Chapter 4—Sensitivity Analysis and the Simplex Method
MULTIPLE CHOICE
1. When a manager considers
considers the effect of changes in an LP model's coefficients he/she is performing
performing
a. a random analysis.
b. a coefficient analysis.
c. a sensitivity analysis.
d. a qualitative analysis.
ANS: C
PTS: 1
2. The coefficients in an LP
LP model
model (c j , a ij , b j ) represent
a. random variables.
b. numeric constants.
c. random constants.
d. numeric variables.
ANS: B
PTS: 1
3. A manager should consider how sensitive
sensitive the model
model is to changes
changes in all of the following except
a. differential coefficients.
b. objective function coefficients.
coefficients.
c. constraint coefficients.
d. right-hand side values for constraints.
ANS: A
PTS: 1
4. Benefits of sensitivity analysis include all the following
following except:
except:
a. provides a better picture
picture of how solutions
solutions change
change as model
model factors change.
b. fosters managerial acceptance of the optimal
optimal solution.
c. overcomes management skepticism of optimal solutions.
d. answers potential
potential managerial
managerial questions
questions regarding the solution
solution to an LP problem.
ANS: B
PTS: 1
5. Risk Solver Platform (RSP) provides sensitivity
sensitivity analysis information
information on all of the following
following except the
a. range of values
values for objective
objective function coefficients which
which do not
not change optimal solution.
solution.
b. impact on optimal objective
objective function value of changes in constrained
constrained resources.
c. impact on optimal
optimal objective
objective function value
value of changes
changes in value
value of decision
decision variables.
d. impact on right hand
hand sides of changes in constraint
constraint coefficients.
coefficients.
ANS: D
PTS: 1
6. The sensitivity
sensitivity analysis provides information
information about which of the following:
following:
a. the impact
impact of a change
change to an objective
objective function
function coefficient.
coefficient.
b. the impact of a change in a resource level.
c. the impact of adding simple upper
upper or lower bounds
bounds on a decision variable.
d. all of these.
ANS: D
PTS: 1
7. Risk Solver
Solver Platform (RSP) provides all of the following
following reports
reports except
a. Answer
b. Sensitivity
c. Cost performance
d. Limits
ANS: C
PTS: 1
8. The Cell Value column
column in the Solver Answer Report
Report shows
shows
a. which constraints are binding.
b. final (optimal) value assumed by each constraint cell.
c. objective function values.
d. Right hand sides of constraints.
ANS: B
PTS: 1
9. Meaningful Risk Solver Platform (RSP) sensitivity report headings can be defined
defined
a. by adding cell notes to spreadsheet cells.
b. by using the Guess button
button in the Risk Solver Platform (RSP) dialog box.
c. by carefully labeling rows and
and columns
columns in the spreadsheet
spreadsheet model.
model.
d. naming cells in the spreadsheet model.
ANS: C
PTS: 1
10. The difference between the right-hand
right-hand side (RHS) values of the constraints
constraints and the final
final (optimal)
value assumed by the left-hand side (LHS) formula for each constraint is called the slack and is found
in the .
a. Status report
b. Slack report
c. Results report
d. Cell Value report
ANS: C
PTS: 1
11. A binding
binding greater than or equal
equal to ( ≥) constraint in a minimization problem means that
a. the variable
variable is up against an upper limit.
b. the minimum requirement for the constraint
constraint has just been met.
c. another constraint is limiting the solution.
d. the shadow price for
for the constraint will be positive.
positive.
ANS: B
PTS: 1
12. A binding
binding less than or equal
equal to ( ≤) constraint in a maximization problem means
a. that all of
of the resource
resource represented by the constraint is consumed in the solution.
b. it is not a constraint that
that the level curve contacts.
c. another constraint is limiting the solution.
d. the requirement
requirement for the constraint has been exceeded.
ANS: A
PTS: 1
13. Binding constraints have
a. zero slack.
b. negative slack.
c. positive slack.
d. surplus resources.
ANS: A
PTS: 1
14. The slope of the level curve for the objective function value can be changed by
a. increasing the value of the decision variables.
b. doubling all the coefficients in the objective function.
c. increasing the right hand sides of constraints.
d. changing a coefficient in the objective function.
