Robust Vehicle Routing under Uncertainty via

Branch-Price-and-Cut

Akang Wang1,2, Anirudh Subramanyam1,2,3, and Chrysanthos E. Gounaris∗1,2

1Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213

2Center for Advanced Process Decision-making, Carnegie Mellon University, Pittsburgh, PA 15213

3Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL 60439

Abstract

This paper contemplates how branch-price-and-cut solvers can be employed along with the

robust optimization paradigm to address parametric uncertainty in the context of vehicle routing

problems. In this setting, given postulated uncertainty sets for customer demands and vehicle

travel times, one aims to identify a set of cost-effective routes for vehicles to traverse, such that

the vehicle capacities and customer time window constraints are respected under any antici-

pated demand and travel time realization, respectively. To tackle such problems, we propose a

novel approach that combines cutting-plane techniques with an advanced branch-price-and-cut

algorithm. Specifically, we use deterministic pricing procedures to generate “partially robust”

vehicle routes and then utilize robust versions of rounded capacity inequalities and infeasible

path elimination constraints to guarantee complete robust feasibility of routing designs against

demand and travel time uncertainty. In contrast to recent approaches that modify the pric-

ing algorithm, our approach is both modular and versatile. It permits the use of advanced

branch-price-and-cut technologies without significant modification, while it can admit a variety

of uncertainty sets that are commonly used in robust optimization but could not be previously

employed in a branch-price-and-cut setting.

Keywords: vehicle routing, demand uncertainty, travel time uncertainty, robust optimiza-

tion, branch-price-and-cut, robust rounded capacity inequalities, infeasible path elimination

constraints

∗Corresponding author: gounaris@cmu.edu

1

1 Introduction

The vehicle routing problem (VRP) is one of the most highly studied combinatorial optimization

problems in the areas of supply chains and distribution logistics. The classic setting is the one

where, given a distribution center, a fleet of vehicles and a group of customers to be served, we aim

to identify a set of minimum-cost routes for vehicles to traverse, such that customer demands are

satisfied and applicable system constraints, including vehicle capacities, delivery time windows, and

route duration limits, among others, are respected. For the past a few decades, many researchers

have dedicated their efforts towards the development of exact and heuristic approaches for solving

VRPs [51]. For the larger part, these studies assume that all the information necessary to instantiate

the corresponding routing problems is known and readily available at the time of decision making. In

practice, however, this assumption does not usually hold due to the uncertainty around the problem

parameters. For example, vehicle travel times can vary due to unexpected events, such as vehicle

breakdowns, or due to factors incuding traffic congestion and bad weather, which sometimes might

be hard to accurately forecast. Making routing decisions while ignoring parameter variability can

potentially cause either cost-prohibitive routing designs or failure to satisfy service commitments.

Hence, it is of great importance for the distributor to take uncertainty into account at the route

planning stage.

The work of [19] identified three types of uncertainty sources that are commonly found in

VRP applications: customer demands, travel/service times, and customer orders. The customer

demand uncertainty applies in a situation where the amount of product that is delivered (or picked

up) at customers are random. The travel/service times are subject to change due to unpredicted

events, e.g., weather conditions, and hence are often not known with precision at the moment

the distributor is designing routes. Finally, customer order uncertainty reflects settings when the

distributor may receive unanticipated orders from customers, which have to be incorporated in

the delivery plan after the original routes have been committed upon. For a treatment of how to

incorporate customer order uncertainty in the context of a robust multi-period routing setting in

which customers might call-in to place orders after the routes have been designed, we refer to [47].

In this work, we focus on customer demand and vehicle travel time uncertainty, which are the two

types of parametric uncertainty most often contemplated in the context of branch-price-and-cut

algorithms.

2

Most contributions for tackling VRPs with uncertainty in demands and travel times employ the

stochastic programming [12] paradigm. The latter considers customer demands and/or vehicle travel

times as random variables that follow a given probability distribution. The work of [19] surveyed

existing research efforts using stochastic programming and classified them into three main modeling

frameworks. The most common one is to model the routing problem under uncertainty as a two-

stage optimization problem: routing decisions are made a priori in the first stage, while in the

second stage, uncertainty is gradually revealed and recourse actions are taken so as to fulfill service

commitments that otherwise would not have been met; hence, this approach is usually referred to

as a priori optimization. The second framework is the re-optimization approach. Its key feature is

that routing decisions are made dynamically in order to benefit from the fact that the information

related to uncertain parameters is revealed over time. The third one is the chance-constrained

approach, which ensures that the feasibility of a routing design is achieved above a prespecified

probability threshold. We refer interested readers to [19] for a comprehensive discussion on these

approaches.

