Optimization Methods: Linear Programming Applications Assignment Problem 1. Module 4 Lecture Notes 3. Assignment Problem

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1 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem Itroductio Module 4 Lecture Notes 3 Assigmet Problem I the previous lecture, we discussed about oe of the bech mark problems called trasportatio problem ad its formulatio. The assigmet problem is a particular class of trasportatio liear programmig problems with the supplies ad demads equal to itegers (ofte ). Sice all supplies, demads, ad bouds o variables are itegers, the assigmet problem relies o a iterestig property of trasportatio problems that the optimal solutio will be etirely itegers. I this lecture, the structure ad formulatio of assigmet problem are discussed. Also, travelig salesma problem, which is a special type of assigmet problem, is described. Structure of assigmet problem As metioed earlier, assigmet problem is a special type of trasportatio problem i which. Number of supply ad demad odes are equal. 2. Supply from every supply ode is oe. 3. Every demad ode has a demad of oe. 4. Solutio is required to be all itegers. The goal of a geeral assigmet problem is to fid a optimal assigmet of machies (laborers) to jobs without assigig a aget more tha oce ad esurig that all jobs are completed. The objective might be to miimize the total time to complete a set of jobs, or to maximize skill ratigs, maximize the total satisfactio of the group or to miimize the cost of the assigmets. This is subjected to the followig requiremets:

2 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 2. Each machie is assiged ot more tha oe job. 2. Each job is assiged to exactly oe machie. Formulatio of assigmet problem Cosider m laborers to whom tasks are assiged. No laborer ca either sit idle or do more tha oe task. Every pair of perso ad assiged work has a ratig. This ratig may be cost, satisfactio, pealty ivolved or time take to fiish the job. There will be N 2 such combiatios of persos ad jobs assiged. Thus, the optimizatio problem is to fid such ma- job combiatios that optimize the sum of ratigs amog all. The formulatio of this problem as a special case of trasportatio problem ca be represeted by treatig laborers as sources ad the tasks as destiatios. The supply available at each source is ad the demad required at each destiatio is.the cost of assigig (trasportig) laborer i to task j is c. It is ecessary to first balace this problem by addig a dummy laborer or task depedig o whether m< or m>, respectively. The cost coefficiet c for this dummy will be zero. Let x 0, if =, if the the j j th th job is ot assiged to the i job is assiged to the i th th machie machie Thus the above model ca be expressed as m i= j= Miimize c x Sice each task is assiged to exactly oe laborer ad each laborer is assiged oly oe job, the costraits are

3 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 3 i= j= x = for j =,2,... x = for i =,2,... m x = 0 or Due to the special structure of the assigmet problem, the solutio ca be foud out usig a more coveiet method called Hugaria method which will be illustrated through a example below. Example : (Taha, 982) Cosider three jobs to be assiged to three machies. The cost for each combiatio is show i the table below. Determie the miimal job machie combiatios. Table Job Machie 2 3 a i b j Solutio: Step : Create zero elemets i the cost matrix (zero assigmet) by subtractig the smallest elemet i each row (colum) from the correspodig row (colum). After this exercise, the resultig cost matrix is obtaied by subtractig 5 from row, 0 from row 2 ad 3 from row 3.

4 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 4 Table Step 2: Repeatig the same with colums, the fial cost matrix is Table The italicized zero elemets represet a feasible solutio. Thus the optimal assigmet is (,), (2,3) ad (3,2). The total cost is equal to 60 (5 +2+3). I the above example, it was possible to obtai the feasible assigmet. But i more complicated problems, additioal rules are required which are explaied i the ext example. Example 2 (Taha, 982) Cosider four jobs to be assiged to four machies. The cost for each combiatio is show i the table below. Determie the miimal job machie combiatios.

5 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 5 Table 4 Job Machie a i b j Solutio: Step : Create zero elemets i the cost matrix by subtractig the smallest elemet i each row from the correspodig row. Table Step 2: Repeatig the same with colums, the fial cost matrix is Table

6 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 6 Rows ad 3 have oly oe zero elemet. Both of these are i colum, which meas that both jobs ad 3 should be assiged to machie. As oe machie ca be assiged with oly oe job, a feasible assigmet to the zero elemets is ot possible as i the previous example. Step 3: Draw a miimum umber of lies through some of the rows ad colums so that all the zeros are crossed out. Table Step 4: Select the smallest ucrossed elemet (which is here). Subtract it from every ucrossed elemet ad also add it to every elemet at the itersectio of the two lies. This will give the followig table. Table This gives a feasible assigmet (,), (2,3), (3,2) ad (4,4) with a total cost of = 2. If the optimal solutio had ot bee obtaied i the last step, the the procedure of drawig lies has to be repeated util a feasible solutio is achieved.

7 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 7 Formulatio of Travelig Salesma Problem (TSP) as a Assigmet Problem A travelig salesma has to visit cities ad retur to the startig poit. He has to start from ay oe city ad visit each city oly oce. Suppose he starts from the k th city ad the last city he visited is m. Let the cost of travel from i th city to j th city be c. The the objective fuctio is m i= j= Miimize c x subject to the costraits i= j= x x mk x = for j =,2,..., i j, i m x = for i =,2,... m, i j, i m = = 0 or Solutio Procedure: Solve the problem as a assigmet problem usig the method used to solve the above examples. If the solutios thus foud out are cyclic i ature, the that is the fial solutio. If it is ot cyclic, the select the lowest etry i the table (other tha zero). Delete the row ad colum of this lowest etry ad agai do the zero assigmet i the remaiig matrix. Check whether cyclic assigmet is available. If ot, iclude the ext higher etry i the table ad the procedure is repeated util a cyclic assigmet is obtaied. The procedure is explaied through a example below.

8 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 8 Example 3: Cosider a four city TSP for which the cost betwee the city pairs are as show i the figure below. Fid the tour of the salesma so that the cost of travel is miimal Table Solutio: Step : The optimal solutio after usig the Hugaria method is show below. Table

9 Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem 9 The optimal assigmet is 4, 2 3, 3 2, 4 which is ot cyclic. Step 2: Cosider the lowest etry 2 of the cell (2,). If there is a tie i selectig the lowest etry, the break the tie arbitrarily. Delete the 2 d row ad st colum. Do the zero assigmet i the remaiig matrix. The resultig table is Table Thus the ext optimal assigmet is 4, 2, 3 2, 4 3 which is cyclic. Thus the required tour is ad the total travel cost is = 24.