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Practical Multiobjective Optimisation

by Chris Lucas

"Now, as there are many actions, arts, and sciences, their ends also are many; ... other arts fall under yet others - in all of these the ends of the master arts are to be preferred to all the subordinate ends; for it is for the sake of the former that the latter are pursued."

Aristotle, Nicomachean Ethics, 350 BCE, Book I,1

"The great decisions of human life have as a rule far more to do with the instincts and other mysterious unconscious factors than with conscious will and well-meaning reasonableness. The shoe that fits one person pinches another; there is no recipe for living that suits all cases. Each of us carries his own life-form - an indeterminable form which cannot be superseded by any other."

Carl Gustav Jung, Modern Man in Search of a Soul, 1933, p. 69

Introduction

In the world around us it is rare for any problem to concern only a single value or objective. Generally multiple objectives or parameters have to be met or optimised before any 'master' or 'holistic' solution is considered adequate. This introduction looks at some of the issues involved when we try to do this, and outlines the technique of Evolutionary Multiobjective Optimisation (EMOO) that can be used to solve Multiobjective Optimisation Problems (MOP) or Multiobjective Combinatorial Optimisation (MOCO) problems by using forms of genetic algorithms called Multiobjective Evolutionary Algorithms (MOEA). This process can become very technical and theoretical, with much of the information on the subject contained within offline PhD thesis documents and papers, but here we will also look at simplified practical ways of employing this technique within our daily lives, without computer assistance. This introduction is more difficult than many of our others, so we recommend that readers new to this subject first read through (in sequence) all our prior introductions or the glossary, in order to gain familiarity with the many technical terms used herein.

One of the most important findings when we pursue these studies is that normally there is no single solution to such problems, no Utopian 'answer' or optimum in the sense traditionally expected. Instead there is a large family of alternative solutions, different balances of the various objectives that all have the same global fitness (sometimes so many as to be in practice infinite). This diversity of solutions precludes isolated 'right' or 'wrong' decisions, we must instead make our decision based upon the full dynamic context of the situation, in other words complex systems force complex choices with complex implications. We can regard the various objectives as resources, so agent populations naturally evolve to minimise competition for these, which leads to the formation of sets of specialist solutions or niches, a type of division of labour or speciation, a symbiosis or synergy generating maximum efficiency in the wider global context. This result implies that multiple values, rather than single ones, drive real world ecologies and societies and that these techniques can equally be applied in those fields. It also implies a multi-level form of selection which operates on populations as well as individuals.

The General Idea

In normal genetic algorithms (GA) we take a population of genomes (individuals) randomly scattered across state space and evaluate the fitness of the results. The best are then retained (selection) and a new population created (reproduction), incorporating mutation and crossover operations to gain a different set of possibilities (variation). Over many generations the population will search state space and hopefully converge on the best solution, the global optimum. In multiobjective genetic algorithms we do much the same, except that in this case we are trying to optimise not for one fitness parameter but against a collection of them. To achieve this we must generate an understanding of the overall fitness of the set of objectives, so that we can compare solutions, and there are many ways of doing this. In traditional multiobjective optimisation it is usual to simply aggregate together (add in some way) all the various objectives to form a single (scalar) fitness function, which can then be treated by classical techniques such as simple GAs, multiple objective linear programming (MOLP), multiple attribute utility theory (MAUT), random search, simulated annealing etc.

A problem that arises however is how to normalise, prioritise and weight the contributions of the various objectives in arriving at a suitable measure, e.g. when choosing a car how do we compare incommensurable values like size and colour ? Is speed more important than cost and by how much ? How can we quantify such fuzzy objectives as beauty or comfort ? Additionally values can interact or conflict with each other, increasing one can reduce others in turn and this can happen in nonlinear ways. This mapping stage is in itself problematical in that the set of solutions produced is highly dependent upon the value sharing function used and how the weights are assigned. Only if this is adequate will the solutions also be adequate and it is very easy to assign prejudicial or inappropriate sharing factors which can lead to apparently quantitatively exact solutions that are sub-optimal or misleading in practice (e.g. traditional 'objective independence' assumptions). Thus a Multiobjective Optimisation Problem requires solving two problems, firstly to establish a statement of the problem in a suitable mathematical form called the 'objective function' (i.e. a definition of the design space) and secondly to somehow search this space to locate acceptable solutions (ideally global optima) in terms of the 'decision variables'.

