Search results for: mathematical programming problems with equilibrium constraints

Authors: Shashi Kant Mishra

Abstract:

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPEC). We establish necessary and sufficient optimality conditions for the SIMPEC, using convexificators. We study the Wolfe type dual problem for the SIMPEC under the ∂∗convexity assumptions. A Mond-Weir type dual problem is also formulated and studied for the SIMPEC under the ∂∗-convexity, ∂∗-pseudoconvexity and ∂∗quasiconvexity assumptions. Weak duality theorems are established to relate the SIMPEC and two dual programs in the framework of convexificators. Further, strong duality theorems are obtained under generalized standard Abadie constraint qualification (GS-ACQ).

Keywords: mathematical programming problems with equilibrium constraints, optimality conditions, semi-infinite programming, convexificators

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Authors: N. Gharbi

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Constraint Satisfaction problems are mathematical problems that are often used to model many real-world problems for which we look if there exists a solution satisfying all its constraints. Table constraints are important for modeling parts of many problems since they list all combinations of allowed or forbidden values. However, they admit practical limitations because they are sometimes too large to be represented in a direct way. In this paper, we present a survey of the different categories of the proposed approaches to compress table constraints in order to reduce both space and time complexities.

Keywords: constraint programming, compression, data mining, table constraints

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Authors: Nur Aidya Hanum Aizam, Vikneswary Uvaraja

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The agenda of showing the scheduled time for performing certain tasks is known as timetabling. It widely used in many departments such as transportation, education, and production. Some difficulties arise to ensure all tasks happen in the time and place allocated. Therefore, many researchers invented various programming model to solve the scheduling problems from several fields. However, the studies in developing the general integer programming model for many timetabling problems are still questionable. Meanwhile, this thesis describe about creating a general model which solve different types of timetabling problems by considering the basic constraints. Initially, the common basic constraints from five different fields are selected and analyzed. A general basic integer programming model was created and then verified by using the medium set of data obtained randomly which is much similar to realistic data. The mathematical software, AIMMS with CPLEX as a solver has been used to solve the model. The model obtained is significant in solving many timetabling problems easily since it is modifiable to all types of scheduling problems which have same basic constraints.

Keywords: AIMMS mathematical software, integer linear programming, scheduling problems, timetabling

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Authors: A. Bari

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Integer programming is a branch of mathematical programming techniques in operations research in which some or all of the variables are required to be integer valued. Various cuts have been used to solve these problems. We have also developed cuts known as NAZ cut & A-T cut to solve the integer programming problems. These cuts are used to reduce the feasible region and then reaching the optimal solution in minimum number of steps.

Keywords: Integer Programming, NAZ cut, A-T cut, Cutting plane method

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Authors: Alina Fedossova, Jose Jorge Sierra Molina

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SIP problems are part of non-classical optimization. There are problems in which the number of variables is finite, and the number of constraints is infinite. These are semi-infinite programming problems. Most algorithms for semi-infinite programming problems reduce the semi-infinite problem to a finite one and solve it by classical methods of linear or nonlinear programming. Typically, any of the constraints or the objective function is nonlinear, so the problem often involves nonlinear programming. An investment portfolio is a set of instruments used to reach the specific purposes of investors. The risk of the entire portfolio may be less than the risks of individual investment of portfolio. For example, we could make an investment of M euros in N shares for a specified period. Let yi> 0, the return on money invested in stock i for each dollar since the end of the period (i = 1, ..., N). The logical goal here is to determine the amount xi to be invested in stock i, i = 1, ..., N, such that we maximize the period at the end of ytx value, where x = (x1, ..., xn) and y = (y1, ..., yn). For us the optimal portfolio means the best portfolio in the ratio "risk-return" to the investor portfolio that meets your goals and risk ways. Therefore, investment goals and risk appetite are the factors that influence the choice of appropriate portfolio of assets. The investment returns are uncertain. Thus we have a semi-infinite programming problem. We solve a semi-infinite optimization problem of portfolio selection using the outer approximations methods. This approach can be considered as a developed Eaves-Zangwill method applying the multi-start technique in all of the iterations for the search of relevant constraints' parameters. The stochastic outer approximations method, successfully applied previously for robotics problems, Chebyshev approximation problems, air pollution and others, is based on the optimal criteria of quasi-optimal functions. As a result we obtain mathematical model and the optimal investment portfolio when yields are not clear from the beginning. Finally, we apply this algorithm to a specific case of a Colombian bank.

