LUDMIL MIKHAILOV∗,‡, HOSEIN DIDEHKHANI†,§
and SOHEIL SADI-NEZHAD†,¶
∗Manchester Business School
The University of Manchester, United Kingdom
†Science and Research Branch
Islamic Azad University, Tehran, Iran
‡ludi.mikhailov@manchester.ac.uk
§h.didehkhani@gmail.com
¶sadinejad@hotmail.com
The paper deals with the group prioritization problem in the fuzzy analytic hierarchy process. We extend the fuzzy preference programming method to fuzzy group prioritization by introducing important weights of decision-makers (DMs). The modified prioritization problem is represented as a weighted fuzzy goal programming model. Additionally, we represent the uncertain DMs’ importance weights as fuzzy numbers and modify the goal programming model by a possibilistic approach. Both proposed models transform the initial prioritization problems with fuzzy or crisp important weights into equivalent crisp linear programs. Unlike the known fuzzy prioritization methods, the proposed approach does not require an additional defuzzification procedure for final ranking of alternatives and can deal with incomplete set of comparison judgments
Keywords: Fuzzy AHP; fuzzy group prioritization; fuzzy decision-making; goal programming; fuzzy linear programming
The multiple criteria decision-making (MCDM) is a very efficient approach to decision-making in complex environments. There exist many MCDM methods, which are widely applied for solving various decision-making problems. As the complex situations require the involvement of many experts, considering different aspects of the decision problem, group versions of the MCDM methods are proposed.
The group MCDM methods are concerned with obtaining aggregated preferences by a group of decision-makers (DMs), who provide individual preferences, regarding a finite set of criteria and alternatives.
Although the group AHP methods are rather efficient in many cases, they cannot handle uncertain decision-making problems. Usually, the complex decision-making problems are characterized by vagueness and ambiguity of the decision elements. A natural way to deal with uncertainty, imprecision, and vagueness in decision-making is to apply the fuzzy set theory.
Several fuzzy versions of AHP have been developed recently. Buckley et al., Chang, Van Laarhoven and Pedrycz, Mikhailov, and some other researchers propose methods for deriving priorities from fuzzy pairwise comparison matrices.
The fuzzy AHP methods have been applied in various areas. Some of the numerous applications reported over the last few years include: supplier selection, evaluation of computer companies, consultant selection, expatriate assignment, multicriteria inventory classification, bridge construction, and many others. Some novel applications of the AHP, as software defect prediction are also suitable
for applying fuzzy modifications of the AHP.
For solving complex managerial and engineering problems, as those mentioned above, group versions of the fuzzy AHP are required. Similar to the standard AHP, the existing fuzzy AHP methods can be extended to group decision-making either by aggregation of the individual fuzzy pairwise comparison matrices or by deriving individual priorities and their consecutive aggregation. This approach yields fuzzy final results and requires additional defuzzification procedure for final ranking of alternatives. However, in some cases different defuzzification methods might give different final ranking.
Mikhailov proposes a method for group prioritization in the crisp AHP, which does not require explicit aggregation of individual judgments or priorities. In this method all individual judgments are represented as constraints and combined into a symmetric linear programming model. The method derives a priority vector that best satisfies all group judgments. However, the model assumes that all DMs have the same weights of importance. It is shown that different weights of DMs can be obtained by varying some deviation parameters, however, the link between the DMs weights and the actual value of those parameters is unclear.
into a weighted additive goal programming model. Second, we introduce fuzzy DMs’ weights and represent the prioritization problem as a crisp multi-objective model.
The proposed models transform the initial prioritization problem into equivalent crisp linear problems. Unlike the known fuzzy prioritization methods, our approach does not need an additional defuzzification procedure for final ranking of alternatives. A further advantage of this approach is that it does not require a full set of judgments, which makes it very suitable for group decision-making under uncertainty.
The paper is organized as follows. The initial version of the FPP method and its use in the fuzzy AHP are briefly discussed in Sec. 2. A group modification of the method is described in Sec. 3 and a modified weighted FPP model is proposed, which is further transformed into a crisp multi-objective linear model in Sec. 4. Numerical examples in Secs. 3 and 4 illustrate the applicability of the proposed approach.
