INEN 420
Team Discovery Channel:
“Extreme Engineering”
Due: December 3, 2004
Dr. Lewis Ntaimo
Members:
Francis Pinero
Todd Miller
Diana Tran
“On my honor, as an Aggie, I have neither given nor received unauthorized
aid on this academic work.”
________________________________________
________________________________________
________________________________________
1
Table of Contents
______________________________________________________
Executive Summary………………………………………3
Problem Description……………………………………....5
Grummin’s Truck …………………………………...5
Wheatie’s ………….……………..............................8
Extreme Engineering ………………………………12
Computation Results……………………………………..16
Grummin’s …………………………………………16
Wheatie’s …………………………………………..20
Extreme Engineering ………………………………24
Conclusions & Recommendations……………………….28
References………………………………………………..29
Appendix-LINDO Code…………...……………………..30
2
Executive Summary
On the Grummins Truck problem, the linear formulation was fairly easy. We
made several assumptions including that Grummins Truck Company was the only truck
producer for this country. This allowed us to calculate the average pollution emissions
per truck sold over the next 3 years.
After putting the linear formulation into standard form and entering it into
LINDO, we found the maximum profit over the next three years to be $3,600,000. It
took 7 iterations and less than a second for the program to give us all our analysis. The
production of the both type of trucks for all three years were below the 320 maximum
production capacity. The only year that sold below maximum demand was year 3 with
truck type 1. No inventory was held for this system.
We recommend for type 1 trucks to sell 100 in year 1, 200 in year 2, and 150 in
year 3. For type 2 trucks, we recommend selling 200 in year 1, 100 in year 2, and 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.
On the Wheatie’s problem, the formulation was slightly harder since this system
needed to include previous, current, and future inventories. After putting it in standard
form, our maximum profit obtained from LINDO was $162. Month 8 and 10 held no
inventories. Many of the months had a higher purchasing cost than selling cost. This
controlled our buying and selling. 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.
On our T-Back Inc. problem, we found that x11, x22, x32, x44, x51, x61, x63,
and x64 were the only variables with traffic. Our maximum profit was $25,391.08. All
our supply and demand nodes were used. 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
3
care of by Houston. The major provider for Las Vegas is New Orleans with support from
Seattle and Houston. Las Vegas is so central to all our ports that it requires shipments
from almost all of them.
We recommend finding another trucking company that can lower the transportation costs.
Also, increasing demand through advertising in our most profitable cities will increase
profits.
4
Problem Description
Problem 1: Grummins Trucks
Description
Grummins Engine Company produces two types of trucks. With the new government
emission standards, Grummins would like to maximize their profit over the next three
years. Inventory can be held at a cost, and all maximum demands are given for the next 3
years.
Assumptions
We assume the maximum demands given are correct. We also assume the maximum
demand is not met for every year. We assume that Grummins is the only truck producer.
Problem Data
The following lists and tables summarize the data given.
o
The emission standards states that the average pollution emissions of all trucks
produced over the next three years cannot exceed 10 grams per truck.
o
Production capacity limits total truck production during each year to at most 320
trucks.
o
Inventory can be held.
Maximum Demand Over Next 3 Years
Year
Type 1
Type 2
1
100
200
2
200
100
3
300
150
5
Truck Specifications (per truck)
Sells for
Cost
Pollution Emissions (grams)
Inventory Cost
Type 1
$20000
$15000
15
$2000
Type 2
$17000
$14000
5
$2000
Decision Variables
Our decision variables are
xi,j= number of type i trucks produced in year j
si,j=number of type i trucks sold in year j
Ii,j= number of type i trucks held in inventory year j
Objective Function
The goal of this problem is to maximize the profit of Grummins Engine Company. Profit
is our total revenue minus our costs of manufacturing our trucks and inventory.
