question paper unsolved -qmb - ii

Quantitative Methods for Business - II April – 2002

60 Marks

Note:(1) Section-I is compulsory.(2) Answer ANY THREE questions from Section-lI.(3) Answer of both Sections should be written in the same answer book.(4) Figures to the right side of questions indicate marks.(5) Graph papers will be supplied on request.(6) Use of Non-programmable calculator is allowed.

Section — I

(1)

(2)

(a) Explain the meaning of degeneracy and infeasibility in a Linear ProgrammingProblem

(b) Explain shadow prices in Linear Programming Problem.(c) How do you solve an Unbalanced Transportation Problem of maximisation

type?(d) Explain the multiple optimal solutions in an Assignment Problem.(e) What is Dummy Activity? Explain its use in Network Analysis.

(a) M/s ABC & Co. is interested in developing an advertising campaign that will reach to the persons belonging to four different age groups. Advertising campaigns can be conducted through media M1, M2 and M3. The following table gives the estimated cost in paise per exposure for each age group according to the media employed. In addition, maximum exposure levels possible in each of the media, namely, M1, M2 and M3 are 40, 30 and 20 mns., respectively. Also, the minimum desired exposures within each age group, namely 16-20, 21-25, 26-35 and 36 and above, are 30, 25, 15 and 10 rnns. The objective is to minimize the cost of obtaining the minimum exposure level in each age group.

(2)

(2)(2)

(2)(2)

(10)

MediaAge Groups

16-20 21-25 26-35 36 and aboveM1 12 7 10 10M2 10 9 12 10M3 14 12 9 12

(i) Formulate the above as a transportation problem, and find the optimal solution.

(ii) Solve this problem if the policy is to provide at least 4 million exposures through M1 in the 16-20 age group and at least 8 million exposures through M1 in the age group 21-25.

TY BMS – Sem VI of 4 QMB - II


60 Marks

(b) Trick and Tack produces several types of glass containers. They have

recently reduced capacity at several of their plants. Glass manufacturing

involves large, expensive machines (including ovens), several of which

were turned off in the capacity reduction. The machines were hard to shut

down and to start up. In the event of a surge in demand, they wanted to

know how quickly they could start one. How quickly can they start a new

oven using normal times? What is the fastest time in which a new oven can

be started, and how much additional cost is involved?

(10)

A Preheat glassB Preheat ovenC Obtain materialsD Check valves

Description

C 8D 12- 4- 4

Cost(Rs.) Per

Unit Time

(hour)reuction

8 -12 -2 4002 200

E Check pressure sealsF Add glass to ovenG Prepare bottlemakerH Run test productionI Examine test quantity and make adjustmentsJ Refill oven with glass

B 2 1A, E 2 2

E 6 3F, G 4 4

H 4 2H 2 2

200-

500-

500-



Section — II

60 Marks

(3)

(4)

(a) State the algorithm of solving an Assignment Problem

(b) Agashe & Co. plans to reach target audiences belonging to two different

monthly income groups, the first with incomes greater than Rs. 15,000 and

the second with income of less than Rs. 15,000. The total advertising budget

is Rs. 2,00,000. Advertising on TV costs Rs. 50,000 for one program, where

as advertising on Radio costs Rs. 20,000 for one program. For contract

reasons at least 3 programmes must be given on TV and the No. of Radio

programmes are limited to 5 only. One TV programme covers 4,50,000

audiences belonging to income group having more than Rs. 15,000 monthly

income where as it reaches to 50,000 audiences belonging to below Rs.

15,000 monthly income group. Similarly one Radio program reaches to

20,000 and 80,000 audiences belonging to above Rs. 15,000 and below

15,000 monthly income groups respectively. Formulate the linear

programming problem and using graphical method determines the media mix

so as to maximize the total number of target audience. Comment on the

Solution.

(a) Explain “Least cost method” to obtain initial feasible solution for a transportation problem. Is this method better than North West Corner rule? Why?

(b) A sales manager has to assign salesman to four territories. He has four candidates of varying experience and capabilities. The manager assesses the possible profit for each salesman in each territory as given below:

Territory

(3)

(7)

(4)

(6)

SalesmanT1 T2 T3 T4

S1 35 27 28 37

S2 28 34 29 40

S3 35 24 32 33

S4 24 32 25 28

Find the assignment of salesman to the territories so that total profit is maximum.

