1. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables.
2. Only binding constraints form the shape (boundaries) of the feasible region.
3. The constraint 5x1 - 2x2 < 0 passes through the point (20, 50).
4. The point (3, 2) is feasible for the constraint 2x1 + 6x2 £ 30.
5. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem.
6. The reduced cost for a positive decision variable is 0.
7. For any constraint, either its slack/surplus value must be zero or its dual price must be zero.
8. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective.
9. For a minimization problem, a positive dual price indicates the value of the objective function will increase.
10. The dual price associated with a constraint is the improvement in the value of the solution per unit decrease in the right-hand side of the constraint.
Part II. Short Problems.
1. Solve the following system of simultaneous equations.
6X + 2Y = 50
2X + 4Y = 20
2. Use this graph to answer the questions.
Max 20X + 10Y
s.t. 12X + 15Y < 180
15X + 10Y < 150
3X - 8Y < 0
X , Y > 0
a. Which area (I, II, III, IV, or V) forms the feasible region?
b. Which point (A, B, C, D, or E) is optimal?
c. Which constraints are binding?
d. Which slack variables are zero?
3. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below.
Fliptop Model Tiptop Model Available
Plastic 3 4 36
Ink Assembly 5 4 40
Molding Time 5 2 30
The profit for either model is $1000 per lot.
a. What is the linear programming model for this problem? (show the
objective function and constraints).
4. Use the following Management Scientist output to answer the questions.
LINEAR PROGRAMMING PROBLEM
Objective Function Value = 763.333
Variable Value Reduced Cost
X1 13.333 0.000
X2 10.000 0.000
X3 0.000 10.889
Constraint Slack/Surplus Dual Price
1 0.000 -0.778
2 0.000 5.556
3 23.333 0.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 30.000 31.000 No Upper Limit
X2 No Lower Limit 35.000 36.167
X3 No Lower Limit 32.000 42.889
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 77.647 90.000 107.143
2 126.000 150.000 163.125
3 96.667 120.000 No Upper Limit
a. Give the solution to the problem.
b. Which constraints are binding?
c. What would happen if the coefficient of x1 increased by 3?
d. What would happen if the right-hand side of constraint 1 increased by 10?