Topic 28
Flexible Assembly
Systems
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Job Shops




Each job has an
unique identity
Make to order, low
volume environment
Possibly complicated
route through
system
Very difficult
July 11, 2017
Flexible Assembly





Limited number of
product types
Given quantity of
each type
Mass production
High degree of
automation
Even more
difficult!
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
Flexible Assembly Systems

Sequencing Unpaced Assembly Systems


Sequencing Paced Assembly Systems


Simple flow line with finite buffers
Conveyor belt moves at a fixed speed
Scheduling Flexible Flow Systems

Flow lines with finite buffers and bypass
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
3
Sequencing Unpaced Assembly
Systems



Number of machines in series
No buffers
Material handling system




When a job finishes moves to next station
No bypassing
Blocking
Can model any finite buffer situation
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
4
Cyclic Schedules

Schedules often cyclic or periodic:

Given set of jobs scheduled in certain order




Contains all product types
May contain multiple jobs of same type
Second identical set scheduled, etc.
Practical if insignificant setup time


Low inventory cost
Easy to implement
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
5
Minimum Part Set

Suppose l product types

Let Nk be target number of jobs of type k

Let z be the greatest common divisor

Then
Nl 
 N1 N 2
N  ,
,..., 
z 
 z z
*
is the smallest set with ‘correct’ proportions

Called the minimum part set (MPS)
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
6
Defining a Cyclic Schedule

Consider the jobs in the MPS as n jobs
1 l
n   Nk
z l 1



Let pij be as before
A cyclic schedule is determined by
sequencing the job in the MPS
Maximizing TP = Minimizing cycle time
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
7
MPS Cycle Time Example
Jobs
1
2
3
p1 j
0
1
0
p2 j
0
0
0
p3 j
1
0
1
p4 j
1
1
0
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Buffer!
8
Sequence: 1,2,3
(ii)
(iii)
(i)
(i)
(ii)
(ii)
(ii)
(iii)
(iii)
(i)
(i)
(i)
(ii)
(ii)
(ii)
(iii)
(i)
(i)
(ii)
(ii)
1
2
3
4
5
6
7
8
Cycle Time = 3
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
9
Sequence: 1,3,2
1
(i)
(ii)
(ii)
(iii)
(iii)
(i)
(i)
(ii)
(ii)
(iii)
(iii)
(i)
(i)
(ii)
(ii)
(iii)
(iii)
(i)
(i)
(ii)
(ii)
(iii)
2
3
4
5
6
7
Cycle Time
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
10
Minimizing Cycle Time

Profile Fitting (PF) heuristic:

Select first job j1



Arbitrarily
Largest amount of processing
Generates profile
i
Di , j1   ph , ji
h 1

Determine which job goes next
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
11
PF: Next Job

Compute for each candidate job



Time machines are idle
Time job is blocked
Start with departure times:

 max  D
D1, j2  max D1, j2  p1c , D2, j1
Di , j2
i 1, j2


 pic , Di 1, j1 , i  2,..., m  1
Dm1, j2  Dm 1, j2  pmc
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
12
Nonproductive Time

Calculate sum of idle and blocked time
m
D
i 1
i , j2
 Di , j1  pic ,

Repeat for all remaining jobs in the MPS
Select job with smallest number

Calculate new profile and repeat

July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
Discussion: PF Heuristic


PF heuristic performs well in practice
Refinement:



Nonproductive time is not equally bad on
all machines
Bottleneck machine
Use weight in the sum
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
14
Discussion: PF Heuristic

Basic assumptions



Setup is not important
Low WIP is important
 Cyclic schedules good
Want to maximize throughput
 Minimize cycle time
 PF heuristic performs well
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
15
Discussion: FMS

Flexible Manufacturing Systems (FMS)




Scheduling




Numerically Controlled machines
Automated Material Handling system
Produces a variety of product/part types
Routing of jobs
Sequencing on machines
Setup of tools
Similar features but more complicated
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
16
Discussion: Solution Methods




Formulated as ‘simple’ sequencing
Can apply branch-and-bound
In general constraints make
mathematical programming formulation
difficult
PF heuristic easy to generalize
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
17
Additional Complications


The material handling system does not wait
for a job to complete
 Paced assembly systems
There may be multiple machines at each
station and/or there may be bypass
 Flexible flow systems with bypass
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
18
Topic 29
Paced Assembly
Systems
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Paced Assembly Systems


Conveyor moves jobs at fixed speeds
Fixed distance between jobs



Spacing proportional to processing time
No bypass
Unit cycle time

time between two successive jobs
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
20
Grouping and Spacing

Attributes and characteristics of each
job


Changeover cost


color, options, destination of cars
Group operations with high changeover
Certain long operations


Space evenly over the sequence
Capacity constrained operations (criticality index)
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
21
Objectives


Minimize total setup cost
Meet due dates for make-to-order jobs


Spacing of capacity constrained
operations


Total weighted tardiness
Pi(l) = penalty for working on two jobs l
positions apart in ith workstation
Regular rate of material consumption
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
22
Grouping and Spacing
Heuristic




Determine the total number of jobs to
be scheduled
Group jobs with high setup cost
operations
Order each subgroup accounting for
shipping dates
Space jobs within subgroups accounting
for capacity constrained operations
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
23
Example




Single machine with 10 jobs
Each job has a unit processing time
Setup cost
c jk  a j1  ak 1
If a j 2  ak 2 there is a penalty cost
P2 (l )  max( 3  l ,0)
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
24
Example Data
Job 1
2
3
4
5
6
7
8
9
10
a1 j 1
1
1
3
3
3
5
5
5
5
a2 j 0
1
1
0
1
1
1
0
0
0
dj 
2




