Multiple Objective Programming Problems Research Articles

Topsis for multiple objective programming with rough decision set

Many optimization problems have competing objectives that require being optimized at the same time. These problems are called ?multiple objective programming problems (MOPPs)?. Real-world MOPPs may have some imprecision (roughness) in the decision set and/or the objective functions. These problems are known as ?rough MOPPs (RMOPPs)?. There is no unique method able to solve all RMOPPs. Accordingly, the decision maker (DM) should have more than one method for solving RMOPPs at his disposal so that he can select the most appropriate method. To contribute in this regard, we propose a new method for solving a specific class of RMOPPs in which all the objectives are precisely defined, but the decision set is roughly defined by its lower and upper approximations. Our proposed method is a modified version of the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). TOPSIS was chosen as the foundation for our method because it is one of the most widely applied methods for solving MOPPs. The basic concept underlying TOPSIS is that the compromise solution is closer to the ideal solution while also being farther away from the anti-ideal solution. The conventional TOPSIS can only solve MOPPs with precise (crisp) definitions of the two main parts of the problem. We extend TOPSIS to optimize multiple precise objectives over an imprecise decision set. The proposed approach is depicted in a flowchart. A numerical example is given to demonstrate the effectiveness of our proposed method to solve RMOPPs with a rough decision set at different values of objectives? weights and using different Lp-metrics.

A common weights multiplicative aggregation approach for composite indicators: the case of Global City Competitiveness Index

We propose a mathematical programming approach to assess a set of entities against a composite index. The weights used to construct the composite index are derived via optimization process. Specifically, we adopt the multiplicative method for the aggregation of the components (sub-indicators) of the evaluated index, which reduces the effect of compensation between the components. Also, we model the problem of identifying a common weighting scheme for the evaluated entities as a multiple objective programming (MOP) problem, with the performance of each entity being a distinct objective-criterion. This allows for comparisons and inferences between the assessed entities as a common ground is established. We stick to the concept of multiplicative aggregation by scalarizing the MOP problem using the weighted product method. Utilizing the multiplicative aggregation method in our modelling approach results in a nonlinear model that is scale invariant. However, using logarithms we show that is straightforwardly converted to a linear equivalent model, which is translation invariant. We examine whether the derived weighting scheme is unique, and in case there are alternative optimal solutions, we employ a max–min achievement function to identify a unique solution. We demonstrate our approach using synthetic datasets and compare it to other multiplicative approaches in the literature. We show that the proposed approach has neglective computational requirements and discriminates to a greater extent than those approaches with which we are compared, while at the same time provides a similar ranking. Similarities and dissimilarities in the results are justified. Since our approach is translation invariant, it enables us to apply it to cases where the data needs translation. We further illustrate our approach for such a case using the Global City Competitiveness Index, developed by Citigroup's Economist Intelligence Unit (EIU), which evaluates the competitiveness of 120 major cities worldwide.

Inverse data envelopment analysis with stochastic data

The inverse Data Envelopment Analysis (InvDEA) is an exciting and significant topic in the DEA area. Also, uncertain data in various real-life applications can degrade the efficiency results. The current work addresses the InvDEA in the presence of stochastic data. Under maintaining the efficiency score, the inputs/outputs-estimation problem is investigated when some or all of its outputs/inputs increase. A novel optimality concept for multiple-objective programming problems, stochastic (weak) Pareto optimality in the level of significance α ∈[0,1], is introduced to derive necessary and sufficient conditions for input/output estimation. Furthermore, the performance of the developed theory in a banking sector application is verified.

Assessment of OECD Better Life Index by incorporating public opinion

Well-being has a multidimensional nature as it depends on multifaceted factors such as material conditions and quality of life. The Organization for Economic Co-operation and Development (OECD) has developed the Better Life Index (BLI) as part of the OECD Better Life initiative to facilitate the better understanding of what drives well-being of people. The BLI is a three-level hierarchical composite indicator that covers several socio-economic aspects. In this paper, considering the entire hierarchical structure of the index, we introduce a bottom-up procedure for the aggregation of the components at each level. We formulate the assessment of BLI as a multiple objective programming (MOP) problem that facilitates the implementation of different concepts to derive different aggregation schemes. We incorporate the data from previous years into the normalization process of the indicators, to take into account the discrepancy on their observed values and smooth their deviations across the years. Also, we consider the public opinion about well-being that is captured from the worldwide responses in the web platform of OECD BLI. We incorporate the public opinion into the assessment models in the form of weight restrictions. In this way, we reduce the effect of compensation that might be imposed by the adopted modelling approach. We apply our methodology to the data of 38 countries (35 OECD and 3 non-OECD economies) for the year 2017. Our findings illustrate that the public opinion in the form of weight restrictions can effectively drive the optimization process and depict the collective preferences to the BLI scores.

