Optimal inventory policies for profit maximizing EOQ models under various cost functions

In this paper, we establish and analyze three EOQ based inventory models under profit maximization via geometric programming (GP) techniques. Through GP, we find optimal order quantity and price for each of these models considering production (lot sizing) as well as marketing (pricing) decisions. We also investigate the effects on the changes in the optimal solutions when different parameters are changed. In addition, a comparative analysis between the profit maximization models is conducted. By investigating the error in the optimal price, order quantity, and profit of these models, several interesting economic implications and insights can be observed.     

A fuzzy stochastic model for telecommunications bandwidth brokers under probabilistic QoS measures

In this paper, we model and solve profit maximization problem of a telecommunications Bandwidth Broker (BB) under uncertain market and network infrastructure conditions. The BB may lease network capacity from a set of Backbone Providers (BPs) or from other BBs in order to gain profit by leasing already purchased capacity to end-users. BB’s problem becomes harder to deal with when bandwidth requests of end-users, profit and cost margins are not known in advance. The novelty of the proposed work is the development of a mechanism via combining fuzzy and stochastic programming methodologies for solving complex BP selection and bandwidth demand allocation problem in communication networks, based on the fact that information needed for making these decisions is not available prior to leasing capacity. In addition, suggested model aims to maximize BB’s decision maker’s satisfaction ratio rather than just profit. As a solution strategy, the resulting fuzzy stochastic programming model is transformed into deterministic crisp equivalent form and then solved to optimality. Finally, the numerical experiments show that on the average, proposed approach provides 14.30% more profit and 69.50% more satisfaction ratio compared to deterministic approaches in which randomness and vagueness in the market and infrastructure are ignored.     

Interactive fuzzy programming for two-level linear and linear fractional production and assignment problems: A case study

In this paper, we deal with actual problems on production and work force assignment in a housing material manufacturer and a subcontract firm. We formulate two kinds of two-level programming problems: one is a profit maximization problem of both the housing material manufacturer and the subcontract firm, and the other is a profitability maximization problem of them. Applying the interactive fuzzy programming for two-level linear and linear fractional programming problems, we derive satisfactory solutions to the problems. After comparing the two problems, we discuss the results of the applications and examine actual planning of the production and the work force assignment of the two firms to be implemented.     

Optimal inventory policies under decreasing cost functions via geometric programming

In this paper, we establish and analyze two economic order quantity (EOQ) based inventory models under total cost minimization and profit maximization via geometric programming (GP) techniques. Through GP, optimal solutions for both models are found and managerial implications on the optimal policy are determined through bounding and sensitivity analysis. We investigate the effects on the changes in the optimal order quantity and the demand per unit time according to varied parameters by studying optimality conditions. In addition, a comparative analysis between the total cost minimization model and the profit maximization model is conducted. By investigating the error in the optimal order quantity of these two models, several interesting economic implications and managerial insights can be observed.     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

利润最大问题与弱DEA有效(C2GS2)

对利润最大问题[1]本文用不同于文[1]的方法来讨论.通过解这一个问题,不但知道在一定条件下有最大利润的决策单元的弱DEA有效性(C2GS2),而且找到所有有最大利润的决策单元.     

Fuzzy measures for profit maximization with fuzzy parameters

Profit maximization is an important issue to the firms that pursue the largest economic profit possible. This paper extends the situation from the deterministic to uncertain, where the coefficients are represented by fuzzy numbers. Intuitively, when the problem has fuzzy parameters, the derived profit value should be a fuzzy number as well. The extension principle is utilized to develop a pair of two-level mathematical programs to calculate the upper and lower bounds of the profit value at α-cuts. Following the duality theorem and a variable separation technique, the two-level mathematical programs are transformed into a class of one-level signomial geometric programs to solve. An example is given to illustrate the idea proposed in this paper.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

Posynomial Parametric Geometric Programming with Interval Valued Coefficient

The article presents solution procedure of geometric programming with imprecise coefficients. We have considered problems with imprecise data as a form of an interval in nature. Many authors have solved the imprecise problem by geometric programming technique in a different way. In this paper, we introduce parametric functional form of an interval number and then solve the problem by geometric programming technique. The advantage of the present approach is that we get optimal solution of the objective function directly without solving equivalent transformed problems. Numerical examples are presented to support of the proposed approach.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

Fair transfer price and inventory holding policies in two-enterprise supply chains

A key issue in supply chain optimisation involving multiple enterprises is the determination of policies that optimise the performance of the supply chain as a whole while ensuring adequate rewards for each participant.In this paper, we present a mathematical programming formulation for fair, optimised profit distribution between echelons in a general multi-enterprise supply chain. The proposed formulation is based on an approach applying the Nash bargaining solution for finding optimal multi-partner profit levels subject to given minimum echelon profit requirements.The overall problem is first formulated as a mixed integer non-linear programming (MINLP) model. A spatial and binary variable branch-and-bound algorithm is then applied to the above problem based on exact and approximate linearisations of the bilinear terms involved in the model, while at each node of the search tree, a mixed integer linear programming (MILP) problem is solved. The solution comprises inter-firm transfer prices, production and inventory levels, flows of products between echelons, and sales profiles.The applicability of the proposed approach is demonstrated by a number of illustrative examples based on industrial processes.     

Multi-period reverse logistics network design for used refrigerators

This paper focuses on the design of a multi-stage reverse logistics network for product recovery. Different recovery options such as product remanufacturing, component repairing and material recycling are simultaneously considered. Initially, we propose a mixed integer linear programming model – with a profit maximization objective – for the network design problem. The structure of the product, by way of bill of materials (BOM), is also incorporated into the proposed model in order to analyze the flow at component and material levels. Sensitivity analysis is carried out to study the effects of variations in the values of the input parameters such as product return quantity, unit transportation cost per unit distance, and unit processing cost. The analysis shows that the design decisions of different facilities considerably change even for 5–20%% variations in input parameter values. This led to the development of a refined mathematical model which incorporates variations in the different input parameter values over time. The new model provides a unified design for the entire planning horizon and has been validated with the design of a used refrigerator recovery network.     

