Expected Profit Function Research Articles

Transient Analysis for a Queuing System with Impatient Customers and Its Applications to the Pricing Strategy of a Video Website

In this paper, we consider a queuing system with impatient customers, which includes infinite servers and two types of customers. During the service process, Type-1 customers may leave the system or upgrade to be Type-2 customers due to their impatience. By solving the partial differential equations, we obtain the generating functions of the transient distribution of the queue length, and many stationary performance measures are further derived. Then, as an application, we formulate an expected profit function for a video website, and maximize it by determining the optimal pricing strategy. Finally, numerical examples are provided to demonstrate the impacts of parameters on the optimal website profit.

Joint optimization of dynamic pricing, quality, and production quantity in the presence of online reviews: A distributionally robust optimization approach

This paper investigates the joint pricing, product quality, and production quantity decisions of a firm in a two-stage scenario, considering potential market demand uncertainty and the existence of online reviews. Two decision models are established based on whether the firm adjusts product quality and prices according to consumer feedback from reviews: a static model (as a baseline) and a dynamic model. In the static model, the firm decides product quality, prices, and production quantity in the first stage, while in the second stage, the firm does not enhance product quality, and prices remain unchanged. In the dynamic model, in the first stage, the firm decides initial product quality and pricing, then in the second stage, adjusts product quality based on consumer feedback and sets pricing accordingly. We propose a distributionally robust optimization (DRO) method to address this problem, obtaining a closed-form solution for the expected profit function under the worst distribution, making it easy to determine the optimal decision. Through numerical simulation analysis, we find that the firm implementing dynamic decisions always make higher profits compared to those using static decisions. This highlights the superiority of dynamic decisions in adapting to market changes and improving product quality. We also analyze the impact of expected and standard deviation of potential market demand, as well as consumer private assessment weight, on business decisions and profits. Our research provides effective support for firms with limited market information, enabling them to better formulate joint pricing, quality, and production decisions.

Determination of optimal warranty length for smart wearable device considering customer complaint

ABSTRACT This study investigates optimal warranty length for a new type of smart wearable device to maximize manufacturer’s profit in a competitive market. Most studies explore optimal warranty length based on product performance without considering customer complaints. In this study, unlike existing models, the optimal warranty length is obtained by maximizing an expected profit function comprising essential factors such as customer complaints, early and late warranty periods, product reliability, and reduced profit, using Bayesian analysis. Managers can effectively integrate these factors, and can be more confident about the accuracy of estimated optimal warranty length. This study’s contribution is to propose a methodology to develop new types of smart wearable devices considering their long-term commercial prospects. Sensitivity analyses indicate that in the early warranty period, managers should be more concerned about average reduced profit due to customer complaints and reduced profit percentage due to complaint handling costs. In the late warranty period, improved profit percentage due to good response to complaints is essential in smart wearable device sales. Consequently, managers should adequately deal with customer complaints about smart wearable devices, whether in the early or late warranty period.

Is the online-offline buy-online-pickup-in-store retail strategy best among other product delivery strategies under variable lead time?

Every industry improves customer satisfaction by either preventing stockout situations or fulfilling the customer's demand by reducing lead time and variance of the lead time with several services offered. This study considers online selling, offline selling, and buy-online-pickup-in-store policies for selling products from retail stores. This study builds a characteristic relation between buy-online-pickup-in-store, backorder, and on-hand inventory by giving an exact total profit function instead of the expected profit function. The marginal values on lead time and variance of the lead time earn more exact outcomes for any retail strategy. The specific transportation cost for different retail strategies is considered and tested. This paper shows the analytical results which prove that the total profit of the sustainable supply chain is a convex function of both lead time and variance of the lead time. As different channel selling is crucial to any industry for increasing customer demand, the retailer sells their products by hybrid channel and advertises their products online to make them more famous. The model is solved through a classical optimization technique to get a semi-closed-form solution. In this method, it is possible to find out the optimum results for the decision variables separately by equating the first order partial derivative of the total profit with respect to the decision variables to zero and the global optimality by the Hessian matrix. A numerical experiment is conducted here, and the result shows that the supply chain system gives 14.48% more profit if advertisement policy is applied. The supply chain system gives 57.31% less profit and 72.72% less profit when only an online or offline channel is adopted, respectively. A sensitivity analysis is done for the managerial effects of cost parameters on the total profit.

