System optimum and pricing for the day-long commute with distributed demand, autos and transit

Transportation Research Part B 55 (2013) 98–117

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System optimum and pricing for the day-long commute with distributed demand, autos and transit Carlos F. Daganzo ⇑ UC Berkeley Center for Future Urban Transport, A Volvo Center of Excellence, United States

a r t i c l e

i n f o

Article history: Received 11 February 2013 Received in revised form 14 May 2013 Accepted 14 May 2013

Keywords: Multi-modal transportation Urban transportation Multi-modal pricing Macroscopic urban modeling

a b s t r a c t The day-long system optimum (SO) commute for an urban area served by auto and transit is modeled as an auto bottleneck with a capacitated transit bypass. A public agency manages the system’s capacities optimally. Commuters are identical except for the times at which they wish to complete their morning trips and start their evening trips, which are given by an arbitrary joint distribution. They value unpunctuality – their lateness or earliness relative to their wish times – with a common penalty function. They must use the same mode for both trips. Commuters are assigned personalized mode and travel start times that collectively minimize society’s generalized cost for the whole day. This includes unpunctuality penalties, queuing delays, travel times and out-of-pocket costs for users, as well as travel supply costs and externalities for society. It is shown that in a SO solution there can be no queuing and that the set of SO solutions forms a convex set. Furthermore, if the schedule penalty that users suffer due to unpunctuality is separable into morning and evening components, then the set of commuters traveling by the same mode arrive at work and depart from work in the order of their wishes. These orders are in general different in the morning and the evening. It is also shown that there always is a SO solution in which users are at all times, and on both modes, either punctual or flowing at capacity. These problem properties are used to identify search methods, both, for SO solutions and for time-dependent tolls and transit fares that preserve the solutions as Nash equilibriums. In every case studied, these prices exist. They must peak concurrently for the two modes in both periods. In special cases involving only one mode, only one period or concentrated demand the solution to the complete problem decomposes by period conditional on the number of transit users, and this facilitates the solution. In these cases the day-long SO cost is the sum of the SO costs for the two peaks considered separately. However, this is not true in general – the solution obtained by combining the two single-period solutions can be infeasible. When this happens, the optimum day-long cost will exceed the sum of the single-period costs. The discrepancy is about 40% of the total schedule penalty for an example representing a large city. Thus, to develop realistic policies the day-long problem must be addressed head on. An approximate method that yields closed form formulas for the case with uniformly distributed wishes is presented.  2013 Elsevier Ltd. All rights reserved.

⇑ Tel.: +1 510 642 3853. E-mail address: [email protected] 0191-2615/$ - see front matter  2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trb.2013.05.004

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1. Introduction This paper examines how travel for a complete workday consisting of two peaks should be optimally organized in a city served by two modes: auto and transit. The number of commuters and their desired trip starting times are given. In the spirit of system optimality (SO) it is assumed that the analyst can assign the travel mode and trip starting times to each commuter. These assignments, and the transit agency’s best way of accommodating the transit trips, are to be found. Also to be found are tolls and fares that would induce selfish, well-informed commuters to travel in the identified SO pattern. To develop insights the spatial distribution of travel shall not be modeled in detail. Instead, the city is macroscopically idealized as in Daganzo (2007); i.e., as a factory or bottleneck that can process car and transit trips at given rates. To model demand recognizing that commuters are spatially distributed and that their work shift durations differ, the commuters’ morning and evening wish times are described with a bivariate distribution. To the author’s knowledge, the bimodal daylong scenario with distributed demand has not yet been studied. Dynamic travel analyses have a rich history, dating back to Vickrey (1969), which examined an idealized version of the morning commute problem. In Vickrey’s model, a population of car commuters wish to pass through a finite-capacity bottleneck at a common time, which would get them to work on time. Since the bottleneck cannot serve everyone simultaneously, queues form. Each commuter chooses the start time of her trip from home to minimize his/her generalized trip cost, including trip time, queuing delay and a ‘‘lateness’’ penalty associated with deviation from the wished passage time. Vickrey (1969) showed that there was a Nash equilibrium distribution of arrivals at the bottleneck where no commuter could reduce his/her generalized cost by changing arrival times and that all the queuing delay could be eliminated with congestion pricing. This model was later extended to distributed wished passage times (Hendrickson and Kocur, 1981), to a wider class of penalty functions (Smith, 1984; Daganzo, 1985), and to allow for elastic demand (Arnott et al., 1993). All these models considered only the morning peak. The evening rush is slightly different because commuters have a desired arrival time at the bottleneck, which is related to the end of their shift, instead of a wished passage time. Despite this difference, the evening problem is very similar. Some of its features have been shown to be the ‘‘mirror image’’ of the morning counterpart unless the commuters are heterogeneous (see; e.g., Vickrey, 1973; Fargier, 1983; De Palma and Lindsey, 2002). All these works, however, consider only one mode and only one peak. Mode choice has also been studied (Tabuchi, 1993; Braid, 1996; Huang, 2000; Danielis and Marcucci, 2002; Qian and Zhang, 2011) but only for the morning peak with a common wished passage time for all commuters, and with an unresponsive transit agency. Gonzales and Daganzo (2012a) allows for distributed wish times and a responsive transit agency but only for the morning commute. The day-long problem appears to have been first examined in Zhang et al. (2005, 2008). These references link the morning and evening commutes for auto travel via work duration and parking location. Only the second reference examines the system optimum. Neither considers distributed wish times or the availability of transit. These two features complicate the day-long problem considerably as we shall see. Gonzales and Daganzo (2012b) also analyzes the day-long commute, and considers transit, but does not analyze jointly distributed wish times. Distributed wish times add realism when the bottleneck model is to represent a congestible urban area. Thus, this restriction is relaxed in this paper. The exposition is organized as follows. Section 2 defines the problem. Section 3 shows that the SO problem can be formulated as a continuum linear programming problem for four multi-variable functions. The rest of the paper is restricted to penalty functions that separate by period. Section 4 shows that there is always a day-long SO solution in which people pass the bottleneck in a first-wish, first-pass (FWFP) order in both peaks. This type of solution then becomes the focus of analysis. The next two sections treat the single-mode problem: Section 5 examines the character of the SO, and Section 6 when and how it can be achieved with pricing. The two-mode problem is treated similarly over the three ensuing sections: Section 7 provides basic results, Section 8 examines pricing issues and Section 9 describes the solution for an important case. Finally, Section 10 concludes the paper, summarizing its key results and suggesting generalizations.

2. The problem Considered are N commuters who wish to travel to/from work in an urban area that can process auto and transit trips at fixed maximum rates, as in Daganzo (2007). Commuters are treated as fluids, so that the commuter index n is a real number in the interval [0, N]. The area is managed to prevent overcrowding, with queues of people wishing to travel occurring outside the network, e.g., either at home or at work. In this way the queues do not affect the system’s trip processing rates and the area roughly behaves as a set of two parallel, fixed-capacity bottlenecks separately serving car and transit trips. These bottlenecks are visualized to be spatially diffused and discharging at the points in space where people leave the network—either their destinations or the network’s periphery. Thus, the bottleneck service rate expresses the rate at which people leave the system. As is customary we time-stamp all relevant commuter events (wished and actual arrivals/departures from work and the actual arrivals and departures from home) at the times they would manifest themselves at the bottleneck in the absence of congestion. This simplification allows us to exclude from the model the travel times between the origins, bottlenecks and destinations. Idealizing the urban area in this way is somewhat drastic but has an important payoff: it allows us to model the area’s travel dynamics without explicitly tracking individual trips in space, which would require detailed data, impede the

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optimization and be obfuscating. With the simplification the only network features in the model are its bottleneck capacities, and two cost functions for transit and car trips. The bottleneck capacities are fixed and denoted: l (pax/h) for auto in the morning, l0 for auto in the evening, lT for transit in the morning and l0T for transit in the evening. This assumes that a consistent level of transit service is offered through both peaks regardless of demand. Although this is reasonable, one could also assume as in Gonzales and Daganzo (2012a) that transit service is eliminated when there is no transit demand, and that the unused transit lanes are then made available to cars, increasing the car capacity. This variant of the problem can be analyzed with nearly identical logic but the results are more cumbersome to describe, with more parameters, cases and sub-cases. Therefore the variant’s results are not described. The reader can verify that they are qualitatively similar to those presented. The total cost function for car trips is the product of the number of car travelers and the generalized cost, zc, of one commuting round trip by car in free-flow, i.e. excluding delay. This cost includes the users’ out-of-pocket and travel-time costs without congestion, plus externalities. Thus, if we denote by NT the number of commuters that use transit, the total cost due to car users is (N  NT)zc. Congestion (queuing) delays are ignored because, as shown at the end of this section, they can always be eliminated from a solution without increasing cost. The only considered cost associated with delay is a ‘‘schedule penalty’’ that captures the effect of people’s unpunctuality in arriving and departing from work. This cost arises regardless of the mode and will be counted separately. The total cost function for transit trips combines the cost to the transit agency including externalities and the travel time of transit users including walking, waiting and riding. As in the case of cars, it is assumed that the transit system is uncongested so users can always catch the first transit vehicle to come. The generalized cost of transit mostly depends on: the number of users NT, the capacities provided, lT and l0T , the spacing between lines and the service headway. Fortunately, the last two variables can be omitted as arguments from the function because for any given {NT, lT, l0T } they only affect the transit agency and the transit riders. Thus, the headway and spacing variables can be optimized conditional on {NT, lT, l0T } considering only the transit system cost, and then internalized into the transit cost function. The result is a function that is denoted Z T ðN T ; lT ; l0T Þ. As is typical of logistics systems, this function should increase with the capacities and be continuous, concave and increasing in NT for any capacities. For most of this paper the transit capacities will be considered fixed and the transit cost function will be abbreviated as ZT(NT). The last cost element to be defined pertains to schedule delay. A user’s schedule delay is the difference between the actual and wished times of departing and arriving at the bottleneck, i.e. arriving and departing from work, in the morning and the evening. To unify the terminology, these two events shall be called from now on ‘‘passages’’ through the bottle  neck. The two wished passage times of commuter n for the morning and the evening are denoted: wn ¼ wn ; w0n . These wished times are given for all n. It is assumed that wn  w0m for every pair (n, m) to ensure that the two peaks do not over  lap. The actual bottleneck passage times of every commuter are denoted t n ¼ tn ; t0n . The schedule delay or ‘‘lateness’’ is the 2-vector, tn  wn. User latenesses are converted into cost by means of a penalty function, P(tn  wn), which is identical for all commuters. The function P is assumed to be positive semidefinite with P(0, 0) = 0, and to increase with its second argument, t0n  w0n , at a smaller rate than the value of people’s time while waiting for a transit vehicle or driving. This second condition is reasonable since most people prefer to do any waiting after work at a location of their choice (say at a coffee shop) rather than queuing at a transit stop or inside a car. The total schedule delay penalty, SD, is the sum across n of P(tn  wn). It is now shown that queuing delays can be eliminated without increasing cost merely by postponing the arrival of every delayed commuter at each bottleneck until his/her departure time. Since the postponements leave the monetary car and transit costs invariant we focus on the costs associated with schedule and queuing delay. These costs cannot increase because (i) morning postponements eliminate queuing delay without affecting lateness, which reduces their combined cost; and (ii) evening postponements reduce queuing delay by the same amount as they increase lateness, which prevents the combined cost of delay from increasing because the schedule penalty function for the evening values queuing delay at least as much as evening lateness. Note that if the penalty function is positive definite then evening postponements, like the morning counterparts, actually reduce cost. Thus, in this case all postponements reduce cost, so that SO solutions with queuing cannot exist. Since queuing is not in the picture, the SO problem consists in choosing the mode and passage times of every commuter in order to minimize society’s cost, SD + ZT(NT) + (N  NT)zc, without exceeding the capacity of any bottleneck. This idea can be expressed as an optimization problem. Note, however, that if NT is given only SD can vary with the remaining decision variables. Thus, much of this paper focuses on the minimization of SD for a fixed NT. 3. The SO optimization problem It is assumed that N is so large that people can be treated as a continuum. Accordingly, people’s wishes {wn, " n} are modeled by a bivariate function D(w) with units of users/time2 and giving the temporal density of wishes in the morning and the evening. The decision variables, i.e., the mode and passage times of every user, are also expressed in terms of continuous temporal densities. This view of the world simplifies the task at hand. We shall look for the joint density of transit and auto travelers that pass the bottleneck at times, t when wishing to pass at times w. These decision variables (actually decision functions) have units of users/time4, and will be denoted: dT(t, w) for transit, and dA(t, w) for auto.

