Solving Fuzzy Assignment Problem Using Fourier Elimination Method

Transcription

1 Global Joural of Pure ad Applied Mathematics. ISSN Volume 3, Number 2 (207), pp Research Idia Publicatios Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio Method Dr. S. Murugaadam ad K. Hema 2 Professor ad Head, Departmet of Mathematics, Imayam College of Egieerig, Kaaur, Thuraiyur, Trichy , Tamiladu, Idia. 2 Associate Professor, Departmet of Mathematics, M.A.M. College of Egieerig ad Techology, Trichy-62 05, Tamiladu, Idia. Abstract Assigmet problem is oe of the most-studied, well kow ad sigificat i mathematical programmig. It is a combiatorial optimizatio problem i the field of operatio research.i this paper, for a give fuzzy assigmet problem, the cost are triagular fuzzy umbers ad the fuzzy assigmet problem is coverted ito crisp assigmet problem by graded mea itegratio represetatio method ad the crisp assigmet problem is coverted ito liear programmig problem which is solved by a proposed method called Fourier Elimiatio method to get the optimal solutio. The proposed method is illustrated with a umerical eample. Keywords: Assigmet problem, Fuzzy assigmet problem, Triagular fuzzy umber, Graded mea Itegratio represetatio method, Liear programmig problem, Fourier elimiatio method. AMS Mathematics Subject Classificatio (200): 03E72, 90C05

2 454 Dr. S. Murugaadam ad K. Hema INTRODUCTION Assigmet problem is a special type of liear programmig problem whose objective is to obtai the optimum assigmet for umber of tasks (jobs) which is equal to the umber of resources (workers) at a miimum cost or maimum profit. However, cosiderig the decisios made i the real world where the objectives, costrais or parameters are ot precise. Therefore, a decisio is ofte made o the basis of vague iformatio. The cocept of fuzzy set theory was first itroduced Bellma & Zadeh[2] ad liear programmig problem was itroduced by Chares ad Cooper W.W, [4]. The approach i which fuzzy assigmet problem is coverted ito multiobjective liear programmig problem where the cost co-efficiet are fuzzy umbers was proposed by De.P.K ad Bharati Yadav[3].Further Kaiappa.P & Thagavel.K,[6] modified the method of Fourier elimiatio method which was developed by Williams.H.P[9]. I this method, solvig liear programmig problems is doe by choosig a variable for elimiatio. Padia.P & Nataraja.G,[8] discussed a Fourier trasportatio method to fid a optimal solutio for trasportatio problems with mied costraits. Geeralizatio of two phase method to solve multiobjective iteger programmig with three objectives was proposed by Athoy Przybylski et.al[]. Farahi et.al [5] used fuzzy set theory ad fuzzy programmig to covert the multiobjective liear bi-level programmig problem to a liear bi-level programmig problems. A approach to solve multi-objective assigmet problem with iterval parameters where a weighted mi-ma method is applied to trasform the multiobjective ito sigle objective optimizatio problem was formulized by Kayva Salehi[0].Ail D.Gotmare et.al[]solved the fuzzy assigmet usig Brach ad Boud techique.i additio,to solve multi-objective assigmet problem with liear ad o liear membership fuctios a algorithm was proposed by Atul Kumar Tiwari et.al[3]. I this paper, Fourier elimiatio method is proposed to fid the optimal solutio to the fuzzy assigmet problem. We orgaize this paper as follows: I sectio, we itroduce the fuzzy assigmet problem. I sectio 2, we costruct the mathematical model for the problem. I sectio 3, we give the methodology for the problem. I sectio 4, we give a solutio procedure for the proposed method to solve the problem. I sectio 5, a umerical eample is give to show the efficiecy of the proposed method ad fially we give a coclusio for the problem.

