University of Mumbai 2008 Post Graduate Diploma SYPGDORM Part - II - Advance Operation Reserch I - Question Paper

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Solve the following parametric programming problem : Maximize z = 3x₁ + 2x₂+ 5x₃ subject to x₁ + 2x₂ + x₃ < 30 - 7t

x. + 2x, + 4x > 5, x₂, x₃> 0

(b) The following information gives distances in kms between 6 cities A, B, C, D, E, F.

Draw the Spanning Tree.

4.    Determine an optimum solution for the following LPP :    20

Maximize Z = 25x₁ + 6x₂ + 8x₃ subject to 2x₁+4x₂-3x₃ < 30 x₁ + 3x₂ + x₃ < 25 x_1f x₂, x_? > 0.

(a)    Find the change in the optimum solution if

c₁ changes from 25 to 30

(b)    If a new contraint, x₁ + 2x₂- x₃< 30 is added find the new optimum solution.

(c)    If the RHS of the constraints changes from (25) *⁰(2o) ^w*^{ia w}'"    ^can9^ecl optimum solution ?

5.    Maximize Z = 30x₁ + 20x₂ + 25x₃ subject to 3x₁ + 2x₂+ x₃ < 120 (supply of Resource I)

5x₁ + 3x₂ + 2x₃ <100 (supply of Resource 2)

Further it was possible to acquire more units of Resources 1 and 2 therefore prefer to buy excess units of Resource 2.

The goods were set as to minimize

(a)    deviation from a target profit of Rs. 2500

(b)    additional units of Resource 2 and

(c)    additional units of Resource 1.

Formulate the problem as a Goal Programming problem and solve it.

6.    (a) Find the maximum flow (units) of an item that can be transported through a network with 10

the paths listed below from source 1 to sink 5.

The paths and their capacities in units (x) are listed below :

Paths	1-2 '	1-3	2-4	2-5	3-4	3-5 2-3
X	9	12	4	11	6	10 4

Indicate the paths fully utilizing their capacities available.

(b) The following table gives activities, durations, in days and manpower requirements for a certain 10 project :

Activity 1-2 1-3 2-4 3-4 4-5 Duration 3 2 1 2 2 Manpower 4 3 2 5 4

Level out the resources without, increasing the project duration.

7. (a) The following table gives optimistic (a) most likely (m), and passimistic (c) durations, in days, 10 for activities of a certain project:

Activity 1-2 1-3 1-4 2-3 2-5 3-4 4-5 a 1 3 2 2 3 1 1 m 3 4 5 6 4 3 2 b 5 5 8 10 5 5 3

(i)    Find the mean and variance of each activity duration.

(ii)    What is the probability that the project will be completed in 5 days more than the expected duration of the project.

(b) The table below gives estimates of Normal Durations (ND), Crash Durations (CD), Normal Cost (NC) and Crash Cost (CC) for each of the project activities.

Activity 1-2 1-3 2-3 3-4 4-5 ND, days 4 5 3 4 6 CD, days 2 3 2 1 2 NC, Rs. 100 50 150 80 110 200 CC, Rs. 100 100 170 110 260 275

Indirect Cost is Rs. 60 per day. Determine : (i)    the minimum cost project duration (ii)    the minimum project duration and the corresponding cost.

8. (a) For the following project, determine EST, LFT, EFT, LST, Critical path and its duration :

Activity 1-2 1-3 2-5 3-4 4-5 Duration (days) 3 2 5 6 4

Explain the following terms with illustration :

(i)    Total Float

(ii)    Gomery's Cutting Plane algorithm

(iii)    Aternative optimum, No feasible solution, unbounded solution and degeneracy in LPP.

BB-8594 [ Total Marks : 100

N.B. : (1)    Answer any five questions.

(2)    Figures to the right indicate full marks.

(3)    Use of Statistical Tables and Non-Programmable calculator is permitted.

(4)    Answers must be brief and to the point

(5)    Intermadiate explanations and calculations must be given.

(6)    Assumptions, wherever necessary, must be clearly stated.

