The Theory of the Simplex

MethodChapter 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.

c=

𝑐1𝑐2⋮𝑐𝑛

, 𝐀 =

𝑎11𝑎21

𝑎12 …𝑎22 …

𝑎1𝑛𝑎2𝑛

⋮ ⋮ ⋮ ⋮𝑎𝑚1 𝑎𝑚2 … 𝑎𝑚𝑛

, x=

𝑥1𝑥2⋮𝑥𝑛

,b=

𝑏1𝑏2⋮𝑏𝑛

, 0=

00⋮0

cT= 𝑐1, 𝑐2, … , 𝑐𝑛

Wyndor Problem in Matrix Form

𝒄 =35, 𝒙 =

𝑥1𝑥2

, 𝑨 =1 00 23 2

, 𝒃 =41218

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

𝑨 =1 00 23 2

, 𝑩 =26, 𝐴𝐵 =

1 ∗ 2 + 0 ∗ 60 ∗ 2 + 2 ∗ 63 ∗ 2 + 2 ∗ 6

=21218

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

Define 𝑨 =2 6 93 5 81 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

𝑨 =1 31 1

, 𝒙 =𝑥1𝑥2

, 𝒃 =84

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 sub-matrices

◦ 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 −21 0

0 0 01 0 0

0 23 2

0 1 00 0 1

Sub-Matrices Cont.

−3 −21 0

0 0 01 0 0

0 23 2

0 1 00 0 1

We can rewrite this matrix as:

−𝒄 𝟎𝑨 𝑰