Profit Maximization Problem Research Articles

An EOQ model of deteriorating item in imprecise environment with dynamic deterioration and credit linked demand

In this paper, an inventory model for a deteriorating item with time dependent deterioration has been developed in imprecise environment. For an order more than a fixed label, the supplier offers a credit period for payments per order to his/her retailers, but an interest is charged by the supplier if the payment period exceeds the credit period. The supplier also offers a partial permissible delay in payments even if the order quantity is less than the said fixed label, i.e., the credit period is offered on a fraction of the amount purchased. Due to these facilities, the retailer also offers a fixed partial credit period to his/her customers to boost the demand. Obviously, demand depends on the duration on which the credit period is offered. Depending upon the different types of imprecise inventory costs (fuzzy, rough or fuzzy-rough), different models are developed. Models are formulated as a profit maximization problem with respect to the retailer and depending upon different imprecise parameters, different models are solved. As the optimization of imprecise objectives are not well defined, their expected values are optimized to find an optimal decision for the decision maker (DM). Non-linear optimization problems for different models are solved using a gradient based optimization technique – Generalized Reduced Gradient (GRG) method (using LINGO software). Numerical experiments are performed to illustrate the models and some parametric studies are made.

Demand side management in a smart grid with multiple electricity suppliers

In future smart grids, the electricity suppliers can modify the customers' load consumption pattern by implementing appropriate DSM (demand side management) programs using smart meters. Most of the existing studies on DSM, only consider one utility company in the supplier side. In this paper, the possibility of existing more than one supplier in the smart grid is addressed by modeling the DSM problem as two non-cooperative games: the supplier side game, and the customer side game. In the first game, the suppliers' profit maximization problem is formulated by applying supply function bidding mechanism. In the proposed mechanism, the electricity suppliers submit their bids to the DSM center, where the electricity price is computed and is sent to the customers. In the second game, the customers aim to determine optimal load profile to maximize their daily payoff. The existence and uniqueness of the Nash equilibrium in the mentioned games are explored and a computationally tractable distributed algorithm is designed to determine the equilibrium. Simulations are performed for a smart grid system with 3 suppliers and 1000 customers. Simulation results demonstrate the superior performance of the proposed mechanism in reducing the peak load and increasing the suppliers' profit and the customers' payoff.

Optimal Offering Strategy of a Virtual Power Plant: A Stochastic Bi-Level Approach

This paper addresses the optimal bidding strategy problem of a commercial virtual power plant (CVPP), which comprises of distributed energy resources (DERs), battery storage systems (BSS), and electricity consumers, and participates in the day-ahead (DA) electricity market. The ultimate goal of the CVPP is the maximization of the DA profit in conjunction with the minimization of the anticipated real-time production and the consumption of imbalance charges. A three-stage stochastic bi-level optimization model is formulated, where the uncertainty lies in the DA CVPP DER production and load consumption, as well as in the rivals’ offer curves and real-time balancing prices. Demand response schemes are also incorporated into the virtual power plant (VPP) portfolio. The proposed bi-level model consists of an upper level that represents the VPP profit maximization problem and a lower level that represents the independent system operator (ISO) DA market-clearing problem. This bi-level optimization problem is converted into a mixed-integer linear programing model using the Karush–Kuhn–Tucker optimality conditions and the strong duality theory. Finally, the risk associated with the VPP profit variability is explicitly taken into account through the incorporation of the conditional value-at-risk metric. Simulations on the Greek power system demonstrate the applicability and effectiveness of the proposed model.

