Multi-Criteria Decision Analysis Techniques Overview
2.3 Typical Non-compensatory Decision Analysis Methods
Non-compensatory decision analysis methods do not permit trade-offs among criteria, that is, a disadvantage in one criterion cannot be offset by an advantage in other criterion. The non-compensatory methods are credited for their simplicity. As summarized in Table 2.1, typical non-compensatory methods are explained in detail in the following subsections.
2.3.1 Conjunctive Method
The DM sets up the acceptable minimal criteria values. Any alternative which has a criterion value less than the standard level will be rejected [58]. When bigger criteria values are preferred, the i-th alternativeAi (i= 1,2, ..., m) is classified as an acceptable alternative only if
xij ≥x0j, j= 1,2, ..., n (2.6)
where x0j is the standard level of the j-th criterion xj. The cutoff values play a key role in eliminating the alternatives; if too high, none is left; if relatively low, several alternatives are left after filtering. Hence, increasing the minimal standard levels in an iterative way, the alternatives can be narrowed down to a single choice.
The Conjunctive method does not require the criteria to be in numerical form, and the relative importance of the criteria is not needed. This method is usually used for dichotomizing alternatives into acceptable and unacceptable categories.
2.3 Typical Non-compensatory Decision Analysis Methods
2.3.2 Disjunctive Method
In the Disjunctive method, an alternative is evaluated on its greatest value of a criterion [58].
When bigger criteria values are preferred, the i-th alternativeAi (i= 1,2, ..., m) is classified as an acceptable alternative only if
xij ≥x0j, j= 1 or 2 or...orn (2.7) wherex0j is the desirable level of the j-th criterion xj.
As with the Conjunctive method, the Disjunctive method does not require the criteria to be in numerical form, and it does not need information on the relative importance of the criteria.
2.3.3 Dominance Method
The Dominance method can be used to screen the alternatives in order to obtain a set of non-dominated solutions before the final choice. The procedures of the Dominance method are described as follows [26].
• Compare the first two alternatives and if one is dominated by the other, discard the dominated one.
• Next, compare the retained alternative with the third alternative and discard any domi-nated alternative.
• Then, compare the fourth alternative and so on.
• After all the alternatives are compared, the non-dominated set is determined.
The Dominance method does not require any assumption or any transformation of crite-ria. The non-dominated set usually has multiple alternatives, hence, the Dominance method is mainly used for initial filtering.
2.3.4 ELECTRE
ELECTRE (Elimination and Choice Translation Reality) methods use the concept of outranking relation introduced by Benayoun [15]. For instance, suppose there aremalternatives based onn evaluation criteria, with weighting factors [w1, w2, ..., wn],xij stands for the value of criterionxj
with respect to alternativeAi. An outranking relation between alternativeAkand alternativeAl
(k, l = 1,2, ..., m, k 6=l) is defined as: Ak is preferred to Al when Ak is at least as good asAl with respect to a majority of criteria and when Ak is not significantly poor regarding any other criteria. After the assessment of the outranking relations for each pair of alternatives,
dominated alternatives can be eliminated and non-dominated alternatives can be obtained for further consideration.
There are several different versions of ELECTRE methods, including ELECTRE I, IS, II, III, IV and TRI [106], [33]. ELECTRE I is the first decision analysis method using the concept of outranking relation, the other versions of ELECTRE methods are extensions of ELECTRE I. In this subsection, the stepwise calculations of ELECTRE I are described in detail and the other ELECTRE methods are briefly introduced.
ELECTRE I is composed of the following nine steps [58].
1. Normalize the decision matrix
2. Calculate the weighted normalized decision matrix.
V =RW =
3. Determine the concordance and discordance sets.
For each pair of alternatives Ak and Al, the set of decision criteria J = (j |j= 1,2, ..., n) is divided into two disjoint subsets. The concordance set Ckl of Ak and Al is composed of all criteria which support that Ak is preferred to Al. The discordance set Dkl is the complementary subset of the concordance set Ckl.
Ckl={j|xkj ≥xlj},(k, l = 1,2, ..., m,andk6=l)
Dkl={j|xkj < xlj}=J−Ckl (2.10) 4. Calculate the concordance matrix C.
Each element of the concordance matrixCis calculated by the sum of the criteria weights which are contained in the concordance set. For example, the elementcklbetweenAk and Al is calculated by Equation 2.11.
