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This note considers a joint inventory-pricing control problem in an infinite-horizon periodic-review system. Demand in a period is random and depends on the posted price. Besides the holding and shortage costs, the system incurs inventoryreplenishment costs that consist of both variable and fixed components. At the beginning of each period, a joint inventory and pricing decision is made. Under the long-run average profit criterion, we show that an optimal policy exists within the class of so-called 4s1 S1p5 policies. This is established based on our algorithmic development, which also results in an algorithm for finding an optimal 4s1 S1p5 policy. Subject classifications: joint pricing and inventory control; setup cost; price dependent demand; stochastic inventory model. Area of review: Manufacturing, Service, and Supply Chain Operations. History: Received May 2003; revisions received May 2004, August 2005, July 2007, May 2009; accepted April 2010.

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... In a multi-period problem setting with a fixed ordering cost, Chen and Simchi-Levi (2004a) establish the optimality of (s, S, p) policies for a finite horizon and for an infinite horizon (Chen and Simchi-Levi 2004b). Computational approach for finding the optimal joint inventory-pricing policies includes Feng and Chen (2011) and Wei and Chen (2011). There are also various extensions under the framework of joint pricing and inventory problem. ...

... Q β , r β and p β are the optimal policy parameters maximizing l β (n Q, r , p) for a given β. Following Chen (2005) and Feng and Chen (2011), without proof we propose the following lemma. ...

In business practice the misalignment between frequent inventory adjustments and a sticky price is common. Motivated by Pipi Milk, a diary manufacturer who commits its customer at a fixed price for a whole year meanwhile weekly replenishes its cheese product stock at a specific batch size, this paper develops an integrated constant pricing and inventory control model in a periodic-review system with batch order consideration. For such a system, our focus is on an (nQ, r)-p policy: the price p is determined and committed at the beginning of the planning horizon; inventories are managed based on an (nQ, r) policy with reorder level r and order quantities at non-negative integer multiples of batch size Q. With a fixed ordering cost included, we develop an efficient algorithm to compute the optimal policy parameters with the purpose to maximize the expected long-run average profit. We also numerically investigate the policy performance and verify its advantages by comparing to a sequential optimization of price and inventory decisions. Finally, managerial insights are generated.

... Relevant literature that puts particular attention on the study of periodic review inventory systems in the infinite horizon scenario includes the classical works by Iglehart [11], [12], Veinott [18], Zheng [21], and more recent works (after year 2000) Chen [4], Chen and Simchi-Levi [5], Feng and Chen [7], van Ryzin and Vulcano [17], Yuan and Cheung [20]. These papers consider inventory models, with some considering joint inventory and pricing models, for either discounted or average profit/cost criteria or both. ...

... where the above equality holds since, by Theorem 3, π ∞ (x 1 , x 2 ) satisfies (7), with (u * 1,∞ , u * 2,∞ ) being the optimal solution to the maximization problem in (7). ...

We consider a periodic review inventory system and present its optimal policy in the infinite horizon setting. The optimal inventory policy that maximizes the infinite horizon expected discounted profit for the model is analytically obtained by relating to the finite horizon setting using results from variational analysis. Results are provided that elucidate the operations of the inventory system in the long run.

... Finally, the heuristic allows us to derive insights regarding the impact of lead time on pricing decisions. As reported in the literature (see, e.g., Federgruen and Heching 1999, Chen et al. 2010, Feng and Chen 2011, the benefit of dynamic pricing in systems with zero lead time is limited. In contrast, when lead time is positive, the system is less responsive to demand through its replenishment process and therefore the value of dynamic pricing can be significant. ...

