Robust Counterpart Optimization

1
Robust Scheduling Optimization Zukui Li,
Marianthi Ierapetritou Department of Chemical
and Biochemical Engineering, Rutgers University

• Scenario-based method
• Model size increase exponentially with the
  number of parameters
• Need statistic distribution of the uncertainty
• Robust counterpart optimization
• Dont need accurate statistic distribution
  information
• Small model size

- Generalize the scheduling model to involve
uncertainty only in the coefficient of the left
hand side (LHS) of the constraints

• Reactive Scheduling
• Rescheduling
• Online scheduling
• Dynamic scheduling

• Disruptive Events
• Rush Order Arrivals
• Order Cancellations
• Machine Breakdowns

Not much information is available
Objective function
Price uncertainty
Demand constraints
Demand uncertainty
- largest feasible region - flexible feasible
region - smallest feasible region

• Preventive Scheduling
• Stochastic scheduling
• Robust scheduling
• Fuzzy scheduling

• Parameter Uncertainty
• Processing times
• Demand of products
• Prices

Duration constraints
Information is available
Processing time uncertainty
nominal
Capacity, Balance Allocation,
flexible
worst-case
- Nominal value - True value - Uncertain
coefficient index set
- Robust scheduling aims to obtain preventive
schedules that minimize the effects of
disruptions on the performance measure, and tries
to ensure that the preventive schedules maintain
a high level of performance
LHS coefficient uncertainty
worst-case feasible region
nominal feasible region
Robust Counterpart Optimization
Variability level
- Ensure that only a given number (budget
parameter) of uncertain parameters can reach
their worst case value
- Ensure the worst-case feasibility
- Ensure the probability of constraint violation
does not exceed a certain level
- Processing time 15 - Price 5 - Demand 50
(LP)
(MILP)
Dual
- Vibration amplitude

• Budget parameter model is the most appropriate
  robust counterpart optimization model for
  uncertain scheduling problems since it has the
  advantages that
• It does not increase substantially the problem
  size
• It maintains the linearity of the model
• It can be used to control the degree of
  conservatism for every constraint.

• Nonlinear model
• With flexibility
• Small number of variables and constraints
• Assume symmetric distribution

• Linear model
• No flexibility, most pessimistic
• Simple model

• Linear model
• Higher flexibility
• Relative larger number of variables and
  constraints

Ben-Tal and Nemirovski (2000). Mathematical
Programming. 88, 411-424. Lin, Janak et al.
(2004). Computers and Chemical Engineering. 28,
1069-1085.
Soyster, A. L.(1973). Operations Research. 21,
1154-1157.
Bertsimas and Sim (2003). Mathematical
Programming. 98, 49-71.
FOCAPO-2008, Boston, Massachusetts, June 29 -
July 2, 2008