On Activity-based Network Design Problems

Transcription

1 UCI-ITS-WP-12-3 On Activity-based Network Design Problems UCI-ITS-WP-12-3 Jee Eun Kang Josep Y. J. Cow Will W. Recker Department of Civil Engineering and Institute of Transportation Studies University of California, Irvine; Irvine, CA , U.S.A Institute of Transportation Studies University of California, Irvine Irvine, CA , U.S.A. ttp://

2 On activity-based network design problems Jee Eun Kang a, Josep Y.J. Cow b, Will W. Recker a a Institute of Transportation Studies, University of California, Irvine, CA, 92697, USA b Department of Civil Engineering, Ryerson University, Toronto, ON, M5B 2K3, Canada Abstract Tis paper examines network design were OD demand is not known a priori, but is te subject of responses in ouseold or user itinerary coices tat depend on subject infrastructure improvements. Using simple examples, we sow tat falsely assuming tat ouseold itineraries are not elastic can result in a lack in understanding of certain penomena; e.g., increasing traffic even witout increasing economic activity due to relaxing of space-time prism constraints, or worsening of utility despite infrastructure investments in cases were ouseold objectives may conflict. An activity-based network design problem is proposed using te location routing problem (LRP) as inspiration. Te bilevel formulation includes an upper level network design and sortest pat problem wile te lower level includes a set of disaggregate ouseold itinerary optimization problems, posed as ouseold activity pattern problem (HAPP) (or in te case wit location coice, as generalized HAPP) models. As a bilevel problem wit an NP-ard lower level problem, tere is no algoritm for solving te model exactly. Simple numerical examples sow optimality gaps of as muc as 5% for a decomposition euristic algoritm derived from te LRP. A large numerical case study based on Soutern California data and setting suggest tat even if infrastructure investments do not result in major canges in itineraries te results provide muc iger resolution information to a decision-maker. Wereas a conventional model would output te best set of links to invest given an assumed OD matrix, te proposed model can output te same best set of links, te same OD matrix, and a detailed temporal distribution of activity participation and travel, given a set of desired destinations and scedules. Keywords: activity based model; network design; location routing problem; HAPP; pickup and delivery problem; bi-level problem 1. Background Network design problems (NDPs) are a class of optimization models related to strategic or tactical planning of resources to manage a network (Magnanti and Wong, 1984). In general, NDPs assume static demand (elastic or not) at a node or trip-based origin-destination demand, even for purposes of improving road networks for commuters (Yang and Bell, 1998) despite te complexity of traveler coices (Recker, 2001). Wile tis assumption is sufficient in many applications, tere is increasing recognition tat explicit consideration of travelers scedules, coices, and time dimension is needed. Tis need as grown in parallel to tree related researc trends in network design in te last few years: (operational) network design wit dynamic assignment considerations wen considering only peak period effects, (tactical) service network design wit scedule-based demand under longer periods of activity, and (planning) facility location problems tat explicitly consider te effects tey ave on routing and sceduling of veicles for a decision-maker. At te planning level, tese NDPs ave often been based on private firm decisions, rater tan on ouseold-based urban transportation planning considerations. Te rationale beind dynamic network design problems is rooted in bi-level NDPs tat feature congestion effects. Tese NDPs operate primarily in civil infrastructure systems, as oter types of

3 networks do not generally sare te same selfis travelers assumptions. In tis paradigm, te performance of infrastructure improvements is assumed to depend primarily on te route coices of travelers (primarily te commuter) during peak periods of travel, wic in turn depend on te coices of oter travelers. Te dynamic component furter allows modelers to assess intelligent transportation systems (ITS) tat require more realistic modeling of traffic propagation obeying pysical queuing constraints and information flow. Some examples include te stocastic dynamic NDP from Waller and Ziliaskopoulos (2001), Heydecker s (2002) NDP wit dynamic user equilibrium (DUE), te linear DUE- NDP from Ukkusuri and Waller (2008), dynamic toll pricing problem wit route and departure time coice (Joksimovic et al., 2005), and te reliability maximizing toll pricing problem wit dynamic route and departure time coice (Li et al., 2007). Altoug tese NDPs are especially useful for ITS evaluation and operational strategies, tey focus primarily on coices made over a single trip. Tactical level NDPs tend to place more empasis on time use and sceduling over congestion effects. Tactical service NDPs (Crainic, 2000) are a specific class used to manage fleets of veicles wit suc temporal decision variables as service frequency. However, most of tese NDPs focus on te scedules of te service being provided, rater tan on incorporating te demand-side scedules of te travelers/users as endogenous elements of te design. Despite te incorporation of temporal effects, most service NDPs assume trip-based demand. Tere as been a surge of researc in scedule-based transit assignment (as opposed to NDP), were travelers departure time coices are andled explicitly. Tong and Wong (1998) formulated suc a model wit eterogeneous traveler values of time. Poon et al. (2004) presented a dynamic equilibrium model for scedule based transit assignment. Hamdouc and Lawpongpanic (2008) developed a scedule-based transit assignment model tat accounts for individual veicle capacities, and proposed one of te few scedule-based service network design problems, in te form of a transit congestion pricing problem tat models passengers departure time coices (Hamdouc and Lawpongpanic, 2010). Teir model uses a time-expanded network and considers fare pricing to optimize te distribution of travelers witin specific capacitated transit veicles. Te origin-destination (OD) demand remains fixed, and not as a linked itinerary. Despite aving te greatest need for suc consideration, at te planning level tere are no NDP models tat consider routing and sceduling coices of travelers.. It as long been acknowledged tat models of traveler activities and time use are muc more accurate tan statistical trip-based approaces (Recker, 2001; Pinjari and Bat, 2011). Activity consideration can bring about a tigter integration of infrastructure investment wit land use planning and demand management strategies. Activity-based models can capture realistic impacts on travelers tat are not limited to single trips but rater to cains of trips and activities forming detailed daily itineraries. Historically, te bulk of activity-based models ave been designed primarily as econometric models tat do not account for network routing and sceduling mecanisms. Te emerging trend in seeking to integrate network caracteristics as been to force an interaction wit a dynamic traffic assignment problem (e.g. Lin et al., 2008; Konduri, 2012), wic extends te planning model toward operational applicability. However, tis approac still ignores te network constraints present in sceduling and selection of activities for a ouseold. Tere ave been two primary exceptions to tis approac. Te first is te disaggregate activity route assignment model (HAPP) pioneered by Recker (1995), wit subsequent studies on dynamic resceduling/rerouting of tose itineraries (Gan and Recker, 2008), and calibration of te activity route assignment models (Recker et al., 2008; Cow and Recker, 2012). Te second is te aggregate time-dependent activity-based traffic assignment model (Lam and Yin, 2001). Bot modeling frameworks address te issue of activity sceduling, altoug Lam and Yin s model gives up disaggregate itinerary route coices and trip cains in favor of capturing congestion effects. Altoug te transportation planning field as not seen any significant NDP researc tat models traveler routing and sceduling, te private logistics field as. One suc model is te location routing problem (LRP), formulated and solved by Perl and Daskin (1985). Te LRP is a set of inter-related problems tat includes a facility location problem. Wat distinguises LRPs from oter NDPs is tat it doesn t assume tat demand for te nodes is given in terms of round trips to demand nodes. Instead, a lower level veicle routing problem is embedded in te model to satisfy demand nodes in te most

