Operations Research Homework #4: The Dual Simplex Method
Due date: November 10, 2005 1. Consider the following Linear Programming problem:
min 2x1+x2 + 3x3
s.t. 5x1+ 2x2+ 7x3 = 420 3x1+ 2x2+ 5x3−x4 = 280 (P1)
x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥0.
(a) Identify all basic dual feasible solutions of the problem.
(b) Using the simplex method to solve the Phase I problem of (P).
(c) Is the solution obtained by (b) a dual feasible solution?
(d) Select {x1, x4} as the basic variables and {x2, x3} as the nonbasic variables and generate a basic solution of the following LP
min 2x1+x2+ 3x3
s.t. 5x1+ 2x2+ 7x3 = 7 3x1+ 2x2+ 5x3−x4 = 4 (P2)
x1 ≥0, x2 ≥0, x3 ≥0, x4 ≥ 0.
Verify that this basic solution is a dual feasible basic solution of (P2).
(e) Select the same basic and nonbasic variables as in (d) and generate a basic solution of (P1). Does the basic solution satisfy the primal and dual feasibility of (P1)?
(f) From the original idea of revised simplex method, in order to find the primal fea- sibility, the basic variable with negative value should be dropped from basic index set and find a nonbasic variable to enter the basic index set. Use the basic solution in (e) to explain which nonbasic variable should be used to keep dual feasibility.
(g) From (f), when you find a new basic solution, is it an optimal solution of the problem (P1)? If not, please perform the iteration in (f) again. If yes, please explain why it is an optimal solution clearly.
2. Consider the following linear program:
min 2x1+x2−x3
s.t. x1+ 2x2+x3 ≤8
−x1+x2 +−2x3 ≤4 (P3)
x1 ≥0, x2 ≥0, x3 ≥0.
In (P3), the optimal solutions are (0,0,8) and the optimal value is −8. The questions (a)-(2) below modify either the objective function or the constraints of (P3). Please check whether or not the solution (0,0,8) is still an optimal solution of the modified LP.
If not, please find an optimal solution of the modified LP using sensitivity analysis.
(a) Let the cost coefficient of x2 be changed from 1 to 6.
(b) Let the coefficient of x2 in the first constraint be changed from 2 to 2.5.
(c) Add one more variable x4 into (P3). Let the new objective function be 2x1+x2− x3+ 4x4 and the new constraints sets be
{x= (x1, x2, x3, x4)≥0|x1+ 2x2 +x3+x4 ≤8,−x1 +x2−2x3+ 2x4 ≤4}. (d) If you try to reduce the optimal value by increasing the right-hand side of the first
or the second constraints, which one would you choose?
(e) Add one more constraint x2 +x3 = 3. (You may need to perform an iteration of Dual Simplex Method, please see (f) in problem 1.)
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