Two Stage Modeling and Risk-I

# Two Stage Modeling and Risk-I - DESIGN AND PLANNING UNDER...

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Unformatted text preview: DESIGN AND PLANNING UNDER UNCERTAINTY Prof. Miguel Bagajewicz CHE 4273 1 Two-Stage Two-Stage Stochastic Optimization Models 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. Profit Net • 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 Two-Stage Programming, uncertainty is modeled through a finite number of independent Scenarios. Scenarios • Scenarios are typically formed by random samples taken from the probability random distributions of the uncertain parameters. 2 Characteristics of Two-Stage Two-Stage Stochastic Optimization Models First-Stage Decisions • Taken before the uncertainty is revealed. They usually correspond to structural 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 of Two-Stage Characteristics Two-Stage Stochastic Optimization Models Second-Stage Decisions • Taken in order to adapt the plan or design to the uncertain parameters realization. • Also 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: 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: Vinyl Chloride Plant Example: Consider the following forecasts: 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 lb-mol/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 two cases. These are 3 different first stage decisions. 6 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 raw materials and product prices using sampling from a Net Present Worth normal distribution. This can be done in Excel Risk & Probability 7 Example: Vinyl Chloride Plant Probability vs. Net Present Worth Histograms and Risk Curves are 0.35 0.3 Probability 0.25 Notice the asymmetry in the distributions. 0.2 0.15 0.1 0.05 0 -4500 Risk at Different Capacity -1500 0 6 1500 NPW (\$10 ) 6.44E9 lb/yr 1 Cummulative Probability -3000 4.09E9 lb/yr 3000 1.05E10 lb/yr 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 2.00E+09 4.00E+09 Net Present Worth (\$) 6.44E9 lb/yr 4.09E9 lb/yr 6.00E+09 41% chance for the 4.09 billion lbs/year capacity. Expected Profits are: 24%, 25% and 20%. 1.05E10 lb/yr 8 Capacity Planning GIVEN: Process Network Forecasted Data DETERMINE: Set of Processes Set of Chemicals B A Demands & Availabilities Costs & Prices Capital Budget Network Expansions 2 C 3 D 1 Timing Sizing Location Production Levels OBJECTIVES: Maximize Net Present Value 9 Capacity Investment Planning Design Variables: to be decided before the uncertainty reveals to x= {Yit , Eit , Qit } Y: Decision of building process i in period t E: Capacity expansion of process i in period t Q: Total capacity of process i in period t Control Variables: selected after uncertain parameters become known. Assume them known for the time being!!!! ys = { S jlt , P jlt , Wit } 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 and W: Operating level of of process i in period t and scenario s 10 Example Project Staged in 3 Time Periods of 2, 2.5, 3.5 years 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 One feasible (not necessarily optimal) solution could be Chemical 5 Period 1 2 years Process 1 Chemical 1 Process 2 Chemical 6 Chemical 2 Period 2 2.5 years Chemical 5 Chemical 1 Process 1 Process 2 Chemical 6 Chemical 2 Chemical 7 Process 3 Chemical 3 Period 3 3.5 years Process 5 Chemical 