ANS: D
PTS: 1
15. The allowable increase for a changing cell (decision variable) is
a. how many more units to produce to maximize profits.
b. the amount by which the objective function coefficient can increase without changing the
optimal solution.
c. how much to charge to get the optimal solution.
d. the amount by which constraint coefficient can increase without changing the optimal
solution.
ANS: B
PTS: 1
16. The allowable decrease for a changing cell (decision variable) is
a. the amount by which the constraint coefficient can decrease without changing final
optimal solution.
b. an indication of how many more units to produce to maximize profits.
c. the amount by which objective function coefficient can decrease without changing the
final optimal solution.
d. an indication of how much to charge to get the optimal solution.
ANS: C
PTS: 1
17. Which of the following statements is false concerning either of the Allowable Increase and Allowable
Decrease columns in the Sensitivity Report?
a. The values equate the decision variable profit to the cost of resources expended.
b. The values give the range over which a shadow price is accurate.
c. The values give the range over which an objective function coefficient can change without
changing the optimal solution.
d. The values provide a means to recognize when alternate optimal solution exist.
ANS: A
PTS: 1
18. The allowable increase for a constraint is
a. how many more units of resource to purchase to maximize profits.
b. the amount by which the resource can increase given shadow price.
c. how much resource to use to get the optimal solution.
d. the amount by which the constraint coefficient can increase without changing the final
optimal value.
ANS: B
PTS: 1
19. The allowable decrease for a constraint is
a. how many more units of resource to purchase to maximize profits.
b. the amount by which the resource can decrease given shadow price.
c. how much resource to use to get the optimal solution.
d. the amount by which constraint coefficient can increase without changing the final optimal
value.
ANS: B
PTS: 1
20. When performing sensitivity analysis, which of the following assumptions must apply?
a. All other coefficients remain constant.
b. Only right hand side changes really mean anything.
c. The X 1 variable change is the most important.
d. The non-negativity assumption can be relaxed
ANS: A
PTS: 1
21. Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of
resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the
objective function value?
a. increase of 50
b. increase of unknown amount
c. decrease of 50
d. increase of 10
ANS: A
PTS: 1
22. If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to
the objective function value?
a. increase of 100
b. increase of 110
c. decrease of 100
d. increases but by unknown amount
ANS: D
PTS: 1
23. The shadow price of a nonbinding constraint is
a. positive
b. zero
c. negative
d. indeterminate
ANS: B
PTS: 1
24. If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the
objective function value?
a. increase of 150
b. increases more than 0 but less than 150
c. no increase
d. increases but by an unknown amount
ANS: C
PTS: 1
25. If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the
optimal solution?
a. increases by an unknown amount
b. increases more than 0 but less than 150
c. no change
d. decreases by an unknown amount
ANS: C
PTS: 1
26. A change in the right hand side of a binding constraint may change all of the following except
a. optimal value of the decision variables
b. slack values
c. other right hand sides
d. objective function value
ANS: C
PTS: 1
27. A change in the right hand side of a constraint changes
a. the slope of the objective function
b. objective function coefficients
c. other right hand sides
d. the feasible region
ANS: D
PTS: 1
28. The absolute value of the shadow price indicates the amount by which the objective function will be
a. improved if the corresponding constraint is loosened.
b. improved if the corresponding constraint is tightened.
c. made worse if the corresponding constraint is loosened.
d. improved if the corresponding constraint is unchanged.
ANS: A
PTS: 1
29. The reduced cost for a changing cell (decision variable) is
a. the amount by which the objective function value changes if the variable is increased by
one unit.
b. how many more units to product to maximize profits.
c. the per unit profits minus the per unit costs for that variable.
d. equal to zero for variables at their optimal values.
ANS: C
PTS: 1
30. All of the following are true about a variable with a negative reduced cost in a maximization problem
except
a. its objective function coefficient must increase by that amount in order to enter the basis.
b. it is at its simple lower bound.
c. it has surplus resources.
d. the objective function value will decrease by that value if the variable is increased by one
unit.
ANS: C
PTS: 1
31. A variable with a final value equal to its simple lower or upper bound and a reduced cost of zero
indicates that
a. an alternate optimal solution exists.
b. an error in formulation has been made.
c. the right hand sides should be increased.
d. the objective function needs new coefficients.