Compromise Solutions

Where multiple objectives or decision variables are treated it is usual that only some values of each are feasible in practice (e.g. a chair cannot be made too small to fit a person). These constraints (not smaller than x, not bigger than y, equal to z) reduce the extent of the design space to be searched in accordance with our goals, i.e. what we wish to achieve overall. In cases where we impose too strict constraints however it is possible to exclude all the solutions i.e. our problem then has no degrees of freedom (it is over-constrained), so we must try to balance design freedom with ideal preferences. In most situations, where values interact, it is quite impossible to optimise all the objectives at the same time, so we must adopt various trade-offs or compromises, e.g. between cost and performance. Many equally good solutions are possible in these cases, e.g. do we want a cheap, low performance car or an expensive, high performance one, or something in-between ? What is the best compromise for such decisions always depends upon the environmental context in which it is to be used, i.e. our wider lifestyle or worldview.

When we have a set of solutions such that we can't improve any objective further without at the same time worsening another then we have what is called the 'Pareto-optimal set' (of resultant fitnesses) or 'Pareto front' (of objective vectors). In such a case all the other lesser solutions are said to be 'dominated' by these better ones and we can discard them, e.g. the set of objective values 5,3,4 dominates the set 4,3,4 (the former improves the first objective without reducing the others), the set 4,8,2 doesn't dominate 4,7,3 however, but is an alternative trade off between the last two objectives. The set of these 'can't do better' trade-offs (the non-dominated set) contains all the acceptable solutions, different combinations of the objectives or 'niches' and we then need to select one or more of these for practical use by more intuitive means - their fitnesses are exactly the same, i.e. the 'Decision Maker' (DM) needs to prioritise the criteria or establish preferences (incorporate subjective values) appropriate to the full contextual situation. This can be done prior to optimisation (giving a scalar fitness function), after (choosing from the full Pareto-optimal set), or interactively (gradual convergence on an acceptable solution). The first is appropriate given static (hard) preferences and changing objectives, the second for static objectives and changing (soft) preferences, and the third where both are dynamically changing (coevolution).

Conventional Approaches

Before we look at GA inspired approaches, let us mention a selection of the more conventional Operational Research methods of obtaining solutions or approaching the Pareto front (many other techniques also exist). These mostly focus on the first stage of ranking the objectives, i.e. trying to reduce the design space to a more easily managed mathematical form (since most such problems are far too complex to enumerate and evaluate all the possible combinations in any reasonable time).

Comparing GA and MOEA Approaches

In standard GAs we take the specifications of a number of parts or attributes (the genotype bits) and combine them to create a function (the phenotype), and it is the fitness of this function that we wish to optimise. In these situations we are not bothered about which parts are used or not used, as long as the required function is met. Technically the 'phenotype' function is an emergent property of the environmentally situated 'genotype' but most GAs simply relate the two mathematically. However generating just a single fitness function in such a linear style, e.g. of the form F = xA + yB + zC (which is then maximised or minimised), can lead us to difficulties. If we take for example A = air, B = food and C = warmth, then 5A + 0B + 0C seems to achieve a fitter organism overall than say 1A + 1B + 1C, yet a creature cannot live without all three needs (objectives), so this would be an erroneous solution. Thus maximising either a single objective or a simple sum of them (where inter-dependencies exist) can lead to invalid (unstable) solutions. The problem is that the apparent overall fitness (5 here) is unsustainable, the population size would decay dynamically P(t+n) goes to 0 (i.e. the creatures would soon starve or freeze to death). We need instead to converge on stable equilibria (which are often short term ones in open, soft preference, dynamical systems). To deal with such problems we can add sets of constraints, based upon our needs and the available resources (e.g. B not less than w, A not greater than v), thus preventing certain of our objectives from escaping sensible bounds (i.e. we discard 'solutions' that arise violating these constraints).

