Assignment and Travelling Salesman Problems with Coefficients as LR Fuzzy Parameters

Transcription

1 Iteratioal Joural of Applied Sciece ad Egieerig., 3: 557 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters Amit Kumar ad Aila Gupta * School of Mathematics ad Computer Applicatios, Thapar Uiversity, Patiala, Idia Abstract: Mukherjee ad Basu (Applicatio of fuzzy rakig method for solvig assigmet problems with fuzzy costs, Iteratioal Joural of Computatioal ad Applied Mathematics, 5,, proposed a ew method for solvig fuzzy assigmet problems. I this paper, some fuzzy assigmet problems ad fuzzy travellig salesma problems are chose which caot be solved by usig the foremetioed method. Two ew methods are proposed for solvig such type of fuzzy assigmet problems ad fuzzy travellig salesma problems. The fuzzy assigmet problems ad fuzzy travellig salesma problems which ca be solved by usig the eistig method, ca also be solved by usig the proposed methods. But, there eist certai fuzzy assigmet problems ad fuzzy travellig salesma problems which ca be solved oly by usig the proposed methods. To illustrate the proposed methods, a fuzzy assigmet problem ad a fuzzy travellig salesma problem is solved. The proposed methods are easy to uderstad ad apply to fid optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma problems occurrig i real life situatios. Keywords: Fuzzy assigmet problem; fuzzy travellig salesma problem; Yager's rakig ide; fuzzy umber.. Itroductio The assigmet problem is a special type of liear programmig problem i which our objective is to assig umber of jobs to umber of persos at a miimum cost (time. The mathematical formulatio of the problem suggests that this is a programmig problem ad is highly degeerate. All the algorithms developed to fid optimal solutio of trasportatio problems are applicable to assigmet problem. However, due to its highly degeeracy ature, a specially desiged algorithm widely kow as Hugaria method proposed by Kuh [] is used for its solutio. Eamples of these types of problems are assigig me to offices, crew (drivers ad coductors to buses, trucks to delivery routes etc. Over the past 5 years, may variatios of the classical assigmet problem are proposed e.g. bottleeck assigmet problem, geeralized assigmet problem, quadratic assigmet problem etc. However, i real life situatios, the parameters of assigmet problem are imprecise umbers istead of fied real umbers because time/cost for doig a job by a facility (machie/perso might vary due to differet reasos. Zadeh [4] itroduced the cocept of fuzzy sets to deal with imprecisio ad vagueess i real life situatios. * Correspodig author; ailasigal@gmail.com Received 6 September Chaoyag Uiversity of Techology, ISSN 7794 Revised February Accepted 6 February It. J. Appl. Sci. Eg.,., 3 55

