Little´s law Skorkovský ,KPH,ESF.MU Based on Factory Physics (Hopp and Spearman) Little´s law - definition (formula) •Fundamental relationships among : –WIP (Work In Process) –Cycle Time (CT) –Throughput (T or sometimes TH) •Formula • •Can be applied to : –Single machine station – Complex production line –Entire plant Relationships among these variables will serve to se clearly precise (quantitative) description of behaviour of the single production line . It helps user to use a given scale to benchmark actual production systems Definition of basic parameters •Throughput (Throughput rate, TH) : production per unit time that is sold (see TOC definition) •If TH is measured in cost dollars rather than in prices it is typically called : • • Cost of good sold (COGS) • •Upper limit of TH in production process is capacity • •If you release more raw material above capacity of the line (machine), system become unstable –> WIP goes up !! • • Definition of basic parameters •WIP (Work In Process) : inventory between start and end points of the product routing • •WIP can be used as one parameter to calculate (measure) an efficiency • •Efficiency can be defined as Turnover Ratio = TH/FGI for warehouses or TH/(FGI+WIP) for production plants where FGI=Finished goods inventory • •WIP : inventory still in line • •FGI : inventory waiting for dispatch (shipping) • • • Definition of basic parameters •CT (Cycle Time, Flow Time or Throughput Time) : average time from release of the job of the beginning of the routing until it reaches an inventory point at the end of the routing or time that part spends as a WIP. • •LT (Lead Time) : managerial constant used for planning of production • •Service Level (especially for MTO lines, where plant have to satisfy orders with specific due dates) : • • Definition of basic parameters •Where –Arrival Rate =r = amount of parts arriving to workstation per time unit –Effective production time is maximum average rate at which the workstation can process parts (considering effects of failures, setup times and so on) •Bottleneck rate (see TOC) –rb = rate (parts per unit time or jobs per unit time) of workstation having the highest long term utilization •Example : taking into consideration only rate, so bottleneck is B for sure A B But having in mind some rejects on A output , so B have to process only y, where 1-y = quantity of rejects. U(A)= r/1=1 and U(B) =y*r/0,5 =2*y*r If r<0,5, that utilization of A is higher than U(B) and hence A workstation is bottleneck ! Processing time=1 min , r=1 Processing time=2 min- so processing r=0,5=1/2 (1-y) % is scraped, so A is heavily loaded r y Definition of basic parameters •T₀ =Row process time of the line is the sum of the long –term average process time of each workstation in the line (single job entering empty line from staring point to the ending one) 7, 8, 9, 7, 6, 10 7,83 5,3, 4, 5, 6, 8 5,16 9, 8, 4, 5, 5, 6 6,1 Definition of basic parameters •Critical WIP (W₀) of the line is the WIP level for which a line with given values of rb (bottleneck rate) and T₀ achieves maximum throughput ( rb ) with minimum cycle time (which is in this case T₀ ) • • • Use of defined parameters •Simple production line that makes giant one-cent pieces •It is as a model very with unrealistic assumption because process times are deterministic (no waiting times after every operation and no queue times before any other operation, 24 hours /day an unlimited market)à balanced line •Any machine can be regarded as the bottleneck (one- half part per hour) • Head stamping Tail stamping Rimming Deburring 2 h 2 h 2 h 2 h rb = rate (parts per unit time or jobs per unit time) of workstation having the highest long term utilization =0.5 penny per hour , which means 24 hour x 0.5=12 pennies /day T₀ =Row process time of the line is the sum of the long –term average process time of each workstation in the line = 8 hours=2+2+2+2 Critical WIP (W₀) of the line is the WIP level for which a line with given values of rb and T₀ achieves maximum throughput ( rb ) with minimum cycle time (which is in this case T₀ ) = rb x T₀ = 0.5 x 8 = 4 pennies Use of defined parameters Head stamping Tail stamping Deburring 2 h 5 h 10 h 3 h Tail stamping Rimming Rimming Rimming Rimming Rimming Rimming Deburring Station number Number of machines Process time (hours) Station capacity (jobs per hour) 1 1 2 0.5 2 2 5 0.4=(1/5)*2 3 6 10 0.6=(1/10)*6 4 2 3 0.67=(1/3)*2 Bottleneck= rb =0,4 It is neither the station that contains the slowest machines nor the one with fewest machines !!! T₀ =Row process time of the line =2+5+10+3 and Critical WIP(W₀) =rb x T₀ = 0.4 x20 = 8 pennies