سفارش بهینه و سیاست های قیمت گذاری برای توابع تقاضا که قابل تفکیک به قیمت و موجودی هستند

We formulate and analyze two models for determining the optimal pricing, order quantity and replenishment period for items whose demand function is separable into components of price and inventory age. The first model assumes a multiplicative demand function. We provide conditions, which are satisfied by most common price-dependent demand functions, to reduce the three-variable profit maximization problem into a single-variable problem, which can be solved using an efficient line-search method. Next, we show that a genuine additive model cannot exist, and instead suggest and analyze a pseudo-additive model. However, this model is more limited than the multiplicative model in its ability to incorporate various combinations of price and inventory age effects, and reduction of the maximization problem into a single-variable problem is more complicated, except in the case of a linear price effect, which is further analyzed. For both models, we show that the optimal solution satisfies the first-order condition for equilibrium under a monopoly, with a modification that includes inventory holding costs. We solve numerical examples to illustrate the solution procedures.

Most inventory management policies involve two decision variables: order quantity and replenishment period. These variables directly affect the inventory level trajectory. For some time, the literature has emphasized the unit selling price as a decision variable that affects the inventory level indirectly, through the demand function. Only recently have researchers begun considering how the demand function is influenced by the inventory age (i.e., the elapsed time measured from the most recent replenishment) in conjunction with the selling price. This is especially important for inventory modeling of perishable items, whose demand might be affected by their freshness (see, for example, You, 2005, Tsao and Sheen, 2008, Avinadav and Arponen, 2009, Valliathal and Uthayakumar, 2011, Maihami and Kamalabadi, 2012 and Avinadav et al., 2013). In this paper we optimize the order quantity, replenishment period and unit selling price to maximize profits, given a demand function that is affected by both price and inventory age. Most of the results pertain to cases in which the demand function can be separated into the latter two factors, taking either a multiplicative form or an additive form. The contribution of this study to the literature is threefold: (i) showing that under common conditions the multiplicative form maximization problem can be reduced into a single-variable problem (the replenishment period), which can be solved using the efficient line-search methods; (ii) showing that a genuine additive demand function cannot exist and suggesting and analyzing a pseudo-additive model instead; and (iii) providing economic interpretations for the optimality conditions. We assume that demand rate (demand per unit of time) is non-increasing in inventory age, and that unit retail price is fixed. These assumptions suit inventory systems of perishable items – i.e., items that physically deteriorate or whose quality decreases over time – in stores, such as supermarkets, that replenish items before expiration but avoid discounts for reduced freshness. Examples of perishable items are food products and beverages (e.g., fresh vegetables and fruit, dairy and meat products), fashion goods, ink cartridges for printers, and batteries. In general, the age of inventory is likely to negatively affect the demand rate for perishable items. Sarker et al. (1997) claim that this effect occurs because consumers tend to feel less confident purchasing perishable items as their expiration dates grow nearer. Disregarding the effect of perishable products’ limited shelf-life on their demand may lead to significant losses. For example, according to van Donselaar and Broekmeulen (2012), the United States Department of Agriculture (USDA) estimates that average annual food losses due to leftover inventory 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. In the next section we present the literature dealing with the influence of price and time on demand rate. In Section 3, we explain the approach we use to find the optimal inventory policy and to evaluate its sensitivity to changes in the decision variables. We then formulate the model and present general properties of the optimal policy. In Section 4, we thoroughly analyze demand functions that take a multiplicative form (i.e., functions in which the components of price and inventory age are multiplied by each other), and in Section 5, we analyze demand functions that take an additive form and discuss their limitations. Each analysis is followed by numerical examples that demonstrate the procedure for obtaining the corresponding optimal solution. We conclude with the main findings and directions for future work.

In this paper we investigated optimal ordering and pricing policies when the demand function is deterministic and is affected by both price and inventory age. We showed that an optimal solution is obtained when the inventory is exhausted exactly at replenishment time, and that the marginal profit of the last sold unit is positive. In order to provide additional insights into the optimal solution, we assumed that the demand function is separable into components of price and inventory age. Two types of separability were analyzed: multiplicative and additive. In order to extract the optimal price for a given replenishment period under each model type, we modified the optimality equation of a monopoly (i.e., at equilibrium, marginal revenue is equal to marginal cost) to include inventory holding costs. We showed that the properties of this optimality equation – specifically, the way in which it is influenced by the effect of inventory age on the demand function – are dependent on the model type. In the multiplicative case, the effect of inventory age influences only the integrated marginal cost, while in the additive model it influences only the marginal revenue. For the multiplicative model, we showed that under common assumptions, the three-variable maximization problem can be reduced into a single-variable problem, in which the variable is the replenishment period. We showed that a genuine additive model does not exist, and suggested a pseudo-additive model instead. In this model, which is considerably more limited than the multiplicative model, reducing the maximization problem into a single-variable problem is not a straightforward task, as p and T are not separable in the optimality equation, except in special cases. One such exception, which we have analyzed, is the case in which price has a linear effect on demand. Owing to the difficulties associated with the additive model, any extension to a non-separable function of demand should be carried out carefully to ensure that the essential properties of a legitimate demand function are maintained. Avinadav et al. (2013) modeled demand that decreases linearly in price and polynomially in inventory age. The current work extends their model to any price-dependent demand function, in which marginal revenue is a quasi-convex function of price (with derivative greater than 0.5), and to any non-increasing time-dependent demand function. Avinadav et al. (2013) obtained a condition for profitability of the optimal solution, given a specific demand function; herein we show that this condition applies to our extended model as well. We conclude with three directions for future work. One direction is relaxing the assumption that price is fixed along the period, and developing a model that allows price to be a function of inventory age or inventory level. A simple case of time-dependent price is giving a discount at a certain time-point within the period to elevate sales of items that are close to their expiration date. A second direction is including additional marketing factors that affect the demand function, such as observed inventory-level or investment in product promotion. Another direction, which seems to be the most important, is relaxing the assumption that demand is deterministic and developing an appropriate model for demand that is a stochastic process. Such a model has to consider ordering-level and shortages, due to uncertainty of demand, and how shortage affects consumers’ behavior.