Refined Execution Cost Estimation for LTL Load Plans

Transcription

1 Refined Execution Cost Estimation for LTL Load Pans Aan Erera, Michae Hewitt, Martin Savesbergh, Yang Zhang Schoo of Industria and Systems Engineering Georgia Institute of Technoogy Abstract A oad pan specifies how freight is routed through a inehau termina network operated by a ess-than-truckoad (LTL) motor carrier. Determining the design of the oad pan, a probem of service network design, is critica to effective operations of such carriers. Operations research techniques currenty support oad pan design, and a modes in use or proposed today approximate transportation costs by using costs per traier dispatched between terminas; often, these costs are determined by mutipying a per traier-mie cost by the mieage between terminas. Furthermore, empty transportation costs are determined by soving a traier re-baancing probem, either within the oad panning mode or as a post-processing step. These approximations ignore two important ideas: (1) short traiers, commony known as pups, can be moved behind tractors in trains of two or three traiers, and the cost of moving a traier train is not inear in the number of traiers; and (2) drivers must be schedued for each dispatch, and driver rues may introduce trave in addition to that minimay required for traier baance. In this paper, we present technoogy designed to more accuratey estimate the operationa execution costs of a oad pan. A computationa study demonstrates that our technoogy produces accurate operationa execution costs estimates, typicay within 2% of actua incurred costs. 1 Introduction The trucking industry provides an essentia service to the U.S. economy by transporting goods from business to business and from business to consumer. Less-than-truckoad (LTL) transportation represents an important, but reativey sma, segment of the trucking industry serving businesses that ship quantities ranging from 150 bs to 10,000 bs, i.e., ess-than-truckoad quantities. A typica shipment occupies ony 5-10% of the traier capacity. LTL carriers therefore coect and corresponding author; emai: aerera@isye.gatech.edu 1

2 consoidate freight from mutipe shippers, and route shipments through a termina network of crossdock transfer terminas. Consoidation of shipments eads to higher traier capacity utiization, but aso to increased trave distance (referred to in the industry as circuity) and freight handing costs. How freight is routed through the termina network, and thus where opportunities for consoidation occur, is determined by a so-caed oad pan, which specifies a fixed sequence of termina transfer points for each shipment entering the system. Consequenty, oad pan design is critica to effective operations of an LTL carrier. Freight fuctuations, whether seasona or caused by changing economic conditions, force LTL carriers to reguary review and adjust their oad pans; sma percentage gains in traier utiization can ead to significant cost savings. Our focus in this paper is not on oad pan design, but instead on accuratey estimating the operationa execution costs of a given oad pan. During oad pan design, transportation costs are usuay approximated using inear cost factors per traier dispatched between termina pairs; often this cost is determined by mutipying a cost per mie by the mieage separating the terminas. However, this can be a crude approximation, since actua transportation costs are affected by the dispatched driver tours, and driver tours are severey restricted by government reguations and company and/or union poicies. These poicies and reguations can impact the amount of empty trave required, and may ead to more empty trave than predicted by empty traier baancing modes. Furthermore, short traiers (often referred to as pups) can be moved by a singe driver in trains of two or three traiers; in this paper, we assume that a traier train contains at most two traiers. Since it is difficut to predict in advance what fraction of traiers dispatched on a ane between two terminas wi trave aone or in a train, it is not easy to determine an appropriate inear cost per traier. As a resut, oad pan design methods may substantiay under- or over-estimate transportation costs. Such cost estimation errors may have unintended and costy consequences. The technoogy we deveop and present in this paper takes a set of shipments for a certain panning horizon and a oad pan to route shipments through the termina network, and then buids driver dispatches with associated dispatch windows (a dispatch corresponds to a combination of up to two traiers and each traier contains one or more shipments) and generates cost-effective driver tours to cover these dispatches and baance empty traiers. The cost of executing these driver tours is then our estimate of the transportation costs incurred when executing the oad pan. Having an accurate estimate of the cost of executing a new oad pan is an essentia part of the oad pan design process. An anciary benefit of our approach is that it buids a set of cost-effective driver tours. First, this set of tours may be usefu in practice. Second, the set aso can be used to hep determine the number of drivers needed at the different terminas. Identifying a set of suggested driver tours for new oad pans is especiay important to speed up impementation, and thus the reaization of any cost savings resuting from the use of the adjusted oad pan. We have conducted a computationa study of the proposed approach using an actua oad pan 2

3 and actua shipment data from a super-regiona LTL carrier operating in the continenta U.S. We compare the execution cost estimates of the oad pan from two different approaches with the actua inehau costs incurred in practice when executing the oad pan. The first estimate comes from SuperSpin (Manhattan Associates (2010)), the current industry standard software for oad pan design. The second estimate is taken from the technoogy presented in this paper. The resuts show that SuperSpin tends to underestimate actua costs, between 88.8% to 90.6% of actua, whie our technoogy provides accurate cost estimates, between 99.6% and 101.7% of actua. Summarizing, this research makes contributions primariy in the context of oad pan design, evauation, and execution for LTL carriers. Specificay, we have deveoped technoogy that improves oad pan execution cost estimates; accuracy improvements on the order of 10-15% are shown for a super-regiona carrier, buids a set of dispatches and generates a set of cost-effective driver duties and tours covering these dispatches; driver duties and tours satisfy government reguations and union and/or company rues and can thus be used in practice, and soves rea-ife instances efficienty; ess than 2 hours for instances representing a week of data with over 140,000 shipments, which equates to over 10,000 oads and approximatey 6,000 driver duties. The remainder of the paper is organized as foows. In Section 2, we describe LTL operations, discuss oad pan design, and present the need for more accurate oad pan cost estimation. In Section 3, we review reevant iterature. In Section 4, we formay state the oad pan costing optimization probem and discuss the modeing issues and choices. In Section 5, we introduce our soution approach. In Section 6 we present the resuts of an extensive computationa study using historica data from a super-regiona LTL carrier in the U.S. Finay, in Section 7, we present some concuding remarks. 2 LTL System Description LTL carriers operate fixed termina networks to provide service. During the day carriers pick up shipments from various shippers in a reativey sma geographica area, say a city, and bring them to a termina serving the area, referred to as an end-of-ine termina. End-of-ine terminas serve as sorting centers and oading faciities for outbound and inbound freight for the geographic area. There usuay is not enough freight coected at an end-of-ine termina to buid fu or neary fu traieroads to other end-of-ines, hence another ayer of consoidation is introduced. Outbound freight from an end-of-ine is sent to a breakbuk termina where it may be consoidated with freight from other end-of-ines. By consoidating freight from and destined for many ocations, 3