2 Amit Kumar ad Aila Gupta Sice the, sigificat advaces have bee made i developig umerous methodologies ad their applicatios to various decisio problems. Fuzzy assigmet problems have received great attetio i recet years [5, 8, 3, 8, ]. Travellig salesma problem is a wellkow NPhard problem i combiatorial optimizatio. I the ordiary form of travellig salesma problem, a map of cities is give to the salesma ad he has to visit all the cities oly oce ad retur to the startig poit to complete the tour i such a way that the legth of the tour is the shortest amog all possible tours for this map. The data cosists of weights assiged to the edges of a fiite complete graph ad the objective is to fid a cycle passig through all the vertices of the graph while havig the miimum total weight. There are differet approaches for solvig travellig salesma problem. Almost every ew approach for solvig egieerig ad optimizatio problems has bee tried o travellig salesma problem. May methods have bee developed for solvig travellig salesma problem. These methods cosist of heuristic methods ad populatio based optimizatio algorithms etc. Heuristic methods like cuttig plaes ad brach ad boud ca optimally solve oly small problems whereas the heuristic methods such as opt, 3opt, Markov chai, simulated aealig ad tabu search are good for large problems. Populatio based optimizatio algorithms are a kid of ature based optimizatio algorithms. The atural systems ad creatures which are workig ad developig i ature are oe of the iterestig ad valuable sources of ispiratio for desigig ad ivetig ew systems ad algorithms i differet fields of sciece ad techology. Particle Swarm Optimizatio, Neural Networks, Evolutioary Computatio, At Systems etc. are a few of the problem solvig techiques ispired from observig ature. Travellig salesma problems i crisp ad fuzzy eviromet have received great attetio i recet years [4, 6, 9, 4, 6, 7, 9, ]. With the use of fuzzy umbers, the computatioal efforts required to solve fuzzy assigmet problems ad fuzzy travellig salesma problem are cosiderably reduced [5]. Moreover, all types of crisp umbers, triagular fuzzy umbers ad trapezoidal fuzzy umbers ca be cosidered as particular cases of fuzzy umbers, thereby etedig the scope of use of fuzzy umbers. Mukherjee ad Basu [5] proposed a ew method for solvig fuzzy assigmet problems. I this paper, some fuzzy assigmet problems ad fuzzy travellig salesma problems are chose which caot be solved by usig the foremetioed method. Two ew methods are proposed for solvig such type of fuzzy assigmet problems ad fuzzy travellig salesma problems. The fuzzy assigmet problems ad fuzzy travellig salesma problems which ca be solved by usig the eistig method, ca also be solved by usig the proposed methods. But, there eist certai fuzzy assigmet problems ad fuzzy travellig salesma problems which ca be solved oly by usig the proposed methods. To illustrate the proposed methods, a fuzzy assigmet problem ad a fuzzy travellig salesma problem is solved. The proposed methods are easy to uderstad ad apply to fid optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma problems occurrig i real life situatios. This paper is orgaized as follows: I Sectio, basic defiitios ad Yager's rakig approach for the rakig of fuzzy umbers are discussed. I Sectio 3, formulatios of fuzzy assigmet problems ad fuzzy travellig salesma problems are preseted. I Sectio 4, limitatios of eistig method [5] are discussed. I Sectio 5, to overcome the limitatios discussed i Sectio 4, two ew methods are proposed to fid optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma 56 It. J. Appl. Sci. Eg.,., 3

3 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters problems. I Sectio 6, the advatages of proposed methods over eistig method are discussed ad illustrated by solvig two eamples. The results are discussed i Sectio 7 ad the coclusios are discussed i Sectio 8.]. Prelimiaries I this sectio, some basic defiitios ad Yager's rakig approach for the rakig of fuzzy umbers are preseted... Basic defiitios I this sectio, some basic defiitios are preseted. Defiitio. [7] A fuctio L :[, [,] (or R :[, [,] is said to be referece fuctio of fuzzy umber if ad oly if (i L ( L ( (or R ( R ( (ii L ( (or R ( (iii L (or R is oicreasig o [,. Defiitio. [7] A fuzzy umber A defied o the uiversal set of real umbers deoted as A (m,,,, is said to be a flat fuzzy umber if its membership fuctio A ( is give by: m L, A ( R, m,,.. Yager's rakig approach A umber of rakig approaches have bee proposed for comparig fuzzy umbers. I this paper, Yager's rakig approach [] is used for rakig of fuzzy umbers. This approach ivolves relatively simple computatioal ad is easily uderstadable. This approach ivolves a procedure for orderig fuzzy umbers i which a rakig ide ( A is calculated for a flat fuzzy umber A (m,,, from its A [m L (, R ( ] cut accordig to the followig formula: ( A ( ( m L ( d ( R ( d Let A ad B be two flat fuzzy umbers the (i A B if ( A ( B (ii (iii A B if ( A ( B A B if ( A ( B... Liearity property of Yager s rakig ide Let A (m ad B (m,,, be two flat fuzzy umbers ad k, k be two o egative real umbers. Usig Defiitio, the cut A ad B correspodig to A ad B are: A [ m L (, R m Defiitio 3. [7] Let A (m,,, be a flat fuzzy umber ad be a real umber i the iterval [,] the the crisp set A { X : A ( } [m L (, R ( ] ( ] ad B [ m L (, R ( ] Usig the property, ( A A ( A ( A, ( is set of real umbers, the cut (k A k B correspodig to k A k B is: is said to be cut of A. It. J. Appl. Sci. Eg.,., 3 57

