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

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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.   n x x x X , , , 2 1     n w w w w , , , 2 1   j w j x   m A A A A , , , 2 1   } ,..., , { 2 1 k e e e E 
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 .   4 3 2 1 , , , A A A A A    5 2 1 , , , x x x X  
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). n m ij t T   ) ~ ( ~ i A ) ( j A x f i , 4 , , 1   i , 5 , , 1   j ) ( j A x f i ) ( j A x f i , } | ) ( , { X x x f x A j j A j i i    
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 T ~ n m ij h H   ) ( ) , , , ( ~ d c b a A  . 6 / ) 2 2 ( ) ~ ( d c b a A p    
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   5 1 , , w w w     n m ij s S     n m ij s S        m i ij ij ij s s s 1  ; ) 1 ln( ) 1 ( ln 2 ln 1 1            m i ij ij ij ij j s s s s m E , , , 2 , 1 n j   . ) 1 ( ) 1 ( 1      n j j j j E E w  . 0.228144 0.202036, , 0.175857 , 0.192146 , 0.201817  w
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  A  A       , | min ; | max ..., , , ) ( ) ( 2 1                      J j h h J j h h h h h A j ij i j j ij i j n         , | min ; | min ..., , , ) ( ) ( 2 1                      J j h h J j h h h h h A j ij i j j ij i j n   . ,.., 2 , 1 , ,.., 2 , 1 n j m i   J    ; ) 0.45 0.45, 0.65, 0.85, ( ), 0.65 0.65, 0.85, 0.65, ( , 0.15) 0.15, 0.15, (0.15, ), 0.45 0.85, 0.65, 0.65, ( ), 0.45 0.65, 0.85, 0.65, (   A  . ) 0.15 0.15, 0.15, 0.15, ( 0.15), 0.15, 0.25, 0.15, ( ), 0.65 0.45, 0.65, 0.25, ( ), 0.15 0.45, 0.15, 0.15, ( ), 0.15 0.15, 0.15, 0.25, (   A J 
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:  i d  i d m i Ai ,.., 2 , 1 ,  , ) ( 1 ) , ( 1 ) ( ) ( 1 1                     l j j j j ij n j j j n j j ij i h h l w w h h d d   , ) ( 1 ) , ( 1 ) ( ) ( 1 1                     l j j j j ij n j j j n j j ij i h h l w w h h d d   . ,.., 2 , 1 , ,.., 2 , 1 n j m i   0.216805 1   d 0.393337 2   d 0.150815 3   d 0.155484 4   d 0.292325 4   d 0.296994 3   d 0.0544715 2   d 0.231004 1   d
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. i RC i A  A . ,.., 2 , 1 , ) ( m i d d d RC i i i i       0.515853 1  RC 0.12164 2  RC 0.663216 3  RC 0.65279 4  RC m i Ai ,.., 2 , 1 ,  i RC  A  A   RC RC  A 2 1 4 3 A A A A    3 A 3 A  A  A
13. Thank You for Attention!
15. 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 A ~ ) , , , ( ~ d c b a A                           , , 0 , , , , 1 , , , , 0 ) ( ~ d x if d x c if c d x d c x b if b x a if a b a x a x if x A d c b a    X A ~ X ] 1 , 0 [ : ) ( ~   X x A ) ( ~ x A  X x  x A ~
16. 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 ) (x f T  , | ) ( , X x x f x T T     ) (x f T ) (x f T   n x x x X ,..., , 2 1  ) (x hE  , | ) ( , X x x h x E E     ) (x hE X x  ) (x hE