INEN 420 Final Project
Rhoda
Daniel
Javier
Grummins Engine Company
•
Facts
– Produces 2 types of diesel
trucks (<300/year)
– Selling prices and
manufacturing costs
– Government regulation:
• Pollution emission
– Cost $2,000.00 to hold a
truck
– Maximum Demand for
Trucks:
Year
Type 1
Type 2
1
100
200
2
200
100
3
300
150
Grummins Engine Company
• Assumptions:
– Zero trucks in stock at the end of the 3rd year
– 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
– Trucks are only produced and sold, not
acquired by any other method such as
auctions, trading, etc.
Grummins Engine Company
• Formulation:
– Decision variables:
• 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.
• For i=1, 2; j=1, 2, 3.
Grummins Engine Company
• Formulation:
– Objective Function:
Max Z= 20 (S11+S12+S13) + 17 (S21+S22+S23)
- 15 (P11+P12+P13) - 14 (P21+P22+P23)
- 2 (R11+R12+R21+R22)
(in $ thousands)
Grummins Engine Company
• Formulation:
– Constraints:
P11+P21
P12+P22
P13+P23
S11
S12
S13
S21
S22
S23
<=320
<=320
<=320
<=100
<=200
<=300
<=200
<=100
<=150
(Production)
(Sale)
Grummins Engine Company
• Formulation:
– Constraints:
R11-P11+S11
=0
(Remain in stock)
R21-P21+S21
=0
R12-P12+S12-R11 =0
R22-P22+S22-R21 =0
P13+R12-S13
=0
P23+R22-S23
=0
5P11+5P12+5P13-5P21-5P22-5P23<=0
(Emissions requirement)
– Pij, Sij, Rij >= 0
Grummins Engine Company
• Optimal Solution (using LINDO/Excel
Solver):
–
–
–
–
–
–
–
Z
S11
S12
S13
S21
S22
S23
= 3,600.00 (in $ thousands)
= 100
= 200
= 150
= 200
= 100
= 150
Grummins Engine Company
• Optimal Solution:
–
–
–
–
–
–
–
–
–
–
P11
P12
P13
P21
P22
P23
R11
R12
R21
R22
= 100
= 200
= 150
= 200
= 100
= 150
=0
=0
=0
=0
Grummins Engine Company
• Optimal Solution:
– 10 iterations
– Company should sell every truck they make each
year
Grummins Engine Company
• Sensitivity Analysis:
Decision
Variables
Current Sale &
Production
($ thousands)
Range of
Increase/Decrease
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
Grummins Engine Company
• Sensitivity Analysis:
Decision
Variables
Current Sale &
Production
($ thousands)
Range of Increase/
Decrease
Objective Function
Coefficient
Range of Decision Variables
P13
15
0<=∆<=2
15<=C16<=17
P21
14
-2<=∆<=8
12<=C24<=22
P22
14
-2<=∆<=2
12<=C25<=16
P23
14
-∞<=∆<= 2
-∞<=C26<= 16
R11
2
-2 <=∆<=+∞
0 <=C1<=+∞
R12
2
-2 <=∆<=+∞
0 <=C2<=+∞
R21
2
-2 <=∆<=+∞
0 <=C3<=+∞
R22
2
-2 <=∆<=+∞
0 <=C4<=+∞
Grummins Engine Company
• Sensitivity Analysis:
Decision
Variable
Current RightHand Side
Range of
Increase/Decrease
Ranges of RightHand 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
Grummins Engine Company
• Sensitivity Analysis:
Decision
Variable
Current RightHand Side
Range of
Increase/Decrease
Ranges of RightHand Sides
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
Wheat Warehouse
• Facts:
– Capacity = 20,000
bushels
– Month # 1: 6,000
bushels
– Sell up to the initial
stock at the current
month’s selling price.
– Buy as much wheat as
wanted (Up to 20K)
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
Wheat Warehouse
• Assumptions:
– Wheat can only be sold or purchased (at the
given rates)
– Ending inventory:
• Ending inventory = beginning inventory – amount
sold + amount purchased
Wheat Warehouse
• Formulation:
– Decision variables:
• 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
Wheat Warehouse
• Formulation:
– Objective Function:
Max Z = 3s1 – 8p1 + 6s2 – 8p2 + 7s3 – 2p3 + s4 –
3p4 + 4s5 – 4p5 + 5s6 – 3p6 + 5s7 – 3p7
+ s8 – 2p8 + 3s9 – 5p9 + 2s10 – 5p10
(in $ thousands)
Wheat Warehouse
• Formulation:
– Constraints:
s1
s2 – e1
s3 – e2
s4 – e3
s5 – e4
s6 – e5
s7 – e6
s8 – e7
s9 – e8
s10 – e9
<= 6
<= 0
<= 0
<= 0
<= 0
<= 0
<= 0
<= 0
<= 0
<= 0
(selling restrictions)
Wheat Warehouse
• Formulation:
– Constraints:
p1 – s1
p2 – s2 + e1
p3 – s3 + e2
p4 – s4 + e3
p5 – s5 + e4
p6 – s6 + e5
p7 – s7 + e6
p8 – s8 + e7
p9 – s9 + e8
p10 – s10 + e9
<= 14 (purchasing restrictions)
<= 20
<= 20
<= 20
<= 20
<= 20
<= 20
<= 20
<= 20
<= 20
Wheat Warehouse
• Formulation:
– Constraints:
e1 + s1 – p1
=6
(ending inventory )
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
Wheat Warehouse
• Optimal Solution (using LINDO/Excel Solver
after 20 iterations):
–
–
–
–
–
–
–
–
–
–
–
–
Z = 162 (in $ thousands)
s3 = 6,000
p3 = 20,000
s5 = 0
p5 = 0
s6 = 20,000
p6 = 20,000
s7 = 20,000
p7 = 0
p8 = 20,000
s9 = 20,000
s10 = 0
Wheat Warehouse
• Sensitivity Analysis:
Decision
Variable
Current
Selling/Purchase
Price
Range of
Increase/Decrease
Objective Function
Coefficient Ranges
of Decision Variables
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 <= ∞
Wheat Warehouse
• Sensitivity Analysis:
Decision
Variable
Current
Selling/Purchase
Price
Range of
Increase/Decrease
Objective Function
Coefficient Ranges of
Decision Variables
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
p9
$5
-2 <= ∆ <= ∞
3 <= c91 <= ∞
s10
$2
-2 <= ∆ <= 1
0 <= c10 <= 3
p10
$5
-5 <= ∆ <= ∞
0 <= c11 <= ∞
Wheat Warehouse
• Sensitivity Analysis:
Decision
Variable
Current RightHand 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 <= ∞
Wheat Warehouse
• Sensitivity Analysis:
Decision
Variable
Current RightHand Side
Range of
Increase/Decrease
Ranges of RightHand Sides
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 <= ∞
Wheat Warehouse
• Sensitivity Analysis:
Decision
Variable
Current RightHand Side
Range of
Increase/Decrease
Ranges of RightHand Sides
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
Power Generation
• Facts
– Puerto Rico Electric Power Authority (PREPA)
accounts for a majority of net electricity generation (5
plants).
