*2.1. Background*

Menu planing has been solved by using computers since early 1960 [2,14]. Many of the proposed formulations are NP-complete, meaning it is quite a complex task [15]. In essence, the classical MPP aims to find a combination of dishes which satisfies certain restrictions involving budget, variety and nutritional requirements for an n-day sequence. Even though there is no consensus on the specific objectives that an MPP formulation should consider, in almost every formulation, it is common for the cost of the menu plan to be considered one of the main objectives to optimise [2–4,10,16,17]. Other options for defining the objective functions are the users' preferences for certain foods, the level of adequacy or the level of acceptance [18–21].

The constraints considered in menu planning problems are usually based on the nutritional requirements that meal plans have to satisfy. As a result a set of constraints is defined that models the recommended minimum and maximum amounts of different nutrients [21–26]. Other constraints consider the variety of the meals, their predominant colour, their consistency, the time required to prepare them and food that cannot be consumed, among others [19,22,25,27]. Two of the most frequently used techniques to handle constraints are based on the application of repairing methods [4,10,19,26,28,29] or penalisation functions [16,30,31]. In the first case, operators are applied to an infeasible solution until it becomes feasible. In the second case, the fitness function is penalised somehow depending on the degree of infeasibility of the corresponding solution. Hence, the higher the degree of infeasibility of the solution, the larger the probability to be discarded.

Both single-objective [3,16,19,32] and multi-objective optimisers [4,10,17,28] have been devised for the MPP. In the single-objective case, most of the formulations consider the cost as the only objective to optimise, while the nutritional requirements are used as constraints. In the case of multi-objective formulations, the cost is always considered as one of the objective functions to optimise [29–31]. Additionally, the seasonal quality, food flavour and food temperature [29], food preferences [30] and the nutritional error [31] are considered as other objective functions. In almost all cases, the nutritional requirements, as well as the users' personal preferences, model the constraints of the multi-objective formulations.

Although there exist many different types of algorithms for solving this problem, a high percentage of published papers use evolutionary algorithms (EAs) or other types of meta-heuristics due to the benefits they offer, such as robustness, reliability, global-search ability and simplicity [16,20,21,26,28]. EAs are approximated methods based on the concept in natural evolution of survival of the fittest individual [33]. Given a population of individuals in some environment with limited resources, the competition for survival causes natural selection, with the fittest individuals more likely to survive and reproduce. In addition, being approximated methods means that although there is no guarantee of obtaining the optimal solution to a problem, high-quality solutions can be found in a reasonable period of time. The classical genetic algorithm (GA) is the most common approach for solving single-objective formulations of the MPP [16,34], while previous multi-objective formulations of the MPP have been frequently addressed by applying the state-of-art NSGA-II, such as the work proposed in [28,30,31].

The lack of ad-hoc operators for the MPP could possibly yield sub-optimal solutions due to the problem of premature convergence. In order to avoid this, problem-specific operators or procedures, such as intensification mechanisms, have been included into EAs, resulting in memetic algorithms (MAs) [35,36]. MAs can be seen as the combination of an EA with an intensification mechanism in order to improve the general performance of the optimiser [37]. An example of a single-objective MA successfully applied to the MPP is that proposed in [3].

Additionally, a software named SCHOOLTHY was proposed in [4], which allows menus to be planned automatically. The tool uses an MPP formulation similar to the one proposed in this work to generate not only affordable plans, but also varied from a nutritional standpoint. Although it was designed for school cafeterias, it could be adapted to other environments, such as hospitals, prisons and retirement homes, among others.

As we previously mentioned in Section 1, in the current work, we present a novel constrained multi-objective formulation of the MPP. It consists of the same set of nutritional daily and n-days (global) constraints presented in [3]. At the same time, the two objective functions proposed in [4], i.e., the cost of the meal plan and its level of repetition of courses and food groups, are also considered. As far as we know, this is the first time that an objective function modelling the level of repetition of specific courses and food groups is optimised together with the cost under a multi-objective formulation of the menu planning problem, which in addition considers the managemen<sup>t</sup> of daily and global nutritional constraints. Those constraints are successfully managed by considering the infeasibility degree of a particular solution, thus giving preference to those solutions with, first, a lower infeasibility degree, and second, a lower fitness in terms of both aforementioned objective functions. Furthermore, the application of a multi-objective memetic approach based on decomposition that includes knowledge about the menu planning problem in the form of a tailored improvement operator and an ad-hoc crossover operator had never been carried out before. The above allows feasible solutions of the multi-objective constrained formulation that we are presenting herein to be attained quickly.
