net21527.pdf

Forty Years of Periodic Vehicle Routing

Ann Melissa CampbellDepartment of Management Sciences, Tippie College of Business, The University of Iowa, Iowa City, Iowa

Jill Hardin WilsonDepartment of Industrial Engineering and Management Sciences, Northwestern University, Chicago, Illinois

The periodic vehicle routing problem (PVRP) firstappeared in 1974 in a paper about garbage collection(Beltrami and Bodin, Networks 4 (1974), 6574). The wideapplicability and versatility of the problem has led to avast body of literature addressing both novel applica-tions and solution methods. This article discusses thewide array of circumstances and settings in which thePVRP has been applied and describes the developmentof solution methods, both exact and heuristic, for thePVRP. As with many core research problems, many vari-ants have been proposed. We will describe additionalproblem variants and extensions, as well as discuss thefuture of research for the PVRP. 2013 Wiley Periodicals,Inc. NETWORKS, Vol. 63(1), 215 2014

Keywords: periodic routing; literature review

1. INTRODUCTION

In the standard periodic vehicle routing problem (PVRP),customers require visits on one or more days within a plan-ning period, and there are a set of feasible visit options ifor each customer i. Customers must be assigned to a feasiblevisit option i i, and a (VRP) is solved for each day in theplanning period. The typical objective is to minimize the totaldistance traveled over the planning period. In Figure 1, weprovide an example of a PVRP with a 2-day planning periodwhich initially appeared in [52]. In this example, customer 1must be visited twice so = {1, 2}. Customers 2 and 3 mustbe visited once so 2 = 3 = {{1}, {2}}. In Figure 1a, weshow the route for day 1, and in Figure 1b, we show the routefor day 2. Here, 1 = {1, 2}, 2 = {1}, and 3 = {2}, and thetotal distance over 2 days is 34.

The PVRP arises in a diverse array of applications,from the collection of recyclables, to the routing of home

Received December 2012; accepted April 2013Correspondence to: A. M. Campbell; e-mail: [email protected] 10.1002/net.21527Published online 1 October 2013 in Wiley Online Library(wileyonlinelibrary.com). 2013 Wiley Periodicals, Inc.

healthcare nurses, to the collection of data in wireless net-works. The wide applicability and versatility of the problem,coupled with the problems difficulty, has led to a vast body ofliterature addressing both novel applications and ever moresuccessful solution methods. The PVRP is truly a global prob-lem. Table 1 was created by recording the listed nationalaffiliation of the lead author of the references that are includedin this article, excluding survey papers and citations notspecifically related to periodic routing. Table 1 demonstratesthat interest in the study of the PVRP and its applicationsarises around the world, and it is not difficult to imaginesome variation of the problem arising in most any country,even if not represented by the set of publications cited here.

The PVRP was first introduced in Networks and VehicleRouting for Municipal Waste Collection by [12]. To the bestof our knowledge, this is the first time the periodicity of cus-tomer deliveries was specifically addressed in combinationwith the consideration of vehicle routing costs. Thus, [12] isthe parent publication of the many papers that focus on rout-ing problems with periodic customer deliveries. Motivatedby their work with the New York City Environmental Protec-tion Agency, the authors look at several interesting problemsrelated to the idea of garbage collection. The first problemthey consider is how to route vehicles to pick up garbage atlarge industrial sites to minimize both the number of vehiclesand the total travel time. This is very different than previ-ously studied vehicle routing problems because of the natureof the demand. Some sites need to be serviced three times aweek, whereas other sites require deliveries six times a week(which is daily in the NYC application). The customers thatare serviced three times a week can be served on Monday,Wednesday, and Friday; or Tuesday, Thursday, and Satur-day. This choice affects the daily routing problems, whichcan in turn impact the travel times and the number of vehi-cles required. Beltrami and Bodin [12] also introduce the firsttwo key heuristics used to solve the problem.

Ref. [12] is highly cited also in part because it introducedone of the earliest examples of the arc routing problem, in thiscase examining how to route street sweepers on the streets ofNew York to minimize deadheading. The arc routing problem

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FIG. 1. A PVRP with a two-day planning period.

also has a very rich history, but we will not address it here.For excellent surveys on arc routing problems, see [8], [36],[37], and [33].

In any search online or scan of review papers, it becomesquickly apparent that there are two other very influentialpapers in the early history of periodic vehicle routing: Anassignment routing problem by R. Russell and W. Igo, whichappeared in 1979, and The periodic routing problem by N.Christofides and J.E. Beasley, which appeared in 1984. Bel-trami and Bodin [12] obviously predates the other two andis cited by both of them, but many papers related to periodicrouting cite [81] or [19] in lieu of [12]. Although Beltrami andBodin [12] introduced the problem, the papers by Christofidesand Beasley [19] and Russell and Igo [81] are highly citeddue to the important roles they played in the development ofthe PVRP.

Beltrami and Bodin [12] introduces the idea of consideringthe periodicity of site visits in conjunction with routing costs,but does not give this new problem a name. Russell and Igo[81] gives the periodic routing problem a name, calling itThe Assignment Routing Problem. They make the deliverytimes a little more flexible than in [12] by specifying thateach customer i receives deliveries on Si different days ina week, where 1 Si 7. There can also be additionalspecifications on which days of the week are acceptable. Thenumber of vehicles are given and capacity constraints mustbe followed.

Christofides and Beasley [19] is also well cited by periodicrouting papers, likely due in part to the fact that it namesthe problem the period routing problem and provides thefirst mathematical formulation for the problem. The authorsgeneralize the problem definition such that customers now

request k deliveries during the planning period and have agiven set of allowable k-day combinations.

The first article that uses the term periodic vehicle rout-ing appears to be [45]. This article is unlike previous onesin that it focuses on minimizing the fleet size. The authorsintroduce yet a different definition of periodicity from theabove three, enforcing that at least Ki days and at most Uidays must elapse between visits to customer i. All of theseearly publications also present heuristics to solve the problemthat are built on by later investigators. These methods will bediscussed further in section 4.

