This is a presentation of some MCDM methods.

1. Training on
MULTI-CRITERIA
DECISION-MAKING
(MCDM) methods
Lanndon Ocampo
Center for Applied Mathematics
and Operations Research

2. OUTLINE OF THE TRAINING
▰ WHAT IS MULTI-CRITERIA DECISION-MAKING?
▰ CRITERIA IMPORTANCE THROUGH INTERCRITERIA CORRELATION
(CRITIC)
▰ ENTROPY Method
▰ Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)
▰ COMBINATIVE DISTANCE-BASED ASSESSMENT (CODAS)
▰ COMBINED COMPROMISE SOLUTION (CoCoSo)
▰ EVALUATION BASED ON DISTANCE FROM AVERAGE SOLUTION (EDAS)
▰ WORKSHOP
2

3. What is multi-criteria
decision-making?
3

4. MULTI-CRITERIA DECISION-MAKING (MCDM)
▰ Multiple-criteria decision analysis (MCDM) comprises a set of methods
that explicitly considers multiple criteria in decision-making
environments.
▰ If the solution space is explicitly defined, then the MCDM is a multiple
criteria evaluation problem; if the space is implicitly defined (by the
constraints set); then the MCDM is a multiple criteria design problem
▰ Formally structuring complex problems appropriately and considering
multiple criteria explicitly is expected to lead to more informed and
transparent decisions.
▰ MCDA methods are increasing in popularity
4

5. CLASSIFICATION OF MCDM METHODS
Roy (1981) described four problem formulations within MCDM:
(1) Choice problems
(2) Sorting problems
(3) Ranking problems
(4) Description problems
5

6. BASIC STRUCTURE OF MADM PROBLEMS
6

7. 4 Steps necessary to build proper MCDM
1) Deliberation – what is the expected from the analysis, which criteria will be
used for decision-making, what are the valid alternatives for the analysis
2) MCDM - inter-criteria comparison: What is the priority weight of each
criteria?
3) MCDM– alternatives comparison: Evaluate the alternatives under the set of
criteria
4) MCDM– results: Establish ranking/Identity the best among alternatives
7

8. CRITERIA IMPORTANCE THROUGH
INTERCRITERIA CORRELATION
(CRITIC)
Diakoulaki et al. (1995)
8
1

9. THE CRITIC METHOD
9
1. Construct the decision-
making matrix 𝑋 =
𝑥𝑖𝑗 𝑛×𝑚
, where 𝑥𝑖𝑗 denotes
the performance value of the
𝑖th alternative on the 𝑗th
criterion.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 4.33
Mobile 2 $200 16gb 8mp 4
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 3.67
Mobile 5 $225 16gb 16mp 2

10. THE CRITIC METHOD
10
2. Normalize the decision matrix
𝑋 = 𝑥𝑖𝑗 𝑚×𝑛
=
𝑥𝑖𝑗 − 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡
𝑥𝑗
𝑏𝑒𝑠𝑡
− 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 4.33
Mobile 2 $200 16gb 8mp 4
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 3.67
Mobile 5 $225 16gb 16mp 2

11. THE CRITIC METHOD
11
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 250 16 12 4.33
Mobile 2 200 16 8 4
Mobile 3 300 32 16 4
Mobile 4 275 32 8 3.67
Mobile 5 225 16 16 2
Best 200 32 16 4.33
Worst 300 16 8 2
Best-Worst -100 16 8 2.33
2. Normalize the decision matrix
𝑋 = 𝑥𝑖𝑗 𝑚×𝑛
=
𝑥𝑖𝑗 − 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡
𝑥𝑗
𝑏𝑒𝑠𝑡
− 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡

12. THE CRITIC METHOD
12
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 (250-300)/-100 (16-16)/16 (12-8)/8 (4.33-2)/2.33
Mobile 2 (200-300)/-100 (16-16)/16 (8-8)/8 (4-2)/2.33
Mobile 3 (300-300)/-100 (32-16)/16 (16-8)/8 (4-2)/2.33
Mobile 4 (275-300)/-100 (32-16)/16 (8-8)/8 (3.67-2)/2.33
Mobile 5 (225-300)/-100 (16-16)/16 (16-8)/8 (2-2)/2.33
Best 200 32 16 4.33
Worst 300 16 8 2
Best-Worst -100 16 8 2.33
2. Normalize the decision matrix
𝑋 = 𝑥𝑖𝑗 𝑚×𝑛
=
𝑥𝑖𝑗 − 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡
𝑥𝑗
𝑏𝑒𝑠𝑡
− 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡

13. THE CRITIC METHOD
13
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.50 0.00 0.50 1.00
Mobile 2 1.00 0.00 0.00 0.86
Mobile 3 0.00 1.00 1.00 0.86
Mobile 4 0.25 1.00 0.00 0.72
Mobile 5 0.75 0.00 1.00 0.00
2. Normalize the decision
matrix
𝑋 = 𝑥𝑖𝑗 𝑚×𝑛
=
𝑥𝑖𝑗 − 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡
𝑥𝑗
𝑏𝑒𝑠𝑡
− 𝑥𝑗
𝑤𝑜𝑟𝑠𝑡

14. THE CRITIC METHOD
14
3. Calculate the standard
deviation 𝜎𝑗for each criteria
𝜎𝑗 = 𝑖=1
𝑚
𝑥𝑖𝑗−𝑥𝑗
2
𝑚
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.50 0.00 0.50 1.00
Mobile 2 1.00 0.00 0.00 0.86
Mobile 3 0.00 1.00 1.00 0.86
Mobile 4 0.25 1.00 0.00 0.72
Mobile 5 0.75 0.00 1.00 0.00
𝝈𝒋 0.40 0.55 0.50 0.40