Keywords: outer approximation methods, portfolio problem, semi-infinite programming, numerial solution

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Authors: Mohsen Ziaee

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In this paper, a mixed integer linear programming (MILP) model is presented to solve the flexible job shop scheduling problem (FJSP). This problem is one of the hardest combinatorial problems. The objective considered is the minimization of the makespan. The computational results of the proposed MILP model were compared with those of the best known mathematical model in the literature in terms of the computational time. The results show that our model has better performance with respect to all the considered performance measures including relative percentage deviation (RPD) value, number of constraints, and total number of variables. By this improved mathematical model, larger FJS problems can be optimally solved in reasonable time, and therefore, the model would be a better tool for the performance evaluation of the approximation algorithms developed for the problem.

Keywords: scheduling, flexible job shop, makespan, mixed integer linear programming

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Authors: Tomoaki Hashimoto

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Recently, optimal control problems subject to probabilistic constraints have attracted much attention in many research field. Although probabilistic constraints are generally intractable in optimization problems, several methods haven been proposed to deal with probabilistic constraints. In most methods, probabilistic constraints are transformed to deterministic constraints that are tractable in optimization problems. This paper examines a method for transforming probabilistic constraints into deterministic constraints for a class of probabilistic constrained optimal control problems.

Keywords: optimal control, stochastic systems, discrete-time systems, probabilistic constraints

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Authors: El Hassene Osmani, Mounir Haddou, Naceurdine Bensalem

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In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely, the obstacle problem. We present a relaxed formulation for the problem using smoothing functions. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods, we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm (Interior Point Optimizer) are presented to verify the efficiency of our approach.

Keywords: complementarity problem, IPOPT, Lagrange multipliers, mathematical programming, optimal control, smoothing methods, variationally inequalities

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: Nurul Liyana Abdul Aziz, Nur Aidya Hanum Aizam

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Course timetabling problems occur every semester in a university which includes the allocation of resources (subjects, lecturers and students) to a number of fixed rooms and timeslots. The assignment is carried out in a way such that there are no conflicts within rooms, students and lecturers, as well as fulfilling a range of constraints. The constraints consist of rules and policies set up by the universities as well as lecturers’ and students’ preferences of courses to be allocated in specific timeslots. This paper specifically focuses on the preferences of the course timetabling problem in one of the public universities in Malaysia. The demands will be considered into our existing mathematical model to make it more generalized and can be used widely. We have distributed questionnaires to a number of lecturers and students of the university to investigate their demands and preferences for their desired course timetable. We classify the preferences thus converting them to construct one mathematical model that can produce such timetable.

Keywords: university course timetabling problem, integer programming, preferences, constraints

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Authors: Ghussoun Al-Jeiroudi

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Stochastic programming is one of the powerful technique which is used to solve real-life problems. Hence, the data of real-life problems is subject to significant uncertainty. Uncertainty is well studied and modeled by stochastic programming. Each day, problems become bigger and bigger and the need for a tool, which does deal with large scale problems, increase. Interior point method is a perfect tool to solve such problems. Interior point method is widely employed to solve the programs, which arise from stochastic programming. It is an iterative technique, so it is required a starting point. Well design starting point plays an important role in improving the convergence speed. In this paper, we propose a starting point for interior point method for multistage stochastic programming. Usually, the optimal solution of stage k+1 is used as starting point for the stage k. This point has the advantage of being close to the solution of the current program. However, it has a disadvantage; it is not in the feasible region of the current program. So, we suggest to take this point and modifying it. That is by adding to it a vector in the null space of the matrix of the unchanged constraints because the solution will change only in the null space of this matrix.

