1 Capacity Planning under Uncertainty 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 CHE: 5480 Economic Decision Making in the Process Industry Prof. Miguel Bagajewicz University of Oklahoma School of Chemical Engineering and Material Science

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

3 First-Stage Decisions First-Stage Decisions Taken before the uncertainty is revealed. They usually correspond to structural Taken before the uncertainty is revealed. They usually correspond to structural decisions (not operational). decisions (not operational). Also called “Here and Now” decisions. Also called “Here and Now” decisions. Represented by “Design” Variables. Represented by “Design” Variables. Examples: Examples: Characteristics of Two-Stage Stochastic Optimization Models Characteristics of Two-Stage Stochastic Optimization Models −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 the condenser and the reboiler of a column, etc

4 Second-Stage Decisions Second-Stage Decisions Taken in order to adapt the plan or design to the uncertain parameters Taken in order to adapt the plan or design to the uncertain parameters realization. realization. Also called “Recourse” decisions. Also called “Recourse” decisions. Represented by “Control” Variables. Represented by “Control” Variables. Example: the operating level; the production slate of a plant. Example: the operating level; the production slate of a plant. Sometimes first stage decisions can be treated as second stage decisions. Sometimes first stage decisions can be treated as second stage decisions. In such case the problem is called a multiple stage problem. In such case the problem is called a multiple stage problem. Characteristics of Two-Stage Stochastic Optimization Models Characteristics of Two-Stage Stochastic Optimization Models

5 Two-Stage Stochastic Formulation LINEAR MODEL SP s.t. First-Stage Constraints Second-Stage Constraints RecourseFunctionFirst-StageCost First stage variables Second Stage Variables Technology matrix 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. Let us leave it linear because as is it is complex enough.!!!

6 Process Planning Under Uncertainty OBJECTIVES: Maximize Expected Net Present Value Maximize Expected Net Present Value Minimize Financial Risk Minimize Financial Risk Production Levels Production Levels DETERMINE: Network Expansions Network Expansions TimingSizingLocation GIVEN: Process Network Process Network Set of Processes Set of Chemicals A A 1 1 C C 2 2 D D 3 3 B B Forecasted Data Forecasted Data Demands & Availabilities Costs & Prices Capital Budget

7 Process Planning Under Uncertainty Design Variables: to be decided before the uncertainty reveals  x  E it Y,, Q 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 S: Sales of product j in market l at time t and scenario s 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 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  ysys  P jlts S,, W its

8 MODELMODEL LIMITS ON EXPANSION I : Processes i,=1,…,NP J : Raw mat./Products, j=1,…,NC T: Time periods. T=1,…,NT L: Markets, l=1,..NM NEXP t : maximum number of expansions in period t α it : Variable cost of expansion for process i in period t β it : Fixed cost of expansion for process i in period t Lower and upper bounds on a process expansion in period t TOTAL CAPACITY IN EACH PERIOD LIMIT ON THE NUMBER OF EXPANSIONS Y it : An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0) E it : Expansion of capacity of process i in period t. Q it : Capacity of process i in period t. LIMIT ON THE CAPITAL INVESTMENT

9 MODELMODEL UTILIZED CAPACITY IS LOWER THAN TOTAL CAPACITY I : Processes i,=1,…,NP J : Raw mat./Products, j=1,…,NC T: Time periods. T=1,…,NT L: Markets, l=1,..NM MATERIAL BALANCE BOUNDS Y it : An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0) E it : Expansion of capacity of process i in period t. Q it : Capacity of process i in period t. W it : Utilized capacity of process i in period t. P jlt : Amount of raw material/intermediate product j consumed from market l in period t S jlt : Amount of intermediate product/product j sold in market l in period t NONNEGATIVITY INTEGER VARIABLES 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

10 MODELMODEL OBJECTIVE FUNCTION I : Processes i,=1,…,NP J : Raw mat./Products, j=1,…,NC T: Time periods. T=1,…,NT L: Markets, l=1,..NM Y it : An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0) E it : Expansion of capacity of process i in period t. W it : Utilized capacity of process i in period t. P jlt : Amount of raw material/interm. product j consumed from market l in period t S jlt : Amount of intermediate product/product j sold in market l in period t γ jlt : Sale price of product/intermediate product j in market l in period t Γ jlt : Cost of product/intermediate product j in market l in period t δ it : Operating cost of process i in period t α it : Variable cost of expansion for process i in period t β it : Fixed cost of expansion for process i in period t L t : Discount factor for period t DISCOUNTED REVENUES INVESTMENT

11 ExampleExample Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price Total of 400 Scenarios Total of 400 Scenarios Project Staged in 3 Time Periods of 2, 2.5, 3.5 years Project Staged in 3 Time Periods of 2, 2.5, 3.5 years Process 1 Chemical 1 Process 2 Chemical 5 Chemical 2 Chemical 6 Process 3 Chemical 3 Process 5 Chemical 7 Chemical 8 Process 4 Chemical 4

12 Period 1 2 years Period years Period years Process 1 Chemical 1 Chemical 5 Process 3 Chemical 3 Chemical kton/yr kton/yr 5.27 kton/yr kton/yr Process 1 Chemical 1 Chemical 5 Process 3 Chemical 3 Process 5 Chemical 7 Chemical 8 Process 4 Chemical kton/yr kton/yr 4.71 kton/yr kton/yr kton/yr Chemical 1 Process 2 Chemical 5 Chemical 2 Chemical 6 Process 3 Chemical 3 Process 5 Chemical 7 Chemical 8 Process 4 Chemical kton/yr ton/yr kton/yr kton/yr kton/yr kton/yr kton/yr kton/yr kton/yr Process 1 Example – Solution with Max ENPV

13 Period 1 2 years Period years Period years Example – Solution with Min DRisk(  =900) Process 1 Chemical 1 Chemical 5 Process 3 Chemical 3 Chemical kton/yr kton/yr 5.59 kton/yr kton/yr Process 1 Chemical 1 Chemical 5 Process 3 Chemical 3 Process 5 Chemical 7 Chemical 8 Process 4 Chemical kton/yr kton/yr kton/yr 4.99 kton/yr kton/yr kton/yr Process 1 Chemical 1 Process 2 Chemical 5 Chemical 2 Chemical 6 Process 3 Chemical 3 Process 5 Chemical 7 Chemical 8 Process 4 Chemical kton/yr ton/yr kton/yr 7.54 kton/yr 2.39 kton/yr 5.15 kton/yr kton/yr 5.15 kton/yr kton/yr

14 Example – Solution with Max ENPV