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Maharashtra State BoardHSC Commerce (English Medium) 12th Standard Board Exam
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State whether the following statement is True or False: In assignment problem each worker or machine is assigned only one job - Mathematics and Statistics

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Question

State whether the following statement is True or False: 

In assignment problem each worker or machine is assigned only one job

Options

  • True

  • False

MCQ
True or False

Solution

True

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Assignment Problem
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Q 6
Chapter 2.7: Assignment Problem and Sequencing - Q.3

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SCERT Maharashtra Mathematics and Statistics (Commerce) [English] 12 Standard HSC
Chapter 2.7 Assignment Problem and Sequencing
Q.3 | Q 6

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W 41 72 39 52
X 22 29 49 65
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A departmental head has three jobs and four subordinates. The subordinates differ in their capabilities and the jobs differ in their work
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1 10 5 9 18 11
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5 29 62 41 34 39

Find the optimal assignment schedule.


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A dairy plant has five milk tankers, I, II, III, IV and V. These milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.

  I II III IV V
A 150 120 175 180 200
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Choose the correct alternative:

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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:

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    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?


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  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
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Choose the correct alternative:

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From 1 31 62 29 42 15 41
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  I II III IV V
A 150 120 175 180 200
B 125 110 120 150 165
C 130 100 145 160 170
D 40 40 70 70 100
E 45 25 60 70 95

How should the milk tankers be assigned to the chilling center so as to minimize the 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
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Find the optimal assignment to minimize the total processing cost.


A department store has four workers to pack goods. The times (in minutes) required for each worker to complete the packings per item sold is given below. How should the manager of the store assign the jobs to the workers, so as to minimize the total time of packing?

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A 3 11 10 8
B 13 2 12 12
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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.


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