(5) Using simplex method, solve the following linear programming problem and explain the solution.

Maximise Z = 6x1 – 2 x2 ;Subject to:2x1 – x2 ≤ 2

X1 ≤ 4x1, x2 ≥ 0

(10)



60 Marks

(6) M/s. Raj and Bilimoria Associates produce three items ‘X’, ‘Y’ and ‘Z’ each of

which have to be processed through three machines ‘P’, ‘Q’ and ‘R’. Each unit of

the product ‘X’ requires 3,4 and 2 hours on machines ‘P’, ‘Q’ and ‘R’ respectively.

Similarly each unit of product ‘Y’ requires 5, 4 and 4 hours on machine ‘P’, ‘Q’ and

‘R’ respectively, where as for product ‘Z’, these requirements are 2, 4 and 5 hours

on these three machines P, Q and R. Every day 60 hours are available on

machine P, 72 hours on machine ‘Q’ and 100 hours on machine ‘R’. The unit

contribution of these products ‘X’, “Y’ and ‘Z’ are Rs 5, Rs. 10 and Rs. 8

respectively.

(a) Formulate the linear programming problem and using simplex method find the optimal solution for the product mix, also find the unused capacity of machines if any.

(b) What would be the effect on the solution of each of the following:

i. Obtaining an order of 12 units of ‘X’ which has to be met. ii. An increase of 20% in the capacity of machine ‘P’.

(7) A project consists of eight activities with the following relevant information:

(10)

(10)

ActivityImmediate Estimated Duration (Days)

Predecessor Optimistic Most Likely PessimisticA - 2B - 2C - 3D A 2E B 3F C 3G D, E 4H F, G 2

2 85 83 92 26 156 97 163 4

i. Draw the PERT network and find out the expected project completion time. ii. What duration will have 95% confidence for project completion? (Given area

under normal curve from Z = 0 to Z = 1.65 is 0.45)

**********



60 Marks

Note:(1) Both questions in Section-I are compulsory.(2) Answer ANY THREE questions from Section-lI.(3) Answers of both sections should be written in the same answer book.(4) Figures to the right side of questions indicate marks.(5) Graph papers will be supplied on request.(6) Clarity in answers supported by proper working should be maintained.(7) Use of Non-programmable calculator is allowed.

Section — I

(1) Answer the following concept questions in brief:

(a) Basic variables in simplex method of a Linear Programming Problem.(b) Prohibited Transportation Problem.(c) Forward and Backward pass in PERT/CPM.(d) Three time estimates in PERT and their relationship with expected time and

its variance in the project.(e) Restricted Assignment problem, which is an unbalanced problem.

(10)

(2)(a) A television manufacturing firm is planning to produce television sets of

various designs and specifications. The televisions are marketed on the basis of its over all quality appearance and warranty. The market research survey and the firms past experience indicates that all the three types — Flat screen, Black screen and Normal T.V. sets will all be sold which ever are produced. However, the firm plans to test the market response first by manufacturing only 200 sets of all the three types, all of which will definitely be sold because of the reputation of the firm. The manufacturing firm wants to decide; how many of Flat screen and how many of Black screen T.V. sets the firm should produce where as the numbers of T.V. sets of Normal type is automatically decided on the basis of the first two types. All the three types of T.V. sets differ significantly in their quality, tube costs and their other electronic features. The following table summarizes the estimated prices for the three types of T.V. sets and the corresponding expenses for the firm. The manufacturing firm has hired a high-tech plant to manufacture these T.V. sets at a fixed charges of Rs. 2,00,000 for a period of one month.

(10)

Types ofT.V. Sets

PricesRs.

Tube CostRs.

Labour and otherMaterial Expenses

Flat Screen 10,000 3,000 4,750Black Screen 7,000 2,200 2,500Normal 6,500 1,900 2,200

TY BMS – Sem VI Page 1 of 6 QMB - II


60 Marks

In planning the production the following considerations must be taken into account:

(i) The marketing management and manufacturing conditions require that at least

120 T.V. sets be of Flat and Black screen types.

(ii) At least 35% but not more than 70% must be of the Black screen T.V. sets.

(iii) At least 10% of the T.V. sets must be of the Flat screen type.

(iv) At least 30% of the total sets must be of normal type.

