Transcript Review Problem #1 (Text problem #20)

Review Problem #1
•
INTERNATIONAL SHIPPING. The Takahashi Transport Company
(TTC) leases excess space on commercial vessels to the United States
at a reduced rate of $10 per square foot. The only condition is that
goods must be packaged in standard 30-inch-high crates.
• TTC ships items in two standard 30-inch-high crates,
– an eight-square-foot crate (two feet by four feet) and
– a four-square-foot (two feet by two feet) specially insulated crate.
• It charges customers
– $160 to ship an eight-square-foot crate and
– $100 to ship the insulated four-square-foot crate.
• Allowing for the cost of $10 per square foot, TTC makes
– $80 per standard eight-foot crate and (160 – 10*8)
– $60 on the four-foot crate (100 – 10*4)
1
• TTC stores the crates until space becomes available on a
cargo ship, at which time TTC receives payment from its
customers. TTC has been able to lease
– 1200 square feet of cargo space on the Formosa Frigate cargo ship,
which leaves for the United States in two days.
• As of this date, TTC has
– 140 eight-square-foot crates and
– 100 insulated four-square-foot crates awaiting shipment to the
United States.
• It has
– 48 hours to finish loading the crates, and
• it estimates the average loading time to be
– 12 minutes (.2 hour) per eight-square-foot crate and
– 24 minutes (.4 hour) per four-square-foot crate (owing to the
special handling of the insulated crates).
2
• Formulate and solve a linear program for TTC to optimize its profit on
the upcoming sailing of the Formosa Frigate.
8-SQFT 4-SQFT
Profit
80
# of 8 FT
1
<=
140
1
<=
100
0.2
0.4
<=
48
8
4
<=
1200
# of 4 FT
Load Time
Space
60
3
• What are the optimal values of the slack on each constraint in the
optimal solution? Express this result in words.
Constraints
Ce
ll
Name
Cell Value
Formula
Status
$D
$4 # of 8 FT
120
$D$4<=$F$4
Not Binding
20
$D
$5 # of 4 FT
60
$D$5<=$F$5
Not Binding
40
$D
$6 Load Time
48
$D$6<=$F$6
Binding
0
1200
$D$7<=$F$7
Binding
0
$D
$7 Space
Slack
4
Review Problem # 1 - continued
a. Determine the range of optimality for each revenue coefficient.
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Name
Value
Cost
Coefficient
Increase
Decrease
8-SQFT
120
0
80
40
50
4-SQFT
60
0
60
100
20
The ranges of optimality are:
80+40, 80-50
60+100, 60-20
5
Review Problem # 1 - continued
b. Determine the range of feasibility for the number of square feet
available, the loading time, the number of standard containers
available, and the number of insulated containers available.
Constraints
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
# of 8 FT
120
0
140
1E+30
20
# of 4 FT
60
0
100
1E+30
40
Load Time
48
66.66666667
48
12
12
1200
8.333333333
1200
120
480
Space
The ranges of feasibility are:
140+infinity, 140-20
100+infinity, 100-40
48+12, 48-12
1200+120, 1200-480
6
c. Determine the shadow price for each resource for which a range of
feasibility was calculated in part (b). Do you think that any of these
should be treated as a sunk cost? Given your answer, explain the
meaning of each shadow price.
• As long as extra space could be leased for 10 + 8.33 = $18.33 it should be
leased. Explanation: The leasing cost per sq-ft was included in determining
the profit per sq-ft (for an 8-sq ft crate it is 160/8-10). Thus, TCC should be
willing to pay up to $8.33 above $10 per sq ft, but not more).
• Loading time is a sunk cost. Extra loading time will add $66.67 to the profit
per hour added
7
e. Suppose that at the last second, the Formosa Frigate decided to raise its
charge per square foot from $10 to $12. Would the optimal solution
change?
Two coefficients in the objective function are changing.
Percent change: For 8-square foot crate=((80-64)/(80-30))=.32
For 4-square foot crate=((60-52)/(60-40))=.40
Total=.72 < 1.00
The optimal solution will not change.
