INEN 420
Semester Project
Grummins Engine Company
Wheat Warehouse
Power Generation
By
Class of ‘93
Javier Garcia
Dinh Nguyen
Rhoda Read
Class of ’93: Garcia, Nguyen, Read
Page 1
CONTENTS
1.0
Executive Summary ………………………………………….……………………. Page 3
2.0
Problem Description ………………………………………………….……………...…….
3.0
2.1
Problem 1 (Grummins Engine Company) ………………...............……….. Page 3
2.2
Problem 2 (Wheat Warehouse) …………….……………………………… Page 5
2.3
Problem 3 (Power Generation) ………....………………………………….. Page 7
Computational Results ………………………………………………..……………………
3.1
Problem 1 (Grummins Engine Company) ………………...……..……….. Page 14
3.2
Problem 2 (Wheat Warehouse) ………………………….……..………… Page 17
3.3
Problem 3 (Power Distribution) ………………………………………….. Page 21
4.0
Conclusions and Recommendations ………………………..……………………. Page 29
5.0
References ……………………………………………………………………….. Page 32
Class of ’93: Garcia, Nguyen, Read
Page 2
1.0
Executive Summary
This report focuses on three different possible scenarios. First, a diesel truck manufacturing
company wants to maximize profit, but is restricted by industry and government regulations. Its
model fits the profile of an optimal solution but upon testing the ranges produced in the
sensitivity analysis, it became clear that the original LP is degenerate. A similar case surfaced
after analyzing a model to maximize profit for a wheat farmer with a small warehouse and a rigid
selling/purchasing schedule. Again, we deduced that the original LP was degenerate. Lastly,
power generation in Puerto Rico was studied to determine how to minimize costs. The ranges
produced by its sensitivity analysis, again, indicate degeneracy. However, in all three cases the
LPs were functional. The interpretations and recommendations should be made with caution and
understanding of the effects of degeneracy is important.
2.1
Problem 1 (Grummins Engine Company)
The Grummins Engine Co. produces 2 types of diesel trucks that have different selling prices,
manufacturing costs, and pollution emission. We want to formulate an LP that can be used to
determine how to maximize profit during the next three years given the following conditions:
Sales Price, Manufacturing Costs and Pollution Emission:
Sales Price
Manufacturing Costs
Emissions
Type 1
$20,000.00
$15,000.00
15 grams
Type 2
$17,000.00
$14,000.00
5 grams
Maximum Demand for Trucks:
Year
Type 1
Type 2
1
100
200
2
200
100
3
300
150
Class of ’93: Garcia, Nguyen, Read
Page 3
•
Production capacity limits total truck production during each year to at most 300 trucks.
•
It cost $2,000.00 to hold 1 truck (of any type) in inventory for one year.
•
From the table above, at most 300 type 1 trucks can be sold in year 3. Demand may be
met from previous production or the current year’s production.
Grummins Engine Co. wants a plan to help them arrange their product in the next three years to
get the maximum profit. This means that there should be no trucks in stock at the end of the third
year.
Assumptions include that the company should not keep more trucks in inventory than demand
predicts, so production is regulated by the amount of trucks in inventory and the amount of
trucks that can be sold. Also, we assume that trucks are only produced and sold, not acquired by
any other method such as auctions, trading, etc.
We determined our decision variables to be the following:
Pij=
Number of trucks (each type i) produced for each year j
Sij=
Number of trucks (each type i) sold for each year j
Rij=
Number of trucks (each type i) that remain in stock at the end of each year j
With i=1, 2; j=1, 2, 3.
The objective function for the maximum profit is as follows:
Max Z=
s.t.
20 (S11+S12+S13) + 17 (S21+S22+S23) - 15 (P11+P12+P13)
-14 (P21+P22+P23) - 2 (R11+R12+R21+R22)
(in $ thousands)
P11+P21
<=320
P12+P22
<=320
P13+P23
<=320
S11
<=100
S12
<=200
S13
<=300
S21
<=200
S22
<=100
S23
<=150
R11-P11+S11 =0
R21-P21+S21 =0
R12-P12+S12-R11 =0
Class of ’93: Garcia, Nguyen, Read
(Production)
(Sale)
(Remain in stock)
Page 4
R22-P22+S22-R21 =0
P13+R12-S13 =0
P23+R22-S23 =0
5P11+5P12+5P13-5P21-5P22-5P23<=0
Pij, Sij, Rij >=0; i=1,2; j=1,2,3
2.2
(Emissions requirement)
Problem 2 (Wheat Warehouse)
As owner of a wheat warehouse with a capacity of 20,000 bushels, we want to formulate an LP
that can be used to determine how to maximize the profit earned over the next 10 months, given
the following conditions:
•
We begin Month 1 with 6,000 bushels of wheat.
•
Each month, wheat can be bought and sold at the price per 1000 bushels listed in the
following table:
•
Month
Selling Price ($)
Purchase Price ($)
1
3
8
2
6
8
3
7
2
4
1
3
5
4
4
6
5
3
7
5
3
8
1
2
9
3
5
10
2
5
Each month, the sequence of events is as follows:
o Observe the initial stock of wheat.
o Sell any amount of wheat up to the initial stock at the current month’s selling
price.
o Buy (at the current month’s buying price) as much wheat as wanted, subject to the
warehouse size limitation.
Assumptions include that wheat can only be sold or purchased (at the given rates). In this way,
ending inventory is simply determined by the following equation:
Class of ’93: Garcia, Nguyen, Read
Page 5
ending inventory = beginning inventory – amount sold + amount purchased
For Month 1, our beginning inventory is 6,000 bushels of wheat. We can then sell up to 6,000
bushels. Next, we can purchase any amount of wheat up to our capacity (20,000 bushels) minus
the beginning inventory plus the amount previously sold. Finally, our ending inventory is as
stated above (6,000 – amount sold + amount purchased). For Month 2, our beginning inventory
is the ending inventory from Month 1. We can then sell up to our ending inventory from Month
1. Next, we can purchase up to 20,000 minus the beginning inventory plus the amount
previously sold. Finally, our ending inventory for Month 2 is the ending inventory from Month 1
– amount sold during Month 2 + amount purchased during Month 2. The remaining months
follow the same process and are detailed in the constraints listed later.
We determined our decision variables to be the following:
si = amount of wheat (in thousands) sold during month i, i = 1,…,10
pi = amount of wheat (in thousands) purchased during month i, i=1,…,10
ej = # of bushels (in thousands) left at the end of month j, j=1,…,9
For our objective function, we want to maximize profit over the next 10 months, and determined
it to be the following equation:
max z = 3s1 – 8p1 + 6s2 – 8p2 + 7s3 – 2p3 + s4 – 3p4 + 4s5 – 4p5 + 5s6 – 3p6 + 5s7
– 3p7 + s8 – 2p8 + 3s9 – 5p9 + 2s10 – 5p10
Therefore, the LP formulation is as follows:
max z = 3s1 – 8p1 + 6s2 – 8p2 + 7s3 – 2p3 + s4 – 3p4 + 4s5 – 4p5 + 5s6 – 3p6 + 5s7
– 3p7 + s8 – 2p8 + 3s9 – 5p9 + 2s10 – 5p10
s.t.
s1
<= 6
s2 – e1 <= 0
s3 – e2 <= 0
s4 – e3 <= 0
s5 – e4 <= 0
s6 – e5 <= 0
s7 – e6 <= 0
s8 – e7 <= 0
s9 – e8 <= 0
s10 – e9 <= 0
p1 – s1 <= 14
Class of ’93: Garcia, Nguyen, Read
(selling restrictions – up to current inventory)
(purchasing restrictions – up to warehouse capacity)
Page 6
p2 – s2 + e1
<= 20
p3 – s3 + e2
<= 20
p4 – s4 + e3
<= 20
p5 – s5 + e4
<= 20
p6 – s6 + e5
<= 20
p7 – s7 + e6
<= 20
p8 – s8 + e7
<= 20
p9 – s9 + e8
<= 20
p10 – s10 + e9 <= 20
e1 + s1 – p1
=6
(ending inventory for each month)
e2 + s2 – p2 – e1
=0
e3 + s3 – p3 – e2
=0
e4 + s4 – p4 – e3
=0
e5 + s5 – p5 – e4
=0
e6 + s6 – p6 – e5
=0
e7 + s7 – p7 – e6
=0
e8 + s8 – p8 – e7
=0
e9 + s9 – p9 – e8
=0
si, pi >= 0, i = 1,…,10; ei >= 0, i = 1,…,9
2.3
Problem 3 (Power Generation)
GENERAL BACKGROUND
Puerto Rico enjoys a highly diversified economy, a
strong tourist sector, and good trade relations with the
United States, its largest trading partner. Despite
mixed current economic indicators, Puerto Rico’s
short-term economic outlook looks relatively positive
given the strengthening of the U.S. economy, which is
expected to grow over 4% this year. The island’s real
gross domestic product (GDP) is expected to grow
3.3% in 2004 and 2.8% in 2005, and 2.4% over the medium term (2006–10). As consumers take
advantage of low interest rates, private consumption has increased. Over the past year, high oil
prices have had an adverse affect on Puerto Rico's economy and inflation, as the Commonwealth
is heavily dependent on oil imports to meet its domestic energy needs, particularly for electricity
generation.
