Models of Heijunka-levelled Kanban- Systems Kai Furmans Fifth International Conference on ``Analysis of Manufacturing Systems – Production Management’’

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 2 IFL A Kanban System Withdrawal Kanban Or Market Demand Production Kanban Finished Goods Warehouse Production Raw Materials Warehouse TransportCustomer

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 3 IFL C C B B A A A A Kanban System with Levelling Withdrawal Kanban Or Market Demand Production Kanban Finished Goods Warehouse Raw Materials Warehouse TransportCustomer Type A B C Sequence 5 2 ON Next day Overflow 3 Production

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 4 IFL EPEI – „Every Part Every Interval“ Calculation EPEI Product A Setup Product BProduct C Setup Expected downtimes EPEI Product A Setup Product BProduct C Setup Expected downtimes Product A Setup Product BProduct C Setup Expected downtimes EPEI

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 5 IFL No Capacity Limits, iid Demands, no Levelling Assumptions: Periodic demand and review Fixed and constant lead times t r Demand is iid distributed, a demand of j items is described by the probability d j Stock development (up or down) Distribution of stocks in replenishment period

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 6 IFL Basic Model with Unlimited Capacity: Example Necessary base stock level: 6 units Replenishment time: 3 time units Input parameter:

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 7 IFL Basic Model with Unlimited Capacity The necessary base stock level increases linear with coefficient of variation of demand

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 8 IFL Capacity Limits, iid Demands, with Levelling The assembly line has a capacity of c units per EPEI In period n Q n units are sold Kanbans, that exceed capacity c in one period, are collected in the overflow box If in one period, less than c units are sold, Kanbans from the overflow box are moved to the board (if available)  Daily production is never more than c units, but can be less.Difference between Capacity and Demand:  Number of waiting Kanbans in period n + 1 If replenishment interval equals 1, (for assembly systems often the case), then the number of not yet replaced items in stock is:

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 9 IFL Duality: Heijunka Model as GI/D/1-System or D/GI/1-System Stochastic demand – deterministic capacity: GI/D/1-System? Discrete Time GI/D/1-Queueing-System: Distance between requests vary, requests for capacity are homogenous Discrete Time D/GI/1-System Queueing-System: Distance between requests is identical (one period), workload is varying Lindleys Equation in discrete time: Distribution of probabilities can be computed with existing algorithms (i.e. Grassmann / Jain using Wiener-Hopf-Factorization)

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 10 IFL Limited Capacity - Heijunka Controlled: Example Necessary base stock level: 17 units

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 11 IFL Reduction of Variability of Demand y i describes the distribution of the idle capacity per time interval for arriving Kanbans. z i describes the distribution of the idle capacity per time interval. The difference between the available capacity c and the idle capacity z i is the demand of parts which is requested from the supplier to replenish the raw materials. Thus: The requested replenishment quantity at the supplier now is:

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 12 IFL Limited Capacity - Heijunka Controlled: Example The coefficient of variation of supplier demand decreases: v customer demand = 0,27  v demand to supplier = 0,17 Reduction of the bullwhip effect Capacity: 10 If supplier has a capacity of 10 all demands will be fulfilled immediately

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 13 IFL Multiple Product Case, iid Demands, Limited Capacity The Multiple product case can be treated exactly as the single product case, if the capacity c is preallocated on the different products –The calculations can be done for each product separately. If all requests are handled by the same assembly unit, then the single product case has to be applied –With a subsequent stock sizing using the waiting time distribution of Kanbans for the determination of the respective stock sizes. Question: On which level should Levelling be performed? Multiple Product Case, Demands generated from Kanban Loop, Limited Capacity Ongoing work

© Institut für Fördertechnik und Logistiksysteme - Universität Karlsruhe (TH) 14 IFL Conclusions We have not succeeded in answering all of your problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things Sign in a computer shop