**1. Introduction**

Nowadays, due to unhealthy and sedentary lifestyles, a high percentage of the human population suffers from various diseases such as high cholesterol, diabetes and other conditions related to those habits which can cause other serious illnesses, including several types of cancer [1]. As a result, promoting healthy and active habits for young people is becoming a significant duty for governments around the world. Since it is impossible for governments to control what children and teenagers consume in their homes, having a healthy and balanced meal plan for school cafeterias is essential to help mitigate the effects of the remaining meals. The Government of the Canary Islands wants to promote healthy habits in children and teenagers. Generating healthy and balanced menu plans is

key to this effort. This is precisely the main goal of this research, which is part of the "Programa de Eco-comedores Escolares de Canarias" programme, which seeks to generate healthy, balanced and affordable menu plans for regional school cafeterias.

This work presents a novel constrained multi-objective formulation of the well-known menu planning problem (MPP) [2]. Specifically, the version of the MPP considered herein is a multi-objective formulation that includes a set of daily and n-days constraints for several nutrients, as introduced in [3], as well as the two objective functions related to the cost of the menu and the level of repetition of courses and food groups proposed in [4].

Planning and scheduling problems have been successfully solved with meta-heuristics. Particularly, many of the best-known solutions for problems in this area have been achieved with memetic algorithms (MAs) [5–8]. In [3], an MA was proposed to deal with a single-objective constrained formulation of the MPP where the cost of the menu had to be minimised. To do so, a specific iterated local search (ILS) was designed, as well as an ad-hoc crossover operator. Additionally, the MA included an explicit mechanism to promote diversity in the decision variable space in order to avoid premature convergence. The above allowed high-quality solutions to be attained. The working hypothesis herein is that by using our novel multi-objective constrained formulation of the MPP, which considers the cost of the plan and at the same time the level of repetition of the specific courses and food groups contained in the plan, it is possible to find solutions that are similar in terms of the cost to those provided by the aforementioned single-objective MA, but significantly better regarding the level of repetition.

Taking these findings into account, this work presents a novel multi-objective approach based on the well-known multi-objective evolutionary algorithm based on decomposition (MOEA/D) [9]. MOEA/D was applied to the menu planning problem in a previous work carried out by the authors [10]. It was compared to the well-known Non-dominated Sorting Genetic Algorithm II (NSGA-II) [11] and Strength Pareto Evolutionary Algorithm 2 (SPEA2) [12]. Since knowledge about the menu planning problem was not considered, ad-hoc variation operators, as well as an improvement operator, were not incorporated into MOEA/D. Results showed that MOEA/D was outperformed by NSGA-II and SPEA2. In order to improve the performance of MOEA/D when dealing with the menu planning problem, in the current work, an extension of the ILS applied by the single-objective MA proposed in [3] was considered as the improvement operator of MOEA/D. We should note that, in opposition to other approaches, MOEA/D facilitates the incorporation of an improvement operator, which is an important component to quickly yield feasible solutions. Finally, the ad-hoc crossover operator proposed by the same authors was also integrated into MOEA/D.

As a result, our proposal is referred to as ILS-MOEA/D, and in contrast to the single-objective optimiser presented in [3], it does not include an explicit strategy to manage diversity in the decision variable space. The reason is that, as in other problems [13], the promotion of diversity in the objective function space, imposed by the use of different weights in ILS-MOEA/D to decompose the problem, causes an implicit preservation of diversity in the decision variable space. A wide experimental evaluation is carried out herein, where the novel ILS-MOEA/D is compared against the single-objective MA proposed in [3], in terms of the quality of the solutions attained by both optimisers.

Bearing all the above in mind, the main contributions of this work are the following:


The rest of this paper is structured as follows. Section 2 provides a review of related works and presents our new formulation of the MPP. Afterwards, the novel ILS-MOEA/D is detailed in Section 3. Then, the experimental evaluation performed to validate our proposals is presented in Section 4. The results obtained from the experiments are discussed in Section 5. Finally, the conclusions are exposed in Section 6, together with some lines of further research.

#### **2. Menu Planning Problem**