2 A farmer has 300 acres of land for cultivating two crops P₁ and P₂. The cost of cultivating P₁ is Rs. 500 per acre, whereas that of P₂ is Rs. 700 per acre. Budger fir cultivation available is Rs. 80,000.

Each acre of P₁ requires 20 hours of labour and for P₂ 25 hours of labour per acre. Maximum labour hours available are 4000 hours.

The farmer wants to cultivate at least 80 acres of P_r He wants to make a profit of Rs. 200 of P₁ and Rs. 300 of B. He wants to maximize the total profit.

Formulate this problem as a LPP and using Simplex method, find the optimum solution.

3. (a) State the advantages of Revised Simplex Method over Standard Simplex Method, (b) Solve the following LPP using Revised Simplex Method :

Maximize Z = 3x₁ + 5x₂ + 8x₃

5x₁ + 3x₂ + 4x₃ < 20 2x₁ + 4x₃ + 5x₃ < 30

x₃> 0

4. (a) Solve the following LPP by Dual Simplex Method : Minimize 2 subject to

2x₁ + x₂ + x₃ > 6 x, + x₂ + 2x₃ <8, x_r x₂, X₃ > 0 (b) For the following LPP

Minimize Z = 190x₁ + 140x₂ + 200x₃ subject to 2x₁ + x₂ + x₃ > 70

x_t + x₂ + 2x₃ > 80, x_r x₂, x₃ > 0 Write down the Dual Problem and solve it. Hence find the optimum solution for the above given problem.

5. Find the optimum solution for the following LPP :

Maximize Z = 2x₁ + 3x₂ subject to 3x₁ + x₂ < 22, x 3x₂ < 26 x,, x₂ > 0

Find the effect on the optimum solution if :

(i)    C₁ changes from 2 to 4

(ii)    b₂ changes from 24 to 20

(iii)    coefficients of x₁ in the constraints change from j to

(iv)    a new constraint 2x₁ + 2x₂ < 25 is added.

6. (a) The following table indicates duration, in days, and number of workers required for each activity 10 of the project :

Activity 1-2 1-3 2-4 3-5 4-5 Duration No. of Workers 2 3 12 3 4 2 3 4 2

Level out the manpower without increasing the project duration, (b) Consider a street network as shown below :

Numbers on the arcs indicate traffic flow capacities. The problem is to place one-way signals on the streets, not already oriented, so as to maximize the trafic flow from point s to point n.

7. (a) The following table shows for a certain project, Normal Duration (ND), and Crash Duration 10 (CD), in days, Normal Cost (NC) and Crash Cost (CC), in Rs. for activities of a certain project.

Activity 1-2 1-3 2-4 2-5 3-4 4-5 ND 5 3 4 12 7 5 CD 4 2 3 7 3 3 NC 40 60 15 40 50 25 CC 90 170 35 190 90 50

Indirect Cost is Rs. 30 per day.

Determine :

(i)    Minimum cost duration of the project.

(ii)    Minimum Duration cost for the entire project.

(b) Optimistic (a), most likely (m) and pessimestic (b) durations for a certain project are given 10 below :

Activity	1-2	1-3	2-4	3-4	4-5
a	3	6	4	9	5
m	4	9	7	12	8
b	5	12	10	15	11

(i)    Determine the expected duration and variance for each activity.

(ii)    What is the probability that the project will be completed in more them 4 days after its expected duration ?

8. (a) Consider the following pay-off matrix, with Player A as the maximising player :    10

B

	Bi	b2	b3
A,	3	5	8
A,	-3	3	8
A3	4	3	6

Obtain the optimum solution for the above game problem.

(b) For a washing powder manufacturing factory, frequency distribution of contribution (x) 10 (= sales price - variable cost) per unit, annual demand (y) and requirement of investment (z) hence found as follows :

X 3 5 7 9 Prob. 'sf CO 2

y 20 25 30 35 Prob. 1 2 6 -1

z 175 200 300 Prob. 3 5 2

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Using Mont Carlo Simulation of 5 runs, estimate the percentage of return on investment. Use the following random numbers :

3025 1348 0752 3148 1234 5897 6325 1485 2008 3529 7784 3948 2145 6974

( 3 Hours )    [ Total Marks : 100

4

N.B. : (1) Answer any five questions.

