Multi-Objective Optimisation and Pareto Optimum

In a design process a designer typically deals with several, often conflicting, objectives. Finding the right solution or a best approach for each of those objectives, let alone for all of them together is not an easy endeavour. Unfortunately most of the traditional and simpler optimisation techniques usually only operate with just one of these objectives at a time. Differential calculus is such a simple way to find a solution to a straightforward design decision problem. Linear programming, nonlinear programming and dynamic programming are other techniques for introducing optimisation strategies in the design process where a single objective can be clearly formulated and understood (Radford & Gero, 1987). Those traditional tools, methods and techniques for optimisation, which are available to assist the designer with optimisation tasks, can only be used to produce additional design information and are most of the time restricted to support limited decision making in the detailing phase of the design process. As such, design as a goal-oriented and decision-making process can rarely benefit from this kind of approach, this is not how a valid novel design solution can be discovered or explored, nor how a design process should or is developed.

Unfortunately, design problems are never simple and straightforward, and so it is difficult to think of any real world design problem which is not

characterised by the presence of many conflicting objectives, boundary conditions and requirements. In the design problems a designer gets involved with, especially those problems related with environmental performances that are multi-criteria in nature; he usually does have more than one objective to

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achieve. It is therefore possible to look at any design problem as an optimisation problem and those optimisation problems are almost always multi-objective optimisation problems. However, different objectives regarding different aspects of performance are most of the times conflicting.

Improvement in quality of one of the objectives can reduce the performance of another objective. Optimum performance to one objective usually implies unacceptable low performance of other objectives. Therefore, in multi- objective optimisation problems one single best solution may not exist and a possible solution or solutions are usually a trade-off between conflicting criteria which are difficult to compare. Sometimes objective functions can be optimised separately from each other just to gain some more insight into each performance objective and thus gain additional knowledge about the solution space.

In multi-objective optimization problems there is not one single best solution

but a population of solutions. This set of solutions can be graphically

represented showing a Pareto frontier (Fig. 9) of optimum solutions displaying different trade-offs between conflicting criteria (Ciftcioglu & Bittermann, 2009) (Fontes & Gaspar-Cunha, 2010) (Caldas 2008).

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Whenever the optimisation problem involves two conflicting objectives, this set of non-dominated solutions can be depicted by a two dimensional “curve”, such as the Pareto Optimum Frontier depicted and illustrated by the red dots on the graph in figure 9. For a multi-objective optimisation problem with three different objectives, a Pareto Optimum Frontier can be depicted and described by a curved surface in a three dimensional graph.

Pareto optimality uses the concept of dominated and non-dominated

solutions. The result of a multi-objective optimisation process should be a set of non-dominated (Pareto optimal) solutions, this in contrast to a single objective optimisation problem where the result is a single optimum or a set of equivalent optima (Deb, 2001). If between any two solutions none can be

considered better than the other on both objectives, these are called non-

dominated (red dots on the graph in Fig. 9). A solution is Pareto optimal if it is not worse than another solution in all the objectives and better in at least one objective. In other words, a solution is Pareto optimal if it is not dominated by any other solution (Deb, 2001). In Figure 9, for solution B there is a solution A which is better than B for criteria 1 as well as for criteria 2.

Since most of the optimisation problems in design and architectural design are in fact multi-objective and the goal of optimisation in a design process should go beyond the generation of factual information, different methods, tools and algorithms have to be applied. Those methods for building multi-objective optimisation models can be classified in two general approaches (Radford &

Gero, 1987). A non-preference approach is limited to the production of

information on non-dominated performances and on solutions with that performance. The solution chosen by a designer from this Pareto set is often called a ‘best compromise’ solution because there always has to be a trade-off between decisions that would be in favour of one set criteria and decisions

that would point to another set of criteria. In a preference approach, the

designer’s trade-off preferences are placed within the optimisation model. The set of feasible solutions is narrowed down by available information together with rational decisions about preferences by the designer.

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A non-preference method (Pareto method) is thus a bottom-up approach that provides a lot of information for the designer to make his decisions. The preference methods are a kind of top-down approach, which provides much

less information and attempts to determine in advance what information the

designer needs to make decisions (Radford & Gero, 1987). In this model and in this thesis both approaches will be balanced according to the design problem which is researched. This way the design process can find a fine balance of potentiality in a digital process where prospective solutions are no longer possible to grasp by human reasoning alone.