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Analytical modeling of part supply process in a bin-kanban system with logistic trains

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Fabio Bursi
Fabio Bursi

This paper deals with the part supply process in a mixed-model assembly line served by a logistic train. The mixed-model assembly line is made up of a number s of stations each of which is provided with bins of different parts. According to the concept of bin-kanban, when a bin becomes empty it represents a request for a replenishment at that station. In this system the empty bins are retrieved by a logistic train that transports them to a centralized warehouse area where they are refilled (supermarket). The duration of a tour of the logistic train is a stochastic variable depending on the number of stations that require a service and the number of bins that must be retrieved or supplied. We assume Poisson arrival processes and exponential service times. The aim of this paper is to analytically model the system described above as a polling system, that can be basically defined as a collection of queues served by a single server. In particular, we reformulate the model proposed by Blanc (1992a) for polling systems with limited service disciplines in order to model a polling system where the server has a finite capacity.

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Analytical modeling of part supply process in a bin-kanban system with logistic trains Fabio Bursi, Elisa Gebennini, Andrea Grassi, Bianca Rimini
Motivation Part-supply process in a mixed-model assembly line made up of a number s of stations served by a logistic train. 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece bin-kanban: an empty bin represents a request for a replenishment
Goal 3 Objective: to analytically model the system in order to support the dimensioning of the rack lanes at the assembly stations the choice of the capacity of the logistic train Number and dimensions of the wagon Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
Problem statement 4 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Our idea is to model the bin-Kanban system as a capacitated polling system with s queues stations single server logistic train The logistic train follows a single fixed route and visits each station in a cyclic and fixed manner; The duration of a route is a stochastic variable; Focus on the withdrawal of the empty bins; Bins are supposed of identical standardized size; Arrival process of empty bins at each station is a Poisson process; Service times is exponentially distributed;
Assumptions s queues of jobs and a single server; jobs arrive at the queues according to a Poisson process; service times are exponentially distributed; the server inspects the queues in a cyclic and fixed order; as soon as the server complete a cycle, it is able to start the next cycle with the maximum capacity K available (no supermarket); switchover times are neglected; each queue may contain an unbounded number of jobs; for each cycle, the server can process K jobs at most. 5 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
Notation : arrival rate of jobs at queue j; : service rate of jobs at queue j; = load offered at queue j. The total load of the system is = =1 ; = parameter introduced to address the model with the power-series expansions of the state probabilities in terms of the load ; the server capacity per cycle; number of jobs in queue j; 6 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
7 Capacitated polling system Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
8 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
9 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
10 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
11 consequently We introduce a reformulation of the so-called li-limited polling problem: in a li-limited polling system the server can process at most li jobs at each queue i in the proposed model the server can process at most K jobs per cycle (i.e., by considering all the s queues) Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
12 Similarly as in Blanc (1992)* the queue length process is transformed into a Markov process by introducing a polling table * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. 12 K K+1 K+2 2K (i-1)K+1 (i-1)K+2 iK (s-1)K+1 (s-1)K+2 sK Supplementary variable H indicating the actual position on the table; L=sK table length; Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
13 * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. The value of the variable H: is increase by one whenever a service has been completed or when queue l(H) is empty, unless the whole system has become empty or the server capacity is full is set to 1 when the system is empty or the server becomes full Polling Table Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
14 * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. The value of the variable H: is increase by one whenever a service has been completed or when queue l(H) is empty, unless the whole system has become empty or the server capacity is full is set to 1 when the system is empty or the server becomes full New aspect with respect to Blanc (1992)* New variable representing the available capacity of the server before processing a new job at a certain queue. Polling Table Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
15 Thus, let = (1, , ) be the vector of the number of jobs in the queues, with values = (1, , ) H be supplementary variable H indicating the actual position on the polling table, with values h=1,, sK representing the available capacity of the server before processing a new job at a certain queue, with values (depending on K and the position h) The state of the system is = ( , , ) The state probability is denoted by ( , , ) System state Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
16 s=3 K=2 How to get state ((2,1,0),1,K) ? 2 Example Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
17 2 2 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
18 2 2 3 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
19 2 1 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
20 2 2 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
21 How to leave the state ((2,1,0),1,K) ? 2 Example Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece 1 ((1,1,0),2,1) 1 ((3,1,0),1,1) 2 ((2,2,0),1,1) 3 ((2,1,1),1,1)
22 In general, when h=1 Balance equations Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece *Blanc, J. P. C. 1998. The power-series algorithm for polling systems with time limits. Probability in the Engineering and Informational Sciences 12 221237. *Blanc, J.P.C. 1990. A numerical approach to cyclic-service queueing models. Queueing Systems 6(1) 173 188. *Blanc, J.P.C. 1992b. Performance evaluation of polling systems by means of the power-series algorithm. Annals of Operations Research 35(3) 155186. The model proposed can be addressed by means of the so-called power-series algorithm (PSA) as in Blanc (1998, 1990, 1992b)
23 The current analytical formulation of the capacitated polling system suffers of two main assumptions: switchover times are neglected times between consecutive cycles are neglected (it can be also treated as a switchover time) but can provide support for: - the dimensioning of the the logistics train; - the dimensioning of the rack lanes at the assembly stations; and provide approximate system information ( avg waiting time, empty bins in the system, ...) Conclusions Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
24 Thank you for your attention Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece

More Related Content

Analytical modeling of part supply process in a bin-kanban system with logistic trains