Decision making under service‐level contracts: The role of cost saliency

This study examines human ordering behavior in service‐level inventory contracts, a class of contracts important in practice. Studies of wholesale price contracts find that people tend to place orders that are suboptimal and biased toward mean demand. Unlike wholesale price contracts, service‐level contracts can be parameterized such that they have steep expected profit functions, making the expected profit‐maximizing order more salient, in the sense that deviations from optimal ordering are more costly. Utilizing an analytical model and results from existing literature, we hypothesize that people will order closer to optimality under service‐level contracts with steeper expected profit functions. In a laboratory experiment, we find that subjects achieve up to 97.2% supply chain efficiency under a steep service‐level contract, compared with 92.2% under a flat service‐level contract, and steep service contract ordering also exhibits lower variability. Our results suggest that managers can benefit by designing service‐level contracts with higher penalty costs and lower fill rates.

Bidding to procure supply in newsvendor networks

We develop a model to determine the optimal bids a trade agent submits to sellers in a B2B market. The trade agent bids at multiple supply locations and allocates the goods procured after bidding to fulfill the demand of multiple clients. We analyze the structural properties of the trade agent's expected profit function and the optimal bids.

An emergency supplies procurement strategy based on a bidirectional option contract

The occurrence of a disaster and its location, type, intensity, scope and duration are highly uncertain, making it hard to accurately estimate the emergency demand. As the main purchasers and managers of emergency supplies, governments face shortage and surplus risks when pre-purchasing and reserving emergency supplies before disasters occur. To address this dilemma, we design an emergency supplies procurement strategy based on a bidirectional option contract (BOC) by considering the system as a relief supply chain. Expected profit functions of the relief supply chain members include the procurement cost, inventory cost and production cost of emergency supplies, the cost and revenue of option trading, the salvage value and the value of demand satisfaction. We derive the purchaser’s optimal procurement portfolio and demonstrate that the BOC enables the coordination of the relief supply chain. We illustrate the superiority of the BOC procurement strategy over the benchmark procurement scenario without the use of the BOC. As an extension, we also compare the BOC with two unilateral option contracts in emergency supplies procurement and summarise the advantage of the BOC. The research extends the application context of the BOC and provides a feasible strategy for governments to procure emergency supplies.

Study on Incentive Mechanism of Electronic Warehouse Receipt Financing Mode in Online Supply Chain Finance

We take the electronic warehouse receipt business patterns of online supply chain finance as the research object, and based on the perspective of moral risk prevention and adverse selection, with the goal of maximizing the overall benefits of supply chain, we focus on the incentive mechanism between banks and B2B platform and its own logistics enterprises. By constructing the expected profit function, the optimal profit function of the bank and the B2B platform can be deduced, which can directly stimulate the B2B platform and its own logistics enterprises, so as to prevent the moral hazard of the B2B platform. The relevant parameters are assigned and some parameters are simulated and analyzed by MATLAB. Therefore, we get the variation trends of related functions and the influencing factors, and verify the incentive model has certain practical value. Through the establishment of incentive mechanism we find that it is very useful for reducing the risk of principal-agent and credit bank. Risk reduction will greatly improve the enthusiasm of B2B platforms and their own logistics enterprises.

Stability analysis of a standby system with an unreliable server and switching failure

Abstract This paper is devoted to analysing stability of a standby system with an unreliable server and switching failure, where both the time-to-repair of units and the time-to-repair of server follow general distributions. By employing $C_0$ semigroup theory, we show that time-dependent solution of the system converges to steady-state solution under a rather mild condition. Further, if component’ repair rate and server’s repair rate satisfy a reasonable assumption, then the system is exponentially stable. Based on the above theoretical foundation, some steady-state reliability indices are obtained and total expected profit function per unit time is developed. Finally, a sensitivity analysis with numerical illustrations is also provided.

The Generalized First- and Second-Price Auctions: Overbidding, Underbidding, and Optimal Reserve Price

The paper generalizes the models of the standard first- and second-price auctions (FPA and SPA) by using the Cobb-Douglas function to evaluate bidders’ two conflicting bidding preferences for the probability of winning the item (Cpwin) and the profit conditional on winning (Cprofit). Compared with the popular expected profit function, this function has the advantage of allowing the bidders to equally or differently value Cpwin and Cprofit from the point of view of multiple criteria decision making, and then can capture bidders’ bidding behavior in reality. Introduction of the Cobb-Douglas function into the FPA allows us to provide new interpretations for overbidding (underbidding) behavior and the dependence of the optimal reserve price on the number of bidders. Our new interpretations are all attributed to the bidders’ different preferences for Cpwin and Cprofit. In contrast, this introduction into the SPA does not matter. Our findings suggest that the seller should prefer the FPA (SPA) over the SPA (FPA) and set a lower (higher) optimal reserve price if he conjectures that the bidders have a stronger desire to win an auction (to obtain a profit). The above results help the seller to select between the two auctions and set a reserve price in practice.

