Co-Design to Enable User-Friendly Tools to Assess the Impact of Future Mobility Solutions
Gioele Zardini, Nicolas Lanzetti, Andrea Censi, Emilio Frazzoli, Marco Pavone
11 Co-Design to Enable User-Friendly Tools to Assessthe Impact of Future Mobility Solutions
Gioele Zardini , , Nicolas Lanzetti , , Andrea Censi , Emilio Frazzoli , and Marco Pavone Abstract —The design of future mobility solutions (autonomousvehicles, micromobility solutions, etc.) and the design of the mo-bility systems they enable are closely coupled. Indeed, knowledgeabout the intended service of novel mobility solutions wouldimpact their design and deployment process, whilst insightsabout their technological development could significantly affecttransportation management policies. This requires tools to studysuch a coupling and co-design future mobility systems in terms ofdifferent objectives. This paper presents a framework to addresssuch co-design problems. In particular, we leverage the recentlydeveloped mathematical theory of co-design to frame and solvethe problem of designing and deploying an intermodal mobilitysystem, whereby autonomous vehicles service travel demandsjointly with micromobility solutions such as shared bikes and e-scooters, and public transit, in terms of fleets sizing, vehicle char-acteristics, and public transit service frequency. Our frameworkis modular and compositional, allowing one to describe the designproblem as the interconnection of its individual components andto tackle it from a system-level perspective. Moreover, it onlyrequires very general monotonicity assumptions and it naturallyhandles multiple objectives, delivering the rational solutions onthe Pareto front and thus enabling policy makers to selecta policy. To showcase our methodology, we present a real-world case study for Washington D.C., USA. Our work suggeststhat it is possible to create user-friendly optimization tools tosystematically assess the costs and benefits of interventions, andthat such analytical techniques might inform policy-making inthe future.
I. I
NTRODUCTION
Current transportation systems will undergo dramatic mu-tations, arising from the deployment of novel mobility solu-tions, such as autonomous vehicles (AVs) and micromobility( µ M) systems. New mobility paradigms promise to drasticallyreduce negative externalities produced by the transportationsystem, such as emissions, travel time, parking spaces and,critically, fatalities. However, the current design process formobility solutions largely suffers from the lack of clear,specific requirements in terms of the service they will beproviding [3]. Yet, knowledge about their intended service(e.g., last-mile versus point-to-point travel) might dramaticallyimpact how vehicles are designed and significantly ease theirdevelopment process. For instance, if for a given city we knewthat for an effective on-demand mobility system AVs only needto drive up to 30 mph and only on relatively easy roads, theirdesign would be greatly simplified and their deployment could Institute for Dynamic Systems and Control, ETH Z¨urich, Z¨urich (ZH),Switzerland { gzardini,acensi,emilio.frazzoli } @ethz.ch Automatic Control Laboratory, ETH Z¨urich, Z¨urich (ZH), Switzerland, [email protected] Department of Aeronautics and Astronautics, Stanford University, Stanford(CA), United States, { pavone } @stanford.edu A preliminary version of this paper was presented at the 99th AnnualMeeting of the Transportation Research Board [1] and at the 23rd IEEEInternational Conference on Intelligent Transportation Systems [2]. be accelerated. Furthermore, from the system-level perspectiveof transportation management, knowledge about the trajectoryof technology development for new mobility solutions wouldcertainly impact decisions on future infrastructure investmentsand provisions of service. In other words, the design offuture mobility solutions and the design of a mobility systemleveraging them are intimately coupled . This calls for methodsto reason about such a coupling, and in particular to co-design the invidual mobility solutions and the associated mobilitysystems. A key requirement in this context is to be able toaccount for a range of heterogeneous objectives that are oftennot directly comparable (consider, for instance, travel time,public expense, and emissions).Accordingly, the goal of this paper is to lay the foundationsfor a framework through which one can co-design futuremobility systems. Specifically, we show how to leverage therecently developed mathematical theory of co-design [4]–[6],which provides a general methodology to co-design complexsystems in a modular and compositional fashion. This tooldelivers the set of rational design solutions lying on the Paretofront, allowing to reason about the costs and benefits of theindividual design options. The framework is instantiated inthe setting of co-designing intermodal mobility systems [7],whereby fleets of self-driving vehicles provide on-demandmobility jointly with fleets of micromobility vehicles ( µ MVs)such as e-scooters (ESs), shared bikes (SBs), mopeds (Ms)and fuel-cell mopeds (FCMs), and public transit. Aspects thatare subject to co-design include fleet sizes, vehicle-specificcharacteristics for AVs and µ MVs, and public transit servicefrequency.
A. Related Literature
Our work lies at the interface of the design of the publictransportation service and the design of novel mobility solu-tions.The first research stream is reviewed in [8], [9] andcomprises strategic long-term infrastructure modifications and operational short-term scheduling. The joint design of trafficnetwork topology and control infrastructure has been presentedin [10]. Public transportation scheduling has been solvedjointly with the design of the transit network by optimizingpassengers’ and operators’ costs in [11], the satisfied demandin [12], and the energy consumptions of the system in [13].However, these works are non-modular, only focus on thepublic transportation system, and do not consider its jointdesign with new mobility solutions.The research on novel mobility solutions mainly pertainsto AVs, Autonomous Mobility-on-Demand (AMoD) systems,and µ M. The research on AMoD systems is reviewed in [14]and mainly concerns their fleet sizing. In this regard, existing a r X i v : . [ ee ss . S Y ] A ug studies range from simulation-based approaches [15]–[19] toanalytical methods [20]. In [21], the fleet size and the charginginfrastructure of an AMoD system are jointly designed, andthe arising design problem is formulated as a mixed integerlinear program. In [22], the fleet sizing problem is solvedtogether with the vehicle allocation problem. Finally, [23]proposes a framework to jointly design the AMoD fleet sizeand its composition. More recently, the joint design of multi-modal transit networks and AMoD systems was formulatedin [24] as a bilevel optimization problem and solved withheuristics. Overall, the problem-specific structure of existingdesign methods for AMoD systems is not amenable to amodular and compositional problem formulation. Furthermore,key AV characteristics, such as the achievable speed, arenot considered. Research on the design and impact of µ Msolutions has been reviewed in [25], which focuses on theurban deployment of SBs and ESs. In particular, [26] presentsa design framework for a multi-modal public transportationsystem, including various µ M solutions and buses, optimizinguser preferences and social costs. Fleet deployment modelsare analyzed in [27], [28]. The optimal allocation of SBs ina city is studied through mathematical programming modelsin [27], and solved through stochastic optimization in [28].Finally, [29] explores the impact of µ M on urban planningand identifies strategies to increase µ MVs utilization.In conclusion, to the best of the authors’ knowledge, theexisting design frameworks for mobility systems have a fixedproblem-specific structure, and therefore do not permit torigorously design the mobility infrastructure in a modularand compositional manner. Moreover, previous works neithercapture important aspects of future mobility systems, suchas the interactions among different transportation modes, norspecific design parameters of novel mobility solutions, as forinstance the level of autonomy of AVs.