Exercise of Market Power on Ramp Rate in Wind-Integrated Power Systems

With an increasing penetration of wind power, there is likely to be an increasing need for fast-ramping generating units. These generators ensure that no load is lost if supply drops due to the uncertainties in wind power generation. However, it is observed in practice that, in a presence of network constraints, fast-ramping generating units are prone to act strategically and exercise market power by withholding their ramp rates. In this paper we model this gaming behavior on ramp rates. We assume a market operator who collects bids in form of marginal costs, quantities, and ramp rates. He runs a ramp-constrained economic dispatch given the generators' bids, forecasted demand, and contingencies. Following the game-theoretic concepts, we set up a multi-level optimization problem. The lower-level problem is the ramp-constrained economic dispatch and the higher-level represents the profit maximization problems solved by strategic generators. The whole problem is formulated as an equilibrium problem with equilibrium constraints (EPEC). The outcome of the EPEC problem is a set of Nash equilibria. To tackle the multiple Nash equilibria problem, the concept of the extremal-Nash equilibria is defined and formulated. We model the concept of extremal-Nash equilibria as a single-stage mixed-integer linear programming problem (MILP) and demonstrate the application of this mathematical framework on an illustrative case and on a more realistic case study with tractable results.

Multi-item fuzzy-stochastic supply chain models for long-term contracts with a profit sharing scheme

In this study, we develop multi-item multi-objective manufacturer–retailer supply chain models with risk and budget constraints for long-term contracts with a profit sharing scheme in a fuzzy-stochastic environment. The manufacturing costs of the items are fuzzy and the demands for the items are random during each period. We examine two separate models with crisp and probabilistic budget constraints for long-term contracts where the business relationship is repeated, while demand is uncertain in a stochastic sense. The models are formulated as profit maximization problems. Next, the fuzzy compromise programming method, global criteria method, and weighted sum method are used to reduce the multi-objective problems into their corresponding single objective problems. Finally, the problems are solved using a gradient-based nonlinear programming technique: the generalized reduced gradient method. The models are illustrated with numerical examples for different types of stochastic demand and a comparative study is presented.

Analysis of an imperfectly competitive cellulosic biofuel supply chain

We study the strategic behavior in an imperfectly competitive cellulosic biofuel supply chain. An optimization-based supply chain model is used to obtain long-run planning outcomes, based on which we develop market models considering both perfect and imperfect competitions. The equilibrium among stakeholders in the multi-echelon supply chain can be obtained by solving a collection of first-order conditions associated with their profit-maximization problems. For the imperfect competition, the model, additionally, allows firms with significant market share at different segment of the supply chain to exercise market power. We apply the models to an illustrative case study of California.

Inventory policy of a deteriorating item with variable demand under trade credit period

In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailer’s profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.

How to Generate Good Profit Maximization Problems

In this article, the author considers the merits of two classes of profit maximization problems: those involving perfectly competitive firms with quadratic and cubic cost functions. While relatively easy to develop and solve, problems based on quadratic cost functions are too simple to address a number of important issues, such as the use of second-order conditions and the short-run shutdown condition. Problems based on cubic cost functions are mathematically richer but often involve messy arithmetic; furthermore, many are not plausible representations of a firm's costs. Finding cubic functions that do not suffer from these drawbacks can be a time-consuming process. The author addresses this issue by providing a procedure to generate profit maximization problems that are theoretically interesting, economically plausible, and computationally simple.

Financing and reliability analysis for schedules in the project calendar planning problem

We continue our earlier studies of the profit maximization problem in calendar planning of investment projects taking into account reinvesting of the obtained revenue and possible credit financing. We construct the corresponding model and describe a situation when only part of the jobs is financed with own money and credits, and the rest is raised by reinvesting the revenue. We study the calendar planning problem where incomes from various jobs are random values. We study the risks related both to getting a smaller income than expected and to failing a project. We propose an approach to estimate the reliability of job schedules.

The Operation Dispatch of Cogeneration Systems in the Deregulation Environment

In this paper, an optimal strategy was presented to solve the operation dispatch of cogeneration systems in a deregulated market. With the load demand including steam and electricity, the operational model of cogeneration system was derived by considering the bi-lateral contracts and operation constraints. The objective function is formulated the profit-maximizing problem in the searching process. Sequential Quadratic Programming (SQP) was used to solve this problem. Test results verify that SQP can offer an efficient way for cogeneration plants to meet the load growth and promoted the competed ability of cogeneration plants.

