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प्रश्न
State whether the following is True or False :
In assignment problem, each facility is capable of performing each task.
विकल्प
True
False
उत्तर
In assignment problem, each facility is capable of performing each task True.
APPEARS IN
संबंधित प्रश्न
A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job is given in the following table:
|
Jobs
|
Machines |
|||
|
P |
Q |
R |
S |
|
|
Processing Cost (Rs.)
|
||||
|
A |
31 |
25 |
33 |
29 |
|
B |
25 |
24 |
23 |
21 |
|
C |
19 |
21 |
23 |
24 |
|
D |
38 |
36 |
34 |
40 |
How should the jobs be assigned to the four machines so that the total processing cost is minimum?
Solve the following minimal assignment problem and hence find the minimum value :
| I | II | III | IV | |
| A | 2 | 10 | 9 | 7 |
| B | 13 | 2 | 12 | 2 |
| C | 3 | 4 | 6 | 1 |
| D | 4 | 15 | 4 | 9 |
Solve the following minimal assignment problem and hence find minimum time where '- ' indicates that job cannot be assigned to the machine :
| Machines | Processing time in hours | ||||
| A | B | C | D | E | |
| M1 | 9 | 11 | 15 | 10 | 11 |
| M2 | 12 | 9 | - | 10 | 9 |
| M3 | - | 11 | 14 | 11 | 7 |
| M4 | 14 | 8 | 12 | 7 | 8 |
Solve the following maximal assignment problem :
| Branch Manager | Monthly Business ( Rs. lakh) | |||
| A | B | C | D | |
| P | 11 | 11 | 9 | 9 |
| Q | 13 | 16 | 11 | 10 |
| R | 12 | 17 | 13 | 8 |
| S | 16 | 14 | 16 | 12 |
A departmental head has three jobs and four subordinates. The subordinates differ in their capabilities and the jobs differ in their work
contents. With the help of the performance matrix given below, find out which of the four subordinates should be assigned which jobs ?
| Subordinates | Jobs | ||
| I | II | III | |
| A | 7 | 3 | 5 |
| B | 2 | 7 | 4 |
| C | 6 | 5 | 3 |
| D | 3 | 4 | 7 |
Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at 5 stations I, II, III, IV, and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?
| I | II | III | IV | V | |
| 1 | 10 | 5 | 9 | 18 | 11 |
| 2 | 13 | 9 | 6 | 12 | 14 |
| 3 | 3 | 2 | 4 | 4 | 5 |
| 4 | 18 | 9 | 12 | 17 | 15 |
| 5 | 11 | 6 | 14 | 19 | 10 |
The assignment problem is said to be balanced if ______.
The objective of an assignment problem is to assign ______.
Fill in the blank :
When an assignment problem has more than one solution, then it is _______ optimal solution.
Fill in the blank :
An _______ is a special type of linear programming problem.
State whether the following is True or False :
It is not necessary to express an assignment problem into n x n matrix.
Solve the following problem :
A plant manager has four subordinates, and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. This estimate of the time each man would take to perform each task is given in the effectiveness matrix below.
| I | II | III | IV | |
| A | 7 | 25 | 26 | 10 |
| B | 12 | 27 | 3 | 25 |
| C | 37 | 18 | 17 | 14 |
| D | 18 | 25 | 23 | 9 |
How should the tasks be allocated, one to a man, as to minimize the total man hours?
Choose the correct alternative:
Assignment Problem is special case of ______
Choose the correct alternative:
The assignment problem is said to be balanced if ______
In an assignment problem if number of rows is greater than number of columns, then dummy ______ is added
State whether the following statement is True or False:
In assignment problem, if number of columns is greater than number of rows, then a dummy row is added
State whether the following statement is True or False:
In assignment problem each worker or machine is assigned only one job
What is the Assignment problem?
Three jobs A, B and C one to be assigned to three machines U, V and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost.
| Machine | ||||
| U | V | W | ||
| Jobs | A | 17 | 25 | 31 |
| B | 10 | 25 | 16 | |
| C | 12 | 14 | 11 | |
(cost is in ₹ per unit)
A computer centre has got three expert programmers. The centre needs three application programmes to be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minitues required by the experts to the application programme as follows.
| Programmers | ||||
| P | Q | R | ||
| Programmers | 1 | 120 | 100 | 80 |
| 2 | 80 | 90 | 110 | |
| 3 | 110 | 140 | 120 | |
Assign the programmers to the programme in such a way that the total computer time is least.
A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below:
| Tasks | |||||
| 1 | 2 | 3 | 4 | ||
| Subordinates | P | 8 | 26 | 17 | 11 |
| Q | 13 | 28 | 4 | 26 | |
| R | 38 | 19 | 18 | 15 | |
| S | 9 | 26 | 24 | 10 | |
How should the tasks be allocated to subordinates so as to minimize the total manhours?
Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.
| 1 | 2 | 3 | 4 | |
| A | 4 | 7 | 3 | 7 |
| B | 8 | 2 | 5 | 5 |
| C | 4 | 9 | 6 | 9 |
| D | 7 | 5 | 4 | 8 |
| E | 6 | 3 | 5 | 4 |
| F | 6 | 8 | 7 | 3 |
Choose the correct alternative:
In an assignment problem involving four workers and three jobs, total number of assignments possible are
A natural 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 distance(in kilometers) between the cities with a surplus and the cities with a deficit are displayed below:
| To | |||||||
| 7 | 8 | 9 | 10 | 11 | 12 | ||
| From | 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 | 33 | |
| 6 | 72 | 30 | 30 | 50 | 41 | 20 | |
How should the truck be dispersed so as to minimize the total distance travelled?
A job production unit has four jobs P, Q, R, and S which can be manufactured on each of the four machines I, II, III, and IV. The processing cost of each job for each machine is given in the following table:
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 31 | 25 | 33 | 29 |
| Q | 25 | 24 | 23 | 21 |
| R | 19 | 21 | 23 | 24 |
| S | 38 | 36 | 34 | 40 |
Find the optimal assignment to minimize the total processing cost.
Five wagons are available at stations 1, 2, 3, 4 and 5. These are required at 5 stations I, II, III, IV and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?
| I | II | III | IV | V | |
| 1 | 10 | 5 | 9 | 18 | 11 |
| 2 | 13 | 9 | 6 | 12 | 14 |
| 3 | 7 | 2 | 4 | 4 | 5 |
| 4 | 18 | 9 | 12 | 17 | 15 |
| 5 | 11 | 6 | 14 | 19 | 10 |
A job production unit has four jobs P, Q, R, S which can be manufactured on each of the four machines I, II, III and IV. The processing cost of each job for each machine is given in the following table :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 31 | 25 | 33 | 29 |
| Q | 25 | 24 | 23 | 21 |
| R | 19 | 21 | 23 | 24 |
| S | 38 | 36 | 34 | 40 |
Complete the following activity to find the optimal assignment to minimize the total processing cost.
Solution:
Step 1: Subtract the smallest element in each row from every element of it. New assignment matrix is obtained as follows :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 2: Subtract the smallest element in each column from every element of it. New assignment matrix is obtained as above, because each column in it contains one zero.
Step 3: Draw minimum number of vertical and horizontal lines to cover all zeros:
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 4: From step 3, as the minimum number of straight lines required to cover all zeros in the assignment matrix equals the number of rows/columns. Optimal solution has reached.
Examine the rows one by one starting with the first row with exactly one zero is found. Mark the zero by enclosing it in (`square`), indicating assignment of the job. Cross all the zeros in the same column. This step is shown in the following table :
| Job | Machines (Processing cost in ₹) |
|||
| I | II | III | IV | |
| P | 6 | 0 | 8 | 4 |
| Q | 4 | 3 | 2 | 0 |
| R | 0 | 2 | 4 | 5 |
| S | 4 | 2 | 0 | 6 |
Step 5: It is observed that all the zeros are assigned and each row and each column contains exactly one assignment. Hence, the optimal (minimum) assignment schedule is :
| Job | Machine | Min.cost |
| P | II | `square` |
| Q | `square` | 21 |
| R | I | `square` |
| S | III | 34 |
Hence, total (minimum) processing cost = 25 + 21 + 19 + 34 = ₹`square`
A plant manager has four subordinates and four tasks to perform. The subordinates differ in efficiency and task differ in their intrinsic difficulty. Estimates of the time subordinate would take to perform tasks are given in the following table:
| I | II | III | IV | |
| A | 3 | 11 | 10 | 8 |
| B | 13 | 2 | 12 | 2 |
| C | 3 | 4 | 6 | 1 |
| D | 4 | 15 | 4 | 9 |
Complete the following activity to allocate tasks to subordinates to minimize total time.
Solution:
Step I: Subtract the smallest element of each row from every element of that row:
| I | II | III | IV | |
| A | 0 | 8 | 7 | 5 |
| B | 11 | 0 | 10 | 0 |
| C | 2 | 3 | 5 | 0 |
| D | 0 | 11 | 0 | 5 |
Step II: Since all column minimums are zero, no need to subtract anything from columns.
Step III: Draw the minimum number of lines to cover all zeros.
| I | II | III | IV | |
| A | 0 | 8 | 7 | 5 |
| B | 11 | 0 | 10 | 0 |
| C | 2 | 3 | 5 | 0 |
| D | 0 | 11 | 0 | 5 |
Since minimum number of lines = order of matrix, optimal solution has been reached
Optimal assignment is A →`square` B →`square`
C →IV D →`square`
Total minimum time = `square` hours.