Assigning our total profit to be Z, our objective function is the following:
max Z=20000(s11+s12+s13 + 17000(s21+s22+s23) – 15000(x11+x12+x13) 14000(x21+x22+x23) - 2000(I11+I12+I21+I22)
LP Formulation
Given all the information above, we were able to create our LP formulation.
max Z=20000(s11+s12+s13) + 17000(s21+s22+s23) – 15000(x11+x12+x13) 14000(x21+x22+x23) - 2000(I11+I12+I21+I22)
St:
Production Constraints:
x11+x21<=320
x12+x22<=320
x13+x23<=320
Truck 1 Inventory Control:
6
x11>=s11
x11-s11=I11
x12+I11>=s12
x12+I11-s12=I12
x13+I12>=s13
Truck 2 Inventory Control:
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:
(15(x11+x12+x13)+5(x21+x22+x23))/(x11+x12+x13+ x21+x22+x23)<=10
Non-negative Constraints:
x11>=0
x12>=0
x13>=0
x21>=0
x22>=0
x23>=0
7
s11>=0
s12>=0
s13>=0
s21>=0
s22>=0
s23>=0
Problem 2: Wheatie’s
Description
Our wheat warehouse wants to maximize its profit over the next 10 months. Each month,
wheat can be bought and sold given the rules explained in the sections below.
Assumptions
We assume we buy bushels of wheat at the end of each month.
Problem Data
The following lists and tables summarize the data given.
o
We observe our initial stock at the beginning of each month.
o
We can sell any amount of wheat up to our initial stock at the current month’s selling
price.
o
We can buy as much wheat as we want, subject to the warehouse capacity limit.
o
Our warehouse can hold up to 20,000 bushels.
o
We start out with an initial stock of 6000 bushels.
8
Price per 1000 bushels
Selling Price
Month
($)
1
3
2
6
3
7
4
1
5
4
6
5
7
5
8
1
9
3
10
2
Purchase Price
($)
8
8
2
3
4
3
3
2
5
5
Decision Variables
Our decision variables are
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
Objective Function
Assigning our total profit to be Z, our objective function is the following:
max Z=3S1+6S2+7S3+S4+4S5+5S6+5S7+s8+3S9+2S10-8B1-8B2-2B3-3B4-4B5-3B63B7-2B8-5B9-5B10
LP Formulation
Given all the information above, we were able to create our LP formulation.
max Z=3S1+6S2+7S3+S4+4S5+5S6+5S7+s8+3S9+2S10-8B1-8B2-2B3-3B4-4B5-3B63B7-2B8-5B9-5B10
St:
I1=6
S1<=I1
(I1-S1)+B1<=20
9
(I1-S1)+B1=I2
S2<=I2
(I2-S2)+B2<=20
(I2-S2)+B2=I3
S3<=I3
(I3-S3)+B3<=20
(I3-S3)+B3=I4
S4<=I4
(I4-S4)+B4<=20
(I4-S4)+B4=I5
S5<=I5
(I5-S5)+B5<=20
(I5-S5)+B5=I6
S6<=I6
(I6-S6)+B6<=20
(I6-S6)+B6=I7
S7<=I7
(I7-S7)+B7<=20
(I7-S7)+B7=I8
S8<=I8
(I8-S8)+B8<=20
(I8-S8)+B8=I9
S9<=I9
(I9-S9)+B9<=20
10
(I9-S9)+B9=I10
S10<=I10
(I10-S10)+B10<=20
Non-negative Constraints:
I1>=0
I2>=0
I3>=0
I4>=0
I5>=0
I6>=0
I7>=0
I8>=0
I9>=0
I10>=0
B1>=0
B2>=0
B3>=0
B4>=0
B5>=0
B6>=0
B7>=0
B8>=0
B9>=0
B10>=0
S1>=0
S2>=0
S3>=0
S4>=0
S5>=0
11
S6>=0
S7>=0
S8>=0
S9>=0
S10>=0
Problem 3: Extreme Engineering
Description
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.
Assumptions
We assume the estimates for demand are accurate. We assume the diesel price acquired
stay stable through the holiday season. We are assuming no maintenance costs on the
trucks.
Problem Data
The following lists and tables summarize the given data.
o T-Back Inc. has 4 distribution centers located a major ports in New Orleans, Seattle,
Miami, and Houston.
o The major Victoria Secret stores ordering our thongs are located in San Francisco,
New York, Chicago, Dallas, Los Angeles, and Las Vegas.
o T-Back Inc. uses a subcontractor to ship their items. Their trucks can hold up to 100
thongs, but we only pay for the space we take up (shipping costs/thong).
o The national average as of 11/29/04 for diesel gas prices per gallon is $2.11.
