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Hungarian Method

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The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem as minimizing the cost of completing jobs by assigning workers to tasks, where each job is assigned to exactly one worker. It then outlines the steps of the Hungarian method, which involves constructing a cost matrix, subtracting rows and columns to find zeros, and using the zeros to determine the optimal assignment. Finally, it provides an example and lists some applications of the Hungarian method like assigning machines, salespeople, contracts, teachers, and accountants.

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ASSIGNMENT PROBLEM Hungarian Method PRESENTED BY Aritra Kanjilal MT16ENV001
INTRODUCTION  An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.  One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project).  This method was developed by D. Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem.
COST MATRIX IN ASSIGNMENT PROBLEM
SOLUTION OF ASSIGNMENT PROBLEM  Step 1. Determine the cost table for the given problem. (i) If the no. of sources is equal to no. of destinations, go to step 3. (ii) If the no. of sources is not equal to the no. of destination, go to step2.  Step 2. Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero
 Step 3. Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row.  Step 4. In the reduced matrix obtained in the step 3, locate the smallest element of each column and then subtract the same from each element of that column. Each column and row now have at least one zero. 53 96 37 47 87 41 60 92 36           a 16 59 0 6 46 0 24 56 0           a 10 13 0 0 0 0 18 10 0          
 Step 5. In the modified matrix obtained in the step 4, search for the optimal assignment as follows: (a) Examine the rows successively until a row with a single zero is found. Enrectangle this zero and cross off all other zeros in its column. Continue in this manner until all the rows have been taken care of. (b) Repeat the procedure for each column of the reduced matrix. (c) If a row and/or column has two or more zeros and one cannot be chosen by inspection then assign arbitrary any one of these zeros and cross off all other zeros of that row / column. (d) Repeat (a) through (c) above successively until the chain of assigning or cross ends.
10 13 0 0 0 0 18 10 0          
          01018 000 01310
10 13 0 0 0 0 18 10 0          
 Step 6. If the number of assignment is equal to n (the order of the cost matrix), an optimum solution is reached.  If the number of assignment is less than n(the order of the matrix), go to the next step.  Step7. Draw the minimum number of horizontal and/or vertical lines to cover all the zeros of the reduced matrix.  Step 8. Develop the new revised cost matrix as follows:  (a)Find the smallest element of the reduced matrix not covered by any of the lines.  (b)Subtract this element from all uncovered elements and add the same to all the elements laying at the intersection of any two lines.  Step 9. Go to step 6 and repeat the procedure until an optimum solution is attained.
0 3 0 0 0 10 8 0 0          
0 3 0 0 0 10 8 0 0           0 3 0 0 0 10 8 0 0           SOLUTION
Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are in Austin, TX; Boston, MA; and Chicago, IL. You want them to fly to three other cities: Denver, CO; Edmonton, Alberta; and Fargo, ND. The table below shows the cost of airplane tickets in dollars between these cities. From To Denver Edmonton Fargo Austin 250 400 350 Boston 400 600 350 Chicago 200 400 250 Where should you send each of your salespeople in order to minimize airfare?
Hungarian Method
Hungarian Method
APPLICATIONS  In assigning machines to factory orders.  In assigning sales/marketing people to sales territories.  In assigning contracts to bidders by systematic bid- evaluation.  In assigning teachers to classes.  In assigning accountants to accounts of the clients.
Hungarian Method

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Hungarian Method

  • 1. ASSIGNMENT PROBLEM Hungarian Method PRESENTED BY Aritra Kanjilal MT16ENV001
  • 2. INTRODUCTION  An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.  One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project).  This method was developed by D. Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem.
  • 3. COST MATRIX IN ASSIGNMENT PROBLEM
  • 4. SOLUTION OF ASSIGNMENT PROBLEM  Step 1. Determine the cost table for the given problem. (i) If the no. of sources is equal to no. of destinations, go to step 3. (ii) If the no. of sources is not equal to the no. of destination, go to step2.  Step 2. Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero
  • 5.  Step 3. Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row.  Step 4. In the reduced matrix obtained in the step 3, locate the smallest element of each column and then subtract the same from each element of that column. Each column and row now have at least one zero. 53 96 37 47 87 41 60 92 36           a 16 59 0 6 46 0 24 56 0           a 10 13 0 0 0 0 18 10 0          
  • 6.  Step 5. In the modified matrix obtained in the step 4, search for the optimal assignment as follows: (a) Examine the rows successively until a row with a single zero is found. Enrectangle this zero and cross off all other zeros in its column. Continue in this manner until all the rows have been taken care of. (b) Repeat the procedure for each column of the reduced matrix. (c) If a row and/or column has two or more zeros and one cannot be chosen by inspection then assign arbitrary any one of these zeros and cross off all other zeros of that row / column. (d) Repeat (a) through (c) above successively until the chain of assigning or cross ends.
  • 7. 10 13 0 0 0 0 18 10 0          
  • 9. 10 13 0 0 0 0 18 10 0          
  • 10.  Step 6. If the number of assignment is equal to n (the order of the cost matrix), an optimum solution is reached.  If the number of assignment is less than n(the order of the matrix), go to the next step.  Step7. Draw the minimum number of horizontal and/or vertical lines to cover all the zeros of the reduced matrix.  Step 8. Develop the new revised cost matrix as follows:  (a)Find the smallest element of the reduced matrix not covered by any of the lines.  (b)Subtract this element from all uncovered elements and add the same to all the elements laying at the intersection of any two lines.  Step 9. Go to step 6 and repeat the procedure until an optimum solution is attained.
  • 11. 0 3 0 0 0 10 8 0 0          
  • 12. 0 3 0 0 0 10 8 0 0           0 3 0 0 0 10 8 0 0           SOLUTION
  • 13. Example 1: You work as a sales manager for a toy manufacturer, and you currently have three salespeople on the road meeting buyers. Your salespeople are in Austin, TX; Boston, MA; and Chicago, IL. You want them to fly to three other cities: Denver, CO; Edmonton, Alberta; and Fargo, ND. The table below shows the cost of airplane tickets in dollars between these cities. From To Denver Edmonton Fargo Austin 250 400 350 Boston 400 600 350 Chicago 200 400 250 Where should you send each of your salespeople in order to minimize airfare?
  • 16. APPLICATIONS  In assigning machines to factory orders.  In assigning sales/marketing people to sales territories.  In assigning contracts to bidders by systematic bid- evaluation.  In assigning teachers to classes.  In assigning accountants to accounts of the clients.