Keywords: interior point methods, stochastic programming, null space, starting points

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Authors: Mazura Mokhtar, Adibah Shuib, Daud Mohamad

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Portfolio optimization problem has received a lot of attention from both researchers and practitioners over the last six decades. This paper provides an overview of the current state of research in portfolio optimization with the support of mathematical programming techniques. On top of that, this paper also surveys the solution algorithms for solving portfolio optimization models classifying them according to their nature in heuristic and exact methods. To serve these purposes, 40 related articles appearing in the international journal from 2003 to 2013 have been gathered and analyzed. Based on the literature review, it has been observed that stochastic programming and goal programming constitute the highest number of mathematical programming techniques employed to tackle the portfolio optimization problem. It is hoped that the paper can meet the needs of researchers and practitioners for easy references of portfolio optimization.

Keywords: portfolio optimization, mathematical programming, multi-objective programming, solution approaches

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Authors: Li-Zhi Liao

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The central path has played the key role in the interior point method. However, the convergence of the central path may not be true even in some convex programming problems with linear constraints. In this paper, the generalized central paths are introduced for convex programming. One advantage of the generalized central paths is that the paths will always converge to some optimal solutions of the convex programming problem for any initial interior point. Some additional theoretical properties for the generalized central paths will be also reported.

Keywords: central path, convex programming, generalized central path, interior point method

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Authors: Debjani Chakraborty, Abhijit Chatterjee, Aishwaryaprajna

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Posynomials, a special type of polynomials, having singularities, pose difficulties while solving geometric programming problems. In this paper, a methodology has been proposed and used to obtain extreme values for geometric programming problems by nth degree polynomial interpolation technique. Here the main idea to optimise the posynomial is to fit a best polynomial which has continuous gradient values throughout the range of the function. The approximating polynomial is smoothened to remove the discontinuities present in the feasible region and the objective function. This spatial interpolation method is capable to optimise univariate and multivariate geometric programming problems. An example is solved to explain the robustness of the methodology by considering a bivariate nonlinear geometric programming problem. This method is also applicable for signomial programming problem.

Keywords: geometric programming problem, multivariate optimisation technique, posynomial, spatial interpolation

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Authors: Sujeet Kumar Singh, Shiv Prasad Yadav

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This paper develops an approach for solving intuitionistic fuzzy linear fractional programming (IFLFP) problem where the cost of the objective function, the resources, and the technological coefficients are triangular intuitionistic fuzzy numbers. Here, the IFLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming (MOLFP) problem. By using fuzzy mathematical programming approach the transformed MOLFP problem is reduced into a single objective linear programming (LP) problem. The proposed procedure is illustrated through a numerical example.

Keywords: triangular intuitionistic fuzzy number, linear programming problem, multi objective linear programming problem, fuzzy mathematical programming, membership function

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Authors: Filipe Cardoso de Oliveira, Lino Marcos da Silva, Ademar Nogueira do Nascimento, Cristiano Hora de Oliveira Fontes

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This article presents a mathematical formulation for the production Break-Even Point problem in a non-homogeneous system. The optimization problem aims to obtain the composition of the best product mix in a non-homogeneous industrial plant, with the lowest cost until the breakeven point is reached. The problem constraints represent real limitations of a generic non-homogeneous industrial plant for n different products. The proposed model is able to solve the equilibrium point problem simultaneously for all products, unlike the existing approaches that propose a resolution in a sequential way, considering each product in isolation and providing a sub-optimal solution to the problem. The results indicate that the product mix found through the proposed model has economical advantages over the traditional approach used.

Keywords: branch and bound, break-even point, non-homogeneous production system, integer linear programming, management accounting

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Authors: A. A. Behroozpoor, M. M. Mazarei

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In this paper, a fuzzy recurrent neural network is proposed for solving the classical quadratic control problem subject to linear equality and bound constraints. The convergence of the state variables of the proposed neural network to achieve solution optimality is guaranteed.

