Linear Programming Exercises: Proving Optimal Solutions and Solving with Simplex Method, Exams of Programming Methodologies

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 ( ) 1 2 3 48 3 2 11 maxf x x x x= − + + − →x 1 2 3 4 1 2 3 4 1 4 2 3 1 2 2 2 2 3 4 4 6 5 3 5 3 4 7 10 x x x x x x x x x x x x x x − + +  −  − + + −   +   −   +  Prove that ( )0 1, 2, 0, 1= −x is an extreme optimal solution of this problem. Exercise 2. Given linear programming problem (I) as follow: ( ) 1 2 3 1 2 3 4 1 2 3 2 2 max 2 3 6 2 3 6 0 ( 1 4)j f x x x x x x x x x x x j = − + → − + − = − −   =  x and vector ( )0 0, 0, 2, 0=x . a. Prove that 0 x is an extreme solution of (I). b. Solve (I) using simplex method with the initial extreme solution 0 x Exercise 3. Given problem (I) as follow: ( ) 1 2 3 42 4 3 minf x x x x= + + + →x ( ) 1 2 3 4 1 2 3 1 2 3 4 4 3 3 31 2 6 2 3 33 0 1,4j 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: ( ) 1 2 3 42 3 4 2 minf x x x x= + − + →x ( ) 1 2 3 2 3 5 2 3 4 5 4 2 2 3 6 2 3 2 6 0 1,5j 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 1 10x = Exercise 5. Given linear programming problem (I) as follow: ( ) 1 2 3 4 52 5 3 2 minf x x x x x= + + − + →x ( ) 1 2 3 1 2 3 4 5 1 2 3 4 5 2 6 4 2 24 3 ( ) 2 3 33 2 0 1,5j x x x x x x x x I x x x x x x j − −   − + + + − = −   − − − − +     = a. Is vector ( )0 6, 0, 0, 0, 9=x an extreme optimal solution of (I)? b. Find the set of optimal solutions of the dual problem. Exercise 6. Given a problem: ( ) 1 2 3 5 6 711 9 8 8 2 minf x x x x x x= + − + − − →x 1 2 5 6 7 1 3 4 5 6 1 2 3 6 7 1 2 3 5 6 2 3 5 7 2 1 3 5 2 2 6 4 4 3 5 12 , , , 0 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.