# Applied Linear Programming. For the Socioeconomic and by Michael R. Greenberg

By Michael R. Greenberg

By Michael R. Greenberg

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Additional resources for Applied Linear Programming. For the Socioeconomic and Environmental Sciences

Example text

Their b/a ratios are equal), then a degenerate solution in which one of the basis variables assumes a value of zero follows. Theoretically, the solution process could cycle, repeating solutions rather than moving toward an optimum. The causes of and remedies for degenerate solutions are discussed in Chapter 3. c. SUMMARY This chapter has presented the simplex method of solving linear programming problems. All linear programming problems may be solved by the single-phase or two-phase processes.

BV b Cj 1 3 4 5 6 21 12 0 0 0 -2 2 4 2 3 4 5 2 1 0 0 4 0 1 0 -2 0 0 1 50 / 2: ALGEBRAIC METHODS Fourth, the Z, — C, values are established. 1 0 -3 Z J Cj 2 3 4 5 0 0 0 0 6 0 0 0 3 -6 0 0 0 Zj - Cj This is a minimization problem. We are looking for positive Z 7 — Cj values. Vector 1 has the only positive value and will be put into the basis. 5 2 For V5,b/a = ¥ = 3 4 The vector V5 has the lowest positive b/a ratio. We will therefore insert vector 1 and remove vector 5. BV b Cj 3 4 5 6 21 12 0 0 0 Zj - C J 1 2 2 1 0 0 4 0 1 0 [4] -2 0 0 1 -2 2 3 Z =0 3 4 5 -6 0 0 0 First, we convert a51 into 1 by dividing row 5 by 4.

Do the element signs and Zj — Cj values allow the solution to be improved? Choose column to put into the basis. Choose row to remove from the basis. Perform iteration Fig. 2 Flow diagram of single-phase process. 38 / 2: ALGEBRAIC METHODS At this point, one vector could be inserted randomly into the basis and another removed from it. One or two undirected iterations at the beginning would probably move the solution toward the optimum. However, unless the user is somehow psychic, trial-and-error iterations lead only slowly, if at all, to an optimum.