4 efficient manner, subject to were te facilities are located. In essence, it is an integrated NDP tat accounts for responsive routing and sceduling. Numerous studies ave been conducted in variants of te problem or applications in industry. Several literature reviews ave been publised, including one from Min et al. (1998) and a more recent contribution by Nagy and Sali (2007). Problem types developed over te years tat may be applicable to activity-based network design in transportation planning include: stocastic LRP (Laporte and Dejax, 1989), were tere is more tan one planning orizon and customer locations and demands cange over time; LRP wit a mixed fleet (Wu et al., 2002) for multimodal network consideration; location-routing-inventory (Liu and Lee, 2003) for modeling activity types as inventory-based needs tat are fulfilled periodically; and LRP wit nonlinear costs (Melecovsky et al., 2005) tat may provide means to incorporate congestion effects at link or activity node level. Readers are referred to Nagy and Sali s paper for furter details. One direct application of LRP wit truck fleet replaced by ouseold travelers is sown in Kang and Recker (2012a). Tey use HAPP as a routing subproblem in a ydrogen fuel cell refueling station LRP tat allows ouseolds to respond to located facilities to refuel, wic can reflect te beavioral impacts of siting decisions. Given te increasing realization tat transportation planning needs to reflect travelers preferences at te activity level, we make a parallel observation to Perl and Daskin tat in te transportation planning field tere is also a need for integrated NDPs tat feature explicit consideration of travelers tour patterns including trip caining, sceduling, time windows and even destination coice. At te activity-based level, we are concerned more wit tactical and planning level policies, and less so wit suc operational tecnologies as ITS and information flow (ence foregoing congestion effects for now). In essence, we propose to cange te conventional NDP, wit a given OD matrix, to a new class of activity-based NDPs. Tis new problem accounts for a population of travelers wit demand for activities at particular locations and at particular times, wic are fulfilled via calibrated activity routing models. Like te LRP, te activity-based NDP is a set of integrated models. Unlike te conventional NDP, te OD matrix is not given a priori, since it depends on te sceduling coices of ouseolds wic in turn depend on travel impedances. Te solution of tis set of models is a set of infrastructure link investments as well as te resulting optimal itineraries decided by te ouseolds in response to te canges. Te itineraries can ten be aggregated to obtain te final OD matrix resulting from te NDP. In Section 2, several examples and insigtful paradoxes are used to illustrate wy an activitybased approac is necessary at te tactical and planning level NDP. Section 3 introduces te formulation as a bi-level structure wit sortest pat allocation and disaggregated subproblems per ouseold. Wile te inspiration of te formulation is from Perl and Daskin s LRP, key differences are also noted. An alternative model wit activity/destination coice is also provided. Since te problem is nonconvex and NP-ard, Section 4 presents a euristic solution metod and suggestions for meta-euristics, using a simple test network to demonstrate te metod and te sensitivity of underlying assumptions. Section 5 presents a larger-scale case study of te Orange County, California region as a test network to demonstrate te model s practical application to systematic improvement. 2. Motivating Examples Te argument tat we provide ere, muc like Perl and Daskin (1985) did for locating wareouses, is tat te coice of wic element of a network to improve can ave a significant impact on ow impacted ouseolds set teir itineraries eac day. Trip-based (even dynamic ones) or fixed scedules ignore canges tat eac driver/ouseold makes according to te canges made in te network, suc as departure time, sequence of activities, or routing. Te following tree cases demonstrate te influence tat network designs can ave on a ouseold, wic would be unaccountable under trip-based circumstances. For tese examples, te utility maximization framework from Recker (1995) is assumed: ouseolds are multi-objective decision-makers wit teir own sets of objectives wit respective weigts tat dictate ow tey coose to scedule and route teir activities. Tis as been demonstrated empirically in Cow and Recker (2012) were a population of ouseolds were fitted wit eterogeneous sets of

5 objective weigts and desired arrival times to activities suc tat eac of teir observed itineraries were considered optimal to tem Departure Time Coice and Itinerary Re-Sequencing Assume a ouseold as one ouseold member and one veicle, and two activities to perform for te day: a work activity and a grocery sopping activity. Specifications of start and completion time windows and activity duration are sown in Table 1, in units of ours 1. Assume also tat te ouseold v v objective is solely to minimize te lengt of teir itinerary, min Z ( T2 n1 T0 ). Table 1. Case 1 ouseold caracteristics Houseold Location au, b u an, u b nu s u Home Node 0 a0, b0 6,21, a [10,22] 2n1 b2 n1 NA 10,22 8 work activity Node 3 9,9 Grocery Sopping activity Node 1 5,20 vv 6,22 1 Assume a grid network wit four nodes, and network connections as sown in Figure 1-(a). Travel time for eac link is 0.5 ours. Figure 1-(b) sows te optimal pattern if no investment is made. 19:00 Grocery Sopping 17: :00 Work 9: (a) Current Network (b) Optimal Houseold Activity Pattern 1 Here and trougout, te notation used in Recker (1995) is followed.

