Prof. Bhambwani’s

RELIABLE CLASSES / C.A. FINAL / A.M.A. / O.R. / ASSIGNMENT PROBLEMS

Lecture 1: Theory with solution of Q1 (b)

Q1 (b) Cost matrix

Machine

ABCD

I 7 10 8 6

Operator II 9 15 12 10

III 10 12 13 8

IV 7 14 15 8

Lecture 2: Q1(a) & (c)

Q1.Find an optimum assignment solution for the following:

(a) Duration Matrix

Machine

IIIIII

J15810

JobJ26915

J34815

(c) Cost Matrix

Plant

ABCDE

11211131114

21718161519

Product31410121113

41512111014

51112111013

Lecture 3:

Q2.To simulate 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, equity research. Such seminars should be held once per week in the afternoons. However scheduling these seminars (one for each topic and not more that one seminar per afternoon) has to be done carefully so that the number of students unable to attend is kept to the minimum. A careful study indicated that the number of students who can not attend a particular seminar on a specific day is as follows:

Leasing Portfolio Private Mutual Equity

Management Fund Research

Monday5040 60 20

Tuesday4030 40 30

Wednesday6020 30 20

Thursday3030 20 30

Friday1020 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.

Lecture 4:

Q3.A solicitor’s firm employs typists for piece-meal work on an hourly basis. There are five typist available and their charges and speeds are different. According to an earlier understanding only one job is given to one typist and a typist is paid for full hour even if he works for a fraction of an hour. Find the least cost allocation for the following data:

TypistRate per No. of pages Jobs No. of pages

Hour (Rs.) Typed per hr. A 5 12 P 200

B 6 14 Q176

C 3 08 R 150

D 4 10 S300

E 4 11 T180

Lecture 5:

Q4.A Departmental stores agency runs five stores located at different parts of Grater Bombay. Each store is administered by a Manager appointed by the Agency. The managers reside in different parts of the city. The Agency reimburses the car travel expenses incurred by the managers in commuting to work from their residence to the stores to which they are assigned. The basis of reimbursements is:

A fixed sum of Rs.300/- per month for Repair/maintenance. A variable amount at the rate of Rs.1.60 per kilometre of travel incurred during the month.

All stores work for 25 days in a month

The distance in kilometres from a manager’s residence to the stores is displayed in the following matrix:

(a)Find the optimal assignment of managers to stores so that the monthly expenditure to be incurred by the Agency on car travel of managers is minimum. Calculate this expenditure.

(b)After having operated the optimal assignment for some times, manager A request the Agency to assign him store S1, which is the closest to his residence, as he has been medically advised not to take long journeys. If the Agency agrees to this request, how should the present assignments be changed and how much extra will it cost the Agency?

Stores (Distance in Kms.)

ManagersS1S2S3S4S5

A410121817

B716162218

C8691921

D1112151213

E914191814

Lecture 6:

Q5.An airfreight company picks up and deliveries freight where customers require. The company has two types of aircrafts X and Y with equal loading capacities but different operating costs. These are shown below:

Cost per mile (Rs.)

Types of air craftEmptyLoaded

X1.002.00

Y1.503.00

The present four locations of the air craft which the company is having are as shown below

J → X, K → Y, L → Y, M → X

Four customers of the company located at A, B, C and D want to transport nearly the same size of load to their final destinations. The final destinations are at a distance of 600, 300, 1000 and 500 miles from the loading points A, B, C and D respectively.

Distance in miles between location of the aircraft and the loading points are as follows

Loading Points

ABCD

PresentJ300200400100

LocationK300100300200

of air craftL400100100500

M200200400200

Determine the allocations, which minimize the total cost of the transportation.

