Download Linear Programming Exercises: Proving Optimal Solutions and Solving with Simplex Method and more Exams Programming Methodologies in PDF only on Docsity!
LINEAR PROGRAMMING EXERCISES
Exercise 1. Given a problem f ( x )= − 8 x 1 + 3 x 2 + 2 x 3 − 11 x 4 →max
1 2 3 4 1 2 3 4 1 4 2 3 1 2
x x x x x x x x x x x x x x
^ −^ +^ +^ ^ −
+^
+^
Prove that x^0 = ( −1, 2, 0, 1) is an extreme optimal solution of this problem.
Exercise 2. Given linear programming problem (I) as follow:
1 2 3 4 1 2 3
2 2 max 2 3 6 2 3 6 j^0 (^1 4)
f x x x x x x x x x x x j
x
and vector x^0 =( 0, 0, 2, 0 ).
a. Prove that x^0 is an extreme solution of (I). b. Solve (I) using simplex method with the initial extreme solution x^0
Exercise 3. Given problem (I) as follow:
f ( x )= 2 x 1 + 4 x 2 + 3 x 3 + x 4 →min
1 2 3 4 1 2 3 1 2 3 4
j^0 1, 4
x x x x x x x x x x x x j
^ −^ +^ +^
Prove that (I) always has an extreme optimal solution.
Exercise 4. Given problem (I) as follow:
f ( x )= 2 x 1 + 3 x 2 − 4 x 3 + 2 x 4 →min
1 2 3 2 3 5 2 3 4 5
j^0 1,
x x x x x x x x x x x j
^ −^ −^ =
Solve this problem using simplex method and find the set of optimal solutions. Show that the
optimal solution which is not an extreme point has x 1 (^) = 10
Exercise 5. Given linear programming problem (I) as follow:
f ( x )= 2 x 1 + 5 x 2 + 3 x 3 − x 4 + 2 x 5 →min
1 2 3 1 2 3 4 5 1 2 3 4 5
j^0 1,
x x x x x x x x x x x x x I
x j
^ −^ −^
^ =
a. Is vector x^0 =( 6, 0, 0, 0, 9 )an extreme optimal solution of (I)?
b. Find the set of optimal solutions of the dual problem.
Exercise 6. Given a problem:
f ( x )= 11 x 1 + 9 x 2 − 8 x 3 + 8 x 5 − 2 x 6 − x 7 →min
1 2 5 6 7 1 3 4 5 6 1 2 3 6 7 1 2 3 5 6 2 3 5 7
x x x x x x x x x x x x x x x x x x x x x x x x
^ −^ +^ +^ −^ =^ −
+^ −^ −^ −^ ^ −
a. Find the feasible set and extreme points of the dual problem.
b. Prove that the above problem is solvable; Find the set of optimal solution of this problem.
