سیاست های بهینه نشانه گذاری قیمت برای یک مدل موجودی با نوسانات قیمت تصادفی و گزینه برای خرید اضافی

In this paper, based on the analysis of empirical data of dealer and retailer prices and sales of cement – as a prototype of functional products – we develop a probabilistic inventory model for the situation when dealer′s selling price fluctuates and affects the retailer′s selling price. Demand is a function of price markup and retailer′s purchase price. Dynamic price markup policy is proposed and optimal length of replenishment cycle and order quantity is obtained. Concavity of the profit function with respect to price markup is discussed. A procurement policy which considers opportunity of purchase at a low price before the end of the optimal replenishment cycle is proposed and compared with other policies. Algorithm, numerical examples, sensitivity analysis and managerial insights are presented.

Products which have long product life cycle (PLC) and low demand uncertainty, like groceries, casual readymade garments, cement and other building materials, are termed as functional products. These are characterized by exogenous random fluctuations in dealer price (DP), stable demand and low profit margin (Fisher, 1997). Due to heavy competition, demand for such products mainly depends upon the retailer′s selling price (RSP). As the retailer′s purchase price (RPP) is not under the retailer′s control, the end user demand depends mainly on his price markup (PM) or sometimes, the markup proportion. Unlike those competitive situations where retailer may use pricing to control demand, the only control that can be exercised by the retailer on the demand for functional products is a careful choice of a competitive PM. Many retailers follow the retail fixed markup (RFM) policy of charging a fixed amount over and above the dealer selling price (DP) (Liu et al., 2009). However, practical observations indicate that (1) generally retailers replenish inventory for weeks or months, during which DP may fluctuate to values above or below the RPP; (2) retailer is governed by the business ethics that “on any day, RSP cannot be below the DP”; (3) when the DP on a day is lower than the RPP, the retailer may have to sell at a low markup; (4) retailer will not sell below his purchase price. Hence in the presence of fluctuating DP, RFM may not always be practical. In order to develop a realistic model based on which we can suggest price markup strategies to a retailer of a functional commodity in India, we have collected empirical data of price (Fig. 1) and sales (Fig. 2) of cement as an important prototype of functional product, from a retailer located in Indore (India). Full-size image (29 K) Fig. 1. Dealer′s and retailer′s price of cement (per bag). Figure options Full-size image (26 K) Fig. 2. Sales data of cement for a retailer in Indore. Figure options We have analyzed the collected data to gain insights into the important factors in order to build a viable model and made an attempt to work out the best ordering and pricing options for the retailer. Statistical analysis of the empirical data shows that the dealer prices follow two parameter exponential distribution with location parameter γ=200 and inverse scale parameter λ=0.08 (P-value of Kolmogorov–Smirnov test statistic is 0.76436). We were informed that the retailer places order every week, whereas analysis of the sales data of cement, using autocorrelation function and partial autocorrelation function, indicates that during a quarter, the series is stationary. In view of this, and the support of deterministic demand as an approximation to probabilistic demand by Netessine (2006), we consider the demand to be deterministic if length of the planning horizon is such that demand is not affected by seasonal variations. Before going on to the model and its optimization aspects, we present some of the literature related to various aspects of the model considered in this paper viz. a diversity of types of demand functions, pricing decisions and effect of change in price (selling or purchase price) on the demand rate. Time dependent demand is considered in inventory models by Urban and Baker (1997), Teng and Chang (2005), Banerjee and Sharma, 2008, Banerjee and Sharma, 2009 and Banerjee and Sharma, 2010, Bitran and Mondschein (1997), etc. Various authors consider pricing as a means of control over demand. For the newsvendor type problem, Petruzzi and Dada (1999) provide an excellent review with extensions. Other researchers have also considered price dependent demand (Chen et al., 2006, Lau and Lau, 1998, Lau and Lau, 2003, Polatoglu, 1991 and Avinadav et al., 2013). Elmaghraby and Keskinocak (2003) give a comprehensive review of the literature and practices in dynamic pricing. Continuous change in purchase price was considered by some authors. Among them, Erel (1992) considered a compound increasing unit cost due to inflation during a finite planning horizon while Khouja and Park (2003) proposed an extension of EOQ that can be used when unit cost is decreasing. Khouja et al. (2005) developed the joint replenishment problem to analyze the effect of continuous decrease or increase in unit purchasing cost on the optimal ordering frequencies. Arnold et al. (2009) used deterministic optimal control approach for optimizing the procurement and inventory policy of an enterprise that is processing a raw material when the purchasing price, holding cost, and the demand rate fluctuate over time. Banerjee and Meitei (2010) consider linearly declining selling price for the single period problem. Gavirneni (2004) considered periodic review policy for the inventory model where the purchase price of the product undergoes a Markovian transition from one period to the next. However, in Gavirneni′s model the purchase price can take values only from a finite set and the selling price is fixed whereas in practice, RSP often fluctuates due to exogenous factors (e.g., DP) and/or endogenous factors (e.g., PM). PM is one of the major tools often used by retailers for setting selling price. For a detailed discussion on PM one may refer to Liu et al., (2009). In addition to the observations at the beginning of this section regarding demand and price of functional commodities, a common observation is that fluctuating DP may result in fluctuating RSP. If DP falls below the RPP, the retailer may have to sell at a low markup but will still be at a disadvantage since he has to sell at a higher RSP compared to those retailers who purchased at a low RPP and hence sell at a lower RSP thereby generating higher demand. On a positive note, it lends the retailer an opportunity to purchase the commodity at a lower price than the RPP for his current lot and then sell at a lower RSP. This strategy of making purchase at low price while still having positive stock is similar to the practice followed in real life business, especially in share market and is termed as “additional purchase”. In addition to the issues mentioned earlier, a theoretical investigation into such a business strategy may be useful. To the best of our knowledge, fluctuating DP resulting in fluctuations in PM and RSP under the dependence of demand over RPP and PM has not yet been considered in published inventory literature. Further, no theoretical model has as yet considered this type of policy of additional purchase at reduced price when length of replenishment cycle is a decision variable. We make an attempt to address these issues in the present work. In this paper, we determine the optimal values of (1) the time epoch at which inventory level becomes zero, (2) dynamic PM – which in turn gives the optimal values of the order quantity and the RSP, and (3) the earliest time epoch for a given reduced DP at which the opportunity of additional purchase would be beneficial for the retailer. The organization of this paper is as follows: In Section 2, model, notations and assumptions are presented for the model under Policy 1. Analysis of the profit function for general demand and a particular case when demand decreases exponentially with markup proportion is presented in Section 3. In Section 4, the problem of ‘additional-purchase’ at a reduced purchase price within the optimal replenishment cycle is discussed as per Policy 2. For both the policies, Section 5 presents the algorithms for the general demand function, some numerical examples and managerial insights. Sensitivity analysis is presented in Section 6. Finally Section 7 concludes the paper.