4 breakbuk terminas hande enough freight to buid and dispatch cost-effective traieroads. Consoidation at a breakbuk requires freight to be cross-docked which requires time and resuts in handing cost. A typica shipment may trave from an origin end-of-ine to an origin breakbuk, then to a destination breakbuk, and eventuay to a destination end-of-ine. Once it arrives at the destination end-of-ine, it is oaded on to a deivery truck and sent to the consignee. Figure 1: A typica LTL network We focus on the freight movements between terminas, i.e., the inehau operations of an LTL carrier. A oad pan determines how freight is routed through the inehau network by specifying a sequence of transfer points for each shipment entering the system. Each termina-to-termina traier movement without intermediate handing is referred to as a direct move. A oad pan, therefore, determines a unique path of direct moves for a shipments entering the inehau system at a specific origin end-of-ine termina and destined for a specific destination end-of-ine termina. To keep cross-dock operations simpe, traditiona oad pans have a tree-ike structure; at any termina, freight (either originating or transferring) with the same fina destination wi aways be oaded to the same next termina. LTL carriers pack shipments into 28-foot traiers known as pups, or 53-foot vans. Typicay, one tractor pus either a singe van or two pups in a so-caed dispatch. A traier direct move may be executed by a singe dispatch, or may require mutipe dispatches aong the egs of the traier path associated with the direct move. In case a traier path consists of mutipe egs, the freight is reayed at the intermediate terminas. Reaying is frequenty necessary because of the imitations imposed upon drivers by reguations and work rues. In the U.S., reguations dictate that a driver is not aowed to drive for more than 11 hours or be on duty for more than 14 hours before requiring a rest period of at east 10 hours. Therefore, when the trave time between the origin and destination of a direct move is more than a singe driver can cover without a rest period, one or more reays 4

5 are introduced to ensure timey transfer of the freight. Usuay, traier reay operations occur at breakbuk terminas, athough they may aso occur at specia reay faciities. At a reay point, the oad is transferred to another driver and continues with minima deay. For exampe, a direct move for one LTL carrier from Daas to San Francisco requires 27 hours of drive time and invoves two reays and three drivers. It happens frequenty that different direct moves may incude common egs in their traier paths. For exampe, both the Daas-to-San Francisco and the Daas-to-E Paso direct moves incude the Daas-E Paso eg in their respective traier paths. Most carriers today use pups in their inehau operations because a pup fis up more quicky than a van, and by combining pups with different fina destinations it is possibe to buid oads that can be dispatched earier, and thus improve service. For exampe, a pup on the Daas-San Francisco direct move can be paired up with a pup on the Daas-E Paso direct move to form a dispatch on the Daas-E Paso eg. Effectivey expoiting the advantages of using pups requires proper pup matching at breakbuks and reays, i.e., deciding which pups to pair up into dispatched oad. Note that empty and oaded pups can aso be matched. Driver management is a compex task for LTL carriers, since numerous rues govern how drivers can be used and are compensated (e.g., a driver is compensated for a ong rest away from his domicie to cover meas and accommodation). Furthermore, carriers are concerned about the quaity of ife of their drivers and want them resting at their domicies with some frequency. In fact, LTL carriers often execute empty movements in order to return drivers to their domicies, and ignoring this key component of cost distorts traditiona cost estimates. 3 Reated Literature Load pan design eads to chaenging optimization probems. Eary research in this area focused on reativey simpe oca search heuristics for modes based on static networks that do not expicity capture service standard constraints or the timing of consoidation opportunities (see Powe and Sheffi (1983), Powe (1986), Powe and Sheffi (1989), and Powe and Koskosidis (1992)). In Powe and Farvoden (1994), a dynamic mode is presented that more accuratey modes consoidation timing, and an aternative heuristic is deveoped that reies on determining service network arc subgradients by soving arge-scae muti-commodity network fow probems. The approach, however, aows freight for a specific origin-destination pair to be moved over mutipe paths, and does not mode empty equipment baancing decisions. Jarrah et a. (2009) deveop a coumn generation approach to create oad pans where coumns represent freight path trees into a destination. A sope scaing heuristic is used to inearize costs when generating coumns. The approach expicity modes consoidation timing, but ony uses a singe time point per day per termina, so it cannot capture many tighty constrained freight paths usefu for 1- and 2-day service offerings. 5

6 Erera et a. (2009) deveop optimization technoogy to create improved oad pans, incuding day-differentiated pans, for a major super-regiona U.S. LTL carrier. The approach uses a detaied time-space network representation to accuratey mode consoidation in time. However, it reies on severa important simpifying assumptions for tractabiity: 1. Each day in the panning horizon is discretized into 8 decision epochs, varying in ength from 2 hours to 5 hours; 2. Loaded transportation costs are assumed to be inear in the number of traiers required for each direct move; and 3. Empty transportation costs are assumed to be inear in the number of empty traiers dispatched between terminas, and empty dispatches are determined ony to correct traier imbaances created by oaded traier movements. These assumptions, whie reasonabe, sti provide ony a rough approximation of true transportation costs. In the research described in this paper, we mode LTL operations at a finer eve of detai. Probems aso reated to the one we consider in this paper are the focus of Cohn et a. (2007) and Root and Cohn (2008), in which soution approaches are deveoped that integrate empty baancing with a pup matching and routing for sma package express carrier operations. The proposed set partitioning mode uses composite variabes that define compete paths for one or more traiers, and empoys tempates to imit the set of such composites generated. Whie we aso consider a pup oading and matching probem, our matching probem is simpe since we assume that the best traier path is known for each direct. Furthermore, we estimate transportation costs more precisey since we construct feasibe driver tours to cover oads. Erera et a. (2008) empoy greedy approaches to construct driver tours that cover dispatches; in this paper, we deveop an optimization-based set covering heuristic. 4 Mode Formuation As mentioned in the introduction, our focus is not oad pan design, but accuratey estimating the operationa execution costs of a given oad pan. A number of modeing choices were made when formuating the probem. These choices are discussed beow. 1. The probem is formuated on a time-space network. Fat network representations, i.e., networks without an expicit time dimension, such as the ones used in Powe and Sheffi (1983), Powe (1986), Powe and Sheffi (1989), and Powe and Koskosidis (1992), are based on two important assumptions: (1) the tota traier oads needed on a direct during the panning horizon can be determined by assuming that a freight traveing at any time within the panning 6