8
Review Problem #2
BAKERY. Mary Custard's is a pie shop that specializes in
custard and fruit pies. It makes delicious pies and sells
them at reasonable prices so that it can sell all the pies it
makes in a day. Every dozen custard pies nets Mary
Custard's $ 15 and requires 12 pounds of flour, 50 eggs,
and 5 pounds of sugar (and no fruit mixture). Every dozen
fruit pies nets a $25 profit and uses 10 pounds of flour, 40
eggs, 10 pounds of sugar, and 15 pounds of fruit mixture.
On a given day, the bakers at Mary Custard's had 150
pounds of flour, 500 eggs, 90 pounds of sugar, and 120
pounds of fruit mixture.
9
Problem 2 - solution
• Formulate a linear program that will give the optimal
production schedule of pies for the day.
X1 = the number of dozen custard pies baked
X2 = the number of dozen fruit pies baked
MAX 15X1+25X2
S.T. 12X1 + 10X2 <= 150 (Flour)
50X1 + 40X2 <= 500 (Eggs)
5X1 + 10X2 <= 90 (Sugar)
15X2 <= 120 (Fruit mixture)
XI, X2 >= 0
10
Problem 2 - solution
• Solve for the optimal production schedule
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
x1
4.67
0
15
0.5
7
x2
6.67
0
25
14
0.4
Bake 56 (4 2/3 dozen) custard pies and 80 (6 2/3 dozen) fruit pies;
Profit = $236.67
11
c. If Mary Custard's could double its profit on custard pies, should more custard pies be
produced? Explain.
$30 is within the range of optimality - no change.
d. If Mary Custard's raised the price (and hence the profit) on all pies by $0.25
($3.00 per dozen),
optimal
schedule
for the day change?
Coverwould
only if the
the 100%
ruleproduction
was discussed
in class
Would the profit change?
Percent changes: Custard pies 3/16.25 = 18.46%
Fruit pies 3/5 = 60.00%. Total = 78.46%
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
x1
4.67
0
12
16.25
2.5
x2
6.67
0
10
5
13
No change in the quantities produced; change in the profit.
12
e. Suppose Mary Custard's found that 10% of its fruit mixture had been stored in
containers that were not air-tight. For quality and health reasons, it decided
that it would be unwise to use any of this portion of the fruit mixture. How
would this affect the optimal production schedule? Explain.
12 pounds of fruit mixture are not available (10% of 120); yet there is a slack
of 20 pounds and thus the optimal solution will not be affected.
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
Flour
122.67
0
150
1E+30
27.333
Eggs
500
0.083
500
117.143
80
Sugar
90
2.17
90
8
40
100
0
120
1E+30
20
Fruit
13
f.
Mary Custard's currently pays $2.50 for a five-pound bag of sugar from its bakery
supply vendor. (The $0.50 per pound price of sugar is included in the unit profits given
earlier.) Its vendor has already made its deliveries for the day. If Mary Custard's wishes
to purchase additional sugar, it must buy it from Donatelli's Market, a small,
independent grocery store that sells sugar in one-pound boxes for $2.25 a box. Should
Mary Custard's purchase any boxes of sugar from Donatelli's Market? Explain
The shadow price for sugar is $2.17. Since cost of sugar is included, extra sugar is
worth $2.17 + $0.50 = $2.67. Since this is greater than $2.25, Mary Custard's should
purchase it.
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
Flour
122.67
0
150
1E+30
27.333
Eggs
500
0.083
500
117.143
80
Sugar
90
2.17
90
8
40
100
0
120
1E+30
20
14
Fruit
Review Problem # 3
• This problem focuses on modeling a blending problem,
multiple changes and the application of the 100% rule.
• The Party Nut Company has on hand 550 pounds of
peanuts, 150 pounds of cashews, 90 pounds of Brazil nuts.
It packages and sells four varieties of mixed nuts in
standard 8-ounce (half pound) cans. The mix requirements
and the unit profit per can are shown in the table below.