3 Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio Method 455 MATHEMATICAL FORMULATION The geeral assigmet problem Suppose there are people ad jobs. Each job must be doe by eactly oe perso; also each perso ca do, at most, oe job. The problem is to assig jobs to the people so as to miimize the total cost of completig all of the jobs. The geeral assigmet problem ca be mathematically stated as follows: Miimize Z= c i= j= Subject to j for i,2,..., i for j,2,..., 0 or Fuzzy assigmet problem Miimize Z = c i= j= Subject to j= i= = = for i =,2,..., = for j=,2,..., 0 or METHODOLOGY Triagular fuzzy umber A fuzzy umber A is a triagular fuzzy umber deoted by (a, a2, a3) ad its membership fuctio μ A ()is give below:

4 456 Dr. S. Murugaadam ad K. Hema Graded Mea Itegratio Represetatio: We describe graded mea itegratio represetatio as follows: Suppose L - ad R - are iverse fuctios of fuctios L ad R, respectively, ad the graded mea h-level value of geeralised fuzzy umber A = (a, a2, a3, a4 : w) LR is h[l - (h) + R - (h)] / 2. The the defuzzified value P(A) by graded mea itegratio represetatio of geeralised fuzzy umber based o the itegral value of graded mea h-level is P( A) w 0 L h h w 0 R 2 hdh h dh Where h is betwee 0 ad, 0 < w < If A = (a, a2, a3) is a triagular fuzzy umber. Che ad Hsieh [6] already foud the geeral formulae of the represetatio of geeralized triagular fuzzy umber ad it is as follows: P( A) P Theorem: A 2 a Cosider the system 0 h a 4a 6 2 h a If a 0, for all i, the j = 0. a a a ha a hdh dh a + a a b a 2 + a a 2 b 2 a m + a m a m b m... ()

5 Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio Method 457 Proof: Now, elimiatig j from the give system usig Fourier elimiatio method, we have the followig system a + a a j j + a j+ j+ + a b a 2 + a a 2j j + a 2j+ j+ + a 2 b 2... (2) a m + a m a mj j + a mj+ j+ + a b m, 2, j, j+, 0 From the system ()&(2), we observe that there is o role of the variable j.that is j is iactive. Therefore j =0. Mathematical formulatio of Liear Programmig Problem: If j (j=,2, ) are the decisio variables ad if the system is subject to m costraits,the geeral mathematical model ca be writte i the form : Optimize Z = f(, 2, 3, ) Sub to g i (, 2, 3, ), =, b i, (i =,2 m), 2, 3, 0. THE PROPOSED METHOD - FOURIER ELIMINATION METHOD Solutio procedure: Step:. Covert the fuzzy umbers ito crisp umbers usig Graded Mea Itegratio Represetatio Method.. Step:2 Crisp umbers are ito liear programmig problem. Step:3 Select ad elimiate the variables from the previous step usig Fourier Elimiatio Method Step:3. Covert the miimizatio problem ito maimizatio problem as Mi(Z) = Ma( z) = Ma(W). Step:3.2 Trasform the objective fuctio i the form of liear iequalities. For elimiatig the variables, we use the followig procedure:

6 458 Dr. S. Murugaadam ad K. Hema Step:3.3 First write the maimizatio problem havig a iequality i the form of from the variable i a equality costrait i the give problem. Step:3.4 The write the equivalet pure iteger problem to the modified maimizatio problem. Step:3.5 Select ad remove a variable from previous step by Fourier elimiatio method Step:3.6 Write the set of iequalities icludig the objective fuctio after deletig the true statemets ad redudat costraits ad also usig Theorem. Step:3.7 Repeat step util all variables s are elimiated ecept the objective fuctio variable w. Step:3.8 Fid the optimum solutio for w. Step:3.9: The values of all s basic algebraic method. are calculated usig back substitutio method ad Step:3.0: The optimal problem is achieved. solutio ad the optimal allocatio for the assigmet NUMERICAL EXAMPLE Three persos are available to do three differet jobs. The cost (i Rupees) that each perso takes to do each job is kow ad are represeted by triagular fuzzy umbers ad are show i followig Table. Persos ( 7,8,9 ) ( 6,7,8 ) ( 26,27,28 ) Jobs ( 4,6,8 ) ( 8,20,22) ( 9,0,) ( 56,7 ) ( 23,25,27) ( 0, 2,4 )

