Fuzzy Optimization and Decision Making

, Volume 15, Issue 2, pp 177–193

# Probability distribution based weights for weighted arithmetic aggregation operators

Article

## Abstract

One key point in the multiple attribute decision making is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then, motivated by the idea of data distribution, we develop some novel methods for obtaining the weights associated with the weighted arithmetic aggregation operators. The methods can relieve the influence of biased data on the decision results by weighting these data with small values based on the corresponding probability of data. Furthermore, some commonly used probability distribution methods are used to solve the problems in different conditions. Finally, four practical examples are provided to illustrate the weighting method.

## Keywords

Multiple attribute decision making (MADM) Probability distribution Ordered weighted aggregation (OWA) operators Weighted arithmetic aggregation (WAA) operators

## Notes

### Acknowledgments

This research was funded by the National Natural Science Foundation of China (No. 61273209), and the Central University Basic Scientific Research Business Expenses Project (No. skgt201501).

## References

1. Anders, H. (1990). De Moivre and the doctrine of chances, 1718, 1738, and 1756, history of probability and statistics and their applications before 1750., Wiley Series in Probability and Statistics New York: Wiley Interscience.Google Scholar
2. Balakrishnan, N., & Basu, A. P. (1996). The exponential distribution: Theory, methods, and applications. New York: Gordon and Breach.
3. Doob, J. L. (1934). Probability and statistics. Transactions of the American Mathematical Society, 36, 759–775.
4. Emrouznejada, A., & Amin, R. G. (2010). Improving minimax disparity model to determine the OWA operator weights. Information Sciences, 180, 1477–1485.
5. Harsanyi, J. C. (1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy, 63, 309–321.
6. Hlawka, E. (1984). The theory of uniform distribution. Oxon: AB Academic.
7. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (2nd ed., Vol. 2). New York: Wiley.
8. Merigó, J. M. (2010). Fuzzy decision making using immediate probabilities. Computers & Industrial Engineering, 58, 651–657.
9. Merigó, J. M. (2012a). The probabilistic weighted average and its application in multi-person decision making. International Journal of Intelligent Systems, 27, 457–476.
10. Merigó, J. M. (2012b). Probabilities with OWA operators. Expert Systems with Applications, 39, 11456–11467.
11. Muzychuk, A. (2011). OWA weight updating in repeated decision making under the influence of additional information. International Journal of Intelligent Systems, 26, 591–602.
12. Sadiq, R., & Tesfamariam, S. (2007). Probability density functions based weights for ordered weighted averaging (OWA) operators: An example of water quality indices. European Journal of Operational Research, 182, 1350–1368.
13. Wang, Y., & Xu, Z. S. (2008). A new method of giving OWA weights. Mathematics in Practice and Theory, 138, 51–61.
14. Wei, W. X., & Feng, J. (1998). Study on multiobjective weights combination assigning method. Systems Engineering and Electronics, 20, 14–16.Google Scholar
15. Xu, Z. S. (2005). An overview of methods for determining OWA weights. International Journal of Intelligent Systems, 20, 843–865.
16. Xu, Z. S. (2011). Approaches to multi-stage multi-attribute group decision making. International Journal of Information Technology & Decision Making, 10, 121–146.
17. Xu, Z. S., & Cai, X. Q. (2010). Recent advances in intuitionistic fuzzy information aggregation. Fuzzy Optimization and Decision Making, 9, 359–381.
18. Xu, Z. S., & Da, Q. L. (2003). An overview of operators for aggregating information. International Journal of Intelligent Systems, 18, 953–969.
19. Yager, R. R. (1988). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 18, 183–190.
20. Yager, R. R., Engemann, K. J., & Filev, D. P. (1995). On the concept of immediate probabilities. International Journal of Intelligent Systems, 10, 373–397.
21. Zeleny, M. (1982). Multiple criteria decision making. New York: McGraw-Hill.