Speed Limits, Accidents and

Assignment

Mike Maher

University College London

ITS Leeds, 12 December 2016

Overview of the talk

• Describing two pieces of recent work

• First: two projects on speed limits

– effect of 20 mph speed limits

– effect on accidents of increase in HGV speed limits

– both being carried out by Atkins

– both being done for DfT

– my role is providing statistical analysis guidance

– no data as yet on 20 mph limits

• Second: a novel assignment problem

– but not as we generally know it!

HGV speed limit increase

• Increases came into force at start of April 2015

• From 40mph to 50 mph on single carriageways ..

• .. and from 50mph to 60mph on duals

• DfT want to know if any impact on accidents

• Quarterly accident data on affected roads

– for 10 years before the change ..

– .. and then up to three years after

• So far only data for two quarters after

– so just an illustration of what is to be done

– and the modelling to be applied

Time series model (1)

• average 400 accs / month

• average 100 FSCs / month

• 41 obs’ns before

• Clear trend and seasonality

• Fit SARIMA model

• Use auto.arima function in R

• d = 1, D = 1

• AR(1) term

log(all accidents)

actual (blue), fitted (red)

Time series model (2)

• Then fit an intervention model

– using all quarterly data (43 observations)

– include a regressor variable: before / after dummy

– = 0 in before period, and = 1 in after period

– coefficient β estimates step change in mean of log(accidents) following speed limit increase

– allowing for trend and seasonality

– so β < 0 implies a reduction in accidents

– accident rate then factored by exp(β)

Results

• All accidents

– 𝛽 = -0.276 (se = 0.088), so a reduction in accidents

– 95% CI on change: (-36%, -10%)

• FSCs

– 𝛽 = -0.212 (se = 0.110), so a reduction in FSCs

– 95% CI on change: (-35%, 0%)

• But clearly very limited amount of after data so far

– further work to be done in early 2017

Tennis assignment problem

• Midweek men’s doubles group in North Berwick

– around 20 men: retired, semi-retired etc

– each lets me know when available next week ..

– .. and how much they’d like / be willing to play

– pattern changes from week to week

– I put together the groups of four (or eight)

– maximise number of matches, satisfying the constraints

• Used to do it manually: pen and paper

– but wrote an algorithm to automate the process

– integer linear programming problem

– makes it easier for me and fair to everyone

– article in June issue of Mathematics Today

Names Mon Tues Wed Thurs Fri Times

Barry T 0 0 1 1 0 2

Tom B 1 1 0 1 0 3

Gordon B 0 0 0 0 1 1

Peter W 1 1 0 0 0 2

Colin C 1 0 0 1 0 2

Mike M 0 1 1 1 1 3

Keith I 0 1 1 0 0 1

Alan C 1 0 0 1 0 2

John S 0 1 0 0 0 1

Keith B 1 0 1 0 0 2

George StC 1 1 1 1 0 1

Michael L 0 0 1 0 0 1

Phil M 0 1 0 0 0 1

Brian F 1 1 0 0 0 2

Peter K 0 1 0 1 0 2

Willie McM 0 0 0 1 0 1

Ken L 0 1 0 0 0 1

Availability matrix: A

The basic model

Solve in R using the MILP solver lp (part of the lpSolve package)

Generally, many equally-optimal solutions

𝑥𝑖𝑗 = 1 if man 𝑖 plays on day 𝑗 and if 𝐴𝑖𝑗 = 1

𝑔𝑗 = number of 4 − man groups on day 𝑗

𝑗𝑥𝑖𝑗 ≤ 𝑇𝑖 ∀ 𝑖

𝑖𝑥𝑖𝑗 − 4𝑔𝑗 = 0 ∀ 𝑗

Maximise 𝑧 = 𝑖𝑗

𝑥𝑖𝑗 (or number of matches in the week)

Equity issues

}

𝐺1 and 𝐺2 act as secondary criteria, after no. matches

𝑚𝑖 = 𝑗 𝑥𝑖𝑗 = number of matches played by man 𝑖

Solution 1: 𝑚𝐴 = 1 and 𝑚𝐵 = 1

Solution 2: 𝑚𝐴 = 2 and 𝑚𝐵 = 0

𝑦𝑖(𝑘)

= 1 if man 𝑖 gets at least 𝑘 matches in the week

𝐺𝑘 = 𝑖 𝑦𝑖(𝑘)

= number getting at least 𝑘 matches

solution 1 fairer

If – then constraints

If 𝑚𝑖 ≥ 𝑘 then 𝑦𝑖𝑘

= 1 non − linear but imposed by

𝑘𝑦𝑖(𝑘)

≤ 𝑚𝑖 ≤ 𝑘 − 𝜀 1 − 𝑦𝑖(𝑘)

+ 5𝑦𝑖(𝑘)

Maximise 𝑧 = 𝑖𝑗 𝑥𝑖𝑗 + 𝛼1𝐺1 + 𝛼2𝐺2

𝛼1 = 0.01 and 𝛼2 = 0.0001

𝐺1 = 16 𝐺2 = 8

Solution: assigned groups

Day Players

Monday Peter W, Colin C, Keith B, Brian F

Tuesday Tom B, Peter W, Mike M, John S, Phil M, Brian F, Peter K, Ken L

Wednesday Barry T, Keith I, Keith B, Michael L

Thursday Barry T, Tom B, Colin C, Mike M, Alan C, George StC, Peter K, Willie McM

Copy and paste into email message to group members

Two devices in the optimisation

• Integer linear programing problem

– mix of binary and integer variables

• “If-then” conditions: if 𝑚𝑖 ≥ 𝑘 then 𝑦𝑖𝑘

= 1

– intrinsically non-linear

– but implemented via linear constraints

• Hierarchy of criteria: 𝑖𝑗 𝑥𝑖𝑗 , 𝐺1, 𝐺2

– but combined into one objective function

– 𝑧 = 𝑖𝑗 𝑥𝑖𝑗 + 𝛼1𝐺1 + 𝛼2𝐺2

Summary

• Efficient and fair

– algorithm makes life much easier for me

– no favouritism: random permutation of names

• Article in Mathematics Today

– or at http://discovery.ucl.ac.uki/1522020

• Algorithm now produced as a Shiny App

– https://mikemaher.shinyapps.io/TennisApp

• So no need for R or knowledge of the algorithm

– just need to upload the availability matrix

Thank you!

Any questions?