Vector optimisation using fuzzy preference in ...

Int. J. Operational Research, Vol. 16, No. 1, 2013

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Vector optimisation using fuzzy preference in evolutionary strategy based firefly algorithm Surafel Luleseged Tilahun* and Hong Choon Ong School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia Fax: +604 6570910 E-mail: [email protected] E-mail: [email protected] *Corresponding author Abstract: Solving vector optimisation entails the conflict among component objectives. The best solution depends on the preference of the decision-maker. Firefly algorithm is one of the recently proposed metaheuristic algorithms for optimisation problems. In this paper, the random movement of the brighter firefly is modified by using (1 + 1)-evolutionary strategy to identify the direction in which the brightness increases. We also show how to generate a dynamic weight for each component of the vector by using a fuzzy trade-off preference. This dynamic weight will be imbedded in computing the intensity of light of fireflies in the algorithm. From the simulation results, it is shown that using fuzzy preference is promising to obtain solutions according to the given fuzzy preference. Furthermore, simulation results show that the evolutionary strategy based firefly algorithm performs better than the ordinary firefly algorithm. Keywords: vector optimisation; evolutionary strategy.

fuzzy

preference;

firefly

algorithm;

Reference to this paper should be made as follows: Tilahun, S.L. and Ong, H.C. (2013) ‘Vector optimisation using fuzzy preference in evolutionary strategy based firefly algorithm’, Int. J. Operational Research, Vol. 16, No. 1, pp.81–95. Biographical notes: Surafel Luleseged Tilahun received his BSc and MSc degrees in Mathematics from Addis Ababa University, Addis Ababa, Ethiopia, in 2004 and 2007, respectively; and MSc degree in Computational Operations Research in 2010 from Addis Ababa University. Currently, he is a PhD candidate in School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia. His research focuses on computational operations research and its applications. Hong Choon Ong received his BSc degree from Universiti Malaya (UM), in 1980 and his MSc and PhD degrees in Mathematics from Universiti Sains Malaysia (USM), Penang, Malaysia in 1996 and 2003, respectively. He is currently a Senior Lecturer in USM. His research interest includes data mining, statistical modelling and machine learning applications.

Copyright © 2013 Inderscience Enterprises Ltd.

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Introduction

Vector optimisation problem is a problem of finding a member from a non-empty set of candidate solutions which optimise components of a vector function. Vector optimisation problem is also called multi-objective or multi-criteria optimisation problem. The set of candidate solutions is called the feasible set. Vector optimisation problems appear in a wide range of disciplines, mainly in economic and engineering studies. Studying and formulating a solution method for vector optimisation problems is very helpful in solving real-case problems, which can be formulated as a vector optimisation problem. The main challenge in solving these problems is that after a certain level, if one tries to optimise a component leads the worsening of another competing component. The solution set for these problems is defined after the Italian economist Velfredo Pareto as Pareto optimal solutions. A Pareto optimal solution is a member of the feasible set where there is no other feasible member exist which does better at least in one component while doing the same in the rest. Choosing a member among the Pareto solutions depends on the preference of the decision-maker. Many studies have been conducted regarding decision-maker’s preference and on how to represent it. The trade-off method and component ranking are mainly used methods of expressing the preference of the decision-maker. The trade-off method expresses the preference better than component ranking, because it uses numerical values to compare a pair of components, whereas in the case of ranking the difference between preferences of two consecutive components is considered the same for all component of the objective function (Tilahun and Ong, 2011). Even though tradeoff methods express the preference better, it is not an easy task for the decision-maker to give trade-off values (Keeney and Raiffn, 1976). However, it is easier to use a fuzzy number as trade-off than crisp numbers. Among the solution methods used to solve optimisation problems, metaheuristic solution methods are widely used in different studies. Metaheuristic solution methods are methods which try to improve the quality of solution members iteratively with some randomness property. Even though these solution methods do not guarantee optimality, they give a sound and acceptable solutions. Most of these solution methods are not affected by the behaviour of the problem which makes them to be used in different applications. Firefly algorithm is one of the newly introduced metaheuristic solution algorithms by Xin-She Yang in 2009 (Yang, 2010). It is inspired by the flashing light behaviour of fireflies. In the algorithm, randomly generated feasible solutions are considered as a firefly with light intensity or brightness depending on the objective function. The algorithm works in such a way that a firefly will be attracted and moves towards the brighter firefly. Among the three ideal rules used to construct the algorithm, one is that for a firefly to move towards the brighter fireflies and if no brighter firefly exists it will move randomly. In this paper, we will show that modifying this random movement of the brighter firefly by using (1 + 1)-evolutionary strategy to identify the direction in which the brightness increases, improves the performance of the algorithm. The use of metaheuristic solution methods particularly for vector optimisation problems has also been the focus of many researchers. This approach has found many applications in different disciplines. The study of incorporating preference has been proposed and mentions as one of the forefront research area which needs further exploration (Coello, 2009). This paper discuses the imbedding of decision-maker’s preference as a fuzzy trade-off in the evolutionary-based firefly algorithm.

