Analyses of advanced iterated tour partitioning heuristics for generalized vehicle routing problems

Analyses of Advanced Iterated Tour PartitioningHeuristics for Generalized Vehicle Routing Problems

Anupam SethDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois

Diego KlabjanDepartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois

Placid M. FerreiraDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois

Theoretical analyses of a set of iterated-tour partitioningvehicle routing algorithms applicable to a wide varietyof commonly used vehicle routing problem variants arepresented. We analyze the worst-case performance ofthe algorithms and establish tightness of the derivedbounds. Among other variants, we capture the casesof pick-up and delivery and multiple depots. We alsointroduce brand new concepts such as mobile depots,partitioning of customer nodes into groups, and poten-tial opportunistic under-utilization of vehicle capacity byonly partially loading the vehicle, among others, whicharise from a printed circuit board application. The prob-lems studied are of critical importance in many practi-cal applications. © 2012 Wiley Periodicals, Inc. NETWORKS,Vol. 000(00), 000–000 2012

Keywords: vehicle routing; worst-case analysis; printed circuitboard; approximation algorithms

1. INTRODUCTION

Logistics in today’s increasingly global economies hasbecome a more important field than ever before. Althoughlogistics in itself has a very significant business impact, nev-ertheless, the breadth of applications of vehicle routing goeswell beyond the traditional trucking industry. Vehicle routingalgorithms play an important role in vehicle routing relatedinformation systems. Theoretical analyses of vehicle rout-ing problems (VRPs) have led to significant insights into thenature and complexity of the proposed algorithms, and theyhelp in better understanding the potentials and limitations ofsuch algorithms.

Received November 2009; accepted June 2012Correspondence to: A. Seth; e-mail: [email protected] 10.1002/net.21478Published online in Wiley Online Library (wileyonlinelibrary.com).© 2012 Wiley Periodicals, Inc.

A novel and recent area of applications of vehicle routing isproduction planning of printed circuit board (PCB) assemblyon an automated (primarily surface-mount type) equipment,for example, on the collect-and-place or pick-and-place typemachines [27]. More specifically, collect-and-place machinesare typically used for high-volume, high-flexibility produc-tion and are becoming increasingly popular in industrialsettings because of their inherently advantageous characteris-tics and capabilities. The placement sequencing problem, thatis, the problem of planning the pick-up and delivery of elec-tronic components or chips from feeder trays on the side ofthe machine onto a bare PCB in the center on such machinesis an NP-hard problem that turns out to be essentially ageneralization of the VRP in several different dimensions.Several interesting variants appear based on the manufac-turing scenario and configuration of the machine, whichare generalizations of many well-known and standard VRPvariants that have been studied in the past. Thus, existingalgorithms and research do not directly apply to the prob-lems motivated by the PCB context, but they are interestinggeneralizations of many important standard VRP variants,and hence a study of these problems is of great value.

PCB manufacturing plays a very important role in today’seconomy. Global revenues for the PCB industry exceeded$50 billion in 2007 and are expected to reach more than$76 billion in 2012 [22]. A PCB is a flat board that car-ries the chips and various other electronic components. Thisboard is made up of alternating layers of copper and plastic,with the etching process performed on the copper layers toprovide interconnects. These boards are capable of holdingseveral components depending on the required specifica-tions and produce complex interconnections. PCBs can beof different sizes and varying densities and are manufacturedin automated assembly lines where high-speed placementmachines put components on the boards. A line can assemble

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FIG. 1. A collect-and-place machine.

components on multiple types of PCBs, and it has one or sev-eral high-speed machines to perform the actual placement ofoperations [26]. The assembly of PCBs is a complex processinvolving the placement of hundreds (even up to a thousandor more) of electronic components in different shapes andsizes at specific locations on the board. These assembly lines,and more specifically, collect-and-place machines (see Fig.1) form the bottleneck in production, thus even small sav-ings in total assembly time can significantly improve thebottom-line.

These machines have one or more rotary heads (see Fig.3) that work in tandem to pick up components or chips froma feeder tray (or an array of feeder slots each feeding oneparticular type of component) and deliver them to specificlocations on the bare circuit board as dictated by the specificdesign requirements (illustrated in Fig. 2). Each rotary headcontains typically between 5 and 20 spindles, or placeholdersfor the chips, thus limiting the capacity of the “vehicle.” Thereare additional complicating manufacturing constraints andspecifications that make the problem much more complexand harder to solve than just a basic VRP.

In this article, we study the placement sequencing problemon a single machine processing a single board at a time witha single machine head. We assume that the arrangement ofcomponents or chips on the feeder magazine is given, that is,the supply characteristics of each depot are fixed beforehand,which is an assumption frequently used in the PCB literature.Under these specific constraints posed by the scoped config-uration of the machine, the placement sequencing problemis very similar to a complex, yet striking version of the VRPwith a single capacitated vehicle (in our case, the rotationalrobotic head with multiple spindles loaded onto a dual gantry)performing multiple trips from the depot (in our case, thefeeder magazine) to satisfy customer demand (in our case,the design requirements for the board being populated, stip-ulating the specific component type to be inserted at eachplacement location).

As already alluded to, the problem in its general formin the given setting captures many important VRP variants.The first major difference from all standard VRP variants is

FIG. 2. A schematic of the collect-and-place machine [13].

the introduction of a moving depot that utilizes the time thatvehicle is away to reposition itself to a more advantageouslocation. Thus, classical VRP is a special case when the depotmovement velocity is fixed to zero. The second major exten-sion lies in the generalization of the underlying network touse groups of customer locations, only one location in eachpredefined group needs to be visited by the vehicle. We takethis generalization one step further and consider also groupsof depot locations such that a vehicle visits only one locationin a group. Such problems have been commonly studied inthe context of the traveling salesman problem (TSP) wherethey are known as generalized TSP (GTSP) [11,14], but not somuch in the VRP context, though they have several importantpractical applications. Apart from these differences, in themodeled problem, if we consider only one possible pick-upand delivery point for each type of a component, the problem

FIG. 3. Detailed view of a rotary collect-and-place head [28]. [Colorfigure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

2 NETWORKS—2012—DOI 10.1002/net

essentially becomes a static dial-a-ride problem in which allrequests for transportation are known at the beginning. Thisscenario can also be analyzed as a VRP with pick-up anddelivery, because there are no extra side-constraints typicallyfound in the dial-a-ride problem. To this scenario, if we fur-ther add the restriction that all pick-ups precede all drop-offsin a route, it can be seen that the problem reduces to a VRPwith backhauls type scenario. If we further simplify the prob-lem and assume only one component type, that is, a singlecommodity problem, we get the standard multidepot VRP.Finally, if we assume a single pick-up point for all compo-nents in addition to the previous simplifications, we end upwith the standard capacitated VRP.

Other interesting VRP variants that have not been stud-ied in the literature ever before also arise as special cases,for instance, the VRP with moving depots and the general-ized VRP (GVRP), which is an extension of the VRP withan underlying GTSP replacing the standard TSP, potentiallyboth on the pick-up and on the delivery side. A GVRP is aVRP where the underlying network is a GTSP, that is, thereare clusters of customer and depot locations, and the vehicleneeds to only visit one location in each cluster. We stress thaton the customer side, every cluster would need to be visitedexactly once, but such a restriction is absent on the depot side.

We next formally state GVRP where it is assumed thateach customer and depot is associated with a unique location.We are given clusters Ci, i = 1, . . . , n of customers, withcluster Ci having ni customers, and each customer exhibitinga unit demand. We also have clusters Dj, j = 1, . . . , m ofdepots, each cluster Dj having mj depot locations and eachindividual depot storing a single unit of demand. A vehiclehas capacity of q. Before starting a route to deliver goodsto customers, the vehicle must pick the items to be deliveredalong the route by visiting depots within a single cluster Dj fora j in 1, . . . , m. As each depot holds a single unit of demand,it must be the case that |Dj| ≤ q. GVRP is the problemof finding the minimum cost route through all the customerclusters and (some) depot clusters by visiting exactly onelocation in each cluster of customers. After the items aredelivered along a route, the vehicle must return to a clusterof depots to pick up the items for the next route. Variousspecial cases of this problem yield well-studied variations tothe classical VRP problem. In addition, GVRP is applicableto a variety of automation- and robotics-related applications,and a well-designed algorithm could lead to potentially largesavings in such situations.

