28/07/2011
Rogo puzzle
How can the
Operational Research
Society support teachers
of
Decision Maths?
Louise Orpin – Education Officer, The Operational Research Society
www.LearnAboutOR.co.uk
Rogo puzzle
Rogo - the New Puzzle Game
• Rogo is a completely new, fun, puzzle that uses adding and
problem-solving skills.
• The object is to collect the biggest score possible using a given
number of steps in a loop around a grid. The best possible score for
a puzzle is given with it, so you can easily check that you have
solved the puzzle. Rogo puzzles can also include forbidden
squares, which must be avoided in your loop.
Rogo is a puzzle based on the Travelling Salesperson Problem
• Rogo is a special case of the TSP. To start with, you are limited in
the distance you can travel by the number of steps or squares you
can use. Secondly you can’t go to all the destinations, so you need
to choose which ones to visit. Thus it is a “subset selection” TSP.
And the destinations have different reward values (the numbers in
the squares), so it is a www.LearnAboutOR.co.uk
“Prize-collecting, subset selection TSP.”
28/07/2011
1
The O.R. Society
Example of a Rogo puzzle
• The Operational Research (O.R.) Society is the professional
membership body of the Operational Research community.
• The O.R. Society provides training, conferences, publications and
information to those working in Operational Research (O.R.).
Good score = 11
• The Society encourages its members to continue their professional
development through accreditation which enables members to
certify their achievements in their job.
Best score = 14
A great teaching resource
• Visit the Rogo website, www.rogopuzzle.co.nz, for more
information on Rogo, teaching tips and puzzles for your class!
• The Society also promotes O.R. in education and business, and as
a career.
• The Society organises an annual careers fair in November at the
University of Birmingham where students can meet O.R. employers.
28/07/2011
www.LearnAboutOR.co.uk
2
28/07/2011
www.LearnAboutOR.co.uk
3
1
28/07/2011
Operational Research (O.R.)
Operational Research (O.R.)
O.R. is the discipline of applying appropriate, often advanced,
analytical methods to help make better decisions.
•
•
•
•
•
•
• Operational Research (O.R.) methods were developed during the
Second World War as analysts undertook a number of crucial
projects that aided the war effort.
Provides information for decision makers
Structures problems to aid understanding
Wide-ranging in application and technique
Provides tools for experimenting with potential scenarios
Measures performance
Optimises the use of limited resources
28/07/2011
www.LearnAboutOR.co.uk
• After the war it soon became evident that O.R. techniques could be
applied to similar problems in industry.
• O.R. techniques such as network analysis, linear programming,
scheduling, Game theory and Decision theory are studied in
Decision Maths.
4
Why study Decision Maths
28/07/2011
www.LearnAboutOR.co.uk
5
How can the Society help?
• It provides useful background for studying O.R., business, computer
sciences, electronics, statistics (and even some maths courses).
• Better examples for how to teach the techniques?
• More problems for students to practise on once they have learnt the
technique?
• It is probably the most widely used branch of maths in the “real
world”.
• Resources showing the techniques used in real life?
• It is an area of Maths that many students will meet when they go into
work.
28/07/2011
www.LearnAboutOR.co.uk
6
• More help to show the point of studying Decision Maths?
28/07/2011
www.LearnAboutOR.co.uk
7
2
28/07/2011
O.R. in the ‘real world’
What is O.R. video
• O.R. is maths applied to help solve many everyday problems faced
by businesses and organisations. The following companies use
O.R. techniques that are studied in Decision Maths to help make
better decisions ...
• A video production showcasing a wide range of real life O.R.
applications.
• O.R. has a very positive influence on the success of an organisation.
– Unipart Rail – Critical Path Analysis
– Virgin Media – Integer Programming
– Steer Davies Gleave & Emirates Stadium – Network Flows
• Many more examples of O.R. in everyday life at
www.LearnAboutOR.co.uk
• Web link for video is
www.LearnAboutOR.co.uk/learn/flash/flash_video.htm
• OR Insight – Isle of Wight Library Service & TSP
28/07/2011
www.LearnAboutOR.co.uk
8
28/07/2011
www.LearnAboutOR.co.uk
9
3
A GUIDE TO OPERATIONAL RESEARCH
MAKING A
REAL DIFFERENCE
IT’S ALL AROUND US AND CAN BE FOUND IN JUST
ABOUT EVERYTHING WE DO
WHAT IS IT?
OPERATIONAL RESEARCH: THE SCIENCE OF BETTER
“
OPERATIONAL RESEARCH
IS VITAL TO OUR BUSINESS.
IT’S CRITICAL IN HELPING
US UNDERSTAND HOW WE
CAN IMPROVE AND HOW WE
CAN BE MORE SUCCESSFUL
IN WHAT ARE VERY
COMPETITIVE MARKETS.
CHRIS DALE
SITE MANAGER
CROWN PAINTS
“
CONTENTS
O.R. INTRODUCTION
O.R. IN TRANSPORT
O.R. IN MANUFACTURING
O.R. IN SPORT
O.R. IN GOVERNMENT
O.R. IN SUPPLY CHAINS
O.R. EDUCATION FOR A CAREER
2
OPERATIONAL RESEARCH (O.R.)
THE SECRET OF BETTER DECISION MAKING
IN A COMPLEX WORLD
In a nutshell, O.R. is the discipline of applying
appropriate, often advanced, analytical methods
to help make better decisions.
By using techniques such as problem structuring methods (sometimes known
as ‘soft O.R.’) and mathematical modelling to analyse complex situations,
O.R. gives executives the power to make more effective decisions and build
more productive organisations.
O.R. is all around us and can be found in just about everything we do.