15. THE CRITIC METHOD
15
4. Determine the symmetric
matrix 𝑅 =
𝑟𝑗𝑘 𝑚×𝑚
where 𝑟𝑗𝑘
denotes the linear
correlation coefficient of
two vectors 𝑥𝑗 and 𝑥𝑘,
𝑗, 𝑘 ∈ 1, … , 𝑚 , using the
formula:
𝑟𝑗𝑘
=
𝑖=1
𝑚
(𝑥𝑖𝑗−𝑥𝑗)(𝑥𝑖𝑘−𝑥𝑘)
𝑖=1
𝑚
𝑥𝑖𝑗 − 𝑥𝑗
2
𝑖=1
𝑚
𝑥𝑖𝑘 − 𝑥𝑘
2
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.50 0.00 0.50 1.00
Mobile 2 1.00 0.00 0.00 0.86
Mobile 3 0.00 1.00 1.00 0.86
Mobile 4 0.25 1.00 0.00 0.72
Mobile 5 0.75 0.00 1.00 0.00

16. THE CRITIC METHOD
16
4. Determine the symmetric
matrix 𝑅 =
𝑟𝑗𝑘 𝑚×𝑚
where 𝑟𝑗𝑘
denotes the linear
correlation coefficient of
two vectors 𝑥𝑗 and 𝑥𝑘,
𝑗, 𝑘 ∈ 1, … , 𝑚 , using the
formula:
𝑟𝑗𝑘
=
𝑖=1
𝑚
(𝑥𝑖𝑗−𝑥𝑗)(𝑥𝑖𝑘−𝑥𝑘)
𝑖=1
𝑚
𝑥𝑖𝑗 − 𝑥𝑗
2
𝑖=1
𝑚
𝑥𝑖𝑘 − 𝑥𝑘
2
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Price ($) 1 -0.8660 -0.3162 -0.2857
Storage (gb) -0.8660 1 0 0.2321
Camera (mp) -0.3162 0 1 -0.4517
Looks -0.2857 0.2321 -0.4517 1

17. THE CRITIC METHOD
17
5. Compute the amount of
information 𝑧𝑗 as follows:
𝑧𝑗 = 𝜎𝑗
𝑘=1
𝑛
(1 − 𝑟𝑗𝑘)
where the higher value
of 𝑧𝑗 implies that the criterion 𝑗
contains more information.
EXAMPLE
Attribute Price ($)
Storage
(gb)
Camera
(mp)
Looks sum(𝟏 − 𝒓𝒋𝒌) 𝝈𝒋 𝒛𝒋
Price ($) 1 -0.8660 -0.3162 -0.2857 4.4679 0.3953 1.7661
Storage
(gb)
-0.8660 1 0 0.2321 3.6339 0.5477 1.9904
Camera
(mp)
-0.3162 0 1 -0.4517 3.7679 0.5000 1.8839
Looks -0.2857 0.2321 -0.4517 1 3.5052 0.3967 1.3906
Sum 7.0310

18. THE CRITIC METHOD
18
6. Determine the criteria
weights according to the
following formula
𝑤𝑗 =
𝑧𝑗
𝑘=1
𝑛
𝑧𝑘
EXAMPLE
Attribute 𝐳𝐣
𝐳𝐣
𝐤=𝟏
𝐧
𝐳𝐤
𝐰𝐣
Price ($) 1.7661 1.7661/7.0310 0.2512
Storage (gb) 1.9904 1.9904/7.0310 0.2831
Camera (mp) 1.8839 1.8839/7.0310 0.2679
Looks 1.3906 1.3906/7.0310 0.1978
7.0310

19. ENTROPY Method
Shannon (1948)
19
2

20. THE ENTROPY METHOD
20
1. Construct the decision-
making matrix 𝑋 =
𝑥𝑖𝑗 𝑛×𝑚
, where 𝑥𝑖𝑗
denotes the performance
value of the 𝑖th alternative
on the 𝑗th criterion.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 5
Mobile 2 $200 16gb 8mp 3
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 4
Mobile 5 $225 16gb 16mp 2

21. THE ENTROPY METHOD
21
2. Normalize the decision-
making matrix as follows:
𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
=
𝑥𝑖𝑗
𝑗 𝑥𝑖𝑗
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 0.20 0.14 0.20 0.28
Mobile 2 0.16 0.14 0.13 0.17
Mobile 3 0.24 0.29 0.27 0.22
Mobile 4 0.22 0.29 0.13 0.22
Mobile 5 0.18 0.14 0.27 0.11

22. THE ENTROPY METHOD
22
3. Compute the entropy of
each criterion as follows:
𝑒𝑗 = −
𝑖 𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
ln 𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
ln 𝑚
for all 𝑗
EXAMPLE
Price Storage Camera Looks
0.9938 0.9630 0.9718 0.9737

23. THE ENTROPY METHOD
23
4. Compute the weights as
follows: 𝑤𝑗 =
1−𝑒𝑗
𝑛− 𝑗 𝑒𝑗
for all
𝑗
EXAMPLE
Price Storage Camera Looks
0.0639 0.3788 0.2879 0.2693

24. Technique for Order Preference by
Similarity to Ideal Solution (TOPSIS)
Hwang and Yoon (1981)
24
3

25. THE TOPSIS METHOD
25
1. Construct the decision-
making matrix 𝑋 =
𝑥𝑖𝑗 𝑛×𝑚
, where 𝑥𝑖𝑗
denotes the performance
value of the 𝑖th alternative
on the 𝑗th criterion.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 5
Mobile 2 $200 16gb 8mp 3
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 4
Mobile 5 $225 16gb 16mp 2

