LINEAR PROGRAMMING EXERCISES
Exercise 1. Given a problem f ( x ) = −8 x1 + 3x2 + 2 x3 − 11x4 → max
 2 x1 − x2
−3x + x
2
 1
x
 1

3x2

 4 x1 + 7 x2
+ x3
+ 4 x3
+ 2 x4
− x4
+ 6 x4
− 5 x3
 −2
 4
 5
 3
 10
Prove that x0 = ( − 1, 2, 0, 1) is an extreme optimal solution of this problem.
Exercise 2. Given linear programming problem (I) as follow:
f ( x ) = 2 x1 − x2
2 x1 − x2
2 x1 − 3x2
+ 2 x3
+ 3x3
− x3
xj  0
− x4
→ max
=
6

6
( j =1  4)
and vector x0 = ( 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 x1 + 4 x2 + 3x3 + x4 → min
4 x1 − 3x2 + 3x3
 x − 2x + x
2
3
 1
2 x − x
+ 3x3
2
 1
x j  0
j = 1, 4

(
)
+ x4
+ x4
Prove that (I) always has an extreme optimal solution.
Exercise 4. Given problem (I) as follow:
 31
 6
 33
f ( x ) = 2 x1 + 3x2 − 4 x3 + 2 x4 → min
 x1 − 4 x2 − 2 x3

3x2 + x3


2 x2 + 3x3

x j  0
j = 1,5

(
)
+ x4
+ x5
+ 2 x5
= 2
 6
 6
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 x1 = 10 .
Exercise 5. Given problem (I) as follow:
f ( x ) = −2 x1 − 5 x2 − 3x3 + x4 − 2 x5 → max

 x1 − x2

− x1 + 4 x2

− x1 − 2 x2

− 2 x3
+ x3
+
−
−
x3
x4
3
x4
2
( j = 1,5)
xj  0
− 2 x5
 6
= −24
+ 3x5

(I )
33
a. Solve this problem using simplex method.
b. Find the set of optimal solutions and an optimal solution which is not extreme point.
Exercise 6. Given problem (I) as follow:
f ( x ) = 2 x1 − x2 + 3x3 + x4 − 3x5 → max
4 x1
x
 1


 x1 +
− 2 x3
x2
xj  0
− 6 x3
+ x3
+ x4
+ 2 x4
+ x4
+ x4
( j = 1,5)
− 4 x5
− x5
+ 3x5
− 2 x5
= 8
 −20
= 15
= 14
a. Solve this problem using simplex method.
()
b. Find a feasible point x which has x3  0 and f x = −27 .
Exercise 7. Given problem (I) as follow:
f ( x ) = x1 + x2 + c3 x3 + 2 x4 + 2 x5 → min
3x1 + x2

6 x1


xj  0
+ x3
+ 2 x3
− 2 x3
+ 2 x4
+ 5 x4
+ 2 x4
+ 3x5
− 3x5
+ 3x5
= 18
 42
 14
(j = 1,5)
a. For c3 = 2, solve this problem using simplex method.
()
b. Find c3 such that the problem has extreme point x and f x = 10.
Exercise 8. Given linear programming problem (I) as follow:
f ( x ) = 2 x1 + 5 x2 + 3x3 − x4 + 2 x5 → min
 x1 − x2 − 2 x3
− x + 4 x + x
2
3
 1

− x − 2 x − x
2
3
 1

j = 1,5
 x j  0
(
)
+
−
x4
3
x4
2
− 2 x5
 6
= −24
+ 3x5

33
(I )
a. Is vector x0 = ( 6, 0, 0, 0, 9 ) an extreme optimal solution of (I)?
b. Find the set of optimal solutions of the dual problem.
Exercise 9. Given a problem:
f ( x ) = 11x1 + 9 x2 − 8 x3 + 8 x5 − 2 x6 − x7 → min
+ x5
 2 x1 − x2
− x
+ x3 + x4 + 3x5
 1
 x1 + x2 − 2 x3
x
− 4 x2 + 4 x3
+ 3x5
 1
x2 , x3 , x5 , x7  0

+ x6
− x6
− x6
+ 5 x6
−
x7
− 2 x7
= −1
= 5
 −6
= 12
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.
Exercise 10. Given problem (I):
f ( x ) = −8 x1 + 6 x2 + 4 x3 + 5 x4 → min
 x1
−2 x + x
 1
2

