operations research and optimization: a · pdf fileoperations research and optimization: a...
TRANSCRIPT
Operations Research and Optimization: A Primer
Ron Rardin, PhDNSF Program Director, Operations Research and
Service Enterprise Engineeringalso Professor of Industrial Engineering,
Purdue University
Introduction
• Operations Research (OR) is the study of math modeling tools for complex, usually large-scale engineering and management design/planning/control problems
• Major components include optimization methods, stochastic/probability modeling, and event-oriented simulation
• Purpose here is to present an elementary primer on the optimization part to acquaint those not trained in OR with some fundamental concepts and definitions• How do optimization researchers think about planning
problems?
A Toy Conformal Therapy Example
• To begin, I will ask you to suspend reality and consider a massively over-simplified, toy example based loosely on Conformal Radiotherapy
• No claim of correctness in the application, but it allows us to discuss optimization issues in a familiar context
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Parameters and Decisions
• Parameters of an optimization problem are values taken as given• Here dose limits 60, 80, 100,
and the limit of 2 beams
• Decisions (variables in our models) are what we get to decide/control• Discrete are logical/on-off
type (here which beams on)
• Continuous take on numeric values (here beam intensities)
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Constraints and Feasible Solutions
• Constraints of an optimization problem define the applicable limits on decision choice• Here 2-beam and healthy
tissue total dose limits
• Feasible solutions are those that satisfy all constraints• B1=B2=30 Feasible
• B2=110, B3=20 Infeasible
• B1=B2=B3=10 Infeasible
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Objective Functions and Optimal Solutions
• Objective or criterion function is a numerical measure of preference among decision choices• Here max total tumor dose
• Optimal solution is a feasible solution with best objective value• B1=B2=30, TD=60
Feasible but not Optimal
• B2=60, B3=40, TD=100Optimal
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Some Implications
• Parameters (given constants)
• Decisions (discrete or continuous choices)
• Constraints (limits on decision choice)
• Feasible solutions (satisfy all constraints)
• Objective function(quantifies preference)
• Optimal solution (feasible and best in objective)
• Optimal is a well-defined mathematical concept• Too often used casually
• Every optimal solution has the same objective value• Can be multiple optimal solns
• Infeasible solutions can have better than optimal obj values
• Computing an optimum implies search over the decision choices • Parameters are fixed• Looking for feasible solns with
good objective values
Challenge of Multiple Criteria
• To apply optimization or talk about an optimal solution, must reduce to a single preference measure
• Extremely common to encounter multiobjectiveplanning problems were more than one criterion should be made as big or small as possible• E.g. in our toy problem, max
tumor dose and min purple dose
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Challenge of Multiple Criteria
• Common approach: make all but one constraints• E.g. toy prob with tumor dose• Could have been any single one
• Can refine with sensitivity analysis = multiple runs with different values of the parameters• E.g. try purple <= 55, 60, 65
• Tune in to Eva Lee tomorrow morning for more refined options
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Inverse Methods
• Inverse methods make everything a constraint and minimize the violation• E.g. add min TD restriction
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60• Does give single objective• Challenge: how to weight
violations?• There is usually no solution
that satisfies all reqs• Balancing violation by
weighting may produce critical infeasibilities
TD >= 150
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Models & Tractability
• To apply formal optimization methods, need to represent decisions as variables, and both constraints and the objective as functions of those variables in a mathematical model, e.g.max B1 + B2 + B3 (tumor dose)B1 + B3 <= 80 (green limit). . . .
• Math forms are critical to tractability = convenience for solution
• Search strategies determine what is tractable
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Hillclimbing (Local Search)
• First consider unconstrained search with only an objective• Can draw an image with a surface
representing objective value at different choices of vbls x1 & x2
• Maximizing goal is to find the values that correspond to the top of the highest mountain
• Hillclimbing process:• Survey the nearby neighborhood• Find an up-hill search direction• Follow it while it helps & repeat• Stop when no such direction exists
• E.g. gradient, conjugate gradient x1 value -><- x2 value
objectivevalue
optimalsolution
Local and Global Optimal Solutions
• Local optimum is a feasible solution that is best in the neighborhood of current one
• Global optimum is overall best• Easy to see that hillclimbing
can lead us to a local optimum that is not global• Search’s “vision” does not extend
beyond the immediate neighborhood
• Implication: tractability is enhanced if the objective has no local optima not global
x1 value -><- x2 value
objectivevalue
optimalsolutionlocal
optimum
Hillclimbing with Constraints
• For models with constraints hillclimbing usually tries to stay feasible• Search from one feasible solution to
another with better objective value
• Constraints introduce barriers • If the constraints have
irregular form can easily block the search at a local optimum
• Implication: tractability is enhanced if constraint functions are smooth and regular x1 value ->
<- x2 value
objectivevalue
optimalsolution