On a fully fuzzy framework for minimax mixed integer linear programming

In this work, we present a modeling framework for minimax mixed 0–1 fuzzy linear problems. It is based on extending the usual rewriting of crisp minimax problems via auxiliary variables to model the maximum of a finite set of fuzzy linear functions. We prove that the considered problem can be equivalently formulated as a (crips) multiple objective mixed integer programming problem. The framework is applied to a fully fuzzy version of the capacitated center facility location problem.

Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.

Portfolio selection problem: a review of deterministic and stochastic multiple objective programming models

The literature on portfolio selection mostly concentrates on computational analysis rather than on modelling efforts. In response, this paper provides a comprehensive literature review of multiple objective deterministic and stochastic programming models for the portfolio selection problem. First, we summarize different concepts related to portfolio selection theory, including pricing models and portfolio risk measures. Second, we report the mathematical models that are generally used to solve deterministic and stochastic multiple objective programming problems. Finally, we present how these models can be used to solve the portfolio selection problem.

A Class of Multiple Objective Mathematical Programming Problems in a Rough Environment

This paper presents a set of multiple objective programming problems in a rough environment. These problems are classified into five classes according to the location of the roughness in the objective functions or the feasible set. We study the class in which all of the objective functions are crisp and the feasible region is a rough set and, in particular, discuss the properties of the completely and efficient (Pareto optimal) solutions of rough multiple objective programming problems. In order to obtain these solutions we need certain theorems which we derive. Finally, we illustrate our results by examples

Rough multiple objective programming

In this paper, we focused on characterizing and solving the multiple objective programming problems which have some imprecision of a vague nature in their formulation. The Rough Set Theory is only used in modeling the vague data in such problems, and our contribution in data mining process is confined only in the “post-processing stage”. These new problems are called rough multiple objective programming (RMOP) problems and classified into three classes according to the place of the roughness in the problem. Also, new concepts and theorems are introduced on the lines of their crisp counterparts; e.g. rough complete solution, rough efficient set, rough weak efficient set, rough Pareto front, weighted sum problem, etc. To avoid the prolongation of this paper, only the 1st-class, where the decision set is a rough set and all the objectives are crisp functions, is investigated and discussed in details. Furthermore, a flowchart for solving the 1st-class RMOP problems is presented.

Inverse DEA under inter-temporal dependence using multiple-objective programming

This paper deals with the inverse Data Envelopment Analysis (DEA) under inter-temporal dependence assumption. Both problems, input-estimation and output-estimation, are investigated. Necessary and sufficient conditions for input/output estimation are established utilizing Pareto and weak Pareto solutions of linear multiple-objective programming problems. Furthermore, in this paper we introduce a new optimality notion for multiple-objective programming problems, periodic weak Pareto optimality. These solutions are used in inverse DEA, and it is shown that these can be characterized by a simple modification in weighted sum scalarization tool.

An interactive algorithm for Large Scale Bi level Linear Multiple Objective programming problems through TOPSIS approach

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Bi-level Linear Multiple Objective Programming (LS-BL-LMOP) problems. In order to obtain a compromise ( satisfactory) solution to the LS-BL-LMOP problems using the proposed TOPSIS approach, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions of both the first and the second levels. An interactive decision making algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is provided where the first level decision maker (FLDM) is asked to specify the relative importance of the objectives. Finally, a numerical example is given to clarify the main results developed in the paper.

From ranking fuzzy numbers to solving fuzzy linear programming: a comprehensive review

Solving fuzzy linear programming (FLP) requires the employment of a consistent ranking of fuzzy numbers. Ineffective fuzzy number ranking would lead to a flawed and erroneous solving approach. This paper presents a comprehensive and extensive review on fuzzy number ranking methods. Ranking techniques are categorised into six classes based on their characteristics. They include centroid methods, distance methods, area methods, lexicographical methods, methods based on decision maker's viewpoint, and methods based on left and right spreads. A survey on solving approaches to FLP is also reported. We then point out errors in several existing methods that are relevant to the ranking of fuzzy numbers and thence suggest an effective method to solve FLP. Consequently, FLP problems are converted into non-fuzzy single (or multiple) objective linear programming based on a consistent centroid-based ranking of fuzzy numbers. Solutions of FLP are then obtained by solving corresponding crisp single (or multiple) objective programming problems by conventional methods.

Global maximization of UTA functions in multi-objective optimization

The UTAs (UTilité Additives) type methods for constructing nondecreasing additive utility functions were first proposed by Jacquet-Lagrèze and Siskos in 1982 for handling decision problems of multicriteria ranking. In this article, by UTA functions, we mean functions which are constructed by the UTA type methods. Our purpose is to propose an algorithm for globally maximizing UTA functions of a class of linear/convex multiple objective programming problems. The algorithm is established based on a branch and bound scheme, in which the branching procedure is performed by a so-called I-rectangular bisection in the objective (outcome) space, and the bounding procedure by some convex or linear programs. Preliminary computational experiments show that this algorithm can work well for the case where the number of objective functions in the multiple objective optimization problem under consideration is much smaller than the number of variables.