Transportation Cost Estimations
2.11
Cost/Gallon
8
Miles/Gallon
0.26375
Cost/Mile
12
Demand Point Specifications
Demand
San Francisco (1)
100
New York (2)
250
Chicago (3)
50
Dallas (4)
75
Los Angeles (5)
150
Las Vegas (6)
300
Supply Point Specifications
Supply
Limits
Seattle (1)
Miami (2)
New Orleans (3)
Houston (4)
Selling Price ($)
40
42
36
20
33
19
300
300
150
175
Milage Chart
SF
NY
Chicago
Dallas
LA
LV
Seattle
807
2858
2061
2199
1136
1257
Miami
3120
1298
1456
1364
2741
2678
NO
2348
1308
926
520
1894
1834
Houston
1928
1640
1201
239
1549
1555
Cost Estimates per Truck Chart
Seattle
Miami
NO
Houston
212.84625
822.9
619.285
508.51
SF
753.7975 342.3475 344.985
432.55
NY
543.58875
384.02
244.2325 316.76375
Chicago
579.98625 359.755
137.15
63.03625
Dallas
299.62
722.93875 499.5425 408.54875
LA
331.53375 706.3225 483.7175 410.13125
LV
13
Cost Estimates per Thong Chart
Seattle
Miami
NO
Houston
2.1284625
8.229 6.19285
5.0851
SF
7.537975 3.423475 3.44985
4.3255
NY
3.8402 2.442325 3.1676375
Chicago 5.4358875
5.7998625
3.59755
1.3715 0.6303625
Dallas
2.9962 7.2293875 4.995425 4.0854875
LA
3.3153375 7.063225 4.837175 4.1013125
LV
Decision Variables
Our decision variables are the following:
Xi,j=number of thongs shipped to demand point i to supply point j
Objective Function
Assigning our total profit to be Z, our objective function is the following:
max Z=
40(X11+X12+X13+X14)+42(X21+X22+X23+X24)+36(X31+X32+X33+X34)+20(X41
+X42+X43+X44)+33(X51+X52+X53+X54)+19(X61+X62+X63+X64)-2.1285X118.229X12-6.1929X13-5.0851X14-7.5380X21-3.4235X22-3.4499X23-4.3255X245.4359X31-3.8402X32-2.4423X33-3.1676X34-5.7999X41-3.5976X42-1.3715X43.6304X44-2.9962X51-7.2294X52-4.9954X53-4.0855X54-3.3153X61-7.0632X624.8372X63-4.1013X64
LP Formulation
Given all the information above, we were able to create our LP formulation.
max Z=
40(X11+X12+X13+X14)+42(X21+X22+X23+X24)+36(X31+X32+X33+X34)+20(X41
+X42+X43+X44)+33(X51+X52+X53+X54)+19(X61+X62+X63+X64)-2.1285X118.229X12-6.1929X13-5.0851X14-7.5380X21-3.4235X22-3.4499X23-4.3255X245.4359X31-3.8402X32-2.4423X33-3.1676X34-5.7999X41-3.5976X42-1.3715X43.6304X44-2.9962X51-7.2294X52-4.9954X53-4.0855X54-3.3153X61-7.0632X624.8372X63-4.1013X64
14
St:
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
X11+X21+X31+X41+X51+X61<=300
X12+X22+X32+X42+X52+X62<=300
X13+X23+X33+X43+X53+X63<=150
X14+X24+X34+X44+X54+X64<=175
Non-negative Constraints
Xi,j for i=1…6 and j=1…4 are >=0
15
Computational Results
Problem 1: Grummins Truck
Optimal Solution:
Our maximum profit for Grummins Truck Company during the next three years was
$3,600,000. The table below summarized the specific decision variable values.
