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Bp Cheatsheet

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Submitted By threefinger
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Waiting Time Formula, Multi Parallel Resources (Servers)
Utilization, u = (1/a)/(m x 1/p) = p/am
Tq = (p/m) x (u^[√(2(m+1)-1)-1]/(1-u) x (CVa2+CVp2)/2
Iq = (1/a)*Tq
Ip = u*m = (1/a)*p
I = Iq + Ip
Flow Time = Tq + p

Waiting Time Formula, Multi Parallel Resources (Servers)
Utilization, u = (1/a)/(m x 1/p) = p/am
Tq = (p/m) x (u^[√(2(m+1)-1)-1]/(1-u) x (CVa2+CVp2)/2
Iq = (1/a)*Tq
Ip = u*m = (1/a)*p
I = Iq + Ip
Flow Time = Tq + p

Waiting Time Formula, Single Server
Utilization, u = Flow Rate/Capacity = (1/a)/(1/p) = p/a
Inventory Related Measures
Iq = (1/a) x Tq
Ip = u
I = Iq + Ip
Time in queue, Tq = Activity Time (p) x (u)/(1-u) x (CVa2+CVp2)/2
Flow Time = Tq + p

Waiting Time Formula, Single Server
Utilization, u = Flow Rate/Capacity = (1/a)/(1/p) = p/a
Inventory Related Measures
Iq = (1/a) x Tq
Ip = u
I = Iq + Ip
Time in queue, Tq = Activity Time (p) x (u)/(1-u) x (CVa2+CVp2)/2
Flow Time = Tq + p

Inter-Arrival Time, a = amount of time between two arrivals to a process
Arrival process is Stationary over a period of time if the number of arrivals on any sub interval depends only on length of interval, not on when interval starts a = average inter-arrival time (IAi+1 – IAi)
Sa = standard deviation (absolute measure of variability)
CVa = Standard deviation/average inter-arrival time (Sa/a) – relative variability
CVp = Standard deviation/Average activity time (Sp/p)

Abnormal peaks of processing time Would be more efficient to automate it
Variability leads to waiting time (even if p/a < 100%)
To predict waiting time even if u<100%
Waiting Time Formula, Single Server
Utilization, u = Flow Rate/Capacity = (1/a)/(1/p) = p/a
Inventory Related Measures
Iq = (1/a) x Tq
Ip = u
I = Iq + Ip
Time in queue, Tq = Activity Time (p) x (u)/(1-u) x (CVa2+CVp2)/2
Flow Time = Tq + p
Inter-Arrival Time, a = amount of time between two arrivals to a process
Arrival process is Stationary over a period of time if the number of arrivals on any sub interval depends only on length of interval, not on when interval starts a = average inter-arrival time (IAi+1 – IAi)
Sa = standard deviation (absolute measure of variability)
CVa = Standard deviation/average inter-arrival time (Sa/a) – relative variability
CVp = Standard deviation/Average activity time (Sp/p)

Abnormal peaks of processing time Would be more efficient to automate it
Variability leads to waiting time (even if p/a < 100%)
To predict waiting time even if u<100%
Waiting Time Formula, Single Server
Utilization, u = Flow Rate/Capacity = (1/a)/(1/p) = p/a
Inventory Related Measures
Iq = (1/a) x Tq
Ip = u
I = Iq + Ip
Time in queue, Tq = Activity Time (p) x (u)/(1-u) x (CVa2+CVp2)/2
Flow Time = Tq + p

Inventory Related Cost: C = (Q/2)(Ch) + (D/Q)(Co)
Total Cost Per Year: C = (Q/2)(Ch) + (D/Q)(Co) + DC
Where Ch = Cost of Product x Cost of Capital, C x r
Minimum Cost: Holding cost = Ordering Cost, (Q/2)(Ch) = (D/Q)(Co)
EOQ, Q* = (2DCo/Ch)^1/2
Where R = D, K = Co, h = Ch

Average Level of Inventory = Q*/2
Cycle Time, Order cycle: T* = Q*/D

Finite Production Rate
Production Time: Q/p
Order Cycle: Q/D
Maximum Level of Stock = Q(1-D/p)
Production Rate = (p-D)
Inventory Related Cost: C= (Q/2)(1-D/p)(Ch) + (D/Q)(Co)
EOQ, Q* = 2DCo/[Ch(1-D/p)]^1/2
Inventory Related Cost: C = (Q/2)(Ch) + (D/Q)(Co)
Total Cost Per Year: C = (Q/2)(Ch) + (D/Q)(Co) + DC
Where Ch = Cost of Product x Cost of Capital, C x r
Minimum Cost: Holding cost = Ordering Cost, (Q/2)(Ch) = (D/Q)(Co)
EOQ, Q* = (2DCo/Ch)^1/2
Where R = D, K = Co, h = Ch

Average Level of Inventory = Q*/2
Cycle Time, Order cycle: T* = Q*/D

Finite Production Rate
Production Time: Q/p
Order Cycle: Q/D
Maximum Level of Stock = Q(1-D/p)
Production Rate = (p-D)
Inventory Related Cost: C= (Q/2)(1-D/p)(Ch) + (D/Q)(Co)
EOQ, Q* = 2DCo/[Ch(1-D/p)]^1/2

