Mixed-integer linear-fractional programming model and it’s linear analogue for reducing inconsistency of pairwise comparison matrices
Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency bel...
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| Published in: | Information sciences Vol. 592; pp. 192 - 205 |
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| Abstract | Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency below an acceptable threshold. In order to support the decision maker (DM) in making a rational decision, we must keep in mind the following: Our suggestion should be as close to the DM’s original result as possible, moreover it should have as low inconsistency as possible. We have studied various linear programming (LP) models that are used for reducing the inconsistency of pairwise comparison matrices (PCM) (Bozóki et al., 2011, 2015). These models, however aim at fulfilling only one of the previously mentioned two goals at a time. Therefore, the optimal solutions given may differ from each in the other respect, which is not taken as an objective but as a constraint in the model. So, they cannot be considered as equally good optimal solutions from a wider perspective. Based on our experiences concerning these models, we have developed a mixed-integer linear-fractional programming (MILFP) model that takes both mentioned goals as objectives by combining them into a linear-fractional objective function. We also provide the linear analogue (LA) of our MILFP model using an appropriate adaptation and combination of the Charnes-Cooper transformation and Glover’s linearization scheme. |
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| AbstractList | Pairwise comparison matrices are often used in multicriteria decision making (MCDM). The most critical part of this technique is the inconsistency, which emerges for logical reasons but can cause significant problems during the decision making process. Hence it is necessary to keep inconsistency below an acceptable threshold. In order to support the decision maker (DM) in making a rational decision, we must keep in mind the following: Our suggestion should be as close to the DM’s original result as possible, moreover it should have as low inconsistency as possible. We have studied various linear programming (LP) models that are used for reducing the inconsistency of pairwise comparison matrices (PCM) (Bozóki et al., 2011, 2015). These models, however aim at fulfilling only one of the previously mentioned two goals at a time. Therefore, the optimal solutions given may differ from each in the other respect, which is not taken as an objective but as a constraint in the model. So, they cannot be considered as equally good optimal solutions from a wider perspective. Based on our experiences concerning these models, we have developed a mixed-integer linear-fractional programming (MILFP) model that takes both mentioned goals as objectives by combining them into a linear-fractional objective function. We also provide the linear analogue (LA) of our MILFP model using an appropriate adaptation and combination of the Charnes-Cooper transformation and Glover’s linearization scheme. |
| Author | Rácz, Anett |
| Author_xml | – sequence: 1 givenname: Anett surname: Rácz fullname: Rácz, Anett email: anett.racz@inf.unideb.hu organization: Department of Applied Mathematics and Probability Theory, Faculty of Informatics, University of Debrecen, 4028 Debrecen, Hungary |
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| Cites_doi | 10.1007/s10100-010-0136-9 10.1002/mcda.428 10.1002/aic.14185 10.1007/BF00056121 10.1007/978-94-015-9799-9_2 10.1016/0895-7177(93)90059-8 10.1016/j.mcm.2011.03.026 10.1016/0270-0255(82)90013-6 10.1016/0270-0255(87)90473-8 10.1007/s10898-007-9236-z 10.1016/j.ins.2017.08.082 10.1037/h0046162 10.1016/j.ins.2019.08.042 10.1037/h0070288 10.2307/1907264 10.1016/0377-2217(90)90057-I 10.1287/mnsc.22.4.455 10.1002/nav.3800090303 10.1007/s10100-014-0346-7 |
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| Keywords | Decision support systems Pairwise comparison Linear-Fractional programming Consistency |
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| References_xml | – volume: 48 start-page: 9 year: 1990 end-page: 26 ident: b0105 article-title: How to make a decision: the analytic hierarchy process publication-title: European journal of operational research – volume: 4 start-page: 113 year: 1991 end-page: 134 ident: b0045 article-title: Nontransitive preferences in decision theory publication-title: Journal of Risk and Uncertainty – volume: 64 start-page: 153 year: 1957 ident: b0115 article-title: On the psychophysical law publication-title: Psychological review – volume: 18 start-page: 79 year: 1993 end-page: 84 ident: b0065 article-title: A new definition of consistency of pairwise comparisons publication-title: Mathematical and computer modelling – volume: 22 start-page: 455 year: 1975 end-page: 460 ident: b0050 article-title: Improved linear integer programming formulations of nonlinear integer problems publication-title: Management Science – volume: 9 start-page: 161 year: 1987 end-page: 176 ident: b0100 article-title: The analytic hierarchy process–what it is and how it is used publication-title: Mathematical modelling – start-page: 111 year: 1950 end-page: 141 ident: b0075 article-title: Rational behavior, uncertain prospects, and measurable utility publication-title: Econometrica: Journal of the Econometric Society – reference: Poesz, A. 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hierarchy process publication-title: Expert Systems with Applications – volume: 59 start-page: 4255 year: 2013 ident: 10.1016/j.ins.2022.01.077_b0130 article-title: Global optimization of large-scale mixed-integer linear fractional programming problems: a reformulation-linearization method and process scheduling applications publication-title: AIChE Journal doi: 10.1002/aic.14185 – ident: 10.1016/j.ins.2022.01.077_b0090 – volume: 4 start-page: 113 year: 1991 ident: 10.1016/j.ins.2022.01.077_b0045 article-title: Nontransitive preferences in decision theory publication-title: Journal of Risk and Uncertainty doi: 10.1007/BF00056121 – ident: 10.1016/j.ins.2022.01.077_b0110 doi: 10.1007/978-94-015-9799-9_2 – volume: 18 start-page: 79 year: 1993 ident: 10.1016/j.ins.2022.01.077_b0065 article-title: A new definition of consistency of pairwise comparisons publication-title: Mathematical and computer modelling doi: 10.1016/0895-7177(93)90059-8 – volume: 54 start-page: 789 year: 2011 ident: 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| SubjectTerms | Consistency Decision support systems Linear-Fractional programming Pairwise comparison |
| Title | Mixed-integer linear-fractional programming model and it’s linear analogue for reducing inconsistency of pairwise comparison matrices |
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