6 19:00 Grocery Sopping 17:30 17:42 Work 8:18 Work 9:00 7:00 9:00 7:30 Grocery sopping (c) t Time Adjustment (d) t Activity Sequence and Time Adjustment Figure 1: Te Optimal Houseold Activity Patterns for Case 1 Even in tis simplest case, two types of scedule responses can be observed for standard link investments wic would be ignored in conventional NDPs. If link {0,3} is constructed wit travel time of 0.7 ours as sown in Figure 1-(c), te ouseold would now be able to delay its departure time from 8AM to 8:18AM. Alternatively, if link {3,0} is instead constructed wit travel time of 0.7 ours as sown in Figure 1-(d), te optimal itinerary results in a re-sequence of activities as well as an adjustment in departure times Trip Caining Trade-Offs A paradoxical consequence of considering elastic itineraries in network design is tat it is possible to evaluate a link investment tat generates traffic witout any increase in economic activity. Traditionally, te argument made wit elastic demand considerations is tat improving infrastructure may result in additional trips made to fulfill latent demand between an OD pair. However, exceptions can also exist if travel is viewed as a way of acieving objectives wile constrained witin a space-time prism. By relaxing some of tose constraints troug network improvements, we may observe only increased trips due to untangling of less desired travel patterns witin te tigter constraints. Tis can result in more trips made if it improves te overall objective of te ouseold but would not contribute in any way to economic demand because te ouseold may be reconfiguring te same itinerary witout adding new destinations to visit. Tis occurrence can be best illustrated wit a ouseold wit activities tat ave very strict time windows. We consider te same activity agenda as in te previous section, but wit bot activities aving strict constraining start time windows as in Table 2. Bot activities require te ouseold member to be at te respective locations at a specific time, wic is often quite a realistic assumption. Assume also tat tis particular ouseold as two potentially conflicting objectives: to minimize te travel time wit weigt T, and to minimize delay from returning ome after an activity, wit weigt. Te delay from returning ome objective represents te desire of te ouseold to minimize te duration of any particular activity period away from ome, as discussed in Recker (1995) and calibrated empirically in Cow and Recker (2012). Te iger te weigt of tis objective relative to travel time, te more likely it is tat a ouseold would not want to trip cain. Ten te objective function becomes:

7 min Z v T tuw Xuw C ( Tu n Tu ) vv wn un un were te weigts are assumed to be and. Te optimal solution on te base network is sown in Figure 2-(a), wit te objective function travel disutility of and a total of tree trips made. Due to te time windows, te ouseold traveler is constrained to trip cain from te work activity to te social activity. Table 2. Case 2 ouseold caracteristics Houseold Location au, b u anu, b nu su Home Node 0 a0, b0 6,21 a2nu, b2n u [10,22] NA 10,22 8 work activity Node 3 9,9 social activity Node , , :45 Social 19:45 18:15 Waiting 17:45 17:30 time 17:00 17:00 17:42 Social 18:15 Work Work 9:00 8:00 9:00 8:00 (a) Optimal Houseold Activity Pattern (b) t extra trip generation Figure 2: Te Optimal Houseold Activity Patterns for Case 2 Now consider a link addition {3,0} wit travel time of 0.7 ours. Because te ouseold can now return ome immediately after work and still make te social activity in time, tey do so for an improved travel disutility of Te result is not only a cange in trip ODs (due to re-sequence in a tour), but one extra trip is also created as sown in Figure 2-b (4 trips). Essentially a trip as been added witout adding a new non-ome destination to visit, but te ouseold sees an improvement in travel disutility because of te relaxation of spatial-temporal constraints tat were binding before te network improvement. A conventional trip-based approac, or even a fixed scedule approac, would miss suc a response altogeter.

8 2.3. Increasing Travel Disutility If we consider a continuous link improvement (in wic a route travel time is improved), ten anoter counterintuitive situation can occur. Consider te ouseold in Table 2 again, but in tis case let s assume tat te ouseold seeks to minimize idle time. Idle time is defined as te extent of te travel day tat is not used in performing activities or traveling suc tradeoffs are similar to studies comparing values of in-veicle travel time against out-of-veicle access or idle/wait time. Te potential for conflict between te two objectives is not immediately apparent; owever, in te presence of strict time windows it is possible tat improving travel times can result in increasing idle time. min Z ( ) t X ( T T ) v v v T W uw uw W 2n 1 0 vv wnu N vv Were T 1 and W 1.5. Te durations of te activities are not included because tey are constant and drop out. In te base case sown in Figure 2-(a), te disutility under tis new objective is instead of If a continuous improvement is made to link {3,1} suc tat travel time improves from 0.5 ours to 0.25 ours (e.g. repaving, lane expansion), ten due to time window constraints tere are no oter alternative routes and te ouseold would still ave to follow te same scedule. However, tis results in a direct trade-off between travel time and idle time. If a ouseold values idle time minimization more tan travel time minimization, ten suc an improvement can result in a paradoxically iger disutility, even witout considering congestion effects. Te travel time improvement simply results in a decrease in travel time objective of 0.25 but a direct increase in idle time of Since, te disutility actually increases from to Effects suc as tis would be completely ignored if NDPs were applied witout considering teir effect on ouseold sceduling. However, explicitly incorporating ouseold sceduling mecanisms into te NDP allow paradoxes suc as tis to be avoided. We ave presented tree scenarios tat can arise from network improvements wen realistically considering te effects tey ave on ouseold sceduling and planning. Network canges can cause significant resaping of temporal /spatial constraints for ouseolds wic result in canges in teir trip patterns. We argue tat tese effects sould not be ignored wen considering NDPs at te tactical or planning level. 3. Proposed NDP-HAPP Model 3.1. Definitions As a kernel activity-based NDP, te NDP-HAPP is formulated using te simplest structure. Essentially, te activity-based NDP using HAPP subproblems to address ouseold scedule response to network canges is ere designated as NDP-HAPP. More complex formulations tat explore link capacities, veicle and ouseold member interactions, multimodal networks, or congestion effects will be explored in future researc. Te kernel formulation is first presented as a set of multiple subproblems, and ten furter modified to consider activity coice in cases wit non-compulsory activities. Tere are two distinct types of networks in tis problem: an infrastructure network were canges can actively be made, and a responsive activity network tat represents te routing and sceduling decisions made at te ouseold level. Assume an infrastructure network layer L, and te following parameters for te infrastructure network system: I N E F set of all nodes in te analysis set of all direct links in te analysis fixed arc design costs