(v) The maximum no. of Flat screen T.V. sets that can be manufactured at the plant

is restricted to 60 only.

The manufacturing firm wishes to determine the number of T.V. sets to produce for each type, so as to maximize the profits.

(a) Formulate the above as the Linear Programming Problem (L.P.P.)

(b) Rewrite the above L.P.P. in terms of two decision variables, taking advantage of

the fact that all 200 T.V. sets produced will be sold.

(c) Find the optional solution using graphical method for the restated Linear

Programming Problem in (b). Interpret your results.

(b) Choice distributes a variety of food products that are sold through grocery

store and supermarket outlets. The company receives orders directly from

the individual outlets, with a typical order requesting the delivery of several

cases of anywhere from 20 to 50 different products. Under the company’s

current warehouse operation, warehouse clerks dispatch order-picking

personnel to fill each order and have the goods moved to the warehouse

shipping area. Because of the high labor costs and relatively low

productivity of hand order-picking, management has decided to automate

the warehouse operation by installing a computer-controlled order-picking

system, along with a conveyor system for moving foods from storage to the

warehouse shipping area. Choice’s director of material management Mr.

Gautam Shah has been named the project manager in charge of the

automated warehouse system. After consulting with members of the

Engineering staff and warehouse management personnel, the director has

compiled a list of activities associated with the project. The optimistic, Most

probable, and pessimistic times (in weeks) have also been provided for

each activity.

(10)



60 Marks

Activity Description ImmediatePredecessors

A Determine equipment needsB Obtain vendor proposalsC Select vendorD Order systemE Design new warehouse layoutF Design warehouseG Design Computer interfaceH Interface computerI Install system

- A A B B C E

D, F G, H

Activity A B C D E F G H IOptimistic Time 4 5 4 15 10 8 4 1 6Most Probable Time 6 7 8 20 18 9 8 2 7Pessimistic Time 8 15 12 25 26 16 12 3 8

(a) Choice’s top management has established a required 52-week completion time for the project. Can this completion time be achieved? Include probability information in your discussion. What recommendations do you have if the 52 week completion time is required?

(b) If the management requests that activity times be shortened to provide an 80% chance of meeting the 50-week completion time. If the variance in the project completion time is the same as you found in part (a) above, how much should the expected project completion time be shortened to achieve the goal of an80% chance of completion within 50 weeks?

(4)

(6)

Activity A1-2

B2-3

C2-4

D3-6

E3-5

F4-6

G5-7

H6-7

I7-8

Crashed

Activity

(weeks)

Normal

Cost (Rs.)

Crashed

Cost (Rs.)

Note:

4 6 4 15 15 8 6 1 5

1000 1000 1500 2000 5000 3000 8000 5000 10000

1900 1800 2700 3200 8000 4100 10250 6400 12400

(i) The area for S.N.V. Z =0 and Z = 1.4302 is given as 0.4236.(ii) The area for S.N.V. Z =0 and Z = 0.845 is given as 0.3009.



60 Marks

Section — II

(3)(a) Compare Transportation Problem and Assignment Problem.

(b) For the data given in the table below, draw the network. Crash systematically the activities and determine the optional project duration and cost.

(3)

(7)

Activity 1-2 2-3 2-4 3-6 3-5 4-6Normal Time (days) 8 4 2 10 5 3

Normal Cost (Rs.) 100 150 50 100 100 80

Crash Time (days) 6 2 1 5 1 1

Crash Cost (Rs.) 200 350 90 400 200 100

Indirect cost is Rs.70/ day

(4)

(a) State and critically examine the uses of Post Optimality Analysis in a Linear

Programming Problem and its solution.

(b) ‘YOURS OWN’ garment manufacturing firm of Mumbai wishes to develop a

monthly production schedule for the next three months. Depending on sales

commitments, the company can either keep the production constant, and

allowing the fluctuations in inventory or maintained inventories at a constant

level, with fluctuating production. The fluctuating production necessitates,

working overtime, the cost of which is estimated to be double the normal

production cost of Rs.10 per unit. Fluctuating inventories result in inventory

carrying cost of Rs.4 per unit. If the company fails to fulfill its sales

commitment, it incurs a shortage cost of Rs.5 per unit per months. The

production capacities for the next three months are shown in the following

table:

(3)

(7)

Production CapacityMonths

1 2 3

Regular 50 50 60

Overtime 30 00 50

Sales 60 120 40

Formulate it as a Transportation Problem to obtain an optional production schedule.