Class of ’93: Garcia, Nguyen, Read
Page 7
ENERGY OVERVIEW
Puerto Rico lacks domestic hydrocarbon reserves (including oil, natural gas, and coal) and relies
on imports to meet its energy needs. Imported oil, mainly from U.S. and Caribbean suppliers, is
the source of about 90% of Puerto Rico's power. Although consumption of natural gas has been
increasing over the past few years, Puerto Rico still relies overwhelmingly on oil. Many industry
analysts agree that gas- or coal-fired facilities are needed to supplement oil-burning power
plants. However, plans to widen and/or diversify the electric power supply through cogeneration and agreements with independent power producers have barely progressed due to
opposition from environmental groups and powerful labor unions.
In 2004, Puerto Rico generated an estimated 22.1 billion kilowatt hours (Bkwh) of electricity,
predominantly from five oil-fired generators, with a fraction coming from small hydroelectric
dams. Also, the country consumed about 223,000 barrels per day (bbl/d) of oil, all imported,
primarily for transportation and electric power generation. As of 2003, installed generation
capacity was 4.9 gigawatts. The five oil-fired plants are: the Costa Sur plant (1,090 MW); the
Aguirre plant (900 MW); the Palo Seco plant (602 MW); the San Juan plant (400 MW); and the
Arecibo plant (248 MW). The Puerto Rico Electric Power Authority (PREPA) accounts for a
majority of net electricity generation, and is the Commonwealth's sole distributor of electric
power. PREPA also purchases excess power generation from co-generators, primarily in the
cement industry, and from independent power producers. Early this year, Puerto Rico began
importing liquefied natural gas (LNG) to supply the EcoEléctrica facility, a 540-megawatt (MW)
natural gas-fired power plant that will supply power generation under a contract to the island at
the end of this year. Also, they begin importing coal to the new 454-MW coal-fired plant in
Guayama (ASE-PR).
With power consumption increasing more than 3% per year for more than a decade, both PREPA
and independent power producers have been investing in new capacity in order to meet growing
demand and to diversify energy sources. The following table provides the power plant capacity,
expected demand for each substation for next year and the cost of shipping power from a plant to
a substation.
Class of ’93: Garcia, Nguyen, Read
Page 8
Cost of Shipping to Substation #
FROM
(As of 30 SEP 2004)
1
2
3
4
5
6
7
8
9
10
11
12
6
5
6
7
9
10
9
12
14
17
11
10
(Plant)
San Juan
Supply (MW)
400
Palo Seco
6
6
5
8
10
11
10
11
13
16
10
9
602
Aguirre
10
10
11
12
11
13
9
7
8
10
12
13
900
Costa Sur
15
14
15
13
18
20
17
9
6
8
9
10
1090
Arecibo
9
8
9
10
12
15
11
11
9
10
6
4
248
ASE-PR
9
8
9
9
9
9
7
5
6
7
6
10
454
14
13
14
13
16
17
14
7
5
8
7
8
507
190
175
175
195
165
190
200
190
200
175
145
190
2190
Eco
Electrica
Expected
Demand in
MW (2005)
Last September, the tropical storm Jeanne hit the island. PREPA initial system-operations report
shows the island’s largest electricity-generating plants collapsed between noon and 1:00 p.m.
while operating at their average capacity, causing expensive damage to the equipment. Not only
did the thermoelectric plants collapse, but so did the island’s gas and steam turbines and the cogeneration plants. The failures meant the system was producing only some 607 megawatts
(MW) to power the entire island out of a maximum capacity of 3,240 MW, or less than 19%. If
a plant is operating at more than 50% of its peak capacity when objects such as broken trees and
flying debris strike or knock transmission poles down, the resultant shock to the power grid can
be destructive. "When the technicians went to inspect the systems on Thursday, the filed reports
indicated cracks in equipment, failed start-ups, a damaged generator in Costa Sur No. 3, and a
cracked boiler in Costa Sur No. 4," said the PREPA source. "Aguirre’s steam turbines alone had
an oil leak in the electrohydraulic system, a broken cooling fan, and a damaged rotor. The
damage to Puerto Rico’s power grid was estimated at $60 million.
PREPA wants to add two (2) new substations, ASE-PR and Eco Electrica, to the power grid
beginning next year and want them to provide at least 20% of the power demand during high
peak demand. Both ASE-PR and Eco Electrica are privately-owned and will be operational in
November 2004. Also, the Palo Seco plant will be able to supply only 50% of its maximum
output and Costa Sur only 70% of its maximum output during high peak demand beginning next
Class of ’93: Garcia, Nguyen, Read
Page 9
year due to an upgrade to their plants. The peak power demand occurs at the same time (around
0100 PM) on each substation. Assumptions include that all plants operate at 90% of their
maximum capacity and that all power supply to the substation is only being used by the intended
sources. Also, in the event of any bad weather, we will assume that at most two plants, Palo
Seco and Aguirre, will be disconnected.
We want to formulate two LPs with different
conditions:
The first condition is to minimize the cost of meeting each substation’s peak power demand for
next year during high peak demand. The second condition is to minimize the cost of meeting
each substation’s peak power demand if Palo Seco and Aguirre are disconnected due to bad
weather.
11
12
4
1
3
2
5
6
7
10
9
8
EcoElectrica
ASE-PR
New Power Plant
Substations
Class of ’93: Garcia, Nguyen, Read
Page 10
FORMULATION (Both Conditions):
First, we defined our decision variables by determining how much power is sent to each
substation during high peak hour (0100 PM). We define seven plants (i = 1, 2, …., 7). Plant 1 is
San Juan, Plant 2 is Palo Seco, Plant 3 is Aguirre, Plant 4 is Costa Sur, Plant 5 is Arecibo and
Plants 6 and 7 are the two new facilities (ASE-PR and Eco Electrica). The twelve substations are
defined j = 1,2,3,4,…,12. See map for locations.
