MATH2901/3901-8
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Exercise 8
1. Consider the transportation problem described in Table 1.
Destination
Source
1 2 3 Supply
1
4 3 5
10
2
6 8 9
20
3
2 5 4
25
Demand 15 35 5
Table 1: problem
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH3903 Network Models in Operations Research
Assignment 2
1. The following is the adjacency matrix of a simple connected graph G of 11 nodes
and 21 edges. (Note: adjacency matrix is the n x n zeroo
AS/3903/2/SC/lp/09-10
MATH3903 Network Models in Operations Research
Suggested Solutions to Assignment 1
1. This question can be answered equivalently by giving an MST in the graph.
The Prim algorithm goes as follows.
4
1
6
4
2
1
1
4
9
7
3
5
54
7
5
10
3
1
AS/3903/4/SC/lp/09-10
MATH3903 Network Models in Operations Research
Suggested Solutions to Assignment 2
1. (a) The planar drawing of G is:
3
1
4
5
2
6
7
9
8
10
11
(b)
i. There are only three exposed nodes left: 4, 8, and 10. However, any two
nodes of the
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH3903 Network Models in Operations Research
Suggested Solutions to Assignment 5
1. We rst transform the given directed graph into a ow network with source node
3 and sink node t, with every are ha
MA1505
PRACTICE TEST
VERY IMPORTANT: This practice test is just a sample for the students reference only. Students should NOT expect to nd all the questions in the actual
test to be of exactly the same type as the questions in the practice test. The
level
Superstition: Psychological bane or boon?
Name: Chan Wai Yen
Tutor: Ms. Ethel Chong
Class: E21
One of the assumptions made by Hutson in the introductory paragraph of his article In Defense of
Superstition (The New York Times Sunday Review: The Opinion Pag
MATH 2901/3901
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Assignment 1
Due Date: 16 Feb. 2016 (11:30 pm).
There are 6 exercises. The full mark is 100. The weight of each exercise (full mark) is
shown at the right bottom of each
MATH2901/3901-1
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Exercise 1
1. Solve the following linear program graphically.
minimize
s.t.
z = 4x1 + 5x2
3x1 + 2x2 24
x1 5
3x1 x2 6
x1 , x2 0
Solution. See tutorial.
2. Given the foll
MATH2901/3901-6
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Exercise 6
1. Let A be an m-by-n matrix of rank m, c Rn and b Rm . Prove that in each of
the following pairs of systems, one and only one of the two systems has solutio
Resource-Allocation Problem:
v0 = max
s.t.
c>x
Ax b
x0
(1)
Optimal Simplex Tableau:
Basic
z
xB
0
xB
I
xN
> 1
cB B N
c>
N
Solution
1
c>
BB b
B 1 N
B 1 b
Table 1: Optimal Simplex Tableau of (1).
Perturbed Problem:
v(t) = max
s.t.
(c + tc0)>x
Ax b
x0
(3)
If
MS-E2140 Linear Programming
Exercise 8
Fri 02.10.2015
Y346
Week 4
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=39963 is due no
later than Tuesday 13.10.2015 23:55.
Exercise 8.1 Sensitivity tableau
Course book Exercise 5.5
While sol
MATH2901/3901-7
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Exercise 7
1. Consider the following resource-allocation problem and the accompanying optimal
tableau (x5 , x6 and x7 are the respective slack variables)
max
s.t.
x2
x6
MATH 2901/3901
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Assignment 3
Due Date: 1 April 2016 (11:30 pm).
There are 5 exercises. The full mark is 100. The weight of each exercise (full mark) is
shown at the right bottom of each
MATH2901/3901: Operation Research I
Instructor: Dr. Zheng QU (Room 419, Run Run Shaw Building, Email: [email protected]).
Consultation Hours: WED 09:0012:00
Tutor: Jiejun LU (A320B, Run Run Shaw Building, Email: [email protected] ).
Consultation Hours: MON 09:
Classical examples of linear programming
Example 1.1. (The diet problem). How can we determine the most economical diet that
satisfies the basic minimum nutritional requirements for good health? Such a problem
might, for example, be faced by the dietician
MATH2901/3901-3
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operation Research I
Exercise 3
1. Use simplex tableau to solve the following linear program.
maximize
s.t.
z = 2x1 + 3x2
x1 2x2 + x3 = 4
2x1 + x2 + x4 = 18
x2 + x5 = 10
x1 , x2 , x3 , x4 ,
-y"6rtal
-Tm5
11
for ( raa
2- ,syvnttan
3.
7*4i"nt)
( ar I
?PE b o DE
1. f.*'3
Ci/Qn o'
Pllwftql
,f
VqrhhLu
rl)
f,nc/.ton
PDE (or)
.
fu P oE r CI+)
,te$"r A u #te solubion
wly
t 3t"^
Tie chsiftetton
PDE
"f
w qve
L h lYy* ^ Y:r,
t cfw_ttT) = O
ft, o ) = O
MATH2901
Operations Research I Ideas of Solutions for Quiz / Class Test 2
1. (15%)
(a) The given constraints can be re-written as
n
n
aij xj bi
and
j =1
aij xj bi
(i = 1, . . . , m) .
j =1
(b) The given objective function can be linearized by adding const