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Section 2 discusses the literature review and in Section 3, basic preliminaries will be explained. This is followed by our proposed modified algorithm in the Section 4. In Section 5, simulation results will be shown and discussed. Finally, in Section 6, we give the conclusion and future works.

2

Literature review

Multi-objective optimisation model describes the reality better than the single objective one. Hence, many real problems have been modelled as a vector optimisation problem. The use of a vector optimisation model and solution procedure to find a solution is common in different disciplines. It has been used in engineering problems (Tavana et al., 2008), management decision-making problems (Chaabane et al., 2010; Fernandez et al., 2009), biological and chemical problems (Mokeddem and Khellaf, 2010; Ortiz et al., 2005), medicine science (Petrovski and McCall, 2001), transportation problems (Yin, 2002), etc. Once a problem is formulated as a vector optimisation problem, the next issue is solving it. Many solution methods have been proposed for vector optimisation problems. Perhaps weighting method is one of the easiest and used in many engineering applications (Ehrgott, 2005). Lexicographic method (Emelichev et al., 1995), goal programming (Schniederjans, 1995), Benson method (Ehrgott, 2005), utility function method (Keeney and Raiffn, 1976) are other methods to solve vector optimisation problems. Most of these deterministic methods use the conversion of the problem into single-objective optimisation problem. After the introduction of metaheuristic algorithms for optimisation problems, the extension to solve vector optimisation problems has been done (Abido, 2009; Chaudhuria and Deb, 2010; Jaeggi et al., 2008). Jin et al. (2001) use preference order and discuss the dynamic weight aggregation evolutionary algorithm for vector optimisation problem. Similar studies of extending metaheuristic algorithms for vector optimisation problems have been done depending on the decision-maker’s preference (Ishibuchi et al., 2006; Jin and Sendhoff, 2002; Ong and Tilahun, 2011). Coello (2009), in his review paper on evolutionary multi-objective optimisation, mentioned that incorporating decision-maker’s preference is one of the areas which need further exploring. Some researches have been done on that aspect. In general, metaheuristic solution algorithms for optimisation problems have become popular and many new methods are still being proposed. Harmony search algorithm was introduced in 2004 by Z.W. Geem et al., honey bee algorithm by Craig A. Torey in 2004, cuckoo search algorithm and firefly algorithm in 2009 by Xin-She Yang and bat algorithm in 2010 again by Xin-She Yang (Yang, 2010). Firefly algorithm is an algorithm inspired by the flashing light behaviour of fireflies. It has three basic ideal rules. Among those rules one is that a brighter firefly will move randomly. In this paper, we will modify the random movement of the brighter firefly, in firefly algorithm, by identifying a better direction, in which the intensity increases, using the (1 + 1)-evolutionary strategy. Furthermore, the fuzzy trade-off of each couple of components will be collected and an appropriate probability distribution will be constructed depending on the membership function of the cumulative fuzzy trade-offs. Then the dynamic weight, which is expressed using a probability distribution, will be put into intensity measurement stage of firefly algorithm. Simulation will also be done on selected vector optimisation problems.

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Preliminaries

3.1 Vector optimisation and dynamic preference A vector optimisation, which is also called a multi-objective optimisation problem or a multi-criteria optimisation problem, is an optimisation problem with more than one component objectives. The components which need to be optimised are called objective functions. A vector maximisation problem can be converted to minimisation by multiplying the objective functions by negative one. So, in this paper, we consider vector minimisation problems as: min F ( x ) = ( f1 ( x ), f 2 ( x ), … , f k ( x ) )

x ∈S ⊆ ℜ n

(1)