Xij = number of megawatts produced at plant i and sent to substations j
For our objective function, we want to minimize the cost of meeting each substation’s peak
power demand:
Min Z = 6X11 + 5X12 + 6X13 + 7X14 + 9X15 + 10X16 + 9X17 + 12X18 + 14X19 +
17X110 + 11X111 + 10X112 + 6X21 + 6X22 + 5X23 + 8X24 + 10X25 + 11X26 +
10X27 + 11X28 + 13X29 +16X210 + 10X211 + 9X212 + 10X31 + 10X32 + 11X33 +
12X34 + 11X35 + 10X36 + 9X37 + 6X38 + 7X39 + 9X310 + 12X311 + 13X312 +
15X41 + 14X42 + 15X43 + 13X44 + 18X45 + 20X46 + 17X47 + 9X48 + 5X49 + 8X410
+ 9X411 + 10x412 + 9X51 + 8X52 + 9X53 + 10X54 + 12X55 + 15X56 + 11X57 +
11X58 + 9X59 + 9X510 + 6X511 + 4X512 + 9X61 + 8X62 + 9X63 + 9X64 + 9X65 +
9X66 + 7X67 + 5X68 + 6X69 + 7X610 + 6X611 + 10X612 + 14X71 + 13X72 + 14X73
+ 13X74 + 16X75 + 17X76+ 14X77 + 7X78 + 5X79 + 8X710 + 7X711 + 8X712
PREPA faces three types of constraints. First, the total power supplied by each plant cannot
exceed the plant capacity. Examples are: amount of power sent from San Juan to the twelve
substations cannot exceed 360 megawatts (=90% MAX). Also, we have restrictions to Palo Seco
and Costa Sur power generation. Palo Seco can only supply 50% of its maximum capacity and
Costa Sur 70%. The formulation problem contains the following constraints for supply:
X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112 <= 360
X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212 <= 301
X31+X32+X33+X34+X35+X36+X37+X38+X39+X310+X311+X312 <= 810
X41+X42+X43+X44+X45+X46+X47+X48+X49+X410+X411+X412 <= 743
X51+X52+X53+X54+X55+X56+X57+X58+X59+X510+X511+X512 <= 224
X61+X62+X63+X64+X65+X66+X67+X68+X69+X610+X611+X612 <= 409
X71+X72+X73+X74+X75+X76+X77+X78+X79+X710+X711+X712 <= 451
Class of ’93: Garcia, Nguyen, Read
Page 11
The following constraint ensures the ASE-PR and Eco Electrica supply at least 20% of total
power demand (3,240 MW):
4X61 + 4X62+ 4X63 + 4X64 + 4X65 + 4X66 + 4X67 + 4X68 + 4X69 + 4X610 + 4X611
+ 4X612+ 4X71 + 4X72 + 4X73 + 4X74 + 4X75 + 4X76 + 4X77 + 4X78 + 4X79 +
4X710 + 4X711 + 4X712 - X11 - X12 - X13 - X14 - X15 - X16 - X17 - X18 -X19 X110 - X111 - X112 - X21 - X22 - X23 - X24 - X25 - X26 - X27 - X28 - X29 -X210 X211 - X212 - X31 - X32 - X33 - X34 - X35 - X36 - X37 - X38 - X39 - X310 -X311 X312 - X41 - X42 - X43 - X44 - X45 - X46 - X47 - X48 - X49 - X410 - X411 -X412 X51 - X52 - X53 - X54 - X55 - X56 - X57 - X58 - X59 - X510 - X511 - X512 >= 0
Also, we need to ensure that each substation will receive sufficient power to meet its peak
demand. The demand constraint is as follows:
X11+X21+X31+X41+X51+X61+X71 >= 190
(Substation 1)
X12+X22+X32+X42+X52+X62+X72 >= 175
(Substation 2)
X13+X23+X33+X43+X53+X63+X73 >= 175
(Substation 3)
X14+X24+X34+X44+X54+X64+X74 >= 195
(Substation 4)
X15+X25+X35+X45+X55+X65+X75 >= 165
(Substation 5)
X16+X26+X36+X46+X56+X66+X76 >= 190
(Substation 6)
X17+X27+X37+X47+X57+X67+X77 >= 200
(Substation 7)
X18+X28+X38+X48+X58+X63+X78 >= 190
(Substation 8)
X19+X29+X39+X49+X59+X69+X79 >= 200
(Substation 9)
X110+X210+X310+X410+X510+X610+X710 >= 175
(Substation 10)
X111+X211+X311+X411+X511+X611+X711 >= 145
(Substation 11)
X112+X212+X312+X412+X512+X612+X112 >= 200
(Substation 12)
Sign Restrictions: Xij >= 0
Therefore, the LP formulation for condition 1 is as follows:
Min Z = 6X11 + 5X12 + 6X13 + 7X14 + 9X15 + 10X16 + 9X17 + 12X18 + 14X19 +
17X110 + 11X111 + 10X112 + 6X21 + 6X22 + 5X23 + 8X24 + 10X25 + 11X26 +
10X27 + 11X28 + 13X29 +16X210 + 10X211 + 9X212 + 10X31 + 10X32 + 11X33 +
12X34 + 11X35 + 10X36 + 9X37 + 6X38 + 7X39 + 9X310 + 12X311 + 13X312 +
15X41 + 14X42 + 15X43 + 13X44 + 18X45 + 20X46 + 17X47 + 9X48 + 5X49 + 8X410
+ 9X411 + 10x412 + 9X51 + 8X52 + 9X53 + 10X54 + 12X55 + 15X56 + 11X57 +
11X58 + 9X59 + 9X510 + 6X511 + 4X512 + 9X61 + 8X62 + 9X63 + 9X64 + 9X65 +
9X66 + 7X67 + 5X68 + 6X69 + 7X610 + 6X611 + 10X612 + 14X71 + 13X72 + 14X73
+ 13X74 + 16X75 + 17X76+ 14X77 + 7X78 + 5X79 + 8X710 + 7X711 + 8X712
Class of ’93: Garcia, Nguyen, Read
Page 12
s.t.
X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112 <= 360
X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212 <= 301
X31+X32+X33+X34+X35+X36+X37+X38+X39+X310+X311+X312 <= 810
X41+X42+X43+X44+X45+X46+X47+X48+X49+X410+X411+X412 <= 743
X51+X52+X53+X54+X55+X56+X57+X58+X59+X510+X511+X512 <= 224
X61+X62+X63+X64+X65+X66+X67+X68+X69+X610+X611+X612 <= 409
X71+X72+X73+X74+X75+X76+X77+X78+X79+X710+X711+X712 <= 451
4X61 + 4X62+ 4X63 + 4X64 + 4X65 + 4X66 + 4X67 + 4X68 + 4X69 + 4X610 + 4X611
+ 4X612+ 4X71 + 4X72 + 4X73 + 4X74 + 4X75 + 4X76 + 4X77 + 4X78 + 4X79 +
4X710 + 4X711 + 4X712 - X11 - X12 - X13 - X14 - X15 - X16 - X17 - X18 -X19 X110 - X111 - X112 - X21 - X22 - X23 - X24 - X25 - X26 - X27 - X28 - X29 -X210 X211 - X212 - X31 - X32 - X33 - X34 - X35 - X36 - X37 - X38 - X39 - X310 -X311 X312 - X41 - X42 - X43 - X44 - X45 - X46 - X47 - X48 - X49 - X410 - X411 -X412 X51 - X52 - X53 - X54 - X55 - X56 - X57 - X58 - X59 - X510 - X511 - X512 >= 0
X11+X21+X31+X41+X51+X61+X71 >= 190
X12+X22+X32+X42+X52+X62+X72 >= 175
X13+X23+X33+X43+X53+X63+X73 >= 175
X14+X24+X34+X44+X54+X64+X74 >= 195
X15+X25+X35+X45+X55+X65+X75 >= 165
X16+X26+X36+X46+X56+X66+X76 >= 190
X17+X27+X37+X47+X57+X67+X77 >= 200
X18+X28+X38+X48+X58+X63+X78 >= 190
X19+X29+X39+X49+X59+X69+X79 >= 200
X110+X210+X310+X410+X510+X610+X710 >= 175
X111+X211+X311+X411+X511+X611+X711 >= 145
X112+X212+X312+X412+X512+X612+X112 >= 200
Xij >= 0, i=1,……..,7; j = 1,………..,12
(Substation 1)
(Substation 2)
(Substation 3)
(Substation 4)
(Substation 5)
(Substation 6)
(Substation 7)
(Substation 8)
(Substation 9)
(Substation 10)
(Substation 11)
(Substation 12)
The LP formulation for the condition 2 is as follows:
Min Z = 6X11 + 5X12 + 6X13 + 7X14 + 9X15 + 10X16 + 9X17 + 12X18 + 14X19 +
17X110 + 11X111 + 10X112 + 6X21 + 6X22 + 5X23 + 8X24 + 10X25 + 11X26 +
10X27 + 11X28 + 13X29 +16X210 + 10X211 + 9X212 + 10X31 + 10X32 + 11X33 +
12X34 + 11X35 + 10X36 + 9X37 + 6X38 + 7X39 + 9X310 + 12X311 + 13X312 +
15X41 + 14X42 + 15X43 + 13X44 + 18X45 + 20X46 + 17X47 + 9X48 + 5X49 + 8X410
+ 9X411 + 10x412 + 9X51 + 8X52 + 9X53 + 10X54 + 12X55 + 15X56 + 11X57 +
11X58 + 9X59 + 9X510 + 6X511 + 4X512 + 9X61 + 8X62 + 9X63 + 9X64 + 9X65 +
9X66 + 7X67 + 5X68 + 6X69 + 7X610 + 6X611 + 10X612 + 14X71 + 13X72 + 14X73
+ 13X74 + 16X75 + 17X76+ 14X77 + 7X78 + 5X79 + 8X710 + 7X711 + 8X712
Class of ’93: Garcia, Nguyen, Read
Page 13
s.t.