26. THE TOPSIS METHOD
26
2. Normalize the decision-
making matrix as follows:
𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
=
𝑥𝑖𝑗
𝑗 𝑥𝑖𝑗
2
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 0.4428 0.3015 0.4286 0.5976
Mobile 2 0.3542 0.3015 0.2857 0.3586
Mobile 3 0.5314 0.6030 0.5714 0.4781
Mobile 4 0.4871 0.6030 0.2857 0.4781
Mobile 5 0.3985 0.3015 0.5714 0.2390

27. THE TOPSIS METHOD
27
3. Multiply 𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
with the
corresponding weight of
the criteria: 𝑣𝑖𝑗 =
𝑥𝑖𝑗
𝑁𝑂𝑅𝑀
𝑤𝑗, wherein 𝑤𝑗 is the
weight of criteria 𝑗.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 0.1107 0.0754 0.1071 0.1494
Mobile 2 0.0886 0.0754 0.0714 0.0896
Mobile 3 0.1328 0.1508 0.1429 0.1195
Mobile 4 0.1218 0.1508 0.0714 0.1195
Mobile 5 0.0996 0.0754 0.1429 0.0598

28. THE TOPSIS METHOD
28
4. Obtain the ideal solutions
as follows: 𝑣𝑗
+
=
max
𝑖
𝑣𝑖𝑗 , 𝑣𝑗
−
= min
𝑖
𝑣𝑖𝑗
whenever 𝑗 is a benefit
criterion, else 𝑣𝑗
+
=
min
𝑖
𝑣𝑖𝑗 , 𝑣𝑗
−
= max
𝑖
𝑣𝑖𝑗
whenever 𝑗 is a cost criterion
EXAMPLE
Price Storage Camera Looks
𝑣𝑗
+
0.0886 0.1508 0.1429 0.1494
𝑣𝑗
−
0.1328 0.0754 0.0714 0.0598

29. THE TOPSIS METHOD
29
5. Compute the distance of
the alternatives to the ideal
solutions as follows: 𝑆𝑖
+
=
𝑗 𝑣𝑗
+
− 𝑣𝑖𝑗
2
, 𝑆𝑖
−
=
𝑗 𝑣𝑗
−
− 𝑣𝑖𝑗
2
EXAMPLE
Attribute 𝑆𝑖
+
𝑆𝑖
−
Alternative
Mobile 1 0.0863 0.0990
Mobile 2 0.1198 0.0534
Mobile 3 0.0534 0.1198
Mobile 4 0.0842 0.0968
Mobile 5 0.1176 0.0788

30. THE TOPSIS METHOD
30
6. Compute the relative
closeness of the
alternatives to the ideal
solution as follows: 𝐶𝑖
∗
=
𝑆𝑖
−
𝑆𝑖
+
+𝑆𝑖
−, where the alternatives
with 𝐶𝑖
∗
value closer to 1 are
desired.
EXAMPLE
Attribute 𝐶𝑖
∗
𝑅𝑎𝑛𝑘
Alternative
Mobile 1 0.5343 3
Mobile 2 0.3084 5
Mobile 3 0.6916 1
Mobile 4 0.5347 2
Mobile 5 0.4010 4

31. COMBINATIVE DISTANCE-BASED
ASSESSMENT (CODAS)
Ghorabaee et al. (2016)
31
4

32. THE CODAS METHOD
32
1. Construct the decision matrix
𝐴 which is represented as
follows:
𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
=
𝑎11 𝑎12 … 𝑎1𝑛
𝑎21 𝑎22 … 𝑎2𝑛
⋮ ⋮ ⋮ ⋮
𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛
,
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Alternative
Mobile 1 250 16 12 4.33
Mobile 2 200 16 8 4
Mobile 3 300 32 16 4
Mobile 4 275 32 8 3.67
Mobile 5 225 16 16 2

33. THE CODAS METHOD
33
2. Determine normalized decision
matrix 𝑁 = 𝑛𝑖𝑗 𝑚×𝑛
according
to the type of each criterion using
the linear normalization of
performance values as follows:
𝑛𝑖𝑗 =
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗
if 𝑗 ∈ 𝑃
min
𝑖
𝑎𝑖𝑗
𝑎𝑖𝑗
if 𝑗 ∈ 𝐶
where 𝑃 and 𝐶 represent the
sets of benefit and cost criteria
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Alternative
Mobile 1 250 16 12 4.33
Mobile 2 200 16 8 4
Mobile 3 300 32 16 4
Mobile 4 275 32 8 3.67
Mobile 5 225 16 16 2

34. THE CODAS METHOD
34
2. Determine normalized decision
matrix 𝑁 = 𝑛𝑖𝑗 𝑚×𝑛
according
to the type of each criterion using
the linear normalization of
performance values as follows:
𝑛𝑖𝑗 =
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗
if 𝑗 ∈ 𝑃
min
𝑖
𝑎𝑖𝑗
𝑎𝑖𝑗
if 𝑗 ∈ 𝐶
where 𝑃 and 𝐶 represent the
sets of benefit and cost criteria
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Alternative
Mobile 1 200/250 16/32 12/16 4.33/4.33
Mobile 2 200/200 16/32 8/16 4/4.33
Mobile 3 200/300 32/32 16/16 4/4.33
Mobile 4 200/275 32/32 8/16 3.67/4.33
Mobile 5 200/225 16/32 16/16 2/4.33
Max 300 32 16 4.33
Min 200 16 8 2