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It is shown in this section that, conditional on NT, the SO problem is a feasible and bounded linear program. Thus, the SO solution exists for any NT, all its local minimums are global and the optimum solutions form a convex set. The unconditional solution is also shown to exist. Let us first examine the objective function, which combines the congestion-free car and transit costs with the sum of all schedule delay penalties. The latter is just the integral of the penalties experienced by the people assigned to every possible combination of (t, w). Thus, the minimization objective is:

ZðNT ; dT ; dA Þ ¼ zc ðN  NT Þ þ Z T ðNT Þ þ

Z

dt dwfPðt  wÞðdT þ dA Þg:

ð1Þ

Here and in what follows, a single boldfaced integral sign denotes a multiple integral. The number of simple integrals can be inferred from the number of differentials in the expression: four in the case of (1). If no limits of integration are shown, as occurs above, the integrals range from 1 to +1. Note that dT and dA are functions of (t, w), and that these arguments have been dropped to simplify the notation. Finally note that (1) is linear in these decision variables. The decision variables are subject to constraints. To express and later analyze these constraints it will be convenient to introduce as auxiliary decision variables the wish density functions by mode: DT(w) and DA(w). Therefore, the decision variables are dT, dA, DT and DA. The first set of constraints stipulates that for any given w, the chosen allocations must be non-negative and add up to the given density of people, D(w):

dT ; dA P 0; DT ; DA P 0; Z

8t; w; 8w;

dtfdT g ¼ DT ðwÞ;

ð2aÞ ð2bÞ Z

DA ðwÞ þ DT ðwÞ ¼ DðwÞ;

dtfdA g ¼ DA ðwÞ;

8w;

8w:

ð3Þ ð4Þ

A second set expresses the capacity constraints. Note that the flow of users passing a modal bottleneck (morning or evening) at a given time is the marginal density of passages for the given mode and period, which is a triple integral of the quadrivariate decision density for the corresponding mode. Thus, the capacity constraints can be expressed as follows:

Z

Z dw Z Z Z dt dw Z Z Z 0 dt dw Z Z Z dt dw dt

0

Z

0

dw fdT g 6 lT ; 8t; 0

dw fdT g 6 l0T ; 0

dw fdA g 6 l; 0

dw fdA g 6 l0 ;

8t 0 ;

ðtransit-morningÞ ðtransit-eveningÞ

ð5aÞ ð5bÞ

8t;

ðauto-morningÞ

ð5cÞ

8t 0 ;

ðauto-eveningÞ

ð5dÞ

A final constraint defines NT:

06

Z

dwfDT g ¼ NT 6 N:

ð6Þ

Note how constraints (2)–(6) are linear in the decision variables. Therefore, problem (1)–(6), conditional on NT, is a linear program as claimed. The program is feasible, as can be verified by setting DA = dA = 0, DT = D; and then choosing a sufficiently large value M and setting: dT = D(w)/M2 if t, t0 2 [M/2, M/2], and dT = 0 otherwise. With this choice, (2)–(4), (5c), and (5d) are satisfied; (6) yields the definitional N = NT; and the left hand sides of (5a) and (5b) are equal to N/M. Thus, for sufficiently large M they are satisfied too. Thus, the LP is feasible. Since the objective function to be minimized is bounded from below (it is non-negative) the optimum solution is bounded. Thus, the optimum solution exists for any NT 2 [0, N]. Consider now the optimum across all possible NT obtained by minimizing the new objective function SD⁄(NT) + ZT(NT) + (N  NT)zc over the interval [0, N], where SD⁄(NT) is the optimum cost of our feasible LP. An optimum NT exists because the new objective function is continuous and is being minimized over a closed interval – the first term, SD⁄(NT), is continuous because NT is a right hand side parameter of an LP. The above shows that a SO solution always exists. It can be found approximately by discretizing (1)–(6) and solving the resulting LP numerically for different values of NT. Although the above arguments rule out multiple local optima and show that the solution can be approximately obtained, they do not shed much insight. The results do not show for example, if prices exist that would encourage selfish and wellinformed commuters to accept a SO pattern. These issues and other simplifications are explored in the sections that follow.

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4. Separable penalties: the FWFP property The general solution of Section 3 is obscure because it is described by two surfaces in 5-dimensions, dA and dT, and two surfaces in 3-dimensions, DA and DT. Fortunately, there are special cases in which the solution simplifies. One case is the flex-time model examined in Gonzales and Daganzo (2012b). Another case, which is the focus from now on, involves penalty functions that can be separated by period. These penalties will be called ‘‘separable’’ and defined to be of the form:

  Pðt n  wn Þ ¼ Pðtn  wn Þ þ P0 t0n  w0n ; 0

ð7Þ 0

where P and P are positive definite and convex, and P increases everywhere at a smaller rate than the value of queuing time. Note, since P and P0 are positive definite and convex they must be non-decreasing for positive arguments and non-increasing for negative arguments. From now on penalties are assumed to be separable unless something is stated to the contrary. It will be shown in this section that if a problem’s penalty function is separable then there always is a SO solution in which people pass all four bottlenecks in the order of their wishes (first wish first pass, FWFP, order). Sections 5–9 will exploit this property. The FWFP property is proven by demonstrating that any SO densities dT and dA can be rearranged into FWFP order without increasing cost. The following lemmas will be used. Lemma 1. If C is a convex function of a real variable in an interval [a, b], and 0 < c < (b  a), then C(a) + C(b) P C(a + c) + C(b  c). Furthermore, if C is strictly convex, the inequality is strict. Proof. For the convex case the chord above the curve property ensures: C(a + c) 6 [(b  a  c)/(b  a)]C(a) + [c/(b  a)]C(b) and C(b  c) 6 [c/(b  a)]C(a) + [(b  a  c)/(b  a)]C(b). Now add these two inequalities to get: C(a + c) + C(b  c) 6 C(a) + C(b). For strictly convex functions strict inequalities can be used with the same logic. The result is a strict inequality as claimed. h Lemma 1 helps establish the following additional fact: Lemma 2. If two people traveling by the same mode pass their morning or evening bottleneck in reverse order of their wishes, cost does not increase if their passage times are swapped. Furthermore, if the penalty function is strictly convex the swap reduces cost. Proof. The people involved in the swap are labeled n = 1 and n = 2. Since the labeling order is arbitrary it suffices to examine the effect of morning reversals in which t2 < t1 when w2 > w1; or else evening reversals where t02 < t01 when w02 > w01 . Since the proof for the evening is identical to that for the morning, only the former is given. We need to show (i) that the before-swap penalty is equal or greater than the after-swap penalty i.e. that P(t1  w1) + P(t2  w2) P P(t2  w1) + P(t1  w2); and (ii) that this inequality is strict if P is strictly convex. To see this, let a = t2  w2, b = t1  w1, c = t1  t2, and note that these three constants satisfy: 0 < c < b  a. Lemma 1 can now be used to write: P(t1  w1) + P(t2  w2) = P(a) + P(b) P P(a + c) + P(b  c) = P(t2  w1) + P(t1  w2), which establishes (i). Since Lemma 1 also guarantees that the last inequality is strict if P is strictly convex, (ii) is also established. h Lemmas 1 and 2 are now used to show that every problem has a SO solution with commuter densities of FWFP form, and for strictly convex penalties that all SO solution densities are FWFP. The proof of this result is a bit lengthy because in our continuum framework, which is useful for other purposes, the number of customers is not countable. Before we begin note that for separable P the quadruple integral of (1) becomes:

integral; SD ¼

Z

dt dwf½Pðt  wÞ þ P0 ðt0  w0 ÞðdT þ dA Þg;

ð8Þ

which can be separated into four double integrals for the morning/evening periods and auto/transit modes. The first of these integrals for morning/transit is:

Z

dt dwPðt  wÞdT ðt; wÞ;

where dT ðt; wÞ ¼

Z

0

0

dt dw dT :

ð9Þ

The function dT(t, w) is the bivariate density of morning transit riders wising to pass at w and passing at t. The marginal Rx R Ry R cumulative densities, dt dwdT ðt; wÞ and dw dtdT ðt; wÞ are the cumulative curves of actual and wished passages, which will be respectively denoted TT(x) and WT(y). Similar definitions are used for the other three mode/period bottlenecks. The FWFP result is now proven. It will first be shown that the morning/transit density dT(t, w) can be concentrated along a non-decreasing curve CT on the (t, w) plane without increasing cost by rescheduling passages, and that the other three bivariate densities can be similarly concentrated. The rescheduling preserves the cumulative passage curves of all four bottlenecks. Lemma 3. The passage times of every feasible solution can be rearranged into FWFP form without increasing cost and while preserving the cumulative marginal curves W and T of each mode/period bottleneck. Furthermore, the rearrangement reduces cost if P is strictly convex.

