A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM

A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM *Kore B. G. Departmet Of Statistics, Balwat College, VITA - 415 311, Dist.: Sagli (M. S.). Idia *Author for Correspodece ABSTRACT I this paper I have proposed a ew approach to solve a ubalaced assigmet problem (UBAP). This approach icludes two parts. First is to obtai a iitial basic feasible solutio (IBFS) ad secod part is to test optimality of a IBFS. I have proposed two ew methods Row Pealty Assigmet Method (RPAM) ad Colum Pealty Assigmet Method (CPAM) to obtai a IBFS of a UBAP. Also I have proposed a ew method No-basic Smallest Effectiveess Method (NBSEM) to test optimality of a IBFS. We ca solve a assigmet problem of maximizatio type usig this ew approach i opposite sese. By this ew approach, we achieve the goal with less umber of computatios ad steps. Further we illustrate the ew approach by suitable examples. Key Words: BAP, UBAP, IBFS, RPAM, C PAM, NBSEM INTRODUCTION The assigmet problem is a special case of the trasportatio problem where the resources are beig allocated to the activities o a oe-to-oe basis. Thus, each resource (e.g. a employee, machie or time slot) is to be assiged uiquely to a particular activity (e.g. a task, site or evet). I assigmet problems, supply i each row represets the availability of a resource such as a ma, machie, vehicle, product, salesma, etc. ad demad i each colum represets differet activities to be performed such as jobs, routes, factories, areas, etc. for each of which oly oe ma or vehicle or product or salesma respectively is required. Etries i the square beig costs, times or distaces. The assigmet method is a special liear programmig techique for solvig problems like choosig the right ma for the right job whe more tha oe choice is possible ad whe each ma ca perform all of the jobs. The ultimate objective is to assig a umber of tasks to a equal umber of facilities at miimum cost (or maximum profit) or some other specific goal. Let there be m resources ad activities. Let c ij be the effectiveess (i terms of cost, profit, time, etc.) of assigig resource i to activity j (i = 1, 2,., m; j = 1, 2,., ). Let x ij = 0, if resource i is ot assiged to activity j ad x ij = 1, if resource i is assiged to activity j. The the objective is to determie x ij s that will optimize the total effectiveess (Z) satisfyig all the resource costraits ad activity costraits. 1. Mathematical Formulatio Let umber of rows = m ad umber of colums =. If m = the a AP is said to be BAP otherwise it is said to be UBAP. A) Case 1: If m < the mathematically the UBAP ca be stated as follows: Miimize m Z C ij X i 1 j 1 ij (1.1) Subject to X 1, i = 1, 2,.., m (availability costraits), (1.2) j 1 ij X ij i 1 = 0 or 1, j = 1, 2,, (requiremet costraits), (1.3) Ad x ij = 0 or 1, for all i ad j. (1.4) 46

B) Case 2: If m > the mathematically the UBAP ca be stated as follows: Miimize Subject to m Z C X ij j 1 X ij X ij i 1 i 1 j 1 ij (1.5) 0 or 1, i = 1, 2,, m (availability costraits), (1.6) = 1, j = 1, 2,, (requiremet costraits), (1.7) Ad x ij = 0 or 1, for all i ad j. (1.8) Presetly a AP ca be solved by usig oe of the four methods, (i) Eumeratio method, (ii) Simplex method (iii) Trasportatio method ad (iv) Hugaria method. Amog these four methods Hugaria method ca be used as a efficiet method for fidig a optimal solutio of a AP. But this method also requires more umber of computatios ad steps. For usig Hugaria method to solve UBAP it is require to covert it ito BAP, Hadley (1997), Taha (2008), Kati Swarup et al., (2008),Gupta ad Hira (2010), Sharma (2010). This leads to cosider a assigmet table of higher order tha the iitial assigmet table, to do more umber of computatios, iteratios ad steps to get a optimal solutio. Also, to solve a AP of maximizatio type, it is require to covert it ito miimizatio type. This leads to do more umber of computatios ad steps to get a optimal solutio. Kore (2008) made a attempt to solve ubalaced trasportatio problem without balacig it. As AP is a particular case of TP i this paper, I have proposed a ew approach to solve a UBAP, which overtakes the problem of degeeracy of trasportatio method. Usig our ew approach we get, optimal solutio of a UBAP without balacig it, with less umber of computatios, steps ad cosiderig iitial assigmet table, without chagig its order. We ca illustrate the compariso betwee our ew approach ad Hugaria Method by solvig various types of UBAPs. 2. A) Algorithms of the ew methods to obtai a IBFS 2.1: Row Pealty Assigmet Method (RPAM) Step1: For each row determie row pealty by takig differece betwee smallest ad ext smallest effectiveess. Step2: Observe the maximum row pealty, select smallest effectiveess correspodig to that row, ad ecircle it, cross out correspodig row ad colum. If there is a tie i maximum row pealty the select the largest effectiveess of the smallest effectiveess correspodig to them. If there is a tie i the largest effectiveess of the smallest effectiveess the select that largest effectiveess correspodig to which ext to ext smallest effectiveess i the row is maximum. If there is agai tie the select oe of them radomly, Step3: Repeat step1 ad step2 util oly oe row is remaied ucrossed. Select smallest effectiveess i the last row, ecircle it ad cross out correspodig row ad colum. If there is a tie i smallest effectiveess the select that smallest effectiveess correspodig to which ext smallest effectiveess i the colum is miimum. 2.2: Colum Pealty Assigmet Method (CPAM) Step1: For each colum determie colum pealty by takig differece betwee smallest ad ext smallest effectiveess. Step2: Observe the maximum colum pealty, select smallest effectiveess correspodig to that colum, ecircle it, cross out correspodig row ad colum. If there is a tie i maximum colum pealty the select the largest effectiveess of the smallest effectiveess correspodig to them. If there is a tie i the largest effectiveess of the smallest effectiveess the select that largest effectiveess correspodig to which ext to ext smallest effectiveess i the colum is maximum. If there is agai tie the select oe of them radomly. 47

