  In this paper, we introduce an improved upper bound for the efficiency of
Nash equilibria in utilitarian scheduling games on related machines. The
machines have varying speeds and adhere to the Shortest Processing Time (SPT)
policy as the global order for job processing. The goal of each job is to
minimize its completion time, while the social objective is to minimize the sum
of completion times. Our main result provides an upper bound of
$2-\frac{1}{2\cdot(2m-1)}$ on the price of anarchy for the general case of $m$
machines. We improve this bound to 3/2 for the case of two machines, and to
$2-\frac{1}{2\cdot m}$ for the general case of $m$ machines when the machines
have divisible speeds.