35. THE CODAS METHOD
35
2. Determine normalized decision
matrix 𝑁 = 𝑛𝑖𝑗 𝑚×𝑛
according
to the type of each criterion using
the linear normalization of
performance values as follows:
𝑛𝑖𝑗 =
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗
if 𝑗 ∈ 𝑃
min
𝑖
𝑎𝑖𝑗
𝑎𝑖𝑗
if 𝑗 ∈ 𝐶
where 𝑃 and 𝐶 represent the
sets of benefit and cost criteria
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Alternative
Mobile 1 0.80 0.50 0.75 1.00
Mobile 2 1.00 0.50 0.50 0.92
Mobile 3 0.67 1.00 1.00 0.92
Mobile 4 0.73 1.00 0.50 0.85
Mobile 5 0.89 0.50 1.00 0.46

36. THE CODAS METHOD
36
3. Calculate the weighted
normalized decision matrix 𝐵 =
𝑏𝑖𝑗 𝑚×𝑛
calculated as follows:
𝑏𝑖𝑗 = 𝑤𝑗𝑛𝑖𝑗
wherein 𝑗=1
𝑛
𝑤𝑗 = 1, 𝑤𝑗 denotes the
weight of the 𝑗th criterion, and 0 <
𝑤𝑗 < 1. 𝑤𝑗 can be determined through
any priority weight generation method
(e.g., CRITIC).
EXAMPLE
Attribute
Price ($)
𝑤𝑗=0.2512
Storage (gb)
𝑤𝑗=0.2831
Camera (mp)
𝑤𝑗=0.2679
Looks
𝑤𝑗=0.1978
Mobile 1 0.80*0.2512 0.50*0.2831 0.75*0.2679 1.00*0.1978
Mobile 2 1.00*0.2512 0.50*0.2831 0.50*0.2679 0.92*0.1978
Mobile 3 0.67*0.2512 1.00*0.2831 1.00*0.2679 0.92*0.1978
Mobile 4 0.73*0.2512 1.00*0.2831 0.50*0.2679 0.85*0.1978
Mobile 5 0.89*0.2512 0.50*0.2831 1.00*0.2679 0.46*0.1978

37. THE CODAS METHOD
37
3. Calculate the weighted
normalized decision matrix 𝐵 =
𝑏𝑖𝑗 𝑚×𝑛
calculated as follows:
𝑏𝑖𝑗 = 𝑤𝑗𝑛𝑖𝑗
wherein 𝑗=1
𝑛
𝑤𝑗 = 1, 𝑤𝑗 denotes the
weight of the 𝑗th criterion, and 0 <
𝑤𝑗 < 1. 𝑤𝑗 can be determined through
any priority weight generation method
(e.g., CRITIC).
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.22 0.14 0.19 0.19
Mobile 2 0.27 0.14 0.13 0.18
Mobile 3 0.18 0.28 0.26 0.18
Mobile 4 0.20 0.28 0.13 0.16
Mobile 5 0.24 0.14 0.26 0.09

38. THE CODAS METHOD
38
4. Determine the negative ideal
solution 𝑏∗ as follows:
𝑏∗ = 𝑏𝑗∗ 1×𝑛
𝑏𝑗∗ = min
𝑖
𝑏𝑖𝑗
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.22 0.14 0.19 0.19
Mobile 2 0.27 0.14 0.13 0.18
Mobile 3 0.18 0.28 0.26 0.18
Mobile 4 0.20 0.28 0.13 0.16
Mobile 5 0.24 0.14 0.26 0.09
𝒃∗ 0.18 0.14 0.13 0.09

39. THE CODAS METHOD
39
5. Calculate the Euclidean 𝑑𝑖 and
Taxicab 𝑡𝑖 distances of
alternatives from the negative
ideal solution using the following:
𝑑𝑖 = 𝑗=1
𝑚
𝑏𝑖𝑗 − 𝑏𝑗∗
2
𝑡𝑖 =
𝑗=1
𝑚
𝑏𝑖𝑗 − 𝑏𝑗∗
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.22 0.14 0.19 0.19
Mobile 2 0.27 0.14 0.13 0.18
Mobile 3 0.18 0.28 0.26 0.18
Mobile 4 0.20 0.28 0.13 0.16
Mobile 5 0.24 0.14 0.26 0.09
𝒃∗ 0.18 0.14 0.13 0.09

40. THE CODAS METHOD
40
5. Calculate the Euclidean 𝑑𝑖 and
Taxicab 𝑡𝑖 distances of
alternatives from the negative
ideal solution using the following:
𝑑𝑖 = 𝑗=1
𝑚
𝑏𝑖𝑗 − 𝑏𝑗∗
2
𝑡𝑖 =
𝑗=1
𝑚
𝑏𝑖𝑗 − 𝑏𝑗∗
EXAMPLE
Alternatives 𝒅𝒊 𝒕𝒊
Mobile 1 0.0159 0.2027
Mobile 2 0.0159 0.1785
Mobile 3 0.0440 0.3570
Mobile 4 0.0255 0.2307
Mobile 5 0.0201 0.1885

41. THE CODAS METHOD
41
6. Construct the relative assessment
matrix 𝐻 = ℎ𝑖𝑘 𝑚×𝑚 presented
as follows:
ℎ𝑖𝑘 = 𝑑𝑖 − 𝑑𝑘 + 𝛽 𝑑𝑖 − 𝑑𝑘 × 𝑡𝑖 − 𝑡𝑘
where 𝑘 ∈ 1,2, … , 𝑚 and 𝛽
is a threshold function which can be set
by the decision-maker, and defined as
follows:
𝛽 =
1 𝑖𝑓 𝑑𝑖 − 𝑑𝑘 ≥ 𝜌
0 𝑖𝑓 𝑑𝑖 − 𝑑𝑘 < 𝜌
EXAMPLE
Alternatives Mobile 1 Mobile 2 Mobile 3 Mobile 4 Mobile 5
Mobile 1 0 -0.00001 -0.02809 -0.00958 -0.00416
Mobile 2 0.00001 0 -0.02807 -0.00957 -0.00415
Mobile 3 0.03242 0.03308 0 0.01850 0.02795
Mobile 4 0.00958 0.00957 -0.01850 0 0.00542
Mobile 5 0.00416 0.00415 -0.02392 -0.00542 0