Resource Capacity = Batch Size / ( Setup Time + Batch Size x Processing Time)

Recommended Batch Size = (Flow Rate x Setup Time) / (1 – Flow Rate x Processing Time) – Using Flow Rate of next Bottleneck
Time taken to complete x units in Batches of B = (B x P1) + (B x P2) + (X x P3) - Assuming batches must be whole in each process

Resource Capacity = Batch Size / ( Setup Time + Batch Size x Processing Time)

Recommended Batch Size = (Flow Rate x Setup Time) / (1 – Flow Rate x Processing Time) – Using Flow Rate of next Bottleneck
Time taken to complete x units in Batches of B = (B x P1) + (B x P2) + (X x P3) - Assuming batches must be whole in each process

Definition
• Flow units (Often correspond to as “jobs”)
Performance Metric
• Inventory (number of flow units in the process) units (No. of units at any given point of time)
• Flow Time (time a flow unit spends in the process) min/unit
• Flow Rate (or throughput rate) units/min

Little’s Law: Inventory (I) = Flow Rate (R) * Flow Time (T)
Per-unit inventory Costs (%)
(Annual inventory costs ($) as a percentage of COGS)
= Annual Inventory Costs / COGS
= Annual Inventory Costs * I ($) / R ($/year)
= Annual Inventory Costs (%) * T (years)
= Annual Inventory Costs (%) / Inv Turns
Inventory Turns = Flow Rate (R) / Inventory (I) = 1/Flow Time (T)
Inventory Turns = COGS / Inv
Average Inventory Costs = Annual Inv Costs / Inv Turns

Lead Time = Time taken for entire process to complete
Definition
• Flow units (Often correspond to as “jobs”)
Performance Metric
• Inventory (number of flow units in the process) units (No. of units at any given point of time)
• Flow Time (time a flow unit spends in the process) min/unit
• Flow Rate (or throughput rate) units/min

Little’s Law: Inventory (I) = Flow Rate (R) * Flow Time (T)
Per-unit inventory Costs (%)
(Annual inventory costs ($) as a percentage of COGS)
= Annual Inventory Costs / COGS
= Annual Inventory Costs * I ($) / R ($/year)
= Annual Inventory Costs (%) * T (years)
= Annual Inventory Costs (%) / Inv Turns
Inventory Turns = Flow Rate (R) / Inventory (I) = 1/Flow Time (T)
Inventory Turns = COGS / Inv
Average Inventory Costs = Annual Inv Costs / Inv Turns

Lead Time = Time taken for entire process to complete
Bottleneck = Resource with lowest capacity
Process Capacity = Cap of Bottleneck (units/min)
Flow Rate = Min (Input, Process Cap, Demand)
Utilization of Process = Flow Rate/Process Cap = 1 (if flow rate = process cap)
Utilization of Resource = Flow Rate / Resource Cap
Utilization not at 100% - Mismatch of Demand and Supply at process level

An = activity time
Direct Labour Content = A1 + A2 + A3 + A4
Assuming, A4 = Bottleneck
Direct Idle Time = (A4 – A1) + (A4 – A2) + (A4 – A3)
Average Labour Utilization = Labour Content / (Labour Content + Direct Idle Time)
Cost of Direct Labour = Total wages per unit time / Flow rate per unit time
Idle time = Cycle Time x No of workers at resource i – activity time at resource i
Cycle time = 1 / Flow rate (R)
Capacity of resource = Number of resources / Processing Time
Time to finish X Units = Time through an empty process + (X-1) units / Flow rate
Time through empty Worker-paces process = sum of activity times
Time through empty machine-paced (conveyer belt) = Number of resources x processing time of bottleneck
Takt Time = Average unit production Time

Bottleneck = Resource with lowest capacity
Process Capacity = Cap of Bottleneck (units/min)
Flow Rate = Min (Input, Process Cap, Demand)
Utilization of Process = Flow Rate/Process Cap = 1 (if flow rate = process cap)
Utilization of Resource = Flow Rate / Resource Cap
Utilization not at 100% - Mismatch of Demand and Supply at process level

An = activity time
Direct Labour Content = A1 + A2 + A3 + A4
Assuming, A4 = Bottleneck
Direct Idle Time = (A4 – A1) + (A4 – A2) + (A4 – A3)
Average Labour Utilization = Labour Content / (Labour Content + Direct Idle Time)
Cost of Direct Labour = Total wages per unit time / Flow rate per unit time
Idle time = Cycle Time x No of workers at resource i – activity time at resource i
Cycle time = 1 / Flow rate (R)
Capacity of resource = Number of resources / Processing Time
Time to finish X Units = Time through an empty process + (X-1) units / Flow rate
Time through empty Worker-paces process = sum of activity times
Time through empty machine-paced (conveyer belt) = Number of resources x processing time of bottleneck
Takt Time = Average unit production Time

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