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Capacity Planning under Uncertainty CHE: 5480 Economic Decision Making in the Process Industry Prof. 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 Two-Stage Stochastic Optimization Models Philosophy • Maximize the Expected Value of the Characteristics of Two-Stage Stochastic Optimization Models Philosophy • Maximize the Expected Value of the objective over all possible realizations of uncertain parameters. • Typically, the objective is Profit or Net Present Value. • Sometimes the minimization of Cost is considered as objective. Uncertainty • Typically, the uncertain parameters are: market demands, availabilities, prices, process yields, rate of interest, inflation, etc. • In Two-Stage Programming, uncertainty is modeled through a finite number of independent Scenarios. • Scenarios are typically formed by random samples taken from the probability distributions of the uncertain parameters. 2

Characteristics of Two-Stage Stochastic Optimization Models First-Stage Decisions • Taken before the uncertainty is Characteristics of 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 Stochastic Optimization Models Second-Stage Decisions • Taken in order to adapt Characteristics of 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

Two-Stage Stochastic Formulation Let us leave it linear because as is it is complex Two-Stage Stochastic Formulation Let us leave it linear because as is it is complex enough. !!! Technology matrix LINEAR MODEL SP s. t. Recourse Function First-Stage Cost First-Stage Constraints Second Stage Variables 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. First stage variables Recourse matrix (Fixed Recourse) Sometimes not fixed (Interest rates in Portfolio Optimization) 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 Forecasted Data DETERMINE: Set of Processes Set Process 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 6

Process Planning Under Uncertainty Design Variables: to be decided before the uncertainty reveals x= Process Planning Under Uncertainty Design Variables: to be decided before the uncertainty reveals 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 W: Operating level of of process i in period t and scenario s 7

MODEL LIMITS ON EXPANSION TOTAL CAPACITY IN EACH PERIOD LIMIT ON THE NUMBER OF MODEL LIMITS ON EXPANSION TOTAL CAPACITY IN EACH PERIOD LIMIT ON THE NUMBER OF EXPANSIONS LIMIT ON THE CAPITAL 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. 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 MATERIAL BALANCE BOUNDS NONNEGATIVITY INTEGER VARIABLES MODEL UTILIZED CAPACITY IS LOWER THAN TOTAL CAPACITY MATERIAL BALANCE BOUNDS NONNEGATIVITY 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 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 Total of 400 Scenarios Project 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 11

Example – Solution with Max ENPV Period 1 3 2 3. 5 2. 5 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. 27 kton/yr 4. 71 kton/yr 29. 49 kton/yr Chemical 1 44. 44 kton/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 80. 77 Process 3 Chemical 33 Chemical 3 Process 2 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 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( =900) Period 1 3 2 3. 5 2. Example – Solution with Min DRisk( =900) Period 1 3 2 3. 5 2. 5 years 2. 39 kton/yr Chemical 5 5 Chemical 1 7. 54 kton/yr 4. 99 kton/yr Process 1 Chemical 1 10. 85 kton/yr 5. 59 kton/yr Chemical 5 4. 99 kton/yr 5. 15 kton/yr 5. 59 Process 2 kton/yr Process 1 Chemical 3 41. 70 kton/yr Chemical 3 43. 54 kton/yr Process 3 22. 37 kton/yr Chemical 3 Process 4 19. 30 kton/yr Process 4 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 14 Example – Solution with Max ENPV 14