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Proof. It will be shown that if a feasible solution has FWFP violations, these can be eliminated without increasing cost (or actually reducing it in the strictly convex case) by rescheduling the passage times without changing TT so the density is concentrated on a monotonic increasing curve on the (t, w) plane. The rescheduling is only demonstrated for the morning/transit bottleneck because the process is identical for the other bottlenecks. If the SO solution has FWFP violations in the morning/transit bottleneck it should be possible to tessellate the (t, w) plane into squares of sufficiently small area, a, to obtain two squares (‘‘1’’ and ‘‘2’’) with positive average densities dT1, dT2 > 0, such that every commuter pair in the squares violates FWFP. Such squares must be situated on different rows and different columns, centered on times (t1, w1) and (t2, w2), and such that w1 > w2 for the rows but t1 < t2 for the columns; i.e., such that the line joining the centers of the two squares has a negative slope. This is shown in Fig. 1a. By being on different rows and columns, every commuter pair on the two squares violates the FWFP order. : Define dTm ¼ min{dT1, dT2} > 0 and consider what happens if dTm  a pairs of people in the two squares swap passage times. As illustrated by the arrows of Fig. 1a, the swap is accomplished by moving dTma people from the square centered at (t1, w1) to the square (t2, w1) situated on the same row, and then moving the same number of people from the square at (t2, w2) to the one at (t1, w2). The moves are feasible because they do not create negative densities in any square, so (2a) is satisfied; they conserve the people quantities across all rows, preserving the wish distribution so (2b), (3), (4) and (6) are satisfied; and they conserve the people quantities across columns, which leaves the bottleneck flows invariant so (5a)–(5d) are satisfied. Thus, the swap is feasible. Because the people in the original two squares violate FWFP, Lemma 2 guarantees that this feasible swap does not increase cost, and that in the strictly convex case it reduces cost. Note that the commuter moves eliminate all FWFP violations in the two original squares without changing the number of FWFP violations in the other squares. Although a move can create a violation in some squares, its counter move always cancels it. Thus, the moves effectively eliminate all the violations in one pair of squares without increasing them in any other. It should also be noted that because the people quantities along all rows are conserved, the moves do not change the marginal distributions of dT(t, w): WT(w) and TT(t). This people moving process can be repeated for all square pairs with different rows and different columns until only violations within a row or within a column remain. The marginal curves WT(w) and TT(t) remain invariant. For finite domains, the procedure terminates in fewer steps than there are square pairs. If the domain is infinite, the procedure converges quickly if one always chooses the square pair that reduces violations the most. Once finished, the remaining squares’ centers must form a non-decreasing monotonic sequence on the (t, w)-plane for otherwise there would be square pairs that could be joined by a declining line, and could be interchanged. Thus, all the remaining squares can be covered by a continuous swath of alternating horizontal rows and vertical columns (both of unit width), which form an increasing ‘‘ladder’’ shown by the shaded area in Fig. 1b. This swath/ladder is denoted S1. Obviously, one can construct a continuous non-decreasing curve, denoted t = C1(w), that is entirely within S1, as illustrated in Fig. 1b. This is the end of iteration k = 1. Now, set k = 2, divide each square of S1 into four equal squares of area a/4k1 and repeat the process; i.e., identify square pairs with negative slopes and execute the interchanges to create a new ‘‘non-decreasing’’ ladder of smaller squares, S2, and then identify a continuous non-decreasing curve C2 within S2. As before, this process preserves the marginal curves WT(w) and TT(t), does not increase cost, and actually reduces it in the strictly convex case. Note the interchanged squares must always be in the same row or column of S1 because S1 forms an increasing ladder. As a result, the four squares involved in every interchange are also in S1. Since all the interchanges take place within S1, it follows that S2  S1. It also follows that the separation in the 45 direction between C1 and any curve within S1, pffiffiffiffiffimaximum ffi such as C2, ispffiffiffiffiffi bounded ffi k1 from above by 2a. And of course, the maximum separation between C2 and any curve in S2 is bounded by 2a=2 .

Fig. 1. The assignment problem: (a) people swap; (b) ladder including all remaining violations.

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Repeat now the division and interchange process ad infinitum, so that at the kth step we’d have obtained: Sffik  Sk1 and pffiffiffiffiffi Ck  Sk. Furthermore, the maximum separation between Ck and every curve in Sk is bounded from above by pffiffiffiffiffi 2affi=2k1 . Since the sequence of ladders is nested, the curves (Ck+1, Ck+2, . . .) 2 Sk and are separated from Ck by no more than 2a=2k1 . Thus, : the sequence of curves {Ck} satisfies the Cauchy criterion and converges uniformly to a limit curve C1 ¼ CT, where all the demand is concentrated in FWFP form. This is done without increasing cost, or actually reducing it in the strictly convex case. And, since the marginal curves WT(w) and TT(t) remain invariant throughout the process the result is proven for the morning/ transit bottleneck. The above arguments apply equally to the other three bottlenecks (morning/auto, evening/transit and evening/auto) and the conclusion is the same. This concludes the proof. h We are now ready for the main result. Result 1 (FWFP–Existence and Necessity). Every problem has a FWFP SO. Furthermore, if P is strictly convex, only FWFP SO exist.

Proof. Section 3 showed that a SO solution always exists, and Lemma 3 guarantees that any such solution can be transformed into one that is FWFP without increasing cost. Therefore this new FWFP solution is also SO, which proves the first part of the result. For the second part it suffices to show that non-FWFP solutions cannot be SO if P is strictly convex. This is obviously true because Lemma 3 guarantees that the cost of non-FWFP solutions can be reduced. h   The C T ; C 0T ; C A ; C 0A curves of Lemma 3 can be thought of as scheduling tools. For each mode and period, they assign passage times to people with specific wish times. Clearly then, the information needed to specify when and by what mode everyone passes in a FWFP solution is embodied in the combination of these assignments and the modally split wish density functions DT and DA. An expression for the solution densities of a FWFP solution in terms of this information is now derived. A dot subscript will signify a mode, T or A. Since the FWFP solution density for a given mode d.(t, t0 , w, w0 ) is concentrated along t = C.(w) and   t0 ¼ C 0: ðw0 Þ, it must be of the form: d: ðt; t 0 ; w; w0 Þ ¼ F : ðw; w0 ÞDðt  C : ðwÞÞD t0  C 0: ðw0 Þ , where D is the Dirac delta function (with units of time1) and F. a bivariate density with units of (people/time2). The functions F. are to be determined. However, by inserting the expression just derived as the integrands in the LHS of (3) we find that F.(w, w0 ) = D.(w, w0 ). Thus, the FWFP solution density for a generic mode is:

  d: ðt; t 0 ; w; w0 Þ ¼ D: ðw; w0 ÞDðt  C : ðwÞÞD t 0  C 0: ðw0 Þ ;

for : ¼ T; A

ð10Þ

As anticipated (10) only involves the assignment curves and the bivariate modal split density functions. This reformulation simplifies the searchfor a solution considerably for it now suffices to identify two bivariate densities (DT, DA) and four monotonic planar curves C T ; C 0T ; C A ; C 0A , whereas before one had to identify the same bivariate densities and two quadrivariate densities (dT, dA).   It is now shown that, instead of the C-curves, we can search for the four curves of cumulative passage times T T ; T 0T ; T A ; T 0A on a queuing diagram because the latter uniquely identify the former if DT and DA are fixed. To see this refer to Fig. 2, which pertains to the morning/transit bottleneck and therefore illustrates the correspondence between TT and CT. Note first that in any well posed problem every user, n0, must have a unique wish time, w0. Thus, a function, w = wT(n), that returns the wish time of the nth user to pass the transit/morning bottleneck exists. This is illustrated by the cumulative wish curve of the figure. Note as well that the function wT is fixed because it can be obtained from the also fixed bivariate wish density for transit; Rt R 0 i.e., because wT(n) is the inverse of W T ðtÞ ¼ dw dw DT ðw; w0 Þ.

Fig. 2. The function TT and its relation to the curve CT.