It forms the building blocks for our every day world – from buying a flight to
queueing for our weekly groceries. It helps save lives, improves sporting events,
drives business and government. Without O.R. the modern world we take for
granted wouldn’t exist.
This is not maths in theory but maths in the real world making a real difference.
THIS GUIDE WILL SHOW YOU HOW O.R. HELPS
This booklet describes some examples of O.R. being used to help solve
real-world problems.
GETTING THE NUMBERS RIGHT
CAN MAKE PASSENGERS MORE
SATISFIED WITH THE SERVICE AND
MILLIONS OF POUNDS WORTH OF
DIFFERENCE TO THE BUSINESS
[Full story page 4]
3
O.R. IN TRANSPORT
NEVER BEFORE HAVE SO MANY PEOPLE OR PRODUCTS NEEDED TO BE
MOVED AROUND THE WORLD. AS PASSENGERS AND CONSUMERS WE
EXPECT EVERYTHING TO RUN SMOOTHLY WHETHER WE ARE FLYING
ON HOLIDAY OR TRAVELLING TO SCHOOL, UNIVERSITY OR WORK.
At British Airways, getting the numbers right can make passengers more
satisfied with the service as well as making millions of pounds worth of
difference to the business.
As with all airlines, British Airways’ staff have to be ‘rostered’ to take into
account things like number of hours they work per shift, minimum rest periods
between flights, breaks at home but they also need to make sure crews spend as
much time as possible working in the air.
Research gets involved in a whole
“ Operational
range of decisions here at BA.
Starting with the whole booking process where
O.R. is involved in helping to set the ticket prices
and calculate the seat availability through to the
flights logistics on the day. O.R. has a massive part
to play in a whole range of day to day decisions.
You get the opportunity to be involved in a whole
different range of aspects of an airline… a whole
different range of problems.
One day you could be looking at something that is
very short term, very tactical and the next day or
the next week you could be looking at something
strategic for twenty years time.
Neil Cottrell
Head of Fleet Planning
British Airways
4
“
TICKET PRICING
There’s one challenge that every airline faces: Seats on any flight are perishable –
once the plane has taken off, there is no possibility of selling any empty seats.
This being so, it pays an airline to fill a seat, even at a very low fare, rather than
have it take off empty. But obviously it isn’t viable to sell every seat at a low price,
so a ‘model’ has to be found for selling seats at different prices.
The real skill comes in working out how many tickets to sell at each fare. Ideally,
on any flight, the airline would first like to see how many people are willing to pay
the highest price, sell as many tickets as possible to them, then sell as many as
possible at the next highest price, and so on, filling up any remaining seats at the
cheapest price. Unfortunately they can’t do it that way because the kind of people
who are willing to pay the higher fares often want to book at the last minute.
So the airline usually sells the cheaper tickets first and holds back some places
at the higher prices. The problem is knowing how many seats to hold back.
This is where the airline uses O.R. By observing the day to day variations in the
number of high priced tickets sold, the number of seats that need to be reserved
to give high fare passengers the best chance of being able to get on the flight can
be estimated. In addition the profile of bookings – how bookings come in over
time - is monitored on a continuous basis, compared with the typical profile for
the flight, and the number of seats held back is adjusted according to whether
bookings are heavier or lighter than the typical profile.
To do all this accurately and in such a way as to produce the best achievable
results is difficult, and calls for some sophisticated ‘Yield Management’ analysis,
which the O.R. model provides.
5
O.R. IN MANUFACTURING
Operational Research is often at the heart of what
makes a business run efficiently. In a manufacturing
plant, everything must be planned and timed precisely
to avoid bottlenecks.
At Crown Paints there are literally hundreds of different colours made and
well over a hundred varieties of paint. So knowing what colours can be made
where and when and how many of each variety will be needed is crucial to
the success of the business.
people built a model of the plant that allows
“ O.R.
us to understand better the complexities of the
manufacturing process by breaking down a
complex problem, to make it very visual.
O.R. is vital to our business. It’s critical in helping
us understand how we can improve and how we
can be more successful in what are very
competitive markets.
“
Chris DALE
Site Manager
Crown Paints
6
SIMULATION MODEL
The simulation model in some ways looks like a computer game. It allows
managers to visualise exactly where any problems might occur. It’s like a map
of the plant which shows all the tanks, the mixers and all the different routings
and filling areas.
The model starts running and as soon as it hits the start time the process begins
in the premix and the manufacturing areas. The batch that was made earlier can
be seen running through into the plant and going into the filling line. It’s possible
to see the batches run through a step by step process until they reach the filling
line and the filled cans are shown moving through into the warehouse.
In a paint manufacturing plant there are a huge number of variables – hundreds
of colours, varieties and can sizes – and the simulation model allows managers
to structure the problem, visualise it and understand what is really happening.
This piece of software allows the company to schedule production more
efficiently and helps them to run the plant and use the people more effectively.
7
O.R. IN SPORT
The world of sport isn’t an obvious place where
Operational Research might be used. Yet, many of our much
loved activities rely on O.R. – in fact, without O.R. some of
the games we know and love wouldn’t be recognisable.
For example, in limited-overs cricket, when weather interrupts play,
an O.R. model – the Duckworth Lewis method – is used to calculate the
fairest run target for the team batting second. O.R. is also behind football’s
Actim Index stats that help calculate the best player in each position.
O.R. had a big part to play in the design of the magnificent Emirates Stadium
in North London – home of Arsenal F.C. Operational Research was used to
help make the running of the site on match days safe and efficient.
imagine a match day situation…
“ Just
when you arrive at the station O.R. starts to
feature immediately. You get off the train,
people marshal you toward the stadium. That
part of the process has already got O.R. taken
into account, trying to find the safest way for you
to move. Then when you reach the stadium, the
number of turnstiles, for example, will have been
determined by an O.R. technique – so O.R.’s really
important to everything about this stadium.
