Group Assignment Job Constrained Three Dimensional Model

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1 Group Assigmet Job Costraied Three Dimesioal Model Vijayalakshmi R #1, Revathi P #2, Sriivasulu Y #3 Sudara Murthy M #4 #1 Research Scholar, Dept., of Mathematics, Sri Vekateswara Uiversty, Tirupati, Idia # 2Assistat professor, Dept., of Mathematics, Brahmmaiah Egieerig College, Kovur, Nellore Dist. Idia #3 Research Scholar, Dept., of Mathematics, Sri Vekateswara Uiversty, Tirupati, Idia #4 Professor Dept., of Mathematics, Sri Vekateswara Uiversity, Tirupati, Idia Abstract - I a classical assigmet problem the goal is to fid a optimal assigmet for agets with tasks without assigig a aget more tha oce ad esurig that all tasks should complete with miimum cost. I this paper we study the problem called Group Assigmet Job Costraied Three Dimesioal Model. Let us cosider a set of workers W={1,2,,w}, a set of jobs J={1,2,,} ad K={1,2,..,k} represets facilities which ifluece the cost as a third dimesio. The set of W workers due to their idetical skills are subdivided ito p differet groups, i th group is havig w i = W i workers as a result total workers are w, the set of J jobs are subdivided ito q differet groups such that i each group have j = J i jobs with total jobs. Let there are s products which require the jobs as compoets for its fiishig. Let the products be called frames ad they are F 1, F 2,,F s ad their frequecies are l 1, l 2,,l s. Let f ij be the umber of jobs required from j th group for the i th frame the s i 1 f. l j. The umber of all jobs from the q groups required are ij i = j 1 j is the umber of jobs required for the s frames from the j th group of jobs ad j 1 q j 1 1 j There is a restrictio that the jobs from same group should use same facility. Where C(i,k) is the assigmet cost of doig a job i j th group by a worker i i th group usig k th facility. The problem is to assig the total umber of jobs 0 which are required for the s frames subjected to the coditio such that the total assigmet cost is miimum. A Lexi Search algorithm is developed usig Patter ecogitio Techique to get a optimal assigmet. Key words- Lexi Search algorithm, Patter Recogitio Techique, Alphabet table, Word, Search table. 0. I. INTRODUCTION Assigmet problem is amog the first liear programmig problems to be studied extesively. It is a particular case of a trasportatio problem where the sources are assigees ad the destiatios are tasks. I a classical assigmet problem the goal is to fid a optimal assigmet of agets to tasks without assigig a aget more tha oce ad esurig that all tasks are completed. I this paper we study the problem called Group Assigmet Job Costraied Three Dimesioal Model. Let us cosider a set of workers W, aother set of jobs J. The third dimesio which is a idepedet factor which iflueces cost is cosidered as facility which is deoted by K. Agai workers W are cosidered as p groups, i th group is havig w i workers as a result total workers are w, jobs J are cosidered as q groups such that i each group have j jobs with total jobs ad facility K, C(i,k), is the cost of doig a job i j th group whe assiged to a worker i i th group usig facility k. Let the jobs J={1,2,3,.,} be grouped as J 1, J 2,., J q such that J= J 1 U J 2 U.. U J q with IJ j I = j, ljl =, i.e., q =. Similarly W = {1, 2,.., m} be the set of m workers grouped as W 1, W 2,, W p such that W = W 1 U W 2 U U W p with lw i l = w i ad lwl = w, that is w 1 + w w p = w. ISSN: Page 85

