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CA. B L Chakravarti – For Solution SMS Your Name & Mail Id at 9810993376 & 9910993376 2.1

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The Assignment Problems

Past Examinations Coverage:

Year M-13 N-12 M-12 N-11 M-11 N-10 N-10* M-10 M-10* N-09

Q No. - 4 (a) 7 (b) 2 (b) 3 (a) 7 (d) 3 (b) 4 (b)

Marks - 8 4 6 10 4 8 4#

Year N-09* J-09 J-09* N-08 N-08* M-08 N-07 M-07 N-06 M-06

Q No. 4 (b) 1 (b) 4 (a) 2 (a)

Marks 7 8 11 6

Year N-05 M-05 N-04 M-04 N-03 M-03 N-02 M-02 N-01 M-01

Q No. 6 (b) 2 (a) 2 (b) 2 (c) 2 (c) 2 (c)

Marks 7 10 10 10 10 10

Year N-00 M-00 N-99 M-99 N-98 M-98 N-97 M-97 N-96 M-96

Q No. 2 (b) 2 (c) 3 (c) 2 (b) 2 (b) 2 (b) 3 (b) 2 (b) 6 (b) 3 (b)

Marks 10 10 10 10 10 10 10 10 10 10

* means questions from old syllabus

# means theory questions

Learning Objectives:

Understanding the feature of Assignment Problems.

Formulate an Assignment problem.

Hungarian Method

Unbalanced Assignment Problems

Profit Maximization Problems

Advance Management Accounting

1. Email: [email protected] Page 12.2

The Assignment Problem:

The assignment problem, like the transportation problem, is a special case of the linear programming problem.

In general items, it is concerned with a one to one assignment. One person to one machine, one machine to one

job etc. The cost (or profit) of each person-machine, machine-job, or other assignment is known; the objective is

to minimize the total cost or to maximize the total profit of the resource.

Suppose there are 5 jobs to be performed and 5 persons are available for doing these jobs. Each persons can do

each job but with different efficiency. Now the problem is to find an assignment for allocation of jobs to the

available persons i.e. which jobs should be assigned to which person so that the total cost of performing all jobs

is minimum. thus in Assignment Problem, the objective is to assign a no. of person to the same number of

jobs in such a way that the total cost is minimum. The assignment is to be made on one to one basis i.e. each

person can associate with one and only one job.

The Assignment Algorithms (Hungarian Assignment Method)

The assignment problem can be solved by applying the following steps:

Step 1(Row Operation):

In the Cost Matrix, subtract the lowest (minimum) element of each row from every elements of the same row.

Step 2(Column Operation):

In the Cost Matrix obtained after Step 1, subtract the lowest (minimum) element of each column from every

elements of the same column. The resulting matrix is the starting matrix for the following procedure.

Step 3:

In the Cost Matrix obtained after Step 2, draw the minimum number of horizontal and vertical lines that cover all

the zeros.

(i) If number of lines (N) = Order of the matrix (n), optimal assignment can be made and follow Step 4.

(ii) If number of lines (N) ≠ Order of the matrix (n), then for optimal assignment, we have to follow Step

3A until we obtain the matrix with number of lines (N) = Order of the matrix (n) and then follow

Step 4

Step 3A:

Select the smallest element out of the remaining elements which do not lie on any line. Subtract this element

from all such (uncovered) elements and add it to the elements which are placed at the intersections of the

horizontal and vertical lines. Do not alter the elements through which only one line passes. Repeat this steps

until we get the number of lines (N) = Order of the matrix (n)

Step 4:

(i) Starting with first row, examine all rows of matrix in step 3 until a row containing exactly

one zero is found. Surround this zero by () , indication of an assignment there. Draw a

vertical line through the column containing this zero. This eliminates any confusion of

making any further assignments in that column. Process all the rows in this way.


