A Shortest-Path Algorithm for the Departure Time and Speed Optimization Problem

Transcription

1 A Shortest-Path Algorithm for the Departure Time an Spee Optimization Problem Anna Franceschetti Dorothée Honhon Gilbert Laporte Tom Van Woensel Noember 2016 CIRRELT

2 A Shortest-Path Algorithm for the Departure Time an Spee Optimization Problem Anna Franceschetti 1,*, Dorothée Honhon 2, Gilbert Laporte 1, Tom Van Woensel 3 1 Interuniersity Research Centre on Enterprise Networks, Logistics an Transportation (CIRRELT) an Department of Decision Sciences, HEC Montréal, 3000 Côte-Sainte-Catherine, Montréal, Canaa H3T 2A7 2 Naeen Jinal School of Management, Uniersity of Texas at Dallas, Richarson, Texas , USA 3 School of Inustrial Engineering, Einhoen Uniersity of Technology, P.O. Box 513, The Netherlans 5600 MB Einhoen Abstract. We present a shortest-path algorithm for the Departure Time an Spee Optimization Problem (DSOP) uner traffic congestion. The objectie of the problem is to etermine an optimal scheule for a ehicle isiting a fixe sequence of customer locations in orer to minimize a total cost function encompassing emissions cost an labor cost. We account for the presence of traffic congestion which limits the ehicle spee uring peak hours. We show how to cast this problem as a shortest path problem by exploiting some structural results of the optimal solution. We illustrate the solution metho an iscuss some properties of the problem. Keywors: Spee optimization, shortest path, scheuling. Acknowlegements. The work was partly supporte by the Dutch Institute for Aance Logistics uner the project 4C4D an by the Natural Sciences an Engineering Research Council of Canaa (NSERC) uner grant This support is gratefully acknowlege. Results an iews expresse in this publication are the sole responsibility of the authors an o not necessarily reflect those of CIRRELT. Les résultats et opinions contenus ans cette publication ne reflètent pas nécessairement la position u CIRRELT et n'engagent pas sa responsabilité. * Corresponing author: anna.franceschetti@cirrelt.net Dépôt légal Bibliothèque et Archies nationales u Québec Bibliothèque et Archies Canaa, 2016 Franceschetti, Honhon, Laporte, Van Woensel an CIRRELT, 2016

3 1 Introuction In recent years there has been a growing interest in moels an algorithms for routing an scheuling problems in which fuel consumption an relate CO 2e emissions are taken into account. This line of research is roote in the works of Fagerholt et al. (2010), Norsta et al. (2011) an Hattum et al. (2013) in the fiel of maritime transportation, an in those of Bektaş an Laporte (2011) an Demir et al. (2012) in the fiel of roa transportation. The first set of papers seek to optimize the routing an scheuling of a essel isiting a set of ports on a path subject to arrial time restrictions. In particular, Hattum et al. (2013) propose an exact algorithm to optimize the cruise spee of the essel in orer to minimize the fuel consumption while ensuring that ships arrie within the time winows. This problem is referre to as the Spee Optimization Problem (SOP). The secon set of papers tackles a similar problem of optimizing the routing an scheuling of a fleet of ehicles that must sere a set of customers on a path with har time winows for the start of serice at customer locations. In this ariant, the ehicles may arrie before the opening of the time winows. These authors consier a more global objectie function that inclues labor an fuel costs. In this context, Demir et al. (2012) aapte the algorithm propose by Hattum et al. (2013) to optimize the trael spee of a ehicle on each arc of the path. More recently Kramer et al. (2015) eelope a quaratic-time algorithm to sole this problem to optimality. Franceschetti et al. (2013) extene the moel from Demir et al. (2012) to allow for traffic congestion, which limits the ehicle spee uring peak hours. They refer to the problem as the Departure Time an Spee Optimization Problem (DSOP). The consieration of traffic congestion greatly complicates the analysis since it may now be optimal for the ehicle to wait ily at the epot or at customer locations in orer to mitigate the negatie impact of slow congestion spee on emission costs. These authors propose a polynomial-time heuristic algorithm which buils upon the close-form solution they eelop for the single-arc ersion of the problem. In this paper, we eelop a shortest-path base algorithm for the DSOP uner traffic congestion, which is exact an efficient. Hence we exten the work from Kramer et al. (2015) by consiering traffic congestion an that of Franceschetti et al. (2013) by computing an optimal solution. Note that our algorithm can be use a subroutine to optimize iniiual routes in some ehicle routing an scheuling problems with spee consierations. The remainer of the article is organize as follows. In 2 we present our moel an iscuss feasibility conitions. In 3 we establish key structural CIRRELT

4 results of the optimal solution, which we exploit to buil our shortest path formulations, presente in 4. We illustrate our solution metho in 5 an conclue with a summary of our contributions in 6. All proofs are presente in the Appenix. 2 Moel A single ehicle eparts from an origin location calle the epot to isit a number of customer locations accoring to a fixe route. For example, a eliery ehicle leaes from a central warehouse to isit retail locations an elier merchanise, or a plumber leaes her office to isit customer homes in orer to make repairs. Let 0 enote the epot, let locations 1 to n represent customers, an let n + 1 enote a copy of the original location, which represents the return of the ehicle to its starting point. 1 Hence, the fixe route is (0, 1,..., n + 1). We refer to arc (i, i + 1) as arc i for i = 0,..., n. Let i enote the istance on arc i an h i enote the serice time at location i for i = 0,..., n. In the eliery ehicle example, the serice time woul correspon to the time spent loaing the ehicle with merchanise an the serice time at each customer location woul be the unloaing an eliery time. Each customer location i {1,..., n} has a har time winow [l i, u i ] within which serice must start. The ehicle may arrie at the customer location earlier than l i but serice may not start until that time. Let µ i be the arrial time of the ehicle at location i for i = 0,..., n+1. We set µ 0 = 0, which correspons to the start of the planning horizon. Note that this alue oes not necessary correspon to the time at which the rier arries to the epot. We set l 0 = l n+1 = 0 an u n+1 = for notational conenience. We refer to the time between the arrial time at a customer location an the start of serice as the pre-serice waiting time. This waiting time can be iie into the manatory pre-serice waiting time, which is equal to (l i µ i ) +, an the oluntary pre-serice waiting time, which is enote by y i. If the ehicle arries at the customer location before the lower time winow limit, i.e., µ i < l i, then the manatory pre-serice waiting time is l i µ i, an the oluntary pre-serice waiting time is equal to the ifference between the start of serice at location i an l i. If the ehicle arries at location i after the lower time winow limit, i.e., µ i l i, then the manatory preserice waiting time is zero, an the oluntary pre-serice waiting time is 1 All of our results woul also apply with minor moifications if the last noe is also a customer noe an there is no return to the epot or if the ehicle must return to the epot within a certain time perio. 2 CIRRELT