ANS: A
PTS: 1
32. For a minimization problem, if a decision variable's final value is 0, and its reduced cost is negative,
which of the following is true?
a. Alternate optimal solutions exist.
b. There is evidence of degeneracy.
c. No feasible solution was found.
d. The variable has a non-negativity constraint.
ANS: D
PTS: 1
33. What is the value of the objective function if X 1 is set to 0 in the following Limits Report?
Cell
Target
Name
Value
$E$5
Unit profit: Total Profit:
3200
Cell
Adjustable
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
a.
b.
c.
d.
Value
Lower
Limit
Target
Result
Upper
Limit
Target
Result
80
20
0
0
800
2400
79.99999999
20
3200
3200
80
800
2400
3200
ANS: B
PTS: 1
34. When the allowable increase or allowable decrease for the objective function coefficient of one or
more variables is zero it indicates (in the absence of degeneracy) that
a. the problem is infeasible.
b. alternate optimal solutions exist.
c. there is only one optimal solution.
d. no optimal solution can be found.
ANS: B
PTS: 1
35. To convert ≤ constraints into = constraints the Simplex method adds what type of variable to the
constraint?
a. slack
b. dummy
c. redundant
d. spreading
ANS: A
PTS: 1
36. The solution to an LP problem is degenerate if
a. the right hand sides of any of the constraints have an allowable increase or allowable
decrease of zero.
b. the shadow prices of any of the constraints have an allowable increase or allowable
decrease of infinity.
c. the objective coefficients of any of the variables have an allowable increase or allowable
decrease of zero.
d. the shadow prices of any of the constraints have an allowable increase or allowable
decrease of zero.
ANS: A
PTS: 1
37. When a solution is degenerate the reduced costs for the changing cells
a. is always equal to zero.
b. may not be unique.
c. may be set to any value the manager needs.
d. is equal to infinity.
ANS: B
PTS: 1
38. When a solution is degenerate the shadow prices and their ranges
a. may be interpreted in the usual way but they may not be unique.
b. must be disregarded.
c. are always valid and unique.
d. are always understated
ANS: A
PTS: 1
39. What is the value of the slack variable in the following constraint when X 1 and X 2 are nonbasic and
only non-negativity is used as simple bounds?
X 1 + X 2 + S 1 = 100
a.
b.
c.
d.
0
50
100
can't be determined from the given information
ANS: C
PTS: 1
40. How many basic variables are there in a linear programming model which has n variables and m
constraints?
a. n
b. m
c. n + m
d. n − m
ANS: B
PTS: 1
41. A solution to the system of equations using a set of basic variables is called
a. a feasible solution.
b. basic feasible solution.
c. a nonbasic solution.
d. a nonbasic feasible solution
ANS: B
PTS: 1
42. The Simplex method works by first
a. identifying any basic feasible solution.
b. choosing the largest value for X 1 .
c. setting X 1 at one-half of the its maximum value.
d. going directly to the optimal solution.
ANS: A
PTS: 1
43. The Simplex method uses which of the following values to determine if the objective function value
can be improved?
a. shadow price
b. target value
c. reduced cost
d. basic cost
ANS: C
PTS: 1
44. The optimization technique that locates solutions in the interior of the feasible region is known as
_____?
a. sub-optimal optimization
b. sensitivity analysis
c. robust optimization
d. USET optimization
ANS: C
PTS: 1
45. Why might a decision maker prefer a solution in the interior of the feasible region of a linear
programming problem?
a. Such a solution has a better objective function value than any other solution
b. Such a solution is likely to remain feasible if some of the coefficients in the problem
change
c. The decision maker is not sure if he/she wants to maximize or minimize the objective
d. Such a solution has more binding constraints
ANS: C
PTS: 1
46. When automatically running multiple optimizations in Risk Solver Platform (RSP), what spreadsheet
function indicates which optimization is being run?
a. =PsiOptNum()
b. =PsiOptValue()
c. =PsiOptIndex()
d. =PsiCurrentOpt()
ANS: D
PTS: 1
PROBLEM
47. Given the following Risk Solver Platform (RSP) sensitivity output what range of values can the
objective function coefficient for variable X1 assume without changing the optimal solution?
