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Unformatted text preview: DESIGN
DESIGN
UNDER UNCERTAINTY
Prof. Miguel Bagajewicz
CHE 4273 1 TwoTwoStage
Stochastic Optimization Models
Stochastic
Philosophy
Philosophy
• Maximize the Expected Value of the objective over all possible realizations of
Expected
uncertain parameters.
• Typically, the objective is Profit or Net Present Value.
Typically the objective is Profit or Net Present Value
• Sometimes the minimization of Cost is considered as objective.
Cost Uncertainty
• Typically, the uncertain parameters are: market demands, availabilities,
market
prices, process yields, rate of interest, inflation, etc.
• In TwoStage Programming, uncertainty is modeled through a finite number
Programming
is modeled thro
finite
of independent Scenarios.
Scenarios
• Scenarios are typically formed by random samples taken from the probability
random
distributions of the uncertain parameters.
th
2 Characteristics
Characteristics of TwoStage
TwoStochastic Optimization Models
Stochastic FirstFirstStage Decisions
• Taken before the uncertainty is revealed. They usually correspond to structural
decisions (not operational)
decisions (not operational).
• Also called “Here and Now” decisions.
• Represented by “Design” Variables.
• Examples:
−To build a plant or not. How much capacity should be added, etc.
−To place an order now.
−To sign contracts or buy options.
−To pick a reactor volume, to pick a certain number of trays and size
the condenser and the reboiler of a column, etc 3 Characteristics
Characteristics of TwoStage
TwoStochastic Optimization Models
SecondSecondStage Decisions
• Taken in order to adapt the plan or design to the uncertain parameters
realization.
• Also called “Recourse” decisions.
called Recourse decisions
• Represented by “Control” Variables.
• Example: the operating level; the production slate of a plant.
• Sometimes first stage decisions can be treated as second stage decisions.
In such case the problem is called a multiple stage problem. 4 Example:
Example: Vinyl Chloride Plant HCl recycle Air or O2
Oxychlorination
Ethylene Cl2 Light ends EDC
purification
Direct
chlorination EDC
pyrolysis VCM
purification VCM EDC recycle
Heavy ends 5 Example:
Example: Vinyl Chloride Plant
Consider the following forecasts:
Forecasted prices of raw materials product
Forecasted prices of raw materials product
Year Forecasted excess demand over current capacity Ethylene Chlorine Oxygen VCM $/ton $/ton $/ft3 $/ton 2004 492.55 212.21 0.00144 499.19 2004 3602 2005 499.39 214.14 0.00144 506.19 2005 5521 2006 506.22 216.07 0.00143 513.18 2006 7355 2007 513.06 218.00 0.00142 520.18 2007 9551 2008 519.90 219.93 0.00141 527.18 2008 11888 2009 526.73 221.86 0.00140 529.17 2009 14322 2010 533.57 223.79 0.00139 535.17 2010 16535 2011 540.41 225.72 0.00138 543.17 2011 18972 Std. Dev 24.17 10.56 0.00010 26.15 Year VCM
lbmol/hr Consider building (in 2004) for three capacities to satisfy excess demand at 2004,
2006 and 2011 Plants will operate under capacity until 2006 or 2011 in the last
2006 and 2011. Plants will operate under capacity until 2006 or 2011 in the last
two
two cases. These are 3 different first stage decisions.