60 Marks

Section — II

(5) A bread distribution Van of Santosh Bakery has to supply bread at different outlets A, B and C in the morning, it collects the bread from Bakery and distributes to outlets A, B and C only once in the mornings. The van has to visit outlets once only and all the outlets have to be supplied with morning fresh bread. The distances of the outlets A, B and C from the Bakery is given in the following table. The van starts from Bakery and has to come back to the Bakery after visiting each outlet only once. Which route should be selected by the Van so that the total distance traveled by it is minimized? What is the total distance traveled by the Van? Find the alternate route, if any.

(10)

Activity ToBakery Outlet A Outlet B Outlet C

Bakery - 4 7 3

Outlet A 4 - 6 3

Outlet B 7 6 - 7

Outlet C 3 3 7 -

(6) Solve the following Linear Programming problem by Simplex

Method without using the artificial variables.

Maximise Z = 3x1 + 5x2

Subject to:

(10)

x1 + x3

x2 + x4

3x1 + 2x2 + x5

x1 , x2 , x3 , x4 , x5

= 4

= 6

= 12

≥ 0

Does the degeneracy occur in this problem?

(7) Zigma Electronics produces two models of electronic products using Resistors,

Capacitors and Chips. The following table gives the entire Technological and

other details in this regard:

(10)

ResourceUnit resource requirement

Model 1 Model 2

Maximum

Availability

Resistor 2 3 1200

Capacitor 2 1 1000

Chips 0 4 800

Unit Profit (Rs.) 3 4

After formulating the above problem as a Linear Programming Problem the following optimal Simplex Solution table is obtained.



60 Marks

Profit Basis Solution C: 3 4 0 0 0Coefficient Variables Values X: X1 X2 S1 S2 S3

Cβ Xβ b3 X1 450 1 0 - ¼ ¾ 00 S3 400 0 0 -2 2 14 X2 100 0 1 1/3 -1/2 0

Z = Rs. 175 Z 3 4 5/4 ¼ 0∆ = C - Z 0 0 -5/4 -1/4 0

(i) Determine the value of each resource.

(ii) In terms of optimal profit, determine the worth of one Resistor, one Capacitor and one Chip.

(iii) Determine the range of the applicability of the shadow prices (dual prices) for each resource.

(iv) If the available number of chips is reduced to 350 units, will you be able to determine the new optimum solution directly from the given information? Explain.

**********



60 Marks

Note:

(1) All questions in section I are compulsory.(2) Answer any three questions from section II.(3) Answers of both sections are to be written in the same answer book.(4) Figure in bracket to the right side of the questions indicates marks.(5) Graph papers will be supplied on request.(6) Clarity in answer supported by proper working should be maintained.(7) Use of non-programmable calculator is allowed.(8) Use of mobile phone calculators is prohibited.

Section — I

(1) Answer the following briefly: (10)

(2)

(a) Write the major differences between Simple and Dual Simplex method of solving a L.P.P.

(b) Looping in Transportation problem.(c) Uses of Slack, Surplus and Artificial variables in solving the Linear

Programming Problem.(d) Difference between Assignment Problem & Transportation Problem.(e) Dangling event and Dummy activity in Network Diagram.

(10)(a) Peculiar Outsourcing Company Ltd. has production centres at Mumbai, Chennai,

and Kolkata. The company has its distribution centres at Ahmedabad, Bhopal, Bangalore and Goa. Production costs are equal and fixed at all three production centres, however the variable cost are only the transportation costs. The monthly productions at Mumbai, Chennai and Kolkata are 10,000 units, 12,000 units and

5000 units- respectively. The monthly demand at company’s four distribution

centers viz. Ahmedabad, Bhopal, Bangalore and Goa are 12000 units, 8000 units,

4000 units and 30000 units respectively. The transportation cost per unit

from different production centres to different distribution centres are given

in the following table:

Production Distribution CentresCentre Ahmedabad Bhopal Banglore Goa

Mumbai 6 4 14 12Chennai 14 10 4 6Kolkota 4 10 8 10

a) Obtain a optimum transportation schedule so as to minimise the transportation cost. b) If the transportation cost from production centre Kolkata to distribution centre

Bangalore is changed from Rs. 8 per unit to Rs. 12 per unit, what will be the effecton the transportation schedule? Will it change? If yes, state the reason.

c) If the company wants to meet the requirement of at least 2000 units at its Goa distribution centre only from Mumbai, will the optimum solution obtained in a change? If so, find the new optimum transportation schedule and its effects on total cost?



