Lecture Capacity Investment Planning Stochastic Model

8/20/2019 Lecture Capacity Investment Planning Stochastic Model

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Capacity PlanningCapacity Planning

under Uncertaintyunder Uncertainty

CHE: 5480CHE: 5480

Economic Decision Making in the Process IndustryEconomic Decision Making in the Process Industry

Prof. Miguel BagajewiczProf. Miguel Bagajewicz

University of OklahomaUniversity of Oklahoma

School of Chemical Engineering and Material ScienceSchool of Chemical Engineering and Material Science


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Characteristics of Two-StageCharacteristics of Two-StageStochastic Optimization ModelsStochastic Optimization Models

PhilosophyPhilosophy• Maximize theMaximize the Expected Value Expected Value of the objective over all possible realizations of of the objective over all possible realizations of

uncertain parameters.uncertain parameters.

• Typically, the objective isTypically, the objective is Profit Profit oror Net Present Value Net Present Value..

• Sometimes the minimization ofSometimes the minimization of Cost Cost is considered as objective.is considered as objective.

UncertaintyUncertainty• Typically, the uncertain parameters are:Typically, the uncertain parameters are: market demands, availabilities,market demands, availabilities,

prices, process yields, rate of interest, inflation, etc. prices, process yields, rate of interest, inflation, etc.

• n T!o"Sta#e Pro#rammin#, uncertainty is modeled throu#h a finite numbern T!o"Sta#e Pro#rammin#, uncertainty is modeled throu#h a finite number

of independentof independent Scenarios Scenarios..

• Scenarios are typically formed byScenarios are typically formed by random samplesrandom samples ta$en from the probabilityta$en from the probability

distributions of the uncertain parameters.distributions of the uncertain parameters.


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%irst"Sta#e &ecisions%irst"Sta#e &ecisions• Ta$en before the uncertainty is revealed. They usually correspond to structuralTa$en before the uncertainty is revealed. They usually correspond to structural

decisions 'not operational(.decisions 'not operational(.

• )lso called *+ere and o!- decisions.)lso called *+ere and o!- decisions.• epresented by *&esi#n- /ariables.epresented by *&esi#n- /ariables.

• 0xamples:0xamples:


−To build a plant or not. +o! much capacity should be added, etc.To build a plant or not. +o! much capacity should be added, etc.

−To place an order no!.To place an order no!.

−To si#n contracts or buy options.To si#n contracts or buy options.

−To pic$ a reactor volume, to pic$ a certain number of trays and sizeTo pic$ a reactor volume, to pic$ a certain number of trays and size

the condenser and the reboiler of a column, etcthe condenser and the reboiler of a column, etc


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Second"Sta#e &ecisionsSecond"Sta#e &ecisions

• Ta$en in order to adapt the plan or desi#n to the uncertain parametersTa$en in order to adapt the plan or desi#n to the uncertain parameters

realization.realization.

• )lso called *ecourse- decisions.)lso called *ecourse- decisions.

• epresented by *1ontrol- /ariables.epresented by *1ontrol- /ariables.• 0xample: the operatin# level2 the production slate of a plant.0xample: the operatin# level2 the production slate of a plant.

• Sometimes first sta#e decisions can be treated as second sta#e decisions.Sometimes first sta#e decisions can be treated as second sta#e decisions.

n such case the problem is called a multiple sta#e problem.n such case the problem is called a multiple sta#e problem.



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Two-Stage Stochastic Formulation Two-Stage Stochastic Formulation

30)30) M4&03 SPM4&03 SP

xc yq p Max T

s s

T s s −∑ xc yq p Max

T

s s

T s s −∑

b Ax=b Ax=

X x x ∈≥0 X x x ∈≥0

s s s hWy xT =+ s s s hWy xT =+

0≥ s

y 0≥ s y

s.t.

%irst"Sta#e 1onstraints%irst"Sta#e 1onstraints

Second"Sta#e 1onstraintsSecond"Sta#e 1onstraints

ecourseecourse

%unction%unction

%irst"Sta#e%irst"Sta#e

1ost1ost

%irst sta#e variables

Second Sta#e /ariables

Technolo#y matrix

ecourse matrix '%ixed ecourse(

Sometimes not fixed 'nterest rates in Portfolio 4ptimization(

Complete recourse: therecourse cost (or profit) for

every possible uncertainty

realization remains finite,

inepenently of the first!sta"e

ecisions (#)$

Relatively complete recourse:

the recourse cost (or profit) is

feasible for the set of feasible

first!sta"e ecisions$ %his

conition means that for every

feasible first!sta"e ecision,

there is a &ay of aaptin" the

plan to the realization ofuncertain parameters$

'e also have foun that one

can sacrifice efficiency for

certain scenarios to improve

ris mana"ement$ 'e o notno& ho& to call this yet$

3et us leave it linear

because as is it is

complex enou#h.555


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Process Planning Under UncertaintyProcess Planning Under Uncertainty

46701T/0S46701T/0S** Maximize 0xpected et Present /alueMaximize 0xpected et Present /alue

Minimize %inancial is$ Minimize %inancial is$

Production 3evelsProduction 3evels

&0T0M0&0T0M0** et!or$ 0xpansionset!or$ 0xpansionsTiming Timing

Sizing Sizing

Location Location

8/0:8/0: Process et!or$ Process et!or$ Set of ProcessesSet of Processes

Set of ChemicalsSet of Chemicals)) 99

11

&&;;

66

%orecasted &ata%orecasted &ata eman!s " A#ailabilities eman!s " A#ailabilities

Costs " PricesCosts " Prices

Capital $%!get Capital $%!get


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Process Planning Under UncertaintyProcess Planning Under Uncertainty