The PVRP is a challenging problem, with a rich his-tory of research spanning the last 40 years. In this article,we will describe the evolution of periodic vehicle routingas it has progressed since the 1974 paper, focusing on thebreadth of contexts in which it has been applied and solution

TABLE 1. National affiliations of lead authors.

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methods which have been proposed. In section 2, we willprovide problem descriptions for the PVRP and its moreprominent variants. Applications for periodic routing extendfar beyond garbage collection, and we will describe the vari-ety of applications where periodic routing occurs in section 3.We describe the development of solution methods, both exactand heuristic, for the PVRP in section 4. As with many coreresearch problems, many variants have been proposed. Wewill describe additional problem variants and extensions insection 5 and discuss the future of the PVRP in section 6.

2. PROBLEM DEFINITION

As noted above, the wide variety of papers on the PVRPdiffer in how strictly they define the allowable delivery sched-ule alternatives and, consequently, in the number of feasiblecombinations that can be generated. Three main representa-tions have been proposed in the literature for definining thedelivery options to a set of customers V . Some papers includea predetermined set of allowable alternatives as in [12] and[19], whereas others, such as [18] and [30], specify that deliv-eries to a customer i V must occur every ri days. Othersenforce constraints on the minimum and maximum requiredspacing between deliveries, as in [45]. We will consider allof these definitions of periodicity in our discussion of thePVRP. In fact, all of these representations can be viewed asspecific cases of the PVRP definition introduced earlier, inwhich there is a planning period T , and for each customeri V there is a set of feasible visit options i over T . Theaim, then, is to assign each customer to one of its allowablesets of service combinations i i, and to create dailyroutes subject to the following set of constraints:

all vehicles begin and end their day at a single depot, if a product is to be delivered or collected, the quantity will

be known and will be fully satisfied by a single vehicle, the number of vehicles is given, each vehicle has a limited capacity, and there is a limitation on the total travel time for each route.

Most papers on the PVRP have an objective of minimizingtravel costs subject to these limitations. We will focus on thisobjective, with minor variations in how travel costs are com-puted, in sections 3 and 4, but discuss alternative objectivesin section 5.

A few basic variants of the PVRP should be introducedearly in the article, as innovations in solving them are tied toinnovations in solving the PVRP. The periodic traveling sales-man problem (PTSP) is a special case of the PVRP restrictedto one vehicle. The PVRP with time windows (PVRPTW)generalizes the PVRP to include time windows for deliv-eries to the customers. In the multidepot VRP (MDVRP),each vehicle is assigned to one from a set of depots where itbegins and end each delivery route, and the planning horizonis restricted to a single day. Cordeau et al. [30] demonstratethat the MDVRP can be viewed as a special case of the PVRPby considering each of T depots to be a day on a T -dayplanning horizon, and each customer to require one deliv-ery over that horizon. The PVRP (MDPVRP), on the other

hand, adds periodic customer deliveries to the multidepotproblem. If the customers and vehicles are preassigned todepots, the problem decomposes into a set of PVRPs. If theproblem requires assigning customers to depots, solving theproblem now involves making these assignment decisions,representing a generalization of the PVRP.

The literature on the PVRP considers a number of variantsof the PVRP beyond these. Some of these variants requiresubstantial modifications to solution methods; these will bediscussed further in section 5. These include papers with sig-nificant differences in the objective function, constraints, orboth.

3. APPLICATIONS

The applications for the PVRP are quite diverse. We willreview many of these and discuss the nature of the periodic-ity where it is not obvious. We will categorize them primarilyby whether a product is being picked up, a product is beingdelivered, or a person or machine is being routed to per-form an on-site service. The applications we will discuss aresummarized in Table 2.

3.1. Pickup

As stated earlier, the motivation for Beltrami and Bodin[12] is the collection of industrial garbage in New York City,and many of the papers related to the PVRP also focus onthe collection of garbage and other wastes. Many model theresidential garbage collection as an arc routing problem as allcustomers have the same frequency and all customers on astreet are usually served together, but there are exceptions thattake more of a PVRP approach. For example, Nuortio et al.[67] solve a waste collection problem in Eastern Finland forresidential customers with varying collection requirementsas a PVRP. This article justifies the use of node routing, ver-sus arc routing, due to the sparsely distributed customers inthe service area. Matos and Oliveira [63] describe efforts toimprove residential garbage collection among 8087 sites inViseu, Portugal. Instead of using an arc routing approach,the authors reduce the 8087 sites to 202 localities and solvea PVRP among these localities. Angelelli and Speranza [7]use PVRP solution methods to evaluate the operational costsof different technologies for garbage collection. For exam-ple, they compare traditional pickup of garbage at individualhouses with having citizens put their waste into big streetcontainers that can be picked up by more expensive trucks.

The number of papers that involve the PVRP has definitelygrown as the emphasis on recycling has increased. As societalinterest in recycling grows, more recyclable products need tobe collected from consumers, creating higher costs that com-panies and cities want to minimize. Bommisetty et al. [15], forexample, consider the problem of collecting recyclable mate-rials at Northern Illinois University. The problem is modeledas a PVRP in which different buildings on campus requiredifferent numbers of visits per week and one truck is avail-able for collecting all types of recyclables. Baptista et al. [11]

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TABLE 2. Applications of PVRP.

present a case study of the collection of recycling paper con-tainers in a city in Portugal. Feasible combinations for eachcustomer are generated such that the containers would beat least 50% full on collection day. Unlike the traditionalPVRP, the objective function considered here is one of profitmaximization. Becuase recyclable products can be sold, thecost/benefit tradeoff of collecting a recyclable can be incorpo-rated in the problem. Teixeira et al. [88] look at how to collectthree types of waste (glass, paper, and plastic/metal) at 1642urban locations of varying delivery frequency in a regionin Portugal. Their problem becomes a PVRP for each typeof waste with a planning horizon of 4 weeks. Hemmelmayret al. [56] consider the periodic collection of recyclables, butthese recyclables are delivered to one of a set of intermediatefacilties, rather than the depot. This problem variant will bediscussed in some detail in section 5.