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Next note that in any feasible solution the cumulative number of passages must increase at a finite rate with time; thus, a function associating a unique user number n0 with each time t0 must exist. This is the cumulative function, n = TT(t), defined earlier and illustrated by the cumulative passage curve of Fig. 2. These two functions, wT and TT can be composed into a function w = wT(TT(t)) that yields a unique wish time, w0, for each passage time, t0, as illustrated by the arrow path starting at t0 and ending at w0 in the figure. The arrow path is labeled CT because the locus of all the pairs (w0, t0) on the (w, t) plane is the CT-curve that connects the wish times and passage times of our morning/transit bottleneck. Obviously, since the function wT is fixed and given, TT uniquely identifies CT, as claimed. Since the morning/transit logic also holds for the other bottlenecks, the same can be said in these cases. This confirms that    we can look for the familiar queuing diagram passage curves T T ; T 0T ; T A ; T 0A instead of the assignment curves C T ; C 0T ; C A ; C 0A . These results are now applied. Sections 5 and 6, immediately following, examine the day-long problem for a single mode. 5. The single-mode, day-long problem: basic results Section 5.1 shows how the formulation of the day-long, single-mode problem with separable penalties can be simplified and decomposed by period into two conventional problems. Thus, the morning and evening rushes can be treated separately and then combined together. This means that all the insights and methods of the single-mode, single-period theory extend to the day-long problem. Section 5.2 discusses an additional feature of these single-period problems: the existence of a FWFPSO solution in which every person passes the bottleneck either on time or at capacity. 5.1. Decomposition by period The simplification and decomposition for general penalties are now examined. This is done first for general (convex) penalties and then for bilinear penalties. The day-long, single mode problem formulation with separable penalties uses the constraints ((2)–(6)) and objective function ((1), (8) and (9)) of the two-mode problem, but setting dT,DT,NT  0 and dropping the subscript A. Thus, the only unknown is d, since DA  D. Furthermore, since the penalty function is separable the results of Section 4 hold. Therefore, it suffices to look for a density of form (10): d = D(w, w0 ) D(t  C(w))D(t0  C0 (w0 )), or equivalently for the curves {C, C0 } or {T, T0 }. If this expression for the density is inserted in the simplified formulation of the problem we find that constraints ((2–4,6)) are automatically satisfied. The remaining constraints (5) can be expressed in terms of {T, T0 } after integration w.r.t t and t0 . They become:

_ 6 l; 8t; ðmorningÞ 0 6 TðtÞ 0 _ 0 6 T ðtÞ 6 l0 ; 8t0 : ðeveningÞ

ð11aÞ ð11bÞ

Alternatively, these can be written in terms of the inverse functions:

_ tðnÞ P 1=l; n 2 ½0; N; ðmorningÞ _t0 ðnÞ P 1=l0 ; n 2 ½0; N: ðeveningÞ

ð12aÞ ð12bÞ

The objective function (8) can be simplified similarly. It reduces to:

Objective function integral ¼

Z

N

dnfPðtðnÞ  wðnÞÞg þ 0

Z

N

0

dn fP0 ðt0 ðn0 Þ  w0 ðn0 ÞÞg:

ð13Þ

0

The above shows that our SO problem is merely the minimization of (13) subject to (12) with t(n) and t0 (n) as the decision functions. Now note how t only appears in the first term of the objective function and in (12a), and how t0 only appears in the second term of the objective function and in (12b). Thus, the SO problem decomposes into two separate minimizations: a minimization of the first term of (13) subject to (12a), which yields t and a minimization of the second term of (13) subject to (12b), which yields t0 . These individual minimizations are conventional single-period, single mode SO problems with a wish curve W (or W0 ) and penalty function P (or P0 ). They can be easily solved with calculus of variations. In the strictly convex case the objective function (13) is strictly convex and the constraints are linear. Thus, the FWFP solution is unique. Bilinear penalty reformulation: For bilinear penalties, and only for bilinear penalties, the problem can be reformulated on a time basis. Bilinear penalties are special because the user cost of a unit of earliness or lateness is fixed, independent of the total earliness or lateness experienced by the user. The cost is e for each person-h of earliness and L for each person-h of lateness. Thus, the total penalty associated with a differential of time can be written without reference to information outside the differential. This means that the grand total (13) can be rewritten as an integral over time:

Objective function integral ¼

Z

dtfPðWðtÞ  TðtÞÞ þ P0 ðW 0 ðtÞ  T 0 ðtÞÞg;

ðbilinear caseÞ:

ð14Þ

This is the sum of the costs associated with ‘‘vertical slices’’ of the (t, n) plane instead of the sum of the costs associated with ‘‘horizontal’’ slices given by (13). Vertical slices can be used because in the bilinear case the functions P and P0 take on constant values when t (or t0 ) is above/below w (or w0 ).

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Thus, in the bilinear case the optimization problem can be time-based and defined by (11) and (14), with T and T0 as the decision functions. This is the conventional form. The formulation, however, is not valid for other penalty functions. 5.2. The CMP property of the passage curves We say that a cumulative passage curve T has the CMP (capacity or matching passages) property if fT_ ¼ l or T = W, " t}. In other words, under CMP passages occur either at capacity or matching the cumulative wishes. Conditions for the existence, necessity and uniqueness of SO solutions with the CMP property are now presented. This property can be exploited in solution methods. It is proven below for the single-mode, single-period problem but it holds for the single-mode, day-long problem as well by virtue of decomposition. Result 2 (Existence, Necessity and Uniqueness of a FWFP-CMP-SO). The result pertains to single-mode, single period problems and has three parts: (i) Every problem has a FWFP-CMP-SO; (ii) if P is bilinear every SO is CMP, albeit not necessarily FWFP, and the optimum T-curve is unique; and (iii) if P is strictly convex the SO is unique and FWFP-CMP. Proof. Since a FWFP solution always exists, part (i) is proven by showing that all lateness and earliness violations of the property in any such solution can be eliminated without increasing cost. Parts (ii) and (iii) are then proven by showing that violations lead to contradictions. If a user were both late and under capacity (a lateness violation) in a FWFP solution we could reschedule all following users, as shown in Fig. 3a: (i) reschedule late users at capacity until one became punctual; (ii) from then on, reschedule them at a rate equal to the smallest of the wish rate or the capacity rate if they are punctual, or at capacity rate if they are late, stopping when a user passes at the original time; (iii) leave all ensuing users on their original schedule. The effect of steps (i) and (ii) is shifting a portion of the T-curve to the left without violating the capacity constraint and while remaining to the right side of the W-curve; see Fig. 3a. The shift eliminates all the violations in the shifted portion without introducing other violations. It also reduces the (positive) latenesses for all switched users without inducing earliness. Since P is positive semidefinite, the rescheduling is guaranteed to not increase cost. Since all the rescheduled users flow either at capacity or on time, the rescheduling ensures that all the users switched satisfy the CMP property; see Fig. 3a. If the rescheduling does not remove all the lateness violations, it can be repeated until none remains. Thus, all the lateness violations can be removed without increasing cost or affecting the users who pass early. The same thing but in reverse (working backwards in time) can now be done to eliminate all earliness violations without increasing cost. When done, all the violations will have been eliminated without increasing cost. Fig. 3b shows the result of applying the complete procedure to a hypothetical example. This proves part (i) of the result. Consider now part (ii). Assume that a non-CMP solution is SO. Lemma 3 guarantees that associated with any such solution there is a FWFP solution with the same T curve and equal (or less) cost. Assume the cost is equal to avoid a contradiction, and now apply the rescheduling moves used in the proof of part (i) to this FWFP-SO solution. However, if P is bilinear every rescheduling move reduces cost, which is impossible. Thus, the SO must be CMP. Now, to see that the T curve is unique, note that any local perturbation to a CMP passage curve obtained by changing slightly the position of its capacity episodes invariably increases cost. This means that any SO T-curve must be an isolated local optimum. Since the SO solutions of a problem form a convex and therefore connected set (as per Section 3) it follows that there can be no other optima for they would have to be disconnected. This establishes part (ii).

Fig. 3. Restoring CMP: (a) removing a lateness violation; (b) final solution.

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Finally, consider part (iii). In this case, the SO solutions must be FWFP as per Result 1. Since Section 5.1 showed that there can be only one FWFP solution, it follows that the SO is unique and FWFP. And, since part (i) guarantees that at least one SO is CMP, it also follows that the unique SO is FWFP-CMP. h The bilinear SO solution is not unique and not necessarily FWFP because departure time swaps between users in the same earliness or lateness episode leave cost invariant. Thus, although there is only one FWFP SO solution in the bilinear case, the ordering of users within its earliness and lateness episodes can be arbitrarily changed. Finally, recall that the day-long problem decomposes by period. Therefore we can state the following corollary of Result 1: Corollary 1 (Existence, Necessity and Uniqueness of FWFP-CMP-SO for day-long problems). Result 2 also holds for singlemode, day-long problems. This corollary is put to use in the following section. 6. The single-mode, day-long problem: pricing This section shows that any FWFP passage curve satisfying the CMP property can be achieved by pricing. Since there always is a SO solution that is FWFP-CMP (Corollary 1), it follows that pricing can achieve SO. In what follows we only consider single period problems because prices can be set separately by period to achieve any desired temporal day long pattern.

Fig. 4. Equilibrium prices: (a) definitions; (b) relationship with the indifference price curves.

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Section 6.1 shows that SO prices exist and gives SO pricing formulae for general, differentiable penalty functions. Applied to the morning and evening rushes, these formulas achieve the day-long SO. Section 6.2 derives similar (and simpler) expressions for bilinear penalty functions. 6.1. Optimal prices with differentiable penalty functions It is shown here that if the penalty function is separable and differentiable then prices exist that can achieve any FWFPCMP passage pattern. That is, with those prices no user has an incentive to change his/her passage time in either period from that of the pattern. Formulas are given. To begin, consider a wish curve and FWFP passage pattern satisfying the CMP property with a single capacitated episode; : see Fig. 4a. The capacitated episode spans the user interval ½n0 ; n00  ¼ N . The first and last users pass punctually; thus: ðt n0 ; tn00 Þ ¼ ðwn0 ; wn00 Þ. Users not in N also pass punctually. It is now shown that this type of passage pattern can always be preserved by pricing without inducing queuing. A formula for the prices is given. Let pm be a curve on the (t, n)-plane associated with user m and expressing the prices pm(n) at which said user is indifferent across all positions n. The sum of the indifference price paid and the penalty received by user m for departing in position n, i.e., the quantity pm(n) + P(tn  wm), is the user’s indifference cost, which by definition cannot depend on n and is therefore denoted cm. It follows that the expression for the curve is pm(n) = cm  P(tn  wm). It will be convenient to simplify this expression for n 2 N by placing the origin of coordinates on the passage curve at a level, n, that is in N ; e.g., as in Fig. 4a. Then tn = n/l, for n 2 N , so we can write:

pm ðnÞ ¼ cm  Pðtn  wm Þ; 8n; ¼ cm  Pðn=l  wm Þ; 8n 2 N ;

