This document contains a question bank with multiple choice questions related to operations research. It covers topics in linear programming problems (LPP) such as the definition of an LPP, feasible solutions, graphical representation, simplex method, objective functions, constraints, unbounded/bounded solutions, degenerate solutions, and slack and surplus variables. There are 27 total multiple choice questions across two units related to modeling problems as LPPs and using the simplex method to solve LPPs.

1.
Question
Question Bank
Bank US05FBCA01-
US05FBCA01- Operations
Operations Research
Research
Page
Page 1
1 of
of 24
24
TYBCA
TYBCA
US05FBCA01 – OPERATIONS RESEARCH
US05FBCA01 – OPERATIONS RESEARCH
QUESTION BANK
QUESTION BANK
MULTIPLE
MULTIPLE CHOICE QUESTIONS
CHOICE QUESTIONS
UNIT 1
UNIT 1
1.
1. Operations research is the application of ____________methods to arrive at the optimal
Operations research is the application of ____________methods to arrive at the optimal
solutions to the problems.
solutions to the problems.
A.
A. economical
economical C.
C. a
a and
and b
b both
both
B.
B. scientific
scientific
D. artistic
D. artistic
2.
2. Feasible solution satisfies __________
Feasible solution satisfies __________
A.
A. Only
Only constraints
constraints C.
C.
[a] and [b] both
[a] and [b] both
B.
B. only
only non-negative
non-negative restriction
restriction D.
D. [a],[b]
[a],[b] and
and Optimum
Optimum solution
solution
3.
3. In Degenerate solution value of objective function _____________.
In Degenerate solution value of objective function _____________.
A.
A. increases
increases infinitely
infinitely C.
C. basic
basic variables
variables are
are nonzero
nonzero
B.
B. decreases
decreases infinitely
infinitely D.
D. One or more
One or more basic variables are
basic variables are zero
zero
4.
4. Minimize Z =
Minimize Z = ______________
______________
A.
A. –maximize(Z)
–maximize(Z) C.
C. -maximize(-Z)
-maximize(-Z)
B.
B. maximize(-Z)
maximize(-Z) D.
D. none
none of
of the
the above
above
5.
5. In graphical method the restriction on number of
In graphical method the restriction on number of constraint is __________.
constraint is __________.
A.
A. 2
2 C.
C. not
not more
more than
than 3
3
B.
B. 3
3 D.
D. none of the above
none of the above
6.
6. In graphical representation the bounded region is known as _________ region.
In graphical representation the bounded region is known as _________ region.
A.
A. Solution
Solution C.
C. basic
basic solution
solution
B.
B. feasible solution
feasible solution
D. optimal
D. optimal
7.
7. Graphical optimal value
Graphical optimal value for Z
for Z can be obtained from
can be obtained from
A.
A. Corner points of feasible region
Corner points of feasible region
C.
C. Both
Both a
a and
and c
c
B.
B. corner
corner points
points of
of the
the solution
solution region
region D.
D. none
none of
of the
the above
above
8.
8. In LPP the condition to be satisfied
In LPP the condition to be satisfied is
is
A.
A. Constraints
Constraints have
have to
to be
be linear
linear C.
C. both [a ]and [b]
both [a ]and [b]
B.
B. Objective
Objective function
function have
have to
to be
be linear
linear D.
D. none
none of
of the
the above
above
9.
9. The solution to LPP give below is,
The solution to LPP give below is,
1
1 2
2 1
1 2
2 1
1 2
2 1
1 2
2
3
3 1
14
4 1
1,
, 2
2 ,
, 0
0
Max
Max Z
Z x
x x
x subject
subject to
to x
x x
x x
x x
x where
where x
x x
x
=
= +
+ −
− ≥
≥ −
− +
+ ≥
≥ ≥
≥
A.
A. Unbounded
Unbounded solution
solution C.
C. Max
Max Z
Z =
= 3
3
B.
B. Max
Max Z
Z =
= 14
14 D.
D. Infeasible solution
Infeasible solution
10.
10. The solution to LPP give below is ______________
The solution to LPP give below is ______________
1
1 2
2 1
1 2
2 1
1 2
2 1
1 2
2
3
30
0 1
15
5 2
2 2 2
2 2,
, 2
2 2 2
2 2 , 0
, 0
Max
Max Z
Z x
x x
x subject
subject to
to x
x x
x x
x x
x where
where x
x x
x
=
= −
− −
− ≤
≤ −
− +
+ ≤
≤ ≥
≥
A.
A. Unbounded solution
Unbounded solution
C.
C. Max
Max Z
Z =
= 15
15
B.
B. Max
Max Z
Z =
= 30
30 D.
D. Infeasible
Infeasible solution
solution
11.
11. In the definition of LPP
In the definition of LPP m stands for number
m stands for number of constraints and n
of constraints and n for number
for number of
of
variables, then which of the following relations hold
variables, then which of the following relations hold
A.
A. m
m =
= n
n C.
C. m
m ≥
≥ n
n
B. m
B. m ≤
≤
n
n D.
D. None
None of
of them
them

2.
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US05FBCA01- Operations
Operations Research
Research
Page
Page 2
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12.
12. The linear function of variables which is to be maximized or minimized is called
The linear function of variables which is to be maximized or minimized is called
________
________
A.
A. constraints
constraints C.
C. objective function
objective function
B. basic
B. basic requirements
requirements D.
D. none
none of
of them
them
EXTRA
EXTRA
13.
13. Operation research approach is
Operation research approach is
A.
A. Multi-disciplinary
Multi-disciplinary
C.
C. Intuitive
Intuitive
B.
B. Artificial
Artificial D.
D. All of the above
All of the above
14.
14. Operation research analysts do not
Operation research analysts do not
A.
A. Predict future operation
Predict future operation
C.
C. Collect
Collect the
the relevant
relevant data
data
B.
B. Build
Build more
more than
than one
one model
model D.
D. Recommend
Recommend decision
decision and
and accept
accept
15.
15. Mathematical model of Linear Programming is important because
Mathematical model of Linear Programming is important because
A.
A. It helps in converting the verbal description and numerical data into
It helps in converting the verbal description and numerical data into
mathematical expression
mathematical expression
B.
B. decision
decision makers
makers prefer
prefer to
to work
work with
with formal
formal models
models
C.
C. it
it captures
captures the
the relevant
relevant relationship
relationship among
among decision
decision factors
factors
D.
D. it
it enables
enables the
the use
use of
of algebraic
algebraic techniques
techniques
16.
16. A constraint in an LP model restricts
A constraint in an LP model restricts
A.
A. value
value of
of the
the objective
objective function
function C.
C. use
use of
of the
the available
available resources
resources
B.
B. value of the decision variable
value of the decision variable
D.
D. all
all of
of the
the above
above
17.
17. In graphical method of linear
In graphical method of linear programming problem if the ios-cost line coincide with a
programming problem if the ios-cost line coincide with a
side of region of basic feasible solutions we get
side of region of basic feasible solutions we get
A.
A. Unique
Unique optimum
optimum solution
solution C. no
C. no feasible
feasible solution
solution
B.
B. unbounded
unbounded optimum
optimum solution
solution D.
D. Infinite number of optimum
Infinite number of optimum
solutions
solutions
18.
18. A feasible solution of LPP
A feasible solution of LPP
A.
A. Must satisfy all the constraints simultaneously
Must satisfy all the constraints simultaneously
B.
B. Need
Need not
not satisfy
satisfy all
all the
the constraints,
constraints, only
only some
some of
of them
them
C.
C. Must
Must be
be a
a corner
corner point
point of
of the
the feasible
feasible region
region
D.
D. all
all of
of the
the above
above
19.
19. The objective function for a L.P model is 3x
The objective function for a L.P model is 3x1
1+2x
+2x2
2, if x
, if x1
1=20 and x
=20 and x2
2=30, what is the value
=30, what is the value
of the objective function?
of the objective function?
A.
A. 0
0 C.
C. 60
60
B.
B. 50
50 D.
D. 120
120
20.
20. Maximization of objective function in LPP means
Maximization of objective function in LPP means
A.
A. Value
Value occurs
occurs at
at allowable
allowable set
set decision
decision
B.
B. highest value is chosen among allowable decision
highest value is chosen among allowable decision
C.
C. none
none of
of the
the above
above
D.
D. all
all of
of the
the above
above
21.
21. Alternative solution exist in a linear programming problem when
Alternative solution exist in a linear programming problem when
A.
A. one
one of
of the
the constraint
constraint is
is redundant
redundant
B.
B. objective function is parallel to one of the constraints
objective function is parallel to one of the constraints
C.
C. two
two constraints
constraints are
are parallel
parallel
D.
D. all
all of
of the
the above
above

3.
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Operations Research
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22.
22. Linear programming problem involving only two variables can be solved by
Linear programming problem involving only two variables can be solved by
______________
______________
A.
A. Big
Big M
M method
method C.
C. Graphical method
Graphical method
B.
B. Simplex
Simplex method
method D.
D. none
none of
of these
these
23.
23. The linear function of the variables which is to be maximize or minimize is called
The linear function of the variables which is to be maximize or minimize is called
_________
_________
A.
A. Constraints
Constraints C.
C. Decision
Decision variable
variable
B.
B. Objective function
Objective function
D.
D. None
None of
of the
the abov
abov
24.
24. A physical model is an example of
A physical model is an example of
A.
A. An iconic model
An iconic model
C.
C. A
A verbal
verbal model
model
B.
B. An
An analogue
analogue model
model D.
D. A
A mathematical
mathematical model
model
25.
25. If the value of the objective function z can be increased or decreased indefinitely, such
If the value of the objective function z can be increased or decreased indefinitely, such
solution is called _____________
solution is called _____________
A.
A. Bounded
Bounded solution
solution C.
C. Solution
Solution
B.
B. Unbounded solution
Unbounded solution
D.
D. None
None of
of the
the above
above
26.
26. A model is
A model is
A.
A. An
An essence
essence of
of reality
reality C.
C. An
An idealization
idealization
B.
B. An
An approximation
approximation D.
D. All of the above
All of the above
27.
27. The first step in formulating a
The first step in formulating a linear programming problem is
linear programming problem is
A.
A. Identify
Identify any
any upper
upper or
or lower
lower bound
bound on
on the
the decision
decision variables
variables
B.
B. State
State the
the constraints
constraints as
as linear
linear combinations
combinations of
of the
the decision
decision variables
variables
C.
C. Understand
Understand the
the problem
problem
D.
D. Identify the decision variables
Identify the decision variables
UNIT 2
UNIT 2
1.
1. In the simplex method for solving of
In the simplex method for solving of LPP number of variables can be ______________.
LPP number of variables can be ______________.
A.
A. Not
Not more
more than
than three
three C.
C. at least two
at least two
B. at
B. at least
least three
three D.
D. none
none of
of them
them
2.
2. In the simplex method the variable enters t
In the simplex method the variable enters the basis if __________________.
he basis if __________________.
A.
A. Z
Z j
j – C
– C j
j
≥
≥
0
0 C.
C. Z
Zj
j – C
– Cj
j < 0
< 0
B.
B. Z
Z j
j – C
– C j
j
≤
≤
0
0 D.
D. Z
Z j
j – C
– C j
j = 0
= 0
3.
3. In the simplex method the variable leaves the
In the simplex method the variable leaves the basis if the ratio is
basis if the ratio is
A.
A. maximum
maximum C.
C. 0
0
B.
B. minimum
minimum D.
D. none
none of
of them
them
4.
4. The _________ variable is added to the constraint of less t
The _________ variable is added to the constraint of less than equal to type.
han equal to type.
A.
A. slack
slack
C. artificial
C. artificial
B.
B. surplus
surplus D.
D. basic
basic
5.
5. For the constraint of greater than equal to type we make use of ______________
For the constraint of greater than equal to type we make use of ______________
variable.
variable.
A.
A. slack
slack C.
C. artificial
artificial
B.
B. surplus
surplus
D.
D. basic
basic
6.
6. The coefficient of slack variable in the objective function
The coefficient of slack variable in the objective function is _________________.
is _________________.
A.
A. -M
-M C.
C. 0
0
B.
B. +M
+M D.
D. none
none of
of them
them

