The Theory of the Simplex Method

1. The Theory of the Simplex Method Chapter 5: Hillier and Lieberman Dr. Hurley’s AGB 328 Course

2. 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 Solution, Basic Variables, Non-Basic Variables, Vector of Basic Variables, Basis Matrix

3. 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

4. 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

5. The Simplex Method in Matrix Form Cont. c=, cT=

6. Wyndor Problem in Matrix Form

7. 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

8. 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

9. Important Rules/Facts of Matrices Cont. • Example of matrix multiplication using Wyndor’s constraints evaluated at Wyndor’s optimal

10. 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

11. 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

12. 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

13. 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

14. 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

15. Quick Matrix Exercise • Define • 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 to many rows or columns before you press F2 when you attempt to find these answers in Excel?

16. Another Matrix Example • Suppose we had the following: • x1+ 3x2= 8 • x1+ x2= 4 • We could put this problem in the following matrix notation • , • 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

17. 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

18. Sub-Matrices Cont. • We can rewrite this matrix as: