Advances in Operations Research http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Resource Constrained Project Scheduling Subject to Due Dates: Preemption Permitted with Penalty Wed, 16 Apr 2014 14:08:13 +0000 http://www.hindawi.com/journals/aor/2014/505716/ Extensive research works have been carried out in resource constrained project scheduling problem. However, scarce researches have studied the problems in which a setup cost must be incurred if activities are preempted. In this research, we investigate the resource constrained project scheduling problem to minimize the total project cost, considering earliness-tardiness and preemption penalties. A mixed integer programming formulation is proposed for the problem. The resulting problem is NP-hard. So, we try to obtain a satisfying solution using simulated annealing (SA) algorithm. The efficiency of the proposed algorithm is tested based on 150 randomly produced examples. Statistical comparison in terms of the computational times and objective function indicates that the proposed algorithm is efficient and effective. Behrouz Afshar-Nadjafi Copyright © 2014 Behrouz Afshar-Nadjafi. All rights reserved. Insensitive Bounds for the Stationary Distribution of a Single Server Retrial Queue with Server Subject to Active Breakdowns Mon, 24 Mar 2014 07:29:23 +0000 http://www.hindawi.com/journals/aor/2014/985453/ The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results. Mohamed Boualem Copyright © 2014 Mohamed Boualem. All rights reserved. Benchmarking in Data Envelopment Analysis: An Approach Based on Genetic Algorithms and Parallel Programming Thu, 13 Feb 2014 12:42:30 +0000 http://www.hindawi.com/journals/aor/2014/431749/ Data Envelopment Analysis (DEA) is a nonparametric technique to estimate the current level of efficiency of a set of entities. DEA also provides information on how to remove inefficiency through the determination of benchmarking information. This paper is devoted to study DEA models based on closest efficient targets, which are related to the shortest projection to the production frontier and allow inefficient firms to find the easiest way to improve their performance. Usually, these models have been solved by means of unsatisfactory methods since all of them are related in some sense to a combinatorial NP-hard problem. In this paper, the problem is approached by genetic algorithms and parallel programming. In addition, to produce reasonable solutions, a particular metaheuristic is proposed and checked through some numerical instances. Juan Aparicio, Jose J. Lopez-Espin, Raul Martinez-Moreno, and Jesus T. Pastor Copyright © 2014 Juan Aparicio et al. All rights reserved. Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems Wed, 27 Nov 2013 15:16:28 +0000 http://www.hindawi.com/journals/aor/2013/708979/ Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on the -convex, -convex, and -convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving -convex, -convex, and generalized -convex functions. Xiaomin Zhang and Zezhong Wu Copyright © 2013 Xiaomin Zhang and Zezhong Wu. All rights reserved. An Optimal Number-Dependent Preventive Maintenance Strategy for Offshore Wind Turbine Blades Considering Logistics Wed, 31 Jul 2013 13:35:15 +0000 http://www.hindawi.com/journals/aor/2013/205847/ In offshore wind turbines, the blades are among the most critical and expensive components that suffer from different types of damage due to the harsh maritime environment and high load. The blade damages can be categorized into two types: the minor damage, which only causes a loss in wind capture without resulting in any turbine stoppage, and the major (catastrophic) damage, which stops the wind turbine and can only be corrected by replacement. In this paper, we propose an optimal number-dependent preventive maintenance (NDPM) strategy, in which a maintenance team is transported with an ordinary or expedited lead time to the offshore platform at the occurrence of the th minor damage or the first major damage, whichever comes first. The long-run expected cost of the maintenance strategy is derived, and the necessary conditions for an optimal solution are obtained. Finally, the proposed model is tested on real data collected from an offshore wind farm database. Also, a sensitivity analysis is conducted in order to evaluate the effect of changes in the model parameters on the optimal solution. Mahmood Shafiee, Michael Patriksson, and Ann-Brith Strömberg Copyright © 2013 Mahmood Shafiee et al. All rights reserved. A