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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