Changing Cells
Cell
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
9.49
1.74
0
0
5
6
1.54
1.5
1
1.47
Constraints
Cell
Name
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
$D$8
$D$9
$D$10
Used
Used
Used
42
132
24
0
0.24
1.24
48
132
24
1E+30
12
1.33
6
12
2
ANS:
4 − 6.54
PTS: 1
48. Given the following Risk Solver Platform (RSP) sensitivity output how much does the objective
function coefficient for X 2 have to increase before it enters the optimal solution at a strictly positive
value?
Cell
Name
Final
Value
$B$4
$C$4
$D$4
X1
X2
X3
9.52
0
10.79
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
0
−500.01
0
2100
899.99
1050
1E+30
500.01
210
350
1E+30
375.01
ANS:
500.01
PTS: 1
49. What is the optimal objective function value if X 1 is at its lower limit in the following Risk Solver
Platform (RSP) sensitivity output?
Cell
Target
Name
$E$5
Unit profit: OBJ. FN. VALUE
Cell
Adjustable
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Value
58
Value
Lower
Limit
Target
Result
Upper
Limit
Target
Result
6
4
0
0
16
42
6
4
58
58
ANS:
16
PTS: 1
50. What are the objective function coefficients for X 1 and X 2 based on the following Risk Solver
Platform (RSP) sensitivity output?
Cell
Target
Name
$E$5
Unit profit: OBJ. FN. VALUE
Cell
Adjustable
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Value
58
Value
Lower
Limit
Target
Result
Upper
Limit
Target
Result
6
4
0
0
16
42
6
4
58
58
ANS:
Coefficient for X 1 is 7 and coefficient for X 2 is 4
PTS: 1
51. Which of the constraints are binding at the optimal solution for the following problem and Risk Solver
Platform (RSP) sensitivity output?
MAX:
Subject to:
7 X1 + 4 X2
2 X 1 + X 2 ≤ 16
X 1 + X 2 ≤ 10
2 X 1 + 5 X 2 ≤ 40
X1, X2 ≥ 0
Changing Cells
Cell
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
6
4
0
0
7
4
1
3
3
0.5
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
3
1
0
16
10
40
4
1
1E+30
2.67
2
8
Constraints
Cell
Name
Final
Value
$D$8
$D$9
$D$10
Used
Used
Used
16
10
32
ANS:
X 1 = 6, X 2 = 4
2 * 6 + 4 = 16
6 + 4 = 10
2 * 6 + 5 * 4 = 32
binding
binding
non-binding
PTS: 1
52. Is the optimal solution to this problem unique, or is there an alternate optimal solution? Explain your
reasoning.
MAX
Subject to:
5 X1 + 2 X2
3 X 1 + 5 X 2 ≤ 15
10 X 1 + 4 X 2 ≤ 20
X1, X2 ≥ 0
Changing Cells
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
2
0
0
0
5
2
1E+30
0
0
1E+30
Final
Cell
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Constraints
Cell
Name
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
$D$8
$D$9
Used
Used
6
20
0
0.5
15
20
1E+30
30
9
20
ANS:
Alternate optimal solutions exist because variable X 2 has a final value of 0 and a reduced cost of 0.
PTS: 1
53. Consider the following linear programming model and Risk Solver Platform (RSP) sensitivity output.
What is the optimal objective function value if the RHS of the first constraint increases to 18?
MAX:
Subject to:
7 X1 + 4 X2
2 X 1 + X 2 ≤ 16
X 1 + X 2 ≤ 10
2 X 1 + 5 X 2 ≤ 40
X1, X2 ≥ 0
Changing Cells
Cell
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
6
4
0
0
7
4
1
3
3
0.5
Constraints
Cell
Name
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
$D$8
$D$9
$D$10
Used
Used
Used
16
10
32
3
1
0
16
10
40
4
1
1E+30
2.67
2
8
ANS:
Shadow price of first constraint is 3 with an allowable increase of 4. A 2-unit increase in RHS value
increases objective function by 6. New objective function value is 6 * 7 + 4 * 4 + 2 * 3 = 64.
