Lot-Sizing and Lead Time Performance in a Manufacturing Cell

1. 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

2. 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

3. 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% !

4. 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

5. Three models for the Kodak Job Shop • In-house: EOQ Inventory model • Simulation commissioned by Kodak • Q-Lots by Karmarkar et al.

6. 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

7. 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

8. 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.

9. 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.

10. 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

11. 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

12. 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

13. 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