4 Amit Kumar ad Aila Gupta k A k k m k m ( k k ( B [ L (, k k ( k k R ( ] Usig Sectio., the Yager's rakig ide ( k A k B correspodig to fuzzy umber k A k is: ( B ( B k A k k [ ( m L ( d ( R ( d k [ ( m L ( d ( R ( d] k ( A k ( B Similarly, it ca be proved that ( k A k B k ( A k ( B k k R 3. Liear programmig formulatios of fuzzy assigmet problems ad fuzzy travelig salesma problems I this sectio, liear programmig formulatios of fuzzy assigmet problems ad fuzzy travellig salesma problems are preseted [5]. 3.. Liear programmig formulatio of fuzzy assigmet problems Suppose there are jobs to be performed ad persos are available for doig these jobs. Assume that each perso ca do oe job at a time ad each job ca be assiged to oe perso oly. Let c be the fuzzy cost th i th (paymet if j job is assiged to perso. The problem is to fid a assigmet so that the total cost for performig all the jobs is miimum. The chose fuzzy assigmet problem may be formulated ito the followig fuzzy liear programmig problem (FLPP: Miimize subject i j to i j c j,, i,, or i, j P where, c is a trapezoidal fuzzy umber. i j c Total fuzzy cost : for performig all the jobs. ( 3.. Liear programmig formulatio of fuzzy travellig salesma problems Suppose a salesma has to visit cities. He starts from a particular city, visits each city oce ad the returs to the startig poit. th The fuzzy travellig costs from i city to th j city is give by c. The objective is to select the sequece (tour i which the cities are visited i such a way that the total travellig cost is miimum. The chose fuzzy travellig salesma problem may be formulated ito the followig FLPP: Miimize subject i j to i j c j,, ad j i ( i,, ad i j ( P i j (3 ji, i j k (4 ip jk p p ki p p3 p i ( 58 It. J. Appl. Sci. Eg.,., 3

5 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters, i p p, ( Optimal solutio of fuzzy travellig salesma problems where, c is a trapezoidal fuzzy umber. c : i j Total fuzzy travellig cost of completig the tour. if the salesma visits city j immediately after visitig city i ad otherwise. Costraits ( ad ( esure that each city is visited oly oce. Costrait (3 is kow as subtour elimiatio costrait ad elimiates all city subtours. Costrait (4 elimiates all 3city subtours. Costrait (5 elimiates all ( city subtours. For a feasible solutio of travellig salesma problem, the solutio should ot cotai subtours. So, for a 5city travellig salesma problem, we should ot have subtours of legth, 3 ad 4. For a 6city travellig salesma problem, we should ot have subtours of legth 3, 4 ad 5. Similarly, for a city travellig salesma problem, we should ot have subtours of legth to (. The optimal solutio of the fuzzy assigmet problem ( P is the set of oegative itegers { } which satisfies the followig characteristics: (i j,,, j i ad i i,,, i j ad also j satisfies subtour elimiatio costraits. (ii If there eist ay set of oegative itegers {' } such that ' j,,, j i ad i ' i,,, i j ad also j satisfies subtour elimiatio costraits, the ( c ( c '. i j i j 4. Limitatios of eistig method 3.3. Optimal solutio of fuzzy assigmet problems The optimal solutio of the fuzzy assigmet problem ( P is the set of oegative itegers { } which satisfies the followig characteristics: (i j,, ad i i,,. j (ii If there eist ay set of oegative itegers {' } such that ' j,, i ad ' i,,, j the ( i j c ( i j c '. I this sectio, the limitatios of eistig method are discussed. (i The eistig method [5] ca be applied oly to solve the followig type of fuzzy assigmet problems: Eample 4.. The fuzzy assigmet problem, solved by Mukherjee ad Basu [5], may be formulated ito the followig FLPP: Miimize (3,5,6,7 (5,8, (9,,5 3 (5,8,, 4 (7,8,, (3,5,6,7 (6,8,, (5,8,9, 4 (,4,5,6 3 (5,7,, 3 (8,3,5 33 (4,6,7, (6,8,, 4 (,5,6,7 4 (5,7,, 43 (,4,5,7 44 It. J. Appl. Sci. Eg.,., 3 59