Lecture 7:

Q7. Middlesam Management consultants have six consultants available to be assigned to do six jobs for clients. However, because of technical difficulties in particular area, consultant B cannot do job 3, D cannot do job 4 and consultant F cannot do job 6. The cost incurred to complete the available assignments are given in the table below:

Job

123456

A77361011

B 89 -5810

ConsultantsC 910 11 1313 8

D 6 6 8 - 12 13

E 5 5 9 10 10 12

F 8 4 10 12 9 --

Assign the available consultants to the different jobs in such manner that the total cost are kept at minimum.

Lecture 8:

Q8.WELDONE company has taken the third floor of a multi-storeyed building for rent with view to locate one of the zonal offices. There are five main rooms on this floor to be assign to five managers. Each room has its own advantages and disadvantages. Some have windows, some are closer to wash room or to the canteen or to the secretarial pool. The rooms are of different sizes and shapes. Each of the five managers were asked to rank their preferences amongst the rooms 301, 302, 303, 304,305. Their preferences were recorded in a table as indicated below:

MANAGER

M1M2M3M4M5

Room302302303302301

303304301305302

304305304304304

301305303

302

Most of the managers did not list all the five rooms since they were not satisfied with some of these rooms. They have left off these from the list. Assuming that their preferences can be quantified in numbers, find out as to which manager should be assign to which room, so that their total preference ranking is minimum.

Lecture 9:

Q9.A student has to select one and only one elective in each semester and the same elective should not be selected in different semesters. Due to various reasons, the expected grades in each subject, if selected in different semesters, vary and they are given below:

AdvancedAdvancedGraphDiscrete

Semesters O.R.StatisticsTheoryMathematics

I F E D C

II E E C C

III C D C A

IV B A H H

The grade points are: H= 10, A=9, B=8, C=7, D=6, E=5 and F=4. How will the student select the electives in order to maximize the total expected points and what will be his maximum expected total points?

Lecture 10:

Q1(d) Sales Matrix

Machine

ABC

1121018

Job2171116

3141115

Lecture 11:

Q1(e) Production Matrix

Plant

XYZ

A100110108

ProductB111115112

C116120111

Lecture 12:

Q6.A firm is contemplating the introduction of three products 1, 2 and 3 in its three plants A, B and C, only a single product can be introduced in each of the plants. The unit cost of producing ith product in jth plant, Cij, is as given in the following matrix:

Plant

ABC

Product1812M*

21064

3765

(M* indicates that Product 1 cannot be produced in Plant C)

(a)How should the products be assigned so that the total unit cost is minimsed?

(b)If the quantity of different products be produced is as follows then what assignment shall minimise the aggregate production cost.

Product Quantity (Units)

12,000

22,000

310,000

(c)What would be your answer if the three products are to be produced in equal quantities?

(d)If is expected that the selling price of the products produced by different plants would be different as shown in the following table:

PlantABC

11518-

2181610

312108

Assuming the quantities mentioned in (b) should be produced and sold. How should the products be assigned to the plants to obtain maximum profits?

Lecture 13:

Q10.The marketing Director of multi-unit company, Ms.X is faced with the problem of assigning 5 Senior Marketing Managers to six zones. From past experience she knows that the efficiency percentage judge by sales, operating cost and increases in market share depend a lot on marketing manager – zone combination given in the table below:

Marketing Zones

Manager IIIIIIIVVVI

Mr. A718985807678

Mr. B798367747283

Mr. C737081827689

Mr. D919484898180

Mr. E888977876774

As an adviser to Ms. X, recommend which zone should be manned by a Junior Manager so as to maximised the overall efficiency of the company.

Lecture 14:

Q11.Parul Corporation has four plants, each of which can manufacture any one of four products. Production costs differ from one plant to another as do sales revenue. Given the revenue and cost data below, obtain which product each plant should produce to maximise profit:

Sales revenue (Rs.’000s)Production cost (Rs.’000)

ProductProduct

Plant1 2 3 41 2 3 4

A50 68 49 6249 60 45 61

B60 70 51 7455 63 45 69

C55 67 53 6052 62 49 58

D58 65 54 69 55 64 48 66

Lecture 15:

Q12.Smitsubhashi PLC within the frame work of its corporate plan is considering closure of one of the UK factory. The cost of closure will be the same for each factory. Currently, it operates five factories, each of which can produce any of four products. It is envisaged that in the future, individual factories will achieve production and other economies by concentrating on one product alone. The production cost data for existing operations is summarised below.