ð15aÞ ð15bÞ

Note from both Fig. 4a and (15b) that the lateness argument of P in (15b) increases with n linearly for n 2 N ; and that therefore (15b) is concave in n for n 2 N . Let us now look for the actual prices p(n) that discourage position swaps within N . The considered set of positions will be extended to the totality later. Thus, we use (15b) for now. According to this equation, {pm} is a 1-parameter family of concave curves in n parameterized by m; e.g., as shown by Fig. 4b. Customer m will stay with its assigned FWFP position n = m if the actual price curve p(n) touches the indifference price curve pm at n = m, but p P pm for n – m, as shown by the middle curve of the figure. If this happens for all m, then curve p achieves equilibrium. Another way of stating this condition is that the actual price curve p should be the upper envelope of pm, denoted pE, and such that the points of tangency for each curve m occur at n = m, as with the middle curve of the figure. Because the individual curves are concave, an upper envelope exists and is given by: pE(n) = maxm{pm(n)}. Now, requiring that the points of tangency occur at n = m for all m is the same as requiring that curve m = n be tangent to the envelope at n for all n; i.e.,: arg maxm{pm(n)} = n, "n. This can be expressed as the following condition involving the {cm}:

arg maxfcm  Pðn=l  wm Þg ¼ n; m

8n 2 N :

ð16Þ

Eq. (16) is now used to determine the {cm} and the actual price curve p(n). First, it is rewritten using the first order condition for a maximum, i.e., specifying that the function’s derivative should vanish at m = n. The condition becomes dcm  _ ¼ Pðn= l  wm Þðw_ m Þj , where an overdot denotes the function’s derivative. This expression is more streamlined m¼n

dm m¼n

if m is used instead of n as the variable:

dcm _ ¼ Pðm= l  wm Þðw_ m Þ: dm

ð17Þ

Eq. (17) is an ODE that can be integrated to determine {cm} up to an additive constant, k. Thus:

cm ¼ k 

Z

m

_ Pðx= l  wx Þðw_ x Þdx;

8m 2 N :

ð18Þ

0

Since the upper envelope occurs at the points where n = m for each curve pm, the actual prices are given by p(m) = pm(m), where pm(m) is given by (15b), with cm given by (18); i.e.:

pðmÞ ¼ k 

Z

m

_ Pðx= l  wx Þðw_ x Þdx  Pðm=l  wm Þ;

8m 2 N :

ð19Þ

0

These are the prices that maintain equilibrium within N . The last two terms on the RHS are bounded, so there always is a sufficiently large k to guarantee that pðmÞ P 0; 8n 2 N . Note that the cost experienced by user m at equilibrium is the sum of the price and the schedule penalty experienced by the user, which is p(m) + P(m/l  wm) = cm, as can be seen from (18) and (19). Thus, the constants cm express not just the indifference costs, but the actual costs users experience including schedule penalties and tolls. It will be of interest for bimodal problems to know how the user cost cm varies with the user’s wish time wm. This idea can be expressed by introducing the pertinent function c(w) = cm through the change of variable w = wm,m = W(w) and noting

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: _ m . Now, introducing this relation in that c_ ðwÞ¼dcðwÞ=dw ¼ dcm =dm  dm=dw ¼ ðdcm =dmÞ=w _ _ m  wm Þ, which in terms of the new independent variable, w, becomes: c_ ðwÞ ¼ Pðm= l  wm Þ ¼ Pðt

_  wÞ; c_ ðwÞ ¼ PðtðWðwÞÞ

for any w in a capacitated episode:

(17)

we

find

ð20Þ

These results are now extended to the complete set of positions, including the complement of N ; N . It is shown that the 00 extension {p(m) = p(n0 ) if m < n0 ; p(m) = p(n ) if m > n00 } defines an equilibrium for all m. This extension sets the price function as a continuous curve with zero derivative outside the capacitated intervals; and since outside these intervals there is no schedule delay, it also sets the user cost as a continuous function with zero derivative in these intervals; i.e.:

c_ ðwÞ ¼ 0;

for any w not in a capacitated episode:

ð21Þ

To see that these statements are true note that the choice of prices in N discourages swaps from N to N because each of these swaps costs more than a switch to the intervening end point, which being part of N cannot reduce cost. Likewise, there is no incentive to switch from N to N because the cost for any m 2 N is the same as that of the closest end point of N (both positions exhibit on time passage times and have the same price), but the schedule penalty for switching into N is greater for m. Now, since a switch from the end point into N cannot decrease cost, neither can the more costly switch from m. Finally, we note that switches within N cannot reduce cost either: this is true for switches that remain on the same side of N because then the price stays the same but the schedule penalty increases; and it is also true for switches to the far side of N because these switches are even more costly than switches to the far end point, which cannot reduce cost as they are switches from N to N . Thus, switches of no type can reduce cost. As claimed, the chosen prices are an equilibrium for all m. The above establishes that equilibrium prices exist for any FWFP-CMP passage pattern that consists of a single capacitated episode. The reader can verify that prices also exist for any sequence of capacitated episodes. To do this use the above recipe for each episode, choosing the constants k to ensure that the two prices calculated for each sub-capacity interval from its two neighboring capacitated episodes match. Then, for the reasons given in the previous paragraph, users will have no incentive to leapfrog across intervals, or to switch positions within theirs. 6.2. Optimal prices and costs with bilinear penalties This subsection derives expressions for the price and cost curves in the bilinear case. This is necessary because for punc_ is not defined. tual passages Pð0Þ In intervals of unpunctuality our formulas can be used. In this case, P_ ¼ L if there is lateness and P_ ¼ e if there is earliness. Making this substitution in (19) we find that the price of user m is p(m) = k  P(m/l) = p(m) = k  P(tm). The price only depends on the user’s passage time relative to the origin, which is the point when earliness changes into lateness. Thus, this is the only feature of the wish curve that needs to be determined in order to set prices. In intervals of punctuality (whether passages are at capacity or not) the prices of Section 6.1 are constant. It is shown below that in the bilinear case these prices can vary with time, provided that dp/dt 2 [L, e]. As in the differentiable case the overall price function must be continuous, and this condition can be satisfied by properly choosing the values of the constants k associated with each capacitated interval. The reason for being able to choose dp/dt 2 [L, e] is that with such choice nobody can win by switching departure times. People passing in a punctuality interval, where t = w, would experience a non-negative cost change for any perturbation  in their passage time, since the cost change is ðL þ p_ Þ P 0 if  P 0 and ðe þ p_ Þ P 0 if  6 0. Thus, these people would stay put. Now, for people with departure (and wish) times before the punctuality interval cost increases with t at a rate L þ p_ P 0 in the punctuality interval. Thus, they have no incentive to depart in this interval and for the reasons given in Section 6.1, they would stay put. The same can be said of people who wish to depart after the interval since their cost decreases with t at a rate e þ p_ 6 0 in the unpunctuality interval. Thus, nobody can win by switching departure times. Let us now examine the parallel relations to (20) and (21). We find from (20) that:

c_ ðwÞ ¼ L; ¼ e;

if there is lateness so that T < W;

if there is earliness so that T > W:

ð22aÞ ð22bÞ

And, of course, in any punctuality interval c = p. Thus, c˙(w) = dp/dt so that:

c_ ðwÞ 2 ½L; e;

if there is punctuality so that T ¼ W:

ð23Þ

As before, c(w) is continuous. 7. The two-mode, day-long problem: basic results The two-mode day-long problem with separable penalties is examined over the next three sections. This section will present some basic results, Section 8 will then discuss pricing and Section 9 the solution for a case that can be solved approximately. The basic results are presented in four parts: Section 7.1 shows that conditional on the modal split distribution the SO optimization problem decomposes into four simple single-mode single-period problems, and then proposes an approximate

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solution method that exploits this decomposition; Section 7.2 shows that the decomposition result does not hold if one only conditions on NT; Section 7.3 examines the special case with concentrated demand; and Section 7.4 the case with distributed demand, albeit only for the single-period problem. 7.1. Conditional decomposition and approximate solution method Unlike with a single mode, the problem with two modes does not decompose by period. However, if one conditions on the modal split functions, DA and DT, one effectively changes the problem into two single-mode problems so decomposition is possible. This is now demonstrated by reexamining the pertinent optimization problem; i.e., the separable penalty version of (1)– (6). For separable penalties the integral in the objective function (1) is given by (8), so the pertinent objective function is:

Z ¼ zc ðN  NT Þ þ Z T ðNT Þ þ

Z

dt dw½Pðt  wÞ þ P0 ðt 0  w0 ÞdT þ

Z

dt dw½Pðt  wÞ þ P0 ðt 0  w0 ÞdA :

ð24Þ

The pertinent constraints continue to be (2)–(6). Now, inspection of (24) and (2)–(6) reveals that conditional on a modal split (DA, DT) that satisfies (2b) and (4), and therefore fixes NT feasibly as per (6), the problem decomposes by mode. This is true because the two decision functions dT and dA appear in separate terms of the objective function (24), and never together in the same constraint. Thus, the problem can be conditionally decomposed by mode into two single-mode problems of the type studied in Sections 5 and 6. Recall, these decomposed by period. Therefore, conditional on the modal split function, the problem decomposes into four elementary single-mode, single-period scheduling problems. This decomposition allows us both to characterize the solution with a corollary of Result 2, and then to find it approximately. The corollary follows: Corollary 2 (Existence of a FWFP-CMP-SO for two-mode, day-long problems). Every two-mode, day-long problem has a FWFP-CMP-SO.