Usually, in MOP, function fitness is made a sum of the objective fitnesses, but if instead we use the product of those fitnesses then we can avoid this problem, e.g. 5x0x0 = 0 (unfit), 1x1x1 = 1 (some fitness). This gives us the idea that interactions between objectives imply competitive (negative-sum) or cooperative (positive-sum) behaviours and that finding the optimum overall fitness in these cases will require cooperation (at least in a weak sense) and not competitive exclusion. The notion of a population of cooperative niches (diverse Pareto front solutions) is more naturally modelled by MOEAs. In these, instead of selecting and breeding just the best overall solution we wish to maintain a set of solutions based upon the values of all the separate objectives. These are only replaced by better solutions if they become dominated in the Pareto sense. In this way we avoid the situation where solutions that improve one variable at the expense of another come to dominate the population. This approach naturally allows niches to persist and in the end delivers a set of alternative optimised compromises from which the decision maker can make a choice, or alternatively allow a diversity of 'lifestyles' to coexist. For example, in a village we should have a balance between the various trades - farmers, carpenters, blacksmiths, shopkeepers, doctors, innkeepers, etc. None of these occupations are 'better' than the others, and they can all only exist within the wider community, within a population-dependent mutual support arrangement, i.e. if we say 'doctor' is more 'profitable' then we should maximise that occupation - but that 'profitability' collapses if there are no patients, only doctors - all the fitnesses are mutually dependent upon the whole, the set of niches is collectively of higher fitness than any isolated occupation (this is the meaning of synergy and is a form of frequency-dependent selection).

Evolutionary Multiobjective Approaches

In traditional single-objective (scalar) GA approaches, to fully search state space, we can vary the weights or treat most of the objectives as varying constraints and optimise just the main objective, employing multiple runs to generate Pareto-optimal solutions sequentially. For MOEAs we wish instead to generate all the Pareto-optimal solutions in a single run, for efficiency. A number of approaches are possible, often combined (we won't try to fully explain these here):

One of the most useful techniques in practice is 'elitism' where we preserve the best individuals from each generation, but in multiobjective cases this can lock the system on a local optimum (premature convergence). We need in this case to decide how many solutions to include in the elite set and whether to keep them within the population (evolving) or as an external reference set used only in selection.

Problem Areas

Several problems occur when we try to search the space of possible solutions. First we wish to cover all of the search space (i.e. find all the good solutions) and not get stuck on local optima, secondly we wish to approach the Pareto front as closely as possible (ensuring optimal solutions), thirdly we wish to ensure a good spread of objective values (i.e. a diverse choice of niches) and fourthly we wish to achieve convergence in a reasonable time. These depend in turn upon how we choose to define the fitness or utility function and carry out the search, which can be problematical (again we won't go into the details, just give some outlines):

Fitness function (decision variable space) related problems include:

Search (objective function space) related problems include:

Once we have a set of solutions our problems are not over. We then have no information as to how these solutions arose, so cannot say how critical they are, i.e. if we vary one of the parameters slightly does the solution persist ? Adopting a 'perfect' solution that turns into a disaster if some parameter varies by say 1% would be unwise ! Attempts to clarify this, i.e. provide some 'sensitivity assessment', require us to maintain and analyse an history of the search trajectories, e.g. by using such techniques as case-based reasoning (CBR). These analyses can also help improve search performance by generating a better understanding of the structure of design space. An example of just how complicated nonlinear optimisation can be is illustrated by the following graph of how the fitness value of just one objective (out of seven) changes when plotted against variations in two of the other variables within the system. The narrowness of the fitness peaks reflects a sensitivity to initial conditions and the number of such peaks emphasises the difficulties in finding a representative spread of good solutions, i.e. objective nonlinear interactions generates a rugged 'fitness landscape':

