1. Summary Machine Scheduling
2. ELSP (one item, multiple items)
3. Arbitrary Schedules
4. More general ELSP
Operational Research & Management Operations Scheduling

Summary Machine Scheduling Problems
Operational Research & Management Operations Scheduling

Machine Scheduling
 Single machine
1|| C max
1||

C j
1 ||
 w j
C j
1| r j
|

C j
1| j
, |

C j
 Parallel machine
|| max
Pm ||

C j
| | max
Operational Research & Management
1|| L max
1| | max
1| r L j
| max
1| j
, | max
3-partition
P 2 || C partition max
P

| | max
Operations Scheduling
1||

U j
1||
 w U j
1||

T j
1||
 w T j
3

Why easy?
 Show polynomial algorithm gives optimal solution
 Often by contradiction
– See 1 ||
 w j
C j week 1
– See 1| | max week 1
 By induction
– E.g. 1||

U j
Operational Research & Management Operations Scheduling 4

Why hard?
Operational Research & Management Operations Scheduling 5

Why hard?
Operational Research & Management Operations Scheduling 6

Why hard?
Operational Research & Management Operations Scheduling 7

Why hard?
 Some problem are known to be hard problems.
– Satisfiability problem
– Partition problem
– 3-Partition problem
– Hamiltonian Circuit problem
– Clique problem
 Show that one of these problems is a special case of your problem
– Often difficult proof
Operational Research & Management Operations Scheduling 8

Partition problem
 Given positive integer a
1
, …, a t and b with ½  j a j
= b do there exist two disjoint subsets S
1 and S
2 such that
 j

S a j
= b ?
 Iterative solution:
– Start with set {a
1
} and calculate value for all subsets
– Continue with set {a
1
, a
2
} and calculate value for all subsets; Etc.
– Last step: is b among the calculated values?
 Example: a = (7, 8, 2, 4, 1) and b = 11
Operational Research & Management Operations Scheduling 9

Some scheduling problems
 Knapsack problem 1| d j
 d |
 w U j
Translation: n = t ; p j
= a j
; w j
= a j
; d = b ; z = b ;
Question: exists schedule such that
 j w j
U j
 z ?
 Two-machine problem P 2 || C max
Translation: n = t ; p j
= a j
; z = b ;
Question: exists a schedule such that C max
 z ?
Operational Research & Management Operations Scheduling 10

3-Partition problem
 Given positive integer a
1
, …, a
3t and b with ¼b < a j
< ½ b and
 j a j
= subsets S i t b do there exist pairwise such that disjoint three element
 j

S a j
= b ?
Operational Research & Management Operations Scheduling 11

Economic Lot Scheduling Problem:
One machine, one item
Operational Research & Management Operations Scheduling

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
 Applications
– Continuous manufacturing: chemical, paper, pharmaceutical, etc.
– Service industry: retail procurement
Operational Research & Management Operations Scheduling 13

Objective
 Minimize total cost
– setup cost
– inventory holding cost
 Trade-off
 Cyclic schedules but acyclic sometimes better
Operational Research & Management Operations Scheduling 14

Scheduling Decisions
 Determine the length of runs
– gives lot sizes
 Determine the order of the runs
– sequence to minimize setup cost
 Economic Lot Scheduling Problem (ELSP)
Operational Research & Management Operations Scheduling 15

Overview
 One type of item / one machine
– without setup time
 Several types of items / one machine
– rotation schedules
– arbitrary schedules
– with / without sequence dependent setup times / cost
 Generalizations to multiple machines
Operational Research & Management Operations Scheduling 16

Minimize Cost
 Let x denote the cycle time
 Demand over a cycle
 Length of production run needed
 Maximum inventory level
= Dx
= Dx / Q =
 x
= ( Q D ) Dx / Q = ( Q - D )
 x
= ( 1 -

