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

By Michael R. Greenberg

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1969). " Addison-Wesley, Reading, Massachusetts. Faddeeva, V. N. (1959). " Dover, New York. Hohn, F. E. (1964). " MacMillan, New York. , and Mine, H. (1965). " MacMillan, New York. School Mathematics Study Group (1960). " Yale Univ. Press, New Haven. Chapter 3 Beyond the Global Optimum Linear and Nonlinear Programming Options The purpose of this chapter is to extend your knowledge beyond the optimal solution to a linear programming problem. The first section focuses on extensions of linear programming: sensitivity analysis, parametric programming, spot testing, and the dual problem.

2) from Xx + 2X2 + X3 = 10 to Xt + 2X2 + X3 = 9, the optimal value of the solution will remain at lOf. The amount of slack from activity X3 will be reduced. Since the slack variable has no value, the optimal value of the problem is intact. If, however, we change either constraints (3) or (4), the optimal solution will change. For example, if we change Eq. (4) from 3X1 + X2 + X5 = 15 to 3Xl + X2 + X5 = 16, the value of the optimal solution will increase from lOf to lOf, an increase off. If the right-hand-side value goes from 15 to 18, the value of the solution changes from 10f to 12.

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. BV b Cj 1 2 3 4 5 3 4 5 6 21 3 0 0 0 -2 2 1 2 1 0 0 4 0 1 0 1 1 2 0 0 4 Zj - Cj ~ 3 -6 0 0 0 Second, we convert a31 into 0 by multiplying row 5 by 2 and adding it to row 3. MATHEMATICAL SOLUTION OF LINEAR PROGRAMMING PROBLEMS / 51 BV b 1 3 5 6 6 -2 2 3 12 0 3 4 5 2 1 0 -1 0 0 0 1 0 1 2 2 1 1 2 Third, we transform element a4l into 0 by multiplying row 5 by —2 and adding it to row 4.

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