MO fitness landscape

Diversity, Creativity and Stability

We noted at the start that multiple objectives or values implies a possible diversity of specialist niches and that part of the function of state space searches is to locate and exploit these possibilities. This is a form of creativity, where we become aware of ways of operating that go beyond individualistic success criteria and embody a form of group selection, a 'building block' approach to cooperative social fitness exemplified by the 'credit assignment' mode of learning classifier systems (LCS). This relates to changing competing objectives into supporting ones, e.g. 'work' and 'fun' are often opposed, but by re-organising our approach (escaping the 'rules') work can also be fun - the two become mutually supporting. In order for these higher-level attractor niches to persist we must have a selection pressure opposing homogeneity (the lowest common denominator), i.e. keeping partial solutions (specialist functions) intact and restoring them if perturbed. This generally means that fitness is defined in terms of the whole, i.e. the mix of partial solutions present, similar to our MOEA focus on preserving all objectives. Thus as any partial solution grows in the fixed GA population, the others upon which its fitness depends must shrink, compensating and self-stabilising the whole - which has been called in Reinforcement Learning (RL) and LCS literature a strong cooperation mode or 'symbiotic evolution', which relates to the multiple environments of ecological GAs. This balance reflects what has been called the 'edge of chaos', the upward causation of the part fitness functions being countered by the downward causation of the fitness of the whole population. An alternative way of implementing this is to select using different criteria for each specialist population, i.e. we use differential selection to both create and maintain niche diversity. One way of doing this is to subdivide state space so that each sub-population is optimised in a different environment and thus finds alternative optima or niches.

The idea of competition for multimode resources can be illustrated by taking the three economic resources of Money, Time and Ideas (thought). These all interrelate, i.e. generating money needs time and ideas, generating ideas needs time and perhaps money, generating time needs ideas and money. A simple dynamic model for this would be:

Strange Attractors

M( t+1 ) = a * M(t) + b * I ( t ) + c * T ( t )
I ( t+1 ) = d * I ( t ) + e * T ( t ) + f * M( t )
T ( t+1 ) = g * T (t) + h * M( t ) + j * I ( t )

Models of this form are notoriously nonlinear, the cross coupling of the variables often leads to chaotic behaviour, solutions that take the form of strange attractors (example illustrated) where the trade-offs partition state space into semi-stable niches. This is a result of feedback, the system interconnections form causal loops which generate semi-stable equilibria, areas of balance in state space, i.e. dynamic compromises. In such systems the stability of analyses are highly suspect and we would be advised to take great care, employing risk analysis methods as far as we possibly can. Little work has yet taken place in understanding MOP in these unstable dynamical attractor terms.

State Of The Art

A few of the more advanced MOEA approaches used recently can be listed (there are many variations):

A sister field to MOEAs, Multiple Criteria Decision Making (MCDM), also has many variants - outranking, value & utility based, group negotiation, and multiple objective programming. In addition to approaches using firm values we can also allow fuzziness in the objective values and weights. This reduces the need for accurate function definitions, and accounts for uncertainty in real world values and for incomplete information. A fuzzification of MCDM also allows interdependence of objectives, something few of the other MCDM techniques support well. Most of this class of techniques don't employ GA (evolving population) techniques, i.e. they concentrate on mapping from objectives to fitness function, rather than on searching state space, so a combination of FMCDM with MOEAs may well prove advantageous and initial work on such combinations is now starting to get underway. At the time of writing improvements to the techniques listed above, e.g. NSGA-II, SPEA2 and PESA-II, and for fast operation Coello & Pulido's micro-GA, seem to be the current state-of-the art for MOP. For MOCO problems, progress has been made in combining recombination with local search techniques in the form of Memetic algorithms - Jaszkiewicz's Random Directions Multiobjective Genetic Local Search (RD-MOGLS) or Pareto Simulated Annealing (PSA) and Knowles & Corne's M-PAES are the current state-of-the-art here, although another new technique, the ALife related Ant Colony Optimisation (ACO), a form of Swarm Intelligence, also shows promise.