Other types of recyclables include waste vegetable oil([1]). New technologies are being developed to use wasteoil in the production of biodiesel. Unlike a regular PVRP,this problem also incorporates the selection of which poten-tial source points to include on pickup routes. Their workis motivated by a biodiesel production facility in Istanbul.Potential waste sources include restaurants, hotels, and hos-pitals. The periodicity emerges from the different rates atwhich the source points generate waste oil. A predeterminedproduction plan dictates the amount of input needed eachday, so service at all potential source points may not beneeded.

Besides garbage and recyclables, collection is needed forother waste products that are generated periodically. Shihand Lin[85] and Shih and Chang [84] model the collection ofinfectious waste at 384 hospitals and clinics in Taiwan as aPVRP. The motivation for this work is the lack of incinerators,or access to incinerators, and the importance of carefully dis-posing of infectious waste. Thus, the collection of this wastebecame an emerging need. The periodicity enters the prob-lem because hospitals are only able to hold waste for a certain

length of time (one week) and some require multiple visitsper week due to the volume of waste generated. Coene et al.[26] examine the collection of animal waste from slaughter-houses, butchers, and supermarkets in Belgium and northernFrance. The rate at which this waste is generated and thestorage capacity available are used to determine the numberof visits needed. For example, supermarkets tend to requirevisits every day, while smaller butchers require only weeklypickups. Legislation that originated due to the outbreak ofmad cow disease in the 1990s led to the categorization ofanimal waste into high-risk and low-risk categories. Vehi-cles assigned to collecting high-risk waste cannot be usedto collect low-risk waste and vice versa. Thus, the problemdecomposes into two different instances of the PVRP.

Other products are collected periodically besides wasteproducts. For example, Claassen and Hendriks [23] exam-ine how to improve goat milk collection in the Netherlands.Dairies only want to buy specified kinds of milk on specifieddays, so both the customers (farmers) and the dairies haveacceptable visit days. Alegre et al. [2] consider the collectionof parts for use in auto parts manufacturing in Northern Spain.The production schedules of the suppliers and the manufac-turer determine feasible visit sequences for each supplier. Thefocus is on how to solve instances with long time horizons ofup to 90 days. Goncalves et al. [50] model how to extract oilfrom wells in Brazil using mobile units. Each well needs aspecified number of days between visits for a sufficient quan-tity to be collected. Le Blanc et al. [61] examine the factorygate pricing problem where goods are not delivered from adistribution center (DC) but are picked up by the retailer atthe factory gates of the suppliers. The authors try to exam-ine the potential benefits of factory gate pricing for Dutchretailers. Retailers would have to make decisions about thefrequency of these pickups and the modes of transportation.In the proposed solution process, as different frequency allo-cations are considered, PVRP subproblems are created thatare solved with heuristics.


3.2. Delivery

The first application, besides garbage collection, of thePVRP to appear in the literature is in [49] where the authorsexamine the delivery of Coca-Cola products to stores. Thedrivers have their own territories that include the storeswhere they routinely sell their products. Some customersreceive deliveries once per week, where others, due to shelfspace, require deliveries twice per week. Grocery stores alsomotivate the study by Gaur and Fisher [46] of the peri-odic inventory routing problem. The maximum time betweendeliveries to stores is defined, and trucks must be routedfrom a single DC to satisfy these requirements with mini-mum transportation cost. Stores are grouped into clusters, sothat all stores in a cluster are replenished together.

Ronen and Goodhart [79] examine how to replenish overone thousand stores from several DC. Due to changes indemand patterns over the year, the frequency at which storesare visited may change, thus new routes need to be created.Most stores are visited between two and six times per week,and these deliveries need to be evenly spaced over the weekfor storage reasons. The workload at the DC to prepare forthese deliveries must also be considered due to limited pick-ing and loading capacity. Because each store is preassignedto a DC, the problems can be solved for each DC separately.Rademeyer and Benetto [78] also consider delivery to retailstores, where stores are visited between 1 and 6 days a week.These stores represent one of South Africas largest retailchains with over 275 stores.

Banerjea-Brodeur et al. [10] give another example ofdeliveries that are modeled using the PVRP. The authorsconsidered the delivery of hospital linens such as bedsheets,pillow cases, gowns, and towels to the 58 different clinicswithin the Jewish General Hospital in Montreal. The authorswere not concerned with the collection of linens, as theywere sent to the laundry via an elaborate chute system. Afterlooking at the historical usage by different departments inconjunction with the space limitations in each department,delivery frequencies could be established for each unit ofthe hospital. Once these frequencies were established, theproblem could be solved as a PVRP.

One of the biggest applications of the PVRP is in solvinginventory routing problems. In vendor-managed inventorysystems, the vendor decides when to visit his or her customersand how much to deliver to prevent the customer from run-ning out of product. Many of the papers on the inventoryrouting problem, particularly the early ones, decompose theproblem such that the first step is to use usage rate informa-tion about customers to determine a delivery frequency. Thisperiod between visits should be as long as possible to mini-mize delivery costs, but should be sufficiently small to keepthe customer from running out of product. Once this periodis known, the remaining problem can be solved as a PVRPfor a defined planning period. For example, Rusdiansyah andTsao [80] transform an inventory routing problem inspiredby the replenishment of vending machines into an instanceof the PVRPTW. The delivery frequencies for the vending

machines are selected to minimize a balance of inventorycosts and travel costs. In [13], the authors compute a fre-quency for each customer based on a modified economicorder quantity calculation and use these frequencies to definea PVRP. Hemmelmayr et al. [55] transform an inventory rout-ing problem for the blood bank for the Austrian Red Crossinto a PVRP. They evaluate the performance of their solu-tion techniques using data from the Austrian Red Cross from2003. For a thorough review of approaches used in solvingthe inventory routing problem, see [25].

The only application of the PVRP that involves bothpickup and delivery that we encountered is in [43]. Here, theauthors consider the distribution problem motivated by inter-library loan services for the North Suburban Library Systemin northern Chicago. Vans leave a sorting facility and peri-odically visit libraries to deliver interlibrary loan items andpick up items that need to be returned to the depot.

