Exercises for Linear Programming (Cht. 7)
BSNS2120, J. Wang
1. There is one, and only one, objective function in a linear program.
a. True
b. False
2. Which are the non-negative constraints of a linear program?
a. X1, X2 >0
b. X1, X2>=0
c. X1+X2>0
d. X1+X2>=0
3. Every linear program has "non-negative constraints".
a. True
b. False
4. Equation X1+2X2=5 corresponds to a __________ in a two-dimensional coordinate system.
a. point
b. straight line
c. circle
d. parabola
5. In a linear program, number of variables must be less than number of constraints.
a. True
b. False
6. A constraint must be an inequality (either "<=" or ">="). A constraint cannot be an equation
("=").
a. True
b. False
7. A solution is feasible if it makes
a. all the constraints hold.
b. at least one constraint hold.
8. RHS stands for "right-hand-side".
a. True
b. False
9. There is no variable term in RHS.
a. True
b. False
10. Does (X1=2, X2=1) satisfy the constraint 2X1-3X2<=2?
a. Yes
b. No
11. Is (X1=5, X2=10) feasible to the LP {Min X1+X2, s.t. X1+0.4X2>=5, X1+X2>=8)?
a. Yes
b. No
12. If a linear program is infeasible then
a. the objective function value can be as large as possible in the feasible region.
b. the feasible region is empty.
c. there is no feasible solution at all for this program.
d. there is no solution satisfying all the constraints.
e. All of b, c, and d.
13. Suppose a maximization linear program is infeasible. If we change the objective from
maximization to minimization, then it is still infeasible.
a True
b. False
14. Linear program {Max 3X1+2X2, s.t. X1+X2>=10}
a. is infeasible.
b. has unbounded optimal solution.
c. has multiple optimal solution.
d. (X1=10,000, X2=20,000) is the optimal solution.
15. Suppose (X1=3, X2=1) is a feasible solution of a linear program. If the linear program’s
objective function changes while its constraints remain unchanged, then, in the new linear
program,
a. (X1=3, X2=1) is no longer a feasible solution.
b. (X1=3, X2=1) is still a feasible solution.
c. Neither of above. (X1=3, X2=1) can be, or can be not, a feasible solution.
16. Suppose the optimal objective function value of a maximization linear program problem is
470. Suppose X=(X1=5, X2=7) is a feasible solution of the same linear program, but it is not
optimal. Let Z denote the objective function value associated with X. Then
a. Z<470.
b. Z>470.
c. Z=470.
d. Given information is not sufficient to decide.
For #17 - #20
Given definitions of variables:
X1 = number of Product 1 to produce
X2 = number of Product 2 to produce
17. Write a constraint representing “at least 38 Product 1’s must be produced”.
18. Write a constraint representing “the amount of Product 2’s cannot be more than 245.”
19. Write a constraint representing “amount of Product 1 cannot be less than Product 2”.
20. Write a constraint representing “for each Product 2 produced, at least three Product 1’s must
be produced.”
Answers:
1.a 2.b 3.a 4.b 5.b 6.b 7.a 8.a 9.a 10.a 11.b 12.e 13.a 14.b 15.b 16.a
17. X1 ≥ 38
18. X2 ≤ 245
19. X1 ≥ X2, that is X1X2 ≥ 0
20. X1/X2 ≥ 3, that is X1 ≥ 3X2, that is X13X2 ≥ 0.