) D x
Inventory
( Q

)
Q
Time x
Operational Research & Management idle time
Operations Scheduling 17

 Solve
Optimizing Cost min book c x

1
2

 
2
D x
Q

 shorter min c x

1
2 h

1
  
Dx
 Optimal Cycle Time x

2 Qc
(

D ) x

2 c h (1
 
) D
 Optimal Lot Size
Operational Research & Management gx

2 cDQ
(

D ) gx

2 cD h (1
 
)
Operations Scheduling 18

With Setup Time
 Setup time s
 If s
 x (1-

) above optimal
  g q
 utilizatio n
 Otherwise cycle length x

1
 s
 is optimal
Operational Research & Management Operations Scheduling 19

Economic Lot Scheduling Problem:
Lot Sizing with Multiple Items
With Rotation Schedule
Operational Research & Management Operations Scheduling

Multiple Items
 Now assume n different items
 Demand rate for item j is D j
 Production rate of item j is Q j
 Setup independent of the sequence
 Rotation schedule : single run of each item
Operational Research & Management Operations Scheduling 21

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
Operational Research & Management Operations Scheduling 22

Optimal Cycle Length
 Average total cost j n 

1

 c j x


 with a j

1
2 h j
(1
  j
) D j

 j c j x


 j a
 j x
 Solve as before x
 j n 

1 c j


 j n 

1 a j



Operational Research & Management Operations Scheduling 23

Example
Items 1 2 3 4
_ q j
400 400 500 400
_ g j
50 50 60 60
_ h j
20 20 30
_ c j
2000 2500 800
70
0
Operational Research & Management Operations Scheduling 24

Data of example 7.3.1
c(j) h(j)
D(j)
Q(j) rho(j) a(j) t(j)
1
2000
20
50
400
0.125
437.5
2.14
2
2500
20
50
400
0.125
437.5
2.39
3
800
30
60
500
0.12
792
1.01
4
0
70
60
400
0.15
1785
0.00
 j c
 j
5300
 j a
 j
3452
Operational Research & Management Operations Scheduling 25

Solution x
 j n  

1 c j


 j n

1
½ h j
(1
  j
) g j




1




1
7
2 10. .50 15.
8

44
50

34
40
.60
5300
 
3452
 
1
5300

1.5353

1.24 months


 j j c j a j
Operational Research & Management Operations Scheduling 26

Solution
 The total average cost per time unit is
2155

2559

1627

2213

$ 8554
(alternative 1:
 a j

3452
 c j

5300
(alternative 2: AC

1.24
AC or
2
 
 How can we do better than this?
Operational Research & Management Operations Scheduling
)
)
27

With 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
Operational Research & Management Operations Scheduling 28

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 possible x j n 

1 s j

1
 j n 

1
 j
Operational Research & Management Operations Scheduling 29

Job Dependent Setup Times
 Now there is a sequencing problem
 Objective: minimize sum of setup times
 Equivalent to the Traveling Salesman Problem (TSP)
Operational Research & Management Operations Scheduling 30

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
Operational Research & Management Operations Scheduling 31

Arbitrary Schedules
Operational Research & Management Operations Scheduling

Arbitrary Schedules
 Sometimes a rotation schedule does not make sense
(remember problem with no setup cost)
 For the 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
Operational Research & Management Operations Scheduling 33

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 met between production runs
Operational Research & Management Operations Scheduling 34

 Setup cost and setup times
Notation c jk
 c k
, s jk
 s k
.
 All possible sequences
S


( j
1
, j
2
,..., j h
) : h
 n

 Item k produces in l -th position q l  q j l
 q k
 Setup time s l , run time (production) t l , and idle time u l
Operational Research & Management Operations Scheduling 35

Inventory Cost
 Let x be the cycle time
 Let v be the time between production of k v
 q l t l g l
 q k t l g k
 Total inventory cost for k is
1
2 h l
( q l  g l

)
 q l g l

 t
 
2
Operational Research & Management Operations Scheduling 36

Mathematical Program mi n
S min x ,
 l
, u l
1 x



 l


1
1
2 l ( l 
D l )