X11+X12+X13+X14+X15+X16+X17+X18+X19+X110+X111+X112 <= 360
X21+X22+X23+X24+X25+X26+X27+X28+X29+X210+X211+X212 <= 0
X31+X32+X33+X34+X35+X36+X37+X38+X39+X310+X311+X312 <= 0
X41+X42+X43+X44+X45+X46+X47+X48+X49+X410+X411+X412 <= 743
X51+X52+X53+X54+X55+X56+X57+X58+X59+X510+X511+X512 <= 224
X61+X62+X63+X64+X65+X66+X67+X68+X69+X610+X611+X612 <= 409
X71+X72+X73+X74+X75+X76+X77+X78+X79+X710+X711+X712 <= 451
4X61 + 4X62+ 4X63 + 4X64 + 4X65 + 4X66 + 4X67 + 4X68 + 4X69 + 4X610 + 4X611
+ 4X612+ 4X71 + 4X72 + 4X73 + 4X74 + 4X75 + 4X76 + 4X77 + 4X78 + 4X79 +
4X710 + 4X711 + 4X712 - X11 - X12 - X13 - X14 - X15 - X16 - X17 - X18 -X19 X110 - X111 - X112 - X21 - X22 - X23 - X24 - X25 - X26 - X27 - X28 - X29 -X210 X211 - X212 - X31 - X32 - X33 - X34 - X35 - X36 - X37 - X38 - X39 - X310 -X311 X312 - X41 - X42 - X43 - X44 - X45 - X46 - X47 - X48 - X49 - X410 - X411 -X412 X51 - X52 - X53 - X54 - X55 - X56 - X57 - X58 - X59 - X510 - X511 - X512 >= 0
X11+X21+X31+X41+X51+X61+X71 >= 190
X12+X22+X32+X42+X52+X62+X72 >= 175
X13+X23+X33+X43+X53+X63+X73 >= 175
X14+X24+X34+X44+X54+X64+X74 >= 195
X15+X25+X35+X45+X55+X65+X75 >= 165
X16+X26+X36+X46+X56+X66+X76 >= 190
X17+X27+X37+X47+X57+X67+X77 >= 200
X18+X28+X38+X48+X58+X63+X78 >= 190
X19+X29+X39+X49+X59+X69+X79 >= 200
X110+X210+X310+X410+X510+X610+X710 >= 175
X111+X211+X311+X411+X511+X611+X711 >= 145
X112+X212+X312+X412+X512+X612+X112 >= 200
Xij >= 0, i=1,……..,7; j = 1,………..,12
3.1
(Substation 1)
(Substation 2)
(Substation 3)
(Substation 4)
(Substation 5)
(Substation 6)
(Substation 7)
(Substation 8)
(Substation 9)
(Substation 10)
(Substation 11)
(Substation 12)
Problem 1 (Grummins Engine Company)
The optimal solution obtained through Lindo is as follows:
Z
S11
S12
S13
S21
S22
S23
P11
P12
= 3,600.00 (in $ thousands)
=100
=200
=150
=200
=100
=150
=100
=200
Class of ’93: Garcia, Nguyen, Read
Page 14
P13
P21
P22
P23
R11
R12
R21
R22
=150
=200
=100
=150
=0
=0
=0
=0
Lindo determined that 10 iterations were necessary to find this optimal solution. The solution
indicates that the best way to get the maximum profit is to not have remaining inventory at the
end of each year. Therefore, the company should sell every truck they make each year, for both
types of trucks.
After examining the sensitivity analysis report, we determined the ranges for our decision
variables could be as follows, and still remain the optimal solution:
Decision
Variables
Current Sale &
Production
Range of Increase/Decrease
($ thousands)
Objective Function Coefficient
Range of Decision Variables
S11
20
0<=∆<= +∞
20<=C11<= +∞
S12
20
0<=∆<= +∞
20<=C12<= +∞
S13
20
-5<=∆<= 0
15<=C13<= 20
S21
17
-8<=∆<= +∞
9<=C21<= +∞
S22
17
-8<=∆<= +∞
9<=C22<= +∞
S23
17
-8<=∆<= +∞
9<=C23<= +∞
P11
15
-2<=∆<=0
13<=C14<=15
P12
15
-2<=∆<=0
13<=C15<=15
P13
15
0<=∆<=2
15<=C16<=17
P21
14
-2<=∆<=8
12<=P24<=22
P22
14
-2<=∆<=2
12<=P25<=16
P23
14
-∞<=∆<= 2
-∞<=P26<= 16
R11
2
-2 <=∆<=+∞
0 <=C1<=+∞
R12
2
-2 <=∆<=+∞
0 <=C2<=+∞
R21
2
-2 <=∆<=+∞
0 <=C3<=+∞
R22
2
-2 <=∆<=+∞
0 <=C4<=+∞
Class of ’93: Garcia, Nguyen, Read
Page 15
The selling price for a type 1 truck (which emits 15 gm of pollution) can vary from $20,000 up to
+∞ for the first two years and the manufacturing price can also increase up to $20,000 more. In
other words, the company can increase the selling price C11 but due to the constraint in truck
sale S11 <= 100, it will not affect the current basis. By increasing C11, the current basis remains
optimal and the only change will be the optimal Z (Cbv B-1b). However, the company has to
reduce the price for the type 1 truck at the third year down to $15,000 to guarantee that all type 1
trucks will be sold. It is similar for the manufacturing costs for type 1 trucks. The company can
make up to $2,000.00 for each type 1 truck for the first two years, but then they have to reduce
its price on the third year. When we change the value of S11 or S12 of the objective function
coefficient out of range, that variable will leave the basis and change the optimal solution, thus
decreasing the profit for Grummins Engine Company. Similarly, when we increase the value of
S13 out of range (above 20), the optimal solution will include less trucks sold in year 2 (plus
leftover inventory) in order to sell as many trucks as possible during year 3, when the selling
price is higher.
For type 2 trucks, the selling price can vary from $9,000.00 up to +∞ for all three years. The
manufacturing costs are also the same; their price can vary from $6,000.00 to $16,000.00 for the
first year and produce more profit for each year later. But we have the same situation as we
explain in the prior paragraph (Constraint in selling trucks will not affect the current basis, only
the optimal Z). Also as type 1 trucks, if we change the objective function coefficient (S21, S22,
S23) out of the range, that variable also will leave the basis and change the optimal solution, thus
decreasing the company’s profit.
Similarly, Grummins Engine Company could reduce the manufacturing prices (P11, P12, P13,
P21, P22, P23) out of range to less than the minimum value and the maximum profit will
increase. Otherwise, the maximum profit will decrease when manufacturing costs increase above
the range maximum. However, upon closer inspection of the variable P22, the optimal solution
changes if P22 = 16, which is in the allowable range. Similarly, if P21 = 12, which is within the
allowable range, the optimal solution changes. These particular occurrences indicate that our
original LP is degenerate.
The ranges for the right-hand side variables are listed in the table below:
Class of ’93: Garcia, Nguyen, Read
Page 16
Decision
Current Right-Hand Side
Range of Increase/Decrease
Ranges of Right-Hand Sides
b1
320
-20 <= ∆ <= ∞
300 <= b1 <= ∞
b2
320
-20 <= ∆ <= ∞
300 <= b2 <= ∞
b3
320
-20 <= ∆ <= ∞
300 <= b3 <= ∞
b4
100
-20 <= ∆ <= 20
80 <= b4 <= 120
b5
200
-20 <= ∆ <= 20
180 <= b5 <= 220
b6
300
-150 <= ∆ <= ∞
150 <= b6 <= ∞
b7
200
-150 <= ∆ <= 20
50 <= b7 <= 220
b8
100
-100 <= ∆ <= 20
0 <= b8 <= 120
b9
150
-150 <= ∆ <= 10
0 <= b9 <= 160
b10
0
-20 <= ∆ <= 20
-20 <= b10 <= 20
b11
0
-20 <= ∆ <= 150
-20 <= b11 <= 150
b12
0
-20 <= ∆ <= 20
-20 <= b12 <= 20
b13
0
-20 <= ∆ <= 100
-20 <= b13 <= 100
b14
0
-150 <= ∆ <= 150
-150 <= b14 <= 150
b15
0
-150 <= ∆ <= 10
-150 <= b15 <= 10
b16
0
-750 <= ∆ <= 100
-750 <= b16 <= 100
Variable
According to this analysis, the range for b1 could be increased to any amount & the solution
would remain optimal. That is, our limit for total truck production could be any amount over
300 trucks per year. Also, the range of optimality for the amount of each truck we can sell each
year is given by the ranges of b4 – b9. For instance, the ranges indicate that if the right-hand side
of the constraint for the amount of type 1 trucks sold in year 1 (b4) is less than 80 or more than
120, we should receive a new optimal solution. However, upon testing this range, we see that
the optimal value stays the same for any value chosen greater than 120. This, too, seems to
indicate that our original LP is degenerate.