What mix of products (how many cans of each product)
should be produced and sold?
15
Data
Mix
Contents
Profit per can
1
Peanuts only
$0.26
2
No more than 50%
peanuts
At least 15% cashew
$0.40
3
Cashew only
$0.51
4
At least 30% cashew
At least 20% brazil nuts
$0.52
16
Solution
• Definitions:
• M1, M2, M3, M4 = the number of cans produced of
mix 1, 2, 3, 4 respectively.
• Pi, Ci, Bi = the amount (in pounds) of peanuts,
cashew, brazil use in mix ‘i’.
17
The model
– Max .26M1+.40M2+.51M3+.52M4
ST.
P1 = .5M1
P2  (50%)(.5M2)
C2  (15%)(.5M2)
C3 = .5M3
C4  (30%)(.5M4)
B4  (20%)(.5M4)
P1 + P2  550
C2 + C3 + C4 150
B4  90
Required quantities
Available resources
18
WINQSB solution and sensitivity
analysis
•
•
M=infinity
Assume the profit drops by 10 cents per can for mix 1, increases by 10 cents
per can for mix 2, and by 20 cents per can for mix 3. Would the optimal
production plan change?
(.16-.26)/(0-.26)+ (.50-.40)/(M-.40)+(.71-.51)/(2.67-.51) = .4772 < 1.
The optimal solution will not change.
19
WINQSB solution and sensitivity
analysis
•
•
•
Assume the available amount of all the materials used decreases by 10% each.
Would the optimal solution change. Would the total profit change? By how
much?
Observe the last three constraints, and check the 100% rule: –55/(0-550)+
(-15)/(0-150)+(-9)/(0-90) = .3 < 1. The shadow prices will remain the same,
but the solution will change, because the changes are made in binding
constraints (no slack).
The new profit = old profit + S[shadow price(change in the constraint’s RHS)]
= 1086 + (52)(-55)+5.33(-15)+(0)(-9) = 977.45
20
Modeling and Sensitivity
Analysis – Example 1
APPAREL INDUSTRY. Exclaim! Jeans is setting up a
production schedule for the coming week. Exclaim! Can
make four jean products: men’s and women’s jackets and
pants. Although it can make different sizes of each, the
variation in material usage and labor between sizes is
negligible. Each jacket and pair of pants goes through
cutting and stitching operations before being boxed. The
following table gives the profit, denim, cutting time,
stitching time, and boxing time required per 100 items, as
well as the total resource availabilities during the week.
21
APPAREL INDUSTRY
Item
Profit
($)
Denim
(yd.)
Cutting
(hr.)
Stitching
(hr.)
Boxing
(hr.)
Men’s
Jacket
Women’s
Jacket
Men’s
Pants
Women’s
Pants
$2,000
150
3
4
.75
2,800
125
4
3
.75
1200
200
2
2
.50
1500
150
2
2
.50
2500
36
36
8
Available
22
APPAREL INDUSTRY Solution
• Develop and solve a linear programming model. For Exclaim! Jeans
which will maximize its profit for the week.
Decision variables and objective function
Max 2000MJacket+2800WJacket+1200MPants+1500WPants
Constraints
Material constraint
150MJacket+125WJacket+200MPants+150WPants 2500
Cutting time constraint
3MJacket+ 4WJacket+ 2MPants+ 2WPants 36
Stitching time constraint
4MJacket+ 3WJacket+ 2MPants+ 2WPants 36
Boxing time constraint
.75MJacket+ .75WJacket+ .50MPants+ .50WPants 7.5
23
APPAREL INDUSTRY Solution
• Develop and solve a linear programming model. For Exclaim! Jeans
which will maximize its profit for the week.
