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Lecture 3: Manufacturing Scheduling ConceptsPowerPoint Presentation

Lecture 3: Manufacturing Scheduling Concepts

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Lecture 3: Manufacturing Scheduling Concepts. Outline. Jobs & Operations Characteristics & notation Resources/machines Setup/transition cost Objective functions Complexity. M 1. M 2. M 3. S ij. C ij. S ij – starting time of job j on machine i C ij – completion time of job j.

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### Lecture 3: Manufacturing Scheduling Concepts

© J. Christopher Beck 2005

Outline

- Jobs & Operations
- Characteristics & notation

- Resources/machines
- Setup/transition cost

- Objective functions
- Complexity

© J. Christopher Beck 2005

M1

M2

M3

Sij

Cij

Sij – starting time of job j on machine i

Cij – completion time of job j

Jobsrj

dj

wj

pij

pij – processing time of job j on machine i

rj – release date of job j

dj – due date of job j

wj – weight of job j

© J. Christopher Beck 2005

Jobs & Operations

- Often jobs are made up of a set of operations

rj

dj

wj

p2j

p0j

p3j

p1j

precedence constraints

© J. Christopher Beck 2005

Example: House Building

Excavate

Floor joists

Foundations

…

4 wks

2 wks

3 wks

Exterior plumbing

…

3 wks

© J. Christopher Beck 2005

Resources/Machines

- Jobs may need resources
- Mixing machine, back-hoe, cement mixer

- May be multiple similar resources are available and you need to choose one

© J. Christopher Beck 2005

House Building Resources

Excavate

Floor joists

Foundations

…

4 wks

2 wks

3 wks

requires

Carpenter

Exterior plumbing

Backhoe

Backhoe operator

Dump truck

…

…

3 wks

© J. Christopher Beck 2005

Resources & Setup

- If 2 jobs need the same resource (and the resource can only do 1 thing at a time), then the jobs must be sequenced
- May be a time or cost for a resource to change jobs (“sequence dependent setup”)

© J. Christopher Beck 2005

Objectives

- Minimize maximum completion time (aka “makespan”)
- Min Cmax
- Cmax = max(C1, … Cn)

- Minimize maximum lateness
- Min Lmax
- Lmax = max(C1 – d1, … Cn – dn)

© J. Christopher Beck 2005

Objectives

- Minimize total weighted tardiness
- Min ΣwjTj
- Tj = max(Cj – dj, 0)

© J. Christopher Beck 2005

Hard Problems vs. Easy Problems

- Exercise 2.1b was “easy”
- Adding resources would have made it hard
- Hard & easy have precise mathematical definitions
- You need to have, at least, an intuitive understanding of what this means

© J. Christopher Beck 2005

Hard vs Easy

- Easy:
- Sort n numbers
- Solve a system of linear equations

- Hard:
- Schedule a factory, deliver packages, schedule buses, …

© J. Christopher Beck 2005

Hard vs Easy

- f (n): the number of “basic operations” needed to solve the problem with input size n
- Easy: f (n) is polynomial in n
- O(n), O(n log n), O(n2), …

- Hard: f (n) is exponential in n
- O(2n), …

© J. Christopher Beck 2005

Hard vs Easy

- 10301 operations required in worst case
- Age of universe: 1018 seconds
- Fastest Computer today: 1014 op/sec
- Let’s say we get a computer 1018 times faster (a sextillion times faster)
- 1033 op/sec

- It may still take 10250 times longer than the age of the universe to solve the problem!

© J. Christopher Beck 2005

Hard vs Easy

- If it is going to take 10250 times the age of the universe to schedule a factory, why bother?

© J. Christopher Beck 2005

Hard vs Easy

- If it is going to take 10250 times the age of the universe to schedule a factory, why bother?
- May be we can do it in a reasonable time in most cases?
- May be we can get a good (but not necessarily best possible) solution in a reasonable amount of time?

© J. Christopher Beck 2005

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