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