The manufacturing systems have many configurations. The hybrid flow-shop configuration is one of the most
manufacturing systems that have many applications. One of these applications is the metal nails factory. This
system consists of different stages and each stage has different parallel machines. The machines are unrelated
parallel or multi-machines. The speed of the machines can be uniform or variable. With respect to the jobs, the
stream of material flows in the same direction among the stages and the jobs can be split into batches to utilize the
machines. In this paper, a system of hybrid flow-shop of machines, which are unrelated, have variable speed, and
have eligibility constraints is considered. The jobs have variable sub-lot sizes. The variability of the sub-lots sizes
can be achieved by splitting the sub-lots into smaller sub-lots from one stage to another. A real case for this system
is solved using a non-linear mixed-integer programming model and the sub-lot size and schedule that minimize the
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