3.2
Problem 2 (Wheat Warehouse)
The optimal solution obtained through Lindo is as follows:
z = 162
s3 = 6
p3 = 20
s5 = 0
p5 = 0
s6 = 20
p6 = 20
(in thousands $)
(in thousands)
”
”
”
”
”
Class of ’93: Garcia, Nguyen, Read
Page 17
s7 = 20
p7 = 0
p8 = 20
s9 = 20
s10 = 0
”
”
”
”
”
Lindo determined that 20 iterations were necessary to find this optimal solution. The solution
indicates that we should hold our initial stock of wheat (6,000 bushels) until Month 3, when we
should sell the entire stock at $7 per bushel and then purchase 20,000 bushels at $2 per bushel.
Again, this solution suggests that we should hold this stock until Month 6, when we should sell
all of it at $5 per bushel and purchase 20,000 more bushels at $3 per bushel. Then, in Month 7,
we should sell all 20,000 bushels at $5 per bushel and then have zero bushels in inventory until
we could purchase 20,000 bushels in Month 8 at $2 per bushel. Finally, the solution indicates we
should sell the entire stock in Month 9 at $3 per bushel, leaving the wheat warehouse empty
through Month 10. This solution makes sense because it encourages us to sell the wheat at a
higher rate than we can purchase it, thus maximizing profit.
After examining the sensitivity analysis report, we determined that the ranges for our decision
variables could be as follows, and still maintain the optimal solution:
Objective Function
Decision
Current Selling/Purchase
Variable
Price
s1
$3
-∞ <= ∆ <= 4
-∞ <= c1 <=7
p1
$8
-1 <= ∆ <= ∞
7 <= c11 <= ∞
s2
$6
-∞ <= ∆ <= 1
-∞ <= c2 <= 7
p2
$8
-1 <= ∆ <= ∞
7 <= c21 <= ∞
s3
$7
-1 <= ∆ <= 1
6 <= c3 <= 8
p3
$2
-1 <= ∆ <= 1
1 <= c31 <= 3
s4
$1
-∞ <= ∆ <=1
-∞ <= c4 <= 2
p4
$3
-1 <= ∆ <= ∞
2 <= c41 <= ∞
s5
$4
-2 <= ∆ <=0
2 <= c5 <= 4
p5
$4
0 <= ∆ <= ∞
4 <= c51 <= ∞
s6
$5
-1 <= ∆ <= ∞
4 <= c6 <= ∞
p6
$3
-∞ <= ∆ <= 2
-∞ <= c61 <= 5
s7
$5
-2 <= ∆ <= ∞
3 <= c7 <= ∞
p7
$3
-1 <= ∆ <= 2
2 <= c71 <= 5
s8
$1
-∞ <= ∆ <= 1
-∞ <= c8 <= 2
p8
$2
-1 <= ∆ <= 1
1 <= c81 <= 3
s9
$3
-1 <= ∆ <= 2
2 <= c9 <= 5
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Coefficient Ranges of
Decision Variables
Page 18
Objective Function
Decision
Current Selling/Purchase
Variable
Price
p9
$5
-2 <= ∆ <= ∞
3 <= c91 <= ∞
s10
$2
-2 <= ∆ <= 1
0 <= c10 <= 3
p10
$5
-5 <= ∆ <= ∞
0 <= c11 <= ∞
Range of Increase/Decrease
Coefficient Ranges of
Decision Variables
That is, the selling price of wheat c1 during Month 1 can vary from –∞ (we pay someone to take
our wheat – which is not profitable) to $7 (up $4 from the given selling price). In the same way,
the purchase price of wheat during Month 1 can vary from $7 (down $1 from the given purchase
price) to ∞ (again, this is not profitable), etc. We see that for the basic variables in the optimal
solution, the range of possible increases/decreases is much smaller, and therefore, more sensitive
to change(s). When we increase/decrease the objective function coefficients outside of the
range, that variable will enter/leave the basis & change the optimal solution. We tested this by
changing the coefficient of c1 to $8 & solving the LP again. As expected, s1 entered the basis,
was part of the optimal solution, and increased the z-value of the LP. By changing the range for
c31, the optimal solution changed as expected when c31 = 4, which is out of the allowable range.
Our new basis doesn’t include p3 as basic variable, and the new optimal Z = 142,000. Also, the
allowable range for c9 was found to be between 2 and 5, but setting c9 = 7, the current basis
changed but remains optimal and the only change will be the optimal Z (Cbv B-1b) with Z =
222,000. This happen is because a constraint on the selling quantity s8– e8 <= 0. Event if we
increased the selling price to a value greater than $5, we can only sell what we have on inventory
(20,000 bushels). Similarly, picking values outside the ranges of c3 and c9 will only affect the Zvalue, but will not affect the current basis. However by changing c5 , the basis will remain the
same but s5 will be equal to 20,000. Originally s5 was a basic variable equal to 0. In other
words, this leads us to believe that the original LP is degenerate, especially since there are at
least one basic variable in the optimal solution equal to zero.
In looking at the sensitivity ranges for the right-hand side, we see that the only major affectation
is to increase/decrease the storage capacity of the warehouse. Then, we could sell more wheat
and purchase more as determined by a new optimal solution. The ranges for the right-hand side
are listed in the table below:
Class of ’93: Garcia, Nguyen, Read
Page 19
Decision
Current Right-Hand Side
Range of Increase/Decrease
Ranges of Right-Hand Sides
b1
6
-6 <= ∆ <= ∞
0 <= b1 <= ∞
b2
0
-6 <= ∆ <= ∞
-6 <= b2 <= ∞
b3
0
-6 <= ∆ <= ∞
-6 <= b3 <= ∞
b4
0
-20 <= ∆ <= ∞
-20 <= b4 <= ∞
b5
0
-20 <= ∆ <= ∞
-20 <= b5 <= ∞
b6
0
-20 <= ∆ <= ∞
-20 <= b6 <= ∞
b7
0
0 <= ∆ <= ∞
0 <= b7 <= ∞
b8
0
0 <= ∆ <= ∞
0 <= b8 <= ∞
b9
0
0 <= ∆ <= ∞
0 <= b9 <= ∞
b10
0
0 <= ∆ <= ∞
0 <= b10 <= ∞
b11
14
-14 <= ∆ <= ∞
0 <= b11 <= ∞
b12
20
-14 <= ∆ <= ∞
6 <= b12 <= ∞
b13
20
0 <= ∆ <= ∞
20 <= b13 <= ∞
b14
20
0 <= ∆ <= 0
20 <= b14 <= 20
b15
20
-20 <= ∆ <= 0
0 <= b15 <= 20
b16
20
-20 <= ∆ <= ∞
0 <= b16 <= ∞
b17
20
-20 <= ∆ <= ∞
0 <= b17 <= ∞
b18
20
-20 <= ∆ <= ∞
0 <= b18 <= ∞
b19
20
-20 <= ∆ <= ∞
0 <= b19 <= ∞
b20
20
-20 <= ∆ <= ∞
0 <= b20 <= ∞
b21
6
-6 <= ∆ <= 14
0 <= b21 <= 20
b22
0
-6 <= ∆ <= ∞
-6 <= b22 <= ∞
b23
0
0 <= ∆ <= 20
0 <= b23 <= 20
b24
0
0 <= ∆ <= ∞
0 <= b24 <= ∞
b25
0
-20 <= ∆ <= ∞
-20 <= b25 <= ∞
b26
0
-20 <= ∆ <= ∞
-20 <= b26 <= ∞
b27
0
-20 <= ∆ <= 0
-20 <= b27 <= 0
b28
0
-20 <= ∆ <= ∞
-20 <= b28 <= ∞
b29
0
-20 <= ∆ <= 0
-20 <= b29 <= 0
Variable
According to this analysis, it is interesting to note that the optimal solution will not change
regardless of how large our warehouse storage capacity may be, as seen in the range for b1. On
the other hand, the range for b15 indicates that the original warehouse capacity is the maximum
value that will keep the current solution optimal. We tested this in Lindo & any increase in the
right-hand side of constraint #16 changes the optimal solution where the z-value is greater and p6
increases, as expected.