60 Marks

(b) M/s ABC are in jewellery business and are specialised in making of Rings and

Bracelets of silver and gold. Making of one Bracelet requires one unit of silver

and 2 units of gold whereas making of one Ring require 3 units of silver and I

unit of gold. M/s ABC have 9 units of silver and 8 units of gold. They earn profit

of Rs.40 on each Ring and Rs.50 on each Bracelet. Formulate it as a Linear

programming problem and obtain its optimal solution using Simplex method.

Based on the optimum solution Simplex table answer the following:

(i) What will be the optimal solution of one until extra gold is made available to M/sABC?

(ii) What will be the new optimum profit if the profit contribution of each Ring is increased by Rs 10?

(iii) It is claimed that the production gets double whenever the silver and gold availability is doubled. Justify this claim using appropriate technique.

Section — II

(3) Four warehouses with capacities of 85, 35, 50 and 45 tons were receiving the

materials from 3 factories with their supply capacity as 70, 55 and 90 tons on

regular basis. The transportation costs per ton from factories to warehouses are

given in the following table:

(10)

(10)

FactoryWarehouse

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

A feasible solution states that from Factory I, 25 tons have to be transported to

Warehouse 3 and 45 tons to Warehouse 4. Similarly 30 tons and 25 tons were

transported from Factory II to Warehouse 1 and Warehouse 3 respectively. However,

from Factory Ill, 55 tons and 35 tones were transported to warehouse 1 and

warehouse 2 respectively.

Is this transportation schedule optimum? If not, modify it and obtain optimum

solution and optimum cost.



60 Marks

(4) Nagaria & Associates are preparing for laying the foundation of State Computer

Centre to be inaugurated by the Chief Minister by the end of August 2004.

Following are the abbreviated activities and their predecessor activities with their

three time estimates of completion time.

Activities A B C D E F G H I J K

Predecessor activities - - A B C C C,D F,G E I H

Optimistic Time estimate 2 8 7 6 9 10 11 6 4 3 1

Presumptive Time estimate 4 8 11 6 11 18 11 14 6 5 1

Most likely Time estimate 3 8 9 6 10 14 11 10 5 4 1

(10)

(a) Draw the PERT Network diagram.

(b) Compute the slack for each activity and determine the critical path.

(c) As per the contract a penalty of Rs. 5000/— is to be charged for any

delay beyond 37 weeks. What is the probability that Nagaria & Associates will

have to pay a maximum penalty of Rs.15000/- ? (Note: Area under

standard normal variate z = 0 to z = 1.4795 is 0.4306)

(5) (A) Following are the details of estimated times of activities for a certain

project.

Activity A B C D E F

Immediate Predecessor Activity - A A B,C - E

Estimated Time (Weeks) 2 3 4 6 2 8

(4)

(a) Find the critical path and expected time of the project.

(b) Calculate the earliest start time and earliest finish time for each activity.

(c) Calculate the slack for each activity.



60 Marks

(5) (B) PQR Ltd. produces 4 different products viz, pen, ink, pencil and rubber using 4 workers viz. Alok, Satish, Vaze and Rathod, who are capable of producing any of the four products and they work effectively for 7 hours a day. The time (in minutes) required for producing each of the product are given in the following matrix along with the profit (Rs per unit):

(6)

WorkersProducts

Pen Ink Pencil Rubber

Alok 6 10 14 12

Satish 7 5 3 4

Vaze 6 7 10 10

Rathod 20 10 15 15

Profit (Rs./unit) 3 2 4 1

(6) (A) Explain the updating of network in PERT and CPM analysis

(B) Using Dual Simplex Method, the optimum solution table for the LinerProgramming Problem. Minimize: Z = 3x1 + 6x2 + x3

Subject to:

(3)

(7)