8 Chemical 4 Same flowsheet different production rates 12 MODEL SETS VARIABLES I : Processes i,=1,…,NP J : Raw materials and Products, j=1,…,NC T: Time periods. T=1,…,NT L: Markets, l=1,..NM 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 ηij : Amount of raw material/intermediate product j used by process i µij : Amount of product/intermediate product j consumed by process i 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 δ : Operating cost of process i in period t PARAMETERS it 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 L U E it , E it :Lower and upper bounds on a process expansion in period t a L , a U : Lower and upper bounds on availability of raw material j in market l in period t jlt jlt d L , d U : Lower and upper bounds on demand of product j in market l in period t jlt jlt CI t : Maximum capital available in period t NEXPt: maximum number of expansions in period t 13 MATHEMATICAL PROGRAMMING MODEL OBJECTIVE FUNCTION Max NPV = NT ∑ t =1 NP NP NT NM NC Lt ∑∑ (γ jlt S jlt − Γjlt Pjlt ) − ∑ δitWit − ∑∑ (α it Eit + βitYit ) l =1 j =1 i =1 i =1 t =1 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 14 MODEL LIMITS ON EXPANSION L L Yit Eit ≤ Eit ≤ Yit Eit i =1,K, NP t =1,K, NT TOTAL CAPACITY IN EACH PERIOD Qit = Qi ( t −1) + E it i =1,K, NP t =1,K, 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,K, NP t =1,K, 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 L U E it , E it : Lower and upper bounds on a process expansion in period t 15 MODEL UTILIZED CAPACITY IS LOWER THAN TOTAL CAPACITY Wit ≤ Qit NM l =1 a L ≤ Pjlt ≤ a U jlt jlt BOUNDS NONNEGATIVITY NP ∑P MATERIAL BALANCE i =1,K, NP t =1,K, NT jlt NM NP i =1 l =1 i =1 + ∑ ηijWit = ∑ S jlt + ∑ µ ijWit d L ≤ S jlt ≤ d U jlt jlt Eit , Qit , Wit , Pjlt , S jlt ≥ 0 ∀i, j , l , t i =1,K, NP t =1,K, NT j = 1,..., NC , t = 1,..., NT , l = 1,..., NM INTEGER VARIABLES Yit ∈{0,1} i =1,K, NP t =1,K, 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. 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 a L , a U : Lower and upper bounds on availability of raw material j in market l in period t jlt jlt d L , d U : Lower and upper bounds on demand of product j in market l in period t jlt jlt 16 MODEL NM ∑P MATERIAL BALANCE l =1 k NP jlt NM NP i =1 l =1 i =1 i =1,K, NP t =1,K, NT + ∑ η ijWit ≤ ∑ S jlt + ∑ µ ijWit C A i D p B NM ∑P l =1 Blt NM + ηkAWkt = ∑ S Clt + µ iDWit l =1 Reference Component is C ηkA ,µ iD “Stoichiometric” Coefficients 17 Two-Stage Stochastic Formulation Two-Stage Let us leave it linear because as is it is complex enough.!!! LINEAR MODEL SP T Max ∑ ps qs ys − cT x s Technology matrix s.t. Recourse Function First-Stage Cost Ax = b First-Stage Constraints Ts x +Wys = hs Second-Stage 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 first-stage decisions (x). Relatively complete recourse: the recourse cost (or profit) is feasible for the set of feasible first-stage decisions. This condition means that for every feasible first-stage 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. 