6 Example:
Example: Vinyl Chloride Plant
The different investment costs are:
Plant Capacity 4090
MMlb/yr 6440
MMlb/yr 10500
MMlb/yr TCI $47,110,219 $68,886,317 $77,154,892 Consider the following calculation procedure
Random Number
Generation Total Product
Cost Income from
selling VCM Random numbers are Raw material
Cost Gross Income Cash Flow Net Profit obtained for each year for
obtained for each year for
raw
raw materials and product
prices using sampling from a Net Present
Worth normal distribution. This can
be done in Excel Risk &
Probability
7 Example:
Example: Vinyl Chloride Plant
Probability vs. Net Present Worth Histograms and Risk Curves are 0.35
0.3
Pro b ability 0.25 Notice the asymmetry in the distributions. 0.2
0 .15
0.1 pec(j)
pec(j) 0.05
0
 4500 Risk at Different Capacity 3000 1500
0 6 1500
NPW ($10 ) 6.44E9 lb/yr 4.09E9 lb/yr 3000 1.05E10 lb/yr Cummulative Probability 1
0.8
0.6 The risk curves show a 36% chance of losing 0.4 money for the 10.5 billion lbs/year capacity, 0.2 31.7% for the 6.44 billion lbs/yr capacity and 0
6.00E+09 4.00E+09 2.00E+09 0.00E+00 41% chance for the
41% chance for the 4.09 billion lbs/year capacity.
billion lbs/year capacity
2.00E+09 4.00E+09 Net Present Worth ($) 6.44E9 lb/yr 4.09E9 lb/yr 6.00E+09 Expected Profits are: 24%, 25% and 20%. 1.05E10 lb/yr 8 SCENARIO
SCENARIO GENERATION SCENARIO GENERATION
Consider each parameter’s probability distribution.
di Discretize
Discretize it.
Option 1: pick values of
probabilities.
probabilities. For example, for
3 values, pick 25%, 50% and
values,
25% probability and find the
va
values. Use the cumulative
th
curve
curve to locate the numbers. P(θ) θ1 θ2 θ3 θ4 θ5 θ SCENARIO GENERATION
Consider each parameter’s probability distribution.
di Discretize
Discretize it.
Option 2: pick values
pick
(equidistantly or randomly)
and
and find the probability that
corresponds to them from the
area
area they “span”. Use the
cumulative
cumulative curve for this. P(θ) θ1 θ2 θ3 θ4 θ5 θ SCENARIO GENERATION
Consider each parameter’s probability distribution.
di Discretize
Discretize it.
Option 3: pick equal probability
va
values and find parameter
fi
values.
values. For example, for 3
values, pick 33% and locate the
points Use
points. Use the cumulative curve
to
to do this. P(θ) θ1 θ2 θ3 θ4 θ5 θ FOR
FOR A LARGE NUMBER OF
SAMPLES
SAMPLES WE USE THIS OPTION SCENARIO GENERATION
Each scenario is constructed by picking
one realization for each parameter. EXAMPLE:
EXAMPLE:
2 parameters (θ1, θ2). If each parameter
parameters
is discretized in three instances
(θi,low, 25%, θi,avg 50%, θi,hig 25%) SCENARIO GENERATION Scenario
Probability Scenario Probability
θ1,low, θ2,low
6.25%
θ1,hig, θ2,low
6.25%
θ1,low, θ2,avg
12.5%
12.5%
θ1,hig, θ2,avg
12.5%
12.5%
θ1,low, θ2,hig
6.25%
θ1,hig, θ2,hig
6.25%
θ1,avg, θ2,low
12.5%
12.5%
θ1,avg, θ2,avg
25.0%
SUM OF ALL
1,avg, 2,avg
θ1,avg, θ2,hig
12.5%
12.5%
PROBABILITIES=1 SCENARIO GENERATION
Effect of Small Number of Samples
1.2
SAMPLING 1 ACTUAL 20 Scenarios 0.8
0.6
0.4
0.2
0
75 85 95 105 115 125 1.2
SAMPLING 1 ACTUAL
0.8 500 Scenarios 0.6
0.4
0.2
