8/1/2013
Solving LPs Geometrically
Solving LPs Geometrically
The Blending Problem of Case Chemical
Case Chemicals produces two solvents:
CS-01 and CS-02. Each 1000 gallons of
CS-01 yields a net profit of $300 and
requires 2 hours in the Blending
Departmen

8/6/2012
A Question for Case Chem.
What-Happens-If Questions
Consider the optimal solution to the following modified
problem of Case Chemicals:
max 3CS1 5CS 2
(profit in $ 00' s)
Recall that one of the advantages of a math model
is that, after solving a p

8/1/2013
The Simplex Algorithm
Step 0 (Initialization): Find an initial bfs,
x, or determine there is none.
Step 1 (TFO): If cN cB B1N 0, stop,
the current bfs x is optimal. If not,
Step 2 (Move): (a) Use any
j with (cN cB B1N)j < 0
to compute the directi

8/1/2013
Phase 1
The Phase 1 LP
Phase 1 is a systematic method for attempting to find
an initial bfs. If successful, then you can use phase 2
that you have already learned to find an optimal
solution or determine that the LP is unbounded.
To create the p

8/1/2013
The Geometric Simplex Alg.
Finding an Initial bfs
Eventually, a systematic
Step 0 (Initialization): Find an initial
extreme point, x, or determine there is none. procedure is needed. For now,
either you will be given an
Step 1 (TFO): Perform a te

8/1/2013
The OR Methodology
OPRE 411
OPTIMIZATION MODELING
Identify a Problem
Formulate a Math Model and Collect Data
PART 1
LINEAR PROGRAMMING
Identify a Solution Procedure and
Obtain a Solution from the Computer
Daniel Solow
Copyright 2013
Implement Sol

Linear Algebra in Linear Prog.
Linear Algebra in Linear Prog.
Multiplying a Matrix by a Vector
Scalar Notation
Min 3x1 + x2 x3 + 2x4
x = (x1 , x2 , x3 , x4 ) c = (3, 1, 1, 2 )
For a (p x n) matrix A and a column n-vector x, you have:
s.t.
2
1 0 1 2
A
, b

8/1/2013
The Investment Problem of PP
The Investment Problem of PP
You, the Portfolio Manager of Pension Planners (PP), have
been asked to invest $1,000,000. The Research Department
has identified six different mutual funds (each mutual fund
is a collecti

8/1/2013
Problem Formulation
The Problem of MTV Steel
The Make-or-Buy Problem of MTV Steel
Table 1: Data for the Problem of MTV Steel
MTV Steel produces three sizes of tubes. Table 1
provides the amount of machine time and welding
material, as well as the

8/1/2013
Solving LPs By Computer
The Problem of Case Chemicals
Recall the following problem of Case Chemicals, with one
additional constraint.
The Problem of Case Chemicals
You want to find values for the variables:
CS1 = number of thousands of gallons of