Topic 31
Lot Sizing
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Lot Sizing

Domain:




large number of identical jobs
setup time/cost significant
setup may be sequence dependent
Terminology


jobs = items
sequence of identical jobs = run
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
Applications

Continuous manufacturing


chemical, paper, pharmaceutical, etc.
Service industry

retail procurement
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
3
Objective

Minimize total cost


setup cost
inventory holding cost

Trade-off

Cyclic schedules
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
4
Scheduling Decisions

Determine the length of runs


Determine the order of the runs


lot sizes
sequence to minimize setup cost
Economic Lot Scheduling Problem
(ELSP)
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
5
Overview

One type of item/one machine

Several types of items/one machine



rotation schedules
arbitrary schedules
Generalizations to multiple machines
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
6
Problem Description

Single machine
Single item type
Production rate Q/time
Demand rate D/time

Problem: determine the run length



June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
7
Minimize Cost



Let x denote the cycle time
Demand over a cycle = Dx
Length of production run needed = Dx/Q
(Q  D) Dx
Q
Inventory
1
D2 x 
  x
AREA   Dx 
2
Q 
Time
x
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
8
Costs



Setup cost per run
Average setup cost c/x
Average inventory holding cost
1 
D2 x 

h Dx 
2 
Q 
Total cost
2

1
D x c
 
h Dx 
2 
Q  x
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Per item holding cost
9
Optimizing Cost

Derivative
d 1 
D2 x  c 1  D  c
   hD1    2
h Dx 
dx 2 
Q  x 2  Q x

Solve
June 28, 2016
1  D c
hD1    2  0
2  Q x
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
10
Optimal Cycle Time
1  D c
hD1    2
2  Q x
2Qc
x 
hD(Q  D)
2
2Qc
x
hD(Q  D)
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
11
Optimal Lot Size

Total production
2cQD
Dx 
h(Q  D)

When unlimited production capabilities
2cQD
2c
Q 

h(Q  D)
hD

Economic Order Quantity (EOQ)
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
12
Setup Time

Setup time s

If s  x(1-r) above optimal

D
r   utilizatio n
Q
Otherwise cycle length
s
x
1 r
is optimal
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
Numerical Example




Production
Demand
Setup cost
Holding cost
Q = 90/month
D = 50/month
c = $2000
h = $20/item
2  90  2000
3600
36
x


3
20  50(90  50)
10  40
4
June 28, 2016
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Prepared by Siggi Olafsson
14
Optimal Schedule

Cycle time = 3 months

Lot size = 150 items

Idle time = 3(1-5/9)=1.33 months
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
15
Example: Setup Times



Now assume setup time
If < 1.33 months then 3 month cycle
still optimal
Otherwise the cycle time must be
s
longer
x
1 r
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
16
Inventory Levels
Inventory
120
Month
1
2
3
4
5
6
Inventory
180
Month
1
June 28, 2016
2
3
4
5
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
6
17
Example

A plant needs to produce 10000 car chassis
per year
The plant capacity is 25000 chassis/year
Each chassis costs $2000
It costs $200 to set up a production run
Holding cost is $500/chassis/year

What is the optimal lot size?




June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
18
Solution

The optimal lot size is
2cQD
2  25000  200 10000
Dx 

h(Q  D )
500(25000  1000)
2  25  200 100
1000000


 115.5
5(25  10)
75
which means we should make
runs in a year.
June 28, 2016
10000
 86.6
115.5
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
19
Discussion



Notice that the preceding result does not tell
us how to produce those chassis in detail
Lot size models are used for planning
Time horizon usually a few months (short
range planning)
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
20
Topic 33
Lot Sizing with Multiple
Items
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Multiple Items

Only considered one item type before

Now assume n different items
Demand rate for item j is Dj
Production rate of item j is Qj
Setup independent of the sequence

Rotation schedule: single run of each item



June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
22
Scheduling Decision

Cycle length determines the run length for
each item

Only need to determine the cycle length x

Expression for total cost/time unit
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
23
Inventory Holding Cost


Average inventory level for the j-th item
2

Dj x 
1

Dj x 
2 
Q j 
Average total cost
2
n 

 cj 
D
x
1
j
 h D x
 

j
j
 x

Q
j 1  2
j



June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
24
Optimal Lot-Size

Solve as before
 n h j D j (Q j  D j ) 

x  
 j 1

2
Q
j



1
n
c
j 1
j
Limiting case (infinite production rate)
 hj Dj 

x   

2
 j 1

n
June 28, 2016
1
n
c
j 1
j
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
25
Example
Items
1
2
3
4
_Qj
400
400
500
400
_Dj
50
50
60
60
_hj
20
20
30
70
2000 2500
800
0
_cj
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
26
Solution
 n h j D j (Q j  D j ) 

x  
 j 1

2
Q
j


1
n
c
j 1
j
1
 10  350 18  440 42  340 
 2


 5300
8
10
8 

1
 10  350 18  440 42  340 
 


 5300
10
8 
 4

June 28, 2016
3452
1
5300  1.5353  1.24 months
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
27
Solution

The total average cost per time unit is
2155  2559 1627  2213  $8554

How can we do better than this?
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
28
Topic 34
Lot Sizing with Setup
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Setup Times


With sequence independent setup costs
and no setup times the sequence within
each lot does not matter
 Only a lot sizing problem
Even with setup times, if they are not
job dependent then still only lot sizing
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
30
Job Independent Setup Times


If sum of setup times < idle time then
our optimal cycle length remains
optimal
Otherwise we take it as small as
n
possible
sj

j 1
x
n
1  r j
j 1
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
31
Job Dependent Setup Times



