Chapter 8 – LP Sensitivity Analysis
VARIABLE CURRENT
COEFFICIENT ALLOWABLE
INCREASE ALLOWABLE
DECREASE
X1 12.000000 INFINITY 4.000000
X2 10.000000 5.000000 10.000000
X3 9.000000 INFINITY 4.000000
RIGHTHAND SIDE RANGES
ROW CURRENT
RHS ALLOWABLE
INCREASE ALLOWABLE
DECREASE
2 60.000000 4.000000 INFINITY
3 80.000000 INFINITY 5.000000
a. What is the solution to the problem?
b. Which constraints are binding?
c. Interpret the reduced cost for x1.
d. Interpret the dual price for constraint 2.
e. What would happen if the cost of x1 dropped to 10 and the cost of x2 increased to 12?
51. The LP problem whose output follows determines how many necklaces, bracelets, rings, and earrings a jewelry store
should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1
measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are
marketing restrictions.
LINEAR PROGRAMMING PROBLEM
MAX 100X1+120X2+150X3+125X4
S.T.
1) X1+2X2+2X3+2X4<108
2) 3X1+5X2+X4<120
3) X1+X3<25
4) X2+X3+X4>50
OPTIMAL SOLUTION
Objective Function Value = 7475.000
Variable Value Reduced Cost
Chapter 8 – LP Sensitivity Analysis
X1 8.000 0.000
X2 0.000 5.000
X3 17.000 0.000
X4 33.000 0.000
Constraint Slack/Surplus Dual Price
1 0.000 75.000
2 63.000 0.000
3 0.000 25.000
4 0.000 −25.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 87.500 100.000 No Upper Limit
X2 No Lower Limit 120.000 125.000
X3 125.000 150.000 162.500
X4 120.000 125.000 150.000
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 100.000 108.000 123.750
2 57.000 120.000 No Upper Limit
3 8.000 25.000 58.000
4 41.500 50.000 54.000
Use the output to answer the questions.
a. How many necklaces should be stocked?
b. Now many bracelets should be stocked?
c. How many rings should be stocked?
d. How many earrings should be stocked?
e. How much space will be left unused?
f. How much time will be used?
g. By how much will the second marketing restriction be exceeded?
h. What is the profit?
i. To what value can the profit on necklaces drop before the solution would change?
j. By how much can the profit on rings increase before the solution would change?
k. By how much can the amount of space decrease before there is a change in the profit?
l. You are offered the chance to obtain more space. The offer is for 15 units and the total price is 1500. What should
you do?
Chapter 8 – LP Sensitivity Analysis
52. The decision variables represent the amounts of ingredients 1, 2, and 3 to put into a blend. The objective function
represents profit. The first three constraints measure the usage and availability of resources A, B, and C. The fourth
constraint is a minimum requirement for ingredient 3. Use the output to answer these questions.
a. How much of ingredient 1 will be put into the blend?
b. How much of ingredient 2 will be put into the blend?
c. How much of ingredient 3 will be put into the blend?
d. How much resource A is used?
e. How much resource B will be left unused?
f. What will the profit be?
g. What will happen to the solution if the profit from ingredient 2 drops to 4?
h. What will happen to the solution if the profit from ingredient 3 increases by 1?
i. What will happen to the solution if the amount of resource C increases by 2?
j. What will happen to the solution if the minimum requirement for ingredient 3 increases to 15?
LINEAR PROGRAMMING PROBLEM
MAX 4X1+6X2+7X3
S.T.