5 equal to the ifference between the start of serice an the arrial time µ i. Let y 0 enote the time elapse between the start of serice at the epot an the start of the planning horizon. Upon completion of serice at a location, the rier may ecie to wait before leaing; let z i enote the post-serice waiting time at location i = 0,..., n. The trip ens with the arrial at location n + 1, that is, with the return to the epot. Finally, let ρ i enote the eparture time from location i for i = 0,..., n. We hae ρ 0 = µ 0 + y 0 + h 0 + z 0 an ρ i = max{µ i, l i } + y i + h i + z i for i = 1,..., n. Figure 1: Sequence of eents at customer location i. Figure 1 epicts the sequence of eents at customer i. In this example, the ehicle arries before l i, so that there is a positie manatory pre-serice waiting time. Also, because serice oes not start immeiately at l i, there is a positie oluntary pre-serice waiting time. Finally the ehicle oes not leae the location immeiately after the completion of serice, so there is also a positie post-serice waiting time. As in Franceschetti et al. (2013) an Jabali et al. (2012), we assume that the planning horizon starts with an initial perio of traffic congestion lasting a time units uring which the ehicle traels at a constant spee. After this perio of traffic congestion ens, the rier can choose its ehicle s spee between the limits min an max. We refer to this perio as the free-flow perio. Let i enote the chosen free-flow spee on arc i for i = 0,..., n. Note that i is only efine if location i + 1 is reache past the en of the congestion perio, that is, if µ i+1 a. Let T i (ρ i, i ) enote the total trael time to traerse arc i if the ehicle leaes location i at time ρ i an chooses a free-flow spee i (if applicable). We iie this time between the time spent riing in congestion an the time spent riing in free-flow as follows: we set T i (ρ i, i ) = Ti con (ρ i ) + (ρ i, i ) where Ti con (ρ i ) = min{(a ρ i ) +, i / } is the time spent traeling in congestion on arc i an T free i (ρ i, i ) = [ i (a ρ i ) + ] + / i is the time spent traeling at the free-flow spee i on arc i when eparting at time ρ i from location i. The arrial time at location i+1 is then calculate T free i CIRRELT

6 as µ i+1 = ρ i + T i+1 (ρ i, i ). The ecision maker makes two types of ecision: how fast to rie on each arc uring the free-flow perio an how much to wait at each location, either pre- or post-serice. We assume that waiting can only take place at the locations, not between them. Hence an optimal scheule is fully characterize by three ectors: y = (y 0,..., y n ), z = (z 0,..., z n ) an = ( 0,..., n ), which are the oluntary pre-serice waiting time ector, the post-serice waiting time ector an the free-flow spee ector, respectiely. Table 2 summarizes our notation. Notation Description i arc i istance l i lower time winow limit at location i u i upper time winow limit at location i U i effectie upper time winow limit at location i (see 2.3) h i serice time at location i y i oluntary pre-serice waiting time at location i z i post-serice waiting time at location i µ i arrial time of the ehicle at location i ρ i eparture time from location i µ i earliest possible arrial time at location i (see 2.3) i chosen free-flow spee on arc i congestion spee min minimum free-flow spee max maximum free-flow spee Table 1: Notation 2.1 Objectie function The ecision maker aims at minimizing total cost, which is the sum of: (i) the labor cost, i.e. the cost of paying the rier of the ehicle an (ii) the cost of carbon ioxie equialent (CO 2e ) emissions, which are irectly proportional to the amount of fuel use uring the trip. As such, our objectie function is ientical to that of Franceschetti et al. (2013). In line with Franceschetti et al. (2013), we consier two ways of calculating the total time for which the rier is pai. In the early policy the rier is pai starting from the start of the planning horizon, i.e., time zero, until the arrial time at location n+1; in the late policy the rier is pai starting from the start of the serice at the epot until the arrial time at location 4 CIRRELT

7 n + 1. Uner the early policy, the rier reports to the epot at the start of the planning horizon, which coincies with the start of a typical work ay, but may hae to wait before starting her serice an riing uties (uring this time she may be aske to perform some aministratie uties). Uner the late policy, the rier arries at the epot right on time to start serice. We assume that the labor cost on arc i is measure from the arrial time at customer location i to the arrial time at location i + 1, i.e., from µ i to µ i+1. Formally, the labor cost, L i, for traersing arc i, gien an arrial time of µ i at customer location i, a pre-serice waiting y i, a post-serice waiting z i an a free-flow spee i is gien by L i(µ i, y i, z i, i) = D [ (l i µ i) + + y i + h i + z i + T i(max{µ i, l i} + y i + h i + z i, i) ], where D is the labor cost. For arc 0, the labor cost is calculate ifferently epening on the rier wage policy. Uner the early policy, we hae L 0 (0, y 0, z 0, 0 ) = D [y 0 + h 0 + z 0 + T 0 (y 0 + h 0 + z 0, 0 )]. Otherwise, uner the late policy, we hae L 1 (0, y 0, z 0, 0 ) = D [h 0 + z 0 + T 0 (y 0 + h 0 + z 0, 0 )]. The ifference between the two rier wage policies is the pre-serice waiting time at the epot Dy 0, which is pai uner the early policy but not uner the late policy. To measure the emissions cost, we use the moel of Barth et al. (2005) an Scora an Barth (2006) who assume that the amount of CO 2e emissions prouce by a ehicle is irectly proportional to the amount of fuel consume. Accoring to their moel, the emissions costs E i, for traersing arc i gien a chosen free-flow spee i an a eparture time from location i of ρ i, is gien by: [ E i (ρ i, i ) = A i + BT i (ρ i, i ) + C T con i (ρ i )( ) 3 + T free (ρ i, i )i 3 where A, B an C are non-negatie constants which epen on features of the ehicle such as frontal surface area an curb weight as well as on the roa conitions. (see Franceschetti et al. (2013) for how to calculate these alues). The total cost for traersing arc i, measure from the arrial time at location i to the arrial time at location i + 1, i.e., from µ i to µ i+1, is enote by g i an is gien by g i (µ i, y i, z i, i ) = E i (max{µ i, l i } + y i + h i + z i, i ) + L i (µ i, y i, z i, i ). i ], CIRRELT

