Comparing Rankings from using TODIM and a Fuzzy Expert System Valério A. P. Salomon Luís A. D. Rangel Sao Paulo State University (UNESP)Fluminense Federal.

Comparing Rankings from using TODIM and a Fuzzy Expert System

Valério A. P. Salomon Luís A. D. RangelSao Paulo State University (UNESP) Fluminense Federal University (UFF)[email protected]

Comparing ranks from TODIM and Fuzzy Expert System 2Salomon & Rangel (2015)

Outline

1. Introduction 2. Theory background

Correlation between ranks3. Illustrative case

Real Estate in Rio State4. Discussion and conclusionsAcknowledgmentsReferences

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1. Introduction

Multi-Criteria Decision Analysis (MCDA) methods [1]AHP, ANP, ELECTRE, MACBETH, MAUT, TOPSIS

Decision problemsContinuous (large number of alternative solutions, even, infinite) Discrete (small number of alternatives, perhaps, two)

Choice, Sort, Ranking and Description [2]

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1. Introduction

Different MCDA methods may yield different results[11]: rank correlation [12]TODIM is an MCDA method developed to Ranking problems [13]Fuzzy Sets Theory (FST) was proposed to Classification problems [20]The use of FST in MCDA is slightly controversial [27]: FST may result in loss of information [28]Our aim is to prove that TODIM can provide a better solution than FST for Ranking problems

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2. Theory background

2.1. Correlation between ranksRank correlation coefficient [12]

2.2. TODIM methodProspect Theory [14]

2.3. Fuzzy expert systemsIf-Then rules [36], Mamdani model [39]

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2. Theory background (Edmond-Mason coefficient)

τ 𝑥=∑𝑖=1

𝑛

∑𝑗=1

𝑛

𝑎𝑖𝑗𝑏𝑖𝑗

𝑛 (𝑛−1 )

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2. Theory background (examples)

𝐴= (1 ,2 ,3 ,4 ) ,𝐵=(1 ,3 ,2 ,4 ) ,𝐶=(4 ,3 ,2 ,1)

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2. Theory background (TODIM’s value function)

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2. Theory background (TODIM elements)Matrix of evaluation: composed by the numerical evaluation for the alternatives regarding to all the criteriaThe matrix must be normalized, for each criterion

Matrix of normalized alternatives: P = [pnm]Number of criteria: m Number of alternatives: nReference criterion, r, usually the highest weighted really

Vector of weights: w = [wrc] = wc/wr

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2. Theory background (TODIM results)Dominance (Equation 3)d(Ai, Aj) = SF(Ai, Aj)

Overal value (Equation 3) xi = (Sd(Ai, Aj) - min Sd(Ai, Aj)) / (max Sd(Ai, Aj) - min Sd(Ai, Aj))

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2. Theory background (Fuzzy set)

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2. Theory background (Fuzzy expert system)

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3. Illustrative case (data)

Volta Redonda is a city in the South of the State of Rio de Janeiro, Brazil. It has approximately 260,000 inhabitants. There are a large number of properties, residential and commercial, rented or available for rent. The major steel plant installed in the city in the 1940’s is a landmark of Brazilian industrialization.

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3. Illustrative case (data)Criterion Weight Normalized weight

Localization (C1) 5 0.25

Construction area (C2) 3 0.15

Construction quality (C3) 2 0.10

State of conservation (C4) 4 0.20

Garage spaces (C5) 1 0.05

Rooms (C6) 2 0.10

Attractions (C7) 1 0.05

Security (C8) 2 0.10

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3. Illustrative case (matrix of evaluation)Residential property C1 C2 C3 C4 C5 C6 C7 C8

A1 3 290 3 3 1 6 4 0A2 4 180 2 2 1 4 2 0A3 3 347 1 2 2 5 1 0A4 3 124 2 3 2 5 4 0A5 5 360 3 4 4 9 1 1A6 2 89 2 3 1 5 1 0A7 1 85 1 1 1 4 0 1A8 5 80 2 3 1 6 0 1A9 2 121 2 3 0 6 0 0

A10 2 120 1 3 1 5 1 0A11 4 280 2 2 2 7 3 1A12 1 90 1 1 1 5 2 0A13 2 160 3 3 2 6 1 1A14 3 320 3 3 2 8 2 1A15 4 180 2 4 1 6 1 1

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3. Illustrative case (normalized matrix of evaluation)Residential property C1 C2 C3 C4 C5 C6 C7 C8