Step3: Repeat step1 ad step2 util oly oe colum is remaied ucrossed. Select smallest effectiveess i the last colum, ecircle it ad cross out correspodig row ad colum. If there is a tie i smallest effectiveess the select that smallest effectiveess correspodig to which ext smallest effectiveess i the row is miimum. B) Algorithm of the Method to obtai a Optimal Solutio 2.3: No-basic Smallest Effectiveess Method (NBSEM) Step1: Select o-basic cell havig smallest effectiveess. Step 2 : a) Form a loop which starts ad eds at selected o-basic cell cosiderig two basic cells ad two o-basic cells such that, the o-basic cells ad basic cells are alterate i the loop, o more tha two cells i the loop are i the same row or colum. b) Make the total of effectiveess i the o-basic cells (T) ad the total of effectiveess i the basic cells (T ). c) If T = T the it idicates that there exists a alterative solutio to the give AP. d) If T < T the it idicates that improvemet i the preset IBFS is possible. If there is a tie i smallest effectiveess i the o-basic cells the select that smallest effectiveess which provides maximum improvemet. Iterchagig o-basic cells ad basic cells i the row. Select agai smallest effectiveess i o-basic cells ad go to step 2. If T > T the go to (e) e) Icrease the umber of basic cells ad o-basic cells oe by oe up to mi (m, ), form all possible loops oe by oe satisfyig the coditios of formig a loop as stated i (a), go to (b). Note: 1) If a UBAP is of maximizatio type the use RPAM to obtai a IBFS of the UBAP of case1 ad CPAM, to obtai a IBFS of the UBAP of case 2, i opposite sese. 2) For usig the NBSEM to test a optimality of IBFS of the UBAP of case 1 or case 2, select o-basic smallest effectiveess correspodig to which row ad colum have a basic cell. 3) For a UBAP of maximizatio type, to test a optimality of a IBFS we ca use the NBSEM i opposite sese. C) Algorithm of the New Approach to solve a UBAP Step 1: Express the give AP i tabular form. Step 2: Check whether the AP is BAP or UBAP. Step 3: If a AP is UBAP of case 1 the obtai a IBFS usig RPAM. If a AP is UBAP of case 2 the obtai a IBFS by usig CPAM. Step 4: Optimize a IBFS of UBAP by usig NBSEM to get a optimal solutio of give UBAP. Step 5: Write optimal solutio ad the optimum value of objective fuctio (Z). 3. Applicatios of the Approach method We illustrate the effectiveess of the ew approach by solvig various types of APs. Example 3.1: Cosider the followig AP of miimizatio type, Sharma (2010): 1 2 3 4 5 6 1 80 140 80 100 56 98 2 48 64 94 126 170 100 3 56 80 120 100 70 64 4 99 100 100 104 80 90 5 64 90 90 60 60 70 Here, m = 5 ad = 6 i.e. m <, the problem is UBAP of case 1. Usig the RPAM to obtai a IBFS ad testig its optimality by usig the NBSEM we get, 48

