MACHINE SCHEDULING WITH TRANSPORTATION CONSIDERATIONS

1
MACHINE SCHEDULING WITH TRANSPORTATION
CONSIDERATIONS
Chung-Yee Lee and Zhi-Long Chen
2
OUTLINE

• Introduction
• Problems and Notation
• Problems with Type-1 Transportation
• Problems with Type-2 Transportation

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INTRODUCTION

• Most of the literature assumes either
• infinite number of transporters or
• instantaneous transportation of jobs without
  transportation time involved
• Considering scheduling and job transportation
  jointly is more realistic

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INTRODUCTION

• Scheduling of machine and material handling
  operations
• In these problems following issues must be
  addressed
• simultaneously
• Sequencing that specifies the order in which jobs
  are processed at machining centers
• Scheduling that makes time-phased routing and
  dispatching of transporters for job pick-up and
  delivery
• Facility layout and flowpath design that makes
  efficient operations possible
• Due to combinatorial nature of the problems,
  finding an
• optimal solution that addresses all these issues
  at the same
• time is very difficult.

5
INTRODUCTION
m machining centers
Transporters and machines can hold 1 job at a time
n jobs

Input buffer
K identical transporters
Flowpath
Output buffer
...
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INTRODUCTION

• Recent work can be divided into
• Robotic cell scheduling
• Scheduling of Automated Guided Vehicles (AGVs)
• Cyclic scheduling of hoists subject to
  time-window constraints
• Differ mainly in the structure of their
  constraints
• Robotic cell scheduling problem has the fewest
  constraints
• while cyclic scheduling of hoists with time
  windows is the
• most restrictive

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INTRODUCTION

• Robotic Cell Scheduling Problem

...
Main concern is to find the job input sequence
and the robot move sequence with respect to a
certain objective.
8
INTRODUCTION
Scheduling of AGVs Deals with automated job
shop with non-zero buffers at machining centers
and multiple AGVs travelling on a shared
network. Main concern is how to schedule the
moves of AGVs in a traffic network so that
traffic collusions are eliminated and the risk of
machine blocking is minimized.
9
INTRODUCTION
Flowpaths i- Unidirectional( ) ii-
Bidirectional( ) Higher control and
implementation cost, greater potential to improve
productivity, fewer AGVs, reduced travel
time Network Configurations i- Single-loop
All machines accessible via the loop, avoiding
AGV collusions easy ii- Multi-loop AGV
collusion and machine blocking are the major
concerns in scheduling.
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INTRODUCTION

• The Hoist Scheduling Problem
• Most distinct feature is that the job processing
  time at each machine is strictly limited by a
  lower and an upper bound.
• Hoist schedule that causes a hoist not to pick up
  a job within the time window is infeasible.
• Also the traffic collusions must be eliminated
• Single hoist scheduling problem with numerical
  processing times can be viewed as a special case
  of robotic cell scheduling problem
• Multiple hoist problem can be considered as a
  special class of AGV scheduling problems.

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INTRODUCTION
Intermediate buffers
Customer or warehouse
M1
Mk
M2
...
Transporters
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INTRODUCTION

• Objectives
• makespan
• maximum lateness
• cycle time
• Buffers (for all or some of the m/cs)
• infinite
• no buffers
• finite capacity buffers
• Transporters
• 1, finite or infinite of transporters
• capacity may be 1, finite or infinite

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INTRODUCTION

• Transportation time
• Job dependent
• Depending on the location of the m/cs
• Number of m/cs
• 1, 2 or k
• Several machines at each stage of a flowshop
• Special assumptions
• Completion time of job is when the job arrives to
  the customer
• Some jobs must be scheduled consecutively
• Simultaneous scheduling of jobs and transporters

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PROBLEMS NOTATION
Type-1 Transportation
m m/cs
n jobs
Mk, pjk
M1
Mm
...
Mk1
...
...
tk,k1
...
...
c jobs
v transporters
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PROBLEMS NOTATION

• Cj Completion time of job j
• Objectives considered
• Makespan
• Total completion time
• abg Notation
• TF in a field denotes flowshop with
  transportation
• Example TF2vx,cyCmax

