Assignment:
Mathematics Optimization
Q1. Consider the LP Max z=c1x1+2x2+c3x3
Subject to x1 + 5x2+a1x3 ≤ b1
X1-5x2+a2x3≤ b2
X1, x2, x3 ≥ 0
The optimal tableau for this LP is
1 d1 2 1 0 30
0 d2 -8 -1 1 10
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0 d3 -7 d4 0 z – 150
Without using the simplex method, determine all unknown constants in this problem
(i.e., a1,a2,b1,b2, c1, c3, d1,d2, d3, d4)
Q2. Consider the following problem:
Maximize z=6x1+8x2
Subject to 5x1+2x2 ≤ 20
x1+2x2≤ 12
x1, x2 ≥ 0
(a) Sketch the feasible set and solve the program geometrically
(b) Determine all the basic solutions for this program and indicate them on the sketch in (a). Which ones are feasibke? (there are 6 basic solutions, not all feasible).
(c) Solve the program by the simplex method. For each tableau, indicate which feasible solution corresponds to it
(d) Write the dual of this program. For each primal basic solution, determine the corresponding dual basic solution that satisfies the complementary slackness principle. Which ones are dual feasible?
Q3. Consider the system of equations –x1+x2-x3=3, -x1+2x2-x4=2, and x1+x2+x5=2
(a) by converting the equations into inequalities in two variables, show geometrically that there are no nonnegative solutions to this system.
(b) use the simplex method to show the same thing
Q4. Use the Karush-Kuhn- tucker conditions to solve:
Maximize z=2x1+x2+3x3
Subject to x1+2x2+x3≤ 12
X1,x2,x3 ≥ 0
Provide complete and step by step solution for the question and show calculations and use formulas.