3.3. Routing for On-Site Service

Periodic routing problems emerge in the planning of routesfor salespeople and for those doing repair and maintenancework. For example, planning the routes for sales representa-tives for the Missouri lottery via the PVRP was the subjectof [58]. The 39 sales representatives for the lottery visit the5043 retailers that sell lottery tickets. Most retailers are vis-ited every 2 weeks, but the high volume retailers are visitedweekly. At these visits, the sales representative checks oninventory, replenishes stock, collects returned tickets, andinspects the equipment. One issue considered by the authorsis that many sales representatives have long standing rela-tionships with their retailers, so they preserved the currentassignment of retailers to territories. This decomposes theproblem into a series of single vehicle problems. Blakeleyet al. [14] optimize periodic maintenance of elevators andescalators for Schindler, one of the worlds largest escalatorand elevator companies. The intervals for different main-tenance tasks range from monthly to yearly. Unlike mostapplications of the PVRP, the skillsets for the technicians varyand must be considered in the assignment process. Prior to thedevelopment of the new routing and scheduling system, thecompany developed tours at their 250 offices in the U.S. andCanada using spreadsheets and roadmaps. Their new system,based on solving PVRPs, is estimated to save the companyover $1 million annually. Hadjiconstantinou and Baldacci[54] examine how a utility company provides preventivemaintenance to 162 customers using a fleet of 17 depot-basedgangs. Service to the utility customers ranges from once perday to once every 4 weeks. The gangs are assigned to differ-ent depots, and these depots represent the starting and endingpoint of the tours. Thus, the problem is an application of theMDPVRP. Clatanoff [24] looks at how to allocate and routequality assurance inspectors among the 26 locations wheretheir services are required. The inpectors are based at twolocations and travel by car to the inspection sites, and the plan-ning period considered is an entire year. The authors focusis more on the minimization of the number of inspectors.


Determining the best way to visit students and patientsat home can also be viewed as an application of the PVRP.Maya et al. [64] examine the problem of how to assign teach-ing assistants to help disabled students. Every morning, theteaching assistants depart from their homes and travel fromschool to school to visit the pupils that have been assigned tothem for that day. At the end of the working day, the teachingassistants return to their homes. Thus, the problem involvesmultiple depots as in the MDPVRP. The government is inter-ested in reducing the total mileage for the teaching assistantsto visit the students since that is the primary driver of theircompensation. The periodicity comes from the type of dis-ability the students have. Students with moderate disabilitiesare visited once per week, whereas students with more severedisabilties receive two visits per week. A related problem iscovered by [6] who consider the routing of home healthcarenurses in Korea. The nurses will begin and end each day atthe hospital, and the intervals between visits will vary for dif-ferent patients depending on the seriousness of the illness. Atthe company in Korea that motivated this work, each patientis assigned to a nurse, so the problem becomes a series ofPTSP problems.

An emerging application of the PVRP is in the collectionof data. Almiani et al. [4] consider a periodic routing problemmotivated by the growth in wireless sensor networks. Mobilemechanisms (e.g., a vehicle equipped with a data transceiver)visit each sensor and collect data. As the data generation ratemay be different for different sensors, the frequency of visitsto each sensor will vary. For example, sensors at industriallocations must be visited more often than residential loca-tions. The problem is to route the mobile mechanisms tosatisfy these frequencies. Giger [48] examines planning toolsfor use with unmanned underwater vehicles. Such vehiclesare used in the oil and gas industry and in several branches ofthe military. These vehicles have sensors that perform differ-ent kinds of readings, and these readings must occur at sitesperiodically.

4. SOLUTION METHODS

Section 3 describes a wide variety of real-world contextsin which the PVRP and its variants can be applied. Yet suc-cessful application of the problem requires solution methodscapable of producing good solutions in a reasonable amountof time. Next, we turn to a discussion of solution methods forperiodic routing problems with careful note of key datasetsused for testing.

Although the presentation of solution methods here isorganized according to the type of method, it is interest-ing to note that the historical arc of publications followsthis organization rather closely. Early approaches for thePVRP essentially focused on construction and improve-ment heuristics, sometimes assigning customers to daysbefore routing them, and sometimes creating routes andthen attempting to assign these routes to days. As applica-tions of the PVRP became more prevalent and computationalpower increased, these relatively straightforward approaches

gave way to the development of metaheuristics, which havedemonstrated considerable success on the problem. Althoughseveral researchers developed exact (i.e., mathematical pro-gramming) approaches, these are often unable to solve thelarge instances that arise in many applications. Only recentlyhave computational power and innovative mathematical pro-gramming methods combined to result in tractable exactapproaches for the PVRP.

A number of publications present heuristics designedspecifically to deal with problems arising in applications.These heuristics often exploit special characteristics of theproblem in question, thus they are not applied to generalPVRP instances. Of particular note are [85], [84], [79], [26],and [6] who all utilize information based on integer pro-gramming (IP) to drive their solution procedures, whereasTeixeira et al. [88] and Nuortio et al. [67] use special-purposeheuristics and metaheuristics. More detailed information onthe applications driving these approaches can be found insection 3. We note, however, that many of the most success-ful solution methods in the literature for the standard PVRPwere also developed in response to interesting applications.Though these applications have been described in section 3,we mention some of them again here to point out their inno-vations in solving the PVRP without further comment on theapplications themselves.