Table 3.1: IBFS 1 2 3 4 5 6 R.P. 1 80 140 80 100 56 98 (24) 2 94 126 170 100 48 64 (16) 16) 3 80 56 120 100 70 64 (8) (8) (16) 4 99 100 100 104 80 90 (10) (9) (10) (0) 5 64 80 90 60 60 70 (0) (4) (10) (20) Here, T= 200 ad T = 212 i.e. T < T, improvemet i the preset solutio is possible by itroducig the o-basic smallest effectiveess 56 i the basis. The improved solutio is, Table 3.2: Optimal Solutio 1 2 4 5 6 1 80 140 100 56 98 2 48 64 126 170 100 3 56 80 100 70 64 4 99 100 104 80 5 64 80 60 60 70 Here, Improvemet i the preset solutio is ot possible by itroducig the o-basic smallest effectiveess 48 i the basis. The preset solutio is optimal solutio to the give UBAP. The optimal solutio is assig, 1 5, 2 2, 3 1, 4 6 ad 5 4, job 3 remaied uassiged. The optimum value of Z is, Z mi = 56 + 64 + 56 + 90 + 60 = 326. Now, we solve the above UBAP by usig Hugaria method, covertig the give UBAP ito BAP by itroducig dummy row havig zero effectiveess i each cell we get, Table 3.3: BAP 1 2 3 4 5 6 1 80 140 80 100 56 98 2 48 64 94 126 170 100 3 56 80 120 100 70 64 4 99 100 100 104 80 90 5 64 80 90 60 60 70 6 0 0 0 0 0 0 Subtractig the smallest effectiveess i each row from each effectiveess of that row ad drawig he miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the reduced matrix obtaied, we get, Table 3.4 1 2 3 4 5 6 1 24 84 24 44 0 42 2 0 16 46 78 122 52 3 0 24 64 44 14 8 4 19 20 20 24 0 10 5 4 20 30 0 0 10 6 0 0 0 0 0 0 90 49

Here, the umber of lies draw the umber of rows or colums, the optimal solutio caot be obtaied. Subtractig the smallest ucovered effectiveess from each ucovered effectiveess ad addig ito the effectiveess which lies at the itersectio of two lies, ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the ew reduced matrix obtaied we get, Table 3.5 Here, the umber of lies draw the umber of rows or colums, the optimal solutio caot be obtaied. Subtractig the smallest ucovered effectiveess from each ucovered effectiveess ad addig ito the effectiveess which lies at the itersectio of two lies, ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the ew reduced matrix obtaied we get, Table 3.6 Here, the umber of lies draw the umber of rows or colums, the optimal solutio caot be obtaied. Subtractig the smallest ucovered effectiveess from each ucovered effectiveess ad addig ito the effectiveess which lies at the itersectio of two lies, ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the ew reduced matrix obtaied we get, Table 3.7 Here, the umber of lies draw the umber of rows or colums, the optimal solutio caot be obtaied. Subtractig the smallest ucovered effectiveess from each ucovered effectiveess ad addig ito the effectiveess which lies at the itersectio of two lies, ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the ew reduced matrix obtaied we get, 50

Table 3.8 Here, the umber of lies draw = the umber of rows or colums. The optimal solutio ca be obtaied. Table 3.9: Optimal Solutio Here, the optimal solutio is assig, 1 5, 2 2, 3 1, 4 6 ad 5 4. The optimum value of Z is, Z mi = 56 + 64 + 56 + 90 + 60 = 326. Note: For above example we get a optimal solutio by usig our ew approach i 2 steps, ad cosiderig assigmet table of order 5 6 ad 5 5 i each step respectively. By usig Hugaria method we get a optimal solutio i 7 steps, ad cosiderig assigmet table of order 6 6 from first step to last step. Sice steps are more, computatios are more, ad order of assigmet table is higher, the time required to solve the above UBAP by usig Hugaria method is more tha the time required by usig our ew approach. Example 3.2: Cosider the followig AP of maximizatio type, Kati Swarup et al., (2008): 1 2 3 4 1 3 6 2 6 2 7 1 4 4 3 3 8 5 8 4 6 4 3 7 5 5 2 4 3 6 5 7 6 4 Here, m = 6 ad = 4 i.e. m >, the problem is UBAP of case 2.Usig the CPAM i opposite sese to obtai a IBFS ad usig the NBSEM i opposite sese to test its optimality we get, 51

