Unformatted text preview: Capacity Planning under Uncertainty
CHE: 5480
Economic Decision Making in the Process Industry
Prof. Miguel Bagajewicz
University of Oklahoma
School of Chemical Engineering and Material Science 1 Characteristics of TwoStage
Stochastic Optimization Models Philosophy
Philosophy
• Maximize the Expected Value of the objective over all possible realizations of
Expected
uncertain parameters.
uncertain
• Typically, the objective is Profit or Net Present Value.
Profit Net
• Sometimes the minimization of Cost is considered as objective.
Cost Uncertainty
Uncertainty
• Typically, the uncertain parameters are: market demands, availabilities,
market
prices, process yields, rate of interest, inflation, etc.
prices,
• In TwoStage Programming, uncertainty is modeled through a finite number
In
of independent Scenarios.
Scenarios
• Scenarios are typically formed by random samples taken from the probability
random
distributions of the uncertain parameters.
distributions
2 Characteristics of TwoStage
Stochastic Optimization Models FirstStage Decisions
FirstStage
• Taken before the uncertainty is revealed. They usually correspond to structural
decisions (not operational).
• Also called “Here and Now” decisions.
Also
• Represented by “Design” Variables.
Represented
• Examples:
Examples:
−To build a plant or not. How much capacity should be added, etc.
To
−To place an order now.
To
−To sign contracts or buy options.
To
−To pick a reactor volume, to pick a certain number of trays and size
To
the condenser and the reboiler of a column, etc 3 Characteristics of TwoStage
Characteristics of TwoStage
Stochastic Optimization Models SecondStage Decisions
SecondStage
• Taken in order to adapt the plan or design to the uncertain parameters
realization.
realization.
• Also called “Recourse” decisions.
Also
• Represented by “Control” Variables.
Represented
• Example: the operating level; the production slate of a plant.
Example:
• Sometimes first stage decisions can be treated as second stage decisions.
In such case the problem is called a multiple stage problem. 4 TwoStage Stochastic Formulation
Let us leave it linear
because as is it is
complex enough.!!! LINEAR MODEL SP
LINEAR MODEL
T
Max ∑ ps qs ys − cT x
s Technology matrix Recourse
Function s.t. FirstStage
Cost Ax = b FirstStage Constraints Ts x +Wys = hs SecondStage Constraints x≥0
First stage variables x∈ X Second Stage Variables ys ≥ 0
Recourse matrix (Fixed Recourse)
Sometimes not fixed (Interest rates in Portfolio Optimization) Complete recourse: the
recourse cost (or profit) for
every possible uncertainty
realization remains finite,
independently of the firststage
decisions (x). Relatively complete recourse:
the recourse cost (or profit) is
feasible for the set of feasible
firststage decisions. This
condition means that for every
feasible firststage decision,
there is a way of adapting the
plan to the realization of
uncertain parameters. We also have found that one
can sacrifice efficiency for
certain scenarios to improve
risk management. We do not
know how to call this yet. 5 Process Planning Under Uncertainty
GIVEN: Process Network
Process Forecasted Data
Forecasted DETERMINE: Set of Processes
Set of Chemicals B
A Demands & Availabilities
Costs & Prices
Capital Budget Network Expansions
Network 2 C 3 D 1 Timing
Sizing
Location Production Levels
Production OBJECTIVES: Maximize Expected Net Present Value
Maximize
Minimize Financial Risk
Minimize
6 Process Planning Under Uncertainty
Design Variables: to be decided before the uncertainty reveals
to
Design
x= {Yit , Eit , Qit } Y: Decision of building process i in period t
Y:
E: Capacity expansion of process i in period t
E:
Q: Total capacity of process i in period t
Q: Control Variables: selected after the uncertain parameters become known
selected
ys = { Sjlts , Pjlts , Wits} S: Sales of product j in market l at time t and scenario s
P: Purchase of raw mat. j in market l at time t and scenario s
