مشترک قیمت گذاری و کنترل موجودی برای آیتم های رو به وخامت غیر آنی با برگشت به سیستم جزئی و تقاضای وابسته به قیمت و زمان

In this paper, a joint pricing and inventory control for non-instantaneous deteriorating items is developed. We adopt a price and time dependent demand function. Shortages is allowed and partially backlogged. The major objective is to determine the optimal selling price, the optimal replenishment schedule and the optimal order quantity simultaneously such that, the total profit is maximized. We first show that for any given selling price, optimal replenishment schedule exists and unique. Then, we show that the total profit is a concave function of price. Next, we present a simple algorithm to find the optimal solution. Finally, we solve a numerical example to illustrate the solution procedure and the algorithm.

Pricing and inventory policy are two important factors in success of business of different items. When the items are deteriorating the importance of these factors is increased. As declared in literature, deterioration is defined as decay, damage, spoilage, evaporation, obsolescence, pilferage, loss of utility or loss of marginal value of commodity that results in decreasing usefulness. Most of physical goods undergo decay or deterioration over time, the examples being medicine, volatile liquids, blood banks, and others (Wee, 1993). Consequently, the pricing and inventory control problem of deteriorating items has been extensively studied by researchers. The first attempt to describe the optimal ordering policies for deteriorating items was made by Ghare and Schrader (1963). They presented an EOQ model for an exponentially decaying inventory. Later, Covert and Philip (1973) formulated the model by considering variable deteriorating rate with two-parameter Weibull distribution. Goyal and Giri (2001) provided an excellent and detailed review of deteriorating inventory literatures. Balkhi (2011) developed and solved a general finite trade credit economic ordering policy for an inventory model with deteriorating items under time value of money. In some inventory systems, such as fashionable items, the length of the waiting time for next replenishment would determine whether the backlogging will be accepted or not. Therefore, the backlogging rate is variable and dependent on the waiting time for the next replenishment (Geetha and Uthayakumar, 2010). Chung, 2009 and Dye, 2007 considered a pricing and lot-sizing problem for a product with variable rate of deterioration, allowing shortages and partial backlogging. Dye (2007) developed a joint pricing and ordering policy for a deteriorating inventory with partial backlogging. The demand is known and linear function of price. Later, Dye et al. (2007) presented an inventory and pricing strategy for deteriorating items with shortages. Demand and deterioration rate are continuous and differentiable function of price and time, respectively. Chang et al. (2006) established an EOQ model for deteriorating items for a retailer for determine its optimal selling price and lot-sizing policy with partial backlogging and log-concave demand. Cai et al., (2011) studied qthe problem of pricing and ordering policy in two-stage supply chains by considering the partial lost sales and using the game theory. In the deteriorating items inventory literatures that been mentioned above, all researchers assume that the deteriorating of the items in inventory starts from the instant of their arrival in stock. In fact, most goods would have a span of maintaining quality or original condition, namely, during that period, there is no deterioration occurring. Wu et al. (2006) defined the phenomenon as “non-instantaneous deterioration”. They developed a replenishment policy for non-instantaneous deteriorating items with stock-dependent demand such that the total relevant inventory cost per unit time had a minimum value. In the real world, this type of phenomenon exists commonly such as firsthand vegetables and fruits have a short span of maintaining fresh quality, in which there is almost no spoilage. Afterward, some of the items will start to decay. For this kind of items, the assumption that the deterioration starts from the instant of arrival in stock may cause retailers to make inappropriate replenishment policies due to overvalue the total annual relevant inventory cost. Therefore, in the field of inventory management, it is necessary to consider the inventory problems for non-instantaneous deteriorating items. Ouyang et al. (2006) present an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Although their inventory model is correct and interesting, processes of arguments to derive those theorems and the easy-to-use method to search for the optimal replenishment cycle time are not complete. Chung (2009) considers this issue and present complete proofs for Ouyang et al. (2006) model. Yang et al. (2009) developed an inventory system for non-instantaneous deteriorating items with price-dependent demand. In their model, shortages are allowed and partially backlogged. The major objective is to determine the optimal selling price, the length of time in which there is no inventory shortage, and the replenishment cycle time simultaneously. Geetha and Uthayakumar (2010) proposed EOQ based model for non-instantaneous deteriorating items with permissible delay in payments. In this model demand and price is constant and shortages are allowed and partially backlogged. Musa and Sani (in press) developed a mathematical model for inventory control of non-instantaneous deteriorating items with permissible delay in payments. An appropriate pricing and inventory model for a non-instantaneous deteriorating item is presented in this paper. In the sum of all works on pricing and inventory control, models that consider non-instantaneous deteriorating items are very small portion. on the other side, in all papers that consider pricing and inventory control for non-instantaneous deteriorating items, the demand functions are simple and dependent on price, stock or time, separately. But in non-deteriorating items the price and the time should be considered jointly. Because, this form of demand function reflect a real situation: i.e. the demand may increase when the price decreases, or it may vary through time. In this work, we consider time and price dependent demand function. Also, Shortages are allowed and partially backlogged. The backlogging rate is variable and dependent on the waiting time for the next replenishment. The major objective is to determine the optimal selling price, the optimal replenishment cycle time and the order quantity simultaneously. This is the first work that has the above assumptions. The rest of the paper is follows. In Section 2, assumptions and notations used throughout this paper are present. In Section 3, we establish the mathematical model and the necessary condition for finding an optimal solution. For any given selling price, we then show that the optimal solution is exist and unique. Moreover, we prove that the total profit is a concave function of selling price. Next, in Section 4, we present a simple algorithm to find the optimal selling price and inventory control variables. In Section 5, we use a numerical example and finally, we make a summary and provide some suggestions for future in Section 6.

