MA3252 Linear and Network Optimization, 2013-14 Semester 2.
Tutorial 4
1. Consider the standard form polyhedron
0.
Suppose
AR
mn
P = cfw_x Rn | Ax = b, x
has linearly independent rows. For each of the
following statements, state whether it is true or fal

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 1
1. A small bank is allocating a maximum of $30,000 for personal and car
loans during the next month. The bank charges an annual interest rate
of 14% for personal loans and 12% for car

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 7
Clarification: The table
min
constraints
=
0
variables
0
free
max
0
0
variables
free
constraints
=
will be included in your final exam. There is no need to memorize it.
1. Consider th

N > 0, where N is the index set of all nonbasic variables at
1(a) Suppose c
x and B is the index set of the basic variables. For any feasible solution y, let
d = y x. Then Ad = 0 implies dB = B1 AN dN . The change in cost is
equal to
TN dN .
cT d = cTB dB

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 2
1. (a) Reformulate the following problem as a linear programming problem:
max min(x1 , x2 )
s.t. |2x1 + x2 | 7
3x1 x2
0.5
1 + x1 + x 2
x1 , x2 0.
(b) Consider the problem of minimizin

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 11
[Note: You would need topic 7 to solve Q4. Knowledge in topic 7 would
give some intuition for Q3, and perhaps Q2.]
1. In the following two maximum flow problems with source s = 1 and

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 3
1. (a) For each of the following sets, decide whether it is representable as
a polyhedron.
(i) The set of all (x, y) R2 satisfying the constraints
x cos + y sin 1 for all [0, /2]
x 0
y

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 10
[Note: You can do the first question after the lecture on week 10, but you will need
the lecture on week 11 to do the rest of the tutorial.]
1. Adam and Eve are planning a drive from

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 9
1. Consider the following linear programming problem.
min
s.t.
5x1
x1
12x1
5x2
+ x2
+ 4x2
13x3
+ 3x3
+ 10x3
x1 , x2 , x3
20
90
0
Let x4 and x5 denote the slack variables for the resp

1 0
1 0
1(a) The matrix B is
, and B1 is
. When the RHS of the first constraint is changed to b1 = 30,
4 1
4 1
30
then B1 30
90 becomes 30 . The updated tableau becomes
Basic
c
x2
x5
c
x2
x3
c
x4
x3
x1
x2
0
0
1
1
16
0
16
0
23
1
8
0
20.6
0.2
4.6 0.2
1.2

1. (a) The dual problem is
max
s.t.
p1
3p1
p1
+
+
+
4p2
p2 2
2p2 1
p2 1
p1 , p2 0
By solving it graphically, we find that the optimal solution is p1 = 0, p2 = 21 .
(b) When p2 = 12 , it means that the 2nd primal constraint is tight. The first
and third du

1. Let x1 be the amount of personal loans, and x2 be the amount of car
loans. Then
max
(1.14 0.97 1)x1 + (1.12 0.98 1)x2 = 0.1058x1 + 0.0976x2
s.t.
x1 + x2 30, 000
2x1 x2 0
x1 0, x2 0.
x2
2x1 x2 <=0
0.0976
0.1058
x1 + x2<=30k
x1 >=0
x1
x2 >=0
2. (a) x1 is

The diagram in the right can be
1. The set S equals cfw_1, 2, 3, 4, 5\S.
broken up with the following two steps:
2
4
1
4
6
2
2
8
6
3
4 4
5
4
4
For various possibilities of S and the two diagrams, we have the following
table for the capacities of (S, S).