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PROBLEMS NOTATION
Type-2 Transportation
Customer or warehouse
n jobs
t2
c jobs
t1
v transporters
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PROBLEMS NOTATION
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PROBLEMS NOTATION
Frequently cited NP-hard Problems 3-Partition
(3-PP) Given H1,2,,3h, each item j?H has a
positive integer such that b/4ltajltb/2, and
?ajhb, for some integer b. Are there h disjoint
subsets H1,H2,,Hh such that each subset has
three items and its total size is equal to
b? Equal-size Partition(ESPP) Given
H1,2,,2h, each item j?H has a positive
integer such that ?j?Haj2A for some integer A.
Is there a subset G?H such that ?j?GajA and
there are h items in G.
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TYPE-1 TRANSPORTATION

• All the jobs transported in one shipment are
  called a batch.
• Bk is the kth batch
• dk departure time of Bk.
• Cj1,Cj2 completion times of job j on m/c1 and
  m/c 2.
• t1 (t2) Transportation time from m/c1 (m/c2) to
  m/c 2 (m/c 1)

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TYPE-1 TRANSPORTATION
Property 1 TF2v ? 1, c ? 1f(C1,,Cn), where f
is regular (i) Jobs are processed on m/c 1
without idle time, (ii) Jobs transported in the
same batch are processed consecutively without
idle time on both m/cs, (iii) Jobs finished
earlier on m/c 1 delivered earlier to m/c 2.
Furthermore, it is a permutation schedule.
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TYPE-1 TRANSPORTATION
(iv) Vehicle k carries batches with index kqv,
q ?0 (v) Two consecutive batches delivered by
the same vehicle k satisfy either
dk(q1)vdkqvt1t2 or dk(q1)v is the
completion time of the last job in Bk(q1)v
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TYPE-1 TRANSPORTATION

• TF2v n, c ? 1Cmax
• Simplest among all TF2
• Whenever a job is finished there is always a
  transporter available
• Generalized Johnsons rule solves the problem

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TYPE-1 TRANSPORTATION

• TF2v 1, c 1Cmax
• With general t1 and t2 NP-hard
• General t1 and t20, identical with F3pj2pCmax
  and strongly NP-hard
• t1t2t is strongly NP-hard.

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TYPE-1 TRANSPORTATION

• TF2v 1, c 2Cmax Open
• TF2v 1, c ? 3Cmax
• Strongly NP-hard even if t1t2
• Reduction from 3-PP. Given a 3-PP instance
  construct an instance of the problem as follows
• Let each element of H be a job to be scheduled.
  Also add one dummy job called (3h1)th job
• pj1pj22aj for j?H, p3h1,11, p3h1,22b
• t1t2b, c ? 3, threshold value y1(2h3)b
• Is there a schedule so that Cmax ? y?

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TYPE-1 TRANSPORTATION
If there is a solution to 3-PP, there is a
schedule to our problem with Cmax ? y
...
H1
3h1
H2
Hh
...
...
3h1
H1
H2
Hh
0
1
1b
13b
15b
1(2h3)b
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TYPE-1 TRANSPORTATION

• If there is a schedule with Cmax ? y 1(2h3)b
• Total processing time on m/c 2 is 2(h1)b.
• In order to not to exceed y, jobs must start no
  later than y-2(h1)b1b Only if job 3h1 is
  the first job this is achieved. Then,
• Job 3h1 must be the first job on both m/cs
• It must be transported at time 1, B1 contains
  only this job.
• There is no idle time on m/c 2 after it starts
  processing.

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TYPE-1 TRANSPORTATION

• Assume there are q batches
• di Departure time of batch i1,..,q
• ri Arrival time to m/c 2
• Ci Completion time on m/c 2
• d11, r11b, C113b r2 ? 13b
• t1t22b dk1 ? dk2b d2 ? 12b
  r2?13b
• Thus, r213b and d212b
• ?i?B2pj1 ?2b.