Besides the above-mentioned importances of this prob-lem in the PCB assembly planning context, and its valuein studying and deriving bounds for important VRP vari-ants, this problem also is of striking similarity and crucialimportance to related research areas including, but not lim-ited to, seaport container problems for ship loading andunloading [30], point-to-point motion planning of robots[19] used in automated assembly lines and for operationssuch as painting and welding, warehouse operations [33]utilizing automated storage and retrieval systems [25] requir-ing the routing of autonomous guided vehicles [4], whichare also used as an integral part of flexible manufacturing

systems [29], and hard disk drive data retrieval and storageoptimization problems [24].

1.1. Contributions and Outline

This article presents analyses of advanced iterated-tourpartitioning (ITP) algorithms for generalized VRPs, whichencompass many standard versions of the VRP and extendthem to a generalized setting with the ability to deal withgroups or clusters of customer and depot sites and a maximumattribute arising in the objective function as a result of mobiledepots.

The ITP heuristic in the standard VRP setting finds a TSPtour in the network and then traverses the tour. Each time thetruck following the tour delivers all loaded items, it detoursfrom the tour by going to the close-by depot, picks up newitems, and returns back to the tour at the next node on the tour.We generalize this heuristic to accommodate our settings. Inthe most general case, we set up an auxiliary network basedon the computed TSP, and the final solution consists of findinga shortest path on this auxiliary network.

The most striking and unique feature of our problem isthe notion of moving depots. Another innovative feature ofour model is the pick-up and delivery aspect. It turns outthat this can be modeled on a network with only customernodes without explicitly modeling depot nodes using twoarc weights (one capturing the cost of the movement of thevehicle and the other one reflecting the cost of pick-ups). Thecost of a VRP solution is the combination of these two arcson the routes. We call such a network the generalized andexpanded 2-cost edge network. Two arcs per pair of nodesare sufficient if each “truck” must leave fully loaded. If thiscondition is relaxed, we need more than two arcs between twonodes and the resulting network is termed the generalizedand expanded multicost edge network. The reason we candeal with the depots in such an implicit manner is due to theweaker condition of not requiring to visit every depot exactlyonce as is the case on the delivery side.

The second major contribution is the extension of the stan-dard VRP into the generalized setting wherein the vehicleneeds to visit only one customer in each predefined set ofcustomers.

These models provide the underlying framework we useto derive a generalized ITP heuristic that covers the variantsexamined. We present tight approximation ratios for the ITPheuristics for

• the generalized full-load case, in which the vehicles areassumed to always run on a full-load and return empty tothe depot,

• the generalized partial-load case, in which vehicles areallowed to carry only as much as they intend to deliveron a given route, and

• the generalized partial-load case with waiting cost, inwhich an additional cost component is present in theobjective function to capture moving depots.

In analyzing the algorithms, we use a unique lower bound-ing procedure based on the structure of the optimal solutions

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to such problems coupled with an upper bound derived fromITP. To do so in a more general context, we need to defineq auxiliary networks with cost structures deduced from theoriginal problem. The entire analysis and the heuristics arebased on the multicost network.

Our contributions first lie in the development of the mod-eling framework. The novel idea of introducing multicostnetworks to model pick-up operations is an important con-tribution toward a unified framework. Second, we provideapproximation and tightness analyses for our ITP-heuristicsfor the generalization of standard VRP scenarios, specificallythe VRP with pick-up and delivery. Another important con-tribution is the new and novel generalization of the heuristic.We generalize the problem to solve it on groups of customersites only one of which needs to be visited by a vehicle ratherthan individual customers all of which must be on the vehi-cle’s route. Furthermore, we allow for multiple depots (andalso generalize the concept of a depot to a cluster of depots),each of which is capable of moving when not replenishinga vehicle leading to the introduction of a waiting cost. Andfinally, we generalize the standard VRP scenario to exploitopportunities where under-utilization of the vehicle’s capac-ity in the short-term leads to shorter routes. One of the majoraccomplishments of our work is to keep the problem generaland present a unified model, so that approximation ratios forwell-known VRP variants (for many of which there does notexist any theoretical analysis so far) can easily be reducedfrom our model.

The article is structured as follows. We first finish the intro-duction with a literature review. Section 2 provides a broadoverview of the problem, including the variants of the prob-lem addressed in this work, and those that can be directlyreduced from them. In Section 3, we present the ITP-heuristicand develop approximation ratios for each of the above-mentioned three scenarios. Tightness of the derived ratiosis also proven. Finally, in Section 4, we present a numericalstudy.

1.2. Literature Review

To the best of our knowledge, there is no previously pub-lished research studying and presenting a model and analysisof the generalized VRP problems addressed in this article. Afew purely computational studies (see, e.g., [13, 17, 18, 32])have been carried out, but are exclusively empirical in nature,and do not provide a significant insight into the analysis oftheir underlying algorithms or theoretical implications of theunderlying mechanisms used to solve the problem.

ITP algorithms have been studied and applied to manydifferent VRP scenarios. Most of this literature exists in thetheoretical computer science domain, and the algorithms arewell known for their amenability to theoretical analysis. Theunderlying concept and the first ITP algorithm was presentedby Haimovich et al. in their path-breaking article [15] (actu-ally, they published an earlier version of their work in Ref.[16]). The worst-case bound developed in Ref. [15] wasshown to be tight by Li and Simchi-Levi [20]. This work

was extended and modified later by Altinkemer and Gavish[1] and Beasley [3], but still applicable to only the standardVRP scenario. Subsequent research has seen applications andtransformations to a few different hypothetical and practicalsettings, however, none of these have addressed the problemin its general form as we do. For example, there is individ-ual work on ITP heuristics for multiple depot problems [20]and for problems that have both pick-up and delivery aspects[21], but there is none that addresses both aspects.

Recently, Bompadre et al. [5] presented lower bounds forthe VRP with and without split deliveries, improving the well-known bound of Haimovich et al. These bounds were thenutilized in a design of best-to-date approximation algorithms.Chawla [8] presented new approximation algorithms as wellas hardness of approximation results for several planning andpartitioning problems. She gave the first approximation algo-rithms for problems such as orienteering, deadlines-TSP, andtime-windows-TSP, as well as results for planning in stochas-tic graphs (Markov decision processes). However, her workis limited to TSPs, or uncapacitated VRPs. More importantly,the moving depot and the GVRP aspects have not yet beenstudied.

Bramel and Simchi-Levi [7] developed an asymptoticallyoptimal heuristic method called the location-based heuris-tic for the VRP with time windows by transforming it intoa capacitated location problem with time windows. Theyuse Lagrangian relaxation coupled with probabilistic anal-ysis to prove their result. Earlier, they proved a variant of thelocation-based heuristic to be also asymptotically optimal forthe standard capacitated VRP [6].

Daganzo [9] presents some analytical insight into dis-tribution patterns for the VRP with pick-up and deliverytype problems at the system design level so as to poten-tially provide guidelines for districting and cost estimation bycharacterizing broad routing strategies in terms of problemcharacteristics independent of specific customer locations.For the VRP with pick-up and delivery, the first probabilisticanalysis of a simple algorithm was presented by Stein [31].Later on, Psaraftis [23] showed his adaptation of the mini-mum spanning tree heuristic for the TSP to have a worst-caseperformance ratio of 3.0 while also demonstrating its supe-rior practical performance as compared to Stein’s method.Gendreau et al. [12] analyzed the single unit VRP with pick-up and delivery (or, in other words, the TSP with pick-up anddelivery) on trees and cycles showing that these problems canbe solved optimally in linear time, and that a tight worst-caseperformance ratio of 2, 3 can be achieved on undirected, com-plete, and triangular networks by the tree- and cycle-basedheuristic, respectively.

In summary, we find that there is no prior work thataddresses a practical and unified VRP scenario where thereare

• simultaneous considerations of both the pick-up and thedelivery aspects in routing,

• allowing only one or more locations out of mutually exclu-sive and exhaustive groups of several possible customer


FIG. 4. Example of a typical “route”. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

sites to be visited, that is, delivering to groups of customersites (such that the underlying structure of the problembecomes a GTSP rather than a vanilla TSP),

• exploiting the availability of multiple depots,• the depots are mobile and can reposition themselves into

advantageous locations, and• allowing for vehicles to have partially utilized capacity in

favor of opportunistic pick-ups at a later stage in the route.