Variable Truck Type
S11
Type 1
S12
Type 1
S13
Type 1
S21
Type 2
S22
Type 2
S23
Type 2
X11
Type 1
X12
Type 1
X13
Type 1
X21
Type 2
X22
Type 2
X23
Type 2
I11
Type 1
I12
Type 1
I21
Type 2
I22
Type 2
Action
Sold
Sold
Sold
Sold
Sold
Sold
Produced
Produced
Produced
Produced
Produced
Produced
held in Inventory
held in Inventory
held in Inventory
held in Inventory
Slack or Surplus:
ROW SLACK OR SURPLUS DUAL PRICES
2)
20.000000
0.000000
3)
20.000000
0.000000
4)
20.000000
0.000000
5)
0.000000
-2000.000000
6)
0.000000 -18000.000000
7)
0.000000
0.000000
8)
0.000000 -20000.000000
9)
0.000000 -20000.000000
10)
0.000000
0.000000
11)
0.000000
-9000.000000
16
Year
in Year 1
in Year 2
in Year 3
in Year 1
in Year 2
in Year 3
in Year 1
in Year 2
in Year 3
in Year 1
in Year 2
in Year 3
in Year 1
in Year 2
in Year 1
in Year 2
Value
100
200
150
200
100
150
100
200
150
200
100
150
0
0
0
0
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
0.000000
0.000000
0.000000
0.000000
0.000000
150.000000
0.000000
0.000000
0.000000
0.000000
100.000000
200.000000
150.000000
200.000000
100.000000
150.000000
100.000000
200.000000
150.000000
200.000000
100.000000
150.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-9000.000000
-9000.000000
0.000000
0.000000
0.000000
8000.000000
8000.000000
8000.000000
1000.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Running Time: 00:00:01
Iterations: 7
Software: LINDO
Sensitivity Report (as Given in LINDO):
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE
CURRENT
ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
S11 20000.000000
INFINITY
0.000000
S12 20000.000000
INFINITY
0.000000
S13 20000.000000
0.000000
5000.000000
S21 17000.000000
INFINITY
8000.000000
S22 17000.000000
INFINITY
8000.000000
S23 17000.000000
INFINITY
8000.000000
17
X11
X12
X13
X21
X22
X23
I11
I12
I21
I22
-15000.000000
-15000.000000
-15000.000000
-14000.000000
-14000.000000
-14000.000000
-2000.000000
-2000.000000
-2000.000000
-2000.000000
2000.000000
0.000000
2000.000000
0.000000
0.000000
2000.000000
2000.000000
8000.000000
2000.000000
2000.000000
9000.000000
2000.000000
2000.000000
INFINITY
2000.000000
INFINITY
2000.000000
INFINITY
2000.000000
INFINITY
RIGHTHAND SIDE RANGES
ROW
CURRENT
ALLOWABLE
ALLOWABLE
RHS
INCREASE
DECREASE
2
320.000000
INFINITY
20.000000
3
320.000000
INFINITY
20.000000
4
320.000000
INFINITY
20.000000
5
0.000000
0.000000
INFINITY
6
0.000000
20.000000
0.000000
7
0.000000
0.000000
INFINITY
8
0.000000
20.000000
0.000000
9
0.000000
150.000000
150.000000
10
0.000000
0.000000
INFINITY
11
0.000000
20.000000
0.000000
12
0.000000
0.000000
INFINITY
13
0.000000
20.000000
0.000000
14
0.000000
10.000000
150.000000
15
100.000000
20.000000
20.000000
16
200.000000
20.000000
20.000000
17
300.000000
INFINITY
150.000000
18
200.000000
20.000000
150.000000
19
100.000000
20.000000
100.000000
20
150.000000
10.000000
150.000000
21
0.000000
100.000000
750.000000
22
0.000000
100.000000
INFINITY
23
0.000000
200.000000
INFINITY
24
0.000000
150.000000
INFINITY
25
0.000000
200.000000
INFINITY
26
0.000000
100.000000
INFINITY
27
0.000000
150.000000
INFINITY
28
0.000000
100.000000
INFINITY
29
0.000000
200.000000
INFINITY
30
0.000000
150.000000
INFINITY
31
0.000000
200.000000
INFINITY
32
0.000000
100.000000
INFINITY
33
0.000000
150.000000
INFINITY
18
Interpretations of Results:
Based on our slack/surplus analysis from LINDO, we found that the production capacity
constraints all had a slack of 20. This meant that production was not the maximum at 320
but instead produced 300 trucks a year. Type 1 trucks sold in year 3 were below the
yearly demand of 300. Instead, we only sold 150 trucks. All inventory levels were zero
for every year; all other decision variables were greater than zero.
