Multiobjective Optimization
Carlos A. Santos Silva
Motivation
•Usually, in optimization problems, there is more than one objective:
• Minimize Cost
• Maximize Performance
•The objectives are often conflicting:
• Minimize Cost implies minimizing performance
• Maximize Performance implies maximize cost
Wind
Wind
15
Cost(€)
5
Performance(kW/year)
20
10
25
Solar
What is the best solution?
7,5
Solar
A possible approach is…
•To transform multiple objectives into a single objective
• Maximize Profit (with performance measured in terms of cost)
• Profit is a simple sum of cost and performance
If objectives are equally important....
Wind
(10)
Profit (€)
(10)
(17,5)
Solar
What is the best solution?
Are objectives equally important?
•If objectives are not equally important
• What is the relative importance between them?
Wind
(25)
One has to decide a priori the relative importance of objectives
Is Performance more important than Cost?
Profit (€)
(30)
(42,5)
Solar
Is Performance more important than Cost?
How much? 2 times? 10 times?
2x Cost = Performance
Wind
Wind
(5)
Profit (€)
(0)
(10)
Profit (€)
35
80
Solar
Cost = 2x Performance
50
Solar
Cost = 10xPerformance
What is the best solution?
What if objectives are not comparable?
•Often, objectives are often non- commensurable
• Expressing performance in monetary units might be impossible
• Example:2 star hotel by 50€ or 4 star hotel by 150€
• Is each star valued as 50€? Does a 1 star hotel worth 0€?
• Is it the same pay 100€ by a 3 star hotel, 150€ by a 4 star or 200 by a 5 star
• Even if cost of stars is not linear, is it possible to compare both objectives in
the same unit?
Why not compare solutions?
Another approach is to evaluate solutions for both objectives and let someone
(Decision Maker) choose the best solution
Performance
Best performance
10
Best Cost
5
15
20
Cost
Decision Maker decides is paying extra 5 is worth to have an extra 5 in performance!
MULTIOBJECTIVE OPTIMIZATION
General Description
•Multiobjective optimization
• Choosing the best solution considering different, usually contradictory objectives
• Usually, there is no single best solution, but a set of solutions that are equally good
•Methodology
• A posteriori (Decision Maker defines preferences based on optimization)
• Modeling
• Optimizing
• Deciding
• A priori (DM defines preferences before optimization)
Also know as…
• Multicriteria decision Making (MCDM)
• Multicriteria decision aiding (MCDA)
• Multatribute decision making (MADM)
• If all functions are linear
• Multiobjective Linear Programming (MOLP)
Definition
•Domain
• x = (x1, x2, …, xn)
•Cost function
• f(x) = f1(x) ○ f2(x) ○… ○ fk(x))
Multi-objective problem:
minimize f (x)
subject to g m (x )  0, m  1, , ng
hm (x )  0, m  ng  1, , ng  nh
x  [x min , x max ]
What is an optimum in this case?
•Improving in one objective may deteriorate another…
•Balance in trade-off solutions is achieved when…
• A solution cannot improve any objective without degrading one or more of the other objectives.
f1
f2
A
B
Pareto Optimum
•Pareto improvement
• change from one allocation to another that can make at least one individual better
off without making any other individual worse off is called a Pareto improvement
•Pareto Optimum
• An allocation is defined as Pareto efficient or Pareto optimal when no further
Pareto improvements can be made
• These solutions are called non-dominated solutions.
• The set of these solutions is a non-dominated set or the Pareto-optimal set.
• The corresponding objective vectors are referred to as the Pareto-front.
•Weak Pareto Optimum
• there are alternative allocations where at least one objective would be worse
Vilfredo Pareto
1848-1923
Multiobjective Optimization
All Pareto optimal can be regarded as equally desirable and we need a decision
maker to identify the most desirable among them
Types of Approaches
•Non interactive
• Basic
• NonPreference
• Others
•Iterative
• Trade-off
• Reference Point
• Classification Based
•Evolutionary
• Evolutionary algorithms
• Ant Colonies
• Particle Swarm
• Have proven to be the best methodologies
NON INTERACTIVE
Basic Methods
“Not really” multioptimization methods
Weighted method
• Only works well in convex problems
• It can be used a priori or a posteriori (DM defines weights afterwards)
• It is important to normalize different objectives!
