for multi-objective optimization, the Pareto-based multi-objective optimization method is the effective solution. It returns the best solutions as a Pareto front set. From the Pareto set, using one of the Multi-Criteria Decision Making (MCDM) algorithms to select the one best solution.

I understand that you have two objective functions that require to be optimized, with one being maximized and the other being minimized.

When both objectives need to be maximized and minimized, it is appropriate to utilize a Pareto-based Particle Swarm Optimization (PSO) algorithm.

However, if you only have access to a single-objective PSO implementation (which is common), you can combine the two objective functions into a single objective. Let F1 be the function that needs to be maximized and F2 be the function that needs to be minimized. In this single-objective context, your objective function would be defined as:

F = -w1 * F1 + w2 * F2, where the optimization problem aims to minimize F. You can assign weights to each function by changing the values of w1 and w2 (try w1 = w2 = 1).

Hello Oussama,

from your question, it is not very clear with what you need help. Do you need help implementing a multi-objective problem based on a real-world scenario, or is the problem that one objective needs to be minimized and one maximized?

For the second case, one possible solution is to transform one objective so that both are minimized. You can do this just by multiplying the fitness value with -1. Then both objectives can be minimized. When you have the Pareto front, you just have to transform the objective back so that it maximizes.

Rafael Otávio Alves Abreu

Thank you very much for your answer Sir. Rafael

I will try the second solution, but is it possible for you to advise me of the scientific sources that prove the validity of the second solution in general?

Waiting for your response.

Best Regards.

Formulating a multi-objective problem as a single-objective one is widely applied in optimization tasks. Below are some examples of papers that use this approach in different fields:

• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (Raissi et al., 2019) - See Equations 4 and 6
• Single‐and multi‐objective genetic algorithm optimization for identifying soil parameters (Papon et al., 2012) - See Equation 11
• Optimization techniques for identifying soil parameters in geotechnical engineering: Comparative study and enhancement (Yin et al., 2018) - See Equation 8

Converting a maximization problem into a minimization problem is a trivial task that is described in many textbooks on optimization algorithms, such as 'Numerical Optimization' by Nocedal and Wright (2006, page 4).

Rafael Otávio Alves Abreu Thank you very much Sir .

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