The Theory of the Simplex MethodChapter 5: Hillier and LiebermanChapter 5: Decision Tools for AgribusinessDr. Hurley’s AGB 328 Course

Terms to KnowConstraint 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 SolutionsGiven 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 FormA general maximization problem

can be written more succinctly in the following matrix notation:

Maximize Z = cTxSubject to: Ax ≤ bx ≥ 0

The Simplex Method in Matrix Form Cont.

c=,

cT=

Wyndor Problem in Matrix Form

𝒄=[35] ,𝒙=[𝑥1𝑥2] ,𝑨=[1 00 23 2 ] ,𝒃=[ 41218 ]

Important Rules/Facts of MatricesMatrices 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∗6

3∗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 FunctionsTranspose()

◦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 inputtedMinverse()

◦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 ExerciseDefine 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 ExampleSuppose we had the following:x1+ 3x2= 8x1+ x2= 4We could put this problem in the following

matrix notation, Hence we could write the problem as:Ax = bWe 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-MatricesA 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

Sub-Matrices Cont.

We can rewrite this matrix as: