Assignment 7



Assignment 7

(Hungarian Method – Assignment Problem)

1. A computer centre has three expert programmers. The centre wants three applications programmes to be be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minutes required by the experts for the application programmes as follows:

Programmers

A B C

|120 |100 |80 |

|80 |90 |110 |

|110 |140 |120 |

Align the programmers to the programmes in such a way that the total computer time is minimum.

2. A department has five employees with five jobs to be performed. The time (in hours) each men will take to perform each job is given in the effectiveness matrix. How should the jobs be allocated, one per employee, so as to minimize the total man hours?

| |I |II |III |IV |V |

|A |10 |5 |13 |15 |16 |

|B |3 |9 |18 |13 |6 |

|C |10 |7 |2 |2 |2 |

|D |7 |11 |9 |7 |12 |

|E |7 |9 |10 |4 |12 |

3. Five men are available to do five different jobs. From past records, the time (in hours) that each man takes to do each job is known and given in the following matrix.

Find the assignment of men to jobs that will minimize the total time taken.

| |I |II |III |IV |V |

|A |2 |9 |2 |7 |1 |

|B |6 |8 |7 |6 |1 |

|C |4 |6 |5 |3 |1 |

|D |4 |2 |7 |3 |1 |

|E |5 |3 |9 |5 |1 |

4. A pharmaceutical company is producing a single product and is selling it through five agencies situated in different cities. All of a sudden, there is a demand for the product in another five cities not having any agency of the company. The country is faced with the problem of how to assign the existing agencies to dispatch the product to needed cities in such a way that the travelling distance is minimized. The distance between the surplus and deficit cities (in km) is given in the following table.

| |I |II |III |IV |V |

|A |160 |130 |115 |190 |200 |

|B |135 |120 |130 |160 |175 |

|C |140 |110 |125 |170 |185 |

|D |50 |50 |80 |80 |110 |

|E |55 |35 |80 |80 |105 |

Determine optimum assignment schedule.

5. A national truck rental service has a surplus of one truck in each of the cities, 1, 2, 3, 4, 5 and 6; and a deficit of one truck in each of the cities 7, 8, 9, 10, 11 and 12. The distances (in km) between the cities in surplus and cities in deficit are displayed in the table:

| |7 |8 |9 |10 |11 |12 |

|1 |31 |62 |29 |42 |15 |41 |

|2 |12 |19 |39 |55 |71 |40 |

|3 |17 |29 |50 |41 |22 |22 |

|4 |35 |40 |38 |42 |27 |33 |

|5 |19 |30 |29 |16 |20 |23 |

|6 |72 |30 |30 |50 |41 |20 |

How should the trucks be displayed so as to minimize the total distance travelled?

Annexure – 8

Assignment 8

(Game Theory – Zero Sum Game)

1. Determine the best strategy for players A & B for the following pay-off matrix. Also, determine the value of the game. Is this game fair?

| |B1 |B2 |B3 |

|A1 |-1 |2 |-2 |

|A2 |6 |4 |-6 |

2. A company management and the labor union are negotiating a new three year settlement. Each of the four strategies are; 1. Hard and aggressive bargaining; 2. Reasoning and logical approach; 3. Legalistic strategy; 4. Conciliatory approach.

What strategy will the two sides adopt? What is the value of the game?

| |I |II |III |IV |

|I |20 |15 |12 |35 |

|II |25 |14 |8 |10 |

|III |40 |2 |10 |5 |

|IV |-5 |4 |11 |0 |

3. Solve the following games by using maxmin (minimax) principle whose payoff matrix are given below: Include in your answer – (i) strategy selection for each player, (ii) the value of the game to each player. Does the game have a saddle point?

(a)

| |I |II |III |IV |

|I |1 |7 |7 |4 |

|II |5 |6 |3 |5 |

|III |7 |2 |0 |3 |

b)

| |B1 |B2 |B3 |B4 |B5 |

|A1 |-2 |0 |0 |5 |3 |

|A2 |3 |2 |1 |2 |2 |

|A3 |-4 |-3 |0 |-2 |6 |

|A4 |5 |3 |-4 |2 |6 |

c)

| |B1 |B2 |B3 |B4 |

|A1 |3 |-5 |0 |6 |

|A2 |-4 |-2 |1 |2 |

|A3 |5 |4 |2 |3 |

d)

| |B1 |B2 |B3 |

|A1 |-2 |15 |-2 |

|A2 |-5 |-6 |-4 |

|A3 |-5 |20 |-8 |

e)

| |B1 |B2 |B3 |B4 |

|A1 |-5 |3 |1 |10 |

|A2 |5 |5 |4 |6 |

|A3 |4 |-2 |0 |-5 |

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