Q l


D l
 (
 l ) 2 
 l


1 c l



Subject to Q k
  k j 
D x k
,

 l
(

 j
(  S

1
)
(
 j j s j u j
)


Q l


D l

 l
, s u j
)
 x k

1,..., n l

1,...,

S
Operational Research & Management Operations Scheduling 37

 Master problem
– finds the best sequence
Two Problems
 Subproblem
– finds the best production times, idle times, and cycle length
 Key idea: think of them separately
Operational Research & Management Operations Scheduling 38

Subproblem (lot sizing) min x ,
 l
, u l
1 x


 l



1
1
2 l ( l 
D l )

Q l


D l


2  l



1 c l



Subject to

 l
(
 j



1
(
 j j s j u j
)


Q l


D l

 l
,
  u j
)
 x l

But first: determine a sequence
Operational Research & Management Operations Scheduling 39

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
Operational Research & Management Operations Scheduling 40

Step 1: Computing Relative Frequencies
 Let y k denote the number of times item k is produced in a cycle
 We will
– simplify the objective function by substituting
– a k

1
2
( k k
 g k
)
 k drop the second constraint
 sequence no longer important
Operational Research & Management Operations Scheduling 41

Rewriting objective function min x ,
 l
, u l
1 x


 l



1
1
2 k y k a k k substitute:
 k l ( l 
D l )

Q l


D l


2  l



1 c l




1 x

 k n 

1 y k
1
2 k
( k

D k
)


Q
D k k


 k
2  k n 

1 c y k k



1 x

 k n 

1 a y k k
 k
2
ρ 2 k
 k n 

1 c y k k


 k n 

1 a k x y k
 k n 

1 c k y k x
Assumption: for each item production runs of equal length and evenly spaced
Operational Research & Management Operations Scheduling 42

Mathematical Program min k k n 

1 a x y k
 k n 

1 c y x k
Subject to k n 

1 s y k k x
Remember: for each item production runs of equal length and evenly spaced
Operational Research & Management Operations Scheduling 43

 Using Lagrange multiplier:
Solution y k
 x c k a k
  s k
 Adjust cycle length for frequencies
 Idle times then

= 0
 No idle times, must satisfy k n 

1

 s k c k a k
  s k

 
Operational Research & Management Operations Scheduling 44

Step 2: Adjusting the Frequencies
 Adjust the frequencies such that they are
– integer
– powers of 2
– cost within 6% of optimal cost
– e.g. such that smallest y k
=1 y k
' , and
 k
'
 New frequencies and run times
Operational Research & Management Operations Scheduling 45

Step 3: Sequencing


Variation of LPT
Calculate y
'  max max
 y
1
'
,..., y
' n


 y
' y k
'
Consider the problem with machines in parallel and jobs of length
 k
'
(
' k
,

'
List pairs in decreasing order
 Schedule one at a time considering spacing
 Only if for all machines assigned processing time < x y ' max equal lot sizes possible
Operational Research & Management Operations Scheduling then
46

Lot Sizing on Multiple Machines
Operational Research & Management Operations Scheduling

Multiple Machines
 So far, all models single machine models
 Extensions to multiple machines
– parallel machines
– flow shop
– flexible flow shop
Operational Research & Management Operations Scheduling 48

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
Operational Research & Management Operations Scheduling 49

Decision Variables
 Same as previous multi-item problem
 Addition: assignment of items to machines
 Objective: balance the load
 Heuristic: LPT with
 k
 g k q k
Operational Research & Management Operations Scheduling 50

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
Operational Research & Management Operations Scheduling 51

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
Operational Research & Management Operations Scheduling 52

Further Generalizations
 Sequence dependent setup
 Must consider
– preferred cycle time
– machine balance
– setup times
 Unsolved
 General schedules
 even harder!
 Research needed :-)
Operational Research & Management Operations Scheduling 53

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 single machine problem c k
 i m 

1 c ik
Operational Research & Management Operations Scheduling 54

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
Operational Research & Management Operations Scheduling 55