Technique for Order Performance by Similarity to Idea Solution

(TOPSIS)

2

TOPSIS METHOD

• Technique of Order Preference by Similarity to Ideal Solution

• TOPSIS is based on an aggregating function representing ‘‘closeness to the ideal’’

• This method considers three types of attributes or criteria– Qualitative benefit attributes/criteria

– Quantitative benefit attributes

– Cost attributes or criteria

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TOPSIS METHOD

• In this method two artificial alternatives arehypothesized:

• Ideal alternative: the one which has the best levelfor all attributes considered.

• Negative ideal alternative: the one which has theworst attribute values.

TOPSIS selects the alternative that is theclosest to the ideal solution and farthestfrom negative ideal alternative.

Why TOPSIS?

• Each criterion can be taken into consideration in making a final ranking– the concept of TOPSIS is rational– the computation involved is easy– It allows objective weights to be incorporated into the

comparison process.

• TOPSIS method while determining ‘‘ideal” and ‘‘anti-ideal” solutions computes the weighted distances to measure the relative distances away from the ideal and anti-ideal solutions for each alternative (i.e., decision rule).

• Not only the best alternative should be as close as possible to the ideal solution but also it should be as far as possible away from the anti-ideal solution.

Idea of TOPSIS method

Components of TOPSIS

• TOPSIS is composed of two components:

– Weights

– Distances.

• The most commonly used weights

– Mean weight

– standard deviation weight

• Among the most commonly used distances is Euclidean distance

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Positive and negative ideal solution

• Positive ideal solution– Maximizes the benefit criteria and minimizes the

cost criteria

– Best values attainable from the criteria

• Negative ideal solution– Maximizes the cost criteria and minimizes the

benefit criteria

– Worst values attainable from the criteria

Steps of multi criteria decision making technique

a) Establishing system evaluation criteria that relatesystem capabilities to goals

b) Developing alternative systems for attaining the goals(generating alternatives)

c) Evaluating alternatives in terms of criteria (the valuesof the criterion functions)

d) Applying a normative multi criteria analysis method

e) Accepting one alternative as ‘‘optimal’’ (preferred)

f) If the final solution is not accepted, gather newinformation and go into the next iteration of multicriteria optimization

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Input to TOPSIS

• TOPSIS assumes that we have m alternatives(options) and n attributes/criteria and we havethe score of each option with respect to eachcriterion.

• Let xij score of option/alternative i with respect tocriterion j. We have a matrix X = (xij) mn matrix.

• Let J be the set of benefit attributes or criteria(more is better)

• Let J' be the set of negative attributes or criteria(less is better)

mnmm

n

n

xxx

xxx

xxx

X

........

.......

.......

21

22221

11211

Criteria

Alternatives

Decision matrix for ranking

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Steps of TOPSIS

• Step 1: Construct normalized decisionmatrix.

• This step transforms various attributedimensions into non-dimensionalattributes, which allows comparisons acrosscriteria.

• Normalize scores or data as follows:

rij = xij/ (x2ij) for i = 1, …, m; j = 1, …, n

i

Normalized decision matrix

mnmm

n

n

rrr

rrr

rrr

N

........

.......

.......

21

22221

11211

Criteria

Alternatives

2

1

2

21

2

11

11

11

........m

xxx

xr

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Steps of TOPSIS

• Step 2: Construct the weighted normalized decision matrix.

• Assume we have a set of weights for each criteria wj for j = 1,…n.

• Multiply each column of the normalized decision matrix by its associated weight.

• An element of the new matrix is:

vij = wj rij

Normalized decision matrix

mnmm

n

n

rrr

rrr

rrr

N

........

.......

.......

21

22221

11211

Criteria

Alternatives

w1 w2 ….. wn

Weights

Weighted normalized decision matrix

mnmm

n

n

vvv

vvv

vvv

V

........

.......

.......

21

22221

11211

11111rwv

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Steps of TOPSIS

• Step 3: Determine the ideal and negative ideal solutions.

• Ideal solution.A* = { v1

* , …, vn*}, where

vj* ={ max (vij) if j J ; min (vij) if j J' }

i i

• Negative ideal solution.A' = { v1' , …, vn' }, wherev' = { min (vij) if j J ; max (vij) if j J' }

i i

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Steps of TOPSIS

• Step 4: Calculate the separation measures for each alternative.

• The separation from the ideal alternative is:

Si * = [ (vj

*– vij)2 ] ½ i = 1, …, m

j

• Similarly, the separation from the negative ideal alternative is:

S'i = [ (vj' – vij)2 ] ½ i = 1, …, m

j

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Steps of TOPSIS

• Step 5: Calculate the relative closeness to the ideal solution Ci

*

Ci* = S'i / (Si

* +S'i ) , 0 Ci* 1

Select the option with Ci* closest to 1.

• Cj* index value lies between 0 and 1

• The larger the index value means the better the performance of the alternatives

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Applying TOPSIS Method to Example

Weight 0.1 0.4 0.3 0.2

Style Reliability Fuel Eco.

Saturn

Ford

7 9 9 8

8 7 8 7

9 6 8 9

Civic

Mazda 6 7 8 6

Cost

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Applying TOPSIS to Example

• m = 4 alternatives (car models)

• n = 4 attributes/criteria

• xij = score of option i with respect to criterion j

X = {xij} 44 score matrix.

