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?

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

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?

बेरीज

उत्तर

We can express the matrix form

`[(31,25,33,29), (25,24,23,21), (19,21,23,24), (38,36,34,40)]`

Subtracting the smallest element in each row from every element of it,

`[(6,0,8,4), (4,3,2,0), (0,2,4,5), (4,2,0,6)]`

Subtracting the smallest element In each column from very element of it.

`[(6,0,8,4), (4,3,2,0), (0,2,4,5), (4,2,0,6)]`

All the zeros of the above matrix are covered with minimum number of lines as below :

No. of lines = No. of rows/columns

Assignment of jobs :

A → Q, B → S, C → P, D → R

Minimum cost = 25 + 21 + 19 + 34

= Rs 99

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Assignment Problem

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 6.2.3 Q 6.2.2 Next

2014-2015 (March)

APPEARS IN

2014-2015 (March) (with solutions)

Q 6.2.3 | 4 marks

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Assignment Problem

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संबंधित प्रश्‍न

Determine `l_92 and l_93, "given that" l_91 = 97, d_91 = 38 and q_92 = 27/59`

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
Machines	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

In a factory there are six jobs to be performed each of which should go through two machines A and B in the order A - B. The processing timing (in hours) for the jobs arc given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T. What is the value of T? Also find the idle time for machines · A and B.

Jobs	J₁	J₂	J₃	J₄	J₅	J₆
Machine A	1	3	8	5	6	3
MAchine B	5	6	3	2	2	10

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 for each machine is given in the following table:

Jobs	Machines (Processing Cost in ₹)
Jobs	P	Q	R	S
A	31	25	33	29
B	25	24	23	21
C	19	21	23	24
D	38	36	34	40

Find the optimal assignment to minimize the total processing cost.

The assignment problem is said to be balanced if ______.

Choose the correct alternative :

The assignment problem is said to be balanced if it is a ______.

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.

Choose the correct alternative:

Assignment Problem is special case of ______

State whether the following statement is True or False:

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

What is the difference between Assignment Problem and Transportation 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)

Choose the correct alternative:

The purpose of a dummy row or column in an assignment problem is to

A car hire company has one car at each of five depots a, b, c, d and e. A customer in each of the fine towers A, B, C, D and E requires a car. The distance (in miles) between the depots (origins) and the towers(destinations) where the customers are given in the following distance matrix.

	a	b	c	d	e
A	160	130	175	190	200
B	135	120	130	160	175
C	140	110	155	170	185
D	50	50	80	80	110
E	55	35	70	80	105

How should the cars be assigned to the customers so as to minimize the distance travelled?

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 ₹)
Job	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 ₹)
Job	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 ₹)
Job	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 ₹)
Job	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 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

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	a	b	c	d	e
A	160	130	175	190	200
B	135	120	130	160	175
C	140	110	155	170	185
D	50	50	80	80	110
E	55	35	70	80	105

	a	b	c	d	e
A	160	130	175	190	200
B	135	120	130	160	175
C	140	110	155	170	185
D	50	50	80	80	110
E	55	35	70	80	105

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

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उत्तर

	a	b	c	d	e
A	160	130	175	190	200
B	135	120	130	160	175
C	140	110	155	170	185
D	50	50	80	80	110
E	55	35	70	80	105