Operational Research Assignment

Ques1. Solve the assignment problem for optimal solution using HAM. The information is reproduced in table 1.

Table 1. Time taken (in minutes) by 4 workers.

Workers Job

A B C D

1 45 40 51 67

2 57 42 63 55

3 49 52 48 64

4 41 45 60 55

Ques.2 Using the following cost matrix, determine (a)optimal job assignment, and (b) the cost of assignments.

Machinist Job

1 2 3 4 5

A 8 1 1 0 6

B 7 5 6 0 5

C 5 3 4 0 2

D 1 3 6 0 2

E 3 4 3 0 4

Ques 3. You are given the information about the cost of performing different jobs by different persons. The job-person making x indicates that the individuals involved cannot perform the particular job. Using the information, state (i) the optimal assignments of jobs, and (ii) the cost of such assignments.

Person Job

J1 J2 J3 J4 J5

P1 27 18 x 20 21

P2 31 24 21 12 17

P3 20 17 20 x 16

P4 22 28 20 16 27

Balancing the problem and assigning a high cost to the pairing P1-J3 and P3-J4, we have the cost table given in table.

Ques. 4 Solve the following assignment problem and obtain the minimum cost at which all the jobs can be performed.

Worker Job (cost in ’00 Rs)

1 2 3 4 5

A 25 18 32 20 21

B 34 25 21 12 17

C 20 17 20 32 16

D 20 28 20 16 27

Ques 5. A company plan to assign 5 salesmen to 5 districts in which it operates. Estimate of sales revenue in thousands of rupees for each salesman in different districts are given in the following table. In your opinion, what should be the placement of the salesman if the objective is to maximise the expected sales revenue?

Expected Sales Data

District

Salesman

D1 D2 D3 D4 D5

S1 40 46 48 36 48

S2 48 32 36 29 44

S3 49 35 41 38 45

S4 30 46 49 44 44

S5 37 41 48 43 47

Ques 6. To stimulate interest and provide an atmosphere for intellectual discussion, the 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 would be held once per 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:

Leasing portfolio mgmt. Pvt Mutual funds Swaps & 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

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

Ques 7. A solicitor’s firm employs typist on hourly piece-rate basis for their daily work. These are five typists and their charges and speed are different. According to an earlier

understanding, only one job is given to one typist and the typist is paid for a full hour even when he works for a fraction of an hour. Find the least one allocation for the following data:

Typist Rate/hour ( Rs.) No. of pages Typed/Hr. job no. of Pgs.

A 5 12 P 199

B 6 14 Q 175

C 3 8 R 145

D 4 10 S 298

E 4 11 T 176

Ques 8. Welldone company has taken the third floor of a multi-storeyed building for rent with a view to locate one of their zonal offices. There are five main rooms in this office to be assigned to five managers. Each room has its own advantage and disadvantages.

Some have windows; some are closer to the washrooms or the canteen or secretarial pool. The rooms are of all different sizes and shapes. Each of the five managers were asked to rank their five preferences amongst the rooms 301, 302, 303, 304 and 305. The 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 manager did not list all the five rooms since they were not satisfied with some of these rooms and they have left these from the list. Assuming that there 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.

Ques9. A company has four sales representatives who are to be assigned to four different sales territories. The monthly sales increase estimated for each sales representative for different sales territories (in lakhs of Rs.), are shown in the following table:

Sales territories

Sales Representatives

I II III IV

A 200 150 170 220

B 160 120 150 140

C 190 195 190 200

D 180 175 160 190

Suggest optimal assignment and the total maximum sales increase per month.

If for certain reasons, sales representatives B cannot be assigned to sales territory III, will the optimal assignment schedule be different? If so, find that schedule and the effect on total sales.

Ques 10. A firm produces four products. There are four operators who are capable of producing any of these four products. The firm records 8 hrs. a day and allows 30 minutes for lunch. The producing time in minutes and the profit for each of the products are given below:

Products

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.) 8 6 5 4

Per unit

Find the optimal assignment of product to operators.

Ques 11. Solve the following assignment problem by (a) enumeration method, and (b) Hungarian assignment method.

Time (in minutes)

Worker Job 1 Job 2 Job 3

A 4 2 7

B 8 5 3

C 4 5 6

Ques 12. Five employees of a company are to be assigned to five jobs which can be done by any of them. Because of different number of years with the fir, the worker gets different wages per hour. These are: Rs. 15 per hour for A, B and C each, and Rs. 13 per hour for D and E each. The amount of time taken (in hours) by each employee to do a given job is given in the following table. Determine the assignment pattern that (a) minimise the total time taken, and (b) minimise the total cost of getting five units of work done.

Employee

Job

A B C D E

1 7 9 3 3 2

2 6 1 6 6 5

3 3 4 9 10 7

4 1 5 2 2 4

5 6 6 9 4 2

Ques. 13. There are five jobs to be assigned to five machines. Cost of completion of the jobs on the respective machine are as given in the table below:

Machine

Job

M1 M2 M3 M4 M5

J1 65 40 90 80 90

J2 65 35 100 85 85

J3 60 38 105 90 95

J4 70 45 120 90 100

J5 65 40 105 87 90

Employ the Hungarian method of optimal assignment of jobs to the machine so as to minimise the total copy of all the jobs. Is the solution obtained by you unique? If not, work out an alternative solution and calculate the total cost implied by it.

Ques 14. A dispatcher of the police department has received four request for police assistance. Currently, six patrol cars are available for the assignment and the estimated response time (in minutes) are shown in the table that follows:

Patrol unit

Incident

1 2 3 4 5 6

I 6 5 3 4 5 6

II 8 6 2 3 7 6

III 4 4 7 6 5 5

IV 3 7 9 8 4 7

(A) Which patrol units should respond?

(B) What will be the average response time?

Ques 15. A fast-food chain wants to build four stores. In the past, the chain has used six different construction companies, and satisfied with each, has invented them to bid on each job. The final bids (on lakhs of rupees) were as shown in the following table:

Construction Companies

1 2 3 4 5 6

Step 1 85.3 88.0 87.5 82.4 89.1 86.7

Step 2 78.9 77.4 77.4 76.5 79.3 78.3

Step 3 82.0 81.3 82.4 80.6 83.5 81.7

Step 4 84.3 84.6 86.2 83.3 84.4 85.5

Since the fast-food chain wants to have each of the new stores ready as quickly as possible, it will award at most one job to one company. What assignment results in minimum total cost to the fast-food chain?

Ques 16. In the following situation calling for assignment of a-truck-to-a-territory, what assignment would be involve the lowest aggregate cost?

Territory

Truck

1 2 3 4 5

T1 200 190 180 190 110

T2 150 210 190 200 100

T3 80 90 200 90 70

T4 220 80 90 140 60

T5 170 110 90 150 40

T6 90 100 120 90 60

T7 120 200 130 80 60

T8 160 90 160 110 80

Ques 17. A company solicits bid on each of four projects from five contractors. Only one project may assign to any contractor. The bid received (in thousands of rupees) is given in the accompanying table. Contactor D feels unable out project 3 and, therefore, submits no bid.

Contractor

Project

A B C D E

1 18 25 22 26 25

2 26 29 26 27 24

3 28 31 30 - 31

4 26 28 27 26 29

(i) Use Hungarian method to find the set of assignments with the smallest possible total cost.

(ii) what is minimum total achievable total cost?