aamas14 taxi

An Efficient Algorithm for Taxi System Optimization

(Extended Abstract)

Jiarui GanUniversity of Chinese Academy of SciencesThe Key Laboratory of Intelligent InformationProcessing, ICT, CAS, Beijing 100190, China

ganjr@ics.ict.cn

Bo An, Chunyan MiaoNanyang Technological University

Singapore 639798{boan,ascymiao}@ntu.edu.sg

ABSTRACTTaxi service is an important mode of modern public trans-portation. However, operated by a large number of self-controlled and prot-driven taxi drivers, taxi systems arequite in ecient and dicult to analyze and regulate. Whilethere has been some work on designing algorithms for im-proving taxi system eciency, the state of the art algorith-m, unfortunately, cannot scale up eciently. To addressthe inadequacy, we propose a novel algorithmFLORAinthis paper. Using convex polytope representation conversiontechniques, FLORA provides a fully compact representationof taxi drivers strategy space, and avoids enumerating anytype of schedules. Experimental results show orders of mag-nitude improvement of FLORA in terms of the complexity.

Categories and Subject DescriptorsI.2.11 [Artificial Intelligence]: Intelligent agents, Multia-gent systems

General TermsAlgorithms, Performance

KeywordsTaxi System, Game Theory, Optimization

1. INTRODUCTIONTaxi service has long been an indispensable part of public

transportation in modern cities due to its high exibility,great comfortableness, and easy accessibility. However, op-erated by a large number (e.g., around 66,000 in Beijing [2])of self-controlled and prot-driven taxis, taxi systems arequiet inecient and dicult to analyze and regulate. Be-sides, taxi systems can be aected by many factors rangingfrom road condition, customer demand, to fare price setting,which not only depend on each other in a very complex way,but also vary with time. How to analyze, regulate and op-timize the taxi system are thus important but challengingproblems, which have attracted many research interests overthe past decades [4, 1, 3]. Among these problems, the taxisystem optimization problem aims at improving eciency of

Appears in: Alessio Lomuscio, Paul Scerri, Ana Bazzan,and Michael Huhns (eds.), Proceedings of the 13th Inter-national Conference on Autonomous Agents and MultiagentSystems (AAMAS 2014), May 5-9, 2014, Paris, France.Copyright c 2014, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.

taxi systems though adjusting the taxi fare. Related workon this topic found in the transportation science community(e.g., [7, 8, 6]) mostly focused on modeling the interdepen-dencies among factors in the system but ignored taxi driversbehaviors which contribute the most to the characteristicsof taxi systems. Specically, prot-driven taxi drivers com-pete with each other for their maximum prots. These com-peting behaviors determines how taxi drivers react to farestructure changes and cannot be ignored in the analysis.In order to include the taxi drivers behaviors, Gan et al.recently proposed a game theoretical approach to model atime-varying taxi market [5]. The key to such an approach isto address the scalability issue encountered in computing thetaxi drivers best strategy. A compact representation of thetaxi drivers strategy space, called Atom Schedule Method(ASM), was proposed to address this issue. Unfortunately,ASM is still inecient in dealing with large-scale problems.In our work, a more ecient algorithmFLORAis pro-

posed to address the scalability issue. FLORA provides anovel compact representation of the taxi drivers pure strat-egy space by utilizing convex polytope representation con-version techniques. Experiment were conducted to evaluateFLORA. The results show that FLORA can produce ordersof magnitude improvement over existing algorithm in bothtime and space complexities.

2. TAXI SYSTEM OPTIMIZATIONDerived from a multi-period model based on existing trans-

portation research (e.g., [9, 10, 5]), the taxi system optimiza-tion problem is dened by the following bilevel program.

maxf

E (f ,p(x)) (1)

s.t. x argmaxx

U (f ,p(x)) (2)

The term bilevel refers to the two levels of optimization pro-grams. In the rst level program (Eq. (1)), we maximize theeciency E of the taxi system through adjusting the fareprice f . E is a function of f and the percentage p of work-ing taxis, and p is furthermore a function of taxi driversstrategy (we assume that all the taxis are identical, so thatthe taxi divers choose the same strategy). Note that in or-der to study a time-varying taxi system, the model dividesthe optimization horizon (e.g., a whole day) equally into aset of n periods, and treat the system in each period as auniform system. Thus, f and p are vectors f = (f1, . . . , fn)and p = (p1, . . . , pn) with f i and pi corresponding to the ith

period. The second level program (Eq. (2)) indicates thattaxi drivers choose the best strategy with respect to their u-

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tility U (which also depends on f and p). A strategy x is amixed strategy, i.e., a distribution over a set of pure strate-gies. Each pure strategy is a working schedule denoted bya vector s {0, 1}n, where si = 1 (resp. si = 0) representsworking (resp. not working) in the ith period. We requirethat a schedule satisfy the following constraints.