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TYPE-1 TRANSPORTATION

• If ?i?B2pj1 lt 2b ?i?B2pj2 lt 2b C2lt15b
• On the other hand, r3 ? r22b15b.
• Idle time on m/c 2 b/t B2 and B3.
• Thus, ?i?B2pj1 2b must hold.
• Generalize this for all k, then ?i?Bkpj1 2b.
• qh1 and B2, B3,,Bh1 form a solution to 3-PP

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TYPE-1 TRANSPORTATION

• TF2 pj1p1, v ? 1, c ? 1 Cmax
• Polynomially solvable by DP.
• (i) Non-increasing order of pj2 on both m/cs
• (ii) if tt1t2 ? vp1, each job is transported
  from m/c 1 to m/c 2 immediately after it is
  completed on m/c 1.
• Sequencing of jobs is trivial
• if tt1t2 ? vp1 then starting time of each trip
  is also trivial

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TYPE-1 TRANSPORTATION

• Otherwise, when a transporter returns from m/c 2
  to m/c 1 it either immediately transports a batch
  of jobs or wait until completion time of a job.
  So possible departure times are Cj1, Cj1t,
  Cj12t,,Cj1qt where q ? n-j, j1,,n
• Example Let n5, t20, p115. Then possible
  departure times are
• 15, 35, 55, 75, 95 30, 50, 70, 90 45, 65,
  85
• 60, 80 75

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TYPE-2 TRANSPORTATION
1?D v ? 1, c ? 1 Cmax Polynomially solvable
in O(nlogn) time (i) Nondecreasing order of
processing time (ii) Batches contain consecutive
jobs (iii) Earlier processed jobs delivered no
later than later processed ones (iv) n/cinteger,
then n/c batches with c jobs (v) o.w. (?n/c?)1
batches. First batch contains un-(?n/c?)c, all
others c jobs.
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TYPE-2 TRANSPORTATION
Example n10, v2, c3, t120, t215
n/c10/33.33, B11 job, Bh3 jobs,
h2,4 B11, r14 B22,3,4,
r224 B35,6,7, r355 B48,9,10,
r499
v1
v2
4
24
39
55
59
90
99
119
Cmax119
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TYPE-2 TRANSPORTATION

• 1?D v ? 1, c ? 1 ?Cj
• Polynomially solvable by DP similar to previous
• Non-decreasing order of pj
• Finite number of candidate departure times

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TYPE-2 TRANSPORTATION

• Pm?D v 1, c ? 1 ?Cj
• Pm ?Cj ?Non-decreasing order of processing
  times
• P2?D v 1, c ? 1 ?Cj ?NP-hard even if t1t2
• Reduction from ESPP.

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TYPE-2 TRANSPORTATION
Example 2 parallel m/cs, p15, p28, p318,
t1t24 1, 2, 3 ?Cj53 1, 3, 2 ?Cj51
m/c1
m/c 2
vehicle
5
13
21
25
18
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TYPE-2 TRANSPORTATION

• F2?D v 1, c 1 Cmax
• F2 Cmax ? Johnsons algorithm
• Reduction from 3-PP
• Example p115, p128, p217, p225, t1t24

m/c1
m/c 2
vehicle
5
13
21
25
18
m/c1
m/c 2
vehicle
7
12
24
20
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TYPE-2 TRANSPORTATION

• F2?D v 1, c ? 4 fixed Cmax strongly
  NP-hard
• F2?D v 1, c 2 (3) Cmax open
• Property For F2?D v 1, c k Cmax
• earlier processed jobs delivered earlier
• first trip delivers n-(?n/k?-1)k jobs, all others
  deliver k jobs
• c ?n deliver all jobs at the same time,
  Johnsons algorithm
• cn-1 fix 1st job and deliver, Johnsons
  algorithm for remaining and deliver all at the
  same time

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TYPE-2 TRANSPORTATION

• Following same argument,
• F2?D v 1, c ? n-k Cmax is polynomially
  solvable
• F2?D v 1, c n/2 Cmax
• NP-hard even t1t2, reduction from ESPP
• ? optimal schedule with two trips (each carrying
  n/2 jobs)
• Use Johnsons algorithm then partition into two
  s.t. 1st group is carried by 1st trip and 2nd
  group by 2nd trip.

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TYPE-2 TRANSPORTATION
Heuristic for F2?D v 1, c k
Cmax Example n10, v1, c4, t110,
t28 ?n/k? ?10/4?3, 10-4(3-1)2 then B12
jobs, Bh4 jobs, h2,3 B11,2,
B23,4,5,6, B37,8,9,10
m/c 1
m/c 2
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TYPE-2 TRANSPORTATION
cH ? (1(2n-2k)/(2n-k))C cH ?
(1(20-8)/(20-4))C cH ? (7/4)C
m/c1
m/c 2
vehicle
1,2
3,4,5,6
7,8,9,10
15
49
33
65
67
77
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Questions-Comments?