CA. B L Chakravarti – For Solution SMS Your Name & Mail Id at 9810993376 & 9910993376 Page 12.3

(ii) Apply the same treatment to columns also. Starting with the first column, examine all

columns until a column containing exactly one zero is found. Mark around this zero and

draw a horizontal line through the row containing this marked zero.

(iii) Repeat these steps, until one of the following situations arises:

a) No unmarked ( ) or uncovered (by a line) zero is left.

b) There may be more than one unmarked zero in one column or row. In this case, put

around one of the unmarked zero arbitrarily and pass 2 lines in the cells of the

remaining zeros in its row and column. Repeat the process until no unmarked zero is left

in the matrix.

Step 5:

After Step 4, we have a matrix which contains exactly one marked zero in each row and each column. The

assignment corresponding to these marked zeros will give the optimal assignment. After obtaining the optimal

assignment, the total actual cost can be calculated by adding the values of the corresponding cost in the assigned

cell from the optimal cost matrix.

Unbalanced Assignment Problems (No. of row ≠ No. of column)

An assignment problem is called an unbalanced assignment problem if number of sources is not equal to

number of destinations or the number of jobs is not equal to the number of persons. In such cases, dummy row

or dummy columns with zero costs are added in the matrix so as to form a square matrix.

Maximization Problems

When the objective of the assignment problem is to maximize the profits or Revenues, such types of assignment

problems is called Maximization Problems. To solve such problems, we first convert the maximization problem

to minimization problem and then apply the normal procedure of assignment algorithm.

Conversion of Maximization Problem to Minimization Problem can be done either of any following two ways:

(a) Identify the highest element in the given matrix from all the elements of the matrix and then subtract all

the elements of the matrix from the highest element. OR

(b) Put a negative sign before each of the elements in the given matrix and then subtract all the elements of

the matrix from the highest element.

Prohibited Assignment:

Prohibited Assignments are the cells where no assignment is to be made. Hence we put a very large value M or

∞ for the prohibited cells.



Theoretical Questions Q 1: [N 09, 4b, 4M]

In an assignment problem to assign jobs to men to minimize the time taken, suppose that one man does not

know how to do a particular job, how will you eliminate this allocation from the solution?

Practical Questions P1: [SM-1] *[Maximization-balanced]

An Accounts Officer has 4 subordinates and 4 tasks. The subordinates differ in efficiency. The tasks also differ in

their intrinsic difficulty. His estimates of the time each would take to perform each task is given in the matrix

below. How should the tasks be allocated one to one man, so that the total man hours are minimized?

Sub-Ordinates Task

I II III IV

1 8 26 17 11 2 13 28 4 26 3 38 19 18 15 4 19 26 24 10

[Ans.: 1-I; 2-III; 3-II; 4-IV; Minimum Time is 42 Hr]

P2: [SM-2]

A manager has 5 jobs to be done. The following matrix shows the time taken by the j-th job (j = 1,2...5) on the i-th

machine (i = 1,2,3...5). Assign 5 jobs to the 5 machines so that the total time taken is minimized.

Jobs Manager

1 2 3 4 5

I 9 3 4 2 10 II 12 10 8 11 9 III 11 2 9 0 8 IV 8 0 10 2 1 V 7 5 6 2 9

[Ans: I-2; II-3; III-4; IV-5; V-1; Minimum Time = 19]

P3: [SM-3]

5 salesmen are to be assigned to 5 districts. Estimates of sales revenue in thousands of rupees for each sales man

are given below.

District Sales Man

A B C D E

1 32 38 40 28 40 2 40 24 28 21 36 3 41 27 33 30 37 4 22 38 41 36 36 5 29 33 40 35 39

Find the assignment pattern that maximizes the sales revenue.

