Little\'s law - Parte 2. Little\'s Law. PDF

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Parte 2. Little's Law....

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Little’s law Little law allow to understand relationships among WIP, Throughput TH and Lead Time LT of clients (i.e. parts) flowing in a generic system for discrete production, given the input rate of clients (i.e. parts) is constant (steady status). Little law is independent of the configuration of the system, of the type of distribution of processing times, of routing and of the distribution of inter-arrival times. If we increase the input rate of parts to the system, then WIP will increase progressively and TH will increase linearly, while LT will remain constant, but: • When we reach a critical value WIP*, TH will achieve its maximum and will never increase more, while LT will start increasing; • If we want to decrease LT, while maintaining TH constant, we have to reduce the WIP; In a steady (stationary) system, the following law applies: 𝑇𝐻# = #𝑊𝐼𝑃 #/#𝐿𝑇 where: • TH = Throughput (TH): the average output of a production process (machine, workstation, line, plant) per unit time (e.g., parts per hour) is defined as the system's throughput, or sometimes throughput rate. • WIP = Work in process (WIP): The inventory between the start and end points of a product routing is called work in process (WIP). Since routings begin and end at stock points, WIP is all the product between, but not including, the ending stock points. • LT = Lead time (or throughput-time or flow-time): the lead time of a given routing or line is the time allotted for production of a part on that routing.

Rb = (Bottleneck Rate) Rate (parts/unit time or jobs/unit time) of the station having the highest long-term utilization. T0 Raw Process Time: Sum of the long-term average process times (working time) of each station in the line. Congestion: different cases may happen. • Best-case performance (zero variability, zero randomness); • Practical worst-case performance (maximum randomness); • Worst-case performance (zero randomness but batch moves); Critical WIP (i.e., W0): the WIP level in which a line having no congestion would achieve maximum throughput (i.e., rb) with minimum throughput time (i.e., T0). 𝑊𝐼𝑃 ∗# (𝑜𝑟 #𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙#𝑊𝐼𝑃) # = #𝑡ℎ𝑒#𝑊𝐼𝑃#𝑙𝑒𝑣𝑒𝑙#𝑎𝑡 #𝑚𝑎𝑥. 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 #𝑇𝐻𝑚𝑎𝑥 W0 = rb *T0

Best case performance

The maximum throughput (i.e., THbest) for a given WIP level, w, is given by:

TTPbest

if w £ W0 ìT0 , =í î w / rb , otherwise.

The minimum throughput time (i.e., TTPbest) for a given WIP level, w, is given by:

TH best

ì w / T0 , if w £ W0 =í otherwise. î rb ,

Worst case

The worst case throughput (i.e., THworst) for a given WIP level, w, is given by: THworst = 1 / T0

The worst case throughput time (i.e., TTPworst) for a given WIP level, w, is given by:

TTPworst = w * T0

Practical worst-case performance Let w = jobs in the line, N = number of stations in the line, and t = working time at each station: TTP (through the single station) = (1 + (w-1)/N) * t TTP (through the line) = N * [1 + (w-1)/N] * t = N * t + (w-1) * t = T0 + (w-1)/rb Using the Little’s law 𝑇𝐻 # = #𝑊𝐼𝑃/𝑇𝑇𝑃# = # [𝑤/(𝑊0 + 𝑤 − 1)]# ∗ #𝑟𝑏

The practical worst case throughput (i.e., THPWC) The practical worst case throughput time (i.e., TTPPWC) for a given WIP level, w, is given for a given WIP level, w, is given by: by:

TH PWC

w w -1 rb , TTP = PWC = T0 + W0 + w - 1 r b

where W0 is the critical WIP.

Relationship to performance cases (for balanced lines): • LV -> Between Best-case and Practical worst-case; • MV -> Practical worst-case; • HV -> Between Practical worst-case and Worst-case.

The penny Fab example: Features • 4 identical machines in line (4 stations) • Process time of each station 2 h • Process time are fixed (no variability = steady state) • Inter-arrival time of parts (Delta T) is lowered step by step • Elapsed time (Time) is registered

Best case For Penny Fab 1, rb = 0,5 and T0 = 8, so W0 = 0,5 ´ 8 = 4. Thus:

TTPbest

if w £ 4 ì8, =í î2 w, otherwise.

TH best

ì w / 8, if w £ 4 =í î 0,5, otherwise....