A timed Petri net, an extended model of an ordinary Petri net with introduction of discrete time delay in firing activity, is practically useful in performance evaluation of real-time systems and so on. Unfortunately though, it is often too difficult to solve (efficiently) even most basic problems in timed Petri net theory. This motivates us to do research on analyzing complexity of Petri net problems and on designing efficient and/or heuristic algorithms. The minimum initial marking problem of timed Petri nets (TPMIM) is defined as follows: “Given a timed Petri net, a firing count vector X and a nonnegative integer π, find a minimum initial marking (an initial marking with the minimum total token number) among those initial ones M each of which satisfies that there is a firing scheduling which is legal on M with respect to X and whose completion time is no more than π, and, if any, find such a firing scheduling.” 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 TPMIM. The subject of the paper is to propose two pseudo-polynomial time algorithms TPM and TMDLO for TPMIM, and to evaluate them by means of computer experiment. Each of the two algorithms finds an initial marking and a firing sequence by means of algorithms for MIM (the initial marking problem for non-timed Petri nets), and then converts it to a firing scheduling of a given timed Petri net. It is shown through our computer experiments that TPM has highest capability among our implemented algorithms including TPM and TMDLO.

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 M0, with the minimum total token number, for which there is a sequence δ of transitions such that each transition t appears exactly X(t) times in δ, the first transition is enabled at M0 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.

The minimum initial marking problem MIM of Petri nets is described as follows: "Given a Petri net and a firing count vector X, find an initial marking M0, with the minimum total token number, for which there is a sequence δ of transitions such that each transition t appears exactly X(t) times in δ, the first transition is enabled at M0 and the rest can be fired one by one subsequently." This paper proposes two heuristic algorithms AAD and AMIM + and shows the following (1) and (2) through experimental results: (1) AAD is more capable than any other known algorithm; (2) AMIM + can produce M0, with a small number of tokens, even if other algorithms are too slow to compute M0 as the size of an input instance gets very large.

This paper proposes a new heuristic algorithm FMDB for the minimum initial marking problem MIM of Petri nets: "Given a Petri net and a firing count vector X, find an initial marking M0, with the minimum total token number, for which there is a sequence δ of transitions such that each transition t appears exactly X(t) times in δ, the first transition is firable on M0 and the rest can be fired one by one subsequently. " Experimental results show that FMDB produces better solutions than any known algorithm.