B. Statement of contributions
In this paper we lay the foundations for the systematicstudy of the design of future mobility systems. Specifically,we leverage the mathematical theory of co-design [4] todevise a framework to study the design of intermodal mobilitysystems in terms of mobility solutions characteristics andfleet characteristics, enabling the computation of the rational solutions lying on the Pareto front of minimal travel time,transportation costs, and emissions. Our framework paves theway to structure the design problem in a modular way, inwhich each different transportation option can be “pluggedin” in a larger model. Each model has minimal assumptions:Rather than properties such as linearity and convexity, weask for very general monotonicity assumptions. For example,we assume that the cost of automation of an AV increasesmonotonically with the speed achievable by the vehicle. Weare able to obtain the full Pareto front of rational solutionsor, given policies, to weigh incomparable costs (such as traveltime and emissions) and to present actionable information tothe stakeholders of the mobility ecosystem. We consider thereal-world case study for Washington D.C., USA, to showcaseour methodology. We illustrate how, given the model, wecan easily formulate and answer several questions regardingthe introduction of new technologies and investigate possibleinfrastructure interventions. A preliminary version of this paper was presented in [2]. This revised and extended versionis novel in four ways. First, we broaden the presentation ofthe mathematical theory of co-design and its application inthis work. Second, we extend the discussion of the literature,including recent research pertaining to the co-design of futuremobility systems. Third, we show the modularity of ourframework by including the design of µ M solutions (bothat the vehicle and at the fleet level) in the future mobilityco-design problem, and evaluate their impact on the trans-portation system for the real-world case study of WashingtonD.C., USA. Finally, we extend our case studies with furtherscenarios and provide new managerial insights.
C. Organization of the paper
The remainder of this paper is structured as follows: Sec-tion II and Section III review the mathematical backgroundon which our framework is based. Section IV presents the co-design problem for future mobility systems, by introducing thesingle design problems (DPs) and their interconnection form-ing a co-design problem (CDP). We showcase our methodol-ogy with several real-world case studies for Washington D.C.,USA, in Section V. Section VI concludes the paper with adiscussion and an overview on future research directions.II. M
ATHEMATICAL B ACKGROUND
This paper applies the mathematical theory of co-design,first introduced in [4]. In this section, we present a non-exhaustive review of the main concepts needed for this work.
A. Orders
To define a DP, we will use basic facts from order theory,which we review in the following. The interested reader isreferred to [30].
Definition II.1 (Poset) . A partially ordered set (poset) is atuple (cid:104) P , (cid:22) P (cid:105) , where P is a set and (cid:22) P is a partial order,defined as a reflexive, transitive, and antisymmetric relation. Definition II.2 (Opposite of a poset) . The opposite of aposet (cid:104) P , (cid:22) P (cid:105) is the poset (cid:104) P op , (cid:22) op P (cid:105) , which has the sameelements as P , and the reverse ordering. Definition II.3 (Chains) . Given a poset (cid:104) P , (cid:22) P (cid:105) , a chain isa sequence of points p i ∈ P where two successive points arecomparable, i.e., i ≤ j ⇒ s i (cid:22) s j . Given a poset, we can formalize the idea of “Pareto front”via antichains.
Definition II.4 (Antichains) . A subset S ⊆ P is an antichain iff no elements are comparable: For x , y ∈ S, x (cid:22) y impliesx = y. We denote by A P the set of all antichains in P . We hereafter introduce properties of posets which are in-strumental for the formal definition of DPs.
Definition II.5 (Directed set) . A subset S ⊆ P is directed ifeach pair of elements in S has an upper bound: For all a , b ∈ S,there exists c ∈ S such that a (cid:22) c and b (cid:22) c. Definition II.6 (Completeness) . A poset is a complete partialorder (CPO) if each of its directed subsets has a supremumand a bottom.
Example II.7.
For instance, the poset (cid:104) R + , ≤(cid:105) , with R + : = { x ∈ R | x ≥ } , is not complete, as its directed subset R + ⊆ R + does not have an upper bound (and therefore a supremum).Nonetheless, we can make it complete by artificially adding atop element (cid:62) , i.e., by defining (cid:104) R + , ≤(cid:105) with R + : = R + ∪ {(cid:62)} and a ≤ (cid:62) for all a ∈ R + . Similarly, we can complete N to N . Remark II.8.
Let ( P , (cid:22) P ) and ( Q , (cid:22) Q ) be CPOs. Then, ( P × Q , (cid:22) P × Q ) , with ( p , q ) (cid:22) P × Q ( p , q ) iff p (cid:22) P p and q (cid:22) Q q , is a CPO. When defining DPs, monotone maps will play a key role.
Definition II.9 (Monotonicity) . A map f : P → Q betweentwo posets (cid:104) P , (cid:22) P (cid:105) , (cid:104) Q , (cid:22) Q (cid:105) is monotone iff x (cid:22) P y impliesf ( x ) (cid:22) Q f ( y ) . Note that monotonicity is a compositionalproperty. III. M
ATHEMATICAL T HEORY OF C O -D ESIGN
We can now formally define a DP.