The design of this stadium is obviously
magnificent. It’s a sixty thousand capacity
stadium and without a simulation model of people
moving in and out you’d never know whether
during a match situation if there was a fire, for
example, if those people could be evacuated
safely within the eight minute guide line.
“
Danielle Czauderna
Principal Consultant
STEER DAVIES GLEAVE
8
FORMULA 1
In the exciting world of Formula 1 racing, split second decisions are often
the difference between success and failure. Often these decisions rely on
mathematical models, developed by skilled O.R. experts.
think the first thing we’ll see with the direct
“ Iapplication
of O.R. with the race strategy is the
fuel load. Estimating competitor’s fuel load and
when they’e going to stop and when we are going to
hit traffic during a race and when it’s good to go in
for a pit stop and tyre degradation… these are all
estimations we need to do as the race develops.
I strongly believe that O.R. has a greater role to
play in F1 for the future…
“
Israel Vieira
F1 Team Race Strategy
Mathematical Modeller
9
O.R. IN GOVERNMENT
Central government is the biggest business in Britain and,
to help government, O.R. gets involved in just about
everything that affects our day to day lives.
The Prime Minister has his own team of advisors that works in 10 Downing Street
but they rely on information that comes to him from every one of the government
departments. One of the roles in providing an O.R. service at the centre of
government is to ‘quality assure’ the analysis and evidence that gets sent to the
PM so he knows that what he’s looking at is accurate, timely and effective.
Whatever the issue – whether working out the most efficient way of treating
us when we’re ill, policing our streets and improving transport networks or
even what is taught in the classroom, an operational researcher helps monitor
government performance and finds the most effective ways of putting policy
into practice.
passionate about maths and you want
“ Iftoyou’re
make a difference, the Government O.R.
Service gives you the perfect opportunity to
solve problems… to solve puzzles… and actually
improve the quality of life right the way across
the public agenda. If your analysis can make
a differenceand make the right decision and
improve the effectiveness then the country’s
going to be a better place overall.
Tony O’Connor CBE
Chair
Government O.R. Service
10
“
Helping to run the National Health Service is a major task for the analysts in
the Government O.R. Service. In a hospital, for example, using O.R. techniques
can literally be the difference between life and death.
One of the really big things that hospitals have to think about is how long patients
stay – so, therefore, how many beds and how many wards and doctors you have
to have look after them.
really important for us to get as far as we can
“ It’s
the right capacity for the patients who are going
to come through the doors so that we can give the
treatment that they need when they need it.
Patients, when they are referred by their GP,
go either into A and E or Out Patients and enter a
series of queues. One of the things that we really
have to do – and something we’ve been working
very hard on – is getting those queues as short as
we can. And making sure we don’t have any bottle
necks in our system. Therefore, we need queueing
theory and other O.R. techniques to help us make
sure we get that right.
“
Dr Trudi Kemp
Interim Director of Strategic Development
St George’s Hospital
11
O.R. IN SUPPLY CHAINS
We all expect the things we want and need to be readily
available. That means shelves to be always stocked and
deliveries to arrive on time.
This is what is known as the ‘supply chain’ of a business and O.R. techniques lie at
the very heart of getting this process to function efficiently. At WaverleyTBS, one of
the UK’s leading distribution companies, O.R. is key to the forecasting process.
take a mathematical forecast of sales going
“ We
forward and then we use that forecast to order
stock from suppliers. That stock then comes into
these warehouses where we manage, store and
deliver stock from here to end customers.
Once the forecast has been created we then have
teams of people who order stock against that
forecast with the suppliers and they manage
that stock into the warehouses. We then use O.R.
mathematical models to work out manning levels,
to control our costs in the operation and, more
importantly, manage the space as our warehouses
only have a finite capacity.
12
Bob Wigglesworth
Director
Waverly TBS
“
O.R AND THE SUPPLY CHAIN
Sales in large grocery retailers in the UK follow a weekly cycle – more is sold
on Fridays and Saturdays than on other days of the week. So, on Thursdays and
Fridays their distribution centres supplying stores have to work very hard.
In fact, they often have to employ temporary staff and sub-contract trucks to
cope with the peaks.
Truck-loads of products received from suppliers have to be broken down into
store orders and delivered to the stores. O.R. people analyse sales patterns
and develop replenishment policies. One major retailer detected irregularity in
suppliers’ production schedules which led to either too much or too little product
being available for delivery to the warehouse. This situation, caused by the
‘bullwhip effect’, resulted in poor customer satisfaction due to the unavailability
of products.
O.R. was used in the construction of a simulation model of the replenishment
system. The model helped the retailer to alter the replenishment rules to
incorporate a feedback system to their suppliers. The model also showed that
the distribution workload could be spread more evenly over the whole week for
certain products by smoothing the sales cycle.
The simulation led to the implementation of a small change to the replenishment
algorithms, resulting in considerable smoothing of daily variability. The company
trialled the new replenishment model for three months in a single store, it proved
so successful that the implementation rolled out across the entire UK business.
As a result, the savings achieved by not having to employ temporary staff in the
distribution centres and subcontract transport ran into millions of pounds per
year. Crucially, service levels improved as distribution centres could keep up
with demand, while stock was directed where needed rather than too much
being held in stores.
13
O.R. EDUCATION FOR A CAREER
If you are interested in a career in operational research
take a look at the interesting University Degree courses
that are the best route into the profession.