0
75 85 95 105 115 125 SCENARIO GENERATION
Effect of the Number of Samples on Results (Gas in Asia)
1.0 1s
4.666 0.8 10s
4.357
100s
4.565 0.6 200s
4.678 0.4 50s
4.793 0.2 0.0
0 1 2 3 4 5 6 7 8 9 10 REGRET
REGRET ANALYIS 17 MINIMAX
MINIMAX REGRET ANALYSIS
Motivating Example
s1
H igh Maximize Average…select A Optimistic decision maker s3
L ow Average A Traditional way s2
M edium 19 14 3 10 B 16 7 4 9 C 20 8 4 8 D 10 6 5 7 M ax 20(C) 14(A) 5(D) 10(A) MaxiMax … select C 20 Pessimistic decision maker C
$ Million MaxiMmin … select D A 15
10
5 B
D 0
1
5 s
1 2
s2 3
s3 18 MINIMAX
MINIMAX REGRET ANALYSIS
Motivating Example A … regret = 8 @ low market
C … regret = 9 @ low market
D … regret = 10 @ high market s1
High s2
Medium s3
Low Maximum
Regret A 1 0 8 8 B Calculate regret:
find maximum regret 4 7 1 7 C 0 6 9 9 D 10 8 0 10 10
8 B In general, gives conservative decision
general gives conservative
but not pessimistic. $ Million B … regret = 7 @ medium market
MINIMAX D
B 6 C 4
2 A 0
s1
1 s2
2 s3
3
19 MINIMAX
MINIMAX REGRET ANALYSIS
TwoTwoStage Stochastic Programming Using Regret
Theory
Theory
s1
d1
19.01
d2
11.15
d3
12.75
d4
5.41
d5
15.09
Max 19.01 s1
d1
d2
d3
d4
d5 0.00
7.86
6.26
13.60
3.92 NPV
s4
15.48
20.54
22.25
32.02
12.48
32.02 s2
10.38
14.47
7.81
9.91
7.40
14.47 s3
10.57
8.87
16.02
12.63
8.81
16.02 s5
10.66
10.58
9.16
8.08
15.05
15.05 ENPV
13.22
13.12
13.60
13.61
11.77
13.61 s2
4.09
0.00
6.66
4.56
7.07 Regret
s3
s4
s5
5.45 16.54 4.39
7.15 11.48 4.47
0.00 9.77 5.89
3.39 0.00 6.97
7.21 19.54 0.00
Min Max Min
19.01 10.38
20.54 8.87
22.25 7.81
32.02 5.41
15.09 7.40
32.02 10.38 Max
16.54
11.48
9.77
13.60
19.54
9.77 20 SAMPLING
SAMPLING ALGORITHM
Generate
h(s)
d=d+1 d=1 Outer Loop yes
h=h(d)
EXIT Fi Fi
Fix FirstStage
Variables s=1 h=h(s) Solve
Deterministic
Model no d<=d_max Solve
Deterministic
Model no Inner Loop
yes
s= s+1 s<=s_max Store
Store
NPV(d,s) This generates several solutions
21 UPPER
UPPER AND LOWER BOUNDS
Risk
1
D
0.8 C
B 0.6
Lower
Bound
0.4 Upper
Bound
Bound A 0.2 0
0.0 2.0 4.0 6.0 8.0 10.0 This is very useful because it allows nice decomposition,
That is, there is no need to solve the full stochastic problem
22 UPSIDE
UPSIDE POTENTIAL
Point measure for the upside
Risk 1.0
0.9
0.8
0.7
OV =3.075 0.6 OV =0.75 0.5
E(Pro f it ) = 3.4
E(Pro f it ) = 3.0 0.4
0.3
0.2
0.1 VaR =0.75
VaR =1.75 0.0
0 1 2 3 4 5
Profit 6 7 8 9 10 23 AREA
AREA RATIO
Comparison measure
Risk 1.0
0.9 Risk(x2,NPV) O_Area 0.8
0.7 Risk(x1,NPV) 0.6
0.5
0.4
R_Area 0.3
0.2
0.1
0.0
0 1 ENPV2 ENPV1
2
3
4 5 6 7 8 9 NPV
10 24 CONCLUSIONS
CONCLUSIONS
• Regret Analysis can help in identifying good
solutions
solutions (It can also fail)
• The sampling Algorithm is an important tool to
identify upper bounds and good solutions.
• The upper potential is important to be
Th
considered.
considered. 25 ...
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This note was uploaded on 08/31/2011 for the course CHE 4273 taught by Professor Staff during the Spring '10 term at Oklahoma State.
 Spring '10
 staff
 Chemical Engineering, pH

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