We apply the firefly algorithm to the uncapacitatcd facility location problem which is one of optimization problems and investigate the optimum number of the fireflies. The light absorption coefficient parameter $gamma$ of the firefly algorithm is examined to obtain better performance and suitable values of $gamma$ are explored for the uncapacitated facility location problem. Effectiveness of local search in the firefly algorithm is also investigated. In addition, we investigate the optimum number of fireflies for the firefly algorithm

Fazekas, Szilárd Zsolt

Takaya, Mayumi

Tsuya, Kohei

Yamamura, Akihiro

Kyoto University Research Information Repository

TitleFirefly Algorithm for Uncapacitated Facility Location Problemand Number of Fireflies (Developments of Language, Logic,Algebraic system and Computer Science)Author(s)Tsuya, Kohei; Takaya, Mayumi; Fazekas, Szilárd Zsolt;Yamamura, AkihiroCitation数理解析研究所講究録 = RIMS Kokyuroku (2017), 2051:149-157Issue Date2017-10URL http://hdl.handle.net/2433/237104RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityFirefly Algorithm for UncapacitatedFacility Location Problem and Numberof  $\Gamma$irefliesKohei Tsuya, Mayumi Takaya, Szilárd Zsolt Fazekas \backslash Akihiro YamamuraAkita UniversityAbstractWe apply the firefly algorithm to the uncapacitatcd facility locationproblem which is one of optimization problems and investigate the opti‐mum number of the fireflies. The light absorption coefficient parameter $\gamma$ of the firefly algorithm is examined to obtain better performance andsuitable values of  $\gamma$ are explored for the uncapacitated facility locationproblem. Effectiveness of local search in the firefly algorithm is also in‐vestigated. In addition, we investigate the optimum number of fireflies forthe firefly algorithm.1 IntroductionThe method of simulating swarm intelligence is inspired by the movement andforaging behavior in herd animals [1, 11, 21]. As typical examples, particleswa,rm optimiza,tion was inspired by the behavior of groups of birds and fish,ant colony optimization was inspired by the foraging behavior of ants. Numer‐ous swarm intelligence algorithms have been proposed and investigated: particleswarm optimization [10, 11], ant colony optiinization [6], artificial bee colonyalgorithm [9], bat algorithm [22], cuckoo search [23], genetic algorithm [5, 8]and so on. These have been applied to a wide range of computational problemslike data mining and image processing [1] in addition to numerous optimiza‐tion problems like the traveling salesman problem and the flow shop schedulingproblem.Firefly algorithm (FA for short) is one of metaheuristic algorithms and hasbeen studied by many researchers. Recently, effectiveness of FA and suitablevalues of parameters of FA are investigated for some optimization problem in[18]. We introduce the results in [18] and add some results on effect.iveness oflocal search,数理解析研究所講究録第2051巻 2017年 149-157149Efficient supply chain management has led to increased profit, increasedmarket share, reduced operating cost, and improved customer satisfaction formany businesses [13, 16, 17]. For this purpose, it is getting more and more im‐portant in information and communications technologies to solve optimizationproblem such as the uncapacitated facility location problem (UFI P) which is acombinatorial optimization problem. The objective of the UFLP is to optimizethe cost of transport to each customer and the cost associated to facility open‐ing, when the set of potential locations of facilities and the customers are given,whereas \mathrm{U} $\Gamma$ \mathrm{L}\mathrm{P} is known to be NP‐hard (see [13]). In the context of performingeconomic activities efficiently, various objects have been considered as