Keywords: REFERENCES [1] Xia, Y, A new neural network for solving linear and quadratic programming problems. IEEE Transactions on Neural Networks, 7(6), 1996, pp.1544–1548. [2] Xia, Y., & Wang, J, A recurrent neural network for solving nonlinear convex programs subject to linear constraints. IEEE Transactions on Neural Networks, 16(2), 2005, pp. 379–386. [3] Xia, Y., H, Leung, & J, Wang, A projection neural network and its application to constrained optimization problems. IEEE Transactions Circuits and Systems-I, 49(4), 2002, pp.447–458.B. [4] Q. Liu, Z. Guo, J. Wang, A one-layer recurrent neural network for constrained seudoconvex optimization and its application for dynamic portfolio optimization. Neural Networks, 26, 2012, pp. 99-109.

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Authors: E. Koyuncu

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The problem of Order Acceptance and Scheduling (OAS) is defined as a joint decision of which orders to accept for processing and how to schedule them. Any linear programming model representing real-world situation involves the parameters defined by the decision maker in an uncertain way or by means of language statement. Fuzzy data can be used to incorporate vagueness in the real-life situation. In this study, a fuzzy mathematical model is proposed for a single machine OAS problem, where the orders are defined by their fuzzy due dates, fuzzy processing times, and fuzzy sequence dependent setup times. The signed distance method, one of the fuzzy ranking methods, is used to handle the fuzzy constraints in the model.

Keywords: fuzzy mathematical programming, fuzzy ranking, order acceptance, single machine scheduling

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Authors: T. G. Kassem, A. K. Adunchezor, J. P. Chollom

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This paper describes a mathematical model developed to predict the dynamics of Hepatitis B virus (HBV) infection and to evaluate the potential impact of vaccination and treatment on its dynamics. We used a compartmental model expressed by a set of differential equations based on the characteristic of HBV transmission. With these, we find the threshold quantity R0, then find the local asymptotic stability of disease free equilibrium and endemic equilibrium. Furthermore, we find the global stability of the disease free and endemic equilibrium.

Keywords: hepatitis B virus, epidemiology, vaccination, mathematical model

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Authors: Matthew T. Wilson

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This paper covers a selection of research utilizing grammar-based genetic programming, and illustrates how context-free grammar can be used to constrain genetic programming. It focuses heavily on grammatical evolution, one of the most popular variants of grammar-based genetic programming, and the way its operators and terminals are specialized and modified from those in genetic programming. A variety of implementations of grammatical evolution for general use are covered, as well as research each focused on using grammatical evolution or grammar-based genetic programming on a single application, or to solve a specific problem, including some of the classically considered genetic programming problems, such as the Santa Fe Trail.

Keywords: context-free grammar, genetic algorithms, genetic programming, grammatical evolution

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Authors: F. Hamidi, N. Amiri, H. Mishmast Nehi

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The Bilevel Programming (BP) model has been presented for a decision making process that consists of two decision makers in a hierarchical structure. In fact, BP is a model for a static two person game (the leader player in the upper level and the follower player in the lower level) wherein each player tries to optimize his/her personal objective function under dependent constraints; this game is sequential and non-cooperative. The decision making variables are divided between the two players and one’s choice affects the other’s benefit and choices. In other words, BP consists of two nested optimization problems with two objective functions (upper and lower) where the constraint region of the upper level problem is implicitly determined by the lower level problem. In real cases, the coefficients of an optimization problem may not be precise, i.e. they may be interval. In this paper we develop an algorithm for solving interval bilevel linear fractional programming problems. That is to say, bilevel problems in which both objective functions are linear fractional, the coefficients are interval and the common constraint region is a polyhedron. From the original problem, the best and the worst bilevel linear fractional problems have been derived and then, using the extended Charnes and Cooper transformation, each fractional problem can be reduced to a linear problem. Then we can find the best and the worst optimal values of the leader objective function by two algorithms.