A list of universities offering degrees in O.R. can be found at
www.LearnAboutOR.co.uk
are many different academic paths into
“ There
operational research although most people tend
to have undergraduate degrees with mathematics,
statistics, O.R., computer science or management
content. Specialist Master’s degrees in O.R. and
related subjects are also available.
Students will also learn a number of O.R. techniques
such as computer simulation, optimisation, queueing
theory, inventory control, and problem structuring
methods to name but a few. You’ll also get a good
balance between the theoretical work and an
appreciation of the application of these techniques,
and this will be through case studies and projects
which might involve working with a company on a real
world problem.
“
14
Paul Harper
Professor of Operational Research
Cardiff University
“
IF YOU’RE PASSIONATE ABOUT
MATHS AND YOU WANT TO MAKE
A DIFFERENCE, THE GOVERNMENT
O.R. SERVICE GIVES YOU THE
PERFECT OPPORTUNITY TO SOLVE
PROBLEMS... to solve puzzles...
AND ACTUALLY IMPROVE THE
QUALITY OF LIFE RIGHT THE
WAY ACROSS THE PUBLIC
AGENDA. WITH THE HELP OF
YOUR O.R. ANALYSIS, THE
COUNTRY’S GOING TO BE A
BETTER PLACE OVERALL.
TONY O’CONNOR, CBE
CHAIR
GOVERNMENT O.R. SERVICE
“
15
THE SECRET
OF BETTER
DECISION MAKING
IN A COMPLEX WORLD
IN A NUTSHELL, OPERATIONAL RESEARCH (O.R.) IS THE
DISCIPLINE OF APPLYING APPROPRIATE, OFTEN ADVANCED,
ANALYTICAL METHODS TO HELP MAKE BETTER DECISIONS.
By using techniques such as problem structuring methods (sometimes
known as ‘soft O.R.’) and mathematical modelling to analyse complex
situations, O.R. gives executives the power to make more effective
decisions and build more productive organisations.
O.R. is all around us and can be found in just about everything we do.
It forms the building blocks for our every day world – from buying a
flight to queueing for our weekly groceries. It helps save lives, improves
sporting events, drives business and government. Without O.R. the
modern world we take for granted wouldn’t exist.
This is not maths in theory but maths in the real world – making a
real difference.
IF YOU’D LIKE TO LEARN MORE ABOUT O.R.,
VISIT WWW.LEARNABOUTOR.CO.UK
You’ll find: E
xamples of O.R. helping to solve real-world problems
O.R. problems for you to solve
Details of Universities offering degrees in O.R.
Careers in O.R.
A video of O.R. in action
SEYMOUR HOUSE, 12 EDWARD STREET, BIRMINGHAM B1 2RX UK.
TEL: +44 (0)121 233 9300 www.theorsociety.com
Original Article
Look, here comes the library van!
Optimising the timetable of the
mobile library service on the Isle
of Wight
Tanutr Rienthonga, Andrew Walkerb and Tolga Bektas¸a,*
a
School of Management and Centre for Operational Research,
Management Science and Information Systems (CORMSIS),
University of Southampton, Highfield, Southampton SO17 1BJ, UK.
b
Isle of Wight Council, Library HQ, 5 Mariners Way, Cowes, Isle of
Wight PO31 8DP, UK.
*Corresponding author.
Abstract
This article describes an approach taken to optimise the
timetable of the mobile library service operating on the Isle of Wight. The
mobile library visits over 90 communities on the island, offering books,
DVDs, videos and CDs, and operates on a periodic timetable. The optimisation problem is formulated as a multiple travelling salesmen model
with additional time-balancing constraints on route durations. The article
also shows ways in which data required for the model, in particular travel
times, were gathered, and discusses practical issues arising in pre-processing the data to fit the purposes of the case study. The model is used to
produce an improved timetable over the current one that implies driving
time reductions of up to 25 per cent and yields routes that are better
balanced in terms of time spent on the visits made each day. The model is
also used to test various scenarios differing with respect to the number of
locations visited and days over which the service operates.
OR Insight (2011) 24, 49–62. doi:10.1057/ori.2010.17;
published online 26 January 2011
Keywords: multiple travelling salesman problem; routing; integer
programming; optimisation
Received November 2010; accepted November 2010 after one revision
& 2011 Operational Research Society Ltd 0953-5543 OR Insight
www.palgrave-journals.com/ori/
Vol. 24, 1, 49–62
Rienthong et al
Introduction
The Isle of Wight Council’s Library Service serves a population of around
140 000, and among unitary authorities is in the top quartile for visits and
issues. The service has a stock of just under a quarter of a million books
and issues close to 1 million books, CDs, DVDs and videos every year. There
are currently 28 000 active borrowers out of a population of 140 000, but the
service is also used for its internet-based services, access to other council
services and for a wide variety of enquiries.
The Isle of Wight Council’s Library Services has 11 branches providing
service for Bembridge, Brighstone, Cowes, East Cowes, Freshwater, Newport,
Niton, Ryde, Sandown, Shanklin and Ventnor. The main library is the Lord Louis
Library located in Newport. Figure 1 shows a map of the Isle of Wight and the
main centres of population on the island. The Council also provides additional
library services, namely the Mobile Library Service that visits locations on
the island where there are no static libraries, and the Home Library Service
that delivers to the homes of people who are unable to visit their local library
because of being housebound. In addition, they also have a delivery van in
order to provide facilities for library branches, pre-school, mid-school and
services for each of the island’s three prisons.
The mobile library service offers services to locations where there are no
libraries. The mobile service operates on a 3-week timetable. Over weeks
Figure 1: A map of the Isle of Wight.
Source: Contains Ordnance Survey data r Crown copyright and database rights 2010.
