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Performance Evaluation Of Flow Lines With Multiple Products POLITECNICO DI MILANO Dipartimento di Meccanica Colledani M., Matta A. and Tolio T.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Outline System: description and assumptions Analytical model: building block Analytical model: decomposition Numerical results Conclusions and future developments

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione System description K machines, i=1,…,K z products, q=1,…,z Homogeneous buffers Discrete material/discrete time Deterministic and equal processing times Saturated system

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione System description Machines can fail with multiple failures, j=1,…,F i Machines can be failed in only one mode at the same time period MTTF and MTTR are geometrically distributed The production rule at machine is local and stochastic (production parameters i,q ) Buffers have finite capacity, N i,q Blocking Before Service

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione System description Production parameters of machine M i at time t are adjusted depending on the state of the immediately upstream and downstream buffers. (Nemec, 1999) and (Syrowicz, 1999) deal with lines with two products and priority rules. (Colledani et al., 2003 and 2005) deals with lines with two products and same assumptions.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Two-machine line with z products Production parameters may change because of the emptying and/or filling of buffers. Production parameters are adjusted as follows:

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione ZP2M: two-machine line with z products The state of the system is represented by: (n 1,…, n z,x u,x d ). The total number of possible states is: The corresponding Markov Chain is too complex to be solved numerically with traditional techniques.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique The behaviour of z-1 products can be modelled in an approximate way by considering an equivalent aggregate product. a a Aggregate product

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique … … … … The system with z products is represented by a set of equivalent z systems, each one crossed by 2 products: product q and the corresponding aggregate product. (Baynat and Dallery, 1995) first proposes this technique for analyzing multiclass queuing systems.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique The production probabilities in the original system are adjusted as follows: However, the aggregation of all products except q does not allow to recognize the buffer levels of single aggregated products in the analysis of the two-machine two-product system, thus we are forced to find new values for the production parameters.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique For each system it is necessary to calculate the parameters: Buffer capacities: N q (q) and N a (q) Upstream production parameters: u q (q) and u a (q) Downstream production parameters: d q (q) and d a (q) There are totally 6z unknowns. (Colledani et al., 2003 and 2005) is used to calculate the performance.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique The sum of production parameters of each machine must be equal to one: The buffer capacity of the product q corresponds to that in the original system: The buffer capacity of the aggregate product is the sum of the buffer capacities in the original system of the single aggregated products:

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique We define u q as the set containing all the combinations, in the two-machine line original system, of buffers full and not full obtained without considering buffer B q : The new value of the upstream production probability of product q is calculated as a weighted combination of the adjusted u q values overall the possible combinations belonging to the set u q : probability associated to the occurrence of combination u q

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique We define d q as the set containing all the combinations, in the original system, of buffers empty and not empty obtained without considering buffer B q : The new value of the upstream production probability of product q is calculated as a weighted combination of the adjusted d q values overall the possible combinations belonging to the set d q : probability associated to the occurrence of combination d q

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Long lines with Z products … …

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Long lines with Z products Local failures Remote failures (starvation) Remote failures (blocking) (Tolio and Matta, 1998) The new production parameters are calculated as follows:

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Numerical results: 3P3M CASE1 (buffer 4-4-4/4-4-4)2 (buffer 4-4-4/4-4-4)3 (buffer 6-6-6/6-6-6)4 (buffer 4-4-4/4-4-4) MAC H M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 M1M1 M2M2 M3M3 p0.0490.0920.0090.1040.090.0640.1040.090.0640.120.0390.105 r0.620.160.1020.620.120.1020.70.280.320.420550.37 0.6 0.4 0.5 0.3 0.2 0.35 0.4 0.1 0.2 0.25 0.1 PARTP1P1 P2P2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 P3P3 E sim0.3660.1910.0680.2710.110 0.2960.2610.1900.3690.3010.084 E anal0.3620.1980.0640.2650.114 0.2960.2610.1900.3680.3040.094 Err-0.591.31-0.70-1.310.69 -0.03-0.02-0.01-0.160.41.37 n 1 sim3.383.583.753.433.62 4.985.025.111.691.671.6 n 1 anal3.063.243.463.113.18 4.264.334.451.601.581.9 Err-8.03-8.51-7.2-8.0-10.7 -11.9-11.4-11-2.27-2.057.55 n 2 sim0.670.50.371.751.71 1.791.761.682.252.262.29 n 2 anal 0.870.710.561.81.86 2.112.262.192.152.211.94 Err5.215.224.881.123.763.725.346.698.5-2.45-1.28-8.75

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Numerical results: 4P3M Test on 50 cases: 0.66 % error on average throughput 6.4 % error on average buffer levels.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione Conclusions and future developments New model to estimate the performance of multiple product flow lines. Ongoing work Split/merge systems with z different products To be developed The continuous model with different processing times Closed systems with z different products Different production policies

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione System description E q is the average production rate of the system related to product q, with q=1,…,z E is the overall average production rate of the system

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique Some relationships: 2P2M: two- product two- machine system

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique Algorithm Step 1: Initialization of 2P2M systems. Set the production probabilities of 2P2M systems to some initial value and the buffer capacities. Step 2: Solve 2P2M systems. For q=1,…,Z solve the 2P2M system producing products q and a(q) using the method in (Colledani et al., 2005). Step 3: Calculate alphas. For q=1,…,Z calculate the new values of production probabilities u q (q), u a (q), d q (q) and d a (q). Step 4. Check convergence. The algorithm converges if all the production probabilities do not significantly change from one iteration to another, otherwise go back to Step2.

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique

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Dipartimento di Meccanica Sezione Tecnologie Meccaniche e Produzione The aggregation technique

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