7 horizon can be consoidated; and (2) service standard constraints can be modeed by using a proxy, e.g., by ensuring a minimum traier frequency on a direct per day. In today s LTL market where 1-day and 2-day service have become the norm, these assumptions are no onger vaid. It is necessary to use a representation that can expicity represent time. A detaied time-space network mode aows consoidation timing and service standards to be modeed accuratey. Given a time discretization of the panning horizon, mutipe nodes are created for each termina, one for each time point, so that each node represents a ocation and a point in time. For each eg in the inehau network, we create mutipe transportation arcs in the timespace network, each representing the possibiity to move freight at a particuar time. Each node in the time-space network is connected with an arc to the node representing the same termina at the next time point, thus modeing the possibiity to hod freight at a termina. See Figure 2 for an iustration. Figure 2: Time-Space Network 2. The panning horizon considered is a week. The freight voumes within a week often exhibit marked variabiity by day-of-week, but freight patterns tend to be simiar across weeks. As a resut, carriers have started to expore day-differentiated oad pans, i.e., oad pans that aow for different freight routing decisions on different days of the week. Carriers often out-source a portion of their transportation needs to third-party carriers, a practice referred to for the remainder of the paper as purchased transportation. Usuay, the third-party carriers are rairoads, but occasionay aso trucking companies are used. Transporting freight by rai is cheaper, but sower than by truck. Since weekend days do not count 7

8 against service, carriers often utiize rai transportation over the weekend. In fact, most rai options are ony avaiabe near the end of the week. Purchased transportation schedues tend to repeat weeky. The above discussion suggests that a week-ong panning horizon is appropriate. To accuratey capture daiy freight voume fuctuations, we mode freight originating at a termina on a given day and destined for another termina on another day as a commodity, with a size measured in fractiona traieroads. Arcs representing purchased transportation options are ony created at their schedued time of the week. 3. Time is discretized in hours. Time must be modeed at a fine eve of granuarity for two reasons: (1) to be abe to accuratey mode the driver rues discussed in Section 2, and (2) to be abe to propery mode freight paths between origin-destination pairs with tight service standards. Consider the freight path encountered at a super-regiona carrier for freight avaiabe in Lexington, KY at 7 pm and due in Graying, MI at 8 am the next day shown in the top part of Tabe 1. Trave from Lexington, KY to Cincinnati, OH for 2 hours Spend 0.5 hour being handed at Cincinnati, OH Trave from Cincinnati, OH to Toedo, OH for 3.75 hours Spend 0.5 hour being handed at Toedo, OH Trave from Toedo, OH to Graying, MI for 4.46 hours Leave Lexington, KY at 19:00, arrive at Cincinnati, OH at 21:00 Finish handing at Cincinnati, OH at 21:30 Leave Cincinnati, OH at 22:00, arrive at Toedo, OH at 01:45 Finish handing at Toedo, OH at 02:15 Leave Toedo, OH at 03:00, arrive at Graying, MI at 07:27 Tabe 1: Modeing Freight Paths An houry time discretization, i.e., constructing a node at every hour, aows us to accuratey mode this freight path by timing the dispatches as shown in the bottom part of Tabe Freight enters the inehau system at 7 p.m. and eaves the inehau system at 8 a.m.. A freight picked up during the day is assumed to be ready to be send into the inehau system at 7 p.m. oca time. A freight to be deivered during the day must arrive at its destination termina at 8 a.m. oca time. Thus, we mode the freight that enters the inehau network at termina t 1 on day d 1 and is due at termina t 2 on day d 2 as originating in the time-space network at node n 1 = (t 1, 7 p.m. d 1 ) and is destined to node n 2 = (t 2, 8 a.m. d 2 ). 8

9 5. Handing 1-day service freight takes 30 minutes and handing a other freight takes 2 hours. A certain amount of time is required for handing freight at intermediate breakbuks. Specia handing procedures are generay used at breakbuks to prioritize the processing of 1-day freight to ensure that it can meet its service expectation. Therefore, a short handing time for 1-day freight is appropriate. We note that handing of freight can ony occur during business hours, which typicay start at 12 am on Monday morning, and end Saturday at noon. The terminas are, however, accessibe to drivers arriving/departing a weekend ong. 6. Modeing fu truckoad freight. To diversity their offerings, and because it is more profitabe, many LTL carriers run a sma fu truckoad operation as we. Fu truckoad shipments do not require any intermediate handing, but the traiers used for fu truckoad service are reayed aong egs according to the oad pan. LTL carriers frequenty fi truckoad traiers with LTL shipments to expoit any remaining traier capacity. Therefore, fu truckoad freight shoud be considered when estimating the execution cost of a oad pan. We can now state the LTL oad pan cost estimation probem as the probem of determining a freight path for each commodity in the time-space network, conforming to the oad pan, and creating vaid driver tours to cover the resuting dispatches with minima tota cost over the week. Since handing costs are fixed given a oad pan, minimizing tota cost is equivaent to minimizing the transportation cost required to move empty and oaded traiers. As we wi describe in the foowing section, tota transportation cost in this probem is assumed to be the sum of the costs of executing the set of driver tours necessary to move a empty and oaded traiers. 5 Soution Approach We have designed and impemented a three-phase soution approach for the LTL oad pan cost estimation probem. 5.1 Phase I: Loading and Matching Traiers In the first phase, we determine a timed path for each commodity to generate oaded traiers and dispatches. As an approximation, we assume that the freight for each commodity can be spit arbitrariy across mutipe traiers; this assumption is not overy optimistic since the tota fow for a commodity is comprised of many shipments, and since in practice this fow is rarey spit across more than two traiers. We deveop a GRASP heuristic to determine the timed paths. The GRASP heuristic chooses a path for each commodity sequentiay using a shortest path agorithm; note that since the oad pan is fixed, determining the path for a commodity is simpy seecting the dispatch times for each 9