50
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
one and two, communities in towns, villages and rural areas are visited. In
week three, it provides service to shelter housing residents. The mobile library
currently serves 98 locations over a 3-week period consisting of 38 locations
in week one, 42 locations in week two and 18 locations in week three. Over
the 3 weeks, the mobile library spends around 93 hours and makes about
282.5 miles in travelling around the island. This time is a combination of
driving and service times.
The Isle of Wight Library Service is currently undertaking a review of their
mobile and home library service provision in terms of route revision, vehicle
suitability, service provision and staffing. The aim of the review is to maintain
the delivery of the high-quality service to customers, but at a significantly
reduced cost.
The main objective of this article is to develop alternative timetables
for the Mobile Library to improve and enhance the service on the Isle of
Wight in terms of both the total mileage and the time spent on the routes.
Through the use of mathematical modelling and optimisation, this article
will first show how the current schedule can be improved. The model will
also be used to generate alternative timetables with differing characteristics.
The article will describe how travel time data, required as an input to
the proposed model, were collected, and will discuss some practical
issues arising in pre-processing the data to fit to the needs of the case
study. More specifically, we will show that travel times as estimated through
an online map database needed to be modified before being used in the
model. The next section describes the modelling and optimisation approach
for the problem and discusses the requirements for data collection and
processing.
Modelling and Optimisation
The first part of this section provides a detailed description of the current
practice and discusses data requirements. The second part presents a formal
definition of the problem and describes the model used to produce the new
timetables.
Description of the current practice and data collection
The mobile library service visits 98 locations on the island over a time horizon
of 16 days, and this timetable is repeated throughout the year. Tours start
daily from the Library Headquarters (HQ) located in Cowes and return to the
same location at the end of each day. To give the reader an idea of the existing
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
51
Rienthong et al
Table 1: Summary statistics for the timetable in use
Day
Distance
(miles)
Scheduled
travel time
(hours)
Service
time
Total
time
Week 1
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
26.0
12.5
17.1
20.6
16.6
12.6
2:45
1:55
2:30
3:00
2:00
0:45
2:25
4:45
2:30
4:25
4:20
4:00
5:10
6:40
5:00
7:25
6:20
4:45
Week 2
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
25.4
25.6
19.9
9.00
20.1
7.3
2:50
2:40
2:20
2:10
2:40
0:20
2:40
3:15
2:55
3:40
4:30
4:00
5:30
5:55
5:15
5:50
7:10
4:20
Week 3
Monday
Tuesday
Wednesday
Thursday
Friday
28.5
8.9
13.8
18.3
Off-road (Admin day)
2:55
3:35
2:15
4:00
2:10
3:30
2:15
3:00
6:30
6:15
5:40
5:15
Total
282.5
35:30
—
93:00
timetable in place, we first present some detailed statistics in Table 1 as to the
total distance traversed by the mobile library, as well as the associated travel
and service times for each day.
The service time for each location is dependent on its population. These
data are extracted from the current schedule and a summary is presented in
Figure 2.
As Figure 2 shows, there are 16 locations with service times of up to 10 min
and 39 locations with service time of around 20 min. Of the remaining
locations, there are 38 with service times ranging between 30 and 80 min.
Finally, only four of the locations visited have service times higher than
100 min, with the maximum being 4 hours (240 min).
There are two fundamental sets of data required for this study: (i) driving
times between every pair of locations, and (ii) location durations (service
times) at each location visited. The latter was readily available from the
existing timetable as is as shown in Figure 2. The former set was collected
using an online map database (Google Maps, 2010). This meant that
(992–99)/2 ¼ 4851 individual distances were manually entered and the
corresponding estimated driving times were extracted using the postcodes
52
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
40
Number of locations
35
30
25
20
15
10
5
0
Service time (minutes)
Figure 2: Histogram showing the spread of the service times for the 98 locations.
Table 2: Discrepancies between the collected data and the existing schedule
Week 1 Friday
PO31
PO30
PO30
PO30
PO30
PO30
PO30
PO30
PO30
8PD
3JT
4AA
3LH
4EH
4LD
4JD
5ST
5RQ
Location
Library HQ
Shorwell
Limerstone
Yafford
Hulverstone
Brook
Calbourne
Kinchington Rd
Marlborough Rd
Driving time from the previous location
Data from the
online map
Data from the
actual schedule
—
0:21
0:03
0:02
0:08
0:01
0:08
0:10
0:01
—
0:30
0:05
0:05
0:20
0:05
0:20
1:20a
0:05
a
includes 1-hour lunch break.
available to us, yielding a 99 99 travel time matrix. However, when
these driving times given by the online database were compared with those
available in the schedule, some discrepancies were found. Table 2 shows these
discrepancies in greater detail for a section of a particular route.
A closer look at the above-mentioned discrepancy revealed that the mobile
library bus was driving slower than an average car, owing to its size, and hence
required higher travel times. The figures presented in Table 1 and the rest of
the timetable indicated that the scheduled travel times were about twice as
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
53
Rienthong et al
Table 3: An overall summary of the current timetable
Total driving time
(hours) based on
online map data
Total driving time
based on the
actual schedule
Total
service
time
Week 1 (6 days)
Week 2 (6 days)
Week 3 (4 days)
13:38
10:43
8:20
12:55
13:00
9:35
22:25
21:00
14:05
Total (16 days)
32:31
35:30
57:30
much as the estimations given through the online database. To reflect this
difference, we have multiplied all elements of the 99 99 driving time matrix
by two and used this revised data set in the computations. We note that the
modification performed on the data set will not affect the resulting tours as
all elements are magnified by the same amount. However, the modification
yields a better estimation of the actual times spent travelling, and therefore
will have an impact on the resulting schedules and the way in which they would
be implemented.