4.
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Operations Research
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7.
7. The coefficient of artificial variable in the objective function of maximization problem is
The coefficient of artificial variable in the objective function of maximization problem is
_________.
_________.
A.
A. -M
-M
C.
C. 0
0
B.
B. +M
+M D.
D. none
none of
of them
them
EXTRA
EXTRA
8.
8. The role of artificial variables in the
The role of artificial variables in the simplex method is
simplex method is
A.
A. to
to aid
aid in
in finding
finding an
an initial
initial solution
solution
B.
B. to
to find
find optimal
optimal dual
dual prices
prices in
in the
the final
final simplex
simplex table
table
C.
C. to
to start
start with
with Big
Big M
M method
method
D.
D. all of these
all of these
9.
9. For a minimization problem, the objective function coeff
For a minimization problem, the objective function coefficient for an artificial variable is
icient for an artificial variable is
A.
A. + M
+ M C. Zero
C. Zero
B.
B. -M
-M D.
D. None
None of
of these
these
10.
10. For maximization LPP, the simplex method is terminated
For maximization LPP, the simplex method is terminated when all values
when all values
A.
A. cj –zj
cj –zj ≤
≤ 0
0
C.
C. cj
cj –zj
–zj =
= 0
0
B.
B. cj
cj –zj
–zj ≥
≥
0
0 D.
D. zj
zj ≤
≤ 0
0
11.
11. If any value in b
If any value in b - column of final simplex
- column of final simplex table is negative, then the solu
table is negative, then the solution is
tion is
A.
A. unbounded
unbounded C.
C. optimal
optimal
B.
B. infeasible
infeasible
D.
D. None
None of
of these
these
12.
12. To convert
To convert ≥
≥ inequality constraints into equality constraints, we must
inequality constraints into equality constraints, we must
A.
A. add
add a
a surplus
surplus variable
variable
B.
B. subtract
subtract an
an artificial
artificial variable
variable
C.
C. subtract a surplus variable and an add artificial variable
subtract a surplus variable and an add artificial variable
D.
D. add
add a
a surplus
surplus variable
variable and
and subtract
subtract an
an artificial
artificial variable
variable
13.
13. In the optimal simplex table cj
In the optimal simplex table cj –zj = 0 value indicates
–zj = 0 value indicates
A.
A. unbounded
unbounded solution
solution C.
C. alternative solution
alternative solution
B.
B. cycling
cycling D.
D. None
None of
of these
these
14.
14. At every iteration of simplex method, for minimization problem, a variable in the current
At every iteration of simplex method, for minimization problem, a variable in the current
basis is replaced with another variable that has
basis is replaced with another variable that has
A.
A. a
a positive
positive cj
cj –zj
–zj value
value C.
C. cj
cj –zj
–zj =
= 0
0
B.
B. a negative cj –zj value
a negative cj –zj value
D.
D. None
None of
of these
these
15.
15. A variable which does not appear in the basis variable (B) column of
A variable which does not appear in the basis variable (B) column of simplex table is
simplex table is
A.
A. never
never equal
equal to
to zero
zero C.
C. called
called basic
basic variable
variable
B.
B. always equal to zero
always equal to zero
D.
D. None
None of
of these
these
16.
16. To formulate a problem for solution by the simplex method, we must add artificial
To formulate a problem for solution by the simplex method, we must add artificial
variable to
variable to
A.
A. only
only equality
equality constraints
constraints C.
C. both A & B
both A & B
B.
B. only
only >
> constraints
constraints D.
D. None
None of
of these
these
17.
17. If all xij values in t
If all xij values in the incoming variable column of the simplex table are
he incoming variable column of the simplex table are negative, then
negative, then
A.
A. solution is unbounded
solution is unbounded
C.
C. there
there exist
exist no
no solution
solution
B.
B. there
there are
are multiple
multiple solution
solution D.
D. None
None of
of these
these
18.
18. If an artificial variable is present in the basic variable column of optimal simplex table,
If an artificial variable is present in the basic variable column of optimal simplex table,
then the solution is
then the solution is
A.
A. unbounded
unbounded C.
C. optimal
optimal
B.
B. infeasible
infeasible
D.
D. None
None of
of these
these

5.
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5 of
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19.
19. If for a given solution,
If for a given solution, a slack variable is equal to zero, then
a slack variable is equal to zero, then
A.
A. the solution is optimal
the solution is optimal
C.
C. there
there exist
exist no
no solution
solution
B.
B. the
the solution
solution is
is infeasible
infeasible D.
D. None
None of
of these
these
20.
20. Linear programming problem involving more than two variables can be solved by
Linear programming problem involving more than two variables can be solved by
______________
______________
A.
A. Simplex
Simplex method
method C.
C. Matrix
Matrix minima
minima method
method
B. Graphical
B. Graphical method
method D.
D. None
None of
of these
these
UNIT 3
UNIT 3
1.
1. The TP
The TP is said
is said to be un
to be unbalanced if
balanced if _______________.
_______________.
A.
A.
≠
≠
∑
∑ ∑
∑
i
i j
j
a
a b
b
C. m=n
C. m=n
B.
B.
∑
∑ ∑
∑
i
i j
j
a
a =
= b
b
D.
D. m+n-1=
m+n-1= no.
no. of
of allocated
allocated cell
cell
2.
2. The initial solution of a transportation problem can be obtained by applying any known
The initial solution of a transportation problem can be obtained by applying any known
method. However, the only condition is that
method. However, the only condition is that
A.
A. Rim condition should be satisfied
Rim condition should be satisfied
C.
C. one
one of
of the
the X
Xij
ij < 0
< 0
B.
B. cost
cost matrix
matrix should
should be
be square
square D. None
D. None of
of them
them
3.
3. In non-degenerate solution number of allocated cell is____________.
In non-degenerate solution number of allocated cell is____________.
A.
A. Equal
Equal to
to m+n-1
m+n-1 C.
C. Equal
Equal to
to m+n+1
m+n+1
B.
B. Not equal to m+n-1
Not equal to m+n-1
D.
D. Not
Not equal
equal to
to m+n+1
m+n+1
4.
4. From the following methods ___________ is a method to obtain initial solution to
From the following methods ___________ is a method to obtain initial solution to
Transportation Problem.
Transportation Problem.
A.
A. North-West
North-West
C. Hungarian
C. Hungarian
B.
B. Simplex
Simplex D.
D. Newton
Newton Raphson
Raphson
5.
5. The Penalty in VAM represents difference between _________ cost of respective
The Penalty in VAM represents difference between _________ cost of respective
row / column.
row / column.
A.
A. Two
Two Largest
Largest C.
C. largest
largest and
and smallest
smallest
B.
B. smallest two
smallest two
D.
D. none
none of
of them
them
6.
6. Number of basic allocation in any row or column
Number of basic allocation in any row or column in Assignment Problem can be
in Assignment Problem can be
A.
A. Exactly one
Exactly one
C.
C. at
at least
least one
one
B.
B. at
at most
most one
one D.
D. none
none of
of them
them
7.
7. North – West corner refers to ____________.
North – West corner refers to ____________.
A.
A. top left corner
top left corner
C.
C. both
both of
of them
them
B. top
B. top right
right corner
corner D.
D. none
none of
of them
them
8.
8. The ______________ method's solution for transportation problem is sometimes an
The ______________ method's solution for transportation problem is sometimes an
optimal solution itself.
optimal solution itself.
A.
A. NWCM
NWCM C.
C. LCM
LCM
B.
B. VAM
VAM
D.
D. Row
Row Minima
Minima
9.
9. In Assignment Problem the value of decision variable x
In Assignment Problem the value of decision variable xij
ij is____.
is____.
A.
A. no
no restriction
restriction C.
C. one or zero
one or zero
B.
B. two
two or
or one
one D.
D. none
none of
of them
them

6.
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10.
10. If number of sources is not equal t
If number of sources is not equal to number of destination in Assignment problem then
o number of destination in Assignment problem then it
it
is called ___________.
is called ___________.
A.
A. unbalanced
unbalanced
C. unsymmetric
C. unsymmetric
B.
B. symmetric
symmetric D.
D. balanced
balanced
11.
11. The _____ method used to obtain optimum
The _____ method used to obtain optimum solution of travelling salesman problem.
solution of travelling salesman problem.
A.
A. Simplex
Simplex C.
C. dominance
dominance
B.
B. Hungarian
Hungarian
D. graphical
D. graphical
EXTRA
EXTRA
12.
12. The initial solution of a transportation problem can be obtained by applying any known
The initial solution of a transportation problem can be obtained by applying any known
method. However, the only condition is that
method. However, the only condition is that
A.
A. the
the solution
solution be
be optimal
optimal
B.
B. the rim condition are satisfied
the rim condition are satisfied
C.
C. the
the solution
solution not
not be
be degenerate
degenerate
D.
D. all
all of
of the
the above
above
13.
13. The dummy source or destination in a transportation problem is added t
The dummy source or destination in a transportation problem is added to
o
A.
A. satisfy rim condition
satisfy rim condition
B.
B. prevent
prevent solution
solution from
from becoming
becoming degenerate
degenerate
C.
C. ensure
ensure that
that total
total cost
cost does
does not
not exceed
exceed a
a limit
limit
D.
D. all
all of
of the
the above
above
14.
14. The occurrence of degeneracy while solving a transportation problem means that
The occurrence of degeneracy while solving a transportation problem means that
A.
A. total
total supply
supply equals
equals total
total demand
demand
B.
B. the solution so obtained is not feasible
the solution so obtained is not feasible
C.
C. the
the few
few allocations
allocations become
become negative
negative
D.
D. none
none of
of the
the above
above
15.
15. An alternative optimal solution to a minimization transportation problem exists whenever
An alternative optimal solution to a minimization transportation problem exists whenever
opportunity cost corresponding to unused routes of transportation is:
opportunity cost corresponding to unused routes of transportation is:
A.
A. positive
positive and
and greater
greater than
than zero
zero
B.
B. positive with at least one equal to zero
positive with at least one equal to zero
C.
C. negative
negative with
with at
at least
least one
one equal
equal to
to zero
zero
D.
D. all
all of
of the
the above
above
16.
16. One disadvantage of using North-West Corner Rule to find initial solution to the
One disadvantage of using North-West Corner Rule to find initial solution to the
transportation problem is that
transportation problem is that
A.
A. it
it is
is complicated
complicated to
to use
use
B.
B. it does not take into account cost
it does not take into account cost of transportation
of transportation
C.
C. it
it leads
leads to
to degenerate
degenerate initial
initial solution
solution
D.
D. all
all of
of the
the above
above
17.
17. The solution to a transportation problem with m-rows and n-columns is feasible if
The solution to a transportation problem with m-rows and n-columns is feasible if
number of positive allocations are
number of positive allocations are
A.
A. m+n
m+n C.
C. m + n -1
m + n -1
B.
B. m
m x
x n
n D.
D. all
all of
of the
the above
above
18.
18. The calculation of opportunity cost in the MODI
The calculation of opportunity cost in the MODI method is analogous to a
method is analogous to a
A.
A. cj –zj value for non-basic
cj –zj value for non-basic variable columns in the simplex method
variable columns in the simplex method
B.
B. value
value of
of a
a variable
variable in
in b-column
b-column of
of the
the simplex
simplex method
method
C.
C. variable
variable in
in xb-column
xb-column
D.
D. all
all of
of the
the above
above