feasiblesolutions
Penalty Methods
• Can avoid dealing with constraints by weighting violations in the objective and searching unconstrained• Objective terms = penalty functions
• Frees the search to move• Lots of potential problems
• Can make objective have local optima when it did not originally
• Have to choose the penalty weightsbig enough to make sure any unconstrained optimum is feasible
• Choosing the penalties too high will lose search freedom of movement x1 value ->
<- x2 value
objectivevalue
optimalsolution
feasiblesolutions penalized
region
Discrete Decisions and Enumeration
• Continuous decisions have infinitely many choices• When decisions are discrete, we can think of solution by
enumeration = trying all (or many) of the combinations• E.g B1&B2, then B2&B3, then B1&B3 in our toy example and keep best
• Enumeration quickly becomes impractical with problem size• 2 yes/no decisions gives 4 combinations• 10 yes/no decisions makes 2048 combinations• 100 yes/no decisions would occupy a computer evaluating a
trillion per second for about 402 million centuries• Real methods do careful partial enumeration of choices• Implication: discrete elements in a model decrease
tractability
Math Forms and Tractability
• Linear functions are weighted sums of variables• E.g. 3x1 + 2x2 +1.9x3• Much easier to deal with in both the objective and the
constraints• Nonlinear functions are everything else
• E.g. 3x1*x2 + 1.9x3 + sqrt(x2)• Can lead to local optima and difficult searches
• Discrete decisions are usually modeled by integerdecision variables (restricted to whole numbers)• Leads to more difficult searches and need for some
enumeration
Classes of Optimization Models
NLP
LP
MINLP
MIPco
ntinu
ous
varia
blesdis
crete
(integ
er)va
riable
slinear constraints
nonlinear constraints
linear objective
nonlinear objective •LP = Linear Program (highly tractable)•NLP = Nonlinear Program (some tractable)•MIP = Mixed Integer Program (some tractable)•MINLP = Mixed Integer Nonlinear Program (tough)
Strategies:Relaxations & Bounds
• Relaxations are easier forms of optimization models obtained by weakening some constraints• E.g. let discrete variables
deciding which beams be continuous (allow fractions)
• Now LP gives a solution with all beams part on and TD=120 vs. MIP optimum of TD=100
• Relaxations yield bounds• Easier problem can only have
better answer (120 >= 100)
Beam1
Beam
2
Beam 3Target
dose <= 80
dose <= 100
dose <= 60
•May use at most two of the beams•Beam intensity is controllable•Tissues considered as single points•If a beam intersects a tissue, it adds dose equal to beam intensity•Goal is to maximize tumor dose•Limit dose on healthy tissues
Strategies: Using Relaxation Bounds
• Bounds from relaxations can be used to narrow the search• If the bound for one part of the feasible region is
poorer than a known, fully feasible solution elsewhere, we do not have to search that region (the idea of Branch and Bound)
• Bounds can also help evaluate solutionsobtained by means not assuring global optima• Compare what was obtained to what might be
possible
Strategies: Heuristics
• So far we have dealt mainly with exact optimization• Goal to find a mathematically optimal solution (or very close)
• Heuristic methods seek only a good feasible solution• Many heuristic strategies
• Rounding = solve a relaxation and adjust to a nearby feasible solution (often in the context of an MIP)
• Constructive = make decisions one by one in sequence, each time making the choice that seems best at the moment (rare in radiation therapy planning)
• Improving = mimic local search in moving to neighboring (and better) feasible solutions (examples include Simulated Annealing and Genetic Algorithms)
• Expert judgment or past experience with similar instances
Concerns with Heuristics
• Heuristics are often the only way to get a usable solution to an poorly tractable optimization model
• One issue: how near are solutions to optimal?• Desirable to have a a bound on error (suboptimality) in the
heuristic solution (automatic if a relaxation was solved)• Methods like Simulated Annealing provide no guarantees at all
(may eventually find an optimum but won’t know it has done it, must rely on historical experience)
• Another issue is handling of constraints• Many improving search heuristics (e.g. Simulated Annealing,
Genetic Algs) can really only do unconstrained optimization• Constraints must be weighted with penalty functions which raises
issues of what weights to choose and whether the solutions that result will satisfy all constraints
Stochastic Optimization
• Everything so far is deterministic optimization = parameters know with certainty
• This is an obvious over-simplification because almost everything is estimated and has some uncertainty• Especially where the system changes
through time
• Stochastic optimizationmethods assume probability distributions on parameters to model this uncertainty
param value
prob
Tractability of Stochastic Opt
• Stochastic usually implies much tougher and more limited math
• Often leads to Monte Carlo sampling of possibilities • Can be slow and misled by sampling
error• Another issue: output values
will have prob distributions• Raises issue of how to compare and
choose a best decision choice• Implication: stochastic
modeling reduces tractability
obj value
prob
Themes
• Optimal is a mathematically precise concept = a best feasible solution for a single measure of preference
• Constant balancing of tractability vs. usefulness of results in choice of optimization methods and models• Model must be somewhat tractable to get any results• Too many assumptions lead to useless outcomes
• Users need to be aware of limitations of various methods• Are methods prone to local optima?• How critical are any needed penalty weights?• Do methods at least guarantee a feasible solution?• If a solution is not guaranteed to be optimal, is error bounded?• Can stochastic effects be neglected?