[Ans: 1-B; 2-A; 3-E; 4-C;5-D; Maximum Sales Revenue =R 191 Thousands]



P4: [SM-4] [M 99, 2b, 10M]

To stimulate interest and provide an atmosphere for intellectual discussion, a finance faculty in a management

school decides to hold special seminars on four contemporary topics– leasing, portfolio management, private

mutual funds, swaps and options. Such seminars should be held once a week in the afternoons. However,

scheduling these seminars (one for each topic, and not more than one seminar per afternoon) has to be done

carefully so that the number of students unable to attend is kept to a minimum. A careful study indicates that the

number of students who cannot attend a particular seminar on a specific day is as follows:

Topics Days

Leasing Portfolio Management

Private Mutual Funds

Swaps and Option

Monday 50 40 60 20 Tuesday 40 30 40 30 Wednesday 30 20 30 20 Thursday 60 30 20 30 Friday 10 20 10 30

[Ans: Mon – Swaps; Tue – No Seminar; Wed- Portfolio; Thu – Pvt. MF; Fri- Leasing; No. of Missing Student = 70]

P5: [SM-5] [N 96, 6b, 10M] [RTP N 10] [RTP N 10*]

A private firm employs typists for piecemeal work on an hourly basis. There are five typists available 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/Hour Pages/Hour Job No. of Pages A R 5 12 P 199 B R 6 14 Q 175 C R 3 8 R 145 D R 4 10 S 298 E R 4 11 T 178

[Ans: A-T; B-R; C-Q; D-P; E-S; Total Cost R 399]

P6: [SM-6] [Nov-1990] [SA-11] [RTP N 09] [Prohibited Assignment]**

WELLDONE Company has taken the third floor of a multi- storied building for rent with a view to locate one of

their zonal offices. There are five main rooms in this floor to be assigned to five managers. Each room has its

own advantages and disadvantages. Some have windows; some are closer to the washrooms or to the canteen or

secretarial pool. The rooms are of all different sizes and shapes. Each of the five managers was asked to rank

their room preferences amongst the rooms 301, 302, 303, 304 and 305. Their preferences were recorded in a

table as indicated below:

Manager M1 M2 M3 M4 M5 302 302 303 302 301 303 304 301 305 302 304 305 304 304 304

* 301 305 303 * * * 302 * *



Most of the managers did not list all the five rooms since they were not satisfied with some of these rooms and

they have left off these from the list. Assuming that their preferences can be quantified by numbers, find out as

to which manager should be assigned to which room so that their total preference ranking is a minimum.

[Ans.: M1 302, M2 304, M3 303, M4 305 and M5 301 with total minimum ranking as 7]

P7: [SM-7] [M 00, 2c, 10M] [M 97,2b, 10M] [SA-18]

XYZ airline operating 7 days a week has given the following timetable. Crews must have a minimum layover of 5

hours between flights. Obtain the pairing flights that minimize layover time away from home. For any given

pairing the crew will be based at the city that results in the smaller layover.

Hyderabad Delhi Delhi Hyderabad Flignt No Departure Arrival Flignt No Departure Arrival

A1 6 A.M 8 A.M B1 8 A.M 10 A.M A2 8 A.M 10 A.M B2 9 A.M 11 A.M A3 2 P.M 4 P.M B3 2 P.M 4 P.M A4 8 P.M 10 P.M B4 7 P.M 9 P.M

[Ans: A1-B3; A2-B4; A3-B1; A4-B2; Total Layover time = 40 hours]

P8: [SM-8] [M 96, 3b, 10M] [SA-33] [N-97 (Adp)] [RTP M 11] [RTP M 09*]

An organization producing 4 different products viz. A, B, C, and D having 4 operators viz. P, Q, R and S, who are

capable of producing any of the four products, works effectively 7 hours a day. The time (in minutes) required

for each operator for producing each of the product are given in the cells of the following matrix along with

profit (Rs. Per unit). Find the optimal assignment of operators to products which will maximize the profit.

Products Operators

A B C D

P 6 10 14 12 Q 7 5 3 4 R 6 7 10 10 S 20 10 10 15 Profit (Rs./Unit) 3 2 4 1

[Ans: P-A; Q-C; R-B; S-D; Maximum Profit = ₹ 918]

P9: [SM-9] [N 97, 3b, 10M] [SA-34] **

A firm produces four products. There are 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 and allow 30

minutes for lunch. The processing time in minutes and the profit for each of the products are given below:

Products Operators

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

Find the optimal assignment of operators to products.