Now there is a sequencing problem
Objective: minimize sum of setup times
Equivalent to the Traveling Salesman
Problem (TSP)

A salesman must visit n cities exactly once with
the objective of minimizing the total travel time,
starting and ending in the same city
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
32
Equivalence to TSP

Item = city
Travel time = setup time

TSP is NP-hard


If best sequence has
sum of setup times < idle time
 optimal lot size and sequence
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
33
Long Setup

If sum of setups > idle time, then the
optimal schedule has the property:


Each machine is either producing or being
setup for production
An extremely difficult problem with
arbitrary setup times
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
34
Arbitrary Schedules



Sometimes a rotation schedule does not
make sense
(remember problem with no setup cost)
For example, we might want to allow a cycle
1,4,2,4,3,4 if item 4 has no setup cost
No efficient algorithm exists
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
35
Problem Formulation


Assume sequence-independent setup
Formulate as a nonlinear program
min
min COST
 sequences  lot sizes
s.t.
demand met over the cycle
demand is meet between production runs
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
36
Notation

Setup cost and setup times
c jk  ck , s jk  sk .



All possible sequences
S  ( j1 , j2 ,..., jh ) : h  n
Item k produces in l-th position
l
Q  Q jl  Qk
Setup time sl, run time tl, and idle time
ul
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
37
Inventory Cost



Let x be the cycle time
Let v be the time between production of
k
Q lt l Qkt l
v l 
D
Dk
Total inventory cost for k is
l

1 l l
Q
l
h (Q  D ) l
2
D
June 28, 2016
 l
 t

 
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
38
Mathematical Program
l

1 h 1 l l
Q
  h (Q  D l ) l
min min
D
S x ,t l ,u l x 
2

 l 1
Subject to
 l 2 h l
(t )   c 

l

1


j
Q
t
 k  Dk x,
k  1,..., n
jI k
Q
(t  s  u )   l

jLl
D
l
j
j
j
l
t ,

k  1,..., n
h
j
j
j
(
t

s

u
)x

June 28, 2016
j 1
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
39
Two Problems

Master problem


Subproblem


finds the best sequence
finds the best production times, idle times,
and cycle length
Key idea: think of them separately
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
40
Subproblem
l

1 h 1 l l
Q
  h (Q  D l ) l
min
D
x ,t l ,u l x 
2

 l 1
 l 2 h l
(t )   c 

l

1


Subject to
l

Q
j
j
j
(t  s  u )   l

jLl
D
 l
t ,

k  1,..., n
h
j
j
j
(
t

s

u
)x

j 1
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
41
Master Problem




Sequencing complicated
Heuristic approach
Frequency Fixing and Sequencing (FFS)
Focus on how often to produce each
item



Computing relative frequencies
Adjusting relative frequencies
Sequencing
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
42
Computing Relative
Frequencies


Let yk denote the number of times item
k is produced in a cycle
We will

simplify the objective function by
substituting
1
ak  hk (Qk  Dk ) r k
2

drop the second constraint
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
43
Mathematical Program
n
n
ak x
ck yk
min 

yk , x
x
k 1 yk
k 1
Subject to
n
sk y k
 1 r

x
k 1
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
44
Solution




Using Lagrangean multiplier:
ak
yk  x
ck  lsk
Adjust cycle length for frequencies
Idle times l = 0
No idle times, must satisfy
n 

a
k


s
 1 r


ck  lsk 
k 1  k
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
45
Adjusting the Frequencies

Adjust the frequencies such that they
are




integer
powers of 2
cost within 6% of optimal cost
New frequencies and run times
'
'
yk , t k
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
46
Sequencing


Variation of LPT
Calculate
y
'
max

 max y ,..., y
'
1
'
n


'
y
Consider the problem with max machines in
'
'
y
parallel and k jobs of length t k
'
'
List pairs ( yk , t k ) in decreasing order

Schedule one at a time considering spacing

June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
47
Topic 35
Lot Sizing on Multiple
Machines
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Multiple Machines


So far, all models single machine
models
Extensions to multiple machines



parallel machines
flow shop
flexible flow shop
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
49
Parallel Machines

Have m identical machines in parallel
Setup cost only
Item process on only one machine

Assume




rotation schedule
equal cycle for all machines
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
50
Decision Variables

Same as previous multi-item problem

Addition: assignment of items to machines

Objective: balance the load

Dk
Heuristic: LPT with r k 
Qk
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
51
Different Cycle Lengths




Allow different cycle lengths for machines
Intuition: should be able to reduce cost
Objective: assign items to machines to
balance the load
Complication: should not assign items that
favor short cycle to the same machine as
items that favor long cycle
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
52
Heuristic Balancing




Compute cycle length for each item
Rank in decreasing order
Allocation jobs sequentially to the
machines until capacity of each
machine is reached
Adjust balance
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
53
Further Generalizations


Sequence dependent setup
Must consider






preferred cycle time
machine balance
setup times
Unsolved
General schedules  even harder!
Research needed
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
54
Flow Shop




Machines configured in series
Assume no setup time
Assume production rate of each item is
identical for every machine
 Can be synchronized
Reduces to singlem machine problem
ck   cik
i 1
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
55
Variable Production Rates




Production rate for each item not equal
for every machine
Difficult problem
Little research
Flexible flow shop: need even more
stringent conditions
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
56
Discussion

Applicability of lot sizing models


short range planning
demand assumed known


make-to-stock systems



determines throughput
due date of little importance/not available
extensions to mixed systems
Multiple facilities in series

supply chain management
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
57