The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015
An Implementation of Genetic Algorithm to Generate the Most
Sartono. B., Rahardiantoro. S., “An Implementation of Genetic Algorithm to Generate the Most Compromised Decision when Information of the Alternatives is Incomplete”
ISBN : 978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology
The basic idea of the approach is that the preference rank vector should have high agreement to the criteria. Since all criteria are measured in at least an ordinal scale, we propose to use a rank correlation coefficient to represent the degree of agreement.
Suppose that Rk is the correlation between criterion k and the preference rank vector s, or Rk = corr(zk, s). For all k = 1, …, p, we would like to have high values of Rk’s. To ensure that each of criteria has high value of Rk, we propose to the restate the problem of finding preference rank vector s as follow: “Find s so that the minimum value of Rk’s is maximum” or “Find s which maximize min{Rk}”.
3. Proposed Approach
As described, the problem of choosing the best alternative can be seen as an optimization problem. We could obviously view that it is a combinatorial problem since the solution basically is the permutation of n different things labeled as 1, 2, …, n. Since it is a combinatorial optimization problem, we could employ a genetic algorithm methodology [4] to find an optimal solution.
The main components of the algorithm could be summarized as follow. First, a preference rank vector could be seen as a gene consisting n chromosomes with the value as one of the element of {1,
…, n} and be distinguished each other. Second, the fitness value of each gene or the objective function is the min{Rk}. Each of the genes has a single fitness value, and the gene with higher fitness value is the preferable one.
As previously mentioned, the Rk value is the rank correlation between the score of the k-th criterion of decision making and the preference rank. It is possible that one or more objects do not have a score of the criterion. Whenever it happened, the correlation was calculated by involving m (<n) pair observations whose available score.
The pseudocode of the proposed algorithm is as follow 1. set n = the number of alternatives
set p = the number of criteria
set g = the maximum number of iteration of the genetic algorithm set npop = number of genes in a population
2. initiate a population ofnpop preference rank vectors,S = {s | s = (s1, s2, …, sn)} by randomly permute {1, 2, … n}} and n(S) = npop
3. for t = 1 to g
a. select as many as nsel genes from S whose higher fitness value
b. do a cross-over procedure appropriate for combinatorial case problem and do mutation
c. collect the new genes resulted by cross-over and mutation to a new population S d. if the fitness value improves, then repeat a – c, otherwise set t = g
4. Illustrative Example
As an illustration, we implemented the proposed approach to the following case. Suppose a man would like to purchase a car. At that time, he had ten alternatives of cars. To be able to find the best, he considered nine attributes that describe the quality of the alternatives: X1 Attractive , X2 Quiet , X3 Reliable, X4 Well Built , X5 Comfortable , X6 Roomy , X7 High Prestige, X8 Unique, and X9 Worth Value. Table 1 provides the score of the attributes for each car where an alternative whose a higher score means more preferred. Suppose that there are several attribute scores are not available.
As a note, the data was taken from the discussion of Positioning the Nissan Infiniti G20 in[5].
We implemented the algorithm and a program written in SAS/IML language was developed to perform the genetic algorithm approach. Readers who are interested to have could have it upon request to the authors.
A result that we obtained from the algorithm yields the order of the preference of the alternatives as follow: Nissan G20, BMW 318i, SAAB 900, Honda Prelude, Toyota Supra, Audi 90, Ford T-Bird, Mercury Capri, Eagle Talon, andPontiac Firebird. This order has the fitness value of minimum correlation coefficient as large as 0.728. In other word, the preference order has high correlation to all of attributes by 0.728 or larger.
Table 1. The attribute score of the ten alternative cars
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Sartono. B., Rahardiantoro. S., “An Implementation of Genetic Algorithm to Generate the Most Compromised Decision when Information of the Alternatives is Incomplete”
Cars X1 X2 X3 X4 X5 X6 X7 X8 X9
Toyota Supra 5.6 4.2 7.0 6.9 5.3 3.5 5.3 6.1 5.5 SAAB 900 5.3 4.8 5.3 6.2 6.2 5.1 5.7 7.1 6.1 Pontiac Firebird 3.9 2.8 5.1 4.6 4.7 3.3 3.8 4.7 4.7 Nissan G20 5.6 6.3 6.1 7.4 6.6 5.6 5.4 5.5 5.6 Mercury Capri 3.9 3.3 5.0 4.7 4.6 3.6 . 5.1 5.2 Honda Prelude 5.2 5.4 5.8 6.2 5.7 3.9 4.7 5.1 6.4 Ford T-Bird 4.0 3.6 . 4.8 5.0 3.9 3.5 5.4 4.7 Eagle Talon 4.0 3.5 4.7 4.7 5.0 3.6 2.8 4.7 5.4 BMW 318i 5.7 5.0 6.7 7.2 5.5 4.3 6.4 6.2 5.7 Audi 90 4.6 5.2 5.3 6.4 . 5.3 5.6 5.6 4.7
To compare how good is the result of the proposed algorithm, we ran 2000 repetitions of random permutation. Each repetition evaluated 1000 random order and selected the best order. Figure 1 presents the distribution of the best order from the random order. We could see that the best result using this procedure is around 0.713 which is worse than the one that we obtained using a genetic algorithm.
In addition that we could get a better solution, the proposed algorithm ran faster than the greedy search using random permutation. To obtain best preference order with minimum correlation of 0.713, the computer took 375 seconds, while the genetic algorithm needed 6 seconds only to obtain the best of 0.728.
5. Conclusion
The proposed method deals with a problem of ordering several alternatives and expects that the ordering optimally agree with all criteria. It is utilizing a genetic algorithm approach and proved to be able to provide a good decision to rank alternatives based on multicriteria information which some information is incomplete. Not only it results excellent solutions, the method also ran quickly. By this fact, we recommend scientists and decision makers to adopt the method for the cases that they might face to handle.
Figure 1. The distribution of minimum correlation of 1000 random preference order
Sartono. B., Rahardiantoro. S., “An Implementation of Genetic Algorithm to Generate the Most Compromised Decision when Information of the Alternatives is Incomplete”
ISBN : 978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology
References
[1] Gass, S.; Saaty, T. "Parametric Objective Function Part II".Operations Research 3: 316–319. (1955).
[2] Saaty, T.L. “The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation”. New York: McGraw-Hill, 1980.
[3] Rezaei, J. (2015) "Best-Worst Multi-Criteria Decision-Making Method", Omega, 53, 49-57.
[4] Mitchell, M. “An Introduction to Genetic Algorithms”. Cambridge, MA: MIT Press, 1996
[5] Lilien GL, Rangaswamy A. “Marketing Engineering”. Create Space Independent Publishing Platform.
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The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015