2 Let there be S frames that is F i =(i=1,2,,s) these frames require various jobs as compoets. Let f ij be the umber of jobs required as compoets from the j th group of job for the i th frame. Let l i be the umber of frames of the i th frame that is l i is the frequecy of the frame F i. The total umber of jobs required from j th group as compoets for the S frames alog with their frequecies is give by s i 1 f ij. l That is 1 j are the total umber of jobs from j th group is required for all the frames alog with their frequecies. The total umber of jobs required for all the frames is give by q j 1 1 j 0 i 1 j j ad whe jobs are assiged to workers from same group they have to use same facility. Whe the cost C(i,k) is cosider i the process of assigmet if w i ad j are respectively the workers i the i th group ad jobs i the j th group the the miimum of mi ( w i, j ) = α ij will be iteger if it is o-zero ad i this case the α ij is the umber of workers will be assiged to α ij umber of jobs with the cost C(i,k)* α ij where the k th facility is used ad a fixed umber of jobs( 1 j ) has to perform from each job group ( j ) Now our problem is to assig 0 jobs for the workers subject to the coditios such that total assigmet cost is miimum. We develop a Lexi-Search algorithm usig patter recogitio techique for gettig a optimal assigmet with total least cost. II. MATHEMATICAL FORMULATION Miimize Z (X) = i W j J k K c ( i, k). ij. x( i, k) (1) Where α ij = Miimum ( w i, j ) (i.e., i=1,2,3,.,p & j=1,2,3.,q {Where ( w i, j ) are the workers i i th group ad Jobs i the j th group uassiged i the process ad α ij is a iteger} Subject to the costraits i j k.. (2) s j k i s k ij x ( i, k) 0 fori 1,2,..., p; j 1, 2,... q, k sj x ( s, k) Ws, j (1,2,..., q), k K (3) is x( i, s, k) s, i (1,2,..., p), k K..(4) x(i 1, j 1, k 1 ) = x(i 2, j 2, k 2 ) = 1, where i 1, i 2 є (1,2,,p) j 1 =j 2 & k 1 =k 2...(5) x(i, k) =0 or 1,... (6) i=1,2,.p ; j=1,2,.q & kєk The costrait (1) is the objective fuctio of the problem i.e., total miimum cost for assiged 0 jobs uder the give costraits. The costrait (2) describes the restrictio that the total umber of assiged jobs 0, less tha. The costrait (3) states the umber of assiged workers i a group is less tha or equal to its capacity. The costrait (4) shows the umber of jobs i a group is less tha or equal to its capacity. The costrait (5) illustrates that same job group whe assiged to workers group should use the same facility. The costrait (6) describes if the i th worker is assiged to j th group of job with k th facility the X(i, k) = 1 otherwise 0 III NUMERICAL ILLUSTRATION The cocepts developed will be illustrated by a umerical example for which the total umber of workers w = 15, the total umber of jobs =20 ad the facilities k = 2. Agai these 20 workers made as 4 groups,15 jobs made as 6 groups i.e.,p=4 ad q=6 groups respectively. Let the umber of workers i each group be w 1 =3, w 2 =4, w 3 =5, w 4 = 3 ad jobs as J 1 =2, J 2 =4, J 3 =3, J 4 =4, J 5 =4, J 6 =3. To make l i frequecy of s frames of F a fixed umber of jobs has to be assiged from each group that is =1, 2 =3, 1 3 =2, 1 4 =3, 1 5 =4, 1 6 =2 ad if a job i oe set used oe of facilities the the remaiig jobs i that set should use the same facility. For this problem we took four groups of workers, six groups of jobs with two facilities ad the umber of jobs to be assiged * are = = 15 (i.e., 0 =15) K ISSN: Page 86

3 Let there be two frames F 1, F 2 ad the frequecies of frames are l 1 =2, l 2 =1. For these two frames job compoets from differet groups are take as product compoet f ij. The followig figure-1 represets frames, job compoets ad their frequecies. FIGURE-1 F 1 F 2 C(i, j,1) = Table Table C(i, j, 2) = I table-1, C(3, 2, 1) = 10 meas that the cost of assigig a job i 2 d group by ay idividual worker o 3 rd group usig 1 st facility is 10 uits. IV FEASIBLE SOLUTION Cosider a ordered triple set {(1, 1, 1), (2, 5, 1), ( 1, 3, 2), ( 3, 6,, 2), ( 3, 4, 1), (4, 2,2)} represets a feasible solutio metioed below l 1 =2 l 2= 1 I the above figure F 1, F 2 represets frames, l 1 =2, l 2 =1 represets frequecies of frames. Frame F 1 ivolves 6 job compoets ad they are represeted as 1 1 =0, 1 2 =1, 1 3 =1 1 4 =0, 1 5 =2, 1 6 =1. f 2 frame ivolves 6 job compoets ad they are represeted as 1 1 =1, 1 2 =1, 1 3 =0, 1 4 =3, 1 5 =0, 1 6 =0. As a result total umber of job compoets i two frames is 15 i.e., =15 also sum of 1 i i two frames are satisfyig the restrictio that they should be equal to the fixed umber of jobs assiged from each group. For this problem we took four groups of workers, six groups of jobs usig two facilities ad the umber of jobs to be assiged is fiftee (i.e., 1 =15) I the followig umerical example, C(i, k) s are take as positive itegers but it ca be easily see that this is ot a ecessary coditio. C(i, k) meas the cost of assigig of that i th worker o j th job with facility k. The followig table represets the requiremet of the cost to do the job with respect to correspodig worker. The the cost array C(i, k) is give table-1. The above figure-2, represets a feasible solutio. The rectagle shape represets worker group, hexago shape represets job group, diamod shape represet umber of jobs assiged, octago shape represets facility, parallelogram shape represets the ISSN: Page 87