7.
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Operations Research
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7 of
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24
19.
19. If we were to use opportunity cost value f
If we were to use opportunity cost value for an unused cell to test optimality,
or an unused cell to test optimality, it should be
it should be
A.
A. equal
equal to
to zero
zero C.
C. most
most positive
positive number
number
B.
B. most negative number
most negative number
D.
D. all
all of
of the
the above
above
20.
20. An assignment problem is considered as a particular case of a Transportation problem
An assignment problem is considered as a particular case of a Transportation problem
because
because
A.
A. the
the number
number of
of rows
rows equals
equals columns
columns
B.
B. all
all x
xij
ij= 0
= 0
C.
C. all
all rim
rim conditions
conditions are
are 1
1
D.
D. all of above
all of above
21.
21. The purpose of a dummy row or column in
The purpose of a dummy row or column in an assignment problem is to
an assignment problem is to
A.
A. obtain balance between total activities and total resources
obtain balance between total activities and total resources
B.
B. prevent
prevent a
a solution
solution from
from becoming
becoming degenerate
degenerate
C.
C. provide
provide a
a means
means of
of representing
representing a
a dummy
dummy problem
problem
D.
D. none
none of
of the
the above
above
22.
22. The
The Hungarian method for solving an assi
Hungarian method for solving an assignment problem can also be used to solve
gnment problem can also be used to solve
A.
A. a
a transportation
transportation problem
problem C. both
C. both A
A and
and B
B
B.
B. a traveling salesman problem
a traveling salesman problem
D.
D. only
only B
B
23.
23. An optimal of an assignment problem can be
An optimal of an assignment problem can be obtained only if
obtained only if
A.
A. each
each row
row and
and column
column has
has only
only one
one zero
zero element
element
B.
B. each
each row
row and
and column
column has
has at
at least
least one
one zero
zero element
element
C.
C. the
the data
data are
are arrangement
arrangement in
in a
a square
square matrix
matrix
D.
D. none of the above
none of the above
24.
24. The method used for so
The method used for solving an assignment probl
lving an assignment problem is
em is called
called
A. reduced
A. reduced matrix
matrix method
method C.
C. Hungarian method
Hungarian method
B. MODI
B. MODI method
method D.
D. none
none of
of the
the above
above
UNIT 4
UNIT 4
1.
1. Dynamic programming is a mathematical technique dealing with the optimization of
Dynamic programming is a mathematical technique dealing with the optimization of
_______ stage decision process
_______ stage decision process.
.
A.
A. multi
multi
C.
C. both
both A
A and
and B
B
B.
B. single
single D.
D. none
none of
of them
them
2.
2. In sequencing if smallest time for a job belongs to machine-1 then that job has to placed
In sequencing if smallest time for a job belongs to machine-1 then that job has to placed
___________ of the sequence.
___________ of the sequence.
A.
A. in
in the
the middle
middle C.
C. in the starting
in the starting
B.
B. at
at end
end D.
D. none
none of
of them
them
3.
3. In sequencing the time involved in moving jobs from one machine to another is
In sequencing the time involved in moving jobs from one machine to another is
________.
________.
A.
A. negligible
negligible
C.
C. positive
positive number
number
B.
B. significant
significant D.
D. none
none of
of them
them
4.
4. ___________ operation is carried ou
___________ operation is carried out on a machine at a time.
t on a machine at a time.
A.
A. Two
Two C.
C. atleast
atleast one
one
B.
B. only one
only one
D.
D. none
none of
of them
them
5.
5. Processing
Processing time
time M
Mij
ij’s are __________of order of processing the jobs.
’s are __________of order of processing the jobs.
A.
A. dependent
dependent C.
C. negligible
negligible
B.
B. independent
independent D.
D. none
none of
of them
them

8.
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Operations Research
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24
6.
6. Activity which starts only after finishing other activity is called _____________.
Activity which starts only after finishing other activity is called _____________.
A.
A. dummy
dummy C.
C. successor
successor
B.
B. Predecessor
Predecessor D.
D. none
none of
of them
them
7.
7. Burst and Merge are types of ___________ in networking.
Burst and Merge are types of ___________ in networking.
A.
A. event
event C. arrow
C. arrow
B.
B. activity
activity D.
D. tools
tools
8.
8. Activity which does not require any resources or time is called __________.
Activity which does not require any resources or time is called __________.
A.
A. dummy
dummy
C. successor
C. successor
B.
B. Predecessor
Predecessor D.
D. none
none of
of them
them
9.
9. Event indicates ____________ of activity.
Event indicates ____________ of activity.
A.
A. starting
starting C.
C. both A and B
both A and B
B.
B. ending
ending D.
D. none
none of
of them
them
10.
10. _____________ is indicated by dotted arrow
_____________ is indicated by dotted arrow.
.
A.
A. burst
burst event
event C.
C. dummy activity
dummy activity
B.
B. merge
merge event
event D.
D. none
none of
of them
them
11.
11. ______ event represents beginnin
______ event represents beginning of more than one activities.
g of more than one activities.
A.
A. burst
burst
C.
C. dummy
dummy
B.
B. merge
merge D.
D. none
none of
of them
them
12.
12. Merge event represents ___________ of two or more events.
Merge event represents ___________ of two or more events.
A.
A. beginning
beginning C.
C. splitting
splitting
B.
B. completion
completion
D.
D. none
none of
of them
them
13.
13. Activity which is completed before starting new activity is called___________.
Activity which is completed before starting new activity is called___________.
A.
A. dummy
dummy C.
C. successor
successor
B.
B. predecessor
predecessor
D.
D. none
none of
of them
them
EXTRA
EXTRA
14.
14. The Objective of network analysis to
The Objective of network analysis to
A.
A. Minimize total project duration
Minimize total project duration
B.
B. Minimize
Minimize total
total project
project cost
cost
C.
C. Minimize
Minimize production
production delays,
delays, interruption
interruption and
and conflicts
conflicts
D.
D. All
All of
of the
the above
above
15.
15. Network models have advantage in terms of project
Network models have advantage in terms of project
A.
A. Planning
Planning C.
C. Controlling
Controlling
B.
B. Scheduling
Scheduling D.
D. All of the above
All of the above
16.
16. The slack for an activity is equal t
The slack for an activity is equal to
o
A.
A. LF-LS
LF-LS C.
C. LS-ES
LS-ES
B.
B. EF-ES
EF-ES D.
D. None
None of
of the
the above
above
17.
17. The Another term commonly used for activity slack time is
The Another term commonly used for activity slack time is
A.
A. Total
Total float
float C.
C. independent
independent float
float
B.
B. Free
Free float
float D.
D. All of the above
All of the above
18.
18. If an activity has zero slack, it
If an activity has zero slack, it implies that
implies that
A.
A. It lies on the critical path
It lies on the critical path
B.
B. It
It is
is a
a dummy
dummy activity
activity
C.
C. The
The project
project progressing
progressing well
well
D.
D. None
None of
of the
the above
above

9.
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9 of
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24
19.
19. A dummy activity is used in the net
A dummy activity is used in the network diagram when
work diagram when
A.
A. Two
Two parallel
parallel activities
activities have
have the
the same
same tail
tail and
and head
head events
events
B.
B. The chai
The chain of
n of activities
activities may hav
may have a
e a common e
common event y
vent yet be
et be independent
independent by
by
themselves
themselves
C.
C. Both A and B
Both A and B
D.
D. None
None of
of the
the above
above
20.
20. While drawing the network diagram , for each activity project,
While drawing the network diagram , for each activity project, we should look
we should look
A.
A. What activities precede this activity?
What activities precede this activity?
B.
B. What
What activities
activities follow
follow this
this activity?
activity?
C.
C. What
What activities
activities can
can take
take place
place concurrently
concurrently with
with this
this activity?
activity?
D.
D. All
All of
of the
the above
above
21.
21. The critical path satisfy the condition that
The critical path satisfy the condition that
A.
A. E
Ei
i = L
= Li
i & E
& E j
j = L
= Lj
j
C.
C. L
L j
j-E
-Ei
i=L
=Li
i-E
-E
j
j=c(constant)
=c(constant)
B.
B. L
L
j
j-E
-Ei
i=L
=Li
i-L
-L j
j
D.
D. All
All of
of the
the above
above
22.
22. If there are in jobs to be performed, one at a time, on each of m machines, the possible
If there are in jobs to be performed, one at a time, on each of m machines, the possible
sequences would be
sequences would be
A.
A. (n!)
(n!)m
m
C. (n)
C. (n)m
m
B. (m!)
B. (m!)n
n
D.
D. (m)
(m)n
n
23.
23. Total elapsed time to process all jobs through t
Total elapsed time to process all jobs through two machine is given by
wo machine is given by
A.
A. n
n n
n
Σ
Σ
M
M1j
1j +
+ Σ
Σ
M
M2j
2j
j=1
j=1 j=1
j=1
C.
C. n
n
Σ
Σ
(M
(M1j
1j + I
+ I1j
1j)
)
j=1
j=1
B.
B. n
n n
n
Σ
Σ
M
M2j
2j +
+ Σ
Σ
M
M1j
1j
j=1 j=1
j=1 j=1
D.
D. None
None of
of the
the above
above
24.
24. The minimum processing time on machine M
The minimum processing time on machine M1
1 and M
and M2
2 are related as
are related as
A.
A. Min
Min t
t1j
1j = Max t
= Max t2j
2j
C.
C. Min t
Min t1j
1j ≥
≥ Max t
Max t2j
2j
B.
B. Min
Min t
t1j
1j ≤
≤ Max t
Max t2j
2j
D.
D. Min
Min t
t2j
2j ≥
≥ Max t
Max t1j
1j
25.
25. You Would like to assign operators to the equipment t
You Would like to assign operators to the equipment that has
hat has
A.
A. Most
Most jobs
jobs waiting
waiting to
to be
be processed
processed
B.
B. Job
Job with
with the
the earliest
earliest due
due date
date
C.
C. Job
Job which
which has
has been
been waiting
waiting longest
longest
D.
D. All of the above
All of the above
SHORT QUESTIONS (Each of 2 Marks)
SHORT QUESTIONS (Each of 2 Marks)
UNIT 1
UNIT 1
1.
1. Define Operation research.
Define Operation research.
2.
2. Write down any two scopes of operation research.
Write down any two scopes of operation research.
3.
3. State phases for formulating the operations research problems.
State phases for formulating the operations research problems.
4.
4. Define
Define i)
i) Solution
Solution ii)
ii) Basic
Basic solution.
solution.
5.
5. Define
Define i)
i) Unbounded
Unbounded solution
solution ii)
ii) Optimum
Optimum solution
solution
6.
6. Define feasible solution.
Define feasible solution.
7.
7. Define LPP in the mathematical form.
Define LPP in the mathematical form.
8.
8. Give any four models of operations research.
Give any four models of operations research.
Extra
Extra
9.
9. What is Linear programming problem?
What is Linear programming problem?
10.
10. Write the advantages of LPP.
Write the advantages of LPP.
11.
11. Write the limitations of
Write the limitations of LPP
LPP
12.
12. How will you plot inequalities of
How will you plot inequalities of a LPP?
a LPP?