ε - constrained method
• Only one objective is optimized, the other are constraints
• Works for convex or non-convex problems
Non-preference methods
•DM opinion is only listened after solving the problem
• There is no DM or he is not expecting any special result
•Global criteria
• Minimize distance to some reference solution
• Depends on distance metric
•Neutral compromise solution
• Try to find the “middle” point of all solutions
Others
•Weighted metrics
• The distance to different objectives is weighted
•Goal Programming / Goal Attaining
• Define a set of aspiration goals
• Minimize distance to goals
x
non-preference
x
a priori
x
x
a posteriori
x
x
x
x
(x)
can find any Pareto
Optimal
solutions always
Pareto Optimal
(x)
(x)
Information
weights
bounds
goal programming
weighted
metric
neutral
solution
weighted
ε-constrained
global criteria
Comparison
x
(x)
(x)
x
(x)
weights
reference point
order
INTERACTIVE METHODS
General
•Decision Maker expresses preferences during the optimization process
• Only a part of Pareto solutions are found and evaluated
• DM does not need a global structure view of preferences
• Saves time and makes comparison between solutions easier
• Implies an active participation during optimization process
•Algorithm
1. Initialize (e.g. Neutral Solution)
2. Ask DM for preference
3. Evaluate a new set of solutions
Usually has two phases
• Learning phase for DM
• Real Decision Making phase
Trade-off Methods
•Trade-off
• Rate of exchange between two objectives (how much you win / how much you loose)
• Trade-off computation helps DM to know which region should be explored
•Zionts-Wallenius or ISWT methods
• Ask DM to express preferences and evaluate trade-off values
•Geoffrion-Dyer-Feinberg (GDF) or SPOT or GRIST methods
• DM provides subjective trade-off values
Reference Point Approaches
Decision maker provides:
Preference values for the outcomes (reference points)
Relative order between objectives
DM may change reference points
Based in three principles:
1. Considers separation between preferential and substantive methods
2. Objective aggregation is nonlinear (different from weighted basic approach)
3. Holistic perception of objectives
• Signal substantive changes in objective values
Stopping criteria
When the DM is satisfied with solution
Classification-Based Methods
•DM chooses which objective functions should be improved and which ones
can be maintain the value
• DM can also indicate intervals of improvements
• Similar to reference point methods
•Step method
• At each iteration, DM indicates acceptable values and unacceptable values
• DM gives up a little bit on acceptable values to improve unacceptable
•Satisficing Trade-off method
• DM is asked to define the objectives into three classes:
• acceptable, to relax, to improve
• DM defines bounds for trade-offs (aspiration levels)
•NIMBUS method
• DM defines 5 classes of objectives
• DM receives up to 4 Pareto Optimal solutions
EVOLUTIONARY
MULTIOPTIMIZATION (EMO)
Ideal Multiobjective Optimization
•The strength is the fact that parallel solutions are computed at the same time
EVOLUTIONARY ALGORITHMS
Approaches
•Vector Evaluated GA (VEGA), (Shaffer, 1985).
•Multi-Objective GA (MOGA), (Fonseca & Fleming, 1993)
•Non-dominated Sorting GA (NSGA), (Deb et al., 1994).
•Niched Pareto GA (NPGA), (Horn et al., 94)
•Target Vector approaches, (several authors)
•NSGA II, (Deb et al., 2002).
• Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm:
NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): 182-197 Apr 2002.
VEGA
With M objectives to be handled, population is divided by the objectives. Each
subpopulation has its own fitness.
• Advantages: only selection mechanism is modified, so it is easy to implement and efficient
(computational complexity is the same).
• Drawbacks: difficult to find good compromise solutions, as each solution is looking only to
individual objective function. It can happen that few points of the Pareto front are found.