42. THE CODAS METHOD
42
7. Calculate the assessment score (𝑆𝑖)
of each alternative shown as
follows:
𝑆𝑖 = 𝑘=1
𝑚
ℎ𝑖𝑘
EXAMPLE
Alternatives Mobile 1 Mobile 2 Mobile 3 Mobile 4 Mobile 5 𝑺𝒊
Mobile 1 0 -0.00001 -0.02809 -0.00958 -0.00416 -0.04184
Mobile 2 0.00001 0 -0.02807 -0.00957 -0.00415 -0.04178
Mobile 3 0.03242 0.03308 0 0.01850 0.02795 0.11196
Mobile 4 0.00958 0.00957 -0.01850 0 0.00542 0.00607
Mobile 5 0.00416 0.00415 -0.02392 -0.00542 0 -0.02104

43. THE CODAS METHOD
43
8. Rank the alternatives. The
alternative with the highest 𝑺𝒊
value is the most desirable.
EXAMPLE
Alternatives 𝑺𝒊 Rank
Mobile 1 -0.04184 5
Mobile 2 -0.04178 4
Mobile 3 0.11196 1
Mobile 4 0.00607 2
Mobile 5 -0.02104 3

44. COMBINED COMPROMISE
SOLUTION (CoCoSo)
44
5
for ranking alternatives

45. THE CoCoSo METHOD
45
1. Construct the decision matrix
𝐴 which is represented as
follows:
𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
=
𝑎11 𝑎12 … 𝑎1𝑛
𝑎21 𝑎22 … 𝑎2𝑛
⋮ ⋮ ⋮ ⋮
𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛
,
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Alternative
Mobile 1 250 16 12 4.33
Mobile 2 200 16 8 4
Mobile 3 300 32 16 4
Mobile 4 275 32 8 3.67
Mobile 5 225 16 16 2

46. THE CoCoSo METHOD
46
2. Normalize the decision matrix
using:
𝑟𝑖𝑗 =
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for benefit criterion
𝑟𝑖𝑗 =
max
𝑖
𝑎𝑖𝑗−𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for cost criterion
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 250 16 12 4.33
Mobile 2 200 16 8 4
Mobile 3 300 32 16 4
Mobile 4 275 32 8 3.67
Mobile 5 225 16 16 2
Max 300 32 16 4.33
Min 200 16 8 2
Max-Min 100 16 8 2.33

47. THE CoCoSo METHOD
47
2. Normalize the decision matrix
using:
𝑟𝑖𝑗 =
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for benefit criterion
𝑟𝑖𝑗 =
max
𝑖
𝑎𝑖𝑗−𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for cost criterion
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 (300-250)/100 (16-16)/16 (12-8)/8 (4.33-2)/2.33
Mobile 2 (300-200)/100 (16-16)/16 (8-8)/8 (4-2)/2.33
Mobile 3 (300-300)/100 (32-16)/16 (16-8)/8 (4-2)/2.33
Mobile 4 (300-275)/100 (32-16)/16 (8-8)/8 (3.67-2)/2.33
Mobile 5 (300-225)/100 (16-16)/16 (16-8)/8 (2-2)/2.33
Max 300 32 16 2
Min 200 16 8 2
Max-Min 100 16 8 2.33

48. THE CoCoSo METHOD
48
2. Normalize the decision matrix 𝐴
into 𝑅 = 𝑟𝑖𝑗 𝑚×𝑛
using:
𝑟𝑖𝑗 =
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for benefit criterion
𝑟𝑖𝑗 =
max
𝑖
𝑎𝑖𝑗−𝑎𝑖𝑗
max
𝑖
𝑎𝑖𝑗−min
𝑖
𝑎𝑖𝑗
for cost criterion
EXAMPLE
Attribute Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.50 0.00 0.50 1.00
Mobile 2 1.00 0.00 0.00 0.86
Mobile 3 0.00 1.00 1.00 0.86
Mobile 4 0.25 1.00 0.00 0.72
Mobile 5 0.75 0.00 1.00 0.00

49. THE CoCoSo METHOD
49
3. The sum of the weighted
comparability sequence is
calculated using:
𝑆𝑖 =
𝑗=1
𝑛
𝑤𝑗 𝑟𝑖𝑗
where 𝑤𝑗 is obtained using
the CRITIC method.
EXAMPLE
Attribute
Price ($)
𝑤𝑗 = 0.2512
Storage (gb)
𝑤𝑗 = 0.2831
Camera (mp)
𝑤𝑗 = 0.2679
Looks
𝑤𝑗 = 0.1978
Mobile 1 0.50*0.2512 0.00*0.2831 0.50*0.2679 1.00*0.1978
Mobile 2 1.00*0.2512 0.00*0.2831 0.00*0.2679 0.86*0.1978
Mobile 3 0.00*0.2512 1.00*0.2831 1.00*0.2679 0.86*0.1978
Mobile 4 0.25*0.2512 1.00*0.2831 0.00*0.2679 0.72*0.1978
Mobile 5 0.75*0.2512 0.00*0.2831 1.00*0.2679 0.00*0.1978