8 The ecision maker s problem is to etermine the spee ector an waiting time ectors y an z in orer to minimize the total cost C incurre oer the entire ehicle trip. The problem can be written as follows: minimize,z,y C(, y, z) = n g i (µ i, y i, z i, i ) i=0 subject to µ 0 = 0, µ i = max {l i 1, µ i 1 } + y i 1 + h i 1 + z i 1 +T i 1 (max {l i 1, µ i 1 } + y i 1 + h i 1 + z i 1, i 1 ) for i = 1,..., n + 1, µ i u i for i = 1,..., n, min i max for i = 0,..., n. The first two sets of constraints follow from the efinition of the arrial time into a location. The thir set correspons to the upper time winows constraints an the last set is where we impose the minimum an maximum free-flow spee constraints. Note that, for fixe, z an y, the total cost uner the early policy is higher than uner late policy by a quantity equal to Dy 0, which correspons to the cost of paying the rier uring the pre-serice waiting time at the epot. 2.2 Important spee alues Gien a fixe eparture time ρ i from noe i, the spee that minimizes the emissions costs on arc i, i.e., E i (ρ i, i ), is equal to = ( B 2C )1/3. Similarly, gien fixe alues for µ i, y i an z i, the spee alue that minimizes the total cost of traersing arc i, i.e., g i (µ i, y i, z i, i ) is equal to = ( B+D 2C )1/3. We always hae an we further assume that min an max. These two spee alues are useful in our subsequent analysis an hae preiously been ientifie by Demir et al. (2012). 2.3 Feasibility conitions an effectie upper time winow limits In this section we establish the conitions uner which the problem is feasible an we introuce the concept of effectie upper time winow limits, which is useful in our subsequent eriations. Let µ i enote the earliest possible arrial time at location i, which is calculate by assuming the ehicle ries at the maximum free-flow spee wheneer possible until reaching location i 6 CIRRELT

9 an neer waits (pre- or post-serice) at any location, i.e., setting j = max, z j = 0 for j = 0,..., i 1 an y j = 0 for j = 0,..., i 1. We obtain µ i for i = 0,..., n + 1, recursiely starting from location 0 as follows: µ 0 = 0, µ i = max {µ i 1, l i 1 } + h i 1 + T i 1(max{µ i 1, l i 1} + h i 1, max ) for i = 1,..., n + 1. The problem is feasible if µ i u i for i = 1,..., n. In what follows we assume that these conitions are satisfie. For i = 1,..., n + 1, let U i enote the latest possible arrial time at location i so that it is still possible to meet all the time winows at locations i + 1,..., n + 1. By construction, we always hae U i u i. We call U i the effectie upper time winow limit at location i: if the ehicle was to arrie at location i between U i an u i, then it woul not iolate the upper time winow limit of location i, but it woul certainly iolate the upper time winow limit of at least one of the subsequent locations. We set U n+1 = u n+1 = + an calculate U 1,..., U n recursiely starting from location n, assuming maximum spee an no postserice waiting time, as follows: U n = u n, U i = min { u i, U i+1 min{(u i+1 a) +, i/ max } [ i (U i+1 a) + max ] + / h i } for i = n 1,..., 1. Note that the feasibility of the problem implies that l i U i for i = 1,..., n. We use these effectie upper time winow limit U i, rather than u i, throughout our analysis, an we refer to [l i, U i ] as the effectie time winow at location i {1,..., n + 1}. (1) 3 Structural Results In this section we present some structural properties of the optimal solution to the DSOP with traffic congestion. When there is traffic congestion, the optimal riing scheule may be such that it is optimal for the ehicle to wait ily at the epot or at customer locations because oing so can reuce the riing time in congestion, thereby possibly reucing GHG emissions. Howeer, waiting may increase labor costs: the ile time at a customer location is always costly, while the ile time at the epot is only costly if the rier is pai from the start of the planning horizon, that is, uner the early policy. Our first result is about the oluntary pre- an post-serice waiting times at the epot an customer locations uner both rier wage policies. CIRRELT