Partial Backlogging Inventory Model for Deteriorating Item under Fuzzy Inflation and Discounting over Random Planning Horizon: A Fuzzy Genetic Algorithm Approach Thu, 25 Jul 2013 12:02:26 +0000 http://www.hindawi.com/journals/aor/2013/973125/ An inventory model for deteriorating item is considered in a random planning horizon under inflation and time value money. The model is described in two different environments: random and fuzzy random. The proposed model allows stock-dependent consumption rate and shortages with partial backlogging. In the fuzzy stochastic model, possibility chance constraints are used for defuzzification of imprecise expected total profit. Finally, genetic algorithm (GA) and fuzzy simulation-based genetic algorithm (FSGA) are used to make decisions for the above inventory models. The models are illustrated with some numerical data. Sensitivity analysis on expected profit function is also presented. Scope and Purpose. The traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However, to keep sales higher, the inventory level would need to remain high. Of course, this would also result in higher holding or procurement cost. Also, in many real situations, during a longer-shortage period some of the customers may refuse the management. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging is diminishing with the length of the waiting time. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But in the past, the economic situation of most of the countries has changed to such an extent due to large-scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any more. The purpose of this paper is to maximize the expected profit in the random planning horizon. Dipak Kumar Jana, Barun Das, and Tapan Kumar Roy Copyright © 2013 Dipak Kumar Jana et al. All rights reserved. Partnership Selection Involving Mixed Types of Uncertain Preferences Thu, 27 Jun 2013 09:25:02 +0000 http://www.hindawi.com/journals/aor/2013/323468/ Partnership selection is an important issue in management science. This study proposes a general model based on mixed integer programming and goal-programming analytic hierarchy process (GP-AHP) to solve partnership selection problems involving mixed types of uncertain or inconsistent preferences. The proposed approach is designed to deal with crisp, interval, step, fuzzy, or mixed comparison preferences, derive crisp priorities, and improve multiple solution problems. The degree of fulfillment of a decision maker’s preferences is also taken into account. The results show that the proposed approach keeps more solution ratios within the given preferred intervals and yields less deviation. In addition, the proposed approach can treat incomplete preference matrices with flexibility in reducing the number of pairwise comparisons required and can also be conveniently developed into a decision support system. Li-Ching Ma Copyright © 2013 Li-Ching Ma. All rights reserved. Mathematical Programming Approach to the Optimality of the Solution for Deterministic Inventory Models with Partial Backordering Thu, 20 Jun 2013 16:41:09 +0000 http://www.hindawi.com/journals/aor/2013/272648/ We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models. Irena Stojkovska Copyright © 2013 Irena Stojkovska. All rights reserved. An Inventory Model with Price and Quality Dependent Demand Where Some Items Produced Are Defective Thu, 13 Jun 2013 15:24:40 +0000 http://www.hindawi.com/journals/aor/2013/795078/ This paper analyzes an inventory system for joint determination of product quality and selling price where a fraction of items produced are defective. It is assumed that only a fraction of defective items can be repaired/reworked. The demand rate depends upon both the quality and the selling price of the product. The production rate, unit price, and carrying cost depend upon the quality of the items produced. Quality index is used to determine the quality of the product. An algorithm is provided to solve the model with given values of model parameters. Sensitivity analysis has also been performed. Tapan Kumar Datta Copyright © 2013 Tapan Kumar Datta. All rights reserved. On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits Mon, 13 May 2013 15:51:17 +0000 http://www.hindawi.com/journals/aor/2013/680539/ We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotes of its capacity to the first queue and the remaining to the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable. Charles Knessl and Haishen Yao Copyright © 2013 Charles Knessl and Haishen