C1: total working time should be at most nw periods.

C2: continuous working time should be at most nc periods.

Let the set of n-period schedules satisfying C1 and C2 beS . It follows that p can be calculated as p(x)=sS xss,where x = xs, and xs is the probability of schedule s.To resolve the bilevel program, we discretize the fare price

space into a set F of candidate prices. For each price f F ,we rst solve the second level program to calculate x, andthen take it to the function of E to calculate the eciencyunder price f . In such a way, the price with maximumeciency is the optimal price for the problem. Therefore,the key is to solve the second level program, where thereis a scalability issue as the number of variables (i.e., |S|) isexponentially large in terms of the number of periods.To address this issue, we propose an algorithm called

FLORA (FuLly cOmpact RepresentAtion), which is in con-trast with the partially compact representation of ASM.FLORA reformulates the bilevel program as

maxf

E (f ,p) (3)

s.t. p argmaxpP

U (f ,p) (4)

The idea is to compute the p = p(x) directly without com-puting the best strategy x rst. To guarantee that the p

obtained from the second level program can be implementedby schedules satisfying C1 and C2, we utilize polytope rep-resentation conversion techniques (indeed, the feasible set ofp is a convex polytope of the set S of n-dimensional points),and dene the feasible set P by the following inequality set.

In

In1 . . . 1

1 . . . 1. . .

. . .1 . . . 1

p

1

0

nw

nc...nc

nc + 1

,

where In denotes an nn identity matrix. In such a way, theFLORA formulation is equivalent to the original formulationin Eqs. (1) and (2). The proof of the equivalence is omitteddue to page limit. The new formulation has only n variablesand less than 3n constraints, and can be easily solved.

3. EVALUATION AND CHALLENGESExperiments were conducted to compare the scalability of

FLORA with ASM. Figure 1 depicts the runtime and mem-ory use for solving the second level program. The resultsshow that FLORA produces orders of magnitude improve-ment in both time and space complexities in comparison toASM. FLORA is able to solve optimization problem with100 periods very eciently and still has the potential tohandle even larger problems.

0

50

100

150

200

10 20 30 40 50 60 70 80 90 100

Run

time

(sec

ond)

Number of periods

Atom Schedule MethodFLORA

0

50

100

150

10 20 30 40 50 60 70 80 90 100

Mem

ory

use

(MB

)

Number of periods

Atom Schedule MethodFLORA

Figure 1: Runtime and memory use scalability.

A limitation of both ASM and FLORA is that they canonly solve problems with constraintsC1 andC2, and cannotdeal with situations where other types of practical schedul-ing constraints (e.g., minimum continuous working/restingtime constraint or constraints resulted from market regula-tions) exist. Algorithm capable of handling additional con-straints are needed to calculate more accurate optimal faresin these situations.

4. ACKNOWLEDGEMENTSThis research is supported in part by Interactive and Dig-

ital Media Programme Oce (IDMPO), National ResearchFoundation (NRF) hosted at Media Development Authority(MDA) of Singapore under Grant No.: MDA/IDM/2012/8/8-2 VOL 01, Singapore MOE AcRF Tier 1 grant MOE RG33/13,and NSFC grant No. 61202212.

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[4] G. W. Douglas. Price regulation and optimal servicestandards: The taxicab industry. Journal of TransportEconomics and Policy, pages 116127, 1972.

[5] J. Gan, B. An, H. Wang, X. Sun, and Z. Shi. Optimalpricing for improving eciency of taxi systems. InProceedings of the 23th International Joint Conferenceon Artificial Intelligence, pages 28112818, 2013.

[6] Y.-J. Kim and H. Hwang. Incremental discount policyfor taxi fare with price-sensitive demand. InternationalJournal of Production Economics, 112(2):895902,2008.

[7] B. Schaller. Issues in fare policy: Case of the new yorktaxi industry. Transportation Research Record:Journal of the Transportation Research Board,1618(1):139142, 1998.

[8] H. Yang, C. Fung, K. Wong, and S. Wong. Nonlinearpricing of taxi services. Transportation Research PartA: Policy and Practice, 44(5):337348, 2010.

[9] H. Yang, S. C. Wong, and K. Wong. Demandsupplyequilibrium of taxi services in a network undercompetition and regulation. Transportation ResearchPart B: Methodological, 36(9):799819, 2002.

[10] H. Yang, M. Ye, W. H.-C. Tang, and S. C. Wong. Amultiperiod dynamic model of taxi services withendogenous service intensity. Operations research,53(3):501515, 2005.

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