7 Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio Method 459 Fid a optimal assigmet of persos to jobs that will miimize the total cost. Solutio: Fuzzy assigmet problem is coverted ito crisp assigmet problem usig Graded mea itegratio represetatio method. Jobs Persos Liear programmig model for the assigmet problem is give by Mi Z = Sub to = = = = = = 0, 2 0, 3 0, 2 0, 22 0, 23 0, 3 0, Mi Z= - Ma(-Z) = - Ma W = () Sub to = (2) 2 = (3) 3 = (4) 2 = (5) 3 = (6)

8 460 Dr. S. Murugaadam ad K. Hema Substitutig equatios (2) to (6) i equatio() costrait, we get ad rewritig the objective as W (7) (8) (9) (0) () Elimiatig 22, we get (3) (4) (5) (6) Applyig Fourier elimiatio method, we get Elimiatig a variable from the above equatios,we get w By back substitutio we get 33 =, 22 =, = & w = 40 Mi Z = -Ma(w)= 40 Mi Z = 40. Optimal assigmet:, 2 2, 3 3 Optimal assigmet cost =40

9 Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio Method 46 CONCLUSION Fuzzy umbers are coverted ito crisp values usig Graded mea itegratio represetatio method. Assigmet problem is writte i terms of pure liear programmig problem by cosiderig sum of each costrait equal to. Fourier elimiatio method is a ovel method used to elimiate the variables to get the optimal allocatio of workers ito jobs ad to get the optimal solutio for the give fuzzy assigmet problem. REFERENCES [] Ail, D.Gotare., Khot, P.G, 206, Solutio of fuzzy assigmet problem by usig Brach ad Boud Techique with applicatio of liguistic variable, Iteratioal Joural of Computers ad Techology.,5(4),pp [2] Athoy Przybylski, Xavier Gadibleua, Matthias Ehrgott, 200, A two phase method for multi-objective iteger programmig ad its applicatio to the assigmet problem with three objectives, Discrete Optimizatio (7)pp [3] Atul Kumar et.al, 206, Proposal of a algorithm for solvig multi-objective assigmet problem (MOAP) with liear ad o liear membership fuctios, Iteratioal Joural of Idustrial Egieerig ad Techology,(6),pp.-2 [4] Bellma, R.E.,Zadeh, L.A., 970, Decisio Makig i a Fuzzy Eviromet. Maagemet Sciece, 7, (4), B:4: B:64. [5] Chares,A.,Cooper,W.W.,Hederso,A.,953 A Itroductio to Liear Programmig problem. Joh Wiley & Sos, New York. [6] Che,S.H., ad Hsieh,C.H., 999, Graded mea itegratio represetatio of geeralized fuzzy umbers, Joural of Chiese fuzzy systems,(5).pp.-7. [7] De,P.K.,Bharati Yadav.,202, A Geeral approach for solvig assigmet problems ivolvig with fuzzy cost coefficiets,moder Applied Sciece, 6(3),pp.2-0 [8] Farahi M.H,, Asari.E,200, A ew approach to solve Multi-objective liear bi-level programmig problems, TJMCS e (4),pp [9] Kaiappa, P.,Thagavel,K., 998, Modified Fourier method for solvig liear programmig problems OPSEARCH,35,pp [0] Kayva Salehi., 998, A approach for solvig multi-objective assigmet problem with iterval parameters, Maagemet Sciece Letters, 4,pp

10 462 Dr. S. Murugaadam ad K. Hema [] Padia.P., Nataraja,G.,200, Fourier method for solvig Trasportatio problem with mied costraits, It. J. Cotemp. Math. Scieces, 5(28),pp [2] Williams,H.P.,986, Fourier s method of liear programmig ad its Dual, America Mathematical Mothly,93,pp