Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS

The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. In VIKOR linear normalization and in TOPSIS vector normalization is used to eliminate the units of criterion functions. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an individual regret for the “opponent”. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. A comparative analysis of these two methods is illustrated with a numerical example, showing their similarity and some differences.     

Extended VIKOR method in comparison with outranking methods

The VIKOR method was developed to solve MCDM problems with conflicting and noncommensurable (different units) criteria, assuming that compromising is acceptable for conflict resolution, the decision maker wants a solution that is the closest to the ideal, and the alternatives are evaluated according to all established criteria. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and on proposing compromise solution (one or more). The VIKOR method is extended with a stability analysis determining the weight stability intervals and with trade-offs analysis. The extended VIKOR method is compared with three multicriteria decision making methods: TOPSIS, PROMETHEE, and ELECTRE. A numerical example illustrates an application of the VIKOR method, and the results by all four considered methods are compared.     

Extension of VIKOR method for decision making problem based on hesitant fuzzy set

The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are all based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent”, which is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know to express his/her preference at the beginning of system design. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. And, the hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we develop the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. Firstly, the hesitant fuzzy set information and corresponding concepts are described, and the basic essential of the VIKOR method is introduced. Then, the problem on multiple attribute decision marking is described, and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. Finally, a numerical example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared.     

A comparison of multiple criteria analysis techniques for water resource management

Multiple criteria analysis (MCA) is a framework for evaluating decision options against multiple criteria. Numerous techniques for solving an MCA problem are available. This paper applies MCA to six water management decision problems. The MCA methods tested include weighted summation, range of value, PROMTHEE II, Evamix and compromise programming. We show that different MCA methods were in strong agreement with high correlations amongst rankings. In the few cases where strong disagreement between MCA methods did occur it was due to presence of mixed ordinal-cardinal data in the evaluation matrix. The results suggest that whilst selection of the MCA technique is important more emphasis is needed on the initial structuring of the decision problem, which involves choosing criteria and decision options.     

On the efficiency test in multi-objective linear fractional programming problems by Lotfi et al. 2010

The solution concept in multi-objective programming is represented by a program which reaches “the best compromise”. Many solving methods find a good compromise feasible solution and then check whether the solution is efficient or not. In this paper, using the efficiency test introduced by Lotfi et al. (2010) [12], we propose two procedures for deriving weakly and strongly efficient solutions in multi-objective linear fractional programming problems (MOLFPP) starting from any feasible solution, and present their possible applications in multiple criteria decision-making process. Then, we discuss a shortcoming of some fuzzy approaches to solving MOLFPP and modify them in order to guarantee the efficiency of the optimal solution.     

Weight determination for consistently ranking alternatives in multiple criteria decision analysis

One of the most difficult tasks in multiple criteria decision analysis (MCDA) is determining the weights of individual criteria so that all alternatives can be compared based on the aggregate performance of all criteria. This problem can be transformed into the compromise programming of seeking alternatives with a shorter distance to the ideal or a longer distance to the anti-ideal despite the rankings based on the two distance measures possibly not being the same. In order to obtain consistent rankings, this paper proposes a measure of relative distance, which involves the calculation of the relative position of an alternative between the anti-ideal and the ideal for ranking. In this case, minimizing the distance to the ideal is equivalent to maximizing the distance to the anti-ideal, so the rankings obtained from the two criteria are the same. An example is used to discuss the advantages and disadvantages of the proposed method, and the results are compared with those obtained from the TOPSIS method.     

Solving a multiobjective possibilistic problem through compromise programming

Real decision problems usually consider several objectives that have parameters which are often given by the decision maker in an imprecise way. It is possible to handle these kinds of problems through multiple criteria models in terms of possibility theory.Here we propose a method for solving these kinds of models through a fuzzy compromise programming approach.To formulate a fuzzy compromise programming problem from a possibilistic multiobjective linear programming problem the fuzzy ideal solution concept is introduced. This concept is based on soft preference and indifference relationships and on canonical representation of fuzzy numbers by means of their α-cuts. The accuracy between the ideal solution and the objective values is evaluated handling the fuzzy parameters through their expected intervals and a definition of discrepancy between intervals is introduced in our analysis.     

