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March 22, 2019
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# Fuzzy TOPSIS based Facility Location Selection Problem

MACSPro'2019 - Modeling and Analysis of Complex Systems and Processes, Vienna
21 - 23 March 2019

Irina Khutsishvili, Gia Sirbiladze, Bezhan Ghvaberidze

Conference website http://macspro.club/

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March 22, 2019

## Transcript

1. Fuzzy TOPSIS based Facility Location Selection
Problem
Authors: Irina Khutsishvili, Gia Sirbiladze and Bezhan Ghvaberidze
Date: March 22, 2019

2. This study develops a decision support methodology for Facility
Location Selection Problem.
The methodology is based on the Technique for Order Performance
by Similarity to Ideal Solution (TOPSIS) approach in the hesitant
fuzzy environment and implies using experts' evaluations.
We consider a special case of facility location selection problem,
namely, location planning for service centers.

3. Facility Location Selection Problem
Facility Location Selection Problem, in fact, represents MCGDM problem and can be briefly
described as follows:
Suppose, we have
- a set of possible locations of facilities (alternatives);
- the group of DMs evaluating each alternative
- a set of weighted criteria influencing the decision;
- the vector of criteria weights must be also known, its
component shows the importance level of criterion in decision
making process.
The main goal of the problem is to make ranking of feasible alternatives in decreasing order to
choose implementation of a single alternative that is the best with respect to all criteria.
This formulation is also suitable for Service Centers Location Planning Problem.
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4. Service Centers Location Planning Problem
The proposed approach for location planning of candidate centers involves:
• Selection of location criteria to assess potential locations for candidate centers
• Selection of potential locations for implementing service centers
• Locations evaluation using fuzzy TOPSIS
The selection of location criteria, as well as the selection of potential locations is based on DMs
knowledge and experience, their discussions with the city transportation group members, municipal
administration members, logistics specialists, and other experts in the field.
For locations evaluation our approach proposes hesitant fuzzy TOPSIS model with entropy weights.
The idea of TOPSIS method is to choose an alternative with the nearest distance from the so-called
fuzzy positive ideal solution (FPIS) and the farthest distance from the fuzzy negative ideal solution
(FNIS). Or, in general, TOPSIS makes ranking of alternatives in accordance with the proximity of
their distances to the both FPIS and FNIS.

5. Service Centers Location Planning Problem
Presume, in the service centers location planning problem the following five main criteria have been
determined:
THE CRITERIA FOR LOCATION CENTERS SELECTION
There are four locations of candidate centers – decision making alternatives.
The group of experts consists of four members. They evaluate locations with respect to the five
criteria .
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6. Initial data for Fuzzy TOPSIS Method
Experts are giving the evaluations over criteria in a form of lingual expressions – linguistic terms that
are the most natural representations of decision makers' assessments. The linguistic variable
“criterion" can have the values, such very low (VL), low (L), medium (M), high (H), very high (VH).
Presume, the criteria assessments given by all DMs looks like:
EXPERTS INITIAL ASSESSMENTS - RATINGS OF ALTERNATIVES
There are different ways to process lingual assessments. One of them is transformation to trapezoidal
fuzzy numbers (TrFNs), by using, for instance, 5-point linguistic scale
LINGUISTIC TERMS FOR CRITERIA RATINGS

7. Initial data for Fuzzy TOPSIS Method
After converting lingual assessments to the relevant TrFN, we receive the following
aggregate hesitant trapezoidal fuzzy decision matrix
because each row of this matrix - an alternative - we consider as HTrFS:
For a reference set X, a hesitant trapezoidal fuzzy set on X is defined in terms of a function as follows:
where is a set of several trapezoidal fuzzy numbers, representing the possible membership degrees of the
element x j
∈ X to the HTrFS; is called a hesitant trapezoidal fuzzy element (HTrFE).
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8. Locations evaluation using hesitant fuzzy TOPSIS approach
Then we transform aggregate hesitant trapezoidal decision matrix into aggregate hesitant decision
matrix by using Graded Mean Integration Representation Method:
For any trapezoidal fuzzy number we can get its representation by formula
After calculations we obtain
THE HESITANT FUZZY DECISION MATRIX H
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9. Locations evaluation using hesitant fuzzy TOPSIS approach
In the our problem we consider a case when the criteria weights are unknown. At this stage of
TOPSIS method we determine the criteria weights .
The criteria weights definition method based on the De Luca-Termini entropy can be described as
follows:
Step 1: Calculate the score matrix of hesitant decision matrix H by averaging values of each
hesitant fuzzy element;
Step 2: Calculate the normalized score matrix , where ;
Step 3: By using De Luca-Termini normalized entropy in context of hesitant fuzzy sets
the definition of the criteria weights is expressed by the formula
Using this algorithm, we obtain the weighting vector of criteria as
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10. Locations evaluation using hesitant fuzzy TOPSIS approach
Following the hesitant fuzzy TOPSIS method, we compute the hesitant FPIS and the hesitant
FNIS by formulas:
where is associated with a benefit criteria, and - with a cost criteria.
By these formulas for our task we obtain

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11. Locations evaluation using hesitant fuzzy TOPSIS approach
We compute the distances of each alternative from the hesitant FPIS and the hesitant FNIS.
To calculate the separation measures and of each alternative from the
FPIS and the FNIS the hesitant weighted Hamming distance is used as follows:
On this stage we obtain following results:

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12. Locations evaluation using hesitant fuzzy TOPSIS approach
Then we compute the relative closeness coefficients of each alternative to the hesitant
FPIS using equation below
Calculations give us results of four closeness coefficients
Finally, we perform the ranking of the alternatives according to the relative
closeness coefficients . The ranking is made in decreasing order by the rule: from two alternatives
and
≽ , if , where ≽ is a preference relation on . The best alternative will be the
closest to the FPIS and farthest from the FNIS.
In our case we obtain
This means that in accordance with the common opinion of the experts, TOPSIS method prefers the
alternative , i.e., is the best location for service centers.
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13. Thank You for Attention!

14. Definition 1: A trapezoidal fuzzy number can be determined by a quadruple
and its membership function is defined as
where .
Definition 2: Let be a reference set. A fuzzy set on is defined by a membership
function , where , , indicates the degree of
membership of in .
On the Trapezoidal Fuzzy Numbers and Hesitant Fuzzy Sets
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15. Definition 2: Let be a reference set, a hesitant fuzzy set E on X is
defined in terms of a function , which when applied to X returns a subset of [0,1]:
where is a set of some different values in [0,1], representing the possible
membership degrees of the element to E; is called a hesitant fuzzy element
(HFE).
Definition 3: For a reference set X, a hesitant trapezoidal fuzzy set T on X is defined in
terms of a function as follows:
where is a set of several trapezoidal fuzzy numbers, representing the possible
membership degrees of the element x ∈ X to the HTFS; is called a hesitant
trapezoidal fuzzy element (HTFE).
On the Hesitant Fuzzy Set and Hesitant Trapezoidal Fuzzy Set
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