Optimal Procurement Management by Reverse Auctions for a Price-Setting Newsvendor

In this paper, we study the optimal procurement management by reverse auctions for a price-setting newsvendor (retailer) in a single period setting. The retailer facing price-dependent stochastic demand first designs a procurement contract and then invites the suppliers to bid for this contract in the reverse auction. The winning supplier produces and delivers the demanded quantity. The retailer obtains the procurement quantity and simultaneously determines the retail price. By using the price elasticity of the lost-sales rate, we show that the retailer’s expected profit (excluding the procurement cost) is a concave function of the purchased quantity, which can be used to obtain the optimal procurement and retail pricing decisions for the retailer. Further, when the underlying random term of demand function is normally distributed under left-truncation (at 0), we get the analytical expressions of the purchased quantity and expected profit function for the retailer. Moreover, some numerical examples are given.

A newsvendor problem with markup pricing in the presence of within-period price fluctuations

We consider a single-item single-period joint inventory management and pricing problem of a retailer selling an item that has selling price uncertainties. Unlike most of the literature on the newsvendor problem, we assume that price-dependent demand arrives randomly according to a stochastic arrival process whose rate depends on the fluctuating market input price process. The retailer’s problem is to choose the order quantity and a proportional price markup over the input price to maximize the expected profit. This setting is mostly encountered by retailers that trade in different currencies or have to purchase and convert commodities for seasonal sales. For this setting, we characterize both the optimal inventory and markup levels. We present monotonicity properties of the expected profit function with respect to each decision variable. We also show that more volatile input price processes lead to lower expected profits.

Pricing and inventory control decisions in the stochastic hybrid production systems with multiple recovery options

Although pricing and inventory control are crucial decisions in each production system, these decisions are usually investigated separately. This paper considers pricing and inventory control decisions simultaneously in a hybrid production system. The hybrid production system has two recovery options, remanufacturing and refurbishing sites. The demand follows Poisson distribution, which depends on the sale price of each product. Returned products arrive according to a Poisson process. Each returned product can be remanufactured, refurbished, or disposed. The time to manufacturing, refurbishing, and remanufacturing a product also follows an exponential distribution. By modeling the system as a Markov chain, the long-run expected profit function is obtained in terms of the dispose-down-to level of returned products and the order-up-to level and the sale price of serviceable products. A three-dimensional state space of the Markov Chain dependent to the sale price is developed considering pricing and inventory control decisions simultaneously with remanufacturing and refurbishing returned products. Since the model is a mixed integer nonlinear programming and known as complex models, the Artificial Bee Colony (ABC) algorithm, simulation and complete search method are used to solve the problem. The results show that by increasing the purchase price of the returned products, the amount of returned products will increase. If the refurbishing cost of the returned products is high or the disposal cost is low, less inventory should be kept in the system with a high price of serviceable products. If the lost sale cost is high, the more inventory should be maintained. Moreover, by decreasing the price elasticity of demand, the customer’s demand increases, and then more inventory should be maintained in the system.

Analysis of a Variant Working Vacation Queue with Customer Impatience and Server Breakdowns

This paper analyses an infinite buffer single server variant working vacation queueing model in which the arriving customer may balk and the server is subject to breakdown. The service times during regular busy period and working vacation period, vacation times, breakdown times and repair times are assumed to be exponentially distributed and are mutually independent. In working vacation period, the customer may renege due to impatience. We derive the probability generating function of the steady-state probabilities and obtain the closed form expressions of the system size. In addition, we obtain some performance measures and a total expected profit function per unit time is designed to determine the optimal values at the maximum profit. We employ the particle swarm optimisation method to solve the profit optimisation problem.

Supply Chains Involving a Mean‐Variance‐Skewness‐Kurtosis Newsvendor: Analysis and Coordination

The classical newsvendor problem seeks to minimize the expected inventory cost or maximize the expected profit. But optimizing an expected value alone does not fully capture the stochastic nature of the newsvendor problem. Inspired by the higher‐moment analyses explored in finance literature, we conduct a mean‐variance‐skewness‐kurtosis (MVSK) analysis for the newsvendor problem. We first derive the analytical expressions for the profit’s mean, variance, skewness, and kurtosis in the standard newsvendor setting, and reveal their structural properties. We then establish various MVSK optimization problems and find the solution to each of them. We show that kurtosis aversion always induces the newsvendor to order less, while skewness seeking can induce the newsvendor to order either more or less depending on the specific structure of the profit’s skewness, which is affected by the symmetric and asymmetric properties of the demand distribution. Finally, based on the Pareto‐optimality concept, we address the challenge of supply chain coordination (SCC) in the presence of MVSK agents in two specific cases: (i) each agent maximizes its MVSK‐objective‐function and (ii) each agent maximizes its expected profit function, subject to given constraints on the profit’s variance, skewness, and kurtosis. In each case, we explore whether and how the supply chain can be coordinated. We find that considering the MVSK preferences of supply chain agents will affect the achievability of SCC and flexibility of the coordinating contract. We also uncover that if we assume an individual MVSK agent to be an MV one, the achievability of SCC by contracts will be very much negatively affected.

