International Journal of Nonlinear Analysis and Applications

Simultaneous use of two concepts of equitable efficiency and efficiency in solving multi-objective optimization problems

Document Type : Research Paper

Authors

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Abstract

The two requirements of impartiality and equitability expressed with the principle of transfers are fulfilled by all objective functions in equitable multi-objective optimization. However, in some practical situations, the decision-maker believes these requirements should only be satisfied by a subset of objective functions. To solve the problem in this paper, we first divide the set of objective functions into two subsets, the subset given by the decision maker and its complement. Then, we apply the concepts of equitable efficiency and efficiency for these two subsets, respectively. Furthermore, we apply the mean and inequality measures for these subsets of objective functions and present the new mean-equity models for solving the location problem. We investigate the relationship between $2$-efficient solutions of the new mean-equity models and efficient solutions of the location problem.

Keywords

[1] A.B. Atkinson, On the measurement of inequality, J. Econ. Theory 2 (1970), 244–263.
[2] M. Ehrgott, Multicriteria Optimization, Springer Verlag, Berlin, 2005.
[3] E. Erkut, Inequality measures for location problems, Location Sci. 1 (1993), 199–217.
[4] M.M. Kostreva and W. Ogryczak, Linear optimization with multiple equitable criteria, RAIRO Oper. Res. 33 (1999), no. 3, 275–297.
[5] M. M. Kostreva, W. Ogryczak, and A. Wierzbicki, equitable aggregations in multiple criteria analysis, Eur. J. Oper. Res. 158 (2004), no. 2, 362–377.
[6] M.O. Lorenz, Methods of measuring the concentration of wealth, Publ. Amer. Stat. Assoc. 70 (1905), 209–219.
[7] M.C. Lopez-de-los-Mozos and J.A. Mesa, The maximum absolute deviation measure in location problems on networks, Eur. J. Oper. Res. 135 (2001), 184–194.
[8] M.C. Lopez-de-los-Mozos and J.A. Mesa, The sum of absolute differences on a network: algorithm and comparison with other equality measures, INFOR. 41 (2003), 195–210.
[9] O. Maimon, The variance equity measure in locational decision theory, Ann. Oper. Res. 6 (1986), no. 5, 147–160.
[10] M.B. Mandell, Modelling effectiveness-equity trade-offs in public service delivery systems, Manag. Sci. 37 (1991), 467–482.
[11] A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
[12] M.T. Marsh, and D.A. Schilling, Equity measurement in facility location analysis: A review and framework, Eur. J. Oper. Res. 74 (1994), 1–17.
[13] G. F. Mulligan, Equity measures and facility location, Papers in Regional Sci. 70 (1991), no. 4, 345–365. 
[14] W. Ogryczak, Inequality measures and equitable approaches to location problems, Eur. J. Oper. Res. 122 (2000), no. 2, 374–391.
[15] W. Ogryczak, Inequality measures and equitable locations, Ann. Oper. Res. 167 (2009), 61–86.
[16] W. Ogryczak and M. Zawadzki, Conditional median: a parametric solution concept for location problems, Ann. Oper. Res. 110 (2002), 167–181.
[17] A. Sen, On Economic Inequality, Clarendon Press, Oxford, 1973.
[18] H.P. Young, Equity in Theory and Practice, Princeton University Press, Princeton, 1994.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 3
March 2025
Pages 63-76
  • Receive Date: 21 August 2023
  • Accept Date: 01 February 2024