MenJackets
WomenJackets
MenPants
WomenPants
Total
Profit
Denim-Yds
Cutting-Hrs
Stitching-Hrs
Boxing-Hrs
2000
150
3
4
0.75
MenJackets
0
Profit
Denim-Yds
Cutting-Hrs
Stitching-Hrs
Boxing-Hrs
2000
150
3
4
0.75
2800
125
4
3
0.75
WomenJackets
6
2800
125
4
3
0.75
1200
200
2
2
0.5
MenPants
1500
150
2
2
0.5
0
0
0
0
0
<=
<=
<=
<=
2500
36
36
7.5
Total
25800
1650
36
30
7.5
<=
<=
<=
<=
2500
36
36
7.5
WomenPants
0
1200
200
2
2
0.5
6
1500
150
2
2
0.5
24
APPAREL INDUSTRY Solution
• How much should the profit for Men’s Jacket increase before it
becomes part of the production plan?
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Name
MenJackets
WomenJackets
MenPants
WomenPants
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase
Decrease
0
-250
2000
1E+30
+ 250
6
0
2800
200
550
0
-300
1200
300
1E+30
6
0
1500 366.6666667
100
Constraints
Cell
$F$5
$F$6
$F$7
$F$8
Final Shadow Constraint Allowable Allowable
Name
Value
Price
R.H. Side
Increase
Decrease
Denim-Yds Total
1650
0
2500
1E+30
850
Cutting-Hrs Total
36
550
36
4
6
Stitching-Hrs Total
30
0
36
1E+30
6
Boxing-Hrs Total
7.5
800
7.5 1.214285714
0.75
25
APPAREL INDUSTRY Solution
• Assume the unit profit of women’s pants increases by 20%. If all pants
produced can still be sold should Exclaim change its production plan in
order to sell more women’s pants? Explain.
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Name
MenJackets
WomenJackets
MenPants
WomenPants
Constraints
Cell
$F$5
$F$6
$F$7
$F$8
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase
Decrease
0
-250
2000
250
1E+30
6
0
2800
200
550
0
-300
1200
300
1E+30
6
0
1500 366.6666667
100
No.
20%(1500)=
300 < 366.67
Allowable
Final Shadow Constraint Allowable
Name
Value
Price
R.H. Side
Increase
Decrease
Denim-Yds Total
1650
0
2500
1E+30
850
Cutting-Hrs Total
36
550
36
4
6
Stitching-Hrs Total
30
0
36
1E+30
6
Boxing-Hrs Total
7.5
800
7.5 1.214285714
0.75
26
APPAREL INDUSTRY Solution
• Management is considering allocating one hour of overtime in one
department. Where should this hour be allocated? How much should
be paid per hour?
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell
Costat the
Coefficient
Increase
Decrease
Both Cutting
andName
Boxing areValue
candidates
shadow price
contribution
shown.
MenJackets
0
2000
250 $800
1E+30
Prefer$B$2
the boxing
department because
1-250
hour of overtime
contributes
to the
6
0
2800
200
550
profit$C$2
(moreWomenJackets
than $550 in the cutting
department).
$D$2 MenPants
0
-300
1200
300
1E+30
$E$2 WomenPants
6
0
1500 366.6666667
100
Constraints
Cell
$F$5
$F$6
$F$7
$F$8
Final Shadow Constraint Allowable Allowable
Name
Value
Price
R.H. Side
Increase
Decrease
Denim-Yds Total
1650
0
2500
1E+30
850
Cutting-Hrs Total
36 +1
550
36
4
6
Stitching-Hrs Total
30 +1
0
36
1E+30
6
Boxing-Hrs Total
7.5 +1
800
7.5 1.214285714
0.75
27
APPAREL INDUSTRY Solution
• Suppose that, in addition to the existing restrictions,management
wishes to produce at least 300 of each item. Add these constraints to
your linear program and re-solve the problem. What is the result?
To what do you attribute this result?