Proof. Since a FWFP-SO always exists (Result 1), we consider any such solution and its (optimal) modal split. Clearly, any optimum solution to the problem conditional on this modal split will also be a SO. However, conditional on the modal split, the problem decomposes into four independent single-mode, single period problems for which there always are FWFP-CMPSO solutions. Since these solutions collectively solve the conditional problem, the collective solution is a SO. And since the collective solution is FWFP-CMP, it is a FWFP-CMP-SO. h The above establishes that finding a SO is relatively easy if an optimum modal split function is given. However, finding an optimum split function is generally difficult. Fortunately, approximations can be easily found. As in the calculus of variations, it is possible to obtain very good solutions by searching for the best among a broadly defined family of splits, defined parametrically in order to reduce the problem to a conventional optimization problem (for the parameters). In some cases, this type of solution can be obtained analytically. For example, if D is uniform in a rectangle R of the (w, w0 )-plane, we could specify that the transit (or auto) density should be itself uniform in an equally oriented rectangle, RT # R. Five-parameters then suffice to identify a member of the family: the first four, p1 . . . p4, could be used to define two opposite corners of RT, and the fifth the density’s value, or even better the number of transit users, p5 = NT. It is then easy to write feasibility conditions for the vector p based on the constraints. In addition, since the conditional problem decomposes four-ways, one can solve the four scheduling problems associated with each modal split, p, to obtain the conditional optimum: Z⁄(p). If the penalty functions are bilinear the conditional solutions are piecewise linear and can be expressed analytically. Thus, Z⁄(p) can be minimized relatively easily. This will be illustrated in Section 9. 7.2. Non-decomposition of the single-period solutions conditional on NT Although the SO problem decomposes conditional of the modal split function, the same is not true if one only conditions on NT. This will be proven by showing that there are feasible bimodal solutions with the same NT for the morning and evening periods that cannot be pieced together to form a feasible solution for the day-long problem; i.e., that the two single-period solutions are mutually incompatible. Consider the following example: D(w, w0 ) is uniform with density 1 in the unit square with corners at the origin and point _ _ 0 ðtÞ ¼ 1 if t 2 [0, 1]. Assume now that the modal split is NT = 0.5, and for each of the single-period (1, 1), so N = 1 and WðtÞ ¼W problems (in the morning and evening) the first half of the time interval [0, 1] includes only transit wishers and the second _ T ðtÞ ¼ W _ 0 ðtÞ ¼ 1 if t 2 [0, 0.5) and W _ T ðtÞ ¼ W _ 0 ðtÞ ¼ 0 if t 2 (0.5, 1]. Clearly, these wish densihalf only auto wishers. Thus, W T T _ T ðtÞ 6 WðtÞ _ _ 0 ðtÞ 6 W _ 0 ðtÞ. Thus, they are feasible for the single-period problems. and W ties satisfy for all t: W T T Yet, a joint density that will generate both marginals does not exist. To see this note that the transit evening density (with 0.5 people wishing to travel in the first 0.5 time units and nobody in the second 0.5) can only arise if everyone in the

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‘‘bottom’’ half of the square travels by transit, i.e. DT(w, w0 ) = 1 if w0 < 0.5, and everyone on the top by car, i.e. DT(w, w0 ) = 0 if _ w0 > 0.5. However, this would imply that the morning transit density was uniform, WðwÞ ¼ 0:5; 8w, and therefore different from the marginal density of the morning solution. Thus, the morning and evening solutions are incompatible. These incompatibility issues do not arise if demand is concentrated. In that case, conditioning on NT is akin to conditioning on the modal split function, and the problem decomposes as in Section 7.1. Closed form formulas are developed below using this property. 7.3. Two-mode, bilinear day-long problems with concentrated demand   If the demand is concentrated at a point so that everyone has the same i.e., Dðw; w0 Þ ¼ NDðw  wo ÞD w0  w0o ,  0 wishes,  0 0 then the modal split function is of the form DT ðw; w Þ ¼ N T Dðw  wo ÞD w  wo , so that it is fully defined by NT. Thus, if we condition on NT the problem must decompose as stipulated in Section 7.1 into four single-period, single-mode problems. As in Section 7.1, this allows us to characterize the SO solution with another corollary of Result 2, and to streamline the solution method. Corollary 3 (Existence, Necessity and Uniqueness of a FWFP-CMP-SO for two-mode, day-long problems with concentrated demand). Result 2 holds for two-mode, day-long problems with concentrated demand and a given NT. Proof. For given NT the problem decomposes into four single-mode, single-period problems for which Result 2 holds. The logic then is the same as for Corollary 1: since Result 1 holds for the individual solutions it also holds for the collective solution. h The four individual solutions that can be pieced together to form the collective solution are very easy to find because the demand is concentrated. For bilinear penalty functions with earliness and lateness rates equal to (e, L, e0 , L0 ) the solution’s curves are piecewise linear and the system optimum schedule penalty can be expressed in closed form. The formula is:

N2T

lT

þ

ðN  NT Þ2

!

l

!   eL N2T ðN  NT Þ2 e0 L0 þ ; þ 0 2ðe þ LÞ l0T l0 2ðe0 þ L Þ

ð25Þ

where the first term corresponds to the morning and the second term to the evening. Both terms include a penalty for transit and a penalty for auto. Expressions such as (25) for the optimum penalty can be combined with the operating cost, zc(N  NT) + ZT(NT), and then minimized with respect to NT to figure out the optimum modal split. If desired one can include capital costs into the objective function, which would depend on the capacities, in order to obtain the optimum capacities too. Note that the single period problem only includes one of the terms (25) and that the optimum schedule penalty both for the day-long and single period problems is of the form

c1 N2T þ c2 ðN  NT Þ2 ;

ð26Þ

with suitable choices of the constants c1 and c2. Given the similarity of these two problems we should expect the solution to the day-long problem to be qualitatively similar to the conventional single-period solution. For example, it turns out, as it is easy to see from the geometry of the queuing diagrams leading to (25), that in all the problems both modes should be at capacity when used. This is true regardless of NT. If NT turns out to be very low, e.g., because the provision of transit is expensive, the result simply means that transit trips should be focused in the middle of the rush and not last very long. This makes sense, since the middle of the rush is the time when shifting users from auto to transit reduces the total schedule penalty by the greatest amount. So this is the system optimum thing to do. 7.4. Two-mode, bilinear single-period problems with distributed demand Although the bimodal day-long SO problem with distributed demand is complicated, its single-period version can be simplified. It is shown below that if the penalty function is bilinear then the two-mode, single-period problem can be cast as a calculus of variations problem. For the same reasons introduced in Section 5.1, the objective function in the bilinear case can be rewritten as a single integral over time involving the functions WT, WA, TT and TA. Since the constraints only involve these functions and their derivatives, the optimization problem becomes indeed a standard calculus of variations problem. Its objective function is:

Z

½PðW A ðtÞ  T A ðtÞÞ þ PðW T ðtÞ  T T ðtÞÞdt;

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and the constraints are:

ðW A ; W_ A ; W T ; W_ T ; T T ; T_T ; T A ; T_A Þ P 0; W A þ W T ¼ W;

8t;

8t;

T T ð1Þ ¼ T A ð1Þ ¼ 0; W T ð1Þ ¼ T T ð1Þ ¼ NT ; W A ð1Þ ¼ T A ð1Þ ¼ N  NT ; T_ T ðtÞ 6 l ; 8t; T

T_ A ðtÞ 6 l;

8t;

This problem can be solved with standard numerical methods. For uniform wish densities, i.e., Z-shaped wish curves, the solution can be obtained analytically. Formulas are not given because they only generalize slightly those given in Gonzales and Daganzo (2012a), which apply for infinite transit capacity, and because more general (day-long) formulas will be presented in Section 9.

8. The two-mode, day-long problem: pricing This section extends the SO and pricing results of Section 6 to two-modes. It is assumed that the penalty function is either continuously differentiable and strictly convex or bilinear. The extension is not trivial because the equilibrium prices must now dissuade users from switching not just passage times but also modes. Section 8.1 first derives a sufficient condition for a day-long passage pattern to be achievable with pricing; i.e., to be ‘‘priceable’’. Section 8.2 then shows that a priceable SO pattern always exists if P is either differentiable and strictly convex or bilinear. Thus, in these cases prices that achieve the SO exist. 8.1. The priceability condition A sufficient condition for a priced day-long equilibrium is now derived. We must guarantee that users do not benefit by switching their two passage times, and the mode they use for both periods. However, since we are dealing with sufficiency we can make the condition more restrictive. It will be convenient to stipulate that users should not be able to benefit by switching both the passage time and the mode in either period. This stricter condition preserves an equilibrium even if users could use a different mode for the morning and the evening. It has the advantage that it can be applied to the morning and the evening separately. Using separability, we now derive priceability conditions for the morning curves. This suffices since the same ideas apply to the evening. For (morning) equilibrium, each of the two modes must be in temporal equilibrium, i.e. their prices and user costs must satisfy either ((18) and (19)) or the equivalent relations for bilinear penalties, and users must be discouraged from switching modes. This must happen for everyone regardless of their wished passage time w. To express this idea mathematically we will use subscripts ‘‘.’’, T and A to identify the functions corresponding to each mode, and the parameter zt to denote the user cost of a (one-way) punctual transit trip without pricing. A sufficient condition for (morning) equilibrium is then: ((18) and (19)) satisfied for both modes and cT(w) + zt = cA(w) + zc, "w. Now, if the initial prices at w = 1 are set so that this last condition is satisfied then the sufficient condition can be reduced to: c˙T(w) = c˙A(w),"w. Eqs. ((20)–(24)) can now be used. For continuously differentiable and strictly convex penalty functions, (20) implies that the equilibrium condition at _ T ðW T ðwÞÞ  wÞ ¼ Pðt _ A ðW A ðwÞÞ  wÞ; 8w. Since P_ is times w when both modes are in capacitated intervals is Pðt continuous and monotonically decreasing, this equality is satisfied if and only if the arguments of P_ are equal; i.e., if and only if users that pass at the same time by either mode have the same wish times and the same earliness or lateness. This sufficient condition implies that the crossing points of the capacitated episodes (i.e. when there is no schedule delay) have to be synchronized across modes. Thus, the punctuality intervals must coincide for both modes. In summary, the sufficient condition for priceability is that at every passage time t the cumulative wish and passage curves for each mode must either coincide for both (with TA = WA and TT = WT) or exhibit the same earliness/lateness. In the bilinear case, consideration of (22) and (23) shows that the earliness/lateness amounts don’t have to match but there still is a synchronicity requirement. Note from the formulas that if one of the two users with a given wish time w is early then the other cannot be late, and vice versa. Graphically this means that there can be no times when one wish curve is strictly above its corresponding passage curve (so there is lateness for that mode) while the other is strictly below (so there is earliness). Earliness must be synchronous with earliness (or punctuality), and so must lateness. As in the strictly convex case, the overall pattern is one where episodes without lateness for either mode are separated from episodes without earliness for either mode by time intervals (or instants) where both modes are punctual. We are now ready to look into the priceability of SO passage patterns.