2. PROBLEM STATEMENT

In the PCB context, the problem can be stated as one offinding the least cost sequence of picking-up componentsfrom the feeder magazine and placing them on the boardsubject to the following conditions:

1. all components required by the design must be placed,2. every component that is placed must have been picked

up previously (and not yet placed),3. every spindle may contain only one component during a

“route”, and4. at no time in a route must the capacity of the head be

violated.

At the same time, we want to fully exploit opportunitiesoffered by simultaneous motion, spindle jumping (a phe-nomenon caused by allowing bidirectional and multisteppedindexing motion of the placement head), and feeder magazinemovement.

This can be restated in a generic VRP context as follows.Assuming

• there are n customers (i.e., board locations in our case) and• there are m depots (i.e., feeder slots in our case) to pick up

the items from,

the problem characteristics entail the following restrictions:

1. every “customer” requires only one item of a specifictype, that is, there is a single-unit demand; and

2. each depot stocks a single item type, implying that travelamong depots is required as the vehicle moves fromdepot to depot to pick up each of the required items.

The objective is to find the least cost sequence of pick-up ofitems from the depots and delivering them to the “customers”subject to the following conditions:

• all items required by the customers must be delivered,• every item delivered must have been picked up previously

(and not yet delivered),• the capacity limit of the vehicle must not be violated, and• the cost implications of the loading and unloading

sequence for the vehicle to be formalized later areaccounted for.

A typical route is depicted in Figure 4. Each of the cus-tomer locations, as well as depot locations, can be served byany one of the spindles on the head. The choice of the spindlesfor the different locations is akin to the assignment of a cus-tomer on a vehicle with multiple seats, and correspondinglydifferent “occupancy costs" for the seats, as in a commercialairliner, for example. This can be modeled by expanding thenetwork to accommodate groups of customer and depot loca-tions, only one of which is finally chosen in any given route.(Each node corresponds to a seat occupied by the customer,and as the customer must sit in exactly one seat, a single nodein a pick-up group must be selected, similarly when a cus-tomer alights, only a single seat becomes vacant, and thusonly a single node in the delivery group must be selected.)


Another analogy can be found in the freight industry with thenotion of load positions for cargo, for example, cargo in anaircraft belly is stored by load positions. The problem is fur-ther complicated by the introduction of mobility for depots,allowing them to reposition themselves to a more suitablelocation when eventually intercepted by the vehicle duringthe period that the vehicle is out on its tour delivering cus-tomers. This is handled by discretizing the possible locationsthat the depot can move to, and correspondingly adding athird dimension to the network model and consequent extralayers of nodes in the clusters, where each of the pairs ofdepot location-spindle combinations is now supplementedby real-time depot position information.

2.1. Network Modeling Concepts

In this section, we present the modeling concepts usedto define our network models based on which the analysishas been carried out for the different cases. Based on thedescription aforementioned, it becomes clear that we need ageneralized network model comprising of groups of nodes torepresent the problem, because each pick-up or drop-off loca-tion can be potentially visited by any one of the spindles on thehead. Thus, each group represents a physical site or location,and the nodes within a group represent the different possibil-ities (of which only one can exist in a feasible route solution)of visiting that location with the spindles (based on Fig. 4, wewould have to have a set of board and feeder nodes). How-ever, such a network would be thus, extremely complicatedobliterating the possibility of an analytical study. Therefore,to simplify the network model we developed an innovativeidea of using a double-cost arc representation, which we laterextend to a multiple-cost arc representation. This implies thatthere is no need for the depot nodes and their groups, and thus,they can be disregarding completely for the sake of the anal-ysis. It is important to stress that this new network nicelycaptures the depot movements through the unconventionalcost structure defined later on in this section and does notcompromise our results due to no additional or restrictiveassumptions being enforced in it.

2.1.1. Formal Definition of the Generalized and ExpandedTwo-Cost Edge Network Model. In this section, wepresent a generalized and expanded the two-cost edge net-work (GE2-CEN) formulation. We begin by assuming thateach time the vehicle is dispatched, it must be fully loadedwith exactly q items. Consider a complete undirected graphG(X, E), where X is the set of nodes, and E is the set ofedges. Let G1, G2, ., GK be a partition of X into K mutuallyexclusive and exhaustive subsets or groups. Every node i ∈ Xhas a unique index g(i) ∈ {1, 2, . . . , K} such that i ∈ Gg(i).Each group is defined for a customer subset (board loca-tion) having multiple “cities” or locations (board locationand pick-up spindle combinations). We assume throughoutthat n ≥ K ≥ q, that is, ζ = K/q ≥ 1, where q is the capac-ity of the vehicle. Any two nodes i and j in this network areconnected by an edge with two types of costs on it — the

FIG. 5. GE2-CEN formulation. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]

primary cost, which is the “direct” cost to go from node i tonode j, and the secondary cost, which is the “indirect” cost togo from node i, visiting exactly q depots en route, filling upthe entire vehicle, which always is assumed to return empty,and returning to node j (see Fig. 5). The direct cost di,j is the“true” cost of the vehicle moving from node i to node j. Theindirect cost ri,j models the case of the truck finishing the lastdelivery at node i, next visiting q depots (not modeled explic-itly), and then moving to node j with q items to start the nextdelivery. Cost ri,j is the cost of performing these operationsin the most cost effective way.

In standard VRP settings, di,j is the cost of going directlyfrom i to j. On the other hand, ri,j would typically be thecost of finishing a route at node i, transitioning to the lowestcost depot and starting a new route at node j. Note that thereare groups only corresponding to the customers, and thus,the depots are not represented either by nodes or groups.We define for each edge (i, j) ∈ E, the “direct” cost di,j =dj,i between nodes i and j, and the “indirect” cost ri,j = rj,i

between nodes i and j.We also define rmin

u,v = mini∈Gu,j∈Gv ri,j to be the smallestdistance between any two nodes of customer groups Gu andGv.

For ease of notation, we define di,i = ri,i = 0. We alsomake the following assumptions:

di,j ≤ ri,j i ∈ X, j ∈ X, (1)

di,j ≤ di,k + dk,j i ∈ X, j ∈ X, k ∈ X, (2)

ri,j ≤ ri,k + dk,j i ∈ X, j ∈ X, k ∈ X. (3)

Assumption (1) states that the direct cost is lower than orequal to the indirect cost, which is clearly the case in VRPsettings. In the PCB setting, this is also the case as argued inRef. [27]. Inequality (2) is the standard triangle inequality ofthe direct cost. Finally, (3) links direct and indirect costs. Assuch, it holds in VRP settings and also in the PCB setting asshown in Ref. [27].


FIG. 6. GEM-CEN formulation. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]

Note that (3) and (1) imply the triangle inequality ofindirect cost. For future analysis, we rewrite (3) as

ri,j ≤ di,i′ + rj,i′ ≤ di,i′ + ri′,j′ + dj′,j

i ∈ X , i′ ∈ X , j ∈ X , j′ ∈ X . (4)

2.1.2. Formal Definition of the Generalized and ExpandedMulti-Cost Edge Network Model. We also define a spe-cially formulated generalized and expanded multicost edgenetwork (GEM-CEN) formulation. This network is used inthe cases when trucks are not required to be fully loaded. TheGEM-CEN network is a complete undirected graph G(X, E)

as before, where X is the set of nodes, and E is the set ofedges. Let us also have G1, G2, . . . , GK as a partition of Xinto K mutually exclusive and exhaustive groups as before.Likewise, every node i has a unique index g(i) ∈ {1, 2, . . . , K}such that i ∈ Gg(i). In this formulation, q+1 costs are definedfor each edge, that is, there is a direct cost and q indirectcosts between any two nodes (see Fig. 6). The indirect costrh

i,j between node i and node j corresponds to the smallestindirect cost of going from node i to node j while visiting hdepots en route, computed by solving a constrained shortest-path problem. While this might be computationally hard, inpractice this limitation can be overcome using abundant fastheuristics to solve such problems. We note that r1

i,j refers tothe indirect cost between nodes i and j through a single depot.We also define rh,min

u,v = mini∈Gu,j∈Gv rhi,j.