2. Deriving Group Priorities in the Fuzzy AHP
2.1. Fuzzy pairwise comparison judgments
Consider a group of K DMs, comparing pairwise n elements at the same level of the AHP hierarchy. The kth DM, k = 1, 2, . . .,K, provides a set of mk fuzzy comparison judgments. Contrary to the standard AHP requirement, we do not need a complete set of all pairwise comparison judgments, i.e. mk ≤ n(n − 1)/2.
The sets of the fuzzy comparison judgments provided by all DMs are
where ˜aijk represents the fuzzy relative importance of the ith decision element over the jth one, provided by the kth DM, with respect to an upper level element. The group prioritization problem is to find a crisp group priority vector W = (w1, w2, . . ., wn)T that best satisfies the initial individual judgments.
In this paper we assume that the fuzzy comparisons are represented as normalized fuzzy numbers ˜aijk = (lijk,mijk, uijk ), where lijk and uijk are the lower and upper bounds, and mijk is the mean value of the fuzzy number, corresponding to the maximum degree of membership, equal to one.
Using the concept of α-level sets (or simply α-cuts), each fuzzy judgment ˜aijk can be decomposed into a sequence of interval sets aijk (αt) = [lijk (αt), uijk (αt)], where lijk (αt) and uijk (αt) are the lower and upper bounds of the corresponding intervals at level αt, t = 1, 2, . . ., T, and T is the number of all α-levels.
The FPP method represents the interval judgments at each α-level as fuzzy linear constraints and defines a convex fuzzy feasible area of all possible solutions. Then the assessment of the priorities is formulated as an optimization problem, maximizing the decision-maker’s satisfaction with a specific crisp priority vector.
For a given α-level, the FPP method tries to find a crisp priority vector W = (w1, w2, . . ., wn)T , which approximately satisfies all interval constraints:
where the symbol ≤ denotes the statement “fuzzy less or equal to”.
The inequality (2) can be represented as two single-side fuzzy constraints:
Let us denote by m the overall number of comparison judgments at a given level of the AHP hierarchy. Therefore, by the above transformation we obtain a set of 2m fuzzy constraints of type (3).
The fuzzy constraints can be represented in a matrix form as a system of fuzzy linear inequalities:
The qth fuzzy inequality of (4) can be denoted by RqW ≤0, q = 1, 2, . . . , 2m. For a given priority vector W, the degree of satisfaction of this fuzzy inequality is measured by a linear membership function of the type:
where dq is a deviation parameter, defining the allowed range of approximate satisfaction
of the soft inequality constraints (4).
The fuzzy feasible area ˜ P is defined as an intersection of all fuzzy constraints, characterized by the following membership function:
The FPP method tries to find “a maximizing solution” W* that has a maximum degree of membership λ∗ in ˜ P, such that
Max λ
Since the membership functions μq(RqW) are linear with regard to the decision variablesW = (w1, w2, . . . , wn)T, the prioritization problem (8) is a linear program. By taking into account (5), this linear program can be presented as
3. Weighted FPP Method
3.1. DMs importance weights
In the FPP method, presented in the previous section it is assumed that all DMs have equal importance in the group decision process. However, in some complex situations, it is not realistic to assume that all DMs have equal expertise. In such situations it is reasonable to assign different weights to the DMs, reflecting their
level of expertise or importance in the group decision.
If the DMs have different importance, their judgments need to have different weights of importance as well. Therefore, it is reasonable to assign the importance weights of DMs to the fuzzy constraints that correspond to their individual judgments.
Additionally, the fuzzy feasible area ˜ P defined by (6) can be decomposed into K fuzzy feasible areas ˜ Pk, defined as intersection of membership functions, corresponding to the judgments of the kth DM. Therefore, we may represent the max–min problem (8) as K optimization problems of the type:
where the decision variable λk measures the degree of membership of a given priority vector in the fuzzy feasible area ˜ Pk, and μkq (Rk qW) are membership functions of type (5), corresponding to the soft constraints of the kth DM.
To combine all individual models (10) into a group model, which takes into consideration the importance weights of the DMs, we can apply the goal programming additive method, proposed by Tiwari.
