Inferring efficient weights from pairwise comparison matrices. (English) Zbl 1132.90007
Summary: Several multi-criteria-decision-making methodologies assume the existence of weights associated with the different criteria, reflecting their relative importance. One of the most popular ways to infer such weights is the analytic hierarchy process, which constructs first a matrix of pairwise comparisons, from which weights are derived following one out of many existing procedures, such as the eigenvector method or the least (logarithmic) squares. Since different procedures yield different results (weights) we pose the problem of describing the set of weights obtained by “sensible” methods: those which are efficient for the (vector-) optimization problem of simultaneous minimization of discrepancies. A characterization of the set of efficient solutions is given, which enables us to assert that the least-logarithmic-squares solution is always efficient, whereas the (widely used) eigenvector solution is not, in some cases, efficient, thus its use in practice may be questionable.
MSC:
| 90B50 | Management decision making, including multiple objectives |
| 90C20 | Quadratic programming |
| 90C32 | Fractional programming |
References:
| [1] | Bryson N (1995) A goal programming method for generating priority vectors. J Oper Res Soc 46:641–648 · Zbl 0830.90001 |
| [2] | Carrizosa E, Conde E, Fernández FR, Muñoz FR, Pareto J (1995) Pareto optimality in linear regression. J Math Anal Appl 190:129–141 · Zbl 0820.62062 · doi:10.1006/jmaa.1995.1067 |
| [3] | Chankong V, Haimes Y (1983) Multiobjective decision making. North-Holland, Amsterdam · Zbl 0622.90002 |
| [4] | Choo EU, Wedley WC (2004) A common framework for deriving preference values from pairwise comparison matrices. Comput Oper Res 31:893–908 · Zbl 1043.62063 · doi:10.1016/S0305-0548(03)00042-X |
| [5] | Connell A, Corless RM (1993) An experimental interval arithmetic package in Maple. Interval Comput 2:120–134 · Zbl 0829.65149 |
| [6] | Cook WD, Kress M (1988) Deriving weights from pairwise comparison ratio matrices: an axiomatic approach. Eur J Oper Res 37(3):355–362 · Zbl 0652.90002 · doi:10.1016/0377-2217(88)90198-1 |
| [7] | Hoffman AJ (1960) Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Proceedings of symposia in applied mathematics, Vol 10. Bellman R, Hall M, Jr (eds), American Mathematical Society, Providence, pp113–127 |
| [8] | Lootsma FA (1996) A model for the relative importance of the criteria in the multiplicative ahp and smart. Euro J Oper Res 94:467–476 · Zbl 0947.90598 · doi:10.1016/0377-2217(95)00129-8 |
| [9] | Martos B (1975) Nonlinear Programming. Theory and methods. North-Holland, Amsterdam · Zbl 0357.90027 |
| [10] | McCormick ST (1997) How to compute least infeasible flows. Math Program Ser B 78(2):179–194 · Zbl 0889.90065 |
| [11] | Monagan M, Geddes K, Heal K, Labahn G, Vorkoetter S (1997) Maple V programming guide for release 5. Springer, Berlin Heidelberg New York · Zbl 0877.68070 |
| [12] | Ramanathan R (1997) A note on the use of goal programming for the multiplicative ahp. J Multi-criteria Decis Anal 6:296–307 · Zbl 0889.90009 · doi:10.1002/(SICI)1099-1360(199709)6:5<296::AID-MCDA152>3.0.CO;2-G |
| [13] | Saaty TL (1977) A scaling method for priorities in hierarchical structures. J Math Psychol 15:234–281 · Zbl 0372.62084 · doi:10.1016/0022-2496(77)90033-5 |
| [14] | Saaty TL (1980) Multicriteria decision making: the Analytic hierarchy Process. McGraw-Hill, New York |
| [15] | Saaty TL (1990) Eigenvector and logaritmic least squares. Euro J Oper Rese, 48:156–160 · Zbl 0707.90003 · doi:10.1016/0377-2217(90)90073-K |
| [16] | Saaty TL (1994) Fundamentals of decision making. RSW Publications, Pittsburg |
| [17] | Saaty TL (1994) How to make a decision: The analytic hierarchy process. Interfaces 24:19–43 · doi:10.1287/inte.24.6.19 |
| [18] | Schaible S (1995) Fractional programming. Horst R, Pardalos PM, (eds), In: Handbook of global optimization, pp 495–608 |
| [19] | Steuer R (1986) Multiple criteria optimization: theory, computation, application. Wiley, New York · Zbl 0663.90085 |
| [20] | Zeleny M (1982) Multiple criteria decision making. McGraw-Hill, New York · Zbl 0588.90019 |
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