MenJackets WomenJackets
MenJackets WomenJackets
3
Profit
Profit
Denim-Yds
Denim-Yds
Cutting-Hrs
Cutting-Hrs
Stitching-Hrs
Stitching-Hrs
Boxing-Hrs
Boxing-Hrs
M-Jacket>3
M-Jacket>3
W-Jacket>3
W-Jacket>3
M-Pants>3
M-Pants>3
W-Pants>3
W-Pants>3
2000
2000
150
150
33
44
0.75
0.75
11
Additional constraints
3
2800
2800
125
125
44
33
0.75
0.75
MenPants
MenPants
3
1200
1200
200
200
22
22
0.5
0.5
11
11
WomenPants
WomenPants
3
Total
Total
1500
225000
1500
150
18750 <=
150
<=
22
330 <=
<=
22
330 <=
<=
0.5
7.5
<=
0.5
0 <=
30 >=
>=
30 >=
>=
30 >=
>=
11
30 >=
>=
2500
2500
36
36
36
36
7.5
7.5
3
5
3
5
3
5
3
5
28
APPAREL INDUSTRY Solution
• If the minimum required of 300 women’s pants is increased to 350,
what will happen to the profit?
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Final
Value
Reduced
Cost
Name
MenJackets
3
WomenJackets
3.666666667
MenPants
Cover only if the3100%
WomenPants
3
rule
Objective Allowable
Allowable
Coefficient Increase
Decrease
0
2000
800
1E+30
0
2800
1E+30
550
0
1200 666.6666667
1E+30
was
discussed
in class
0
1500 366.6666667
1E+30
Constraints
Cell
$F$5
$F$6
$F$7
$F$8
$F$9
$F$10
$F$11
$F$12
Name
Denim-Yds Total
Cutting-Hrs Total
Stitching-Hrs Total
Boxing-Hrs Total
M-Jacket>3Total
W-Jacket>3 Total
M-Pants>3 Total
W-Pants>3 Total
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side
Increase
1958.333333
0
2500
1E+30
35.66666667
0
36
1E+30
35
0
36
1E+30
8 3733.333333
8
0.0625
3
-800
3 0.666666667
3.666666667
0
3 0.666666667
3 -666.6666667
3
1
3
1
3.53 -366.6666667
Allowable
Decrease
541.6666667
0.333333333
1
0.5
0.333333333
1E+30
0.529
0.5
APPAREL INDUSTRY Solution
• Managements wants to relax one of the constraints on the minimum of
300 units production by 30 units. Which constraint should be selected?
What is the new profit?
Discussion:
Material constraint
2600
150MJacket+125WJacket+200MPants+150WPants £ 2500
Minimum required constraint
MJacket  32
Cover only if the 100% rule was discussed in class
30
APPAREL INDUSTRY Solution
• Managements wants to relax one of the constraints on the minimum of
300 units production by 30 units. Which constraint should be selected?
What is the new profit?
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Name
MenJackets
WomenJackets
MenPants
WomenPants
Final
Value
3
3.666666667
3
3
Reduced
Cost
0
0
0
0
Objective Allowable
Coefficient Increase
2000
800
2800
1E+30
1200 666.6666667
1500 366.6666667
Allowable
Decrease
1E+30
550
1E+30
1E+30
Constraints
Cell
$F$5
$F$6
$F$7
$F$8
$F$9
$F$10
$F$11
$F$12
Name
Denim-Yds Total
Cutting-Hrs Total
Stitching-Hrs Total
Boxing-Hrs Total
M-Jacket>3Total
W-Jacket>3 Total
M-Pants>3 Total
W-Pants>3 Total
Final
Shadow
Constraint Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
1958.333333
0
2500
1E+30 541.6666667
35.66666667
0
36
1E+30 0.333333333
35
0
36
1E+30
1
8 3733.333333
8
0.0625
0.5
3
-800
3 0.666666667 0.333333333
3.666666667
0
3 0.666666667
1E+30
3 -666.6666667
3
1
0.5 31
3 -366.6666667
3
1
0.5
APPAREL INDUSTRY Solution
• Managements wants to relax one of the constraints on the minimum of
300 units production by 30 units. Which constraint should be selected?
What is the new profit?
Change in objective value =
(Shadow price)(Change in the right hand side) =
(-800)(-.3) = +240.
The new objective value is: 22,500+240 = 22,740.