4.1. Early Approaches

Beltrami and Bodin [12] introduces the first two keyheuristics used to solve the PVRP. The first idea is to routecustomers using a Clarke and Wright procedure, then assignroutes to days. The second idea is to randomly assign cus-tomers to delivery days and create routes for each day basedon this assignment. In the heuristic presented by [81], theauthors cluster customers that are close together and that havethe same weekly delivery requirements to reduce the prob-lem size. Once the problem is sufficiently small, they considerthree heuristics. The first assigns all daily customers to eachday of the week, then schedules the remaining customersbased on the estimated costs of combining with customersalready scheduled on those days. These estimates are basedon values such as average distance to the centroid of cus-tomers serviced on a given day. Customers are inserted inorder of highest to lowest delivery frequency. Their secondapproach uses link exchanges to improve the initial solutioncreated by the first heuristic. Their third heuristic uses Clarkeand Wright ideas, modified to enforce the spacing of peri-odic deliveries throughout the week. Russell and Igo [81] alsointroduce a 490 customer dataset to test their approaches forthe periodic routing problem. In their approach, the 490 cus-tomers are clustered to create a 126 customer problem. Thisdataset has been used by many others to test their solutionapproaches.

Christofides and Beasley [19] offer an exact formula-tion, but solve the problem via a heuristic. Their heuristicassigns customers to days and then solves the resulting dailyVRPs. The initial assignment is based on an initial ordering


of customers in which customers with the fewest deliverycombinations and the largest delivery quantities are sched-uled first, as these create the biggest challenge for creatingfeasible solutions. Customers are inserted in this order usingthe allowable delivery combination that yields the smallestincrease in total costs. An improvement method is proposedthat considers swapping customers from one allowable com-bination to another. Because this improvement would resultin solving many new VRPs, they propose simplifications suchas solving the new daily problems as median problems and astravelling salesman problems (TSPs). These simplificationsare also solved with heuristics. The authors test their approachwith the dataset from [81] as well as 10 new instances derivedfrom existing datasets for the VRP in [35]. The Eilon-baseddatasets all involve smaller numbers of customers than theone in [81], but a planning horizon of up to 10 days andinvolving up to six vehicles. Their tests with the 126 customerRussell and Igo dataset yielded improved solution values withboth of their solution approaches. In general, they find theTSP-based simplification to be the most successful heuristic.Nearly 20 years later, Baptista et al. [11] modify this approachto address their own waste collection problem, a testament tothe influence that [19] had on subsequent efforts to addressPVRP and related problems.

Tan and Beasley [87] and Russell and Gribbin [82] bothutilize information provided by solutions to an IP formula-tion of the problem to guide solution development. Tan andBeasley [87] solve a seed-based IP (see [40]) to assign cus-tomers to delivery days. The authors then utilize the approachoutlined in [19] to create daily routes over the planninghorizon. Russell and Gribbin [82] extend this approach byconsidering additional local improvement heuristics. BothTan and Beasley [87] and Russell and Gribbin [82] test theirresults on the ten instances provided in [19] and improve onthe results presented there. Russell and Gribbin [82] also con-sider the instance presented in [81] and present two additionalinstances, one based on data in [27].

The next substantial improvement to the best-known solu-tions for this set of 13 instances appeared in Chao et al. [18].In particular, [18] aimed to enhance previous heuristics byproviding a means of escaping local optima. An initial solu-tion is generated by solving a linear program that seeks tolevel the number of customer deliveries across the planninghorizon. This linear programming solution is then roundedto create an initial integer solution. Local improvements, inpart based on record-to-record improvement (see [62]), arethen sought on the current solution. To allow for more flex-ibility in these local moves, vehicle capacity is relaxed toallow for moves that would otherwise be infeasible. Solutionsare then post processed to remedy any resulting infeasi-bilities (Pourghaderi et al. [73] later implement a similarcapacity relaxation). This approach led to improved best-known solutions for all thirteen instances introduced by [19](Instances 110), [81] (Instance 11), and [82] (Instances1213). Additionally, Chao et al. [18] introduced 19 newinstances (Instances 14-32) that, together with the previous13, became the canonical set of instances against which all

future approaches to PVRP have been compared. These arecollectively known as the old data set. Table 3 shows theinstances in this data set, and how the best-known value foreach of these instances changed over time as researchersexplored solution approaches for PVRP and applied themto these instances. Data values included in the table reflectattainment of a new best-known value at the time of publica-tion. Since publications report values with varying levels ofprecision, presentation, and comparison here ignore anythingbeyond one decimal place.

4.2. Metaheuristics

4.2.1. Tabu Search. One of the most influential solutionapproaches for the PVRPprimarily because of the degreeto which it has been modified for a variety of applica-tions and extensions of the PVRPappeared in [30]. Theauthors present the first tabu search algorithm designed tosolve PVRP and two of its special cases: the PTSP andthe MDVRP. The heuristic begins by randomly assigningcustomers to combinations of delivery days. Considering cus-tomers according to some ordering, each customer is thenassigned a route on each of its assigned delivery days usinginsertion heuristics to determine how the customer shouldbe routed. Neighborhood moves are made by either movinga customer to a new route on one of its assigned deliv-ery days, or by changing the combination of delivery daysfor a customer and reinserting them into appropriate newroutes. If a customer is removed from a route on a given day,tabu search prevents this customer from being reinserted onthat route for a specified number of moves. Their imple-mentation allows for infeasible solutions, which are thendiscouraged via penalty terms in the objective function. Thepenalty function also helps to diversify the search by dis-couraging solution attributes that are seen frequently in thecourse of the search. The authors note that their tabu searchimplementation requires fewer user-controlled parameters (anumber of which are used to control the penalty function)than other implementations of tabu search, making it easierto use. They test a number of parameter settings, and on theold data set, they produced new best solutions on the major-ity of the 32 instances. Cordeau et al. [30] also introduced10 new instances collectively known as the new dataset. Although these are often used for comparison in subse-quent publications, they are not as frequently solved as theinstances in the old data set.