10 of the direct moves in the oad pan path (represented by arcs in the time-space network). The sequentia nature of the search enabes us to estimate the margina cost of adding the commodity under consideration to a possibe dispatch arcs, and thus to minimize the margina cost increase that resuts from seecting a set of feasibe dispatch times. Dispatch costs for purchased transportation direct moves are inear in the number of traiers oaded. For reguar moves, dispatch costs are inear in the number of two-pup trains oaded aong each eg of the traier path. We now iustrate how to compute the margina cost of adding a commodity of size c (measured in fractiona traieroads) to arc a representing a eg of the traier path of non-purchased direct d 0 ; a simiar computation is used for computing the margina cost on a purchased transportation direct. Suppose that d 0,..., d m are the directs whose traier paths incude arc a as a eg and that the dispatch cost (per traier pair) on arc a is p a. Furthermore, et w 0 be the existing freight (measured in fractiona traieroads) on arc a for direct d 0 and et e i be the current number of traiers on eg a for direct d i, i = 0,..., m. Finay, et U be the capacity of a traier. Adding commodity c to arc a changes the required number of traiers on eg a for direct d 0 from e 0 to w0+c U. Hence the margina dispatch cost is ( w 0+c U + m 2 i=1 e i m i=0 e i 2 ) p a. (1) The sequence in which the commodities are processed impacts the paths chosen, hence we decided to impement a GRASP heuristic. Let a commodity s sack time be defined as the maximum ength of time it can be hed at its origin such that it can sti be dispatched aong a path that satisfies the service deadine. A arge sack time is an indication of more fexibiity for choosing dispatch times aong the path, and hence more opportunities for taking advantage of avaiabe capacity on traiers aong the way that have been opened for transporting other commodities. On the other hand, a sma sack time for a commodity impies that there is itte or no fexiity for choosing dispatch times. Therefore, for a given direct in its oad pan path, uness there is a traier with sufficient remaining capacity dispatched at the exact time required by this commodity, a new traier must be opened to accommodate this commodity. Ceary, it is better to open such new traiers earier in the heuristic to aow other commodities to fi in any remaining unused capacity therein. This suggests that we process commodities with sma sack times first. Furthermore, a commodity with smaer size c is more ikey to be abe to take advantage of remaining capacity in open traiers; thus, we break sack time ties between commodities by choosing those with arger sizes first. The GRASP heuristic is described in Agorithm 1. Note that it is simpe to determine the actua dispatches moving on each arc representing a eg in time, given the method of seecting the timed paths. Since we assume that each commodity is spittabe across traiers, we simpy create traiers (and two-pup traier trains) whenever a traier (or train) becomes fu in the process above. These traier pairs (and singe traiers, when necessary) 10

11 Agorithm 1 GRASP for Traier Loading and Matching Sort the commodities in order of increasing sack time. In case of ties, sort the commodities in order of decreasing weight. for i = 1 to N do Create a copy of the commodity ist whie the ist is not empty do Seect a commodity from the ist biased towards the top, i.e., the k-th commodity c k with probabiity λ (1 λ) k 1, k = 1, 2,... Find the east-margina-cost path for c k that conforms to the oad pan, using the cost in (1) Remove c k from the commodity ist end whie if an improved soution is found then Update the best soution end if end for wi be referred to as dispatches, and the commodities contained within each dispatch wi be used in Phase II to determine its time fexibiity (if any). 5.2 Phase II: Determining Dispatch Windows Note that in Phase I, a timed path is seected for each commodity, and thus the dispatches that have been constructed a have been assigned specific dispatch times. However, the shipments comprising a dispatch may not be tighty constrained by service, and thus there may be some fexibiity in the seection of actua dispatch time of each dispatch. Such fexibiity wi be beneficia when buiding driver tours. Therefore in Phase II, we use a inear program to determine dispatch windows for each dispatch constructed in Phase I. Let P be the set of dispatches buit in Phase I corresponding to purchased transportation. Since purchased transportation takes pace on fixed schedues (reca that purchased transportation is typicay provided by rairoad companies), these dispatches are not atered. Let L be the set of dispatches, or oads, buit in Phase I corresponding to transportation provided by company drivers. Our goa is to find for each oad i L an eariest and a atest dispatch time, denoted by α i and β i respectivey, such that if a oads are dispatched between their eariest and atest dispatch times, a freight can be moved feasiby from origin to destination, i.e., every shipment reaches its destination at or before its due time and can make feasibe connections at transfer and reay points. See Figure 3 for an iustration of a dispatch window. We introduce some additiona notation before discussing a inear programming mode for this 11

12 Figure 3: Dispatch Window probem. Let o c and d c denote the ready time at the origin and the deadine at the destination for commodity c. Furthermore, et p c 1,..., p c n c denote the sequence of dispatches for commodity c. Finay, et dt p c k, tt p c k, ht p c k, and ft p c k, be the dispatch time, trave time, handing time, and finish time, respectivey, of dispatch p c k, where the dispatch time and the finish time are given by the timed path seected for commodity c in Phase I. Our goa is to determine oad dispatch windows that provide the most fexibiity, and therefore the objective function of the inear program must be chosen consistent with this goa. We have chosen to maximize the sum of the widths of the individua dispatch windows for each oad. An aternative is to maximize the minimum width of any dispatch window. However, since there typicay are a few dispatches without any fexibiity, this objective does not produce usefu information in this case. The inear program to determine dispatch windows is as foows Maximize (β i α i ) i L subject to α p c 1 o c c, p c 1 L (2) β p c n c + tt p c nc dc c, p c n c L (3) β p c + tt p c + ht p c α p c +1 (4) c, 1 n c 1, p c L, p c +1 L β p c + tt p c + ht p c dt p c +1 (5) c, 1 n c 1, p c L, p c +1 P ft p c α p c +1 (6) c, 1 n c 1, p c P, p c +1 L α p c dt p c β p c c, 1 n c, p c L (7) Constraints (2) ensure that the first dispatch occurs no earier than the atest ready time at the 12