Class of ’93: Garcia, Nguyen, Read
Page 20
3.3
Problem 3 (Power Generation)
The optimal solution obtained through Lindo for condition 1 is as follows:
Z = 14192
X12 = 175
X14 = 185
X15 = 0
X21 = 190
X22 = 0
X24 = 10
X25 = 101
X35 = 30
X36 = 190
X37 = 0
X38 = 15
X49 = 200
X410 = 175
X511 = 24
X512 = 200
X63 = 175
X65 = 34
X67 = 200
X79 = 0
X710 = 0
X711 = 121
Lindo determined that 26 iterations were necessary to find the optimal solution. The solution
also highlighted the power supply from each plant. For example the San Juan (1) plant will
supply power to the substations 2 and 4, total 360 MW. Palo Seco (2) supplies power to
substations 1, 4 and 5, total 301 MW (100% capacity). ASE-PR (6) and Eco Electrica (7) supply
530 MW (24% of the total high peak demand). Also Costa Sur (4) supplies 375 MW (less than
the 70% max capacity of the plant). The optimal solution holds the constraints.
After examining the sensitivity analysis, we determined that the ranges for our decision variables
could be as follows, and still maintain the optimal solution:
Decision
Cost of Shipping
Variables
Power ($)
X11
6
-1 <= ∆ <= ∞
5 <= C11<= ∞
X12
5
-7 <= ∆ <= 0
-2 <= C12 <= 5
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Page 21
Decision
Cost of Shipping
Variables
Power ($)
X13
6
-3 <= ∆ <= ∞
3 <= C13 <= ∞
X15
9
0 <= ∆ <= ∞
9 <= C15 <= ∞
X16
10
-2 <= ∆ <= ∞
8 <= C16 <= ∞
X17
9
-2 <= ∆ <= ∞
7 <= C17 <= ∞
X18
12
-8 <= ∆ <= ∞
4 <= C18 <= ∞
X19
14
-11 <= ∆ <= ∞
3 <= C19 <= ∞
X110
17
-11 <= ∆ <= ∞
6 <= C110 <= ∞
X111
11
-6 <= ∆ <= ∞
5 <= C111 <= ∞
X112
10
-2 <= ∆ <= ∞
8 <= C112 <= ∞
X21
6
-7 <= ∆ <= ∞
1 <= C21 <= ∞
X22
6
0 <= ∆ <= ∞
6 <= C22 <= ∞
X23
5
-1 <= ∆ <= ∞
4 <= C23 <= ∞
X25
10
-2 <= ∆ <= 0
8 <= C25 <= 10
X26
11
-2 <= ∆ <= ∞
9 <= C26 <= ∞
X27
10
-2 <= ∆ <= ∞
8 <= C27 <= ∞
X28
11
-6 <= ∆ <= ∞
5 <= C28 <= ∞
X29
13
-9 <= ∆ <= ∞
4 <= C29 <= ∞
X210
16
-9 <= ∆ <= ∞
7 <= C210 <= ∞
X211
10
-4 <= ∆ <= ∞
6 <= C211 <= ∞
X212
9
-5 <= ∆ <= ∞
4 <= C212 <= ∞
X31
10
-3 <= ∆ <= ∞
7 <= C31 <= ∞
X32
10
-3 <= ∆ <= ∞
7 <= C32 <= ∞
X33
11
-6 <= ∆ <= ∞
5 <= C33 <= ∞
X34
12
-3 <= ∆ <= ∞
9 <= C34<= ∞
X35
11
-1 <= ∆ <= 0
10 <= C35 <= 11
X36
10
-10 <= ∆ <= 1
0 <= C36 <= 11
X39
8
-2 <= ∆ <= ∞
6 <= C39 <= ∞
X310
10
-1 <= ∆ <= ∞
9 <= C310 <= ∞
X311
12
-5 <= ∆ <= ∞
7 <= C311 <= ∞
X312
13
-8 <= ∆ <= ∞
5 <= C312 <= ∞
X41
15
-8 <= ∆ <= ∞
7 <= C41 <= ∞
X42
14
-7 <= ∆ <= ∞
7 <= C42 <= ∞
X43
15
-10 <= ∆ <= ∞
5 <= C43 <= ∞
X44
13
-4 <= ∆ <= ∞
9 <= C44 <= ∞
X45
18
-7 <= ∆ <= ∞
11 <= C45 <= ∞
X46
20
-10 <= ∆ <= ∞
10 <= C46 <= ∞
X47
17
-8 <= ∆ <= ∞
9 <= C47 <= ∞
X48
9
-3 <= ∆ <= ∞
6 <= C48 <= ∞
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Page 22
Decision
Cost of Shipping
Variables
Power ($)
X49
6
-5 <= ∆ <= 0
1 <= C49 <= 6
X410
8
-8 <= ∆ <= 0
0 <= C410 <= 8
X411
9
-2 <= ∆ <= ∞
7 <= C411 <= ∞
X412
10
-5 <= ∆ <= ∞
5 <= C412 <= ∞
X51
9
-3 <= ∆ <= ∞
6 <= C51 <= ∞
X52
8
-2 <= ∆ <= ∞
6 <= C52 <= ∞
X53
9
-5 <= ∆ <= ∞
4 <= C53 <= ∞
X54
10
-2 <= ∆ <= ∞
8 <= C54 <= ∞
X55
12
-2 <= ∆ <= ∞
10 <= C55 <= ∞
X56
15
-6 <= ∆ <= ∞
9 <= C56 <= ∞
X57
11
-3 <= ∆ <= ∞
8 <= C57 <= ∞
X58
11
-6 <= ∆ <= ∞
5 <= C58 <= ∞
X59
9
-5 <= ∆ <= ∞
4 <= C59 <= ∞
X510
10
-2 <= ∆ <= ∞
8 <= C510 <= ∞
X511
6
-1 <= ∆ <= 1
5 <= C511 <= 7
X512
4
-5 <= ∆ <= 1
-1 <= C512 <= 5
X61
9
-4 <= ∆ <= ∞
5 <= C61 <= ∞
X62
8
-3 <= ∆ <= ∞
5 <= C62 <= ∞
X63
9
-5 <= ∆ <= 1
4 <= C63 <= 10
X64
9
-2 <= ∆ <= ∞
7 <= C64 <= ∞
X65
9
0 <= ∆ <= 1
9 <= C65 <= 10
X66
9
-1 <= ∆ <= ∞
8 <= C66 <= ∞
X67
7
-9 <= ∆ <= 0
-2 <= C67 <= 7
X68
5
-7 <= ∆ <= ∞
-2 <= C68 <= ∞
X69
6
-3 <= ∆ <= ∞
3 <= C69 <= ∞
X610
7
-1 <= ∆ <= ∞
6 <= C610 <= ∞
X611
6
-1 <= ∆ <= ∞
5 <= C611 <= ∞
X612
10
-7 <= ∆ <= ∞
3 <= C612 <= ∞
X71
14
-7 <= ∆ <= ∞
7 <= C71 <= ∞
X72
13
-6 <= ∆ <= ∞
7 <= C72 <= ∞
X73
14
-9 <= ∆ <= ∞
5 <= C73 <= ∞
X74
13
-4 <= ∆ <= ∞
9 <= C74 <= ∞
X75
16
-5 <= ∆ <= ∞
11 <= C75 <= ∞
X76
17
-7 <= ∆ <= ∞
10 <= C76 <= ∞
X77
14
-5 <= ∆ <= ∞
9 <= C77 <= ∞
X78
7
-1 <= ∆ <= ∞
6 <= C78 <= ∞
X711
7
-1 <= ∆ <= 1
6 <= C711 <= 8
X712
8
-8 <= ∆ <= ∞
0 <= C712 <= ∞
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Page 23
Looking at Plant 1, the ranges for X11 (NBV) are 5 <= C11<= ∞. If we decrease the cost of
shipping in X11 to $4, then X11 will enter the basis with C11 = 190 and a new optimal solution
Z = 14,002. But, by increasing the shipping cost to any value greater than $5, the current basis
will remain optimal and the values of the decision variables will remain the same. Another
example is the basic variable X12. X12 holds for -2 <= C12 <= 5. But if we increase the
shipping cost C12 to more than $6, the solution is no longer optimal and we will get another
optimal solution (Z = $14,256 IAW Lindo). Also, X15 (BV = 0) holds for 9 <= C15 <= ∞. But
if we increase it to $10, X15 will exit the basis, but the solution will remain unchanged because
of the degenerate LP. Because we are dealing with a real-life problem, it’s unrealistic to bring
the shipping price to zero ($0). By increasing the shipping price on any BV above the range will
help to get another solution and a NBV to enter the basis. Any variation on the shipping cost on
any of the NBV (decreasing) or BV (increasing) will change by increasing or decreasing the
shipping cost, and we can get a new optimal solution.