Definition III.1 (DP) . Given the posets F , R , a DP is amonotone function (feasibility relation) of the formd : F op × R → Bool , (1) and we represent it by an arrow d : F R . Remark III.2 (Intended semantics for DPs) . The poset F represents the functionality to be provided, while the poset R represents the resources required. The object d is a relationwhich describes which combinations of functionality and re-sources are feasible for each f ∗ ∈ F op and r ∈ R : d ( f ∗ , r ) isa truth value, which we call feasibility of f given r. Remark III.3 (Intended semantics for monotonicity of DPs) . The intended semantics for the monotonicity of a designproblem d are: • If functionality f is feasible with resource r, then anyfunctionality f (cid:48) (cid:22) F f is feasible with r. • If functionality f is feasible with resource r, then f isfeasible with any resource r (cid:48) (cid:23) R r. Definition III.4 (DPI) . A design problem with implementation(DPI) is a tuple of the form ( I d , prov , reqs ) , where I d is theimplementations set, and prov , reqs are functions from I d to F and R , respectively: F prov ←−− I d reqs −−→ R . (2) Furthermore, one defines d : F R asd : F op × R → P ( I d ) (cid:104) f ∗ , r (cid:105) (cid:55)→ { i ∈ I d : ( prov ( i ) (cid:22) F f ) ∧ ( reqs ( i ) (cid:23) R r ) } . (3) Remark III.5 (Intended semantics for DPIs) . The semantics of F , R is the same as in Definition III.1. The expression d ( f ∗ , r ) maps functionality f and resource r to a set of implementationsi ∈ I d which provide f given r. Definition III.6 (Functionality to resources map) . For anyDPI d, one can define a monotone function h d which sends Fig. 1: The intermodal AMoD network consists of road (AVsand µ MVs), public transportation, and walking digraphs. Thecolored circles represent stops or intersections and the blackarrows denote road links, public transit arcs, or pedestrianpathways. The grey arrows represent the mode-switching arcsconnecting them. a functionality f ∈ F to the minimum antichain of resourceswhich provide f :h d : F → A R f (cid:55)→ min { reqs ( i ) ∈ R | i ∈ I ∧ d ( prov ( i ) , reqs ( i )) = T } . (4)Individual DPIs can be composed in series (i.e., the func-tionality of a DPI is the resource of a second DPI), and inparallel to obtain a co-design problem with implementation(CDPI), and can be solved with the PyMCDP solver [31].Notably, such compositions preserve monotonicity and, thus,all related algorithmic properties. For further details we referto [4].IV. C O -D ESIGN OF F UTURE M OBILITY S YSTEMS
In this section, we detail our co-design framework for futuremobility systems, and instantiate it for the specific case of anintermodal transportation network.
A. Intermodal Mobility Framework1) Multi-Commodity Flow Model:
The transportation sys-tem and its different modes are modeled using the digraph G = ( V , A ) , sketched in Figure 1. It is described through aset of nodes V and a set of arcs A ⊆ V × V . Specifically,it is composed of four layers: The road network layer G R =( V R , A R ) , consisting of an AVs layer G R , V = ( V R , V , A R , V ) anda µ MVs layer G R , M = ( V R , M , A R , M ) , the public transportationlayer G P = ( V P , A P ) , and a walking layer G W = ( V W , A W ) . TheAVs and the µ MVs networks are characterized by intersections i ∈ V R , V , i ∈ V R , M and road segments ( i , j ) ∈ A R , V , ( i , j ) ∈ A R , M , respectively. Similarly, public transportation lines are modeled through station nodes i ∈ V P and line segments ( i , j ) ∈ A P . The walking network describes walkable streets ( i , j ) ∈ A W , connecting intersections i ∈ V W . Mode-switchingarcs are modeled as A C ⊆ V R , V × V W ∪ V W × V R , V ∪ V R , M × V W ∪ V W × V R , M ∪ V P × V W ∪ V W × V P connecting the AVs,the µ MVs, and the public transportation layers to the walkinglayer. Consequently, V = V W ∪ V R , V ∪ V R , M ∪ V P and A = A W ∪ A R , V ∪ A R , M ∪ A P ∪ A C . Consistently with the structuralproperties of transportation networks in urban environments,we assume G to be strongly connected.We represent customer movements by means of travelrequests. A travel request refers to a customer flow startingits trip at a node o ∈ V and ending it at a node d ∈ V . Definition IV.1 (Travel request) . A travel request ρ is a triple ( o , d , α ) ∈ V × V × R + , described by an origin node o ∈ V , adestination node d ∈ V , and the request rate α > , in otherwords, the number of customers who want to travel from o tod per unit time. To ensure that a customer is not forced to use a given trans-portation mode, we assume all requests to appear on the walk-ing digraph, i.e., o m , d m ∈ V W for all m ∈ M : = { , . . . , M } .The flow f m ( i , j ) ≥ ( i , j ) ∈ A and satisfying a travelrequest m . Furthermore, f , V ( i , j ) ≥ f , M ( i , j ) ≥ µ MVs on AVs arcs ( i , j ) ∈ A R , V and µ MVs arcs ( i , j ) ∈ A R , M , respectively. This accounts forrebalancing flows of AVs and µ MVs between a customer’sdrop-off and the next customer’s pick-up. Assuming AVs and µ MVs to carry one customer at a time, the flows satisfy ∑ i : ( i , j ) ∈ A f m ( i , j ) + j = o m · α m = ∑ k : ( j , k ) ∈ A f m ( j , k ) + j = d m · α m ∀ m ∈ M , j ∈ V (5a) ∑ i : ( i , j ) ∈ A R , V f tot , V ( i , j ) = ∑ k : ( j , k ) ∈ A R , V f tot , V ( j , k ) ∀ j ∈ A R , V (5b) ∑ i : ( i , j ) ∈ A R , M f tot , M ( i , j ) = ∑ k : ( j , k ) ∈ A R , M f tot , M ( j , k ) ∀ j ∈ A R , M , (5c) where j = x denotes the boolean indicator function, f tot , V ( i , j ) : = f , V ( i , j ) + ∑ m ∈ M f m ( i , j ) , and f tot , M ( i , j ) : = f , M ( i , j ) + ∑ m ∈ M f m ( i , j ) . Specifically, (5a) guaranteesflows conservation for every transportation demand, (5b)preserves flow conservation for AVs on every road node,and (5c) preserves flow conservation for µ MVs on everyroad node. Combining conservation of customers (5a) withthe conservation of AVs (5b) and µ MVs (5c) guaranteesrebalancing AVs and µ MVs to match the demand.
