The first studies on the GCM, which is included in the classification of the ‘preference information- of the MODM methods, were carried out by Yu (1973) and Zeleny (1973). GCM is given mathematically below

Subject to (2)

The benefit of this approach is that it is possible to derive the solution from the decision-making itself, and not from the designer (Shih & Chang, 1995). In this method, the distance is minimized between some reference points and the feasible objective region. Additionally, it is accepted that all objective functions have equal importance in the GCM (Miettinen, 2012). According to the method, more than one objective function is transformed into a single-objective optimization problem (Tabucanon. 1988). Within the global objective function, each objective function is expressed as a ratio. Non-dimensionalization is necessary because there are different dimensions to objective functions. The global objective value should be within the range of [0;1], as normalization is done on the basis of the constructive ideal solutions of objectives. The best solution for the problem varies according to the p-value chosen (Umarusman, 2019). Umarusman and Turkmen (2013) extended the GCM for problems with goals of maximization and minimization. From this point of view, GCM is given mathematically below.

Subject to (2.1)


Zk(x): Maximization-directed objective function,

Zp. Positive ideal solution of &-th objective function,

Ws (x): Minimization directed objective function,

W*: Positive ideal solution of .v-tli objective function,

A: Matrix of technical coefficients, b: Right-hand-side constants for the constraints, p (1 < p < ~).

Although a GCM can be a mathematical function with no association of preferences, a weighted GCM is a kind of utility function in which system parameters are used to model preferences (Marler & Arora, 2004). Weight factors for objective functions have not been used in this study.

Fuzzy Global Criterion Method

Fuzzy GCM could be seen as an extension of the GCM approach. Conflicting objectives resolve on the basis of the fuzzy-PIS only by deriving a collection of compromise solutions. In this case, a fuzzy set (Chang & Pires, 2015) is formed only by a PIS. A compromise solution is sought with respect to distance formulation by reducing the distance from a particular aspect of the fuzzy-PIS (Chang & Lu, 1997). This procedure has the benefit that the solution can be derived from the decision-making itself, not from the planner (Shih & Chang, 1995). The steps of the Fuzzy GCM proposed by Leung (1984) are as follows.

Step 1: Find the PIS /** = (f1*, /2*w'here /** is the solution of the problem below':

Subject to (3)

where no tolerance value ii is considered.

Step 2: Acquire a compromise solution equivalent to /**. This compromise solution can be acquired by solving the problem below.

Subject to (4)

Where [/*'* - fk (x)] = [l - pk (/* (x))] if Цк {fk (x)) = ^ (here /*'* is crisp), and p can be any nature number.

For simplicity, we might only consider the following cases p = 1 and Subject to (4.1)

whose solution is assumed to be x", and Subject to (4.2)

whose solution is assumed to be x'~. Both compromise solutions x" and x'°° (/" and /'”) serve as bounds of the compromise solution set for

1 < p< ~.

Step 3: Like Step 2, find the PIS within the constraints with maximum tolerances. That is, total avaible resources are b, + p, Vk. The PIS f°* = ,fi* ’•■■’fk*) where fk is the solution of the problem below:

Subject to (5)

Step 4: Like Step 2, find the compromise solution with respect to f°* solving the problem below:

Subject to (6)

Again, we might just solve two boundary situations г/, and d„ with the corresponding solution ,v01 and x°“.

By stretching the limitsoftheconstraintsfromfr, to/?, + />„V„ maximum values of individual objectives functions ft (x), V„ shift from /*’ to fk Consequently, the positive ideal solution moves from /** to fk*. The positive idal becomes fuzzy

and can be identidied as j/j (x),..., fk (x), |/*' k (x) < fk к = 1.....A'j. For

each single objective, we have fk (x)>[fk fk*], V, within fuzzy constraints with various and different tolerance values.

Step 5: By solving the following К single-objective fuzzy linear programming obtain the most appropriate PIS.

Subject to (7)

For convenience, let’s say membership functions are linear, with fuzzy objectives and constraints like:

for V„ and

for V,. By using Bellman and Zadeh’s max-min operator, we have (for each single objective):


Subject to (10)

whose solution is assumed to be fk = fk (.v') and ak. After solving К problems of Equation (10), the most suitable PIS can be /* = (/Г,/2*,) w'ith a = max*a*.

Step 6: Acquire a compromise solution from original MOP when the most suitable PIS is / and a. With a, new constraints wil be

Therefore, the compromise solutions of (1) can be obtained by using the global criterion (distance metric family) by solving the problem below:

Subject to (13)

Where,/) can be any nature number. Again, we may only chose to solve boundary situations of /2 = 1 and Leung (1984) stressed that such a fuzziness of constraints is versatile to allow one to create alternatives to resolve disputes that cannot otherwise be resolved or dissolved within permissible limits of tolerance. This explains that, in a dispute settlement, DMs don't need to confine themselves to a single point-valued PIS. The method of defining differing values and solutions to compromise is tractable.

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