inventory had been attracted by many researchers' attention. Ghare and Shrader (1963) were the first to propose the inventory model where the factor of deterioration considered was significantly affected. Later, there are many papers presented in the analysis of deteriorating inventory, such as Dave and Patel (1981), Hariga (1996), Teng et al. (1999), and others. Goyal and Giri (2001) presented a review of the inventory literature for deteriorating items since the early 1990s. Recently, there still has many articles explored for the deterioration inventory model by taking different affecting factors into account, such as quantity discount, pricing, partial backlogging, etc. In practical, the deterioration rate for some items, such as steel, glassware hardware and toys, is so low that there is almost no need to consider deterioration in determining the inventory lot-size. However, some items deteriorated with age, such as the leakage failure of batteries, life expectancy of drugs, etc., the longer the items remained unused, the higher the rate at which they failed. Covert and Philip (1973) developed an inventory model for deteriorating items with variable rate of deterioration. They used the two-parameter Weibull distribution to represent the deterioration distribution. Misra (1975) also used the two-parameter Weibull distribution to fit the deterioration rate for production lot-size model. Wee et al. (2005) proposed a two-warehouse inventory model for deteriorating item with two-parameter Weibull distribution and constant partial backlogging rate under inflation. Lo et al. (2007) proposed an integrated production-inventory model with imperfect production processes under inflation and the product deteriorated with two-parameter Weibull distribution. Recently, Skouri and Konstantaras (2009) considered the order level inventory models for deteriorating seasonable/fashionable products with two-parameter Weibull deterioration rate, time dependent demand and shortages. Concurrently, Skouri et al. (2009) considered an inventory model with general ramp-type demand rate, two-parameter Weibull deterioration rate and partial backlogging following two different replenishment policies. However, the two-parameter Weibull distribution may not be always applicable in the real life. Some of the items begin to deteriorate only after a period of time in storage, not at the initial. Thus, a three-parameter Weibull distribution is more practical. Philip (1974) generalized the model proposed by Covert and Philip (1973) with three-parameter Weibull distribution. Chakrabarty et al. (1998) extended Philip's model with shortages and linear trend demand. Giri et al. (2003) developed an EOQ model with a three-parameter Weibull deterioration distribution, shortages and ramp-type demand. The major assumptions mentioned in the selected articles are summarized in Table 1. Table 1. Major characteristic of inventory models on selected articles. Author(s) and published (year) Demand rate Single/Two warehouse Deterioration rate Allow for shortages With partial backlogging Under inflation Chakrabarty et al. (1998) Linear demand Single warehouse Three-parameter Weibull distribution Yes No No Covert and Philip (1973) Constant Single warehouse Two-parameter Weibull distribution No No No Dave and Patel (1981) Time proportional Single warehouse Constant No No No Ghare and Shrader (1963) Constant Single warehouse Constant No No No Giri et al. (2003) Ramp-type demand Single warehouse Three-parameter Weibull distribution Yes No No Hariga (1996) Time varying (log concave) Single warehouse Constant Yes No No Lo et al. (2007) Constant Single warehouse Two-parameter Weibull distribution Yes Yes Yes Misra (1975) Constant Single warehouse Two-parameter Weibull distribution No No No Philip (1974) Constant Single warehouse Three-parameter Weibull distribution No No No Skouri and Konstantaras (2009) Time dependent Single warehouse Two-parameter Weibull distribution Yes Yes No Skouri et al. (2009) Ramp-type Single warehouse Two-parameter Weibull distribution Yes Yes No Teng et al. (1999) Time varying Single warehouse Constant Yes No No Wee et al. (2005) Constant Two warehouses Two-parameter Weibull distribution Yes Yes Yes Yang (2006) Constant Two warehouses Constant Yes Yes Yes Present paper (2010) Constant Two warehouses Three-parameter Weibull distribution Yes Yes Yes Table options Note that (1) if the location parameter is zero, then the three-parameter Weibull distribution is reduced to be a two-parameter Weibull distribution. (2) If the shape parameter is equal to 1, then the Weibull distribution is reduced to be an exponential form. Thus, the three-parameter Weibull distribution is a generalized distribution to be considered. Therefore, I here consider the models which introduced in Yang (2006) to develop a two-warehouse partial backlogging inventory model under inflation in which the deterioration rate is a three-parameter Weibull distribution. The deterioration distribution in the two warehouses is independent. Two alternatives are compared based on the minimum cost approach. The study shows that the optimal solution not only exists but also is unique. In general, Model 2 is still less expensive to operate than Model 1 in the proposed model. Finally, some numerical examples for illustration and sensitivity analysis on parameters are provided.

In this paper, a two-warehouse partial backlogging inventory model in which three-parameter Weibull distribution deterioration under inflation is proposed. The partial backlogging rate is time varying and the deterioration distribution in the two warehouses is independent. The study shows that the optimal solution not only exists but also is unique. It also reveals that Model 2 is still less expensive to operate than Model 1 in the proposed model. Moreover, the percentage saved is much affected by the deterioration distribution, especially by the scale parameter in OW and shape parameter in RW. The proposed model can be extended in numerous ways. For example, we may extend the constant demand to a more generalized demand pattern that fluctuates with time or stock-dependent demand rate. Also, we could extend the model to incorporate some more realistic features, such as quantity discount or the related costs are fluctuating with time.