10.
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Operations Research
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10
10
of
of
24
24
UNIT 2
UNIT 2
1.
1. Define slack variables.
Define slack variables.
2.
2. Define surplus variables.
Define surplus variables.
3.
3. Define artificial variables.
Define artificial variables.
4.
4. When is Big M method useful
When is Big M method useful ?
?
5.
5. What is the condition for optimality in simplex table ?
What is the condition for optimality in simplex table ?
6.
6. What is the condition for entering
What is the condition for entering variable in simplex table ?
variable in simplex table ?
7.
7. Write the standard form of LPP for
Write the standard form of LPP for the following LPP:
the following LPP:
1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
M
M a
a x
x i
im
m i
iz
z e
e 1
1 3
3 2
2 5
5
S
S u
u b
b j
je
e c
c t
t t
to
o 2
2 1
1 3
3 4
4 0
0 ,
, 5
5 2
2 7
7 ,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
8.
8. Write the standard form of LPP for
Write the standard form of LPP for the following LPP:
the following LPP:
1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
M
M a
ax
x i
im
m i
iz
ze
e 3
3 5
5
S
S u
u b
b j
je
ec
ct
t t
to
o 2
2 3
3 4
4,
, 3
3 2
2 7
7 ,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≥
≥ ≥
≥
Extra
Extra
9.
9. What is NER while solving LPP by Simplex method?
What is NER while solving LPP by Simplex method?
10.
10. What is the criterion for the
What is the criterion for the entering variable and outgoing variable?
entering variable and outgoing variable?
11.
11. What is Replacement ratio while solving LPP by Simplex method
What is Replacement ratio while solving LPP by Simplex method
12.
12. What is the criterion for test of optimality?
What is the criterion for test of optimality?
13.
13. Represent the form of a simplex t
Represent the form of a simplex table.
able.
14.
14. How do you transform key row in a simplex table? Or How will you find the new solution
How do you transform key row in a simplex table? Or How will you find the new solution
in the Simplex table.
in the Simplex table.
15.
15. State the situation in simplex table
State the situation in simplex table which will represent
which will represent
(i) Unbounded solution (ii) unique optimum solution (iii) alternate
(i) Unbounded solution (ii) unique optimum solution (iii) alternate optimum solution.
optimum solution.
UNIT 3
UNIT 3
1.
1. What is transportation problem?
What is transportation problem?
2.
2. Write mathematical form of t
Write mathematical form of transportation problem.
ransportation problem.
3.
3. What is feasible solution and non degenerate solution in
What is feasible solution and non degenerate solution in transportation problem?
transportation problem?
4.
4. What do you mean by balanced transportation problem?
What do you mean by balanced transportation problem?
5.
5. What is the Assignment problem?
What is the Assignment problem?
6.
6. Give mathematical form of assignment problem.
Give mathematical form of assignment problem.
7.
7. What is travelling salesman problem?
What is travelling salesman problem?
Extra
Extra
8.
8. What is the difference between Assignment Problem and Transportation Problem?
What is the difference between Assignment Problem and Transportation Problem?
9.
9. Write steps for North-West Corner Method.
Write steps for North-West Corner Method.
10.
10. Write steps for Matrix Minima Method.
Write steps for Matrix Minima Method.
UNIT 4
UNIT 4
1.
1. State Bellman’s principle of optimality in
State Bellman’s principle of optimality in dynamic programming.
dynamic programming.
2.
2. In brief explain problem of sequencing.
In brief explain problem of sequencing.
3.
3. How will you allocate jobs in a sequence if two jobs on first machine have same
How will you allocate jobs in a sequence if two jobs on first machine have same
processing time?
processing time?
4.
4. Write down any two assumptions used for solving sequencing problem.
Write down any two assumptions used for solving sequencing problem.
5.
5. Define two types of events used in
Define two types of events used in network analysis.
network analysis.
6.
6. What is dummy activity?
What is dummy activity?
7.
7. What is total float?
What is total float?
8.
8. What is free float and
What is free float and independent float?
independent float?
9.
9. What is successor activity?
What is successor activity?

11.
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11
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24
24
Extra
Extra
10.
10. Define terms: Activity, Event, Merge Event, Burst Event, Total float, Free float,
Define terms: Activity, Event, Merge Event, Burst Event, Total float, Free float,
Independent
Independent float,
float, Critical
Critical path
path
11.
11. State Rules for Network Diagram.
State Rules for Network Diagram.
12.
12. Write disadvantages of Network t
Write disadvantages of Network techniques.
echniques.
13.
13. What is Dynamic Programming
What is Dynamic Programming Problem?
Problem?
14.
14. State the principle of optimality in
State the principle of optimality in dynamic programming.
dynamic programming.
15.
15. Write assumptions of sequencing
Write assumptions of sequencing problems.
problems.
LONG QUESTIONS (>=3 Marks)
LONG QUESTIONS (>=3 Marks)
UNIT 1
UNIT 1
1.
1. Explain the history of operations research.
Explain the history of operations research.
2.
2. Write the algorithm to
Write the algorithm to solve LPP using Graphical method for maximization of profit.
solve LPP using Graphical method for maximization of profit.
3.
3. Give the limitations of
Give the limitations of operations research.
operations research.
4.
4. Note down the applications of operations research.
Note down the applications of operations research.
5.
5. Write down meanings of operations research.
Write down meanings of operations research.
6.
6. A person requires at least 10 and 12 units of chemicals A and B respectively, for his
A person requires at least 10 and 12 units of chemicals A and B respectively, for his
garden. A liquid product contains 5 and 2 units of A and B respectively per bottle. A dry
garden. A liquid product contains 5 and 2 units of A and B respectively per bottle. A dry
product contains 1 and 4 units of A and B respectively per box. If the liquid product sales
product contains 1 and 4 units of A and B respectively per box. If the liquid product sales
for Rs. 30 per bottle, dry product sales for Rs. 40 per box. How many of each should be
for Rs. 30 per bottle, dry product sales for Rs. 40 per box. How many of each should be
purchased in order to minimize the cost and
purchased in order to minimize the cost and meet the requirements? Formulate the L.P.P.
meet the requirements? Formulate the L.P.P.
7.
7. A firm manufactures two types of products A and B and sells them at a profit of Rs. 200
A firm manufactures two types of products A and B and sells them at a profit of Rs. 200
on type A and Rs. 300 on type B. each product is processed on two machines G and H.
on type A and Rs. 300 on type B. each product is processed on two machines G and H.
type A requires 1 minute of processing time on G and 2minutes on H; Type B requires 1
type A requires 1 minute of processing time on G and 2minutes on H; Type B requires 1
minute on G and 1 minute on H. the machine G is available for not more than 6 hours, 40
minute on G and 1 minute on H. the machine G is available for not more than 6 hours, 40
minutes while H is available for 10 hours during any working day. Formulate this problem
minutes while H is available for 10 hours during any working day. Formulate this problem
as a linear programming problem.
as a linear programming problem.
8.
8. A firm manufactures headache pills in two sizes A and B. size A contains 2 grains of
A firm manufactures headache pills in two sizes A and B. size A contains 2 grains of
aspirin , 5 grains of bicarbonate and 1 grain of codeine. Size B contains 1 grain of aspirin,
aspirin , 5 grains of bicarbonate and 1 grain of codeine. Size B contains 1 grain of aspirin,
8 grains of bicarbonate and 6 grains of codeine. It is found by users that it requires at
8 grains of bicarbonate and 6 grains of codeine. It is found by users that it requires at
least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing
least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing
immediate effect. It is required to determine the least number of pills a patient should take
immediate effect. It is required to determine the least number of pills a patient should take
to get immediate relief. Formulate the problem as a LPP.
to get immediate relief. Formulate the problem as a LPP.
9.
9. A carpenter produces two products chairs and tables. Processing of these products is
A carpenter produces two products chairs and tables. Processing of these products is
done on two machines A and B. Chair requires 2 hours on machine A and 6 hours on
done on two machines A and B. Chair requires 2 hours on machine A and 6 hours on
machine B. A table
machine B. A table requires 5 hours on
requires 5 hours on machine A and no hours on
machine A and no hours on machine There are
machine There are
16 hours of time per day available on machine A and 30 hours on machine B. Assuming
16 hours of time per day available on machine A and 30 hours on machine B. Assuming
that the profit per chair is Rs. 20
that the profit per chair is Rs. 20 and Rs. 35 for table. Formulate the pro
and Rs. 35 for table. Formulate the problem as LPP in
blem as LPP in
order to determine the number of chairs and tables to be produced so as to maximize the
order to determine the number of chairs and tables to be produced so as to maximize the
profit.
profit.
10.
10. A manufacturer has two machines A and B. He manufactures two products P and Q on
A manufacturer has two machines A and B. He manufactures two products P and Q on
these two machines. For manufacturing product P he has to use machine A for 3 hours
these two machines. For manufacturing product P he has to use machine A for 3 hours
and machine B for 6 hours, and for manufacturing product Q he has to use machine A for
and machine B for 6 hours, and for manufacturing product Q he has to use machine A for
6 hours and machine B for 5 hours. On each unit of P he earns Rs. 14 and on each unit
6 hours and machine B for 5 hours. On each unit of P he earns Rs. 14 and on each unit
of Q he earns Rs. 10. How many units of P and Q should be manufactured to get the
of Q he earns Rs. 10. How many units of P and Q should be manufactured to get the
maximum profit? Each machine cannot be used for more than 2100 hours. Formulate as
maximum profit? Each machine cannot be used for more than 2100 hours. Formulate as
LPP.
LPP.

12.
Question
Question Bank
Bank US05FBCA01-
US05FBCA01- Operations
Operations Research
Research
Page
Page
12
12
of
of
24
24
11.
11. Vitamin A and B are found in foods F
Vitamin A and B are found in foods F1
1 and F
and F2
2. One unit of food F
. One unit of food F1
1 contains three unit of
contains three unit of
vitamin A and four unit of vitamin B. One unit of food F
vitamin A and four unit of vitamin B. One unit of food F2
2 contains six unit of vitamin A and
contains six unit of vitamin A and
three unit of vitamin B. One unit of food F
three unit of vitamin B. One unit of food F1
1 and F
and F2
2 cost Rs 14 and Rs 20 respectively. The
cost Rs 14 and Rs 20 respectively. The
minimum daily requirement (for person) of vitamin A and B is of 80 and100 units.
minimum daily requirement (for person) of vitamin A and B is of 80 and100 units.
Assuming excess of vitamin is not harmful to health, formulate LPP to obtain optimum
Assuming excess of vitamin is not harmful to health, formulate LPP to obtain optimum
mixture of food F
mixture of food F1
1 and F
and F2
2 required to meet the daily demand such that the total cost is
required to meet the daily demand such that the total cost is
minimized.
minimized.
12.
12. An electronic company is engaged in production of two components C
An electronic company is engaged in production of two components C1
1 and C
and C2
2 used in
used in
radio sets. The availability of different aspects and the prices are given below. Formulate
radio sets. The availability of different aspects and the prices are given below. Formulate
as LPP to determine number of components C
as LPP to determine number of components C1
1 and C
and C2
2 to be produced so as to maximize
to be produced so as to maximize
the profit.
the profit.
Resources/Constraint
Resources/Constraint Components
Components Total
Total Availability
Availability
C
C1
1
C
C2
2
Budget(Rs)
Budget(Rs) 10
10 /
/ unit
unit 40
40 /
/ unit
unit 4000
4000
Machine
Machine time
time 3
3 hr
hr /
/ unit
unit 2
2 hr
hr /
/ unit
unit 2000
2000 hrs
hrs
Assembly
Assembly time
time 2
2 hr
hr /
/ unit
unit 3
3 hr
hr /
/ unit
unit 1400
1400 hrs
hrs
Profit
Profit Rs.22
Rs.22 Rs.
Rs. 40
40
13.
13. A firm can produce two types of cloth, say: A and B. Three kinds of wool are required for
A firm can produce two types of cloth, say: A and B. Three kinds of wool are required for
it, say : red, green and blue wool. One unit length of type A cloth needs 4 meters of red
it, say : red, green and blue wool. One unit length of type A cloth needs 4 meters of red
wool and 3 meters of green wool; whereas one unit length of type B cloth needs 3 meters
wool and 3 meters of green wool; whereas one unit length of type B cloth needs 3 meters
of red wool, 2 meters of green wool and 5 meters of blue wool. The firm has only a stock
of red wool, 2 meters of green wool and 5 meters of blue wool. The firm has only a stock
of 10 meters of red wool, 6 meters of green wool and 15 meters of blue wool. It is
of 10 meters of red wool, 6 meters of green wool and 15 meters of blue wool. It is
assumed that the profit obtained from one unit length type A cloth is Rs. 13 and of type B
assumed that the profit obtained from one unit length type A cloth is Rs. 13 and of type B
cloth is Rs. 25. Formulate as
cloth is Rs. 25. Formulate as LPP.
LPP.
14.
14. A company makes two type varieties, Alpha and Beta, of pens. Each Alpha pen needs
A company makes two type varieties, Alpha and Beta, of pens. Each Alpha pen needs
twice as much labour time as a Beta pen. If only Beta pens are manufactured, the
twice as much labour time as a Beta pen. If only Beta pens are manufactured, the
company can make 500 pens per day. The market can take only up to 150 alpha pens
company can make 500 pens per day. The market can take only up to 150 alpha pens
and 250 Beta pens per day. If alpha and Beta pens yield profits of Rs. 8 and Rs. 5
and 250 Beta pens per day. If alpha and Beta pens yield profits of Rs. 8 and Rs. 5
respectively per pen, determine the number of Alpha and Beta pens to be manufactured
respectively per pen, determine the number of Alpha and Beta pens to be manufactured
per day so as to maximize the
per day so as to maximize the profit. Formulate as L.P.P.
profit. Formulate as L.P.P.
15.
15. Graphical
Graphical Method
Method
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
2
2 1
1 2
2 1
1 2
2
Solve the following LLP by graphical method.
Solve the following LLP by graphical method.
(
(1
1)
) M
Ma
ax
xi
im
mi
iz
ze
e 2
25
5 2
20
0 (
(2
2) M
) Ma
ax
xi
im
mi
iz
ze
e 3
3 2
2
S
Su
ub
bj
je
ec
ct
t t
to
o 1
16
6 1
12
2 1
10
00
0 S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 3
3 9
9,
,
8
8 1
16
6 8
80
0,
, 5
5 2
20
0,
,
2
2,
, ,
, 0
0.
. ,
, 0
0.
.
z
z x
x x
x z
z x
x x
x
x
x x
x x
x x
x
x
x x
x x
x x
x
x
x x
x x
x x
x x
x
=
= +
+ =
= +
+
+
+ ≤
≤ −
− +
+ ≤
≤
+
+ ≤
≤ −
− +
+ ≤
≤
≥
≥ ≥
≥ ≥
≥
(
( )
) 1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
3
3 M
Ma
ax
xi
im
mi
iz
ze
e 6
6 8
8 (
(4
4) M
) Ma
ax
xi
im
mi
iz
ze
e 5
5 7
7
S
Su
ub
bj
je
ec
ct
t t
to
o 5
5 1
10
0 6
60
0,
, S
Su
ub
bj
je
ec
ct
t t
to
o 4
4,
,
4
4 4
4 4
40
0,
, 1
10
0 7
7 3
35
5,
,
,
, 0
0.
. ,
, 0
0.
.
Z
Z x
x x
x z
z x
x x
x
x
x x
x x
x x
x
x
x x
x x
x x
x
x
x x
x x
x x
x
=
= +
+ =
= +
+
+
+ ≤
≤ +
+ ≤
≤
+
+ ≤
≤ +
+ ≤
≤
≥
≥ ≥
≥