13 origin of a of the constituent commodities, and constraints (3) ensure that ast dispatch is such that a freight arrives at the destination before its deadine. Constraints (4), (5), and (6) ensure feasibe connections at transfer and reay points; note that a commodities carried within the dispatches must make feasibe connections. Finay, constraints (7) forces the dispatch times on the timed path seected in Phase I to be feasibe. The inear program presented above ignores one important probem characteristic: terminas operate ony imited hours over the weekend. Therefore, we may have produced dispatch windows that require handing to take pace during weekend hours. A post-processing step is added to fix such situations. More specificay, the predecessor s atest dispatch time is pushed back or the successor s eariest dispatch time is pushed forward, whichever appicabe, to their dispatch times on the timed path seected in Phase I to be feasibe, since a connections are feasibe with these dispatch times. The post-processing steps are given in Agorithm 2. Agorithm 2 Post-Processing Dispatch Windows for a commodity c do for = 1 to n c 1 do if p c L and c requires a handing after traveing on pc and the time period from β p c + tt p c to β p c + tt p c + ht p c does not fa entirey in the business hours then τ start of next business day + ht p c if p c +1 L and τ > α p c +1 then β p c dt p c α p c +1 dt p c +1 ese if p c +1 P and τ > dt p c +1 then β p c dt p c end if end if end for end for 5.3 Phase III: Constructing Driver Tours In the third phase, we determine driver tours to cover a dispatches in the set L; note that the dispatches in the purchased transportation set P are easy to cost, and thus are ignored in this phase. Each driver tour can be performed by a singe driver. It begins and ends at a driver domicie, and thus forms a timed cyce, and consists of one or more duties. A duty is a feasibe sequence of timed dispatches that can be performed in a singe 24-hour period and conforms to hours-of-service reguations. If a tour contains mutipe duties, the duties are separated by a required rest period of 13

14 appropriate duration. We use the current (2009) U.S. hours-of-service reguations that impose the foowing restrictions on drivers: a driver is aowed to drive up to 11 hours in a duty, a duty must not exceed 14 hours, and a driver must rest for at east 10 hours between duties. Note that duties may incude empty dispatches. If we assume that drivers are aways dispatched with two traiers (either oaded or empty), empty traier baance over time is impied at a terminas. For this reason, neither Phase I or Phase II above incudes any consideration of empty traier baancing decisions. LTL companies must compensate drivers for ong rests (between duties) spent away from their domicies, referred to as ay-downs. Lay-down costs typicay incude hote room stays and meas. Most companies ike to have their drivers resting at their domicies with some frequency. Singe-man drivers typicay do not rest away from their domicie two nights consecutivey. Therefore, a tour consists of either one or two duties. If a tour contains two duties (the first ending away from the domicie, and the second returning to the domicie), the ong rest that separates the two duties shoud not exceed 14 hours (and must be at east 10 hours). A duty typicay contains no more than four dispatches. Since we want to buid company driver tours in this phase, we modify the time-space network by removing a purchased transportation arcs, and by adding ay-down arcs from every node to the nodes representing the same termina 10 to 14 hours into the future. This network is then used to generate coumns for a set-covering mode. For each i L, et A(i) be the subset of arcs in the time-space network associated with oad i that fa within its dispatch window. Since A(i) and A(i ) with i i may contain common arcs, an arc does not uniquey identify a oad. For each arc a, et L(a) = {i L a A(i)} be the set of oads that can potentiay use a. In the set covering mode the goa is to seect a subset of tours covering a the dispatches at minimum cost. Let T be the set of a tours, c t be the cost of executing tour t T, a it be the number of times tour t T covers oad i, and z i be the number of dispatches required for oad i. If x t represents the number of times tour t T is executed, then we have the foowing integer programming formuation: Minimize subject to c t x t (8) t T a it x t z i t T i L x t Z + t T Since the set of tours is too arge in practice to consider expicity, we rey on coumn generation to sove the inear programming reaxation. Given a dua soution π to the inear programming reaxation of a restricted master probem, the pricing probem seeks to identify a tour with negative reduced costs. More specificay, the pricing probem seeks a tour minimizing a t (c a max i L(a) π i ). 14

15 Note that because mutipe oads may be covered with the same arc, we must use max i L(a) π i to determine the dua vaue to use for an arc. The pricing probem is a resource-constrained shortest path probem with arc cost c a max i L(a) π i. We track four resources to ensure the feasibiity of the tour identified by the constrained shortest path procedure: the duty time, the driving time in a duty, the number of dispatches in a duty, and the number of ay-downs in a duty. Let d a be the driving time on arc a and H be the ay-down cost. Then the resource extension functions for the various arc types and the resource imits are summarized in Tabe 2. Initia vaue Resource extension functions Transportation arc a Waiting arc Lay-down arc Resource imits at a node Duty time 0 + d a + 1 reset to 0 [0,14] Driving time in duty 0 + d a unchanged reset to 0 [0,11] Num of dispatches in duty unchanged reset to 0 [0,4] Num of ay-downs in duty 0 unchanged unchanged + 1 [0,1] Cost 0 + (c a max i A(a) π i) unchanged + H Tabe 2: Resource Extension Functions and Resource Limits We round up the driving time when updating the duty time abe because dispatches ony occur at points in the time discretization, i.e., whoe hours. Note that the duty time abe is ony used in the dynamic programming agorithm. Actua duty times may be sighty smaer, and these are aso computed and used when reporting duty times in our computationa study. We sove the resource-constrained shortest path probem using a standard dynamic programming approach (see Irnich and Desaunier (2005) and Desrosiers et a. (1995) for discussions of dynamic programming approaches for constrained shortest path probems). One sight modification is that in the path extension step, a waiting arc and a ay-down arc are not aowed to immediatey foow each other in order to prevent undesirabe ong rests. The foowing ideas were used to acceerate the coumn generation process: We do not sove the pricing probem competey, but terminate the search as soon as a feasibe tour with a negative reduced cost is found, and then add the newy identified coumn to the restricted master probem, which is then re-soved. We restrict the search to tours that start with a oaded dispatch. This does not precude good soutions, but speeds up the search consideraby. Furthermore, we sort the oads in order of increasing cost c i π i and seect oads in that order to start a tour. Because ony one coumn is added in each iteration, many dua prices wi not change between successive pricing probem soves. It is thus reasonabe to assume that a oad that faied to produce a tour with a negative reduced cost wi ikey continue to do so in the near future. 15

16 Hence, we excude such oads from consideration for a number of iterations to speed up the search. The agorithm to find a ow-cost set of driver tours covering a dispatches is provided in detai in Agorithm 3. Agorithm 3 Load Covering Generate a set of initia coumns repeat Sove the inear programming reaxation of the restricted master probem Retrieve the dua prices Sort the oads in increasing order of (c i π i ) for i = 1 to L do if oad i is not excuded from consideration then Invoke a dynamic programming search for a tour that starts with oad i whie a tour with a negative cost is not found and there are feasibe extensions do Perform a dominance check and a path extension end whie if a desirabe tour is found then ese Add a coumn representing the tour break oop Excude oad i from consideration for M iterations end if end if end for unti a coumn with negative reduced cost is not found Sove a set covering probem over the coumns in the restricted master probem Meet-and-Turns and Initia Coumns It is we-known that a good set of initia coumns can reduce the running time of a coumn generation agorithm. However, before presenting our approach for creating initia coumns, we must describe meet-and-turns, which are used by LTL carriers on ong egs to reduce ay-down costs. A meet-and-turn can occur when two drivers move oads in opposite directions on a eg that is onger than haf of the maximum aowed driving time in a duty, i.e., 5.5 hours. Without intervention, the drivers moving these oads wi be unabe to return to their domicies at the end of the day because they woud vioate the driving time imit. A meet-and-turn, iustrated in Figure 4, instead has the 16

17 two drivers meet at a ocation aong the eg, exchange their oads, and then return to their respective starting ocation. This ensures that both oads arrive at their destination on time, and both drivers get back to their domicies at the end of the day. A parking ot or a rest area usuay suffices as a meet-and-turn ocation. Executing meet-and-turns reduces ay-down expenses for the carriers and improves the quaity of ife for the drivers. Figure 4: Meet-and-Turn To generate a set of initia coumns, i.e., driver tours, we use tempates of desirabe driver tours. Some of these tempates invove meet-and-turns. The agorithm for creating initia coumns is described in Agorithm Short Driving and Duty Times Hours-of-service reguations, which are motivated by safety considerations, ony restrict maximum driving and duty times. Therefore, short driver duties with short driving times are ega, but may not be cost-effective. When non-unionized carriers pan short driver duties, they often attempt to do so by aowing for dua-use of the drivers by empoying them aso for oading and unoading traiers (so-caed dock work), and by reducing the staff of dedicated dock workers accordingy. In genera, short driver duties are therefore acceptabe, but typicay ess desirabe than onger driver duties. In this section, we propose a penaty-based approach that enabes the anaysis of the tradeoff between the quaity of the tours (in terms of duty time and driving time) and the execution costs. The approach penaizes short duties with a term in the objective function that is proportiona to the difference between the actua and the maximum aowed driving time in a duty, i.e., 11 hours. The reason that we penaize short driving times rather than short duty times is because waiting between dispatches is counted towards duty time; penaizing short duty times woud create incentives for adding waiting time to duties, and this shoud not be encouraged. Let n t be the number of duties in tour t (either 1 or 2), et α be a parameter indicating the weight assigned to the penaty term, and et LD be the set of ay-down arcs. Then the cost of a 17

18 Agorithm 4 Creation of Initia Coumns Let SL contain oads that require ess than or equa to 5.5 hours of driving and et LL contain oads that require more than 5.5 hours of driving for a i, j SL, i j do if i and j can form a feasibe out-and-back tour without a ay-down then Create a coumn representing the tour end if end for for a i, j LL, i j do if i and j can form a feasibe meet-and-turn then Create a coumn representing both tours in the meet-and-turn end if end for for a i, j LL, i j do if i and j can form a feasibe out-and-back tour with a ay-down then Create a coumn representing the tour end if end for for a i I = SL such that i has not been covered by any tour do Create a coumn representing an out-and-back tour with an outbound dispatch moving i and empty inbound dispatch end for for a i I = LL such that i has not been covered by any tour do Create a coumn representing a meet-and-turn consisting of i and an empty dispatch in the opposite direction end for 18

19 tour t becomes ( ) ( c a max π i + α 11 n t ) d a i A(a) a t a t = α 11 (1 + ) 1 + ( ) c a max π i α d a i A(a) a t LD a t = α 11 + ( ) c a max π i α d a + 1 LD (a) α 11 i A(a) a t which is sti additive on arcs. Therefore, the same soution methodoogy can be appied to the probem with a short driving time penaty with ony minor modifications. A that is required is to adapt a few eements in the ast row of Tabe 2, as depicted in Tabe 3. Initia vaue Transportation arc a Waiting arc Lay-down arc Cost α 11 + (c a max i A(a) π i) α d a unchanged + H + α 11 Tabe 3: Cost Extension with Penaty 6 Computationa Study In this section, we present the resuts of a set of computationa experiments conducted to tune and anayze the performance of our proposed LTL oad pan cost estimation technoogy. We use four instances, each representing an actua week of shipment data of a super-regiona LTL carrier in the U.S. The carrier s inehau network consists of 253 terminas (end-of-ines, breakbuks, and reays) and 8,152 inehau egs. The carrier transports over 140,000 shipments every week. Each week begins on a Sunday at 12:00 a.m., and concudes the foowing Saturday at 11:59 p.m. Tabe 4 gives the start and end dates of the weeks used in our computationa experiments. Instance Start date End date W 1 March 01, 2009 March 07, 2009 W 2 March 08, 2009 March 14, 2009 W 3 March 15, 2009 March 21, 2009 W 4 March 22, 2009 March 28, 2009 Tabe 4: Weeks Used in Our Computationa Study A computationa experiments were executed on a Linux system with a 2.66 GHz Inte Xeon processor and 4 GB of RAM, and use CPLEX 11.1 as the optimization engine. 19