We make the following assumptions:

di,j ≤ rhi,j i ∈ X , j ∈ X , 1 ≤ h ≤ q, (5)

di,j ≤ di,k + dk,j i ∈ X , j ∈ X , k ∈ X , (6)

rhi,j ≤ rh

i,k + dk,j i ∈ X , j ∈ X , k ∈ X , 1 ≤ h ≤ q, (7)

rh1i,j ≤ rh2

i,j i ∈ X , j ∈ X , 1 ≤ h1 ≤ h2 ≤ q. (8)

Assumptions (5)–(8) are similar to the already discussedassumptions (1)–(3). We derive

rhi,j ≤ di,i′ + rh

i′,j′ + dj′,j i ∈ X, i′ ∈ X, j ∈ X, j′ ∈ X (9)

for every h. Condition (9) recognizes that the value of theshortest path through h depots is lower or equal to the valueof the shortest path through h + 1 depots. In Ref. [27], weargue that this is the case for PCB manufacturing.

2.2. Variants of the Problem

We examine three specific variants of the problem. To thebest of our knowledge, the GVRP, even by itself (without thegeneralizations motivated by the PCB setting), has not yetbeen studied from an analysis point of view.

The first variant assumes fully loaded trucks, the nextallows partially loaded trucks, and the last variant takes theaspect of moving depots into account.

2.2.1. The Generalized Full-Load Case. In this case, weassume that the vehicle always runs a full load on its wayout from the depots. The problem is modeled based on GE2-CEN. In other words, we assume that we fill the vehicle tocapacity and the vehicle returns to a depot empty. We alsoassume that the depots are stationary. This is the most basicscenario that we analyze, and it provides a building block forthe remaining two scenarios.

Under the assumptions listed earlier, the problem beinginvestigated here can be restated as that of finding a least costGTSP-like (i.e., each node is visited at most once) sequenceof edges in this specially formulated two-cost edge network.This problem differs from GTSP in that it uses a special costfunction to compute the total cost of a GTSP-like “tour”. Weuse both the primary and the secondary costs to compute thetotal cost, where the cost used on an edge depends on theposition of that edge in the tour. This special cost functionis designed to reflect the actual vehicle routing type problemscenario.

Let us define a “segment” as a sequence of q consecutiveedges on a GTSP-like tour. The cost of a segment is definedas the sum of the indirect cost ri,j of the first edge and thetotal direct costs di,j of the last q−1 consecutive edges of thesegment (see Fig. 7).

The GTSP-like tour we are interested in is a sequence of ζ

consecutive segments of q edges each such that exactly onenode in each G1, . . . , GK is visited.

The problem is to find the least cost GTSP-like tour R ={u1, u2, . . . , uζ } with segments u1, . . . , uζ and the cost of thetour as determined by

C(R) =ζ∑

s=1

rf (us) +

∑e∈F(us)

de

. (10)

Given segment u, we denote by f (us) the first edge in thesegment, and by F(us) the remaining edges in the segment.The orientation of each segment is defined with respect to theorientation of R.


FIG. 7. Cost of a segment for the generalized full-load case. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

2.2.2. The Generalized Partial-Load Case. In the gen-eralized partial-load case, we relax the assumption of thevehicle always carrying a full load from the depots.

The problem being investigated can thus be restated asthat of finding the least cost GTSP-like sequence of edges inthe GEM-CEN.

A segment is now defined to be a sequence of q or lessconsecutive edges on a GTSP-like tour.

The cost of a segment with h edges is now computed asthe sum of rh

i,j of the first edge of the segment and the directcost components di,j of the last h − 1 consecutive edges ofthe segment.

Let us assume we have η segments in a GTSP-like tourwith lengths h1, . . . , hη. The cost of the GTSP-like tour isnow determined as

C(R) =η∑

s=1

rhs

f (us)+

∑e∈F(us)

de

.

2.2.3. The Generalized Partial-Load Case with WaitingCost. In this case, we relax the assumption of stationarydepots and introduce the maximum component in the com-putation of the objective function. We assume that the depotsare linked and move together. This is common place in mostapplications motivating such a setting, for example, robotics,automation, and rail-cars. An alternative way to state thisassumption is to say that we are given a set of depots andtheir locations, and we have the flexibility of moving theentire ‘plane’ (all depots will move by the same amount).This captures the PCB setting since by moving the feeder allfeeder slots move in sync by the same amount. Thus, there isa finite, discrete set of positions P taken on by the depots inrelation to the default “home" or base position.

We redefine the GEM-CEN to have nodes of the form(i, p) where i refers to a customer location and p refers to theposition of the depots when the vehicle is at customer i. Thismakes a discrete approximation to the underlying continuousmotion of the depots, but it is not anticipated to be a badapproximation in practical VRP settings. Note that the qualityof the approximation can be controlled by expanding the setof all depot positions P (clearly at the expense of a largernetwork). Then, the problem being investigated can still bestated as that of finding the least cost GTSP-like sequence ofedges in the GEM-CEN. However, the special cost functionto compute the total cost of a GTSP-like tour is now modifiedto account for the moving depots. Note that the vehicle andthe depots must synchronize at points of convergence, thatis, either the vehicle has to wait for the depot to move to alocation or the other way around. For the remainder of thisarticle, we continue to denote a customer node by a singleindex i assuming that the position information is implicitlystored in each node.

A segment is still defined to be a sequence of q or lessconsecutive edges on a GTSP-like tour, but the cost definitionof a segment now includes a special cost between the twonodes corresponding to the start and end of a segment (seeFig. 8). To this end, let si,j be the special cost correspondingto the cost of moving the depots from positions defined by p1

and p2 corresponding to the nodes i and j, respectively. Valuessi,j capture any waiting time penalties (in the PCB setting, thecost is actually measured in time units). The cost of a segmentwith h edges is now computed as the sum of rh

i,k of the firstedge of the segment, and the maximum of the special cost si,j

between the start and end points of the segment and the directcost components dk,m of last (h−1) consecutive edges of thesegment. The choice of the maximum operator captures thetime/waiting interpretation of the cost. We note that again thesegment is of length h rather than of length q.


FIG. 8. Cost of a segment for the generalized partial-load case with waiting cost. [Color figure can be viewed inthe online issue, which is available at wileyonlinelibrary.com.]

For the partial load case with waiting cost, we need toimpose the following additional assumption on the extentof the violation of the triangular inequality caused by theinclusion of the maximum component. We assume that

si,j

�≤ di,k + dk,j i ∈ X , j ∈ X , k ∈ X (11)

for a fixed finite value �. Requirement (11) states that s andd values might not satisfy the triangle inequality (if they do,there is no need for the maximum component), instead it canbe violated by at most �.

Let us assume we have η segments R = {u1, . . . , uη} in aGTSP-like tour with lengths h1, . . . , hη. The cost of the touris now determined as

C(R) =η∑

t=1

rht

f (ut)+ max

∑

e∈F(ut)

de, sf2(ut),f1(ut)

where we denote by f1(ut) the head node of the last edge inthe segment, and f2(ut) the head node of the first edge in thesegment.

3. ANALYSIS OF THE ITP HEURISTIC

Let us introduce the following notation.X = {x1, x2, . . . , xn} is the set of nodes in the networkXw = subset of nodes visited by the wth segment of a

GTSP-like tourR∗ = optimal GTSP-like tour over set X using C(R∗) to

determine its costG∗ = optimal GTSP tour using direct costs on edges over

set X with cost C̄(G∗) = ∑e∈G∗ de

r̄G∗ = ∑a∈G∗ ra/K

r̄hG∗ = ∑

a∈G∗ rha/K

r̄minG∗ = ∑

{i,j}∈G∗ rming(i),g(j)/K

r̄h,minG∗ = ∑

{i,j}∈G∗ rh,ming(i),g(j)/K

3.1. The Generalized Full-Load Case

We start by formally describing the iterative tour partition-ing heuristic. The heuristic first computes the optimal GTSPtour G∗ by considering only direct cost d. Next a node andan orientation of G∗ are fixed. We follow G∗ based on theselected orientation by starting at the node and each q − 1edges, we design the next edge to bear the indirect cost. Afterreaching back the original node, we obtain a GTSP-like tour.The entire process is repeated for each possible starting nodeand the lowest cost solution among the q solutions is returned.(It actually suffices to only take q consecutive nodes as start-ing nodes.) The main result of this section is to establish aworst-case ratio for this heuristic.

