Pareto Optimal Solution

6.3.1 Introducing the Pareto Solution

Thus far, we have discussed how to formulate a multiobjective optimization design problem. The next important question is: how do we solve this optimization problem? It indeed does not resemble the various single objective problems that we have seen thus far. Does it have a single optimum solution or multiple solutions? What is the value (or values) of x that seeks to minimize μ₁ and μ₂ simultaneously, while providing a desirable tradeoff between the two? One of the interesting features of multiobjective optimization is that the solution to the problem is generally not unique as different tradeoff levels may be desirable (with each tradeoff level yielding a different solution). A set of solutions called Pareto Optimal Solutions form the complete solution set of the optimization problem.

To understand the concept of Pareto optimality, we review the folloing example. For simplicity, we consider an unconstrained problem involving two design objectives, μ₁ and μ₂, which are functions of a single design variable x. We are interested in minimizing both design objectives simultaneously. Figure 6.2 provides the plot of each objective function on the same vertical axis and the design variable x on the horizontal axis.

Figure 6.2. Multiobjective Optimization

If we minimize each objective function independently, ignoring the other objective, we will obtain the point that corresponds to the minimum of the objective being minimized. These two points are indicated by stars in Fig. 6.2. Suppose you are at the minimum of Objective 1 (i.e. Point M1), and you decide that you want a design that has a lower value for Objective 2 than what you have at Point M1. To achieve this, you will have to move to the right of Point M1, say to a point B. In doing so, Objective 2 has decreased in value, but what happened to the value of Objective 1? It increased as we moved from Point M1 to Point B. Remember, we are trying to minimize both objectives. Thus, to improve the value of μ₂, we had to compromise on the performance of μ₁. In fact, this is true for all the points between M1 and M2. We call these points Pareto optimal solutions or non-dominated solutions.

Definition: Pareto optimal solutions are those for which any improvement in one

objective will result in the worsening of at least one other objective [6]. That is, a tradeoff will take place.

Thus, if a point is Pareto optimal, we can be assured that there cannot be simultaneous improvement in all the objectives. In Fig. 6.2, if we move to the right of Point A, both objectives decrease simultaneously. Therefore, we call points, such as A, non-Pareto or dominated solutions. The same it true with other points such as C. The hatched region in the central portion is the set of all design variable values that are Pareto optimal. As an optimization engineer, if you are solving a multiobjective optimization problem, Pareto points are what you should be looking out for.

6.3.2 The Pareto Frontier

In the previous subsection, we determined how to identify Pareto solutions in the design variable space; that is, x space. We introduce the concept of design objective space. This is a plot with a design objective plotted on each axis. We are particularly interested in what happens if we plot the design objective values of the Pareto solutions in this objective space. The pattern of points that you see in the objective space is called the Pareto frontier. The name might sound highly technical, but it is simply a plot of all Pareto solutions in the objective space. Figure 6.3 is a plot of all the Pareto points that we identified in Fig. 6.2.

Figure 6.3. Pareto Frontier

All the points (M1, M2, A, B, and C) illuatrated in Fig. 6.2 are plotted with their respective objective function values in Fig. 6.3. M1 and M2, in particular, form the end points of the Pareto frontier, also known as anchor points. Point M1 is where Objective 1 has the least value, while M2 is where Objective 2 has the least value. Points A and C, as dominated points, do not lie on the Pareto frontier.

You can appreciate why such a plot can be of immense use to a designer who is trying to optimize a particular system. By looking at the Pareto frontier, one can clearly see the tradeoffs associated with each Pareto point. For example, if Objective 1 and Objective 2, respectively, denote stress and deflection, then one can immediately identify regions of the Pareto frontier corresponding to the low values of stress, or regions with low values of deflection, whatever the preference of the designer. The designer can then select a particular Pareto point (say Point B), and map it back to the design variable space (Fig. 6.2) to determine what values of the design variables (x) yield Point B. Thus, the concept of Pareto frontier is central to the understanding and application of multiobjective decision making.

6.3.3 Obtaining Pareto Solutions

From the above discussion, it is clear that to solve a multiobjective problem, we need to obtain a Pareto optimal point or a set of points on the Pareto frontier. We cannot always plot the design objectives as shown in Fig. 6.2. In most real-life design problems, the design objectives are functions of several variables, and we need more methodical techniques to obtain Pareto solutions. Note that most optimization algorithms are developed for single objective problems. As such, one intuitive way to solve a multiobjective problem is to combine all the objectives into a single Aggregate Objective Function (AOF) in such a way that, when the AOF is optimized, a Pareto solution is obtained.

6.3.4 Aggregate Objective Function

Definition: An Aggregate Objective Function (AOF), generally denoted by J, is a function

that combines the design objectives into a scalar function.

The AOF typically contains parameters to be selected by the designer. These parameters reflect the relative importance of each design objective. An objective with higher importance will be given priority during the optimization process. That is a form of inter- criteria preference. In addition, as effective AOF should provide the ability to express the relative preference for different values of a given objective (intra-criteria preference). Note that the most commonly used AOFs do not effectively provide both of these crucial attributes. Keep in mind that the final solution that can be achieved will depend on the type of AOF that is used. An important consideration in selecting the AOF to be used is its ability to allow the designer to impose his/her design objective preferences in an unambiguous manner. Below are some popular AOF formulations used by designers in industry. You will develop a deep appreciation for these crucial practical issues as you study the material in this chapter, and practice solving the problems at the end.