&esi#n /ariables:&esi#n /ariables: to be ecie before the uncertainty revealsto be ecie before the uncertainty reveals

{ }

x E it Y it , , it

* -ecision of builin" process* -ecision of builin" process ii in perioin perio t t

.* /apacity e#pansion of process.* /apacity e#pansion of process ii in perioin perio t t

* %otal capacity of process* %otal capacity of process ii in perioin perio t t

1ontrol /ariables:1ontrol /ariables: selecte after the uncertain parameters become no&nselecte after the uncertain parameters become no&n

** ales of prouctales of prouct & & in maretin maret l l at timeat time t t an scenarioan scenario s s

** urchase of ra& mat$urchase of ra& mat$ & & in maretin maret l l at timeat time tt an scenarioan scenario s s

'*'* peratin" level of of processperatin" level of of process ii in perioin perio t t an scenarioan scenario s s{ }

ys P !lts S !lts , , " its


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MODELMODEL

3MTS 4 0


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MODELMODEL

UT3>0& 1)P)1T= S

34?0 T+) T4T)31)P)1T=

' ( Processes i)*+),)-P

( /a0 mat.1Pro!%cts) &*+),)-C

T( Time perio!s. T*+),)-T

L( Mar2ets) l*+)..-M

-T t -P i ,,1 ,,1

==

-T t -P i ,,1 ,,1 ==M)T0)3 6)3)10

64U&S

8 it ( An expansion of process ' in perio! t ta2es place :8it*+;) !oes not ta2e place :8it*


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MODELMODEL

46701T/0 %U1T4

' ( Processes i)*+),)-P

( /a0 mat.1Pro!%cts) &*+),)-C

T( Time perio!s. T*+),)-T

L( Mar2ets) l*+)..-M

8 it ( An expansion of process ' in perio! t ta2es place :8it*+;) !oes not ta2e place :8it* &lt ( Sale price of pro!%ct1interme!iate pro!%ct & in mar2et l in perio! t

6 &lt ( Cost of pro!%ct1interme!iate pro!%ct & in mar2et l in perio! t

?it ( @perating cost of process i in perio! t

4it ( 5ariable cost of expansion for process i in perio! t

6 it ( 7ixe! cost of expansion for process i in perio! t Lt ( isco%nt factor for perio! t

'SC@=-T3 /353-=3S '-53STM3-T


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ExampleExample

Uncertain Parameters:Uncertain Parameters: &emands, )vailabilities, Sales Price, Purchase Price&emands, )vailabilities, Sales Price, Purchase Price

Total of @AA ScenariosTotal of @AA Scenarios

Project Sta#ed in ; Time Periods of , .B, ;.B yearsProject Sta#ed in ; Time Periods of , .B, ;.B years

Process 91hemical 9 Process

1hemical B

1hemical

1hemical C

Process ;

1hemical ;

Process B

1hemical D

1hemical E

Process @

1hemical @


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Period 9Period 9

years years

Period Period

.B years.B years

Period ;Period ;

;.B years;.B years

Process 91hemical 9

1hemical B

Process ;

1hemical ;

1hemical D

10$23 ton7yr

22$+3 ton7yr

5$2+ ton7yr

5$2+ ton7yr

1$0 ton7yr

1$0 ton7yr

Process 91hemical 9

1hemical B

Process ;

1hemical ;

Process B1hemical D

1hemical E

Process @

1hemical @

10$23 ton7yr

22$+3 ton7yr

22$+3 ton7yr

22$+3 ton7yr

4$+1 ton7yr

4$+1 ton7yr

41$+5 ton7yr

20$+ ton7yr

20$+ ton7yr

20$+ ton7yr

1hemical 9 Process

1hemical B

1hemical

1hemical C

Process ;

1hemical ;

Process B

1hemical D

1hemical E

Process @

1hemical @

22$+3 ton7yr

22$+3 ton7yr 22$+3 ton7yr

0$++ ton7yr 0$++ ton7yr 44$44 ton7yr

14$5 ton7yr

2$4 ton7yr

2$4 ton7yr

43$++ ton7yr

2$4 ton7yr

21$ ton7yr

21$ ton7yr

21$ ton7yr

Process 9

Example – Solution with Max ENPVExample – Solution with Max ENPV


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Period 9Period 9

years years

Period Period

.B years.B years

Period ;Period ;

;.B years;.B years

Example – Solution with Min DRisk(Example – Solution with Min DRisk( =900)=900)

Process 91hemical 9

1hemical B

Process ;

1hemical ;

1hemical D

10$5 ton7yr

22$3+ ton7yr

5$5 ton7yr

5$5 ton7yr

1$30 ton7yr

1$30 ton7yr

Process 91hemical 9

1hemical B

Process ;

1hemical ;

Process B1hemical D

1hemical E

Process @

1hemical @

10$5 ton7yr

22$3+ ton7yr

22$3+ ton7yr

22$43 ton7yr

4$ ton7yr

4$ ton7yr

41$+0 ton7yr

20$5 ton7yr

20$5 ton7yr

20$5 ton7yr

Process 91hemical 9 Process

1hemical B

1hemical

1hemical C

Process ;

1hemical ;

Process B

1hemical D

1hemical E

Process @

1hemical @

22$3+ ton7yr

22$3+ ton7yr 22$++ ton7yr

10$5 ton7yr 10$5 ton7yr +$54 ton7yr

2$3 ton7yr

5$15 ton7yr

5$15 ton7yr

43$54 ton7yr

5$15 ton7yr

21$++ ton7yr

21$++ ton7yr

21$++ ton7yr


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Example – Solution with Max ENPVExample – Solution with Max ENPV

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250

P/ 'MF(

is$

22 solution

.8 -P5 9 : 1140 ;

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Lecture Capacity Investment Planning Stochastic Model