50. THE CoCoSo METHOD
50
3. The sum of the weighted
comparability sequence is
calculated using:
𝑆𝑖 =
𝑗=1
𝑛
𝑤𝑗 𝑟𝑖𝑗
where 𝑤𝑗 is obtained using
the CRITIC method.
EXAMPLE
Attribute
Price ($)
𝑤𝑗 = 0.2512
Storage (gb)
𝑤𝑗 = 0.2831
Camera (mp)
𝑤𝑗 = 0.2679
Looks
𝑤𝑗 = 0.1978 𝑺𝒊
Mobile 1 0.13 0.00 0.13 0.20 0.46
Mobile 2 0.25 0.00 0.00 0.17 0.42
Mobile 3 0.00 0.28 0.27 0.17 0.72
Mobile 4 0.06 0.28 0.00 0.14 0.49
Mobile 5 0.19 0.00 0.27 0.00 0.46

51. THE CoCoSo METHOD
51
4. The sum of the power weight of
comparability sequences for each
alternative is calculated using:
𝑃𝑖 =
𝑗=1
𝑛
𝑟𝑖𝑗
𝑤𝑗
where 𝑤𝑗 is obtained using
the CRITIC method.
EXAMPLE
Attribute
Price ($)
𝑤𝑗 = 0.2512
Storage (gb)
𝑤𝑗 = 0.2831
Camera (mp)
𝑤𝑗 = 0.2679
Looks
𝑤𝑗 = 0.1978
Mobile 1 0.50^0.2512 0.00^0.2831 0.50^0.2679 1.00^0.1978
Mobile 2 1.00^0.2512 0.00^0.2831 0.00^0.2679 0.86^0.1978
Mobile 3 0.00^0.2512 1.00^0.2831 1.00^0.2679 0.86^0.1978
Mobile 4 0.25^0.2512 1.00^0.2831 0.00^0.2679 0.72^0.1978
Mobile 5 0.75^0.2512 0.00^0.2831 1.00^0.2679 0.00^0.1978

52. THE CoCoSo METHOD
52
4. The sum of the power weight of
comparability sequences for each
alternative is calculated using:
𝑃𝑖 =
𝑗=1
𝑛
𝑟𝑖𝑗
𝑤𝑗
where 𝑤𝑗 is obtained using
the CRITIC method.
EXAMPLE
Attribute
Price ($)
𝑤𝑗 = 0.2512
Storage (gb)
𝑤𝑗 = 0.2831
Camera (mp)
𝑤𝑗 = 0.2679
Looks
𝑤𝑗 = 0.1978 𝑃𝑖
Mobile 1 0.59 0.00 0.58 0.73
1.90
Mobile 2 0.71 0.00 0.00 0.70
1.41
Mobile 3 0.00 0.70 0.70 0.70
2.11
Mobile 4 0.50 0.70 0.00 0.68
1.88
Mobile 5 0.66 0.00 0.70 0.00
1.36

53. THE CoCoSo METHOD
53
5. Relative weights of the
alternatives using the following
aggregation strategies are
computed.
i. 𝑘𝑖𝑎 =
𝑃𝑖+𝑆𝑖
𝑖=1
𝑚
𝑃𝑖+𝑆𝑖
EXAMPLE
Attribute 𝑆𝑖 𝑃𝑖 𝑃𝑖 + 𝑆𝑖 𝒌𝒊𝒂
Mobile 1 0.46 1.90 2.36 0.2107
Mobile 2 0.42 1.41 1.83 0.1635
Mobile 3 0.72 2.11 2.83 0.2523
Mobile 4 0.49 1.88 2.37 0.2111
Mobile 5 0.46 1.36 1.82 0.1621
𝑖=1
𝑚
𝑃𝑖 + 𝑆𝑖 11.20

54. THE CoCoSo METHOD
54
5. Relative weights of the
alternatives using the following
aggregation strategies are
computed.
ii. 𝑘𝑖𝑏 =
𝑆𝑖
min
𝑖
𝑆𝑖
+
𝑃𝑖
min
𝑖
𝑃𝑖
EXAMPLE
Attribute 𝑆𝑖 𝑃𝑖 𝒌𝒊𝒃
Mobile 1 0.46 1.90 2.4857
Mobile 2 0.42 1.41 2.0373
Mobile 3 0.72 2.11 3.2609
Mobile 4 0.49 1.88 2.5392
Mobile 5 0.46 1.36 2.0840
min 0.42 1.36

55. THE CoCoSo METHOD
55
5. Relative weights of the
alternatives using the following
aggregation strategies are
computed.
iii. 𝑘𝑖𝑐 =
𝜆𝑆𝑖+(1−𝜆)𝑃𝑖
𝜆 max
𝑖
𝑆𝑖+(1−𝜆) max
𝑖
𝑃𝑖
where 𝜆 0 ≤ 𝜆 ≤ 1 is chosen
by decision-makers. Here,
𝜆=0.05.
EXAMPLE
Attribute 𝑆𝑖 𝑃𝑖 𝒌𝒊𝒄
Mobile 1 0.46 1.90
0.8349
Mobile 2 0.42 1.41
0.6480
Mobile 3 0.72 2.11
1.0000
Mobile 4 0.49 1.88
0.8368
Mobile 5 0.46 1.36
0.6425
max 0.72 2.11

56. THE CoCoSo METHOD
56
6. The final ranking of the
alternatives is determined based
on 𝑘𝑖 values (as more significant
as better):
𝑘𝑖
= 𝑘𝑖𝑎𝑘𝑖𝑏𝑘𝑖𝑐
1
3 +
1
3
𝑘𝑖𝑎 + 𝑘𝑖𝑏 + 𝑘𝑖𝑐
EXAMPLE
Alternative 𝑘𝑖𝑎 𝑘𝑖𝑏 𝑘𝑖𝑐 𝑘𝑖
Mobile 1 0.2107 2.4857 0.8349 1.9362
Mobile 2 0.1635 2.0373 0.6480 1.5495
Mobile 3 0.2524 3.2609 1.0000 2.4416
Mobile 4 0.2112 2.5392 0.8368 1.9613
Mobile 5 0.1622 2.0840 0.6425 1.5640

