## 1 Introduction
We can find multi-model lines somewhere in the evolution from single-model assembly lines to mixed-model lines. A line that spends all year assembling the same model may evolve and spend several weeks on model X and then several more weeks on model Y. From the point of view of LM, large batches of X and Y are sources of _muda_ such as overproduction and inventory, besides conveying a lot of unevenness. Since the evolution of assembly lines is based on the reduction in lot size, the reduction in time devoted to each model and the introduction of more models, the following step is to produce, each day, smaller batches of X and Y, and this is the situation that we analyze in the greatest part of this paper. The last step, so far, is the mixed-model line, where the batch size is one and the assembly line tries to meet the mixed-model demand that companies encounter every day. Currently, in the dawn of Industry 4.0 [8; 10], assembly lines continue to be widely used. They have many advantages but drawbacks too. Based on the principles of specialization and work division [11], the main benefit of assembly lines comes from the specialization of workers and machines. They perform repeatedly specific tasks that do not require the ability of a craftsman. Although little training is generally required--which offers greater employment opportunities--this system allows mass production of complicated goods such as cars or airplanes, while keeping labor costs low. Although the total assembly time may be long, the cycle time is usually short. For example, manufacturing a vehicle may take 10 h but we can see that every hour a vehicle leaves the assembly line [12]. These features are the basis of a highly efficient system with low costs per unit. In consequence, assembly lines have been adopted by lean companies because they have proved to be very efficient in the production of standardized products and even in low volume production of customized products [13]. In addition, one-piece flow avoids different types of _muda_ such as dispensable transport, waiting, unnecessary movements, excessive work-in-process inventory and lack of quality, while saving floor space and allowing visual control [5; 6; 14]. Furthermore, since LM emphasizes the exigency to manufacture a variety of products in small batches to satisfy customer demand, in times of fierce competition, both mixed-model and multi-model assembly line problems become highly relevant for academia and industry [13]. [PERSON] et al. [15] specified the main drawbacks of assembly lines:  
1. Low flexibility with respect to changes in demand. In addition, when models change, it is necessary to re-balance the line. Traditional lines were completely inflexible because they were designed for a single product but, currently, we may have multi-model and mixed-model lines as described in this paper. They allow production to meet demand. 2. Balance loss due to an imperfect balance of the line and stochastic task times, resulting in blockage (parts have to wait before a workstation) and starvation (workers have to wait) [11]. 3. Poorly skilled people, poor work environment and poor quality. Luckily, LM gives a lot of importance to people engagement, which results in cross-training, teamwork and people committed to quality (or _jidoka_ in Japanese) [4; 5]. 4. High costs of material handling (although this depends on the characteristics of the product) and high work-in-process, if lines are long. However, work-in-process is always smaller than in traditional batch-and-queue systems [12]. Additionally, repetitive motions, workers' postures (especially upper limbs), lifting efforts and the pace of work may become important threads for the safety and health of assembly workers [10; 16], not to mention that such repetitive and monotonous work is not motivating, does not favor creativity and may result in alienation. Furthermore, especially in lean factories, continuous work and lack of buffers result in time pressure and stress [17].