In our sensitivity analysis, changes in the non-basic variable coefficients of our objective
function such as all the inventory level coefficients could decrease infinitely or increase
by 2000 to still keep the same basis. Decreasing infinitely would mean increasing the
cost to hold inventory, which would still keep our optimal solution. Increasing to 2000
will just make our inventory costs equal to zero.
Our sensitivity analysis 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. For example, the
coefficient of the variable for type 1 trucks sold in year 1 can increase to infinity or
decrease 0 to keep the basis the same. All the coefficients for variables representing all
trucks sold except for type 1 year 3 can increase infinitely. This makes sense since the
basis will remain optimal, even though the max profits will increase infinitely.
19
Problem 2: Wheatie’s
Optimal Solution:
Our maximum profit for the wheat warehouse during the next 10 months was $162. The
table below summarizes the specific decision variable values.
Quantity
(X1000)
Reduced
Cost
Sold
0
4
Sold
0
1
Sold
6
0
Sold
0
2
Sold
0
0
Sold
20
0
Sold
20
0
Sold
0
1
Sold
20
0
Sold
0
0
Bought
0
1
Bought
0
1
Bought
20
0
Bought
0
0
Bought
0
0
Bought
20
0
Bought
0
1
Bought
20
0
Bought
0
3
Bought
0
5
Held
6
0
Held
6
0
Held
6
0
Held
20
0
Held
20
0
Held
20
0
Held
20
0
Held
0
0
Held
20
0
Held
0
0
Variable
Action
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
I1
I2
I3
I4
I5
I6
I7
I8
I9
I10
20
Slack or Surplus:
ROW SLACK OR SURPLUS DUAL PRICES
2)
0.000000
7.000000
3)
6.000000
0.000000
4)
14.000000
0.000000
5)
0.000000
-7.000000
6)
6.000000
0.000000
7)
14.000000
0.000000
8)
0.000000
-7.000000
9)
0.000000
5.000000
10)
0.000000
1.000000
11)
0.000000
-3.000000
12)
20.000000
0.000000
13)
0.000000
1.000000
14)
0.000000
-4.000000
15)
20.000000
0.000000
16)
0.000000
1.000000
17)
0.000000
-5.000000
18)
0.000000
2.000000
19)
0.000000
2.000000
20)
0.000000
-5.000000
21)
0.000000
3.000000
22)
20.000000
0.000000
23)
0.000000
-2.000000
24)
0.000000
0.000000
25)
0.000000
1.000000
26)
0.000000
-3.000000
27)
0.000000
1.000000
28)
20.000000
0.000000
29)
0.000000
-2.000000
30)
0.000000
2.000000
31)
20.000000
0.000000
Running Time: 00:00:01
Iterations: 27
Software: LINDO
Sensitivity Report (as Given in Lindo)
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE
CURRENT
ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
S1
3.000000
4.000000
INFINITY
21
S2
S3
S4
S5
S6
S7
S8
S9
S10
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
I1
I2
I3
I4
I5
I6
I7
I8
I9
I10
6.000000
7.000000
1.000000
4.000000
5.000000
5.000000
1.000000
3.000000
2.000000
-8.000000
-8.000000
-2.000000
-3.000000
-4.000000
-3.000000
-3.000000
-2.000000
-5.000000
-5.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1.000000
1.000000
1.000000
0.000000
INFINITY
INFINITY
2.000000
2.000000
1.000000
1.000000
1.000000
1.000000
1.000000
0.000000
INFINITY
1.000000
INFINITY
2.000000
5.000000
INFINITY
1.000000
1.000000
1.000000
INFINITY
INFINITY
INFINITY
1.000000
INFINITY
1.000000
INFINITY
1.000000
INFINITY
INFINITY
1.000000
2.000000
INFINITY
1.000000
2.000000
INFINITY
INFINITY
1.000000
INFINITY
1.000000
2.000000
2.000000
1.000000
INFINITY
INFINITY
INFINITY
4.000000
1.000000
1.000000
2.000000
1.000000
2.000000
INFINITY
1.000000
INFINITY
RIGHTHAND SIDE RANGES
ROW
CURRENT
ALLOWABLE
ALLOWABLE
RHS
INCREASE
DECREASE