[Ans: 1-D; 2-B; 3-C; 4-A; Total Profit ₹ 1140]



P10: [SM-10] [M 98, 2b, 10M] [N 07 Adapt] [SA-32]**

A manufacturing company, four has zones A, B, C, D and 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:

Zone A : R 4,20,000

Zone B : R 3,36,000

Zone C : R 2,94,000

Zone D : R 4,62,000

The engineers are having different sales ability. Working under the same conditions, their yearly sales are

proportional to 14, 9, 11 and 8 respectively. The criteria of maximum expected total sales is to be met by

assigning the best engineer to the richest zone, the next best to the second richest zone and so on.

Find the optimum assignment and the maximum sales.

[Ans: A-R; B-Q; C-S; D-P; Maximum Sales = ₹ 3,92,000]

P11: [SA-7] Five swimmers are eligible to compete in a relay team which is to consist of four swimmers

swimming four different swimming styles; back stroke, breast stroke, free style and butterfly. The times taken

for the five swimmers – A, B, C, D and E – to cover a distance of 100 meters in various swimming styles are given

below in minutes: second. A swims the back stroke in 1:09, the breast stroke in 1:15 and has never competed in

the free style or butterfly. B is a free style specialist averaging 1:01 for the 100 meters but can also swim the

breast stroke in 1:16 and butterfly in 1:20. C swims all styles – back stroke 1:10, butterfly 1:12, free style 1:05

and breast stroke 1:20. D swims only the butterfly 1:11 while E swims the back stroke 1:20, the breast stroke

1:16, the free style 1:06 and the butterfly 1:10.

Which swimmer should be assigned to which swimming style? Who will not be in the relay?

[Ans: Anand – Breast; Bhaskar – Free; Chandru – Back; Dorai – do not participate; Eashwar – Butterfly; Min time =

276 Min.]

P12: [SA-12]A manufacturer 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 bids and the cost estimates (in ₹ 10,000’s) for

doing the jobs internally. Note that no more than 2 jobs can be performed internally.

Job Firm

1 2 3 4 5 6

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

[Ans: Firm 1- 3; 2-1; 3-4; 4-5; Internal A -6; Internal B-2; Total Cost ₹ 232]



P13: [SA-8]Four Operators O1, O2, O3 and O4 are available to a manager who has t 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:

Job Firm

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 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: (i) O1-J3; O2-J2; O3-J4; O4-J1 Total Time = 34

(ii) O1-J2; O2-J4; O3-J1; O4-J3 Total Time = 36]

P14: [SA-13] A machine operator process five types of items on his machine each week, and must choose a

sequences for them. The set-up cost per change depends on the item presently n the machine and the set-up to

be made, according to the following table:

To Items For Item

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 each type of item once and only once each week, how should he sequence the items on his

machine in order to minimize the total set-up cost?

[Ans: A-D; B-A; C-E; D-B; E-C; Total Cost ₹ 20]

P15: [SA-27] [Maximization]

Johnson & Johnson Ltd. has four plants each of which can manufacture any one of the four products. Production

costs and sales 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 (₹) Product Production Cost (₹) Product

Plant 1 2 3 4 1 2 3 4

A 65 78 83 85 33 40 43 45 B 85 52 59 73 45 28 31 37 C 83 56 69 78 42 29 36 41 D 49 80 85 73 27 42 44 37 Required: Obtain the optimal assigned pattern that maximizes the sales revenue.

[Ans: A-2; B-1; C-4; D-3 Maximum Profit ₹ 156 or A-2; B-4; C-1; D-3]



P16: [SA-24] [Maximization – Prohibited Route] [May-95]

Imagine yourself to be the Executive Director of a 5-star Hotel which has four banquet halls that can be used for

all functions including weddings. The halls were all about the same size and the facilities in each hall differed.

