Topic 5: Sensitivity Analysis.
March 15, 2013
Suppose after solving an LP, one of these happens:
Change in b
Change in c
Change in a nonbasic column of A
Adding a new variable
Add a new inequality constraint aT x b.
Sensitivity analysis allows us to analy

MA3252 Linear and Network Optimization
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 loans. Both types of

1(a) Suppose cN > 0, where N is the index set of all nonbasic variables at
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
N
cT d = cT dB + cT d

MA3252 Linear and Network Optimization
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
1. A sports factory manufactures three types of footballs which require
o

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
x is a BFS. Then all other BFS are of the form x + d. If there are 3 distinct
BFS, then one of them

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

MA3252 Linear and Network Optimization
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 variable is positive, the

MA3252 Linear and Network Optimization
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 false. If true, provide

1(a) The rst set is a quarter circle in the positive orthant of radius 1.
Since an innite number of linear constraints are needed to describe (i), it is
not a polyhedron. The second set can be expressed as x R satisfying the
two linear constraints x 3, x

MA3252 Linear and Network Optimization
Tutorial 7
Clarication: The table
min
constraints
=
0
variables
0
free
max
0
0
variables
free
constraints
=
will be included in your nal exam. There is no need to memorize it.
1. Consider the following linear progra

MA3252 Linear and Network Optimization
Tutorial 8
1. Consider the following primal LP problem:
min 2x1
s.t. 3x1
x1
+ x2 + x3
+ x2
+ 2x2 + x3
x1 , x2 , x3
1
4
0
(a) Determine the dual problem and solve it graphically.
(b) Use the Complementary Slackness Op

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.
2. (a) x1 is rst product. x2 is second product.
max
(6 3)x1 + (5.4 2

1. The set S equals cfw_1, 2, 3, 4, 5\S. For various possibilities of S and the
two diagrams, we have the following table for the capacities of (S, S).
S
Diagram 1
Diagram 2
cfw_1
u12 + u13
37 u12 + u13 + u14
10
cfw_1, 2
u13 + u23 + u24 + u25 77 u13 + u1

MA3252 Linear and Network Optimization
Tutorial 11
1. In the following two maximum ow problems with source s = 1 and
sink t = 5, determine all s t cuts. Find the minimum s t cut in
each case.
2
25
1
4
20
1
3
12
5
22
(8,0)
(4,6)
3
(6,0)
(2,0) (0,2)
18
15
1

1(a) Adam needs to solve the shortest path problem from A to E. Eve
needs to solve the minimum cost ow problem from A to E with the cost on
each edge the negative of whatever is written on the table. More precisely,
for Adam:
(0.1)
min cT x
x
s.t.
1
Ax =

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

1(a) The matrix B is
10
41
, and B1 is
10
4 1
the rst constraint is changed to b1 = 30, then B1
The updated tableau becomes
. When the RHS of
30
90
30
30
becomes
.
Basic
x1
x2 x3 x4
x5 Solution
c
0
0
2
5
0
x2
1
1
3
1
0
30
x5
16
0 2 4
1
30
c
16
0
0
1
2
x2

MA3252 Linear and Network Optimization
Tutorial 10
1. Adam and Eve are planning a drive from location A to E. The time to travel
and the scenic rating for the roads in the network are given below. All roads are
one-way. Adam wants to reach location E as f

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

MA3252 Linear and Network Optimization
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]
x0
y0
(ii) The set of all

MA3252 Linear and Network Optimization
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 minimizing a cost function of

(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

MA3252: Linear and Network
Optimization
Topic 1: Introduction to Linear Programming
January 11, 2013
Linear Programming (LP) (or Linear Optimization):
A mathematical modeling technique for minimizing a
linear cost function subject to a nite set of equalit

Topic 2: Geometry of Linear Programming
January 16, 2013
A polyhedron or polyhedral set is a set of the form
cfw_x Rn | Ax b, where A Rmn and b Rm .
Geometrically, a polyhedron is a nite intersection of half
m
spaces
i =1
cfw_x | aT x bi .
i
A bounded pol

(After the week 2 lecture, I realized that making you all copy long
proofs is not a good idea. Therefore, I have put the long proofs
into this set of lecture notes handout. I also made corrections to
the extreme pt = BFS proof of Theorem 2.3)
Theorem 2.3

Topic 3: The simplex method.
January 25, 2013
Consider the standard form LP
min cT x
s.t.
Ax = b
x 0.
Assume A Rmn , rank(A) = m with m n. Let
P = cfw_x Rn | Ax = b, x 0.
P does not contain a line.
Hence, if P = , then P has an extreme point.
Therefore, e

Theorem 2.5
A vector x Rn is a basic solution of the standard form LP i
1. Ax = b and
2. There exist indices B (1), B (2), . . . , B (m) such that:
(a) The columns AB (1) , . . . , AB (m) are linearly independent;
(b) x = 0 for i = B (1), B (2), . . . , B

A remark on notation:
11210
8
0 1 6 0 1 x = 12 , x 0.
10000
4
A
b
For standard LPs, A Rmn , b Rm , x Rn , with m n.
Here m = 3, n = 5.
B cfw_1, . . . , n is a basis (set of basic indices) s.t. |B | = m.
N cfw_1, . . . , n is cfw_1, . . . , n\B , i.e., al