© J. Christopher Beck 20051 Lecture 3: Manufacturing Scheduling Concepts

© J. Christopher Beck 2005 1

Lecture 3: Manufacturing Scheduling Concepts


Outline Jobs & Operations

Characteristics & notation Resources/machines

Setup/transition cost Objective functions Complexity


Jobs

pij – processing time of job j on machine i

rj – release date of job jdj – due date of job jwj – weight of job j

pij

rj dj

wj

M1

M2

M3

Sij Cij

Sij – starting time of job j on machine i

Cij – completion time of job j


Jobs & Operations

Often jobs are made up of a set of operations

p2j

rj dj

wj

p0j p3j p1j

precedence constraints


Example: House Building

…

Excavate Foundations Floor joists

…

Exterior plumbing

4 wks 2 wks 3 wks

3 wks


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


House Building Resources

…

Excavate Foundations Floor joists

…

Exterior plumbing

4 wks 2 wks 3 wks

3 wks

BackhoeBackhoe operatorDump truck…

requires Carpenter


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”)


Objectives

Minimize maximum completion time (aka “makespan”) Min Cmax

Cmax = max(C1, … Cn)

Minimize maximum lateness Min Lmax

Lmax = max(C1 – d1, … Cn – dn)


Objectives

Minimize total weighted tardiness Min ΣwjTj

Tj = max(Cj – dj, 0)

# Job Duration (weeks)

Predecessor(s)

1 Excavation 4 –

2 Foundations 2 1

3 Floor joists 3 2

4 Exterior Plumbing 3 1

5 Floor 2 3,4

6 Power On 1 2

7 Walls 10 5

8 Wiring 2 6,7

9 Communication Lines 1 8

10 Inside Plumbing 5 7

11 Windows 2 10

12 Doors 2 10

13 Sheetrock 3 9,10

14 Interior Trim 5 12,13

15 Exterior Trim 4 12

16 Painting 3 11,14,15

17 Carpeting 1 16

18 Inspection 1 17

Exercise 2.1a) Draw precedence graphb) Calculate makespan


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


Hard vs Easy

Easy: Sort n numbers Solve a system of linear equations

Hard: Schedule a factory, deliver packages,

schedule buses, …


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), …

Hard vs EasyO(n) O(n log

n)O(n2) O(2n)

1 0 1

10 10 100

20 26 400

50 85 2500

100 200 10000

1000

3000 1,000,000


n)O(n2) O(2n)

1 0 1 2

10 10 100 1024

20 26 400

50 85 2500

100 200 10000

1000

3000 1,000,000


n)O(n2) O(2n)

1 0 1 2

10 10 100 1024

20 26 400 1048576

50 85 2500 1,125,899,906,842,624

100 200 10000

1000

3000 1,000,000


n)O(n2) O(2n)

1 0 1 2

10 10 100 1024

20 26 400 1048576

50 85 2500 1,125,899,906,842,624

100 200 10000 1.268 X 1030

1000

3000 1,000,000

1.072 X 10301


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 universeuniverse to solve the problem!


Hard vs Easy

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


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?

Documents

© J. Christopher Beck 20051 Lecture 3: Manufacturing Scheduling Concepts