# GATE Questions & Answers of Operations Research Mechanical Engineering

#### Operations Research 50 Question(s)

Processing times (including setup times) and due dates for six jobs waiting to be processed at a work centre are given in the table. The average tardiness (in days) using shortest processing time rule is ___________ (correct to two decimal places).

 Job Processing time (days) Due date (days) A 3 8 B 7 16 C 4 4 D 9 18 E 5 17 F 13 19

The arrival of customers over fixed time intervals in a bank follow a Poisson distribution with an average of 30 customers/hour. The probability that the time between successive customer arrival is between 1 and 3 minutes is _______ (correct to two decimal places).

The arc lengths of a directed graph of a project are as shown in the figure. The shortest path length from node 1 to node 6 is _______.

Two models, P and Q, of a product earn profits of Rs. 100 and Rs. 80 per price, respectively. Production times for P and Q are 5 hour and 30 hours, respectively, while the total production time available is 150 hours. For a total batch size of 40, to maximize profit, the number of units of P to be produced is ________

Following data refers to the jobs (P, Q, R, S) which have arrived at a machine for scheduling. The shortest possible average flow time is ______ days.

 Job Processing Time (days) P 15 Q 9 R 22 S 12

For a single server with Poisson arrival and exponential service time, the arrival rate is 12 per hour. Which one of the following services rates will provide a steady finite queue length?

A product made in two factories, P and Q, is transported to two destinations, R and S. The per unit costs of transportation (in Rupees) from factories to destinations are per the following matrix:

Factory P produces 7 units and factory Q produce 9 units of the product. Each destination requires 8 units. If the north-west corner method provides the total transportation cost as X  (in Rupees) and the optimized (the minimum) total transportation cost is Y  (in Rupees), then $(x-y)$, in Rupees, is

A project start with activity A and ends with activity F. The precedence relation and duration of the activites are as per the following table:

 Activity Immediate predecessor Duration (days) A - 4 B A 3 C A 7 D B 14 E C 4 F D. E 9

The minimum project completion time (in days) is___________

Maximize $Z=5{x}_{1}+3{x}_{2}$,

Subject to

${x}_{1}+2{x}_{2}\le 10$,

${x}_{1}-{x}_{2}\le 8$,

${x}_{1}$, ${x}_{2}\ge 0$.

In the starting Simplex tableau, $x_1$ and $x_2$ are non-basic variable and the value of $Z$ is zero. The value of $Z$ in the next Simplex tableau is___________

Maximize $Z=15X_1 + 20X_2$
subject to
$12X_1 + 4X_2 ≥ 36$
$12X_1 − 6X_2 ≤ 24$
$X_1, X_2 ≥ 0$
The above linear programming problem has

In a single-channel queuing model, the customer arrival rate is 12 per hour and the serving rate is 24 per hour. The expected time that a customer is in queue is _______ minutes.

A project consists of 14 activities, A to N. The duration of these activities (in days) are shown in brackets on the network diagram. The latest finish time (in days) for node 10 is __________

In PERT chart, the activity time distribution is

A firm uses a turning center, a milling center and a grinding machine to produce two parts. The table below provides the machining time required for each part and the maximum machining time available on each machine. The profit per unit on parts I and II are Rs. 40 and Rs. 100, respectively. The maximum profit per week of the firm is Rs._________

 Type of machine Machining time required for  the machine part (minutes) Maximum machining time available   per week (minutes) I II Turning Center 12 6 6000 Milling Center 4 10 4000 Grinding Machine 2 3 1800

Following data refers to the activities of a project , where, node 1 refers to the start and node 5 refers to the end of the project.

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

The critical path (CP) in the network is

A project consists of 7 activities. The network along with the time durations ( in days) for various activities is shown in figure.

The minimum time( in days) for completion of the project is ______.

In the notation (a/b/c) : (d/e/f) for summarizing the characteristics of queueing situation, the letters ‘b’ and ‘d’ stand respectively for

For the linear programming problem:
Maximize   Z = 3X1 + 2X2
Subject to
– 2X1 + 3X2 ≤ 9
X1 – 5X2 ≥ – 20
X1, X2 ≥ 0

The above problem has

The jobs arrive at a facility, for service, in a random manner. The probability distribution of number of arrivals of jobs in a fixed time interval is

Jobs arrive at a facility at an average rate of 5 in an 8 hour shift. The arrival of the jobs follows Poisson distribution. The average service time of a job on the facility is 40 minutes. The service time follows exponential distribution. Idle time (in hours) at the facility per shift will be

If there are m sources and n destinations in a transportation matrix, the total number of basic variables in a basic feasible solution is

A component can be produced by any of the four processes I, II, III and IV. The fixed cost and the variable cost for each of the processes are listed below. The most economical process for producing a batch of 100 pieces is

 Process Fixed cost (in Rs.) Variable cost per piece (in Rs.) I 20 3 II 50 1 III 40 2 IV 10 4

A project has four activities P, Q, R and S as shown below.