2.3 Typical Non-compensatory Decision Analysis Methods
5. Calculate the discordance matrixD.
Each element of the discordance matrix D reflects the degree to which one alternative is worse than the other. For instance, the element dkl between Ak and Al is calculated by Equation 2.12.
It should be noticed that differences among weighting factors are contained in the concor-dance matrix C, while differences among criteria values are reflected in the discordance matrixD.
6. Determine the concordance dominance matrix.
A concordance threshold c needs to be chosen to perform the concordance test. Alter-native Ak possibly dominates alternative Al, if the element ckl exceeds at least a certain thresholdc, that is,ckl ≥c.
In ELECTRE I, a Boolean matrix is used to convert the concordance test into numerical values (0 or 1). If the concordance test is passed (ckl≥c), then the element is 1. Otherwise, if the concordance test is failed (ckl< c), the element is 0.
7. Determine the discordance dominance matrix.
A discordance threshold d needs to be chosen to perform the discordance test. Alter-native Ak possibly dominates alternative Al, if the element dkl is smaller than a certain thresholdd, that is, dkl ≤d.
As with the case of the determination of the concordance dominance matrix, the discor-dance test is converted into numerical values (0 or 1) by a Boolean matrix. The element is 1 when the discordance test is passed (dkl≤d), and it is 0 when the discordance test is failed (dkl> d).
8. Aggregate the dominance matrix.
An outranking relation can be justified only if both the concordance test and the discor-dance test are passed. That is, ckl ≥ c and dkl ≤ d. The aggregated dominance matrix is calculated by an element-to-element product of the concordance dominance matrix and the discordance dominance matrix.
9. Eliminate the dominated alternatives.
The aggregated dominance matrix gives the partial preference of the alternatives. In the aggregated dominance matrix, the element 1 in the column indicates that this alternative is dominated by other alternatives. Thus, any alternative which has at least one element of 1 in the column can be eliminated.
ELECTRE I is widely used because of its simple logic and refined computational procedures.
However, the two concordance and discordance threshold values have significant impact on the final results. Additionally, the calculation procedures will become more complex as the size of decision matrix increases.
An Aircraft Selection Example using ELECTRE I
An aircraft selection example is presented to show how to use ELECTRE I in this subsection.
Suppose that the DMs of an airline consider to purchase an aircraft among three competing aircraft, with the consideration of three criteria: comfort, cost, and environmental friendliness.
Smaller value of cost is preferred, while bigger values of comfort and environmental friendliness are preferred. A ten-point score is assigned to the three criteria for each alternative, respec-tively. The weighting factors among the three criteria are [0.3 0.4 0.3]. The decision matrix is summarized in Table 2.5.
Given the decision matrix shown in Table 2.5, going through the described nine-step calcu-lations of ELECTRE I, the aggregated dominance matrix is shown in matrix M.
2.3 Typical Non-compensatory Decision Analysis Methods
Table 2.5: Decision Matrix of an Aircraft Selection Example using ELECTRE I Criteria
C1: Comfort C2: Cost C3: Environmental friendliness Alternatives w1: 0.3 w2: 0.4 w3: 0.3
Aircraft A 8 7 10
Aircraft B 9 6 5
Aircraft C 6 7 8
In the aggregated dominance matrix M, the element 1 in the column indicates that this alternative is dominated by other alternatives. Thus, Aircraft C is dominated by Aircraft A and Aircraft B. In another words, Aircraft A and Aircraft B are non-dominated alternatives.
Therefore, in this aircraft selection example using ELECTRE I, Aircraft C should be eliminated from the candidate alternatives, Aircraft A and Aircraft B can be recommended for further consideration.
ELECTRE IS
ELECTRE IS is similar to ELECTRE I, except that in Step 6 (Determine the concordance dominance matrix), instead of Boolean numbers (0 or 1), interval values between 0 and 1 are used [106], [33]. In order to discriminate the alternatives, two thresholds have to be defined for each criterion: indifference threshold and strict preference threshold.
ELECTRE II
ELECTRE II is also similar to ELECTRE I. The main difference is the definition of two out-ranking relations: strong outout-ranking and weak outout-ranking [106]. For each criterion, two strong outranking thresholds and one weak outranking threshold have to be defined.