6 Amit Kumar ad Aila Gupta + = = 3 4= = = + = 4 + = 33 + = = or i=, 3, 4 ad j =, 3, 4. The eistig method [5] caot be used for solvig the followig type of fuzzy assigmet problems: Miimize subject i j to i j j,, c i,, P or i, j where, c ( m,,, Fuzzy paymet to i i j c : th : perso for doig j th ( 3 job. Total fuzzy cost for performig all the jobs. Eample 4.. The fuzzy assigmet problem, for which the fuzzy costs are represeted by fuzzy umbers, may be formulated ito the followig FLPP: Miimize ((93,,, ((6,8,3,5 ((8,9,3 ((9,,,4 3 ((4,5,3 3 ( (,3,4 ((,3, ((7,8,3 ( ( 7,8,,3 33 3= = = 3= 3= 3 33= 3 = or i =, 3 ad j =, 3. where, L( maimum {, } ad R( maimum {, } (i The eistig method [5] ca be used for solvig followig type of fuzzy travellig salesma problems: Eample 4.3. The fuzzy travellig salesma problem, solved by eistig method [5], may be formulated ito the followig FLPP: Miimize (5, 8,, (9,,5 3 4 ( 7,8,, ( 5,8,9, 4 3 ( 5,7,, 3 ( 6,8,, 4 4 ( 5,7,, 43 ( 5,8,, (6,8,, (,4,5,6 ( 4,6,7, (,5,6,7 3+4= 3 4= + 4= 3 4= 3 3+ = 3 43= = = ,, = or i =,, 3, 4 ad j =,, 3, 4. The eistig method [5] caot be used for solvig the followig type of fuzzy travellig salesma problems: Miimize subject i j to i j c j,, ad j i i,, ad i j P i j ji, i j k jk ki ( 4 6 It. J. Appl. Sci. Eg.,., 3

7 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters ip p p p p3 p i, i p p, where, c (m,,, : Fuzzy travellig cost from i th city to j th city. c 5.. Method based o fuzzy liear programmig formulatio (FLPF : i j Total fuzzy travellig cost of completig the tour. if the salesma visits city j immediately after visi tig city i ad otherwise. Eample 4.4. The fuzzy travellig salesma problem, for which the fuzzy costs are represeted by fuzzy umbers, may be formulated ito the followig FLPP: Miimize I this sectio, two ew methods (based o fuzzy liear programmig formulatio ad classical methods are proposed to fid the optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma problems. ((9,,3 ((6,8,3,5 3 ( ( 8,9,3 4 ( ( 9,,,4 ( (,3, ( ( 4,5,3 4 ( ( 7,8,3 3 ( (,3,4 3 ( ( 7,8,,3 (( 9,, 3,5 4 (( 9, 3,4 4 (( 6, 8, = 3 4= + 4= 3 4= = 3 43= = 4 4 = I this sectio, a ew method (based o FLPF is proposed to fid the optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma problems occurrig i real life situatios. The steps of proposed method are as follows: Step Check that the chose problem is fuzzy assigmet problem or fuzzy travellig salesma problem. Case (i If the chose problem is fuzzy assigmet problem, the formulate it as ( P3. Case (ii If the chose problem is fuzzy travellig salesma problem, the formulate it as ( P4. Step Covert FLPP ( ( P3 or ( P4, obtaied i Step ito the followig crisp liear programmig problem: Miimize (c i j respective costraits. Step 3 Solve crisp liear programmig problem, obtaied i Step, to fid the optimal solutio {} ad = or i =,3,4 ad j =,3,4. where, Yager's rakig ide L ( maimum {, } ad R ( maimum {, } correspodig to miimum total fuzzy cost. 5. Proposed methods to fid the optimal solutio of fuzzy assigmet problems ad fuzzy travellig salesma problems (c i j 5.. Method based o classical methods I this sectio, a ew method (based o classical methods is proposed to fid the It. J. Appl. Sci. Eg.,., 3 6