18 Capacity Planning Under Uncertainty GIVEN: Process Network Forecasted Data DETERMINE: Set of Processes Set of Chemicals B A Demands & Availabilities Costs & Prices Capital Budget Network Expansions 2 C 3 D 1 Timing Sizing Location Production Levels OBJECTIVES: Maximize Expected Net Present Value Minimize Financial Risk 19 Process Planning Under Uncertainty Design Variables: to be decided before the uncertainty reveals to x= {Yit , Eit , Qit } Y: Decision of building process i in period t E: Capacity expansion of process i in period t Q: Total capacity of process i in period t Control Variables: selected after the uncertain parameters become known 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 and W: Operating level of of process i in period t and scenario s 20 MODEL LIMITS ON EXPANSION L L Yit Eit ≤ Eit ≤ Yit Eit i =1,K, NP t =1,K, NT TOTAL CAPACITY IN EACH PERIOD Qit = Qi ( t −1) + E it i =1,K, NP t =1,K, 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,K, NP t =1,K, 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 L U E it , E it : Lower and upper bounds on a process expansion in period t 21 MODEL UTILIZED CAPACITY IS LOWER THAN TOTAL CAPACITY NM ∑P MATERIAL BALANCE l =1 a L ≤ Pjlts ≤ aU jlts jlts BOUNDS NONNEGATIVITY Wits ≤ Qit NP jlts NM NP i =1 l =1 i =1 + ∑ ηijWits ≤ ∑ S jlts + ∑ µ ijWits d L ≤ S jlts ≤ d U jlts jlts Eit , Qit ,Wits , Pjlts , S jlts ≥ 0 Yit ∈{0,1} i =1,K, NP t =1,K, NT ∀i, j , l , t i =1,K, NP t =1,K, NT , ∀s j = 1,..., NC , t = 1,..., NT , l = 1,..., NM , ∀s 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 a L , aU :Lower and upper bounds on availability of raw material j in market l in period t, scenario s jlts jlts d L , d U : Lower and upper bounds on demand of product j in market l in period t, scenario s jlts jlts 22 MODEL OBJECTIVE FUNCTION NP NT NM NC NP NT Max NPV = ∑ ps ∑ Lt ∑∑ (γ jlts S jlts − Γjlts Pjlts ) − ∑ δ itsWits − ∑∑ (α it Eit + β itYit ) t =1 l =1 j =1 s i =1 i =1 t =1 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 23 Example Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price Total of 400 Scenarios Project Staged in 3 Time Periods of 2, 2.5, 3.5 years 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 24 Example – Solution with Max ENPV Period 1 2 years Chemical 5 5.27 kton/yr Chemical 1 5.27 kton/yr Process 1 10.23 kton/yr Chemical 7 19.60 kton/yr Process 3 Chemical 3 22.73 kton/yr 19.60 kton/yr 25 Example – Solution with Max ENPV Period 2 2.5 years Chemical 5 4.71 kton/yr Chemical 1 4.71 kton/yr Process 1 10.23 kton/yr 20.87 kton/yr Process 3 Chemical 3 41.75 kton/yr Chemical 7 22.73 kton/yr Process 5 22.73 kton/yr Chemical 8 20.87 kton/yr Process 4 22.73 kton/yr Chemical 4 20.87 kton/yr 26 Example – Solution with Max ENPV Period 3 3.5 years 14.95 kton/yr Chemical 5 29.49 kton/yr Chemical 1 44.44 kton/yr Process 2 Process 1 Chemical 6 29.49 kton/yr 80.77 kton/yr 80.77 kton/yr Chemical 2 29.49 kton/yr 21.88 kton/yr Chemical 7 Process 5 Process 3 Chemical 3 43.77 kton/yr Chemical 8 21.88 kton/yr 22.73 kton/yr 22.73 ton/yr Process 4 22.73 kton/yr Chemical 4 21.88 kton/yr 27 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 1 250 1 500 1 750 2000 2250 2500 2750 3000 3250 NPV (M\$) 28 Example – Risk Management Solutions Risk 1 .0 0.9 0.8 Ω Ω increases 0.7 PP 0.6 500 600 0.5 700 800 0.4 900 1 000 0.3 1 00 1 1 200 0.2 1 300 1 400 0.1 1 500 0.0 250 500 750 1 000 1 250 1 500 1 750 2000 2250 2500 2750 3000 3250 NPV (M\$) 29 Example – Risk Management Solutions Risk 1 .0 0.9 Ω = 9 00 Ω = 1 100 ENPV = 908 0.8 PP ENPV =1140 ENPV = 1074 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 250 500 750 1 000 1 250 1 500 1 750 2000 2250 2500 2750 