اندازه گیری مشترک تکمیل دوباره موجودی و تلاش برای فروش همراه با واکنش های نامشخص بازار

An optimal joint operational and marketing decision is crucial for robust supply chain management. This paper addresses concurrent determination of inventory replenishment and sales effort decisions such as price, incentives to salesforce, and short-term promotions, or a combination of them. Market responses to sales efforts are typically highly uncertain, and demand in each period has its distribution dependent on the selected sales effort. In each period a replenishment order may be issued, which incurs both fixed and variable ordering costs, and at the same time the sales effort is also determined, the execution of which may incur costs. For such a model, the previously developed methods which are used for the joint inventory-pricing models become inadequate. A computational procedure for obtaining an optimal joint policy is addressed, and the conditions for the optimality of that policy are identified.

Demand uncertainty is one of the most challenging but important issues in supply chain management. Traditional stochastic demand models are often assumed to be determined exogenously. Models taking price into consideration formulate demand with its distribution parameterized and controlled by the endogenous decision of price. The impact of price on demand is often either in an additive or a multiplicative form, or occasionally in a combination of both. In the additive form, demand is the summation of a deterministic component which is price-sensitive, and a random component which is independent of price, while in the multiplicative form, demand is modeled as a product of the two components (Yano and Gilbert, 2004). In the retail setting, however, such uncertainty is compounded because demands depend on the marketing decisions such as price, efforts exerted by sales personnel, advertising campaigns, and competitions. In this paper, we focus on sale effort. Firms exert various sales efforts, and often at a combination of them to increase sales and profits. For example, price changes, incentives to the sales people, and promotions with short-term effects such as in-store displays and product-service bundling. Clearly, the relationship between the market response and the combination of these sales efforts can be much more complex as compared to a single one. In this paper we allow a general relationship of the sales efforts and the market responses without specifying any form. Under such a setting, our interests lie in how to coordinate inventory and sales effort decisions. Specifically, we consider a single item, periodic-review system as follows. At the beginning of each period, the firm decides the order replenishment and sales effort jointly. Ordering incurs both fixed and variable costs, so does the execution of the sales effort, e.g., rentals for in-store displays, and advertisement costs for newspaper promotion. All replenishment arrives immediately, i.e., the leadtime is zero, and all unmet demand is fully backlogged. The inventory holding and shortage cost is charged based on the inventory leftover at the end of the period. The objective is to find a joint optimal inventory replenishment and sales effort policy to maximize the long-run average profit over an infinite horizon. For such a model, the previously developed methods which are used for the joint inventory-pricing models become inadequate. It is therefore difficult to characterize the optimal inventory and sales effort policy. One of the joint policy forms is thus of special interest, the (s,S,z)(s,S,z) form. Under an (s,S,z)(s,S,z) policy, whenever the inventory level falls to or below s , an order is placed to bring it up to S ; when the inventory level is at x , i.e., s<x≤Ss<x≤S, no order is issued. The choice of sales effort z is determined based on the inventory level x . In other words, the stock is replenished according to a min–max ordering policy, and the decision on the sales effort is specified by the inventory level. This policy is a generalization of the popular (s,S,p)(s,S,p) form, if the sales effort decision is replaced with the price decision. The task of this paper is thus twofold: to develop an efficient approach for searching for the optimal (s,S,z)(s,S,z) policy; and to identify the optimality conditions under which the (s,S,z)(s,S,z) policy is globally optimal. Our model is closely related to the literature on the coordination between marketing and production/inventory management in general and to that on multi-period stochastic pricing and inventory control in particular. Here we briefly review the most relevant work, beginning with the literature on joint pricing and inventory control. Interested reader can refer to Yano and Gilbert (2004) for detailed reviews. For single period models, see, for example, Karakul (2008), Serel (2008), and Webster and Weng (2008). In a multi-period setting, when there is no ordering setup cost, Federgruen and Heching (1999) show that the pricing/inventory decision is the “base stock list price policy”, which generalizes or extends many of the earlier models. Recently, several papers have considered models with fixed ordering costs and identified the conditions under which an (s,S,p)(s,S,p) policy is optimal. One key condition is related to a newsvendor-type profit function, which is defined as the resulting expected one-period profit with price being optimized for every inventory level. In a finite horizon periodic-review setting, Chen and Simchi-Levi (2004a) show the optimality of the (s,S,p)(s,S,p) policy or a variation of such a policy. Chen and Simchi-Levi (2004b) extend the optimality of a stationary (s,S,p)(s,S,p) policy to an infinite horizon setting. Both models require the newsvendor-type profit function to be concave in the inventory level. With a discounted profit criterion, Huh and Janakiraman (2008) further allow the newsvendor-type profit function to be unimodal. Feng and Chen (2004) propose a computational approach to calculate the optimal joint inventory-pricing control policy in an infinite-horizon periodic-review system, assuming that the newsvendor-type profit function is quasi-concave. However, such a condition can easily be violated in the current setting of our paper. Assuming unmet demands are lost, an earlier work of Polatoglu and Sahin (2000) analyzes a similar framework and characterizes the structure of optimal policy which is very complicated, for example, there may be multiple order-up-to levels. Though some sufficient conditions are provided under which the simple (s,S,p)(s,S,p) policy is optimal, these conditions are difficult to verify and non-intuitive. There are also some studies investigating the impact of various sales efforts on operational inventory decisions. To name a few, Balcer (1983) for the joint inventory and advertising strategy problem, Cheng and Sethi (1999) for the joint inventory-promotion problem with Markov-dependent demand state, Porteus and Whang (1991) and Chen (2000) for the impact of incentive schemes of salesforce compensation on manufacturing decisions. Ernst and Kouveils (1999) study the joint decision of goods bundling and inventory control in a newsvendor setting. In a continuous-review setting, Chen et al. (2005) show that the (s,S)-type(s,S)-type policy is optimal for product inventory control, and an inventory level-based service package composite is optimal for service offerings. Zhang et al. (2008) discuss the joint optimization of inventory and pricing, and promotion, and ignore the fixed ordering cost in their setting. The rest of this paper is organized as follows. Section 2 describes the model and the long-run average profit function of a given stationary (s,S,z)(s,S,z) policy. Section 3 proposes an efficient computing procedure to find an optimal (s,S,z)(s,S,z) policy. Section 4 further identifies the conditions under which the optimal (s,S,z)(s,S,z) policy is globally optimal. Section 5 reports numerical findings, and Section 6 concludes the paper. All technical proofs are included in the Appendix.