A Pareto optimal solution, x’, to this problem is a member of the feasible set, S, such that there does not exist another member in S which does the same as x’ in all objectives and better at least in one of the objectives. Usually for a given vector optimisation problem, there are many Pareto optimal solutions. Choosing a solution among the Pareto optimal set depends on the subjective judgement of the decision-maker. Giving preference in the form of trade-off is one among many ways in which the decision-maker gives his preference. A trade-off of objective function j for a unit decrease of objective i is the amount of objective function j that the decision-maker is willing to give up to increase. This trade-off can be better expressed fuzzily in the interval say [wij, wij + dij], with high membership function near wij. This means that the willingness of the decision-maker in increasing the jth objective for a unit decrease of the ith objective decreases, when we go from wij to wij + dij and after wij + dij, the decision-maker is not willing to go any further. The willingness can be considered as a membership function and wij as a fuzzy number. Since it is meaningless to compute wij , we put it to be 0, for all i. From this, it is possible to compute the average weight, a weight with high membership function value and average width, where the membership function value is zero beyond that point from the average weight as follows: wp =

∑

k i =1

wpi

∑ ∑ k

j =1

k

w ij i =1

is the average weight for the pth objective function

⎛ w1 ⎞ ⎜w ⎟ ⎜ 2⎟ ⎜ . ⎟ w = ⎜ ⎟ is the average weight for the vector objective function. . ⎜ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ wk ⎠

(2)

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The fuzzy width for the weight of the pth objective function is also given by:

∑ d = ∑ ∑ k

dp

i =1

pi

k

k

j =1

i =1

(3)

d ij

⎛ d1 ⎞ ⎜ ⎟ ⎜ d2 ⎟ ⎜. ⎟ d = ⎜ ⎟ is the normalised average fuzzy width. ⎜. ⎟ ⎜. ⎟ ⎜ ⎟ ⎜⎝ d ⎟⎠ k Hence for each component of vector objective i, it is possible to compute the range of weights [ wi , wi + di ] , in which the acceptability of these weights by the decision-makers keep decreasing when we go to the right and become unacceptable after wi + d i (Ong and Tilahun, 2011).

3.2 Firefly algorithm Firefly algorithm is a metaheuristic algorithm to solve optimisation problems. It was introduced by Xin-She Yang in 2009 at Cambridge University (Yang, 2010). The algorithm is inspired by the flashing behaviour of fireflies at night. The algorithm is constructed using three basic ideal rules. The first rule is that all fireflies are unisex which means any firefly can be attracted to any other brighter one. The second is the brightness of a firefly is determined from the encoded objective function. The last rule is attractiveness is directly proportional to brightness but decreases with distance, and a firefly will move towards the brighter one and if there is no brighter one it will move randomly. The intensity of a light is inversely proportional to the square of the distance, r, from the source. Let I0 be the intensity at the source.

I (r ) =

I0

(4)

r2

Furthermore, when light passes through a medium with light absorption coefficient of λ , the light intensity varies with distance as:

I (r ) = I0 e − λ r

(5)

The combined effect of these two can be approximated as shown is Equation (6): I ( r) = I 0 e − λ r

2

(6)

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The attractiveness can also be defined in a similar way; hence, the attractiveness of a firefly r distance away can be expressed as: Α(r ) = A0 e − λ r

2

(7) 2

−λr , it is possible to rewrite the Since 1/(1 + λr2) is easier to compute than e attractiveness function as:

A(r ) =

A0

(8)

1+ λr2

Here A0 is found from the coded objective function. Consider two fireflies located at x = ( x1 , x2 ,… , xn ) and y = ( y1 , y2 ,… , yn ). If the firefly at y is brighter then the firefly at x will change its position by moving towards y as: 2

x ← x + A0 e − λ r ( y − x) + αε xy

(9)

The second term is because of the attraction of x towards y and the third term is a randomisation term with α randomisation parameter, and ε xy is a vector of random numbers. Furthermore, r can be taken as the Euclidean distance between x and y is given by: n

r=

∑( y − x ) i

2

i

(10)

i =1

Firefly algorithm can be summarised as shown in Figure 1.

3.3 Evolutionary strategies Evolutionary strategy is an approach to solve optimisation problems which is inspired by natural evolution. Unlike genetic algorithm, it uses only the mutation operator. In this paper, we consider (1 + 1)-evolutionary strategy. (1 + 1)-Evolutionary strategy is an evolutionary strategy in which a parent gives birth to only one child (Negnevitsky, 2005). Basically, evolutionary strategy has the following steps: 1

Generate random set of solutions, {x1 , x2 ,…, xm }.

2

Calculate the functional values of the solutions, f ( xi ).

3

Perform mutation as xi' = xi + a, where a is from a normal distribution, N (0, δ ), δ is pre-assigned algorithm parameter.