We study a joint inventory and pricing problem in a single-stage system with a positive lead time. We consider both additive and multiplicative demand forms. This problem is, in general, intractable due to its computational complexity. We develop a simple heuristic that resolves this issue. The heuristic involves a myopic pricing policy that generates each period’s price as a function of the initial inventory level and a base-stock policy for inventory replenishment. In each period, the firm monitors its so-called price-deflated inventory position and places an order to reach a target base-stock level. The price-deflated inventory position weights the on-hand and pipeline inventory according to a factor that reflects the sensitivity of price to the net inventory level. To assess the effectiveness of our heuristic, we construct an upper bound to the exact system. The upper bound is based on an information-relaxation approach and involves a penalty function derived from the proposed heuristic. A numerical study suggests that the heuristic is near-optimal. The heuristic approach can be applied to a wide variety of inventory systems, such as systems with fixed ordering costs or fixed batch sizes. The heuristic enables us to explore the use of price as a lever to balance supply and demand. Our findings indicate that a responsive strategy (that effectively reduces the replenishment lead time) leads to a more stable pricing policy and that the value of dynamic pricing increases with lead time.
This paper was accepted by Martin Lariviere, operations management.

... (2) Despite the observation above, the optimal price is not necessarily always decreasing with respect to the initial inventory level. In fact, the same phenomenon has been observed for the joint pricing and inventory control problem for nonperishables in the presence of fixed ordering cost (Chen and Feng 2010;Polatoglu and Sahin 2000). This can be explained as follows. ...

I n this article, we study the joint pricing and inventory control problem for perishables when a retailer does not sell new and old inventory at the same time. At the beginning of a period, the retailer makes replenishment and pricing decisions, and at the end of a period, the retailer decides whether to dispose of ending inventory or carry it forward to the next period. The objective of the retailer is to maximize the long-run average profit. Assuming zero lead time, we pro-pose an efficient solution approach to the problem, which is also generalized to solve three extensions to the basic model. A feature of the present study is that we consider explicitly the influence of perishability on the demand. Among the insights gathered from the numerical analysis, we find that dynamic pricing aids extending shelf life and when disposal incurs a lower cost, or even a positive salvage value, the retailer is induced to dispose earlier since the benefit of selling new inventory offsets the loss due to disposal. We also observe that the faster the perceived rate of deterioration, the lower the threshold of the ending inventory for disposal. Perhaps a bit counter-intuitive, maximizing profits does not mean eliminating disposals or expirations.

... However, Eigenfactor's AI uses a five-year time window while Xu et al.'s QI uses a ten-year time window. All in all, we posit that an article's impact may not be fully realized within the timeframe because articles can take longer than 2 years for their influence to be observed since journals typically have lengthy submissionreview-revision-acceptance cycles (see Feng and Chen 2011). That as a result, the influence of articles cited after their two-year rolling timeframe is not only ignored, it appears to favor journals that have faster turnaround cycles, and also ignores articles that have longitudinal impact. ...

The purpose of this research is to furnish the OR/MS research community with an updated assessment of the discipline's journals set with refinements that also highlight the various characteristics of OR/MS journals. More specifically, we apply a refined PageRank method initially proposed by Xu et al. (2011) to evaluate the top 31 OR/MS journals for 2010, and report our findings. We also report the shifts in the rankings that span 5 years, from 2006 to 2010. We observe that Manufacturing and Service Operations Management, indexed by the SCI only in 2008, is a specialized journal that is consistently highly regarded within the discipline. The rankings also suggest that Management Science is more established as a generalized journal as it has more external impact. In general, our ranking results correlate with expert opinions, and we also observe, report and discuss some interesting patterns that have emerged over the past 5 years from 2006 to 2010. Keywords Journals evaluation Á Citations analysis Á Journal influence Á Journal impact Á Impact factor Á Influence index Á OR/MS Á PageRank Andrew Lim is currently on no pay leave from

Considering the restriction of supply risk on optimal profit realization in advance selling, we discuss three selling strategies of a seller who produces and sells a seasonal product to a consumer under uncertain supply and demand: all advance selling, partial advance selling, and non‐advance selling. The robust newsvendor model is designed to solve the problem. Our results show that implementing an advance selling strategy is always beneficial from the demand uncertainty perspective to the seller. However, sellers should choose advance selling carefully from the standpoint of supply uncertainty: sellers will non‐advance selling under certain conditions. This condition is contingent on the market, capacity level, selling price, supply–demand correlation, consumer characteristics, and seller's pricing power. Interestingly, pricing power is the key driver to stimulate advance selling under supply uncertainty. In addition, the impact of supply and demand uncertainty and supply–demand correlation on these strategies are related to price discounts.

