مدل موجودی تحت تورم برای وخیم تر کردن آیتم ها همراه با نرخ مصرف وابسته به سهام و کمبود نسبی بازگشت از سیستم

In this paper, we extend Teng, J.T., Chang, H.J., Dye, C.Y., Hung, C.H. [2002. An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. Operations Research Letters 30(6), 387–393.] and Hou, K.L. [2006. An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting. European Journal of Operational Research 168(2), 463–474.] by considering an inventory lot-size model under inflation for deteriorating items with stock-dependent consumption rate when shortages are partial backlogging. The proposed model allows for (1) partial backlogging, (2) time-varying replenishment cycles, and (3) time-varying shortage intervals. Consequently, the proposed model is in a general framework that includes numerous previous models as special cases. We then prove that the optimal replenishment schedule exists uniquely, and provide a good estimate for finding the optimal replenishment number. Furthermore, we briefly discuss some special cases of the proposed model related to previous models. Finally, numerical examples to illustrate the solution process and some managerial implications are provided.

In many real-life situations, for certain types of consumer goods (e.g., fruits, vegetables, donuts, and others), the consumption rate is sometimes influenced by the stock-level. It is usually observed that a large pile of goods on shelf in a supermarket will lead the customer to buy more and then generate higher demand. The consumption rate may go up or down with the on-hand stock level. These phenomena attract many marketing researchers to investigate inventory models related to stock-level. The related analysis on such inventory system with stock-dependent consumption rate was studied by Levin et al. (1972), Baker and Urban (1988), Mandal and Maiti, 1997 and Mandal and Maiti, 1999, Balkhi and Benkherouf (2004), etc. Recently, Alfares (2007) proposed the inventory model with stock-level dependent demand rate and variable holding cost. As a matter of fact, some products (e.g., fruits, vegetables, pharmaceuticals, volatile liquids, and others) deteriorate continuously due to evaporation, obsolescence, spoilage, etc. Ghare and Schrader (1963) first derived an economic order quantity (EOQ) model by assuming exponential decay. Next, Covert and Philip (1973) extended Ghare and Schrader's constant deterioration rate to a two-parameter Weibull distribution. Shah and Jaiswal (1977) and Aggarwal (1978) then discussed the EOQ model with a constant rate of deterioration. Thereafter, Dave and Patel (1981) considered an inventory model for deteriorating items with time-proportional demand when shortages were prohibited. Sachan (1984) further extended the model to allow for shortages. Later, Hariga (1996) generalized the demand pattern to any log-concave function. Teng et al. (1999) and Yang et al. (2001) further generalized the demand function to include any non-negative, continuous function that fluctuates with time. Recently, Goyal and Giri (2001) wrote an excellent survey on the recent trends in modeling of deteriorating inventory since early 1990s. The characteristic of all of the above articles is that the unsatisfied demand (due to shortages) is completely backlogged. However, in reality, demands for foods, medicines, etc. are usually lost during the shortage period. Montgomery et al. (1973) studied both deterministic and stochastic demand inventory models with a mixture of backorder and lost sales. Later, Rosenberg (1979) provided a new analysis of partial backorders. Park (1982) reformulated the cost function and established the solution. Mak (1987) modified the model by incorporating a uniform replenishment rate to determine the optimal production-inventory control policies. For fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging during a shortage period is diminishing with the length of the waiting time. Hence, the longer the waiting time, the smaller the backlogging rate. To reflect this phenomenon, Chang and Dye (1999) developed an inventory model in which the proportion of customers who would like to accept backlogging is the reciprocal of a linear function of the waiting time. Concurrently, Papachristos and Skouri (2000) established a partially backlogged inventory model in which the backlogging rate decreases exponentially as the waiting time increases. Teng et al., 2002 and Teng et al., 2003 then extended the fraction of unsatisfied demand backordered to any decreasing function of the waiting time up to the next replenishment. Teng and Yang (2004) further generalized the partial backlogging EOQ model to allow for time-varying purchase cost. Yang (2005) made a comparison among various partial backlogging inventory lot-size models for deteriorating items on the basis of maximum profit. Lately, Hou (2006) developed an inflation model for deteriorating items with stock-dependent consumption rate and completely backordered shortages by assuming a constant length of replenishment cycles and a constant fraction of the shortage length with respect to the cycle length. San Jose et al. (2006) proposed an inventory system with exponential partial backordering. Recently, Teng et al. (2007) compared two pricing and lot-sizing models for deteriorating items with partial backlogging. Moreover, the effects of inflation and time value of money are vital in practical environment, especially in the developing countries with large scale inflation. Therefore, the effect of inflation and time value of money cannot be ignored in real situations. To relax the assumption of no inflationary effects on costs, Buzacott (1975) and Misra (1975) simultaneously developed an EOQ model with a constant inflation rate for all associated costs. Bierman and Thomas (1977) then proposed an EOQ model under inflation that also incorporated the discount rate. Misra (1979) then extended the EOQ model with different inflation rates for various associated costs. Later, Yang et al. (2001) established various inventory models with time varying demand patterns under inflation. Recently, Chern et al. (2008) proposed partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation. The major assumptions used in the above research articles are summarized in Table 1.

In this paper, a partial backlogging inventory lot-size model for deteriorating items with stock-dependent demand has been proposed. We have shown that not only the optimal replenishment schedule exists uniquely, but also the total profit associated with the inventory system is a concave function of the number of replenishments. We also have simplified the search process by establishing an intuitively good starting value for the optimal number of replenishments. From the numerical results and sensitivity analysis, we have provided several managerial phenomena. The proposed model can be further extended in several ways. For example, we may add pricing strategy into consideration. Also, we could extend the deterministic model into a stochastic model. Finally, we could generalize the model to allow for quantity discounts, trade credits, or others.