Ten Keys to Success in Optimization Modeling

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Ten Keys to Success in Optimization Modeling

• Richard E. Rosenthal
• Operations Research Department
• Naval Postgraduate School

INFORMS Atlanta, October 2003
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Theme

• Optimization is valuable and pervasive (no need
  to preach to the converted here)
• Practical optimization applications continue to
  be
• bigger
• more complex
• closer to real-time (if the situation warrants)
• less dependent on OR gurus
• more depended upon by companies, and
• taken for granted or
• taken over and claimed credit for by non-ORs

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Theme (continued)

• As we all know, this has been made possible by
  remarkable improvements in computers, modeling
  systems, and solvers (algorithms and their
  implementations). We have many great researchers
  and commercial implementers to thank.
• But, there is also another, important, sometimes
  overlooked piece of the story
• GOOD MODELING PRACTICE

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Acknowledgements

• My ideas on good modeling practice have been
  greatly influenced by my NPS colleagues Jerry
  Brown, Matt Carlyle, Rob Dell, Kevin Wood, and
  other great modelers I have observed in the
  practice of their art, such as Harlan Crowder,
  Terry Harrison, Karla Hoffman, David Ryan, Linus
  Schrage, Julie Ward, Andres Weintraub, Kirk Yost.
• Many great ideas have come from NPS students.
• Todays talk owes a special debt to Jerry Brown.

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Can You Teach Modeling?
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Key 1 Communicate Early and Often

• Mathematical formulation kept up to date
• Verbal description of formulation
• Executive summary in the right language

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Mathematical Formulation

• Index use
• Given data (and units)
• in lower case
• Decision Variables (and units)
• in UPPER CASE
• Objectives and constraints

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Verbal Description of Formulation

• Constraints 3 ensure that one service facility
  is assigned responsibility for each product line
  p.
• You wonder why I mention this, but look at our
  applied literature.

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Non-mathematical Executive Summary

• Jerry Browns Five Essential Steps
• What is the problem?
• Why is the problem important?
• How will the problem be solved without you?
• How will you solve the problem?
• How will the problem be solved with your results,
  but without you?

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Refining Your Executive Summary

• Have a non-OR read your summary out loud
• Ask the reader to explain what is going on
• Listen well
• Revise and repeat
• If you dont learn how to speak in the
  executives language, then someone less qualified
  than you will be entrusted with solving their
  problems.

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Non-mathematical Executive Summary

• Thats it? Thats peer review?

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Key 2 Bound all Decisions

• A trivial concept, too-often ignored
• Remember all the formal neighborhood
  assumptions underlying your optimization method?
  
• Bob Bixby tells of real customer MIP with only 51
  variables and 40 constraints that could not be
  solved until bounds were added, and then it
  solved in a flash.

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Bound all Decisions

• Optimization is an excellent way to find data
  errors, but it really exploits them
• Moderation is a virtue
• Bonus! You never have to deal with the
  embarrassment (or the theory) of
• unbounded models.

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Bound all Decisions
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Key 3 Expect any Constraint to Become an
Objective, and Vice Versa

• Real-world models are notorious for multiple,
  conflicting objectives
• Expert guidance from senior leaders is often
  interpreted as constraints
• These constraints are often infeasible
• Discovering what can be done changes your concept
  of what should be done
• Contrary to impression of textbooks, alternate
  optima are the rule, not the exception.

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Key 4 Sensitivity Analysis in the Real World Is
Nothing Like Textbook SA

• LP Sensitivity Analysis, Textbook Style
• Disappointing in practice because theory creates
  limits.
•  
• Textbooks have sleek algorithms for one
  modification at a time, all else held constant,
  e.g.,
• minimize Sj cj Xj dXk
• Not very exciting in practice. So why is this
  stuff in all the textbooks? What is worth
  talking about?

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LP Sensitivity, Practitioner Style
Operations Research, Jul-Aug 2002
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Sensitivity Analysis, Practitioner Style
Large-scale LP for optimizing airlift -- multiple
time-space muticommodity networks, linked
together with non-network constraints .
Initial results on realistic scenario only 65
of required cargo can be delivered on time.
Analysis of result revealed most of the
undelivered cargo was destined for City A from
City B, so.. what if we redirect some of this
cargo to City A? Sensitivity analysis add
12,000 new rows and 10,000 columns... on-time
delivery improves to 85.
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Forget Textbook Sensitivity Analysis, Plan on
Lots of Model Excursions

• The beauty of this is that it is only of
  theoretical importance, and there is no way it
  can be of any practical use whatsoever!