4 correspodig C(i, k) cost ad oval shape represets multiple of umber of jobs assiged ad cost. The values i rectagle idicate group of the worker ad i brackets represets umber of uassiged workers i that group, values i hexago idicates group of jobs ad i brackets represets the umber of uassiged jobs i that group, values i diamod idicates umber of jobs assiged, values i octago idicates facility, value i parallelogram idicates cost ad value i oval idicates (umber of jobs assiged x cost0f each job). Here { w 1 (3), j 1 (1), α(1), k(1), C(1), αxc(1) } represets that w 1 (3) meas i 1 st group umber of uassiged workers is 3, j 1 (1) meas i 1 st group umber of uassiged jobs is 1, α(1) meas the umber of jobs of j 1 assiged to w 1 workers is 1 (which is required), k(1) represets the facility for performig particular job is 1,C(1) represets the cost of that particular job performed by the worker usig facility is 1 ad αxc(1) represets the product of umber of assiged jobs ad cost. SOLUTION PROCEDURE: I the above figure-2, for the feasible solutio we observed that there are 6 ordered triples (1, 1, 1), (2, 5, 1), ( 1, 3, 2), ( 3, 6, 2), ( 3, 4, 1), (4, 2, 2) take alog with the value from the cost matrices i the umerical example i table-1. The 6 ordered triples are selected such that they represets a feasible solutio accordig to costraits of mathematical formulatio ad is represeted i figure-2. So the problem is that we have to select 6 ordered triples from the cost matrices alog with values such that the total cost is miimum ad represets a feasible solutio. For this selectio of 6 ordered triples from cost matrices we arraged 41 ordered triples with the icreasig order of their values ad call this formatio as alphabet table ad we will develop a algorithm for the selectio of six ordered triples alog with the checkig for the feasibility. feasible if X is a feasible solutio. The patter represeted i the table-2, is a feasible patter. Now V(X) the value of the patter X is defied as V ( X ) C( i, k). X ( i, k) i p j q ij The value V(X) gives the total cost of the feasible solutio represeted by X. Thus the value of the feasible patter gives the total cost represeted by it. I the algorithm, which is developed i the sequel, a search is made for a feasible patter with the least value. Each patter of the solutio X is represeted by the set of ordered triples (i, k)for which X (i, k) =1, with the uderstadig that the other X (i, k) s are zeros. VI ALPHABET TABLE AND A WORD There are p q k ordered triples i the threedimesioal array C. For coveiece these are arraged i ascedig order of their correspodig costs ad are idexed from 1 to pxqxk (Sudara Murthy-1979). Let SN= 1, 2, 3 be the set idices. Let C be the correspodig array of costs. If a, b SN ad a<b the C (a) C(b). Also let the arrays R, C, K be the array of idices of the ordered triples represeted by SN, J ad K. CC is the array of cumulative sum of the elemets of C. The arrays SN, C, CC, R, C, ad K for the umerical example are give i the table-3. If p SN the (R(p),C(p),K(p)) is the ordered triple ad C(a)=C(W(a),J(a),K(a)) is the value of the ordered triple. Table-3. V THE CONCEPTS AND DEFINITIONS 5.1 Defiitio of a Patter: A idicator three-dimesioal array which is associated with a assigmet is called a patter. A patter is said to be ISSN: Page 88

