Gujarat University
MCA Semester IV
CBOM
Date: 5th March, 1990
Time: 3 hrs. Marks:50
Instructions:
1. Answer to the two sections should be written in separate answer books.
2. Make suitable assumption, wherever necessary.
3. Figures to the right indicate full marks.
Section I
Q-1 Attempt any four of the following: [9]
1. Discuss the importance of optimization techniques in modern decision making.
2. Explain sensitivity analysis in brief with example.
3. Define Dual of linear programming problem with its general mathematical formulation.
4. Explain in brief any one of obtaining initial feasible solution for a transportation problem.
Q-2 (a) Explain the use of ‘ Artificial Variable’ LPP. [2]
(b) Alpha radio mfg. Co. must determine production quantities for this month for two different models A and
B. Data per unit are given in the following table:
Model Revenue Sub Assembly Time Final Assembly Quality Inspection
(Rs.) (Hrs.) time (Hrs.) time (Hrs.)
A 150 1.0 0.8 0.5
B 200 1.2 2.0 0
The maximum time available for these products is 1200 hours for subassembly, 1600 hours for final
assembly and 500 hours for quality inspection.
Orders outstanding requires that atleast 200 units of A and 100 units of B be produced. Determine the
quantities of A and B that maximize total revenue.
OR
Q-2 (a) Explain the Dual primal relationship in brief, write the Dual of following LPP. [3]
Min Z= x1 + x2 + x3
Subject to, x1 + 2x2 + 3x3 = 7
X1 + 3x2 <=5
2x1 + x3 >=6
x1, x2 >= 0 and x3 unrestricted.
(b) Write the dual of following LPP: [5]
Min Z = 12x1 + 8x2
Subject to x1 + x2 >= 3
3x1 + x2 >= 7
x1, x2 >= 0
Solve the dual problem by simplex method and read the solution of dual from simplex table.
Q-3 (a) Write an algorithm to store unbalanced transportation problem using lowest cost entry method. [2]
(b) A company has 4 plants P1 to P4 from which it ships its product units to four dealers. Transportation costs
per unit between various plants and dealers locations are as given below: [6]
D1 D2 D3 D4 Capacity
P1 48 60 56 58 140
P2 45 55 53 60 160
P3 50 65 60 62 360
P4 52 64 55 61 220
Requirement 200 320 250 210
(i) Find out ‘initial solution’ using lowest cost entry method to maximize cost.
(ii) Find out ‘Optimal Solution’ using MODI method.
OR
Q-3 (a) Find out the total minimum cost of the following Assignment Problem: [4]
R L M N
A 8 10 10 15
B 14 21 17 10
C 14 18 22 25
D 16 19 20 24
(b) Find out the total maximum sales of the following Assignment problem: [4]
Sales Person
Districts A B C D E
1 32 38 40 28 40
2 40 24 28 21 36
3 41 27 33 30 37
4 22 38 41 36 36
5 29 33 40 35 39
Section II
Q-4 Attempt any four of the following: [9]
1. What is queuing problem? Explain some of the basic characteristics of a queuing system.
2. What is the difference between transient state and steady state? Explain.
3. Explain the essential difference between CPM and PERT.
4. What are the three time estimates needed for PERT analysis and what do they represent?
5. Explain briefly the Branch and Bound techniques.
Q-5 (a) Derive the difference equations for the queuing model (M/M/1) : (a/ FCFS). [6]
(b) Customer arrives at a store in Poisson fashion at a rate of 10/hour. The only cashier can process the
customers at the average rate of 20/hour. [2]
(i) Find the proportion of idle time of the cashier.
(ii) Obtain average number of customers in the queue and in the system at any time.
OR
Q-5 (a) Obtain the steady state equation for the model (M/N/S): (a/ FCFS) and show that: [6]
Pn = { P0 (l/m )n/ n! if n = 0,1,2……S-1}
{ P0 (l/m)n/ (S!Sn-1) if n = S, S+!……….}
(b) A supermarket has two girls ringing up sales at the counters. If the service time for each customer is
exponential with mean 4 minutes, and if people arrive in a Poisson fashion at the counter at the rate of 10
per hour, then calculate. [2]
(i) The probability to wait for service.
(ii) Expected % of idle time for each girl.
Q-6 (a) What is pure birth process and pure death process in queuing theory. [2]
(b) Determine early start (Ts) and late start (TL) in respect of all node points and identify critical path in
respect the network in Figure A (using slack time). [6]
OR
Q-6 (a) Write down the rules for numbering the events of network with example. [3]
(b) A construction company is considering a large building project. The mean time estimates, the variance of
these time estimates, manpower requirements and precedence requirement for the various activities are
given in the following table: [5]
Activity Immediate Mean time estimates Variance (days squared)
Predecessors (days)
A - 7 3
B A 5 1
C B 8 5
D A 20 5
E A 11 9
F C, D 7 6
G D 2 4
H D, E 6 4
I F,G,H 10 9
(i) Draw a PERT network for the building project.
(ii) Assuming manpower resources are unlimited, find a critical path, as well as the minimum expected completion time.