9 c B t v, operational per unit link routing costs total budget for te network system travel time between te direct link from node i to node j c personal travel cost for veicle v of ouseold, between te direct link from node i to node j Variables concerning te infrastructure network system are: f flow on direct link ( i, j ) z binary decision variable tat indicates weter or not link ( i, j ) is cosen as part of te network s design Assume also an activity layer L A, and te following parameters for te activity network system P set of all activity nodes in te analysis. It is a subset of te node set from te infrastructure network, N. ( u, w), u, wp route from activity point u to activity point w. Its connectivity is derived from L I. H{,,...,,..., H } set of ouseolds using on te activity nodes P in te analysis. 1 2 Altoug teir pysical locations are te same, te two sets of networks operate in a bi-level fasion. Tis bi-level property of NDP-HAPP, togeter wit it unfolding in te time-space dimension, can be conceptually depicted in Figure 3. Suc separation of networks, a supernetwork approac, as been used widely in activity-based transportation networks, mainly concerning various modal coices and teir specific networks (e.g., TRANSIMS, 2012; Arentze and Timmermans, 2003). However, an optimizationbased routing and sceduling procedure as never been applied to te activity layer in response to infrastructure canges.

10 H H1 H2 Figure 3. Bi-level interactions between te infrastructure and activity networks Following te notation of Recker (1995), we define te following sets and parameters tat are specific for eac ouseold, H : a b β {,,...} te set of relative weigts for different travel disutility terms for ouseold A te set of out-of-ome activities to be completed by travelers in ouseold,. V : te set of veicles used by travelers in ouseold to complete teir sceduled activities. n A te number of activities to be performed by ouseold P P : te set designating location at wic eac assigned activity is performed for ouseold,. Eac activity and te pysical location is different for eac ouseold. P P : te set designating te ultimate destination of te return to ome trip from out-of-ome activities to be completed by travelers in ouseold,. (Note: te pysical location of eac element of P is ome.) A au, b u : te time window of available start times for activity I for ouseold. an, u b nu : te time windows for te return ome arrival from activity I of ouseold. (Note: b i must precede bn uby an amount equal to or greater tan te duration of te activity) a0, b 0 : te departure window for te beginning of te travel day for ouseold. a2n 1, b 2n1 : te arrival window by wic time all members of te ouseold must complete teir travel.

11 s u : te duration of activity u of ouseold. t uw : te travel time from te location of activity u to te location of activity w., c uw : travel cost for ouseold, from location of activity u to te location of activity w by veicle. B : te travel cost budget for ouseold. C, B T : te travel time budget for te ouseold s member using veicle. P = P P : te set of nodes comprising completion of te activities of ouseold. Q 0, P,2n 1 : te set of all nodes for ouseold, including tose associated wit te initial departure and final return to ome. Tis is a subset of P. Here i, j are used to refer to nodes in te infrastructure layer, and link ( i, j ) refers to te direct link connecting tose two nodes. Notation uw, are used to refer to activity nodes in te activity layer, and it is not necessarily a direct infrastructure link but rater a pat between uw., Pat information suc as travel time and travel cost are passed onto te activity layer from te infrastructure layer, but te connectivity data of te pat needs to be drawn from te infrastructure layer. Te ouseold-specific decision variables are:, X v uv, u, wq, vv, H binary decision variable equal to one if veicle travels from activity u to activity w, and zero oterwise. T, u P, H time at wic participation in activity u of ouseold begins. u v, v, T0, T2 n 1, u P, H times at wic veicle from ouseold first departs from ome and last returns to ome, respectively Y, u P, H total accumulation of eiter sojourns or time (depending on te selection u of D and d u ) of ouseold on a particular tour immediately following completion of activity u. Variables connecting te infrastructure network, L I, and te activity network system, L A, are: uw, uw, () z binary indicator variable weter route uw in te activity network uses link in L I. Assuming te sortest cost pat is used between two activity nodes, te design variables determine te connectivity of nodes in L. If link is not constructed, z 0, uw, is automatically 0, and oterwise, it can be identified by solving a sortest pat problem between te origin and te destination, uw. 0 z 0 uw, uw, () z * ( uw, ) z 1 * were ( ) is te solution of a sortest pat problem for eac activity link uw,, i.e. uw, I

