Chapter 5 , Operations Research ( OR ) Kent Andersen

where the quantities involved in the above equalities are given as follows. Recall that AB denotes the (column induced) submatrix of A generated by B. (i) The basic solution x for the problem (P). (ii) The reduced costs r j := cj − (a.j) T y, where y is the dual basic solution, and a.j denotes the j column of A. (iii) The one-to-one mapping j() : {1, 2, . . . ,m} → B. Given i ∈ {1, 2, . . . ,m}, j(i) denotes the basic variable j(i) ∈ B such that (AB) a.j(i) is the i th unit vector. (iv) For every non-basic variable j ∈ N , the m-dimensional vector ā.j := (AB) a.j . Observe that the mapping j() essentially answers the question Which basic variable is involved in the i row of the simplex tableau? The answer to this question is, of course, the variable xj(i). We now give the geometric intuition behind the simplex tableau. Suppose we add the following redundant/obvious equalities to the system (1)-(2).