During a heavy marriage season, 4 parties approached you to reserve a hall for the marriage to be celebrated on

the same day. These marriage parties were told that the first choice among these 4 halls would cost Rs. 10000

for the day. They were also required to indicate the second, third and fourth preferences and the price that they

would be willing to pay. Marriage party A & D indicated that they won’t be interested in Halls 3 & 4. Other

particulars are given in the following table: (₹ in ‘000)

Revenue per Hall Marriage Party I II III IV A 10 9 * * B 8 10 8 5 C 7 10 6 8 D 10 8 * *

Where * indicates that the party does not want that hall. Decide on an allocation that will maximize the revenue

to your hotel.

[Ans.: A Hall II ₹ 9000; B Hall III ₹ 8000; C Hall IV ₹ 8000; D Hall I ₹10000]

P17: [SA-26] [Maximization]

The captain of a cricket team has to allot five middle batting positions to five batsmen. The average runs scored

by each batsman at these positions are as follows:

Position Batsman

I II III IV V

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

Required:

(i) Find the assignment of batsmen to positions, which would give the maximum number of runs.

(ii) If another batsman ‘U’ with the following average runs in batting positions as given as follows is

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

Batting Positions I II III IV V

Average runs: 45 52 38 50 49

[Ans: (i) P-V; Q-I; R-IV; S-III; T-II Total Run 232; (ii) P-V; Q-I; R-IV; S-VI; T- III; U-II Total Run 263]

P18: [SA-22] [Maximization - Unbalanced]*



A chartered accountant firm has four chartered accountants each of whom can be assigned any of the three audit

assignments. Because of the varying work experience of the chartered accounts, the net surplus (professional

fess minus expenses to be incurred by the CA firm) varies as under:

Audit Assignments

CA X Y Z

A 65 78 83 B 85 52 59 C 83 56 69 D 49 80 85

Required: Find the maximum net surplus which can be obtained.

[Ans: A-Y; B-W; C-Z; D-X; Total Profit ₹ 248 or A-Y; B-W; C-Z; D-Y]

P19: [SA-25] [Maximization]*

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

each):

Weekly Output in Lathes

Operator L1 L2 L3 L4 L5 P 20 22 27 32 36 Q 19 23 29 34 40 R 23 28 35 39 34 S 21 24 31 37 42 T 24 28 31 36 41

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

[Ans: P- L1; Q- L5; R- L3; S- L4; T- L2; Maximum Output 160]

P20: [SA-28] [Maximization- Unbalanced]

A Production manager wants to assign one of the five new methods to each of four operatations. The following

table summarizes the weekly output in units:

Weekly Output Operator

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 in ₹ 10, Selling Price per unit ₹ 35. Find the maximum profit per month.

[Ans: A- M5; B- M4; C- M3; D- M2; Maximum Profit = R 5,500]

P21: [SA-30] [Maximization]*** A firm is contemplating the introduction of three products 1,2 and 3 in its plants A, B and C. Only a single

product is decided to be introduced in each of the plants. The unit cost of producing ith product in jth plants, is

given in the following matrix:

Plant

Product A B C



1 8 12 * 2 10 6 4 3 7 6 6

Required:

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

(ii) If the quantity of different products is as follows, then what assignment shall minimize the aggregate

production cost?

Product Quantity (in units)

1 2000

2 2000

3 10000

(iii) It is expected that the selling prices of the products produced by different plants would be different;

as shown in the following table:

Plant

Product A B C

1 15 18 * 2 18 16 10 3 12 10 8

Assuming that the quantities mentioned in (ii) above would be produced and sold, how should the products be

assigned to the plants to obtain maximum profits?