B. Travel Time and Travel Speed
With the variable t i j we denote the time needed to traversean arc ( i , j ) of length s i j . We assume a constant walking speedon walking arcs and infer travel times on public transportationarcs from the public transportation schedules. Assuming thatthe public transportation system at node j operates with thefrequency ϕ j , switching from a pedestrian vertex i ∈ V W to apublic transit station j ∈ V P takes, on average, t i j = t WS + ϕ j ∀ ( i , j ) ∈ A P , (6) where t WS represents a constant sidewalk-to-station travel time.We assume that the average waiting time for AVs vehiclesis t WV , the average time needed to reach a µ MV is t WM ,and switching from the AVs graph, the µ MVs graph, and thepublic transit graph to the pedestrian graph takes the transfertimes t VW , t MW , and t SW , respectively. While each AVs arc ( i , j ) ∈ A R , V is characterized by a speed limit v L , V , i j , AVssafety protocols impose a maximum achievable velocity v V , a .In order to prevent too slow and therefore dangerous drivingbehaviors [32], we only consider AVs arcs through which theAVs can drive at least at a fraction β of the speed limit: Arc ( i , j ) ∈ A R , V is kept in the road network iff v V , a ≥ β · v L , V , i j , (7)where β ∈ ( , ] . We set the velocity of all arcs fulfillingcondition (7) to v V , i j = min { v V , a , v L , V , i j } and compute thetravel time to traverse them as t i j = s i j v V , i j ∀ ( i , j ) ∈ A R , V . (8)Similarly, µ MVs are allowed to drive at speed v M , i j ≤ min { v M , a , v L , M , i j } , where v M , a is the achievable speed for µ MVs and v L , M , i j is the speed limit for µ MVs on arc ( i , j ) ∈ A R , M . The corresponding travel time reads then t i j = s i j v M , i j ∀ ( i , j ) ∈ A R , M . (9) C. Road Congestion
We assume that each road arc ( i , j ) ∈ A R , V is subject to abaseline usage u i j , capturing the presence of exogenous traffic(e.g., private vehicles), and that it has a nominal capacity c i j .Furthermore, we assume that the central authority operatesthe AMoD fleet such that vehicles travel at free-flow speedthroughout the road network of the city, meaning that thetotal flow on each road link must be below the link’s capacity.Therefore, we capture congestion effects with the thresholdmodel f tot , V ( i , j ) + u i j ≤ c i j ∀ ( i , j ) ∈ A R , V . (10)Finally, we assume µ M to not significantly contribute to roadcongestion [33].
D. Energy Consumption
We compute the energy consumption of AVs via an urbandriving cycle. In particular, the cycle is scaled so that itsaverage speed v avg , cycle matches the free-flow speed on thelink. The energy consumption of road link ( i , j ) is then scaledas e i j = e cycle · s i j s cycle ∀ ( i , j ) ∈ A R , V . (11)For µ MVs we consider a distance-based energy consumption.For the public transportation system we assume a constantenergy consumption per unit time. This approximation isreasonable in urban environments, where the operation of thepublic transportation system is independent from the numberof customers serviced, and its energy consumption is thereforecustomers-invariant.
E. Fleet Sizes
We consider a fleet of n V , max AVs and a fleet of n M , max µ MVs. In a time-invariant setting, the number of vehicles onarc ( i , j ) is expressed as the multiplication of the total vehiclesflow on the arc and its travel time. Therefore, we upper boundthe number of AVs employed as n V , u = ∑ ( i , j ) ∈ A R , V f tot , V ( i , j ) · t i j ≤ n V , max , (12)and the number of µ MVs employed as n M , u = ∑ ( i , j ) ∈ A R , M f tot , M ( i , j ) · t i j ≤ n M , max . (13) F. Discussion
First, the demand is assumed to be time-invariant andflows are allowed to have fractional values. This assumptionis in line with the mesoscopic and system-level planningperspective of our study. Second, we model congestion effectsusing a threshold model. This approach can be interpretedas a municipality preventing mobility solutions to exceed thecritical flow density on road arcs. AV and µ MVs can thereforebe assumed to travel at free flow speed [34]. This assumption isrealistic for an initial low penetration of new mobility systemsin the transportation market, especially when the AV and µ MVfleets are limited in size. Finally, we allow AVs and µ MVs totransport one customer at a time [35].
G. Co-Design Framework
We integrate the intermodal framework presented in Sec-tion IV-A in the co-design formalism, allowing the decouplingof the CDPI of a complex system in the DPIs of its individ-ual components in a modular, compositional, and systematicfashion. In order to achieve this, we decouple the CDPI inthe DPIs of the individual AV (Section IV-G1), the AVs fleet(Section IV-G4), the individual µ MV (Section IV-G2), the µ MVs fleet (Section IV-G4), and the public transportationsystem (Section IV-G3). Their interconnection is presentedin Section IV-G5. We aim at computing the antichain ofresources, quantified in terms of costs, average travel time pertrip, and emissions required to provide the mobility service toa set of customers.
1) The AV Design Problem:
The AV DPI consists of select-ing the maximal speed of the AVs. Under the rationale thatdriving safely at higher speed requires more advanced sensingand algorithmic capabilities, we model the achievable speedof the AVs v V , a as a monotone function of the vehicle fixedcosts C V , f (resulting from the cost of the vehicle C V , v and thecost of its automation C V , a ) and the mileage-dependent opera-tional costs C V , o (accounting for maintenance, cleaning, energyconsumption, depreciation, and opportunity costs [36]). In thissetting, the AV DPI provides the functionality v V , a and requiresthe resources C V , f and C V , o . Consequently, the functionalityspace is F V = R + , the resources space is R V = R + × R + , andthe implementations space I V consists of specific instancesof the AVs.