Issues of Accuracy

Given that in 'real life' complex systems we generally do not have accurate measurements of our variables and that we are more interested in dynamics (the unknown future) rather than statics (the known present) then what use are MOEAs ? Applications have already been found in many engineering and organisational situations, where science has been able to generate stable functions linking the objectives. In these (usually) highly artificial systems we have a great deal of control over the aspects included and excluded, we specify both the parts and their interactions. What about more natural complex systems, like ecologies and societies, where these aspects are not generally under our control ? Systems that themselves evolve bring to the fore the soft preferences mode of DM choice, in other words we are likely (in such open systems) only to be able to guess at both the set of objectives and the interaction of their values. Experience counts for a great deal here, the larger the number of similar situations (historical cases) we have seen then the better our choices are likely to be. In nonstationary (evolving) EA systems genome diversity is essential, if crossover is to track the moving optima (assuming low mutation rates), since a converged population generates only identical genomes with crossover. Techniques to deal with dynamic optima are still rare, but Wineberg's Shifting Balance Genetic Algorithm (SBGA) is worth a mention, as are approaches using Cooperative Coevolutionary Genetic Algorithms (CCGA), also called Coevolutionary Computation (Potter & De Jong) or symbiotic CGAs - all are ways of implementing continuous partial testing or life-time fitness evaluation (LTFE). New approaches, based upon compositional evolution, add in symbiotic and synergistic effects, e.g. Watson's Symbiogenetic Evolutionary Adaptation Method (SEAM).

But even if we make good choices, those are only likely to be viable for a short time, we cannot assume that an optimisation exercise will give a set of solutions that can be applied indiscriminately to future situations. Thus a mathematical analysis can only be a guide, not a guarantee. As we widen the scope of our multiobjective systems to enter the human and cultural areas (the most difficult cases) our problems change from being 'well-defined' to being 'ill-defined', we no longer have firm data from which to plan which option to choose. This is the area of coevolutionary systems, where the optima or attractors are transient, they do not persist long enough for us to fully optimise any solution. The best that we can do here is to find solutions that are an improvement, that go in the right direction, and continue to try to find better ones for as long as the problem is recognisable, i.e. we attempt to track the moving situation. Thus we need a technique that better fits these situations, a less rigid approach that nethertheless can give us better information, more informed options, than current single objective evaluations (e.g. 'Arab' versus 'Jew'). We need a dynamic multiobjective mode of day-to-day reasoning, a conscious equivalent to what is sometimes called 'intuition' or 'wisdom', i.e. our subconscious method of parallel constraint satisfaction. Here however we confront the well known limitation of conscious thought, the 7 item limit to the complexity we can keep in short-term memory, the problem of potential information overload. How can we overcome this ?

An Informal Approach

Whilst the EMOO techniques outlined are good in more formal situations (where adequate resources, expertise and time are available) they clearly cannot easily be applied to our day-to-day requirement to make decisions within casual multivalued situations. We require a cut-down version of this trial and error (heuristic) technique. We need to do four things:

  1. Identify the objectives involved (the set of values that interest us)
  2. Determine how they interact (the fitness mapping or interconnections)
  3. Generate viable alternatives (the population of possible choices or niches)
  4. Identify the best compromise (the Pareto-optimum) in the current context

This more informal approach doesn't look to perform exhaustive mathematical testing, just tries to correctly analyse a complex dynamical situation in order to identify any good way forward and to highlight interdependencies which may otherwise be overlooked (i.e. the connectivity of the complex system). In this sense it is a single solution at a time form of MOO, close to the minimal EMOO technique of (1+1)-PAES, i.e. we try to move from our current situation to a better one following a gradual improvement strategy (or walk) through local state space. This process merges standard GA and OR techniques, and is close to hybrid genetic algorithms (HGA), also called genetic local search (GLS) or memetic algorithms (MA). These produce one suggested solution at a time (often creatively combining or 'crossing-over' two earlier solutions), which the DM then accepts or rejects (or puts to one side as a possible if nothing better shows up later). In other words, for each solution proposed we compare the effects on all the other objectives and reject solutions that make other aspects of our situation (our Quality of Life or utility function) worst, only accepting Pareto dominating solutions.

