A New Approach to Obtain an Optimal Solution for the Assignment Problem

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1 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 A New Approach to Obtai a Optimal Solutio for the Assigmet Problem A. Seethalakshmy 1, r. N. Sriivasa 2 1 Research Scholar, epartmet of Mathematics. St. Peter s Uiversity, Avadi, Cheai, Idia 2 Professor, epartmet of Mathematics, St. Peter s Uiversity, Avadi, Cheai, Idia Abstract: As a method to brig efficiecy i assigmet problem herewith we propose a ew techique amely SS method of maximizatio/ miimizatio of assigmet problem. Here each row is discussed with 1 s assigmet method with the systematic procedure. The proposed method has the systematic procedure, easy to apply ad less calculatio time. A example usig matrix 1 s assigmet methods is discussed ad the result is compared with Hugaria method. I order to give a better uderstadig of this method, we have provided with some of the illustrated examples by ed of this research paper. Keywords: Assigmet for a salesma, SS method, cost matrix, maximizatio, profit, miimizatio, optimizatio 1. Itroductio To assig a umber of origis to a equal umber of destiatios or based o the skills to assig the work for workers i such a coditio to allocate oe job to oe worker is called assigmet problem. The deftess with which this method is employed is such that it maximizes the profit or miimizes the cost/ time. The optimized way of solutio is obtaied by assigmet problem with the variables beig efficietly used to assig resources to m activities. The ideal purpose of usig assigmet problem is to optimize/ reduce the total cost ivolved ad efficietly use the time/ ma hours or to maximize the profit i sales. These assigmet problems ca be applied i the followig cases (but ot limited to it): 1) Allocatio of salesma to respective sales territory(s) 2) Arrival / departure of flights i respective Gates of termial 3) Supply of midday meals from cetralized kitche to various govermet schools of the State o or before 12 oo There are umerous researches doe here ad articles available with varied methods of applicatio towards this objective, the Hugaria method amogst these is most popular but it seems to be tedious compared the Iterative method. This iterative method ivolves fidig a maximum (miimum) elemet i every row ad divide that row usig the maximum(miimum) elemet so as to create some 1 s i the give cost matrix, after which derive a complete assigmet i terms of allocatig 1 s based o its positio. The utility of this method is to determie the optimum allocatio for origis to its destiatios. A uique method herewith adopted o fidig a approach for solvig assigmet problem which differs from the already existig oes. models. I sectio III algorithms has bee discussed ad i sectio IV some of the umerical examples have bee discussed ad i sectio V deals with the coclusio ad brief discussio of the results. 2. Mathematical form of Assigmet problem For origis (rows) ad destiatios (colums), we eed to assig origis to a equal umber of destiatios so as to be paired sigularly. Let P ij deote the assigmet of salesma i to district j such that P ij = 1, if the ith salesma is assiged to the j th district 0, if the i th salesma is ot assiged to the j th district The assigmet problem ca be mathematically defied as the objective fuctio is to, Miimize Z = K ij P ij Subject to the costraits P i i=1 For all i The arragemet of the research paper is as follows; i sectio II we preset the mathematical form of assigmet P i Licesed Uder Creative Commos Attributio CC BY For all j Paper I: ART

2 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 Where P ij = 0, if the i th resource is ot assiged to the j th activity. Where P ij = 1 if the i t h resource is assiged to the j th activity. Ad K ij represets the cost of assigmet of worker i to job j. 3. Algorithm Costruct a matrix for the assigmet problem. Ideally, the matrix should be a square matrix, if ot; we brig it to be a square matrix. For each row i assigmet problem, the maximum or miimum value (say m i ) from the row is picked depedig upo the ature of the problem. Ad the chose elemet (say m i ) is divided i each row resultig i a uity (1), at least oce. Each row is to be discussed Cosider the 1 s of (i, j) th positio ad cosider the distict positio of the matrix i the colum. The assigmet is give for that distict positio. elete the correspodig rows ad colums. The remaiig table is the discussed. The process is cotiued to the remaiig table util the completio of the assigmet. Step 4 I the cotrary for the above coditio, if there is idetical colum for more tha oe row, selectio of the colum employs the calculatio of the differece betwee two uit costs, oe beig the largest uit cost ad the other oe beig the peultimate largest uit cost, for maximizatio, ad similarly for miimizatio the smallest ad peultimate smallest uit costs are cosidered while calculatig the differece. From the outcome of the above calculatio, the colum with the maximum differece gets assiged. Ad the correspodig rows & colum are deleted. Step 5 For the differece value to be a tie, the calculatio employs uit costs which are largest ad atepeultimate largest value (smallest ad atepeultimate smallest value) for the colum. Colum with the maximum differece value gets assiged by cacelig the correspodig row ad colum. Step 6 A uique assigmet of a row is obtaied by iteratig steps 2 to step 5 util all rows get assiged. Step 7 Fial step ivolves the calculatio of total cost as below: Total cost = K ij P ij i=1 4. Numerical Examples Example.4.1 (miimizatio problem) A work mager has to allocate four differet drivers to four schools to supply the luch for studets. epedig o the efficiecy ad the time take by the idividual differ by the capacity as show i the table. rivers Schools A B C How should the drivers be assiged to school so as to miimize the total ma-hours? A B C Mi value (m i ) A B C rivers Schools A C B Assig 1 A, 2 C, 3 B, 4 Optimal solutio= = 38 Compariso of Result Licesed Uder Creative Commos Attributio CC BY Paper I: ART

Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 231-7064 Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 Example.4.2 (miimizatio problem) A departmetal head has four subordiates, ad four jobs to be performed.

3 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 Example.4.2 (miimizatio problem) A departmetal head has four subordiates, ad four jobs to be performed. The subordiates differ i efficiecy, ad the jobs differ i their itrisic difficulty. His estimate of the time each ma would take to perform each task is give i the matrix below. Job Me ifferece * Jobs Me A B C J How should the jobs be allocated so as to miimize the total ma-hours? Solutio A B C Mi value (m i ) J A B C J Assig elete the assiged rows ad colum. B C Mi value (m i ) B C Mi value(m i ) Task Me ifferece C * C 1 Assig C ad B elete the assiged rows ad colum. Hece assig C, J 2 A, B, Ad the optimal solutio is = 5 Compariso of Result Job J 2 Me A Assig J 2 A, elete the assiged rows ad colum. B C Mi Value (m i ) Example.4.3 (Maximizatio problem) A Marketig maager has five salesma five sales districts cosiderig the capability of the salesma ad the ature of districts, the marketig maager estimates that sales per moth for each district would be as follows B C Salesma istrict A B C E S S S Licesed Uder Creative Commos Attributio CC BY Paper I: ART

4 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 How should the salesma be allocated so as to maximize the profit? Solutio Fid the maximum value i each row ad write it i the right-had side ad divide by the value Salesma istrict Max. A B C E Value (m i ) S S S Salesma Salesma istrict A B C E S S S istrict ifferece Salesma istrict ifferece C, E S 3 E S 4 C 3* C 1 Assig S 4 C ad delete the correspodig row ad colum Salesma istrict Max. Value (m i ) B E S Salesma istrict Max. Value (m i ) B E S C, E S 2 A (4) (12)* S 3 A (4) (11) S 4 C C Assig S 2 A colum ad delete the correspodig row ad Salesma istrict Max. Value (m i ) B C E S S Salesma istrict B C E S S Assig S 3 E. elete the correspodig row ad colum Step 4 Salesma istrict Max. Value (m i ) B Salesma Assig B, istrict B Hece assig B, S 2 A, S 3 E, S 4 C, Maximum sales = = 11 Compariso of Result Licesed Uder Creative Commos Attributio CC BY Paper I: ART

Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 231-7064 Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 5.

5 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Idex Copericus Value (2013): 6.14 Impact Factor (2015): Coclusio Herewith brigig i a ew method amely SS method for addressig assigmet problems. All kids of assigmet problems ca be addressed usig this method. A systematic ad easy way of approach is iheret i this method. To draw a coclusio based o this research paper, it provides to be a optimal solutio with few direct steps ivolved by assigig the positio of 1 s for the assigmet problem. With its aim of providig optimal solutios with lesser steps ivolved, this method proves to be a real boo for the decisio makers for its applicability. I coherece to results obtaied optimally as through Hugaria method, this SS method comes i play to propose a ew way of addressig assigmet problem with its uique approach ot applied/ employed i the precedig methods. Refereces [1] H.A.Taha, operatios research-itroductio, pretice hall of Idia New elhi, 8 th editio [2] P.K.Gupta,.Shira, operatio Research, S. Chad & Compay Limited,14 th Editio 1. [3] J.K.Sharma, operatios Research-Theory ad applicatio, Macmillia Idia LT, New elhi-2005 [4] A.Thirupathi,.Iraia, A iovative method for fidig optimal solutio to assigmet problems. Iteratioal joural of Iovative Research i Sciece, Egieerig ad Techology, Vol 6, Issue 8, August [5] N.Sriivasa,.Iraia, A ew approach for solvig assigmet problem with optimal solutio, Iteratioal joural of Egieerig ad maagemet research, Volume 6 Issue 3 may Jue [6] Shweta Sigh, G.C.ubey, Rajesh Shrivastava- Compararative aalysis of Assigmet problem, IOSR Joural of Egieerig (IOSRJEN)vol 2, Issue 8 (Aug 2012) pp 1-15 [7] P.Padia ad G.Nataraja A New method for fidig ad optimal solutio for trasportatio problem, IJMSEA, Vol 4,pp 5-65,2010. [8] N.M.eshmukh A iovative method for solvig TP- Iteratioal Joural of Physics ad Mathematical Scieces ISSN: Vol.2 (3) July-Sep pp: Licesed Uder Creative Commos Attributio CC BY Paper I: ART

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