2) The µ MV Design Problem:
The µ M DPI comprises theselection of the maximal speed of the µ MVs. Following therationale that different µ MVs can reach different speeds andhave different prices, we model the achievable speed of the µ MV v M , a as a monotone function of the µ MV fixed costs C M , f and the mileage-dependent operational costs C M , o . Therefore,the µ MV DPI provides the functionality v M , a and requires theresources C M , f and C M , o . Consequently, the functionality spaceis F E = R + , the resources space is R E = R + × R + , and theimplementations space I E consists of instances of the µ MVs.
3) The Subway Design Problem:
We design the publictransit infrastructure by considering its service frequency,introduced in Section IV-B. Specifically, we assume the servicefrequency ϕ j to scale linearly with the size of the train fleet n S as ϕ j ϕ j , base = n S n S , base . (14)We relate a train fleet of size n S to the fixed costs C S , f (account-ing for train and infrastructural costs) and to the operationalcosts C S , o (accounting for energy consumption, vehicles de-preciation, and train operators’ wages). Given the passengers-independent public transit operation in today’s cities, we as-sume the operational costs C S , o to be mileage independent andto only vary with the size of the fleet. Formally, the number ofacquired trains n S , a = n S − n S , baseline is a functionality, whereas C S , f and C S , o are resources. The functionality space is F S = N ,the resources space is R S = R + × R + , and the implementationsspace I S consists of specific instances of the subway system.
4) The Intermodal Optimization Framework Design Prob-lem:
The intermodal mobility system DPI considers demandsatisfaction as a functionality. Formally, F O = V × V × R + withthe partial order (cid:22) F O defined by D : = { ( o i , d i , α i ) } M i = (cid:22) F O { ( o i , d i , α i ) } M i = = : D iff for all ( o , d , α ) ∈ D there issome ( o , d , α ) ∈ D with o = o , d = d , and α i ≥ α i . Inother words, D (cid:22) F O D if every travel request in D is in D as well. To successfully satisfy a given set of travel requests,we require the following resources: (i) the achievable speed ofthe AVs v V , a , (ii) the number of available AVs per fleet n V , max ,(iii) the achievable speed of the µ MVs v M , a , (iv) the numberof available µ MVs per fleet n M , max , (v) the number of trains n S , a acquired by the public transportation system, and (vi) theaverage travel time of a trip t avg : = α tot · ∑ m ∈ M , ( i , j ) ∈ A t i j · f m ( i , j ) , (15)with α tot : = ∑ m ∈ M α m , (16)(vii) the total distance driven by the AVs per unit time s V , tot : = ∑ ( i , j ) ∈ A R , V s i j · f tot , V ( i , j ) , (17)(viii) the total distance driven by the µ MVs per unit time s M , tot : = ∑ ( i , j ) ∈ A R , M s i j · f tot , M ( i , j ) , (18)(ix) the total AVs CO emissions per unit time m CO , V , tot : = γ · ∑ ( i , j ) ∈ A R , V e i j · f tot , V ( i , j ) , (19) and (x) the total µ MVs CO emissions per unit time m CO , M , tot : = γ · ∑ ( i , j ) ∈ A R , M e i j · f tot , M ( i , j ) , (20)where γ relates the energy consumption and the CO emis-sions. We assume that AVs and µ MVs are routed to maximizethe customers’ welfare, defined without loss of generalitythrough the average travel time t avg . Hence, we link thefunctionality and resources of the mobility system DPI throughthe following optimization problem:min { f m ( · , · ) } m , f , V ( · , · ) , f , M ( · , · ) t avg = α tot ∑ m ∈ M , ( i , j ) ∈ A t i j · f m ( i , j ) s . t . Eq . (5) , Eq . (10) , Eq . (12) , Eq . (13) . (21)Formally, F O = R + , and R O = R + × N × R + × N × N × R + × R + × R + × R + × R + . Furthermore, I O consists of specificintermodal scenarios. Remark IV.2.
The optimization problem (21) might possessmultiple optimal solutions, making the relation between re-sources and functionality ill-posed. To overcome this subtlety,if two solutions share the same average travel time, we selectthe one incurring the lowest mileage.5) The Mobility Co-Design Problem:
The functionality ofthe system is to satisfy the customers’ demand. Formally, thefunctionality provided by the CDPI is the set of travel requests.To provide the mobility service, three resources are required.First, on the customers’ side, we require an average travel time,defined in (15). Second, on the side of the central authority,the resource is the total transportation cost of the intermodalmobility system. Assuming an average AV’s life of l V , anaverage µ MV’s life of l M , an average train’s life of l S , and abaseline subway fleet of n S , baseline trains, we express the totalcosts as C tot = C V + C M + C S , (22)where C V is the AVs-related cost C V = C V , f l V · n V , max + C V , o · s V , tot , (23) C M is the µ MV-related cost C M = C M , f l M · n M , max + C M , o · s M , tot , (24)and C S is the public transit-related cost C S = C S , f l S · n S , a + C S , o . (25)Third, on the environmental side, we consider the total CO emissions m CO , tot = m CO , V , tot + m CO , M , tot + m CO , S · n S , (26)where m CO , S represents the constant CO emissions of asingle train. Formally, the set of travel requests { ρ m } m ∈ M is the CDPI functionality, whereas t avg , C tot , and m CO , tot areits resources. Consistently, the functionality space is F = R + and the resources space is R = R + × R + × R + . Note that theresulting CDPI (Figure 2) is indeed monotone, since it consistsof the composition (series and parallel) of monotone DPIs [4]. I-AMoDVehicle Micromobility Subway (cid:22) (cid:22) (cid:22)×(cid:22) ×(cid:22) (cid:22) (cid:22)(cid:22)×(cid:22) (cid:22) (cid:22)×(cid:22) (cid:22) ×(cid:22) (cid:22) + (cid:22) (cid:22) + (cid:22) (cid:22)(cid:22) + (cid:22) × (cid:22) ++ (cid:22)(cid:22) v V , a v M , a s V , tot s M , tot C V , o C M , o C V , f l V n V , max C M , f l M n M , max n S , a n S C S , o C S , f l S C tot m CO , V , tot m CO , M , tot m CO , S m CO , tot t avg α tot co-designconstraint total cost averagetravel time totalemissionstotalrequest rate Fig. 2: Schematic representation of the CDPI. In solid greenthe provided functionalities and in dashed red the requiredresources. The edges represent co-design constraints: Theresources required by a first design problem are the lowerbound for the functionalities provided by the second one.