PTS: 1
54. What is the smallest value of the objective function coefficient X 1 can assume without changing the
optimal solution?
MAX:
Subject to:
7 X1 + 4 X2
2 X 1 + X 2 ≤ 16
X 1 + X 2 ≤ 10
2 X 1 + 5 X 2 ≤ 40
X1, X2 ≥ 0
Changing Cells
Cell
Name
$B$4
$C$4
Number to make: X1
Number to make: X2
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
6
4
0
0
7
4
1
3
3
0.5
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
3
1
0
16
10
40
4
1
1E+30
2.67
2
8
Constraints
Cell
Name
Final
Value
$D$8
$D$9
$D$10
Used
Used
Used
16
10
32
ANS:
Coefficient − allowable decrease = 7 − 3 = 4
PTS: 1
55. Constraint 3 is a non-binding constraint in the final solution to a maximization problem. Complete the
following entry for the Risk Solver Platform (RSP) sensitivity report. Cell labels are included to ease
of reference.
Constraints
Cell
Name
$D$8
Constraint 3
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
6
??
10
??
??
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
6
0
10
1E+30
4
ANS:
Constraints
Cell
Name
$D$8
Constraint 3
PTS: 1
56. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of
Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each
crop. Each crop also requires fertilizer and irrigation water both of which are in short supply. The
following table summarizes the data for the problem.
Crop
Corn
Pumpkin
Beans
Profit per
Acre ($)
Yield per
Acre (lb)
Maximum
Demand (lb)
Irrigation
(acre ft)
Fertilizer
(pounds/acre)
2,100
900
1,050
21,000
10,000
3,500
200,000
180,000
80,000
2
3
1
500
400
300
Based on the following Risk Solver Platform (RSP) sensitivity output, how much can the price of Corn
drop before it is no longer profitable to plant corn?
Changing Cells
Cell
Name
Final
Value
$B$4
$C$4
$D$4
Acres of Corn
Acres of Pumpkin
Acres of Beans
9.52
0
10.79
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
0
−500.01
0
2100
899.99
1050
1E+30
500.01
210
350
1E+30
375.00
ANS:
The allowable decrease for corn is 350.
PTS: 1
57. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of
Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each
crop. Each crop also requires fertilizer and irrigation water which are in short supply. The following
table summarizes the data for the problem.
Crop
Corn
Pumpkin
Beans
Profit per
Acre ($)
Yield per
Acre (lb)
Maximum
Demand (lb)
Irrigation
(acre ft)
Fertilizer
(pounds/acre)
2,100
900
1,050
21,000
10,000
3,500
200,000
180,000
80,000
2
3
1
500
400
300
Suppose the farmer can purchase more fertilizer for $2.50 per pound, should he purchase it and how
much can he buy and still be sure of the value of the additional fertilizer? Base your response on the
following Risk Solver Platform (RSP) sensitivity output.
Changing Cells
Cell
$B$4
$C$4
$D$4
Name
Acres of Corn
Acres of Pumpkin
Acres of Beans
Final
Value
9.52
0
10.79
Reduced
Cost
0
−500.01
0
Objective
Coefficient
2100
899.99
1050
Allowable
Increase
1E+30
500.01
210
Allowable
Decrease
350
1E+30
375.00
Final
Value
200000
0
37777.78
29.84
8000
Shadow
Price
0.017
0
0
0
3.5
Constraint
R.H. Side
200000
180000
80000
50
8000
Allowable
Increase
136000
1E+30
1E+30
1E+30
3619.04
Allowable
Decrease
152000
180000
42222.22
20.15
3238.09
Constraints
Cell
$E$8
$E$9
$E$10
$E$11
$E$12
Name
Corn demand Used
Pumpkin demand Used
Bean demand Used
Water Used
Fertilizer Used
ANS:
Yes, because the cost of $2.50 is less than the shadow price of $3.50. The allowable increase is
3619.04 pounds.
PTS: 1
58. Jones Furniture Company produces beds and desks for college students. The production process
requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing.
Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry
time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of
profit. Demand for desks is limited so at most 8 will be produced.
How much can the price of Desks drop before it is no longer profitable to produce them? Base your
response on the following Risk Solver Platform (RSP) sensitivity output.
