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Lot-Sizing and Lead Time Performance in a Manufacturing Cell. Article from Interfaces (1987) by U. Karmarkar, S. Kekre, S. Kekre, and S. Freeman Illustrates application of M/M/1 Waiting Line Model to complex manufacturing problem at Kodak. The Job Shop. 10 major & 3 minor work centers

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Lot-Sizing and Lead Time Performance in a Manufacturing Cell

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Lot sizing and lead time performance in a manufacturing cell l.jpg

Lot-Sizing and Lead Time Performance in a Manufacturing Cell

  • Article from Interfaces (1987) by U. Karmarkar, S. Kekre, S. Kekre, and S. Freeman

  • Illustrates application of M/M/1 Waiting Line Model to complex manufacturing problem at Kodak


The job shop l.jpg

The Job Shop

  • 10 major & 3 minor work centers

    • work center houses 1 or more machines with similar functions

  • Each of 13 distinct parts processed in the job shop

  • Each part has a routing through the shop; some include re-circulation and multiple visits to machines


Key performance measure lead time l.jpg

Key performance measure: Lead Time

  • Lead time is the total time a part spends in the system = job shop

  • Includes time in processing (~service time) and waiting for processing

  • Karmarkar et al denote it as T, but it corresponds to W in the M/M/1 queue

  • Waiting time can be  90% !


Key decision variable lot batch size q l.jpg

Key Decision Variable: Lot (batch) size Q

  • Consider one of the 13 types of parts

  • Have a monthly demand of D parts

  • Job shop can process them at a rate of P parts/month

  • a Batch or Lot of Q parts are processed together, hence D/Q total batches per month

  • It takes  months to set up machine for each batch


Three models for the kodak job shop l.jpg

Three models for the Kodak Job Shop

  • In-house: EOQ Inventory model

  • Simulation commissioned by Kodak

  • Q-Lots by Karmarkar et al.


Eoq model l.jpg

EOQ Model

  • Batch corresponds to order size

  • EOQ minimizes Total Cost = c*D/Q + h*Q/2

  • D is total demand over planning period

  • Q is the order quantity ~ batch

  • c is the unit order cost ~ setup

  • h is the unit holding cost per unit time ~processing cost


Eoq scorecard l.jpg

EOQ Scorecard

  • + Well known model, easy to implement and solve

  • - Relies on estimates of cost of processing and cost of setup instead of time

    • not a good predictive model

    • not focused on lead time


Simulation model l.jpg

Simulation Model

  • Key Assumption: Lots released at uniform intervals

  • Key inputs (parameters)

    • monthly demand

    • lot sizes

  • Key Outputs

    • lead time and time spent in waiting for each batch

    • number of setups

    • Work in process (W.I.P.) Inventory

  • Search for best lot size by “trial and error” -- running simulation for many different lot sizes.


Simulation scorecard l.jpg

Simulation Scorecard

  • +captures complexities of job shop, including complex routings; good predictive model

  • - computationally intensive, including trial and error search for best lot size; expensive to develop and maintain; has unrealistic assumption about uniform batch releases.


Q lots model l.jpg

Q-Lots Model

  • Key Assumption: Job Shop behaves like M/M/1 waiting line model & time in the system T (our W) is a function of Q, the lot or batch size.

  • Key Inputs

    • avg arrival rate  = D/Q

    • avg service time =  + Q/P

  • Key Output: Time in system

    • T(Q) = ( + Q/P)/(1 - D/P -D/Q)

    • minimal batch Qmin size below which avg. arrival rate exceeds avg. service rate


Q lots numerical example l.jpg

Q-Lots: Numerical Example

  • Demand D = 750 parts/mo

  • Processing speed P = 1000 parts/mo.

  • Setup time  = .02 mo.

  • Qmin = 60 parts

  • Q* = 129 parts

  • T(Q*) = 1.114 mo. = 33.43 days


Q lots scorecard l.jpg

Q-Lots Scorecard

  • +well known & easily solved analytical model; captures random arrivals of batches; good predictive model; can solve for optimal lot size Q*

  • - possibly too simple: no representation of complex flow patterns; entire job shop as one channel


Conclusions l.jpg

Conclusions

  • Q-Lots very simple but successful model of very complex system

    • correct focus on lead time

    • correct key variable: batch or lot size

  • Simulation expensive, but provides valuable cross-validation of Q-Lots

  • EOQ somewhat out of context here


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