Topic 36
Reservation Systems
without Slack
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Overview of
Service Industry Models

Reservation systems

Timetabling

Workforce scheduling
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
Reservation without Slack





Reservation system with m machines, n jobs
Release date rj, due date dj, weight wj
No slack
p j  d j  rr
If processed job must start at time rj
Should we process the job?
 Maximize number of jobs processed
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
3
Example: A Car Rental




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Four types of cars: subcompact,
midsize, full size, and sport utility
Fixed number of each
Machine = type of car
Job = customer requesting a car
May request certain machine(s)

Job can be processed on a subset of
machines
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
4
Problem Formulation


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
Integer Programming Problems
Fixed time periods
Assume H periods
Let xij be 1 if job j assigned to ith machine
(and zero otherwize)
m
Job requiring
processing in
period l
June 28, 2016
x
i 1
ij
x
jJ l
ij
 1, j  1,..., n
 1 i  1,..., m,
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l  1,..., H
5
Feasibility Problem



Can we process every job?
Can we assign jobs to machines such
that job j is assigned to a set Mj of
machines
Relatively easy to solve
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
6
Optimization Problem

Maximize the total profit
m
n
 w x
i 1 j 1


ij ij
In general NP-hard
Special solvable cases


All processing times = 1
Decomposition into separate time units
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
7
Identical Weights and
Machines




Assume wj = 1 and Mj = {1,2,…,m}
Do not have a time decomposition
Simple algorithm maximizes number of
jobs that are processed
Order jobs in increasing release date
order
r1  r2  ...  rn
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
8
Algorithm
Step 1
Set J= and j=1
Step 2
If a machine available at time rj assign job j to the machine, include it
in J and go to Step 4
Step 3
Select j* such that
C j *  max Ck  max rk  pk
kJ
kJ
C j  rj  p j  C j* do not include j in J and go to Step 4
If
Else delete job j* from J, assign j to freed machine & include in J
Step 4
If j=N STOP; otherwise set j=j+1 and go back to Step 2
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
9
Unlimited Machines

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No slack, arbitrary processing times,
equal weights, identical machines
Infinitely many machines in parallel
Minimize the number of machines used
Easily solved
Order jobs as before
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
10
Algorithm

Assign job 1 to machine 1
Suppose first j-1 jobs have been
processed
Try to assign jth job to a machine in use
If not possible assign to a new machine

Graph theory: node coloring problem



June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
11
Node Coloring Problem



Consider a graph with n nodes
If an arc (j,k) connects nodes j and k
they cannot be colored with the same
color
How many colors do we need to color
the graph?
Number of colors needed = Number of machines needed
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
12
Topic 37
Reservation Systems
with Slack
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Reservation with Slack


Now allow slack
p j  d j  rr
Trivial case


all processing times = 1, identical weights,
identical machines
decomposition in time
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
14
General Equal Processing
Times



Now assume processing times are equal
to some p  2
Interaction between time units
Barriers algorithm

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wait for critical jobs to be released
start the job with the earliest deadline
Slot number l = lth job to start
S(l) starting time of lth slot
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
15
Barrier Algorithm

Barrier
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ordered pair (l,r)
constraint
Release
time
Slot
number
A k-partial schedule
Starting with a k-partial schedule


construct a (k+1)-partial schedule
add a new barrier to the barrier list Lb and
start over from scratch
June 28, 2016
Lecture Notes for Planning and Scheduling
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16
Earliest Deadline with Barriers

Selects machine h for job in (k+1)st slot
m
k  1 multiple of m

h
otherwise
(k  1) mod m

Compute the starting time
  max t1 , t2 , t3 , t4 
June 28, 2016
Lecture Notes for Planning and Scheduling
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17
Computing the Starting Time

Let  be the earliest time the job can
  max t1 , t2 , t3 , t4 
start
Minimum
release date
of jobs left
t1  max S (1),..., S (k )
0
k 1  m

t3  
S (k  1  m)  p otherwise
t4  max r : (k  1, r )  Lb 
June 28, 2016
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Prepared by Siggi Olafsson
18
Selecting a Job




