THE MULTIPLE CHOICE PROBLEM WITH INTERACTIONS BETWEEN CRITERIA

Gomes, Luiz Flavio Autran Monteiro; Machado, Maria Augusta Soares; Rangel, Luis Alberto Duncan

doi:10.1590/0101-7438.2015.035.03.0523

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Articles • Pesqui. Oper. 35 (3) • Sep-Dec 2015 • https://doi.org/10.1590/0101-7438.2015.035.03.0523 copy

THE MULTIPLE CHOICE PROBLEM WITH INTERACTIONS BETWEEN CRITERIA

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

An important problem in Multi-Criteria Decision Analysis arises when one must select at least two alternatives at the same time. This can be denoted as a multiple choice problem. In other words, instead of evaluating each of the alternatives separately, they must be combined into groups of n alternatives, where n = 2. When the multiple choice problem must be solved under multiple criteria, the result is a multi-criteria, multiple choice problem. In this paper, it is shown through examples how this problemcan be tackled on a bipolar scale. The Choquet integral is used in this paper to take care of interactions between criteria. A numerical application example is conducted using data from SEBRAE-RJ, a non-profit private organization that has the mission of promoting competitiveness, sustainable developmentand entrepreneurship in the state of Rio de Janeiro, Brazil. The paper closes with suggestions for future research.

Keywords:
Multi-Criteria Decision Analysis; bipolar Choquet integral; multiple choice problem; formation of portfolios; fuzzy logic

Criterion	Alternative
Criterion	Student A	Student B	Student C	Student D
Subject 1	4	3	1	2
Subject 2	6	5	6	5
Subject 3	3	5	3	2

Criterion	Alternatives
Criterion	Student A	Student B	Student C	Student D
Subject 1	4μ({1}) = 4 · 0.35 = 1.4	3μ({1}) = 3 · 0.35 = 1.05	μ({1}) = 0.35	2μ({1}) = (2) · 0.35 = 0.7
Subject 2	2μ({1, 2}) = (6 − 4) · 0.34 = 0.68	2μ({1, 2}) = (5 − 3) · 0.34 = 0.68	5μ({1, 2}) = (6 − 1) · 0.34 = 1.7	3μ({1, 2}) = (5 − 2) · 0.34 = 1.02
Subject 3	3μ({2, 3}) = 3 · 0.33 = 0.99	(5 − 5)μ({2, 3}) = 0	(6 − 3)μ({2, 3}) = (3) · 0.33 = 0.99	7μ({2, 3}) = (5 − 2) · 0.33 = 0.99
Choquet integral	1.4 + 0.68+ 0.99 = 3.07	1.05+ 0.68 + 0 = 1.73	0.35+ 1.7 + 0.99 = 3.04	0.7 + 1.02+ 0.99 = 2.71
Ordering	1	4	2	3

Criterion	Alternative
Criterion	Apartment # 1	Apartment # 2	Apartment # 3	Apartment # 4
Near downtown	5	5	2	2
Near subway station	4	5	5	4
Near subway station	3	4	4	3

Criterion	Alternative
Criterion	Apartment # 1	Apartment # 2	Apartment # 3	Apartment # 4
Near downtown	5μ({1}) = 5 · 0.39 = 1.95	5μ({1}) = 5 · 0.39 = 1.95	3μ({1}) = 3 · 0.39 = 0.78	3μ({1}) = 3 · 0.39 = 0.78
Near subway station	μ({1, 2}) = (5 − 4) · 0.33 = 0.33	μ({1, 2}) = (5 − 5) · 0.33 = 0	μ({1, 2}) = (5 − 2) · 0.33 = 0.99	μ({1, 2}) = (4 − 2) · 0.33 = 0.66
Low price	(4 − 3)μ({2, 3}) = 1 · 0.31 = 0.31	(5 − 4)μ({2, 3}) = 1 · 0.31 = 0.31	(5 − 4)μ({2, 3}) = 1 · 0.31 = 0.31	(4 − 3)μ({2, 3}) = 1 · 0.31 = 0.31
Min operator	1.95 × 0.33× 0.31 = 0.2	1.95× 0 × 0.31 = 0	0.78× 0.99× 0.31 = 0.24	0.78× 0.66 × 0.31 = 0.16
Max operator	1.95 + 0.33+ 0.31 = 2.59	1.95+ 0 + 0.31 = 2.26	0.78+ 0.99+ 0.31 = 2.08	0.78+ 0.66 + 0.31 = 1.75
Choquet integral	2.59	2.26	2.08	1.75
Rank Ordering	1	2	3	4

