LN/MATH2901/CKC/MS/2008-09
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operations Research I
Denition. (Linear Programming)
A linear programming (LP) problem is characterized by linear functions of the unknowns,
called the decision variables. It
MATH2901 Operations Research I
Game Theory
p.1
GAME THEORY
A game represents a competitive or conicting situation between two or more players .
Each player has a number of choices, called moves (or pure strategies ). A player selects his
moves without any
Numerical Example for Fundamental Theorem of Linear Programming
Given a feasible solution x = (1, 2, 0, 1)T (i.e. Ax = b, x 0), with p = 3 pos entries.
2110
4
A = (a 1 , a 2 , a 3 , a 4 ) =
, b=
.
1201
6
The corresponding p column vectors cfw_a1 , a2 , a4
Pivot Operation (wrt element ars > 0)
Rules: ars > 0 is the pivot element, row r pivot row, and column s is pivot column.
(a) In pivot row, arj arj /ars j .
(b) In pivot column,
ars 1, ais 0 i = r .
(c) For all other elements, aij aij arj ais /ars .
x1
x2
An Example on Duality Relationship of Dual LP Pairs
If Primal (Max) LP is.
then Dual (Min) LP is.
Feasible
Bounded S
Finite opt solution (1:P)
Finite opt solution (1:Q)
(S = )
1. Unbdd S
Finite opt solution (2:P)
Finite opt solution (2:Q)
2. Unbdd S
Unbdd
MATH2901/AS2/SC/
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operational Research I
Problems Set 2
1. Consider the following LP problem :
Min
Subject to
2x1
x1
2x1
x1
+ 3x2
+ x2
+ 3x2
, x2
+
+
+
,
2x3
x3
x3
x3
= 2
= 3
0
a) Find out the BFS correspo
AS4/MATH2901/MCS/
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
Operations Research I
Problems Set 4
1. Write down the duals of the following problems:
(a)
Max 4x1
Subject to
x1
2x1
3x1
+ 3x2
+ x2
x2
+ 7x2
+
+
2x3
x3
x3
+
+
x4
x4
+
10x4
5
= 6
8
MATH2901/AS3/SC/
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Problems Set 3
1. Solve the following LP by the two-phase method. Perform pivots in LP compact tableau.
Max
subjec to
4x1
x1
x1
2x1
+ 5x2 3x3
+ x2 + x3 =
MATH2901/AS1/SC/
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Problems Set 1
Q1. A rm manufactures chicken feed by mixing three dierent ingredients. Each ingredient contains three
key nutrients protein, fat and vita
AS/MATH2901/SC/
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Problems Set 5
1. (a) Find the saddle point(s) of the following pay-o matrix.
3 6 2
4 9 4
3 5 3
8
5
5
(b) Show that if a pay-o matrix has exactly two sadd
MATH2901/AS3/MS/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 / MATH3901 Operations Research I
Assignment 3 - Suggested Solution
1. Solve the following given LP by two-phase method.
Max x0 = 4x1 + 5x2 3x3
s.t.
x1 + x2 + x3 = 10
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 constra
AS/MATH2901/ms/SC/
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Problems Set 5 - Suggested Solution
Q1. (a)
3
4
3
col.max. 4
Maximin = Max cfw_2, 4, 3 = 4 ;
6
9
5
2
4
3
8
5
5
9 4
row min.
2
4
3
8
Minimax = Min cfw_
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Quiz / Class Test 2
November 14, 2013 (Thursday)
1. (15%) Consider a mathematical programming problem, in which both the constraints
and the objective function as given appea
MATH2901
Operations Research I Ideas of Solutions for Quiz / Class Test 1
1. (10%)
(a) P = Mincfw_x0 = cT x|Ax = b , x 0.
Let B be an m m non-singular submatrix of A. Denote A = (N, B) and x
(xN , xB )T accordingly. Then Ax = b can be written as (N, B)(xN
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901 Operations Research I
Quiz / Class Test 1
October 03, 2013 (Thursday)
1. (10%) Let (P) denote a linear (min) program in standard form with m constraints,
n (> m) variables and full row rank. Assum
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901: Operations Research I
Starting Tableaus for Simplex Computation with Articial Variables Techniques
Max cfw_ x0 = cT x | Ax = b (b 0), x 0 .
Two-Phase Method:
Min cfw_ y0 = 1T y | Ax + Iy = b (b
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2901: Operations Research I
An Illustration on Simplex Computations with Articial Variables Techniques
Max cfw_ x0 = 3x1 + 4x2 | 3x1 + 2x2 = 6, x1 , x2 0
Phase-I
Setting up articial problem and calcu