Let
X 1 = Number of Beds to produce
X 2 = Number of Desks to produce
The LP model for the problem is
MAX:
Subject to:
30 X 1 + 40 X 2
6 X 1 + 4 X 2 ≤ 36 (carpentry)
4 X 1 + 8 X 2 ≤ 40 (varnishing)
X 2 ≤ 8 (demand for X 2 )
X1, X2 ≥ 0
Changing Cells
Cell
$B$4
$C$4
Name
Number to make: Beds
Number to make: Desks
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
4
3
0
0
30
40
30
20
10
20
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
Constraints
Cell
Name
$D$8
$D$9
$D$10
Carpentry Used
Varnishing Used
Desk demand Used
36
40
3
2.5
3.75
0
36
40
8
24
26.67
1E+30
16
16
5
ANS:
The allowable decrease is 20.
PTS: 1
59. Jones Furniture Company produces beds and desks for college students. The production process
requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing.
Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry
time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of
profit. Demand for desks is limited so at most 8 will be produced.
Suppose the company can purchase more varnishing time for $3.00, should it be purchased and how
much can be bought before the value of the additional time is uncertain? Base your response on the
following Risk Solver Platform (RSP) sensitivity output.
Let
X 1 = Number of Beds to produce
X 2 = Number of Desks to produce
The LP model for the problem is
MAX:
Subject to:
30 X 1 + 40 X 2
6 X 1 + 4 X 2 ≤ 36 (carpentry)
4 X 1 + 8 X 2 ≤ 40 (varnishing)
X 2 ≤ 8 (demand for X 2 )
X1, X2 ≥ 0
Changing Cells
Cell
$B$4
$C$4
Name
Number to make: Beds
Number to make: Desks
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
4
3
0
0
30
40
30
20
10
20
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
36
40
3
2.5
3.75
0
36
40
8
24
26.67
1E+30
16
16
5
Constraints
Cell
$D$8
$D$9
$D$10
Name
Carpentry Used
Varnishing Used
Desk demand Used
ANS:
Yes, because the cost of $3.00 is less than the shadow price of $3.75. The allowable increase is 26.67
hours.
PTS: 1
Exhibit 4.1
The following questions are based on the problem below and accompanying Risk Solver Platform
(RSP) sensitivity report.
Carlton construction is supplying building materials for a new mall construction project in Kansas.
Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period.
Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery,
and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of
the total delivered by the end of week two, and the entire amount delivered by the end of week three.
Contracts in place with the transportation companies call for at least 45% of the total delivered be
delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the
total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of
each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week
1
2
3
Costs ($ per 1000 tons)
Trucking Limits
45
50
55
$200
Railway Limits
60
55
45
$140
Air Cargo Limits
15
10
5
$400
The following is the LP model for this logistics problem.
Let
X ij = amount shipped by mode i in week j
where i = 1(Truck), 2(Rail), 3(Air)
and j = 1, 2, 3
Let
WL ij = weekly limit of mode i in week j (as provided in above table)
MIN:
200(X 11 + X 12 + X 13 ) + 140(X 21 + X 22 + X 23 ) + 500(X 31 + X 32 + X 33 )
Subject to:
Weekly limits by mode
X ij ≤ WL ij for all i and j
Total at end of three weeks
X 11 + X 12 + X 13 + X 21 + X 22 + X 23 + X 31 + X 32 + X 33 ≥ 250
Total at end of two weeks
X 11 + X 21 + X 31 + X 12 + X 22 + X 32 ≥ 200
Total at end of first week
X 11 + X 21 + X 31 ≥ 120
Truck mix requirement
X 11 + X 12 + X 13 ≥ 0.45*250
Rail mix requirement
X 21 + X 22 + X 23 ≥ 0.40*250
Air mix limit
X 31 + X 32 + X 33 ≤ 0.15*250
X ij ≥ 0 for all i and j
Cell
Name
$D$6
$E$6
$F$6
$D$7
$E$7
$F$7
$D$8
$E$8
$F$8
Week 1 by Truck
Week 1 by Rail
Week 1 by Air
Week 2 by Truck
Week 2 by Rail
Week 2 by Air
Week 3 by Truck
Week 3 by Rail
Week 3 by Air
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
45
60
15
50
55
0
13
12
0
0
0
0
0
0
360
0
0
360
200
140
500
200
140
500
200
140
500
360
360
1E+30
0
0
1E+30