6) Discussion:
First, we lump the AV’s autonomy in itsachievable velocity. We leave to future research more elab-orated AV models, accounting, for instance, for accidentsrates [37] and for safety levels, e.g., comparing specific percep-tion pipelines [38]. Second, we assume the service frequencyof the subway system to scale linearly with the number oftrains. We inherently assume that the existing infrastructurecan homogeneously accommodate the acquired train cars. Tojustify the assumption, we include an upper bound on thenumber of potentially acquirable trains in our case studydesign in Section V. Third, we highlight that the intermodalmobility framework is only one of the many feasible waysto map total demand to travel time, costs, and emissions.Specifically, practitioners can easily replace the correspondingDPI with more sophisticated models (e.g., simulation-basedframeworks like MATSim [39]), as long as the monotonicity ofthe DPI is preserved. In our setting, we conjecture customersand vehicles routes to be centrally controlled by the centralauthority in a socially-optimal fashion. Fourth, we assumea homogeneous fleet of AVs and µ MVs. Nevertheless, ourmodel is readily extendable to capture heterogeneous fleets.Finally, we consider a fixed travel demand, and compute theantichain of resources providing it. Nonetheless, our formal-ization can be easily extended to arbitrary demand modelspreserving the monotonicity of the CDPI to account, forinstance, for elastic effects.V. D
ESIGN OF E XPERIMENTS AND R ESULTS
In this section, we showcase the co-design frameworkpresented in Section IV on the real-world case of WashingtonD.C., USA. We detail our experimental design in Section V-Aand present numerical results in Section V-B.
A. Design of Experiments
Our study is based on the real-world case of the urbanarea of Washington D.C., USA. The city road network and its features are imported from OpenStreetMap [40], whilstthe public transit network together with its schedules areextracted from GTFS [41]. Original demand data is obtainedby merging origin-destination pairs of the morning peak ofMonday 1 st May, 2017, provided by taxi companies [42]and the Washington Metropolitan Area Transit Authority(WMATA) [43]. On the public transportation side, we focusour studies on the MetroRail system and its design. To takeaccount of the recently increased presence of ride-hailingcompanies, the taxi demand rate is scaled by a factor of 5 [44].The complete demand dataset includes 16 ,
430 distinct origin-destination pairs, describing travel requests. To account forcongestion effects, the nominal road capacity is computed asin [45] and an average baseline usage of 93 % is assumed, inline with [46]. We assume an AV fleet composed of batteryelectric BEV-250 mile AVs [47]. We summarize the mainparameters characterizing our case studies together with theirbibliographic sources in Table I. In the remainder of this sec-tion, we solve the co-design problem presented in Section IVwith the PyMCDP solver [31]. Beside our basic setting (S1),we evaluate the sensitivity of the design strategies to differentmodels of automation costs of AVs (S2–S4), and assess theimpact of emerging µ M solutions (S5). We summarize theconsidered mobility solutions and their complementarity inTable II.
S1 - Basic setting:
We consider the co-design of the mobilitysystem by means of AMoD and public transportationsystems, and do not include µ M solutions (cf. S5).Specifically, we co-design the system by means of the AVfleet size, achievable free-flow speed, and subway servicefrequency: The municipality is allowed to (i) deployan AV fleet of size n V , max ∈ { , , , , . . . , , } vehicles, (ii) choose the single AV achievable speed v V , a ∈ {
20 mph ,
25 mph , . . . ,
50 mph } , and (iii) increase thesubway service frequency ϕ j by a factor of 0 %, 50 %,or 100 %. In line with recent literature [48]–[52], weassume an average achievable-velocity-independent costof automation. S2 - Speed-dependent automation costs:
To relax the po-tentially unrealistic assumption of a velocity-independentautomation cost, we consider a performance-dependentcost structure, detailed in Table I. The large variance insensing technologies available on the market and theirperformances [65] suggests that AV costs are, in fact,performance-dependent. Indeed, the technology currentlyrequired to safely operate an autonomous vehicle at50 mph is substantially more sophisticated, and thereforemore expensive, than the one needed at 20 mph. Fur-thermore, the frenetic evolution of automation techniqueswill inevitably reduce automation costs: Experts forecasta massive automation cost reduction (up to 90 %) in thenext decade, principally due to mass-production of AVssensing technology [66], [67]. Therefore, we performour studies with current (2020) automation costs aswell as with their projections for the upcoming decade(2025) [47], [67].
S3 - High automation costs:
We assess the impact of highautomation costs. In particular, we assume a performance-independent automation cost of 0 . / car, captur-ing the extremely high research and development costs that AVs companies are facing today [68]. Indeed, nocompany has shown the ability to safely and reliablydeploy large fleets of AVs on the market yet. S4 - MoD setting:
We analyze the current Mobility-on-Demand (MoD) case. The cost structure of MoD systemsis characterized by lower vehicle costs (due to lack ofautomation) and higher operation costs, mainly due todrivers’ salaries.
S5 - Impact of new transportation modes:
We evaluatethe impact of µ M solutions on urban mobility. Weconsider ESs (e.g., Lime in DC), SBs (e.g., CapitalBikeshare in DC), Ms (e.g., Revel in DC), andFCMs. In addition to the design parameters introducedin the basic setting, we design the specific µ Msolution M ∈ { ES,SB,M,FCM } and the µ M fleet size n M , max ∈ { , , , . . . , } vehicles. We studythe joint deployment of µ M solutions and AVs, andtherefore consider the extended settings of 2020 and2025.
B. Results1) Basic setting:
Figure 3a reports the solution of the co-design problem through the antichain consisting of the totalCO emissions, average travel time, and total transportationcost. The design solutions are rational (and not comparable),since there exists no instance which simultaneously yieldslower emissions, average travel time, and cost. In the interest (a) Left: Three-dimensional representation of antichain elementsand their projection in the cost-time space. Right: Two-dimensionalprojections.(b) Results for constant automation costs. On the left, the two-dimensional representation of the antichain elements: In red are theunfeasible strategies, in orange the feasible but irrational solutions,and in green the Pareto front. On the right, the implementationscorresponding to the highlighted antichain elements, quantified interms of achievable vehicle speed, AVs fleet size, and train fleet size . Fig. 3: Solution of the CDPI: Basic setting.
