This section develops an IM-ORESTE method to solve the MCDM problems with the evaluations expressed as IMNs. We first present an objective weight determining method based on the correlation coefficient, which can avoid the misleading results caused by highly-related criteria. Then, the classical ORESTE method is improved by introducing a new global score function. Finally, the IM-ORESTE method is proposed based on the distance measure between IMNs.
3.2. The Correlation Coefficient-Based Weighting Method
This part presents an objective weight determining method based on the correlation coefficient of criteria with the IMNs information.
Most practical decision-making problems are defined on multiple criteria. The “conflict” is the fundamental concept in MCDM problems in that different criteria should provide independent information on the performance of alternatives [
32]. However, there are always highly interactive influence, strong repeatability, and linear crossover among criteria because of the interrelated and interactive objective things. For example, when evaluating a teacher’s ability, the criterion, respecting the dignity of students, has a causative effect on the criterion, students’ satisfaction. If the same weights are assigned to these highly-related criteria, the results will be biased, because the same information existing in more than one criterion is reconsidered when aggregating the performance of each alternative under all criteria. Merging interdependent criteria by some methods, such as principle component analysis [
33], is complex and may mislead the final results. However, deleting these criteria is impolite and may lose useful information to some extent. Therefore, assigning a small weight to the highly-related criterion is a reasonable way to overcome this problem [
32]. Inspired by this idea, we used the correlation coefficient to denote the correlation degree of one criterion to others. The larger the correlation coefficient is, the smaller is the weight of the criteria. The steps of the correlation coefficient-based weighting method are constructed as follows:
Step 1. Aggregate the preference values
of alternative
compared with others
under criterion
.
where
is an IMN of alternative
with respect to criterion
. The IM decision matrix can be established as
.
Step 2. Calculate the Pearson correlation coefficient between criteria
and
based on the distance measure of IMNs.
where
,
.
Step 3. Compute the subjective weights by:
There are other kinds of objective weighting methods, such as the standard deviation-based weighting method [
34] and the entropy measure-based weighting method [
35]. They are based on the dispersion degrees of evaluations in that the criterion with great dispersion degree of alternatives’ performances is assigned a big weight. They have less influence on the final rankings, since the gaps between alternatives are widened. More importantly, however, they fail to handle the highly-related criteria. In the following, we take a simple example to illustrate the effectiveness of our proposed method.
Example 1. Suppose that we rank three teachers
, , and based on respecting the dignity of students , students’ satisfaction and stimulating students’ initiative . The preference matrices are We can find that criteria
and
are highly correlated since similar information is presented by them. By Equation (10), we obtain the IM decision matrix as:
The scores
(
,
) calculated by Equation (2) are shown as
We rank the alternatives based on the scores of IMNs. The comprehensive score of each alternative can be calculated by:
In the following, we use two methods to calculate the objective weights of criteria and then derive the rankings of the three teachers.
Rank the teachers based on the correlation coefficient-based weighting method
From Equations (3)–(5), we obtain the weights of criteria are , , and . By Definition 2, we obtain , , and , thus .
Rank the teachers based on the standard deviation-based weighting method
The dispersion degree-based weighting methods aim to determine the criteria weights based on the variation of evaluations under each criterion. A small weight is assigned when the evaluations under this criterion are close, while a big weight is assigned when the gaps of the evaluations are large. We use a representative method of this type, the standard deviation-based weighting method, to solve Example 1. Standard deviation
under IM context can be defined as:
In this Example, by Equation (14), we obtain , , and . After normalization, we obtain , , and . From Definition 2, we get , , and , thus . The same weight is assigned to each criterion since these criteria maintain the same variation of evaluations.
Comparative analyses: From the preference matrices, we can find that is highly related to since similar information is composed in these two criteria. We can integrate them as one criterion to describe the performance of alternatives. By the correlation coefficient-based weighting method, small weights are assigned to them. The correlation coefficient-based weighting method is effective in avoiding the misleading results caused by the highly-related criteria. We find that are derived by the standard deviation-based weighting method. However, both alternatives perform equally in total since one of and should be considered in decision making in this case. We can conclude that the misleading results caused by some highly-related criteria cannot be avoided by the dispersion degree-based weighting methods.
3.3. The IM-ORESTE Method
This part improves the ORESTE method by introducing the objective criteria weights and extends it to solve the MCDM problems with the evaluation values expressed as IMNs. In IM-ORESTE, both the preferences on criteria evaluated by experts subjectively and the preferences on pairwise alternatives under each criterion are expressed as IMNs. Like the classical ORESTE method, the process of the IM-ORESTE method is divided into two stages.