13.
Question
Question Bank
Bank US05FBCA01-
US05FBCA01- Operations
Operations Research
Research
Page
Page
13
13
of
of
24
24
(
( )
) (
( )
)
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
1
1 2
2 1
1
1
1 2
2 2
2
1
1 2
2
5 M
5 M a
ax
xi
im
m i
iz
ze
e Z
Z 3
30
00
0x
x 4
40
00
0x
x 6 M
6 M a
ax Z
x Z 4
40
0x
x 3
30
0x
x
S
Su
ub
bj
je
ec
ct
t t
to 5
o 5x
x 2
2x
x 1
18
80
0,
, s
su
ub
bj
je
ec
ct
t t
to
o 3
3x
x x
x 3
30
0
3
3x
x 3
3x
x 1
13
35
5 x
x 8
8
x
x ,
, x
x 0
0.
. x
x 1
12
2
x
x ,
, x
x 0
0.
.
=
= +
+ =
= +
+
+
+ ≤
≤ +
+ ≤
≤
+
+ ≤
≤ ≤
≤
≥
≥ ≤
≤
≥
≥
(
( )
) 1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
1
1 2
2 1
1 2
2
2
2 1
1 2
2
1
1 2
2
7
7 M
Ma
ax
xi
im
mi
iz
ze
e 3
3 5
5 (
(8
8)
)M
Ma
ax
xi
im
mi
iz
ze
e 2
2 3
3
S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 2
20
0,
, S
Su
ub
bj
je
ec
ct
t t
to
o 1
1,
,
1
15
5,
, 3
3 6
6,
,
8
8,
, ,
, 0
0.
.
,
, 0.
0.
Z
Z x
x x
x z
z x
x x
x
x
x x
x x
x x
x
x
x x
x x
x x
x
x
x x
x x
x
x
x x
x
=
= +
+ =
= +
+
+
+ ≤
≤ +
+ ≤
≤
+
+ ≤
≤ +
+ ≤
≤
≤
≤ ≥
≥
≥
≥
Extra
Extra
16.
16. State the different scope of
State the different scope of operation research.
operation research.
17.
17. What are various phases of
What are various phases of operation research?
operation research?
18.
18. What is model? List out the model.
What is model? List out the model.
19.
19. Write the steps for solving Linear Programming Problem by Graphical method. State its
Write the steps for solving Linear Programming Problem by Graphical method. State its
limitations.
limitations.
20.
20. The ABC Company has been a producer of picture tubes for television sets and certain
The ABC Company has been a producer of picture tubes for television sets and certain
printed circuits for radios. The company has just explained into full scale production and
printed circuits for radios. The company has just explained into full scale production and
marketing of AM and AM-FM r
marketing of AM and AM-FM radios.
adios. It has built a new plant that can o
It has built a new plant that can operate 48 hours
perate 48 hours
per week. Production of an AM radio in the new plant will require 2 hours and production
per week. Production of an AM radio in the new plant will require 2 hours and production
of an AM-FM radio will require 3 hours. Each AM radio will contribute Rs. 40 to profits
of an AM-FM radio will require 3 hours. Each AM radio will contribute Rs. 40 to profits
while an AM-FM radio will contribute Rs. 80 to profits. The marketing departments, after
while an AM-FM radio will contribute Rs. 80 to profits. The marketing departments, after
extensive research, have determined that a maximum of 15 AM radios and 10 AM-FM
extensive research, have determined that a maximum of 15 AM radios and 10 AM-FM
radios can be sold each week. Formulate the LPP.
radios can be sold each week. Formulate the LPP.
21.
21. Sudhakant has two iron mines. The production capacities of the mines are different. The
Sudhakant has two iron mines. The production capacities of the mines are different. The
iron ore can be classifies into good, mediocre and bad varieties after certain process. The
iron ore can be classifies into good, mediocre and bad varieties after certain process. The
owner has decided to supply 12 or more tons of good iron, 8 or more tons of mediocre
owner has decided to supply 12 or more tons of good iron, 8 or more tons of mediocre
iron and 24 or more tons of bad iron per week. The daily expense is Rs.2000 and that of
iron and 24 or more tons of bad iron per week. The daily expense is Rs.2000 and that of
the second mine is Rs.1600. The daily production of each type of iron is given in the
the second mine is Rs.1600. The daily production of each type of iron is given in the
table.
table.
Mine
Mine Daily
Daily production
production
Good
Good Mediocre
Mediocre Bad
Bad
I
I 6 2
6 2 4
4
II 2 2 12
II 2 2 12
Formulate the LPP
Formulate the LPP
22.
22. Solve the following LP problems graphically
Solve the following LP problems graphically
1)
1) Minimize Z = 3
Minimize Z = 3 x
x1
1 +
+ 2
2 x
x2
2
s.t.
s.t. 5
5 x
x1
1 +
+ x
x2
2
≥
≥
10
10
x
x1
1 +
+ x
x2
2
≥
≥
6
6
x
x1
1 +
+ 4
4 x
x2
2
≥
≥
12
12
x
x1
1 ,
, x
x2
2
≥
≥
0
0
2)
2) Maximize Z = 30
Maximize Z = 30 x
x1
1 + 20 x
+ 20 x2
2
s.t.
s.t. 3
3 x
x1
1 +
+ 3
3 x
x2
2
≥
≥
40
40
3
3 x
x1
1 +
+ x
x2
2
≥
≥
40
40
2
2 x
x1
1 +
+ 5
5 x
x2
2
≥
≥
44
44
x
x1
1 ,
, x
x2
2
≥
≥
0
0

14.
Question
Question Bank
Bank US05FBCA01-
US05FBCA01- Operations
Operations Research
Research
Page
Page
14
14
of
of
24
24
3) Max z=3x
3) Max z=3x1
1+5x
+5x2
2
x
x1
1 + 2x
+ 2x2
2
≤
≤ 2000
2000
x
x1
1 + 2x
+ 2x2
2 ≤
≤ 1500
1500
x
x2
2 ≤
≤ 600
600
x
x1
1 , x
, x2
2 ≥
≥ 0
0
23.
23. A company produces two articles X and Y. These are two departments through which the
A company produces two articles X and Y. These are two departments through which the
articles are processed assembly and finishing. The potential capacity of the assembly
articles are processed assembly and finishing. The potential capacity of the assembly
department is 48 hours a week and that of the finishing department is 60 hours a week.
department is 48 hours a week and that of the finishing department is 60 hours a week.
Production of each of X requires 2 hours of assembly and 4 hours of finishing. Each unit
Production of each of X requires 2 hours of assembly and 4 hours of finishing. Each unit
of Y requires 4 hours in assembly and 2 hours in finishing department. If profit is Rs.6 for
of Y requires 4 hours in assembly and 2 hours in finishing department. If profit is Rs.6 for
each unit of X and Rs.7 for each unit of Y, find out the number of units of X and Y to be
each unit of X and Rs.7 for each unit of Y, find out the number of units of X and Y to be
produced each week to obtain maximum profit. (use graphical method).
produced each week to obtain maximum profit. (use graphical method).
UNIT 2
UNIT 2
1.
1. (
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
1 M
1 M a
ax
xi
im
mi
iz
ze
e 3
3 5
5
S
Su
ub
bj
je
ec
ct
t t
to
o 4
4,
, 3
3 2
2 1
18
8,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
2.
2. (
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
2
2 M
M a
ax
xi
im
mi
iz
ze
e 7
7 5
5
S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 6
6,
, 4
4 3
3 1
12
2,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
3.
3.
(
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
3 M
3 Ma
ax
xi
im
mi
iz
ze
e 5
5 7
7
S
Su
ub
bj
je
ec
ct
t t
to
o 4
4,
, 1
10
0 7
7 3
35
5,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
4.
4.
5.
5. (
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
5
5 M
Ma
ax
xi
im
mi
iz
ze
e 3
3 4
4
S
Su
ub
bj
je
ec
ct
t t
to
o 6
6,
, 2
2 4
4 2
20
0,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
6.
6. (
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
6
6 M
Ma
ax
xi
im
mi
iz
ze
e 5
5 3
3
S
Su
ub
bj
je
ec
ct
t t
to 3
o 3 5
5 1
15
5,
, 5
5 2
2 1
10
0,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥
BIG M Method
BIG M Method
7.
7. (
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
1
1 M
M a
ax
xi
im
mi
iz
ze
e 3
3
S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 2
2,
, 3
3 3
3,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= −
−
+
+ ≥
≥ +
+ ≤
≤ ≥
≥
8.
8. (
( )
) 1
1 2
2 3
3
1
1 2
2 3
3 1
1 2
2 1
1 2
2
2
2 M
M a
ax
xi
im
mi
iz
ze
e 3
3 4
4
S
Su
ub
bj
je
ec
ct
t t
to
o 3
3 2
2,
, 5
5 2
2 3
3,
, ,
, 0
0
=
= −
− +
+
−
− +
+ +
+ ≥
≥ −
− −
− ≤
≤ ≥
≥
Z
Z x
x x
x x
x
x
x x
x x
x x
x x
x x
x x
x
9.
9. (
( )
) 1
1 2
2 3
3
1
1 2
2 3
3 1
1 2
2 2
2 3
3 1
1 2
2 3
3
3 M
3 Ma
ax
xi
im
mi
iz
ze
e 5
5 2
2
S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 2
2 2
2,
, 3
3 4
4 3
3,
, 3
3 5
5 ,
, ,
, 0
0
=
= −
− −
−
+
+ −
− ≥
≥ −
− ≥
≥ +
+ ≥
≥ ≥
≥
Z
Z x
x x
x x
x
x
x x
x x
x x
x x
x x
x x
x where
where x
x x
x x
x
10.
10. (
( )
) 1
1 2
2 3
3
1
1 2
2 3
3 1
1 2 3
2 3 1
1 2
2 3
3
4 M
4 Ma
ax
xi
im
mi
iz
ze
e 2
2 9
9
S
Su
ub
bj
je
ec
ct
t t
to
o 4
4 2
2 5
5,
, 3
3 2
2 4
4,
, ,
, ,
, 0
0
=
= −
− −
− −
−
+
+ +
+ ≥
≥ +
+ +
+ ≥
≥ ≥
≥
Z
Z x
x x
x x
x
x
x x
x x
x x
x x
x x
x x
x x
x x
x
(
( )
) 1
1 2
2
1
1 2
2 1
1 2
2 1
1 2
2
4
4 M
Ma
ax
xi
im
mi
iz
ze
e 3
3 2
2
S
Su
ub
bj
je
ec
ct
t t
to
o 2
2 5
5,
, 3
3,
, ,
, 0
0
Z
Z x
x x
x
x
x x
x x
x x
x x
x x
x
=
= +
+
+
+ ≤
≤ +
+ ≤
≤ ≥
≥