2
6.000000
14.000000
6.000000
3
0.000000
INFINITY
6.000000
4
20.000000
INFINITY
14.000000
5
0.000000
6.000000
14.000000
6
0.000000
INFINITY
6.000000
7
20.000000
INFINITY
14.000000
8
0.000000
6.000000
INFINITY
9
0.000000
INFINITY
6.000000
10
20.000000
INFINITY
0.000000
11
0.000000
0.000000
20.000000
12
0.000000
INFINITY
20.000000
13
20.000000
0.000000
20.000000
22
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
0.000000
0.000000
20.000000
0.000000
0.000000
20.000000
0.000000
0.000000
20.000000
0.000000
0.000000
20.000000
0.000000
0.000000
20.000000
0.000000
0.000000
20.000000
20.000000
INFINITY
INFINITY
20.000000
INFINITY
INFINITY
20.000000
INFINITY
INFINITY
20.000000
0.000000
INFINITY
20.000000
INFINITY
INFINITY
20.000000
INFINITY
INFINITY
0.000000
20.000000
0.000000
INFINITY
20.000000
20.000000
INFINITY
0.000000
20.000000
0.000000
20.000000
20.000000
INFINITY
0.000000
20.000000
0.000000
0.000000
20.000000
Interpretation of Results
Based on our slack/surplus analysis, we found that the amount sold in month 1, 2, 4, and
5 was less than the initial inventory. For example, in month 1, we sold 0 bushels since
the slack variable is equal to 6. For month 1, 2, 7, 9, and 10, we held less than our 20,000
bushel warehouse capacity.
In our sensitivity analysis, changes in the range of the non-basic variable coefficients can
be seen in our output above. For example, 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. Our current basis is
based on that detail.
Our sensitivity analysis table also 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. For example, in row 4, the
RHS can increase infinitely or decrease by 14.
23
Problem 3: Extreme Engineering
Optimal Solution:
Our maximum profit for T-Back Inc. during the holiday season was $25,391.08. The
table below summarized the specific decision variable values.
Variable
X11
X12
X13
X14
X21
X22
X23
X24
X31
X32
X33
X34
X41
X42
X43
X44
X51
X52
X53
X54
X61
X62
X63
X64
Demand Pt
San Francisco
San Francisco
San Francisco
San Francisco
New York
New York
New York
New York
Chicago
Chicago
Chicago
Chicago
Dallas
Dallas
Dallas
Dallas
Los Angeles
Los Angeles
Los Angeles
Los Angeles
Las Vegas
Las Vegas
Las Vegas
Las Vegas
Supply Pt
Seattle
Miami
New Orleans
Houston
Seattle
Miami
New Orleans
Houston
Seattle
Miami
New Orleans
Houston
Seattle
Miami
New Orleans
Houston
Seattle
Miami
New Orleans
Houston
Seattle
Miami
New Orleans
Houston
Slack and Surplus Data:
ROW SLACK OR SURPLUS DUAL PRICES
2)
0.000000
22.186798
3)
0.000000
26.639700
4)
0.000000
20.223001
5)
0.000000
4.470900
6)
0.000000
14.319099
7)
0.000000
0.000000
24
Value (# of Thongs)
100
0
0
0
0
250
0
0
0
50
0
0
0
0
0
75
150
0
0
0
50
0
150
100
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
0.000000
0.000000
0.000000
0.000000
100.000000
0.000000
0.000000
0.000000
0.000000
250.000000
0.000000
0.000000
0.000000
50.000000
0.000000
0.000000
0.000000
0.000000
0.000000
75.000000
150.000000
0.000000
0.000000
0.000000
50.000000
0.000000
150.000000
100.000000
15.684700
11.936800
14.162800
14.898701
0.000000
0.000000
0.000000
-2.170597
0.000000
0.000000
0.000000
-3.863901
0.000000
0.000000
-0.828100
-2.289301
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.477299
-0.303299
0.000000
0.000000
0.000000
0.000000
Running Time: 00:00:01
Iterations: 16
Software: LINDO
Sensitivity Report (as Given in Lindo):
