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**Flow, Inventory, Throughput, and Little’s Law**

Models in IE: Lecture 6 Flow, Inventory, Throughput, and Little’s Law

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**Today’s Core Concepts Flow, Flow Unit Flowtime Throughput**

WIP, Inventory Little’s Law Bottleneck

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**Georgia Tech as a flow process**

1 student = 1 flow unit

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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).

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IE 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

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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)

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**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

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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?

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**Throughput: rate of flow units through 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

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**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

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**Georgia Tech Little’s Law at Georgia Tech**

12,500 Students 2500/year How long does it take to graduate?

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**Simple example: all students take 5 years**

1 2 3 4 5 6 7 8 Simple example: all students take 5 years

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**Better example: some take 4, some take 6 years**

1 2 3 4 5 6 7 8 Better example: some take 4, some take 6 years

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**IE 2030 Lecture 6 Little’s Law Measurement**

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

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**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!

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**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?

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**Little’s Law works even if**

System has Variability

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**P[4 years]=.4 P[5 years]=.2 P[6 years]=.4**

1 2 3 4 5 6 7 8 P[4 years]=.4 P[5 years]=.2 P[6 years]=.4

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**Random number of students arriving/year**

1 2 3 4 5 6 7 8 Random number of students arriving/year

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**Little’s Law still works**

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

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**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

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**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?

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**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?)

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**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

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