facilities,such as manufacturing plants, storage facilities, warehouses, libraries, fire sta‐tions, hospitals or wireless service stations. Several techniques have been appliedto the UFLP such as the swarm intelligence algorithms or a meta‐heuristics al‐gorithm; particle swarm optimization (PSO) [7], ant colony optimization (ACO)[12], artificial bee colony algorithm (AUC) [19], and genetic algorithm [14].UFLP is described as follows. Given a set F of fa,cilities and a set C ofcustomers, a fixed opening cost f_{i} \in \mathbb{R}_{+} for each facility i\in F , and a transportcost c_{ij}\in \mathbb{R}_{+} from each facility i\in F to each customer j\in C , where \mathbb{R}_{+} standsfor the set of positive real numbers, \mathrm{I}\mathrm{T} $\Gamma$ \mathrm{L}\mathrm{P} asks to find a combination of openingfacilities to minimize cost of transportation and opening facilities.Each customer j is expected to select a facility i from the opened facilitiesso that the cost c_{ii} is lowest. It is asked to find a subset X of facilities to beopened and the assignment  $\sigma$ :  C \rightarrow  X of each customer to an appropriatefacility so \mathrm{t}\mathrm{h}\mathrm{a}_{1}\mathrm{t} these minimize the sum of the opening costs of the facilities andthe transport costs given by the formula (1).\displaystyle \sum_{i\in X}f_{i}+\sum_{j\in C}c_{ $\sigma$(j)g} (1)Several swarm intelligence algorithms have been applied to UFLP until now.For example, particle swarm optimization, ant colony optimiza,tion, artificialbee colony algorithm (ABC for short), and genetic algorithm are studied in[7, 12, 14, 19, 20]. In [18], FA is applied to \mathrm{U} $\Gamma$ \mathrm{L}\mathrm{P} and explore suitable value ofparaÌneters of FA. Effectiveness of equipping a local search mechanisni to FA isalso discussed to minimize cost (1). FA is also compared with ABC algorithmto find a solution of \mathrm{U} $\Gamma$ \mathrm{L}\mathrm{P}. Experiments are carried out to answer these issues.In this paper, we ir the optimum number of fireflies for FA in additionto the results in [18].2 Firefly algorithm for the uncapacitated facil‐ity location problem2.1 Applying to UFLPEach firefty k is given an open facility vector Y_{k} (= [y_{k1}, y_{k2}, y_{k3}, \ldots, y_{k\mathrm{z} $\iota$}]) rep‐resenting a potential solution of opening facilities, where n is the number of150facilities. If the i‐th facility is open, y_{ki}=1 . For the open facility vector \mathrm{Y}_{k} of afirefly k, X(k) is defined to be the set of facilities i in F such that y_{ki}=1 , thatis, X(k) is the set of the facilities that are opened. For the open facility vector\mathrm{Y}_{k} , assignment function  $\sigma$ :  C\rightarrow X(k) from C to X(k) is defined by the trans‐port costs c_{\dot{ $\eta$}j} as follows. For each j in C,  $\sigma$(j) is defined to be i \in X(k) suchthat c_{ij} is the smallest among \{c_{hj} |h\in X(k)\} . If there are several candidatesh , one of them is selected randomly. The total cost T(\mathrm{Y}_{k}) for each firefly k iscomputed as the sum of the cost of opening facilities determined by the openfacility vector \mathrm{Y}_{k} and the transport cost determined by the assignment function $\sigma$ :  C\rightarrow X(k) for \mathrm{Y}_{k} . The cost T(\mathrm{Y}_{k}) for the open facility vector Y_{k} is definedby the formula (1) and is rewritten as follows.T(Y_{k})=\displaystyle \sum_{i\in X(k)}f_{i}+\sum_{j\in C}c_{ $\sigma$(j)j} (2)The notations used in this paper is summarized in Table 1.Table 1: NotationsAn example of a problem instance of \mathrm{U} $\Gamma$ \mathrm{L}\mathrm{P} is given in Table 2. Supposethat \mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E} are facilities, \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} are