Post on 30-Mar-2015
Models in IE: Lecture 6
Flow, Inventory, Throughput,
and Little’s Law
Today’s Core Concepts
• Flow, Flow Unit
• Flowtime
• Throughput
• WIP, Inventory
• Little’s Law
• Bottleneck
GeorgiaTech
Georgia Tech as a flow process
1 student = 1 flow unit
IE 2030 Lecture 6
• Flow unit• Throughput: rate of flow units through a
point per unit time• Input rate, output rate, and steady state• Flow time: on average, amount of time a
flow unit spends within the system• WIP, inventory: number of units in system
(within system boundaries).
IE 2030 Flow Unit Examples
• Kitchen in restaurant: flow unit=1food order
• Gas station pump: flow unit = 1 gallon of gasoline
• Gas station: flow unit = 1 customer (1 car)
• Clothes store: flow unit = 1 article of clothing
IE 2030 Lecture 6: Inventory
• Inventory: number of flow units within system boundaries
• At Tech: number of students who have matriculated but not graduated (ignoring dropouts)
• Number of cars waiting for or getting gas
• Number of food orders waiting or cooking
• OR, # of food orders brought to kitchen, not cooked and taken by waiters (different system boundary)
Flow unit, inventory
• Input: many different materials and parts
• Output: many different electronics components
• What is a flow unit?– Filled order– One component– materials to make a
component??– $ of materials
IE 2030 Lecture 6: Flowtime
• Flowtime for a particular item in a system = time it leaves system - time it enters system
• Flowtime usually means: on average, the amount of time a flow unit spends in system
How long does a dollar remain in your checking account?
Throughput: rate of flow unitsthrough a point
• Kitchen in restaurant: # food orders arriving OR started cooking OR finished cooking...
• Gas pump:# gallons pumped out/hour
• Gas station: # customers served/hour
• # clothes sold/week
IE 2030: Little’s Law
• Little’s Law is for a system in steady state– input rate = output rate
• Similar to rate × time = distance
• Applies to most systems, even those with variability
• Uses AVERAGE values
• throughput rate × flowtime = inventory
GeorgiaTech
Little’s Law at Georgia Tech
How long does it take to graduate?
25002500/year/year
12,500 Students
Simple example: all students take 5 years
1 2 3 4 5 6 7 8
Better example: some take 4, some take 6 years
1 2 3 4 5 6 7 8
IE 2030 Lecture 6 Little’s LawMeasurement
• In the first example, if you ask students how long they will be at Tech, they say…
• In the second example, some say 4, some say 6, but on average they say….
5 years
5.2 years
Little’s Law,Measurement, and Sampling
• Visit a prison and ask inmates the lengths of their sentences until probation
• Find the time served of inmates who died or were released on probation
• Do you believe statistics reported in the news by honest, well-meaning reporters?
• In general, should sample flow units passing a point in the system. More work!
Steady State vs. Startup
• Flow time defined for stable system– Input rate = Output rate
– Inventory doesn’t • Startup or transient behavior can be
important, especially if change is frequent– Does the economy ever reach equilibrium?
Little’s Law works even if
System has
Variability
P[4 years]=.4 P[5 years]=.2 P[6 years]=.4
1 2 3 4 5 6 7 8
Random number of students arriving/year
1 2 3 4 5 6 7 8
Variability
Little’s Law still works• Randomness in arrival rate
• Randomness in arrival type
• Randomness in service or production rates
System must be stable
Dependence can be a problem
Bottlenecks• Definition: reduce rate, reduce throughput
• Why not defined in terms of increase?
• Semester conversion at Tech --- Chem labs a bottleneck
• Flowlines usually have bottlenecks. Line balancing.
• Jobshops are more complex; idea of bottleneck less easily applicable.
• Bottlenecks are often unclear when there is variability
Example: Insight from Little’s Law(L. McGinnis)
• We put orders into the production system 1 month before their deadlines, but they are taking 1 month to be produced on average. More than half are late (why need it not be exactly half?)
• Response: we put orders in 2 months before deadline. What happens?
Example: Insight from Little’s Law (L. McGinnis)
• We think we’ve changed rate, but output rate and future input rate are the same.
• We’ve doubled inventory
doubled flowtime• Now orders take 2 months to produce, on
average• In fact, orders now take more than 2 months
on average! (Why?)
Some Objectives for a System
• Throughput (max.)
• Cost per unit, including inventory (min)
• flowtime (min)
• total flowtime for a set of jobs (min)
• makespan for a set of jobs (min)– example: 6 jobs time 2; 4 time 3; 3 time 4, 2
time 6. On 4 machines, minimizing makespan is not the same as minimizing total flowtime