## 1 Introduction  
Dyes are colored substances, usually organic compounds (natural or synthetic), with affinity to the substrate to which they are applied. Nowadays, more than one hundred thousand kinds of commercial dyes are utilized, with an annual production of over nine million tones [1]. Dyes have a wide range of applications: paints, plastics, textiles, leather, food, and cosmetics [1; 2; 3], and a large amount of these dyes are wasted during manufacturing and dying procedures, which generates a large volume of dye-containing wastewater [4; 5]. Therefore, the measurement of dye concentrations is a key factor in the control of dyeing processes. Tartrazine (TZ), with a molecular formula of \(C_{16}H_{9}N_{4}Na_{3}O_{9}S_{2}\), is an orange water-soluble dye with applications in food, cosmetic, textile, and pharmaceutical industries [6]. Amarnath red (AR), \(C_{20}H_{11}N_{2}Na_{3}O_{10}S_{3}\), is a synthetic colorant that can be found in common food products such as beverages, dry mix products, candy, and bakery products [3; 7]. Brilliant Blue FCF (B1), \(C_{37}H_{36}N_{2}O_{9}S_{3}Na_{2}\), is used as a food dye in ice cream, candies,drinks, and other products [8]. These dye contents are strictly controlled since they can prompt allergies and other diseases in sensitive people [9; 10; 11]. The chemical structures of TA, AR, and B1 are illustrated in Figure 1.  
Dye mixture concentration levels in water or wastewater can be determined by high-performance liquid chromatography (HPLC) [12; 13; 14], gas chromatography/mass spectrometry [15], and liquid mass spectrometry/chromatography [16]. However, chromatographic determination of dyes in mixtures is a laborious and time-consuming procedure [16]. The UV-Vis spectrophotometric technique can also be used for this purpose because dyes are highly absorbent species in the visible region, and the technique has a low operating cost and is a relatively simple procedure [13; 17; 18]. One of the common challenges for the spectrophotometric technique in multi-component dye mixtures is the fact that, usually, mixtures present the superposition or overlapping of the absorption spectra, which prevents them from being determined by direct measurement by UV-Vis spectrophotometry [17]. In recent years, novel algorithms calibrated using UV-Vis spectrophotometry data for the simultaneous estimation of chemical constituents in binary and ternary aqueous mixtures have been developed. For example, [PERSON] and co-workers [18] focused on a genetic algorithm (GA) along with partial least-squares (PLS) regression for Nitrobenzene, aniline, and phenol concentrations for the determination of individual components in ternary synthetic mixtures. [PERSON] et al. [19] reported the usage of a multivariate curve resolution (MCR-ALS) alternating with PLS regression for the determination of Beta-Antagonists in complex matrices, and [PERSON] and [PERSON] [20] used derivative spectrophotometry for the simultaneous determination of Paracetamol, Caffeine, and Ibuprofen in their ternary mixture. In these investigations, the authors were able to analyze the mixtures with high overlapping absorption spectra without the need for the prior separation of the species. On the other hand, very recently, the use of machine learning models like artificial neural networks (ANN) have been implemented to solve problems in very diverse areas, like engineering and science [21], because ANNs are considered tools that can help us  
Figure 1: Chemical structure of (**a**) TZ, (**b**) AR and (**c**) B1. to analyze variable relationships in complex systems from large datasets [22]. However, the performance of conventional ANNs depends strongly on the size of the input data and large variables in the data, which may contain redundant information, which would increase the complexity of the computation.