• J = set of benefit attributes: style, reliability, fuel economy (more is better)

• J' = set of negative attributes: cost (less is better)

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Steps of TOPSIS

• Step 1(a): calculate (x2ij )

1/2 for each column

Style Rel. Fuel

Saturn

Ford

49 81 81 64

64 49 64 49

81 36 64 81

Civic

Mazda

Cost

xij2

i

(x2)1/2

36 49 64 36

230 215 273 230

15.17 14.66 16.52 15.17

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Steps of TOPSIS

• Step 1 (b): divide each column by (x2ij )

1/2

to get rij

Style Rel. Fuel

Saturn

Ford

0.46 0.61 0.54 0.53

0.53 0.48 0.48 0.46

0.59 0.41 0.48 0.59

Civic

Mazda 0.40 0.48 0.48 0.40

Cost

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Steps of TOPSIS

• Step 2 (b): multiply each column by wj to get vij.

Style Rel. Fuel

Saturn

Ford

0.046 0.244 0.162 0.106

0.053 0.192 0.144 0.092

0.059 0.164 0.144 0.118

Civic

Mazda 0.040 0.192 0.144 0.080

Cost

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Steps of TOPSIS

• Step 3 (a): determine ideal solution A*.

A* = {0.059, 0.244, 0.162, 0.080}

Style Rel. Fuel

Saturn

Ford

0.046 0.244 0.162 0.106

0.053 0.192 0.144 0.092

0.059 0.164 0.144 0.118

Civic

Mazda 0.040 0.192 0.144 0.080

Cost

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Steps of TOPSIS

• Step 3 (a): find negative ideal solution A'.

A' = {0.040, 0.164, 0.144, 0.118}

Style Rel. Fuel

Saturn

Ford

0.046 0.244 0.162 0.106

0.053 0.192 0.144 0.092

0.059 0.164 0.144 0.118

Civic

Mazda0.040 0.192 0.144 0.080

Cost

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Steps of TOPSIS

• Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} Si

* = [ (vj*– vij)

2 ] ½for each rowj

Style Rel. Fuel

Saturn

Ford

(.046-.059)2 (.244-.244)2 (0)2 (.026)2Civic

Mazda

Cost

(.053-.059)2 (.192-.244)2 (-.018)2 (.012)2

(.053-.059)2 (.164-.244)2 (-.018)2 (.038)2

(.053-.059)2 (.192-.244)2 (-.018)2 (.0)2

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Steps of TOPSIS

• Step 4 (a): determine separation from ideal solution Si

*

(vj*–vij)

2 Si* = [ (vj

*– vij)2 ] ½

Saturn

Ford

0.000845 0.029

0.003208 0.057

0.008186 0.090

Civic

Mazda 0.003389 0.058

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Steps of TOPSIS

• Step 4 (b): find separation from negative ideal

solution A' = {0.040, 0.164, 0.144, 0.118}

Si' = [ (vj'– vij)2 ] ½ for each row

j

Style Rel. Fuel

Saturn

Ford

(.046-.040)2 (.244-.164)2 (.018)2 (-.012)2Civic

Mazda

Cost

(.053-.040)2 (.192-.164)2 (0)2 (-.026)2

(.053-.040)2 (.164-.164)2 (0)2 (0)2

(.053-.040)2 (.192-.164)2 (0)2 (-.038)2

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Steps of TOPSIS

• Step 4 (b): determine separation from negative ideal solution Si'

(vj'–vij)2 Si' = [ (vj'– vij)

2 ] ½

Saturn

Ford

0.006904 0.083

0.001629 0.040

0.000361 0.019

Civic

Mazda 0.002228 0.047

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Steps of TOPSIS

• Step 5: Calculate the relative closeness to the ideal solution Ci

* = S'i / (Si* +S'i )

S'i /(Si*+S'i) Ci

*

Saturn

Ford

0.083/0.112 0.74 BEST

0.040/0.097 0.41

0.019/0.109 0.17

Civic

Mazda 0.047/0.105 0.45

Selection of 3PL provider

• XYZ company is the industry leader in the manufacture of automobile safety products

• Produces airbags, seatbelts, steering wheels, and safety electronics

• Company’s managers believe that establishing a flexible and scalable logistics outsourcing network with 3PL providers will be a key element in achieving lower costs, responsiveness to market and enhanced flexibility

• To find best among A1, A2, A3, A4, and A5 3PL providers

Methodology

• A two-phase AHP and TOPSIS methodology is developed to realize the evaluation.

• The weights that are gained from AHP calculations are considered and used in TOPSIS calculations.

• Then TOPSIS is operated for the evaluation problem and the final ranking of the preference order for the potential five 3PL providers are gained

Relative weights of five alternativeswith respect to each weights of sub-criterion (from AHP)

The Normalized Decision Matrix

2

1

2

21

2

11

11

11

........m

xxx

xr

The Normalized Decision Matrix

0.048 0.045 0.124 0.062 0.055 0.105 0.056 0.154 0.131 0.092 0.046 0.083Weights

The Weighted Normalized Decision Matrix

The ideal solution A* and negative ideal solution A-

Results of TOPSIS