Cordeau et al. [30] inspired a host of similar tech-niques, with many authors seeking to modify the tabusearch approach to address related problems and applica-tions. Cordeau et al. [29] modify tabu search to addressPVRPTW, Angelelli and Speranza [7] adapt tabu search toincorporate intermediate facilities (PVRPIF), and Alonsoet al. [5] incorporate multiple trips per vehicle along withaccessibility constraints. Banerjea-Brodeur [10] apply it tooptimize laundry delivery within a hospital, and Parthanadeeand Logendran [69] apply it to MDPVRP in a food ser-vice delivery setting. Francis et al. [44] define metrics that


TABLE 3. Best-known values for the Old data set produced over time.

quantify the operational flexibility of a solution and the oper-ational complexity and then modify tabu search to accountfor these characteristics. They also modify potential neigh-borhood moves so that moving a customer from one deliveryschedule to another is contingent upon one of its geographicalneighbors also being assigned to the same schedule. Cordeauand Maischberger [28] increase the computational power ofthe algorithm by alternating local search moves with diver-sification moves to escape local optima and by parallelizingthe algorithm.

4.2.2. Variable Neighborhood Search. The most signif-icant improvement in best-known solutions since those ofCordeau et al. [30] was produced by the variable neigborhoodsearch (VNS) technique of Hemmelmayr et al. [57]. VNSessentially works by performing neighborhood search, but itchanges the neighborhood when local search stagnates. Whenlocal search is unable to improve the current incumbent, VNSselects the next neighborhood in a series and performs ashaking step by randomly selecting a solution from this newneighborhood, and then performing local search. Basic VNSmoves through neighborhoods in the series until an improving

solution is found. The implementation of Hemmelmayr et al.[57] accepts inferior solutions in a manner typical of sim-ulated annealing. It provides improved solutions on mostinstances in both the old and new data sets; the authors indi-cate that those that are not improved are largely de factomultiple TSP problems. Pirkwieser and Raidl [71] extendVNS by adding multilevel refinement (essentially abstract-ing or coarsening the problem to a simpler one), solving thesimplified problem, then refining the solution by extend-ing the simplified solution to the original problem. Pirkwieserand Raidl [71] claim to produce new best-known solutionsto some of the instances in the canonical data sets, but Pirk-wieser and Raidl [71] only provide aggregate data (see also[70]).

4.2.3. Other Metaheuristics. Perhaps inspired by the suc-cesses with tabu search, a number of other metaheuristicswere developed for PVRP. Ochi and Rocha [68] offer an evo-lutionary approach based on genetic algorithms and localsearch, a parallel version of which is presented in [34](although they present results for instances in the old datasets, a number of later references question the accuracy of


the reported results). Vidal et al. [91] further improve geneticsearch by increasing the diversity in the gene pool when iden-tifying survivors from the current population of solutions.They combine this with local improvement to create a hybridalgorithm that performs well on instances in the data set,though data are only provided in aggregate form. This work isextended in [92] to a number of problems with time windows,including PVRPTW and MDVRPTW.

Matos and Oliveira [63] describe an ant colony optimiza-tion (ACO) algorithm for the PVRP. Nodes are replicated torepresent the required number of deliveries, and the authorsuse ACO to construct good routes for these nodes. After localimprovement on the resulting solution, graph coloring andexchanges are used to finalize the periodic assignment. Theauthors test their algorithm on some of the largest instancesin the canonical data set but do not report their best results.Their aim appears to be to compare their algorithm with priorACO implementations for routing problems, rather than tocompare with previous algorithms designed specifically forPVRP.

Goncalves et al. [50] develop a greedy-randomized searchprocedure (GRASP) and test it on instances derived from anoil extraction application. Alegre et al. [2] describe a scattersearch technique designed to address an auto parts applicationwhich has a longer planning horizon. Although this approachwas designed for a certain class of PVRP instances, it is com-petitive with [30] on the canonical instances and on occasionoutperforms [30].

Gulczynski et al. [52] instigate another paradigm shift byinverting the typical approach to local search (see also [53]).Although local search typically aims to search small neigh-borhoods a large number of times, the approach presented in[52] searches large neighborhoods a small number of times.They use mixed-integer programming to schedule customersin an attempt to evenly spread deliveries across the plan-ning horizon, and solve the resulting VRPs with a Clarke andWright algorithm. They then utilize IP to consider simultane-ously moving a large number of customers to new days and/orroutes. This IP is alternated with a record-to-record improve-ment procedure until no substantial improvement is found.They compare their algorithm to the approaches publishedby [30], [18], [57], and [2], implementing these with theparameter settings suggested by those sources. Although theapproach of [52] outperforms these algorithms under theserecommended settings, it does not produce any new best-known solutions as compared to the best solutions recordedin those sources. Gulczynski et al. [52] also provide exten-sions to the algorithm for handling reassignment constraintsand balance constraints (these are described in Section 5).

4.2.4. Metaheuristics for Problem Variants. Use ofmetaheuristics to solve the PVRP has also run parallel to theiruse in approaching its variants. Polacek et al. [72] use variableneighborhood search to solve the PTSP, allowing deviationfrom periodicity with a penalty. Yu and Yang [16] describean ACO algorithm applied to PVRPTW. Hadjiconstantinouand Baldacci [54] use tabu search to solve the resulting VRPs

in the MDPVRP once customers have been assigned to bothdepots and delivery days, while Vahed et al. [89] apply apath relinking algorithm to MDPVRP. A recent paper byLahrichi et al. [60] on integrative cooperative search seeksto harness the power of the best decomposition and meta-heuristic approaches by using several solution methods intandem and then sharing the information among them toimprove the solutions produced. They apply their approachto the MDPVRP, decomposing it into both a set of PVRPs(by assigning customers to depots) and a set of MDVRPs (byscheduling customers a priori). They then utilize tabu searchand the techniques of Vidal et al. [91] along with initial solu-tions to these subproblems to generate good feasible solutionsfor the MDPVRP.

4.3. Progress Toward Exact Solutions

Foster and Ryan [41] make one of the earliest attempts tomodel the PVRP as a mathematical program. In particular,they provide a linear programming formulation for the VRPand discuss modifications to the formulation to incorporateperiodic delivery requirements as well as other extensions.The linear formulation utilizes variables xj [0, 1] to rep-resent the probability that route j is utilized in an optimalsolution. Thus, the formulation is exponential in size. Fosterand Ryan [41] discuss column generation techniques, notingthat they are slow to converge, as well as relaxations of theproblem that are used to drive heuristic solutions. Results areonly presented for VRP instances.

