Download The Simplex Algorithm: A Matrix Approach to Linear Programming and more Exams Mathematics in PDF only on Docsity! 9 Matrix form of the simplex algorithm. 9.1 Initial tableau. The primal problem max when 0 T Ax b c x x ≤ ≥ , with nx ∈ , mb ∈ , so that A is m n× , yields an augmented system of the form x A I by = . To keep the notation simple, continue to write 0 for each matrix with all entries 0, and also I for each identity matrix. The order of any matrix can be deduced from its relative position in the system. For instance, the I in the augmented system must be m m× . With this in mind, the initial tableau has the form 0 0T A I b c . The simplex algorithm assumes that 0 b is a feasible solution of the augmented system. Each ix in x is a non-basic variable, and each jy in y is a basic variable. It is of course in general not true that 0b ≥ , but in this case the first phase of the two-phase method generates a tableau in the form consistent with the simplex algorithm. 9.2 Subsequent tableaux. The application of elementary row operations to convert a non-basic variable to a basic variable is referred to as pivoting. Each elementary row operation corresponds to a matrix, and if several operations are applied in succession, then some matrix again represents the combined effect. To avoid loosing track of the meaning of the various matrices use the following example. Example Consider the primal problem 1 21 2 1 2 3 1 9 51 1 max 2 1 when 0 0 x xx x x x ≤ ≥ . Here 2 1Tc = , 9 5b = , and 3 1 1 1 A = . The vertices of the feasible set are ( )0, 0 ,( )3, 0 , ( )2,3 and ( )0,5 . The initial tableau 0 0T A I b c is given by 3 1 1 0 9 1 1 0 1 5 2 1 0 0 0 . Here 1 2,x x are non-basic variables, and the slack variables 1 2,y y are basic variables. The values are given by 1 20, 0x x= = , and 1 29, 5y y= = . Four different pivots are possible. Two of the pivots turn 1x into a basic variable. The pivot on the first row yields 1 1/ 3 1/3 0 3 0 2/ 3 1/ 3 1 2 0 1/ 3 2/ 3 0 6 − − − . Now 1 2,x y are basic variables and 2 1,x y are non-basic variables. The values are given by 1 23, 0x x= = , and 1 20, 2y y= = . Geometrically the pivot turns the attention away from the vertex ( )0, 0 and directs it to the vertex ( )3, 0 . Since the identity matrix has been replaced by 1/ 3 0 1/ 3 1 − , it follows that this matrix represents the combined effect of the elementary row operations. A quick check confirms that 1/ 3 0 3 1 1 1/ 3 1 11/ 3 1 0 2/ 3 = − . Instead of observing the change of the identity matrix in the initial tableau, examine how the identity matrix in the new tableau is created. It must be that 1/ 3 0 3 0 1 0 1 1 0 11/ 3 1 = − , and hence the combined effect of the elementary row operations is represented by the inverse of the matrix 3 0 1 1 . This matrix is given by the columns in the initial tableau corresponding to the new basic variables in the appropriate order. Let B denote this matrix and let N denote the matrix consisting of all the remaining columns in A I in order from left to right. In the present example N is given by 1 1 1 0 N = .