Production cost per unit (£)

Products

NIPQ

171789376

269788774

372808976

473808678

565849272

The selling price for each of the products is consistent, irrespective of the factory in which it is produced.

Product:NIPQ

Selling price (£):809010085

Required:

Recommend to the board of Smitsubhashi PLC which factory should be closed.

Lecture 16:

Q13.A company has four zones open and four salesmen available for assignment. The zones are not equally rich in their sale potentials. It is estimated that a typical salesman operating in each zone would bring in the following annual sales: Zone A: 1,26,000; Zone B: 1,05,000; Zone C: 84,000; Zone D: 63,000

The four salesmen are also considered to differ in ability. It is estimated that working under the same conditions their yearly sales would be proportionately as follows: Salesman P: 7; Salesman :5, Salesman R: 5; Salesman S:4.

If the criterion is Maximum expected total sales, the intuitive answer is to assign the best salesman to the richest Zone, the next best to the second richest zone and so on. Verify this by the method of assignment.

Lecture 17:

Q14.An air line that operates seven days a week has time table shown below. Crews must have a minimum layover of 5 hours between flights. Obtain the paring of fights that minimise layover time away from home. For any given pair the crew will be based at the city that results in smaller layover.

DELHI-JAIPURJAIPUR-DELHI

Flight No.DepartureArrivalFlightDepartureArrival

1017.00 A.M.8.00 A.M. 2018.00 A.M.9.15 A.M.

1028.00 A.M.9.00 A.M. 2028.30 A.M.9.45 A.M.

1031.30 P.M.2.30 P.M. 20312.00 Noon1.15 P.M.

1046.30 P.M.7.30 P.M. 2045.30 P.M.6.45 P.M.

For each pair also mention the town where the crew should be based.

Lecture 18

Q15.Prachi Airlines that operates seven days a week has a time table shown below. Crews must have a minimum layover of 6 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing the crew will be based at the city that results in the smaller layover:

Flight Delhi-CalcuttaFlightCalcutta-Delhi

No.Depart Arrive No. Depart Arrive

17.00 A.M. 9.00 A.M. 1019.00 A.M. 11.00 A.M.

29.00 A.M. 11.00 A.M. 10210.00 A.M. 12.00 Noon

31.30 P.M. 3.30 P.M. 1033.30 P.M. 5.30 P.M.

47.30 P.M. 9.30 P.M. 1048.00 P.M. 10.00 P.M.

For each pair also mention the town where the crew should be based.

Lecture 19

Q16.A salesman’s requirement is to complete a minimum time circuit of all cities in his itinerary of cities without any doubling back. Distance matrix is given below:

To City

ABCDE

A-10252510

B1-10152

FormC89-2010

D141024-15

E1082527-

Determine the optimal route.

Lecture 20 & 21

Q17.Solve the following travelling salesman problem so as to minimise the cost per cycle: To

From ABCDE

A-3623

B3-523

C65-64

D226-6

E3346-

Lecture 22

Q18.Solve the travelling salesman problem given by the following data:

C12 = 20, c13 = 4, c14 = 10, c23 = 5, c34 = 6

C25 = 10, c35 = 6, c45 = 20, where cij = cji

And there is no route between cities i and j if a value for cij is not shown.

Lecture 23

Q19.A machine operator processes five types of items on his machine each week, and must choose a sequence for them. The set-up cost per change depends on the item presently in the machine and the set-up to be made, according to the following table: To item

For itemABCDE

A-4734

B4 -634

C 76-75

D337-7

E4457 -

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?

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