# Self Test Questions

#### Theory

1. Give the mathematical formulation of an assignment problem.

2. Describe the Hungarian Method of solving the assignment problem.

3. Explain the difference between a transportation and an assignment problem.

4. Explain Unbalanced Assignment Problem with the help of an example.

5. Explain Maximization In An Assignment Problem with the help of an example.

#### Practical

1. In a textile sales emporium, four salesmen A, B, C and D are available to four counters W, X, Y and Z. Each salesman can handle any counter. The service in (hour) of each counter when manned by each salesman is given below:

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Salesman
Counter A B C D
W 41 32 39 52
X 22 29 49 65
Y 27 39 60 51
Z 45 50 48 52

How should the salesman be allocated appropriate counters so as to minimize the service time?

2. Five wagons are available at five stations 1, 2, 3, 4 and 5. These are required at five stations I, II, III, IV and V. The mileages between various stations are given by:

Station
Wagon I II III IV V
1 10 5 9 18 11
2 13 19 6 12 14
3 3 2 4 4 5
4 18 9 12 17 15
5 11 6 14 19 10

3. A department head has four subordinates and four tasks. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimate of the time each man would take to perform each task is given below. How should the tasks be allocated, one to a man, so as to minimize the total man hours?

Man
Job A B C D
W 8 26 17 11
X 13 28 4 26
Y 38 19 18 15
Z 19 26 24 10

4. Four different buildings W, X, Y and Z are to be constructed in a Maternity Hospital by four different contractors A, B, C and D. It is also assumed that one contractor is required to build one building. Each contractor has submitted bids for the four buildings. The bids have been shown below. The problem is to determine which building is to be awarded to each contractor to keep the cost of construction of the four buildings at a minimum.

Contractor
Building A B C D
W 48 48 50 44
X 56 60 60 68
Y 96 94 90 85
Z 42 44 54 46

#### 5. Solve the following assignment problems:

(a)
Man
Job I II III IV
1 8 26 17 11
2 13 28 4 26
3 38 19 18 15
4 19 26 24 10

(b)
Man
Job I II III IV V
1 11 17 8 16 20
2 9 7 12 6 15
3 13 16 15 12 16
4 21 24 17 28 26
5 14 10 12 11 15

(c)
Man
Job I II III IV V
1 1 3 2 3 6
2 2 4 3 1 5
3 5 6 3 4 6
4 3 1 4 2 2
5 1 5 6 5 4

(d)
Man
Job I II III IV
1 12 30 21 15
2 18 33 9 31
3 44 25 24 21
4 23 30 28 14

(e)
Man
Job I II III IV
1 2 3 4 5
2 4 5 6 7
3 7 8 9 8
4 3 5 8 4

(f)
Man
Job I II III IV V
1 12 8 7 15 4
2 7 9 1 14 10
3 9 6 12 6 7
4 7 6 14 6 10
5 9 6 10 10 6

6. One car is available at each of the station 1, 2, 3, 4, 5, 6 and one car is required at each of the stations 7, 8, 9, 10, 11, 12. The distances between the various stations are given in the matrix below. How should the cars be dispatched so as to minimize the total mileage covered ?

7 8 9 10 11 12
1 41 72 39 52 25 51
2 22 29 49 65 81 50
3 27 39 60 51 32 32
4 45 50 48 52 65 43
5 29 40 39 26 30 33
7 82 40 40 60 51 30

7. The Manager of a company wants to assign X, Y and Z to regional offices Delhi, Mumbai, Kolkata and Chennai. The cost of reallocation (in Rupees) of the three officers at the four regional offices are given below:

Office
Officer Delhi Mumbai Kolkata Chennai
X 16000 22000 24000 20000
Y 10000 32000 26000 16000
Z 10000 20000 46000 30000

How should the officers be allocated appropriate regional offices so as to minimize the cost?

8. A company has four sales territories and four salesmen available for assignment. The sales territories are not equally rich in their sales potential and the salesmen also differ in their selling ability. The following matrix gives the sales (in thousand rupees) for each salesman to be assigned to each territory. How should the territories be assigned to salesmen so as to maximize the total sales?

Territory
Salesman A B C D
W 42 35 28 21
X 30 25 20 15
Y 30 25 20 15
Z 24 20 16 12

9. One-Ride Airlines that operates seven days a week has the time-table given below. Crews must have a minimum layover of 6 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in smaller layover.

Delhi - Jaipur
Flight No. Departure Arrival
101 7.00 AM 8.15 AM
102 8.30 AM 9.45 AM
103 1.00 PM 2.15 PM
104 6.00 PM 7.15 PM
Jaipur - Delhi
Flight No. Departure Arrival
201 8.00 AM 9.15 AM
202 9.00 AM 10.15 AM
203 12.00 Noon 1.15 PM
204 5.45 PM 7.00 PM

For each pair also mention the city where crew should be based.

10. Railways that operates seven days a week has the time-table given below. Crews must have a minimum layover of 2 hours between trains. Obtain the pairing of trains that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in smaller layover.

Delhi - Karnal
Train No. Departure Arrival
1010 7.00 AM 9.00 AM
1020 8.30 AM 10.30 AM
1030 1.00 PM 2.00 PM
1040 6.50 PM 8.50 PM
Karnal - Delhi
Train No. Departure Arrival
2010 8.00 AM 10.00 AM
2020 9.00 AM 11.00 AM
2030 12.20 PM 2.20 PM
2040 5.45 PM 7.45 PM

For each pair also mention the city where crew should be based.

11. One ride airline operating 7 days a week has given the following timetable. Crews must have a minimum layover of 5 hours between flights. Obtain the pairing flights that minimizes layover time away from home. For any pairing the crew will be based at the city that results in the smaller layover.

Chennai - Mumbai
Flight No. Departure Arrive
A1 6 AM 8 AM
A2 8 AM 10 AM
A3 2 PM 4 PM
A4 8 PM 10 PM
Mumbai - Chennai
Flight No. Departure Arrive
B1 8 AM 10 AM
B2 9 AM 11 AM
B3 2 PM 4 PM
B4 7 PM 9 PM