Yao. All rights reserved. Well-Posedness and Primal-Dual Analysis of Some Convex Separable Optimization Problems Mon, 15 Apr 2013 13:49:54 +0000 http://www.hindawi.com/journals/aor/2013/279030/ We focus on some convex separable optimization problems, considered by the author in previous papers, for which problems, necessary and sufficient conditions or sufficient conditions have been proved, and convergent algorithms of polynomial computational complexity have been proposed for solving these problems. The concepts of well-posedness of optimization problems in the sense of Tychonov, Hadamard, and in a generalized sense, as well as calmness in the sense of Clarke, are discussed. It is shown that the convex separable optimization problems under consideration are calm in the sense of Clarke. The concept of stability of the set of saddle points of the Lagrangian in the sense of Gol'shtein is also discussed, and it is shown that this set is not stable for the “classical” Lagrangian. However, it turns out that despite this instability, due to the specificity of the approach, suggested by the author for solving problems under consideration, it is not necessary to use modified Lagrangians but only the “classical” Lagrangians. Also, a primal-dual analysis for problems under consideration in view of methods for solving them is presented. Stefan M. Stefanov Copyright © 2013 Stefan M. Stefanov. All rights reserved. A Generalized Learning Curve Adapted for Purchasing and Cost Reduction Negotiations Wed, 13 Feb 2013 09:54:56 +0000 http://www.hindawi.com/journals/aor/2013/584762/ This paper presents the use and validation of a generalized learning curve in the economies of scale purchasing experience. The model, based on Wright’s curve, incorporates two extra degrees of freedom to accommodate initial purchases of multiple (instead of single) units and a finite asymptotic price at high volumes. The study shows that each time the part purchase quantity is doubled, the price is reduced either by a constant percentage (a learning rate) or by an approach to an asymptotic plateau rate indicating a point of diminishing returns. Supplier price quotations at multiple purchase quantities were obtained for a pool of 17 critical parts. The data were fitted with the generalized learning curve by the method of least squares regression. The regressed learning rate, first unit price, and the asymptotic price can be used to infer supplier pricing strategies. Coupled with a “should-cost” analysis based on estimates of standard time and material, a system cost reduction task was carried out by the supply chain organization. Lam F. Wong Copyright © 2013 Lam F. Wong. All rights reserved. A Cost Model for Integrated Logistic Support Activities Sun, 10 Feb 2013 16:16:59 +0000 http://www.hindawi.com/journals/aor/2013/127497/ An Integrated Logistic Support (ILS) service has the objective of improving a system’s efficiency and availability for the life cycle. The system constructor offers the service to the customer, and she becomes the Contractor Logistic Support (CLS). The aim of this paper is to propose an approach to support the CLS in the budget formulation. Specific goals of the model are the provision of the annual cost of ILS activities through a specific cost model and a comprehensive examination of expected benefits, costs and savings under alternative ILS strategies. A simple example derived from an industrial application is also provided to illustrate the idea. Scientific literature is lacking in the topic and documents from the military are just dealing with the issue of performance measurement. Moreover, they are obviously focused on the customer’s perspective. Other scientific papers are general and focused only on maintenance or life cycle management. The model developed in this paper approaches the problem from the perspective of the CLS, and it is specifically tailored on the main issues of an ILS service. M. Elena Nenni Copyright © 2013 M. Elena Nenni. All rights reserved. A Newton-Type Algorithm for Solving Problems of Search Theory Thu, 31 Jan 2013 08:10:58 +0000 http://www.hindawi.com/journals/aor/2013/513918/ In the survey of the continuous nonlinear resource allocation problem, Patriksson pointed out that Newton-type algorithms have not been proposed for solving the problem of search theory in the theoretical perspective. In this paper, we propose a Newton-type algorithm to solve the problem. We prove that the proposed algorithm has global and superlinear convergence. Some numerical