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VTU Mechanical Engineering (Semester 7)
Operations Research
December 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) The following are the two constraints for the LPP under consideration.
x1-x2≥1; x1+x2≥7; and x1, x2≥0
using the graphical method, answer the following questions:
i) What are the extreme points for the feasible region?
ii) If the problem has an objective function maximize z=3x1+x2, what is the optimal point?
iii) If the problem has an objective function, minimize z=3x1+x2, what is the optimal point.
10 M
1 (b) Solve the following problem using graphical method maximize Z=2x1+3x2 Subject to constraints: 2x1+z2≤6; x1-x2≥3; x1, x2 ≥0.
6 M
1 (c) List any four characteristics of a good model.
4 M

2 (a) Solve the following LPP by Big-Method.
Maximize
X= 30000x1 + 20000x2
Subject to:
x2≤x1+3; x2≤6; x2≥2; x1+2x2≤18; 2x1+x2≤24 and x1, x2 ≥ 0.
14 M
2 (b) Write the dual for the following primal.
Minimize Z=25000x1+35000x2
Subject to:
50x1+60x2=2500; 80x1+60x2≥3000; 100x1+200x2≥7000
Non-negativity constraints: x1, x2≥0.
6 M

3 (a) A product is produced by four factories A, B, C & D. The unit production costs in them are ₹2, ₹3, ₹1 and ₹5 respectively. Their production capacities are: Factory A: 50 units; B: 70 units; C=30 units and D: 50 units. These factories supply the product to four stores, the demand of which are 25, 35, 105 and 20 units respectively. Unit transportation cost in rupees from each factory to each store is given in the table below:
Determine the extent of deliveries from of the factories to each of the stores so that the total production transportation cost is minimum.
1 2 3 4
Stores
Factories
 1 2 3 4 A 2 4 6 11 B 10 8 7 5 C 13 3 9 12 D 4 6 8 3
12 M
3 (b) Four machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E. Owing to the limitations machine M2 cannot be placed at C and M3 cannot be placed at A. The assignment cost of machine i to place j in rupees (1000) is shown below.
Find the optimal assignment schedule.
 A B C D E M1 4 6 10 5 6 M2 7 4 - 5 4 M3 - 6 9 6 2 M4 9 3 7 2 3
8 M

4 Solve the below given integer programming problem:
Maximize
Z=4x1+6x2+2x3
Subject to
4x1-4x2≤5
-x1+6x2≤5
-x1+x2+x3≤5
x1, x2, x3≥0 and x1, x3 are integers.
20 M

5 Following data refer to a project:
a) Draw the network diagram
b) Find out the ES, EF, LS, LF and slack for each activity.
c) Find out variance standard deviation for the critical path.
d) Determine the probability of completing the project in 24 hrs.
 Activity Immediate Predecessor Optimistic Time (Hrs) Most likely Time (Hrs) Pessimistic Time (Hrs) A B C D - - A A 4 1 3 4 6 4.5 3 5 8 5 3 6 E F G H A B, C B, C E, F 0.5 3 1 5 1 4 1.5 6 1.5 5 5 7 I J K E, F D, H G, I 2 2.5 3 5 2.75 5 8 4.5 7
20 M

6 (a) Mention and discuss seven elements of queuing system.
7 M
6 (b) Define: (i) Balking (ii) Renegging (iii) Jockeying.
3 M
6 (c) On an average 96 patient per 24-hours day require the service of an emergency clinic. On an average a patient requires 10 minutes of active attention. The facility can handle only one emergency at a time. Suppose that it costs the clinic ₹100 per patient treated to obtain an averaging service time of 10 minutes and that each minute and that each minute of decrease in this average time would cost the clinic ₹10 per patient treated. How much would have to be budgeted by the clinic to decrease the average queue size from $$1\dfrac {1}{3}$$ patients as to $$\dfrac {1}{2}$$ patient?
10 M

7 (a) What are the characteristics of games?
4 M
7 (b) Solve the following game by the dominance rule.
Can we formulate the above game as LPP and solve it by simplex/Big-M method? If yes discuss how?
B's Strategy
A's Strategy
 b1 b2 b3 a1 12 -8 -2 a2 6 7 3 a3 -10 -6 2
8 M
7 (c) Solve the following game using graphical approach:
 A's Strategy B's Strategy b1 b2 b3 b4 a1 8 5 -7 9 a2 -6 6 4 -2
8 M

8 (a) You are given the following data regarding the processing times of some job on three machines I, II and III. The order of processing is I-II-III. Determine the sequence that minimizes the total elapsed time required to complete all the jobs. Mention clearly the total elapsed time and the idle time of machine II and III.
 Job Machine I II III A 3 4 6 B 8 3 7 C 7 2 5 D 4 5 11 E 9 1 5 F 8 4 6 G 7 3 12
10 M
8 (b) Madan Mathur is the supervisor of legal-copy-express, which provides copy services for downtown Los Angeles law firms. Five customers submitted their orders at the beginning of the week and are as follows. Schedule the jobs as per i) EDD rule ii) SPT rule and iii) Slack-time remaining per operation rule. Which rule gives the best result in terms of mean flow-time?
 Job (In order of arrival) A B C D E Processing Time (Days) 3 4 6 2 1 Due date (Days hence) 5 6 7 9 2
10 M

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