## Multiobjective Optimization

*This is the third of a three-part series on calibration and optimization. The first part can be accessed here. The second part can be accessed here.*

Sometimes optimizing one objective does not give the best solution, for example, when judgment is needed to determine which solution between many equally optimal solutions is truly the best, given the tradeoffs between them. While it is possible to approximate the tradeoffs by assigning different weights to each objective in a single-objective optimization, it is quite likely this will not find the best solution for your situation. In these cases, it is best to use multiobjective optimization, i.e., optimize for multiple objectives at the same time.

But what does that mean in practice? While I can find the optimal value of each objective, what does it mean to optimize all of them simultaneously? It certainly isn’t true that they will all be at their individual optimums when all are optimized. In fact, when we perform multiobjective optimization, we find a set of solutions that are all optimal according to the following definition of *non-dominated*:

A solution that outperforms another solution in at least one objective is

non-dominatedby the other solution.

The set of non-dominated solutions in the population are called the *non-dominated front*. A non-dominated front (Pareto rank 1) for a minimization problem is shown in the figure below. Note there are also two dominated fronts in the set of solutions (Pareto ranks 2 and 3). The characteristic convex shape clearly shows the tradeoffs between the two objectives with the best compromise landing at the middle of the bend in the curve (called the *knee*). Beyond the knee in either direction, you experience diminishing returns.

The goal of a multiobjective optimization is to push all of these solutions toward the non-dominated *Pareto front*, which is the ideal set of optimum solutions. Be aware that you cannot be sure – without knowing the Pareto front – that the final set of non-dominated solutions from a multiobjective algorithm reached the Pareto front. However, you can test if running for additional generations improves the set of solutions (i.e., moves closer to the Pareto front).

While it is possible to optimize for many objectives simultaneously, it becomes somewhat impractical to optimize for more than three objectives: a) as the number of objectives increase, the population size must increase exponentially and the number of generations must increase significantly (the computation time increases combinatorially as the number of objectives increase) and b) it is difficult to visualize the non-dominated fronts when you exceed three dimensions. The discussion in this post will be restricted to two objectives.

**The Problem**

In the *Limits to Growth* series, we explored the Bass diffusion model. The model is shown again below and can be downloaded here (with all the changes described in this post).

As you may recall, customers are converted by both advertising and word of mouth. In the closing post of that series, I discussed the importance of marketing spend to get the ball rolling. Specifically, without it, growth is very slow. Additionally, as more customers are converted, the word of mouth effect dominates how fast the product is adopted, making the marketing spend less and less effective. By effective, I specifically mean that it results in conversions. The graph below shows the two components of *adopting* and how the dominance shifts from marketing spend to word of mouth only six months after the product launch.

Note how the number of adoptions through advertising decreases over time. This suggest it is not necessary to continue advertising spend at the same level indefinitely. There is perhaps an optimal point where we could start to phase it out. [Warning: This scenario represents the adoption of a single durable good that is never replaced or updated. The situation is clearly different for both consumable goods and durable goods that are regularly replaced.]

If we are launching this unique product, say, the Pet Rock, we would be very interested to know whether we can increase our product lifetime profits by investing less in advertising. We can run an optimization that seeks to maximize conversions and minimize advertising spend over the life of the product.