– ASE-PR and Eco Electrica will provide at least 20%
of the power demand during high peak demand
(0100 PM).
– Plant 2 will supply only 50% of its maximum output.
– Plant 4 will supply only 70% of the maximun output.
– Power Grid: Total Seven Plants beginning 2005.
– Tropical Storm Jeanne: Cause damage to Puerto
Rico’s power grid ($60 million).
Power Generation
1
1
1
2
4
1
3
2
7
1
0
9
8
EcoElectrica
New Power Plant
Substations
ASE-PR
5
6
Power Generation
Cost of Shipping to Substation #
FROM
1
2
3
4
5
6
7
(As of 30 SEP 2004)
8
9
10
11
12
Supply (MW)
(Plant)
San Juan
6
5
6
7
9
10
9
12
14
17
11
10
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
Eco
Electrica
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
Expected
Demand in
MW (2005)
Power Generation
• Assumptions:
– Plants operate at 90% of their maximum
capacity.
– Power supply to the substation is only being
used by the intended sources.
– In the event of any bad weather, at most two
plants will be disconnected.
Power Generation
• Formulation:
– Two different Conditions:
• Minimize the cost of meeting each substation’s
peak power demand for next year
• Minimize the cost of meeting each substation’s
peak power demand if Plant 2 and 4 are
disconnected due to bad weather
Power Generation
• Formulation:
– Decision variables (both conditions):
• Xij= number of megawatts produced at plant i and sent to
substations j (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.
Power Generation
• Formulation:
– Objective Function (Both Conditions)
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
Power Generation
• Formulation:
– Constraints (Condition 1): (Supply Constraints – 90%)
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
Power Generation
• Formulation:
– Constraints (Condition 2): (Supply Constraints – 90%)
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
Power Generarion
• Formulation:
– Constraints (Both Conditions): (Supply Constraints for 2 add plants)
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
Power Generarion
• Formulation:
– Constraints (Both Conditions):
(Demand Constraints)
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
(Substation 1)
(Substation 2)
(Substation 3)
(Substation 4)
(Substation 5)
(Substation 6)
(Substation 7)
(Substation 8
(Substation 9)
(Substation 10)
(Substation 11)
(Substation 12)
– Sign Restrictions: Xij >= 0, for i=1,……..,7; j = 1,………..,12
Power Generation
Condition 1 (26 iterations)
• Optimal Solution
(using LINDO):
Condition 1:
Z = $14,912
Condition 2:
Z = $ 17,237
Condition 2 (32 iterations)
Plant i to
Substation j
Power
Transmitted
Plant i to
Substation j
Power
Transmitted
X12
175
X11
174
X14
185
X12
175
X15
0
X15
0
X21
190
X112
11
X22
0
X21
0
X24
10
X36
0
X25
101
X44
195
X35
30
X49
200
X36
190
X410
175
X37
0
X51
16
X38
15
X52
0
X49
200
X55
30
X410
175
X512
178
X511
24
X63
175
X512
200
X65
44
X63
175
X66
190
X65
34
X75
91
X67
200
X77
200
X79/710
0
X78
15
X711
121
X711
145
Power Generation
• Sensitivity Analysis:
Decision
Variables
Cost of
Shipping
Power ($)
Range of
Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
X11 (NBV)
6
-1 <= ∆ <= ∞
5 <= C11<= ∞
X12 (BV)
5
-7 <= ∆ <= 0
-2 <= C12 <= 5
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
Decrease
C11 to $4,
X11 enter
Increase
basis.
C12
to $7,
X11=190
X12 leaves
basis.
Enter
Increase
X15=101
C15 to $10,
X15 leaves
basis.
Z= 14,192
Power Generation
• Sensitivity Analysis:
RHS
Current Value
Range of
Increase/Decrease
Objective Function Coefficient
Ranges of Decision Variables
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
Increase b3
to 190,
New Z =
$14,323
Increase b9
to 292,
New Z =
$14,472