VEGA implementation on TSP
•Optimize Distance (Z1) and Time (Z2)
1. Initialize
Z1=69, Z2=3
Z1=64,Z2=3
1
2 3
4 5
6 7
7
3 5
6 4
2 1
5
6 7
1 2
4 3
1
3 7
5 6
4 2
2. Separate to Selection
Z1=65, Z2=2,5
Z1=66,Z2=2
Z1=64
3. Shuffle to crossover and mutate
Z2=2,5
7
3
5
6
4
2
1
5
6
7
1
2
4
3
5
6
7
1
2
4
3
1
3
7
5
6
4
2
Z1=65
Z2=2
Z1=64
Z1=65
7
3
5
6
4
2
1
5
6
7
1
2
4
3
Z2=2,5
5
1
6
3
7
7
1
5
Z2=2
2
6
4
4
3
2
MOGA
Differs from VEGA in the way fitness is assigned to a
solution:
• A rank is assigned to each solution ri = 1 + ni, where ni is
the number of solutions that dominate solution i.
• Fitness is related to the inverse of ranking.
This simple procedure does not assure diversity among
non-dominated solutions.
• A niche-formation method was introduced to distribute the
population over the Pareto-optimal region.
Advantages:
• fitness assignment scheme is simple.
• Can find spread Pareto-optimal solutions.
Drawbacks:
• introduce unwanted bias towards some solutions.
• May be sensitive to the shape of Pareto-optimal front.
Example
•Objectives:
• minimise internal temperature gradient,
• minimise heat loss,
• minimise area of the evaporator
Design variables:
• height of evaporator bottom,
• evaporator depth.
• evaporator thickness,
• evaporator width,
• insulation thickness
•Geometric constraints:
• each parameter has a minimum and a maximum bound
•Fixed dimensions:
• outside dimensions of the fridge, size of the condenser
•Design evaluators:
• STAR-CD CFD/Heat Transfer Commercial Code
NGSA II (Elitist Non-Dominating Sorting GA)
•This method differs from previous in:
• Uses an elitist principle (sort by fitness
before selection)
• Uses an explicit diversity preserving
mechanism (Crowding distance)
• Emphasizes non-dominated solutions
(classify solutions in three fronts)
ANT COLONY OPTIMIZATION
ACO approaches (MOACO)
Multi-colony algorithms
Multiple pheromone matrices algorithms.
Multiple heuristic functions algorithms
Multi Colony Algorithm
Each colony optimizes one objective.
Having k objectives, a total of k colonies is used.
Colonies cooperate by sharing information about the solutions found by each
colony.
• Local sharing: is performed after next node is added to current path of a new partial
solution. Solutions are grouped into non-dominance solutions.
• Fitness value fij is calculated for the best solution so far.
• Global sharing: similar process but now it is performed after completion of paths.
Multiple pheromone and/or heuristic matrices
•Two objectives: two pheromone matrices and two heuristic matrices (Iredi, 2001):
•Having Kobjectives (Doerner, 2004):
Single pheromone function and several heuristics information functions (Barán and
Schaerer, 2003):
COMPARISON BETWEEN
EA AND ACO
Example: TSP
Traveling Salesman Problem with multiple objectives:
• cost,
• length,
• travel time,
• tourist attractiveness.
•Used approaches:
Results for KROAB50
Results for KROAB100
PARTICLE SWARM
MO Particle Swarm Optimization (MOPSO)
•Uses Archive Mechanism (A)
• List of non-dominated solutions
•Use a swarm like for single objective
• Evaluate each solution to see if it is nondominated or not
• Evaluate pbest and gbest for each of the
objectives
•Similar to VEGA approach
SOFTWARE
Matlab
•Goal Programming / Goal Attain
• x = fgoalattain(fun,x0,goal,weight)
•Evolutionary MultiObjective Optimization
• http://www.mathworks.com/matlabcentral/fileexchange/10351
READINGS
•Energy Systems
• Two objective functions
• Cost
• Emissions
• NonInteractive Approaches
• ε – Constrained and Goal Attained
•Green Building Design
• Two objective functions
• Lyfe Cycle Cost
• Lyfe Cycle Environment Impact
• EA approach
• MOGA