6



wj 0
4
0
0
0
0
4
0
0
0
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
25
Grouping

Group A: Jobs 1,2, and 3
Group B: Jobs 4,5, and 6
Group C: Jobs 7,8,9, and 19

Best order: A  B  C


July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
26
Grouped Jobs
A
B
C
Job 1
2
3
4
5
6
7
8
9
10
a1 j 1
1
1
3
3
3
5
5
5
5
a2 j 0
1
1
0
1
1
1
0
0
0
dj 
2




6



wj 0
4
0
0
0
0
4
0
0
0
Due
date
 Order A  C  B
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
27
Capacity Constrained
Operations
A
C
B
Job 2
1
3
8
7
9
10 5
4
6
a1 j 1
1
1
5
5
5
5
3
3
3
a2 j 1
0
1
0
1
0
0
1
0
1
dj 2



6





wj 4
0
0
0
4
0
0
0
0
0
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
28
Topic 30
Flexible Flow Systems
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Flexible Flow System with Bypass
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
30
Flexible Flow Line Algorithm

Objectives



Minimizes the makespan of a day’s mix


Maximize throughput
Minimize work-in-process (WIP)
Actually minimization of cycle time for MPS
Reduces blocking probabilities
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
31
Flexible Flow Line Algorithm

Three phases:

Machine allocation phase


Sequencing phase



assigns each job to a specific machine at
station
orders in which jobs are released
dynamic balancing heuristic
Time release phase

July 11, 2017
minimize MPS cycle time on bottlenecks
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
32
Machine Allocation




Bank of machines
Which machine for which job?
Basic idea: workload balancing
Use LPT dispatching rule
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
33
Sequencing




Basic idea: spread out jobs sent to the
same machine
Dynamic balancing heuristic
For a given station, let pij be processing
time of job j on ith machine
Let
n
n
Wi   pij
j 1
July 11, 2017
W  Wi
j 1
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
34
Dynamic Balancing Heuristic

Let Sj be the jobs released before and
including job j
Define

Target

 
*
j
July 11, 2017
pik
 ij  
 0,1
kS j Wi
m
n
m
 p /  p
kS j i 1
ik
k 1 i 1
ik

p
kS j
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
k
/W
35
Minimizing Overload

Define the overload of the ith machine
oij  pij  p jWi / W

The cumulative overload is
Oij 

Minimize
 oik 
kS j
kS j
 max O ,0
n
m
i 1 j 1
July 11, 2017
*
p


 ik jWi
ij
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
36
Release Timing

MPS workload of each machine known



Highest workload = bottleneck
MPS cycle time  Bottleneck cycle time
Algorithm



Step 1: Release all jobs as soon as possible
Step 2: Delay all jobs upstream from
bottleneck as much as possible
Step 3: Move up all jobs downstream from
the bottleneck as much as possible
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
37
Example
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
38
Data
Jobs 1
2
3
4
5
p1' j
6
3
1
3
5
'
2j
3
2
1
3
2
p3' j
4
5
6
3
4
p
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
39
Machine Allocation
Jobs 1
2
3
4
5
p1 j
6
0
0
3
0
p2 j
0
3
1
0
5
p3 j
3
2
1
3
2
p4 j
4
5
0
3
0
p5 j
0
0
6
0
4
July 11, 2017
Lecture Notes for Planning and Scheduling
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40
Workload
W1  9
W2  9
W3  11
W4  12
W5  10
p1  13
p2  10
p3  8
p4  9
p5  11
W  51
July 11, 2017
Lecture Notes for Planning and Scheduling
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41
Overload
o11  6  13  9
o21  0  13  9
51
 3.71
 2.29
51
o31  3  13 11  0.20
51
o41  4  13 12  0.94
51
o51  0  13 10  2.55
51
July 11, 2017
Lecture Notes for Planning and Scheduling
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42
Overload Matrix
3.71
-2.29
0.20
0.94
-2.55
July 11, 2017
-1.76
1.24
-0.16
2.65
-1.96
-1.41
-0.41
-0.73
-1.88
4.43
1.41
-1.59
1.06
0.88
-1.76
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
-1.94
3.06
-0.37
-2.59
1.84
43
Dynamic Balancing
3.71
0.00
0.20
0.94
0.00
0.00
1.24
0.00
2.65
0.00
0.00
0.00
0.00
0.00
4.43
1.41
0.00
1.06
0.88
0.00
0.00
3.06
0.00
0.00
1.84
4.84
3.88
4.43
3.35
4.90
First Job
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
44
Selecting the Second Job

Calculate the cumulative overload
O11   p1k  1*W1 
kS1

k{4,1}
p1k  0.43  9
 (3  6)  0.43  9  5.13
where
1* 
p
kS j
k
/W 
p
k{4 ,1}
k
/ 51
 (9  13) / 51  0.43
July 11, 2017
Lecture Notes for Planning and Scheduling
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45
Cumulative Overload
Oi1  (5.11,3.88,1.26,1.82,4.32)
Oi 2  (0.35,0.36,0.90,3.52,3.72)
Oi 3  (0.00,2.00,0.33,1.00,2.67)
Oi 5  (0.53,1.47,0.69,1.71,0.08)
Selected next
July 11, 2017
Lecture Notes for Planning and Scheduling
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46
Final Cycle


Schedule jobs 4,5,1,3,2
Release timing phase



Machine 4 is the bottleneck
Delay jobs on Machine 1, 2, and 3
Expedite jobs on Machine 5
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
47