For the right hand side of the constraints:
Objective Function Coefficient
RHS
Current Value
Range of Increase/Decrease
b1
0
-∞ <= ∆ <=625
-∞ <= b1< = 625
b2
360
-101 <= ∆ <=10
259 <= b2 <= 370
b3
301
-101 <= ∆ <= 30
200 <= b3 <= 331
b4
810
-575 <= ∆ <= ∞
235 <= b4 <= ∞
b5
743
-368 <= ∆ <= ∞
375 <= b5 <= ∞
b9
190
-30 <= ∆ <= 101
160 <= b9 <= 291
b11
175
-30 <= ∆ <= 15
145 <= b11 <= 190
b20
200
-121 <= ∆ <= 24
79 <= b20 <= 224
Ranges of Decision Variables
In this part, we examined the supply and demand constraints, which are listed in the table above.
For example, if Plant 3 (b3) decreases its output by less than 101 MW or increases by more than
30 MW, then we get a new solution with a new constraints and a new optimal solution. For
example, if we decrease its output to 190 MW, the new Z-value we obtain is 14323. Likewise,
for a demand constraint, if substation 1 (b9) increases his demand over 291 MW, the new
solution is 14472. Even if the current basis remains optimal (between the ranges) the values of
the decision variables and Z change.
Class of ’93: Garcia, Nguyen, Read
Page 24
The optimal solution obtained through Lindo for condition 2 is as follows:
Z = 17237.00
X11 = 174
X12 = 175
X15 = 0
X112 = 11
X21 = 0
X36 = 0
X44 = 195
X49 = 200
X410 = 175
X51 = 16
X52 = 0
X55 = 30
X512 = 178
X63 = 175
X65 = 44
X66 = 190
X67 = 0
X75 = 91
X77 = 200
X78 = 15
X711 = 145
Lindo determined that 32 iterations were necessary to find the optimal solution. The power
supply by ASE-PR and Eco Electrica increases dramatically due to the shutdown of Palo Seco
and Aguirre Plant. ASE-PR (6) supplies power to substations 3, 5, and 6, for a total of 314 MW
(70% capacity). Eco Electrica (7) supplies 451 MW (100 % capacity). The solution holds the
constraints.
After examining the sensitivity analysis, we determined that the ranges for our decision variables
could be as follows, and still maintain the optimal solution:
Decision
Cost of Shipping
Variables
Power ($)
X11
6
0 <= ∆ <= 0
X11 will not change
X12
5
-13 <= ∆ <= 0
-8 <= C12 <= 5
X13
6
-5 <= ∆ <= ∞
1 <= C13 <= ∞
X15
9
0 <= ∆ <= ∞
9 <= C15 <= ∞
X16
10
-1 <= ∆ <= ∞
9 <= C16 <= ∞
X17
9
-2 <= ∆ <= ∞
7 <= C17 <= ∞
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Page 25
Decision
Cost of Shipping
Variables
Power ($)
X18
12
-12 <= ∆ <= ∞
0 <= C18 <= ∞
X19
14
-17 <= ∆ <= ∞
-3 <= C19 <= ∞
X110
17
-17 <= ∆ <= ∞
0 <= C110 <= ∞
X111
11
-11 <= ∆ <= ∞
0 <= C111 <= ∞
X112
10
-1 <= ∆ <= 1
9 <= C112 <= 11
X21
6
∞ <= ∆ <= 1
∞ <= C21 <= 7
X22
6
-1 <= ∆ <= ∞
5 <= C22 <= ∞
X23
5
-4 <= ∆ <= ∞
1 <= C23 <= ∞
X25
10
-1 <= ∆ <= ∞
9 <= C25 <= ∞
X26
11
-2 <= ∆ <= ∞
9 <= C26 <= ∞
X27
10
-3 <= ∆ <= ∞
7 <= C27 <= ∞
X28
11
-11 <= ∆ <= ∞
0 <= C28 <= ∞
X29
13
-16 <= ∆ <= ∞
-3 <= C29 <= ∞
X210
16
-16 <= ∆ <= ∞
0 <= C210 <= ∞
X211
10
-10 <= ∆ <= ∞
0 <= C211 <= ∞
X212
9
-8 <= ∆ <= ∞
1 <= C212 <= ∞
X31
10
-3 <= ∆ <= ∞
7 <= C31 <= ∞
X32
10
-4 <= ∆ <= ∞
6 <= C32 <= ∞
X33
11
-9 <= ∆ <= ∞
2 <= C33 <= ∞
X34
12
-6 <= ∆ <= ∞
6 <= C34<= ∞
X35
11
-1 <= ∆ <= ∞
10 <= C35 <= ∞
X36
13
-∞ <= ∆ <= 1
-∞ <= C36 <= 14
X39
7
-9 <= ∆ <= ∞
-2 <= C39 <= ∞
X310
9
-8 <= ∆ <= ∞
1 <= C310 <= ∞
X311
12
-11 <= ∆ <= ∞
1 <= C311 <= ∞
X312
13
-11 <= ∆ <= ∞
2 <= C312 <= ∞
X41
15
-1 <= ∆ <= ∞
14 <= C41 <= ∞
X42
14
-1 <= ∆ <= ∞
13 <= C42 <= ∞
X43
15
-6 <= ∆ <= ∞
9 <= C43 <= ∞
X44
13
-13 <= ∆ <= 1
0 <= C44 <= 14
X45
18
-1 <= ∆ <= ∞
17 <= C45 <= ∞
X46
20
-3 <= ∆ <= ∞
17 <= C46 <= ∞
X47
17
-2 <= ∆ <= ∞
15 <= C47 <= ∞
X48
9
-1 <= ∆ <= ∞
8 <= C48 <= ∞
X49
6
-5 <= ∆ <= 1
1 <= C49 <= 7
X410
8
-8 <= ∆ <= 1
0 <= C410 <= 9
X411
9
-1 <= ∆ <= ∞
8 <= C411 <= ∞
X412
10
-1 <= ∆ <= ∞
9 <= C412 <= ∞
Class of ’93: Garcia, Nguyen, Read
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Page 26
Decision
Cost of Shipping
Variables
Power ($)
X51
9
0 <= ∆ <= 0
9 <= C51 <= 9
X52
8
0 <= ∆ <= ∞
8 <= C52 <= ∞
X53
9
-5 <= ∆ <= ∞
4 <= C53 <= ∞
X54
10
-2 <= ∆ <= ∞
8 <= C54 <= ∞
X55
12
-1 <= ∆ <= 0
11 <= C55 <= 12
X56
15
-3 <= ∆ <= ∞
12 <= C56 <= ∞
X57
11
-1 <= ∆ <= ∞
10 <= C57 <= ∞
X58
11
-8 <= ∆ <= ∞
3 <= C58 <= ∞
X59
9
-9 <= ∆ <= ∞
0 <= C59 <= ∞
X510
10
-4 <= ∆ <= ∞
4 <= C510 <= ∞
X511
6
-3 <= ∆ <= ∞
3 <= C511 <= ∞
X512
4
-0.5 <= ∆ <= 0.5
3.5 <= C512 <= 4.5
X61
9
-3 <= ∆ <= ∞
6 <= C61 <= ∞
X62
8
-3 <= ∆ <= ∞
5 <= C62 <= ∞
X63
9
-9 <= ∆ <= 4
0 <= C63 <= 13
X64
9
-4 <= ∆ <= ∞
5 <= C64 <= ∞
X65
9
-1 <= ∆ <= 0
8 <= C65 <= 9
X66
9
-1 <= ∆ <= 1
7 <= C66 <= 8
X67
7
0 <= ∆ <= ∞
7 <= C67 <= ∞
X68
5
-13 <= ∆ <= ∞
-8 <= C68 <= ∞
X69
6
-9 <= ∆ <= ∞
-3 <= C69 <= ∞
X610
7
-7 <= ∆ <= ∞
0 <= C610 <= ∞
X611
6
-6 <= ∆ <= ∞
0 <= C611 <= ∞
X612
10
-9 <= ∆ <= ∞
1 <= C612 <= ∞
X71
14
-1 <= ∆ <= ∞
13 <= C71 <= ∞
X72
13
-1 <= ∆ <= ∞
12 <= C72 <= ∞
X73
14
-6 <= ∆ <= ∞
8 <= C73 <= ∞
X74
13
-1 <= ∆ <= ∞
12 <= C74 <= ∞
X75
16
0 <= ∆ <= 1
15 <= C75 <= 17
X76
17
-1 <= ∆ <= ∞
16 <= C76 <= ∞
X77
14
-15 <= ∆ <= 0
-1 <= C77 <= 14
X78
7
-4 <= ∆ <= 1
3 <= C78 <= 8
X711
7
-8 <= ∆ <= 1
-1 <= C711 <= 8
X712
8
-9 <= ∆ <= ∞
-1 <= C712 <= ∞
Range of Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
Looking at Plant 7, if we decrease the cost of shipping in C71 (NBV) to less than $13, then X71
will enter the basis and the solution is no longer optimal. Now that the shipping cost is less, the
Class of ’93: Garcia, Nguyen, Read
Page 27
new solution will include X71 (Plant 7 supplies 135 MW to Substation 1) and the new optimal
solution is Z = 17,102. For X711 (BV), if the shipping cost (C711) increases to more than $9,
the solution is no longer optimal and X67 enters the basis with 44 MW. Also the optimal Zvalue will change ($17,516 IAW Lindo). Any variation on the shipping cost of any of the NBV
(decreasing) or BV (increasing) will change by increasing or decreasing the shipping cost, and
we will get a new optimal solution.
