The Theory of the Simplex Method - Agribusiness...
Transcript of The Theory of the Simplex Method - Agribusiness...
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The Theory of the Simplex
MethodChapter 5: Hillier and Lieberman
Chapter 5: Decision Tools for Agribusiness
Dr. Hurley’s AGB 328 Course
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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
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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
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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
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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
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Wyndor Problem in Matrix Form
𝒄 =35, 𝒙 =
𝑥1𝑥2
, 𝑨 =1 00 23 2
, 𝒃 =41218
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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
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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
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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