20 6.1 GRASP Heuristic Parameters The first experiment is designed to determine the parameters λ and N for the GRASP heuristic for oading and matching traiers. For λ = 0.1, 0.2,..., 0.9, 1.0, we et Agorithm 1 run for 100 iterations and monitor the progress of the vaue of the best soution found. Note that when λ = 1, the agorithm reduces to a greedy heuristic and the behavior of the agorithm is deterministic, so there is no benefit of performing more than one iteration. Each iteration takes approximatey 4 minutes to run. Figure 5 and 6 show the progress over time for weeks W 1 and W 4 (simiar behavior was observed for weeks W 2 and W 3). Figure 5: Progress of GRASP Heuristic for Instance W 1 The resuts indicate that athough a greater eve of randomization, i.e., a smaer vaue of λ, tends to ead to a sighty better soution over time, the benefit is minima; the difference between the overa best soution vaue and the one found by the greedy heuristic is ess than 0.40%. Hence, for the rest of the computationa experiments we wi use the greedy heuristic. 6.2 Dispatches and Dispatch Windows Next, we present a few statistics reated to the dispatches buit in Phase I and the dispatch windows determined in Phase II. (Note that the dispatch windows are determined using a reativey sma inear program, and thus are computed in a matter of seconds.) In Figure 7, we show the number of dispatches occurring at particuar times during the day as determined by Phase I. We see that most dispatches occur between 7 p.m. and 6 a.m. This is not unexpected, and in ine with what happens in practice, as a significant portion of shipments have 20

21 Figure 6: Progress of GRASP Heuristic for Instance W 4 1-day service guarantees, which impies that they have to be moved between 7 p.m. and 8 a.m. If possibe, freight with onger service standards is aso moved within this time frame so that it may be consoidated with the 1-day freight. In Figure 8, we show the distribution of the widths of dispatch windows as determined by Phase II. As is evident from the figure, a few dispatches have itte or no fexibiity and must be dispatched according to a specific schedue to make service; most ikey these represent shipments in reativey ong corridors with a 1-day service guarantee. At the other end of the spectrum are a few dispatches that have quite a bit of fexibiity; most ikey these represent shipments on origin-destination pairs that are reativey cose, but have a 5-day service guarantee. From an operationa perspective, the most reevant information is that most dispatches have some fexibiity, and this fexibiity can be expoited to buid ow-cost driver tours. It is aso insightfu examine the dispatch windows on a singe inehau eg in more detai. Figure 9 shows a the Markham-Chicago dispatches and their dispatch windows; note that Markham is another ocation in Iinois, not far from Chicago. A few interesting observations can be made. First, the dispatches occurring at 7 p.m. and 8 p.m., which ikey incude a substantia portion of the shipments picked up during the day, have itte or no fexibiity. These dispatches are the used to move shipments with a 1-day service standard. Furthermore, we note that the dispatches between 10 a.m. and 6 p.m. have the most fexibiity. These ikey contain shipments bound for Chicago or for a further termina with service eves that can reativey easiy be achieved. 21

22 Figure 7: Dispatch Pattern 6.3 Coumn Generation and IP Optimization Next, we consider parameter tuning for the coumn generation and IP optimization processes at the heart of Phase III. Reca that during the coumn generation process, if an attempt to buid a tour with a negative reduced cost starting with a particuar oad fais, we excude that oad from consideration for the next M iterations to hopefuy avoid wasting computing time on an unsuccessfu search. The tradeoff between the computing time and the vaue of the fina LP soution when we vary M is shown in Figure 10; for M = 50, 100, 1, 000, 10, 000,. We see that re-visiting oads provides a sma benefit, but it comes at a very high price in terms computing time. Hence, for the remaining computationa experiments, we use M =, i.e., we wi not re-visit a oad again once our attempt to buid a tour with a negative reduced cost starting from that oad fais. Next, we provide more detais about the initia coumns generated using structured tempates; the tempates are summarized in Tabe 5. Figure 11 shows the composition of the coumns in the initia Tempate code Dispatch ength Type Lay-down Loaded/empty dispatch SHORT-OB-LD ess than or equa to 5.5 hours out-and-back No both oaded LONG-MT-LD more than 5.5 hours meet-and-turn No both oaded LONG-OB-LD more than 5.5 hours out-and-back Yes both oaded SHORT-OB-EMT ess than or equa to 5.5 hours out-and-back No 1 oaded and 1 empty LONG-MT-EMT more than 5.5 hours meet-and-turn No 1 oaded and 1 empty Tabe 5: Tempate Types LP soution and the fina LP soution in terms of their structure, i.e., the tempate corresponding to their structure. Of course, in the fina LP soution we encounter structures that were not present 22

23 Figure 8: Dispatch Window Widths in the initia LP soution. These structures are umped together under the tempate COLGEN. For exampe, coumns representing tours with duties invoving more than two dispatches wi end up under this tempate. This incudes, for exampe, trianguar duties, i.e., duties of the form A-B-C-A (dispatches AB, BC, and CA), which can be quite effective. Coumn generation is used precisey to generate such duties if desirabe. The figure demonstrates that using these more compicated structures substantiay reduces the use of inefficient structures with empty dispatches. Finay, and most importanty, in Tabe 6 we report the vaue of the fina LP soution and the vaue of the IP soution generated using the coumns in the fina LP soution (where the stopping criterion for the IP sove was an optimaity gap of ess than 0.1%). Since IP soution vaues are very cose to LP soution vaues, it appears that it is reasonabe not to perform additiona coumn generation within the branch-and-bound process. Vaue LP soution Vaue IP soution W 1 3,247,583 3,249,398 W 2 3,267,249 3,269,375 W 3 3,311,176 3,314,513 W 4 3,350,560 3,353,409 Tabe 6: Comparison of LP and IP Soutions 23

