Transcript Floudas_poster.ppt

Integrated Framework for Operational Planning and Scheduling under Uncertainty
CMMI 0856021
Peter M. Verderame and Christodoulos A. Floudas
Department of Chemical and Biological Engineering, Princeton University, Princeton NJ 08544 USA
Motivation - 2
Motivation - 1
The integration of planning and scheduling under uncertainty has proven
to be a formidable task due to their disparate time scales
Operational Planning: 1-3 months
Medium-Term Scheduling: 2-4 weeks
Short-Term Scheduling: 1-7 days
System
Desired Framework Characteristics:
Multiproduct and multipurpose batch plant having the capability of
producing hundreds of different states
Planning
Portfolio of Products
87 final products are considered
Several different types of process operations
Aggregate Production Profile
Uncertainty associated with processing times
13 bottleneck units (i.e., reactors)
Desired subset of products
Provides tight upper bound on production capacity of plant
Industrial case studies have indicated that the effective integration of
planning and scheduling under uncertainty can lead to an increase in
profit levels
Minimize Inventory
Maximize Customer Satisfaction Levels
Daily Production Profile
Three-month time horizon
Uncertainty
Plant produces both made-to-order (bulk) and made-to-store (packed)
goods
Outlines how much of each product should be produced on a daily basis
Takes into account the uncertainty associated with customer demand
Determine feasible daily production profile
Provide aggregate production bounds for planning level
Ascertain feasible allocation of plant resources (i.e., the run times of process
units over the course of each day within the time horizon)
Takes into account the uncertainty associated with processing times
Integration
Novel Operational Planning Model
Validated Medium-Term Scheduling Model
Effective Rolling Horizon Framework
Allows the scheduling level’s results to refine the planning level’s production
targets
Binary variables (0-1) – allocation and activation
Continuous variables – process and production
Mixed-Integer Linear
Programming (MILP)
Model
Linear constraints and objective function
Model Characteristics
Unit aggregation - express production capacity through bottleneck units
Discrete-time model - capture continuous-time nature of plant through
processing time carryover
Encourage high reactor utilization
Demand enforcement (Underproduction and Overproduction)
Determines daily production profile (Requisite Profile)
Objective Function:
Inventory penalization
Underproduction penalization
Unit underutilization penalization
Gross profit maximization
Demand Due Date Parameter
Normal Distribution
Mean is nominal demand value
Standard deviation increases
proportionately with time horizon
d-1
(33%)
d
d+1
(67%) (100%)
Value-at-Risk (VaR)
Maximum loss that is expected to be exceeded with a probability of (1-ω)
Provides an ‘optimistic’ level of loss
Difficult to implement within an optimization model when the probability
profile is not normal or lognormal (Krokhmal 2007) - NONCONVEX
Conditional Value-at-Risk (CVaR)
The expected loss given that the loss is greater than the VaR
for a confidence level ω
Provides a more ‘pessimistic’ level of loss
Can be implemented within an optimization model regardless
of the probability distribution (Krokhmal 2007) - CONVEX
Tail of Loss Probability Distribution
APPLY CVaR
CVaR
Scheduling - 2
Start
Augment nominal model with robust constraint(s)
Planning Application
d
Apply Optimization Model
Scenario-based approximation:
Demand uncertainty is represented through the generation of a
finite number of scenarios for each demand due date parameter
Integral replaced by summation
Computational Issues
Model Aggregation:
Bulk and Packed
Weekly and monthly enforcement
Production Profile and Xi Values
Reduce RHS
of Certain
Risk Constraints
Generate Larger Sample of Uncertain Parameter Scenarios
Calculate Avg_LHS and Stdev_LHS for Risk Constraints
Calculate z_trans Values
z_trans=(Orig_RHS-Avg_LHS)/Stdev_LHS
Are All z_trans ≥2?
Replace original demand constraints with CVaR constraints
in operational PPDM
Terminate
Scheduling Level Uncertainty
Nominal Demand Constraints
Underproduction
Production + slack1 ≥ Demand
Overproduction
Production – slack2 ≤ Demand
Probabilistic Demand Constraints
Underproduction
Pr{Production + slack1 < Demand} ≤ (1–Pr{Demand})
Overproduction
Pr{Production – slack2 > Demand} ≤ (1–Pr{Demand})
Amount
Day
Underproduction
Production + slack1 – f(.) ≥ Demand
Overproduction
Production – slack2 + f(.) ≤ Demand
Generate Uncertain Parameter Scenarios
Express demand constraints in the form of loss functions and then
encapsulate the loss functions in CVaR constraints
Operational Planning
under Uncertainty
Robust Optimization - 2
Scheduling - 1
SAA
General Approach
Integration
Daily Production
Profile
Robust Demand Constraints
Random variables encompass demand amount uncertainty
Sample Average Approximation (SAA) Wang and Ahmed (2007)
Probability
(1- ω)
Transform into deterministic robust
counterpart constraint(s)
Reliability level captures due date uncertainty
Normal Distribution
Mean is nominal demand value
Standard deviation increases
proportionately with time horizon
Maximum
Loss
VaR
Identify uncertain parameter(s)
Demand Due Date Parameter
CVaR - 2
CVaR - 1
Specify:
distribution
relative uncertainty level
reliability level
Feasible for nominal conditions and robust
Packed Products
Operational
PPDM
Problem Statement: Determine the daily production profile and allocation
of resources for a chemical plant over the three-month time horizon
with the overall objectives being the satisfaction of projected
demand and maximization of gross profit while concurrently taking
into account the various forms of uncertainty in an explicit manner
General Methodology
Uniform discrete distribution
Bounds of +/- one day from
the nominal
Amount
Intermediate Demand
Due Dates
Uncertainty associated with amount
Formulate probabilistic constraint(s)
Day
Weekly Demand
Totals
Made-to-store goods have weekly demand requirements
Bulk Products
Amount
Bulk Products
(Made-to-Order)
Uncertainty associated with amount and day realization
Robust Optimization - 1
Planning Uncertainty
PPDM - 2
Packed Products
(Made-to-Store)
Made-to-order goods have intermediate demand due dates
Scheduling
In response to these incentives and the lack of effective integration
approaches, a novel framework for the integration of planning and
scheduling under uncertainty for a multipurpose and multiproduct
batch plant will be presented
Planning with Production
Disaggreation Model – (PPDM - 1)
Iteratively refine planning model
Results - Robust 1 Iteration
A 3-level scheme which decomposes the scheduling horizon
into sub-horizons which are then solved within a rolling
horizon framework Janak (2006a,b)
Two-Level Decomposition
Level 1
MILP model which determines duration of a sub-horizon and the
set of products to be considered within a sub-horizon
Level 2
MILP model which determines if additional products should be
considered in order to ensure that the plant’s reactors are
highly utilized
Scheduling
Continuous-time unit-specific event-based short-term MILP
scheduling model (Floudas and co-workers)
Determines allocation of plant resources for the sub-horizon
Determines feasible production profile for the sub-horizon
Rolling Horizon Framework
Schedule each sub-horizon in successive fashion
Results - Robust Rolling
First Iteration
Planning
Data Input
Processing time is modeled as a
normal random variable
Campaign Mode
Production
Relax task duration constraints
Decomposition Model
Scheduling Model
Solution Satisfactory?
Yes
Results Output
No
Period Finished
Yes
No
Reformulate probabilistic task
duration constraints into their
robust counterparts
Add robust task duration constraints
to nominal medium-term
scheduling model
End
Results - CVaR 1 Iteration
Production
Targets
Second Iteration
Production
Bounds
Daily Production Profile
for Planning Period
Scheduling
Planning
Production
Targets
Third Iteration
Fixed
Medium-Term Scheduling
under Uncertainty
Scheduling
Production
Targets
Fourth Iteration
Terminate if entire
horizon is scheduled
Planning
Fixed
Fixed
Fixed
Fixed
Scheduling
Terminate
Results - CVaR Rolling
Fixed
Results - Synopsis
Conclusions
Developed two different modeling approaches in order to deal
with demand uncertainty at the planning level
Robust-PPDM
CVaR-PPDM
Applied robust optimization techniques in order to take into
account processing time uncertainty at the scheduling level
Transformed medium-term scheduling model into its robust
counterpart
Utilized rolling horizon framework with a production feedback
loop in order to integrate the planning and scheduling levels
Integrated-CVaR and Integrated-Robust results indicate that the
proposed framework effectively integrates planning and
scheduling levels while explicitly taking into account
uncertainty at the planning and scheduling levels