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE
CURRENT
ALLOWABLE
ALLOWABLE
COEF
INCREASE
DECREASE
X11
37.871498
INFINITY
2.170597
X12
31.771000
2.352598
INFINITY
X13
33.807098
2.542500
INFINITY
X14
34.914902
2.170597
INFINITY
X21
34.461998
7.862402
INFINITY
25
X22
X23
X24
X31
X32
X33
X34
X41
X42
X43
X44
X51
X52
X53
X54
X61
X62
X63
X64
38.576500
38.550098
37.674500
30.564100
32.159801
33.557701
32.832401
14.200100
16.402401
18.628500
19.369600
30.003799
25.770599
28.004601
28.914501
15.684700
11.936800
14.162800
14.898701
INFINITY
2.252401
3.863901
5.343601
INFINITY
0.828100
2.289301
5.955500
0.005299
0.005199
INFINITY
INFINITY
0.485300
0.477299
0.303299
0.303299
0.828100
INFINITY
0.005199
2.252401
INFINITY
INFINITY
INFINITY
0.828100
INFINITY
INFINITY
INFINITY
INFINITY
INFINITY
0.005199
0.303299
INFINITY
INFINITY
INFINITY
5.343601
0.005299
0.005199
0.303299
RIGHTHAND SIDE RANGES
ROW
CURRENT
ALLOWABLE
ALLOWABLE
RHS
INCREASE
DECREASE
2
100.000000
50.000000
0.000000
3
250.000000
0.000000
0.000000
4
50.000000
0.000000
0.000000
5
75.000000
100.000000
0.000000
6
150.000000
50.000000
0.000000
7
300.000000
INFINITY
0.000000
8
300.000000
0.000000
50.000000
9
300.000000
0.000000
0.000000
10
150.000000
0.000000
150.000000
11
175.000000
0.000000
100.000000
12
0.000000
100.000000
INFINITY
13
0.000000
0.000000
INFINITY
14
0.000000
0.000000
INFINITY
15
0.000000
100.000000
0.000000
16
0.000000
0.000000
INFINITY
17
0.000000
250.000000
INFINITY
18
0.000000
0.000000
INFINITY
19
0.000000
100.000000
0.000000
20
0.000000
0.000000
INFINITY
21
0.000000
50.000000
INFINITY
22
0.000000
50.000000
0.000000
23
0.000000
50.000000
0.000000
24
0.000000
0.000000
INFINITY
26
25
26
27
28
29
30
31
32
33
34
35
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
75.000000
150.000000
0.000000
150.000000
100.000000
50.000000
0.000000
150.000000
100.000000
INFINITY
0.000000
INFINITY
INFINITY
INFINITY
0.000000
0.000000
INFINITY
INFINITY
INFINITY
INFINITY
Interpretations of Results:
Based on our slack/surplus analysis, we found that x11, x22, x32, x44, x51, x61, x63, and
x64 were all greater than zero. For example, the amount of thongs from Seattle to San
Francisco is greater than zero. All other nodes from demand to supply are equal to zero.
We used all our supply and demand points.
In our sensitivity analysis, changes in the range of the non-basic variable coefficients can
be seen in our output above. For example, the coefficient of x24 can increase by 3.86 or
decrease indefinitely. This is the profit of sending from Houston to New York. Limiting
our maximum increase value for this variable keeps our current basis the same. If the
profit margin is too large, x24 will not equal zero; therefore changing the basis.
Our sensitivity analysis table also 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. For 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.
27
Conclusions and Recommendations
For the Grummins Trucking problem, we conclude that the maximum profit is
$3,600,000. We recommend for type 1 trucks to sell 100 in year 1, 200 in year 2, and
150 in year 3. For type 2 trucks, we recommend selling 200 in year 1, 100 in year 2, and
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.
For the Wheatie’s problem, we conclude that the maximum profit for the system
is $162. Month 8 and 10 held no inventories. 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.
