Assignment Problems

                                                                                                  

Q1.      Carefree Corporation has four plants each of which can produce any one of the four products. Product cost differs from one plant to another as follow.

Plants	Product 1	Product 2	Product 3	Product 4
A	33	40	43	32
B	45	28	31	23
C	42	29	36	29
D	27	42	44	38

            You are required:

T o obtain which product each plant should produce to minimize cost,

Ans :    A-4, B-3, C-2, D-1 & TOTAL COST = 119

Q2.      The XYZ company has five jobs A, B, C, D, & E to be done and five men L,M,N,O, P to do these jobs. The number of hours each man would take to accomplish each job is given by the following table

	L	M	N	0	P
A	4	6	11	16	9
B	5	8	16	19	9
C	9	13	21	21	13
D	6	6	9	11	7
E	11	11	16	26	11

Find the optimal schedule with time for the above assignment problem.

Ans :    A-N, B-M, C-L, D-O, E-P & Total Cost= 50

Q3.      Solve the assignment problem represented by the following effective matrix

	A	B	C	D	E	F
A	9	22	58	11	19	27
B	43	78	72	50	63	48
C	41	28	91	37	45	33
D	74	42	27	49	39	32
E	36	11	57	22	25	18
F	3	56	53	31	17	28

Ans:     Total cost= 142

Q4.      A transport corporation has three vehicles in three Cities. Each of vehicles cab be assigned to any of the four other cities. The distance differs from one city to another as under

	W	X	Y	Z
A	33	40	43	32
B	45	28	31	23
C	42	29	36	29

You are required to assign a vehicle to a city in such a way that the total distance travelled is minimized

Ans:     Total cost =85            

Q5.      Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs

	1	2	3	4	5
1	6	2	5	2	6
2	2	5	8	7	7
3	7	8	6	9	8
4	6	2	3	4	5
5	9	3	8	9	7
6	4	7	4	6	8

Ans:     Total cost =16

 Q6.     Four Operators O₁,  O₂, O₃ and O₄ are available to a manager who has to get four jobs J₁,J₂, J₃ and J₄  done by assigning one job to each operator. Given the time needed by different operators for different jobs in the matrix below:

	J1	J2	J3	J4
O1	12	10	10	8
O2	14	12	15	11
O3	6	10	16	4
O4	8	10	9	7

I)                   How should manager assign the jobs so that the total time needed for all four jobs is minimum?

II)                 If job J2 is not to be assigned to operator O2, what should be the assignment over how much additional total time will be required?

Ans:     1)         Total Cost=34,            2)Total Cost=36

Q7.      Solve the following assignment problem

	Leasing	Portfolio Management	Private Mutual Funds	Swaps and Options
Monday	50	40	60	20
Tuesday	40	30	40	30
Wednesday	60	20	30	20
Thursday	30	30	20	30
Friday	10	20	10	30

Ans:     Total cost =70

           

Q8.      A solicitor firm employs typists on hourly piece basis for their daily work. There are five typists for service and their charges and speeds are different. According to an earlier understanding only one job is given to one typist and the typist is paid for full hours even if he works for a fraction of an hour. Find the least cost allocation for the following data:

Typist	Rate per hours (Rs.)	No. of Pages Typed/hour	job	No. of Pages
A	5	12	P	199
B	6	14	Q	175
C	3	8	R	145
D	4	10	S	298
E	4	11	T	178

Ans:     Total cost =399

Q9.      The secretary of a school is taking bids on city’s four school bus routes. Four companies have made the bids as detailed in the following table

Company	Route I	Route II	Route III	Route  IV
1	Rs. 4000	Rs. 5000	-	-
2	-	Rs. 4000	-	Rs. 4000
3	Rs. 3000	-	Rs. 2000	-
4	-	-	Rs.4000	Rs. 5000

Ans:     Total cost 15,000

 Q10     A machine operator processes five type of items on his machine each week, and must choose a sequence for them. The setup per change depends on the item presently are the machine and setup to be made , according to the following table

	A	B	C	D	E
A	-	4	7	3	4
B	4	-	6	3	4
C	7	6	-	7	5
D	3	3	7	-	7
E	4	4	5	7	-

            If he processes type of item once and only once each week how should be sequence the items on his machine in order to minimize the total setup cost?