[Ans: 1-B; 2-C; 3-A; Maximum Profit R 70,000 ]

P 22: [RTP N 12] [SM-4 Adp] [M 99, 2b, 10M]

The ICAI decides to hold special seminars on four contemporary topics for its members: Ind. AS, Goods and

Service Tax (GST), Negative list in Service tax and Direct Tax Code (DTC). Such seminars should be held once in a

week in the afternoons. However, scheduling these seminars (one for each topic, and not more than one seminar

per afternoon) has to be done carefully so that the number of members unable to attend is kept to a minimum. A

careful study indicates that the number of members who cannot attend a particular seminar on a specific day is

as follows:

Topics Days

Ind. AS GST Negative List DTC

Monday 40 30 50 20 Tuesday 30 20 30 30 Wednesday 50 10 20 20 Thursday 20 30 10 30 Friday 10 20 10 30

Find an optimal schedule of the seminars. Also find out the total number of members who will be missing at least

one seminar.


1. Email: [email protected] .12

P 23: [RTP M 12] [RTP M 10*]

Output of 5 operators when they worked with different 4 machines are given in the table below:

Machines Operator

M1 M2 M3 M4

A 10 5 7 8 B 11 4 9 10 C 8 4 9 7 D 7 5 6 4 E 8 9 7 5

Use assignment technique to solve it.

P 25: [RTP N 11] [RTP M 10]

The following table gives the past performance of five sales man in different regions in terms of their sales

achievement in rupee lakh. Find the optimum assignment

Machines Salesman

R1 R2 R3 R4 R5

S1 26 14 10 12 9 S2 31 27 30 14 16 S3 15 18 16 25 30 S4 17 12 21 30 25 S5 20 19 25 16 10

P 26: [RTP N 09*]

Methods engineer wants to assign four new methods to work centers. The assignment of new methods will

increase Production (in units) and they are given below. If only one method can be assigned to a work center,

determine the optimum assignment.

Work Centre Method

A B C

1 10 7 8 2 8 9 7 3 7 12 6 4 10 10 8

P 27: [RTP M 09]

A company is faced with the problem of assigning six different machine to 5 different jobs. The cost are

estimated as follows (hundred of rupees).

Jobs Machines

1 2 3 4 5

1 2.5 5 1 6 1 2 2 5 1.5 7 3 3 3 6.5 2 8 3 4 3.5 7 2 9 4.5 5 4 7 3 9 6 6 6 9 5 10 6



P 28: [RTP M 09*]

A can hire company has one car at each of 5 depots a, b, c, d and e. A customer in each of the 5 towns A, B, C, D

and E requires a car the distance (in miles) between the depots (origins) and the towns (destination) where the

customers are (is given in the following distance matrix).

Depots Towns

a b c d E

A 160 130 175 190 200 B 135 120 130 160 175 C 140 110 125 170 185 D 50 50 80 80 110 E 55 35 80 80 105

How should the cars be assigned to the customer so is to minimize the distance travelled.

P 29: [RTP N 08]

Find the assignment of salesman to district that will result in maximum sales.

Districts Salesman

A B C D E

1 32 38 40 28 40 2 40 24 28 21 36 3 41 27 33 30 37 4 22 38 41 36 36 5 29 33 40 35 39

P 30: [RTP N 08*]

A company has 5 jobs to be done. The following matrix shows the written in Rs. Of assigning of ith machine (i = 1, 2, 3, 4, 5) to the jth job (j = 1, 2, 3, 4, 5) assign the 5 jobs to the 5 machines. So as to maximize the total expected profit.

Jobs Operator

1 2 3 4 5

A 5 11 10 12 4 B 2 4 6 3 5 C 3 1 5 14 6 D 6 14 4 11 7 E 7 9 8 12 5



Practical Questions – Previous Examinations

PE1: [N 12, 4a, 8M]

A production supervisor is considering how he should assign five jobs that are to be performed to five operators.

He wants to assign the jobs to the operators in such a manner that the aggregate cost to perform the job is the

least. He has the following information about the wages paid to the operators for performing these jobs:

Jobs

Operators 1 2 3 4 5

A 10 3 3 2 8 B 9 7 8 2 7 C 7 5 6 2 4 D 3 5 8 2 4 E 9 10 9 6 10

Required: Assign the job to the operators so that the aggregate cost is least.

PE 2: [M 12, 7b 4M]

The following matrix was obtained after performing row minimum operations on rows R1 and R2 an assignment

problem for minimization. Entries “xx” represent some positive numbers. (It is not meant that all “xx” numbers

are equal). State two circumstances under which an optimal solution is obtained just after the row minimum

and column minimum operations

(Candidate may use cell references as CjRj for uniformity e.g. C1R1 represent the cell at the intersection of

Column1 (C1) and Row 1 (R1) etc.)

C1 C2 C3

R1 0 xx xx R2 xx 0 xx R3 xx xx xx

[Ans:]

PE 3: [N 11, 2b 6M]**

A city corporation has decided to carry out road repairs on 4 main roads in the city. The Government has agreed

to make a special grant of ₹ 50 lacs towards the cost with the condition that the repairs should be carried out at

lowest cost. Five contractors have sent their bids. Only road will be awarded to one contractor. The bids are

given below:

Cost of Repairs (₹ in lacs)

Roads Contractors

R1 R2 R3 R4

C1 9 14 19 15 C2 7 17 20 19 C3 9 18 21 18 C4 10 12 18 19 C5 10 15 21 16

You are informed that C2 should get R1 and C4 should get R2 to minimize costs.

(i) What is the minimum cost allocation?



(ii) How much is the minimum discount that the eliminated contractor should offer for meriting a

contract?

(iii) Independent of (ii) above, if the corporation can negotiate to get a uniform discount rate from each

contractor, what is the minimum rate of discount so that the cost is within the grant amount?

[Ans:]

PE 4: [M 11, 3a, 10M]***

A manager was asked to assign tasks to operators (one task per operator only) so as to minimize the time taken.

He was given the matrix showing the hours taken by the operators for the tasks.

First, he performed the row minimum operation. Secondly, he did the column minimum operation. Then, he

realized that there were 4 tasks and 5 operators. At the third step he introduced the dummy row and continued

with his fourth step of drawing lines to cover zeros. He drew 2 vertical lines (under operator III and operator IV)

and two horizontal lines (aside task T4 and dummy task T5 ) At step 5, he performed the necessary operation

with the uncovered element, since the number of lines was less than the order of the matrix . After this, his

matrix appeared as follows:

Operator Task

I II III IV V

T1 4 2 5 0 0 T2 6 3 3 0 3 T3 4 0 0 0 1 T4 0 0 5 3 0 T5 (Dummy) 0 0 3 3 0

i. What was the matrix after step II? Based on such matrix, ascertain (ii) and (iii)given below.

ii. What was the most difficult task for operators I, II and V?

iii. Who was the most efficient operators?

iv. If you are not told anything about the manager’s errors, which operator would be denied any task? Why?

v. Can the manager go ahead with his assignment to correctly arrive at the optional assignment, or should

he start afresh after introducing the dummy task at the beginning?

[Ans:]

PE 5: [N 10*, 7d, 4M]*

Three different salesmen X, Y and Z are to be assigned three different regions A, B and C so that the company’s

revenue is maximized. The following matrix gives the sales revenue:

X Y Z

A 10 60 30

B 20 30 15

C 60 40 10

You are required to use the assignment technique to maximize revenue.

[Ans: A-Y; B-Z ; C-X; Total Sales Revenue₹ 135 ]



PE 6: [M 10*, 3b, 8M] [SA-37]**

A hospital has to pay nurses for 40 hours a week. One nurse is assigned to one patient. The cost per hour for

each of the nurses is given below:

Nurse\Patient W X Y

K 10 10 30

L 30 10 20

M 20 30 20

Suppose that a new patient Z is admitted, and that a new nurse N is appointed. The new patient is charged

Rs. 40 per hour by each of the existing nurses. The new nurse charges Rs. 50 per hour irrespective of the patient.

i. Find the nurse-patient combination to minimize cost to the hospital.

ii. How much does each nurse earn per week?

iii. What would be your revised calculations?

iv. Comment on the new solution.