Table 3.10: IBFS 1 2 3 4 1 3 6 2 6 2 7 1 4 4 3 3 8 5 4 6 4 3 7 5 5 2 4 3 6 5 7 6 4 C. P. (1) (1) (1) (1) (1) (2) (1) (1) (1) Here, T= 19 ad T = 18 i.e. T > T, improvemet i the preset solutio is possible by itroducig the o-basic largest effectiveess 8 i the basis. The improved solutio is, Table 3.11 1 2 3 4 2 7 1 4 4 3 3 8 5 4 6 4 3 6 5 7 6 2 Here, T= 15 ad T = 12 i.e. T > T, improvemet i the preset solutio is possible by itroducig the o-basic largest effectiveess 8 i the basis. The improved solutio is, 8 8 7 Table 3.12 1 2 3 4 2 7 1 4 4 3 3 8 5 8 4 6 4 3 6 5 7 6 2 7 Here, improvemet i the preset solutio is ot possible by itroducig the o-basic largest effectiveess 8 i the basis. The preset solutio is a optimal solutio to the give UBAP. The optimal solutio is assig, 2 1, 3 2, 4 4, ad 6 3, job 1ad 5 remaied uassiged. The optimum value of Z is, Z max = 7 + 8 + 7 + 6 = 28. Now, we solve the above UBAP by usig Hugaria method covertig the give UBAP of maximizatio type ito miimizatio type by subtractig all the effectiveess from largest effectiveess we get, 52

Table 3.13: Miimizatio Problem 1 2 3 4 1 5 2 6 2 2 1 7 4 4 3 5 0 3 0 4 2 4 5 1 5 3 6 4 5 6 3 1 2 4 Covertig the give UBAP ito BAP by itroducig two dummy colums havig zero effectiveess i each cell we get, Table 3.14: BAP 1 2 3 4 5 6 1 5 2 6 2 0 0 2 1 7 4 4 0 0 3 5 0 3 0 0 0 4 2 4 5 1 0 0 5 3 6 4 5 0 0 6 3 1 2 4 0 0 Subtractig the smallest effectiveess i each colum from each effectiveess of that colum ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all the zeros i the reduced matrix obtaied we get, Table 3.15 Here, the umber of lies draw the umber of row or colums the optimal solutio caot be obtaied. Subtractig the smallest ucovered effectiveess from each ucovered effectiveess ad addig ito the effectiveess which lies at the itersectio of two lies, ad drawig the miimum umber of vertical ad horizotal lies ecessary to cover all zeros i the ew reduced matrix obtaied we get, Table 3.16 Here, the umber of lies draw = the umber of rows or colums. The optimal solutio ca be obtaied. 53

Table 3.17: Optimal Solutio 1 2 3 4 5 6 1 3 1 3 1 0 0 2 0 7 2 4 1 1 3 4 0 1 0 1 1 4 0 3 2 0 0 0 5 1 5 1 4 0 0 6 2 1 0 4 1 1 The optimal solutio is assig, 2 1, 3 2, 4 4, ad 6 3. The optimum value of Z is, Z max = 7 + 8 + 7 + 6 = 28. Note: For above example we get a optimal solutio by usig our ew approach i 3 steps ad cosiderig assigmet table of order 6 4, 4 4 ad4 4 respectively. By usig Hugaria method we get a optimal solutio i 5 steps ad cosiderig assigmet table of order 6 6 from first step to last step. Sice steps are more, computatios are more ad order of assigmet table is higher the time required to solve the above UBAP by usig Hugaria method is more tha the time required by usig our ew approach. CONCLUSIONS 1) It is ot required to covert UBAP i the form of BAP to get a optimal solutio. 2) If a IBFS of the UBAP of case 1ad case 2is obtaied by usig RPAM ad CPAM respectively, without balacig it ad it is optimized by usig NBSEM method the the least possible optimum value of Z is achieved. 3) Usig our ew approach to solve the UBAP we get, optimal solutio fastly, without chagig the order of assigmet table, with less umber of steps, iteratios ad computatios. ACKNOWLEDGEMENTS I express my deepest sese of gratitude to Dr. L. B. Thakur, Research Guide, (Rtd.) Professor of Statistics, Dr. B. A. M. Uiversity, Auragabad (M.S.), INDIA, for his ivaluable guidace ad cotiuous ecouragemet. I pay my respect ad gratitude to Dr. A. V. Dattareya Rao, Professor, Departmet of Statistics, Nagarjua Uiversity, Nagarjua Nagar (A.P.), INDIA, for his valuable suggestios ad guidace. REFERENCES Gupta PK, ad Hira DS (2010). Operatios Research, S. Chad ad Compay Ltd., New Delhi, Reprit 2010. Hadley G (1997). Liear Programmig, Narosa Publishig House, Delhi. Kati Swarup, Gupta PK, Ma Moha (2008). Operatios Research, Fourteeth Editio. Kore BG (2008). A New Approach To Solve Ubalaced Trasportatio Problem, Joural of Idia Academy of Mathematics 30 (1) 43 54. Sharma JK (2010). Operatios Research Theory ad Applicatios, Macmilla Idia Limited, 54

Delhi. Taha HA (2008). Operatios Research A Itroductio, Pretice Hall of Idia Private Limited, New Delhi, Eighth Editio. 55