On employing pythagorean fuzzy processing time to minimize machine rental cost

The aim of this paper is to obtain the minimal rental cost of the three-phases flow shop scheduling problems. A novel strategy to tackle this issue using Pythagorean fuzzy processing time is introduced. It depends on converting the three stages machine into two stages when the minimum value of processing time of the first machine is greater than the maximum value of processing time of the second machine. The vague processing time does not convert to its crisp form. The jobs sequencing in machines is obtained using Johnson procedure. The zero element of the Pythagorean set is defined as, ${\tilde{\mathrm{O}}}^{\mathrm{p}}=\left(0,1\right)$ i.e., it has zero membership and one nonmembership values. A numerical example include Pythagorean rental cost is delivered to demonstrate the reliability of the suggested strategy. The idle time, utilization time, and the overall cost are calculated. The idle time of all machines is zero, which minimize the required time and hence, minimize the total rental cost.