Keywords: best and worst optimal solutions, bilevel programming, fractional, interval coefficients

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Authors: B. Güney, Ç. Teke

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In recent years, rapid and correct decision making is crucial for both people and enterprises. However, uncertainty makes decision-making difficult. Fuzzy logic is used for coping with this situation. Thus, fuzzy linear programming models are developed in order to handle uncertainty in objective function and the constraints. In this study, a problem of a factory in food industry is investigated, required data is obtained and the problem is figured out as a fuzzy linear programming model. The model is solved using Zimmerman approach which is one of the approaches for fuzzy linear programming. As a result, the solution gives the amount of production for each product type in order to gain maximum profit.

Keywords: food industry, fuzzy linear programming, fuzzy logic, linear programming

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Authors: S. Periyadharshini, K. Ramkumar, S. Jayalalitha, M. GuruPrasath, R. Manikandan

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This paper presents a formulation and solution for industrial load management and product grade problem. The formulation is created using linear programming technique thereby optimizing the electricity cost by scheduling the loads satisfying the process, storage, time zone and production constraints which will create an impact of reducing maximum demand and thereby reducing the electricity cost. Product grade problem is formulated using integer linear programming technique of optimization using lingo software and the results show that overall increase in profit margin. In this paper, time of use tariff is utilized and this technique will provide significant reductions in peak electricity consumption.

Keywords: cement industries, integer programming, optimal formulation, objective function, constraints

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Authors: Derkaoui Orkia, Lehireche Ahmed

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The topic of this article is to exploring the potentialities of a powerful optimization technique, namely Semidefinite Programming, for solving NP-hard problems. This approach provides tight relaxations of combinatorial and quadratic problems. In this work, we solve the maximum clique problem using this relaxation. The clique problem is the computational problem of finding cliques in a graph. It is widely acknowledged for its many applications in real-world problems. The numerical results show that it is possible to find a maximum clique in polynomial time, using an algorithm based on semidefinite programming. We implement a primal-dual interior points algorithm to solve this problem based on semidefinite programming. The semidefinite relaxation of this problem can be solved in polynomial time.

Keywords: semidefinite programming, maximum clique problem, primal-dual interior point method, relaxation

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Authors: Xiang Zhang, David Rey, S. Travis Waller

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Stochastic User Equilibrium (SUE) model is a widely used traffic assignment model in transportation planning, which is regarded more advanced than Deterministic User Equilibrium (DUE) model. However, a problem exists that the performance of the SUE model depends on its error term parameter. The objective of this paper is to propose a systematic method of determining the appropriate error term parameter value for the SUE model. First, the significance of the parameter is explored through a numerical example. Second, the parameter calibration method is developed based on the Logit-based route choice model. The calibration process is realized through multiple nonlinear regression, using sequential quadratic programming combined with least square method. Finally, case analysis is conducted to demonstrate the application of the calibration process and validate the better performance of the SUE model calibrated by the proposed method compared to the SUE models under other parameter values and the DUE model.

Keywords: parameter calibration, sequential quadratic programming, stochastic user equilibrium, traffic assignment, transportation planning

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Authors: Mandeep Kaur, Fatehgarh Sahib

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The purpose of economic load dispatch is to allocate the required load demand between the available generation units such that the cost of operation is minimized. It is an optimization problem to find the most economical schedule of the generating units while satisfying load demand and operational constraints. The multiobjective optimization problem in which the engineer’s goal is to maximize or minimize not a single objective function but several objective functions simultaneously. The purpose of multiobjective problems in the mathematical programming framework is to optimize the different objective functions. Many approaches and methods have been proposed in recent years to solve multiobjective optimization problems. Weighting method has been applied to convert multiobjective optimization problems into scalar optimization. MATLAB 7.10 has been used to write the code for the complete algorithm with the help of genetic algorithm (GA). The validity of the proposed method has been demonstrated on a three-unit power system.