Cr	Projects
Cr	P₁	P₂	P₃	P₄	P₅	P₆	P₇	P₈	P₉	P₁₀	P₁₁
C₁	775.79	393.06	343.93	1572.24	327.55	884.38	2354.2	884.39	929.54	982.67	3684.94
C₂	0.08	0.06	0.63	0.10	0.10	0.19	0.09	0.09	0.12	0.31	0.06
C₃	4	6	4	6	6	8	10	4	10	6	4
C₄	10	10	10	8	10	6	10	10	10	10	10
C₅	10	8	10	10	10	10	10	8	10	10	10
C₆	8	10	10	8	10	6	10	10	10	8	10
C₇	10	8	10	10	8	10	10	8	10	8	10
C₈	10	10	10	10	10	10	10	10	10	10	10
C₉	10	10	10	10	10	10	10	10	10	10	10

Project	Membership values
P₁	0.93
P₂	0.89
P₃	0.89
P₄	0.98
P₅	0.89
P₆	0.93
P₇	0.01
P₈	0.94
P₉	0.94
P₁₀	0.95
P₁₁	0.97

Project	m₁	m₂	m₃	m₄	m₅	m₆	m₇
P₁	0.93	0.85	0.60	0.90	0.90	0.55	0.91
P₂	0.89	0.79	0.99	0.90	0.10	0.66	0.45
P₃	0.89	0.02	0.60	0.90	0.90	0.66	0.77
P₄	0.98	0.91	0.99	0.50	0.90	0.55	0.77
P₅	0.89	0.91	0.99	0.90	0.90	0.66	0.45
P₆	0.93	0.99	0.70	0.02	0.90	0.01	0.77
P₇	0.91	0.88	0.21	0.90	0.90	0.66	0.77
P₈	0.94	0.91	0.60	0.90	0.10	0.66	0.45
P₉	0.94	0.95	0.21	0.90	0.90	0.66	0.77
P₁₀	0.95	0.68	0.99	0.90	0.90	0.55	0.45
P₁₁	0.97	0.79	0.60	0.90	0.90	0.66	0.77

Criterion	Projects
Criterion	P₁	P₂	P₃	P₄	P₅	P₆	P₇	P₈	P₉	P₁₀	P₁₁
C₁	0.93	0.89	0.89	0.98	0.89	0.93	0.91	0.94	0.94	0.95	0.97
C₂	0.85	0.79	0.02	0.91	0.91	0.99	0.88	0.91	0.95	0.68	0.79
C₃	0.60	0.99	0.60	0.99	0.99	0.70	0.21	0.60	0.21	0.99	0.60
C₄	0.90	0.90	0.90	0.50	0.90	0.00	0.90	0.90	0.90	0.90	0.90
C₅	0.90	0.10	0.90	0.90	0.90	0.90	0.90	0.10	0.90	0.90	0.90
C₆	0.55	0.66	0.66	0.55	0.66	0.01	0.66	0.66	0.66	0.55	0.66
C₇	0.77	0.45	0.77	0.77	0.45	0.77	0.77	0.45	0.77	0.45	0.77

