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) (a) Explain the meaning of degeneracy and infeasibility in a Linear Programming Problem (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.

(2) (2) (2) (2) (2)

(2) (a) M/s ABC & Co. is interested in developing an advertising campaign that will (10) 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. Media M1 M2 M3

16-20 12 10 14

Age Groups 21-25 26-35 7 10 9 12 12 9

36 and above 10 10 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

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QMB - II

Quantitative Methods for Business - II April – 2002 60 Marks (b) Trick and Tack produces several types of glass containers. They have (10) 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?

Cost (Rs.) Per Unit Time (hour) reuction

Description

A B C D E F G H I J

Preheat glass Preheat oven Obtain materials Check valves Check pressure seals Add glass to oven Prepare bottlemaker Run test production Examine test quantity and make adjustments Refill oven with glass

TY BMS – Sem VI

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C D -

8 12 4 4

8 12 2 2

400 200

B A, E E F, G H H

2 2 6 4 4 2

1 2 3 4 2 2

200 500 500 -

QMB - II

Quantitative Methods for Business - II April – 2002 60 Marks

Section — II (3)

(3) (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

(7)

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. (4)

(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 Salesman T1 T2 T3 T4 S1

35

27

28

37

S2

28

34

29

40

S3

35

24

32

33

S4

24

32

25

28

(4)

(6)

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.

(10)

Maximise Z = 6x1 – 2 x2 ;Subject to: 2x1 – x2 ≤ 2 X1 ≤ 4 x1, x2 ≥ 0

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2002 60 Marks (6) M/s. Raj and Bilimoria Associates produce three items ‘X’, ‘Y’ and ‘Z’ each of

(10)

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: Activity A B C D E F G H

Immediate Predecessor A B C D, E F, G

(10)

Estimated Duration (Days) Optimistic Most Likely Pessimistic 2 2 8 2 5 8 3 3 9 2 2 2 3 6 15 3 6 9 4 7 16 2 3 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)

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TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2003 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:

(10)

(a) (b) (c) (d)

Basic variables in simplex method of a Linear Programming Problem. Prohibited Transportation Problem. Forward and Backward pass in PERT/CPM. 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.

(2) (a) A television manufacturing firm is planning to produce television sets of (10) 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. Types of T.V. Sets Flat Screen Black Screen Normal

TY BMS – Sem VI

Prices Rs. 10,000 7,000 6,500

Tube Cost Rs. 3,000 2,200 1,900

Page 1 of 6

Labour and other Material Expenses 4,750 2,500 2,200

QMB - II

Quantitative Methods for Business - II April – 2003 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 (10) 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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2003 60 Marks Activity A B C D E F G H I

Description

Immediate Predecessors

Determine equipment needs Obtain vendor proposals Select vendor Order system Design new warehouse layout Design warehouse Design Computer interface Interface computer Install system

Activity Optimistic Time Most Probable Time Pessimistic Time

A 4 6 8

B 5 7 15

C 4 8 12

A A B B C E D, F G, H

D 15 20 25

E 10 18 26

F 8 9 16

G 4 8 12

H 1 2 3

I 6 7 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 an 80% chance of completion within 50 weeks? Activity Crashed Activity (weeks) Normal Cost (Rs.) Crashed Cost (Rs.)

(4)

(6)

A 1-2

B 2-3

C 2-4

D 3-6

E 3-5

F 4-6

G 5-7

H 6-7

I 7-8

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

Note: (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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2003 60 Marks

Section — II (3)

(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.

Activity Normal Time (days) Normal Cost (Rs.) Crash Time (days) Crash Cost (Rs.)

1-2

2-3

2-4

3-6

3-5

4-6

8 100 6 200

4 150 2 350

2 50 1 90

10 100 5 400

5 100 1 200

3 80 1 100

(7)

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

(3) (7)

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: Production Capacity

Months 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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2003 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. Activity

(10)

To Bakery

Outlet A

Outlet B

Outlet C

4 7 3

4 6 3

7 6 7

3 3 7 -

Bakery Outlet A Outlet B Outlet C

(6) Solve the following Linear Programming problem by Simplex Method without using the artificial variables. Maximise

(10)

Z = 3x1 + 5x2

Subject to: x1 + x3

= 4

x2 + x4

= 6

3x1 + 2x2 + x5 x1 , x2 , x3 , x4 , x5

= 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

(10)

other details in this regard:

Resource

Unit resource requirement

Maximum

Model 1

Model 2

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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2003 60 Marks

Profit Coefficient Cβ 3 0 4

Basis

Solution

C:

3

4

0

0

0

Variables

Values

X:

X1

X2

S1

S2

S3

Xβ X1 S3 X2 Z = Rs. 175 ∆=C-Z

b 450 400 100 Z

1 0 0 3 0

0 0 1 4 0

-¼

¾

-2 1/3 5/4 -5/4

2 -1/2

¼

-1/4

0 1 0 0 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.

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TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2004 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)

(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.

(2)

(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 Centre Mumbai Chennai Kolkota

Ahmedabad 6 14 4

Distribution Centres Bhopal Banglore 4 14 10 4 10 8

Goa 12 6 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 effect on 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?