Is as below:

x1 + x2 + x3

x1 + 5x2 - x3

x1 + 5x2 + x3

x1 , x2 , x3

≥ 6

≥ 4

≥ 24

≥ 0

CB XB B -3 -6 -1 0 0 0X: X1 X2 X3 S1 S2 S3

0 S1 18 0 4 0 1 1 -1-3 X1 14 1 0 0 0 -1/2 -1/2-1 X3 10 1 5 1 0 ½ -1/2

Z = -52 -3 -5 -1 0 0 0∆ = C - Z 0 -1 0 0 -1 -2

Discuss the effects of changing the requirement from [6, 4 , 24] to [6, 2, 12]

(7) Mr. A. P. Ravi wants to invest Rs. 1,00,000 in two companies ‘A’ and ‘B’ so as not

to exceed Rs. 75,000 in either of the company. The company ‘A’ assures average

return of 10% whereas the average return for company ‘B’ is 20%. The risk factor

rating of company ‘A’ is 4 on 0 to 10 scale whereas the risk factor rating for ‘B’ is 9

on similar scale. As Mr. Ravi wants to maximise his returns, he will not accept an

average rate of return below 12% or a risk factor above 6. Formulate this as a

Linear Programming Problem and solve it graphically.

(10)

**********



60 Marks

Note:(1) Both questions in Section-I are compulsory.(2) Answer any three questions from Section- II(3) Answer of both Sections should be written in the same answer book.(4) Figures to the right side of questions indicate marks.(5) Graph papers will be supplied on request.(6) Clarity in answers supported by proper working should be maintained.(7) Use of Non-programmable calculator is allowed.(8) Use of mobile phone calculators is prohibited.

Section — I

(1) Answer the following briefly:

(a) Distinguish between degeneracy and cycling in LPP.(b) What do you mean by Shadow prices in LPP?(c) Explain Unbalanced Transportation Problem.(d) Write briefly on Multiple Optional Solutions in an Assignment Problem.(e) Distinguish between Free float and independent float.

(2)(a) Product A offers a profit of Rs. 25/- per unit and product B yields a profit of Rs.

40/- per unit. To manufacture the products - leather, wood and glue are required

in the amount shown below:

(10)

(10)

Resources require for one unitProduct Leather

(in Kg)Wood

(in Sq. Mts)Glue

(in ltrs)A 0.50 4 0.2B 0.25 7 0.2

Available resources include 2200 kgs. of leathers, 28,000 sq. metres of wood and 1,400litres of glue:(i) State the objective function and constraints in mathematical form.(ii) Find the optiw .im solution.(iii) Which resources are fully consumed? How much of each resource remains

unutilized?(iv) What are the shadow prices of resources?

(2) (b) The following table shows all necessary information on the availability of supply to each warehouse, the requirement of each market and the unit transportation cost (in Rs.) from each warehouse to each market:

(10)

Warehouse Market SupplyP Q R S

A 6 3 5 4 22B 5 9 2 7 15C 5 7 8 6 8Requirements 7 12 17 9


60 Marks

The Shipping clerk has. worked out the following schedule from experience: 12 units

from A to Q, 1 unit from A to R, 9 units from A to S, 15 units form B to R, 7 units from

C to P and 1 unit from C to R.

(i) Check and see if the clerk has the optimal schedule.

(ii) Find the optimal schedule and minimum total transport cost.

(iii) If the clerk is approached by a courier to route C to Q, who offers to reduce

his rate in the hope of getting some business, by how much, the rate should

be reduced such that the clerk will offer him the business?

Section — II

(3) (a) Explain the procedure involved in solving an assignment problem using

Hungarian Method.

(3) (b) A Company has four districts, I, II, III and IV to sell its product and four

salesmen A, B, C and D for it. The District-wise sales record of each salesman is as

given in the table. Determine the area allocation so as to make the sales maximum.

(3)

(7)

SalesmanDistricts

I II III IVA 420 350 280 210B 300 250 200 150C 300 250 200 150D 240 200 160 120

What will be the total Maximum sale?

(4) A project has the following activities and other characteristics. (10)

Activity Preceding ActivityTime Estimates (in weeks)

Optimistic Most Likely Pessimistic

A - 4 7 16

B - 1 5 15

C A 6 12 30

D A 2 5 8

E C 5 11 17

F D 3 6 15

G B 3 9 27

H E, F 1 4 7

I G 4 19 28



60 Marks

(i) Draw the PERT network diagram.