 Activity Normal duration (days) Predecessor Cost slope (Rs./day) P 3 - 500 Q 7 P 100 R 4 P 400 S 5 R 200

The normal cost of the project is Rs. 10,000/- and the overhead cost is Rs. 200/- per day. If the project duration has to be crashed down to 9 days, the total cost (in Rupees) of the project is _______

A minimal spanning tree in network flow models involves

Consider the given project network,where numbers along various activities represent the normal tim me. The free  float on activity 4-6 and the project duration, respectively, are

The total number of decision variables in the objective function of an assignment problem of size n × n (n jobs and n machines) is

The precedence relations and duration (in days) of activities of a project network are given in the table. The total float (in days) of activities e and f , respectively, are

 Activity Predecessors Duration (days) a - 2 b - 4 c a 2 d b 3 e c 2 f c 4 g d,e 5

At a work station, 5 jobs arrive every minute. The mean time spent on each job in the work station is 1/8 minute. The mean steady state number of jobs in the system is _______

Customers arrive at a ticket counter at a rate of 50 per hr and tickets are issued in the order of their arrival. The average time taken for issuing a ticket is 1 min. Assuming that customer arrivals form a Poisson process and service times are exponentially distributed, the average waiting time in queue in min is

A linear programming problem is shown below.
Maximize  3x + 7y
Subject to 3x + 7y ≤ 10
4x + 6y ≤ 8
x, y ≥ 0
It has

For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below.

 Activity Predecessors Duration (days) a - 3 b a 4 c a 5 d a 4 e b 2 f d 9 g c,e 6 h f,g 2

The critical path for the project is

For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below.

 Activity Predecessors Duration (days) a - 3 b a 4 c a 5 d a 4 e b 2 f d 9 g c,e 6 h f,g 2

If the duration of activity f alone is changed from 9 to 10 days, then the

Cars arrive at a service station according to Poisson’s distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is

One unit of product P1 requires 3 kg of resource R1 and 1 kg of resource R2. One unit of product P2 requires 2 kg of resource R1 and 2 kg of resource R2. The profits per unit by selling product P1 and P2 are Rs. 2000 and Rs. 3000 respectively. The manufacturer has 90 kg of resource R1 and 100kg of resource R2

The unit worth of resource R2, i.e., dual price of resource R2 in Rs. Per kg is

One unit of product P1 requires 3 kg of resource R1 and 1 kg of resource R2. One unit of product P2 requires 2 kg of resource R1 and 2 kg of resource R2. The profits per unit by selling product P1 and P2 are Rs. 2000 and Rs. 3000 respectively. The manufacturer has 90 kg of resource R1 and 100kg of resource R2

The manufacturer can make a maximum profit of Rs.

Little’s law is relationship between

Simplex method of solving linear programming problem uses

The project activities, precedence relationships and durations are described in the table. The critical path of the project is

 Activity Precedence Duration (in days) P - 3 Q - 4 R P 5 S Q 5 T R,S 7 U R,S 5 V T 2 W U 10

Four jobs are to be processed on a machine as per data listed in the table.

 Job Processing time (in days) Due date 1 4 6 2 7 9 3 2 19 4 8 17

If the Earliest Due Date (EDD) rule is used to sequence the jobs, the number of jobs delayed is

Four jobs are to be processed on a machine as per data listed in the table.

 Job Processing time (in days) Due date 1 4 6 2 7 9 3 2 19 4 8 17

Using the Shortest Processing Time (SPT) rule, total tardiness is

The expected time (${t}_{e}$) of a PERT activity in terms of optimistic time (${t}_{o}$), pessimistic time (${t}_{p}$) and most likely time (${t}_{l}$) is given by

Consider the following Linear Programming Problem (LPP):

Maximize z = 3x1 + 2x2
Subject to x1 ≤ 4
x2 ≤ 6
3x1 + 2x2 ≤ 18
x1 ≥ 0, x2 ≥ 0

Six jobs arrived in a sequence as given below:

 Jobs Processing Time (days) I 4 II 9 III 5 IV 10 V 6 VI 8

Average flow time (in days) for the above jobs using Shortest Processing Time rule is

Consider the following PERT network:

The optimistic time, most likely time and pessimistic time of all the activities are given in the table below

 Activity Optimistic time (days) Most likely time (days) Pessimistic time (days) 1-2 1 2 3 1-3 5 6 7 1-4 3 5 7 2-5 5 7 9 3-5 2 4 6 5-6 4 5 6 4-7 4 6 8 6-7 2 3 4

The critical path duration of the network (in days) is

Consider the following PERT network:

The optimistic time, most likely time and pessimistic time of all the activities are given in the table below

 Activity Optimistic time (days) Most likely time (days) Pessimistic time (days) 1-2 1 2 3 1-3 5 6 7 1-4 3 5 7 2-5 5 7 9 3-5 2 4 6 5-6 4 5 6 4-7 4 6 8 6-7 2 3 4

The standard deviation of the critical path is

In an M/M/1 queuing system, the number of arrivals in an interval of length T is a Poisson random variable (i.e. the probability of there being n arrivals in an interval of length T is $\frac{{e}^{-\lambda T}{\left(\lambda T\right)}^{n}}{n!}$). The probability density function f(t) of the inter-arrival time is given by

For the standard transportation linear programme with m sources and n destinations and total supply equaling total demand, an optimal solution (lowest cost) with the smallest number of non-zero xij values (amounts from source i to destination j) is desired. The best upper bound for this number is

For the network below, the objective is to find the length of the shortest path from node P to node G. Let dij be the length of directed are from node i to node j.

Let  sj be the length of the shortest path from P to node j. Which of the following equations can be used to find sG ?

Consider the Linear Programme (LP)

Max 4x + 6y

subject to

3x + 2y ≤ 6

2x +3y ≤ 6

x,y ≥ 0

After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0).

 -4 -6 0 0 0 s 3 2 1 0 6 t 2 3 0 1 6 x y s t RHS

After some simplex iterations, the following table is obtained

 0 0 0 2 12 s 5/3 0 1 -1/3 2 y 2/3 1 0 1/3 2 x y s t RHS

From this, one can conclude that

Consider the Linear Programme (LP)

Max 4x + 6y

subject to

3x + 2y ≤ 6

2x +3y ≤ 6

x,y ≥ 0

The dual for the LP in Q 74 is