The change in the right hand side is (-.3) because:
(i) Relaxing this constraint means a reduction(!) in the right hand side
(ii) A change of 30 units translates to .3
32
Modeling and Sensitivity
Analysis– Example 2
• AGRICULTURE: BP Farms is a 300-acre farm located
near Lawrence, Kansas, owned and operated exclusively
By Bill Phashley. For the upcoming growing season, Bill
will grow wheat, corn, oats, and soybeans. The following
table gives relevant data concerning expected crop yields,
labor required, expected expenses, and water required.
Also included is the price per bushel Bill expects to
r4eceivre when the crops are harvested.
33
AGRICALTURE
Yield
Labor Expenses
(bush./acr (hr/acre ($/acre)
e
)
•
•
•
•
Water
(acreft./acre)
Price
($/bushel
)
Wheat
210
4
$50
2
$3.20
Corn
300
5
$75
6
$2.55
Oats
180
3
$30
1
$1.45
Soybeans
240
10
$60
4
$3.10
Bill wishes to produce at least 30,000 bushels of wheat and 30,000
bushels of corn, but not more than 25,000 bushels of oats.
Bill has $25,000 to invest in his crop.
Bill plans to work 12 hours a day during the 150-day season.
He does not wish to exceed the base water supply of 1200acre-feet
allocated to him.
34
AGRICALTURE – Solution
• How many acres should Bill allocate to each crop?
Discussion:
• Since the return depends on the acres allocated to each crop, the decision
variables are Wheat, Corn, Oats, and Soybeans expressed in acres.
• The objective function represents the return, that is sales minus expenses.
In general for each crop:
Return from selling a crop =
{[Price per bushel][Yield in bushels per acre] – [Expenses in $ per
acre]}[Acres]
For example: Return from selling wheat =
[(3.20$/bu.)(210bu./acre) – 50](“Wheat” in acres)
35
AGRICALTURE – Solution
• How many acres should Bill allocate to each crop?
Wheat
Corn
Oats
142.8571 142.8571
Soybeabs
Soybeans
0 14.28571
Total
Total
Sales
Sales
Acres
Acres
Wheat>30K
Wheat>30K
Corn>30K
Corn>30K
Oats<25K
Oats<25K
Budget
Budget
Water
Water
Labor
Labor
622
622
1
1
210
1
690
690
1
1
50
684
684
11
300
1
50
2
2
4
4
231
231
11
75
75
6
6
5
5
180
1
30
301
13
3
60
604
4
10
10
197200
0
300
<=
0 <=
30000 >=
0 >=
42857.14 >=
0 >=
0 <=
0
18714.29 <=
<=
0
<=
1200 <=
0 <=
1428.571
<=
0=
300
300
30000
30000
30000
30000
25000
25000
25000
25000
1200
1200
1800
1800
622=(3.2)(210)-50
622=(3.2)(210)-50
36
AGRICALTURE – Solution
• If the selling price of oats remains $1.45 a bushel, (1) to what level
must the yield increase before oats should be planted? (2) If the yield
remains 180 bu./acre, to what level would the price of oats have to rise
before oats should be planted?
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
We Name
need to useValue
the range ofCost
optimality
for the coefficient
Coefficient
Increase
Decrease
Wheat
0
of Oats in the 142.8571429
objective function. For
Oats 672
to be50.99999999
planted its 1E+30
Corn
142.8571429
0
765 50.99999999
21
return per acre should0be-451.5000055
at least 261+451.5
$712.5.
Oats
260.9999967 =451.5000055
1E+30
Soybeabs
0
21
25.5
(1) If the price14.28571429
remains $1.45 a bushel,
the744
yield
Constraintsshould increase to 712.5/1.45 = 491.38 bushels per acre.