57. THE CoCoSo METHOD
57
6. The final ranking of the
alternatives is determined based
on 𝑘𝑖 values (as more significant
as better):
𝑘𝑖
= 𝑘𝑖𝑎𝑘𝑖𝑏𝑘𝑖𝑐
1
3 +
1
3
𝑘𝑖𝑎 + 𝑘𝑖𝑏 + 𝑘𝑖𝑐
EXAMPLE
Alternative 𝑘𝑖 Rank
Mobile 1 1.9362 3
Mobile 2 1.5495 5
Mobile 3 2.4416 1
Mobile 4 1.9613 2
Mobile 5 1.5640 4

58. EVALUATION BASED ON
DISTANCE FROM AVERAGE
SOLUTION (EDAS)
Ghorabaee et al. (2015)
58
6

59. THE EDAS METHOD
59
1. Construct the decision-
making matrix 𝑋 =
𝑥𝑖𝑗 𝑛×𝑚
, where 𝑥𝑖𝑗
denotes the performance
value of the 𝑖th alternative
on the 𝑗th criterion.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 4.33
Mobile 2 $200 16gb 8mp 4
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 3.67
Mobile 5 $225 16gb 16mp 2

60. THE EDAS METHOD
60
2. Determine the average
solution 𝐴𝑉 = 𝐴𝑉
𝑗 1×𝑚
according to all criteria,
shown as follows:
𝐴𝑉
𝑗 =
𝑖=1
𝑛
𝑥𝑖𝑗
𝑛
EXAMPLE
Price ($) Storage (gb) Camera (mp) Looks
AV 250.00 22.40 12.00 3.60

61. THE EDAS METHOD
61
3. Calculate the positive distance from
average 𝑃𝐷𝐴 = 𝑃𝐷𝐴𝑖𝑗 𝑛×𝑚
matrix
through:
if 𝑗th criterion is beneficial,
𝑃𝐷𝐴𝑖𝑗 =
max 0, 𝑋𝑖𝑗 − 𝐴𝑉
𝑗
𝐴𝑉
𝑗
and if the 𝑗th criterion is non-beneficial,
𝑃𝐷𝐴𝑖𝑗 =
max 0, 𝐴𝑉
𝑗 − 𝑋𝑖𝑗
𝐴𝑉
𝑗
EXAMPLE
PDA Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.00 0.29 0.00 0.00
Mobile 2 0.00 0.29 0.33 0.00
Mobile 3 0.20 0.00 0.00 0.00
Mobile 4 0.10 0.00 0.33 0.00
Mobile 5 0.00 0.29 0.00 0.44

62. THE EDAS METHOD
62
4. Calculate the negative distance from
average N𝐷𝐴 = 𝑁𝐷𝐴𝑖𝑗 𝑛×𝑚
matrix
through:
if 𝑗th criterion is beneficial,
𝑁𝐷𝐴𝑖𝑗 =
max 0, 𝐴𝑉
𝑗 − 𝑋𝑖𝑗
𝐴𝑉
𝑗
and if the 𝑗th criterion is non-beneficial,
𝑁𝐷𝐴𝑖𝑗 =
max 0, 𝑋𝑖𝑗 − 𝐴𝑉
𝑗
𝐴𝑉
𝑗
EXAMPLE
NDA Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.00 0.00 0.00 0.20
Mobile 2 0.20 0.00 0.00 0.11
Mobile 3 0.00 0.43 0.33 0.11
Mobile 4 0.00 0.43 0.00 0.02
Mobile 5 0.10 0.00 0.33 0.00

63. THE EDAS METHOD
63
5. Determine the weighted sum of
PDA and NDA for all alternatives,
shown as follows:
𝑆𝑃𝑖 =
𝑗=1
𝑚
𝑤𝑗𝑃𝐷𝐴𝑖𝑗
𝑆𝑁𝑖 =
𝑗=1
𝑚
𝑤𝑗𝑁𝐷𝐴𝑖𝑗
where, 𝑤𝑗 is the weight of the jth
criterion
EXAMPLE
SPi SNi
Mobile 1 0.0809 0.0403
Mobile 2 0.1702 0.0722
Mobile 3 0.0502 0.2326
Mobile 4 0.1144 0.1250
Mobile 5 0.1688 0.1144

64. THE EDAS METHOD
64
6. Normalize the values of 𝑆𝑃 and 𝑆𝑁
for all alternatives, shown as follows:
𝑁𝑆𝑃𝑖 =
𝑆𝑃𝑖
max
i
(𝑆𝑃𝑖)
𝑁𝑆𝑁𝑖 = 1 −
𝑆𝑁𝑖
max
i
(𝑆𝑁𝑖)
EXAMPLE
NSPi NSNi
Mobile 1 0.4752 0.8268
Mobile 2 1.0000 0.6896
Mobile 3 0.2952 0.0000
Mobile 4 0.6724 0.4627
Mobile 5 0.9917 0.5081

65. THE EDAS METHOD
65
7. Calculate the appraisal score 𝐴𝑆 for
all alternatives, shown as follows:
𝐴𝑆𝑖 =
1
2
𝑁𝑆𝑃𝑖 + 𝑁𝑆𝑁𝑖
where 0 ≤ 𝐴𝑆𝑖 ≤ 1.
Rank the alternatives according to the
decreasing values of appraisal score AS.
The alternative with the highest AS is the
best choice among the candidate
alternatives.
EXAMPLE
ASi Rank
Mobile 1 0.6510 3
Mobile 2 0.8448 1
Mobile 3 0.1476 5
Mobile 4 0.5675 4
Mobile 5 0.7499 2