8 Amit Kumar ad Aila Gupta optimal solutio of fuzzy assigmet problems or fuzzy travellig salesma problems occurrig i real life situatios. The steps of the proposed method are as follow: Step Check that the chose problem is fuzzy assigmet problem or fuzzy travellig salesma problem. Case (i If the chose problem is fuzzy assigmet problem, the formulate it as ( P 3. Represet ( P 3 ito tabular form as show i Table. Case (ii If the chose problem is fuzzy travellig salesma problem, the formulate it as ( P 4. Represet ( P 4 ito tabular form as show i Table. Step Covert the fuzzy assigmet problem or fuzzy travellig salesma problem obtaied i Step ito crisp problem as follows: Case (i For ( P 3 costruct a ew Table 3 as show below: Case (ii For ( P 4 costruct a ew Table 4 as show below: Step 3 Solve crisp liear programmig problem, obtaied i Step, to fid the optimal solutio } ad { Yager's rakig ide ( c i j correspodig to miimum total fuzzy cost. Job Perso P Table. Fuzzy assigmet costs J J J j J c c c j c P j c j c j c jj c j P c c c j c where, c ( m,,, Table. Fuzzy travellig costs City j c c j c j c j c j c j c c c j where, c ( m,,, 6 It. J. Appl. Sci. Eg.,., 3

9 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters Table 3. Crisp assigmet costs Job Perso J J Jj J P ( c ( c (c j (c Pj (cj (cj ( c jj (cj P (c (c ( cj (c where, ( c ( ( m L ( d ( R ( d Table 4. Crisp travellig costs City j ( c (c j (c j (cj (cj (cj (c (c ( cj where, ( c ( ( m L ( d ( R ( d 6. Advatage of the proposed methods over eistig method I this sectio, the advatage of the proposed methods over eistig method are discussed: The eistig method [5] ca be used for solvig oly such fuzzy assigmet problems wherei all the cost parameters are represeted by trapezoidal or triagular fuzzy umbers. However, i real life situatios, it is ot possible to represet all the parameters by trapezoidal or triagular fuzzy umbers i.e., there may eist certai fuzzy assigmet problems i which some ucertai parameters are represeted by fuzzy umbers give by model ( P3. The eistig method [5] ca ot be used for solvig such fuzzy assigmet problems ( P3. The mai advatage of both the proposed methods is that these ca be used for solvig both type of fuzzy assigmet problems ad i additio, both types of fuzzy travellig salesma problems. To show the advatage of proposed methods over eistig methods [5], fuzzy assigmet ad fuzzy travellig salesma problems chose i Eample 4. ad Eample 4.4, which caot be solved by usig the eistig method [5], are solved by usig the proposed methods. It. J. Appl. Sci. Eg.,., 3 63

10 Amit Kumar ad Aila Gupta 6.. Optimal solutio of fuzzy assigmet problem I this sectio, fuzzy assigmet problem chose i Eample 4., which caot be solved by usig the eistig method, is solved by usig the proposed methods Optimal solutio usig the method based o FLPF The optimal solutio of fuzzy assigmet problem, chose i Eample 4., may be obtaied by usig the followig steps of the proposed method: Step The FLPF of fuzzy assigmet problem, chose i Eample 4. is: Miimize (( 9,, 3 ((6,8,3,5 ((8,9,3 3 ((9,,,4 ((,3, ((4,5,3 ((7,8,3 3 ((,3,4 3 ((7,8,, = 3 = =, 3 = =, 3 33 = = or i=, 3 ad j=, 3. Step Usig Step of proposed method, the formulated FLPP is coverted ito the followig crisp liear programmig problem: Miimize ( (9,, 3 ( (6,8,3,5 ( (8,9,3 3 ( (,3, ( (7,8,3 3 3 ( (7,8,,3 33 ( (9,,,4 ( (4,5,3 ( (,3,4 = = 3 = 3 33 = = = 33 = or i=, 3 ad j=, 3. Step 3 Usig Defiitio ad Sectio., the values of c, i, j are , c 7.5, c , c , c 9.75, c , c , c 3.5, c c 3 3 Usig the values of c the crisp liear programmig problem obtaied i Step, may be writte as: Miimize ( = = 3 = 3 33 = = = 33 = or i=, 3 ad j=, 3. Step 4 Solvig the crisp liear programmig problem, obtaied i Step 3, the optimal solutio is 3 ad miimum total fuzzy cost is (7,5,. Yager's rakig ide correspodig to miimum total fuzzy cost is Optimal solutio usig method based o classical assigmet method The optimal solutio of fuzzy assigmet problem, chose i Eample 4., by usig the method based o classical assigmet method, proposed i Sectio 5., may be obtaied as follows: Step The tabular represetatio of fuzzy assigmet problem chose i Eample 4. is: Step Usig Step of proposed method the It. J. Appl. Sci. Eg.,., 3

