• Technique f O d Preference b Si il i to h i for Order f by Similarity Ideal Solution • Yoon and Hwang introduced the TOPSIS method based on the idea that the best alternative should have the shortest distance from the positive ideal solution and farthest distance from the negative ideal l i id l solution. • (K. Paul Yoon and Ching‐Lai Hwang, “Multiple Attribute Decision Making: An Introduction”, Sage Publications, USA, 1995). Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (1/6)

• Step 1. Calculate the normalized decision matrix R = rij mxn . The normalized value rij is calculated as rij = xij

2 xij ∑ i =1 m

i = 1, 2,..., m; j = 1, 2,..., n

The Th number of alternatives b f lt ti The Th number of attributes b f tt ib t

• Th normalization i d The li ti is done f convenience of for i f comparison by converting different units of attributes to an unified unit. Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (2/6)

• Step 2. Calculate the weighted normalized g decision matrix V = vij mxn . The weighted normalized value vij is calculated as vij = ( w j )( rij ) i = 1, 2,..., m; j = 1, 2,..., n

where wj is the weight of the jth attribute and

∑w

j =1

n

j

=1

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (3/6)

• Step 3. Determine the positive ideal solution ( ) (PIS) A+ and negative ideal solution (NIS) A‐. g ( )

} ) )( {( }{ A = {( min v | j ∈ J ) , ( max v | j ∈ J ′ ) , i = 1, 2,..., m} = {v , v ,..., v } A+ =

− + + max vij | j ∈ J , min vij | j ∈ J ′ , i = 1, 2,..., m = v1+ , v2 ,..., vn i i − 1 − 2 − n i ij i ij

where J is a set of benefit attributes and J’ is a set of cost attributes. t f t tt ib t

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (4/6)

• Step 4. Calculate the separation measures, using the n‐dimensional Euclidean distance. • The separation of each alternative from the positive ideal solution

Si+ =

(vijj − v + ) 2 ∑ j

j=1

n

i = 1, 2,..., m

• The separation of each alternative from the negative ideal solution

Si− =

(vij − v − ) 2 ∑ j

j =1

n

i =1 2 m 1, 2,...,

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (5/6)

• Step 5. Calculate the relative closeness to the ideal solution. Si− Ci = + Si + Si− i = 1, 2,..., m; 0 ≤ Ci ≤ 1

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The steps of TOPSIS method (6/6)

• Step 6. Rank the alternatives with respect to Ci in decending order. g • The preferred alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution, where a h h Ci l l h higher would mean higher preference.

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

Example: Choosing an Automobile

• S Suppose you are b i a car, and you are i buying d interested d in both price and life span. • T conflicting objectives: Two fli i bj i – A long expected life span – A low price l i

• Three alternatives:

– The Portalo ( relatively expensive sedan with a reputation h l (a l i l i d ih i for longevity) – The Norushi (renowed for its reliability) – The Standart Motors car (a relatively inexpensive domestic automobile) Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

Example: Choosing an Automobile

Attribute Price ($1000s) Life Span (Years) Portalo 17 12 Norushi 10 9 Standard 8 6

k P = 0 714 0.714 k L = 0.286

Decision Analysis 2009 Fall ‐ Dr. Bahar Sennaroğlu

The normalized decision matrix

Alternatives ( i =1,2,3) Attributes ( j =1 2) 1,2) Price ($1000s) Life Span (Years) Portalo 17 12 Norushi 10 9 Standard 8 6 P L

17 12 X = 10 9 8 6

R = rij mxn

rij =

xij

2 xij ∑ i =1 m

i = 1, 2,..., m; j = 1, 2,..., n

17 2 2 2 17 + 10 + 8 10 R= 2 2 2 17 + 10 + 8 8 2 2 2 17 + 10 + 8

2 2 2 12 + 9...