A tabulated summary of the current timetable in terms of driving and service
times is given in Table 3. In this table, we present statistics for the total driving
time with respect to two sources; one is based on the data collected through
the online database, and the other based on the actual schedule of the mobile
library service. These figures are presented under columns two and three
in Table 3. The total service time remains constant for each location for which
there is only one set of statistics, and these are presented in the last column
of Table 3.
Formal definition of the problem and the optimisation model
The problem of finding an optimal timetable for the mobile library service
corresponds to distributing the set of 98 locations to be visited over 3 weeks.
The first 2 weeks comprise 12 days including Saturdays but not Sundays,
whereas the third week is only 4 days long. The problem also involves finding,
for each day, the order in which the locations will be visited. In other words,
the problem involves finding the optimal routes to traverse in each day. An
additional aspect of the problem involves balancing the total time spent in each
day by the service, including the travel time from one location to another and
service time at each location.
Routing problems have long been studied in the literature. The Travelling
Salesman Problem (TSP) is a well-known member of this class of problems,
which consists of finding a lowest-cost tour among a set of cities such that each
54
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
city is visited exactly once and the tour starts from and ends at a so-called
‘depot’ or a ‘home’ node (see Laporte, 1992, 2010 for overviews). In our study,
the routing problem of the mobile library is modelled using an extension of the
TSP named as the multiple Travelling Salesman Problem (mTSP). The mTSP
consists of finding routes for m salesmen who all depart from and return to a
depot such that each city is visited exactly once and that the total length of
the m tours formed is minimised. There are several variations of the mTSP,
ranging from those with single to multiple depots, and those with additional
restrictions such as bounds on the number of cities visited (Bektas¸, 2006).
In our context, each of the m tours in the mTSP corresponds to one of the 16
days to be planned for and each tour itself will yield the order of the locations in
a tour to be visited on the corresponding day.
The model used in this study is in the form of a 0–1 mixed integer linear
programming formulation. We denote the number of locations by n and the
number of days to plan for by m. The set N ¼ {1, 2, y, n} is the index set of
all locations (or nodes) to be visited in which ‘home’ (that is, Library HQ) is
represented by node 1. The remaining indices correspond to the 98 locations
to be visited. Cij represents the driving time from location i to location j (iaj) in
minutes. The service time spent in a given location iAN is represented
by Si. The total time spent on a tour is composed of the driving time and
service times spent at each of the locations. As for the balancing aspect of
the problem, an upper bound of T minutes is imposed on the total time spent
by the mobile library service on each day. Similarly, it is required that
the service spends at least a lower bound of L minutes every day. Under this
definition, the more the values of L and T approach one another, the more
balanced the resulting tours will be.
The proposed model makes use of a binary variable Xij that takes the value 1
if the service travels from location iAN to location jAN, and 0 otherwise.
An additional (continuous) variable Vi is defined to represent the arrival time of
the service at node iAN. The model is presented below.
Minimise
n X
n
X
Cij Xij
i¼1 j¼1
Subject to
n
X
X1j ¼ m;
ð1Þ
Xj1 ¼ m;
ð2Þ
j¼2
n
X
j¼2
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
55
Rienthong et al
n
X
Xij ¼ 1
j ¼ 2; . . . ; n;
ð3Þ
Xij ¼ 1
i ¼ 2; . . . ; n;
ð4Þ
i¼1
n
X
j¼1
Vi Vj þ ðT þ Cij þ Si ÞXij þ ðT Cij Sj ÞXji pT
Vi þ ðCi1 þ Si ÞXi1 pT
i ¼ 2; . . . ; n;
Vi ðL Ci1 Si ÞXi1 X0
Vi C1i X1i X0
i ¼ 2; . . . ; n;
i ¼ 2; . . . ; n;
Vi C1i X1i þ TX1i pT
Xij 2 f0; 1g
i; j ¼ 2; . . . ; n; i 6¼ j;
ð5Þ
ð6Þ
ð7Þ
ð8Þ
i ¼ 2; . . . ; n;
ð9Þ
i; j ¼ 2; . . . ; n; i 6¼ j:
ð10Þ
In the model presented above, constraints (1) and (2) ensure that
the timetable of the mobile library service covers m days, with each tour
starting and ending at the Library HQ (node 1). Constraints (3) and (4) ensure
that each location appears exactly once in any tour, that is, it is visited
only once over the planning horizon of m days. Constraint (5) is used to
prevent sub-tours, which are tours that are formed within the locations
not connected to Library HQ. These constraints also help to define the
variables Vi in such a way that if the service visits location j immediately
after visiting location i, then the arrival time in location j will be equal to
the arrival time in location i added to the service time in location i and the
travel time between these two nodes. Constraints (6), (7), (8) and (9) are
used to guarantee that the total time spent on each tour is between L and T
minutes.
The above model is an extension of the standard mTSP model found in the
literature (see, for example, Kara and Bektas¸, 2006). As far as we are aware,
constraints (6), (7), (8) and (9) used to balance the routes in terms of travel
time are the novel features of this model.
56
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
Results
It is well known that routing problems are difficult to solve optimally, and
that heuristics offer reasonably good solutions within relatively short computation times (see, for example, Salhi and Currie, 2009). However, it is not the
intention of this article to describe a new solution method for the model but
rather to make use of the proposed model in obtaining solutions that improve
upon the current practice. To this end, we used GUROBI version 3.0.1 (Gurobi
Optimization, 2010), a state-of-the-art optimiser for solving the proposed
model. All experiments were conducted on a 2.4 GHz MacBook.
The model (1)–(10) was coded using the parameters of the current
timetable and run on a pre-defined set of scenarios generated in consultation
with the Isle of Wight Library HQ in line with their review. Further details on
the scenarios generated and the associated results are given below.