4

Calculate the functional values, select the best, update the solution population and stop if termination criteria is fulfilled else go to step 2. The termination criteria could be a pre-specified number of iterations or when no more improvement occurs.

Vector optimisation using fuzzy preference in evolutionary strategy Figure 1

4

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Flowchart of firefly algorithm

Fuzzy preference imbedded evolutionary strategy based firefly algorithm

This study focuses on two issues which are imbedding preference as a dynamic weight and modifying the firefly algorithm. Once we construct the weight interval from fuzzy trade-off given by the decisionmaker, it is possible to construct a probability density function. Suppose the interval [ wi , wi + di ] with high acceptability or membership function around wi and zero after wi + di is given for one of the objective function as shown in Figure 2.

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Figure 2

Weight vs. acceptability

To generate a random weight under the line, it is necessary to construct a probability density function which agrees with degree of acceptability. But for a probability density function, the area under the curve should be 1. Adjusting the point ( wi ,1) to make the area under the line 1 does not affect the degree of acceptability. Hence, by adjusting the end point of the curve, the probability distribution for the weight of ith objective function can be expressed as: g ( wi ) =

2 bi 2

( wi − wi ) −

2 bi

(11)

Hence wi , the weight of objective function i, is generated from the probability density function g (wi ). This dynamic weight will be incorporated in the intensity or attractiveness computation stage of the algorithm. A given firefly i will have k components (k is the number of objective functions) of attractiveness and by taking the weighted sum of those attractiveness, where the weights are randomly generated from the given probability distribution, g ( wi ), one can easily reduce the vector attractive form into a scalar, I0 . k

I0 =

∑w I i

j

j 0

(12)

j =1

The second issue is modifying the random movement of the brightest firefly by identifying the best direction in which the brightness increases. In identifying the best direction, we use (1 + 1)-evolutionary algorithm. Firstly, a set of vector directions will be generated and according to their fitness, evolutionary mutation will be performed. If a direction in which the brightness increases is not found, the firefly will stay in the current position. This will help the algorithm not to jump over optimal solutions. The movement of the brighter firefly, suppose at position x, in the evolutionary strategy based firefly algorithm can be summarised as follows: 1

Generate random set of directions, {d1 , d 2 ,… , d q } .

2

Use (1 + 1)-evolutionary strategy to identify the best direction, d.

3

Compare the brightness of the firefly at best result, x + d, with the current position, x, and take the best.

Vector optimisation using fuzzy preference in evolutionary strategy

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Simulation results

For the simulation purpose, we use a two-dimensional vector optimisation problem, where optimise can be either minimise or maximise. optimise ( f1 ( x ), f 2 ( x ) )

(13)

x ∈S

A MATLAB code for the performance-based modified firefly algorithm with 50 iterations is run on selected test problems. The preference is taken as w1 = 1 , d1 = 1 and w2 = 2 , d2 = 0.8 . Furthermore, the algorithm parameters are set as λ = 0.5 and α = 0.1 . The number of initial population is set to be 12. The simulation results are recorded as follows: 1

The first test problem The first test problem is a minimisation problem, as given in Equation (14), and has a convex and continuous Pareto front. min

0 ≤ x1 , x2 ≤1

( f1 ( x ), f 2 ( x ) )

x12 + x2 2 and 2 ( x − 2.0) 2 + ( x2 − 2.0) 2 f 2 ( x) = 1 2 f1 ( x ) =

After running the MATLAB code, the result can be seen as in Figure 3. Figure 3

Simulation results for the first test problem (minimisation problem), as given in Equation (14): (a) initial population; (b) final population after running the program (see online version for colours)

(14)

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Figure 3

Simulation results for the first test problem (minimisation problem), as given in Equation (14): (a) initial population; (b) final population after running the program (see online version for colours) (continued)

The given expected weight of f1 and f2 are 1 and 2, respectively. This shows that the movement of the points tends to move in the direction of the improvement of f2 than f1 until a certain level. Since the problem is minimisation, improving f1 means moving downwards, which in other word means moving to the right on the Pareto front. Furthermore, improving the solutions of f1 means moving the points to the left on the Pareto front, but that results in the increment of f2. After running the MATLAB code, it is shown that the points converge to the advantage of f2, as shown in Figure 3(b). 2

The second test problem

The second test problem is a maximisation problem with discontinuous Pareto front. It is given as: max

− π ≤ x1 , x2 ≤ π

( f1 ( x), f 2 ( x))

2 2 f1 ( x) = − ⎡1 + ( A1 − B1 ) + ( A2 − B2 ) ⎤ ⎣ ⎦ 2 2 f 2 ( x) = − ⎡( x1 + 3) + ( x2 + 1) ⎤ ⎣ ⎦ where: A1 = 0.5sin1 − 2 cos1 + sin 2 − 1.5cos 2

A2 = 1.5sin1 − cos1 + 2sin 2 − 0.5cos 2 B1 = 0.5sin x1 − 2 cos x1 + sin x2 − 1.5cos x2 B2 = 1.5sin x1 − cos x1 + 2sin x2 − 0.5cos x2

After running the MATLAB code, the result can be seen as shown in Figure 4.