1E+30
60
1E+30
1E+30
1E+30
360
1E+30
1E+30
360
0
0
360
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
45
60
15
−360
45
60
15
13
15
1E+30
0
0
0
Constraints
Cell
Name
$D$18
$E$18
$F$18
Week 1 by Truck
Week 1 by Rail
Week 1 by Air
−360
0
$D$19
$E$19
$F$19
$D$20
$E$20
$F$20
$D$9
$E$9
$F$13
$F$9
$G$6
$G$7
$G$8
Week 2 by Truck
Week 2 by Rail
Week 2 by Air
Week 3 by Truck
Week 3 by Rail
Week 3 by Air
Shipped by Truck
Shipped by Rail
Total Shipped Tons
Shipped by Air
Week 1 Totals
Week 2 Totals
Week 3 Totals
50
55
0
13
12
0
108
127
250
15
120
225
250
0
0
0
0
0
0
60
0
140
0
360
0
0
50
55
10
55
45
5
108
100
250
37.5
120
200
250
13
12
1E+30
1E+30
1E+30
1E+30
12
27
33
1E+30
0
25
0
25
25
10
42
33
5
13
1E+30
0
22.5
15
1E+30
1E+30
60. Refer to Exhibit 4.1. The Week 1 by Truck and Week 1 by Rail constraints each have a shadow price
of −360. What do these values imply?
ANS:
Increase the weekly limits on these two modes to reduce total cost by $360 per unit increase in limit.
PTS: 1
61. Refer to Exhibit 4.1. Of the three percentage of effort constraints, Shipped by Truck, Shipped by Rail,
and Shipped by Air, which should be examined for potential cost reduction?
ANS:
The percentage by Truck, Shipped by Truck, should be examined. Decreasing the percentage by truck
(from 45%) will decrease cost as the shadow price is 60.
PTS: 1
62. Refer to Exhibit 4.1. Are there alternate optimal solutions to this problem?
ANS:
Cannot tell because we cannot rule out degeneracy according to our guidelines due to the zero values
in the Allowable Increase and Allowable Decrease columns of the constraint portion of the Risk Solver
Platform (RSP) sensitivity report.
PTS: 1
63. Refer to Exhibit 4.1. Should the company negotiate for additional air delivery capacity?
ANS:
No. The shadow prices for each week of air delivery are zero.
PTS: 1
Exhibit 4.2
The following questions correspond to the problem below and associated Risk Solver Platform (RSP)
sensitivity report.
Robert Hope received a welcome surprise in this management science class; the instructor has decided
to let each person define the percentage contribution to their grade for each of the graded instruments
used in the class. These instruments were: homework, an individual project, a mid-term exam, and a
final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the
instructor complicated Robert's task somewhat by adding the following stipulations:
•
•
•
•
homework can account for up to 25% of the grade, but must be at least 5% of the grade;
the project can account for up to 25% of the grade, but must be at least 5% of the grade;
the mid-term and final must each account for between 10% and 40% of the grade but
cannot account for more than 70% of the grade when the percentages are combined; and
the project and final exam grades may not collectively constitute more than 50% of the
grade.
The following LP model allows Robert to maximize his numerical grade.
Let
W 1 = weight assigned to homework
W 2 = weight assigned to the project
W 3 = weight assigned to the mid-term
W 4 = weight assigned to the final
MAX:
Subject to:
75W 1 + 94W 2 + 85W 3 + 92W 4
W 1 + W2 + W3 + W4 = 1
W 3 + W 4 ≤ 0.70
W 3 + W 4 ≥ 0.50
0.05 ≤ W 1 ≤ 0.25
0.05 ≤ W 2 ≤ 0.25
0.10 ≤ W 3 ≤ 0.40
0.10 ≤ W 4 ≤ 0.40
Adjustable Cells
Cell
Name
$F$5
Mid Term to grade
$F$6
Final to grade
$F$7
$F$8
Project to grade
Homework to grade
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
0.40
0.25
0.25
0.10
10.00
0.00
2.00
0.00
85
92
94
75
1E+30
2
1E+30
10
10
17
2
1E+30
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
0.65
0.5
1.00
0
17
75.00
0.7
0.5
1
1E+30
0.05
0.15
0.05
0.15
0.05
Constraints
Cell
Name
$E$14
$E$15
$F$9
Both Exams Total
Final & Project Total
100% to grade
64. Refer to Exhibit 4.2. Constraint cell F9 corresponds to the constraint, W 1 + W 2 + W 3 + W 4 = 1, and
has a shadow price of 75. Armed with this information, what can Robert request of his instructor
regarding this constraint?