Criterion	Project # 1	Project # 2	Project # 1	Project # 3
C₁	0.93μ({1}) = 0.93 · 0.56 = 0.52	0.89μ({1}) = 0.89 · 0.56 = 0.50	0.93μ({1}) = 0.93 · 0.56 = 0.52	0.89μ({1}) = 0.89 · 0.56 = 0.50
C₂	(0.93 − 0.85)μ({1, 2}) = 0.14	(0.89 − 0.79)μ({1, 2}) = 0.19	(0.93 − 0.85)μ({1, 2}) = 0.14	(0.89− 0.02)μ({1, 2}) = 1.61
C₃	(0.85 − 0.6)μ({2, 3}) = 0.13	(1 − 0.79)μ({2, 3}) = 0.11	(0.85 − 0.6)μ({2, 3}) = 0.13	(0.6 − 0.02)μ({2, 3}) = 0.31
C₄	(0.9 − 0.6)μ({3, 4}) = 0.15	(1 − 0.9)μ({3, 4}) = 0.04	(0.9 − 0.6)μ({3, 4}) = 0.15	(0.9 − 0.6)μ({3, 4}) = 0.15
C₅	(0.9 − 0.9)μ({4, 5}) = 0	(0.9 − 0.1)μ({4, 5}) = 0.38	(0.9 − 0.9)μ({4, 5}) = 0	(0.9 − 0.9)μ({4, 5}) = 0
C₆	(0.9 − 0.55)μ({5, 6}) = 0.15	(0.66 − 0.1)μ({5, 6}) = 0.25	(0.9 − 0.55)μ({5, 6}) = 0.15	(0.9 − 0.66)μ({5, 6}) = 0.11
C₇	(0.77 − 0.55)μ({6, 7}) = 0.09	(0.66 − 0.45)μ({6, 7}) = 0.09	(0.77 − 0.55)μ({6, 7}) = 0.09	(0.77− 0.66)μ({6, 7}) = 0.05
Min	Min(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 0.19, 0.11,0.04, 0.38, 0.25,0.09, 0.24, 0.24) = 0		Min(0.52,0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 1.61, 0.31,0.15, 0, 0.11, 0.05, 0.10, 0.10) = 0
Choquet integral (Max)	Max(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 0.19, 0.11,0.04, 0.38, 0.25,0.09, 0.24, 0.24) = 0.52		Max(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 1.61, 0.31,0.15, 0, 0.11, 0.05, 0.10, 0.10) = 1.61

A two combination choice	C₁	C₂	C₃	Observations
P₃ − P₇	343.93 and 2354.2 mean = 1349.07	0.63 and 0.092 mean = 0.36	4 and 10 mean = 7	low cost; high generated revenue; high synergy
P₃ − P₅	343.93 and 327.55 mean = 335.74	0.63 and 0.10 mean = 0.36	4 and 6 mean = 5	low cost; high generated revenue; medium synergy
P₃ − P₆	775.79 and 343.92 mean = 559.86	0.08 and 0.63 mean = 0.36	4 and 8 mean = 6	low cost; high generated revenue; medium synergy
P₃ − P₉	343.93 and 929.54 mean = 636.74	0.63 and 0.122 mean = 0.375	4 and 10 mean = 7	low cost; high generated revenue; medium synergy
P₃ − P₈	343.93 and 884.39 mean = 1349.07	0.63 and 0.096 mean = 0.362	4 and 4 mean = 4	low cost; high generated revenue; low synergy
P₂ − P₃	393.06 and 343.93 mean = 368.5	0.061 and 0.09 mean = 0.345	6 and 4 mean = 5	low cost; high generated revenue; low synergy
P₃ − P₄	343.93 and 1572.24 mean = 958.09	0.63 and 0.1 mean = 0.36	4 and 6 mean = 5	low cost; high generated revenue; low synergy
P₁ − P₃	343.93 and 343.93 mean = 614.16	0.63 and 0.192 mean = 0.41	4 and 4 mean = 4	low cost; high generated revenue; low synergy
P₃ − P₁₀	343.93 and 982.67 mean = 663.3	0.63 and 0.305 mean = 0.47	4 and 6 mean = 5	low cost; high generated revenue; low synergy
P₃ − P₁₁	343.93 and 3684.94 mean = 2014.44	0.63 and 0.064 mean = 0.346	4 and 4 mean = 4	low cost; high generated revenue; low synergy

A two-combination choice	C₄	C₅	C₆	Observations
P₃ - P₅	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P₃ - P₇	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P₃ - P₉	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P₃ - P₁₁	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P₁ - P₃	10 and 10 mean = 10	10 and 10 mean = 10	8 and 10 mean = 9	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; medium capacity to generate employment and income

A two-combination choice	C₇	Observations
P₁ - P₃	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P₃ - P₄	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P₃ - P₆	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P₃ - P₇	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P₃ - P₉	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P₃ - P₁₁	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income

Two-combination Portfolios	Results obtained using the bipolar Choquet Integral
P₃ - P₇	1
P₃ - P₉	2
P₃ - P₆	3
P₃ - P₉	4
P₃ - P₁₁	5

Three-combination Portfolios	Results obtained using the bipolar Choquet Integral
P₁ - P₂ - P₃	1
P₁ - P₃ - P₆	2
P₁ - P₃ - P₈	3
P₁ - P₃ - P₉	4
P₁ - P₃ - P₁₀	5
P₁ - P₃ - P₁₁	6
P₂ - P₃ - P₄	7
P₂ - P₃ - P₇	8

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