1) 3X1+2X2+5X3<120
2) 1X1+3X2+3X3<80
3) 5X1+5X2+8X3<160
4) +1X3>10
OPTIMAL SOLUTION
Objective Function Value = 166.000
Variable Value Reduced Cost
X1 0.000 2.000
X2 16.000 0.000
X3 10.000 0.000
Constraint Slack/Surplus Dual Price
1 38.000 0.000
2 2.000 0.000
3 0.000 1.200
4 0.000 −2.600
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 No Lower Limit 4.000 6.000
X2 4.375 6.000 No Upper Limit
X3 No Lower Limit 7.000 9.600
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 82.000 120.000 No Upper Limit
2 78.000 80.000 No Upper Limit
Chapter 8 – LP Sensitivity Analysis
3 80.000 160.000 163.333
4 8.889 10.000 20.000
53. The LP model and LINDO output represent a problem whose solution will tell a specialty retailer how many of four
different styles of umbrellas to stock in order to maximize profit. It is assumed that every one stocked will be sold. The
variables measure the number of women’s, golf, men’s, and folding umbrellas, respectively. The constraints measure
storage space in units, special display racks, demand, and a marketing restriction, respectively.
MAX 4 X1 + 6 X2 + 5 X3 + 3.5 X4
SUBJECT TO
2) 2 X1 + 3 X2 + 3 X3 + X4 <= 120
3) 1.5 X1 + 2 X2 <= 54
4) 2 X2 + X3 + X4 <= 72
5) X2 + X3 >= 12
END
OBJECTIVE FUNCTION VALUE
1) 318.00000
VARIABLE VALUE REDUCED COST
X1 12.000000 .000000
X2 .000000 .500000
X3 12.000000 .000000
X4 60.000000 .000000
ROW SLACK OR SURPLUS DUAL PRICE
2) .000000 2.000000
3) 36.000000 .000000
4) .000000 1.500000
5) .000000 −2.500000
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ. COEFFICIENT RANGES
VARIABLE CURRENT
COEFFICIENT ALLOWABLE
Chapter 8 – LP Sensitivity Analysis
INCREASE ALLOWABLE
DECREASE
X1 4.000000 1.000000 2.500000
X2 6.000000 .500000 INFINITY
X3 5.000000 2.500000 .500000
X4 3.500000 INFINITY .500000
RIGHTHAND SIDE RANGES
ROW CURRENT
RHS ALLOWABLE
INCREASE ALLOWABLE
DECREASE
2 120.000000 48.000000 24.000000
3 54.000000 INFINITY 36.000000
4 72.000000 24.000000 48.000000
5 12.000000 12.000000 12.000000
Use the output to answer the questions.
a. How many women’s umbrellas should be stocked?
b. How many golf umbrellas should be stocked?
c. How many men’s umbrellas should be stocked?
d. How many folding umbrellas should be stocked?
e. How much space is left unused?
f. How many racks are used?
g. By how much is the marketing restriction exceeded?
h. What is the total profit?
i. By how much can the profit on women’s umbrellas increase before the solution would change?
j. To what value can the profit on golf umbrellas increase before the solution would change?
k. By how much can the amount of space increase before there is a change in the dual price?
l. You are offered an advertisement that should increase the demand constraint from 72 to 86 for a total cost of $20.
Would you say yes or no?
54. Eight of the entries have been deleted from the LINDO output that follows. Use what you know about linear
programming to find values for the blanks.
Chapter 8 – LP Sensitivity Analysis
SUBJECT TO
2) 25 X1 + 35 X2 + 30 X3 >= 2400
3) 2 X1 + 4 X2 + 8 X3 >= 400
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 612.50000
VARIABLE VALUE REDUCED COST
X1 ________ 1.312500
X2 ________ ________
X3 27.500000 ________
ROW SLACK OR SURPLUS DUAL PRICE
2) ________ −.125000
3) ________ −.781250
NO. ITERATIONS= 2
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ. COEFFICIENT RANGES
VARIABLE CURRENT
COEFFICIENT ALLOWABLE
INCREASE ALLOWABLE
DECREASE
X1 6.000000 _________ _________
X2 7.500000 1.500000 2.500000
X3 10.000000 5.000000 3.571429
RIGHTHAND SIDE RANGES
ROW CURRENT
RHS ALLOWABLE
INCREASE ALLOWABLE
DECREASE
2 2400.000000 1100.000000 900.000000
3 400.000000 240.000000 125.714300
55. Portions of a Management Scientist output are shown below. Use what you know about the solution of linear
programs to fill in the ten blanks.