We analyze a finite horizon periodic-review inventory model in the presence of advance purchase discount (APD). In addition to the opportunity to place a contingent order when required, the buyer can also commit to advance orders and receive discounts. Thus the buyer faces the following tradeoff: Contingent orders provide operational flexibility to respond to uncertain demands at a high purchase cost, whereas advance orders incur low purchase cost while sacrificing operational flexibility. We consider a simple and easy-to-implement APD scheme in which the buyer enjoys price discounts by committing at the outset to a fixed size of advanced order for each period along with contingent orders when required. We first show that for any given advance order size, an advance-order dependent (s, S) policy is optimal. We then find the advance order size together with the corresponding (s, S) policy that minimizes the buyer's total cost.

We study sourcing and responsive pricing decisions of a firm with two correlated products and price-dependent demand when supply capacities for these two products are uncertain. Cross-price effects exist between the two products, which means that the demand of each product depends on the prices of both products. First, we apply the tool of convexity and some recent results to establish the convexity of the value-to-go function during each period; then we use KKT conditions and some structural properties given by convexity to derive structural results of the optimal order policy for these two correlated products in a periodic review setting. Contrary to the classical “base-stock” policy derived in the literature for single product or multiple products without uncertain supply capacity, we show that the optimal order policy in our inventory control problem is complicated and it follows the “order-up-to” structural property only under some conditions. Numerical studies are carried out to present the structural properties of the optimal order policy and show the effect of the complementarity or substitutability on the profits.

We study an optimal inventory control and allocation for a seller who uses sequential Internet auctions to sell replenishment products. This problem is faced not only by retailers/distributors in the retail industry but also by manufacturers who use sequential Internet auctions to sell products that can be replenished. We consider the total expected discounted profit criterion in a finite horizon and both the discounted and average profit criteria over an infinite horizon, and show the optimality of the so-called base-stock-allocate-all (BSAA) policies. Then, we simplify the problem of computing the optimal policies and profits for either the discounted or average profit criterion in an infinite horizon to an optimization problem with only one integer decision variable. We show these results for the list price case, and also for the case where the reserve price is a decision variable. Finally, numerical results are given, where the effects of several model parameters on the optimal policies and profits are numerically analyzed.

This paper analyzes a periodic-review, joint inventory and pricing control problem for a firm that faces stochastic, pricesensitive demand under a nonstationary environment with fixed ordering costs. Any unsatisfied demand is backlogged. The objective is to maximize expected profit over a finite selling horizon by coordinating the inventory and pricing decisions in each period. We show that for an additive demand model, an (s1 S1p) policy is optimal when the expected revenue is quasi-concave in price, the inventory cost (of holding and/or backlogging) is quasi-convex, and the nonnegative random demand has a Pólya or uniform density function. For the special case with no fixed ordering cost, the optimality of a base stock list price policy is demonstrated for more general demand distributions and convex inventory cost. These sets of sufficient conditions generalize the existing conditions in the literature that require, for example, the demand and/or revenue functions to be concave or the model parameters to be stationary in time. Our generalization makes the structural results applicable to models broadly supported by economic theory and empirical data. In addition, our proof uses a distinct sequential optimization technique for iteratively establishing the quasi-K-concavity of dynamic optimal value functions.

This paper addresses a joint pricing and inventory control problem for a batch production system with random leadtimes. Assume that demand arrives according to a Poisson process with a price-dependent arrival rate. Each replenishment order contains a single batch of a fixed lot size. The replenishment leadtime follows an Erlang distribution, with the number of completed phases recording the delivery state of outstanding orders. The objective is to determine an optimal inventory-pricing policy that maximizes total expected discounted profit or long-run average profit. We first show that when there is at most one order outstanding at any point in time and that excess demand is lost, the optimal reorder policy can be characterized by a critical stock level and the optimal pricing decision is decreasing in the inventory level and delivery state. We then extend the analysis to mixed-Erlang leadtime distribution which can be used to approximate any random leadtime to any degree of accuracy. We further extend the analysis to allowing three outstanding orders where the optimal reorder point becomes state-dependent: the closer an outstanding order is to its arrival or the more orders are outstanding, the lower selling price is charged and the lower reorder point is chosen. Finally, we address the backlog case and show that the monotone pricing structure may not be true when the optimal reorder point is negative.