12 Sortest Pat Allocation Problem t min uw, ( i, j) E Subject to 1 i u uw, ji uw, 0 i u, w jn jn 1 i w uw, ji (0,1) (1) (2) (3) Te problem is defined for all ouseolds and teir activity routes, u, wq, vv, H. t uw t () uw z : travel time from te location of activity u to te location of activity w. It is a function of te decision variable vector z, and te given network ( NE, ) since te connectivity decision variables of z determine te travel times. t t ( z) t ( z), u, wq, H (4) uw uw uw, je ie c c v, v, uw uw () z travel cost from te location of activity u to te location of activity w for veicle of ouseold. It is a function of te decision variable vector z, and te given network ( NE, ) since te connectivity decision variables of z determine te travel times. c c ( z) c ( z), u, wq (5) v, v, v, uw uw uw, je ie f f ( X ) link flow on direct link. It is a function of te ouseold activity decision variable vector, X, and connects te pat flow on layer LA to te link flow on layer L I. E given network ( NE, ) since te connectivity decision variables of z determine te travel times. f X z X i j E (6) v, ( ) uw, ( ) uw, (, ) H uq wq vv 3.2. Decomposed Formulation of NDP-HAPP Typically, te LRP formulation includes tree parts: location, routing, and allocation. Tis property applies to te NDP-HAPP as well, were te upper level location is te network design variables and te lower level routing part is te HAPP model. Allocation refers to assignment of te activity link impedance from te sortest pat problem in te infrastructure network, already sown in Equation (1) (3). Te objective function of te upper problem in te LRP is to minimize te overall cost, wic is comprised of depot cost and veicle cost. Similarly, NDP-HAPP in te most basic form is decomposed into two models solved as a bi-level problem: NDP (upper) and HAPP (lower). Tere are two sets of decision makers, so te solution can be classified as a leader/follower Stackelberg equilibrium, as described in Yang and Bell (1998). Instead of a traffic equilibrium lower level problem, te NDP-HAPP as a set of ouseold sceduling problems in te lower level, were eac ouseold decides on its activity pattern. Considering te network design

13 problem as te upper level decision and te ouseold activity/sceduling/routing decisions (HAPP) as reactions to te network design, we can express te problem most generally in Equations (7). min G( z, f ( X ( z)), T( z)) ( z) z, f subject to H ( z, f ( X ( z)), T( z)) 0 dndp were min g( X, T( ( f ))) ( X, T( ( f ))) X, T subject to ( z, X, T( ( f ))) 0 dhapp (7a) (7b) were G is te objective function, z is te decision vector, and H is te constraint set of te upper level problem. In te lower level problem, g is te objective function, XT, is te decision vector, and is te constraint set. Te kernel network design problem we present is simply a modified version of te unconstrained multicommodity case of te formulation in Magnanti and Wong (1984). Te formulation minimizes te design cost wile satisfying te given flow demands at origin and destination nodes. Te formulation is in terms of direct links and link flows only, wereas te integrated NDP-HAPP includes pat flows wic are connected to direct link flows, f, by uw,. In order for te OD pairs to be assigned to sequences of direct links, we treat eac OD pair as a commodity as in te case of multicommodity flow problems, i.e., uw uw we define a single commodity f, ( u, w) K were f f, were K is te set of all OD uw, pairs. ( uw, ) K We formulate tis decomposed NDP (dndp) in terms of direct link flows only, and eac OD pair is represented as a commodity. Te demand values are calculated as sown in Equation (14). Tey take ouseold sequence decisions and aggregate tem into origin-destination pairs. Upper Level NDP (dndp) min ( z, f ) F z c f dndp ( i, j) E ( i, j) E Subject to: uw f uw uw f D, i u N, ( u, w) K (9) jn jn jn ji ln il uw uw uw f f D, i u N, ( u, w) K, (10) ln li uw uw f f =0, i N, i u, i w, ( u, w) K (11) ji jn uw uw f D z, ( i, j) E,( u, w) K (12) z (0,1), ( i, j) E (13) were uw D v, X, w i N, ( u, w) K (14) H vv uw (8)

14 Equations (9) (10) require eac pat ( uwk, ) to satisfy te given OD demand. Equations (11) simply sow te conservation of flows for intermediate nodes. Equation (12) constrains flow variables to be on te links tat are built in a manner tat does not exceed te capacity. Because we do not consider cases in wic te capacity of links is exceeded in tis paper, only te sortest pat will be loaded wit flows. As suc, te sortest pat information is provided directly by te f variable. We can implicitly obtain te sortest pat variables for eac OD pair as sown in Equation (15) instead of aving to solve Equations (1) (3) separately. uw uw, uw 0 f 0, ( i, j ) E,( u, w ) K, vv, H (15) 1 oterwise Te decomposed lower-level HAPP (dhapp) problem is sown in Equations (16) (19). It is composed of te set of constraints in te Appendix wic would be equivalent to te original constraints from Case 1 in Recker (1995) if travel time/cost factors are not functions of te allocated sortest pat. Also, eac ouseold can be treated separately since all constraints as well as te objective functions are separable by ouseolds. Wit constant travel times/costs, i.e., witout congestion effects, eac ouseold s dhapp is solved separately. Lower Level HAPP (dhapp) for Eac Houseold min ( X, T) ( T T ) c X T v, v, C v, v, dhapp 2n 1 0 uw uw H vv H uq wq vv Subject to (A1) (A26) (16) Were t ( z) t, u, wq, H (17) uw uw, ( i, j) E c ( z) c, u, wq (18) v, v, uw uw, ( i, j) E uw, uw 0 f 0, ( i, j ) E,( u, w ) K, vv, H (19) 1 oterwise As discussed in oter HAPP model studies, te objective sown in Equation (16) is just one example multiobjective problem. Oters can be specified and estimated using te metod from Cow and Recker (2012). Te process of specifying te multiple objectives and calibrating teir coefficients wit desired arrival times can be tougt of as a confirmatory, normative modeling process tat seeks to fit a ypotesis of ow ouseold travelers beave onto a data set. Fitness of an objective is determined by te significance of its estimated coefficient relative to oter objectives. For example, a data set migt reveal tat Equation (16) results in a lengt of day coefficient (first term) equal to relative to a weigt of 1 for te travel cost objective. In tat case, it would suggest tat te first objective is not very important in te travelers sceduling coices. NDP-HAPP as presented in Section 3.2 differs conceptually from te LRP in two primary ways. First, te LRP as a single decision-maker involved in bot planning and tactical strategic design, wereas te NDP-HAPP as a single decision-maker involved in planning and multiple ouseold decision-makers responding to te plan at a tactical level. Second, te node demand for te upper level