results indicate that the proposed algorithm is promising. Liping Zhang Copyright © 2013 Liping Zhang. All rights reserved. Multiobjective Two-Stage Stochastic Programming Problems with Interval Discrete Random Variables Mon, 17 Dec 2012 08:29:16 +0000 http://www.hindawi.com/journals/aor/2012/279181/ Most of the real-life decision-making problems have more than one conflicting and incommensurable objective functions. In this paper, we present a multiobjective two-stage stochastic linear programming problem considering some parameters of the linear constraints as interval type discrete random variables with known probability distribution. Randomness of the discrete intervals are considered for the model parameters. Further, the concepts of best optimum and worst optimum solution are analyzed in two-stage stochastic programming. To solve the stated problem, first we remove the randomness of the problem and formulate an equivalent deterministic linear programming model with multiobjective interval coefficients. Then the deterministic multiobjective model is solved using weighting method, where we apply the solution procedure of interval linear programming technique. We obtain the upper and lower bound of the objective function as the best and the worst value, respectively. It highlights the possible risk involved in the decision-making tool. A numerical example is presented to demonstrate the proposed solution procedure. S. K. Barik, M. P. Biswal, and D. Chakravarty Copyright © 2012 S. K. Barik et al. All rights reserved. An Inventory Model with Time-Dependent Demand and Limited Storage Facility under Inflation Mon, 12 Nov 2012 13:55:52 +0000 http://www.hindawi.com/journals/aor/2012/321471/ The main objective of this paper is to develop a two-warehouse inventory model with partial backordering and Weibull distribution deterioration. We consider inflation and apply the discounted cash flow in problem analysis. The discounted cash flow (DCF) and optimization framework are presented to derive the optimal replenishment policy that minimizes the total present value cost per unit time. When only rented or own warehouse model is considered, the present value of the total relevant cost is higher than the case when two-warehouse is considered. The results have been validated with the help of a numerical example. Sensitivity analysis with respect to various parameters is also performed. From the sensitivity analysis, we show that the total cost of the system is influenced by the deterioration rate, the inflation rate, and the backordering ratio. Neeraj Kumar, S. R. Singh, and Rachna Kumari Copyright © 2012 Neeraj Kumar et al. All rights reserved. A Hybrid Genetic Programming Method in Optimization and Forecasting: A Case Study of the Broadband Penetration in OECD Countries Tue, 02 Oct 2012 15:25:39 +0000 http://www.hindawi.com/journals/aor/2012/904797/ The introduction of a hybrid genetic programming method (hGP) in fitting and forecasting of the broadband penetration data is proposed. The hGP uses some well-known diffusion models, such as those of Gompertz, Logistic, and Bass, in the initial population of the solutions in order to accelerate the algorithm. The produced solutions models of the hGP are used in fitting and forecasting the adoption of broadband penetration. We investigate the fitting performance of the hGP, and we use the hGP to forecast the broadband penetration in OECD (Organisation for Economic Co-operation and Development) countries. The results of the optimized diffusion models are compared to those of the hGP-generated models. The comparison indicates that the hGP manages to generate solutions with high-performance statistical indicators. The hGP cooperates with the existing diffusion models, thus allowing multiple approaches to forecasting. The modified algorithm is implemented in the Python programming language, which is fast in execution time, compact, and user friendly. Konstantinos Salpasaranis and Vasilios Stylianakis Copyright © 2012 Konstantinos Salpasaranis and Vasilios Stylianakis. All rights reserved. Generalized Differentiable -Invex Functions and Their Applications in Optimization Tue, 02 Oct 2012 13:44:01 +0000 http://www.hindawi.com/journals/aor/2012/175176/ The concept of -convex function and its generalizations is studied with differentiability assumption. Generalized differentiable -convexity and generalized differentiable -invexity are used to derive the existence of optimal solution of a general optimization problem. S. Jaiswal and G. Panda