For the Extreme Engineering problem, we conclude the maximum profit for the
system is $25,391.08. 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.
28
References
Bankrate Cost of Living. Retrieved November 11, 2004, from
www.Bankrate.com
Rand McNally Road Atlas Express. Retrieved November 11, 2204, from
www.RandMcNally.com
U.S. Gasoline and Diesel Fuel Prices. Retrieved November 11, 2004, from
http://tonto.eia.doe.gov/oog/info/gdu/gasdiesel.asp
Winston, W.L. and M. Venkataramana. Introduction to Mathematical Programming.
4th Edition, Duxbury Press: CA, 2003.
29
Appendix
Grummins Truck LINDO CODE:
MAX 20000s11+20000s12+20000s13 + 17000s21+17000s22+17000s23 - 15000x1115000x12-15000x13 -14000x21-14000x22-14000x23 - 2000I11-2000I12-2000I212000I22
ST
x11+x21<=320
x12+x22<=320
x13+x23<=320
x11-s11>=0
x11-s11-I11=0
x12+I11-s12>=0
x12+I11-s12-I12=0
x13+I12-s13>=0
x21-s21>=0
x21-s21-I21=0
x22+I21-s22>=0
x22+I21-s22-I22=0
x23+I22-s23>=0
s11<=100
s12<=200
s13<=300
s21<=200
s22<=100
s23<=150
15x11+15x12+15x13+5x21+5x22+5x23-10x11-10x12-10x13-10x21-10x22-10x23<=0
x11>=0
x12>=0
x13>=0
x21>=0
x22>=0
x23>=0
s11>=0
s12>=0
s13>=0
s21>=0
s22>=0
s23>=0
I11>=0
I12>=0
I21>=0
I22>=0
30
END
Wheatie’s LINDO code:
MAX 3S1+6S2+7S3+S4+4S5+5S6+5S7+s8+3S9+2S10-8B1-8B2-2B3-3B4-4B5-3B63B7-2B8-5B9-5B10
ST
I1=6
S1-I1<=0
I1-S1+B1<=20
I1-S1+B1-I2=0
S2-I2<=0
I2-S2+B2<=20
I2-S2+B2-I3=0
S3-I3<=0
I3-S3+B3<=20
I3-S3+B3-I4=0
S4-I4<=0
I4-S4+B4<=20
I4-S4+B4-I5=0
S5-I5<=0
I5-S5+B5<=20
I5-S5+B5-I6=0
S6-I6<=0
I6-S6+B6<=20
I6-S6+B6-I7=0
S7-I7<=0
I7-S7+B7<=20
I7-S7+B7-I8=0
S8-I8<=0
I8-S8+B8<=20
I8-S8+B8-I9=0
S9-I9<=0
I9-S9+B9<=20
I9-S9+B9-I10=0
31
S10-I10<=0
I10-S10+B10<=20
Extreme Engineering LINDO code:
MAX
40X11+40X12+40X13+40X14+42X21+42X22+42X23+42X24+36X31+36X32+36X33
+36X34+20X41+20X42+20X43+20X44+33X51+33X52+33X53+33X54+19X61+19X6
2+19X63+19X64-2.1285X11-8.229X12-6.1929X13-5.0851X14-7.5380X21-3.4235X223.4499X23-4.3255X24-5.4359X31-3.8402X32-2.4423X33-3.1676X34-5.7999X413.5976X42-1.3715X43-.6304X44-2.9962X51-7.2294X52-4.9954X53-4.0855X543.3153X61-7.0632X62-4.8372X63-4.1013X64
ST
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
X11+X21+X31+X41+X51+X61<=300
X12+X22+X32+X42+X52+X62<=300
X13+X23+X33+X43+X53+X63<=150
X14+X24+X34+X44+X54+X64<=175
X11>=0
X12>=0
X13>=0
X14>=0
X21>=0
X22>=0
X23>=0
X24>=0
X31>=0
X32>=0
X33>=0
X34>=0
X41>=0
X42>=0
X43>=0
X44>=0
X51>=0
X52>=0
X53>=0
X54>=0
X61>=0
32
X62>=0
X63>=0
X64>=0
END
33