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8.2. System optimum prices This subsection shows that every FWFP-SO pattern with either bilinear or continuously differentiable and strictly convex penalty functions satisfies these priceability properties; i.e., that every FWFP-SO is priceable. Since a FWFP-SO always exists (Result 1), this means that a SO can always be achieved with pricing. Result 3. Every FWFP-SO with a bilinear or continuously differentiable and strictly convex penalty function is priceable.

Proof. The proof is by contradiction. This is done for bilinear penalties first and strictly convex penalties second. In each case it will be shown that a contradiction arises if the pertinent synchronicity conditions are violated.

Fig. 5. Customer swap between modes 1 and 2; (a) bilinear case; (b) strictly convex case.

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If the proposition is false in the bilinear case there must be a bilinear SO where at some time t0 one of the modes (labeled ‘‘1’’) is late so that W1(t0) > T1(t0), and the other one (labeled ‘‘2’’) is early so that W2(t0) < T2(t0). An example where this occurs is shown in Fig. 5a. Now define w1 as the latest time before t0 at which W1 = T1. This is shown by point A of the figure. Likewise, define w2 as the earliest time after t0 at which W2 = T2, as shown by point B. Points A and B are significant because they must occur at a location where the wish curves increase. Therefore, we can swap a differential number of customers (dn) from mode 1 and wish times around w1 with an equal number of customers from mode 2 and wish times around w2. This swap lowers W1 by dn units in the interval (w1,w2), and raises W2 by the same amount in the same interval; see the figure. We then reassign the passage times of all the users with wish times between w1 and w2 to leave the passage curves invariant and still in FWFP order, as shown in the figure. Because of the way the curves are shifted the following is always true: (i) before (after) t0, there must be times when both the earliness of one mode and the lateness of the other decrease, and (ii) at all other times the increase in lateness (earliness) is exactly compensated by a decrease in lateness (earliness). Note how this happens in the figure. Obviously then, for bilinear penalties the shift induces a net decrease in the total schedule penalty. This indicates that the original curves cannot be part of a SO, which is a contradiction. Thus, the proposition holds in the bilinear case. Consider now the strictly convex case. The argument is qualitatively similar. If the proposition is false there must be a SO solution in which a positive number of users passing with the two modes in the neighborhood of some time t0 have different wish times w1 (for mode ‘‘1’’) and w2 (for mode ‘‘2’’). We assume without loss of generality that w1 < w2 so that mode 1 has the greater lateness. The situation is illustrated in Fig. 5b. (Note we use the convention that lateness can be positive or negative; hence smaller earliness is a special case of greater lateness.) To prove the result it suffices to show that the total schedule penalty can be reduced by a user swap between modes. Consider a swap that changes the mode used by dn people from mode 1 with wished time immediately after w1 and dn people from mode 2 with wished passage time immediately before w2. As illustrated by Fig. 5b, this lowers the W1-curve by dn units in the interval (w1, w2) and raises the W2-curve by the same amount in the same interval. As part of the swap, we also reassign the passage times of all users with wish time in (w1, w2) to leave both passage curves invariant and in FWFP order, as shown by the figure. Now note that for every wish time differential, dw, the increase in lateness of mode 2 (dn dw) is matched by an equal decrease of mode 1. However, the penalty savings due to the lateness decrease are always strictly greater than the penalty increase due to the lateness increase. This is true because the penalty function is strictly convex and the base lateness from which the penalty change is computed for each mode is always greater for mode 1 (decreasing lateness) than for mode 2 (increasing lateness). To see this refer to the figure and note that all affected users, i.e. with w 2 (w1, w2), pass after t0 with mode 1 and before t0 with mode 2. Hence the lateness of the former must always be greater than that of the latter. This is always true by construction. Thus, the net effect of the two shifts is a decrease in the total schedule penalty. This is a contradiction, which proves the result. h

9. The two-mode, day-long problem: approximate solution for bilinear penalties with uniform demand This section applies the approximate solution method of Section 7.1 to the two-mode, day-long problem, assuming that the penalty function is bilinear and the demand is uniformly distributed. Section 9.1 develops the solution, and Section 9.2 provides formulas and interpretations for the special case where the morning and evening parameters are the same. 9.1. Approximate solution It is assumed in this subsection that P and P0 are bilinear, and that the wished passage times of the N travelers are uni_ ¼ N=R and formly distributed in a R  R0 rectangle, R. Therefore, the marginal densities for the morning and the evening, W _ 0 ¼ N=R0 , are themselves uniform. They shall be relabeled k and k0 for conciseness of notation. We also assume that demand W is high enough to congest the auto system; i.e. that k P l and k0 P l0 . As suggested at the end of Section 7.1, we shall restrict ourselves to looking for uniform modal splits; i.e. for splits where the bivariate wish density for one of the modes is itself uniform in a rectangular subset of R. We also assume without loss of generality that this mode is the transit mode, since the results for car can be obtained by switching subscripts. Accordingly, the rectangular subset is denoted RT  R, and its sides: RT and R0T . We further restrict ourselves to looking for solutions where the capacitated episodes in each period have only two stages: earliness followed by lateness. We also assume that the times in the morning and the evening where earliness changes into lateness must be synchronous across modes as this ensures priceability. These times shall be denoted: t0 and t 00 . These additional restrictions seem reasonable for the uniform demand and transit densities of our special case. Now condition the solution on t0 ; t 00 ; N T and RT to decompose the optimization problem by period and by mode. The four solutions can then be analyzed with cumulative plots of commuter number vs time and expressed analytically. By doing this, the problem can be solved in phases. The following phases will be used: (i) conditional on NT, RT and R0T , find the optimum location of RT , as well as the optimum t0 and t00 ; (ii) using the solution of (i) and conditional on NT, find the optimum dimensions of RT : RT and R0T ; and

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(iii) using the solution of (ii) find the optimum NT. This approach works because the solutions in steps (i) and (ii) can be obtained in closed form. Subproblem (i) is now addressed. Because NT is fixed, the agency and car costs are fixed. Thus, the optimum is obtained by minimizing schedule delay. Geometric considerations of the type used in the single-mode, single-period problem reveal that this optimum is such that the ratio of customers who are early over those who are late is L/e in the morning and L0 /e0 in the evening. This is true for each individual mode and for both modes combined. (The reader is encouraged to verify these facts.) These properties uniquely identify all the curves in the four cumulative plots, and therefore yield the optimum schedule penalties for the given NT, RT and R0T . The formulas for the morning period turn out to be:

#  " 2   1 eL N  NT kl N2T ðmorning-autoÞ  R þ RT N T 2 eþL l kl kl    þ 1 eL 1 RT N2T ðmorning-transitÞ  ST ðRT jNT Þ ¼ 2 eþL lT NT

SA ðRT jNT Þ ¼

ð27Þ ð28Þ

where the ‘‘+’’ superscript denotes the positive part. We also define the combined, bi-modal schedule penalty function for the morning as: S(RTjNT) = SA(RTjNT) + ST(RTjNT). These formulas only apply to the case where the slopes of the wish curve for cars equal or exceed l at all times; i.e., if N/ R  NT/RT P l. This is not a problem because the inequality is a necessary and sufficient condition for the capacitated episode to have only two stages, as we are assuming. The inequality must be included as a constraint, however, together with the restriction that R P RT. The combination of these two inequality constraints can be put in the form:

R P RT P

NT ; N=R  l

where N T 6 N  lR;

ðmorningÞ

ð29Þ

The condition NT 6 N  lR ensures that the RHS of (29) is no greater than its LHS; i.e., that there is a feasible value of RT. Values of NT violating the condition do not have to be analyzed because such high values reduce the number of auto drivers so much that some road capacity goes unused. The violating NT values cannot be optimal. Schedule penalty formulas similar to (27) and (28), with primes where appropriate, and a corresponding constraint hold for the evening. The evening constraint is:

R0 P R0T P

NT ; N=R0  l0

where N T 6 N  l0 R0 ;

ðeveningÞ

ð30Þ

The schedule penalty formulas are not written here for brevity, but the day-long schedule penalty, SD, has the closed form:     SD RT ; R0T jN T ¼ SðRT jN T Þ þ S0 R0T jN T .   Subproblem (ii) can now be addressed. Since NT is fixed, we just need to look for the minimum of SD RT ; R0T jN T across RT 0 and RT . These decision variables  are restricted by (29) and (30). In addition, we must ensure that the available number of commuters in RT , which is N RT R0T =ðRR0 Þ, exceeds the stipulated number of transit commuters. This stipulation leads to the constraint1:

RT R0T P NT RR0 =N:

ð31Þ

Thus, optimization problem (ii) is: minimize SD ðRT ; R0T jN T Þ subject to ((29)–(31)). This is a convex optimization problem because the feasible region defined by these three constraints is convex, and as can be seen from (27) and (28), the objective function is also convex. A solution can be obtained in closed form because the component functions of this objective function are non-decreasing: constant for RT 6 NT/lT and R0T 6 N T =l0T , and linearly increasing for greater values of these arguments. Thus, the objective function is minimized by choosing its smallest possible arguments. The closed form solution is not particularly insightful because it involves six different cases, which depend on the relative values of the different parameters. Thus, it is not given here. Instead, the following special case is analyzed. 9.2. The case with morning/evening symmetry A closed form solution is presented for the symmetric case where the morning and evening parameters are the same; i.e.:

l ¼ l0 ; lT ¼ l0T and R = R0 . This case involves fewer parameters, is still interesting and can be simplified considerably. Since the optimization problem is convex, we know that there is a symmetric optimum with R0T ¼ RT so that constraint (30) becomes redundant. The remaining constraints, (29) and (31), simplify to

R P RT P max



pffiffiffiffiffiffiffiffiffiffiffiffi NT ; R NT =N ; N=R  l

where NT 6 N  lR;

and the objective function reduces to 2(SA + ST), as given by (27) and (28). 1

This constraint links the morning and evening wish rates, and is the reason why the morning and evening solutions cannot always be composed.

