University of Mumbai 2008 Post Graduate Diploma FYPGDORM Part - I - Basic of Operations Reseach I - Question Paper
Kindly obtain the attachment.
ftvsl Hfcvr p\onna'm OpoorrxVVcvnG secnrcb fenr
P4/RT-EX-08-598 j * \
~ no-naacnoerM ,
Con. 2439-08.|CX\JU2, \U -QJcg OPtfVClWoTB e<50Cr7KlrlBB-9343
(3 Hours) [Total Marks : 100
N.B. : (1) Attempt any three questions from each section.
(2) Answers to each section should be written in separate books.
(3) Figures to the right indicate full marks.
(4) Necessary explanations at intermediate stages must be given.
(5) Assumptions, wherever necessary must be stated clearly.
(6) Use of ordinary calculator and statistical table is allowed.
Section I
1. (a) Solve the following LPP 8
Maximise Z = 30X-, + 16x2 + 25x3
Subject to 8x1 + 4x2 + 5x3 < 1000 5x1 + 3x2 + 3x3 < 650 3x1 + 2x2 + 3x3 < 420 xv x2 x3 > 0
(b) Write down the Dual.problem of the following LPP and find out the optimum solutions 8 of both primal and dual problems.
2. Solve the following problem- graphically - 20
Maximise Z = 20x-| + 10x2
Subject to x-, + 2x2 < 40 3x.|+ x2 < 30 4x1 + 3x2 >60 x1, x2 > 0
(a) If the objective function coefficients (20 ,10) change to (10, 20), what will be the optimal solution.
(b) If the RHS coefficient change from (40, 30, 60) to (50, 50, 50) respectively find the new optimal solution.
(c) A new constraint x1 + x2 < 45, be added find the new optimal solution if the present optimal solution is affected.
(d) If product x3 with cost 5 and resource requirements (3, 2, 5) respectively be introduced, find the Optimal solution.
3. (a) Goods are transported from factories Fv F2 and F3 to the warehouse W1( W2, W3 and 10
W4 cost of transportation, in Rs. from each factory to each warehouse, in Rs. given in the table below. Also number of demand units and supply units are given-
|
W1 |
w2 |
w3 |
w4 |
Supply (units) | |
|
Fi |
3 |
5 |
2 |
4 |
100 |
|
F2 |
6 |
3 |
7 |
2 |
80 |
|
F3 |
9 |
4 |
2 |
5 |
40 |
|
Demand |
70 |
50 |
40 |
60 |
Determine how many units from each factory to each warehouse should be transported
so as to minimize the to transportation cost.
r [TURNOVER
|
B | ||||||||||||
|
4. The following table gives completion time, in hours for each worker for each job. 16
|
J2 |
3 | |||
|
W, |
3 |
5 |
2 |
4 |
|
w2 |
6 |
3 |
7 |
2 |
|
w3 |
9 |
4 |
2 |
5 |
|
w4 |
8 |
3 |
2 |
5 |
(a) Determine the optimum assign on solution.
(b) Suppose the completion time of J4 by W2 change from 9 hours to 5 hours, obtain the optimum assignment solution.
5. A project consists of 8 activities. 16 A(3), B(4), C(2), D(3), E(5), F(7), G(8), H(2).
Figures, in brackets, denote durations in days of the activities. The following relationship amongst the activities hold.
(a) A, B and C are the starting activities of the project.
(b) A precedes D, B precedes E and F and C precedes G.
(c) D and F precedes H and
(d) C and F control G
(e) G and H are ending activities.
Draw a network diagram. Find EST, LFT, LST for each activity and determine the critical path and project duration. For each activity find the total float, Free float, Interference float and Independent Float.
6. (a) The following table gives, for each activity of a project. Normal Duration (ND), Crash 8
Duration (CD) in days, Normal Cost (NC), Cost Cash (CC), in Rs. Indirect Cost is Rs. 50 per days :-
|
Activity |
1-2 |
1-3 |
2-4 |
2-5 |
3-4 |
4-5 |
|
ND |
7 |
3 |
2 |
9 |
6 |
3 |
|
CD |
5 |
1 |
1 |
4 |
2 |
2 |
|
NC |
100 |
150 |
50 |
100 |
100 |
80 |
|
CC |
200 |
350 |
90 |
400 |
200 |
100 |
(i) Determine the minimum project duration and the corresponding Project Cost,
(ii) Determine the minimum project cost and the corresponding project duration.
(b) In a municipal hospital, patients arrivals are to be considered as Poisson with an 8 average of interarrival time 10 minutes. The doctor's time for examination plus time of dispensing medicine is distributed negative exponentially with an average of 6 minutes.
(i) What are the chances that a new patient will see the doctor without having to wait ?
(ii) For what percentage of time, the doctor will remain idle ?
(iii) Find the average queue length, average number of patients in the system, average waiting time and average time spent in the system.
7. (a) The following table gives the optimistic, most likely and pessimistic project activity 10 duration, in days.
Find the mean time and variance for each activity of the project.
What is the probability that, the project will be completed in 4 days later than expected duration ?
|
Activity |
1-2 |
1-3 |
2-4 |
2-5 |
3-4 |
4-5 |
|
Optimistic |
3 |
5 |
1 |
1 |
4 |
4 |
|
Most Likely |
4 |
6 |
3 |
4 |
8 |
5 |
|
Pessimistic |
5 |
7 |
5 |
7 |
12 |
6 |
(b) A 2 x 2 pay-off matrix for player A is given below. Then will be a riddle point only if- 6
(i) p<q, p>5
5 6 P q
(ii) P > q, P < 5
(iii) neither of the outer (i) and (ii)
|
8. (a) For an LPP, the optimum simplex table is as follows 10 | |||||||||||||||||||||||||||||||||||||||||||||
|
(i)
Find the missing numbers. Find the original LPP.
(ii)
(b) Write short notes on 6
(i) Alternative optima in a LPP.
(ii) Saddle Point in a game.
(iii) unbounded solution.
|
Attachment: |
| Earning: Approval pending. |
