سیاست های قیمت گذاری و تعیین اندازه دسته تولید برای موارد رو به وخامت گذاشته با پشتیبانی جزئی تحت شرایط تورم

In this paper, we develop an inventory lot-size model for deteriorating items under inflation using a discounted cash flow (DCF) approach over a finite planning horizon. We allow not only a multivariate demand function of price and time but also partial backlogging. In addition, selling price is allowed for periodical upward and downward adjustments. The objective is to find the optimal lot size and periodic pricing strategies so that the net present value of total profit could be maximized. By using the properties derived from this paper and the Nelder–Mead algorithm, we provide a complete search procedure to find the optimal selling price, replenishment number and replenishment timing for the proposed model. At last, numerical examples are used to illustrate the algorithm.

In the past 30 years, analysis of the inventory model allowing the constant demand rate to vary with time over a finite time horizon has extended the field of inventory control in practice. In the earlier period, researchers had discussed different demand patterns fitting the stage of product life cycle. Resh et al., 1976 and Donaldson, 1977 considered the situation of linearly time-varying demand and established an algorithm to determine the optimal number of replenishments and timing. Barbosa and Friedman, 1978 and Barbosa and Friedman, 1979 adjusted the lot-size problem for the cases of different increasing power-form demand functions and declining demand patterns, respectively. Henery (1979) further generalized the demand rate by considering a log-concave demand function increasing with time. Following the approach of Donaldson, Dave (1989) developed an exact replenishment policy for an inventory model with shortages. Yang, Teng, and Chern (2002) extended Barbosa and Friedman’s (1978) model to allow for shortages. To characterize the more practical situation, Chen et al., 2007a and Chen et al., 2007b dealt with the inventory model under the demand function following the product-life-cycle shape. They employed the Nelder–Mead algorithm to solve the mixed-integer nonlinear programming problem and determined the optimal number of replenishment and the optimal replenishment time points. In real life, the deterioration phenomenon is observed on inventory items such as fruits, vegetables, pharmaceuticals, volatile liquids, and others. By considering this phenomenon occurring during the holding period, Dave and Patel (1981) integrated time-proportional demand and a constant rate of deterioration in the inventory model where shortages were prohibited over the finite planning horizon. Sachan (1984) further extended the model to allow for shortages. Later, Hariga (1996) generalized the demand pattern to any log-concave function. Researchers including Murdeshwar, 1988, Goswami and Chaudhuri, 1991, Goyal et al., 1992, Benkherouf, 1995, Benkherouf, 1998, Chakrabarti and Chaudhuri, 1997 and Hariga and Al-Alyan, 1997 developed economic order quantity models that focused on deteriorating items with time-varying demand and shortages. However, the above inventory models unrealistically assume that during stockout period all demand is either backlogged or lost. In reality, some customers are willing to wait until replenishment, especially when the waiting period is short, while others are more impatient and go elsewhere. To reflect this phenomenon, Abad (1996) provided two sets of time-proportional backlogging rates: (i) linear time-proportional backlogging rate and (ii) exponential time-proportional backlogging rate. Chang and Dye (1999) developed a finite time horizon EOQ 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 by assuming that the backlogging rate decreases exponentially as the waiting time increases. Recently, Teng, Chang, Dye, and Hung (2002) and Chern, Yang, Teng, and Papachristos (2008) extended the fraction of unsatisfied demand backordered to any decreasing function of the waiting time up to the next replenishment. Since price is viewed as an important vehicle to influence demand in most of the business environment, many researchers are led to investigate inventory models with a price-dependent demand. Cohen (1977) jointly determined the optimal replenishment cycle and price for deteriorating items with the demand rate dependent linearly on the selling price. Abad, 1996, Abad, 2001, Abad, 2003 and Abad, 2008 incorporated the demand rate described by any convex decreasing function of the selling price into the inventory model, taking a general rate of deterioration and a variable backlogging rate. Ho et al., 2008 and Chang et al., 2009 presented the iso-elastic demand in an integrated supplier–buyer inventory model under the condition of a permissible delay in payment, respectively. The aforementioned studies assume firms have no pricing power to change the selling price periodically and adopt the fixed price policy. As opposed to the conventional fixed price policy, Chen and Chen (2004) presented an inventory model for deteriorating items with a multivariate demand function of price and time, taking account of the effects of inflation and time discounting over multiperiod planning horizon. However, the integer length of replenishment cycle is within certain limits due to the procedure using dynamic programming techniques. It is noted that the literature about a finite time horizon inventory model rarely considers the cases with periodic adjustments of price. In this paper, we investigate the replenishment policies for deteriorating items with partial backlogging by considering a multivariate demand function of price and time. The fraction of unsatisfied demand backordered is any decreasing function of the waiting time up to the next replenishment. In addition, the selling price is allowed periodical upward and downward adjustments and the time value of money is taken into consideration. The objective of the inventory problem here is to determine the number of replenishments, the selling price per replenishment cycle, the timing of the reorder points and the shortage points. Following the properties derived from this paper, we provide a complete search procedure to find the optimal solutions by employing the Nelder–Mead algorithm. Several numerical examples are used to illustrate the features of the proposed model. At last, we make a summary and provide some suggestions for future research.

In this paper, we properly extend the fixed price policy to change selling prices upward or downward periodically. We allow not only for the multivariate demand function of price and time but also general partial backlogging rate. For the mixed-integer nonlinear programming problem with 3n decision variables, the compute of the approximately accurate estimate for the optimal number of replenishment significantly reduces computational complexity. Further, a complete search procedure is provided to find optimal selling prices and replenishment schedule by employing the Nelder–Mead algorithm. The proposed algorithm help us determine periodic selling prices and the optimal replenishment schedule for the cases of demand patterns in the different stage of the product life cycle. Therefore, this lot-size model can be viewed as an extension of numerous previous models, such as Hariga, 1996, Teng et al., 2002, Yang et al., 2002, Chen and Chen, 2004, Chen et al., 2007a, Chen et al., 2007b and Chern et al., 2008. The proposed model can be extended in several ways. For instance, we may consider the permissible delay in payments. Also, we could extend the deterministic demand function to stock-dependent demand patterns. Finally, we could generalize the model to allow for quantity discounts, finite capacity and others.