Description of the Heuristic

1. Compute G∗.2. Choose an arbitrary node p and an arbitrary orientation

along G∗.3. Partition G∗ into ζ segments to get a solution starting

from node p.4. Record the underlying cost.5. Repeat steps 2 – 4 by starting with customers p + 1, p +

2, . . . , p + q − 1 to obtain a solution with respect to eachone of them.

6. Choose the best among the q solutions obtained anddenote it by RH .

3.1.1. Approximation Ratio. We start by establishing alower bound on the optimal value.

Theorem A.1. We have

C(R∗) ≥ max{r̄min

G∗ , C̄(G∗)}. (12)

Proof. It is clear that we have C(R∗) ≥ C̄(G∗) due to(1).

Let us first consider ζ > 1. We also know that, for a givenoptimal GTSP tour G∗ and given edge e = {k, l} ∈ G∗, boththe tail customer group Gg(k) and the head customer group


FIG. 9. Derivation of lower bound. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

Gg(l) are visited in some order by the optimal solution R∗,albeit possibly through some different nodes, say, i ∈ Gg(k)

and j ∈ Gg(l). We know that either one of the two possiblepaths in R∗ between customer groups Gg(k) and Gg(l) mustcontain at least one edge that contributes the indirect costsince ζ ≥ 2. Let us renumber the nodes so that one path ofthe optimal solution between customer group Gg(k) to Gg(l)

(along R∗) is denoted by 1, 2 . . . α, β, . . . , t with edge {α, β}contributing the indirect cost toward C(R∗). Note that thismight not be the only edge contributing the indirect cost. Sim-ilarly, let the nodes along the other path following R∗ betweencustomer group Gg(k) and Gg(l) be denoted by 1′, 2′, . . . , t′.We have from our assumptions (see also Fig. 9),

rmine = rmin

g(k),g(l) ≤ ri,j

≤ ri,β + dβ,j (13)

≤ ri,β + (dt,j + dt−1,t + dt−2,t−1 + dβ,t−2) (14)

≤ di,α + rα,β + (dt,j + dt−1,t + dt−2,t−1 + dβ,t−2)

(15)

≤ (di,1 + d1,2 + dα−1,α) + rα,β

+ (dt,j + dt−1,t + dt−2,t−1 + dβ,t−2) (16)

≤ C(R∗), (17)

where (14) and (16) follow from (2), (13), and (15) followfrom (3), and (17) follows from (1) and the non-negativity ofcosts.

If ζ = 1 and ri,j is the only indirect cost edge in R∗, weclearly have ri,j ≤ C(R∗). If ζ = 1 and ri,j is not the onlyindirect cost edge, it implies that there is a different edge onR∗ contributing its indirect cost and thus we proceed as in thecase ζ ≥ 2.

As re ≤ C(R∗) for every e ∈ G∗, it follows that r̄minG∗ ≤

C(R∗). ■

Next, we obtain an upper bound on C(R∗). Let RH bethe GTSP-like tour solution produced by the heuristic withC(RH) as the underlying cost.

Theorem B.1. We have

C(R∗) ≤ C(RH) ≤ K

qr̄G∗ +

(1 − 1

q

)C̄(G∗). (18)

Proof. We observe that the cumulative length of the qsolutions encountered by the heuristic is equal to

∑e∈G∗

re + (q − 1) · C̄(G∗).


The value C(RH) equals the best solution value among the qsolutions, and thus must be less than or equal to the averagevalue of all encountered solutions. Therefore, we conclude

C(RH) ≤∑

e∈G∗ re

q+

(1 − 1

q

)C̄(G∗),

or equivalently

C(RH) ≤ K

qr̄G∗ +

(1 − 1

q

)C̄(G∗).

The statement C(R∗) ≤ C(RH) clearly holds. ■

From Theorems A.1 and B.1, we obtain the following mainresult.

Theorem C.1. We have

C(RH)

C(R∗)

≤ r̄G∗

max{r̄min

G∗ , C̄(G∗)} K

q+ C̄(G∗)

max{r̄min

G∗ , C̄(G∗)}

(1 − 1

q

).

(19)

If we reduce the problem to a standard VRP scenario wherethere are no clusters and there is no need for the truck to visitseveral depots between two routes, we observe that the abovederived bound reduces to the bounds in Refs. [15, 20].

3.1.2. Tightness of the Bound. In this section, we provethat the derived bound (19) is tight by finding a family ofinstances with K clusters for which (19) is at equality.

Consider network G(X , E) with K customer groups, mnodes each. Consider one node, called the “leader” node fromeach of the K groups being placed at equal intervals alongthe perimeter of a circle of diameter φ and consecutive nodesbeing placed at angular intervals of ε = π/n incrementallyfrom the leader node of the underlying group (see Fig. 10).Thus, we have a total of n = K · m nodes and we assume thatq divides n. Note that by definition, we have ε ≤ 2π/n andthus the groups do not overlap. We fix a given orientationof the circle. Let di,j be the distance along the circle fromcustomer i to customer j and ri,j = di,j + δ where δ = q/K2.Then, the optimal GTSP tour follows the circle itself using,say, the first node or leader from each customer group. Wehave C̄(G∗) = πφ.

In our example, we clearly satisfy the required assump-tion ri,j ≥ di,j. Furthermore, we also satisfy the other twoassumptions di,j ≤ di,k + dk,j and ri,j ≤ ri,k + dk,j (becauseδ + di,j = ri,j). We know that

di,j = φ

2(ρ + (b − a)ε)

for all i, j belonging to consecutive groups on the circle whereρ = 2π/K and a, b are the sequence numbers of nodes i, j intheir respective groups. We note that rmin

u,v = (ρ−mε)φ/2+δ

for two consecutive clusters u and v. Therefore, we have

FIG. 10. Tightness proof instance. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]

r̄minG∗ = 1

K

{K

ρφ

2− Km

εφ

2+ Kδ

}= ρφ

2− mεφ

2+ δ.

On the other hand, r̄G∗ = (πφ + Kδ)/K = ρφ/2 + δ.Solution R∗ visits the nodes in the same sequence along

the perimeter of the circle, and its value is clearly Kρφ2 + n

q ·δ,because r values are selected n/q times.

The heuristic begins with the GTSP tour and chooses,regardless of the starting point due to symmetry, a solutionof equal cost, that is, C(RH) = Kρφ

2 + nq · δ. We conclude

C(RH)/C(R∗) = 1. We also obtain, by substituting valuesderived above

r̄G∗

max{r̄min

G∗ , C̄(G∗)} K

q+ C̄(G∗)

max{r̄min

G∗ , C̄(G∗)}

(1 − 1

q

)

= 1 + 1

Kπφ.

Here, we use the definition of ε and δ, which imply thatmax{r̄min

G∗ , C(G∗)} = C̄(G∗). This can be seen by a straight-forward calculation. Hence, as K → ∞, this shows that thebound is tight.

3.2. The Generalized Partial-Load Case

In this section, we perform the analysis for the case whenthe vehicle can carry less than q units. The major challengeis to adapt the ITP heuristic. Although the heuristic still findsa feasible solution, the resulting encountered set of feasiblesolutions is too stringent (only full truck loads are consid-ered). For this reason we need to generalize it. The act offinding G∗ remains the same.

As before, we fix the orientation of G∗. For simplicity,we label the nodes 1, . . . , K along G∗. Let us create a newauxiliary network where there is a node for every set of q orless consecutive edges on G∗. The number of nodes is

(Kq

).


FIG. 11. Costs related to the auxiliary network. [Color figure can be viewedin the online issue, which is available at wileyonlinelibrary.com.]

Two nodes c1 and c2 in this new auxiliary network areconnected by an arc with cost rh

i,j if the tail node c1 represents asequence of edges (in the original network) ending with nodei and the head node c2 represents a sequence of h edges (inthe original network) starting with node j (see Fig. 11).

A node itself carries an additional cost equivalent to thecumulative cost of traversing the sequence of arcs that it rep-resents in the original network using the direct costs. Thus,if node c1 in the auxiliary network represents the sequenceof nodes a − k, a − k + 1, . . . , a, the cost of the node in theauxiliary network is defined by

∑a−1y=a−k dy,y+1.

