Computational Methods for Management and Economics Carla Gomes

1. Computational Methods forManagement and EconomicsCarla Gomes Module 6d Matrix notation for LP problems and revised simplex method

2. How do we describe an LP problem in matrix notation? • Note on revised Simplex Method

3. How do we describe an LP problem in matrix notation?

4. Brief overview of matrices

5. Matrices and Vectors Matrix – an rectangular array of numbers m x n matrix m rows n columns. m x n is the order of the matrix. The number in the ith row and jth column of A is called the ijth element of A and is written aij.

6. Two matrices A = [aij] and B = [bij] are equal if and only if A and B are the same order and for all i and j, aij = bij. A = B if and only if x = 1, y = 2, w = 3, and z = 4

7. Vectors – any matrix with only one column is a column vector. The number of rows in a column vector is the dimension of the column vector. An example of a 2 X 1 matrix or a two-dimensional column vector is shown to the right. Rm will denote the set all m-dimensional column vectors Any matrix with only one row (a 1 X n matrix) is a row vector. The dimension of a row vector is the number of columns.

8. Vectors appear in boldface type: for instance vector v. Any m-dimensional vector (either row or column) in which all the elements equal zero is called a zero vector (written 0). Examples are shown to the right.

9. Any m-dimensional vector corresponds to a directed line segment in the m-dimensional plane. For example, the two-dimensional vector u corresponds to the line segment joining the point (0,0) to the point (1,2) The directed line segments (vectors u, v, w) are shown on the figure to the right.

10. Scalar multiple of a matrix: Example addition of two matrices (of same order):

11. Example Transpose of a matrix Given any m x n matrix Observe:

12. c11 c12 c21 c22 C = Example Given to matrices A and B, the matrix product of A and B (written AB) is defined if and only if the number of columns in A = the number of rows in B. The matrix product C = AB of A and B is the m x n matrix C whose ijth element is determined as follows: Cij = scalar product of row i of A x column j of B Matrix multiplication AB=C Note: Matrix C will have the same number of rows as A and the same number of columns as B.

13. Matrices can simplify and compactly represent a system of linear equations. A system of linear equations may be written Ax = b and is called it’s matrix representation. The matrix multiplication (using only row 1 of the A matrix for example) confirms this representation thus:

14. c = [c1 c2 … cn ] x = x1 x2 … xn LP problem in matrix notation Max (or Min) Z = c1 x1 + c2x2 + …. + cnxn Max (or Min) Z = cx Subject to: Subject to: a11 x1 + a12 x2 + … + a1n xn≤ b1 a21 x1 + a22 x2 + … + a2n xn≤ b2  Ax≤ b … x≥ 0 Where: am1 x1 + am2 x2 + … + amn xn≤ bm x1, x2, …, xn≥ 0

15. 1 0 • 0 1 • 3 2 • 4 • 12 • 18 A = b = LP problem in matrix notation Max Z = 3 x1 + 5x2Max Z = cx Subject to: Subject to: x1 + ≤ 4 2 x2≤ 12 Ax≤ b 3 x1 + 2 x2≤ 18 x≥ 0 Where: x1, x2≥ 0 c = [3 5] x = x1 x2

16. ≤ x1 x2 x≥ 0 • 1 0 • 0 1 • 3 2 • 4 • 12 • 18 Wyndor Glass LP formulation in matrix notation Max Z = [3 5 ] x1 x2 Subject to:

17. Ax = b can sometimes be abbreviated A|b. For example, given: A|b is written:

18. Revised Simplex Method • Streamlined versions of the simplex method for computer implementations do not follow the original simplex algorithm (algebraic or tabular form)  The Revised Simplex Algorithm explicitly uses matrix manipulations it computes and stores only the information required for each iteration, namely the coefficients of the non-basic variables in Equation 0, the coefficients of the basic entering variable in the other equations, and the RHS’s of the equations.

19. Revised Simplex Method – Wyndor Glass