Ans: Total cost = 20

Q11.    A manufacture of complex electronic equipment has just received a sizable contract and plans to subcontract part of the job. He has solicited bids for 6 subcontracts from 4 firms. Each jobs is sufficiently large that any one firm can take only 1 job. The table below shows the bid and the cost estimate (in Rs. 10,000) for doing the job internally, No that no more than 2 jobs can be performed internally.

firm		Job1	Job2	Job3	Job4	Job5	Job6
	1	48	72	36	52	50	65
	2	44	67	41	53	48	64
	3	46	69	40	45	45	68
	4	43	73	37	51	44	62
Internal		50	65	35	50	46	63

Required

1)      How do you complete this table so that can be solved by Hungarian method?

2)      Solve the problem by Hungarian method for minimizing total cost.

Ans: Total Cost = 232

Q12.    An aero plane that operates 7 days a week follows’ the following time table. Crew must have at least 6 hours layover time before starting a second flight. Crew will be stationed at a city that results in smaller layover. Find the pairing of flights that minimize total layover period away from station and also suggest the planning for stations the crew. 

 

Fight No	Delhi-Departure	Srinagar- Arrival	Flight No	Srinagar- -Departure	Delhi-Arrival
1	0700	0800	5	0800	0900
2	0800	0900	6	0900	1000
3	1400	1500	7	1200	1300
4	1900	2000	8	1700	1800

Ans: Total Time = 52

Q13.    An air line that operates 7 days a week has the time table show below. Crew must have a minimum lay over 5 hour between flights. Obtain the pairing of flights that minimizes layover time away from home assuring that crew flying from Delhi to Jaipur can be based either at Delhi or Jaipur for any given pairing, the crew will be based at the city not results in smaller layover

Fight No	Delhi-Departure	Jaipur Arrival	Flight No	Jaipur-Departure	Delhi Arrival
101	7.00 AM	8.30 AM	201	8.00 AM	9.15 AM
102	8.00 AM	9.00 AM	202	8.30 AM	9.45 AM
103	1.30 PM	2.30 PM	203	12 noon	1.15 PM
104	6.30 PM	7.30 PM	204	5.30 PM	6.45 PM

Q14.    Stayfree corporation has four salesmen each of them can be assigned any of the four sales territories. Estimates of the sales revenues in hundred of rupees for each salesmen are as under.

	Sales Territories
	1	2	3	4
A	65	78	83	60
B	85	52	59	44
C	83	56	69	64
D	49	80	85	84

            You are required to obtain the optimal assigned that maximize the sales revenue

Ans: Total profit= 31,600

Q15.    A Engineer wants to assign four new methods to three work centers. The assignment of the new methods will increase production & they are given below. If only one method cab be assigned to a work force centre, determine the optimal assignment.    

	Increase in production (units) Work centre
	A	B	C
1	10	7	8
2	8	9	7
3	7	12	6
4	10	10	8

Ans: Total Profit = 30

Q16.    A charted accountant has four charter Accountant each of whom can be assigned any of three audit assignment. Because of the varying work experience of charted accountants the net surplus (professional fee Minus expenses) varies as under

Chart Accountant	Audit Assignment
	W	X	Y
A	65	78	83
B	85	52	59
C	83	56	69
D	49	80	85

Ans: Total Profit = 248

Q17.    Solve the following assignment problem. The data given in the table refer to production in certain units   

operators	Machines
	W	X	Y
A	65	78	83
B	85	52	59
C	83	56	69
D	49	80	85

Q17.    Five lathes are to be allotted to give operations (one for each). The following table gives weekly output figures (in each).

Operator	Weekly output in Lathe
	L1	L2	L3	L4	L5
A	22	22	27	32	36
B	19	23	29	34	40
C	23	28	35	39	34
D	21	24	31	37	42
E	24	28	31	36	41

Profit per piece is Rs. 25. Find the maximum profit per week.                 

            Answer:           Total cost:25*160=4000        

Q18.    The captain of a cricket team has to allot five middle batting positions to five batsmen. The average runs scored by each batsman of these positions are as follows:

Batsman	Batting Positions
	1	2	3	4	5
p	40	40	35	25	50
Q	42	30	16	25	27
R	50	48	40	60	50
S	20	19	20	18	25
T	58	60	59	55	53

1)     

To find the assignment of batsmen to position,  which would give the maximum number of runs

2)      If another batsman U with the following average runs in batting positions as give below

Positions              1              2              3              4              5

Runs                      45           52           38           50           49

Is added to the team, should he be included to play in the team? If so, who will be replaced by him?

Ans:     1)         Total Cost= 232,         2)         Total Cost=263

Q19.    Johnson & Johnson ltd. Has four plants each of which can manufacture anyone of the four products. Productions cost & sale revenue differ from one plant to another. Given the revenue and cost data below, obtain which product each plant should produce to Maximize Profit

	Sales Revenue (Rs.) Product
Plants	1	2	3	4
A	65	78	83	85
B	85	52	59	73
C	83	56	69	78
D	49	80	85	73

	Product Cost (Rs) Product
Plants	1	2	3	4
A	33	40	43	45
B	45	28	31	37
C	42	29	36	41
D	27	42	44	37

Ans:-   Total Profit=156

Q20     A Production manager wants to assign one of five new methods to each of the four operations. The following table summarizes the weekly output in units

Operator	Weekly Output
	M1	M2	M3	M4	M5
A	4	6	11	16	9
B	5	8	16	19	9
C	9	13	21	21	13
D	6	6	9	11	7

Cost per unit is Rs. 10, Selling price per unit Rs. 35.

Ans:     Total Profit 7700-2200=5500

Q21.    A company is faced with the problem of assigning 4 machines to 6 different jobs (one machine to one job only) the profits are estimated as follow

	Machine
Jobs	A	B	C	D
1	3	6	2	6
2	7	1	4	4
3	3	8	5	8
4	6	4	3	7
5	5	2	4	3
6	5	7	6	4

Required: Solve the problem to maximize the total profit

Ans :    Total Profit 28

 Q22.    A firm is contemplating the introduction of three products 1,2 &3 in it plants A, B & C. Only a single product in decided to be introduced in each of the plants. The unit cost of producing its products in jth plants, is given in the following matrix.

Products	Plants
	W	X	Y
1	8	12	-
2	10	6	4
3	7	6	6

1)      How should the products be assigned so that the total unit cost is minimized?           

Ans 18

2)      If the Quantity of different products is as follows then what assignment shall minimize the aggregate production cost

Product                        Quantity (in unit)

1                                            2000

2                                            2000

3                                            10,000                        

Ans:-84,000

 Q23.    An organization producing 4 different products viz A,B,C & D having 4 operators viz P Q R S. Who are capable of producing any of the four products, work effectively 7 hour a day. The times (in minutes) required for each operator for producing each of the product are given in the cell of the following matrix along with profit (Rs. Per unit)

	Product
Operator	A	B	C	D
1	6	10	14	12
2	7	5	3	4
3	6	7	10	10
4	20	10	15	15
PROFIT (Rs. /unit)	3	2	4	1

F

Find out the assignment of operators to products which will maximize the profit.

Ans:     Total Profit = 918

Q24.    A firm produces four products. There four operators who are capable of producing any of these four products. The processing time varies from operator to operator. The firm records 8 hours a day & allows 30 minutes for lunch. The processing time in minutes and profit for each of the product are given

	Product
Operator	A	B	C	D
1	15	9	10	6
2	10	6	9	6
3	25	15	15	9
4	15	9	10	10
PROFIT (Rs. /unit)	8	6	5	4

F

Find out the assignment of operators to products which will maximize the profit.

Ans:     Total Profit = 1140

Q25.    A manufacturing company has four zones A B C D & four sales engineers P Q R S respectively for assignment. Since the zones are not equally rich in sales potential, therefore it is estimated that a particular engineer operating in a particular zone will bring the following sales

Zones A:4,20,000       Zones B:3,36,000        Zones C:2,94,000        Zones D:4,62,000

The engineers are having different sales ability. Working under the same conditions, their yearly sale are proportional to 14, 9, 11 & 8 respectively. The criteria of maximum next best to the second richest zone and so find the optimum assignment and the maximum sales

Ans:     Total Profit = 11,76,000

Q26.    A Company has four territories & four salesmen available for assignment. The territories are not equally rich in their potential. It is estimated that a typical salesmen operating in each territory would bring the following sales:

Territory             I             II               III             IV

            Annual Sales  60,000     50,000         40,000   30,000

The four salesmen are also considered to differ in ability. It is estimated that working that under the same conditions their yearly sales would be proportionately as follow

Salesmen A                B                     C                     D

            Proportion7               5                      5                      4

If the criterion is maximum expected total sales, the intuitive answer is to assign the best salesmen to the richest territory, the next best salesmen to the second riches & so on.