Keywords: economic load dispatch, genetic algorithm, generating units, multiobjective optimization, weighting method

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Authors: Aquil Ahmed, Srikant Gupta, Irfan Ali

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In this paper, we study a multi-objective capacitated transportation problem (MOCTP) with mixed constraints. This paper is comprised of the modelling and optimisation of an MOCTP in a fuzzy environment in which some goals are fractional and some are linear. In real life application of the fuzzy goal programming (FGP) problem with multiple objectives, it is difficult for the decision maker(s) to determine the goal value of each objective precisely as the goal values are imprecise or uncertain. Also, we developed the concept of linearization of fractional goal for solving the MOCTP. In this paper, imprecision of the parameter is handled by the concept of fuzzy set theory by considering these parameters as a trapezoidal fuzzy number. α-cut approach is used to get the crisp value of the parameters. Numerical examples are used to illustrate the method for solving MOCTP.

Keywords: capacitated transportation problem, multi objective linear programming, multi-objective fractional programming, fuzzy goal programming, fuzzy sets, trapezoidal fuzzy number

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Authors: Abdullah Ali H. Ahmadini, Firoz Ahmad, Intekhab Alam

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Geometric programming problem (GPP) is a well-known non-linear optimization problem having a wide range of applications in many engineering problems. The structure of GPP is quite dynamic and easily fit to the various decision-making processes. The aim of this paper is to highlight the bounded solution method for GPP with special reference to variation among right-hand side parameters. Thus this paper is taken the advantage of two-level mathematical programming problems and determines the solution of the objective function in a specified interval called lower and upper bounds. The beauty of the proposed bounded solution method is that it does not require sensitivity analyses of the obtained optimal solution. The value of the objective function is directly calculated under varying parameters. To show the validity and applicability of the proposed method, a numerical example is presented. The system reliability optimization problem is also illustrated and found that the value of the objective function lies between the range of lower and upper bounds, respectively. At last, conclusions and future research are depicted based on the discussed work.

Keywords: varying parameters, geometric programming problem, bounded solution method, system reliability optimization

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Authors: Wlodzimierz Ogryczak, Michal Przyluski, Tomasz Sliwinski

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In problems of portfolio selection, the reward-risk ratio criterion is optimized to search for a risky portfolio with the maximum increase of the mean return in proportion to the risk measure increase when compared to the risk-free investments. In the classical model, following Markowitz, the risk is measured by the variance thus representing the Sharpe ratio optimization and leading to the quadratic optimization problems. Several Linear Programming (LP) computable risk measures have been introduced and applied in portfolio optimization. In particular, the Mean Absolute Deviation (MAD) measure has been widely recognized. The reward-risk ratio optimization with the MAD measure can be transformed into the LP formulation with the number of constraints proportional to the number of scenarios and the number of variables proportional to the total of the number of scenarios and the number of instruments. This may lead to the LP models with huge number of variables and constraints in the case of real-life financial decisions based on several thousands scenarios, thus decreasing their computational efficiency and making them hardly solvable by general LP tools. We show that the computational efficiency can be then dramatically improved by an alternative model based on the inverse risk-reward ratio minimization and by taking advantages of the LP duality. In the introduced LP model the number of structural constraints is proportional to the number of instruments thus not affecting seriously the simplex method efficiency by the number of scenarios and therefore guaranteeing easy solvability. Moreover, we show that under natural restriction on the target value the MAD risk-reward ratio optimization is consistent with the second order stochastic dominance rules.

Keywords: portfolio optimization, reward-risk ratio, mean absolute deviation, linear programming

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Authors: D. K. Rao, M. G. M. Khan, K. G. Reddy

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The focus of this paper is to develop a technique of solving a combined problem of determining Optimum Strata Boundaries (OSB) and Optimum Sample Size (OSS) of each stratum, when the population understudy is skewed and the study variable has a Pareto frequency distribution. The problem of determining the OSB is formulated as a Mathematical Programming Problem (MPP) which is then solved by dynamic programming technique. A numerical example is presented to illustrate the computational details of the proposed method. The proposed technique is useful to obtain OSB and OSS for a Pareto type skewed population, which minimizes the variance of the estimate of population mean.

Keywords: stratified sampling, optimum strata boundaries, optimum sample size, pareto distribution, mathematical programming problem, dynamic programming technique

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