  • 1. Analytical modeling of part supply process in a bin-kanban system with logistic trains Fabio Bursi, Elisa Gebennini, Andrea Grassi, Bianca Rimini
  • 2. Motivation Part-supply process in a mixed-model assembly line made up of a number s of stations served by a logistic train. 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece bin-kanban: an empty bin represents a request for a replenishment
  • 3. Goal 3 Objective: to analytically model the system in order to support the dimensioning of the rack lanes at the assembly stations the choice of the capacity of the logistic train Number and dimensions of the wagon Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 4. Problem statement 4 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Our idea is to model the bin-Kanban system as a capacitated polling system with s queues stations single server logistic train The logistic train follows a single fixed route and visits each station in a cyclic and fixed manner; The duration of a route is a stochastic variable; Focus on the withdrawal of the empty bins; Bins are supposed of identical standardized size; Arrival process of empty bins at each station is a Poisson process; Service times is exponentially distributed;
  • 5. Assumptions s queues of jobs and a single server; jobs arrive at the queues according to a Poisson process; service times are exponentially distributed; the server inspects the queues in a cyclic and fixed order; as soon as the server complete a cycle, it is able to start the next cycle with the maximum capacity K available (no supermarket); switchover times are neglected; each queue may contain an unbounded number of jobs; for each cycle, the server can process K jobs at most. 5 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 6. Notation : arrival rate of jobs at queue j; : service rate of jobs at queue j; = load offered at queue j. The total load of the system is = =1 ; = parameter introduced to address the model with the power-series expansions of the state probabilities in terms of the load ; the server capacity per cycle; number of jobs in queue j; 6 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 7. 7 Capacitated polling system Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 8. 8 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
  • 9. 9 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
  • 10. 10 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece Capacitated polling system
  • 11. 11 consequently We introduce a reformulation of the so-called li-limited polling problem: in a li-limited polling system the server can process at most li jobs at each queue i in the proposed model the server can process at most K jobs per cycle (i.e., by considering all the s queues) Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 12. 12 Similarly as in Blanc (1992)* the queue length process is transformed into a Markov process by introducing a polling table * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. 12 K K+1 K+2 2K (i-1)K+1 (i-1)K+2 iK (s-1)K+1 (s-1)K+2 sK Supplementary variable H indicating the actual position on the table; L=sK table length; Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 13. 13 * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. The value of the variable H: is increase by one whenever a service has been completed or when queue l(H) is empty, unless the whole system has become empty or the server capacity is full is set to 1 when the system is empty or the server becomes full Polling Table Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 14. 14 * Blanc, J.P.C. 1992. An algorithmic solution of polling models with limited service disciplines. Communications, IEEE Transactions on 40(7) 11521155. The value of the variable H: is increase by one whenever a service has been completed or when queue l(H) is empty, unless the whole system has become empty or the server capacity is full is set to 1 when the system is empty or the server becomes full New aspect with respect to Blanc (1992)* New variable representing the available capacity of the server before processing a new job at a certain queue. Polling Table Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 15. 15 Thus, let = (1, , ) be the vector of the number of jobs in the queues, with values = (1, , ) H be supplementary variable H indicating the actual position on the polling table, with values h=1,, sK representing the available capacity of the server before processing a new job at a certain queue, with values (depending on K and the position h) The state of the system is = ( , , ) The state probability is denoted by ( , , ) System state Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 16. 16 s=3 K=2 How to get state ((2,1,0),1,K) ? 2 Example Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 17. 17 2 2 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 18. 18 2 2 3 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 19. 19 2 1 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 20. 20 2 2 2 Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 21. 21 How to leave the state ((2,1,0),1,K) ? 2 Example Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece 1 ((1,1,0),2,1) 1 ((3,1,0),1,1) 2 ((2,2,0),1,1) 3 ((2,1,1),1,1)
  • 22. 22 In general, when h=1 Balance equations Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece *Blanc, J. P. C. 1998. The power-series algorithm for polling systems with time limits. Probability in the Engineering and Informational Sciences 12 221237. *Blanc, J.P.C. 1990. A numerical approach to cyclic-service queueing models. Queueing Systems 6(1) 173 188. *Blanc, J.P.C. 1992b. Performance evaluation of polling systems by means of the power-series algorithm. Annals of Operations Research 35(3) 155186. The model proposed can be addressed by means of the so-called power-series algorithm (PSA) as in Blanc (1998, 1990, 1992b)
  • 23. 23 The current analytical formulation of the capacitated polling system suffers of two main assumptions: switchover times are neglected times between consecutive cycles are neglected (it can be also treated as a switchover time) but can provide support for: - the dimensioning of the the logistics train; - the dimensioning of the rack lanes at the assembly stations; and provide approximate system information ( avg waiting time, empty bins in the system, ...) Conclusions Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece
  • 24. 24 Thank you for your attention Fabio Bursi, PhD Candidate 10属 conference on Stochastic Models of Manufacturing and Service Operations - Volos Greece

Editor's Notes

  • #5: Replenishment can be supposed to occur in mask-time
  • #14: Creare animazioni per tenere meglio il filo del discorso
  • #16: Dire che descriviamo analiticamente il sistema attraverso le equazioni di bilanciamento
  • #24: Finally, the critical evaluation of the limitations of the model in representing a real bin-kanban system suggests that further research is required to take into consideration switchover times and the time between consecutive cycles when the logistic train stands still at the supermarket where empty bins are Refilled neverthess we know that other authors have studied. a decomposition relationship between the expected waiting times in the zero- and nonzero-switchover times models. Moreover, in several applications the logistic train stands still at the supermarket, at the end of each tour, for a fixed period of time. Hence, the time between two consecutive cycles can be described as a constant switchover time.