customers, and [1, 0 , 1, 0 , 1 ] is theopen facility vector \mathrm{Y}_{s} given to a firefly s . Note that f_{\mathrm{A}}=3, f_{\mathrm{B}}=4_{:} f_{\mathrm{C}} =6,f_{\mathrm{D}}=7 , and f_{\mathrm{E}}=2 and  $\sigma$(\mathrm{a}) =\mathrm{A},  $\sigma$(\mathrm{b}) =\mathrm{B} (it may be \mathrm{E} as well),  $\sigma$(\mathrm{c})=\mathrm{C},and  $\sigma$(\mathrm{d})=\mathrm{D} for this open facility vector \mathrm{Y}_{s} . Then the total cost T(\mathrm{Y}_{s}) for theopen facility vector \mathrm{Y}_{s} is computed following the equation (2). Similarly, theopen facility vector \mathrm{Y}_{t}=[0 , 1, 1, 1, 1 ] of firefly t is as follows.151T(\mathrm{Y}_{s}) = Opening costs + Transport cost= (3+6+2)+\displaystyle \min(1,2,9)+\min(8,5,2)\min(4,3,6)+\min(9,4,3)= 20 .T(\mathrm{Y}_{t}) = (4+6+7+2)+(2+2+3+2)= 28 .Table 2: Example of open facility vector for a firefly kAttractiveness of a firefly is represented by the light intensity. FYom theassumption 3, the light intensity of the firefly is given by the objective function,that is, the total cost T(Y_{k}) . Light intensity I(\mathrm{Y}_{k}) of a firefly k is representedby the equation (3).I(\displaystyle \mathrm{Y}_{k})=\frac{1}{T(\mathrm{Y}_{k})} . (3)Note that I(\mathrm{Y}_{k}) gets larger as T(\mathrm{Y}_{k}) gets smaller. It can be determined whetherthe firefly has a good solution by comparing I (Yk). The Light intensity of fireflys and t is as follows.I(\displaystyle \mathrm{Y}_{s})=\frac{1}{20}iI(\mathrm{Y}_{t})=\frac{1}{28} . (4)Distance between any two fireflies is represented by the hamming distanceof their open facility vectors. Suppose that the fircfly s and the firefly t haveopen facility vectors below. Then the hamming distance r_{st} between s and t is3.Y_{s}=[1, 0, 1, 0, 1]\mathrm{Y}_{t}=[0, 1, 1, 1, 1]If I(Y_{s})>I(\mathrm{Y}_{t}) holds for two fireflies s and t , then t moves towards s . Movementof the firefly t toward the firefly s is the conversion of the open facility vector oft . Common components of \mathrm{Y}_{s} and \mathrm{Y}_{t} are carried over to the new open facilityvector Y_{t}.new Y_{t} = [?,?, 1,?, 1]152Each of components of Y_{t} different from Yô is replaced with a probability  $\beta$.The probability  $\beta$ is given by the following formula (5). $\beta$=\displaystyle \frac{$\beta$_{0}}{1+ $\gamma$ r_{st}^{2}} (5)where $\beta$_{0} is the probability at r_{st}=0 and ỉio is set 1, and  $\gamma$ is the light absorptioncoefficient. In this paper, suitable values of  $\gamma$ is explored.Let firefly  t be close to firefly s by probability  $\beta$ as follows. The total costof new \mathrm{Y}_{t} at this time is 24, which shows that the cost is decreasing.new \mathrm{Y}_{t} = [1, 1, 1, 0 , 1 ]new T(Y_{t}) = (3+4+6+2)+(1+2+3+3)= 24 .2.2 Pseudocode of FAFA program generates fireỉlies and set Lhe parameters. Fireflies compare theintensity each other and then change their positions, that is, the open facilityvectors according to their distance. Repeating this process, FA program updatesfireflies intensity and their open facility vectors, and then it outputs a solution.A pseudocode of FA program is illustrated below.beginObjective function \displaystyle \min T(Y) ,Initialize positions of fireflies \mathrm{Y}_{k}.(k=1,2, \ldots , K)I(Y_{k}) is defined by the reciprocal of T(\mathrm{Y}_{k}) ,Define parameter  $\gamma$ and  $\beta$while ( \mathrm{r}< Repeat count)for t= lto Kfor s= lto Kif I(\mathrm{Y}_{s})>I(\mathrm{Y}_{t})Move firefly t towards firefly selseMove firefly t randomly;end ifUpdate of intensity and total cost;end for send for tRank the fireflies and find the current best.Local Search.