(2)    Figures to the right indicate full marks.

(3)    Use of Statistical Tables and Non-Programmable calculator is permitted.

(4)    Answers must be brief and to the points.

(5)    intermadiate explanations and calculations must be given.

(6)    Assumptions, wherever, necessary, must be clearly stated.

1.    (a) Solve the following LPP :    10

Maximize Z = 3x₁ + 5x₂+ x₃ subject to 2x₁ + 4x₂ + 3x₃ < 40 x₁ + x₂ < 10

2x₂ + x₃ < 12, x_r x₂, x₃ > 0

(b) Write down the dual problem for the following LPP :    10

Maximize Z = -5x₁ x₂ - 2x₃ subject to 2x₁ + 3x₂ + x₃> 10 x₁ - 2x₂ + 4x₃ = 20 x_1f x₂ > 0 and x₃ unrestricted.

2.    A manufacturer produces 3 products A, B and C by using 2 types of machines Lathes (M and 20 Milling (M2). The necessary information is given in the following table :

Machine Time, minute, per unit Time available A B c per day in min. M, 7 10 4 1000 m2 3 40 1 600 Profit, Rs. per unit 45 100 30

(a)    Formulate the problem as an LPP and obtain the optimum solution.

(b)    Find the effect of changing C₁ from Rs. 45 to Rs. 30 and C₂ from Rs. 100 to Rs, 40.

(c)    Find the effect of changing the total time per day available on M₁ and M₂ to 800 and 500 minutes respectively.

(d)    A new product D requires 15 minutes on M₁ and 10 minutes on M₂ per unit. Will the product D will be produced if its profit, per unit, is Rs. 50 ?

(e)    Each product requires 2, 5 and 3 minutes per unit of time on a third machine, M₃ find the new optimum solution, if available time, per day, on M₃ is 800 minutes.

Maximize Z = 40x₁ + 45x₂

2x₁ + 5x₂ + x₃ < 100 (Availability of Steel)

3x₁ + 3x₂ + 2x₃ < 120 (Availability of Rubber)

x x₂, x₃ > 0

Further it was possible to acquire more steel and rubber at the same original cost. But the location of steel supplier was closer. Management therefore, would prefer to buy excess of steel before buying excess rubber. The goals were set as to minimize :

(a)    deviation from a target profit of Rs. 2250

(b)    additional rubber purchase and

(c)    additional steel purchase.

Formulate the problem as a Goal Programming Problem and find the optimum solution.

6. (a) Solve the following LPP by Dual Simplex Method : Minimize 2 subject to

x₁ + 3x₂ - 2x₃ > 20

7. (a) For the following net-work

Activity T T" 00 I T CM I 2-4 2-5 3-4 4-5 Duration (days) 3 2 6 5 7 2 4

determine the critical path, project duration and Total Float for each activity.

(b) The following table gives activities, their duration and manpower requirements for a certain project.

Activity 1-2 1-3 2-4 3-4 3-5 4-5 Duration (days) 3 2 6 7 5 4 No. of men 2 3 1 3 2 3

Level out the man power without increasing the project duration.

8. (a) A maintenance project consists of the jobs shown below. Normal Duration (ND), Crash 10 Duration (CD) in days, Normal-Cost (NC), Crash Cost (CC) in Rupees, for each job are also given below :

Activity	ND	CD	. NC	CC
1-2	5	2	100	250
1-3	2	1	50	80
2-4	5	3	150	190
3-4	3	1	80	120
4-5	2	1	60	70

Determine :

(i)    the project duration when the total cost is minimum, and

(ii)    the total project cost when the project duration is minimum.

(b) Duration of activities are uncertain. The following table gives optimistic, most likely and 10 pessimistic duration of each activity, in days :

Activity 1-2 1-3 2-3 2-4 3-5 4-5 Optimistic duration 5 15 16 8 7 3 Most likely duration 9 20 21 12 10 6 Pessimistic duration 13 25 26 16 13 9

What is the probability that the project will be completed in 4 days later than the expected project duration ?

Attachment: university-of-mumbai-2008-post-graduate-diploma--51848-1.pdf

Earning:   Approval pending.