Optimal dynamic policies for integrated production and marketing planning in business-to-business marketplaces

This work develops optimal dynamic policies for integrated production and marketing planning in a vertically decentralized single-manufacturer and single-retailer channel over a multi-period planning horizon, subject to deteriorating goods and a multivariate demand function. This work formulates the discount profit maximization problem, and provides inter-enterprise dynamic joint decisions for retail price and replenishment schedule/quantity using a calculus-based formulation combined with dynamic programming. Additionally, two alternatives for doing business, namely, retailer-managed inventory with a price-only contract and vendor-managed inventory (VMI) with a consignment contract, are applied to business-to-business traditional marketplaces (TMs) and electronic marketplaces (EMs), respectively. Numerical results demonstrate that solutions generated in EMs outperform those in TMs in terms of maximizing channel-wide total discount profits and those of manufacturer and retailer. Further, analytical results show that the proposed policy under VMI with a consignment contract in EMs significantly increases system efficiency and simultaneously achieves Pareto improvements using an extra one-part tariff for the decentralized channel.

Dynamic Profit Maximization of Cognitive Mobile Virtual Network Operator

We study the profit maximization problem of a cognitive virtual network operator in a dynamic network environment. We consider a downlink OFDM communication system with various network dynamics, including dynamic user demands, uncertain sensing spectrum resources, dynamic spectrum prices, and time-varying channel conditions. In addition, heterogeneous users and imperfect sensing technology are incorporated to make the network model more realistic. By exploring the special structural of the problem, we develop a low-complexity online control policies that determine pricing and resource scheduling without knowing the statistics of dynamic network parameters. We show that the proposed algorithms can achieve arbitrarily close to the optimal profit with a proper tradeoff with the queuing delay.

From Corporate Market Research to Bank Profit Maximization

The purpose of this article is to study the target segment of the corporate bank market in order to reveal the demand for financial products and to incorporate this demand into solving the problem of bank profit maximization. Instrumental guidelines for conducting market research that allow to study the aspects of the problem specifically dealing with the efficiency of a bank’s cooperation with corporate customers has been developed as a technique for studying target corporate market customers. A target retail trade market sector and a client base of corporate customers including 430 trade network operators functioning in the northwest region of Russia were chosen as objects of research. The article defines the retail market demand potential for bank products, analyses the prospective viability of providing additional bank products to retail market customers, and identifies the corporate customer’s informational portrait. On the basis of obtained results, the problem of maximizing the bank’s extra profit from serving trade network operators. The research is confined to the client base of one major Russian bank operating in the northwest region of Russia; it will be useful to adapt the material of this article to studying target sectorial markets nationwide. The results of market research were used to assess the economic potential (estimated optimal profit) of the commercial bank’s cooperation with trade network operators. This article constitutes the groundwork for organizing effective bank marketing that guarantees a congruity of the bank’s resources and capabilities with the demands and requirements of the market in which it operates

An inventory model for perishable items with stock and advertisement sensitive demand

This paper deals with an inventory model in which the demand rate is a function of both the stock level and the effort expended on advertising the product. The expenditure on advertising is considered to be a quadratic function of the effort of the salesmen. The deterioration of items is assumed to be constant. The decision variables are the initial replenishment lot size level and the investment in advertising for increasing sales effort in order to maximize profit. The paper extends an inventory model over finite and infinite time horizons. The associated profit maximization problem is solved by the ‘Euler–Lagrangian method’. The model is illustrated with the help of a numerical example.

Economic production quantity model with trade credit financing and price-discount offer for non-decreasing time varying demand pattern

As a policy when some suppliers offer trade credit periods and price discounts to the retailers in order to increase the demand of their product, the retailers have to face different types of offer of discounts and credits for which they have to take a decision which is the best offer for them for profit. One thing is important to the retailers that they try to buy good quality items at a reasonable price and if possible, they try to invest returns obtained by selling those items in such a manner that their business is not hampered. In this paper, we intend to develop an economic production quantity (EPQ)-based model with non-decreasing time varying (quadratic) demand pattern in order to investigate the inventory system as a profit maximisation problem when delay in payment and price discount are permitted by the suppliers to the retailers. Mathematical theorems are developed analytically to determine optimal replenishment policies and a lot of managerial phenomena are obtained through numerical examples.