A class of rough multiple objective programming and its application to solid transportation problem

This paper establishes a rough multiple objective programming model for a solid transportation problem. Furthermore, a general model for rough multiple objective programming problem is presented. Properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated. In addition, compromise solutions are obtained by using the interactive fuzzy satisfying method. To solve the rough multiple objective solid transportation problem, the rough simulation-based genetic algorithm is proposed, in which the rough simulation is embedded into the genetic algorithm. Finally, an application of the solid transportation problem at Xiluodu Hydropower Station is provided as an illustration.

An interactive algorithm for large scale multiple objective programming problems with fuzzy parameters through TOPSIS approach

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Multiple Objective Programming problems involving fuzzy parameters. These fuzzy parameters are characterized as fuzzy numbers. For such problems, the ?-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the ?-Level sets of fuzzy numbers. An interactive fuzzy decision making algorithm for generating ?-Pareto optimal solution through TOPSIS approach is provided, where a decision maker (DM) is asked to specify the degree ? and the relative importance of objectives. Finally, a numerical example is given to clarify the main results developed in the paper. <br><br><font color="red"><b> This article has been retracted. Link to the retraction <u><a href="http://dx.doi.org/10.2298/YJOR141008034U">10.2298/YJOR141008034U</a><u></b></font>

A fuzzy random resource-constrained scheduling model with multiple projects and its application to a working procedure in a large-scale water conservancy and hydropower construction project

The aim of this paper is to deal with resource-constrained multiple project scheduling problems (rc-mPSP) under a fuzzy random environment by a hybrid genetic algorithm with fuzzy logic controller (flc-hGA), to a large-scale water conservancy and hydropower construction project in the southwest region of China, whose main project is a dam embankment. The objective functions in this paper are to minimize the total project time (that is the sum of the completion time for all projects) and to minimize the total tardiness penalty of multiple projects, which is the sum of penalty costs for all the projects. After describing the problem of the working procedure in the project and presenting the mathematical formulation model of a resource-constrained project scheduling problem under a fuzzy random environment, we give some definitions and discuss some properties of fuzzy random variables. Then, a method of solving solution sets of fuzzy random multiple objective programming problems is proposed. Because traditional optimization techniques could not cope with the rc-mPSP under a fuzzy random environment effectively, we present a new approach based on the hybrid genetic algorithm (hGA). In order to improve its efficiency, the proposed method hybridized with the fuzzy logic controller (flc) concept for auto-tuning the GA parameters is presented. For the practical problems in this paper, flc-hGA is proved the most effective and most appropriate compared with other approaches. The computer generated results validate the effectiveness of the proposed model and algorithm in solving large-scale practical problems.

Extended VIKOR as a new method for solving Multiple Objective Large-Scale Nonlinear Programming problems

The VIKOR method was introduced as a Multi-Attribute Decision Making (MADM) method to solve discrete decision-making problems with incommensurable and conflicting criteria. This method focuses on ranking and selecting from a set of alternatives based on the particular measure of “closeness” to the “ideal” solution. The multi-criteria measure for compromise ranking is developed from the l – p metric used as an aggregating function in a compromise programming method. In this paper, the VIKOR method is extended to solve Multi-Objective Large-Scale Non-Linear Programming (MOLSNLP) problems with block angular structure. In the proposed approach, the Y-dimensional objective space is reduced into a one-dimensional space by applying the Dantzig-Wolfe decomposition algorithm as well as extending the concepts of VIKOR method for decision-making in continues environment. Finally, a numerical example is given to illustrate and clarify the main results developed in this paper.

Solving a sealed-bid reverse auction problem by multiple-criterion decision-making methods

This study presents a model for solving the sealed-bid, multiple-issue reverse auction problem, using multiple-criterion decision-making approaches, such that the interests of both the buyer and the supplier are satisfied. On the supplier side, the bid construction process is formulated as a fuzzy multiple-objective programming problem, and is solved using an exhausted enumeration algorithm which adjusts the production plan in accordance with the buyer’s demand, based on the current master production schedule (MPS) and the available-to-promise (ATP) inventory. The use of the information of MPS and ATP enables the supplier to make accurate estimates of the production costs associated with specific delivery dates, and thus facilitates the construction of a bid which is both profitable and likely to secure the contract. On the buyer side, the winner determination process is treated as a multiple-attribute decision-making problem, and is solved using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method. The validity of the proposed approach is demonstrated via an illustrative example.

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.

Exact generation of epsilon-efficient solutions in multiple objective programming

It is a common characteristic of many multiple objective programming problems that the efficient solution set can only be identified in approximation: as this set often contains an infinite number of points, only a discrete representation can be computed, and due to numerical difficulties, each of these points itself might, in general, be only approximate to some efficient point. From among the various approximation concepts, this paper considers the notion of epsilon-efficient solutions and proposes several new methods for their generation. Supporting theoretical results are established and a numerical example is provided.