11 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters fuzzy assigmet problem show i Table 5 may be coverted ito crisp assigmet problem as show i Table 6. Step 3 The optimal solutio of the crisp liear programmig problem, obtaied i 3 Step, is ad miimum total fuzzy cost is (7,5,. Yager's rakig ide correspodig to miimum total fuzzy cost is.83. The membership fuctio of the type fuzzy umber represetig the miimum total fuzzy cost of the fuzzy assigmet problem, chose i Eample 4., is show i Figure. Table 5. Fuzzy costs as fuzzy umbers Job Perso J J J3 P (9,,3 (6,8,3,5 (8,9,3 P ( 9,,,4 (,3, ( 4,5,3 P3 (7,8,3 (,3,4 (7,8,,3 Table 6. Crisp costs J J J3 P P P Degree of Membership fuctio Job Perso Left shape fuctio. Right shape fuctio Optimal cost Figure. Membership fuctio of LR fuzzy umber represetig the miimum total fuzzy assigmet cost It. J. Appl. Sci. Eg.,., 3 65

12 Amit Kumar ad Aila Gupta 6.. Optimal solutio of fuzzy travellig salesma problems I this sectio, fuzzy travellig salesma problem chose i Eample 4.4, which caot be solved by usig the eistig method, is solved by usig the proposed methods Optimal solutio usig the method based o FLPF The optimal solutio of fuzzy travellig salesma problem, chose i Eample 4.4, may be obtaied by usig the followig steps of the proposed method: Step The FLPF of the fuzzy travellig salesma problem, chose i Eample 4.4, is: Miimize (( 9,, 3 ((6,8,3,5 ((8,9,3 3 4 ((,3, 4 ((7,8,3 3 3 ((7,8,,3 4 ((9, 3,4 4 ((9,,,4 ((4,5,3 ((,3,4 (( 9,, 3,5 (( 6,8, = 3 4= + 4= 3 4= 3 3+= 3 43= = = ,,, = or i=, 3, 4 ad j=, 3, 4. Step Usig Step of proposed method, the formulated FLPP is coverted ito the followig crisp liear programmig problem: Miimize ( (9,, 3 ( (6,8,3,5 ( (8,9,3 3 4 ( (9,,,4 ( (,3, 4 ( (7,8,3 3 3 ( (7,8,,3 4 ( (9,3,4 4 ( (4,5,3 ( (,3,4 ( (9,,3,5 ( (6,8, = 3 4= + 4= 3 4= 3 3+ = 3 43= = = ,,, = or i=,, 3, 4 ad j=,, 3, 4. Step 3 Usig Defiitio ad Sectio., the values of c, i, j are c , c3 7.5, c , c , c 9.75, c , c , c3.5, c ,, c , c4, c Usig the values of c, the crisp liear programmig problem obtaied i Step, may be writte as: Miimize ( = 3 4= + 4= 3 4= 3 3+ = 3 43= = = ,,, 66 It. J. Appl. Sci. Eg.,., 3