(ii) Identify the critical path.

(iii) Prepare the activity schedule for the project.

(iv) Determine the mean project completion time.

(v) Find the probability that the project is completed in 36 weeks. (Area between Z= 0

and Z = 0.2 is 0.0793)

(5) The Purchase Manager, Mr. Taklu, of the State Road Transport Corporation must

decide on the amounts of fuel to buy from three possible vendors. The corporation

refuels its buses regularly at four depots within the area of its operations.

The three oil companies have said that they can furnish up to the following

amounts of fuel during the coming month:- 2,75,000 litres by Oil Company I,

5,50,000 litres by Oil Company II and 6,60,000 litres by Oil Company III. The

required amount of fuel is 1,10,000 litres by Depot I, 2,20,000 litres at Depot II,

3,30,000 litres at Depot III and 4,40,000 litres at Depot IV.

When the transportation costs are added to the bid price per litre supplied, the

combined cost per litre for fuel from each vendor servicing a specific depot is

shown below:

(10)

Company I Company II Company III

Depot I 5.00 4.75 4.25

Depot II 5.00 5.50 6.75

Depot III 4.50 6.00 5.00

Depot IV 5.50 6.00 4.50

Determine the Optimum Transportation Schedule.

(6) R.K. Steel Manufacturing Company produces two items P1 and P2. It uses sheet metal, equipment and labour. Input - Output relationship. Resources available area as follows:

(10)

InputProduct requirement per unit

P1 P2Availability

Sheet Metal 1 sq. cm 1 sq. cm 50 sq. cm

Labour 1 man hour 2 man hours 80 man hours

Equipment 3 hours 2 hours 140 hours

Profit (Rs.) Rs.4 per unit Rs.3 per unit

How many units of P1 and P2 should be manufactured to maximize the profit of the company? Use Graphical Method.



60 Marks

(7) The time and cost estimates and precedence relationship of the different activities

constituting a project are given below:

Time (in weeks) Cost (in Rs.)

Activity Predecessor Normal Crash Normal Crash

A - 3 2 80 190

B - 8 6 6 10

C B 6 4 100 120

D B 5 2 40 100

E A 13 10 30 90

F A 4 4 150 150

G F 2 1 12 14

H C,E,G 6 4 35 45

I F 2 1 70 70

(10)

(i) Draw a project network diagram and find the critical path.

(ii) If a dead line of 17 weeks is imposed for completion of the project which

activities will be crashed, what should be the additional cost and what would

be the critical activities of the crashed network after crashing?

**********



60 Marks

Note: (1) Section I is compulsory.(2) Answer ANY THREE questions from Section II(3) Answer of both sections should be written in the same answer book.(4) Figures to the right side of the questions (in brackets) indicate marks.(5) Graph papers will be supplied on request.(6) Use of only simple calculator is allowed. Mobile phones are not allowed.(7) Normal Distribution Table is given/attached at the last page.

Section — I

(1) Answer the following questions in brief: (10)

(2)

(a) Necessary and Sufficient conditions for Critical Path in PERT /CPM.(b) Shadow Prices in a Optimal solution of Linear Programming Problem.(c) Uses of Slack and Floats in PERT/CPM.(d) Degeneracy in a Transportation Problem.(e) Importance of Dual Simplex Method.

(a) Standard Manufacturers produce three products P, Q and R which generate profits of Rs.20/-, Rs.12/- and Rs.8/- per unit. Three operations are needed for each product on three machines M1, M2 and M3. The maximum working hours available for each of these three machines are 1200, 900 and 400 respectively. One of the Simplex Method Solutions is given in the following table:

(10)

(b)

20 12 8 0 0 0

c X BX1 X2 X3 S1 S2 S3

0 S1 160 0 0 4/5 1 -4/5 4/512 X2 120 0 1 3/5 0 2/5 -3/520 X1 140 1 0 1/5 0 -1/5 4/5

Z 20 12 56/5 0 4/5 44/5∆ = C - Z 0 0 -16/5 0 -4/5 44/5

On the basis of above table, answer the following questions:(a) Which Machine is not fully utilized? If the balance working hrs. of this machine are shifted to M2, what will be the effect on the solution?

(i) Retaining the optimality, find the range of working hours of the third Machine.

(ii) Within what range of profit of each product, the solution will remain optimal?