In this article, we study the joint pricing and inventory control problem for perishables when a retailer does not sell new and old inventory at the same time. At the beginning of a period, the retailer makes replenishment and pricing decisions, and at the end of a period, the retailer decides whether to dispose of ending inventory or carry it forward to the next period. The objective of the retailer is to maximize the long‐run average profit. Assuming zero lead time, we propose an efficient solution approach to the problem, which is also generalized to solve three extensions to the basic model. A feature of the present study is that we consider explicitly the influence of perishability on the demand. Among the insights gathered from the numerical analysis, we find that dynamic pricing aids extending shelf life and when disposal incurs a lower cost, or even a positive salvage value, the retailer is induced to dispose earlier since the benefit of selling new inventory offsets the loss due to disposal. We also observe that the faster the perceived rate of deterioration, the lower the threshold of the ending inventory for disposal. Perhaps a bit counter‐intuitive, maximizing profits does not mean eliminating disposals or expirations.

The purpose of this research is to furnish the OR/MS research community with an updated assessment of the discipline's journals set with refinements that also highlight the various characteristics of OR/MS journals. More specifically, we apply a refined PageRank method initially proposed by Xu et al. (2011) to evaluate the top 31 OR/MS journals for 2010, and report our findings. We also report the shifts in the rankings that span 5 years, from 2006 to 2010. We observe that Manufacturing and Service Operations Management, indexed by the SCI only in 2008, is a specialized journal that is consistently highly regarded within the discipline. The rankings also suggest that Management Science is more established as a generalized journal as it has more external impact. In general, our ranking results correlate with expert opinions, and we also observe, report and discuss some interesting patterns that have emerged over the past 5 years from 2006 to 2010.

This note studies the optimal dynamic decision-making problem for a retailer in a price-sensitive, multiplicative demand framework. Our model incorporates lost sales, holding cost, fixed and variable procurement costs, as well as salvage value. We characterize the structure of the retailer’s (discounted) expected profit-maximizing dynamic inventory policy for both finite and infinite selling horizon problems.

We study a stationary, single-stage inventory system, under periodic review, with fixed ordering costs and multiple sales levers (such as pricing, advertising, etc.). We show the optimality ofsS� -type policies in these settings under both the backordering and lost-sales assumptions. Our analysis is constructive and is based on a condition that we identify as being key to proving thesSstructure. This condition is entirely based on the single-period profit function and the demand model. Our optimality results complement the existing results in this area. Subject classifications: inventory/production: uncertainty, stochastic; marketing: pricing, advertising. Area of review: Manufacturing, Service, and Supply Chain Operations.

In this paper, a new algorithm for computing optimal (s, S) policies is derived based upon a number of new properties of the infinite horizon cost function c(s, S) as well as a new upper bound for optimal order-up-to levels S* and a new lower bound for optimal reorder levels s*. The algorithm is simple and easy to understand. Its computational complexity is only 2.4 times that required to evaluate a (specific) single (s, S) policy. The algorithm applies to both periodic review and continuous review inventory systems.

The classical proofs for the existence of a stationary (s, S) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary (s, S) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an (s, S) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an (s, S) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.

This paper considers a joint pricing and inventory control problem with setup costs and uncertain demand. Specifically, it develops an infinite horizon model that integrates pricing and inven-tory replenishment in a distribution environment. We allow for dynamically varying prices in response to changes in inventory levels by taking advantage of price-sensitive demand. Assuming that demands follow Poisson processes that are parameterized with prices, we identify a class of pricing and inventory policies which is optimal among all policies. For instance, when there are just two price levels, high and low, to be considered, the optimal joint pricing and ordering policy is characterized by the form of (s, d, D, S), where parameter s refers to reorder level, S refers to order-up-to level, d and D refer to pricing reference levels, and s ≤ d ≤ D ≤ S. This policy works as follows: once the inventory level falls to or below s, an order is placed to bring it up to S; when the inventory level is above D or below d, the product is sold at the low price, while when it is above d and below D, the product is sold at the high price. We also develop a simple yet exact algorithm to compute the optimal joint pricing and ordering policy. Keywords: Joint pricing and inventory ordering strategy, setup cost, price dependent demand, stochastic inventory model.