24 Figure 9: Markham-Chicago Dispatches with their Dispatch Windows 6.4 Tour Structures In this section, we provide more detais about the structure of the driver tours generated. In Tabe 7, we report the number of tours with 1 duty and 2 duties, the number of duties with 1, 2, 3, and 4 dispatches, and the number of duties invoving meet-and-turns, and the number of oaded versus empty dispatches. We see that a reativey sma percentage of duties invove more than 2 egs. Since counts ony provide a partia picture, Figure 12 and 13 present the distribution of the driving and duty time of the duties. We see that the majority of duties have a driving time of more than 7 hours and a duty time of more than 9 hours, which is desirabe. However, a non-trivia fraction corresponds to short duties with short driving times. 6.5 Execution Cost Estimates The primary goa of this research was to deveop technoogy to accuratey estimate the operationa execution cost of a oad pan. To demonstrate that we have achieved our goa, we compare for the four instances, in Figure 14, the actua execution costs incurred by the carrier, the execution cost estimate produced by SuperSpin (the current industry-standard for oad pan design), and our execution cost estimate. The actua execution costs are normaized at 100% and the two estimates are given as a percentage of the actua costs. The figure shows that our technoogy produces remarkaby accurate execution cost estimates, within 1.7% of the actua execution costs incurred for each of the four weeks. The figure aso shows that SuperSpin tends to under-predict execution costs (about 90% of the actua execution costs incurred), primariy due to an over-estimation of the 24

25 Figure 10: Impact of Varying the Number of Iterations to Excude a Load from Consideration consoidation opportunities that ikey resuts from the static network representation. Furthermore, we present in Figure 15 for our cost estimate the breakdown into oaded transportation costs, empty repositioning costs, and ay-down costs. 6.6 Varying Maximum Aowed Number of Dispatches in a Duty During the construction of tours, we imit the number of dispatches in a duty. This restriction is incuded for two primary reasons. First, duties with a sma number of egs are preferred by both drivers and the carrier. Second, imiting the number of dispatches per duty imits the number of feasibe duties and thus simpifies the pricing probem, which reduces the computing time. In the next experiment, we investigate the impact of varying the maximum number of dispatches aowed in a duty. Figure 16 shows the tota inehau cost and the number of coumn generation iterations versus the maximum aowed number of dispatches in a duty. When a duty is aowed to contain more dispatches, the technoogy is abe to generate more compicated and efficient driver tours, and thus to reduce the tota inehau cost. However, we see that the benefits of aowing more than 4 dispatches in a duty is negigibe. 6.7 Short Driving and Duty Times To this point in our computationa study, short driving and duty times were not discouraged. As we observed in Figures 12 and 13, a majority of the duties have a driving and duty time cose to their respective imits, but there are a fair number of duties with sma driving and duty times. 25

26 Figure 11: Tempate Uses in Soutions In Figure 17, we anayze the tradeoff between the quaity of the tours, measured in terms of their driving and duty time, and the operationa execution costs. We see that an increase in the average driving time of 1.5 hours and an increase in the average duty of 1 hour comes at an increase in operationa execution costs of approximatey 1%. 7 Future Research Load pan design technoogies must use simpifying assumptions to ensure computationa tractabiity, and thus may substantiay under- or over-estimate actua operationa execution costs. We designed and impemented technoogy that accuratey estimates the operationa execution costs of a given oad pan. One important next chaenge is to integrate oad panning and execution cost estimation technoogies. Buiding a oad panning methodoogy that expicity recognizes that driver tours wi be used to cover panned dispatches may have substantia promise in further improving LTL oad pan design. References Cohn, A., Root, S., Wang, A., Mohr, D., Integration of the oad matching and routing probem with equipment baancing for sma package carriers. Transportation Science 41, Desrosiers, J., Dumas, Y., Soomon, M. M., Soumis, F., Time constrained routing and schedu- 26

27 W 1 W 2 W 3 W 4 Tours Duties Dispatches 1 duty duties tota eg egs, non-meet-and-turn meet-and-turns egs egs tota oaded empty tota Tabe 7: Tour Structure ing. In: Ba, M. O., Magnanti, T. L., Monma, C. L., Nemhauser, G. L. (Eds.), Network Routing. Vo. 8 of Handbooks in Operations Research and Management Science. Esevier, Amsterdam, pp Erera, A., Hewitt, M., Savesbergh, M., Zhang, Y., Improved oad pan design through integer programming based oca search. Erera, A., Karacík, B., Savesbergh, M., A dynamic driver management scheme for ess-thantruckoad carriers. Computers and Operations Research to appear. Irnich, S., Desaunier, G., Shortest path probems with resource constraints. In: Desauniers, G., Desrosiers, J., Soomon, M. M. (Eds.), Coumn Generation. Springer, pp Jarrah, A. I., Johnson, E., Neubert, L. C., Large-Scae Less-than-Truckoad Service Network Design. Operations Research to appear. Manhattan Associates, ifecyce management/super spin.htm. Powe, W., Farvoden, J., Subgradient methods for the service network design probem. Transportation Science 28, Powe, W. B., A oca improvement heuristic for the design of ess-than-truckoad motor carrier networks. Transportation Science 20 (4), Powe, W. B., Koskosidis, I. A., Shipment routing agorithms with tree constraints. Transportation Science 26 (3),

28 Figure 12: Driving Time Histogram Powe, W. B., Sheffi, Y., The oad panning probem of motor carriers: Probem description and proposed soution approach. Transportation Research A 17A (6), Powe, W. B., Sheffi, Y., Design and Impementation of an Interactive Optimization System for Network Design in the Motor Carrier Industry. Operations Research 37, Root, S., Cohn, A., A nove modeing approach for express package carrier panning. Nava Research Logistics 55,

29 Figure 13: Duty Time Histogram Figure 14: Estimated Linehau Cost as a Percentage of Actua Execution Cost 29

30 Figure 15: Cost Breakdowns Figure 16: Impact of Varying the Maximum Number of Aowed Dispatches in a Duty 30