\mathrm{r}++ ;end whilePost‐process results and visualizationend1533 Experiments and Results3.1 ObjectivesA computer program of FA to solve UFLP is implemented and experiments arecarried out. Suitable values of the light absorption coefficient parameter  $\gamma$ isexplored to accomplish better performance. A program is implemented in C#language using Visual Studio and run on an Intel Core i52.  67\mathrm{G}\mathrm{H}\mathrm{z} Desktopwith 4. 0\mathrm{G}\mathrm{B} memory. Several test data sets of benchmark problems are bor‐rowed from OR‐Library [4], which is a collection of test data sets for numerousoperations research problems and originally described in [2]. \mathrm{U} $\Gamma$ \mathrm{L}\mathrm{P} is calledan uncapacitated warehouse location problem in OR‐Library. There are 15 datafiles provided; the test data sets VII, X. XIII and A to \mathrm{C} from [3]. The test datasets VII (cap71, 72, 73, 74), X (cap 101, 102, 103, 104) and XIII (cap 131, 132,133, 134) are employed for experiments. These test data sets are summarizedin Table 3, in which m stands for the number of customers and n stands forthe number of facilities. The optimal solutions for these test data sets are takenfrom OR‐Library.Table 3: Test data from OR‐Library3.2 Number of firefliesFind the optimum number of fireflies for FA. In this paper, average relativepercent error (ARPE for short) and hit to optimum rate (HR for short) is used forperformance evaluation of the algorithm. ARPE is the average of the differencefrom the optimum expressed in percentages. If ARPE is lower, then we have154more chance to obtain better solutions. ARPE is given by the formula (6).ARPE=\displaystyle \sum_{i=1}^{R}(\frac{H_{i}-U}{U}) \times \frac{100}{R} (6)where H_{i} denotes the i‐th replication solution value, R is the number of replica‐tions, and U is the optimal value provided by [4]. HR represents the number oftimes that the algorithm finds the optimal solutions over all repetitions. If HRis higher, then the probability to obtain a better solution is higher.  $\gamma$ is set tobe e.01, the repeat count 50, and the number of fireflies is one of the values 10,15, 20, 25, 30 or 35 in the experiments. The result is the average value when thealgorithm is executed 100 times. The table 4 and table 5 is summary of ARPEor HR at repeat count is 50.Table 4: ARPE for each firefly number (number of ỉirefly, ARPE)Table 5: HR for each firefly number (number of firefly, HR)4 SummaryWe discussed several aspects of the firefly algorithm for UFLP and obtained tworesults in this paper.155First, optimal values of the light absorption coefficient  $\gamma$ of FA in UFLPare obtained. It is found that FA works better when  $\gamma$ is 0.001, 0.005 or 0.01.These values work well regardless of size of problem instance of UFLP. Suitablevalues for particular test data are obtained by experiments, however, suitablevalues of  $\gamma$ have not been complete]y classified for a general case. Therefore, itis necessary to seek suitable parameters when FA is applied to an optimizationproblem.Second, it is verified that local search iinproves performance of FA and sothe combination of FA and local search algorithm is effective. FA is comparedwith FA with local search and ABC algorithm with respect to average relativepercent error and hit to optimum rate. No superiority of FA over ABC algorithmis recognized, whereas FA with local search works as well as ABC algorithm forUFLP.It seems necessary to compare FA and ABC algorithm for more various sizesof test data and other type of optimization problems in order to comprehendthe strong points of FA. It is also desirable to explore suitable parameters makesFA with local search to work better.References[1] A. Abraham, C. Grosan, V. Ramos, Swarm Intelligence in Data Mining,Springer‐Verlag Berlin Heidelberg, (2006)[2] J.E. Beasley, OR‐Library: distributing test problems Uy electronic mail, J.of the Operational Research Society, 41(11) (1990), 1069‐1072.