10 Proposition 1. There exists an optimal solution such that there is no oluntary pre-serice waiting time at any of the customer locations, that is, y i = 0 for i = 1,..., n. Further, when the rier is pai from the start of the planning horizon (early policy), this optimal solution also has no oluntary pre-serice waiting time at the epot, that is, y 0 = 0. In contrast, when the rier is pai from the start of serice at the initial location (late policy), the optimal solution also has no post-serice waiting time at the epot, that is, z 0 = 0. Base on Proposition 1, one can ignore oluntary pre-serice waiting time at the customer locations in the optimization process an focus solely on post-serice waiting times. This is because, at customer locations, the rier is pai equally for pre- an post-serice waiting times, an hence there is no ifference in costs if, gien a fixe amount of total waiting time, all the waiting takes place after serice is complete. But the situation is ifferent at the epot as it epens on the rier wage policy: pre-serice waiting is free if the rier is pai from the start of serice at the epot (late policy), therefore postponing serice at the epot then leaing immeiately (that is, setting y 0 0 an z 0 = 0) may be an effectie strategy to reuce GHG emissions costs without increasing labor costs. When the rier is pai from the start of the planning horizon (early policy), waiting at the epot is equally costly whether it takes place pre- or post-serice, therefore concentrating all the waiting after the serice has ene (that is, setting y 0 = 0 an z 0 0) oes not affect total costs. 2 Gien Proposition 1, the ecision maker only nees to etermine ( 0,..., n ) an (z 0,..., z n ) uner the early policy, an ( 0,..., n ), y 0 an (z 1,..., z n ) uner the late policy. Any solution to the DOSP can be seen as a subsequence (j 1,..., j m ) of the location inices {0, 1,..., n + 1} with 0 j 1... j m = n + 1 as follows. Arcs 0 to j 1 1 are entirely traerse uring the congestion perio, that is, µ i a for i = 1,..., j 1 an arcs j k to j k+1 1 are traerse at the same free-flow spee for k = 1,..., m, that is, jk = jk +1 =... = jk+1 1. In other wors the set of arcs is partitione into ajacent segments on which the rier keeps a constant free-flow spee: the first segment consists of arcs 0 to j 1 1, the secon one consists of arcs j 1 to j 2 1, an the last segment consists of arcs j m 1 to n. Also, arc j 1 is the first arc to be traerse uring the free-flow perio (if all the arcs are traerse uring the congestion 2 Alternatiely, we coul assume that all the waiting takes place before serice at the epot when the rier is pai from the start of the planning horizon. Howeer, this alternatie complicates the exposition of Proposition 2, which is why we hae opte for the present statement of Proposition 1. 8 CIRRELT

11 perio, we hae m = 1). Our next result, which hols uner both rier wage policies, uses this notation to establish important properties of the optimal solution. Proposition 2. There exists an optimal solution such that (i) all the post-serice waiting (if there is any) takes place at the last location that is reache before the en of the congestion perio, i.e., z j1 0 an z i = 0 for i j 1 where j 1 is the largest inex in {1,..., n + 1} such that µ j1 a; (ii) if the ehicle reaches a customer location i > j 1 strictly within its time winow, i.e., µ i (l i, U i ), then the free-flow spee on arcs i 1 an i must be the same. Proposition 2(i) implies that there exists an optimal solution such that the ehicle oes not wait ily post-serice after the en of the congestion perio, which is intuitie since oing so woul increase labor costs without reucing GHG emissions. Proposition 2 (ii) implies that changes in the optimal free-flow spee can only occur when the ehicle reaches a customer location by its lower time winow, i.e., µ i l i, or exactly at its upper time winow, i.e., µ i = U i. In other wors, the spee on two ajacent arcs must be the same unless the start of serice at the common location is exactly equal to its lower or upper time winow limit. Figure 2 epicts an example of scheuling that satisfies the conitions of Proposition 2, where the x-axis represents the geographical istance an the y-axis represents time. The scheule is represente by iscontinuous black line, where the points of iscontinuity occur at the customer locations. The slope of the increasing portions correspons to the time elapse per unit of istance, which is the inerse of ehicle the spee, while the ertical portions of the line correspon to perios of serice an waiting (with waiting being represente by otte lines). Time winows limits for each locations are marke on the y-axis. In this example, the ehicle traels on arc 0 uring the congestion perio (therefore at a spee of ), then waits at location 1 following the completion of serice. The congestion perio ens while it is traersing arc 1, so the rier switches to a free-flow spee which we enote by 1. The ehicle keeps the same spee on arc 2, arriing at location 3 exactly at time U 3. Finally it ries on arcs 3 an 4 at a constant spee, enote by 2. Formally, in this example, we hae m = 3, j 1 = 1, j 2 = 3 an j 3 = 5. This scheule satisfies Proposition 2(i) since the only postserice waiting occurs at location 1, which is the last customer location CIRRELT

12 that is reache before the en of the congestion perio. It also satisfies (ii) since the ehicle arries at locations 2 an 4 strictly within their time winow an keeps the same free-flow spee before an after reaching these locations. Further, at location 3, which marks a change in free-flow spee, the ehicle s arrial time exactly matches the effectie upper time winow limit. Figure 2: Example of scheuling satisfying the conitions of Proposition 2. Note that ˆ = B+C c 3C c (see A) Gien Proposition 2, there exists an optimal sequence {j 1,..., j m 1, n+1} of locations such that serice starts serice at locations j 2,..., j m 1 exactly at their lower or upper time winow limit. Therefore we can obtain an optimal scheule by consiering all possible sequences in combination with the two possible alues for start of serice at locations which mark a spee change. So suppose we fix a sequence {j 1,..., j m 1, n + 1} an fix the start of serice at locations j 2,..., j m 1. Optimizing the pre- an post serice waiting times an ehicle spees can be one inepenently for each segment of locations: 0 to j 1, j 1 to j 2, etc. By efinition, we know the free-flow spee is constant on each such segment an by Proposition 2, we know that the arrial time at the intermeiate noes is strictly within their time winow. Lemma 1 further establishes that the optimization of each segment amounts to soling a single-arc DSOP problem, which implies that there is no waiting time at the intermeiate noes. 10 CIRRELT

