سیاست های موجودی مطلوب برای یک کالای فاسد شدنی با تابع تقاضای حساس به قیمت و زمان

We formulate a model for determining the optimal pricing, order quantity and replenishment period for perishable items with price-dependent and time-dependent demand. The items have a fixed shelf-life, and the demand rate decreases linearly in the selling price and polynomially over the time after replenishment, until it vanishes either at the reservation price or at expiration time. We prove that the three-variable profit maximization problem can be reduced into a single-variable problem, in which the variable is the duration of the replenishment period. We show that the profit function is strictly pseudo-concave and provide means of obtaining the optimal policy. Three numerical examples are presented to demonstrate the model accompanied by a sensitivity analysis.

The primarily negative relationship between the price of a product and the quantity demanded is well established (see Kocabiyikoglu and Popescu (2011) for examples of some common price-dependent demand functions). Yet, in addition to price, there are several other marketing-related factors that may affect the demand for a product at a given time (hereafter referred to as the “demand rate”). Bose et al. (1995) developed a deterministic model for an item whose demand follows a linear trend over time. Wu et al. (2006), Chang et al. (2010) and Devangan et al. (2013), following Baker and Urban (1988), considered also the influence of the quantity on-hand on retail product demand (an effect associated with the impact of shelf-space allocation). Avinadav and Arponen (2009) suggested a demand rate that is positively affected by the remaining shelf-life of the product, since higher time-to-expiry indicates better freshness and higher quality. In this work we focus on finding optimal pricing and inventory policies while considering the effect of two marketing motivators: price and remaining shelf-life duration. While the effect of price on demand has been thoroughly analyzed in the literature (see, for example, Petruzzi and Dada (1999) and several more examples in the next section), there is a paucity of research on the effect of the remaining shelf-life duration despite its importance in modeling the demand rate of perishable items (including most grocery items, such as dairy products, bread, beverages and many other industrial food products, as well as ammunition, batteries and printer ink). In general, the age of inventory is likely to negatively affect the demand for perishable goods. Sarker et al. (1997) claim that this effect occurs because consumers tend to feel less confident purchasing perishable items whose expiration dates are approaching. Some items, such as range ammunition, suffer a negligible decrease in demand up to their expiration date, since they are used immediately after being purchased. On the other hand, the demand for self-defense ammunition is sensitive to the inventory's remaining shelf-life, since it is (hopefully) less likely to be used before its expiration. Our approach seeks to contribute toward the efficient management of perishable inventory and the prevention of waste. Waste is a key source of food loss in supply chains and in inventory systems with perishable items, and it is influenced by managerial factors as well as by technological factors, including production facilities, preservative materials, packaging, storage, and transportation conditions, all of which are vital for maintaining a product's shelf-life. Prevention of food waste is of particular interest in the discussion of perishable inventory management, as perishable food products are a main source of profit for grocery retailers. According to Donselaar and Broekmeulen (2012), the United States Department of Agriculture (USDA) estimates that average annual food losses in supermarkets in 2005 and 2006 were 11.4% for fresh fruit, 9.7% for fresh vegetables and 4.5% for fresh meat, poultry and seafood. Global trends indicate a future further increase of waste. A possible explanation of this trend is the increasing number of supermarkets (Parfitt et al., 2010), as well as the growth of household incomes, which increases the demand for perishable products (e.g., Parfitt et al. (2010) indicate that the demand for meat in China is increasing rapidly). Retailers incur heavy costs for wasted goods, as in addition to lost sales, they must bear the disposal costs of expired items. This paper proposes a model that can be used as a tool in the management of perishable inventory in general and food products in particular, enabling retailers to compare the expected maximal profits of substitute goods, which are different in their unit purchase cost and in their shelf-life duration, and decide which product is more profitable, given consumers' sensitivity to freshness. Furthermore, the model allows us to calculate the loss of profits to a retailer who ignores the decrease in the demand rate due to loss of freshness, and uses only a price-dependent demand function (as common in the literature). The rest of this paper is arranged as follows: In Section 2 we review the literature on the effects of pricing policy and of time on the demand rate, and we position our model in relation to previous work. Section 3 details the model assumptions and notations. In Section 4, we formulate the model as a profit maximization problem, and in Section 5, we find the optimal price for a given replenishment period. In Section 6, we present a thorough analysis of the profit function and properties of the optimal solution. A detailed algorithm is provided to obtain the optimal pricing and inventory policies efficiently. In Section 7, numerical examples are given to illustrate the model, followed by a sensitivity analysis. We sum up with a discussion of this work and suggest directions for future research.

The novelty of our model is in the formulation of a demand rate that decreases polynomially over time after replenishment (up to expiration, where it vanishes) and linearly in the selling price, where the two factors have a multiplicative effect. We show that a search for the optimal price, quantity to order and replenishment period can be reduced to a search for the replenishment period alone. In addition, we show that the profitability of an inventory system can be tested by computing the profit at one particular point. A search interval for the optimal replenishment period is provided, and we prove that the profit per unit time is strictly pseudo-concave. This property enables us to use a relatively fast-converging line-search algorithm for finding numerically the optimal solution as closely as required. Sensitivity analysis of two numerical examples shows that the sensitivity of the decision variables to changes in the value of a single parameter is dependent on the values of the other parameters. The only exception is the optimal selling price, which is found to be insensitive to changes of up to 50% in the value of a single parameter. This result can be explained by Corollary 3, which restricts the maximal change in the optimal selling price in our model. In our model, in contrast to previous models of perishable inventory, the expiration date is not only a constraint on the replenishment period but a factor that influences the demand over time, and thus the profits. Current technology may increase shelf-life duration of products by several means. The model enables us to evaluate whether the monetary investment in those means is economically justified. However, in order to calculate the optimal, profit-maximizing replenishment period for a given product, our model requires knowledge of the shape of the product's demand over time. In practice, to obtain the value of this parameter, a manager must collect and analyze data on the consumption behavior associated with the product at several levels of freshness. The emergence of automated identification and tracking technologies and network information systems may provide a solution to this problem. Managers in food and pharmaceutical supply chains might be able to use technologies such as radio frequency identification (RFID) and other complementary techniques to track products’ freshness as well as real-time consumption. We encourage extending our model to fit a retailer that sells a variety of perishable products. Though some products have similar prices and holding costs, each product is probably characterized by a unique shape of demand over time. Applying the suggested methodology to each one separately results in diverse replenishment schedules. Such a policy induces high ordering costs, additional logistical costs, and supplementary use of resources. A better and probably more economical policy would be to jointly order several products that are characterized by similar parameters and demand shapes over time. Future work should investigate the effects of relaxing some of our assumptions, e.g., allowing a finite replenishment rate or backlogging, or considering a stochastic demand function. However, such relaxations require additional assumptions, such as how demand rate is determined for a product whose units on-shelf have diverse expiration dates (according to their arrival times during the non-instantaneous replenishment interval); or how backlogged demand is influenced by the time left until the next replenishment. Since parameters such as demand, lead time and quality deterioration can fluctuate, our approach can be applied as an approximation only when the fluctuations are small. Future research is also required in order to extend the current methodology to include other price- and time-dependent demand functions, as well as stochastic expiration time and demand.