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 5
1. Consider a minimization linear programming problem in standard form. Let x be
a BFS associated with the basis B. Prove the following:
(a) If the reduced cost of every nonbasic varia

MA3252 Linear and Network Optimization, 2013-14 Semester 2.
Tutorial 1
1. A small bank is allocating a maximum of $30,000 for personal and car
loans during the next month. The bank charges an annual interest rate
of 14% for personal loans and 12% for car

MA3252 Linear and Network Optimization, 2013-14 Semester 2.
Tutorial 2
1. (a) Reformulate the following problem as a linear programming problem:
max min(x1 , x2 )
s.t. |2x1 + x2 | 7
3x1 x2
0.5
1 + x1 + x 2
x1 , x2 0.
(b) Consider the problem of minimizin

MA3252 Linear and Network Optimization, 2013-14 Semester 2.
Tutorial 3
1. (a) For each of the following sets, decide whether it is representable as
a polyhedron.
(i) The set of all (x, y) R2 satisfying the constraints
x cos + y sin 1 for all [0, /2]
x 0
y

1. Let x1 be the amount of personal loans, and x2 be the amount of car
loans. Then
max
(1.14 0.97 1)x1 + (1.12 0.98 1)x2 = 0.1058x1 + 0.0976x2
s.t.
x1 + x2 30, 000
2x1 x2 0
x1 0, x2 0.
x2
2x1 x2 <=0
0.0976
0.1058
x1 + x2<=30k
x1 >=0
x1
x2 >=0
2. (a) x1 is

(1a) The LP is
max t
s.t. t x1
t x2
2x1 + x2 7
2x1 x2 7
3x1 x2 0.5 + 0.5x1 + 0.5x2
x1 , x2 0.
The problem above can be tidied up to be
max t
s.t. t x1 0
t x2 0
2x1 + x2 7
2x1 x2 7
2.5x1 1.5x2 0.5
x1 , x2 0, (t free).
(1b) The function f can be written as

1(a) (i) The first set is a quarter circle in the positive orthant of radius 1.
Note that
x cos + y sin 1 for all [0, /2]
is an infinite number of linear constraints that describe the set, so it is not
immediately clear that the set is or is not a polyhed

1 (a) True. A direction vector d that connects two BFS lies in the nullspace
of A. Since the rank of A is n 1, the nullspace of A has dimension 1. Suppose
is a BFS. Then all other BFS are of the form x
+ d. If there are 3 distinct
x
BFS, then one of the

1 (a) True. A direction vector d that connects two BFS lies in the nullspace of
A. Since the rank of A is n 1, the nullspace of A has dimension 1. Suppose x1
and x2 are two BFS. The formulas Ax1 = Ax2 = b implies that A(x1 x2 ) = 0.
In other words, x1 x2

MA3252 Linear and Network Optimization
Tutorial 10
1(a) Adam needs to solve the shortest path problem from A to E. Eve needs to solve the minimum cost
flow problem from A to E with the cost on each edge the negative of whatever is written on the table.
Mo

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 8
[Note: You will only be able to solve Questions 2, 3 and 4 after the lecture
on week 9.]
1. Consider the following primal LP problem:
min 2x1
s.t. 3x1
x1
+ x2 + x3
+ x2
+ 2x2 + x3
x1 ,

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 6
1. Solve the following linear program by the big-M method
max 3x1
s.t. 2x1
3x1
+ 2x2
+ x2
+ 4x2
+ 3x3
+ x3
+ 2x3
x1 , x2 , x3
2
8
0
2. A sports factory manufactures three types of foot

MA3252 Linear and Network Optimization, 2015-16 Semester 2.
Tutorial 4
1. Consider the standard form polyhedron P = cfw_x Rn | Ax = b, x
0. Suppose A Rmn has linearly independent rows. For each of the
following statements, state whether it is true or fal

(1a) The LP is
max t
s.t. t x1
t x2
2x1 + x2 7
2x1 x2 7
3x1 x2 0.5 + 0.5x1 + 0.5x2
x1 , x2 0.
The problem above can be tidied up to be
max t
s.t. t x1 0
t x2 0
2x1 + x2 7
2x1 x2 7
2.5x1 1.5x2 0.5
x1 , x2 0, (t free).
(1b) The function f can be written as