Download Overview of the Operations Research Modeling Approach

The Theory of the Simplex
Method
Chapter 5: Hillier and Lieberman
Chapter 5: Decision Tools for Agribusiness
Dr. Hurley’s AGB 328 Course
Terms to Know

Constraint Boundary Equation,
Hyperplane, Constraint Boundary,
Corner-Point Feasible Solution, Defining
Equations, Edge, Adjacent, Convex Set,
Basic Solutions, Basic Feasible Solution,
Defining Equations, Indicating Variable,
Basic Variables, Non-Basic Variables,Vector
of Basic Variables, Basis Matrix
Adjacent CPF Solutions
Given n decision variables and bounded
feasible region, an edge can be defined as
the feasible line segment that is defined by
n-1 constraint boundary equations
 Two CPF solutions are considered
adjacent if the line segment connecting
them is an edge of the feasible region

◦ Hence you get an adjacent point by deleting
one of the n constraints currently defining the
CPF solution
The Simplex Method in Matrix
Form
A general maximization problem can be
written more succinctly in the following
matrix notation:
 Maximize Z = cTx
 Subject to:
 Ax ≤ b
x≥0

The Simplex Method in Matrix
Form Cont.
𝑎11
𝑐1
𝑎21
𝑐2
c=
,𝐀 = ⋮
⋮
𝑐𝑛
𝑎𝑚1
cT= 𝑐1 , 𝑐2 , … , 𝑐𝑛
𝑎12 … 𝑎1𝑛
𝑎22 … 𝑎2𝑛
⋮ , x=
⋮
⋮
𝑎𝑚2 … 𝑎𝑚𝑛
𝑥1
𝑥2
, b=
⋮
𝑥𝑛
𝑏1
𝑏2
, 0=
⋮
𝑏𝑛
0
0
⋮
0
Wyndor Problem in Matrix Form
1 0
4
𝑥1
3
𝒄=
, 𝒙 = 𝑥 , 𝑨 = 0 2 , 𝒃 = 12
5
2
3 2
18
Important Rules/Facts of Matrices


Matrices with the same number of rows and
columns can be added/subtracted
component by component
Matrices can be multiplied together as long
as the first matrix has the same number of
columns as the second matrix has of rows
◦ E.g., C = AB is defined as long as the number of
columns in matrix A is equal to the number of
rows in matrix B
 Matrix C will have the same number of rows as matrix
A and the same number of columns as matrix B
Important Rules/Facts of Matrices
Cont.

Suppose matrix A has r number of rows
and m number of columns, matrix B has
m number of rows and c number of
columns, then a matrix Q, which equals
AB, has r rows and c columns where
each component in the Q matrix is found
by the following method:
◦ qij = ai1*blj + ai2*b2j + ai3*b3j +… +aim*bmj
 Note that this is just the Sumproduct() of the
corresponding row from matrix A to the
corresponding column in matrix B
Important Rules/Facts of Matrices
Cont.

Example of matrix multiplication using
Wyndor’s constraints evaluated at
Wyndor’s optimal
1
𝑨= 0
3
1∗2+0∗6
0
2
2
, 𝐴𝐵 = 0 ∗ 2 + 2 ∗ 6 = 12
2 ,𝑩 =
6
3∗2+2∗6
2
18
Important Rules/Facts of Matrices
Cont.

An important matrix is known as a
identity matrix
◦ This matrix is known as I
◦ The identity matrix can be considered like the
number 1 when it comes to matrix
multiplication because when you multiply the
identity by any matrix A, you get A, i.e.,
A*I=I*A=A
Important Rules/Facts of Matrices
Cont.
While there is no formal division in
matrix algebra, it does have the idea of an
inverse for some matrices, .i.e., certain
square matrices
 Normally this inverse matrix of a matrix
A is denoted by A-1 and has the property
that A*A-1= A-1*A = I

Important Rules/Facts of Matrices
Cont.
The transpose of a matrix takes each
component aij in a matrix and swaps it
with component aji
 Basically this exchanges the rows with the
columns leaving the diagonal intact
 It should be noted that AB does not have
to equal BA or even be defined

Excels Key Matrix Functions

Transpose()
◦ This function takes a columns and swaps them
for the rows or vice-versa

Mmult()
◦ This function will give you the product of the
matrices inputted

Minverse()
◦ This function gives the inverse of a matrix
Excels Key Matrix Functions Cont.

It should be noted that to use these
matrix functions correctly, you need to
first enter the formula in a single cell
◦ Next you need to highlight all the cells that
are needed and press the F2 function
◦ Finally you need to press Control-Shift-Enter
at the same time
Quick Matrix Exercise
2 6 9
 Define 𝑨 = 3 5 8
1 4 7
 Using Excel, what is the inverse of A?
 Using Excel, what is the transpose of A?
 Using Excel, what is AA-1?
 What happens if you select too many
rows or columns before you press F2
when you attempt to find these answers
in Excel?
Another Matrix Example








Suppose we had the following:
x1+ 3x2= 8
x1+ x2= 4
We could put this problem in the following
matrix notation
𝑥1
1 3
8
𝑨=
,𝒙 = 𝑥 ,𝒃 =
1 1
2
4
Hence we could write the problem as:
Ax = b
We can solve for x by pre-multiplying both sides
by A-1 to get x = A-1b
◦ Put this into Excel to see what you get
Sub-Matrices

A matrix can be broken-up into submatrices
◦ A sub-matrix is a smaller matrix inside of a
matrix
◦ When you break-up a matrix into smaller
matrices, you are said to be partitioning it


Recall the Original Wyndor tableaux
−3 −2 0 0 0
1 0
1 0 0
0 2
0 1 0
3 2
0 0 1
Sub-Matrices Cont.
−3 −2 0 0 0
1 0
1 0 0

0 2
0 1 0
3 2
0 0 1
 We can rewrite this matrix as:
−𝒄 𝟎

𝑨 𝑰