3000 3250 NPV (M\$) 30 Example – Risk Management Solutions NPV PDF f (ξ) 0.0026 0.0024 0.0022 Ω = 9 00 0.0020 0.001 8 0.001 6 0.001 4 0.001 2 0.001 0 0.0008 Ω = 1100 PP 0.0006 0.0004 0.0002 0.0000 0 500 1 000 1 500 2000 2500 3000 NPV ( ξ , M \$ ) 31 Example – Solution with Min DRisk(Ω=900) DRisk( Period 1 2 years - Same Flowsheet - Different Capacities Chemical 5 5.59 kton/yr Chemical 1 5.59 kton/yr Process 1 10.85 kton/yr Chemical 7 19.30 kton/yr Process 3 Chemical 3 22.37 kton/yr 19.30 kton/yr 32 Example – Solution with Min DRisk(Ω=900) DRisk( Period 2 2.5 years - Same Flowsheet - Different Capacities Chemical 5 4.99 kton/yr Chemical 1 4.99 kton/yr Process 1 10.85 kton/yr 20.85 kton/yr Process 3 Chemical 3 Chemical 7 22.37 kton/yr Process 5 22.43 kton/yr Chemical 8 20.85 kton/yr 41.70 kton/yr Process 4 22.37 kton/yr Chemical 4 20.85 kton/yr 33 Example – Solution with Min DRisk(Ω=900) DRisk( Period 3 3.5 years - Same Flowsheet - Different Capacities - No Expansion of Process 1 2.39 kton/yr Chemical 5 5.15 kton/yr Chemical 1 7.54 kton/yr Process 1 Process 2 Chemical 6 5.15 kton/yr 10.85 kton/yr 10.85 kton/yr Chemical 2 5.15 kton/yr 21.77 kton/yr Chemical 7 Process 5 Process 3 Chemical 3 43.54 kton/yr Chemical 8 21.77 kton/yr 22.37 kton/yr 22.77 ton/yr Process 4 22.37 kton/yr Chemical 4 21.77 kton/yr 34 RECENT RESULTS Gas Commercialization in Asia Network of Alternatives Suppliers Australia Indonesia Iran Kazakhstan Malaysia Qatar Russia “Transportation” Methods Pipeline Markets China LNG India CNG Japan GTL S. Korea Ammonia Thailand Methanol United States 35 Gas Commercialization in Asia Some solutions Processing Facilities 1.0 OV @ 95%: 1.42 Time Period T1 T2 T3 T4 T5 T6 OV @ 95%: 1.75 0.9 0.8 0.7 0.6 Mala-GTL Ships: 4 & 2 ENPV:4.570 DR@ 4: 0.157 DR@ 3.5: 0.058 0.5 0.4 0.3 Indo-GTL Ships: 5 & 3 ENPV:4.633 DR@ 4: 0.190 DR@ 3.5: 0.086 0.2 0.1 VaR @ 5%: 1.82 1 2 VaR @ 5%: 1.49 3 4 5 6 7 FCI 3.00 0.00 1.89 0.00 0.00 0.00 Cap Flow Feed 0.00 4.57 4.57 7.49 7.49 7.49 0.00 4.47 4.57 7.32 7.49 7.49 0.0 297.9 304.9 488.2 499.6 499.6 Transportation to: China 8 9 10 Time Period T1 T2 T3 T4 T5 T6 Flow Ships Flow Avrg. Ships 0.00 1.16 0.00 0.42 0.00 0.00 0.00 0.98 0.00 0.35 0.00 0.00 0.00 2.79 3.66 5.58 6.00 6.00 0.00 3.49 4.57 6.97 7.49 7.49 0.00 3.95 3.66 6.00 6.00 6.00 4.0 4.0 6.0 6.0 6.0 Indo (GTL) FCI 3.00 0.00 1.90 0.00 0.00 0.00 Cap Flow Feed 0.00 4.43 4.43 7.18 7.18 7.18 0.00 4.25 4.43 7.09 7.18 7.18 0.0 283.1 295.5 472.6 479.0 479.0 Thai Ships Ships Processing Facilities 0.0 0 Mala (GTL) Transportation to: China Thai Ships Ships Flow Ships Flow Avrg. Ships 0.00 1.12 0.00 0.44 0.00 0.00 0.00 0.76 0.00 0.30 0.00 0.00 0.00 3.88 4.94 7.56 8.00 8.00 0.00 3.48 4.43 6.79 7.18 7.18 0.00 5.00 4.94 8.00 8.00 8.00 5.0 5.0 8.0 8.0 8.0 36 Conclusions Risk is usually assessed after a design/plan has been selected but it cannot be Risk managed during the optimization stage (even when stochastic optimization including uncertainty has been performed). If Risk is to be managed, the decision maker has two simultaneous objectives: If • • Maximize Expected Profit. Minimize Risk Exposure There are some good ways of obtaining good solutions (next lecture). There 37 ...
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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.

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