Introducing a nonlinear programming model and using genetic algorithm to rank the alternatives in analytic hierarchy process

Document Type: Research Paper


Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.


As ranking is one of the most important issues in data envelopment analysis (DEA), many researchers have comprehensive studies on the subject and presented different approaches. In some papers, DEA and Analytic hierarchy process (AHP) are integrated to rank the alternatives. AHP utilizes pairwise comparisons between criteria and units, assessed subjectively by the decision maker, to rank the units. In this paper, a nonlinear programming (NLP) model is introduced to derive the true weights for pairwise comparison matrices in AHP. Genetic algorithm (GA) is used in order to solve this model. We use MATLAB software to solve proposed model for ranking the alternatives in AHP. A numerical example is applied to illustrate the proposed model.


[1] Adler, N., Friedman, L., Stern, Z. S., Review of ranking methods in data envelopment analysis context. European Journal of Operational Research. 140, 249–265.

[2] Charnes. A., Cooper. W. W., Rhodes. E., 1978. Measuring the efficiency of decision making units, European Journal of Operational Research 2 (6), (2002) , 428–444.

[3] Crawford, G.B., The geometric mean procedure for estimating the scale of a judgment matrix, Applied Mathematical Modelling 9, ( 1987) , 327–334.

[4] Chu, A.T.W., Kalaba, R.E., Spingarn, K., 1979. A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of Optimization Theory and Applications 27 , 4, (2002) , 531–538.

[5] Cogger, K.O., and Yu, P.L., Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios, Journal of Optimization Theory and Applications 46, (1985)483–491.

[6] Davis, L., Genetic Algorithms and simulated Annealing, Morgan Kaufmann Publishers, San Mateo, CA. (1987)

[7] Golden, B., Wasil, E., Harker, P. The analytic hierarchy process: applications and studies. Berlin, Springer.( 1989).

[8] Hwang, C.L., and Yoon, K Multiple Attribute Decision Making: Methods and Applications, Springer-Verlag, Berlin. .(1981).

[9] Holland, J.H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor(1975). [10]Islei, G., and Lockett, A.G., Judgmental modeling based on geometric least square, European Journal of Operational Research 36, (1988) 27–35.

[11]Keane, A., Genetic Algorithms Digest, 1994. Thursday, May 19, Volume 8, Issue 16.

[12]Mikhailov, L., A fuzzy programming method for deriving priorities in the analytic hierarchy process, Journal of the Operational Research Society 51, (2000) 341–349.

[13]Ramanathan, R., Data envelopment analysis for weight derivation and aggregation in the analytic hierarchy process, Computers and Operations Research 33, (2006)1289–1307.

[14]Saaty, T.L., The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, McGraw-Hill, Proposed York(1980). 

[15]Saaty, T.L., Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, RWS Publications, Pittsburgh.

[16]Wasil, E., and Golden, B., 2003. Celebrating 25 years of AHP-based decision making. Computers and Operations Research; 30,( 2000.)1419–20.

[17]Yao, X and Darwen P., An experimental study of N-person Prisoner's Dilemma Games, Proceedings of the Workshop on Evolutionary Computation, University of Proposed England, November 21-22, (1994) 94-113.