 3x1 − x2
 x1 , x2 , x4  0
− 2 x3
− x3
+ 2 x3
+ x4
+ 3x4
− 6 x4
 7
= −4
 5
And vector x0 = ( 3, 0, −2, 0 ) .
a. Is x 0 an extreme optimal solution of (I)?
b. Find the set of optimal solutions of the dual problem. Find the optimal solution of the
dual problem having y1 = −3 .
Exercise 11. Given problem (I):
f ( x ) = −2 x1 − x2 − 4 x3 − 5 x4 − 2 x5 − 4 x6 → max
x2 + 2 x3 − x4

2x
− x3 + 2 x4
 1
− x + 2 x + 4 x + x
2
3
4
 1
 x j  0, j = 1,6

a. Solve (A) using simplex method.
− 2 x5 + 3x6  −1
+ 3x5 + x6  14
− 2 x5 + 5 x6  7
b. Use the solution in a., find the optimal solution x having x4  0 when we have an extra
()
constraint f x  −5 .
Exercise 12. Solve the following problem using simplex method
f ( x ) = 2 x1 − 6 x2 − 3x3 + 3x4 − x5 → max
 2 x1 − 2 x2 + x3

3x2 + 3x3

4x + x
− 2 x3
2
 1
− x1
+ 2 x3

x 0
j = 1,5

 j
(
)
+ x4
+ 2 x4
+ 4 x4
+ x4
+
x5
= 8
= 51
 0
= 40
Exercise 13. Given problem (I)
f ( x ) = −2 x1 − 4 x2 − 3x3 − x4 → max
 4 x1 − 3x2 + 3x3 + x4  31
− x + 2 x − x
 −6
2
3
 1
2x − x
+ 3x3 + x4  33
2
 1
x j  0
j = 1,4

(
)
a. Solve this problem using simplex method; find the optimal solution having x4 = 28.
b. Find the constraint c4 such that x0 = (0, 1, 8, 10) is the optimal basic feasible solution.
Exercise 14. Given a problem:
f ( x ) = 2 x1 − 2 x2 + x3 + 5 x4 − 3x5 → max
 2x1

− x 1
 4x
 1
+ 3x 2
+ 2x 2
− x3
+ x3
− x4
+ x4
+ 2x 5
− x5
+ 3x 5
x2  0; x3  0; x4  0
 −3
= 7
 4
a. Find x4 such that x0 = (0, 2, 0, x4, 0) is a solution.
b. Then is x0 a basic feasible solution, optimal solution of the problem?
Exercise 15. Given problem (I):
f(x) = - 4x1 + x2 – 4x3 + 3x4 + 2x5  min
 2 x1 − x 2

− x1 + 3x 2

2x 2

+ x3
+ 2x 3
+ 4x 4
+ 2x 4
− x4
− 5x 5
+ x5
 − 10

3
 5
a. Write the dual problem of (I) and its optimal basic feasible solutions.
b. Find the set of optimal solutions of the primal problem, deduce that (I) is solvable for
all b.
c. Prove that (I) is insolvable if either the second condition has the sign () or f(x) => max.
Exercise 16. Given a problem:
f(x) = 11x1 + 9x2 - 8x3 + 8x5 - 2x6 - x7 => min
 2x1
− x
 1

 x1
 x1
−
x2
+ x2
− 4x 2
+ x3
− 2x 3
+ 4x 3
+ x4
+ x5
+ 3x 5
+ 3x 5
+ x6
− x6
− x6
+ 5x 6
−
x7
− 2x 7
= −1
= 5
 −6
= 12
x2, x3, x5, x7  0
a. Find the feasible domain and basic feasible solutions of the dual problem.
b. Prove that the problem above is solvable. Find the optimal basic feasible solutions of
primal-dual problems.
c. In the dual problem, given f ( y ) → min ; find the optimal solution of the new primaldual problems.
Exercise 17. Given problem (A):
f(x) = 4x1 + 7x2 + 2x4 - 6x5 - x6 => min
− 2 x 1

 x1
 3x
 1
− x3
+ 2x 2
+ 5x 4
− 2x 4
− 4x 4
+ 3x 5
+ x6
+ 3x 6
+ 6x 6
 9
 −8
 15
x1, x4, x6  0
a. Prove that (A) always has an optimal basic feasible solution.
b. Find the sets of optimal solutions and optimal basic feasible solutions of the dual
problem.
c. Prove that (A) is solvable for all b = (b1, b2, b3); find the optimal value of the objective
function in this case.