Equilibrium Customer Strategies in the Single-Server Constant Retrial Queue with Breakdowns and Repairs

We consider a single-server constant retrial queueing system with a Poisson arrival process and exponential service and retrial times, in which the server may break down when it is working. The lifetime of the server is assumed to be exponentially distributed and once the server breaks down, it will be sent for repair immediately and the repair time is also exponentially distributed. There is no waiting space in front of the server and arriving customers decide whether to enter the retrial orbit or to balk depending on the available information they get upon arrival. In the paper, Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated. We consider separately the partially observable case where an arriving customer knows the state of the server but does not observe the exact number of customers waiting for service and the fully observable case where customer gets informed not only about the state of the server but also about the exact number of customers in the orbit. Some numerical examples are presented to illustrate the effect of the information levels and several parameters on the customers’ equilibrium and optimal strategies.

A Simultaneous Pricing and Inventory Control Model for a Single Product with Capacitated Working Capital

The simultaneous stochastic pricing and inventory management with capacitated cash-on-hand is a profit maximization problem faced by a firm, which has limited working capital for its liabilities. In addition to this working capital constraint, the company has to dynamically price its products and determine inventory levels to minimize its expected cost. In our study, we tackle this complicated problem by assuming a single product, periodic review inventory model with finite multiple periods, and imperfect market information. We also assume no fixed cost in order to relax the complexity of the problem and perform numerical analysis for providing managerial implications. We investigate several research questions to extend our understanding of this complicated but very practical problem faced by numerous decision-makers in organizations. A discussion of possible optimal policies for this complicated unanswered problem with working capital constraints is explored.

Multicrop Production Models with Multinomial Logit Acreage Shares

The main objective of this paper is to show that the MultiNomial Logit (MNL) framework can be used for deriving simple acreage share models. More specifically we show that MNL acreage choice models can be defined as parts of consistent multicrop models. When coupled with suitably designed crop production models, e.g. a specific parameterization of the standard quadratic yield function, the parameters of these multicrop models are easily interpreted and estimated. Although MNL acreage choice models are usually derived by aggregating crop choices defined at the plot level, we show that these models can also be derived by considering a profit maximisation problem at the farm level. This provides a simple theoretical background for the Standard MNL acreage share model, as well as for simple extensions of these models: the Nested MNL acreage share models. An empirical application on French farm-level data over the period 1996–2007 investigates the empirical performances of the multicrop econometric models based on the proposed MNL framework for modelling acreage choices.

Optimal ordering and pricing policy for demand functions that are separable into price and inventory age

We formulate and analyze two models for determining the optimal pricing, order quantity and replenishment period for items whose demand function is separable into components of price and inventory age. The first model assumes a multiplicative demand function. We provide conditions, which are satisfied by most common price-dependent demand functions, to reduce the three-variable profit maximization problem into a single-variable problem, which can be solved using an efficient line-search method. Next, we show that a genuine additive model cannot exist, and instead suggest and analyze a pseudo-additive model. However, this model is more limited than the multiplicative model in its ability to incorporate various combinations of price and inventory age effects, and reduction of the maximization problem into a single-variable problem is more complicated, except in the case of a linear price effect, which is further analyzed. For both models, we show that the optimal solution satisfies the first-order condition for equilibrium under a monopoly, with a modification that includes inventory holding costs. We solve numerical examples to illustrate the solution procedures.

Dynamic co-opetitive approach of a closed loop system with remanufacturing for deteriorating items in e-markets

This study examines the dynamic economics of closed loop supply chains (CLSCs) that incorporate remanufacturing by developing analytical models under both cooperative and competitive policies in electronic markets. Specifically, this investigation presents the multi-variable profit-maximization problem and performs equilibrium analysis using a co-opetitive approach. Additionally, this study undertakes dynamic joint decisions for both brand-new and like-new versions of deteriorating items in a remanufacturing CLSC. The analytical results demonstrate that the dynamic co-opetitive decision depends on the potential size of the market for brand-new products, manufacturing and remanufacturing costs of the original equipment manufacturer, remanufacturing cost of the third-party independent operator, and intensity of competition between different products in the market.