Due to the technological improvement, fiercer market competition, the product life cycle have been greatly shortened and the innovation accelerated, more and more products (e.g. electronic product, personal computer, and information product) now have the same features as the deteriorating items in the other side, in the real world, most deteriorating items would have a span of maintaining quality or original condition, namely, during that period, there is no deterioration occurring. For this kind of items, the assumption that the deterioration starts from the instant of arrival in stock may cause retailers to make inappropriate replenishment policies due to overvalue the total annual relevant inventory cost. Therefore, in the field of inventory management, it is necessary to consider the inventory problems for non-instantaneous deteriorating items. However, research works have already shown that the replenishment policy without considering the sale price cannot optimize the revenue. On the other hand, the simultaneous determination of price and ordering production quantity can yield substantial revenue increase. The coordination of price decisions and inventory control is thus not only useful but also essential. Coordinating these decisions means optimizing the system rather than its individual elements and not only potentially increases profits but also reduce variability in demand or production, resulting in more efficient supply chain. In this work, we addressed the problem of simultaneously determining a pricing and inventory replenishment strategy for non-instantaneous deteriorating items. We extended a new model that a retailer or a manufacturer can use it to optimize sales price and inventory control variables. The demand is deterministic and depended on time and price simultaneously. Also, shortages is allowed and partially backlogged where, the backlogging rate is variable and dependent on the waiting time for the next replenishment. In the paper, some useful theorems which characterize the optimal solution are framed and an algorithm is presented to determine the optimal price and optimal inventory control parameters. Finally, numerical examples are provided to illustrate the algorithm and the solution procedure. The results show that there is an improvement in total profit from the non-instantaneously deterioration items rather than instantaneously deterioration. Also, when the negative effect of time on demand is high, the total profit is low and vice verse. This is the first work that considers the optimal pricing and inventory control policy for non-instantaneous deteriorating items under partially backlogging shortages and time and price dependent demand. This paper can be extended in several ways, for instance, we could extend model by considering the non- zero lead time. Also, we may consider the permissible delay in payment or promotions in the model. Finally, we could extend the deterministic demand function to stochastic demand function.