13 Lemma 1. Minimizing the total cost of traersing arcs i,..., j at a constant free-flow spee, ignoring the time winows at locations i + 1,..., j 1 such that serice at location j starts exactly at time η {l j, U j } is equialent to minimizing the total cost of traersing a single arc with istance = j 1 k=i k such that serice at the estination location starts exactly at time ˆη = η j 1 k=i+1 h k. In A we show how to sole a single-arc problem with a fixe start of serice time. Note that there results are ifferent from those of Franceschetti et al. (2013) since they o not assume a fixe start of serice time at the final location. Because the final arc represents the return to the initial location an there is no time winow at location n + 1, (i.e., l n+1 = 0 an u n+1 = ) the last segment must be ealt with ifferently. In this case the results from Franceschetti et al. (2013) can be use irectly to show that if the ehicle ries at least partially in free-flow, then the last part of the trip shoul be rien at spee as efine in 2.2. Lemma 2. If the optimal solution is such that the ehicle ries at least part of the trip in free-flow (i.e., m > 1), then it shoul rie at spee on the last arcs, i.e., jm 1 = jm 1 +1 =... = n =. When the rier is pai from the start of serice at the epot (late policy), we hae the following extra result. Proposition 3. Consier the case where the rier is pai from the start of serice at the initial location. If at least one customer location has a finite upper time winow, then there exists a piot location ĵ {1,..., n} efine as follows: ĵ is the lowest inex such that µĵ {lĵ, Uĵ} an µ i (l 1, U i ) for i = 1,..., ĵ 1. Further, if µĵ a an µĵ = Uĵ then ĵ = 1. Also the optimal solution is such that z 0 =... = zĵ 1 = 0. When at least one customer location has a finite upper time winow, Proposition 3 establishes that, uner the late policy, there always exists a customer location that is reache exactly at its lower or effectie upper time winow, without any post-serice waiting time at any of the preceing locations. We refer to the earliest customer location satisfying this conition as the piot customer location an enote it by ĵ. Intuitiely, the role of the piot customer location is to etermine the optimal amount of pre-serice waiting time at the epot. When the rier is pai from the start of serice at the epot, postponing serice at the epot is an effectie way to reuce CIRRELT

14 the total cost. Gien a fixe arrial time at the piot location, one can optimize the pre-serice waiting time at the epot an the (constant) freeflow spee on arcs 0 to ĵ 1 by treating the problem as a single-arc DSOP with fixe arrial time (see A for how to sole such a problem uner the late policy). Note that the piot customer location can be reache either before or after the en of the congestion perio. When it is reache after congestion ha ene, i.e., µĵ > a, Proposition 3 together with Proposition 2(ii) imply that there is no post-serice waiting time at any location. In contrast, if the piot customer location is reache uring the congestion perio, i.e., µĵ a, there coul be post-serice waiting time at one of the customer locations (either location ĵ or a later one). Further, if the arrial time at the piot location matches its effectie upper time winow an occurs before congestion ens, then the piot location is location 1. The structural properties of the optimal solution establishe in Propositions 1, 2 an 3 allow us to recast the DSOP with traffic congestion as a shortest path problem, which we present in etail in the next section. Before oing so, we proie a stronger characterization of the optimal solution in a special case. Proposition 4. Suppose that each customer location has an infinite upper time winow, i.e., u i = for i = 1,..., n + 1. If the rier is pai from the start of serice at the initial location (late policy), it is optimal to set y 0 so that the ehicle leaes the initial location past the congestion perio an aois any manatory pre-serice at any customer location. Further, the ehicle ries in free-flow at spee on all the arcs, without any post-serice waiting. Specifically, ( setting y 0 to a alue greater or equal to max {(a h 0 ) +, max i=1,...,n l i )} i 1 j=0 h j an z 0 = i 1 j=0 j z 1 =... = z n = 0 is optimal. The total cost achiee is equal to C = A n i=0 i + (B + C 2 + D) + D n i=0 h i. n i=0 i Intuitiely, when there are no upper time winows an the rier is pai from the start of serice at the epot, the ecision maker can postpone serice inefinitely without bearing any cost consequences, an therefore she waits until the congestion perio has ene so as to aoi riing at the GHG emissions-causing congestion spee. Also she waits long enough to make sure the ehicle neer arries at any customer location earlier than its lower time winow limit. By oing so she is able to aoi any manatory pre-serice wait time. As a result, the total time for which the rier is pai for is the sum of her trael time an serice time an, in this case, the total 12 CIRRELT

15 cost is minimize when using free-flow spee efine in 2.2 on eery arc. Note that total cost of the optimal solution is a lower boun on all possible alues achieable. 4 Shortest Path Formulation In this section we show how to turn the DSOP with traffic congestion into a shortest path problem. By oing so we exploit two main results proen in the preious section. First, we use the property that an optimal scheule can be broken own into segments of arcs where the ehicle keeps a constant freeflow spee, arries at each intermeiate location strictly within its locations time winows limits an only waits post-serice (if at all) at the starting point location of the segment. Secon, we exploit the property that serice at the locations that mark a change in spee starts either exactly at their lower or effectie upper time winow limits. The esign of the shortest path network iffers for the two rier wage policies, so we present the solutions separately in two subsections below. 4.1 Shortest path formulation uner the early policy Propositions 1 an 2 (i) together imply that when the rier is pai from the start of the planning horizon, locations 0 to j 1 are reache as early as possible gien the congestion spee, that is, µ i = µ i for i = 0,..., j 1, where µ i is efine in 2.3. This is because there is no pre- or post-serice waiting at the epot (y0 = z 0 = 0) an no post-serice waiting at locations 1 to j 1 1 (z1 =... = z j 1 1 = 0). Hence, uner the early policy, the start of serice time at location j i for i = 1,..., m 1 can only take three possible alues: µ ji, l ji or U ji. Note that the arrial time at location j i can be earlier than l ji but the start of serice time cannot be. For use in the esign of the shortest path, we efine location k {0,..., n+1} as the highest inex location that can be reache before the en of the congestion perio (assuming no waiting), i.e., k is the largest alue of i such that µ i a (we set k = 0 if it is impossible to reach location 1 before the en of the congestion perio, that is, if h / c > a). From the original network with n + 1 arcs an n + 2 noes we construct a shortest path network with 2n+k noes as follows: noe 0 (corresponing to the epot), noes i for i = 1,..., k (i.e., one such noe for each of the first k customer locations), noes i l an i u for i = 1,..., n (i.e., two such noes for each customer locations) an location n + 1 (corresponing to the return CIRRELT