5 The V(L 3 ) =V(L 2 )+ (D (a k )) (α ijk ), whereα ijk =α 251 = miimum (w i 1,j i 1,k) miimum(w 2 1,j 5 1,1) = 4 jobs, D(a 3 ) = 1 For example the partial word L 3 = (1, 2, 3) the value of L 3 is V (L 3 ) = = 3 ad V(x) = 5 ad L 3 = V(L 3 ) = V(L 2 )+C(a k )(α i,k ) = x 0 = Lower Boud of A partial word LB (L k ): A lower boud LB (L α ) for the values of the block of words represeted by L k = (a 1, a 2, , a k ) ca be defied as follows. a k+1 ) x ( 0 1 ), { where 0 1 represets the umber of jobs remaiig to be assiged} Cosider the partial word L 3 = (1, 2, 3) ad V ( L 3 ) =5 The LB (L k ) = a k+1 ) ( 0 1 ) LB (L 3 ) = V(L 3 ) + ( C ( a 4 ) (6) ) = = 25 LB (L k ) = V (L k ) + ( a k + j ) = V (L k ) + CC (a k + - k) - CC (a k ) Let us cosider 7 є SN it represets the ordered triple R(7), C(7), K(7) = ( 3, 2, 2) the C(7)=4, CC(7) = Value of the Word: The value of the (partial) word L k, V (L k ) is defied recursively as V (L k ) = V (L k-1 ) + D (a k ) x (α i,k ) with V (L o ) = 0 where D (a k ) is the cost array arraged such that D (a k ) < D (a k+1 ). V (L k ) ad V(x) the values of the patter X will be the same. Sice X is the (partial) patter represeted by L k, (Sudara Murthy 1979). Where CC(a k )= k i 1 C( ). It ca be see that LB(L k ) is the a i value of the complete word, which is obtaied by cocateatig the first (-k) letters of SN (a k ) to the partial word L k. 6.3 Feasibility Criterio of a partial word: A algorithm is developed, i order to check the feasibility of a partial word L k+1 = {a 1, a 2...a k, a k+1 } give that L k is a feasible word. We will itroduce some more otatios which will be useful i the sequel. Cosider the partial word L 3 = (1, 2, 3) ISSN: Page 89

6 IR be a array where IR(i) = 1, idicates that a worker of i th group is doig a job, otherwise IR(i)=0. IC be a array where IC (j) = 1, idicates that i j th group a job is performed by a worker, otherwise IC(j)=0. IK be a array where IK(i)=1, idicates that facility used by j th group of job L be a array where L[i] =a i is the letter i the i th positio of a word.the the values of the arrays IR, IC ad L are as follows IR (RA) = 1, i = 1, 2, , k ad IR (j) = 0 for other elemets of j. (where RA = R(a i ),CA = (a i ) ) IC (CA ) = 1, i = 1, 2, , k ad IC (j) = 0 for other elemets of j L (i) = a i, i = 1, 2, , k, ad L(j) = 0, for other elemets of j NA (a k ) = α ij = [w 1 (RA), 1 (CA)] the umber of jobs assiged to workers at i th positio. For example cosider a sesible partial word L 5 = (1, 2, 4, 5, 6) which is feasible. The array IR, IC, IK ad L takes the values represeted i table 4 give below. The recursive algorithm for checkig the feasibility of a partial word L p is give as follows. I the algorithm first we equate IX = 0, at the ed if IX = 1 the the partial word is feasible, otherwise it is ifeasible. For this algorithm we have RA=R (a i ), CA=C (a i ) ALGORITHMS ALGORITHM 1: (Algorithm for feasible checkig) This recursive algorithm will be used as a subroutie i the lexi-search algorithm. We start the algorithm with a very large value, say, 9999 as a trial value of VT. If the value of a feasible word is kow, we ca as well start with that value as VT. Durig the search the value of VT is improved. At the ed of the search the curret value of VT gives the optimal feasible word. We start with the partial word L 1 = (a 1 ) = (1). A partial word L p =L p-1 (a p ) where idicates chai form or cocateatio. We will calculate the values of V (L p ) ad LB (L p ) simultaeously. The two cases arises (oe for brachig ad other for cotiuig the search). 1. LB (L p ) < VT. The we check whether L p is feasible or ot. If it is feasible we proceed to cosider a partial word of order (p+1), which represets a sub block of the block of words represeted by L p. If Lp is ot feasible the cosider the ext partial word of order p by takig aother letter which succeeds a p i ISSN: Page 90

7 the p th positio. If all the words Sof order p are exhausted the we cosider the ext partial word of order (p-1). 2. LB (L P ) VT. I this case we reject the partial word meaig that the block of words with L p as leader is rejected for ot havig a optimal word ad we also reject all partial words of order p that succeeds L p. Now we are i a positio to develop lexi search algorithm to fid a optimal feasible word ALGORITHM2: (LEXI-SEARCH ALGORITHM) word. The followig algorithm gives a optimal feasible STEP-1: (iitializatio): The arrays SN, D, DC, R, C, K, NA(= 0 ), WA, JA, KA, max, ad VT are made available. RA, CA, KA, L, V, LB, WX, JX, JM, W, N, KN, LN,CJM, LNM ad CJN are iitialized to zero. The values i=1, j=0. o goto 13 Step 13 : STOP ISSN: Page 91

