Operation Research - Operations research and Operations management - Production & Industrial Engineering Previous Year Questions

1

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

In queueing models, M/M/c denotes a Poisson arrival process and

A

exponentially distributed service times and c servers in series

B

constant service times and c servers in series

C

exponentially distributed service times and c servers in parallel

D

constant service times and c servers in parallel

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2

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

In sensitivity analysis of LP models, which of the following holds true?
P – Reduced cost of basic variables are zero at optimality
Q – Constraints are binding when shadow prices are non-zero
R – Constraints are binding when shadow prices are zero
S – Reduced cost is same as shadow price

A

P and Q

B

Q and R

C

P and R

D

Q and S

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3

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

Consider the symmetric dual pair of LPs [P] and [D], where A is an \( m \times n \) matrix, b is an \( m \)-vector and c is an \( n \)-vector.
\[ [P] \quad \text{Min} \quad c^T x \\ \text{s.t.} \quad Ax \ge b \\ x \ge 0 \] \[ [D] \quad \text{Max} \quad b^T y \\ \text{s.t.} \quad A^T y \le c^T \\ y \ge 0 \]
Assume that [P] is feasible. If the optimal values are \( z_1^* \) for [P] and \( z_2^* \) for [D], whenever they exist, then which one of the following is true?

A

If [D] is infeasible, then \(z_1^* \) can be determined and is equal to \(z_2^* \)

B

If [D] is infeasible, then \(z_1^* \) cannot be determined

C

If [D] is feasible, then \(z_1^* \) can be determined and is equal to \(z_2^* \)

D

If [D] is feasible, then \(z_1^* \) can be determined but not equal to \(z_2^* \)

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4

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

In an optimization problem, let \( y \) be a 0–1 variable and \( x \) be a positive real number. Now, the condition that \( x \) can take non-zero values only if \( y = 1 \) can be modeled using the linear constraint

A

\(x \le My \) (M is a large number)

B

\(x \ge y \)

C

\(x \ge My \) (M is a large number)

D

\(xy \ge 0 \)

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5

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

Karmarkar’s algorithm for Linear Programming

A

moves along different extreme point solutions of the feasible region

B

enumerates all possible extreme point solutions

C

divides the feasible region into different parts for function evaluation

D

generates interior point iterates which converges to the optimum solution

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6

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

Consider the symmetric dual pair of LPs [P] and [D], where A is an m×n matrix, b is an m-vector and c is an n-vector.
[P] Min c^T x s.t. Ax ≥ b, x ≥ 0
[D] Max b^T y s.t. A^T y ≤ c^T, y ≥ 0
Assume that [P] is feasible. If the optimal values are z_1^* for [P] and z_2^* for [D], whenever they exist, then which one of the following is true?

A

If [D] is infeasible, then z_1^* can be determined and is equal to z_2^*

B

If [D] is feasible, then z_1^* cannot be determined

C

If [D] is feasible, then z_1^* can be determined and is equal to z_2^*

D

If [D] is feasible, then z_1^* can be determined but not equal to z_2^*

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7

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

In an optimization problem, let y be a 0–1 variable and x be a positive real number. Now, the condition that x can take non-zero values only if y = 1 can be modeled using the linear constraint

A

x ≤ My (M is a large number)

B

x ≥ y

C

x ≥ My (M is a large number)

D

xy ≥ 0

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8

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

For a transportation problem that has a feasible solution, the northwest corner rule gives a possible solution which is

A

a basic feasible solution to the problem

B

a near optimal solution to the problem

C

the optimal solution to the problem

D

one of the many optimal solutions to the problem

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9

2007 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2007

The assignment problem in Linear Programming is also an example of a discrete optimization problem. How many feasible solutions are there to this problem defined on n jobs and n persons?

A

\(n^n\)

B

\(n(n-1)\)

C

\(n^2\)

D

\(n!\)

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10

2008 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2008

Customers arrive at a service counter manned by a single person according to a Poisson distribution with a mean arrival rate of 30 per hour. The time required to serve a customer follows an exponential distribution with a mean of 100 seconds. The average waiting time (in hours) of a customer in the system will be

A

0.138

B

0.166

C

0.276

D

0.332

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11

2008 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2008

Consider the following linear programming problem (LPP): \[\begin{aligned} \text{Maximize } z &= 5x_1 + 3x_2 \\ \text{Subject to the following constraints} \\ x_1 - x_2 &\leq 2 \\ x_1 + x_2 &\geq 3 \\ x_1, x_2 &\geq 0 \end{aligned}\] The above LPP has

A

no solution

B

unique solution

C

two solutions

D

unbounded solution

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12

2008 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2008

Consider the following linear programming problem (LPP): \( \text{Maximize } z = 5x_1 + 3x_2 \) Subject to the following constraints \( x_1 - x_2 \leq 2 \) \( x_1 + x_2 \geq 3 \) \( x_1, x_2 \geq 0 \) The above LPP has

A

no solution

B

unique solution

C

two solutions

D

unbounded solution

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13

2009 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2009

As per Kendall’s notation in M/G/c queuing system, the number of arrivals in a fixed time follows

A

Beta distribution

B

Normal distribution

C

Poisson distribution

D

Uniform distribution

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14

2009 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2009

As per Kendall's notation in M/G/c queuing system, the number of arrivals in a fixed time follows

A

Beta distribution

B

Normal distribution

C

Poisson distribution

D

Uniform distribution

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15

2009 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2009

Basic variables in the optimal solution are

A

s_1 and s_2

B

x_1 and x_2

C

x_1, x_2 and x_3

D

x_3, s_1 and s_2

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16

2009 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2009

Suppose that in the LPP given, the right hand side of constraint 1 changes from 50 to 40. The new objective value is

A

90

B

100

C

110

D

120

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17

2010 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2010

Apart from the non-negativity criteria, the dual problem for the given Linear Programming problem consists of

A

2 constraints and both of them are of ≤ type

B

2 constraints and both of them are of ≥ type

C

3 constraints and all of them are of ≤ type

D

3 constraints and all of them are of ≥ type

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18

2010 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2010

The value of the objective function after solving the dual problem is

A

160

B

170

C

190

D

210

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19

2011 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2011

A dedicated machine receives jobs at a rate of 20 per hour and the processing rate of the machine is 30 jobs per hour. Assume the following:
(i) inter-arrival time and processing time for jobs follow exponential distributions
(ii) queue discipline is first-come-first-served (FCFS)
(iii) queue capacity and job population are infinite
For how much time (in minutes), on an average, does a job have to wait before it gets loaded on to the machine?

A

4

B

3

C

5

D

6

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20

2011 · Production & Industrial Engineering · Operations research and Operations management · Operation Research

Production & Industrial Engineering (PI) 2011

A transporter receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of the 5th day?

A

4

B

5

C

6

D

7

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