Suggest Optimum Solution to the Following Assignment. Problem, Also Find the Total Minimum Service Time. Service Time ( in Hrs.) - Mathematics and Statistics

प्रश्न

Suggest optimum solution to the following assignment. Problem, also find the total minimum service time.
Service Time ( in hrs.)

Counters	Salesmen
Counters	A	B	C	D
W	41	72	39	52
X	22	29	49	65
Y	27	39	60	51
Z	45	50	48	52

बेरीज

उत्तर

This problem is already 4 x 4.
Select the smallest element in each row and subtract it from every element in each row :

Select the smallest element in each column of the above matrix and subtract it form every element in that column.

Draw the minimum lines covering all zeros.

Minimum lines covering all zeros is not equal to the order of the matrix.
Minimum uncovered value 2 is subtracted from uncovered values and added to values at intersection of the lines.

Draw minimum lines covering all the zeros.

Minimum lines covering all the zeros equal to the order of the matrix.
∴ Allocation of counters can be done.

The allocation of the counters to the salesmen is
W → C, X → B, Y → A, Z → D

The minimum time = 39 + 29 + 27 +52 = 147 Hrs.

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Assignment Problem

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 5.B.ii Q 5.B.i Q 5.B.iii

2018-2019 (February) Set 1

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2018-2019 (February) Set 1 (with solutions)

Q 5.B.ii | 4 marks

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Assignment Problem

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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 :

Machines	A	B	C	D	E
M₁	27	18	∞	20	21
M₂	31	24	21	12	17
M₃	20	17	20	∞	16
M₄	21	28	20	16	27

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

Solve the following maximal assignment problem :

Branch Manager	Monthly Business ( Rs. lakh)
Branch Manager	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
Subordinates	I	II	III
A	7	3	5
B	2	7	4
C	6	5	3
D	3	4	7

Five different machines can do any of the five required jobs, with different profits resulting from each assignment as shown below:

Job	Machines (Profit in ₹)
Job	A	B	C	D	E
1	30	37	40	28	40
2	40	24	27	21	36
3	40	32	33	30	35
4	25	38	40	36	36
5	29	62	41	34	39

Find the optimal assignment schedule.

Choose the correct alternative :

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

Choose the correct alternative :

In an assignment problem if number of rows is greater than number of columns then

Fill in the blank :

When an assignment problem has more than one solution, then it is _______ optimal solution.

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?

Solve the following problem :

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
B	125	110	120	150	165
C	130	100	145	160	175
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?

Choose the correct alternative:

The assignment problem is generally defined as a problem of ______

Choose the correct alternative:

Assignment Problem is special case of ______

State whether the following statement is True or False:

The objective of an assignment problem is to assign number of jobs to equal number of persons at maximum cost

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

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.

Find the optimal solution for the assignment problem with the following cost matrix.

		Area
		1	2	3	4
	P	11	17	8	16
Salesman	Q	9	7	12	6
	R	13	16	15	12
	S	14	10	12	11

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:

Number of basic allocation in any row or column in an assignment problem can be

Choose the correct alternative:

North – West Corner refers to ______

Choose the correct alternative:

The solution for an assignment problem is optimal if

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?

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 dairy plant has five milk tankers, I, II, III, IV and V. Three 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
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 ₹)
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

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?

Workers	Packing of
	Books	Toys	Crockery	Cutlery
A	3	11	10	8
B	13	2	12	12
C	3	4	6	1
D	4	15	4	9

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`

Suggest Optimum Solution to the Following Assignment. Problem, Also Find the Total Minimum Service Time. Service Time ( in Hrs.) - Mathematics and Statistics

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संबंधित प्रश्‍न

Select a course

	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

Suggest Optimum Solution to the Following Assignment. Problem, Also Find the Total Minimum Service Time. Service Time ( in Hrs.) - Mathematics and Statistics

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Select a course

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