16 to the epot). Let V enote the set of noes in the shortest path network, i.e., V = {0, 1,..., k, 1 l,..., n l, 1 u,..., n u, n + 1}. For i = 1,..., k, location i correspons to the eent the ehicle starts serice at time max{µ i, l i }. For i = 1,..., n, noe i l correspons to the eent the ehicle arries at location i at the latest at the lower time winow limit l i (an therefore starts serice at time l i ), an noe i u correspons to the eent the ehicle arries at location i exactly at the effectie upper time winow limit U i. Finally, noe n + 1 correspons to the eent the ehicle returns to the epot. Each arc in the shortest path network correspon to a segment of ajacent arcs in the original network. For example, the arc in the shorhest path network from location 0 to location i (or i l or i u ) for i = 1,..., n correspons to arc 0 to i 1 in the original network. The length of each arc is equal to the minimum cost of traersing the corresponing segment gien constraints on the arrial time at each location. This cost can be calculate by treating each segment as a single-arc DSOP as explaine in Lemma 1 an is set to infinity if meeting the constraints is not feasible. More precisely, the length of the arcs in the shortest path network are set as follows: The length of arcs (0, i) for i = 1,..., k is set equal to the (minimum) cost of riing in congestion from the epot to location i, without any waiting at locations 0 to k 1. The length of arcs (0, j l ) (resp. (0, j u )) for j = 1,..., n is set equal to the minimum cost of starting serice at the epot at time zero an starting serice at location j at time l j (resp. U j ), while reaching locations i+ 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); The length of arcs (i l, j l ) (resp. (i l, j u ), (i u, j l ), (i u, j u )) for 1 i < j n is set equal to the minimum cost of starting serice at location i at time l i (resp. l i, U i, U i ) an starting serice at location j at time l j (resp. U i, l i, U i ), while reaching locations i + 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); The length of arcs (i, j l ) (resp. (i, j u )) for 1 i < j n is set equal to the minimum cost of starting serice at location i at time µ i an starting serice at location j at time l j (resp. U j ), while reaching locations i+ 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); 14 CIRRELT

17 The length of arcs (i l, n + 1) (resp. (i u, n + 1) an (i, n + 1)) for 1 i min{k, n} is set equal to the minimum cost of starting serice at location i at time l i (resp. U i an µ i ) an riing to location n + 1, while reaching locations i + 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); The length of arc (0, n+1) is set equal to the minimum cost of starting serice at the epot at time zero, reaching locations 1 to n within their time winow an returning back to the epot (an is infinite if this is not possible). If k = n + 1, the length of arc (n + 1, n + 1) is set equal to zero. All other arcs hae infinite lengths. In particular, all arcs from an i-noe (i.e., either i, i l or i u ) to a jnoe (i.e., either j, j l or j u ) with j < i hae infinite length. So o arcs (i l, j), (i u, j) an (i, j) for j > i. Figure 3: Shortest path network for an example with n = 2 an k = 2. From this is follows that the problem of fining the optimal ectors, z an y when the rier is pai from the start of the planning horizon reuces to a shortest path from noe 0 to noe n + 1 as escribe aboe. 4.2 Shortest path formulation uner the late policy When the rier is pai from the start of serice at the epot, we know from Proposition 1 that it may be optimal for the ecision maker to postpone the CIRRELT

18 start of serice at the initial location in orer to sae on labor costs, that is, set y0 > 0. Further, there can be some aitional waiting, which, if it exists, takes place post-serice at the last location isite before the en of the congestion perio (by Proposition 2(i)). This will be the case if the piot customer location (i.e., the ĵ inex efine in Proposition 3) is isite before the en of the congestion perio. When the piot customer location is isite exactly at its upper time winow, Proposition 3 establishes that it must be the first location; in practice this means that the rier postpones the start of serice at the epot so as to arrie at the first customer location exactly at time U 1. In that case, the arrial time at subsequent locations i = 2,...j 1, enote by ˆµ i, are calculate recursiely as: ˆµ 1 = U 1 ˆµ i = max {ˆµ i 1, l i 1} + h i 1 + T i 1 (max {ˆµ i 1, l i 1} + h i 1, max ) for i = 2,..., j 1. In contrast, when arrial at the piot customer location is at its lower time winow an occurs before the en of the congestion perio, the arrial time at subsequent locations ĵ + 1,..., j 1 is gien by µ i since the ehicle will start serice at location ĵ at the earliest possible time an there is no waiting until possibly at location j 1. Hence, uner the late policy, the start of serice time at location j i for i = 1,..., m 1 can only take four possible alues: ˆµ ji, µ ji, l ji or U ji. In other wors, compare to the early policy, there is one more possible alue for the arrial time at a location which marks the en of a segment. We exploit this structure to moify the shortest path formulation presente in the preious section. As in 4.1, we efine k {0,..., n + 1} be the highest inex location that can be reache before the en of the congestion perio (assuming no waiting), i.e., k is the largest alue of i such that µ i a (again, we set k = 0 if een location i cannot be reache before the en of the congestion perio). Further, we also efine p {0,..., n + 1} to be the highest inex location that can be reache before the en of the congestion perio, when the ehicle waits pre-serice at the epot so as to arrie at location 1 exactly at time U 1, i.e., p is the largest alue of i such that ˆµ i a (we set p = 0 if U 1 > a). Note that we always hae k p since µ i ˆµ i for i = 0,..., n + 1. From the original network with n+1 arcs an n+2 locations, we construct an SP network with 2n+k +p noes. The first 2n+k noes are the same as in the SP network from 4.1 an hae the same interpretation. The extra p noes are labele î for i = 1,..., p an correspon to the eent the ehicle arries at location i at time ˆµ i. All the arcs which exist in the SP network from 4.1 continue to exist an hae the same length. On top of these we 16 CIRRELT