[Ans.(i) K W, L X, M Y, (ii) 400, 400, 800 (iii) K W, L X, M Y, N Z &400, 400,

800,2000]

PE 7: [J 09*, 4b, 7M] [V2 -11] [Unbalanced-Prohibited routes]*

A factory is going to modify a plant layout to install four new machines M1, M2, M3 and M4. There are 5 vacant

places J, K, L, M and N available. Because of limited space machine M2 cannot be placed at L and M3 cannot be

placed at J. The cost of locating machine to place in Rupees is shown below:

Places Machine

J K L M N

M1 18 22 30 20 22 M2 24 18 - 20 18 M3 - 22 28 22 14 M4 28 16 24 14 16

Determine the optimal assignment schedule in such a manner that the total costs are kept at a minimum.

[Ans.: M1 J, M2 K, M3 N, M4 M, with total minimum ranking as 7]

PE 8: [N 08, 1b, 8M] [V2 - 9] [Multiple Optimal Solutions]**

The cost matrix giving selling costs per unit of a product by salesman A, B, C and D in Regions R1, R2, R3 and R4

is given below:

A B C D

R1 4 12 16 8

R2 20 28 32 24

R3 36 44 48 40

R4 52 60 64 56

i. Assign one salesman to one region to minimize selling cost.



ii. If the selling price of the product is Rs. 200 per unit and variable cost excluding the selling cost given

in the table is Rs. 100 per unit, find the assignment that would maximize the contribution.

iii. What other conclusion can you make from the above?

[Ans.: R1->A, R2->B, R3 ->C and R4->D with minimum selling cost as ₹. 136; (ii) R1->A, R2->B, R3 ->C and R4->D

with maximum contribution as ₹. 264]

PE 9: [N 07, 4a, 10M] [V2-8] [M-98(Adapted)]**

A company has four zones open and four marketing managers available for assignment. The zones are not equal

in sales potentials. It is estimated that a typical marketing manager operating in each zone would bring in the

following Annual sales:

Zones Rs.

East 2,40,000

West 1,92,000

North 1,44,000

South 1,20,000

The four marketing managers are also different in ability. It is estimated that working under the same

conditions, their yearly sales would be proportionately as under:

Manager M : 8

Manager N : 7

Manager O : 5

Manager P : 4

Required:

If the criterion is maximum expected total sales, find the optimum assignment and the maximum sales.

[Ans.: M – East ₹ 80,000; N – West ₹ 56,000; O – North ₹ 30,000 P – South ₹ 20,000; Total ₹ 1,86,000]

PE 10: [N 06, 2a, 6M] [V2 -6] [N-95] [SA-10] [Prohibited Routes]*

A BPO Co. is taking bids for 4 routes in the city to ply pick-up and drop cabs. Four companies have made bids as

detailed below: Bids for Routes (Rs.):

Routes Company

R1 R2 R3 R4

C1 4000 5000 - - C2 - 4000 - 4000 C3 3000 - 2000 - C4 - - 4000 5000

Each bidder can be assigned only one route. Determine the minimum cost.

[Ans.: The minimum cost is R 15,000 C1 –R1 4,000; C2 – R2 4,000; C3 – R3 2,000; C4 – R4 5,000;]



PE 11: [N 04, 6b, 7M] [V2-5] (Minimization balanced)*

A Marketing Manager has 4 subordinates and 4 tasks. The subordinates differ in efficiency. The tasks also differ

in their intrinsic difficulty. His estimates of the time each subordinate would take to perform each task is given

in the matrix below. How should the task be allocated one to one man so that the total man-hours are

minimized?

I II III IV 1 16 52 34 22 2 26 56 8 52 3 76 38 38 30 4 38 52 48 20

[Ans.: Minimum time taken = 82 hours]

Important Notes

h a c 12 the assignment problems e...

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