Let us now fix a customer group p. To this auxiliary net-work, we add a source node s and sink node t to facilitate thecomputation of the shortest path through the network. Sources is connected to each node corresponding to a sequence orig-inating at customer group p. The cost of an (s, c1) arc equalsto the cost of picking up as many items as nodes in the originalnetwork represented by c1, and then moving to node p. Sim-ilarly, each node corresponding to a sequence ending with pis connected to t with a cost of 0.

We repeat the construction of the network starting withcustomer groups p + 1, p + 2, . . . p + q − 1 to obtain q suchnetworks. It is easy to see that each s−t path in these networksyields a feasible solution. In addition, each solution followingG∗ corresponds to an s − t path in one of the networks. Bythe definition of costs, the cost of a path in any one of theseauxiliary networks equals to the cost of the correspondingGTSP-like solution. We note that the presented methodologyof finding the shortest s − t path on an auxiliary network isakin to the methodology used by Beasley [3], however, weextend it by (1) using q such networks and (2) modeling botharc and node costs, due to the more complicated nature of theproblem at hand.

The formal description of the heuristic follows.

Description of the Heuristic

1. Compute G∗.2. Choose an arbitrary customer group p and an arbitrary

orientation along G∗.3. Construct the q auxiliary networks.

4. Compute the shortest s− t path in each one of the createdauxiliary networks.

5. Choose the best among the q solutions so obtained anddenote it by RH .

3.2.1. Approximation Ratio. We start by generalizingTheorem A.1


C(R∗) ≥ max{r̄1,min

G∗ , C̄(G∗)}. (20)

Proof. It is clear that we have C(R∗) ≥ C̄(G∗) due to(5).

Let first ζ > 1. We know that, for a given optimal GTSPtour G∗ and given edge e = {k, l} ∈ G∗, both the tail customergroup Gg(k) and the head customer group Gg(l) are visited insome sequence on the optimal solution R∗, albeit possiblythrough some different nodes, say, i and j. We know thateither one of the two possible paths in R∗ between customergroups Gg(k) and Gg(l) must contain at least one edge thatcontributes the indirect cost since ζ > 1.

Let us renumber the nodes so that nodes 1, 2, . . . , α1,β1, . . . , ατ , βτ , . . . , t form one path from node i tonode j along R∗ with τ edges {α1, β1}, . . . , {ατ , βτ }contributing indirect costs rp1

α1,β1, . . . , rpτ

ατ ,βτ, and nodes

1′, 2′, . . . , α1′ , β1′ , . . . , ατ ′ , βτ ′ , . . . , t′ form the other pathwith τ ′ edges {α1′ , β1′ }, . . . , {ατ ′ , βτ ′ } contributing indirectcosts rp1′

α1′ ,β1′ , . . . , rpτ ′ατ ′ ,βτ ′ . We have (see Fig. 12)

C(R∗) = {di,1 + d1,2 + · · · + rp1

α1,β1+ · · ·

+ rp2

α2,β2+ · · · + rpτ

ατ ,βτ+ · · · + dt,j

}+ {

dj,t′ + · · · + rpτ ′ατ ′ ,βτ ′ + · · · + r

p′2

β2′ ,α2′ + · · ·+ r

p′1

β1′ ,α1′ + · · · + d1′,i}

(21)

≥ {di,α1 + rp1

α1,β1+ dβ1,α2 + rp2

α2,β2+ dβ2,α3 + · · ·

+ rpτ

ατ ,βτ+ dβτ ,t + dt,j

}+ {

dj,t′ + dt′,βτ ′ + rpτ ′βτ ′ ,ατ ′ + · · ·

+ rp′

2β2′ ,α2′ + dα2′ ,β1′ + r

p′1

β1′ ,α1′ + dα1′ ,1′ + d1′,i}

(22)

≥ {di,α1 + r1

α1,β1+ dβ1,α2 + r1

α2,β2

+ dβ2,α3 + · · · + r1ατ ,βτ

+ dβτ ,t + dt,j}

+ {dj,t′ + dt′,βτ ′ + r1

ατ ′ ,βτ ′ + · · · + r1β2′ ,α2′

+ dα2′ ,β1′ + r1β1′ ,α1′ + dα1′ ,1′ + d1′,i

}(23)

≥ r1i,j (24)

≥ r1,ming(k),g(l)

where (21) follows from the definition of C(R∗), (22) followsfrom (6), (23) follows from (8), and (24) follows from (6)and (7).


FIG. 12. Derivation of modified lower bound. [Color figure can be viewed in the online issue, which is availableat wileyonlinelibrary.com.]

If ζ = 1 and rhi,j is the only indirect cost edge in R∗, we

clearly have r1i,j ≤ C(R∗). If ζ = 1 and rh

i,j is not the onlyindirect cost edge, it implies that there is a different edge onR∗ contributing its indirect cost and thus we proceed as in thecase ζ ≥ 2.

Since r1,ming(a),g(b) ≤ C(R∗) for every {a, b} ∈ G∗, it follows

that r̄1,minG∗ ≤ C(R∗). ■

Next, we establish an upper bound on C(RH).

Theorem B.2. For each h = 1, 2, . . . , q and K mod h = 0,we have

C(R∗) ≤ C(RH) ≤ K

hr̄h

G∗ +(

1 − 1

h

)C̄(G∗). (25)

For each h = 1, 2, . . . , q and K mod h > 0, we have

C(R∗) ≤ C(RH) ≤ 2K

hr̄h

G∗ +(

1 − 2

h

)C̄(G∗). (26)

Proof. We observe that each of the q solutions is theshortest s−t path in the respective network. Thus, it is shorterthan any other path through the underlying network.

Suppose we only consider paths such that each node onthe path represents a sequence of h customer groups in theoriginal network except for the very last node, which could

correspond to a sequence of length less than h. We derivethat there are only h such unique paths among the q networksdenoted by P1, . . . , Ph, and with lengths L(P1), . . . , L(Ph),respectively.

Let S1, S2, . . . , Sq be the shortest paths through the qnetworks and L(S1), L(S2), . . . , L(Sq) be their lengths.

Imagine that we have q · h networks where each of theq networks is repeated h times. We thus have q repetitionsof each of the paths P1, . . . , Ph and h repetitions of pathscorresponding to S1, . . . , Sq among these networks. As L(Si)

is the shortest path value, it follows that L(Si) ≤ L(Pk) foreach matching k. We thus have

h ·q∑

i=1

L(Si) ≤ q ·h∑

j=1

L(Pj).

Therefore, we conclude that

∑qi=1 L(Si)

q≤

∑hj=1 L(Pj)

h.

Clearly, the value C(RH), which is equal to the best solu-tion value among the q solutions, must be less than or equalto the average value of these q solutions. Therefore, we have

C(RH) ≤∑q

i=1 L(Si)

q≤

∑hj=1 L(Pj)

h.


Let first K mod h = 0. We know that

h∑j=1

L(Pj) =∑e∈G∗

rhe + (h − 1)C̄(G∗)

because of the structure of the chosen paths. As K mod h =0, each vehicle dispatched in Pj carries exactly K/h units.Therefore, we get

C(RH) ≤∑

e∈G∗ rhe

h+

(1 − 1

h

)C̄(G∗)

or

C(RH) ≤ K

hr̄h

G∗ +(

1 − 1

h

)C̄(G∗).

Let now K mod h > 0. In this case, the vehicles do notdispatch the same number of units. It is not difficult to seethat

h∑j=1

L(Pj) =∑e∈E1

(rh

e + (h − 1) · de)

+∑e∈E2

(rh

e + rh̄e + (h − 2) · de

)

for h̄ = K mod h−1 < h. Here G∗ = E1 ∪E2, E1 ∩E2 = φ.As rh̄

e ≤ rhe and de ≤ rh

e for every e, we obtain

h∑j=1

L(Pj) ≤∑e∈E1

(2rh

e + (h − 2) · de)

+∑e∈E2

(2rh

e + (h − 2) · de)

=∑e∈G∗

(2rh

e + (h − 2) · de)

= 2∑e∈G∗

rhe + (h − 2) · C̄(G∗).

