## 1 Introduction  
Agriculture is currently facing a major problem with sustainability. The traditional agricultural system requires a huge amount of land surface, water, and materials, which are necessary to supply the nutrients necessary for the growth of crops. The system will be put under further pressure by the increasing global population in the next decades [1]. Vertical farming (VF) is characterized by the indoor farming of crops in a controlled environment and with the use of artificial lightning as the only source of light [2]. VF systems have many advantages that make them attractive as an alternative to conventional agriculture or suitable for the food self-sufficiency of cities [3]. In fact, the total control of the growing substrate enhances the use of water and avoids the waste of nutrients [2, 4, 5], while the closed environment helps to reduce the presence of pests and the use of pesticides. Temperature and humidity control avoid problems like extreme weather conditions, e.g., droughts or frosts. Moreover, the production is stable and continuous all year, resulting in higher yields with respect to open-field agriculture [6]. Finally, the use of stacked growth layers means the efficient use of space and high scalability, which allow VF placement even in working places [7]. In fact, it has been demonstrated that the addition of a VF system in an office can reduce the CO\({}_{2}\) concentration by up to 34%, and the energy for mechanical ventilation by up to 58%, depending on the number of people and crop growth stage [8]. However, VF systems' implementation requires optimizing every aspect of the growth of crops. The simultaneous management of different objectives is hard to achieve, e.g., the minimization of the number of shelves used, change of shelf configurations, or the unmet amount of crop demands [9]. Despite a generally positive attitude among stakeholders towards VF, the associated amount of energy required for their management and consequent high set-up prices hinder their diffusion [10]. In fact, lightning systems could require in general a high amount of energy. The electricity needed for VF management represents 66% to 85% of the total carbon footprint, which is 2.3 to 16.7 times that of conventional open field farms, depending on the type of crops and on the extension of the open field area [11]. In addition, the trade-off between the soil area saved using VF and the soil necessary for the production of energy through renewable sources (wind, photovoltaic) is not trivial. The amount of land area saved is positive only for some crops, e.g., lettuce or wheat. Otherwise, additional amounts of land are required to produce energy to power VF systems [12]. Therefore, optimization of the artificial lightning is pivotal for the future development of VF. In this context, light-emitting diodes (LED) have many advantages with respect to fluorescent and HPS (high-pressure sodium) lamps, since LEDs are more energy efficient and economical [13; 14]. LEDs also have a longer operational lifetime (30,000-50,000 h) than fluorescent (20,000 h) or incandescent lights (1000 h), and the limited emission of radiant heat allows their placement near the plant canopy [15]. While the use of supplemental artificial lighting in greenhouses accounts for no more than 25-30% of total costs, and the use of LEDs instead of HPS lamps results in economic advantages after 7 years [15], the use of solely artificial light for crops growth poses other challenges. The conversion from LED light to plant biomass (1.2-20.1%) is more efficient than that of sunlight (0.03-0.62%), even when there is the action of other factors that could lower the efficiency to 0.1-5.7%. These factors are mainly the type of plant and the presence of air conditioning in the growth environment [16]. Another advantage of LEDs is that they have tunable wavelengths, so light recipes can be tailored according to the type of plant and its stage of growth. Every wavelength acts differently on the morphology and physiology of plants [17].