ANS:
Nothing. The constraint has the largest shadow price but enforces the total percentages to equal 1, thus
nothing can be changed.
PTS: 1
65. Refer to Exhibit 4.2. Based on the Risk Solver Platform (RSP) sensitivity report information, is there
anything Robert can request of his instructor to improve his final grade?
ANS:
Robert can request an increase in the total weight allowed for the project and final exam combined
since this has a positive shadow price.
PTS: 1
66. Refer to Exhibit 4.2. Based on the Risk Solver Platform (RSP) sensitivity report information, Robert
has been approved by his instructor to increase the total weight allowed for the project and final exam
to 0.50 plus the allowable increase. When Robert re-solves his model, what will his new final grade
score be?
ANS:
88.85 since shadow price of 17 and increase of 0.05 equates to 0.85.
PTS: 1
67. Use slack variables to rewrite this problem so that all its constraints are equality constraints.
MAX:
Subject to:
ANS:
MAX:
Subject to:
2 X1 + 7 X2
5 X1 + 9 X2
9 X1 + 8 X2
X2 ≤ 8
X1, X2 ≥ 0
90
≤ 144
≤
2 X1 + 7 X2
5 X 1 + 9 X 2 + S 1 = 90
9 X 1 + 8 X 2 + S 2 = 144
X2 + S3 = 8
X1, X2 ≥ 0
PTS: 1
68. Identify the different sets of basic variables that might be used to obtain a solution to this problem.
MAX:
Subject to:
8 X1 + 4 X2
5 X1 + 5 X2
6 X1 + 2 X2
X1, X2 ≥ 0
20
≤ 18
≤
ANS:
X1
0
0
3
2.5
PTS: 1
X2
0
4
0
1.5
S1
20
0
5
0
S2
18
10
0
0
69. Use slack variables to rewrite this problem so that all its constraints are equality constraints.
MIN:
Subject to:
ANS:
MIN
Subject to:
2.5 X 1 + 1.5 X 2
4 X 1 + 3 X 2 ≥ 24
2 X 1 + 4 X 2 ≥ 24
X1, X2 ≥ 0
2.5 X 1 + 1.5 X 2
4 X 1 + 3 X 2 − S 1 = 24
2 X 1 + 4 X 2 − S 2 = 24
X1, X2 ≥ 0
PTS: 1
70. Identify the different sets of basic variables that might be used to obtain a solution to this problem.
MIN:
Subject to:
2.5 X 1 + 1.5 X 2
4 X 1 + 3 X 2 ≥ 24
2 X 1 + 4 X 2 ≥ 24
X1, X2 ≥ 0
ANS:
X1
0
0
12
2.4
X2
0
8
0
4.8
S1
24
0
24
0
S2
24
8
0
0
PTS: 1
71. Solve this problem graphically. What is the optimal solution and what constraints are binding at the
optimal solution?
MAX:
Subject to:
8 X1 + 4 X2
5 X1 + 5 X2
6 X1 + 2 X2
X1, X2 ≥ 0
20
≤ 18
≤
ANS:
Obj = 26
X 1 = 2.5
X 2 = 1.5
Both constraints are binding.
PTS: 1
72. Solve this problem graphically. What is the optimal solution and what constraints are binding at the
optimal solution?
MIN:
Subject to:
7 X1 + 3 X2
4 X1 + 4 X2
2 X1 + 3 X2
X1, X2 ≥ 0
40
≥ 24
≥
ANS:
Obj = 30
X1 = 0
X 2 = 10
The constraint 4 X 1 + 4 X 2 = 40 is binding
PTS: 1