The rest of the proof follows the steps from the case K modh = 0. Inequality C(R∗) ≤ C(RH) is trivially true. ■

As q ≤ K , we note that if 1 ≤ h ≤ q−1, then K mod h >

0, but if h = q, then K mod h can either be 0 or positive.From Theorems A.2 and B.2, and defining

H(α)

=

2r̄α

G∗

max{r̄1,min

G∗ , C̄(G∗)} K

α

+ C̄(G∗)max

{r̄1,min

G∗ , C̄(G∗)}

(1 − 2

α

)K mod α > 0

r̄αG∗

max{r̄1,min

G∗ , C̄(G∗)} K

α

+ C̄(G∗)max

{r̄1,min

G∗ , C̄(G∗)}

(1 − 1

α

)K mod α = 0,

we obtain the following main result.


C(RH)

C(R∗)≤ min

α∈{1,2,...,q} H(α). (27)

As both Refs. [15, 20] did not consider the partial loadcase, we cannot compare bound (27) to their results.

3.2.2. Tightness of the Bound. In this section, we provethat the modified bound (27) is tight by finding a family ofinstances with K customer groups, each with m nodes, forwhich (27) is at equality.

We consider the same example as in Section 3.1.2, butwith q indirect-cost edges between every pair of nodes. We letrh

a,b = da,b +δ be the distance along the circle from customera to customer b for 1 ≤ h ≤ q with an added term δ. Allassumptions (5)–(8) are clearly satisfied. Then, the optimalGTSP tour follows the circle itself and C̄(G∗) = πφ.

As in Section 3.1.2, we have

r̄h,minG∗ = ρφ

2− mεφ

2+ δ.

We obtain a series of optimal solutions of equal objectivevalue by visiting the nodes in the same sequence along theperimeter of the circle. An optimal solution R∗ also is usingthe indirect cost r1

i,j on each edge {i, j} and is following theoptimal GTSP tour. Thus, the objective value of the solutionis clearly C(R∗) = Kρφ/2 + n/q · δ regardless of whetherK mod h = 0 or K mod h > 0.

The heuristic begins with the TSP tour and chooses,regardless of the starting point, a solution of equal cost, that is,C(RH) = Kρφ

2 + nq ·δ again regardless of whether K mod h =

0 or K mod h > 0. We conclude C(RH)/C(R∗) = 1 in eitherscenario.

If K mod h = 0, we obtain, by substituting valuesobtained above

H(h) = 1

K· K

h+

(1 − 1

h

)+ 1

Kπφ= 1 + 1

Kπφ

and when K mod h > 0, likewise we obtain

H(h) = 2 · 1

K· K

h+

(1 − 2

h

)+ 1

Kπφ= 1 + 1

Kπφ

Hence, as K → ∞, the bound can be seen to be tight.

3.3. The Generalized Partial-Load Case with Waiting Cost

In this section, we perform the analysis for the case whenthe depots can move in the plane, and thereby a waiting cost isintroduced into the cost computation function for the routesmade by a vehicle. We create a new auxiliary network wherethere is a node for every set of q or less consecutive edgeson G∗ as before. However, in this network, we modify thenode costs to be the maximum of the special cost and the


FIG. 13. Costs related to the auxiliary network. [Color figure can be viewed in the online issue, which is availableat wileyonlinelibrary.com.]

cumulative cost of traversing the sequence of edges that thehead node represents in the original network using the directcosts (see Fig. 13).

We again fix a customer group p and add a source node sand sink node t to facilitate the computation of the shortestpath through the network as before. Again, as before, sources is connected to each node corresponding to a sequence orig-inating at group p. Similarly, each node corresponding to asequence ending with p is connected to t.

We repeat the construction of the network starting withnodes p + 1, p + 2, . . . p + q − 1 to obtain q such networks.It is easy to see that each s − t path in these networks yieldsa feasible solution. In addition, each solution following T∗corresponds to an s − t path in one of the networks.

The heuristic is exactly the same as before except usingthese networks with new definitions of costs. Also in this casewe are able to analyze the heuristic.

Theorem A.3. We have, as before

C(R∗) ≥ max{r̄1,min

G∗ , C̄(G∗)}. (28)

Proof. It is clear from (12) that C(R∗) in this case is onlygoing to take on a higher value than the C(R∗) computed inTheorem A.2. As the conditions and assumptions of TheoremA.2 continue to hold here as well, this completes the proof. ■

To upper bound the cost of the heuristic, we obtain thefollowing result.

Theorem B.3. For each h = 1, 2, . . . , q and K mod h = 0,we have

C(R∗) ≤ C(RH) ≤ K

hr̄h

G∗ + �

(1 − 1

h

)C̄(G∗). (29)

For each h = 1, 2, . . . , q and K mod h > 0, we have

C(R∗) ≤ C(RH) ≤ 2K

hr̄h

G∗ + �

(1 − 2

h

)C̄(G∗). (30)

Proof. For any two nodes i, j and path P connect-ing these two nodes, from (11) and (6) we obtain si,j ≤�

∑e∈P de. In turn it follows that max{si,j,

∑e∈P de} ≤

�∑

e∈P de.We observe as before that each of the q solutions is the

shortest s−t path in the respective network. Thus, it is shorterthan any other path through the underlying network.

Suppose we again only consider paths such that each nodeon the path represents a sequence of h groups in the originalnetwork except for the very last node, which could corre-spond to a sequence of length less than h. We derive asbefore that there are only h such unique paths among theq networks denoted by P1, . . . , Ph with lengths denoted byL(P1), . . . , L(Ph). We know from the proof of Theorem B.2that

∑qi=1 L(Si)

q≤

∑hj=1 L(Pj)

h

and therefore, we have

C(RH) ≤∑q

i=1 L(Si)

q≤

∑hj=1 L(Pj)

h.

Let first K mod h = 0. We know that

h∑j=1

L(Pj) ≤∑e∈G∗

rhe + � · (h − 1)C̄(G∗)

because of our assumption (11) and the structure of the chosenpaths.

Let now K mod h > 0. In this case, as in the proof ofTheorem B.2 the vehicles do not dispatch the same numberof units and we obtain

h∑j=1

L(Pj) ≤∑e∈E1

(rh

e + � · (h − 1) · de)


+∑e∈E2

(rh

e + rh̄e + � · (h − 2) · de

).

The rest of the proof is identical to the proof of TheoremB.2. ■

From Theorems A.3 and B.3, and defining

H(α)

=

2r̄α

G∗

max{r̄1,min

G∗ , C̄(G∗)} K

α

+ C̄(G∗)max

{r̄1,min

G∗ , C̄(G∗)}� ·

(1 − 2

α

)K mod α > 0

r̄αG∗

max{r̄1,min

G∗ , C̄(G∗)} K

α

+ C̄(G∗)max

{r̄1,min

G∗ , C̄(G∗)}� ·

(1 − 1

α

)K mod α = 0,

we obtain the following main result.


C(RH)

C(R∗)≤ min

α∈{1,2,...,q} H′(α). (31)

3.4. Results for Standard VRP

In this section, we summarize the results when the prob-lem is reduced to the standard multidepot VRP case. Recallthat cost ri,j is defined as the cost of the truck finishing the lastdelivery at node i, next visiting q depots (not modeled explic-itly), and then moving to node j with q items to start the nextdelivery, in the most cost effective way. They are obtainedfrom the generalized cases by setting m, the number of nodesin each group to be equal to 1.

In the full-load case [20], we obtain the following knownresults. As this refers to the standard VRP case, we have nopartitions Gi, and thus, by shrinking each partition to containonly a single node, we can replace G∗ with its equivalent ofT∗ defined as

T∗ = optimal TSP tour using direct costs on edges over setX with cost C̄(T∗) = ∑

e∈T ∗ de .

Theorem A.4. We have m = 1, G∗ = T∗, and K = n. Thisgives us the following results:

C(R∗) ≥ max{r̄T ∗ , C̄(T∗)},

C(R∗) ≤ C(RH) ≤∑

e∈T ∗ re

q+

(1 − 1

q

)C̄(T∗),

C(RH)

C(R∗)≤ r̄T ∗

max{r̄T ∗ , C̄(T∗)}n

q+ C̄(T∗)

max{r̄T ∗ , C̄(T∗)}(

1 − 1

q

).

