Quantitative Analysis Extra Credit: Chapter 7 & 8
True/False. Indicate whether the following statements are True or False and why? (you
will not get full credit without an explanation.)
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.
15
A
I
10
B
5
II
III
C
D
V
IV
0
0
5
Max
s.t.
a.
b.
c.
d.
E
10
15
20X + 10Y
12X + 15Y < 180
15X + 10Y < 150
3X - 8Y < 0
X, Y > 0
Which area (I, II, III, IV, or V) forms the feasible region?
Which point (A, B, C, D, or E) is optimal?
Which constraints are binding?
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.
Plastic
Ink
Assembly
Molding
Time
Fliptop Model
3
5
Tiptop Model
4
4
Available
36
40
5
2
30
The profit for either model is $1000 per lot.
a.
4.
What is the linear programming model for this problem? (show the
objective function and constraints).
Use the following Management Scientist output to answer the questions.
LINEAR PROGRAMMING PROBLEM
MAX 31X1+35X2+32X3
S.T.
1) 3X1+5X2+2X3>90
2) 6X1+7X2+8X3<150
3) 5X1+3X2+3X3<120
OPTIMAL SOLUTION
Objective Function Value =
Variable
Value
X1
X2
X3
13.333
10.000
0.000
Constraint
1
2
3
Slack/Surplus
0.000
0.000
23.333
763.333
Reduced
Cost
0.000
0.000
10.889
Dual Price
-0.778
5.556
0.000
OBJECTIVE COEFFICIENT RANGES
Variable
Lower Limit
Current
Upper Limit
Value
X1
X2
X3
30.000
No Lower
Limit
No Lower
Limit
31.000
No Upper
Limit
35.000
36.167
32.000
42.889
RIGHT HAND SIDE RANGES
Constraint
Lower Limit
1
2
77.647
126.000
Current
Value
90.000
150.000
3
96.667
120.000
a.
b.
c.
d.
Upper Limit
107.143
163.125
No Upper
Limit
Give the solution to the problem.
Which constraints are binding?
What would happen if the coefficient of x1 increased by 3?
What would happen if the right-hand side of constraint 1 increased by 10?