15 problem in te LRP is known a priori, but te cost of delivering service to te demand node is not known. Instead, it is derived from te output of te VRP. Alternatively, te NDP-HAPP does not ave OD demand known a priori, but costs between eac node are given. Te OD demand is derived from te output of te HAPP Generalized NDP-HAPP (NDP-GHAPP) Te NDP-HAPP model is extended to include te capability for ouseolds to coose locations for non-primary activities, suc as grocery sopping and refueling. Tis is simply done by relaxing te condition in te HAPP tat eac ouseold needs to visit eac specifically designated location, but rater visits one candidate location from a cluster of suc service types. Tis is similar to te generalized traveling salesman problem (e.g. te E-GTSP in Fiscetti et al., 1997) and generalized veicle routing problem (Giani and Improta, 2000) in te logistics literature, were visits to nodes are modified to visits to single nodes from eac cluster. Te generalized HAPP (GHAPP) as been formulated and applied (Kang and Recker, 2012a; Kang and Recker, 2012b), and a variation of tis approac wit exogenously defined activity utilities and time windows was developed for activity-based traveler information systems (Cow and Liu, 2012). In GHAPP, te constraints in Equation (A1) are modified to Equation (A1-1). Instead of requiring eac node to ave a flow, te generalized formulation instead requires one node from a cluster of nodes to be visited. Compulsory activity types would only ave one node in te cluster, wereas non-primary activities suc as grocery sopping or refueling could ave multiple candidate nodes to coose from. up vv A a wq X 1, A A, H (A1-1) v, uw a were A { A, A,..., A,..., A } te set of different activity types wit unspecified locations 1 2 a m P A a te set designating potential locations at wic activity A a may be performed Integrated wit NDP, GHAPP becomes infeasible if one or more candidate nodes are not connected to te network; constraints in (A7), (A11) also need to be modified to be conditional suc tat te temporal constraints are imposed only wen tere is a visit to tat candidate location. Tis allows aving one or more of unconnected candidate nodes, wic ave infinite travel times. vv wq vv wq X 1 T s t ( z) T, u P, H (A7-1) v, wu u u uw nu X 1 a T b, u P, H v, wu u u u (A11-1) Similarly, wen te objective function involves time variables, te time variables for te unvisited activity nodes need to be constrained. For example: vv wq X 0 T 0, u P, H. v, wu u (A7-2)

16 3.4. Decomposition Solution Algoritm Tere are many different types of solution algoritms developed for LRPs (Nagy and Sali, 2007), and tey can potentially be adopted for NDP-HAPP. However, te iterative metod proposed ere decomposes te problem into several blocks tat actually represent eac decision maker s rationale in tis complex problem. Additionally, tis kind of decomposition does not necessarily require te problem to be formulated in te structure of matematical optimization as long as te drivers response to te network design is captured and updated. Tis means different types of integrated activity-based approaces can be used to model individuals routing/sceduling beavior. Because te majority of tese activity-based models are based on discrete-coice models or simulation-based (e.g., Bowman and Ben- Akiva, 2000; Bat et al., 2004; Balmer et al., 2006), te suggested decomposition metod is igly adaptable to different types of activity-based models. Te decomposed problems remain computationally callenging, particularly te NP-ard HAPP. Because tese problems are widely studied, tere are various metods available. Geoffrion and Graves (1974) are referred for network design problems, and Cordeau and Laporte (2003) are referred for a survey of algoritms for te Pickup and Delivery Problem wit Time Windows (PDPTW), wic te simplest HAPP is based on. Te decomposition proposed ere is comparable to Perl and Daskin (1985) in te context of Location Routing Problems, and te Iterative Optimization Assignment (IOA) algoritm in Yang and Bell (1998) in te context of bi-level Network Design Problems. Perl and Daskin (1985) used tree decomposed models to tackle te wareouse location routing problem: te complete multi-depot veicle-dispatc problem (MDVDP), te wareouse location-allocation problem (WLAP), and te multidepot routing-allocation problem (MDRAP). Te location-allocation and muti-depot routing allocation blocks are in parallel wit dndp and dhapp. For NDP, te iterative optimization-equilibrium in Friesz and Harker (1985) includes similar blocks of Equilibrium Assignment Program and Design Optimization, in line wit dhapp and dndp. Since tere is no congestion in te dhapp model, te issue of aving IOA converge to a Cournot-Nas equilibrium is not relevant ere. 2 An iterative solution algoritm for te NDP-HAPP is depicted in Figure 4. First, te initial 0 0 network decision solution is assumed to use all links, z 1, ( i, j) E. Ten, can be derived from 0 z using te standard sortest pat problem for example, Floyd s Algoritm can be used to efficiently update te travel time matrix. Based on te updated travel times, dhapp is solved independently for eac ouseold since no congestion effect is present. Hypotetically speaking, if congestion is incorporated in future researc (peraps troug integration wit Lam and Yin s (2001) framework), tis framework sould still be feasible. After te travel decisions are made by eac ouseold, supply and demand are updated from Equations (15). dndp can ten be solved as te conventional NDP. Te proposed iterative process continues until tere is no improvement in te objective function. Te implicit sortest pat allocation from te upper level problem and te pat-link conversion conditions in Equations (4) (6) are maintained trougout tis iterative process. Te same algoritm can be applied to NDP-GHAPP. uw, 2 For te examples and case studies presented in tis paper, te overall processes are coded in Java calling a CPLEX library for dhapp and dndp problems.