Select for (k+1)th slot job j’ with the
earliest deadline d j '
If d j '    p
creation of (k+1)
partial schedule was successful
Otherwise a ‘crisis’ occurred and job j’ is
a ‘crisis job’
Crisis routine
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
19
Crisis Routine
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Backtracks to find job with the latest starting
time such that the deadline is greater than
the crisis job
If no such job, optimal schedule does not
include crisis job
Otherwise this job is a pull job
Add new barrier with the minimum release
time r* of all the jobs Jr after the pull job
Start over with the barrier list
June 28, 2016
Lecture Notes for Planning and Scheduling
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20
Example: 2 machines, 4 jobs
Jobs
1
2
3
4
rj
0
2
5
5
dj
20
30
19
30
Processing time p = 10
June 28, 2016
Lecture Notes for Planning and Scheduling
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21
Barriers Algorithm
Machine 1
1
Machine 2
3
2
0
June 28, 2016
10
20
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Prepared by Siggi Olafsson
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22
Barriers Algorithm
Crisis job
Machine 1
1
3
Pull job
Machine 2
2
0
10
20
30
d 3  19
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
23
Barriers Algorithm

Found a crisis job and a pull job exists:

Setup a barrier Lb={(2,5)}

Setup restricted set Jr={3}

Start from scratch
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
24
Barriers Algorithm
Second iteration:
Machine 1
1
Machine 2
0
2
3
4
10
20
30
Optimal Schedule
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
25
Generalizations

Non-identical processing times
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NP-hard
No efficient algorithm exists
Need heuristics
Composite dispatching rule
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Preprocessing: flexibility of jobs and machines
Dispatch least flexible job first on the least flexible
machine, etc
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
26
Preprocessing


Let yjk be the number of jobs that can
be processed on machine j in the slot
[k-1,k]
Let Mj be the number of machines job j
can be processed on
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
27
Priority Indices

Low values =
high priority
Priority index for jobs
I j  f ( w j / p j , | M j |)

Priority index for machines
pj

y i ,k l



l 1
g (y i ,k 1 ,y i ,k  2 ,...,y i ,k  p j )  
pj


max y i ,k 1 ,y i ,k  2 ,...,y i ,k  p j

June 28, 2016
Lecture Notes for Planning and Scheduling
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
28
Algorithm
Step 0
Calculate both priority indices
Order jobs according to job priority index (Ij)
Step 1
Set j=1
Step 2
For job j select the machine and time slot with lowest rank
Discard job j if it cannot be processed at all
Step 3
If j=n STOP; otherwise set j=j+1 and go back to Step 2
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
29
Topic 38
Timetabling
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Timetabling



Infinite identical machines in parallel
All of n jobs must be processed
Tooling constraints
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Many tools
Need one or more tools for each job
Resource constraints

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Single resource of quantity R
Need certain amount for each job
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
31
Tooling Constraints
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One of each tool
Objective: minimize makespan
Special case of resource-constrained
project scheduling problem with one of
each resource, that is, Ri = 1
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
32
Problem Complexity

NP-hard

Assume all processing times = 1

Decomposition?

Equivalent to node coloring problem
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
33
Node Coloring Problem

Job = node
Jobs need same tool = arc between nodes

Feasibility problem:



Can the graph be colored with H colors?
Optimization problem:


What is the lowest number of colors needed?
Chromatic number of the graph
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
34
Relation to Reservation Models


Closely related to reservation problem with
zero slack and arbitrary processing times
Special case of timetabling problem
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tools in common = overlapping time slots
tools in common  nodes connected
colors = machines
minimizing colors = minimizing machines
adjacent time slots vs tools need not be adjacent
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
35
Heuristics

Many heuristics exist for graph coloring



degree of node = number of arcs connected to
node
saturation level = number of colors connected to
node
Intuition:


Color high degree nodes first
Color high saturation level nodes first
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
36
Algorithm
Step 1
Order nodes in decreasing order of degree
Step 2
Use color 1 for first node
Step 3
Choose uncolored node with maximum saturation
level, breaking ties according to degree
Step 4
Color using the lowest possible number color
Step 5
If all nodes colored, STOP; otherwise go back to
Step 3
Lecture Notes for Planning and Scheduling
June 28, 2016
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37
Topic 39
Example: Scheduling
Meetings
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Example

Schedule 5 meetings with 4 people


meetings = jobs and people = tools
all meetings take one hour
June 28, 2016
1
2
3
4
5
Joe
1
1
0
1
1
Lisa
1
1
1
0
0
Jane
1
0
1
0
0
Larry 0
1
0
1
1
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Corresponding Graph
Degree =4
2
Degree =4
1
3
5
4
Degree =2
Degree =3
Degree =3
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40
Select First Job


Can select either job 1 or 2 first
Say, select 1 and color with the first
color
2
1
3
5
June 28, 2016
4
Lecture Notes for Planning and Scheduling
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Saturation level = 1
for every node
 Select 2 because
highest degree
41
Color Second Job

Color Job 2 with the next color
Saturation level = 2
for every node
2
1
3
 Select 4 because
highest degree
5
June 28, 2016
4
Lecture Notes for Planning and Scheduling
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42
Color Third Job
Color Job 4 with the next color
2
1
3
Saturation level =
2 for node 3
3 for node 5
 Select 5 next
5
June 28, 2016
4
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43
Color Fourth Job
Color Job 5 with the next color
2
1
3
5
4
 Select 3 next
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44
Color Final Job
Color Job 3 with same color as Job 4
2
1
3
5
4
Had to use 4 colors  Makespan = 4
June 28, 2016
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45
Example

Suppose now we have a reservation system:
June 28, 2016
Job
1
2
3
pj
2
3
1
rj
0
2
3
dj
2
5
4
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
46
Equivalent Timetabling
Problem
Job 1 must
be processed
in time [0,2]


Job 2 must
be processed
in time [2,5]
Job
1
2
3
Tool 1
1
0
0
Tool 2
1
0
0
Tool 3
0
1
0
Tool 4
0
1
1
Tool 5
0
1
0
Toool 5
June 28, 2016
Lecture Notes for Planning and Scheduling
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47
Topic 40
Timetabling with
Resource Constraints
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Resource Constraints

One type of tool but R units of it (resource)

Job j needs Rj units of this resource

Clearly, if Rj + Rk > R then job j and k cannot
be processed at the same time, etc

Applications

scheduling a construction project (R=crew size)

exam scheduling (R=number of seats)
June 28, 2016
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Prepared by Siggi Olafsson
49
Bin-Packing




Assume all processing times = 1
Unlimited number of machines
Minimize makespan
Equivalent to bin-packing problem



each bin has capacity R
item of size Rj
Pack into the minimum number of bins
June 28, 2016
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
50
Solving the Bin-Packing
Problem



Known to be NP-hard
Many heuristics developed
First fit (FF) heuristic


Always but an item in the first bin it fits
into
Know that
17
Cmax (FF)  Cmax (OPT)  2
10
June 28, 2016
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51
Example

Assume 18 items and R=2100




FF heuristic:




Jobs 1-6 require 301 resources
Jobs 7-12 require 701 resources
Jobs 13-18 require 1051 resources
We assign the first 6 jobs to one interval (3016=1806)
We then assign jobs two at a time to next 3 intervals
(7012=1402)
We then assign just one of the remaining jobs to each
interval
Very poor performance due to order of jobs
June 28, 2016
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Prepared by Siggi Olafsson
52
First Fit Decreasing (FFD)



An improvement of FF:
Order jobs in decreasing order first
Know that
11
Cmax (FFD)  Cmax (OPT)  4
9

FF and FFD can be extended to
different release dates
June 28, 2016
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Prepared by Siggi Olafsson
53
Discussion

Only considered very simple models

In practice:


Dynamic reservation systems
Price considerations (yield management)
June 28, 2016
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Prepared by Siggi Olafsson
54