Deadlock control of flexible manufacturing systems (FMS) using Petri nets has been a very active research topic for the past two decades. Siphon-based deadlock control of FMS suffers from reaching fewer states than those of optimal control. This paper proposes three sets of siphon-based liveness-enforcing supervisors (LES) for a well-known FMS to obtain near optimal solutions. The first set includes 11 control places (monitors) providing 21,363 good states. The second set consists of 14 monitors providing 21,562 good states, the same as that using the method of the first-met bad marking (FBM) (based on an iterative method using reachability graph analysis) but with fewer monitors and control arcs. This result is achieved by refining some monitors into several monitors with smaller controller regions, causing fewer disturbances to the original uncontrolled model. The third set contains 15 monitors providing 21,573 good states - the best suboptimal solution so far in the literature. This result is obtained by further shrinking the controller region of the second set. Formal proof of the correctness is provided. Although reaching eight states fewer than that by the set-covering approach, it does not employ weighted control arcs and runs more efficiently. It also computes all lost states based on invariant without reachability analysis. Application to another well-known example is also illustrated.