Rosenthal Atlanta

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Ten Keys to Success in

Optimization ModelingRichard E. Rosenthal

Operations Research Department

Naval Postgraduate School

INFORMS Atlanta, October 2003


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Theme

Optimization is valuable and pervasive (no need topreach to the converted here)

Practical optimization applications continue to be:

biggermore 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 byremarkable improvements in computers,modeling systems, and solvers (algorithms andtheir implementations). We have many greatresearchers and commercial implementers tothank.

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 beengreatly influenced by my NPS colleaguesJerry Brown, Matt Carlyle, Rob Dell, KevinWood, and other great modelers I haveobserved in the practice of their art, such asHarlan Crowder, Terry Harrison, KarlaHoffman, David Ryan, Linus Schrage, JulieWard, Andres Weintraub, Kirk Yost.

Many great ideas have come from NPSstudents.

Todays talk owes a special debt to JerryBrown.


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Can You Teach Modeling?


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Key #1: Communicate Early and Often

Mathematical formulation kept up to dateVerbal description of formulation

Executive summary in the right language

John Stuart Curry,Tragedy and Prelude - J ohn Brown


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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 facilityis assigned responsibility for each product

linep.

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 yourresults, 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 theexecutives language, then someone lessqualified than you will be entrusted withsolving 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 neighborhoodassumptions underlying your optimizationmethod?

Bob Bixby tells of real customer MIP with only51 variables and 40 constraints that could notbe solved until bounds were added, and thenit solved in a flash.


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Bound all Decisions

Optimization is an excellent way to find dataerrors, but it really exploits them

Moderation is a virtue

Bonus! You never have to deal with the

embarrassment (or the theory) ofunbounded models.


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Bound all Decisions


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Key #3: Expect any Constraint toBecome an Objective, and Vice Versa

Real-world models are notorious for multiple,conflicting objectives

Expert guidance from senior leaders is ofteninterpreted as constraints

These constraints are often infeasible

Discovering what can be done changes yourconcept of what should be done

Contrary to impression of textbooks, alternateoptima are the rule, not the exception.


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Key #4: Sensitivity Analysis in the RealWorld Is Nothing Like Textbook SA

LP Sensitivity Analysis, Textbook Style

Disappointing in practice because theory creates

limits.

Textbooks have sleek algorithms for one modificationat a time, all else held constant, e.g.,

minimize SjcjXj+dXkNot very exciting in practice. So why is this stuff inall 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 withnon-network constraints .

Initial results on realistic scenario: only 65% of

required cargo can be delivered on time.Analysis of result revealed most of the undeliveredcargo was destined for City A from City B, so.. what ifwe 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 theoreticalimportance, and there is no way it can be of any practicaluse 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 futurescenarios, 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 Cartesianproducts of indices

Find rules for eliminating lots of index tuplesbefore they are generated in the model

Sources of rules: mathematical reasoning andcommon 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 deliveringcargo for customer i on route r departing at time t

Allow variable to exist only if

Route r is a direct delivery route from customeris origin to is destinationAircraft type a is available at is origin at tAircraft type a can fly route rs critical leg

Aircraft type a can carry some cargo type thatcustomer i demandsTime t is not before is available-to-load timeTime t is not after is required delivery date+ maxlate travel time


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Example 2 of Variable Elimination

Ann Bixby and Brian Downs of AspenTechnology developed real-timeCapable-to-Promise model for large meatpacking company

One of their major efforts to bring solutiontimes down low enough was variableelimination.


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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 offthird.


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Key #8: Incremental Implementation

In a complex model, add featuresincrementally. Test each new feature onsmall instances and take no prisoners.

When new features dont work, there iseither a bug to be fixed or a new insight to

be gained. Either way, treasure the learningexperiences.


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Incremental Implementation

Eliminate variables corresponding to airliftersswitching from long-haul to shuttle status, ifthere are no foreseeable shuttle opportunities.

Feature tested with small example: removing theoption to make a seemingly foolish decision actuallycaused degradation of objective function.

What happened?

Euro

US SWA FOB


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Key #9: Persistence

Any prescriptive model that suggests a planand then, when used again, ignores its ownprior advice

is bound to advise something needlesslydifferent, and lose the faith of itsbeneficiaries.

Jerry Brown


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Illustration of Persistence

Customer Eqpt Type Site Distance

Cust01 Eqpt-01 HHH 353Cust02 Eqpt-01 HHH 724

Cust03 Eqpt-01 HHH 773

Cust04 Eqpt-01 YYY 707

Cust05 Eqpt-01 YYY 719Cust06 Eqpt-03 RRR 495


Cust08 Eqpt-03 RRR 590

Results for Initial Case with 8 Customers

There are initially 8 customers to serve. Wemust choose serving site and equipment type.


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Just a moment after this solution isannounced, two high-priority customers callin. The model is rerun with the 10customers.

There are not enough assets to cover all 10customers.

The new solution requires a majorreallocation of assets. Major changes in thesolution are highlighted.


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Customer Eqpt Type Site Dist (nm)



Cust03 Eqpt-01 YYY 703Cust04



Cust07 Eqpt-01 HHH 442Cust08



esults for 10 Customers without Persistenc

NOT SERVED

NOT SERVED


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A persistent version of the model isrun to obtain a new optimal solution

that discourages major changes fromthe original announced solution.

Add to objective function: penalties ondeviations from original solution,weighted by severity of disruption.


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Customer Eqpt Type Site Dist (nm)









Results for 10 Customers with Persistence

NOT SERVED

NOT SERVED


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At a cost of 1% in the objective function, thepersistent solution causes no disruption to theannounced plans, other than substitution of thetwo new customers.

Orig.

With persistence? - No Yes

Customers 8 10 10

Objective function valueCustomers not served 0 180.00 180.00

Original objective 206.67 151.75 155.20

Total 206.67 331.75 335.20

Comparison of SolutionsSubsequent


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Persistence

Key reference:

G. Brown, R. Dell, K. Wood, Optimization and

Persistence, Interfaces1997.


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Key #10: Common Sense

Heuristics are easy --- so easy we aretempted 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 todevelop, and solve

But it provides a qualitativebound on eachheuristic solution

Without this bound, our heuristic advice is ofcompletely 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 andconstraints exchangeroles (alt. optima likely)

#4 Forget aboutsensitivity analysis as

you learned it

#5 Elasticize (boundduals)

#6 Model robustly#7 Eliminate variables

avoid generating themwhen you can

#8 Incrementalimplementation

#9 Model persistence

#10 Bound heuristics withoptimization


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Thats it? Thats the Grand Unified Theory?

10 Keys to Success in Optimization Modeling


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Questions and Comments?

Stephen Hansen, Man on a Limb

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Rosenthal Atlanta