Multiple Objective Linear Programming Problems Research Articles

Neuro-Fuzzy Programming to Finding Fuzzy Multiple Objective Linear Programming Problems

Neuro-Fuzzy Programming to Finding Fuzzy Multiple Objective Linear Programming Problems

Application of the Plant Propagation Algorithm and NSGA-II to Multiple Objective Linear Programming

Multiple Objective Linear Programming (MOLP) problems are usually solved by exact methods. However, nature-inspired population based stochastic algorithms such as the plant propagation algorithm are becoming more and more prominent. This paper applies the multiple objective plant propagation algorithm (MOPPA) and nondominated sorting genetic algorithm II (NSGA-II) for the first time to MOLP and compares their outcomes with those of prominent exact methods. Computational results from a collection of 51 existing MOLP instances suggests that MOPPA compares favourably with four of the most prominent exact methods namely extended multiple objective simplex algorithm (EMSA), affine scaling interior MOLP algorithm (ASIMOLP), Benson’s outer-approximation algorithm (BOA) and parametric simplex algorithm (PSA), and returns best nondominated points which are of higher quality than those returned by NSGA-II. However, the nondominated points approximated by NSGA-II are evenly distributed across the nondominated front. The methods compare well with the four exact methods especially on the large instances which the exact methods failed to solve even when given generous amounts of computation times.

A fuzzy based solution to multiple objective LPP

<abstract><p>This study presents a Fuzzy Multiple Objective Linear Programming Problem (FMOLPP) method to solve the Linear Programming Problem (LPP). Initially Multiple Objective Linear Programming Problem (MOLPP) is solved using Chandra Sen's approach along with various types of mean approaches. Furthermore, FMOLPP is solved using Chandra Sen's approach and various categories of fuzzy mean techniques. The simplex form is used to solve the LPP, where the three-tuple symmetric triangular fuzzy number with the constraints of the fuzzy objective function is considered. We have presented a comparative study of optimum values of MOLPP with optimum values of FMOLPP, to show the significance of our proposed method.</p></abstract>

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

A Comparison of Benson’s Outer Approximation Algorithm with an Extended Version of Multiobjective Simplex Algorithm

The multiple objective simplex algorithm and its variants work in the decision variable space to find the set of all efficient extreme points of multiple objective linear programming (MOLP). Other approaches to the problem find either the entire set of all efficient solutions or a subset of them and also return the corresponding objective values (nondominated points). This paper presents an extension of the multiobjective simplex algorithm (MSA) to generate the set of all nondominated points and no redundant ones. This extended version is compared to Benson’s outer approximation (BOA) algorithm that also computes the set of all nondominated points of the problem. Numerical results on nontrivial MOLP problems show that the total number of nondominated points returned by the extended MSA is the same as that returned by BOA for most of the problems considered.

On Multiple Objective Linear Programming Problems with Fuzzy Rough Coefficient. (Dept. P)

On Multiple Objective Linear Programming Problems with Fuzzy Rough Coefficient. (Dept. P)

Crisp-linear-and Models in Fuzzy Multiple Objective Linear Fractional Programming

The aim of this paper is to introduce two crisp linear models to solve fuzzy multiple objective linear fractional programming problems. In a novel manner we construct two piece-wise linear membership functions to describe the fuzzy goal linked to a linear fractional objective. They are related to the numerator and denominator of the fractional objective function; and we show that using the fuzzy-and operator to aggregate them a convenient description of the original fractional fuzzy goal is obtained. Further on, with the help of the fuzzy-and operator we aggregate all fuzzy goals and constraints, formulate a crisp linear model, and use it to provide a solution to the initial fuzzy multiple objective linear fractional programming problem. The second model embeds in distinct ways the positive and negative information, the desires and restrictions respectively; and aggregates in a bipolar manner the goals and constraints. The main advantage of using the new models lies in the fact that they are linear, and can generate distinct solutions to the multiple objective problem by varying the thresholds and tolerance limits imposed on the fuzzy goals.

Finding the strong efficient frontier and strong defining hyperplanes of production possibility set using multiple objective linear programming

Data envelopment analysis (DEA) models use the frontier of the production possibility set (PPS) to evaluate decision making units (DMUs). However, the explicit-form equations of the frontier cannot be obtained using the traditional DEA models. To fill this gap, the current paper proposes an algorithm to generate all strong-efficient DMUs and the explicit-form equations of the strong-efficient frontier and the strong defining hyperplanes for the PPS with the variable returns to scale (VRS) technology. The algorithm is based on a multiple objective linear programming (MOLP) problem in the DEA methodology, which is solved through the multicriteria simplex method. Also, Isermann’s test is employed to specify strong-efficient nonbasic variables in each strong-efficient multicriteria simplex table. Before presenting the algorithm, a theoretical framework is introduced to characterize the relationships between the feasible region in the decision space of the MOLP problem and the PPS with the VRS technology. It is shown that the algorithm which has four phases is finitely convergent and has less computational complexity than other algorithms in the related literature. Finally, two examples are used to illustrate the algorithm.