Parameter Variable Value Units Source
Road usage u ij
93 % [46]
S1 S2 (2020) S2 (2025) S3 S4 S5 (2020) S5 (2025)
AVs operational cost C V , o .
084 0 .
084 0 .
062 0 .
084 0 .
50 0 .
084 0 .
062 USD / mile [47], [48]Vehicle cost C V ,
000 32 ,
000 26 ,
000 32 ,
000 32 ,
000 32 ,
000 26 ,
000 USD / car [47]AV automation cost 20 mph C V , a ,
000 20 ,
000 3 ,
700 500 ,
000 0 20 ,
000 3 ,
700 USD / car [48]–[52]25 mph 15 ,
000 30 ,
000 4 ,
400 500 ,
000 0 30 ,
000 4 ,
400 USD / car [48]–[52]30 mph 15 ,
000 55 ,
000 6 ,
200 500 ,
000 0 55 ,
000 6 ,
200 USD / car [48]–[52]35 mph 15 ,
000 90 ,
000 8 ,
700 500 ,
000 0 90 ,
000 8 ,
700 USD / car [48]–[52]40 mph 15 ,
000 115 ,
000 9 ,
800 500 ,
000 0 115 ,
000 9 ,
800 USD / car [48]–[52]45 mph 15 ,
000 130 ,
000 12 ,
000 500 ,
000 0 130 ,
000 12 ,
000 USD / car [48]–[52]50 mph 15 ,
000 150 ,
000 13 ,
000 500 ,
000 0 150 ,
000 13 ,
000 USD / car [48]–[52]AV life l V per Joule γ .
14 0 .
14 0 .
14 0 .
14 0 .
14 0 .
14 0 .
14 g / kJ [53]Time from G W to G R , V t WV
300 300 300 300 300 300 300 s -Time from G R , V to G W t VW
60 60 60 60 60 60 60 s -Speed limit fraction β / . / . / . / . / . / . / . − [32] ES SB M FCM µ MV operational cost C M , o .
79 1 .
58 2 .
05 1 .
20 USD / mile [54]–[56] µ MV cost C M , f
550 8 ,
860 1 ,
000 3 ,
000 USD / µ MV [55]–[57] µ MV achievable speed v M , ij
15 10 15 15 mph - µ MV life l M .
085 7 . . . µ MV emissions m CO2 , M , tot .
101 0 .
033 0 .
158 0 .
033 kg / mile [55], [58]–[60]Time from G W to G R , M t WM
60 60 60 60 s -Time from G R , M to G W t MW
60 60 60 60 s -Subway operational cost 100 % C S , o , ,
000 USD / year [61]150 % 222 , ,
000 USD / year [61]200 % 295 , ,
000 USD / year [61]Subway fixed cost C S , f , ,
000 USD / train [62]Train life l S
30 year [62]Subway CO emissions per train m CO2 , S ,
000 kg / year [63]Train fleet baseline n S , base
112 train [62]Subway service frequency ϕ j , baseline / / min [64]Time from G W to G P and vice-versa t WS
60 s -
TABLE I: Parameters, variables, numbers, and units for the case studies.
Mobility Type Emissions Cost Speed ReliabilityTaxi Point-to-point High High operational cost, medium fixed cost High Up to availability and congestionAV Point-to-point High Low operational cost, high fixed cost High Up to availability and congestion µ MV Point-to-point Medium Medium operational cost, low fixed cost Low/Medium Up to availabilityWalking Point-to-point No emissions Free Low HighSubway Fixed hubs and routes Low Low Medium High
TABLE II: Comparison of the considered mobility solutions.of clarity, we prefer a two-dimensional antichain representa-tion, where emissions are included in the costs via a conver-sion factor of 40 USD / kg [69]. Note that this transformationpreserves the monotonicity of the CDPI and therefore inte-grates in our framework. The two-dimensional antichain andthe corresponding central authority’s decisions are reportedin Figure 3b. In general, as the municipality budget increases,the average travel time per trip required to satisfy the givendemand decreases, reaching a minimum of about 20 . / month.This configuration corresponds to a fleet of 4 ,
000 AVs able todrive at 50 mph, and to the doubling of the current MetroRailtrain fleet. Furthermore, the smallest rational investment of13 Mil USD / month leads to a 22 % higher average travel time,corresponding to the current situation, i.e., to a non-existentAVs fleet, and an unchanged subway infrastructure. Notably,an expense of 18 Mil USD / month (50 % lower than the highestrational investment) only increases the minimal required traveltime by 8 %, requiring a fleet of 3 ,
000 AVs able to drive at45 mph and no acquisition of trains. Conversely, an expenseof 15 Mil USD / month (just 2 Mil USD / month higher than theminimal rational investment) provides a 2 min shorter travel The description in this caption is valid for all the following figures. time. Finally, it is rational to improve the subway systemstarting from a budget of 23 Mil USD / month, leading to atravel improvement of just 8 %. This trend can be explainedwith the high train acquisition cost and increased operationcosts, related to the reinforcement of the subway system.This phenomenon is expected to be even more marked forother cities, considering the moderate operation costs of theMetroRail subway system, due to its automation [64] andrelated benefits [70].
2) Speed-dependent automation costs:2020:
We report the results in Figure 4a. A comparisonwith our basic setting (cf. Figure 3) confirms the trendsconcerning public expense. Indeed, a public expense of26 Mil USD / month (43 % lower than the highest rational ex-pense) only increases the average travel time by 5 %, requiringa fleet of 2 ,
000 AVs able to reach 30 mph and a subwayreinforcement of 50 %. Nevertheless, our comparison showstwo substantial differences. First, the budget required for anaverage travel time of 13 min is 25 % higher compared to S1.Second, the higher AV costs result in an average AVs fleetgrowth of 9 %, an average velocity reduction of 15 %, andan average train fleet growth of 14 %. The latter suggestsa shift towards poorer AVs performance in favor of fleets (a) Speed-dependent automation costs in 2020.(b) Speed-dependent automation costs in 2025.
