ABSTRACT
In COmbinative Distance-based ASsessment (CODAS) method, the alternatives are evaluated and subsequently ranked based on the computation of their desirability with respect to two measures. Between these two measures, the first one is associated with the Euclidean distance of the alternatives from the negative-ideal solution (NIS) based on the formation of l 2 -norm indifference space for criteria. The secondary measure is the Taxicab distance based on the l 1 -norm indifference space. Thus, in this method, the alternative having the maximum distance from the NIS would be the most suitable choice. If any two alternatives are not comparable according to the Euclidean distance, the Taxicab distance is adopted as the secondary measure. Although the l 2 -norm indifference space is mainly considered in this method, it is always preferred to take into account two types of indifference space for better solution of a decision-making problem. The procedural steps of CODAS method are presented as below [1]:
Step 1: Develop the initial decision matrix (X) consisting of m alternatives and n criteria. https://www.w3.org/1998/Math/MathML"> X = [ x 11 x 12 ... x 1 n x 21 x 22 ... x 2 n ... ... ... ... x m 1 x m 2 ... x 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/math15_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; xij is the performance of i th alternative (i = 1, 2, …, m) with respect to j th criterion (j = 1, 2, …, n).
Step 2: Apply linear normalization technique to formulate the corresponding normalized decision matrix. For normalization, the following equations can be employed:
For Beneficial Criterion: https://www.w3.org/1998/Math/MathML"> y i j = x i j max x i j i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_2_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
For Non-Beneficial Criterion: https://www.w3.org/1998/Math/MathML"> y i j = min x i j i x 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/math15_3_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 3: Calculate the corresponding weighted normalized decision matrix. Using weights of the considered criteria, compute the weighted normalized decision matrix while multiplying elements of the normalized decision matrix by the criteria weights. https://www.w3.org/1998/Math/MathML"> r i j = w j × y 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/math15_4_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; wj is the weight of j th criterion.
Step 4: Determine the NIS (point). https://www.w3.org/1998/Math/MathML"> n s = [ n s j ] 1 × n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_5_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> n s j = min r i j i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_6_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 5: Compute the Euclidean and Taxicab distances of the alternatives from the NIS. https://www.w3.org/1998/Math/MathML"> E i ∑ j = 1 n ( r i j − n s j ) 2 ( i = 1 , 2 , ... , m ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_7_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> T i = ∑ j = 1 n | r i j − n s j | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 6: Develop the following relative assessment matrix: https://www.w3.org/1998/Math/MathML"> R a = [ h i k ] m × m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_9_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where; hik = (Ei – Ek ) + (ψ(Ei – Ek )×(Ti – Tk )) (for k = 1, 2, …, m), and ψ represents a threshold function to provide equality of the Euclidean distances of two alternatives. Its value can be defined as below: https://www.w3.org/1998/Math/MathML"> ψ ( x { 1 i f | x | ≥ τ 0 i f | x | < τ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_10_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
In the above function, the value of the threshold parameter (τ) needs to be specified by the concerned decision maker. The most preferred value of τ usually ranges between 0.01 and 0.05. In this method, if
the difference between the Euclidean distances of two alternatives is less than τ, these two alternatives are also compared using the corresponding Taxicab distance.
Step 7: Compute the assessment score of each alternative. https://www.w3.org/1998/Math/MathML"> H i = ∑ k = 1 m h i k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003377030/d9605bd9-7b4d-4c40-a5a9-8fc61b88ce2e/content/math15_11_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Step 8: Rank the candidate alternatives based on the decreasing values of their assessment scores (Ηi ). The alternative having the maximum Ηi value would be the best choice.