[3] J.E. Beasley, Lagrangean heuristics for location problems, European J. ofOperational Research, 65 (1993), 383‐399.[4] J.E. Beasley, OR‐Library, http: //\mathrm{w}\mathrm{w}\mathrm{w} .brunel.ac.uk/\simmastj j \mathrm{b}/\mathrm{j}\mathrm{e}\mathrm{b}/inf0. html.[5] K.D. Jong, Analysis of the behavior of a class of genetic adaptive systems,Ph.D. Thesis, University of Michigan, 1975.[6] M. Dorigo, optimization, Learning and Natural Algorithms, Ph.D. Thesis,Politecnico di Milano, 1992.[7] A.R. Guner and M. Sevkli, A discrete particle swarm optimization algorithmfor uncapacitated facility location problem, J. of Artificial Evolution andApplications, 2008(10) (2008), 9 pages.[8] J.H. Holland, Adaptation in Natural and Artificial Systems: An IntroductoryAnalysis with Applications to Biology, Control, and Artificial Intelligence,A Uradford Book, 1992.[9] D. Karaboga, An Idea Uased on Honey Bee Swarm for Numerical optimiza‐tion, Technical Report‐TR06, Erciyes University, 2005.156[10] J. Kennedy and R.Eberhart, Particle Swarm optimization, Proceedings ofthe IEEE international conference on neural networks, 1995, 1942‐1948.[11] J. Kennedy, R. EUerhart and Y. Shi, Swarm intelligence, Academic Press,2001.[12] A. Kole, P. Chakrabarti and S. Bhattacharyya, An ant colony optimiLationalgorithm for uncapacitated facility location problem, Artzficial Intelligenceand Applications, 1(1) (2014), 55‐61.[13] B. Korte and J. Vygen, Combinatorial optimization: Theory and Algo‐rithms, Springer‐Verlag, 2007.[14] J. Kratica, D. Tosic, V. Filipovic and I. Ljubic, Solving the simple plantlocation problem by genetic algorithm, RAIRO ‐ Operations Research, 35(2001), 127‐142.[15] S.M. Lewis and C. K. Cratsley, Flash signal evolution, mate choice andpredation in fireflies, Annual Review of Entomology, 53(2) (2008), 293‐321.[16] R. Rahmaniani and A. Ghaderi, A combined facility location and networkdesign problem with multi‐type of capacitated links, Applied MathematicalModelling, 37(9) (2013), 6400‐6414.[17] D. Simchi‐Levi, P. Kaminsky, E. Simchi‐Levi, Designing and Managing theSupply Chain. Concepts, Strategies and Case Studies, McGraw‐Hill, Boston,Mass, USA. (2000)[18] K. Tsuya, M. Takaya, and A. Yamamura, Application of the firefly algo‐rithm to the uncapacitated facility location problem, J. of Intelligent andFuzzy Systems, 32(4) (2017) 3201‐3208.[19] N. Tuncbilek,  $\Gamma$ . Tasgetiren and S. Esnaf, Artificial bee colony optimizationalgorithm for uncapacitated facility location problems, J. of Economic andSocial Research, 14(1) (2012), 1‐24.[20] Y. Watanabe, M. Takaya and A. Yamamura, Fitness function in ABC algo‐rithm for uncapacitated facility location problem, Lecture Notes in ComputerScience, 9357 (2015), 129‐138.[21] X.S. Yang, Biology‐Derived Algorithms in Engineering optimization, Hand‐book of Bioinspired Algorithms and Applications, Chapman & Hall /\mathrm{C}\mathrm{R}\mathrm{C},2005, 589‐600.[22] X.S. Yang, A new metaheuristic bat‐inspired algorithm, Studies in Com‐putational Intelligence, 284 (2010), 65‐74.[23] X.S. Yang and S. Deb, Cuckoo search via Levy flights, Proceedings of theWorld Congress on Nature and Biologically Inspired Computing, IEEE Pub‐lications, 2009, 210‐214.157

Firefly Algorithm for Uncapacitated Facility Location Problem and Number of Fireflies (Developments of Language, Logic, Algebraic system and Computer Science)

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/237104/1/2051-25.pdf

Firefly Algorithm for Uncapacitated Facility Location Problem and Number of Fireflies (Developments of Language, Logic, Algebraic system and Computer Science)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository