Chapter 1
Introduction to Linear Programming
Also see Chapter 3 of the text book.
1
Standard Form of Linear Programming
One of the many standard forms of linear programming problems is the following:
max
x1 ,x2 ,.,xn R
c1 x 1 + c2 x 2 + + cn x n
s.t. a1,1

Chapter 2
The Simplex Method
Also see Chapter 4 of the text book.
1
Basic Concept
1.1
Terminology
We illustrate the basic concept by considering the following problem:
Figure 1: Feasible set of an LP problem.
Let n be the number of decision variables; in

Linear Programming and Integer Programming
6.2-1.
Mid-term Examination 2
(a)
Iteration 0: Since all coefficients are zero, at the current solution ! !, the three
Solution
resources (production time per week at plant 1, 2 and 3) are free goods. This means

Linear Programming and Integer Programming
MATH 3205 / MATH 2230
MATH 3205 / MATH 2230 I
Linear and Integer Programming /
Mid-termInteger ProgrammingOROR I
1 Examination
MATH
Linear and3205 / MATH 2230
/
Mid-term solution
Linear and Integer Programming /

Chapter 4
Duality Theory
Also see Chapter 6 of the text book.
1
Motivation
Consider the LP problem in a standard form:
min
xRn
cx
s.t. Ax = b,
0 x,
where b Rm and A Rmn . Let us call this the primal problem and let x be its feasible
solution; assume that

Chapter 3
The Revised Simplex Method
Also see Chapter 5 of the text book.
1
LP Problems in Matrix Forms
We have shown the following standard form of LP problems in Chapter 1:
c1 x 1 + c2 x 2 + + cn x n
max
x1 ,x2 ,.,xn R
s.t. a1,1 x1 + a1,2 x2 + + a1,n xn

11.5-2.
(a) The dots represent the feasible solutions in the graph below.
Optimal Solution: B" B# # $ ^ &B" B# "$
(b) The optimal solution of the LP relaxation is B" B# #' "' ^ "%'. The
nearest integer point is B" B# $ #, which is not feasible, since % $

Assignment 4: Integer Programming
Due Date: November 25, 2015
1. Solution:
cfw_
Let
Mj =
cfw_
Dj =
1 if j does marketing
0 otherwise
1 if j does dishwashing
0 otherwise
cfw_
Cj =
cfw_
Lj =
1 if j does cooking
0 otherwise
1 if j does laundry
0 otherwise
fo