With demand uncertainty, an optimal joint inventory and marketing decision is crucial for robust management in the retail setting. This paper characterizes the structure of the optimal joint inventory and sales effort control policy for a single-item, periodic-review system with the objective of maximizing the long-run average profit. The optimal policy for each period is of the (s,S,z)(s,S,z) structure. Specifically, the inventory is replenished based on an (s,S)(s,S) policy, and the sales effort is dependent on the inventory level. One of the most salient features of our model is its generality. We allow the sales effort decision to be a single effort or a combination of sales efforts such as price changes, short-term promotions, incentives to salespeople, and different service packages. Demands are random and independently distributed over time, as well as parameterized on the choice of sales effort. We propose a general demand model without specifying any form while including most of the popular demand models prevailing in the literature. In addition to the standard inventory costs, we further allow a cost for executing that sales effort, which is practically prevalence yet often ignored. All of these considerations endow our model with considerable practical relevance. However, a model based on such a setup may easily violate the well-accepted assumption of the unimodality of the newsvendor-type profit function. An efficient algorithm to find the optimal stationary (s,S,z)(s,S,z) policy is highlighted, and the optimality conditions for the optimal (s,S,z)(s,S,z) policy are identified ex post. Finally, it is appropriate to point out that the order replenishment leadtime is assumed to be zero. This is a limitation of the model, though it is taken as a norm for complicated inventory replenishment practice. Readers with interests can refer to Federgruen and Heching (1999), who discuss the impact of non-zero lead time in the joint inventory-pricing context, ignoring the fixed ordering cost.