LINEAR PROGRAMMING PROBLEM
Chapter 8 – LP Sensitivity Analysis
MAX 12X1+9X2+7X3
S.T.
1) 3X1+5X2+4X3<150
2) 2X1+1X2+1X3<64
3) 1X1+2X2+1X3<80
4) 2X1+4X2+3X3>116
OPTIMAL SOLUTION
Objective Function Value = 336.000
Variable Value Reduced Cost
X1 ______ 0.000
X2 24.000 ______
X3 ______ 3.500
Constraint Slack/Surplus Dual Price
1 0.000 15.000
2 ______ 0.000
3 ______ 0.000
4 0.000 ______
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 5.400 12.000 No Upper Limit
X2 2.000 9.000 20.000
X3 No Lower Limit 7.000 10.500
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 145.000 150.000 156.667
2 ______ ______ 64.000
3 ______ ______ 80.000
4 110.286 116.000 120.000
56. A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult
Open Trail, the adult Cityscape, the girl’s Sea Sprite, and the boy’s Trail Blazer. It is assumed that every bike ordered will
be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. There are several
conditions that the store needs to worry about. One of these is space to hold the inventory. An adult’s bike needs two feet,
but a child’s bike needs only one foot. The store has 500 feet of space. There are 1200 hours of assembly time available.
Chapter 8 – LP Sensitivity Analysis
The child’s bike need 4 hours of assembly time; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store
would like to place an order for at least 275 bikes.
a. Formulate a model for this problem.
b. Solve your model with any computer package available to you.
c. How many of each kind of bike should be ordered and what will the profit be?
d. What would the profit be if the store had 100 more feet of storage space?
e. If the profit on the Cityscape increases to $35, will any of the Cityscape bikes be ordered?
f. Over what range of assembly hours is the dual price applicable?
g. If we require 5 more bikes in inventory, what will happen to the value of the optimal solution?
h. Which resource should the company work to increase, inventory space or assembly time?
Chapter 8 – LP Sensitivity Analysis
57. A company produces two products made from aluminum and copper. The table below gives the unit requirements, the
unit production man-hours required, the unit profit and the availability of the resources (in tons).
Aluminum Copper Man-hours Unit Profit
Product 1 1 0 2 50
Product 2 1 1 3 60
Available 10 6 24
The Management Scientist provided the following solution output:
Objective Function Value = 540.000
VARIABLE VALUE REDUCED COST
X1 6.000 0.000
X2 4.000 0.000
CONSTRAINT SLACK/SURPLUS DUAL PRICE
1 .000 30.000
2 2.000 0.000
3 0.000 10.000
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ. COEFFICIENT RANGES
VARIABLE CURRENT
COEFFICIENT ALLOWABLE
INCREASE ALLOWABLE
DECREASE
X1 50.000 10.000 10.000
X2 60.000 15.000 10.000
RIGHTHAND SIDE RANGES
CONSTRAINT CURRENT
RHS ALLOWABLE
INCREASE ALLOWABLE
DECREASE
1 10.000 2.000 1.000
2 6.000 INFINITY 2.000
3 24.000 2.000 4.000
Chapter 8 – LP Sensitivity Analysis
a. What is the optimal production schedule?
b. Within what range for the profit on product 2 will the solution in (a) remain optimal? What is the optimal profit when
c2 = 70?
c. Suppose that simultaneously the unit profits on x1 and x2 changed from 50 to 55 and 60 to 65 respectively. Would the
optimal solution change?
d. Explain the meaning of the “DUAL PRICES” column. Given the optimal solution, why should the dual price for
copper be 0?
e. What is the increase in the value of the objective function for an extra unit of aluminum?
f. Man-hours were not figured into the unit profit as it must pay three workers for eight hours of work regardless of the
number of man-hours used. What is the dual price for man-hours? Interpret.
g. On the other hand, aluminum and copper are resources that are ordered as needed. The unit profit coefficients were
determined by: (selling price per unit) – (cost of the resources per unit). The 10 units of aluminum cost the company $100.