A1 0.068 0.103 0.100 0.075 0.045 0.069 0.174 0A2 0.091 0.064 0.067 0.050 0.045 0.046 0.087 0A3 0.068 0.123 0.033 0.050 0.091 0.057 0.043 0A4 0.068 0.044 0.067 0.075 0.091 0.057 0.174 0A5 0.114 0.127 0.100 0.100 0.182 0.103 0.043 0.143A6 0.045 0.031 0.067 0.075 0.045 0.057 0.043 0A7 0.023 0.030 0.033 0.025 0.045 0.046 0 0.143A8 0.114 0.028 0.067 0.075 0.045 0.069 0 0.143A9 0.045 0.043 0.067 0.075 0 0.069 0 0

A10 0.045 0.042 0.033 0.075 0.045 0.057 0.043 0A11 0.091 0.099 0.067 0.050 0.091 0.080 0.130 0.143A12 0.023 0.032 0.033 0.025 0.045 0.057 0.087 0A13 0.045 0.057 0.100 0.075 0.091 0.069 0.043 0.143A14 0.068 0.113 0.100 0.075 0.091 0.092 0.087 0.143A15 0.091 0.064 0.067 0.100 0.045 0.069 0.043 0.143

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3. Illustrative case (overall values without TODIM)Residential property Overall value Rank

A1 0.301 6A2 0.241 10A3 0.245 9A4 0.257 8A5 0.454 1A6 0.192 11A7 0.159 14A8 0.311 5A9 0.185 12

A10 0.185 12A11 0.351 3A12 0.125 15A13 0.291 7A14 0.366 2A15 0.338 4

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3. Illustrative case (TODIM application)• q = 1• For C1, p11 < p12, then -0.303

• For C2, p12 > p22, then 0.076

• ...• In Equation 3, d (A1, A2) 0.017• In Equation 4, 0.644

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3. Illustrative case (overall values with TODIM)Residential properties Without TODIM With TODIM

A1 0.301 6 0.692 5A2 0.241 10 0.386 10A3 0.245 9 0.399 9A4 0.257 8 0.620 7A5 0.454 1 1 1A6 0.192 11 0.286 11A7 0.159 14 0 15A8 0.311 5 0.441 8A9 0.185 12 0.020 14

A10 0.185 12 0.213 12A11 0.351 3 0.858 3A12 0.125 15 0.107 13A13 0.291 7 0.719 4A14 0.366 2 0.937 2A15 0.338 4 0.673 6

𝜏𝑥≈0.73

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3. Illustrative case (Fuzzy Expert System application)Fuzzy sets for Location (C1), Construction Quality (C3),

State of Conservation (C4), Attractions (C7)

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3. Illustrative case (Fuzzy Expert System application)Fuzzy set for Construction area (C2)(Similar to C5, C6 and C8)

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3. Illustrative case (Fuzzy Expert System application)Fuzzy rules

RuleInput Output

Location Constr. Quality State of conservation Attractions Evaluation

1 Bad Bad Bad Bad Bad

2 Bad Bad Bad Average Bad

3 Bad Bad Bad Good Bad

... ... ... ... ... ...

79 Good Good Good Bad Bad

80 Good Good Good Average Good

81 Good Good Good Good Good

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3. Illustrative case (Fuzzy Expert System application)Residential property Overall value Rank

A1 0 6A2 0 6A3 0 6A4 0 6A5 0.259 3A6 0 6A7 0 6A8 0 6A9 0 6

A10 0 6A11 0.452 2A12 0 6A13 0.259 3A14 0.741 1A15 0.259 3

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4. Discussion and Conclusions• Main contribution of this work: application of Fuzzy Expert System and its

comparison with a TODIM application• TODIM applied only with spreadsheets• Fuzzy Expert System required specific software (fuzzyTECH.com)• Sensitivity Analysis were conducted and did not affect the results• TODIM application considered different weights for the criteria; Fuzzy Expert

System considered the same weight (1/8 for all)• Future research: compare TODIM with other techniques

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AcknowledgmentsAuthors need to thank Prof. Dr. Luiz Flavio Autran Monteiro Gomes for valuable advises, comments, and suggestions

This research has financial support from • Brazilian Council for Scientific and Technological Development (Grant No. CNPQ

302692/2011-8) • Sao Paulo State Research Foundation (Grant No. FAPESP 2013/03525-7)