Solutions of multiple objective linear programming problems by applying T-sets in imprecise environment

In this paper, technique to find Pareto optimal solutions to multiple objective linear programming problems under imprecise environment is discussed. In imprecise environment, we observe that more preferable optimal values may be obtained by allowing membership functions to take arbitrary values, i.e., by removing constraints that impose membership values to fall in range between zero and one, than existing fuzzy optimisation techniques. Further, membership functions are not utilised as per definitions in existing fuzzy optimisation techniques. Also, such constraints may make the model infeasible. Consequently, one set viz. T-set is defined to supersede fuzzy set to represent impreciseness. Next, one general algorithm comprising T-sets, is given to find Pareto optimal solutions to multiple objective linear programming problems in imprecise environment. Numerical examples further illustrate proposed algorithm. Finally conclusions are drawn.

Solutions of multiple objective linear programming problems by applying T-sets in imprecise environment

Solutions of multiple objective linear programming problems by applying T-sets in imprecise environment

A computationally efficient algorithm to approximate the pareto front of multi-objective linear fractional programming problem

The main contribution of this paper is the procedure that constructs a good approximation to the non-dominated set of multiple objective linear fractional programming problem using the solutions to certain linear optimization problems. In our approach we propose a way to generate a discrete set of feasible solutions that are further used as starting points in any procedure for deriving efficient solutions. The efficient solutions are mapped into non-dominated points that form a 0th order approximation of the Pareto front. We report the computational results obtained by solving random generated instances, and show that the approximations obtained by running our procedure are better than those obtained by running other procedures suggested in the recent literature. We evaluated the quality of each approximation using classic metrics.

Preferable Pareto optimal solutions for specified key objective functions to multiple objective linear programming problems using trade-off ratios under fuzzy environment: an iterative process

In this paper, one general iterative process is proposed for obtaining preferable Pareto optimal solutions, based on specified key objective functions, to multiple objective linear programming problems under fuzzy environment. In reality, decision maker usually specifies one key objective function to such problems. But there are known disadvantages in applying existing fuzzy optimisation techniques, in which weights, utility functions etc. are used; whereas in other techniques, none of the objective functions can be specified effectively as key objective function. Moreover, correlation between key objective function and other objective functions may not be exactly known to the decision maker. In existing interactive fuzzy optimisation techniques, initially developed by Sakawa et al. (1984), all such reference levels of fuzzy objective functions are taken as unity. But we may find it unrealistic to expect each of conflicting objective functions to attain individual goals simultaneously. In this paper, we propose to employ trade-off ratios of membership functions of fuzzy objective functions to determine corresponding reference membership levels analytically and develop one iterative process to find preferable Pareto optimal solutions under fuzzy environment. Numerical examples further illustrate our proposed iterative process. Finally conclusions are drawn.

A multiple objective programming approach to linear bilevel multi-follower programming

In this paper, we investigate the relationship between a certain class of linear bilevel multifollower programming problems and multiple objective programming. We introduce two multiple objective linear programming problems with different objective functions and the same constraint region. We show that the extreme points of the set of efficient solutions for both problems are the same as those of the set of feasible solutions to the linear bilevel multi-follower programming problem. Based on this relationship, a new algorithm to find an optimal solution for the linear bilevel multifollower programming problem is developed. Some numerical examples are presented to show the feasibility of the proposed algorithm.

A comparative study of two key algorithms in multiple objective linear programming

Multiple objective linear programming problems are solved with a variety of algorithms. While these algorithms vary in philosophy and outlook, most of them fall into two broad categories: those that are decision space-based and those that are objective space-based. This paper reports the outcome of a computational investigation of two key representative algorithms, one of each category, namely the parametric simplex algorithm which is a prominent representative of the former and the primal variant of Bensons Outer-approximation algorithm which is a prominent representative of the latter. The paper includes a procedure to compute the most preferred nondominated point which is an important feature in the implementation of these algorithms and their comparison. Computational and comparative results on problem instances ranging from small to medium and large are provided.

A dynamic DEA model for resource allocation

In this paper we discuss the inverse data envelopment analysis (DEA) in framework of dynamic DEA. This discussion is based on extending the non-radial enhanced Russell measure (ERM) model to framework of dynamic DEA. Also, necessary and sufficient conditions are established to characterising dynamically efficient paths using dynamic ERM-model. Simultaneous estimation of input and output levels are studied using ERM-model under inter-temporal dependence assumption. Necessary and sufficient conditions for simultaneous estimation of input and output levels are introduced based on Pareto solutions of multiple-objective linear programming problems. In addition, the performance of the proposed method is illustrated through an example in banking sector.