What is the most the company should be willing to pay for extra aluminum?
58. Given the following linear program:
MAX 5x1 + 7x2
s.t. x1 ≤ 6
2x1 + 3x2 ≤ 19
x1 + x2 ≤ 8
x1, x2 ≥ 0
The graphical solution to the problem is shown below. From the graph we see that the optimal solution occurs at x1 = 5, x2
= 3, and z = 46.
Chapter 8 – LP Sensitivity Analysis
a. Calculate the range of optimality for each objective function coefficient.
b. Calculate the dual price for each resource.
59. Consider the following linear program:
MAX 3x1 + 4x2 ($ Profit)
s.t. x1 + 3x2 ≤ 12
2x1 + x 2 ≤ 8
x1 ≤ 3
x1, x2 ≥ 0
The Management Scientist provided the following solution output:
OPTIMAL SOLUTION
Objective Function Value = 20.000
Variable Value Reduced Cost
X1 2.400 0.000
X2 3.200 0.000
Constraint Slack/Surplus Dual Price
Chapter 8 – LP Sensitivity Analysis
1 0.000 1.000
2 0.000 1.000
3 0.600 0.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 1.333 3.000 8.000
X2 1.500 4.000 9.000
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 9.000 12.000 24.000
2 4.000 8.000 9.000
3 2.400 3.000 No Upper Limit
a. What is the optimal solution including the optimal value of the objective function?
b. Suppose the profit on x1 is increased to $7. Is the above solution still optimal? What is the value of the objective
function when this unit profit is increased to $7?
c. If the unit profit on x2 was $10 instead of $4, would the optimal solution change?
d. If simultaneously the profit on x1 was raised to $5.5 and the profit on x2 was reduced to $3, would the current
solution still remain optimal?
60. Consider the following linear program:
MIN 6x1 + 9x2 ($ cost)
s.t. x1 + 2x2 ≤ 8
10x1 + 7.5x2 ≥ 30
x2 ≥ 2
x1, x2 ≥ 0
The Management Scientist provided the following solution output:
OPTIMAL SOLUTION
Objective Function Value = 27.000
Variable Value Reduced Cost
X1 1.500 0.000
X2 2.000 0.000
Constraint Slack/Surplus Dual Price
Chapter 8 – LP Sensitivity Analysis
1 2.500 0.000
2 0.000 −0.600
3 0.000 −4.500
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X1 0.000 6.000 12.000
X2 4.500 9.000 No Upper Limit
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 5.500 8.000 No Upper Limit
2 15.000 30.000 55.000
3 0.000 2.000 4.000
a. What is the optimal solution including the optimal value of the objective function?
b. Suppose the unit cost of x1 is decreased to $4. Is the above solution still optimal? What is the value of the objective
function when this unit cost is decreased to $4?
c. How much can the unit cost of x2 be decreased without concern for the optimal solution changing?
d. If simultaneously the cost of x1 was raised to $7.5 and the cost of x2 was reduced to $6, would the current solution
still remain optimal?
e. If the right-hand side of constraint 3 is increased by 1, what will be the effect on the optimal solution?
Essay
61. Describe each of the sections of output that come from The Management Scientist and how you would use each.
62. Explain the connection between reduced costs and the range of optimality, and between dual prices and the range of
feasibility.
63. Explain the two interpretations of dual prices based on the accounting assumptions made in calculating the objective
function coefficients.
Chapter 8 – LP Sensitivity Analysis
64. How can the interpretation of dual prices help provide an economic justification for new technology?
65. How is sensitivity analysis used in linear programming? Given an example of what type of questions that can be
answered.
66. How would sensitivity analysis of a linear program be undertaken if one wishes to consider simultaneous changes for
both the right-hand-side values and objective function.