W: Operating level of of process i in period t and scenario s
W: 7 MODEL
LIMITS ON EXPANSION L
L
Yit Eit ≤ Eit ≤ Yit Eit i =1,, NP t =1,, NT TOTAL CAPACITY IN
EACH PERIOD Qit = Qi ( t −1) + Eit i =1,, NP t =1,, NT LIMIT ON THE NUMBER
OF EXPANSIONS
LIMIT ON THE CAPITAL
INVESTMENT NT ∑Y
t =1 it NP ∑( α
i =1 it ≤ NEXPi
E it + β it Yit ) ≤ CI t i =1,, NP
t =1,, NT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t. I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM NEXPt: maximum number of expansions in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
Lower and upper bounds on a process expansion in period t
8 MODEL
UTILIZED CAPACITY IS
LOWER THAN TOTAL
CAPACITY i =1,, NP t =1,, NT
NM MATERIAL BALANCE ∑P
l =1 jlts NP NM NP i =1 l =1 i =1 + ∑ηijWits ≤ ∑ S jlts + ∑ µijWits i =1,, NP t =1,, NT BOUNDS
NONNEGATIVITY
Yit ∈{ 0,1} INTEGER
VARIABLES Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/intermediate product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t
I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM Lower and upper bounds on availability of raw material j in market l in period t, scenario s
Lower and upper bounds on demand of product j in market l in period t, scenario s
9 MODEL
OBJECTIVE FUNCTION DISCOUNTED REVENUES INVESTMENT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/interm. product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t
I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM γjlt : Sale price of product/intermediate product j in market l in period t
Γjlt : Cost of product/intermediate product j in market l in period t
δit : Operating cost of process i in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
Lt : Discount factor for period t 10 Example
Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price
Demands,
Total of 400 Scenarios
Total
Project Staged in 3 Time Periods of 2, 2.5, 3.5 years
Project
Chemical 5 Chemical 1 Process 2 Chemical 6 Process 5 Process 1 Chemical 8 Chemical 2 Chemical 7 Process 3
Chemical 4 Chemical 3
Process 4 11 Example – Solution with Max ENPV
Period 1
3
2
3.5
2.5 years
2 years 14.95 kton/yr
Chemical 5 5
Chemical 5
Chemical
5.27 kton/yr
4.71 kton/yr
29.49 kton/yr
Chemical 1
Chemical 1
44.44kton/yr
5.27 kton/yr
4.71 kton/yr 19.60 kton/yr
41.75 kton/yr
43.77 kton/yr Chemical 6
29.49 kton/yr 10.23 kton/yr
10.23 kton/yr
80.77 Process 3
Process 3
Chemical 33
Chemical
Chemical 3 Process 2 Process 1
Process 1
Process 1 80.77 kton/yr
Chemical 2
29.49 7
Chemical kton/yr
21.88 kton/yr
20.87 kton/yr
19.60
Chemicalkton/yr
Chemical 77 22.73 kton/yr
22.73 kton/yr Process 5
22.73 kton/yr
22.73 ton/yr Chemical 8
21.88 kton/yr
20.87 kton/yr Process 4
22.73 kton/yr Chemical 44
Chemical
21.88 kton/yr
20.87 kton/yr 12 Example – Solution with Min DRisk(Ω =900)
Period 1
3
2
3.5
2.5 years
2 years
2.39 kton/yr
Chemical 5 5
Chemical Chemical 1
Chemical 1
7.54 kton/yr
4.99 kton/yr Process 1
Process 1
Chemical 1
10.85 kton/yr
10.85 kton/yr
5.59 kton/yr Chemical 5
4.99 kton/yr
5.15 kton/yr
5.59
Process 2kton/yr
Process 1 Chemical 3
41.70 kton/yr
Chemical 3
43.54 kton/yr Process 3
22.37 kton/yr
22.37 kton/yr
Chemical 3
Process 4
19.30 kton/yr
Process 4
22.37 kton/yr
22.37 kton/yr 5.15 kton/yr
10.85 kton/yr Chemical 2
10.85 kton/yr
5.15 kton/yr
20.85 kton/yr Process 3 Chemical 6 Chemical 7
21.77 kton/yr
Chemical 7
Process 3
22.37 kton/yr Chemical 7
Process 5 kton/yr
19.30 Chemical 8 Process 5
22.43 kton/yr 20.85 kton/yr
Chemical 8
21.77 kton/yr 22.77 ton/yr Chemical 4
20.85 kton/yr
Chemical 4
21.77 kton/yr 13 Example – Solution with Max ENPV
Risk
1
.0 PP solution 0.9
0.8
0.7
0.6
0.5
0.4 E[NPV ] = 1140 M$ 0.3
0.2
0.1
0.0
250 500 750 1
000 1250 1
500 1
750 2000 2250 2500 2750 3000 3250 NPV (M$) 14 ...
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This note was uploaded on 09/05/2011 for the course CHE 5480 taught by Professor Staff during the Spring '11 term at OKCU.
 Spring '11
 Staff
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