For the right hand side of the constraints:
Objective Function Coefficient
RHS
Current Value
Range of Increase/Decrease
b1
0
-∞ <= ∆ <=2286
-∞ <= b1 <= 2286
b2
360
-87 <= ∆ <= 8
273 <= b2 <= 368
b3
0
0 <= ∆ <= 8
0 <= b3 <= 8
b4
0
0 <= ∆ <= 11
0 <= b4 <=11
b5
743
-173 <= ∆ <= ∞
570 <= b5 <= ∞
b7
409
-44 <= ∆ <= 11
365 <= b7 <= 420
b10
175
-8 <= ∆ <= 87
167 <= b10 <= 262
b12
195
-195 <= ∆ <= 173
0 <= b12 <= 368
b15
200
-11 <= ∆ <= 89
189 <= b15 <= 289
b20
200
-11 <= ∆ <= 174
189 <= b20 <= 374
Ranges of Decision Variables
In this part, we examined the supply and demand constraints, which are listed in the table above.
For example, if Plant 7 (b7) decreases his power supply by less than 44 MW or increases by
more than 11 MW, then we get a new optimal solution with new constraints and new optimal
solution. For example, if we decrease its power supply to 360 MW, the new Z-value obtained is
17,634. Likewise, if the power demand on the substation 20 (b20) increases over 374 MW, the
solution is no longer optimal and the new Z = 18,812 MW.
Even if the current basis remains
optimal (between the ranges) the values of the decision variables and Z change. Finally, for
both conditions, due to the fact that there are more than one basic variable equal to zero, we
conclude that this LP is also degenerate.
In all three scenarios, Lindo provided adequate ranges. Even with the sensitivity analysis that
Lindo provided, some of the ranges do not seem feasible to apply to real-world situations. For
instance, we would never pay consumers for using electricity and market demands would not
Class of ’93: Garcia, Nguyen, Read
Page 28
allow us to increase the selling price to infinity, as suggested. By having a constraint that affect
a basic variable, even if the ranges indicated that the solution will be suboptimal if we leave the
ranges, the constraint will force the basis to remain the same and the only change will be the Zoptimal. We observed this situation in all three scenarios. Therefore, it is important to consider
all aspects of the situation before changing the variables.
4.0
Conclusions and Recommendations
In this project, we attempted to maximize or minimize three different situations. The solutions
for all three problems using LINDO show that the companies can gain substantial profit or
minimize their expenses by implementing recommendations developed by each model.
First, our recommendation to Cummings Engine Company is to produce and sell trucks is as
follows:
Produce
st
Truck Type
1 Year
Type 1
100
Type 2
200
nd
2
Year
Sell
rd
st
nd
2
Year
3rd Year
3 Year
1 Year
200
150
100
200
150
100
150
200
100
150
During the formulation, we began by defining the decision variables that would describe each
situation and decision to be made (e.g., determine the number of type 1 trucks to produce during
the first year). The formulation presented a challenge for the group due to a set of conditions
that later became constraints. For example, Grummins Engine Company produces two different
types of trucks. The production of each truck was based on how many trucks can be sold every
year during a three year period. Also, if a truck is not sold, the company incurs an additional
charge for inventory. Because they want to maximize its profit, we have to make sure that our
solution takes into consideration all related costs (production, overhaul, etc). During our
presentation of the model to the company we must state that during the formulation phase, we
assumed that Grummins will sell all trucks produced in that year. Also the government imposes
certain restrictions and specifications that we must follow. These restrictions can cost millions
of dollars to company if they are violated.
Class of ’93: Garcia, Nguyen, Read
Page 29
For the inventory problem, we determined the following:
Purchase
Month
Selling Quantity
1
0
0
2
0
0
3
6
20
4
0
0
5
0
0
6
20
20
7
20
0
8
0
20
9
20
0
10
0
0
Quantity
The above table represented the optimal solution. During the formulation phase, we needed to
determine what real-life facts can affect the situation. The capacity of the warehouse, initial
inventory at the beginning of each month, and the monthly selling limitation are some examples
of a real-life situation. Sometimes, we must determine all constraints and limitations before we
begin the formulation phase. If a limitation is ignored during the formulation phase, the optimal
solution can be completely wrong. The above table clearly confirmed the inventory procedures
for the next 10 months. We can recommend as an option to increase the capacity of the
warehouse in order to increase profits.
In the power distribution case, we were asked to develop an LP for two different conditions. The
first condition was to minimize the cost of meeting each substation’s peak power demand for
next year during high peak demand and the second condition was to minimize the cost of
meeting each substation’s peak power demand if two plants were disconnected due to bad
weather. This model doesn’t include the operating limits of the generators, loads and the
transmission line network. The only two types of critical points that we identified were the
transmission line flow limits and generator capability limits (plants). Our final solution was as
follows:
Class of ’93: Garcia, Nguyen, Read
Page 30
Condition 1
Condition 2
Plant i to
Power
Plant i to
Power
Substation j
Transmitted
Substation j
Transmitted
X12
175
X11
174
X14
185
X12
175
X21
190
X112
11
X24
10
X44
195
X25
101
X49
200
X35
30
X410
175
X36
190
X51
16
X38
15
X55
30
X49
200
X512
178
X410
175
X63
175
X511
24
X65
44
X512
200
X66
190
X63
175
X75
91
X65
34
X77
200
X67
200
X78
15
X711
121
X711
145
Because generating companies and power systems have the problem of deciding how best to
meet the varying demand for electricity, which has a daily and weekly cycle, we must develop a
model that will support the high peak demand. Electricity cannot be stored; it is necessary to
start-up and shut-down a number of generating units at various power stations each day. During
the second condition, the problem is to decide when and which generating units to start-up and
shut-down, in order to minimize the total fuel cost or to maximize the total profit, over a study
period of typically a day, subject to a large number of difficult constraints that must be satisfied.
By assuming that two plants will be disconnected at any time, we can develop an optimal
solution. The most important constraint is that the total generation must equal the forecast
demands for electricity.
Also, most of these transportation problems must take into consideration the supply of imported
power. In the United States, the supply of imported power is price responsive. The quantity of
imported power can increase in the face of higher power prices, dampening market power. In
our case, the market is owned by one source, so they can control any changes on the shipping
Class of ’93: Garcia, Nguyen, Read
Page 31
cost of the decision variables. The source can use our formulation to predict the worst case
scenario and to determine any backup plan to sustain the demand if anything should happen.
Furthermore, the source can save millions of dollars by improving the transmission line networks
and also by obtaining more fuel cells.
5.0
References
Winston, W.L. and M.Venkataramanan, Introduction to Mathematical Programming, 4th Edition,
Duxbury Press, Belmont, CA, 2003.
Gonen, Turan, Electric Power Distribution System Engineering, McGraw-Hill Publishing
Company, Oklahoma City, OK, 1986.
Puerto Rico Electric Power Authority, Tarifas para Servicio de Electricidad, November 1989.
Autoridad de Energia Electrica, www.prepa.com, 2002.
Class of ’93: Garcia, Nguyen, Read
Page 32