17 Network Initialization all links are available Sortest Pat Problem for Initialization dhapp uw 0 f 0 uw, 1 oterwise ( i, j) E,( u, w) K, v V, H 1 min ( X dhapp ) min ( X ) dhapp No canges in variables and Obj. value yes Terminate no D X, i N k( u, i) v, i ui H vv dndp min ( f dndp, z ) no No canges in variables and Obj. value yes Terminate Figure 4. A decomposition solution metod for NDP-HAPP 4. Numerical Examples 4.1. Simple Example wit Known Optimal Solution Assume a grid network wit nine nodes, wit possible link construction as Figure 5. Wen constructed, travel time for eac link is 0.5. Construction cost for eac link is 3, and operating cost per link per flow is 0.5, i.e. min ( z, f ) 3 z 0.5 f. dndp ( i, j) E ( i, j) E

18 Assume two ouseolds, H { 1, 2}, wit one veicle eac, V1 {1}, V2 {1}, and teir activities A1 {work, grocery sopping}, A2 {work, general sopping} to perform. Tese activities locations in te infrastructure layer are sown in Figure 5, and teir activity start/end time windows, activity durations are all sown in Table 3. Except for te activity start time windows of work activities, time windows are not necessarily constraining, leaving some room to explore different pat sequences. Bot ouseolds objective functions are assumed to be minimizing total travel cost only v, v, (ouseolds consider te travel costs 1 for all direct links), min ( X ) c X. Table 3. Simple Example Houseold Caracteristics Location, on L au b u I 1 ome Node 0 a0, b0 6,21 dhapp uw uw uq wq vv a, b n u n u s u, a [10,24] 2n1 b2 n1 NA 10, work activity Node 2 9,9.5 1 grocery sopping activity Node 5 5,22 2 ome Node 5 a0, b0 6,21 a2n 1 b2 n1 2 work activity Node 6 8.5,9 general sopping activity Node 8 5, ,22 1, [10,22] NA 10, ,22 1 L A L I Figure 5. Supernetwork depiction. Because te NDP-HAPP is not a simple problem to ceck for optimality, all possible combinations of ouseold decisions are enumerated and given to dndp, and its objective value combined wit te objective value of corresponding ouseold decision combination is used to derive te true optimal solution value. Figure 6 sows te solution from te proposed metod (6-(a), 6-(b)) and te actual optimal solution (6-(c), 6-(d)). Te decomposition solution converged after one iteration and is 5%

19 worse tan te actual optimal solution, 40, for tis simple example. computational process is available in Table 4. Detailed illustration of te H General sopping 17:30 Grocery sopping 17: H2 Work Work 9:00 8: (a) Decomposition dndp Solution (b) Decomposition dhapp Solution H :30 Grocery sopping Work H2 8:30 Work 9:00 General sopping 6: (c) Optimal dndp Solution (d) Optimal dhapp Solution Figure 6. NDP-HAPP Decomposition Solution Comparison to Enumerated Exact Solution.

20 Table 4. Detailed Computational Illustration of te basic NDP-HAPP Example dhapp1 dhapp2 Iteration 1 Iteration 2 Pat 3 : Home (0) work (2) grocery sopping (5) ome (0) Objective Value: 3 Pat 4 : Home (5) work (6) general sopping (8) ome (5) Objective Value: 3 Network Design Decisions: Z01, Z12, Z25, Z30, Z36, Z43, Z54, Z36, Z78, Z85 dndp objective value: 36 Pat: Home (0) work (2) grocery sopping (5) ome (0) Objective Value: 3 Pat: Home (5) work (6) general sopping (8) ome (5) Objective Value: 3 dndp HH1 Pats link Flows: Home (0) (1) Work (2) Work (2) Grocery Sopping (5) Grocery Sopping (5) (4) (3) Home (0) HH2 Pats link Flows: Home (5) (4) (3) Work (6) Work (6) (7) General Sopping (8) General Sopping (8) Home (5) NA 5 Final Objective Update eac dhapp objective values: HH1: 3 HH2: Simple Example: Generalized HAPP Using te generalized model allows us to include beavioral canges in destination coice as well as routing/sceduling of activities wit respect to network design decisions. Following te example in Section 4.1, assume tat tere are two grocery sopping locations, node 1 and node 5, P {1,5}, and two general sopping locations, node 3 and node 8, {3,8} Aa GrocerySopping P Aa GeneralSopping eac ouseold is required to visit one, and only one, of te candidate locations to perform te sop activity. Here, NDP-GHAPP optimality is cecked in te same way as te previous example, i.e., by comparing to te results of dndp for all possible combinations of ouseold decisions, including te destination coice as well as pat sequence decisions and arrival time decisions to return. Te solution from te iterative metod reaced te true optimal value after tree iterations, sown in Figure 7. Te intuition is tat te flexibility introduced by NDP-GHAPP allows te metod to searc for many different options. Detailed illustration of te computational process of te proposed algoritm is sown in Table 5. 3 Tese pats are based on te assumption tat all links are available. 4 Tese pats are based on te assumption tat all links are available. 5 No canges in variables and objective function value. Terefore aborted after tis iteration.

21 In tis simple example, canges in activity sequence, link level flow in dndp, and dndp network design decisions are sown (a) Optimal Solution (b) Optimal Solution Figure 7. NDP-SHAPP Example Enumerated Optimal Solution Table 5. Detailed Computational Illustration of te NDP-SHAPP Example dhapp1 dhapp2 dndp Iteration 1 Iteration 2 Iteration 3 Iteration 4 Pat 6 : Home (0) grocery sopping (1) work (2) ome (0) Objective Value: 2 Pat 7 : Home (5) work (6) general sopping (8) ome (5) Objective Value: 3 Network Design Decisions: Z01, Z10, Z12, Z21, Z58, Z67, Z76, Z78, Z85, Z87 Pat: Home (0) work (2) grocery sopping (1) ome (0) Objective Value: 2 Pat: Home (5) work (6) general sopping (8) ome (5) Objective Value: 3 Network Design Decisions: Z03, Z10, Z21, Z36, Z52, Z67, Z78, Z85 dndp objective Pat: Home (0) grocery sopping (5) work (2) ome (0) Objective Value: 4 Pat: Home (5) work (6) general sopping (3) ome (5) Objective Value: 4 Network Design Decisions: Z03, Z10, Z21, Z34, Z36, Z45, Z52, Z63 dndp objective Pat: Home (0) grocery sopping (5) work (2) ome (0) Objective Value: 3 Pat: Home (5) work (6) general sopping (3) ome (5) Objective Value: 4 NA 8 6 Tese pats are based on te assumption tat all links are available. 7 Tese pats are based on te assumption tat all links are available. 8 No canges in variables and objective function value. Terefore aborted after tis iteration.

