Two Enhanced Heuristic Algorithms for the Minimum Initial Marking Problem of Petri Nets

The minimum initial marking problem of Petri nets (MIM) is defined as follows: "Given a Petri net and a firing count vector X. find an initial marking M-0. with the minimum total token number. for which there is a sequence delta of transitions such that each transition t appears exactly X(t) times in delta, the first transition is enabled at M-0 and the rest can be fired one by one subsequently." In a production system like factory automation, economical distribution of initial resources, from which a schedule of job-processings is executable. can be formulated as MIM. AAD is known to produce best solutions among existing algorithms. Although solutions by AMIM+ is worse than those by AAD, it is known that AMIM+ is very fast. This paper proposes new heuristic algorithms AADO and AMDLO, improved versions of existing algorithms AAD and AMIM+, respectively. Sharpness of solutions or short CPU time is the main target of AADO or AMDLO, respectively. It is shown, based on computing experiment, that the average total number of tokens in initial markings by AADO is about 5.15% less than that by AAD, and the average CPU time by AADO is about 17.3% of that by AAD. AMDLO produces solutions that are slightly worse than those by AAD, while they are about 10.4% better than those by AMIM+. Although CPU time of AMDLO is about 180 times that of AMIM+, it is still fast: average CPU time of AMDLO is about 2.33% of that of AAD. Generally it is observed that solutions get worse as the sizes of input instances increase, and this is the case with AAD and AMIM+. This undesirable tendency is greatly improved in AADO and AMDLO.