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 Q min 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. Q min = 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