Copyright © 2012 S. Jaiswal and G. Panda. All rights reserved. Numerical Methods for Solving Variational Inequalities and Complementarity Problems Thu, 13 Sep 2012 11:02:15 +0000 http://www.hindawi.com/journals/aor/2012/925915/ Abdellah Bnouhachem and Min Li Copyright © 2012 Abdellah Bnouhachem and Min Li. All rights reserved. The 𝐶max Problem of Scheduling Multiple Groups of Jobs on Multiple Processors at Different Speeds Thu, 06 Sep 2012 10:20:11 +0000 http://www.hindawi.com/journals/aor/2012/748597/ We mainly study the 𝐶max problem of scheduling n groups of jobs on n special-purpose processors and m general-purpose processors at different speeds provided that the setup time of each job is less than 𝛼 times of its processing time. We first propose an improved LS algorithm. Then, by applying this new algorithm, we obtain two bounds for the ratio of the approximate solution 𝑇LS to the optimal solution T* under two different conditions. Wei Ding Copyright © 2012 Wei Ding. All rights reserved. Exact and Heuristic Solutions to Minimize Total Waiting Time in the Blood Products Distribution Problem Mon, 03 Sep 2012 15:25:59 +0000 http://www.hindawi.com/journals/aor/2012/393890/ This paper presents a novel application of operations research to support decision making in blood distribution management. The rapid and dynamic increasing demand, criticality of the product, storage, handling, and distribution requirements, and the different geographical locations of hospitals and medical centers have made blood distribution a complex and important problem. In this study, a real blood distribution problem containing 24 hospitals was tackled by the authors, and an exact approach was presented. The objective of the problem is to distribute blood and its products among hospitals and medical centers such that the total waiting time of those requiring the product is minimized. Following the exact solution, a hybrid heuristic algorithm is proposed. Computational experiments showed the optimal solutions could be obtained for medium size instances, while for larger instances the proposed hybrid heuristic is very competitive. Amir Salehipour and Mohammad Mehdi Sepehri Copyright © 2012 Amir Salehipour and Mohammad Mehdi Sepehri. All rights reserved. Solving the Matrix Nearness Problem in the Maximum Norm by Applying a Projection and Contraction Method Mon, 13 Aug 2012 12:56:28 +0000 http://www.hindawi.com/journals/aor/2012/357954/ Let S be a closed convex set of matrices and C be a given matrix. The matrix nearness problem considered in this paper is to find a matrix X in the set S at which max {|𝑥𝑖𝑗−𝑐𝑖𝑗|} reaches its minimum value. In order to solve the matrix nearness problem, the problem is reformulated to a min-max problem firstly, then the relationship between the min-max problem and a monotone linear variational inequality (LVI) is built. Since the matrix in the LVI problem has a special structure, a projection and contraction method is suggested to solve this LVI problem. Moreover, some implementing details of the method are presented in this paper. Finally, preliminary numerical results are reported, which show that this simple algorithm is promising for this matrix nearness problem. M. H. Xu and H. Shao Copyright © 2012 M. H. Xu and H. Shao. All rights reserved. Deriving Weights of Criteria from Inconsistent Fuzzy Comparison Matrices by Using the Nearest Weighted Interval Approximation Tue, 31 Jul 2012 10:19:42 +0000 http://www.hindawi.com/journals/aor/2012/574710/ Deriving the weights of criteria from the pairwise comparison matrix with fuzzy elements is investigated. In the proposed method we first convert each element of the fuzzy comparison matrix into the nearest weighted interval approximation one. Then by using the goal programming method we derive the weights of criteria. The presented method is able to find weights of fuzzy pairwise comparison matrices in any form. We compare the results of the presented method with some of the existing methods. The approach is illustrated by some numerical examples. Mohammad Izadikhah Copyright © 2012 Mohammad Izadikhah. All rights reserved. Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem Tue, 24 Jul 2012 13:22:28 +0000 http://www.hindawi.com/journals/aor/2012/483479/ This paper presents modified halfspace-relaxation projection (HRP) methods for solving the split feasibility problem (SFP). Incorporating with the techniques of identifying the optimal step length with positive lower bounds, the new methods improve the efficiencies of the HRP method (Qu and Xiu (2008)). Some numerical results are reported to verify the computational preference. Min Li Copyright © 2012 Min Li. All rights reserved. On a System of Generalized Mixed Equilibrium Problems Involving Variational-Like Inequalities in Banach Spaces: Existence and Algorithmic Aspects Thu, 19 Jul 2012 15:33:39 +0000 http://www.hindawi.com/journals/aor/2012/843486/ We study the existence and the algorithmic aspect of a System of Generalized Mixed Equilibrium Problems involving variational-like inequalities (SGMEPs) in the setting of Banach spaces. The approach adopted is based on the auxiliary principle technique and arguments from generalized convexity. A new existence theorem for the auxiliary problem is established; this leads us to generate an algorithm which converges strongly to a solution of (SGMEP) under weaker assumptions. When the study is reduced to the setting of reflexive Banach spaces, then it can be more relaxed by dropping the coercivity condition. The results obtained in this paper are new and improve some recent studies in this field. H. Mahdioui and O. Chadli Copyright © 2012 H. Mahdioui and O. Chadli. All rights reserved. Generalized (𝑑-𝜌-𝜂-𝜃)-Type I Univex Functions in Multiobjective Optimization Wed, 18 Jul 2012 10:48:11 +0000 http://www.hindawi.com/journals/aor/2012/279215/ A new class of generalized functions (𝑑-𝜌-𝜂-𝜃)-type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, converse, and strict converse duality theorems are also derived for Mond-Weir-type multiobjective dual program. Pallavi Kharbanda, Divya Agarwal, and Deepa Sinha Copyright © 2012 Pallavi Kharbanda et al. All rights reserved. Unconstraining Methods in Revenue Management Systems: Research Overview and Prospects Tue, 10 Jul 2012 15:24:45 +0000 http://www.hindawi.com/journals/aor/2012/270910/ Demand unconstraining is one of the key techniques to the success of revenue management systems. This paper surveys the history of research on unconstraining methods and reviews over 130 references including the latest research works in the area. We discuss the relationship between censored data unconstraining and forecasting and review five alternative unconstraining approaches. These methods consider data unconstraining in various situations such as single-class, multi-class, and multi-flight. The paper also proposes some future research questions to bridge the gap between theory and applications. Peng Guo, Baichun Xiao, and Jun Li Copyright © 2012 Peng Guo et al. All rights reserved. Erratum to “Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks” Mon, 28 May 2012 09:08:58 +0000 http://www.hindawi.com/journals/aor/2012/543215/ M. Bouznif and R. Giroudeau Copyright © 2012 M. Bouznif and R. Giroudeau. All rights reserved. Adaptive Control of a Reverse Logistic Inventory Model with Uncertain Deteriorations and Disposal Rates Wed, 23 May 2012 16:28:26 +0000 http://www.hindawi.com/journals/aor/2012/490983/ An adaptive control of a reverse logistic inventory system with unknown deterioration and disposal rates is considered. An adaptive control approach with a feedback is applied to track the inventory levels toward their goal levels. Also, the updating rules of both deterioration and disposal rates are derived from the conditions of asymptotic stability of the reference model. Important characteristics of the adaptive inventory system are discussed. The adaptive controlled system is modeled by a nonlinear system of differential equations. Finally, the numerical solution of the controlled system is discussed and displayed graphically. Ahmad Alshamrani Copyright © 2012 Ahmad Alshamrani. All rights reserved. Optimal Policies for a Finite-Horizon Production Inventory Model Mon, 21 May 2012 10:17:41 +0000 http://www.hindawi.com/journals/aor/2012/768929/ This paper is concerned with the problem of finding the optimal production schedule for an inventory model with time-varying demand and deteriorating items over a finite planning horizon. This problem is formulated as a mixed-integer nonlinear program with one integer variable. The optimal schedule is shown to exist uniquely under some technical conditions. It is also shown that the objective function of the nonlinear obtained from fixing the integrality constraint is convex as a function of the integer variable. This in turn leads to a simple procedure for finding the optimal production plan. Lakdere Benkherouf and Dalal Boushehri Copyright © 2012 Lakdere Benkherouf and Dalal Boushehri. All rights reserved.