Multi-objective geometric programming problem with ∊-constraint method

In multi-objective geometric programming problem there are more than one objective functions. There is no single optimal solution which simultaneously optimizes all the objective functions. Under these conditions the decision makers always search for the most “preferred” solution, in contrast to the optimal solution. A few mathematical programming methods namely fuzzy programming, goal programming and weighting methods have been applied in the recent past to find the compromise solution. In this paper ?$?$-constraint method has been applied to find the non-inferior solution. A brief solution procedure of ?$?$-constraint method has been presented to find the non-inferior solution of the multi-objective programming problems. Further, the multi-objective programming problems is solved by the fuzzy programming technique to find the optimal compromise solution. Finally, two numerical examples are solved by both the methods and compared with their obtained solutions.     

Regularized estimation for preference disaggregation in multiple criteria decision making

Disaggregation methods have been extensively used in multiple criteria decision making to infer preferential information from reference examples, using linear programming techniques. This paper proposes simple extensions of existing formulations, based on the concept of regularization which has been introduced within the context of the statistical learning theory. The properties of the resulting new formulations are analyzed for both ranking and classification problems and experimental results are presented demonstrating the improved performance of the proposed formulations over the ones traditionally used in preference disaggregation analysis.     

Extensions of TOPSIS for large scale multi-objective non-linear programming problems with block angular structure

This paper focuses on multi-objective large-scale non-linear programming (MOLSNLP) problems with block angular structure. We extend the technique for order preference by similarity ideal solution (TOPSIS) to solve them. Compromise (TOPSIS) control minimizes the measure of distance, provided that the closest solution should have the shortest distance from the positive ideal solution (PIS) as well as the longest distance from the negative ideal solution (NIS). As the measure of “closeness” L_P-metric is used. Thus, we reduce a q-dimensional objective space to a two-dimensional space by a first-order compromise procedure. The concept of a membership function of fuzzy set theory is used to represent the satisfaction level for both criteria. Moreover, we derive a single objective large-scale non-linear programming (LSNLP) problem using the max–min operator for the second-order compromise operation. Finally, a numerical illustrative example is given to clarify the main results developed in this paper.     

A multiple criteria decision making approach to manure management systems in the Netherlands

The intensification of livestock operations in the last few decades has resulted in an increased social concern over the environmental impacts of livestock operations and thus making appropriate manure management decisions increasingly important. A socially acceptable manure management system that simultaneously achieves the pressing environmental objectives while balancing the socio-economic welfare of farmers and society at large is needed. Manure management decisions involve a number of decision makers with different and conflicting views of what is acceptable in the context of sustainable development. This paper developed a decision-making tool based on a multiple criteria decision making (MCDM) approach to address the manure management problems in the Netherlands. This paper has demonstrated the application of compromise programming and goal programming to evaluate key trade-offs between socio-economic benefits and environmental sustainability of manure management systems while taking decision makers’ conflicting views of the different criteria into account. The proposed methodology is a useful tool in assisting decision makers and policy makers in designing policies that enhance the introduction of economically, socially and environmentally sustainable manure management systems.     

Compromise programming: A utility-based linear-quadratic composite metric from the trade-off between achievement and balanced (non-corner) solutions

In this paper, compromise programming (CP) is viewed as the maximization of the decision maker’s additive utility function (whose arguments are the criteria under consideration) subject to an efficient frontier of criteria and the non-negativity constraints in a deterministic context. This is equivalent to minimizing the difference (‘distance’) between utility at the ideal point and utility at a frontier point on the criteria map, a meaningful statement as minimizing distances to the utopia is the ethos of compromise programming. By Taylor expansion of utility around the ideal point, the distance to the utopia becomes the weighted sum of linear and quadratic CP distances, which gives us the composite metric. While the linear terms pursue achievement, the quadratic ones pursue balanced (non-corner) solutions. Because some decision makers fear imbalance while others prefer large achievements even to the detriment of balance, the paper defines an aversion to imbalance ratio, so that the composite linear-quadratic metric should conform to this ratio depending on the decision maker’s preferences and attitudes. As the problem of selecting an appropriate metric is an ongoing issue in CP, the paper is a contribution to theory and practice. For the sole purpose of suggesting industrial applications, an example is developed.     

Integer Programming Duality in Multiple Objective Programming

The weighted sums approach for linear and convex multiple criteria optimization is well studied. The weights determine a linear function of the criteria approximating a decision makers overall utility. Any efficient solution may be found in this way. This is not the case for multiple criteria integer programming. However, in this case one may apply the more general e-constraint approach, resulting in particular single-criteria integer programming problems to generate efficient solutions. We show how this approach implies a more general, composite utility function of the criteria yielding a unified treatment of multiple criteria optimization with and without integrality constraints. Moreover, any efficient solution can be found using appropriate composite functions. The functions may be generated by the classical solution methods such as cutting plane and branch and bound algorithms.     