66. THE EDAS METHOD
66
8. Rank the alternatives according to the
decreasing values of appraisal score
AS. The alternative with the highest
AS is the best choice among the
candidate alternatives.
EXAMPLE
ASi
Mobile 1 0.6510
Mobile 2 0.8448
Mobile 3 0.1476
Mobile 4 0.5675
Mobile 5 0.7499

67. Ranking of Alternatives through
Functional mapping of criterion sub-
intervals into a Single-Interval (RAFSI)
Žižović et al. (2020)
67
7

68. THE RAFSI METHOD
68
1. Construct the decision-
making matrix 𝑋 =
𝑥𝑖𝑗 𝑛×𝑚
, where 𝑥𝑖𝑗
denotes the performance
value of the 𝑖th alternative
on the 𝑗th criterion.
EXAMPLE
Attribute Price Storage Camera Looks
Alternative
Mobile 1 $250 16gb 12mp 4.33
Mobile 2 $200 16gb 8mp 4
Mobile 3 $300 32gb 16mp 4
Mobile 4 $275 32gb 8mp 3.67
Mobile 5 $225 16gb 16mp 2

69. THE RAFSI METHOD
69
2. Define ideal and anti-ideal
values. For each criterion
𝐶𝑗 𝑗 = 1,2, … , 𝑛 , the
decision-makers defines the
ideal value 𝑎𝐼𝑗 and anti-ideal
value 𝑎𝑁𝑗.
Obviously, 𝑎𝐼𝑗 > 𝑎𝑁𝑗 for max
criteria and 𝑎𝐼𝑗 < 𝑎𝑁𝑗 for
min criteria.
EXAMPLE
Price Storage Camera Looks
aIj 350 32 24 5
aNj 180 8 4 1

70. THE RAFSI METHOD
70
3. Mapping of elements in 𝑋into criteria
intervals. Based on the defined ideal and
anti-ideal values, functions 𝑓𝐴𝑖
𝐶𝑗 are
defined, which map the criterion intervals
from 𝑋 to the criterion interval 𝑛1, 𝑛2𝑘
using the equation,
𝑓𝐴𝑖
𝐶𝑗 =
𝑛2𝑘 − 𝑛1
𝑎𝐼𝑗
− 𝑎𝑁𝑗
𝑥𝑖𝑗 +
𝑎𝐼𝑗
∙ 𝑛1 − 𝑎𝑁𝑗
∙ 𝑛2𝑘
𝑎𝐼𝑗
− 𝑎𝑁𝑗
Where 𝑛1 and 𝑛2𝑘 represent the ratio that
shows how much the ideal value is better
than the anti-ideal value. It is suggested
that the ideal value is at least six times
better than the anti-ideal, or 𝑛1 = 1 and
𝑛2𝑘 = 6. However, the decision-makers
can use other values.
EXAMPLE
Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 3.8824 3.3333 3.8000 6.8333
Mobile 2 1.8235 3.3333 2.4000 6.2500
Mobile 3 5.9412 8.0000 5.2000 6.2500
Mobile 4 4.9118 8.0000 2.4000 5.6667
Mobile 5 2.8529 3.3333 5.2000 2.7500
In this example, 𝑛1 = 1 and 𝑛2𝑘 = 8.

71. THE RAFSI METHOD
71
4. Calculate arithmetic and harmonic means.
The arithmetic and harmonic means for
minimum and maximum sequence of the
elements n_1 and n_2k are calculated as
follows:
𝐴 =
𝑛1 + 𝑛2𝑘
2
𝐻 =
2
1
𝑛1
+
1
𝑛2𝑘
EXAMPLE
𝐴 =
1 + 8
2
= 𝟒. 𝟓𝟎𝟎𝟎
𝐻 =
2
1
1
+
1
8
= 𝟏. 𝟕𝟕𝟕𝟖

72. THE RAFSI METHOD
72
5. Form normalized decision matrix 𝑆 =
𝑠𝑖𝑗 𝑚×𝑛
such that 𝑠𝑖𝑗 ∈ 0,1 using the
following formula:
𝑠𝑖𝑗 =
𝑠𝑖𝑗
2𝐴
when 𝐶𝑗 is max type
𝑠𝑖𝑗 =
𝐻
𝑠𝑖𝑗
when 𝐶𝑗 is min type
EXAMPLE
Price ($) Storage (gb) Camera (mp) Looks
Mobile 1 0.4579 0.3704 0.4222 0.7593
Mobile 2 0.9749 0.3704 0.2667 0.6944
Mobile 3 0.2992 0.8889 0.5778 0.6944
Mobile 4 0.3619 0.8889 0.2667 0.6296
Mobile 5 0.6231 0.3704 0.5778 0.3056

73. THE RAFSI METHOD
73
6. Calculate criteria functions of the
alternatives 𝑉(𝐴𝑖). Criteria functions of
𝑉(𝐴𝑖) are calculated accordingly:
𝑉 𝐴𝑖 = 𝑤1𝑠𝑖1 + 𝑤2𝑠𝑖2 + ⋯ + 𝑤𝑛𝑠𝑖𝑛
where 𝑤𝑗, 𝑗 = 1,2, … , 𝑛 refers to the
criteria weights. Alternatives are ranked
according to the descending order of the
calculated 𝑉(𝐴𝑖) values.
EXAMPLE
V(Ai) Rank
Mobile 1 0.4832 4
Mobile 2 0.5585 2
Mobile 3 0.6190 1
Mobile 4 0.5385 3
Mobile 5 0.4766 5

74. 74
THANKS!
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