TECHICAL Series MATHEMATICS • INFORMATICS • PHYSICS Series PHILOLOGY Series ECONOMIC SCIENCES Sereies EDUCATIONAL SCIENCES Series LAW AND SOCIAL SCIENCES Series
Optimum Flow in a Transportation Petri Net
Vol LXIII • No. 2/2011
Vasilica Bordea*, Petru Junie**, Octav Dinu***, Cristian Eremia**
* Universitatea Maritimă din Constanţa, Str. Mircea cel Bătrân 104, Constanţa, Romania
** Universitatea Politehnica din Bucureşti, Splaiul Independentei 313, Bucureşti, Romania,
e-mail: junpetre2000@yahoo.com
*** Universitatea Petrol-Gaze din Ploieşti, Bd. Bucureşti 39, Ploieşti, Romania,
e-mail: octavytza@yahoo.com

 Keywords   Petri nets, discrete events system, optimum flow, algorithms.

 Abstract
The problem of the optimum flow at minimum costs applied on a transportation Petri network, capacity restricted, is solved in this paper by means of an algorithm based on the network expressed by the incidence matrix, W and described by position set P, transition set T, capacity vector C and cost vector H. The algorithm ? maximum flow at minimum cost (MFLC), establishes in the first stage the capacity matrix, the flow vectors corresponding to execution sequences (also determined by the algorithm), the matrices associated to cost vector, Hi,., cost matrix Q and the associated matrix [FI,Q]. On the last one are selected the flows corresponding to the minimum costs qi Residual matrices of capacities give the opportunity to determine new components of the flow. The components of the flow vector FI expected at the end of the algorithm process are the maximum values of the flows along the minimum cost execution sequences. An application in the field of maritime transport checks the algorithm efficiency.

 Rezumat
Problema fluxului optim cu cost minim aplicată pe o retea Petri de transport cu restrictie de capacitate este rezolvată în lucrare printr-un algoritm construit pe reteaua exprimată prin matricea de incidentă, W si descrisă prin multimea P, a pozitiilor, multimea T, a tranzitiilor, vectorul capacitate C si vectorul cost, H. Algoritmul ? flux optim cu cost minim(FOCM), determină în prima sa parte, matricea capacitate, vectorii flux corespunzători secventelor de executie (determinate si ele de algoritm), matricile asociate vectorilor cost, Hi, matricea cost Q si matricea asociată [FI,Q]. Pe aceasta din urmă se selectează fluxurile corespunzătoare costurilor qi, minime.Matrici reziduale ale capacitătilor dau posibilitatea determinării de noi componente ale fluxului.Componentele vectorului flux, FI, la finalul derulării algoritmului sunt valorile maxime ale fluxurilor de-a lungul secventelor de executie cu cost minimO aplicatie din domeniul transportului maritim verifică competentele algoritmului.



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