3.2. Weighted fuzzy goal programming model
Consider a problem with P fuzzy goals zi, i = 1, 2, . . ., P, and Q fuzzy constraints gj , j = 1, 2, . . .,Q, which are described by fuzzy membership functions μzi(x) and μgj (x), respectively. The weighted additive model21 combines these fuzzy membership functions into a new fuzzy decision function μD(x):
where x is the vector of decision variables, and ai and bj are weighting coefficients that show the relative importance of the fuzzy goals and fuzzy constraints.
By introducing new decision variables λi and γj , the problem of maximizing μD(x) can be represented as a single objective programming problem:
3.3. Weighted FPP model
We may explore the similarity between the FPP model (8) and (12), in order to develop a new group model, where the DMs have different weights of importance.
The FPP model does not deal with fuzzy goals, but represents the soft constraints (4) by linear fuzzy membership functions (5). By introducing importance weights and considering the specific form of the fuzzy constraints Rk qW ≤0 given in (5), we can apply the weighted additive model (12) and combine all individual models (10) into the following additive weighted FPP model:
In the above model the value of C measures the overall consistency of the judgments, so we can call it a consistency index. When the interval judgments are consistent, the optimal value of C is greater or equal to 1. For inconsistent judgments, C takes a value between 1 and 0 that depends on the degree of inconsistency and the values of the deviation parameters dq.
3.4. Numerical example
There are three DMs (K = 3) for ranking three (n = 3) criteria C1, C2, and C3. The DMs provide the following incomplete set of five fuzzy judgments (m = 5):
Let us assume that the importance weights of DMs are v1 = 0.3, v2 = 0.2, and v3 = 0.5. The prioritization problem is to derive a crisp priority vector W = (w1, w2, w3)T that approximately satisfies the above fuzzy judgments and takes into account the importance weight of each DM.
Consider the solution to the problem at level α = 0.5. From the initial set of judgments, we can obtain the following set of interval judgments at that level:
F = {(1.5, 2.5), (2.5, 3.5), (2, 3), (3.5, 4.5), (2.5, 3.5)}.
Therefore, the unknown crisp criteria weights should approximately satisfy the following constraints:
The importance of constraints (i), (ii), and (iii) correspond to the importance of the DMs, providing those judgments.
This linear programming problem is solved by Maple 9.5. The obtained results are shown in Table 1, together with results for other α-levels.
From the results in Table 1 it can be seen that the consistency index C decreases when the level of α increases. For α ≤ 0.5, C is greater than 1, which means that the interval judgments at lower levels are consistent. For higher values of α, the interval judgments are slightly inconsistent, as C < 1, but close to 1.
Although the set of fuzzy judgments is incomplete and there is no direct comparison between C2 and C3, for each α-level the criteria weights preserve the same ranking w1 > w2 > w3. However, if the third DM provides an additional comparison judgment between those two criteria a323 = (2, 3, 4) (which implies that the third criterion is about three times more important than the second one), the initial ranking at all α-levels is changed, so that w1 > w3 > w2. For example, at level α = 0.5 the obtained criteria weights are w1 = 0.623, w2 = 0.143, and w3 = 0.234. The value of the consistency index is C = 0.748, which shows that this new judgment is rather inconsistent with the initial ones.
From this example we may observe the importance of introducing importance weights of the DMs. It is seen that the judgments of the third DM, which has
Table 1. Results from the FPP method with crisp weights.
Generally, the α-levels reflect the confidence of DMs with their fuzzy judgments, so they can also be considered as confidence levels. A small value of α yields a construction of interval judgments having large spreads, which indicates a high level of pessimism and uncertainty. A larger value of α yields smaller but more optimistic interval judgments, whose upper and lower bounds have greater degrees of membership in the initial fuzzy sets.
To handle the different values of priorities at different α-levels, we may construct an interactive model and ask the DMs to select the best ones with respect to their overall level of optimism/pessimism. Alternatively, we may apply the aggregation procedure, proposed in Ref. 18 and obtain aggregated values of the priorities by a weighted sum of the type
where wj (αt) is the value of the jth weight at level αt, t = 1, 2, . . . , T.
