DS 523
Spring 2004
Exam 1
Name: KEY
Show all your work.
1.
The following linear programming problem has been solved by the Management
Scientist. Use the output to answer the questions
LINEAR PROGRAMMING PROBLEM
MAX 25X1+30X2+15X3
S.T.
1) 4X1+5X2+8X3<1200
2) 9X1+15X2+3X3<1500
OPTIMAL SOLUTION
Objective Function Value = 4700.000
Variable
X1
X2
X3
Constraint
1
2
Value
Reduced Costs
140.000
0.000
80.000
Slack/Surplus
0.000
0.000
0.000
10.000
0.000
Dual Prices
1.000
2.333
OBJECTIVE COEFFICIENT RANGES
Variable
Lower Limit
Current Value
Upper Limit
X1
X2
X3
19.286
No Lower Limit
8.333
25.000
30.000
15.000
45.000
40.000
50.000
1
RIGHT HAND SIDE RANGES
Constraint
Lower Limit
1
2
a.
Upper Limit
1200.000
1500.000
4000.000
2700.000
666.667
450.000
Give the optimal solution.
x1 = 140
x2 = 0
x3 = 80
b.
Current Value
Objective Function Value = 4700
Which constraints are binding?
4(140) + 5(0) + 8(80) ≤ 1200
1200 ≤ 1200 binding
9(140) + 15(0) + 3(80) < 1500
1500 <1500 binding
c.
What is the dual price for the second constraint? What interpretation does
this have?
Dual price 2 = 2.33. A unit increase in the right-hand side of constraint 2
will increase the value of the objective function by 2.33
d.
Over what range can the objective function coefficient of x2 vary before a
new solution point becomes optimal?
As long as c2 < 40 the solution will be unchanged.
e.
By how much can the amount of resource 2 decrease before the dual price
will change?
1500 – 450 = 1050
2
2.
The optimal solution of the linear programming problem is at the
intersection of constraints 1 and 2.
Max
s.t.
2x1 + x2
4x1 + 1x2 ≤ 400
4x1 + 3x2 ≤ 600
1x1 + 2x2 ≤ 300
x1 , x2 ≥ 0
c1/c2 = - 2
a1/a2 = - 4
a1/a2 = - 4/3
a. Over what range can the coefficient of x1 vary before the current
no longer optimal?
- 4 ≤ - c1/1 ≤ -4/3
solution is
4/3 ≤ c1 ≤ 4
b. Over what range can the coefficient of x2 vary before the current
no longer optimal?
- 4 ≤ - 2/c2 ≤ - 4/3
4/3 ≤ 2/c2 ≤ 4
1/4 ≤ c2/2 ≤ 3/4
1/2 ≤ c2 ≤ 3/2
solution is
c. Compute the dual prices for the three constraints.
Optimal solution:
4x1 +x2 = 400
4x1 + 3x2 = 600
Dual price for 1st constraint:
4x1 + x2 = 401
4x1 + 3x2 = 600
x2 = 400 - 4x
4x1 + 1200 - 12x1 = 600
x2 = 400 - 4(75) = 100
Z = 2(75) + 100 = 250
x1 = 75
x2 = 401 - 4x1
4x1 + 1203 - 12x1 = 600
x1 = 75.375
x2 = 401 - 4(75.375) = 99.5
Z1 = 2(75.375) + 99.5 = 250.25
Dual price = 250.25 - 250 = .25
Dual price for the 2nd constraint:
4x1 + x2 = 400
x2 = 400 - 4x1
4x1 + 3x2 = 601
4x1 + 1200- 12x1 = 601
x2 = 400 - 299.5 = 100.5
x1 = 74.875
Z2 = 2(74.875) + 100.5 = 250.25
Dual price = Z2 - Z = 250.25 - 250 = .25
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Multiple Choice Questions
Select the best answer.
1. If a decision variable is not positive in the optimal solution, its reduced cost is
a. what its objective function value would need to improve before it
could become positive.
b. the amount its objective function value would need to improve before
it would become positive.
c. zero.
d. its dual price.
2. The range of feasibility measures.
a. the right-hand-side values for which the objective function value will
not change.
b. the right-hand-side values for which the values of the decision
variables will not change.
c. the right-hand-side values for which dual prices will not change.
d. each of the above is true.
3. the amount that the objective function coefficient of a decision variable would
have to improve before that variable would have a positive value in the
solution is the
a. dual price.
b. surplus variable.
c. reduced cost.
d. upper limit.
4. Which of the following statements is NOT true?
a. A feasible solution satisfies all constraints.
b. An optimal solution satisfies all constraints.
c. An infeasible solution violates all constraints.
d. A feasible solution point does not have to lie on the boundary of the
feasible region.
5. Slack
a. is the difference between the left and right sides of a constraint.
b. is the amount by which the left side of a ≤ constraint is smaller than
the right side.
c. is the amount by which the left side of a ≥ constraint is larger than the
right side.
d. exists for each variable in a linear programming problem.
6. The improvement in the value of the objective function per unit increase in a
right-hand side is the
a. sensitivity value.
b. dual price.
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c. constraint coefficient.
d. slack value.
7. A constraint that does not affect the feasible region is a
a. non-negativity constraint.
b. redundant constraint.
c. standard constraint.
d. slack constraints.
8. In a linear programming problem, the objective function and the constraints
must be linear functions of the decision variables.
a.
True b.
False
9. The standard form of a linear programming problem will have the same
solution as the original problem.
a.
True b.
False
10. An optimal solution to a linear programming problem can be found at an
extreme point of the feasible region for the problem.
a.
True b.
False
11. The point (3, 2) is feasible for the constraint 2x1 + 6x2 ≤ 30.
a.
True b.
False
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