Price competition between green and non green products under corporate social responsible firm

This article deals with a newsvendor inventory model in light of green product marketing of corporate social responsible firms. In this model, comparison between green and non green marketing is analyzed including subsidy and tax implementation by Government where the Government offers higher subsidy and lower tax to the green producer unlike the lower subsidy and higher tax to the non green producer. There is also price contest between green and non green producer as the demands of the products are dependent on sales price, carbon emission and corporate social responsibility index. Assuming the cost and profit parameters, an expected profit function of the systems is formulated and maximized analytically. Finally, numerical examples are illustrated to justify the proposed model.

Paralyzed by Fear: Rigid and Discrete Pricing Under Demand Uncertainty

We propose a new theory of price rigidity based on firms' Knightian uncertainty about their competitive environment. This uncertainty has two key implications. First, firms learn about the shape of their demand function from past observations of quantities sold. This learning gives rise to kinks in the expected profit function at previously observed prices, making those prices both sticky and more likely to reoccur. Second, uncertainty about the relationship between aggregate and industry‐level inflation generates nominal rigidity. We prove the main insights analytically and quantify the effects of our mechanism. Our estimated quantitative model is consistent with a wide range of micro‐level pricing facts that are typically challenging to match jointly. It also implies significantly more persistent monetary non‐neutrality than in standard models, allowing it to generate large real effects from nominal shocks.

Inventory and pricing decisions for imperfect quality items with inspection errors, sales returns, and partial backorders under inflation

In this paper, an optimal replenishment inventory policy for imperfect quality items is presented with a selling price-dependent demand under inflationary conditions using a discounted cash flow (DCF) approach. Due to the presence of defectives in the system, all items go through a 100% inspection process. However, the screening process is also considered to be imperfect and involves errors, namely Type-I and Type-II. In addition, shortages are allowed and are partially backlogged. An optimal solution for the proposed model is derived by maximizing the expected profit function by jointly optimizing three decision variables: selling price, order quantity, and backorder level. To validate the theoretical results, a numerical example along with comprehensive sensitivity analysis is offered. The model has pertinence in industries like textiles, electronics, furniture, footwear, automobiles, and plastics.

Using social work donation as a tool of corporate social responsibility in a closed-loop supply chain considering carbon emissions tax and demand uncertainty

Companies are increasingly realizing the benefits of adopting Corporate Social Responsibility (CSR) and including it into their supply chain strategy and operations. We consider two CSR tools: social work donation (SWD) and recycling investment. We integrate the CSR investments into a closed loop supply chain where the stochastic demand depends on the sales price and social work donation. We consider four decisions jointly: donation amount, recycling investment, pricing, and order quantity. Mathematical models are proposed for expected profit maximization under a carbon emissions tax. A two-stage approach and a distribution-free approach are used if limited information is available on the random demand. Optimality conditions and sensitivity of the expected profit functions are analysed. The investment function reflects the joint effect of two different environmental/social investments allowing to study the interactions of the decisions together with the price decision. The study reveals that a company is motivated to use SWD as a tool of social responsibility if the demand has a higher SWD elasticity parameter than price sensitivity parameter. Our models demonstrate that the increment of customers' SWD elasticity has a positive impact on recycling that will also help to reduce the carbon emission tax.

Analysis of strategies for substitutable and complementary products in a two-levels fuzzy supply chain system

In this paper, the pricing competition models have been studied between two substitutable and one complementary products in a two-echelon fuzzy supply chain system. Here, three manufacturers separately (one each) produce and sell their products through a common retailer in the cooperative and non-cooperative markets. The demand for each product depends linearly on the products’ prices. Here, manufacturing costs, base demands and price elasticity are characterized as fuzzy parameters. The closed-form expressions of optimal wholesale and retail pricing decisions have been derived for four different market situations under game theory to maximize the expected profit function of each participant of the supply chain. Sharing of profits under co-operation mechanism amongst the manufacturers and retailer in different proportions are presented numerically for the fuzzy model. As particular cases, the optimal pricing decisions have been derived for two substitutable products or two complementary products separately and the results of previous investigators are presented. In addition, the traditional deterministic version of the fuzzy problem is numerically solved. The results of fuzzy and crisp models are presented and compared. Finally, the problem is illustrated with some numerical data for four market situations, sensitivity analyses are performed and management decisions are explored. The products’ demands and supply chain profits are graphically presented for different price elasticity of the substitutable and complementary products.