By applying (14), the aggregated priority vector, corresponding to the results shown in Table 1 is W = (0.617, 0.208, 0.175).
4. Deriving Priorities with Fuzzy DM’s Weights
In Sec. 3 we represented the DM’s weights of importance as crisp values. However, in uncertain situations it is more reasonable to consider those weights as fuzzy numbers. In this section we develop a procedure to handle this case.
4.1. A possibilistic additive goal programming model
Let us represent the importance weight of the kth DM, k = 1, 2, . . .,K, as a normalized fuzzy number ˜vk = (ak, bk, ck). Then the FPP model (13) can be represented as
Consider the following fuzzy linear model:
where ˜cj = (cp j, cmj , coj ) are fuzzy coefficients with triangular possibility distribution, cmj is the most possible value, cp j and coj are the most pessimistic and the most optimistic ones.
By maximizing the possibility degree and minimizing the pessimistic one, Lai and Hwang transform the fuzzy linear mode (17) into the following three-objective crisp linear model:
where S is the set of feasible solutions for the decision vector X.
By applying this approach, the FPP model given by Eqs. (15) and (16) can be transformed into the following multi-objective crisp linear model:
4.2. Fuzzy programming solution
There are many approaches to solving multiple objective linear programming models of type (19), such as the utility theory, goal programming, and fuzzy programming. In this paper, we use the Zimmermann fuzzy programming approach to solve the model.
These optimization problems should be solved subject to the constraints, given by Eq. (16).
We can introduce a membership function for each objective. For example, the membership function of z1 is defined as
Then, by applying the FPP approach, we obtain the following single objective model:
where θ is an additional decision variable.
4.3. Numerical example
Consider the problem given in Sec. 3.4. Instead of crisp weights, let us assume that the DMs have the following fuzzy weights of importance: ˜v1 = (1.5, 2, 3), ˜v2 = (2.5, 3, 4), and ˜v3 = (4, 4.5, 5.5).
From the DMs’ weights, by using Eq. (19), we form the objective functions z1, z2, and z3:
Consider again the problem at level α = 0.5, where we have the following interval judgments:
F = {(1.5, 2.5), (2.5, 3.5), (2, 3), (3.5, 4.5), (2.5, 3.5)}.
The values of PIS and NIS of each objective at that level are determined by solving the optimization problems (20).
For example, zPIS 1 is obtained by solving the following optimization problem:
subject to the set of constraints, shown in the example of Sec. 3.4.
After solving six optimization problems of type (20), we obtain
Then, membership functions for each objective should be constructed. For example, the membership function of z1, obtained by using Eq. (21) is
In the same way, we determine the membership functions of z2 and z3. Then, for finding the priority vector W, a model of the type (22) is constructed.
The results obtained for solving the model for three different values of α are shown in Table 2.
We may observe that at each α-level we have the same rating w1 > w2 > w3 as in the previous example in Sec. 3.4. However, the values of the criteria weights are more consistent and vary in smaller ranges for different α-levels. The values of the objective function θ at each level are also very close, which is a result of introducing fuzzy weights of importance.
Table 2. Results from the FPP method with fuzzy weights.
Similar to the example in Sec. 3.4, additional judgments, which contradict to this ranking may result in rank violation. For example, the additional contradictory judgment a323 = (2, 3, 4) of the third DM changes the ranking at level α = 0.5, as the obtained priority vector is W = (0.696, 0.148, 0.156). But if only the first DM, whose importance is about two times less than the third one provides the same additional judgment, i.e. a321 = (2, 3, 4), then the obtained priority vector is W = (0.695, 0.154, 0.151); so, the initial rating is not violated. This shows that judgments of DMs with higher weight of importance influence more strongly the
derived group priorities.
5. Conclusions
In this paper we deal with the problem of deriving priorities in the fuzzy AHP. An extension of the FPP method for group decision-making is proposed, which does not require aggregation of the individual fuzzy judgments. The proposed group FPP takes into consideration the importance of the DMs by introducing two types of weights, crisp and fuzzy. In both cases the prioritization problems are transformed into linear programming models, so the proposed approach is very effective from computational point of view. An additional advantage of our approach is that it does not require a full set of judgments, which makes it very suitable for group decision-making under uncertainty.
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