ð32Þ

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Recall that 2(SA + ST) is constant for RT 2 [0, NT/lT], and it increases thereafter. Thus, if the upper range value of this interval n pffiffiffiffiffiffiffiffiffiffiffiffio

NT ; R N T =N then RT should be as large as possible; i.e., RT ¼ min R; N T =lT is optimal. N=Rl

satisfies (32), i.e., N T =lT P max

Otherwise, the RHS of (32) is optimal. These two statements can be combined as:



NT ; RT ¼ max min R;

lT

 pffiffiffiffiffiffiffiffiffiffiffiffi NT ; R NT =N 6 R; N=R  l

where NT 6 N  lR:

ð33Þ

In many cases (e.g., when NT is small and/or lT is large) the optimum period where transit should be used is strictly smaller than R. This means that it is best if transit trips are not spread out over the rush, but rather concentrated in the middle of both rushes because this is the time where they do the most good. The optimum value of the objective function also has a closed form. It depends on whether or not the transit and auto : systems combined have enough capacity to serve all the demand without delay; i.e., on whether l + lT P k ¼ N/R. This inequality determines which of the first two terms on the RHS of (33) is smaller, and therefore can be dropped from (33). Inserting these simplified versions of (33) into the objective function we find:

" " #þ # 3=2 eL ðN  NT  lRÞ2 N=l RNT RN 2T S ðNT Þ ¼ þ  if l þ lT P N=R eþL N  Rl N  lR N1=2 " " #þ # eL ðN  NT  lRÞ2 N=l  RN 2T N2T RN3=2 N2T T if l þ lT 6 N=R: þ þ  ¼ eþL N  Rl lT lT N1=2 D

ð34aÞ

ð34bÞ

Both expressions apply only for the relevant range of NT:NT 6 N  lR. The above concludes subproblem (ii). So far, by fixing NT the optimization could be focused on schedule delay. In subproblem (iii), however, NT is a variable. Thus, we must minimize the total cost to society, which is: SD(NT) + zc(N  NT) + ZT(NT). Since we have only one decision variable, finding the best NT is easy. This minimization is easy even if we allow the transit capacities lT and l0T to be decision variables, in addition to NT. In this case, we would just recognize explicitly that the expressions for SD and ZT in the objective function depend on the capacities, and proceed accordingly. The results of subproblem (ii), and (34) in particular, have interesting properties and connections with previously developed results. This is now explored. The first feature of interest is accuracy. The obtained solution is not necessarily optimal because, at least in theory, there can be non-rectangular domains for the transit demand (e.g., cross-shaped) that may yield an even lower cost than we obtained with our rectangular modal split family. However, given that our family already has 5 degrees of freedom it is doubtful that any further expansion in these degrees of freedom would yield much of an improvement. High accuracy is also suggested by the method’s excellent performance for problems where constraint (31), which links the two periods, is not binding; i.e., when the []+ terms in (34) vanish. When this happens, the exact SO cost must be exactly twice the cost of the corresponding single-period problem. Reassuringly, this is the prediction of (34a) when it is applied to the day-long version of the single-period problem analyzed in Gonzales and Daganzo (2012a).2 Thus, all indications are that the method is very accurate. The second feature of interest is the consistency of (34) with previously demonstrated decomposition results. As we have just mentioned, if our solution decomposes by period, then the linking terms []+ of (34) must vanish. Note how this happens in both (34a) and (34b) if the demand is concentrated, R = 0. This is consistent with Section 7.3, which showed that any daylong solution for concentrated demand is a combination of two single period solutions. The linking term also vanishes if NT = 0. And this is consistent with Section 5.1, which showed that for a single mode the day-long problem decomposes by period conditional on NT. Of course, as pointed out in Section 7.2, there are cases when the combined solution is infeasible and in those cases the []+ term in (34) is positive. Thus, we conclude by illustrating the considerable effect that the linking constraint can have on the solution.3 For example if N = 105, NT = 0.15  105, R = 1 h, l = 0.5  105 and lT = 0.13  105, as could happen in the center of a large city, then the linking term is about 40% of SD. This shows that the day long commute for a given NT should not be generally analyzed by examining the two peaks in isolation, as the errors can be quite significant. 10. Discussion This paper has examined the day-long commute problem with two modes and distributed demand. Distributed demand is important for the modeling of cities, as it is a way to represent the geographical distribution of origins and destinations. Given are a number of commuters and their joint distribution of wished travel (passage) times for the morning and the evening. The objective is finding a system optimum travel arrangement, which minimizes society’s cost. The SO arrangement identifies who should travel by what mode and when. 2

Note the Gonzales and Daganzo (2012a) only analyzes the case with lT = 1. However, since (34a) is independent of lT, it applies for lT = 1. The effect would be slightly overstated if our solution turned out to be suboptimal; i.e., if some unconsidered modal split assignment patterns turned out to be better than those in our family. 3

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C.F. Daganzo / Transportation Research Part B 55 (2013) 98–117 Table 1 Summary of results: system optimum with separable penalties for a given NT. Problem properties

SO without queuing exists; is given by convex program A first wish first pass (FWFP) SO exists A FWFP-SO with CMP passage curves exists SO passage curves are unique SO is unique SO is achievable with pricing The FWFP-CMP-SO is easily calculable Decomposes by period conditional on modal split function Decomposes by period Decomposes by period and mode

Single mode

Two mode

Single period

Day long

Single period

Day long

g/3 y/4 y/5.2 b, c/5.2 c/5.2 y/6 y/5.1 n/a n/a n/a

g/3 y/4 y/5.2 b, c/5.2 c/5.2 y/6 y/5.1 y/7.1 y/5.1 n/a

g/3 y/4 y/7.1 b0 , c0 /7.3 c0 /7.3 b, c/8.2 b, y0 /7.4, 3 n/a n/a y0 /7.3

g/3 y/4 y/7.1 b0 , c0 /7.3 c0 /7.3 b, c/8.2 y0 /7.3 y/7.1 y0 /7.3 y0 /7.3

g = True, even for non-separable penalties; y = true; y0 = true if demand is concentrated. b = True if penalty is bilinear; b0 = true if penalty is bilinear and demand is concentrated. c = True if penalty is strictly convex; n/a = not applicable. c0 = True if penalty is strictly convex and demand is concentrated. Numbers following the slashes denote sections where the results are covered.

The paper unveils conditions under which SO solutions without queuing and with various other properties exist and are unique. Of interest are the FWFP property, whereby users pass the bottlenecks in first wish first pass order, and the CMP property, which pertains to cumulative passage curves that have capacity slopes when not matching the cumulative wish curve. Other properties are whether the SO pattern can be achieved with pricing (knowing and not knowing the demand), and whether it is easily calculated. Because the conditions are varied, the results are best collectively understood if organized in a table. Table 1 is an attempt at this organization. It is the main contribution of this paper. The paper also proposes an approximate solution method for most complex (two-mode, day-long) problems and illustrates it for the case with bilinear penalties and uniform demand. Using this scenario, it is shown that the optimum day-long cost can be considerably larger than the sum of the optimum costs for the two corresponding single period problems. Thus, it is in general necessary to solve the day-long problem in an integrated way; costs estimated by piecing together single-period theory results can significantly underestimate the real costs. The results of this paper can be generalized in several ways; e.g.: allowing for heterogeneous users with different values of time and affinity for transit and cars; modeling SO patterns that discriminate across users with different trip lengths; and allowing for transit service to be canceled when it is not used. The challenge is adding realism without clouding insight. Acknowledgements Research supported by NSF Grant CMMI – 1161427 and by the U.C. Berkeley Center for Future Urban Transport sponsored by the Volvo Research and Educational Foundations. The author is grateful to Dr. Weihua Gu for his careful comments, which improved readability and helped simplify some of the proofs while removing gaps. References Arnott, R., de Palma, A., Lindsey, R., 1993. A structural model of peak-period congestion: a traffic bottleneck with elastic demand. The American Economic Review 83 (1), 161–179. Braid, R.M., 1996. Peak-load pricing of a transportation route with an unpriced substitute. Journal of Urban Economics 40 (2), 179–197. Daganzo, C.F., 1985. The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science 19 (1), 29–37. Daganzo, C.F., 2007. Urban gridlock: macroscopic modeling and mitigation approaches. Transportation Research Part B 41 (1), 49–62. Danielis, R., Marcucci, E., 2002. Bottleneck road congestion pricing with a competing railroad service. Transportation Research Part E 38 (5), 379–388. De Palma, A., Lindsey, R., 2002. Comparison of morning and evening commutes in the Vickrey bottleneck model. Transportation Research Record 1807, 26– 33. Fargier, P.H., 1983. Effects of the choice of departure time on road traffic congestion: theoretical approach. In: Proceedings of the 8th International Symposium on Transportation and Traffic Theory, University of Toronto Press, Toronto, Canada, pp. 223–263. Gonzales, E., Daganzo, C.F., 2012a. Morning commute with competing modes and distributed demand: user equilibrium, system optimum, and pricing. Transportation Research Part B 46 (10), 1519–1534. Gonzales, E., Daganzo, C.F., 2012b. User Equilibrium for the Combined Morning and Evening Peaks with Cars and Transit. Working Paper Draft. Hendrickson, C., Kocur, G., 1981. Schedule delay and departure time decisions in a deterministic model. Transportation Science 15 (1), 62–77. Huang, H.J., 2000. Fares and tolls in a competitive system with transit and highway: the case with two groups of commuters. Transportation Research Part E 36 (4), 267–284. Qian, Z., Zhang, H.M., 2011. Modeling multi-modal morning commute in a one-to-one corridor network. Transportation Research Part C 19 (2), 254–269. Smith, M.J., 1984. The existence of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science 18 (4), 385–394. Tabuchi, T., 1993. Bottleneck congestion and modal split. Journal of Urban Economics 34 (3), 414–431. Vickrey, W.S., 1969. Congestion theory and transport investment. The American Economic Review 59 (2), 251–260. Vickrey, W.S., 1973. Pricing, metering, and efficiently using urban transportation facilities. Highway Research Record 476, 36–48. Zhang, X., Huang, H.J., Zhang, H.M., 2008. Integrated daily commuting patterns and optimal road tolls and parking fees in a linear city. Transportation Research Part B 42 (1), 38–56. Zhang, X., Yang, H., Huang, H.J., Zhang, H.M., 2005. Integrated scheduling of daily work activities and morning-evening commutes with bottleneck congestion. Transportation Research Part A 39 (1), 41–60.