It seems clear here that rather than one (7 item limited) solution generator/Decision Maker, we should attempt to gain a population of them (equivalent to one super-DM with a set of alternative preferences), each contributing diverse solutions (emulating the MOEA population). This technique is an informal dynamic Pareto version of MOGLS. Thus alternative preferences can then be compared and a compromise chosen by consensus (if a single solution is mandatory), or better still we can allow a diversity of equally good solutions to persist (ideally one non-dominated solution would then match the niche preferences of each DM). This distributed decision making (bottom-up) contrasts with the centralised (top-down) decision making common to our institutions - which appear sub-optimal by comparison, since they fail to search state space adequately and often impose poor local decisions on everyone. Thus complex systems techniques echo the growing transdisciplinary approach of getting all interested parties involved, rather than relying on 'experts' focused on single 'context free' objectives. This 'multiple-autonomous agent' strategy does seem to overcome the limitation of information overload, by distributing the problem processing amongst many concurrent DMs.

Experimental Lessons

Cube

There are five axes (or dimensions) of interest when considering the optimisation of complex systems. First there is the axis of complexity, the number of objectives or decision variables included (e.g. air, food, water, warmth). Second there is the axis of variability, the values or space that each objective can take (e.g. density, volume, temperature). These two taken together define the static objective function. Thirdly there is the axis of time, the dynamics of the system, how the objectives interact and change, their trajectories through state space (e.g. consumption, cooling). These first three axes are commonly used in science, but the final two are almost ignored. The fourth is the fractal structure axis, the multi-level nature of emergence in both time and space (e.g. individual, society, ecology) and finally there is the axis of innovation, the novelty of evolutionary creativity (state space expansion). In almost all forms of science we treat only two or three of these axes at a time, we reduce the system to a subset of its real complexity. We may consider only static systems - ignoring time and evolution; or single aspects - ignoring most objectives; or specialisms - ignoring other levels; or classification - ignoring variability; or stochastic evolution - ignoring predictability. Each of these simplifications have their uses, but if we wish to understand complex systems as a whole then we must treat all of these aspects at once. Difficult as this seems, by living in the real world we must do this daily, for better or for worst, we approximate (as best we can) based on the full complex context of our world. So we can ask, do our formal scientific studies help us in any way to do this better ?

The many experiments that have taken place over the years in MOEAs have yielded a few pointers that can help our informal treatments, as well as helping to direct formal model building.

Conclusion

The area of Multiobjective Optimisation Problems is extremely complex and mathematically difficult, with many under-researched areas and outstanding problems. There is a considerable way to go still before we have an adequate technical understanding of all these issues (for an indication of current progress and details of all the algorithms mentioned herein see these references ). Yet as we gain more knowledge of complex systems, in all walks of life and areas of research, we can see the need to understand how objectives coevolve dynamically, and to determine the attractor structure (alternative optima) of such multidimensional state spaces. Outside the mathematical arena, our human systems relate strongly to such nonlinear interrelating values. Here we have the further complication of multiple levels, e.g. the environment, human physiology, psychology and sociology, where objectives often have interlevel effects as well as intralevel interdependencies (e.g. socially produced pesticides intended for environmental purposes poison our physiology, leading to psychological problems and social negative feedback). This escalation in complexity is currently beyond our abilities to model computationally in any detail.

As our artificial systems integrate more fully with our lives and cultures, and start to adapt to our needs, we must be wary of reducing the complexity of our models in order to obtain quantitative 'solutions'. A solution to the wrong problem can be worst than none at all, especially if we then impose it upon the 'real world' as if it were the 'truth'. We must be clear in specifying the assumptions we have made in formulating the model, since in complex systems we can never be certain that these are valid simplifications. Many real world situations potentially need many more objectives (values) to be taken into account than those few treated in much of the MOO literature (using as few as two objectives is very common). Attempts to employ EMOO techniques outside the realm of closed engineering systems (i.e. in optimising human and ecological issues) seem conspicuous by their absence, yet that is where the most difficult optimisation problems lie and where (presumably) we should concentrate our efforts in applying our most advanced techniques. Yet despite several orders of magnitude improvement in our computational ability over recent decades, the evaluation of complex optimisation situations is still a major limiting factor in decision making at a conscious level, leaving us to rely on poorly understood 'intuition' as a fallback. An informal conscious approach is suggested that can provide an intermediate, incremental improvement, step until this evaluation formalism becomes more possible.

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