**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.

A Computationally Improved Optimal Solution

*U ||*

*<sup>P</sup> ra*

*<sup>P</sup> rb*

for Deadlocked Problems of Flexible Manufacturing Systems Using Theory of Regions 71

*<sup>U</sup> ||*

*<sup>L</sup> ||*

*L*

 *||*

*Algorithm U* in Table 11 since the number of MTSIs and ESSPs are the same. In table 11, here, *Nsep* represents the number of MTSIs, and the *Nsep*/*U, Nsep*/*L* and *Nsep*/*P* represent the number of MTSIs of *Algorithms U, L*, and *P*, respectively. Obviously, the number of ||*U* in the plant model grows quickly from cases 1 to 8. For instance, when *M* (*p*15) = *M* (*p*18) = *M* (*p*19) = 5, ||*U* = 4311, meaning that one must solve 4311 MTSIs when *Algorithm U* is used. However, since ||*P* = 228, only 228 equations (MTSIs) need to be solved under *Algorithm P*. As a

1 282 205 16 59 18 30.5% 2 600 484 27 95 28 29.5% 3 972 870 26 103 26 25.2% 4 570 421 16 107 19 17.8% 5 4011 3711 42 288 42 14.6% 6 27152 26316 84 886 84 9.5% 7 124110 122235 145 2115 145 6.9% 8 440850 437190 228 4311 228 5.3%

**Table 11.** Parametersin the Plant and Partially Controlled Models with Varying Markings: *U* vs. *P*.

**Table 12.** Parametersin the Plant and Partially Controlled Models with Varying Markings: *L* vs. *P*.

In Table 12, the number of MTSIs calculated by *Algorithm L* can be controlled, but *Algorithm P* is more efficient in these cases. For instance, when *M* (*p*15) = *M* (*p*18) = *M* (*p*19) = 5, ||*L* = 192, meaning that one still has to solve 192 MTSIs when *Algorithm L* is used. However, ||*<sup>P</sup>* =108. Only 108 MTSIs need to be solved by using *Algorithm P*. Importantly, the computational cost can be reduced by using our proposed method when it is compared with those in12, 16. In conclusion, *Algorithm P* is more efficient in large reachability graph cases

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

1 282 205 1 8 1 12.5% 2 600 484 1 8 1 12.5% 3 972 870 6 10 6 60.0% 4 570 421 1 8 1 12.5% 5 4011 3711 9 15 9 60.0% 6 27152 26316 28 48 28 58.3% 7 124110 122235 60 105 60 57.1% 8 440850 437190 108 192 108 56.3%

result, *Algorithm P* is more efficient than *Algorithm U* in a large system.

**CASES** *|R| |ML| |RD|*

**CASE** *|R| |ML| |RD|*

than those in12-13.

**6. Conclusion** 

The other one is proposed by Li *et al*. 16, 24 called *Algorithm L* in which only the theory of 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 needed by using *Algorithm P*.


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

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

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 *Algorithm U* in Table 11 since the number of MTSIs and ESSPs are the same. In table 11, here, *Nsep* represents the number of MTSIs, and the *Nsep*/*U, Nsep*/*L* and *Nsep*/*P* represent the number of MTSIs of *Algorithms U, L*, and *P*, respectively. Obviously, the number of ||*U* in the plant model grows quickly from cases 1 to 8. For instance, when *M* (*p*15) = *M* (*p*18) = *M* (*p*19) = 5, ||*U* = 4311, meaning that one must solve 4311 MTSIs when *Algorithm U* is used. However, since ||*P* = 228, only 228 equations (MTSIs) need to be solved under *Algorithm P*. As a result, *Algorithm P* is more efficient than *Algorithm U* in a large system.



**Table 11.** Parametersin the Plant and Partially Controlled Models with Varying Markings: *U* vs. *P*.

**Table 12.** Parametersin the Plant and Partially Controlled Models with Varying Markings: *L* vs. *P*.

In Table 12, the number of MTSIs calculated by *Algorithm L* can be controlled, but *Algorithm P* is more efficient in these cases. For instance, when *M* (*p*15) = *M* (*p*18) = *M* (*p*19) = 5, ||*L* = 192, meaning that one still has to solve 192 MTSIs when *Algorithm L* is used. However, ||*<sup>P</sup>* =108. Only 108 MTSIs need to be solved by using *Algorithm P*. Importantly, the computational cost can be reduced by using our proposed method when it is compared with those in12, 16. In conclusion, *Algorithm P* is more efficient in large reachability graph cases than those in12-13.