(iii) Keeping the Shadow Prices intact, find the range for the working hours of M2.

(iv) Without altering the optimality, is it possible to reduce the availability of the working hours of the M2 to 200 hours?

(v) If it is decided to increase the capacities of all three machines by 25% of their respective present capacities, what will be the new product mix?



60 Marks

(b) Project ‘River Clean’ consists of certain activities whose time required for each

activity is given in the following table:

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

Time 2 2 1 4 1 5 8 4 3 3 5

(10)

On the basis of above data answer the following:(i) Draw the New work Diagram and find the Critical Path.

(ii) Calculate the Floats and determine the Sub-critical Path.

(iii) Activities 2-3, 4-5 and 6-9 each require one unit of key machine to complete it. The cost of machine does not permit to acquire another unit. You are asked to opine that availability of one unit of the machine is enough to complete the activities in question. Justify your opinion.

Section — II

(3) ‘UNIK’ Marketing Co. has three Regional Offices and four Distribution Centers.

The Company has decided to launch a new product simultaneously at all centers.

His distribution and transportation plans were leaked to its competitors that ‘UNIK’

will be able to launch the new product only after twenty days. However, based on

the following Initial Feasible Solution, find transportation schedule which requires

the Least Transportation Time. (10)

(10)

Distribution Centers

Regional Office DC1 DC2 DC3 DC4Support Materials

(Tons) to be supplied

RO110 0

10 3

20 1115

RO21 7 9 20

255 15 5

RO312 14 16 18

55

Requirement of

Materials (tons)

12 8 15 10

Note: Figures in bold indicate allocation of Materials (in tons) from Regional Offices to

Distribution Centers, whereas upper left hand corner numbers indicate the Number of

days required to transport any volume (tons) of materials from RO’s to DC’s.



60 Marks

(4) (a) Explain, in brief, the Three Time Estimates and their significance in PERT.

(4) (b) AB Ltd., a chemical company has two plants with daily chemical production of 6 lakhs

and 9 lakhs litres respectively. The Plants must fulfill the needs of its three distribution

centres which have total chemical requirement of 7, 5 an 3 lakh litres respectively. Cost

of transporting one lakh litres of chemical from each plant to each distribution centre is

given in hundreds of rupees below. Formulate this as a Linear Programming Problem:-

(3)

(7)

SourceDistribution Centers

D1 D2 D3Supply

Plant 1 2 3 11 6

Plant 2 1 9 6 9

Demand 7 5 3

(5) A Five Star Hotel which has four banquet halls used for functions. The halls are of same size

but with varying facilities. Four parties approached to reserve a hall for a function on the

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

10,000/- for the day. They were told to indicate the 2nd, 3rd and 4th preferences and the

price they would be willing to pay. Two parties A and D told that they were not interested in

halls 3 and 4.

The following table shows preference-wise income details. What would be the optimal

assignment to maximize the total revenue? (Figures are in thousands)

(10)

PartiesHall

1 2 3 4

A 10 9 - -

B 8 10 8 5

C 7 10 6 8

D 10 8 - -



60 Marks

(6) (a) Following table shows the seven activities, their preceding activities and their three time estimates.

Activities A B C D E F G

Predecessor activities - - - A A B, D C

Optimistic Time (days) 3 5 4 16 7 6 10

Pessimistic Time

(7)

(days)15 17 28 30 13 20 36

Most likely Time (days) 6 11 19 20 10 10 20

On the basis of above table, answer the following questions:

(i) Draw the Net-work Diagram, and calculate the expected duration of all the activities.

(ii) Find the expected duration of the Project with 50% and 75% chances of its completion.

(iii) If a penalty of Rs. 10,000/- per day is to be imposed, what is the probability that more than Rs. 20,000/- penalty will have to be paid?

(6) (b) Interpreting the meaning of the variables ‘u’, ‘v’ and ‘∆’ in the optimality testing of a Transportation Problem, state in brief, the MODI Method.

(7) Using Simplex Method, solve the following Linear Programming Problem. Is it a

degenerate solution?

Maximize: Z = 3x1 + 5x2

Subject to:

(3)

(10)

x1 + x3

2x2 + x4

x1 + 2x2 + x5

x1 , x2 , x3 , x4 , x5

= 4

= 6

= 12

≥ 0

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question paper unsolved -qmb - ii

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