Christofides and Beasley [19] are often credited as pro-viding the first IP formulation of the PVRP. Altough manysubsequent publications reference that formulation or pro-vide their own, most concede that solving these formulationsbecomes prohibitive for larger instances and resort to heuris-tic or metaheuristic approaches, some in fact based on theinformation provided by these IPs. A few researchers, how-ever, have explored mathematical programming approachesin more depth, particularly because of their flexibility inhandling extensions and variations of the problem.

Francis et al. [43] extend the IP formulation presented in[39] for VRP to accommodate service choice, which gen-eralizes the definition of periodicity in PVRP and will bediscussed further in Section 5. They first relax the prob-lem by separating the decisions which schedule customersfrom those that create routes, creating two different prob-lem relaxations. They then use subgradient optimization on aLagrangian function which incorporates both of these relax-ations to develop lower bounds on the solution. At the endof the Lagrangian phase, with the use of feasible solutionsconstructed along the way, a branch and bound procedure isused to close the gap between the upper and lower bounds.They apply this approach to two interlibrary loan examplesfrom a related application, as well as to one of the instancesfrom the old data set modified to include service choice inthe objective.

Francis and Smilowitz [42] present a continuous approx-imation of the formulation presented in [43] by aggregating


some of the data and approximating the discrete model withcontinuous functions. By utilizing decomposition methods,the continuous approximation can be solved quickly, allow-ing it to be useful for larger problems. Francis and Smilowitz[42] state that this approach is not intended to replace dis-crete approaches. Rather, it is intended to provide quicksolutions that can be used to guide design decisions. Francisand Smilowitz [42] apply this approach to a 100-customerinstance from the old data set. Although they do not producea new best-known value, their approach is competitive.

Mourgaya and Vanderbeck [66] develop a model designedto simultaneously address two objective criteria: balanceof the workload across trucks, and regionalization (creat-ing routes that keep vehicles/drivers in familiar areas). Theauthors then use a DantzigWolfe reformulation and columngeneration to solve the relaxed problem. Insertion heuristicsare used to price out columns, with an eye toward balanc-ing the two objectives under consideration. On completionof the LP solution phase, the resulting solution is roundedto produce a feasible solution to the PVRP by heuristicallyexploring the branch and bound tree. Mourgaya and Van-derbeck [66] test this approach on some of the instancesfrom [30]. Although this approach does not produce routeswith smaller overall cost, it is competitive with the approachof [30] when workload balancing and regionalization areconsidered.

Baldacci et al. [9] have arguably demonstrated the great-est success with exact approaches for PVRP. They present anew IP for the problem and three relaxations based on thisformulation that are used to generate strong lower bounds forthe problem. They then use these lower bounds, along withinformation from a related dual solution, to reduce the solu-tion space in such a way that no optimal integer solutions areeliminated, resulting in a tractable IP. This IP is then solvedexactly. They report both solutions and lower bounds for theold canonical instances. The lower bounds produced are onaverage within 1% of optimality, best-known values are pro-duced for five instances, and the best known at the time ofpublication of [9] is matched for the majority of the remaininginstances.

Kang et al. [59] present an IP formulation and an exactsolution approach for MDPVRP. They assume, however, thateach vehicle can service at most one customer per day, thusthe routing component of the problem is completely removed.

5. PROBLEM VARIANTS AND EXTENSIONS

Many variants of the PVRP have been studied. Here, wewill discuss some of the variants beyond the PTSP, PVRPTW,and MDPVRP.

Intermediate facilities provide a location other than thedepot for a truck to renew its capacity. For example, if atruck is collecting goods, such as in waste collection, anintermediate facility would be a place for a truck to unload,such as at a waste treatment plant, before resuming pickupat other customers. The first paper on the PVRP-IF is by

Angelelli and Speranza [7]. Even with intermediate facili-ties, the objective they consider is still the minimization ofthe total length of the routes. The authors propose a tabusearch approach. The discussion of the application of thePVRP for collection of slaughterhouse waste in [26] alsoinvolves intermediate facilities for disposal of waste due tothe use of small capacity vehicles. Hemmelmayr et al. [56]consider the PVRP-IF as well, offering an exact formulationand proposing a solution method based on variable neigh-borhood search and dynamic programming. The authors alsoconsider variants of the PVRP-IF where capacity limits areplaced on the intermediate facilities.

Another related problem with a location component isthe periodic location-routing problem (PLRP). The PLRPrequires decisions about which depots to open, in additionto the routing and assignment decisions typically made inthe MDPVRP. The motivation for this change in formula-tion is the impact that the location of the depots can haveon the routing costs. The PLRP was introduced by Prodhonwith a memetic algorithm offered in [76] and a hybridizedevolutionary algorithm described in [74] and [75]. The useof variable neighborhood search for the PLRP is studied in[70].

Variants that introduce new constraints include the PVRPwith reassignment constraints (PVRP-RC) introduced in[52]. Companies often have existing routes, and they are solv-ing for new routes because of the addition of new customers.They want to minimize the change in service to the existingcustomers. PVRP-RC limits the number of customers whoare moved from an existing service pattern to another servicepattern. Rademeyer [77] examines the assignment routingproblem with nominated delivery days (ARPNDD) in whichcustomers are assigned to a delivery group and must remainin that delivery group for the entire planning period. Alonsoet al. [5] consider a variation of the PVRP where vehicles canmake multiple trips per day and include limitations on whichvehicles can be assigned to which customers. Parthanadeeand Logendran [69] add limitations to the capacity of thedepots in the MDPVRP. The authors also deal with customerdemand for multiple products, fixed supply at the depot ofeach of these products, and scheduled replenishments of theproducts to the depot. The capacity limitations can poten-tially cause customers to receive deliveries of certain productsfrom different depots. In the MDPVRP in [64], constraintson which resources can be matched with which customers, aswell as resource-dependent work hour restrictions, are alsoincluded.