13 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters Step The tabular represetatio of fuzzy travellig salesma problem, chose i Eample 4.4, is: Step Usig Step of proposed method, the travellig salesma problem show i Table 7 may be coverted ito crisp travellig salesma problem as show i Table 8. Step 3 The optimal solutio of the crisp liear programmig problem, obtaied i Step, is ad miimum total fuzzy cost is ( 6,,4. Yager's rakig ide correspodig to miimum total fuzzy cost is The membership fuctio of the type fuzzy umber represetig the miimum total fuzzy cost of the fuzzy travellig salesma problem, chose i Eample 4.4 is show i Figure. = or i =,3,4 ad j =,3,4. Step 4 The optimal solutio of the crisp liear programmig problem, obtaied i Step 3, is ad miimum total fuzzy cost is ( 6,,4. Yager's rakig ide correspodig to miimum total fuzzy cost is Optimal solutio usig the method based o classical travellig salesma method The optimal solutio of the fuzzy travellig salesma problem chose i Eample 4.4 by usig the method based o classical travellig salesma method, proposed i Sectio 5., ca be obtaied as follows: Table 7. Fuzzy costs as fuzzy umbers City 3 4 (9,,3 (6,8,3,5 (8,9,3 ( 9,,,4 (,3, ( 4,5,3 3 (7,8,3 (,3,4 (7,8,,3 4 (9,, 3,5 (9, 3,4 (6,8,5 Table 8. Crisp costs City It. J. Appl. Sci. Eg.,., 3 67

14 Amit Kumar ad Aila Gupta. Degree of membership fuctio left shape fuctio Right shape fuctio 4 6 Optimal cost Figure. Membership fuctio of LR fuzzy umber represetig the miimum total fuzzy travellig cost 7. Results ad discussio To compare the eistig [5] ad the proposed methods, the results of fuzzy assigmet problems ad fuzzy travellig salesma problems chose i Eample 4. Eample 4., Eample 4.3 ad Eample 4.4 obtaied by usig the eistig ad the proposed methods are show i Table 9. It is obvious from the results show i Table 9 that irrespective of whether we use eistig or proposed methods, same results are obtaied for Eample 4. ad Eample 4.3, while, Eample 4. ad Eample 4.4 ca be solved oly by usig the proposed methods. O the basis of above results, it ca be suggested that it is better to use the proposed methods istead of eistig method [5] to solve fuzzy assigmet problems ad fuzzy travellig salesma problems. limitatio, two ew methods are proposed. By comparig the results of the proposed methods ad eistig method, it is show that it is better to use the proposed methods istead of eistig method. I future, the proposed method may be modified to fid fuzzy optimal solutio of fuzzy assigmet problems, fuzzy travellig salesma problems ad geeralized assigmet problems with ituitioistic fuzzy umbers. 8. Coclusios I this paper, limitatio of a eistig method [5] for solvig fuzzy assigmet problems ad fuzzy travellig salesma problems is discussed ad to overcome this 68 It. J. Appl. Sci. Eg.,., 3

15 Assigmet ad Travellig Salesma Problems with Coefficiets as Fuzzy Parameters Table 9. Compariso of results obtaied by usig eistig ad proposed methods Proposed method Eistig method Proposed method Eamples based o classical [6] based o FLPF methods J P3, J P, J P3, J P, J P3, J P, 4. J 3 P, J 4 P4 ad miimum total (6,,7,35 4. Not applicable ad miimum total (5,5, Not applicable J 3 P, J 4 P4 ad miimum total (6,,7,35 J P3 J 3 P, J 4 P4 ad miimum total (6,,7,35 J P3 J P, J P, J 3 P ad miimum total (7,5, 4 3 ad miimum total (5,5, ad miimum total ( 6,,4 J 3 P ad miimum total (7,5, 4 3 ad miimum total (5,5, ad miimum total ( 6,,4 Refereces [ ] Adreae, T.. O the travellig salesma problem restricted to iputs satisfyig a relaed triagle iequality. Networks, 38: [ ] Blaser, M., Mathey, B., ad Sgall, J. 6. A improved approimatio algorithm for the asymmetric TSP with stregtheed triagle iequality. Joural of Discrete Algorithms, 4: 663. [ 3] Bockehauer, H. J., Hromkovi, J., Klasig, R., Seibert, S., ad Uger, W.. Towards the otio of stability of approimatio for hard optimizatio tasks ad the travellig salesma proble m. Theoretical Computer Sciece, 85:. [ 4] Chadra, L. S. ad Ram, L. S. 7. O the relatioship betwee ATSP ad the cycle cover problem. Theoretical Computer Sciece, 37: 88. [ 5] Che, M. S O a fuzzy assigmet problem. Tamkag J., : 474. [ 6] Crisa, G. C. ad Nechita, E. 8. Solvig Fuzzy TSP with At Algorithms. Iteratioal Joural of Computers, Commuicatios ad Cotrol, III (Suppl. It. J. Appl. Sci. Eg.,., 3 69