Final
Shadow
Constraint
Allowable
Allowable
(2)
If
the
yield
remain
180
bushels/acre
the
price
should
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$F$5 Acres
Total to 712.5/180
300= $3.95 per
702 bushel 300 18.57142857 4.761904762
increase
Cell
$B$2
$C$2
$D$2
$E$2
$F$6
$F$7
$F$8
$F$9
$F$10
$F$11
Wheat>30K Total
Corn>30K Total
Oats<25K Total
Budget Total
Water Total
Labor Total
30000 -0.242857143
42857.14286
0
0
0
18714.28571
0
1200
10.5
1428.571429
0
30000
1500 7090.909091
30000 12857.14286
1E+30
25000
1E+30
25000
25000
1E+30 6285.714286
1200 28.57142857 85.71428571
1800
1E+30 371.4285714
37
AGRICALTURE – Solution
• If there were no constraint on the minimum production of
corn, would corn be planted?
Adjustable Cells
Cell
$B$2
$C$2
$D$2
$E$2
Name
Wheat
Corn
Oats
Soybeabs
Final
Reduced
Objective
Value
Cost
Coefficient
142.8571429
0
672
142.8571429
0
765
0 -451.5000055 260.9999967
14.28571429
0
744
Allowable
Allowable
Increase
Decrease
50.99999999
1E+30
50.99999999
21
451.5000055
1E+30
21
25.5
No constraint on the minimum production of corn translates to a
constraint of the form 300Corn 0. Notice that this is always true
(and in fact already exists as a non-negativity constraint). A change
Constraints
Shadow has
Constraint
Allowable
in the right hand side Final
of this constraint
occurredAllowable
(from 30000
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
down
to
zero).
$F$5 Acres Total
300
702
300 18.57142857 4.761904762
$F$6
Total solution
30000
30000
7090.909091
SinceWheat>30K
the current
calls-0.242857143
for a production
of 42,8571500
bushels,
$F$7 Corn>30K Total
42857.14286
0
30000 12857.14286
1E+30
the
optimal
solution
remains
unchanged,
and
corn
is
still
planted.
$F$8 Oats<25K Total
0
0
25000
1E+30
25000
$F$9 Budget Total
$F$10 Water Total
$F$11 Labor Total
18714.28571
1200
1428.571429
0
10.5
0
25000
1E+30 6285.714286
1200 28.57142857 85.71428571
1800
1E+30 371.4285714
38
AGRICALTURE – Solution
• Bill can lease an adjacent 40-acre parcel for $10000.
Should Bill lease this property for this price?
Adjustable Cells
Final
Reduced
Objective
Value
Cost
Coefficient
142.8571429
0
672
142.8571429
0
765
0 -451.5000055 260.9999967
14.28571429
0
744
Allowable
Allowable
Increase
Decrease
50.99999999
1E+30
50.99999999
21
451.5000055
1E+30
21
25.5
A change occurs in the “Acres Total” constraint. The right hand side
is changing from 300 to 340. The maximum value of the right hand side
for the shadow price to remain unchanged is 300+18.57=318.57.
So the shadow price changes if the total acre is 340.
Constraints
Still, we can answer theFinal
question.Shadow Constraint Allowable Allowable
Name the objective
Value
Price
R.H. Side
Increase
AtCell
318.57 acres
value
(total return)
increases
by Decrease
$F$5 Acres Total
300
702
300 18.57142857 4.761904762
(Shadow
price)(Constraint
change)
=
(702)(18.57)
= $13,036.
$F$6 Wheat>30K Total
30000 -0.242857143
30000
1500 7090.909091
$F$7 amount
Corn>30KisTotal
42857.14286
0
30000
12857.14286
1E+30
This
greater
than the cost of $10,000.
Clearly,
with a change
$F$8 Oats<25K Total
0
0
25000
1E+30
25000
of$F$9
40 acres
the
return
exceeds
the
cost.
Bill
should
lease
the
property.
Budget Total
18714.28571
0
25000
1E+30 6285.714286
Cell
$B$2
$C$2
$D$2
$E$2
Name
Wheat
Corn
Oats
Soybeabs
$F$10 Water Total
$F$11 Labor Total
1200
1428.571429
10.5
0
1200 28.57142857 85.71428571
1800
1E+30 371.4285714
39