An important category of variants is those which changethe objective function, possibly in conjunction with con-straints. Most of the solution approaches described so far seekto minimize total route cost or some related metric, but someconsider very different objectives. Gaudioso and Paletta [45]present a heuristic designed to minimize the fleet size. Vahedet al. [90] present a heuristic designed to minimize fleet sizesubject to a maximum route duration. In [17], the authorstry to identify the minimum number of vehicles required toserve customers with periodic delivery requirements under


the assumption that all customers receive full truck ship-ments. They refer to this as the vehicle minimization forperiodic deliveries problem (VMPD). Related is the workby Delgado et al. [32] where the authors look at minimizingthe labor requirements associated with the periodic supply ofproducts to customers from a warehouse. When a customercomes to the warehouse to pick up product, labor is requiredat the warehouse to load the vehicles. The number of cus-tomers who arrive at the same time affects the amount oflabor required. Thus, the pickup day for each customer mustbe determined along with the timing of the pickup so as tominimize the amount of labor needed.

Some authors change the objective to be one of maxi-mization of profits rather than minimization of costs. Asmentioned in section 3, Goncalves et al. [50] model howto extract oil from wells in Brazil using mobile units. Eachwell has a minimum time between visits to collect a reason-able quantity. Their objective is to maximize the amount ofoil extracted which impacts which customers are visited andwhen. They refer to this variant as the period bump mobileunits routing problem. Baptista et al. [11] also focus on aprofit maximization objective.

Other changes to the objective include the addition ofother costs besides travel costs. Gulczynski et al. [52] pro-pose the PVRP with balance constraints, but balance isactually modeled in the objective. Imbalance is defined bythe difference between the largest number of customers ona route and the smallest number of customers on a routein the solution. The objective becomes the total travel costplus a penalty cost for imbalance to encourage workloadsamong drivers to be similar. The PVRP with service choiceis introduced in [43] and [42]. In this problem, the deliv-ery frequency for customers is chosen by the model. Eachcustomer has a minimum delivery frequency but higher fre-quencies, which translate to better service, are rewarded inthe objective function. A multiobjective PVRPTW is consid-ered in [3] to reflect a competitive situation. In a competitiveenvironment, the arrival time at a customer relative to thearrival time by a competitor may impact the amount soldto the customer. Thus, they consider an objective that mini-mizes travel costs and maximizes sales, as well as balancesthe amount of goods distributed by the different vehicles. Amultiobjective PVRP is also considered in [86] but with moreof a focus on workforce management. Their objective consid-ers factors such as consistency of assigning the same driver toserve customers, as well as mileage costs, and they examinethe tradeoffs. The variant of the PVRPTW in [14] includesovertime costs, costs for violating time windows, and costsfor idle time, in addition to travel costs.

Some papers consider problems with more flexibility thanthe traditional PVRP. For example, in [24], a single vehicle isnot required to serve all of the demand at a customer, and in[23], customers may be visited by more than one vehicle ondifferent routes. Danandeh et al. [31] look at the open PVRPwhere vehicles are not required to return to the depot at theend of the day. Mourgaya and Vanderbeck [66] focus strictlyon the assignment of customers to days for delivery subject to

frequency limitations and do not precisely define routes. Thefocus is more on balancing the quantity collected than on min-imizing route costs or fleet size. Francis et al. [43] preciselyexamine the value of flexibility in periodic routing prob-lems. Besides service choice, they also examine the impact ofrestricting a customer to be served by one driver versus mul-tiple drivers, the impact of a greater number of schedulingoptions, and the impact of flexibility in the delivery quantityto customers, while restricting that all demand is eventuallyserved. They also examine these tradeoffs computationally.

Patrolling and data gathering has created several exten-sions of the PVRP. As mentioned in section 3, Giger [48]examines planning tools for use with unmanned underwatervehicles. Since vehicles have sensors that perform differentkinds of readings, visits by multiple vehicles to the same loca-tion may be required to collect the needed data. This makesthe problem more challenging. Almiani et al. [4] considerthe routing of mobile gateways to collect sensor data as well.They introduce the periodic mobile multigateway schedulingproblem. Unlike the PVRP, the frequency for visiting eachcustomer is determined as part of the problem solution dueto constraints on information gathering needs. Fargeas et al.[38] study the persistent visitation problem, where the goal isto determine the rates at which customers should be patrolled,rather than to assign specific days, and is subject to limitationson the fuel available in the patrolling vehicles.

Very little work on the periodic routing problem specif-ically addresses the stochasticity that can occur in practice.One of the few exceptions is [83] who study the periodic rout-ing problem with stochastic demands (SPRP). They considerthe stochasticity of customer demands in the determination ofvisit frequencies and delivery quantities. The differentiationwith the IRP is that inventory costs and customer capacitiesare not considered.

Related is the area of periodic arc routing problems. Inperiodic arc routing problems, as discussed in [47], eachrequired edge of a graph must be visited a given numberof times over a specified planning period. They often arise inresidential waste collecting and street sweeping applications.Chu et al. [2022], Groves et al. [51], and Mei et al. [65] alsoconsider the periodic arc routing problem.

6. FUTURE DIRECTIONS

As this review has indicated, the body of work on thePVRP is extensive. It has been applied in a diverse array ofcontexts and has been solved with a wide variety of solutionmethods. With the increasing use of recycling, home health-care, and remote sensors, we expect that the countries and thecontexts in which the PVRP can be applied will continue togrow. Recent trends indicate that study of the problem is shift-ing more toward its variants, as many growing applicationsrequire changes to the classic PVRP, and solution methodsmust be developed that deal explicitly with these additionalconstraints or alternate objectives. The number of paperson multiobjective versions of the PVRP reflects the grow-ing importance of considering many cost factors beyond just


mileage in the true evaluation of a route plan. We anticipatethat more complex objectives and more operational flexibilitywill be a growing trend in the PVRP literature. Surprisingly,our review found little work on variants of the problem thatexplicitly address the stochasticity of customer demand ortravel times. We see this as another area rich in possibilitiesfor further study.

Acknowledgments

The authors would like to thank Ning Zhou for herassistance with the bibliography.

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