**6. Conclusion**

70 Petri Nets – Manufacturing and Computer Science

The other one is proposed by Li *et al*.

needed by using *Algorithm P*.

II

**Table 10.** Comparison of the Controlled Systems.

**5. Comparison with existing methods** 

One can attempt to make a comparison with the previous methods12, 16, 24 in terms of efficiency. The first one proposed by Uzam12, called *Algorithm U*, is totally based on the theory of regions. It solves six ESSPs in Example I. Then three control places are added on the net such that the controlled net is live and reversible. As for Example II, it solves 59 MTSIs. Nine control places are obtained. However, the proposed deadlock prevention policy called *Algorithm P* solves only two and 18 CMTSIs in Examples I and II, respectively.

regions is used in Example I. Notice that both the controlled results of *Algorithms L* and *U* are the same in Example I. In Example II, using *Algorithm L,* eight MTSIs are solved and six control places are computed. However*,* under the two-stage control policy, only one set of MTSI is needed by using our new policy to obtain the controlled result that is as the same as *Algorithm L* in Example II. Note that both the definitions of ESSP and MTSI are the same. Hence, ESSP and CMTSI can be regarded as MTSI for the comparison purpose. The detailed comparison results are given in Table 10. However, only 18 MTSIs among 59 MTSIs are

I 11 3 6, 6, 2 3, 3, 3 15

For Example II, eight MTSIs are required to obtain the six control places under *Algorithm L*. Hence, one can infer that its performance is better than that of *Algorithm U*. Only one set of CMTSI is needed to obtain the same control result by *Algorithm P*. As a result, one can

To examine and compare the efficiency of the proposed method with those in16, 24 in a system with large reachability graphs, one can use eight different markings of *p*1, *p*8, *p*15, *p*18, and *p*19: [6, 5, 1, 1, 1]T, [7, 6, 2, 1, 1]T, [7, 6, 1, 2, 1]T , [7, 6, 1, 1, 2]T, [9, 8, 2, 2, 2]T, [12, 11, 3, 3, 3]T, [15, 14, 4, 4, 4]T, and [18, 17, 5, 5, 5]T. Tables 11 and 12 show various parameters in the plant and partially controlled net models, where *M* (*p*15), *M* (*p*18), and *M* (*p*19) vary; |*R*|, |*ML*|, |*RD*|*U*, |*RD*|*L*, indicate the number of reachable markings (states), legal markings, and dead markings under *Algorithms U* and *L,* respectively. Additionally, MTSIs of *Algorithms U, L*, and *P* are symbolized by ||*U*, ||*L* and ||*P*, respectively. The last column is *ra* = ||*P* / ||*U* in Table 11, and *rb* = |*P* / ||*L* in Table 12. Notably, *Algorithm G*13 can be regarded as

EXAMPLE # of Places # of Resource Places MTSI

19 6

II (two stages) , 8, 1 , 6, 6

conclude that our proposed policy is more efficient than the other two methods.

16, 24 called *Algorithm L* in which only the theory of

Control Places *U, L, P* 

Reachable Markings

205

*U, L, P* 

59, ,18 9, , 6

The proposed policy can be implemented for FMSs based on the theory of regions and Petri nets, where the dead markings are identified in its reachability graph. The underlying notion of the prior work is that many inequalities (i.e. MTSIs) must be solved to prevent legal markings from entering the illegal zone in the original PN model. One must generate all MTSIs in a reachability graph and require high computation. This work proposes and uses CMTSI to overcome the computational difficulty. The detail information is also obtained in existing literatures.26-29 The proposed method can reduce the number of inequalities and thus the computational cost very significantly since CMTSIs are much less than MTSIs in large models. Consequently, it is optimal with much better computational efficiency than those existing optimal policies12-13, 16. More benchmark studies will be desired to establish such computational advantages of the proposed one over the prior ones. It should be noted that the problem is still NP-hard the same as other optimal policies due to the need to generate the reachability graph of a Petri net. The future research is thus much needed to overcome the computational inefficiency of all these methods.

A Computationally Improved Optimal Solution

for Deadlocked Problems of Flexible Manufacturing Systems Using Theory of Regions 73

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