Integer Fuzzy Programming Approach in Bi-objective Selective Maintenance Allocation Problem

An attempt has been made to obtain a compromise allocation based on maximization of individual reliabilities of repairable and replaceable components with in the subsystems, using the information of failed and operational components and a non linear cost function with fixed budget. A solution algorithm of fuzzy programming technique is used to solve the Bi-Objective Selective Maintenance Allocation Problem (BSMAP). Also, the problem has been solved by two other suggested methods; “Weighted Criterion Technique” and “Desirability Function Technique”. A numerical example is also presented to illustrate the computational details.     

Constrained optimization using multiple objective programming

In practical applications of mathematical programming it is frequently observed that the decision maker prefers apparently suboptimal solutions. A natural explanation for this phenomenon is that the applied mathematical model was not sufficiently realistic and did not fully represent all the decision makers criteria and constraints. Since multicriteria optimization approaches are specifically designed to incorporate such complex preference structures, they gain more and more importance in application areas as, for example, engineering design and capital budgeting. The aim of this paper is to analyze optimization problems both from a constrained programming and a multicriteria programming perspective. It is shown that both formulations share important properties, and that many classical solution approaches have correspondences in the respective models. The analysis naturally leads to a discussion of the applicability of some recent approximation techniques for multicriteria programming problems for the approximation of optimal solutions and of Lagrange multipliers in convex constrained programming. Convergence results are proven for convex and nonconvex problems.     

A procedure for selection of a multiobjective technique with application to water and mineral resources

The problem of selecting the appropriate multiobjective solution technique to solve an arbitrary multiobjective decision problem is considered. Various classification schemes of available techniques are discussed, leading to the development of a set of 28 model choice criteria and an algorithm for model choice. This algorithm divides the criteria into four groups, only one of which must be reevaluated for each decision problem encountered. The model choice problem is itself modeled as a multiobjective decision problem—strongly influenced, however, by the individual performing the analysis. The appropriate technique is selected for implementation by use of the compromise programming technique. Two example problems are presented to demonstrate the use of this algorithm. The first is concerned with ranking a predefined set of river basin planning alternatives with multiple noncommensurate ordinally ranked consequences. The second deals with coal blending and is modeled by dual objective linear programming. An appropriate multiobjective solution technique is selected for each of these two examples.     

Optimization over the efficient set: Four special cases

Recently, researchers and practitioners have been increasingly interested in the problem (P) of maximizing a linear function over the efficient set of a multiple objective linear program. Problem (P) is generally a difficult global optimization problem which requires numerically intensive procedures for its solution. In this paper, simple linear programming procedures are described for detecting and solving four special cases of problem (P). When solving instances of problem (P), these procedures can be used as screening devices to detect and solve these four special cases.     

Goal programming approaches for data envelopment analysis cross efficiency evaluation

Cross efficiency evaluation has long been proposed as an alternative method for ranking the decision making units (DMUs) in data envelopment analysis (DEA). This study proposes goal programming models that could be used in the second stage of the cross evaluation. Proposed goal programming models have different efficiency concepts as classical DEA, minmax and minsum efficiency criteria. Numerical examples are provided to illustrate the applications of the proposed goal programming cross efficiency models.     

Fuzzy programming and linear programming with several objective functions

In the recent past numerous models and methods have been suggested to solve the vectormaximum problem. Most of these approaches center their attention on linear programming problems with several objective functions. Apart from these approaches the theory of fuzzy sets has been employed to formulate and solve fuzzy linear programming problems. This paper presents the application of fuzzy linear programming approaches to the linear vectormaximum problem. It shows that solutions obtained by fuzzy linear programming are always efficient solutions. It also shows the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution.     

An iterative back substitution algorithm for the solution of tridiagonal matrix systems with fringes

For tridiagonal matrix systems, a simple direct algorithm giving the solution exists, but in the most general case of tridiagonal matrix with fringes, the direct solving algorithms are more complicated. For big systems, direct methods are not well fitted and iterative algorithms are preferable. In this paper a relaxation type iterative algorithm is presented. It is an extension of the backward substitution method used for simple tridiagonal matrix systems. The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR. Its nature suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method. A special emphasis is done on the programming aspect. The solving Fortran subroutines implementing the algorithm have been generated automatically from their specification by using a computer algebra system technique.