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2004 60 Marks (b) M/s ABC are in jewellery business and are specialised in making of Rings and (10) 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/s ABC? (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

(10)

regular basis. The transportation costs per ton from factories to warehouses are given in the following table:

Factory I II III

1 6 11 10

Warehouse 2 3 1 9 5 2 12 4

4 3 8 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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2004 60 Marks (4)

Nagaria & Associates are preparing for laying the foundation of State Computer (10) 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

(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

(4)

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

(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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2004 60 Marks (5) (B) PQR Ltd. produces 4 different products viz, pen, ink, pencil and rubber using 4 (6) 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): Products Workers 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

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

(7)

(B) Using Dual Simplex Method, the optimum solution table for the Liner Programming Problem. Minimize: Z = 3x1 + 6x2 + x3 Subject to: x1 + x2 + x3 ≥ 6 x1 + 5x2 - x3

≥ 4

x1 + 5x2 + x3

≥ 24

x1 , x2 , x3

≥ 0

Is as below: CB 0 -3 -1

XB S1 X1 X3 Z = -52 ∆=C-Z

B

-3 X: X1

18 14 10

0 1 1 -3 0

-6 X2

-1 X3

4 0 5 -5 -1

0 0 1 -1 0

0 S1

0 S2

0 S3

1

1

0 0

-1/2 ½

-1 -1/2 -1/2

0

-1

-2

0

0

0

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 (10) 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.

********** TY BMS – Sem VI

Page 4 of 4

QMB - II

Quantitative Methods for Business - II April – 2005 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) (b) (c) (d) (e)

(10)

Distinguish between degeneracy and cycling in LPP. What do you mean by Shadow prices in LPP? Explain Unbalanced Transportation Problem. Write briefly on Multiple Optional Solutions in an Assignment Problem. 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

(10)

in the amount shown below: Product A B

Resources require for one unit Leather Wood Glue (in Kg) (in Sq. Mts) (in ltrs) 0.50 0.25

4 7

0.2 0.2

Available resources include 2200 kgs. of leathers, 28,000 sq. metres of wood and 1,400 litres 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 (10) each warehouse, the requirement of each market and the unit transportation cost (in Rs.) from each warehouse to each market: Warehouse A B C Requirements

TY BMS – Sem VI

Market P Q 6 3 5 9 5 7 7 12

Supply R 5 2 8 17

Page 1 of 4

S 4 7 6 9

22 15 8

QMB - II

Quantitative Methods for Business - II April – 2005 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

(3)

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

(7)

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.

Salesman A B C D

I 420 300 300 240

Districts II III 350 280 250 200 250 200 200 160

IV 210 150 150 120

What will be the total Maximum sale? (4) A project has the following activities and other characteristics.

Activity

Preceding Activity

A

(10)

Time Estimates (in weeks) Optimistic

Most Likely

Pessimistic

-

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

TY BMS – Sem VI

Page 2 of 4

QMB - II

Quantitative Methods for Business - II April – 2005 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

(10)

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: 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: Input

Product requirement per unit

(10)

Availability

P1

P2

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.

TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2005 60 Marks (7) The time and cost estimates and precedence relationship of the different activities

(10)

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

(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?

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TY BMS – Sem VI

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QMB - II

Quantitative Methods for Business - II April – 2006 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: (a) (b) (c) (d) (e)

(10)

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

(2) (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 (10) 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: (b) c

X

B

0 12 20

S1 X2 X1 Z ∆=C-Z

160 120 140

20

12

8

0

0

0

X1 0 0 1 20 0

X2 0 1 0 12 0

X3 4/5 3/5 1/5 56/5 -16/5

S1 1 0 0 0 0

S2 -4/5 2/5 -1/5 4/5 -4/5

S3 4/5 -3/5 4/5 44/5 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?

TY BMS – Sem VI

Page 1 of 4

QMB - II

Quantitative Methods for Business - II April – 2006 60 Marks (b) Project ‘River Clean’ consists of certain activities whose time required for each

(10)

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

2

2

1

4

1

5

8

4

3

3

5

Time

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.

(10)

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) Distribution Centers Regional Office RO1 RO2 RO3 Requirement of

DC1 10

DC2

DC3

0

10 1

20

11

9

20

5

15

14

16 8

18 15

(Tons) to be supplied

25

5

5

5 12

Support Materials

15

3 7

12

DC4

10

Materials (tons) 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.

TY BMS – Sem VI

Page 2 of 4

QMB - II

Quantitative Methods for Business - II April – 2006 60 Marks (4) (a) Explain, in brief, the Three Time Estimates and their significance in PERT.

(3)

(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

(7)

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:Distribution Centers

Source

Supply

D1

D2

D3

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)

Parties

TY BMS – Sem VI

Hall 1

2

3

4

A

10

9

-

-

B

8

10

8

5

C

7

10

6

8

D

10

8

-

-

Page 3 of 4

QMB - II

(10)

Quantitative Methods for Business - II April – 2006 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

15

17

28

30

13

20

36

6

11

19

20

10

10

20

Pessimistic Time (days) Most likely Time (days)

(7)

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.

(3)

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

(10)

degenerate solution? Maximize: Z = 3x1 + 5x2 Subject to: x1 + x3 = 4 2x2 + x4 = 6 x1 + 2x2 + x5 = 12 x1 , x2 , x3 , x4 , x5

≥ 0

**********

TY BMS – Sem VI

Page 4 of 4

QMB - II