Tools › Operations Research › Decision Analysis › Multi-Criteria Decision (AHP, TOPSIS) › Example

Four-concept trade study: AHP consistency, weighted sum, TOPSIS, ELECTRE I, PROMETHEE II and the weight sensitivity

Tool	Multi-Criteria Decision (AHP, TOPSIS)
Source	Lippiatt, B. C., and Fuller, S. K. (2007). An Analytical Approach to Cost-Effective, Risk-Based Budgeting for Federal Information System Security. NISTIR 7385, National Institute of Standards and Technology, Gaithersburg, MD (Appendix A.3, the consistency ratio); Hwang, C.-L., and Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications. Springer-Verlag, Berlin (SAW, TOPSIS, ELECTRE as mathematics); Brans, J. P., and Vincke, P. (1985). "A Preference Ranking Organisation Method." Management Science, Vol. 31, No. 6, pp. 647–656.
Level	Professional

Problem

Four design concepts are compared on cost ($k, min), mass (kg, min), range (km, max), reliability (%, max) and lead time (weeks, min) with entered weights 0.30 / 0.15 / 0.25 / 0.20 / 0.10, and the criteria are also compared pairwise on the 1 to 9 scale. Find the AHP priorities and the consistency ratio of the comparison matrix, the weighted-sum, weighted-product, TOPSIS, AHP-synthesis, ELECTRE I and PROMETHEE II results, whether the methods agree, and the weight of cost at which the weighted-sum choice changes.

Inputs to Enter

Decision matrix: Concept A 420 / 38 / 150 / 92 / 26; Concept B 350 / 45 / 140 / 95 / 20; Concept C 510 / 32 / 180 / 90 / 34; Concept D 390 / 40 / 135 / 97 / 30; weights from the entered line
Criteria comparisons (upper triangle): 2 1 2 3 / 1/2 1 2 / 1 3 / 2; eigenvector priorities, Saaty's random index
ELECTRE thresholds at the matrix means; PROMETHEE linear preference with q = 0.1 and p = 0.5 of each criterion's range; cost weight swept

Open the Multi-Criteria Decision (AHP, TOPSIS) tool (loaded with this example)

Answers

AHP: λ_max = 5.0555, CI = 0.01387, RI = 1.12, CR = 0.0124 (consistent); eigenvector priorities 0.3013 / 0.1595 / 0.2646 / 0.1856 / 0.0891 (geometric mean 0.3025 / 0.1602 / 0.2634 / 0.1840 / 0.0899); synthesis on the direct scores: B 0.2598, C 0.2488, A 0.2474, D 0.2440
Weighted sum S: B 0.8970, A 0.8513, C 0.8503, D 0.8434; weighted product P: B 0.8886, A 0.8497, D 0.8366, C 0.8344
TOPSIS: A⁺ = (0.1245, 0.0615, 0.1478, 0.1037, 0.0357), closeness B 0.6022, A 0.5052, D 0.4939, C 0.4159
ELECTRE I (c* = 0.500, d* = 0.846): B outranks A, C and D; kernel {B}; PROMETHEE II net flows B 0.1643, D 0.0599, A −0.0730, C −0.1512
Every method puts Concept B first; the weighted-sum leader changes from C to B at a cost weight of 0.1775 (current 0.30), TOPSIS at 0.2045

Notes

The tool's default inputs are this case; press Submit. The concepts are illustrative (stated on the page); the library is checked against an independent numpy implementation and against the public worked examples of NISTIR 7385, the Forest Service AHP paper, the INEL AHP 5.1 manual and the NASA Langley TOPSIS study by test_multi_criteria_decision.php (576 checks).