19 hae the following extra arcs: (0, î) for i = 1,..., p: The length of arc (0, î) is set equal to the (minimum) cost of riing from the epot to location i when the ehicle waits at the epot so as to arrie at location 1 exactly at time U 1 an oes not wait at locations 1,..., p 1; (î, j l ), (î, j u ) for 1 i < j n: The length of arc (î, j l ) (resp. (î, j u )) for 1 i < j n is set equal to the minimum cost of starting serice at location i at time ˆµ i an starting serice at location j at time l j (resp. U j ), while reaching locations i+ 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); (î, n + 1) for 1 i min{p, n}: The length of arc (î, n + 1) for 1 i < j n is set equal to the minimum cost of starting serice at location i at time ˆµ i an riing to location n + 1, while reaching locations i + 1,...,j 1 strictly within their time winow (an is infinite if this is not possible); ( n + ˆ 1, n + 1) if p = n + 1: The length of this arc is set equal to zero. From this it follows that the problem of fining the optimal ectors, z an y when the rier is pai from the start of serice at the epot reuces to a shortest path from noe 0 to noe n + 1 as escribe aboe. 5 Numerical illustrations In this section we present the optimal solution for a numerical example uner both rier wage policies. We o so in orer to illustrate the workings of the shortest path an to present some structural properties of the optimal solution. Consier the route epicte in Figure 4 with 4 customer locations. Figure 4: Example of a route with 4 customer locations CIRRELT

20 All istances reporte in Figure 4 are in km an times are in secons. We assume a congestion perio a of secons, a congestion spee of 10 km/h, a minimum free flow spee min of 10 km/h an a maximum free flow spee max of 90 km/h. Regaring the cost parameters, we use the same alues as in Franceschetti et al. (2013), namely A = , B = , C = an D = Using (1) we calculate the effectie upper time winow limits for each customer location an see that it matches the actual upper time winow limit for each of them, except for location 2, where U 2 = < u 2. Gien the cost parameters, we hae = 55.19km/h an = 75.48km/h. The furthest location that can be reache before the en of the congestion perio is location 2, i.e., we hae k = 2 where k is efine in 4.1. Also, the furthest location that can be reache before the en of the congestion perio when the ehicle arries at location 1 at time U 1 is location 1, i.e. we hae p = 1 where p is efine in 4.2. Figures 5 an 6 epict the SP graph uner the early policy an the late policy, respectiely. Both figures only show the arcs which hae finite length. The shortest path on each graph is marke in bol. Figure 5: SP graph uner the early policy. The optimal scheuling uner the early an late policies are also epicte on Figures 7 an 8, respectiely. From Figure 5, we see that the shortest path from 0 to 5 uner the early policy goes through noes 2 an 4 l. This means that the optimal solution has three segments with spee changes occurring when reaching locations 2 an 4, that is, we hae m = 3, j 1 = 2, j 2 = 4 an j 3 = 5. It also means that the optimal solution is such that the ehicle ries in congestion on arcs 0 an 1, arries at location 2 at time µ 2, an starts serice at location 4 at 18 CIRRELT

21 Figure 6: SP graph uner the early policy. time l 4. Figure 7 proies further information about the optimal scheule: we see that the ehicle waits post serice at location 2 until the en of the congestion perio, then ries on arcs 2 an 3 at spee, reaching location 4 strictly before its lower time winow limit, an causing some manatory preserice waiting time. Upon completion of serice at location 4, the ehicle returns to the epot riing at a spee of. The total cost of this solution is 43.99, which is equal to the sum of the arc lengths on the shortest path. From Figure 6, we see that the shortest path from 0 to 5 uner the late policy goes through noes ˆ1 an 3 u. This means that the optimal solution has three segments with spee changes occurring when reaching locations 1 an 3, that is, we hae m = 3, j 1 = 1, j 2 = 3 an j 3 = 5. It also means that the optimal solution is such that the ehicle ries in congestion on arc 0, arries at location 1 at time U 1 (making location 1 the piot customer location), an at location 3 at time U 3. From Figure 8 we further learn that serice at the epot is elaye until after 4400 secons. Also, we see that, the ehicle waits post-serice at location 1 until the en of the congestion perio, then ries on arcs 1 an 2 at the require spee in orer to arrie at location 3 exactly at time U 3. After completing the serice at location 3 the ehicle ries on arcs 3 an 4 at spee. The total cost of this solution is 32.67, which is equal to the sum of the arc lengths on the shortest path. Tables 3, 4 an 2 respectiely isplay a comparison of the optimal cost alues, spee alues, waiting times between the two rier wage policies. CIRRELT

22 Drier wage policy early policy late policy Driing time in congestion Driing time in free-flow Total riing time Pai waiting time Total labor time 13, Return to epot time Labor cost Emissions cost Total cost Table 2: Times (in secons) an costs for the optimal solutions. Drier wage policy early policy late policy Table 3: Optimal spees alues (in km/h). Drier policy early policy late policy Location pre-serice post-serice pre-serice post-serice Table 4: Optimal (manatory) pre- an (oluntary) post-serice waiting times (in secons). 20 CIRRELT

23 Figure 7: Optimal of scheuling uner the early policy We see that both the labor an emission costs are lower uner the late policy. This can be explaine by the following factors. First, because serice time at the epot is elaye uner the late policy, the ehicle spens significantly more time riing in congestion uner the early policy than uner the late policy. Secon, the optimal ehicle spee is (weakly) larger on each arc uner the late policy. Thir, the rier is pai for significantly more waiting time uner the early policy (specifically an extra minutes). In other wors, by elaying the arrial (an therefore the payment) of the rier at the epot uner the late policy the ecision maker is able to mitigate the negatie impact of congestion, increase riing spee an reuce in-transit waiting times. Note that the rier returns sooner to the epot uner the early policy than uner the late policy. 6 Conclusions We hae stuie the DSOP uner traffic congestion, which consists in fining the optimal scheule for a ehicle isiting a fixe sequence of customer locations in orer to minimize the sum of labor an emissions costs. We hae proie an efficient shortest path formulation to sole this problem optimally in polynomial time. Because traffic congestion creates incenties for oluntary waiting times in the scheule, existing methoologies for the DSOP without traffic congestion cannot be irectly applie or easily aapte. Besie being efficient, our solution methoology proies an intuitie isual CIRRELT

