cab–sharing: an effective, door–to–door, on–demand transportation service

CAB–SHARING: AN EFFECTIVE, DOOR–TO–DOOR, ON–DEMAND TRANSPORTATION SERVICE

Győző GidófalviGeomatic ApSCenter for Geoinformatik

Torben Bach PedersenAalborg University

219/6/2007 ITS 2007, Aalborg, Denmark

Motivation

Transportation is a problem in major citiesPros and cons of existing transportation options:

By carcomfortable, door-to-door, fast, parking problem, stressful, not

too cheap By public transportation

cheap, no parking, less stress, longer travel times, less comfort, pre-adjusted schedules

By taxi / cab:comfortable, door-to-door, fast, no parking, expensive for daily

use


Idea

Share a cab to reduce the cost but keep the prosNot a new idea: airport shuttles

Existing collective, personalized approaches: are offered only for a limited # of origins and/or destinations require manual grouping / scheduling of fares requests need to be made well in advance

Proposed cab-sharing approach: is offered for any origin-destination combination automatically groups / schedules fares serves requests on demand with fast response times


Cab-Sharing Service

Premium Cab-Sharing Service

SMS: cab request GeocodingService

[From/To addr.]

[From/To coord.]

Cab-sharingCab-sharingEngineEngine

[From/To coord., uid]

Cab-share info:[uids, cost shares]

User Accounts DB

[uid, cost]

Funds ok?Cab-scheduling / Cab-scheduling /

Cab-routingCab-routingEngineEngine

Cab-share

infocabid, schedule

SMS: fare info[cost, schedule]

Txt. Msg.To: 1234From addr.To addr.

……

…


The Cab-sharing Problem

Given a set of requests R, a maximum cab-share size K, and a minimum saving required to form a cab-share min_saving,

find a pair-wise disjoint partitioning of R (cab-shares), such that

the number of requests in each cab-share is at most K, and the total transportation cost of serving the requests is

minimized, or equivalently the savings is maximized.

The number of possible cab-shares scales factorially with the number of requests.Instead of enumerating all possibilities and selecting the optimum, the proposed method uses a number of heuristics to derive an near-optimal solution.


A Lazy, Greedy Cab-sharing Algorithm1. Wait with mandatory scheduling of requests until

expiration time (more prob for a good cab-share)2. Calculate pair-wise fractional extra cost (FEC )

between every pair of expiring and valid request3. The best k-sized cab-share for an expiring request

ri is composed of the first k requests with lowest lowest FECFEC for ri.

4. Estimate the amortized cost (AC) of a cab-share as the normalized cumulative sum of FECs

5. Greedily schedule the “best”, maximum K-sized cab-share with minimum AC over all expiring requests

6. Remove scheduled requests from further consideration

7. Repeat 2-7 while min_saving requirement met8. Remaining requests as single “cab-shares”

FEC(r1,r2) = (d2’ + d2’’) / d1FEC(r1,r3) = (d3’ + d3’’) / d1

AC({r1,r2,r3})= (1+FEC(r1,r2)+FEC(r1,r3)) / 3

r1_origr2_orig

r1_dest

r2_dest

d1

d2’’

d2’

r3_dest

d3’

d3’’

r3_orig


Running Example

10014

10014 1883

1883

4581

4581

2219

2219

2000

2000

5 sample requestsin the form of originand destination pairsout of which two areexpiring (10014, 2219)


A Simple SQL Implementation in 4 Steps


Preliminaries

Store requests in an RDBMS as:R_q = <rid,tr,te,xo,yo,xd,yd>Define a 2D distance function between two locations: d(x1,y1,x2,y2)Define the FEC between origin and destination coordinates of two cab requests as:

CREATE FUNCTION e(rixo FLOAT, riyo FLOAT, rixd FLOAT, riyd FLOAT, rjxo FLOAT, rjyo FLOAT, rjxd FLOAT, rjyd FLOAT)RETURNS FLOATBEGIN DECLARE ri_dist, ed FLOAT SET ri_dist = d(rixo, riyo, rixd, riyd) SET ed = d(rjxo, rjyo, rixo, riyo) + d(rjxd, rjyd, rixd, riyd) RETURN (ed / ri_dist)END

(rixo,riyo)(rjxo,rjyo)

(rixd,riyd)(rjxd,rjyd)


STEP 1: Calculating Fractional Extra Costs

Select the expiring cab requests from R_q based on te and the current time into a table R_xCreate table E = <ri,rj,e> to store the pair-wise FEC Calculate the pair-wise FEC between cab requests in R_x and R_q as:

INSERT INTO E (ri, rj, e)SELECT x.rid ri, q.rid rj, e(x.xo,x.yo,x.xd,x.yd,q.xo,q.yo,q.xd,q.yd)FROM R_x x, R_q qWHERE x.rid <> q.rid AND e(x.xo,x.yo,x.xd,x.yd,q.xo,q.yo,q.xd,q.yd) <= 1

Pruning heuristic: avoid storing useless cab-share candidates.