15.
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24
Extra
Extra
11.
11. What is the standard form of
What is the standard form of the LPP? State its characteristics.
the LPP? State its characteristics.
12.
12. Write the steps of the
Write the steps of the algorithm for solving LPP by the Simplex method.
algorithm for solving LPP by the Simplex method.
13.
13. Explain about Big –M method for solving LPP by the
Explain about Big –M method for solving LPP by the Simplex method.
Simplex method.
14.
14. Max z = 18x
Max z = 18x1
1
+
+ 24x
24x2
2
s.t. 4x
s.t. 4x1
1 + 2x
+ 2x2
2
≤
≤
8,
8, 2x
2x1
1 + 5x
+ 5x2
2
≤
≤ 12 where x
12 where x1
1 ,x
,x2
2
≥
≥ 0
0
15.
15. Min
Min Z =
Z = 12x
12x1
1 + 20x
+ 20x2
2
s.t. 6x
s.t. 6x1
1 + 8x
+ 8x2
2 ≥
≥
10,
10, 7x
7x1
1 + 12x
+ 12x2
2
≥
≥ 12
12 where
where x
x1
1 ,x
,x2
2
≥
≥ 0
0
16.
16. Max
Max z
z = 30x
= 30x1
1 + 20x
+ 20x2
2
s.t. 3x
s.t. 3x1
1 + x
+ x2
2
≥
≥
40,
40, 2x
2x1
1 + 5x
+ 5x2
2
≥
≥ 44 where x
44 where x1
1 ,x
,x2
2
≥
≥ 0
0
UNIT 3
UNIT 3
1.
1. What is difference between transportation problem and assignment problem?
What is difference between transportation problem and assignment problem?
2.
2. Give the algorithm of NWCM to obtain basic feasible initial solution to transportation
Give the algorithm of NWCM to obtain basic feasible initial solution to transportation
problem.
problem.
3.
3. Give the algorithm of LCM to obtain basic feasible initial solution to transportation
Give the algorithm of LCM to obtain basic feasible initial solution to transportation
problem.
problem.
4.
4. Give the algorithm of VAM to obtain basic feasible initial solution to transportation
Give the algorithm of VAM to obtain basic feasible initial solution to transportation
problem.
problem.
5.
5. Write the steps for solving a
Write the steps for solving a A.P. by Hungarian method.
A.P. by Hungarian method.
6.
6. Write a short note on travelling salesman problem.
Write a short note on travelling salesman problem.
7.
7. Examples
Examples of
of Transportation
Transportation problem
problem
1)
1)
Obtain the initial solution to above
Obtain the initial solution to above
TP using northwest corner method.
TP using northwest corner method.
D
D1
1 D
D2
2 D
D3
3 D
D4
4 Supply
Supply
O
O1
1 6 4
6 4 1
1 5 14
5 14
O
O2
2 8 9
8 9 2
2 7 16
7 16
O
O3
3 4 3
4 3 6
6 2 5
2 5
Dem.
Dem. 6
6 10
10 15 4
15 4
2)
2)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using least cost method.
using least cost method.
A
A B
B C
C D
D Supply
Supply
I
I 6
6 3
3 5
5 4
4 22
22
II
II 5
5 9
9 2
2 7
7 15
15
III
III 5
5 7
7 8
8 6
6 8
8
Demand 7 12 17 9
Demand 7 12 17 9
3)
3)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using least cost method.
using least cost method.
D
D1
1 D
D2
2 D
D3
3 D
D4
4 Supply
Supply
O
O1
1 1 2 3 4 6
1 2 3 4 6
O
O2
2 4 3 2 0 8
4 3 2 0 8
O
O3
3 0 2 2 1 10
0 2 2 1 10
Demand
Demand 4
4 6
6 8
8 6
6
4)
4)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using northwest corner method.
using northwest corner method.
A B
A B C
C D Supply
D Supply
I
I 1
1 5
5 3
3 3
3 34
34
II
II 3
3 3
3 1
1 2
2 15
15
III
III 0
0 2
2 2
2 3
3 12
12
IV
IV 2
2 7
7 2
2 4
4 19
19
Demand
Demand 21 25 17
21 25 17 17
17

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5)
5)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using Vogel’s approximation method.
using Vogel’s approximation method.
I
I II
II III
III IV
IV Supply
Supply
A
A 21
21 16
16 15
15 13
13 11
11
B
B 17
17 18
18 14
14 23
23 13
13
C
C 32
32 27
27 18
18 41
41 19
19
Demand
Demand 6
6 10
10 12
12 15
15
6)
6)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using Vogel’s approximation method.
using Vogel’s approximation method.
D
D1
1 D
D2
2 D
D3
3 D
D4
4 Supply
Supply
O
O1
1 1 2
1 2 1
1 4 30
4 30
O
O2
2 3 3
3 3 2
2 1 50
1 50
O
O3
3 4 2
4 2 5
5 9 20
9 20
Demand
Demand 20 40 30
20 40 30 10
10
7)
7)
Obtain the optimal solution to above T
Obtain the optimal solution to above TP.
P.
I
I II
II III
III IV Supply
IV Supply
A
A 2
2 3
3 11
11 7
7 6
6
B
B 1
1 0
0 6
6 1
1 1
1
C
C 5
5 8
8 15
15 10
10 10
10
Demand
Demand 7
7 5
5 3
3 2
2 17
17
8)
8)
Obtain the initial solution to above TP
Obtain the initial solution to above TP
using northwest corner method.
using northwest corner method.
a b
a b c
c Supply
Supply
I
I 10
10 9
9 8
8 8
8
II
II 10
10 7
7 10
10 7
7
III
III 11
11 9 7
9 7 9
9
IV
IV 12
12 14
14 10
10 4
4
Demand
Demand 10
10 10
10 8
8
9)
9)
Obtain the optimal solution to above T
Obtain the optimal solution to above TP.
P.
Warehouses
Warehouses
W
W1
1
W
W2
2
W
W3
3
W
W4
4 Supply
Supply
F
F1
1 19 30 50 10 7
19 30 50 10 7
F
F2
2
70
70 30
30 40
40 60
60 9
9
F
F3
3
40
40 8 70
8 70 20
20 18
18
Demand
Demand 5
5 8
8 7
7 14
14
10)
10)
Obtain the optimal solution to
Obtain the optimal solution to above TP.
above TP.
Destination Supply
Destination Supply
Source I
Source I II III
II III IV
IV
A
A 19
19 14
14 23
23 11
11 11
11
B
B 15
15 16
16 12
12 21
21 13
13
C
C 30
30 25
25 16
16 39
39 19
19
Demand
Demand 6
6 10
10 12
12 15
15
8.
8. Examples
Examples of
of Assignment
Assignment problem
problem
Solve the following Assignment Problems.
Solve the following Assignment Problems.
1)
1) P Q R S
P Q R S
A 22 30 21 15
A 22 30 21 15
B 18
B 18 33
33 9
9 31
31
C 44 25 24 21
C 44 25 24 21
D 23 30 28 14
D 23 30 28 14
2)
2) I
I II III
II III IV
IV
1
1 11
11 10
10 18
18 5
5
2
2 14
14 13
13 12
12 19
19
3
3 5 3
5 3 4 2
4 2
4
4 15
15 18
18 17
17 9
9
3)
3) A B C D E F
A B C D E F
1
1 13 13 16
13 13 16 23 19 9
23 19 9
2
2 11 19
11 19 26 16
26 16 17 18
17 18
3
3 12
12 11
11 4
4 9
9 6
6 10
10
4
4 7
7 15
15 9
9 14
14 14
14 13
13
5
5 9
9 13
13 12
12 8
8 14
14 11
11
4)
4) P Q R S
P Q R S
A
A 5
5 3 4
3 4 7
7
B
B 2
2 3 7
3 7 6
6
C
C 4
4 1 5
1 5 2
2
D
D 6
6 8 1
8 1 2
2

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5)
5) Find the assignment of salesmen to various districts which will result
Find the assignment of salesmen to various districts which will result minimum cost.
minimum cost.
Salesman District
Salesman District
1 2 3 4
1 2 3 4
A
A 16
16 10
10 14
14 11
11
B
B 14
14 11
11 15
15 15
15
C
C 15
15 15
15 13
13 12
12
D
D 13
13 12
12 14
14 15
15
6)
6) Solve the following assignment problem so as to minimize the time (in days) required to
Solve the following assignment problem so as to minimize the time (in days) required to
complete all the task.
complete all the task.
person task
person task
1 2 3 4 5
1 2 3 4 5
A
A 6
6 5
5 8
8 11
11 16
16
B
B 1
1 13
13 16
16 1
1 10
10
C
C 16
16 11
11 8
8 8
8 8
8
D
D 9 14
9 14 12
12 10
10 16
16
7)
7) A department has 5 employees and five jobs are to be performed. The time each man
A department has 5 employees and five jobs are to be performed. The time each man
will take to perform each job is given in the following table below. How should the job be
will take to perform each job is given in the following table below. How should the job be
allocated one per employee, so as to m
allocated one per employee, so as to minimize the total man-hours.
inimize the total man-hours.
Cost matrix
Cost matrix
Jobs Employee
Jobs Employee
A B C D E
A B C D E
1
1 9 3 10
9 3 10 13
13 4
4
2
2 9
9 17
17 13
13 20
20 5
5
3
3 5 14
5 14 8 11
8 11 6
6
4
4 11
11 13
13 9
9 12
12 3
3
5
5 12
12 8
8 14
14 16
16 7
7
Extra
Extra
9.
9. Three fertilizers factories X, Y and Z located at different places of the country produce 6,4
Three fertilizers factories X, Y and Z located at different places of the country produce 6,4
and 5 lakh tones of urea respectively. Under the directive of the central government, they
and 5 lakh tones of urea respectively. Under the directive of the central government, they
are to be distributed to 3 States A, B and C as 5, 3 and 7 lakh respectively. The
are to be distributed to 3 States A, B and C as 5, 3 and 7 lakh respectively. The
transportation cost per tones in rupees is given below:
transportation cost per tones in rupees is given below:
A
A B
B C
C
X
X 11 17
11 17 16
16
Y
Y 15 12
15 12 14
14
Z
Z 20 12
20 12 15
15
Find out suitable transportation pattern at minimum cost by North West Corner method
Find out suitable transportation pattern at minimum cost by North West Corner method
and Least Cost method.
and Least Cost method.
10.
10. Determine an IBFS by Vogel’s Approximation method.
Determine an IBFS by Vogel’s Approximation method.
Source
Source D1
D1 D2
D2 D3
D3 D4
D4 Supply
Supply
S1 19 30 50 10 7
S1 19 30 50 10 7
S2 70 30 40 60 9
S2 70 30 40 60 9
S3
S3 40 8
40 8 70 20
70 20 18
18
Demand
Demand 5
5 8
8 7
7 14
14

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11.
11. A departmental has five employees with five jobs to be performed. The time ( in hours)
A departmental has five employees with five jobs to be performed. The time ( in hours)
each men will take to perform
each men will take to perform each job is given in the eff
each job is given in the effectiveness matrix.
ectiveness matrix.
How the jobs should be allocated, one per employee, so as to minimize the total man-
How the jobs should be allocated, one per employee, so as to minimize the total man-
hours.
hours.
Employees
Employees
jobs
jobs 1
1 2
2 3
3 4
4 5
5
a
a 10 5
10 5 13
13 15 16
15 16
b
b 3
3 9
9 18
18 13 6
13 6
c
c 10
10 7
7 2 2
2 2 2
2
d
d 7
7 11
11 9 7 12
9 7 12
e
e 7 9
7 9 10
10 4 12
4 12
12.
12. A machine operator processes five types of items on his machine each week, and must
A machine operator processes five types of items on his machine each week, and must
choose a sequence for them. The set up cost per change depends on the item presently
choose a sequence for them. The set up cost per change depends on the item presently
on the machine and the set-
on the machine and the set- up cost be made according to the following table:
up cost be made according to the following table:
To item
To item
From
From item
item A
A B
B C
C D
D E
E
1 -- 4 7 3 4
1 -- 4 7 3 4
2 4 -- 6 3 4
2 4 -- 6 3 4
3 7 6 -- 7 5
3 7 6 -- 7 5
4 3 3 7 -- 7
4 3 3 7 -- 7
5 4 4 5 7 --
5 4 4 5 7 --
If he processes each type of item once and only once each week, how should he
If he processes each type of item once and only once each week, how should he
sequence the items on his machine in order to
sequence the items on his machine in order to minimize the total set-up cost?
minimize the total set-up cost?
13.
13. A city corporation has decided to carry out road repairs on main four arteries of the city.
A city corporation has decided to carry out road repairs on main four arteries of the city.
The government has agreed to make a special grant of Rs 50 lakh towards the cost with
The government has agreed to make a special grant of Rs 50 lakh towards the cost with
a condition that repairs are done at the lowest cost and quickest time. If the conditions
a condition that repairs are done at the lowest cost and quickest time. If the conditions
warrant, a suppl
warrant, a supplementary token
ementary token grant will al
grant will also be consid
so be considered favorably.
ered favorably. The corporation
The corporation
has floated tenders and five contractors have sent in their bids. In order to expedite work,
has floated tenders and five contractors have sent in their bids. In order to expedite work,
one road will be awarded to only one
one road will be awarded to only one contractor.
contractor.
Cost of Repairs (Rs in lakh)
Cost of Repairs (Rs in lakh)
Contractors R
Contractors R1
1
R
R2
2
R
R3
3
R
R4
4
R
R5
5
C
C1
1
9
9 14
14 19 15 13
19 15 13
C
C2
2
7
7 17
17 20 19 18
20 19 18
C
C3
3
9
9 18
18 21 18 17
21 18 17
C
C4
4
10
10 12
12 18
18 19
19 18
18
C
C5
5
10
10 15
15 21
21 16
16 15
15
Find the best way of
Find the best way of assigning the repair work to the contractors and the costs.
assigning the repair work to the contractors and the costs.
If it is
If it is necessary to seek supplementary grants, what should be the amount sought?
necessary to seek supplementary grants, what should be the amount sought?
14.
14. A company has factories at F1, F2 & F3 which supply warehouses ay W1,W2 and W3.
A company has factories at F1, F2 & F3 which supply warehouses ay W1,W2 and W3.
Weekly factory capacities are 200,160 and 90 units respectively. Weekly warehouses
Weekly factory capacities are 200,160 and 90 units respectively. Weekly warehouses
requirements are 180,120 and 150 units respectively. Unit shipping costs ( in rupees) are
requirements are 180,120 and 150 units respectively. Unit shipping costs ( in rupees) are
as follows:
as follows:
Warehouse
Warehouse
Factory
Factory w1
w1 w2
w2 w3
w3 Supply
Supply
F1 16 20 12 200
F1 16 20 12 200
F2
F2 14
14 8
8 18
18 160
160
F3 26 24 16 90
F3 26 24 16 90
Demand
Demand 180
180 120
120 150
150
Determine the optimum distribution for t
Determine the optimum distribution for this company to minimize the shipping cost.
his company to minimize the shipping cost.

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15.
15. Determine the optimum basic feasible solution to
Determine the optimum basic feasible solution to the following transportation problem.
the following transportation problem.
A
A B
B C
C Available
Available
I
I 50
50 30
30 220
220 1
1
II
II 90
90 45
45 170
170 3
3
III
III 250 200
250 200 50
50 4
4
Required
Required 4
4 2
2 2
2
Hint: Find Initial Basic Feasible Solution using VAM met
Hint: Find Initial Basic Feasible Solution using VAM method.
hod.
16.
16. Determine the optimum basic feasible solution to
Determine the optimum basic feasible solution to the following transportation problem.
the following transportation problem.
D
D1
1 D
D2
2
D
D3
3
D
D4
4 Available
Available
O
O1
1
1
1 2
2 1
1 4
4 30
30
O
O2
2
3
3 3
3 2
2 1
1 50
50
O
O3
3
4
4 2
2 5
5 9
9 20
20
Required
Required 20
20 40
40 30
30 10
10 100
100
Hint: Find Initial Basic Feasible Solution using Lowest cost
Hint: Find Initial Basic Feasible Solution using Lowest cost method.
method.
17.
17. Determine the optimum basic feasible solution to the following transportation problem in
Determine the optimum basic feasible solution to the following transportation problem in
which cell entries represent unit costs.
which cell entries represent unit costs.
To Available
To Available
2
2 7
7 4
4 5
5
From
From 3
3 3
3 1
1 8
8
5
5 4
4 7
7 7
7
1
1 6
6 2
2 14
14
Required
Required 7
7 9
9 18
18 34
34
Hint: Find Initial Basic Feasible Solution using VAM.
Hint: Find Initial Basic Feasible Solution using VAM.
18.
18. Solve the transportation problem where all entries are unit costs.
Solve the transportation problem where all entries are unit costs.
D
D1
1 D
D2
2
D
D3
3
D
D4
4 D
D5
5 a
ai
i
O
O1
1
73
73 40
40 9
9 79
79 20
20 8
8
O
O2
2
62
62 93
93 96
96 8
8 13
13 7
7
O
O3
3
96
96 65
65 80
80 50
50 65
65 9
9
O
O4
4 57 58
57 58 29
29 12
12 87
87 3
3
O
O5
5 56 23
56 23 87
87 18
18 12
12 5
5
b
b j
j 6 8
6 8 10
10 4
4 4
4
Hint: Find Initial Basic Feasible Solution using Lowest cost
Hint: Find Initial Basic Feasible Solution using Lowest cost method.
method.
19.
19. The following table gives the cost for transporting material from supply points A, B, C and
The following table gives the cost for transporting material from supply points A, B, C and
to demand points E, F, G, H and J.
to demand points E, F, G, H and J.
To
To
E
E F
F G
G H
H J
J
A
A 8
8 10 12
10 12 17
17 15
15
B
B 15 13
15 13 18
18 11
11 9
9
From
From C
C 14
14 20
20 6
6 10
10 13
13
D
D 13
13 19
19 7
7 5
5 12
12
The present allocation is as follows:
The present allocation is as follows:
A to E 90, A to F 10, B to F 150, C to F 10, C to G 50, C to J 120, D to H 210, D to J 70.
A to E 90, A to F 10, B to F 150, C to F 10, C to G 50, C to J 120, D to H 210, D to J 70.
Check if this allocation is optimum. If not, find an optimum schedule.
Check if this allocation is optimum. If not, find an optimum schedule.

20.
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20.
20. The following table shows all the necessary information on the available supply to each
The following table shows all the necessary information on the available supply to each
warehouse, the requirement of each market and the unit transportation cost for each
warehouse, the requirement of each market and the unit transportation cost for each
warehouse to each market.
warehouse to each market.
Market
Market
I
I II III
II III IV
IV Supply
Supply
A
A 5
5 2
2 4
4 3
3 22
22
Warehouse
Warehouse B
B 4
4 8
8 1
1 6
6 15
15
C
C 4
4 6
6 7
7 5
5 8
8
Requirement
Requirement 7
7 12 17
12 17 9
9
The shipping clerk has worked out the following schedule from experience: 12 units from
The shipping clerk has worked out the following schedule from experience: 12 units from
A to
A to II
II, 1 unit from A to
, 1 unit from A to III
III, 9 units from A to
, 9 units from A to IV
IV, 15 units from B to
, 15 units from B to III
III, 7 units from C to
, 7 units from C to I
I
and 1 unit from C to
and 1 unit from C to III
III.
.
Check and see if the clerk has the optimum schedule. If not, find the optimum schedule
Check and see if the clerk has the optimum schedule. If not, find the optimum schedule
and minimum total shipping cost.
and minimum total shipping cost.
21.
21. Determine the optimum basic feasible solution to
Determine the optimum basic feasible solution to the following transportation problem.
the following transportation problem.
A
A B
B C
C Available
Available
I
I 50
50 30
30 220
220 1
1
II
II 90
90 45
45 170
170 3
3
III
III 250
250 200
200 50
50 4
4
Required
Required 4
4 2
2 2
2
Hint: Find Initial Basic Feasible Solution using Lowest cost
Hint: Find Initial Basic Feasible Solution using Lowest cost method.
method.
22.
22. Is
Is x
x13
13 = 50, x
= 50, x14
14 = 20, x
= 20, x21
21 = 55, x
= 55, x31
31 = 30, x
= 30, x32
32 = 35 , x
= 35 , x34
34 = 25 an optimum solution of the
= 25 an optimum solution of the
following transportation problem?
following transportation problem?
To
To Available
Available
From
From 6
6 1
1 9
9 3
3 70
70
11 5
11 5 2
2 8
8 55
55
10
10 12
12 4
4 7
7 90
90
Required
Required 85
85 35
35 50
50 45
45
23.
23. Solve following assignment problem.
Solve following assignment problem.
1
1 2 3
2 3 4
4
I
I
12
12 30 21
30 21 15
15
II
II
18
18 33 9
33 9 31
31
III
III
44
44 25 24
25 24 21
21
IV
IV
23
23 30 28
30 28 14
14
24.
24. Solve following cost minimizing problem.
Solve following cost minimizing problem.
Jobs
Jobs
I
I II III
II III IV V
IV V
A
A 45
45 30 65
30 65 40 55
40 55
B
B 50
50 30 25
30 25 60 30
60 30
Machines
Machines C 25
C 25 20
20 15 20
15 20 40
40
D
D 35
35 25 30
25 30 30 20
30 20
E
E 80
80 60 60
60 60 70 50
70 50

