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

Solar

WindCost(€) Performance(kW/year)

Solar

Wind

What is the best solution?

15

1020

5

25 7,5

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....

Solar

Wind

Profit (€)(10)

(10)

(17,5)

What is the best solution?

Are objectives equally important?

•If objectives are not equally important

• What is the relative importance between them?

One has to decide a priori the relative importance of objectives

Is Performance more important than Cost?

Is Performance more important than Cost?

How much? 2 times? 10 times?

Solar

Wind

Profit (€)(5)

(0)

Solar

Wind

Profit (€)(25)

(30)

2x Cost = Performance

Cost = 2x Performance

What is the best solution?

Solar

Wind

Profit (€)35

80

Cost = 10xPerformance

(42,5)

(10) 50

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

Cost20

10

15

5

Best performance

Best 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:

min max

subject to ( ) 0, 1, ,

( ) 0, 1, ,

[ , ]

m g

m g g h

g m n

h m n n n

x

x

x x x

minimize ( )f x

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.

A B

f1

f2

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

Comparison

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-

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non-preference x x

a priori x x x

a posteriori x x x

can find any Pareto

Optimal

x (x) x

solutions always

Pareto Optimal

(x) (x) (x) (x) (x)

Information weights bounds 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

2. Separate to Selection

3. Shuffle to crossover and mutate

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

Z1=69, Z2=3 Z1=64,Z2=3

Z1=65, Z2=2,5 Z1=66,Z2=2

7 3 5 6 4 2 1

5 6 7 1 2 4 3 1 3 7 5 6 4 2

Z1=64

Z1=65 Z2=2

5 6 7 1 2 4 3

Z2=2,5

7 3 5 6 4 2 1 5 6 7 1 2 4 3

1 3 7 5 6 4 2

Z1=64 Z1=65

Z2=2

5 6 7 1 2 4 3

Z2=2,5

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 non-

dominated 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