24 Figure 8: Optimal of scheuling uner the late policy representation of the optimal solution through the shortest path network, which represents the breakown of the optimal scheule into ajacent segments of arcs on which the optimal spee is kept constant. In our moel formulation, we hae consiere two ifferent rier wage policies an we hae illustrate through a numerical example how their optimal solutions iffer. Our solution methoology can be embee as a subroutine within an algorithm to sole a more general ehicle routing an scheuling problem such as the pollution-routing problem with traffic congestion (Franceschetti et al. (2013)). Acknowlegements The work was partly supporte by the Dutch Institute for Aance Logistics uner the project 4C4D an by the Canaian Natural Sciences an Engineering Research Council uner grant This support is gratefully acknowlege. 22 CIRRELT

25 A Single-arc DSOP optimization Gien Lemma 1, the length of each arc in the SP network can be compute by consiering a single-arc problem with a fixe start of serice time. We show how to sole such a problem in this section. Let 0 enote the initial noe (epot) an 1 enote the arrial noe (customer location) with time winow [l 1, u 1 ]. To simplify the notation let enote the length of the arc an enote the free-flow spee on the arc. All other notation is the same as in 2. First we consier the problem of minimizing the total cost (i.e., labor an emissions costs) uner the constraint that the arrial time at location 1 must be exactly equal to µ. The ecision maker nees to ecie how long the ehicle shoul wait at the epot (pre- an post-serice, i.e., y 0 an z 0 ) an how fast to rie on the arc (i.e., spee ), so as to minimize the total cost, which is measure until the arrial time at location 1, that is, we o not consier [ the return { trip to the epot. } For the problem to be feasible we nee µ h 0 + min, (a h 0 ) + + ( (a h 0) + ) +, u max 1 ]. We proie conitions on the optimal solution to this problem in Propositions 5 (early policy) an 6 (late policy). Proposition 5. Consier a single-arc DSOP where the rier is pai from the start of the planning [ horizon {(early policy) an } must arrie at location 1 exactly at time µ h 0 + min, (a h 0 ) + + ( (a h 0) + ) +, u max 1 ]. The optimal solution is such that y0 = 0. Further, if µ a, then z 0 = µ h 0 an the entire arc is rien in congestion. If µ > a, then the optimal free-flow spee takes one of the following 4 alues: (i) (a h 0) + µ max{a,h 0 }, (ii) µ a, (iii) or (i) ˆ B+C 3 con 3C. The optimal alue of z 0 can be foun by soling h 0 + z 0 + T 0 (h 0 + z 0, ) = µ. Since y0 = 0 an the arrial time at location 1 is fixe, the problem reuces to a single ecision ariable problem: gien a chosen free-flow spee, it is possible to calculate the amount of post-serice waiting time at the epot so as to arrie exactly at µ. (a h When µ > a, the first possible optimal spee alue, i.e. 0 ) + µ max{a,h 0 }, correspons to the case where the ehicle leaes the epot immeiately upon completion of serice while the other three inole a positie amount of postserice waiting time at the epot. In particular, with the secon possible optimal spee alue, i.e. µ a, the ehicle waits until the en of the congestion perio, leaing the epot exactly at time a. CIRRELT

26 Proposition 6. Consier a single-arc DSOP where the rier is pai from the start of serice at location [ 0 (early { policy) an} must arrie at location 1 exactly at time µ h 0 + min, (a h 0 ) + + ( (a h 0) + ) +, u max 1 ]. The optimal solution is such that z0 = 0. Further, if µ a, then y 0 = µ h 0 an the entire arc is rien in congestion. If µ > a, then the optimal free-flow spee, takes one of the following 4 alues: (i) (a h 0) + µ max{a,h 0 }, (ii) µ a, (iii) or (i) ṽ B+D+C 3 con 3C. The optimal alue of y 0 can be foun by soling y 0 + h 0 + T 0 (y 0 + h 0, ) = µ. The first possible spee alue when µ a, i.e. (a h 0) + µ max{a,h 0 }, correspons to the case where the ehicle leaes the epot immeiately upon completion of serice while the other three inole a positie amount of post-serice waiting time at the epot. In particular, with the secon spee alue, i.e. µ a, the ehicle waits until the en of the congestion perio, leaing exactly after a time units. Next we consier the problem of minimizing total cost uner the constraint that serice at location 1 must start at a fixe time η. The ecision maker nees to ecie how long the ehicle shoul wait at the epot (prean post-serice, i.e. y 0 an z 0 ), how fast to rie on the arc (i.e., spee ) an how long to wait pre-serice at location 1 (i.e., y 1 ) in orer to minimize the total cost which is measure until the fixe start of serice at location 1. For this problem to be feasible we nee η [l 1, u 1 ]. Proposition 7. Consier a single-arc DSOP where serice at location 1 must start exactly at time η [l 1, U 1 ]. Uner the early policy, there exists an optimal solution such that the arrial time at location 1 is equal to min {max{a, h 0 } + /, η}. Uner the late policy, there exists an optimal solution such that the arrial time at location 1 is equal to η. Accoring to Proposition 7, there will be some pre-serice waiting time at location 1 uner the early policy if it is possible for the ehicle to arrie by η at location 1 after waiting at location 0 until the en of traffic congestion an riing on the arc at spee. This waiting time at location 1 may be manatory or oluntary In contrast, uner the late policy, it is always optimal to start serice immeiately upon arrial into location 1. In our shortest path formulation, we calculate the arc lengths by fixing the start of serice time at the estination location. By Lemma 1, we know that these alues can be calculate by consiering a single arc-problem. Hence the results in this section can be use as follows: first we use Proposition 7 to obtain the optimal arrial time at location 1 gien a fixe start of 24 CIRRELT