(15)

Vector optimisation using fuzzy preference in evolutionary strategy Figure 4

91

Simulation results for the second test problem (maximisation problem), as given in (15) (a) Initial population; (b) final population after running the program (see online version for colours)

Again with the same expected weight for f1 and f2 as in the first test problem, we have a maximisation problem. In this case, we want to increase the value of the objective functions. Hence, since f2’s expected weight is greater than f1’s, the points should tend to move upwards than to the right. The simulation result as shown in Figure 4(b) is in agreement with this concept.

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Hence, incorporating the fuzzy preference of the decision-maker indeed give solutions according to the specified preference reasonably near the Pareto front. The next issue is to compare the performance of the ordinary firefly algorithm with the modified one. We run the algorithm and record the best and worst performance. It is expressed graphically in Figures 5 and 6. It is clearly shown that the modified algorithm is better in giving a better result with a specified number of iterations. The performance is compared using the best and worst results from the solution population of each iteration. It is compared using weighted functional value F, as given in Equation (16). F = w1 f1 ( x ) + w2 f 2 ( x ) Figure 5

(16)

Performance graph of the ordinary firefly and evolutionary-based firefly algorithm using test problem one (minimisation): (a) on the best results; (b) on the worst results (see online version for colours)

Vector optimisation using fuzzy preference in evolutionary strategy Figure 6

93

Performance graph of the ordinary firefly and modified firefly algorithm using test problem two (maximisation): (a) on the best results; (b) on the worst results (see online version for colours)

For the purpose of comparison of the ordinary firefly algorithm with the evolutionary strategy based firefly algorithm, the MATLAB code run for 50 iterations and in each iteration the best and worst performances of the combined function F, as in Equation (16), is recorded. For the minimisation problem, the result is shown in Figure 5. The solid line represents the results of the evolutionary strategy based firefly algorithm, whereas the broken line represents the performance of the ordinary firefly algorithm. The solid line is mostly under the broken line which shows that the evolutionary strategy based firefly algorithm give better minimum results than the ordinary firefly algorithm.

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Similarly, for the second test problem, which is the maximisation problem, the result is recorded in Figure 6. As can be seen in Figure 6, the solid line, which represents the evolutionary based firefly algorithm, is mostly above the broken line, which again represents the ordinary firefly algorithm. Hence, the evolutionary strategy based firefly algorithm performs better than the ordinary firefly algorithm, in maximisation test problem as well.

6

Conclusion

In this paper, we improve the performance of ordinary firefly algorithm by modifying the random movement of the brighter firefly. We use (1 + 1)-evolutionary strategy to identify a direction for the brighter firefly in which the brightness increases. From a simulation result done on selected problems, we show that evolutionary strategy based firefly algorithm performs better than the ordinary firefly algorithm. Furthermore, this paper discusses on how to embed the preference of a decision-maker in the evolutionary-based firefly algorithm. The preference is collected as fuzzy trade-off, from which a dynamic weight will be constructed with appropriate probability density function, which agrees with the membership function of the fuzzy preference. The simulation result shows that embedding the preference by using the appropriate probability density function gives a sound and reasonable solution. In this paper, we consider the situation where the preference of the decision-maker does not vary from point to point, but the preference of the decision-maker may depend on the point or values where the preference is collected. A decision-maker may give different preferences for different points. Further study can be done on the embedding of dynamic fuzzy preference of the decision-maker in the algorithm. We consider the preference of a single decision-maker. Future works can consider multiple decision-makers with different levels of decision power.

Acknowledgements This work is supported in part by Universiti Sains Malaysia (USM) Research University (RU) Grant no. 1001/PMATHS/817037. The first author would like to acknowledge a support from USM-TWAS fellowship and would like to thank Mr Adane Fekadu Wogu for his valuable support. Furthermore, the authors would like to thank the editor and the reviewers for their most helpful comments and suggestions.

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