8 6.5. FLOW CHART: The flow chart for this algorithm is as follows Table-5 Vll SEARCH TABLE: The workig details of gettig a optimal word, usig the above algorithm for the illustrative umerical example are give i the Table-5. The colums (1), (2), (3), (4),. gives the letters i the first, secod, third, fourth so o respectively. The correspodig NA, V ad LB are idicated i the ext three colums. The rows R, C ad K gives the row, colum ad facility idices of the letter. The last colum gives the remarks regardig the acceptability of the partial words. I the followig table A idicates ACCEPT ad R idicates REJECT. At the ed of the search the curret value of VT = 40 ad it the value of the feasible word L 6 =(1,2,4,5,7,8) it is give i 13 th row of the search table 4 ad the correspodig order triples are (1, 1, 1), (2, 5, 1), (1, 3,2), (3, 6, 2), (3, 2, 2), (4, 4, 1). For this optimal feasible word the arrays IR, IC, IK, L ad NA are give i the followig Table- 6. ISSN: Page 92

9 worker usig facility ad αxc(5x3=15) assiged jobs x cost. is the umber of Accordig to the patter represeted i figure-3 is satisfies all the costraits the sectio 3. The ordered tripled set represets the cost of total umber of assiged jobs. The total cost = =40. Vlll CONCLUSION I this chapter, we studied a model of Assigmet problem amely Group Asssigmet Job Costraied Three Dimesioal Model. We have developed a Lexi-Search At the ed of the search table the optimum solutio value of VT is 40 ad is the value of optimal feasible word L = (1,2,4,5,7,8). The the followig figure 3 represets the optimal solutio to the assigmet Algorithm usig Patter Recogitio Techique for gettig a optimal solutio. REFERENCES: 1. AGARWAL, V. The assigmet problem uder categorized jobs, Europea Joural of Operatioal Research 14 (1983) AGARWAL, V., TIKEKAR, V.G., HSU, L.F. BOTTLENECK assigmet problem uder categorizatio, Computers ad Operatios Research 13(1) (1986) BURKARD, R. E. RUDOLF.R, AND WOEGINGER, G.J., Three-dimesioal axial assigmet problems with decomposable cost coefficiets. Techical Report 238, Graz, CARLOS A., OLIVEIRA, S. AND PANOS M. PARDALOS., Radomized parallel algorithms for the multidimesioal assigmet problem. Appl. Numer. Math., 49(2004) , CRAMA, Y. AND F. C. R. SPIEKSMA. Approximatio algorithms for three-dimesioal assigmet problems with triagle iequalities. Europea Joural of Operatioal Research, 60 (3) (1992) Fig-3 (OPTIMAL SOLUTION) I figure-4 { w 1 (3), j 1 (3), α(3), k(1), C(5), αxc(5x3) } represets that w 1 (3) meas i 1 st group of workers umber of uassiged workers is 3, j 1 (3) meas i 1 st group of jobs umber uassiged jobs are 3, α(3) meas the umber of assiged job (which is required), k(1) is the facility used by that job group,c(5) is the cost of that job performed by the 6. FRIEZE, A.M. ad J. YADEGAR. A Algorithm for solvig 3-Dimesioal Assigmet Problems with Applicatio to Schedulig a Teachig Practice. J. Ope. Res., 32(1981) ( ). 7. PANDIT, S.N.N ad SUNDARA MURTHY, M. Eumeratio of all optimal Job sequece - Opsearch, 12, (1975) PANNERSELVAM, R. Operatios Research secod editio, Pretice Hall of Idia private limited, New Delhi, ISSN: Page 93

10 9. PURUSOTHAM, S, SURESH BABU, C. Ad SUNDARA MURTHY, M. Patter Recogitio based Lexi-Search Approach to the Variat Multi-Dimesioal Assigmet Problem, Iteratioal Joural of Egieerig Sciece ad Techology (IJEST). 3(8) (2011) SOBHAN BABU.K, CHANDRA KALA.K, PURUSHOTTAM.S, SUNDARA MURTHY.M, A New Approach for Variat Multi Assigmet Problem. Iteratioal Joural o Computer Sciece ad Egieerig, 02,(05) (2010) SOMASEKHAR SRINIVAS.V.K P-AGENTS SEASONAL JOB COMPLETION MODEL i the Iteratioal Joural of Scietific Research ad Egieerig for publicatio. Vol 2 No (11), November,2013. ISSN: Page 94