ELECTRE III
ELECTRE III uses the same principle of ELECTRE II. For each criterion, an indifference threshold, a preference threshold, and a veto threshold have to be defined in order to compare the alternatives. Both the concordance dominance matrix and discordance dominance matrix are constructed by interval values between 0 and 1. The aggregation of the concordance domi-nance matrix and discordance domidomi-nance matrix is obtained by a credibility matrix. The final classification of alternatives is based on ascending and descending distillations [106], [33].
ELECTRE IV
Unlike the previously described ELECTRE methods, ELECTRE IV does not require criteria weights in the calculation procedures. Instead, it uses the number of criteria in different pref-erence areas [106]. For each criterion, an indiffpref-erence threshold, a prefpref-erence threshold, and a veto threshold are required in order to compare the alternatives. Similar to ELECTRE III, a credibility matrix is calculated, and the classification of alternatives is based on ascending and descending distillations.
ELECTRE TRI
In ELECTRE TRI, some reference alternatives are introduced, all alternatives are compared to these reference alternatives [106]. Similar to ELECTRE III, a credibility matrix is computed with respect to reference alternatives. The outranking relations between candidate alternatives and reference alternatives are established using the credibility matrix and a veto threshold.
ELECTRE TRI can reduce the computational cost of alternative comparisons when the number of alternatives is large.
Summary of ELECTRE Methods
The main characteristics of all versions of ELECTRE methods were summarized by Roy [106], as shown in Table 2.6. Considering different problem statements, some guidelines on how to choose among ELECTRE methods were also suggested. For instance, if it is truly essential to work with a very simple method and it is realistic to have no information on the indifference threshold and preference threshold, ELECTRE I should be selected in order to eliminate the non-dominated alternatives, while ELECTRE II should be used in order to build a partial preorder of alternatives. ELECTRE IV would be convenient only if there exists a good reason to refuse the introduction of importance coefficients. In general, ELECTRE IS, II, III, IV, and TRI do provide powerful support for the classification of the alternatives. However, they require too many threshold definitions from DMs, thus, it is rather complex to implement these methods in real world problems [87].
2.3.5 Elimination by Aspects Method
In this method, the DM is assumed to have minimum cutoffs for each criterion. A criterion is selected, and all alternatives which do not pass the cutoff on that criterion are eliminated. Then another criterion is selected, and so forth. The process continues until all alternatives but one are eliminated [58].
2.3 Typical Non-compensatory Decision Analysis Methods
Table 2.6: Main Characteristics of ELECTRE Methods [106]
ELECTRE methods I IS II III IV TRI
Require indifference no yes no yes yes yes
and preference thresholds
Require criteria weights yes yes yes yes no yes
Outranking relations binary binary strong interval strictly, weakly, interval and weak values hardly preferred, values
or indifferent
The elimination by aspects method eliminates alternatives which do not satisfy some stan-dard level, and it continues until all alternatives except one have been eliminated. However, only small part of the information is used when comparing the alternatives.
2.3.6 Lexicographic Method
In the Lexicographic method, the DM compares the alternatives on the most important crite-rion. If one alternative has a better criterion value than all the other alternatives, the alternative is chosen and the decision process ends. However, if some alternatives are tied on the most im-portant criterion, the subset of tied alternatives is then compared on the second most imim-portant criterion. The process continues sequentially until a single alternative is chosen or until all the criteria have been considered.
The Lexicographic method does not require comparability across criteria, and the preference information on the criteria is not necessarily in numerical values. However, it only utilizes a small part of the available information in making a final decision.
2.3.7 Maximin Method
In the Maximin method, the overall performance of an alternative is determined by the weakest or poorest criterion. The DM examines the criteria values for each alternative, notes the worst value for each alternative, and then selects the alternative with the most acceptable value in its worst criterion. It is the selection of the maximum (across alternatives) of minimum (across criteria) values [58]. Mathematically speaking, the alternativeA∗ is selected such that
A∗ = whererij are normalized criteria values, and bigger criteria values are preferred.
2.3.8 Maximax Method
In contrast to the Maximin method, the Maximax method selects an alternative by its best criterion value rather than its worst criterion value. In this method, the best criterion value for each alternative is identified, then these maximum values are compared in order to select the alternative with the best value [58]. Mathematically speaking, the alternativeA∗ is selected such that whererij are normalized criteria values, and bigger criteria values are preferred.
The Maximin method and the Maximax method utilize only one criterion per alternative in making a final choice. The two methods are widely used in game theory, however, their applicability in other fields is relatively limited.