A multiple criteria model for pricing alcoholic beverages

In this paper we discuss the problem of determining optimal price changes in an alcohol sales monopoly. In Finland, the pricing of alcoholic beverages is entrusted to a State Monopoly (Alko Ltd.). The price decisions are considered to be among the most important alcohol policy measures. The pricing problem is not only a profit maximization problem. Other relevant objectives include the restriction of sales with the intention of reducing harmful effects due to alcohol consumption, and the minimization of the impact of price increases on the consumer price index.A multiple criteria model for finding the most preferred solution for the problem of pricing alcoholic beverages is developed. The model includes three criteria: profit (max.), consumption of absolute alcohol (min.), and impact on the consumer price index (min.). Using logarithms of relative changes instead of absolute criterion values makes it possible to operate with a fully linear model.The model can be solved by using any existing multiple criteria decision method. Reflecting our own bias, we use a visual interactive goal programming method developed by Korhonen and Laakso [3] and further refined by Korhonen and Wallenius [4]. The method is implemented on an IBM/PC1 under the name VIG (Visual Interactive Goal Programming).     

A Crop Planning Problem with Fuzzy Random Profit Coefficients

The main purpose of this paper is to present a crop planning problem for agricultural management under uncertainty. It is significant that agricultural managers assign their limited farmlands to cultivation of which crops in a season. This planning is called the crop planning problem and influences their incomes for the season. Usually, the crop planning problem is formulated as a linear programming problem. But there are many uncertain factors in agricultural problems, so future profits for crops are not certain values. A linear programming model with constant profit coefficients may not reflect the environment of decision making properly. Therefore, we propose a model of crop planning with fuzzy profit coefficients, and an effective solution procedure for the model. Furthermore, we extend this fuzzy model, setting the profit coefficients as discrete randomized fuzzy numbers. We show concrete optimal solutions for each models.     

Generalized Newton method for linear optimization problems with inequality constraints

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.     

Optimization of chance constraint programming with sum-of-fractional objectives – An application to assembled printed circuit board problem

This paper presents a procedure to solve a chance constraint programming problem with sum-of-fractional objectives. The problem and the solution procedure are described with the help of a practical problem – assembled printed circuit boards (PCBs). Errors occurring during assembling PCBs are in general classified into three kinds, viz. machine errors, manual errors and other errors. These errors may lead to the rejection of the major portion of the production and hence result the manufacturer a huge loss. The problem is decomposed to have two objective functions; one is a sum-of-fractional objectives and the other is a non-linear objective with bounded constraints. The interest is to maximize the sum-of-fractional objectives and to minimize the non-linear objective, subject to stochastic and non-stochastic bounded environment. The first problem deals with the maximization of the profit (maximizing sum-of-fractional objectives) and the second deals with the minimization of the loss (errors), so as to obtain the maximum profit after removing the number of defective PCBs.     

Pricing and lot-sizing for a deteriorating item in a periodic review inventory system with shortages

We consider a single product that is, subject to continuous decay, a multivariate demand function of price and time, shortages allowed and completely backlogged in a periodic review inventory system in which the selling price is allowed to adjust upward or downward periodically. The objective of this paper is to determine the periodic selling price and lot-size over multiperiod planning horizon so that the total discount profit is maximized. The proposed model can be used as an add-in optimizer like an advanced planning system in an enterprise resource planning system that coordinates distinct functions within a firm. Particular attention is placed on investigating the effect of periodic pricing jointly with shortages on the total discount profit. The problem is formulated as a bivariate optimization model solved by dynamic programming techniques coupled with an iterative search process. An intensive numerical study shows that the periodic pricing is superior to the fixed pricing in profit maximization. It also clarifies that shortages strategy can be an effective cost control mechanism for managing deterioration inventory.     

可变要素的合理投入范围研究

从利润最大化的角度,对一种可变投入要素生产的三个阶段的两种划分方法所确定的两种不同的合理投入范围进行数学分析,研究结果表明,可变要素的合理投入范围应在最大边际产量所对应的要素投入量到最大总产量所对应的要素投入量之间。     

Complete mixed integer linear programming formulations for modularity density based clustering

Modularity density maximization is a clustering method that improves some issues of the commonly used modularity maximization approach. Recently, some Mixed-Integer Linear Programming (MILP) reformulations have been proposed in the literature for the modularity density maximization problem, but they require as input the solution of a set of auxiliary binary Non-Linear Programs (NLPs). These can become computationally challenging when the size of the instances grows. In this paper we propose and compare some explicit MILP reformulations of these auxiliary binary NLPs, so that the modularity density maximization problem can be completely expressed as MILP. The resolution time is reduced by a factor up to two order of magnitude with respect to the one obtained with the binary NLPs.     

A General Alternative Procedure for Solving Negative Degree of Difficulty Problems in Geometric Programming

The degree of difficulty is an important concept in classical geometric programming theory. The dual problem is often infeasible when the degree of difficulty is negative and little has been published on this topic. In this paper, an alternative procedure is developed to find the optimal solution for the posynomial geometric programming problem with a negative degree of difficulty. First an equivalent problem was constructed with a positive degree of difficulty and the general posynomial geometric programming problem was solved using an original method previously developed by the authors. This method avoids the difficulty of non-differentiability of the dual objective function in the classical methods classified as dual. It also avoids the problem that appears when the feasible region for the dual problem is formed by an inconsistent system of linear equations.