Fig. 4: Results for the speed-dependent automation costs.Fig. 5: Results for large automation costs.reinforcements.
The maximal rational budget is 23 % lower than inthe case of immediate deployment (Figure 4b). Further, thereduction in autonomy costs incentifies the acquisition of moreperformant AVs, increasing the average vehicle speed by 14 %.Hence, AVs and train fleets are 10 % and 13 % smaller.
3) High automation costs analysis:
Figure 5 shows thathigh automation costs yield design strategies and costs dif-ferent from the ones of S1 and S2. First, we observe asubstantial shift towards larger train fleet sizes (65 % largerthan in S1) and smaller AVs fleets (55 % smaller than in S1).Second, minimizing the average travel time entails an expenseof approximately 68 Mil USD / month, basically doubling theinvestments observed in the basic setting.Fig. 6: Results for the MoD case. (a) Impact of micromobility in 2020.(b) Impact of micromobility in 2025. Fig. 7: Results for the impact of micromobility.
4) MoD setting:
We summarize the results for the MoDscenario in Figure 6. In particular, by comparing the MoDcase with the 2025 setting, we can notice the game-changingproperties that AVs introduce in the mobility ecosystem. Inparticular, the average train fleet size and the average vehiclefleet sizes increase by 130 % and 66 %, suggesting a cleartransition in investments from public transit to AVs, andtestifies to the interest in AMoD systems developed in thepast years.
5) Impact of new transportation modes:
To assess the im-pact of µ M solutions, we compare the arising design solutions,reported in Figure 7, with their counterpart in S2 (cf. Figure 4).
Figure 7a, together with Figure 4a, demonstrates anoverall benefit from µ M solutions. For instance, the mosttime-efficient solution in S2 yields an average travel time of20 . / month. The deploymentof µ M solutions lowers the average travel time achievable withthe same expense by 10 % (18 . . / month. Overall,the average AVs fleet size and the average train fleet size are35 % and 6 % smaller, in favor of an average µ M fleet of 2 , µ MVs.
Figure 7b, together with Figure 4b, shows thatthe benefit of µ M solutions is less marked than in 2020.For instance, an expense of 35 Mil USD / month (same asthe maximal expense in Figure 4b) results in an averagetravel time of 19 . µ M. Furthermore, we observe an average AVs fleetsize enlargement of 17 %, and an average train fleet sizereduction of 27 %. Finally, the comparison with the 2020 casehighlights a µ MVs fleet reduction of 23 %, which suggeststhe comparative advantage of AVs in the future. Indeed, thestronger the reduction of the cost of automation, the moreinvestments in AVs are rational. The benefits of employing µ M solutions could therefore just be temporary, and graduallyvanish as the costs of automation of AVs decrease. C. Discussion
First, the presented case studies showcase the ability of ourframework to extract the set of rational design strategies fora future mobility system, including AVs, µ MVs, and publictransit. This way, stakeholders such as mobility providers,transportation authorities, and policy makers can get trans-parent and interpretable insights on the impact of future in-terventions, inducing further reflection on this complex socio-technical problem. Second, we perform a sensitivity analysisthrough the variation of autonomy cost structures, and showthe capacity of our framework to capture various models. Onthe one hand, this reveals a clear transition from small fleetsof fast AVs (in the case of low autonomy costs) to largefleets of slow AVs (in the case of high autonomy costs). Onthe other hand, our studies highlight that investments in thesubway infrastructure are rational only when large budgets areavailable. Indeed, the high train acquisition and operation costslead to a comparative advantage of AV-based mobility. Finally,our case studies suggest that the deployment of µ M solutionsis rational primarily on a short-term horizon: The lowering ofautomation costs could eventually make AVs the predominantactor in the future of urban mobility.VI. C
ONCLUSION
This paper leverages the mathematical theory of co-designto propose a co-design framework for future mobility systems.The nature of our framework offers a different viewpointon the future mobility problem, enabling the modular andcompositional interconnection of the design problems of dif-ferent mobility options and their optimization, given multipleobjectives. Starting from the multi-commodity flow model ofan intermodal mobility system, we designed AVs, µ MVs,and public transit both from a vehicle-centric and fleet-levelperspective. Specifically, we studied the problem of deployinga fleet of self-driving vehicles providing on-demand mobilityin cooperation with µ M solutions and public transit, adaptingthe speed achievable by AVs and µ MVs, their fleet sizes, andthe service frequency of the subway lines. Our frameworkallows stakeholders involved in the mobility ecosystem, fromvehicle developers all the way to mobility-as-a-service compa-nies and central authorities, to characterize rational trajectoriesfor technology and investment development. The proposedmethodology is showcased in the real-world case study ofWashington D.C., USA. Notably, we highlighted how ourproblem formulation allows for a systematic analysis of in-comparable objectives, such as public expense, average traveltime, and emissions, providing stakeholders with analyticalinsights for the socio-technical design of future mobility sys-tems. This work urges the following future research streams:
Modeling:
First, we would like to capture heterogeneousfleets of AVs, with different autonomy pipelines, propulsionsystems, and passenger capacity. Second, we would like to de-sign public transit lines in terms of their location and capacity.Third, we would like to investigate variable demand models.Finally, we would like to analyze the interactions betweenmultiple stakeholders in the mobility ecosystem, characterizedby conflicting interests and different action spaces. It is ad-vantageous to formulate this as a game, and to characterizepotentially arising equilibria.
Algorithms:
We are interested in tailoring general co-designalgorithmic frameworks to the particular case of transportationdesign problems, leveraging their specific structure, and char-acterizing their solutions.A
CKNOWLEDGMENTS
The authors would like to thank Mrs. I. New for herassistance with the proofreading of this work and Ms. S. Montifor the I-AMoD illustration. This research was supportedby the National Science Foundation under CAREER AwardCMMI-1454737, the Toyota Research Institute (TRI), andETH Z¨urich. This article solely reflects the opinions andconclusions of its authors and not NSF, TRI, or any otherentity. R
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