(32)

A bound independent of T∗ reads

C(RH)

C(R∗)≤ n − 1

q+ 1. (33)

Inequality (33) follows from (32) due to

r̄T ∗

max{r̄T ∗ , C̄(T∗)} ≤ 1,C̄(T∗)

max{r̄T ∗ , C̄(T∗)} ≤ 1.

The partial-load case has not been studied previously andthus the results below are new. Here rh

i,j is interpreted as theindirect cost of going from node i to node j while visiting hdepots en route in the most cost-efficient manner. Again, wecan substitute G∗ with T∗ due to the underlying absence ofpartitions as a result of setting m = 1.


C(R∗) ≥ max{r̄1

T ∗ , C̄(T∗)}.

For each h = 1, 2, . . . , q, if n mod h = 0, then

C(R∗) ≤ C(RH) ≤ n

hr̄h

T ∗ +(

1 − 1

h

)C̄(T∗),

and if n mod h > 0,

C(R∗) ≤ C(RH) ≤ 2n

hr̄h

T ∗ +(

1 − 2

h

)C̄(T∗).

We also have

C(RH)

C(R∗)≤ min

α∈{1,2,...,q} H(α).


C(RH)

C(R∗)≤ n − 1

q+ 1

if K mod q = 0, and

C(RH)

C(R∗)≤ n − 2

q+ 2

if K mod q > 0.

We can also derive the following results from the partial-load case with waiting cost when moving depots are intro-duced.


C(R∗) ≥ max{r̄1

T ∗ , C̄(T∗)}.

For each h = 1, 2, . . . , q, if n mod h = 0, then

C(R∗) ≤ C(RH) ≤ n

hr̄h

T ∗ + �

(1 − 1

h

)C̄(T∗),

and if n mod h > 0, then

C(R∗) ≤ C(RH) ≤ 2n

hr̄h

T ∗ + �

(1 − 2

h

)C̄(T∗),


and

C(RH)

C(R∗)≤ min

α∈{1,2,...,q} H′(α).


C(RH)

C(R∗)≤ n − �

q+ �

if K mod q = 0, and

C(RH)

C(R∗)≤ n − 2�

q+ 2�

if K mod q > 0.

4. COMPUTATIONAL RESULTS

We have implemented these heuristics and extensivelyran them on both randomly generated and real-world PCBtest cases. The results as documented below are represen-tative of results obtained on multiple runs of the heuristicover test cases of varied problem complexity and dimensions(measured specifically in two dimensions, namely, the totalnumber of components and the number of spindles presenton the head).

The first set of test instances were generated using arandomized model for a board with key parameters suchas number of components, component diversity, and num-ber of spindles as inputs. Statistical analysis of the boarddesigns for a major electronics manufacturer partner overrecent years (2007–2008) were used to determine the rangesof these parameters. Machine parameters such as velocities,on the other hand, were fixed to those values found most com-monly in such configurations. As a reference point, a mid-sizeinstance with 250 board nodes, 10 spindles, 50 componenttypes, 100 feeder slots, and 150 feeder positions yields acomplete graph with 152,500 nodes and of the order of 11.6billion edges.

The second set of test instances comprised of three real-world PCBs used by a partner electronics manufacturer.These PCBs belong to representative scales of boards typicalin the industry. The smallest instance is a board of low com-plexity containing 41 components belonging to one of the 10different component types. The second instance is a board ofmedium complexity containing 185 components belongingto one of the 13 different types. The final instance is a boardof large complexity containing 540 components belonging toone of the 61 types.

The experiments were carried out on a cluster of 52 nodes,each having two 64-bit 3.2 GHz Pentium 4 Xeon processorssharing 6 GB RAM with InfiniBand, running Red Hat Enter-prise Linux 4 and compiled under GCC version 4.1.2. Theimplementation however, does not exploit parallel proces-sors; it is sequential. We used the Concorde Lin-KerniganTSP Solver [2] to solve TSPs and an in-house implementa-tion of a version of the Dijkstra’s algorithm [10] to solve theshortest path problems.

For typical test-case scenarios, the worst-case ratios pre-sented in this article lie anywhere between 10 and 30depending on the particular parameters and features of eachtest-case instance. In practice, the proposed algorithms tendto perform much better. It is hard to establish the ratio ofthe heuristics, because optimal solutions or lower boundsare unknown. Instead, we compare them against the bestknown solution. The solutions generated by our heuristicswere used as initial starting solutions to a state-of-the-artcomputationally intensive iterative improvement algorithm[27] based on solving integer linear programs at each iter-ation. They were typically found to be between 1.5 and 3times off from the solution obtained as a result of running theadvanced improvement heuristic over long periods of time(of the order of 45 min to 5 h). Any “single-shot” solutionto such a difficult problem could be vastly improved by theapplication of very computationally intensive improvementheuristics. In this work, the goal is the derivation of a goodstarting solution to this extremely complex and challengingproblem that could be further refined by various improvementand exchange heuristics available in the research literature,as is a commonly accepted framework for solution method-ologies in the VRP and TSP world. As a testament to this,we ran the same advanced improvement algorithm with acompletely randomized solution, and it did not approach thesolution quality of the heuristics proposed in this article evenafter 3 h of crunching integer linear programming models (weunfortunately cannot compare our results to the mathematicaloptimum due to the intractability and immense complexity ofsolving even small problems exactly and thus this comparisonis used herein).

To further gauge the quality of the heuristics, we alsoreport the generated solution of the heuristic as being con-sistently about 50% better than the best solution obtained byrunning a genetic algorithm using genetic operators reportedin the literature [18] for such problems for 3 h starting froma completely random solution pool. The running time ofthe heuristic averages from 2 to 10 min for common boardinstances. The heuristics are valuable in their contribution to

FIG. 14. GA and ILPBRH starting with heuristic solution. [Color figure canbe viewed in the online issue, which is available at wileyonlinelibrary.com.]


FIG. 15. ILPBRH starting with random and heuristic solutions. [Colorfigure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

the state of the progress toward obtaining an acceptable solu-tion of such problems as such remain unachievable even byadvanced computational techniques without the foundationof such a strong starting solution.

The performance of the initial heuristic as compared to thebest solution obtained by the iterative improvement algorithmfrom Ref. [27] and the genetic algorithm is depicted in Figure14. In this figure, “GA” stands for the genetic algorithm and“ILPBRH” represents the iterative integer linear program-ming based improvement algorithm. Both algorithms havebeen started from the solution obtained by the heuristic pre-sented herein, and this demonstrates the effectiveness of theadvanced improvement algorithm that is compared againstin lieu of a true optimum. Figure 15 shows this same algo-rithm being started with a randomized solution as opposed toa solution generated by the proposed heuristics.

The computational time of the heuristic grows onlylinearly with the total number of nodes n and it growslogarithmically and flattens out with an increasing vehiclecapacity q. In Tables 1 and 2, the column “Soln. time” showsthe total running time of the heuristic. These trends are in linewith our expectations given the nature of the heuristic. Fromthe PCB perspective, an increasing number of componentsyields a near-linearly increasing network size in terms of thenumber of nodes, whereas an increasing number of spindlescorresponds to an increasing network size for the GTSP, butsimultaneously also causes a decrease in the number of routesrequired, which at some point becomes the dominating factorand causes the flattening effect.

TABLE 1. Effect of increasing n.

n Soln. time (min.)

29 0.5367 1.27

129 3.77241 4.81461 10.44

TABLE 2. Effect of increasing q.

q n Soln. time (min)

6 202 4.5912 202 14.5518 202 45.6624 202 93.53

5. CONCLUSIONS

In this article, we derive worst-case bounds on several ver-sions of the ITP algorithm adapted for a superset of manyimportant generalizations of the VRP and its commonlyfound variants such as the VRP with backhauls, the multi-depot VRP, and the VRP with pick-up and delivery. Thesebounds have also been proved to be tight. The analyses giveus an insight into the development of efficient algorithms thatyield solutions that are provably within a given ratio of theoptimal solution and, furthermore, are tight. The presentedheuristics also have a solid practical performance by being1.5–3 times worse than the best known solution.

An interesting and significant extension of this work mightbe to take into account the multicommodity nature of suchproblems, and develop and analyze ITP heuristics under suchconditions. Another interesting extension might be to allowfor other constraints found in VRP scenarios, such as thetime window constraints, preceedence constraints, and depotstocking capacity constraints among others.

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