Team Discovery Channel
INEN 420
December 2, 2004
Grummins Trucks
• The goal of this problem is to maximize the
profit of Grummins Engine Company over the
next three years
• Decision Variables
– Si,j = number of type i trucks sold in year j
– Xi,j = number of type i trucks produced in year j
– Ii,j = number of type i trucks held in year j
Formulation
Objective Function:
MAX Z=20000(s11+s12+s13) + 17000(s21+s22+s23) – 15000(x11+x12+x13) -14000(x21+x22+x23)
-2000(I11+I12+I21+I22)
Constraints:
Production
Truck 1 Inventory Control
Truck 2 Inventory Control
x11+x21<=320
x12+x22<=320
x13+x23<=320
x11>=s11
x11-s11=I11
x12+I11>=s12
x12+I11-s12=I12
x13+I12>=s13
x21>=s21
x21-s21=I21
x22+I21>=s22
x22+I21-s22=I22
x23+I22>=s23
Demand Constraints
s11<=100
s12<=200
s13<=300
s21<=200
s22<=100
s23<=150
Emissions Constraints
Non-negative
(15(x11+x12+x13)+5(x21+x22+x23))
Si,j=>0
Xi,j=>0
Ii,j=>0
<=10
(x11+x12+x13+ x21+x22+x23)
Solution
•Max Profit: $3,600,000
•7 iterations using LINDO
Sensitivity Analysis
ƒSA table gives the maximum amounts that which the
objective function coefficients and right-hand side values
(objective function and constraints) could increase or
decrease with the current basis remaining optimal.
ƒNBV coefficients could decrease infinitely or increase by
2000 (inventory costs)
ƒAll the coefficients for variables representing all trucks sold
except for type 1 year 3 can increase infinitely (continue to
maximize profits with same basis)
Final Say
•Type 1 recommendations
-Sell 100 in year 1
- 200 in year 2
- 150 in year 3
•Type 2 recommendations
- Sell 200 in year 1
- 100 in year 2
- 150 in year 3.
•We recommend that no inventory be held for this system
•A suggestion would be to decrease the pollution levels of truck
type 1; this might increase our profit sales for year 3.
Wheaties
•Goal is to maximize profits over the next 10 months
•Decision Variable
- Si = number of bushels (x1000) sold in month i
- Bi = number of bushels (x1000) bought in month i
- Ii = initial stock of bushes (x1000) for month i
Formulation
Objective Function:
max Z =3S1+6S2+7S3+S4+4S5+5S6+5S7+s8+3S9+2S10-8B1
-8B2-2B3-3B4-4B5-3B6-3B7-2B8-5B9-5B10
Constraints:
I1=6
S1<=I1
(I1-S1)+B1<=20
(I1-S1)+B1=I2
S2<=I2
(I2-S2)+B2<=20
(I2-S2)+B2=I3
S3<=I3
(I3-S3)+B3<=20
Non-negative
Si=>0
Bi =>0
Ii =>0
(I9-S9)+B9=I10
S10<=I10
(I10-S10)+B10<=20
Solution
•Max Profit is $162
•27 iterations using LINDO
Sensitivity Analysis
•S1 (amount of bushels sold in month 1) could decrease
infinitely or increase by 4 to still keep the same basis.
•This means to increase the selling price up to 7 or decrease
to 0, since the selling price should be non-negative.
•We cannot increase the selling price infinitely on all S
variables because in this problem, some months have a
higher purchasing price than selling price.
•Increasing infinitely on NBV selling coefficients (Si) would
change the basis by making the selling price higher than the
purchasing price for those months
Final Say
•Month 8 and 10 held no inventories.
•Amount sold in month 1, 2, 4, and 5 was less than the
initial inventory.
•We recommend buying when the purchasing price is
less than the selling price, such as month 3 and 6.
•Increasing out warehouse capacity will increase our
profits.
T-Back, Inc.
T-Back Inc. is a supplier of
luxury thongs to Victoria
Secret. Considering
transportation costs,
demand, and selling prices,
T-Back Inc. has hired us to
maximize their profits for
the upcoming holiday
season.
Decision Variables
•
Xi,j=number of thongs shipped to demand point i to supply
point j
Demand
1) San
Francisco
2) New York
3) Chicago
4) Dallas
5) Los Angeles
6) Las Vegas
Supply
(1)New Orleans
(2) Seattle
(3) Miami
(4) Houston
Formulation
-Their trucks can hold up to 100 thongs, but we only pay for the
•space we take up (shipping costs/thong).
-The national average as of 11/29/04 for diesel gas prices per
gallon is $2.11
max Z=
40(X11+X12+X13+X14)+42(X21+X22+X23+X24)+36(X31+X32+X33+X34)+20(X41+X42+X43+X
44)+33(X51+X52+X53+X54)+19(X61+X62+X63+X64)-2.1285X11-8.229X12-6.1929X135.0851X14-7.5380X21-3.4235X22-3.4499X23-4.3255X24-5.4359X31-3.8402X32-2.4423X333.1676X34-5.7999X41-3.5976X42-1.3715X43-.6304X44-2.9962X51-7.2294X524.9954X534.0855X54-3.3153X61-7.0632X62-4.8372X63-4.1013X64
Demand Constraints
X11+X12+X13+X14<=100
X21+X22+X23+X24<=250
X31+X32+X33+X34<=50
X41+X42+X43+X44<=75
X51+X52+X53+X54<=150
X61+X62+X63+X64<=300
Supply Point Constraints
Non-negative Constraints
X11+X21+X31+X41+X51+X61<=300 Xi,j for i=1…6 and j=1…4 are >=0
X12+X22+X32+X42+X52+X62<=300
X13+X23+X33+X43+X53+X63<=150
X14+X24+X34+X44+X54+X64<=175
Solution
•Our Profit was $25,391.08
•16 iterations using LINDO
Sensitivity Analysis
Example:
The basic variable coefficient of x22 can increase indefinitely
and decrease by 2.25. Below 36.32 (38.57-2.25), the profit
margin is too small, changing the basis.
However, increasing the coefficient indefinitely will just increase
our max profit (z) and not change the basis
Final Say
•All our demand was met; and all our supply points were capitalized.
•Seattle fed all of San Francisco’s and Los Angeles’s demand.
This makes sense since the distance is smaller, and therefore the
cost of transportation is reduced. New York’s and Chicago’s
demand is fed by Miami; Dallas demand is taken care of by Houston.
The major provider for Las Vegas is New Orleans with support from
Seattle and Houston.
•Increasing the thong demand of our most profitable cities,
decreasing transportation costs by increasing trucking capacity,
and moving to a more opportune location to reduce mileage
will increase our profits.
Overall Conclusion
•Formulations were the easy part; recognizing what LINDO
gave us in the analysis was a little bit more complicated
•We could have used EXCEL Solver for these problems
•All our variables needed to be discrete (integer) since
divisibility assumption is not met
•All other assumptions were assumed (additivity,
proportionality, and certainty)
•We could have played with the coefficients and the
constraints more to see which variables could have the
biggest impact on profits if small changes were made