This paper studies a periodic-review pricing and inventory control problem for a retailer, which faces stochastic price-sensitive demand, under quite general modeling assumptions. Any unsatisfied demand is lost, and any leftover inventory at the end of the finite selling horizon has a salvage value. The cost component for the retailer includes holding, shortage, and both variable and fixed ordering costs. The retailer's objective is to maximize its discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We show that, under a mild assumption on the additive demand function, at the beginning of each period an (s,S) policy is optimal for replenishment, and the value of the optimal price depends on the inventory level after the replenishment decision has been done. Our numerical study also suggests that for a sufficiently long selling horizon, the optimal policy is almost stationary. Furthermore, the fixed ordering cost (K) plays a significant role in our modeling framework. Specifically, any increase in K results in lower s and higher S. On the other hand, the profit impact of dynamically changing the retail price, contrasted with a single fixed price throughout the selling horizon, also increases with K. We demonstrate that using the optimal policy values from a model with backordering of unmet demands as approximations in our model might result in significant profit penalty. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2006

A complete computational approach for finding optimal (s, S) inventory policies is developed. The method is an efficient and unified approach for all values of the model parameters, including a non-negative set-up cost, a discount factor 0 \leqq \alpha \leqq 1, and a lead time. The method is derived from renewal theory and stationary analysis, generalized to permit the unit interval range of values for \alpha . Careful attention is given to the problem associated with specifying a starting condition (when \alpha < 1); a resolution is found that guarantees an (s, S) policy optimal for all starting conditions is produced by the computations. New upper and lower bounds on the optimal values of both s and S are established. The special case of linear holding and penalty costs is treated in detail. In the final section, a model in which there is a minimum guaranteed demand in each period is studied, and a simplified method of solution is developed.

It has been shown that a class of (s, S) policies is optimal to the single item/location inventory system. However, the computational complexity of finding the optimal (s, S) policy has restricted applications of this inventory system. This paper proposes a new algorithm to search for the optimal pair of s and S. We introduce a dummy cost factor and an auxiliary function into our algorithm. The algorithm searches for the optimal dummy cost through continuously evaluating the auxiliary function. It differs from the approach of Zheng and Federgruen (1991) in several aspects and has certain advantages. First, as it revises the dummy cost based on the sign of the auxiliary function, the primary goal of the search is not to compute the optimal s and S during each iteration. Second, by identifying the non-prospective sets of S, the algorithm further reduces the search effort. Numerical tests show that on the average, the proposed algorithm saves more than 30% of evaluation effort compared with Zheng and Federgruen's method.

We analyze an infinite horizon, single product, continuous review model in which pricing and inventory decisions are made simultaneously and ordering cost includes a fixed cost. We show that there exists a stationary (s,S) inventory policy maximizing the expected discounted or expected average profit under general conditions.

We analyze an infinite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are identically distributed random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to maximize expected discounted, or expected average profit over the infinite planning horizon. We show that a stationary (s,S,p) policy is optimal for both the discounted and average profit models with general demand functions. In such a policy, the period inventory is managed based on the classical (s,S) policy and price is determined based on the inventory position at the beginning of each period. Singapore-MIT Alliance (SMA)

We analyze a finite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon. We show that when the demand model is additive, the profit-to-go functions are k-concave and hence an (s,S,p) policy is optimal. In such a policy, the period inventory is managed based on the classical (s,S) policy and price is determined based on the inventory position at the beginning of each period. For more general demand functions, i.e., multiplicative plus additive functions, we demonstrate that the profit-to-go function is not necessarily k-concave and an (s,S,p) policy is not necessarily optimal. We introduce a new concept, the symmetric k-concave functions and apply it to provide a characterization of the optimal policy. Singapore-MIT Alliance (SMA)