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. c= 𝑐 1 𝑐 2 ⋮ 𝑐 𝑛 , 𝐀= 𝑎 11 𝑎 21 𝑎 12 … 𝑎 22 … 𝑎 1𝑛 𝑎 2𝑛 ⋮ ⋮ ⋮ ⋮ 𝑎 𝑚1 𝑎 𝑚2 … 𝑎 𝑚𝑛 , x= 𝑥 1 𝑥 2 ⋮ 𝑥 𝑛 ,b= 𝑏 1 𝑏 2 ⋮ 𝑏 𝑛 ,0= 0 0 ⋮ 0 cT= 𝑐 1 , 𝑐 2 ,…, 𝑐 𝑛

Wyndor Problem in Matrix Form 𝒄= 3 5 ,𝒙= 𝑥 1 𝑥 2 ,𝑨= 1 0 0 2 3 2 , 𝒃= 4 12 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 𝑨= 1 0 0 2 3 2 ,𝑩= 2 6 , 𝐴𝐵= 1∗2+0∗6 0∗2+2∗6 3∗2+2∗6 = 2 12 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 Define 𝑨= 2 6 9 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 𝑨= 1 3 1 1 , 𝒙= 𝑥 1 𝑥 2 ,𝒃= 8 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 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 −2 1 0 0 0 0 1 0 0 0 2 3 2 0 1 0 0 0 1

Sub-Matrices Cont. −3 −2 1 0 0 0 0 1 0 0 0 2 3 2 0 1 0 0 0 1 We can rewrite this matrix as: −𝒄 𝟎 𝑨 𝑰