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Key 5 Bound the Dual Variables

• Huh?
• Elastic constraints,
• with a linear (or piece-wise linear) penalty per
  unit of violation,
• bound the dual variables
• Im willing to satisfy this restriction
  (constraint),
• as long as it doesnt get too expensive.
• Otherwise, forget it
• Ill deal with the consequences

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Key 6 Model Robustly

• Your analysis should consider alternate future
  scenarios, and render a single robust solution.
• There may be many contingency plans,
• but you only get one chance per year
• to ask for the money to get ready.

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Model Robustly

• This is the part I always hate.

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Key 7 Eliminate Lots of Variables

• Big models get to be big through Cartesian
  products of indices
• Find rules for eliminating lots of index tuples
  before they are generated in the model
• Sources of rules mathematical reasoning and
  common sense based on understanding of the
  problem
• You can often eliminate constraints too!

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Example 1 of Variable Elimination

• XD(a,i,r,t) of type a aircraft direct
  delivering cargo for customer i on route r
  departing at time t
• Allow variable to exist only if
• Route r is a direct delivery route from
  customer i s origin to i s destination
• Aircraft type a is available at i s origin
  at t
• Aircraft type a can fly route r s critical
  leg
• Aircraft type a can carry some cargo type that
  customer i demands
• Time t is not before i s available-to-load
  time
• Time t is not after i s required delivery
  date  maxlate  travel time

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Example 2 of Variable Elimination

• Ann Bixby and Brian Downs of Aspen Technology
  developed real-time Capable-to-Promise model for
  large meatpacking company
• One of their major efforts to bring solution
  times down low enough was variable elimination.

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Eliminate Lots of Variables

• You wont win the Nobel or Lanchester Prize for
  this key idea, but it really, really helps.

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Eliminate Lots of Variables

• In effect, what youre doing is taking a big
  lead off third.

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Key 8 Incremental Implementation

• In a complex model, add features incrementally.
  Test each new feature on small instances and
  take no prisoners.
• When new features dont work, there is either a
  bug to be fixed or a new insight to be gained.
  Either way, treasure the learning experiences.

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Incremental Implementation

• Eliminate variables corresponding to airlifters
  switching from long-haul to shuttle status, if
  there are no foreseeable shuttle opportunities.

Feature tested with small example removing the
option to make a seemingly foolish decision
actually caused degradation of objective
function. What happened?
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Key 9 Persistence

• Any prescriptive model that suggests a plan and
  then, when used again, ignores its own prior
  advice
• is bound to advise something needlessly
  different, and lose the faith of its
  beneficiaries.
• Jerry Brown

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Illustration of Persistence

• There are initially 8 customers to serve. We
  must choose serving site and equipment type.

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Illustration of Persistence

• Just a moment after this solution is announced,
  two high-priority customers call in. The model
  is rerun with the 10 customers.
• There are not enough assets to cover all 10
  customers.
• The new solution requires a major reallocation
  of assets. Major changes in the solution are
  highlighted.

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Illustration of Persistence
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Illustration of Persistence

• A persistent version of the model is run to
  obtain a new optimal solution that discourages
  major changes from the original announced
  solution.
• Add to objective function penalties on
  deviations from original solution, weighted by
  severity of disruption.

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Illustration of Persistence
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Illustration of Persistence

• At a cost of 1 in the objective function, the
  persistent solution causes no disruption to the
  announced plans, other than substitution of the
  two new customers.

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Persistence

• Key reference
• G. Brown, R. Dell, K. Wood, Optimization and
  Persistence, Interfaces 1997.

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Key 10 Common Sense

• Heuristics are easy --- so easy we are tempted to
  use them in lieu of more formal methods
• Heuristics may offer a first choice to assess a
  common sense solution
• But, heuristics should not be your only choice

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Common Sense

• A formal optimization model takes longer to
  develop, and solve
• But it provides a qualitative bound on each
  heuristic solution
• Without this bound, our heuristic advice is of
  completely unknown quality
• This quality guarantee is key

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Common Sense

• Its OK to use a heuristic,
• but you should pair it with a traditional,
  calibrating mathematical model
• With no quality assessment,
• you are betting your reputation
• that nobody else is luckier than you are

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Common Sense

• I tend to agree with you,
• especially since thats my lucky number.

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10 Keys to Success in Optimization Modeling

• 1 Write formulation, communicate with execs
• 2 Bound decisions
• 3 Objectives and constraints exchange roles
  (alt. optima likely)
• 4 Forget about sensitivity analysis as you
  learned it
• 5 Elasticize (bound duals)

• 6 Model robustly
• 7 Eliminate variables avoid generating them
  when you can
• 8 Incremental implementation
• 9 Model persistence
• 10 Bound heuristics with optimization

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10 Keys to Success in Optimization Modeling

• Thats it? Thats the Grand Unified Theory?

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Questions and Comments?
Stephen Hansen, Man on a Limb