MA3252: Linear and Network
Optimization
Topic 6: Introduction to Network Optimization
Department of Mathematics
National University of Singapore
March 15, 2017
NUS: MA3252: 1/63
I
Network flow problems are:
I
I
I
Examples include:
I
I
a special case of LP

Student Name: Alex Mitchell
Matric Number: 1234567U
Tutorial: DW1
NM2213, AY2016/17, Semester 02
Academic Week: 3
Assignment Focus: The Human
Tutorial Assignment
My response:
A system that takes advantage of human capabilities is the draw attention behavi

Tutorial 1: CS3243 Introduction to AI
Semester II, 2016/17
National University of Singapore
School of Computing
CS3243 Introduction to AI
Tutorial 1: Introduction to AI & Intelligent Agents
Issue: January 16, 2017
Due: January 27, 2017
Important Instructi

Student Name: [insert name]
Matric Number: [insert number]
Tutorial: DW[insert number]
NM2213, AY2016/17, Semester 02
Academic Week: [insert week number]
Assignment Focus: [insert focus]
Tutorial Assignment
Information to students: Save a copy of this tem

NM2213
Human-Computer Interaction
Tutorial Week 4 - Focus: The Computer
Identify one example of an interface that takes advantage of the
computer as a medium, and one that does not. For each example,
briefly describe why you chose this example, with refer

(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) 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 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

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

Tutorial 2: CS3243 Introduction to AI
Semester II, 2016/17
National University of Singapore
School of Computing
CS3243 Introduction to AI
Tutorial 2: Uninformed Search
Issue: January 23, 2017
Due: February 3, 2017
Important Instructions:
Your solutions f

NM2213
Human-Computer Interaction
Tutorial Week 3 - Focus: The Human
Identify one example of an interface that takes advantage of human
capabilities and limitations, and one example of an interface that
is problematic because it fails to do so. For each e

N ATIONAL U NIVERSITY OF S INGAPORE
D EPARTMENT OF S TATISTICS & A PPLIED P ROBABILITY
ST2334 P ROBABILITY AND S TATISTICS
S EMESTER II, AY 2016/2017
Tutorial 01
Groups 1 5 will discuss this tutorial sheet in Week 3 while Groups 6 10 will do so in Week 4.

A DUALITY THEORY FOR CONTINUOUS-TIME DYNAMIC
SHORTEST PATHS WITH NEGATIVE TRANSIT TIMES
RONALD KOCH AND EBRAHIM NASRABADI
Abstract. This paper concerns the shortest path problem for a network in which arc costs can
vary with time, each arc has a transit

MA3252: Linear and Network
Optimization
Topic 7: The Network Simplex Method
Department of Mathematics
National University of Singapore
April 4, 2017
NUS: MA3252: 1/24
The network simplex method is faster than the simplex
method for network flow problems.

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

N > 0. Then, c
0 because c
B = 0. By Theorem 3.5, x is
1(a) Suppose c
a minimum solution. 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 + cTN dN = cTB B1 AN dN + cTN dN

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

During Tutorial Session
All students bring their laptops to the tutorial session.
Students sit in their groups and discuss (not read silently!) the
benefits/weaknesses of individual responses.
Group arrives at consensus about the best response (possibly a

MA3252 Linear and Network Optimization
Homework 1, Semester 2, 2016/17.
Deadline: 7 February 2017 by the end of the lecture. Late submission will not be
accepted. Hardcopy only.
(This is the first of 4 homeworks. Each homework is 2.5% of final grade. Tota

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 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
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

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

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) (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

(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. 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