Scenario 1
This scenario is based on the current timetable with n ¼ 99 locations (including
the Library HQ) and m ¼ 16 service days covering 3 weeks. After consultation
with the mobile library service driver and Library Services, and in line with the
current timetable, it was decided to set L and T equal to 4 hours (240 min) and
6.30 hours (390 min), respectively, for this scenario. The optimiser for the
corresponding model was run for over 10 hours and a summary of the resulting
solution is shown in Table 4.
As can be seen from Table 4, the new timetable produced through the proposed model results in a total driving time of 26 hours and 39 min. This is an
improvement of 8 hours and 51 min over the current timetable, translating into
a time saving of 25 per cent. If the comparisons are made on the basis of
data obtained from the online map, then the corresponding time reduction is
5 hours and 52 min.
Scenario 2
Scenario 2 will look at the case where mobile library locations within a 2-mile
radius of the main libraries Newport, Ryde, Cowes, Freshwater, Sandown and
Ventnor are not included in the model. The assumption here is that these
locations would be served by the respective main libraries, rather than by the
mobile library. Without such locations, the number of locations reduces to
n ¼ 72. Given the reduced number of locations, we run the model for two cases
with m ¼ 16 and m ¼ 12 days, respectively, where the latter corresponds to
2 weeks. The reason for testing the m ¼ 12 case is due to the decreased
number of locations to be visited. For this scenario, it was decided to increase
L to 5 hours (300 min), but keep T same at 6.30 hours (390 min).
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
57
Rienthong et al
Table 4: Summary statistics for the new timetable produced under Scenario 1
Day
Scheduled travel
time (hours)
Service time
Total time
Week 1
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
2:08
2:08
1:23
1:12
1:30
1:46
3:15
4:00
2:45
4:20
2:50
3:00
5:23
6:08
4:08
5:32
4:20
4:46
Week 2
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
2:15
0:28
2:08
1:40
1:36
2:18
3:45
3:45
3:20
4:50
3:20
4:00
6:00
4:13
5:28
6:30
4:56
6:18
Week 3
Monday
Tuesday
Wednesday
Thursday
Friday
1:48
1:00
2:13
1:06
Total
Off-road (Admin day)
4:10
3:05
4:05
3:00
26:39
—
5:58
4:05
6:18
4:06
84:09
Table 5: The summary table for Scenario 2 with m=16
Current time table based on
actual schedule
Current time table based
on online data
Scenario 2 (16 days)
a
Total driving
time
Time
savings
Total
service timea
Time
savings
35:30
5:04
57:30
18:45
32:31
8:03
57:30
18:45
40:34
—
36:45
—
Service times do not include lunch breaks.
For the case with m ¼ 16 days, the optimisation engine was run for around
23 hours on the model. We present a summary of the resulting solution
in Table 5.
Table 5 shows a comparison between the current timetable and the new one
with 72 locations and 16 days. It is interesting to note that the total driving
time in this scenario is longer than that of the current timetable, with the
reason being the locations that were removed lead to an increased distance
between locations. However, the increase in the driving time is compensated
58
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
Table 6: The summary table for Scenario 2 with m=12
Current time table based on
actual schedule
Current time table based on
online data
Scenario 2 (12 days)
Total driving time
(hours)
Time
savings
Total service
timea
Time
savings
35:30
14:08
57:30
18:45
32:31
11:09
57:30
18:45
21:22
—
36:45
—
a
Service times do not include lunch breaks.
by the reduction in total service time as shown in the last column of Table 5.
When the changes in total driving and service times are combined, it can be
seen that the new scenario results in a significant reduction of the overall time
spent by the mobile library service for the 72 locations.
The second case of Scenario 2 is where the number of service days is
reduced from 16 days to 12 days while all other parameters remain fixed. The
optimisation in this case was run for 24 hours and a summary of the results
are presented in Table 6.
Table 6 shows that the total driving time is significantly lower than that of
the current timetable when the services operates on a periodic timetable
of 12 days, rather than 16 to serve the 72 locations. The savings in driving time
combined with the reduction in the total service time shows that a 2-week-,
rather than a 3-week-, long timetable results in much less time to be spent in
driving and servicing the locations.
Scenario 3
This scenario aims to create a new timetable that will remove all locations
within a 2-mile radius of every main library on the Isle of Wight, namely
Newport, Ryde, Cowes, Freshwater, Sandown, Ventnor, Bembridge, East
Cowes, Shanklin, Brighstone and Niton. This implies a reduction in the total
number of locations from 99 to 59, with 40 locations removed. The number of
service days in this case is set equal to 12 days while all other parameters
remain constant. The optimisation process for the model corresponding to
Scenario 3 was run for around 2 hours. A summary of the results are presented
in Table 7.
As can be seen from Table 7, the total driving time for Scenario 3 is around
27 hours, which implies a reduction of 8 hours and 23 min over the current
timetable based on the actual schedule and a reduction of 5 hours 24 min over
the current timetable based on online data. Further reductions in the total
service time can be seen in the last column of Table 7, which is due to the
number of reduced locations.
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
59
Rienthong et al
Table 7: The summary table for scenario 2 with m=12
Total driving time
(hours)
Time
savings
Total service
timea
Time
savings
35:30
8:23
57:30
26:50
32:31
5:24
57:30
26:50
27:07
—
30:40
—
Current time table based on
actual schedule
Current time table based on
online data
Scenario 3
a
Service times do not include lunch breaks.
