10 Advanced Topics in Linear Programming

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± ± ± ± ± ± ± ± ± ± ± Advanced Topics in Linear Programming In this chapter, we discuss six advanced linear programming topics: the revised simplex method, the product form of the inverse, column generation, the Dantzig–Wolfe decomposi- tion algorithm, the simplex method for upper-bounded variables, and Karmarkar’s method for solving LPs. The techniques discussed are often utilized to solve large linear programming problems. The results of Section 6.2 play a key role throughout this chapter. 10.1 The Revised Simplex Algorithm In Section 6.2, we demonstrated how to create an optimal tableau from an initial tableau, given an optimal set of basic variables. Actually, the results of Section 6.2 can be used to create a tableau corresponding to any set of basic variables. To show how to create a tableau for any set of basic variables BV, we Frst describe the following notation (assume the LP has m constraints): BV ± any set of basic variables (the Frst element of BV is the basic variable in the Frst constraint, the second variable in BV is the basic variable in the second constraint, and so on; thus, BV j is the basic variable for constraint j in the desired tableau) b ± right-hand-side vector of the original tableau’s constraints a j ± column for x j in the constraints of the original problem B ± m ² m matrix whose j th column is the column for BV j in the original constraints c j ± coefFcients of x j in the objective function c BV ± 1 ² m row vector whose j th element is the objective function coefFcient for BV j u i ± m ² 1 column vector with i th element 1 and all other elements equal to zero Summarizing the formulas of Section 6.2, we write: B ³ 1 a j ± column for x j in BV tableau (1) c BV B ³ 1 a j ³ c j ± coefFcient of x j in row 0 (2) B ³ 1 b ± right-hand side of constraints in BV tableau (3) c BV B ³ 1 u i ± coefFcient of slack variable s i in BV in row 0 (4) This chapter covers topics that may be omitted with no loss of continuity.

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c BV B ± 1 ( ± u i ) ² coefﬁcient of excess variable e i in BV row 0 (5) M ³ c BV B ± 1 u i ² coefﬁcient of artiﬁcial variable a i in BV row 0 (6) (in a max problem) c BV B ± 1 b ² right-hand side of BV row 0 (7) If we know BV, B ± 1 , and the original tableau, formulas (1)–(7) enable us to compute any part of the simplex tableau for any set of basic variables BV. This means that if a com- puter is programmed to perform the simplex algorithm, then all the computer needs to store on any pivot is the current set of basic variables, B ± 1 , and the initial tableau. Then (1)–(7) can be used to generate any portion of the simplex tableau. This idea is the basis of the revised simplex algorithm. We illustrate the revised simplex algorithm by using it to solve the Dakota problem of Chapter 6. Recall that after adding slack variables s 1 , s 2 , and s 3 , the initial tableau (tableau 0) for the Dakota problem is max z ² 60 x 1 ³ 30 x 2 ³ 20 x 3 s.t. 8 x 1 ³ 1. 6 x 2 ³ x 3 ³ s 1 ³ s 2 ³ s 2 ² 48 s.t. 4 x 1 ³ 1. 2 x 2 ³ 1.5 x 3 ³ s 2 ³ s 2
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