Optimal peak load pricing with time-additive consumer preferences

It is common for a firm to make use of multiple suppliers of different delivery lead times, reliabilities, and costs. In this study, we are concerned with the joint pricing and inventory control problem for such a firm that has a quick‐response supplier and a regular supplier that both suffer random disruptions, and faces price‐sensitive random demands. We aim at characterizing the optimal ordering and pricing policies in each period over a planning horizon, and analyzing the impacts of supply source diversification. We show that, when both suppliers are unreliable, the optimal inventory policy in each period is a reorder point policy and the optimal price is decreasing in the starting inventory level in that period. In addition, we show that having supply source diversification or higher supplier reliability increases the firm's optimal profit and lowers the optimal selling price. We also demonstrate that, with the selling price as a decision, a supplier may receive even more orders from the firm after an additional supplier is introduced. For the special case where the quick‐response supplier is perfectly reliable, we further show that the optimal inventory policy is of a base‐stock type and the optimal pricing policy is a list‐price policy with markdowns.

Wait Time Based Pricing for Queues with Customer-Chosen Service Times

The Journal of FinanceVolume 41, Issue 5 p. 1141-1148 Note A Note on Optimal Credit and Pricing Policy under Uncertainty: A Contingent-Claims Approach CHUN H. LAM, CHUN H. LAMSearch for more papers by this authorANDREW H. CHEN, ANDREW H. CHEN Both authors from Edwin L. Cox School of Business, Southern Methodist University. We are indebted to the comments and suggestions of Chris Barry, Sam Dennis, and two anonymous referees on earlier versions of the paper. The authors also acknowledge the financial support from the Center for the Study of Financial Institutions and Markets at Southern Methodist University.Search for more papers by this author CHUN H. LAM, CHUN H. LAMSearch for more papers by this authorANDREW H. CHEN, ANDREW H. CHEN Both authors from Edwin L. Cox School of Business, Southern Methodist University. We are indebted to the comments and suggestions of Chris Barry, Sam Dennis, and two anonymous referees on earlier versions of the paper. The authors also acknowledge the financial support from the Center for the Study of Financial Institutions and Markets at Southern Methodist University.Search for more papers by this author First published: December 1986 https://doi.org/10.1111/j.1540-6261.1986.tb02536.xCitations: 3 Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume41, Issue5December 1986Pages 1141-1148 RelatedInformation

Problem definition: We consider the dynamic pricing problem of multiple products under (asymmetric) reference effects over an infinite horizon. Unlike existing literature, which is mostly focused on the single-product setting, our multiproduct setting takes into account the cross-product effects among substitutes and incorporates the memory-based reference prices into the multinomial logit (MNL) demand model. Even with the single-product logit demand, the structure of the optimal pricing policy is intractable. Therefore, we focus on the long-run patterns of the optimal pricing policy and also discuss the performance of the myopic pricing policy. Methodology/results: We first provide a comprehensive characterization of the myopic pricing policy, including its solution, long-run convergence behavior, and optimality gap. For the optimal pricing policy, we show an intricate connection between its long-run dynamics and types of reference effects. We demonstrate that the presence of any gain-seeking product renders a long-run constant pricing policy suboptimal. Conversely, the constant policy (or optimal steady state) can exist in both loss-neutral and loss-averse scenarios, where we provide a sufficient condition for such existence and give the analytical expression for the optimal steady state. We further show that when pricing perfect substitutes, the true optimal policy under the multiproduct framework is more likely to yield a long-run cyclic pattern than the policy derived from the single-product framework, a phenomenon that aligns well with the periodic discounts in real-world markets. Managerial implications: This discrepancy in the long-run behaviors between multi- and single-product-based policies highlights the importance of employing the multiproduct framework and addressing the cross-product effects, as sticking to the single-product framework while managing multiple substitutes can misrepresent long-run dynamics and result in suboptimality. In the multiproduct domain, our model suggests that retailers are more likely to benefit from appropriate price variations than maintaining a constant pricing policy. Funding: H. Jiang acknowledges support from the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN 2024 05796]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0801 .

Optimal pricing and inventory strategies for fashion products under time-dependent interest rate and demand

<p style='text-indent:20px;'>In the real world, the demand cannot be depicted exactly because of customer behavior cannot be forecasted without error. In this paper, we study the effect of the error of the estimated price-demand parameters by analyzing the sensitivity of the optimal joint pricing and ordering policy on the price-demand parameters based on a periodic-review, multi-period and lost sale inventory model for perishable products with constant quantity decay rate and price-sensitive demand. Firstly, we formulate the joint pricing and inventory control problem and find the optimal ordering quantity and the optimal price for deterministic price-demand function. The optimal solutions show that the retailer tends to set a lower price in early periods of each ordering cycle in order to reduce the inventory holding costs. Furthermore, the sensitivity of the optimal joint pricing and inventory control system with respect to the price-demand parameters is examined analytically and evaluated numerically. The sensitivity analysis reveals that compared to the optimal ordering quantity, the optimal prices are less sensitive in the demand-price parameters. Finally, according to the findings of the sensitivity analysis, a heuristic method of regulating the estimated demand-price parameters is employed to improve the average profit. <b>185</b> words.</p>

We consider a robust version of the classic problem of optimal monopoly pricing with incomplete information. In the robust version, the seller faces model uncertainty and only knows that the true demand distribution is in the neighborhood of a given model distribution. We characterize the optimal pricing policy under two distinct, but related, decision criteria with multiple priors: (i) maximin expected utility and (ii) minimax expected regret. The resulting optimal pricing policy under either criterion yields a robust policy to the model uncertainty. While the classic monopoly policy and the maximin criterion yield a single deterministic price, minimax regret always prescribes a random pricing policy, or equivalently, a multi-item menu policy. Distinct implications of how a monopolist responds to an increase in uncertainty emerge under the two criteria.

We consider a dynamic pricing model for selling a given stock of a perishable product over a finite time horizon. Customers, whose reservation price distribution changes over time, arrive according to a nonhomogeneous Poisson process. We show that at any given time, the optimal price decreases with inventory. We also identify a sufficient condition under which the optimal price decreases over time for a given inventory level. This sufficient condition requires that the willingness of a customer to pay a premium for the product does not increase over time. In addition to shedding managerial insight, these structural properties enable efficient computation of the optimal policy. Numerical studies are conducted to show the revenue impact of dynamic price policies. Price changes are set to compensate for statistical fluctuations of demand and to respond to shifts of the reservation price. For the former, our examples show that using optimal dynamic optimal policies achieves 2.4–7.3% revenue improvement over the optimal single price policy. For the latter, the revenue increase can be as high as 100%. These results explain why yield management has become so essential to fashion retailing and travel service industries.

Designing Shipping Policies with Top-up Options to Qualify for Free Delivery

In this paper we develop an economic order quantity model to investigate the optimal replenishment policies for instantaneous deteriorating items under inflation and trade credit. Demand rate is a linear function of selling price and decreases negative exponentially with time over a finite planning horizon. Shortages are allowed and partially backlogged. Under these conditions, we model the retailer’s inventory system as a profit maximization problem to determine the optimal selling price, optimal order quantity and optimal replenishment time. An easy-to-use algorithm is developed to determine the optimal replenishment policies for the retailer. We also provide optimal present value of profit when shortages are completely backlogged as a special case. Numerical examples are presented to illustrate the algorithm provided to obtain optimal profit. And we also obtain managerial implications from numerical examples to substantiate our model. The results show that there is an improvement in total profit from complete backlogging rather than the items being partially backlogged.

Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease

In cognitive radio networks, one key feature of spectrum trading is its short term or, even, real time, since the spectrum availability, quality, and price keep changing over time. Therefore, a spectrum pricing policy should be dynamically optimal. In this work, we address the real-time optimal pricing problem for primary users. Based on differential game model, we analyze the optimal pricing strategy for QoS-aware dynamic networks in which the secondary users' number and primary users' QoS level keep changing over time. Nash equilibrium is derived and an optimal pricing and QoS setting policy is formulated. Since the Nash equilibrium of our differential game based model deals with optimal pricing in each time instance, the real-time optimal pricing characteristic can be realized.

In this paper we study the effect of dynamic behavior of the market on the optimum pricing policy for a revenue maximizing firm in both monopoly and duopoly cases. We show that the optimum pricing policy in the dynamic market can be very different from the static one, and the most important contributing factors for this difference are: consumer response time, planning period and the discount rate. Furthermore we show that for the linear model that we use, in the limit of infinite planning horizon or market equilibrium the optimum price for the dynamic market reduces to the optimum price for the static market.

This article considers the dynamic pricing of two substitutable products over a predetermined, finite selling season. The initial inventory levels of the products are fixed exogenously and there are no replenishment opportunities during the season. It is assumed that each arriving customer chooses from available products based on the multinomial logit choice model, which captures the effect of prices on consumer choice. Every time a product runs out of stock, the set of choices shrinks, capturing the effect of stockouts on consumer choice. It is shown that, under the optimal pricing policy, the marginal value of an item is increasing in the remaining time and decreasing in its own stock level and the other product's stock level. Despite such non-surprising behavior on the part of marginal values, the optimal price itself is not simply monotonic in the remaining time or the other product's stock level. For example, a product's optimal price may increase if the remaining time decreases or if the total inventory grows. It is shown that such optimal behavior can be understood through alternative gauges such as the optimal price difference between the two products and the optimal purchase probabilities.

We describe aggregate inflation as a stochastic process in which the rate of change of the price level can be positive or zero, where the times spent in each state are of random duration. This class of processes includes Two-State Markov Chains and Renewal Processes as special cases. It is shown that the optimal pricing policy of a monopolistic firm with non-convex costs of price adjustment is (S, s) in its real-price, i.e. its nominal price relative to the price level. A basic certainty-equivalence result is proved: i.e. the firm behaves as if it faces a certain and fixed rate of inflation, higher than the actual expected rate, the difference between the two rates being a risk premium which depends on the real interest rate and the parameters of the stochastic process. One can thus apply previous results from the case of certainty (Sheshinski and Weiss, Review of Economic Studies, 1977) to obtain comparative static results. In particular, one finds that an increase in the variance of expected inflation leads firms to choose a pricing policy with larger amplitude in real price. The paper also addresses the question of consistency in firms' expectations when the price level is determined by the firms' actions. In this paper we consider pricing policies of individual firms in an inflationary environment. Each firm expects the general price level to increase and must determine the rate of increase of its own price. It is assumed that the firm incurs an adjustment cost when it changes its nominal price. Consequently, firms choose to change prices occasionally rather than continuously.