Summary and Comparisons
An important aspect of the model proposed in this study is its feature of
balancing the workload (as measured by the daily time spent by the service)
over a number of days, in addition to minimising the total travel time of
the mobile library service. Table 8 presents a general comparison between
the current timetable and the three scenarios tested in terms of the total
driving time. This table also presents, in the last row, the standard deviation
(SD) for each scenario, as well as that of the current timetable, as an indicator
of how ‘balanced’ the new solutions are.
The figures given in Table 8 clearly show that Scenario 1, which is an
optimised version of the current timetable, produces a more balanced set of
routes as indicated by the reduced standard deviation. The table also shows
that, should Scenario 2 be adopted, then a 12-day time period results in a
better balanced set of routes with an even smaller SD than that of Scenario 1.
Conclusions
This article described a practical routing problem that arises in producing a
timetable for the mobile library service on the Isle of Wight. A mathematical
model in the form of an integer programming formulation is proposed for the
problem that not only minimises the total driving time spent by the service, but
also balances the time spent on the routes each day. The model can be generated
easily and fed into an off-the-shelf optimiser to produce practical solutions. The
model could easily be used by practitioners, and is flexible enough to be adapted
to their own needs. Using the proposed model, we were able to produce a new
timetable for the mobile library service, which not only reduced the driving time
requirements of up to 25 per cent, but also helped to better balance the route
durations by reducing the SD of the set of routes by around 17 per cent. We also
tested several scenarios looking at a reduced number of locations and days, and
investigated the impacts of these changes on the total driving and service time.
60
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
Look, here comes the library van
Table 8: Summary table comparing the current timetable with three scenarios with
regard to the driving time (in minutes)
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
Day
SD
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Current
timetable
Scenario 1
Scenario 2
(m=16)
Scenario 2
(m=12)
Scenario 3
198
100
114
154
144
100
136
116
123
60
140
68
184
78
100
138
128
128
83
72
90
106
135
28
128
100
96
138
108
60
133
66
150
156
102
196
188
198
208
136
201
168
135
68
152
140
160
76
102
113
126
106
106
98
154
146
129
118
68
76
—
—
—
—
176
104
142
158
150
176
172
100
145
160
68
76
—
—
—
—
38.43
31.98
42.62
25.30
38.63
One (expected) challenge encountered in this research was the difficulty of
solving the model to optimality. As mentioned in the previous section, the time to
run the models for the four scenarios varied from 2 hours to 24 hours. The
respective optimality gaps for Scenarios 1, 2 (with m ¼ 16), 2 (with m ¼ 12) and
3 were 28.4 per cent, 40.1 per cent, 12.4 per cent and 29.5 per cent, respectively,
which shows the difficulty of obtaining optimal solutions within reasonable
amount of computational time. The quoted statistics on the optimality gaps shows
the need for theoretical developments in solving these types of problems to
optimality in efficient and effective ways. This is especially the case for modelling
and solving more complex library delivery operations (Apte and Mason, 2006).
One observation we made in our experimentation is that the solution of the
proposed model becomes much more difficult with increasing lower bound L on
the travel time. However, the produced solutions were good enough for practical
purposes of this research and showed improvement over the current practice.
Acknowledgement
This research was based on a summer project organised jointly by the four MSc
programmes run by the Centre of Operational Research, Management Science
and Information Systems (CORMSIS) at the University of Southampton and
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62
61
Rienthong et al
carried out in collaboration with the Isle of Wight Council. At the time of
completing this research, the first author was an MSc student at the University
of Southampton. The authors thank Astrid Davies and Dr Ian Rowley for
helping to set up the project, and the Editor-in-Chief for his constructive
comments on the article.
About the Authors
Tanutr Rienthong holds a Bachelor’s degree in Management Sciences, 2008,
from Kasetsart University, Thailand; and holds a Master’s degree in Management Science and Finance at the University of Southampton, 2009 to 2010.
Andrew Walker has worked as a public library manager since graduating from
Oxford University in 1981. Following several posts in London boroughs,
Andrew moved to the Isle of Wight in 1990. He is now employed as the
Development Librarian in charge of library operations, encompassing staffing,
building maintenance, mobile vehicles and performance monitoring.
Tolga Bektas¸ is a lecturer in Management Science at the University of
Southampton and the Director of the MSc in Business Analytics and Management
Sciences at the School of Management. He has a BSc (1998), MSc (2000) and
PhD (2005) in Industrial Engineering, and postdoctoral research experience at
the University of Montreal. His research interests are in discrete optimisation with
applications to vehicle routeing, service network design, and freight transportation and logistics. His publications appeared in journals such as Transportation
Science, Networks, European Journal of Operational Research, Computers &
Operations Research and Journal of the Operational Research Society.
References
Apte, U.M. and Mason, F.M. (2006) Analysis and improvement of delivery operations at
the San Francisco public library. Journal of Operations Management 24: 325–346.
Bektas¸, T. (2006) The multiple traveling salesman problem: An overview of formulations
and solution procedures. Omega 34: 209–219.
Google Maps. (2010) http://maps.google.co.uk/, accessed July 2010.
Gurobi Optimization. (2010) http://www.gurobi.com/.
Kara, I. and Bektas¸, T. (2006) Integer linear programming formulations of multiple
salesman problems and its variations. European Journal of Operational Research 174:
1449–1458.
Laporte, G. (1992) The traveling salesman problem: An overview of exact and
approximate algorithms. European Journal of Operational Research 59: 231–247.
Laporte, G. (2010) A concise guide to the travelling salesman problem. Journal of the
Operational Research Society 61: 35–40.
Salhi, S. and Currie, R.H. (2009) Heuristics are here to help your online vehicle
scheduling. OR Insight 22: 88–104.
62
& 2011 Operational Research Society Ltd 0953-5543
OR Insight
Vol. 24, 1, 49–62