16 Amit Kumar ad Aila Gupta issue: Proceedigs of ICCCC 8, 8. [ 7] Dubois, D. ad Prade, H. 98. Fuzzy Sets ad Systems: Theory ad Applicatios. New York. [ 8] Feg, Y. ad Yag, L. 6. A twoobjective fuzzy kcardiality assigmet problem. Joural of Computatioal ad Applied Mathematics, 97: 4. [ 9] Fischer, R. ad Richter, K. 98. Solvig a multiobjective travellig salesma problem by dyamic programmig. Optimizatio, 3: 475. [] Kuh, H. W The Hugaria method for the assigmet problem. Naval Research Logistics Quarterly, : [] Li, C. J. ad We, U. P. 4. A labelig algorithm for the fuzzy assigmet problem. Fuzzy Sets ad Systems, 4: [] Liu, L. ad Gao, X. 9. Fuzzy weighted equilibrium multijob assigmet problem ad geetic algorithm. Applied Mathematical Modellig, 33: [3] Majumdar, J. ad Bhuia, A. K. 7. Elitist geetic algorithm for assigmet problem with imprecise goal. Europea Joural of Operatioal Research, 77: [4] Melamed, I. I. ad Sigal, I. K The liear covolutio of criteria i the bicriteria travellig salesma problem. Computatioal Mathematics ad Mathematical Physics, 37: 995. [5] Mukherjee S. ad Basu, K.. Applicatio of fuzzy rakig method for solvig assigmet problems with fuzzy costs. Iteratioal Joural of Computatioal ad Applied Mathematics, 5: [6] Padberg, M. ad Rialdi, G Optimizatio of a 53city symmetric travellig salesma problem by brach ad cut. Operatios Research Letters, 6: 7. [7] Rehmat, A., Saeed H., ad Cheema, M. S. 7. Fuzzy multiobjective liear programmig approach for travellig salesma problem. Pakista Joural of Statistics ad Operatio Research, 3: [8] Sakawa, M., Nishizaki, I., ad Uemura, Y.. Iteractive fuzzy programmig for twolevel liear ad liear fractioal productio ad assigmet problems: a case study. Europea Joural of Operatioal Research, 35: 457. [9] Segupta, A. ad Pal, T. K. 9. Fuzzy Preferece Orderig of Iterval Numbers i Decisio Problems. Berli. [] Sigal, I. K A algorithm for solvig largescale travellig salesma problem ad its umerical implemetatio. USSR Computatioal Mathematics ad Mathematical Physics, 7: 7. [] Wag, X Fuzzy optimal assigmet problem. Fuzzy Math., 3: 8. [] Yager, R. R. 98. A procedure for orderig fuzzy subsets of the uit iterval. Iformatio Scieces, 4: 436. [] Ye, X. ad Xu, J. 8. A fuzzy vehicle routig assigmet model with coectio etwork based o prioritybased geetic algorithm. World Joural of Modellig ad Simulatio, 4: [4] Zadeh, L. A Fuzzy sets. Iformatio ad Cotrol, 8: [5] Zimmerma, H. J Fuzzy Set Theory ad its Applicatio. Bosto. 7 It. J. Appl. Sci. Eg.,., 3