In this paper, based on the analysis of empirical data, we make an attempt to contribute to the existing inventory literature by proposing dynamic policies for setting daily values of RSP and PM as functions of the observed value of random DP on the day which conforms to business ethics. Apart from Policy 1 which is a new approach to the routine replenishment policy due to randomized pricing, we have analytically discussed a policy – which is used quite often in practice – where the retailer considers the opportunity of additional purchase at a reduced DP. For both the policies, demand is a general function of price markup and purchase price. Considering a general demand function, for Policy 1, concavity of the net profit function with respect to length of the replenishment cycle is proved and that with respect to a constant which generates PM is discussed. The insights of the model are extracted for a demand function which is exponential decreasing in markup proportion. Algorithm for obtaining optimal values of the decision variables has been presented and optimal values of selling price in terms of PM have been obtained. These algorithms have been used to obtain minimum values of time epochs as well as formulating decision rules defining – for a given RPP – the pair of values of DP and time epoch when additional purchase would be profitable. Numerical results indicate that lower the value of the reduced purchase price, earlier one can go in for additional purchase as the opportunity arises. Sensitivity analysis shows that the optimal values of decision variables are highly sensitive to the initial purchase price of the retailer and the reduced purchase price at which strategic additional-purchase is made. The values of profit functions for Policy 1 and Policy 2 increase as the demand potential and markup discount factor α increase while these decrease as the values of purchase price, holding cost, ordering cost and β increase. The proposed pricing policies differ from the usual fixed markup policy of the retailer. Numerical comparison of policies proposed in this paper with the RFM policy indicates that either one of them may be better than the RFM policy, depending upon the RPP, DP and the sensitivity of the demand function with respect to PM. This work can be extended to give a plethora of new models that may be useful in practice.