Multiple criteria selection problems (MCSP) solve problems with finite and known number of alternatives, where each alternative is measured by several conflicting criteria which values are also known a priori in the form of a pay-off matrix. The ranking methods we will review here identify the best alternative and rank order all the alternatives from the best to the worst.

Basic concepts

Pay-off matrix: A matrix listing the $n$ alternatives ( $1, 2,…, n$ ) and the p criteria ( $f1, f2,…, fp$ ) for a MCSP, as well as the values for each alternative and criteria.

Best compromise solution (or most preferred alternative): In MCSP, due to the conflicting nature of the objectives, the optimal values of the various criteria do not usually occur at the same alternative. Therefore, instead of an “optimal” solution, we have a “best compromise” solution or a solution with preferred trade-offs among the feasible solutions. The task of MCSP methods is, given a set of preferences expressed by the decision maker (i.e. the preference structure), to rank order the alternatives from the best to the worst.

Dominated alternative: That alternative where the criteria values of another alternative are as good as those of the first alternative and, for at least one criterion, alternative the second alternative is better than the first one.

Non-dominated alternative (pareto optimal, or efficient alternative): An alternative that is not dominated by any other alternatives. In other words, no other alternative is as good as the current alternative for at least one criterion. Moreover, an improvement in any one criterion is possible only at the expense of at least one other criterion.

Ideal solution: Represents the vector of the best values achievable for each criterion. It is called “ideal” since, due to the nature of multiple criteria problems, it is very unlikely that any alternative solution will achieve the best values across all criteria.

Multicriteria Selection Methods - Overview

The most common multiple-criteria selection methods are the following:

Lp metric
Linear weighting methods:

Rating
Borda count (or ranking method)
Analytical Hierarchy Process (AHP)
Analytical Network Process (ANP)

Categorical methods
Multi-Attribute Utility Theory (MAUT)

+++

1.Lp metric: A method used to measure the distance between points in n-dimensional space. Therefore, the mathematical distance between two vectors $x$ and $y$ , where $x, y ^+$ is given by:
$| x - y |$
When $p=1$ we have what is known as the Manhattan, city block, boxcar, or absolute value distance and it represents the distance between two points in a city road grid. The Euclidean distance, which is the most commonly used distance, occurs when $p=2$ . In other words, the Euclidean distance represents the length of the line segment between two points or the root square differences between the coordinates of a pair of objects. The $Lp$ metric method can be applied outside of geospatial contexts. For instance, when applied to supplier rankings, the vector $x$ represents the criteria value for a particular supplier and the vector $y$ represents the best/ideal values of those criteria.

2.Weighting methods: These methods use simple scoring to rank alternatives. Each alternative is rated with respect to several criteria and these ratings are combined into a single score. Most of the weighting methods are compensatory, though some are non-compensatory. In a compensatory model, a low rating on one criterion can be compensated by a high rating on another criterion, whereas in non-compensatory models different minimum levels for each criterion are required (Degraeve, et al., 2000). Although this approach is very simple, it depends heavily on human judgment. Moreover, it is generally difficult to come up with exact criteria weights and it also requires all the criteria values to be scaled properly when incommensurability exists. Among the most widely used weighting techniques we have the following:

Rating simply consists of each participant voting for his/her favorite alternative according to an appropriate rating scale (e.g. from 1 to 10, where 10 is the most important and 1 is the least important selection criteria). Next, using the selected scale, the DM provides a rating $rj$ for each criterion, ${C}_j$ . The same rating can be given to more than one criterion. The ratings are then normalized to determine the weights of the criteria $j$ ( ${W}_j$ ). Therefore, for $n$ criteria:

$W_{j} =$
, so that
$\sum_{j = 1}^{n} W_{j} = 1$

After this, a weighted score of the attributes is calculated for each supplier as follows:
$S_{j} =$ for $i=1,…K$ , where ${f}_{ij}$ ’s are the criteria values for supplier $i$ .

The alternatives are then ranked based on their scores and the choice with the highest score is ranked first. To include the opinion of multiple decision makers, each decision maker k rates each criterion ${C}$ . Then, the average rating for criterion $j$ is obtained by ${}$ and each criterion is weighted using ${}$ {j} = $\overline{W}_{j} = \frac{V _{j}}{\sum _{i = 1}^{p} V _{j}}$ where the most preferred criterion has the highest weight. The final rating of alternatives is assigned according to $${V{i}={i=1}^{p}V{ij}*W_{j}$$, for every alternative $i$ .

Borda count. This method is often used for polls that rank sporting teams or academic institutions. It is also known as simple ranking method. Here, judges rank all the alternatives according to their liking from 1 to N, where N represents the number of items and N is assigned to the least important value. Also, weights for the criteria are calculated as the ratio between the number of points awarded to each option and the sum of points assigned.
Analytic hierarchy process (AHP). Developed by Saaty (1980), is another method used for ranking alternatives. In this approach factors (e.g. goals, attributes, issues, stakeholders) are arranged in a hierarchic structure. By doing this, the method provides an overall view of the complex relationships inherent between factors and helps the decision maker assess whether the issues in each level are of the same order of magnitude, allowing the comparison of elements accurately. An important strength of AHP is that it considers multiple criteria and allows the inclusion of qualitative and quantitative factors. It is a good method to handle intuitive and rational thinking and it does not require that the criteria values be scaled properly. The hierarchies of factors are constructed first, then, pair-wise comparisons among factors in each level of the hierarchy are made with respect to the factor in the level above. Next, a nine-point numerical scale is used to display the relative degree of importance of two factors. A matrix is built to represent such relative importance. Eigenvector analysis is used to calculate the individual and overall influence of factors in the goal. Finally, a consistency index is used to measure the entire consistency of judgments for each comparison matrix and the hierarchy structure.
Analytic network process (ANP) is a generalization of the AHP method. ANP provides a general framework to deal with decisions without making assumptions about the independence of higher-level elements from lower-level elements or about the independence of the elements within a level. ANP uses a network without the need to specify levels as in a hierarchy.

Categorical methods help decision-makers evaluate their alternatives’ performance on a set of criteria using historical data and decision-makers’ experience. Initially, the performance on each criterion is categorized as positive, neutral, or negative. After the set of criteria has been evaluated, a second phase consists of assigning an overall rating to the alternatives again through either positive, neutral, or negative labels. The major advantage of the categorical approach is that it helps structure the evaluation process clearly and systematically. An obvious disadvantage is that it typically does not clearly define the relative importance of each criterion.
Multi-attribute utility theory (MAUT) provides a mathematical-based procedure through which the attractiveness (utility) of each criteria value is measured for a set of alternatives and thus, the optimal balance between multiple competing objectives is found. Unlike other methods, instead of criteria weights, it uses scaling parameters that measure the willingness to trade a specific amount of one criterion (attribute) for another focusing on the marginal utility with respect to the levels of other criterion (attribute). The method also does not require scaling of attributes since preferences reflect trade-offs between attribute values. In this way, the actual attribute values are available throughout the decision process. Also, uncertainty and risk are incorporated throughout the decision process.