Stage 1. Determine the weak rankings
We introduce a global preference score function to consider both the subjective and objective weights of criteria. It is hard to assign a crisp weight to a criterion by experts due to the fuzziness of people’s cognition and the complexity of the objects. However, it is easy to compare the importance between two criteria. The IMN is effective in describing the experts’ uncertain and fuzzy preferences over criteria. Suppose that the preference of
over
is denoted as
(
). From Equation (10), we obtain the IMN
of each criterion and denote it as the fuzzy subjective weight
. Translating the fuzzy subjective weights to the crisp numbers will cause information loss. Thus, we are not supposed to integrate the subjective and objective weights of criterion into a collective one. Motivated by the global score function proposed in Ref. [
29], we aggregate each alternative’s performance and subjective weights of criteria by the weighted Euclidean distance measure of IMNs. Furthermore, considering the objective weights, we introduce a new global score function as:
where
,
indicates the relative importance between the alternative’s performance and the criterion importance in calculating the global preference score of
under
.
and
.
The utility value of each alternative is determined by aggregating the global scores under all criteria:
The weak rankings are determined by the utility values in ascending order.
Stage 2. Determine the PIR relation
It is a strict way to determine the relations among alternatives based on the utility values. If
, then
; if
, then
. However, there is usually a certain amount of error in our evaluations due to the fuzziness of thinking and the limitation of cognition. Therefore, we are supposed to allow a certain range of differences when comparing the two alternatives’ utility values. The utility value is limited to derive the definitive relationship between two alternatives. To overcome this defect, we further conduct the pairwise comparison under each criterion based on the global scores. Compared with the initial preference relation between two alternatives given by experts directly, the global scores integrate criteria weights information. The preference intensity of
over
with respect to
can be defined as:
indicates the superiority of
over
with respect to
, and
. The comprehensive preference intensity of
over
under all criteria can be calculated as:
indicates the comprehensive superiority of
over
, and
. The net preference intensity of
over
can be defined as
determines the overall preference relation between and . , and it satisfies and .
The PIR relations between two alternatives should meet the following conditions:
(1) When the absolute value of the net preference intensity is large enough, we can ensure that and are preference relation. That is, or if .
(2) If , and the performances of and are similar under each criterion, that is to say, , then and is the indifference relation . In this situation, they can replace each other when making a decision.
(3) If , but the performances of and are quite different under some criteria, that is to say, , then and is the incomparability relation . In this case, they cannot replace each other when making a decision. For example, we suppose that two products, and , are opposite with regard to quality and price. If is better than in quality, then is better than in price. If quality and price have the same weight, and are not the preference relation but the indifference relation. If we select , the quality is highlighted, while if we select , the price is highlighted. In this condition, we need to redefine the importance of the criteria to select or .
We define as the preference threshold and as the indifference threshold. We need to determine reasonable values of and , respectively. It is not easy to assign the value of based on the preference intensity . Therefore, we employ the initial preference values given by experts to deduce the value of . In this way, is objective and meets our cognition.
Suppose that there are one criterion and two alternatives. For a small dominance, is slightly preferred to , then and . Aggregating and , the global scores are obtained as and . Thus, . Therefore, we let . If , there is significant preference relation between and .
We determine the PIR relations among alternatives based on the comprehensive preference intensity . We further introduce an incomparability threshold . The thresholds and are determined by the value of through analyses on the PIR relations:
(1) Based on the Pareto optimality theory, if for
criteria,
, and for the
th criterion,
, then
. Suppose that the same objective weight is assigned to each criterion. Then,
. As this is the minimal case that
, we let
to distinguish the preference relation based on the net preference intensities of pairwise alternatives. We give a simple example to shown the preference relation between
and
in
Table 1.
(2) If
and
for all criteria, then
. Suppose that each criterion has the same weight. Then,
if
is odd;
if
is even. Thus, we let
if
is odd, and
if
is even.
Table 2 and
Table 3 show the indifference relation between
and
when
is odd and even respectively.
(3) For the incomparability relation
,
and
.
Table 4 shows the incomparability relation between
and
.
Based on the above analyses, we present the test rule to establish the PIR relations in the IM-ORESTE method:
where
,
if
is odd, and
if
is even with
.
The procedure of the IM-ORESTE method is given as follows.
Algorithm 2 (The IM-ORESTE method)
Step 1. Experts compare pairwise alternatives based on their performances under each criterion. The evaluations are expressed as IMNs , , . Establish the preference matrixes , . The preferences of experts on pairwise criteria based on their importance are also expressed as IMNs , .
Step 2. Integrate the preference values into the IMN under each criterion by Equation (10). Build the decision matrix . are aggregated to by Equation (10).
Step 3. Determine by Equation (4) and calculate the distances , , by Equations (2)–(3). Then compute the Pearson correlation coefficient between two criteria by Equation (11). Determine the object weights , , by Equation (12).
Step 4. Calculate the global score of alternative under criterion by Equation (15). is integrated by the performance of and the subjective and objective weights of .
Step 5. Compute the utility values , , by Equation (16), based on which the weak rankings of alternatives are obtained.
Step 6. Derive the preference intensity by Equation (17), the comprehensive preference intensity by Equation (18) and the net preference intensity by Equation (19).
Step 7. Determine the value of which satisfies , and calculate the values of and . Then derive the PIR relations for pairwise alternatives based on the rules in Equation (20).
Step 8. Derive the strong rankings of alternatives considering both the weak rankings and the PIR relations between pairwise alternatives.