Method of the vector of similarity to ideal solution in alternatives

Abstract

A method for solving multiple-attribute decision-making problems at one criterion level is discussed, followed by a presentation of a method based on compromise programming elements, Lp metrics and the TOPSIS method. Suggestions for forming the initial decision-making matrix have been given as well as for the transformation of multiple criteria values. Compromise solutions are obtained based on the values of Lp metrics functions and their combinations with coefficients – functions of relative credibility depending on solution dimensions. The obtained solutions depend on the parameter p in Lp metrics, parameter p being a balancing factor between the solution with the highest total utility and the solution with minimax deviations of criteria values from the ideal and negative-ideal solutions. If a unique solution is required, it is obtained by encompassing all Lp metrics functions and by applying the vector of similarity to ideal solution, which also includes the negative-ideal solution influence. The article shows a method for obtaining compromise solutions based on the values of elements of partial similarity to ideal vectors and the influence of a subjectivelly determined negative ideal on solutions. The method application has been illustrated by a numerical example.

Introduction

Problems of multiple-attribute decision-making (MADM) are defined as a class of multiple-criteria decision-making (MCDM) problems for which there is a mathematical model formed but no unique optimal solution. MADM methods (’soft’ methods) are used to solve ’poorly structured’ MCDM problems. The procedure for solving MADM problems is based on Lp metrics, compromise programming and the TOPSIS method. Compromise solutions obtained by applying Lp metrics parameters (p) further give the final solution (the best alternative) which is obtained using the values of the elements of the vector of similarity to ideal solution (VSIS) which encompasses the parameters of the relation of alternatives with the ideal and negative-ideal solutions.

Decision-making matrix and spaces of alternatives

Initial decision-making matrix

A decision-making matrix for a MADM problem is given by real criterion values, their estimates or estimates of attributes. A preparation of the decision-making matrix for the application of the MADM method requires a systematic preliminary analysis of a problem in question. The initial decision-making matrix is possible to be formed in many ways, but a usual sequence of phases is: preliminary analysis and determination of MADM goals; problem structuring; data collection, and determination of criteria values.

Primary and secondary space of alternatives

Spaces of alternatives are defined with reference to the chosen referent points: the ideal solution and the negative-ideal solution. The primary space of alternatives (PSA) is formed in the range of the ’perceived’ ideal solution and the negative-ideal solution while the secondary space of alternatives (SSA) is formed within the limits of the ’preferable’ ideal solution and the ’requested’ negative-ideal one. By determining the ideal and negative-ideal solutions subjectivelly, a decision-maker (DM) determines the range of criteria or the interval of acceptable values for each criterion. In reality, it is much more difficult to determine the ideal solution than the negative-ideal solution, so it makes sense to determine also the position of an alternative in relation to the negative-ideal solution. For illustrating the VSIS method with a numerical example, a decision-making matrix with four alternatives and five criteria has been formed.

Transformation of criteria values

Since measurement units are different for different criteria (heterogeneous criteria system), the elements of the initial matrix are transformed into dimensionless parameters in the interval [0,1]. In this article, the author used the transformation based on criteria values range (interval length), i.e. the absolute difference between the best and the worst criterion value: , . Such a  transformation is possible when the criteria are mutually independent or when their mutual dependence is not taken into account. The transformation is not without flaws, since criteria values for criteria with a narrower range become more prominent than those with a wider range, which is due to the choice of the ideal and negative-ideal solutions from known criteria values (’perceived’ values).

Application of Lp metrics in MADM

Lp metrics in MADM is a measure of the distance of alternatives from the ideal solution and the negative-ideal solution; it is the basis for determination of compromise solutions. The application of different values for the Lp metrics parameter results in more than one solution (solution is the best alternative), i.e. compromise solutions of MADM problems. The most frequently applied parameters are , for which, regardless of SA, distances of alternatives from the ideal and negative-ideal solutions are defined, and based on their values up to six different compromise solutions are obtained (the best alternative and a ranking list of alternatives). Which ranking list the decision-maker is going to accept depends on taking into account the parameter p influence on overall effects. The method enables usage of other parameter p values for obtaining compromise solutions. Two new compromise solutions are obtained by ‘reconciling’ non-adapted ranks and by applying the linear combination of Lp metrics function values.

Vector of similarity to ideal solution in alternatives

The vector of similarity to ideal solution (VSIS) in alternatives is, according to all criteria, a multidimensional vector  , the elements (coefficients of similarity to ideal solution – CSIS) of which are determined based on diverse distances  and : ; .  Based on a partial VSIS, compromise solutions based on ’pure’ Lp metrics distances are also possible. The application of the VSIS results in a unique solution of an MADM problem, with the influence of the negative-ideal solution taken into account. By a further definition of the secondary negative-ideal solution and a subsequent extension of the PSA, the drawbacks of the applied decision-making matrix transformation are partially eliminated and more realistic parameters of alternatives are obtained. It is also possible to determine the values of the extension of the space of alternatives, as factors influencing new coefficients of similarity to ideal solutions. The compromise solutions and, based on them, the obtained unique solution of the MADM problem should represent a starting point for decision-makers before they make a final decision. Decision-makers have a good basis as well as arguments for a choice of either one of compromise solutions or a unique solution; however, these solutions do not need to be their final choice. This procedure is to help them in their final decision.

Conclusion

The application of Lp metrics for solving an MADM problem has been shown in the article. Based on the distances between n-dimensional points representing alternatives and referent points (the ideal and negative-ideal solutions), six compromise solutions have been obtained for the characteristic values of the Lp metrics parameter . ’Reconciliation’ of such solutions has been done by forming two linear combinations (based on the ideal solution and the negative-ideal solution) with coefficients representing relative credibility of the Lp metrics distance, thus resulting in two more compromise solutions. The application of the linear combinations of the Lp metrics functions helps the decision-maker to rank alternatives in accordance with requests for overall utility, geometric proximity or minimax deviations of criteria values from the ideal and negative-ideal solutions or their combinations. During the procedure, while choosing coefficients of linear combinations, decision-makers can express their attitude towards the influence of the parameter p on solutions. The final solution is obtained by ranking alternatives according to the values of the elements of the vector of similarity to ideal solution, which encompasses the relation of alternatives towards the negative-ideal solution as well. Software support to the VSIS method enables a choice of one, two or three functions of Lp metrics, a large number of criteria and subcriteria as well as the application of the method in real time under fast changing external conditions. Updating the software with elements from different human activities (military issues, ecology, education, health systems, economy issues, etc.) can enhance decision-making in these fields.