STEP 2: Calculating Amortized Costs

Create table AE = <ri,rj,ae,k> to store the AC of the best k-sized cab-share for each request ri in R_xCalculate the amortized costs using a self-join as:

ri rj e2219 2000 0.43632219 10014 0.76602219 1883 0.91942219 4581 0.9677

10014 1883 0.458210014 4581 0.530210014 2000 0.748210014 2219 0.7627

INSERT INTO AE (ri, rj, ae, k)SELECT a.ri, a.rj,

(SUM(b.e)+1)/(COUNT(*)+1) ae, COUNT(*)+1 k

FROM E a, E bWHERE a.ri = b.ri AND a.e >= b.eGROUP BY a.ri, a.rjHAVING COUNT(*)+1 <= max_k

self-

join

aggregationmax_k = 4

a_ri a_rj a_e b_ri b_rj b_e10014 1883 0.4582 10014 1883 0.458210014 4581 0.5302 10014 1883 0.458210014 4581 0.5302 10014 4581 0.530210014 2000 0.7482 10014 1883 0.458210014 2000 0.7482 10014 4581 0.530210014 2000 0.7482 10014 2000 0.748210014 2219 0.7627 10014 1883 0.458210014 2219 0.7627 10014 4581 0.530210014 2219 0.7627 10014 2000 0.748210014 2219 0.7627 10014 2219 0.7627

2219 2000 0.4363 2219 2000 0.43632219 10014 0.7660 2219 2000 0.43632219 10014 0.7660 2219 10014 0.76602219 1883 0.9194 2219 2000 0.43632219 1883 0.9194 2219 10014 0.76602219 1883 0.9194 2219 1883 0.91942219 4581 0.9677 2219 2000 0.43632219 4581 0.9677 2219 10014 0.76602219 4581 0.9677 2219 1883 0.91942219 4581 0.9677 2219 4581 0.9677

ri rj ae k2219 1883 0.7804 42219 2000 0.7181 22219 10014 0.7341 3

10014 1883 0.7291 210014 2000 0.6841 410014 4581 0.6628 3


STEP 3: Selection the Best Cab-Share

Create table CS = <sid,rid> to store the cab-sharesSelect the best cab-share from AE and, conditionally on the savings, store it in CS as:

SELECT ri, (1-ae), k INTO b_rid, b_savings, b_kFROM AE ORDER BY ae LIMIT 1INSERT INTO CS (sid, rid)SELECT s sid, b_rid rid FROM AEUNIONSELECT s sid, rj rid FROM AE WHERE k <= b_k AND ri = b_rid

The best cab-share is {10014,1883,4581} with savings = 1-.6628 (=33.7%)

ri rj ae k2219 1883 0.7804 42219 2000 0.7181 22219 10014 0.7341 3

10014 1883 0.7291 210014 2000 0.6841 410014 4581 0.6628 3


STEP 4: Pruning the Search Space Discard requests of the best cab-share from further consideration by deleting related records from table E as:

DELETE FROM EWHERE ri IN (SELECT rid FROM CS WHERE sid = s) OR rj IN (SELECT rid FROM CS WHERE sid = s)

ri rj e2219 2000 0.43632219 10014 0.76602219 1883 0.91942219 4581 0.9677

10014 1883 0.458210014 4581 0.530210014 2000 0.748210014 2219 0.7627


The Best Cab-share


Periodic, Iterative Scheduling of Cab Requests

Find all cab-shares by iteratively executing STEPs 2-4, until either E is empty or the best cab-share does not meet the min_saving requirementAssign remaining requests in R_x to their own “cab-shares”

Place STEPs in a stored procedure in the RDBMS, and periodically (every minute), automatically execute as a scheduled task (cron in Linux or Task Scheduler in Windows)


Experimental Setup

Simulated movements of approx. 600,000 individuals in Copenhagen, DK during the course of a day based on a realistic Spatio-Temporal ACTivity Simulator (ST-ACTS)Pool of 251,000 potential cab-requests with a minimum length of 3 and average length of 4.95 kilometersExperiments for:

Maximum cab-share sizes K = [2,8] (default 4) Maximum wait times Δt = [1,20] min (default 15 min) Minimum savings min_saving = 0.3 Cab request densities ρ = [500,30000] requests / day

Measuring: Average savings Service availability Resource (cab space) utilization Service delay


Experimental Results

For 30,000 requests, Δt = 15min and K = 4, the cost of 97.5% of the cab fares can be reduced by two thirds (66%) on average.Savings come at the expense of some service delay

Avg. grouping time = 11.7 min Avg. pickup time = 10.7% length increase or 0.8 min Avg. additional travel time = 7.9% length increase or 0.6 min Average total service delay = 12.1 +/- 7.9 min


Conclusions and Future Work

Presented a Cab-Sharing System (CSS)Formalized the cab-sharing problemGave a greedy cab-sharing algorithm along with a simple but effective SQL implementation:

Web demo: http://www.cs.aau.dk/~gyg/CSS/Extensive experiments show that CSS is a cost- and resource-effective, door-to-door, on-demand, transportation service

Future work will consider: A parallel, high performance, stream-based implementation Different optimization criteria and sharing models Optimization of the Cab–Scheduling / Routing Engine based on

spatio-temporal cab request demand prediction


Thank you for your attention!

Question?


Problem Complexity

Given n cab requests, the number of exactly K-sized disjoint partitionings are:

In case of n=100 and K=4, this number has 155 digits.The complexity becomes even worse when

1. there are m>n valid requests to select the cab-shares from2. smaller than K-sized cab-shares are also allowed

Testing all these possible solutions to find the optimum is computationally unfeasible. Instead, the presented method uses a number of heuristics to derive an approximate answer.


Algorithmic Complexity

n - # of expiring cab requestsm - # of valid cab requestsK – maximum cab-share sizeK << n < me – cost of calculating the fractional extra cost between two requests

Cost of pair-wise fractional extra costs: O(n*m)Cost of amortized costs: O(n*m2) (or O(n*m) with iteration)Cost of selecting best cab-share: O(log(n*K)) (or O(n*K) w/o index)

Cost of grouping the n expiring requests in maximum K-sized groups:

O(n*m) + ceil(n/K) * [ O(n*m) + O(log(n*K)) ] = ceil(n/K) * O(n*m)

cab–sharing: an effective, door–to–door, on–demand transportation service

Documents

cab requests

sized cab

maximum cab

servicepremium cab

greedy cab

denmarkthe cab

advanceproposed cab

single cabsharesfecr1