On fuzzy multiple objective linear programming problems

In recent years, multiple objective decision making (MODM) has become more and more important, and multiple objective linear programming (MOLP) approaches have been widely used for solving MODM problems. Three popular fuzzy mathematical programming approaches, including maxmin, arithmetical average, and two-phase methods, are often employed to solve MOLP problems. However, maxmin and two-phase methods cannot obtain an efficient solution. Ideal and anti-ideal solutions are required in all traditional methods. Such solutions cannot easily be established as a model for a knowledge-based system or expert system. In order to enrich the body of knowledge related to the field of MOLP, a new multi-choice goal programming (MCGP) model is proposed to solve fuzzy MOLP (FMOLP) problems in which the ideal and anti-ideal solutions are no longer required. This reduces the complexity of the solution process for solving MODM. The original feasible region can be expended to a potential feasible region to determine the appropriate aspiration level for decision makers. The proposed methods can systematically solve MODM problem to obtain satisfactory solutions in one step. Thus, a model-based can easily be established for expert systems, knowledge-based system, and artificial intelligence systems. In addition, a revised utility function (UF) is derived to solve qualitative and quantitative MODM problems. Based on illustrative examples, the five methods (maxmin, arithmetical average, two-phase, goal programming (GP), and proposed models) are compared to reveal managerial implications. On the basis of these comparisons, DMs can easily determine the best suited solution for their specific MODM problems. Finally, a realistic example is provided to demonstrate the usefulness of the proposed methods.

Two simple ways to find an efficient solution for a multiple objective linear programming problem

Abstract A number of methods and techniques for determining “effective” solutions for multiple objective linear programming problems (MPP) have been developed. In this study, we will present two simple methods for determining an efficient solution for a MPP that reducing the given problem to a one-objective linear programming problem. One of these methods falls under the category of methods of weighted metrics, and the other is an approach similar to the ε- constraint method. The solutions determined by the two methods are not only effective and are found on the Pareto frontier, but are also “the best” in terms of distance to the optimal solutions for all objective function from the MPP. Obviously, besides the optimal solutions of linear programming problems in which we take each objective function, we can also consider the ideal point and Nadir point, in order to take into account all the notions that have been introduced to provide a solution to this problem

Generation of efficient and ε -efficient solutions in multiple objective linear programming

We develop an algorithm to solve a multiple objective linear programming problem with bounded variables. It is based on the scalarization theorem of optimal solutions of multiobjective linear programs and the single objective adaptive method. We suggest a process for the search for the first efficient solution without having to calculate a feasible solution, and we elaborate a method to generate efficient solutions, weakly efficient solutions, and $\epsilon$-efficient solutions. Supporting theoretical results are established and the method is demonstrated on a numerical example.

Hybrid efficiency measurement and target setting based on identifying defining hyperplanes of the PPS with negative data

The traditional data envelopment analysis (DEA) models can be used for efficiency evaluation of decision making units (DMUs) with nonnegative data. However, in the real world there are DMUs which have negative inputs and/or outputs. Consequently, in the literature of DEA various approaches have been offered in order to deal with negative data. The current research proposes an algorithm concerned with a pure mathematical optimization method to measure hybrid efficiency of DMUs in the presence of negative data, and also a pure mathematical procedure for target setting. In doing so, explicit form equations of strong and weak defining hyperplanes of production possibility set (PPS) based on the multiple criteria decision making methodology are obtained. In characterizing these hyperplanes, a new multiple objective linear programming (MOLP) problem is presented whose feasible region of criterion space is similar to the PPS with variable returns to scale technology on nonnegative and negative data. The MOLP problem is solved using the multicriteria simplex method where its process of solving leads to construct all strong and weak defining hyperplanes. Then, using the strong and weak hyperplanes obtained and also without using any DEA optimization model, a hybrid measure of efficiency and a strong efficient target unit for each inefficient DMU with negative data are obtained. Finally, the results are discussed using two examples, where the first one, in detail, explains the proposed methods and algorithm, and the second one briefly shows the contributions are applicable for the real data.

On Optimization Over the Efficient Set of a Multiple Objective Linear Programming Problem

In this paper, we provide a mixed-integer programming approach for solving the problem of minimizing a real-valued function over the efficient set of a multiple objective linear program problem. Instead of solving the problem directly, we introduce a new problem of minimizing the objective function subject to some linear constraints with additional binary variables. We show under certain conditions that the two problems are equivalent. When the objective function of the original problem is a linear or convex function, the new problem is a linear or convex programming problem, respectively, with some binary variables. These problems can be solved as mixed-integer programs with current state-of-art mixed-integer programming solvers.