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25.
25. Solve following cost minimizing problem.
Solve following cost minimizing problem.
Jobs
Jobs
I
I II III
II III IV V
IV V
A
A 2
2 9
9 2
2 7
7 1
1
B
B 6
6 8
8 7
7 6
6 1
1
Machines
Machines C 4
C 4 6
6 5
5 3
3 1
1
D
D 4
4 2
2 7
7 3
3 1
1
E
E 5
5 3
3 9
9 5
5 1
1
26.
26. A department head has four subordinates and four tasks have to be performed.
A department head has four subordinates and four tasks have to be performed.
Subordinates differ in efficiency and tasks differ in their intrinsic difficulty. Time each man
Subordinates differ in efficiency and tasks differ in their intrinsic difficulty. Time each man
would take to perform each task is given in the efficiency matrix. How the tasks should be
would take to perform each task is given in the efficiency matrix. How the tasks should be
allocated to each person so as to minimize t
allocated to each person so as to minimize the total man-hours?
he total man-hours?
I
I II III
II III IV
IV
A 8
A 8 26 17
26 17 11
11
Tasks
Tasks B 13
B 13 28
28 4
4 26
26
C
C 38
38 19 18
19 18 15
15
D
D 19
19 26 24
26 24 10
10
27.
27. A car hire company has one car at each of five depots a, b, c, d and e. A customer
A car hire company has one car at each of five depots a, b, c, d and e. A customer
requires a car in each town, namely A, B, C, D and E. Distance (in kms) between depots
requires a car in each town, namely A, B, C, D and E. Distance (in kms) between depots
(origins) and towns 9destinations) are given in the following matrix:
(origins) and towns 9destinations) are given in the following matrix:
a
a b
b c
c d
d e
e
A
A 160
160 130
130 175 190
175 190 200
200
B
B 135
135 120
120 130 160
130 160 175
175
C
C 140
140 110
110 155 170
155 170 185
185
D 50 50 80 80 110
D 50 50 80 80 110
E 55 35 70 80 105
E 55 35 70 80 105
How should cars be assigned to customers so as to minimize t
How should cars be assigned to customers so as to minimize the distance travelled?
he distance travelled?
28.
28. A department has five employees with five jobs to be performed. The time (in hours) each
A department has five employees with five jobs to be performed. The time (in hours) each
man will take to perform
man will take to perform each job is given in the eff
each job is given in the effectiveness matrix.
ectiveness matrix.
Employees
Employees
I
I II III
II III IV V
IV V
A
A 10
10 5
5 13
13 15 16
15 16
B
B 3
3 9 18
9 18 13 6
13 6
Jobs
Jobs C
C 10
10 7
7 2 2
2 2 2
2
D
D 7
7 11 9 7 12
11 9 7 12
E
E 7
7 9 10
9 10 4 12
4 12
How should the jobs be allocated, one per employee, so as to minimize the total man-
How should the jobs be allocated, one per employee, so as to minimize the total man-
hours?
hours?
29.
29. A solicitor’s firm employs typists for their daily work. There are five typists and their
A solicitor’s firm employs typists for their daily work. There are five typists and their
charges are different. Only one job is given to one typist. Find least cost allocation for the
charges are different. Only one job is given to one typist. Find least cost allocation for the
following data:
following data:
Jobs
Jobs
P
P Q R
Q R S T
S T
A 85
A 85 75
75 65
65 125
125 75
75
B 90
B 90 78
78 66
66 132
132 78
78
Typists
Typists C 75
C 75 66
66 57
57 114 69
114 69
D 80
D 80 72
72 60
60 120
120 72
72
E 76
E 76 64
64 56
56 112
112 68
68

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30.
30. In a textile sales emporium, four salesman A, B, C and D are available to four counters
In a textile sales emporium, four salesman A, B, C and D are available to four counters
W, X, Y and Z. Each salesman can handle any counter. The service (in hour) of each
W, X, Y and Z. Each salesman can handle any counter. The service (in hour) of each
counter when manned by each salesman is given below:
counter when manned by each salesman is given below:
Salesman
Salesman
A
A B C
B C D
D
W
W 41
41 72
72 39
39 52
52
Counter
Counter X
X 22
22 29
29 49 65
49 65
Y
Y 27
27 39 60
39 60 51
51
Z
Z 45
45 50 48
50 48 52
52
How should the salesman be allocated appropriate counters so as t
How should the salesman be allocated appropriate counters so as to minimize the service
o minimize the service
time? Each salesman must handle only one counter.
time? Each salesman must handle only one counter.
UNIT 4
UNIT 4
1.
1. Write down the procedure for solving problem of sequencing with two m
Write down the procedure for solving problem of sequencing with two machines.
achines.
2.
2. State the rules for drawing network diagram.
State the rules for drawing network diagram.
3.
3. Write down the procedure to obtain optimum
Write down the procedure to obtain optimum completion time using Critical Path method.
completion time using Critical Path method.
4.
4. Find the critical path and calculate the Total float and Free float for the following PERT
Find the critical path and calculate the Total float and Free float for the following PERT
diagram.
diagram.
5.
5. A small maintenance project consists of the following
A small maintenance project consists of the following 12 jobs
12 jobs
Jobs
Jobs
Duration
Duration
in days
in days
Jobs
Jobs
Duration
Duration
in days
in days
Jobs
Jobs
Duration
Duration
in days
in days
1-2
1-2 2
2 3-5
3-5 5
5 6-10
6-10 4
4
2-3
2-3 7
7 4-6
4-6 3
3 7-9
7-9 4
4
2-4
2-4 3
3 5-8
5-8 5
5 8-9
8-9 1
1
3-4
3-4 3
3 6-7
6-7 8
8 9-10
9-10 7
7
Draw the arrow network of the project. Determine the
Draw the arrow network of the project. Determine the critical path.
critical path.
2
2
5
5
4
4
2
2
1
1
2
2
4
4
3
3 8
8
5
5
7
7
6
6
9
9
2
2
7
7
15
15
13
13
5
5
12
12
6
6
7
7
16
16
8
8
9
9
8
8
10
10
1
1
2
2
3
3
4
4
5
5
6
6
7
7
9
9
8
8
10
10

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6.
6. A project has the following time
A project has the following time schedule:
schedule:
Activity Time
Activity Time
In month
In month
Activity Time
Activity Time
In month
In month
Activity Time
Activity Time
In month
In month
1-2
1-2 2
2 3-6
3-6 8
8 6-9
6-9 5
5
1-3
1-3 2
2 3-7
3-7 5
5 7-8
7-8 4
4
1-4
1-4 1
1 4-6
4-6 3
3 7-9
7-9 3
3
2-5
2-5 4
4 5-8
5-8 1
1
Construct PERT network and compute total float for each activity. Find Critical path with
Construct PERT network and compute total float for each activity. Find Critical path with
its duration.
its duration.
7.
7. In a machine shop 8 different products are being manufactured each requiring time on
In a machine shop 8 different products are being manufactured each requiring time on
two different machines A and B are given in
two different machines A and B are given in the table below:
the table below:
Product 1
Product 1 2
2 3 4 5
3 4 5 6 7
6 7 8
8
Machine-A
Machine-A 30
30 45
45 15
15 20
20 80
80 120
120 65 10
65 10
Machine
Machine B
B 20
20 30 50
30 50 35
35 35
35 40
40 50
50 20
20
Find an optimal sequence of processing of different
Find an optimal sequence of processing of different product in order to minimize the total
product in order to minimize the total
manufctured time for all product. Find
manufctured time for all product. Find total ideal time for two machines a
total ideal time for two machines and elapsed
nd elapsed time.
time.
8.
8. In a machine shop 6 different products are being manufactured each requiring time on
In a machine shop 6 different products are being manufactured each requiring time on
two different machines A and B are given in
two different machines A and B are given in the table below:
the table below:
Product 1 2
Product 1 2 3 4 5
3 4 5 6
6
Machine-A
Machine-A 30
30 120
120 50
50 20
20 90
90 110
110
Machine
Machine B
B 80
80 100
100 90
90 60
60 30
30 80
80
Find an optimal sequence of processing of different
Find an optimal sequence of processing of different product in order to minimize the total
product in order to minimize the total
manufctured time for all product. Find
manufctured time for all product. Find total ideal time for two machines a
total ideal time for two machines and elapsed
nd elapsed time.
time.
9.
9. In a printing shop 7
In a printing shop 7 different books are printed and bounded on two different machines A
different books are printed and bounded on two different machines A
and B. Time required on two machines are g
and B. Time required on two machines are given in the table below:
iven in the table below:
Product 1 2
Product 1 2 3 4 5
3 4 5 6 7
6 7
Printing
Printing 3 4
3 4 8 3 6
8 3 6 7 5
7 5
Binding
Binding 8 6
8 6 3 7 2
3 7 2 8 4
8 4
Find an optimal sequence of processing of different
Find an optimal sequence of processing of different product in order to minimize the total
product in order to minimize the total
manufctured time for all product. Find
manufctured time for all product. Find total ideal time for two machines a
total ideal time for two machines and elapsed
nd elapsed time.
time.
Extra
Extra
10.
10. Write similarities and differences between PERT and CPM.
Write similarities and differences between PERT and CPM.
11.
11. Write applications of PERT/CPM techniques.
Write applications of PERT/CPM techniques.
12.
12. Write uses of PERT/CPM techniques.
Write uses of PERT/CPM techniques.
13.
13. Write the steps for Processing n jobs through two machines.
Write the steps for Processing n jobs through two machines.
14.
14. State “the Principle of Optimality” and its mathematical formulation of a dynamic
State “the Principle of Optimality” and its mathematical formulation of a dynamic
programming problem.
programming problem.
15.
15. Draw the Network Diagram for the following activities and f
Draw the Network Diagram for the following activities and find the critical path.
ind the critical path.
Job
Job A
A B
B C
C D
D E
E F
F G
G H
H I
I J
J K
K
Job
Job time(days)
time(days) 13
13 8 10
8 10 9
9 11
11 10
10 8
8 6 7
6 7 14 18
14 18
Immediate
Immediate
predecessors
predecessors
-
- A B
A B C
C B
B E
E D,F
D,F E H
E H G,I J
G,I J

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16.
16. A project has the following time Schedule. Construct a PERT network and compute
A project has the following time Schedule. Construct a PERT network and compute
Critical Path and its duration. Also calculate Total
Critical Path and its duration. Also calculate Total float,Free float.
float,Free float.
Activity
Activity 1-2
1-2 1-3
1-3 2-4
2-4 3-4
3-4 3-5
3-5 4-9
4-9 5-6
5-6
Time
Time in
in Weeks
Weeks 4
4 1
1 1
1 1
1 6
6 5
5 4
4
Activity
Activity 5-7
5-7 6-8
6-8 7-8
7-8 8-9
8-9 8-10
8-10 9-10
9-10
Time
Time in
in Weeks
Weeks 8
8 1
1 2
2 1
1 8
8 7
7
17.
17. A project schedule has the following characteristics. Construct the PERT network and find
A project schedule has the following characteristics. Construct the PERT network and find
the critical path and time duration of the project.
the critical path and time duration of the project.
Activit
Activity
y 1
1 -
- 2
2 1
1 -
- 4
4 1
1 -
- 7
7 2
2 -
- 3
3 3
3 -
- 6
6 4
4 -
- 5
5 4
4 -
- 8
8 5
5 -
- 6
6 6
6 -
- 9
9 7
7 -
- 8
8 8
8 -
- 9
9
Time
Time 2 2 1
2 2 1 4 1 5
4 1 5 8 4 3
8 4 3 3 5
3 5
18.
18. A salesman located in a
A salesman located in a city A decided to travel to c
city A decided to travel to city
ity B. He knew the distance o
B. He knew the distance of
f
alternative routes from city A to city B. He then drew a highway network map as shown in
alternative routes from city A to city B. He then drew a highway network map as shown in
Figure. The city of origin, A, is city 1. The destination city B, is city 10. Other cities through
Figure. The city of origin, A, is city 1. The destination city B, is city 10. Other cities through
which the salesman will have to pass through are numbered 2 to 9. The arrow
which the salesman will have to pass through are numbered 2 to 9. The arrow
representing routes between cities and distances in kilometers are indicated on each
representing routes between cities and distances in kilometers are indicated on each
route. The salesman’s problem is to find the shortest route that covers all the selected
route. The salesman’s problem is to find the shortest route that covers all the selected
cities from A to B.
cities from A to B.
Also the distance in kilometers is given below:
Also the distance in kilometers is given below:
Activity
Activity Distance
Distance travelled.
travelled. Activity
Activity Distance
Distance
travelled.
travelled.
1-2
1-2 4
4 3-7
3-7 4
4
1-3
1-3 6
6 4-5
4-5 6
6
1-4
1-4 3
3 4-6
4-6 10
10
2-5
2-5 7
7 4-7
4-7 5
5
2-6
2-6 10
10 5-8 4
5-8 4
2-7
2-7 5
5 5-9
5-9 8
8
3-5
3-5 3
3 6-8
6-8 3
3
3-6
3-6 8
8 6-9
6-9 7
7
3-7
3-7 4
4 7-8
7-8 8
8
4-5
4-5 6
6 7-9
7-9 4
4
4-6
4-6 10
10 8-10 7
8-10 7
4-7
4-7 5
5 9-10
9-10 9
9
1
1
2
2
3
3
4
4
5
5
6
6
7
7
10
10
8
8
9
9
4
4