ABSTRACT
It has been observed that most of the MCDM techniques suffer from the rank reversal problem, i.e., when an existing alternative is removed from the set or a new alternative is included into the set, the final rankings of all the alternatives usually alter. To avoid this rank reversal problem, Žižović et al. [1] proposed a novel MCDM method in the form of RAFSI. Its application steps are detailed out as below [2]:
Step 1: Develop the following decision matrix consisting of m alternatives and n criteria. https://www.w3.org/1998/Math/MathML"> D = [ a 11 a 12 … a 1 n a 21 a 22 … a 2 n … … … … a m 1 a m 2 … a m n ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_317_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; aij is the performance score of i th alternative with respect to j th criterion.
Step 2: Define the ideal and the anti-ideal values. In the decision matrix, two values, i.e., aIj and aNj are defined for each criterion C j (j = 1, 2, …, n), where aIj is the ideal value of C j criterion and aNj is the anti-ideal value of C j criterion. It is quite obvious that for beneficial criterion aIj > aNj and for non-beneficial criterion aIj < aNj .
Step 3: Map each of the elements of the decision matrix into corresponding criteria intervals. These criteria intervals can be defined as follows:
https://www.w3.org/1998/Math/MathML"> C j ∈ { a N j , a I j } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_318_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for beneficial criterion;
https://www.w3.org/1998/Math/MathML"> C j ∈ { a I j , a N j } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_318_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for non-beneficial criterion.
To make all the criteria of the initial decision matrix equal or convert them into a criteria interval [n 1 , n 2 k ], a sequence of numbers is formed where (k − 1) number of points are inserted between the highest and the lowest values of the criteria interval.
https://www.w3.org/1998/Math/MathML"> n 1 < n 2 ≤ n 3 < n 4 ≤ n 5 < n 6 ≤ … ≤ n 2 k − 1 < n 2 k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
It can be noted that this criteria interval remains constant for all criteria having n 1 and n 2 k fixed points. The sub-intervals of the criteria are then mapped into the corresponding criteria intervals. Thus, the map of minimum value aNj (for beneficial criterion) and aIj (for non-beneficial criterion) is n 1 . In the similar direction, the map of maximum value aIj (for beneficial criterion) and aNj (for non-beneficial criterion) is n 2 k . It is always recommended that the ideal value is at least six times better than the anti-ideal value, i.e., n 1 = 1 and n 2 k = 6. Other values of n 1 and n 2 k , like n 1 = 1 and n 2 k = 9 may also be considered.
A function f s(x) is now defined in Eqn. (27.2) to map sub-intervals into criteria interval [n 1 , n 2 k ]. The endpoints of this interval [n 1 ,n 2 k ] would determine the ratio (to be determined by the concerned decision maker) of a barely acceptable alternative to the ideal alternative.
https://www.w3.org/1998/Math/MathML"> f s ( x ) = n 2 k − n 1 a I j − a N j x + a I j . n 1 − a N j . n 2 k a I j − a N j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; n 2 k and n 1 denote the relation to depict the extent to which the ideal value is preferred over the anti-ideal value.
Eqn. (27.2) can be used to map a part of the interval [aNj , aIj ] into an interval [n 1 , n 2 k ]. Here, all these parts, i.e., all the functions f 1 (x), f 2(x), …, fn (x) denote a function fs (x) to map the entire criterion interval into a defined numerical interval. Besides representing a function to map a part of an interval, Eqn. (27.2) can also map a complete criterion interval into the corresponding numerical interval. Thus, the numbers aIj and aNj can either represent values from inside the criterion interval or endpoints of the criterion interval.
Thus, a standardized decision matrix https://www.w3.org/1998/Math/MathML"> S = [ s i j ] m × n ( i = 1 , 2 , … m ; j = https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_317_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 1, 2, …, n) is formulated where all of its elements are mapped into the interval [n 1 , n 2 k ]. After functional mapping of the elements of the decision matrix into criteria interval [n 1, n 2 k ], the condition n 1 ≤ sij ≤ n 2 k is achieved for every i, j.
The elements of https://www.w3.org/1998/Math/MathML"> S = [ s i j ] m × n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_319_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are obtained employing Eqn. (27.2), i.e., https://www.w3.org/1998/Math/MathML"> s i j = f A i ( C j ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_317_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> It can be noted that:
In case of beneficial criterion, for axj (where axj < aIj ), the equality https://www.w3.org/1998/Math/MathML"> f ( a x j ) = f ( a I j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_319_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> holds.
in case of non-beneficial criterion, for axj (where axj > aIj ), the equality https://www.w3.org/1998/Math/MathML"> f ( a x j ) = f ( a I j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_319_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> holds.
Step 4: Calculate the arithmetic and harmonic means. Based on Eqns. (27.3) and (27.4), the corresponding arithmetic and harmonic means are respectively computed for minimum and maximum sequence of the elements n 1 and n 2 k .
https://www.w3.org/1998/Math/MathML"> A = n 1 + n 2 k 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> H = 2 1 n 1 + 1 n 2 k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 5: Formulate the normalized decision matrix, (i = 1, 2, …, m; j = 1, 2, …, n). Now, employing Eqns. (27.5) and (27.6), all the elements of the standardized decision matrix S are normalized, while transferring them into the interval [0,1].
For Benefit Criterion: https://www.w3.org/1998/Math/MathML"> S ^ i j = S i j 2 A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
For cost criterion: https://www.w3.org/1998/Math/MathML"> S ^ i j = H 2 S i j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Thus, from the standardized decision matrix, a new normalized decision matrix is developed, as shown below: https://www.w3.org/1998/Math/MathML"> S ^ = [ S ^ 11 S ^ 12 … S ^ 1 n S ^ 21 S ^ 22 … S ^ 2 n … … … … S ^ m 1 S ^ m 2 … S ^ m n ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_7_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where;
For the elements of the normalized decision matrix, the following relations must apply.
For Beneficial Criterion: https://www.w3.org/1998/Math/MathML"> 0 < n 1 2 A ≤ S ^ i j ≤ n 2 k 2 A < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
For beneficial criterion: https://www.w3.org/1998/Math/MathML"> 0 < H 2 n 2 k ≤ S ^ i j ≤ H 2 n 1 < 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_9_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 6: Compute the criteria function values of all the alternatives V(Ai ) using the following expression: https://www.w3.org/1998/Math/MathML"> V ( A i ) = w 1 × S ^ i 1 + w 2 × S ^ i 2 + … + w n × S ^ i n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math27_10_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
The candidate alternatives are then ranked from the best to the worst based on the descending order of the calculated V(Ai ) values.
As RAFSI is an entirely new method, developed only in 2020, its application in the real-time manufacturing environment is limited in the literature. Alossta et al. [3] applied AHP and RAFSI methods for solving a location selection problem for emergency medical services considering four criteria and five alternatives. The corresponding criteria weights were determined using AHP method, whereas, the alternative locations were ranked using RAFSI method.
