Lecture-Capacity-Investment-Planning-Stochastic-Model

Lecture-Capacity-Investment-Planning-Stochastic-Model -...

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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 Two-Stage 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 First-Stage Decisions First-Stage • 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 Second-Stage Decisions Second-Stage • 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. 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. 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.

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