22 dndp objective value: 35 value: 32 value: 31 HH1 Pats link Flows: Home (0) Grocery Sopping (1) Grocery Sopping (1) Work (2) Work (2) (1) Home (0) HH2 Pats link Flows: Home (5) (8) (7) Work (6) Work (6) (7) General Sopping (8) General Sopping (8) Home (5) HH1 Pats link Flows: Home (0) (3) (6) (7) (8) (5) Work (2) Work (2) Grocery Sopping (1) Grocery Sopping (1) Home (0) HH2 Pats link Flows: Home (5) (2) (1) (0) (3) Work (6) Work (6) (7) General Sopping (8) General Sopping (8) Home (5) HH1 Pats link Flows: Home (0) (3) (4) Grocery Sopping (5) Grocery Sopping (5) Work (2) Work (2) (1) Home (0) HH2 Pats link Flows: Home (5) (2) (1) (0) (3) Work (6) Work (6) General Sopping (3) General Sopping (3) (4) Home (5) Final Objective Update eac dhapp objective values: HH1: 2 HH2: 3 Update eac dhapp objective values: HH1: 4 HH2: 4 Update eac dhapp objective values: HH1: 3 HH2: Large Network Example: NDP-HAPP Tis case study focuses on a major roadway system located in Orange County, a subsystem of te Los Angeles metropolitan roadway network, to compare te NDP-HAPP wit te conventional NDP. Te base network wit ouseold locations and teir activities trougout te day are sown in Figure 8-(a). We assume tat te network design decision maker is a public agency from Orange County, and its goal is to provide te best mobility for Orange County residents, were te mobility is expressed in terms of total travel times. Hypotetically suggested candidate improvements on te network system are extensions of SR 39, SR 57, SR 55, SR 22, SR 261, and SR 241 as seen in dased red lines in Figure 8- (b). Specifications of eac candidate link are in Appendix B. Te speed is drawn from te average speed for all links on te same facility, and construction cost for eac link is assumed to be proportional to bot average speed and distance.

23 0 SR 14 1 I 405 I 5 US 170 I 210 US 101 SR 134 I 210 SR 134 I I 210 I US 101 I 5 7 I 405 SR SR 19 I 605 SR SR 57 SR 2 US 101 I 10 I 10 I 10 I 10 I SR 71 I 10 I 10 SR I 5 SR SR 60 I I 605 SR 57 SR I SR SR 71 SR 60 SR SR I 605 SR 60 SR 1 SR 42 I 5 39 SR SR 42 I 710 SR 19 SR 42 SR SR 57 I SR 110 I 5 I 605 SR 1 SR 142 SR I 710 SR 19I 105 SR 90 SR 90 I 105 I SR 1 I 405 I SR 110 SR I I 710 I 605 I 5 SR 57 SR SR 39 SR 91 SR 91 SR 91 SR 91 SR SR SR 91 SR 91 SR 110 I 710 SR I 405 I 5 SR 91 SR 1 SR 107 I SR I 605 I 405 SR SR 57 SR 110 I 405 SR I SR 241 SR 19 I 405 SR 22 SR SR 1 SR I 5 SR 1 SR 55 SR 1 SR SR 241 I 405 I 5 SR I 405 SR 1 SR SR I 405 I 5 87 SR SR SR SR SR SR 1 I 5 SR 1 SR SR 133 SR I 5 Link Improvement SR I 5 I 5 78 SR (a) Sample ouseolds and teir activity locations (b) Extracted network Figure 8. Large-Scale Case Study Area A sample of 60 single-member, single-veicle ouseolds residing in Orange County drawn from te California Travel Survey (2001) is used to reflect fairly realistic trip patterns of tis class of ouseolds. Te objective function for dndp is to minimize te total travel times for te system, min ( z, f ) t f, and te objective function for eac dhapp is to minimize its own travel dndp ji ( i, j) E disutility. For tis example, individual ouseold s travel disutility is defined by te linear combination of te total extent of te day, te travel times, and te delay of return ome caused by trip caining for eac E T D of out-of-ome activities by te individual weigts of suc measurements,,, : min ( X, T) ( T T ) ( T T ) t X E v, v, D T v, v, dhapp 2n1 0 wn w uw uw H vv H + wp H uq wq vv Te weigts of tese 60 ouseolds are individually estimated from te inverse optimization calibration process in Cow and Recker (2012). For te ouseolds in te sample, te estimated results ave te values of E 0.84, D 0.74, T 3.45, wic means tat on average tese ouseold decision makers value a minute of travel time savings about 4 times more tan a minute of total extent of te day savings, and about 5 times more tan a minute delay in returning ome caused by trip caining from outof-ome activities. Te values were based on aving te same set of arrival time penalties for all activity types, wit early penalty and late penalty, similar to Cow and Recker (2012). Te correlations from te 60 samples were close to zero for ET, and DT,, altoug te correlation between extent of day savings and return ome delay was ED, Time windows of activities are separately estimated using te metodology from Kang and Recker (2012b), wic adopted te metod from Recker and Parimi (1999) wit sligt modifications. In Table 6, results of NDP-HAPP are compared to conventional NDP solutions tat take te O/D matrix derived from te optimal HAPP results wit current network as an input. Six different budget limits are tested. Te results indicate tat bot dndp and dhapp objective function values improved