The stochastic programming paradigm is a natural choice when dealing with uncertainty, but

it might suffer from two main shortcomings, namely information availability and tractability. The

former refers to cases where the probability distribution governing the uncertain parameters is

either not available or not known precisely at the decision-making moment. At the same time,

tractability might become an issue when there is a large number of random variables, making

the task of optimization coupled with numerical integration (i.e., many scenarios) prohibitive to

carry out. An alternative paradigm is that of robust optimization (RO) [9], which might be a

promising approach for treating uncertainty in some VRP applications. This approach only assumes

that uncertain parameters are random variables falling into a postulated set, and it seeks for the

optimal routing design that is immunized against all anticipated parameter realizations from that

set. Usually, this uncertainty set can be constructed with probabilistic confidence using historical

data. Within the realm of RO, there are many settings in which the computational tractability

is preserved compared with their deterministic counterparts [9]. In this work, we opt for the RO

paradigm to a priori design cost-effective routes that are immunized against infeasibility caused by

variability in customer demands and vehicle travel times. For convenience, we refer to the routing

problems of our interest as robust vehicle routing problems.

3

In recent years, there has been a growing focus in tackling robust VRPs exactly. The work of [50]

considered an RO approach for the capacitated vehicle routing problem (CVRP) with uncertainty

in demand, anticipating however that each customer’s demand may attain its maximum value

simultaneously, leading to an overly conservative formulation. In the work of [22], the authors

considered the robust CVRP with customer demands being supported on a generic polyhedron, and

they derived and compared robust counterparts of several deterministic CVRP formulations. The

computational studies showed that embedding their proposed robust rounded capacity inequalities

into the classic vehicle flow formulation [28] is the most effective method to address the robust

CVRP under demand uncertainty. To deal with travel time uncertainty, the work of [1] applied the

classic dualization technique for RO. This initial work yielded a large formulation that was difficult

to solve for instances with more than 20 customers. Prominent contributions were later made

by [2], in which the authors proposed two novel techniques to handle travel time variability, the

column-and-constraint generation [52] and the cutting-plane method. The former method initially

only considers a subset of anticipated scenarios and then gradually selects and introduces some

neglected extreme scenarios via appending both variables and constraints. This process iterates

until the returned route design is robust feasible with respect to all anticipated scenarios. The

cutting-plane method ensures the immunity of a routing solution by dynamically separating and

enforcing infeasible path elimination constraints. The computational studies in [2] showed that

these two techniques have comparable performance in terms of solving robust VRPs with time

windows. The column-and-constraint generation approach was also employed in the work of [3] for

the treatment of travel time uncertainty in a maritime inventory routing application.

All the aforementioned works are utilizing the popular arc-based formulation, which includes

polynomially many binary variables and is thus solved by the branch-and-cut method. In the

realm of deterministic problems, the route-based formulation (a.k.a. set-partitioning formulation)

is becoming more and more popular due to its tight linear programming (LP) relaxation. This

formulation has exponentially many binary variables, each representing the selection of a feasible

vehicle route. The resulting LP relaxation at each branch-and-bound node is thus solved by column

generation [32], which simply starts with a subset of feasible vehicle routes and entails the solution of

pricing subproblems to dynamically introduce any neglected feasible routes that have the potential

to reduce the objective value. The pricing subproblem is commonly modeled as a shortest path

4

problem with resource constraints (SPPRC) and can be solved efficiently via dynamic programming.

For details, we refer the reader to [24]. Strengthening constraints are usually incorporated to tighten

the LP relaxation, and hence the solution approach for route-based formulations is referred to as a

branch-price-and-cut (BPC) method. For the past decade, considerable progress has been made in

the development of BPC algorithms [14], and today the BPC algorithm has emerged as the state-

of-the-art exact approach for addressing deterministic VRPs [51]. In this work, we aim to explore

and compare the different applicable avenues for extending the state-of-the-art deterministic BPC

framework to solve robust VRPs.

Whereas lots of literature has been devoted on how to address deterministic VRPs via BPC

algorithms, there have been only a few works combining RO with BPC to tackle VRPs under

uncertainty. To the best of our knowledge, the first attempt was made in [29]. The authors focused

on the robust VRP with deadlines under demand and travel time uncertainty and proposed a novel

approach that is able to directly encapsulate demand and/or travel time variability into pricing

subproblems. This approach was recently revisited and improved by [33] for solving the robust

VRP with time windows under demand and/or travel time uncertainty. In both works, the pricing

subproblem is formulated as a robust shortest path problem with resource constraints (RSPPRC) [4],

for generating routes that are robust feasible with respect to demand and/or travel time variability.

The authors of the above works demonstrated that the resulting RSPPRC could be efficiently

solved when uncertain parameters are supported on the commonly used cardinality-constrained

uncertainty set [11]. To handle robust VRPs with demand uncertainty, another approach was

proposed in [31] and [38], where the authors transformed a RSPPRC into polynomially many

deterministic SPPRCs, avoiding the burden of solving the RSPPRC directly. The latter work

showed that the transformation could be achieved for two popular polyhedral uncertainty sets from

the RO literature, namely the cardinality-constrained set and the budget set.

Although the above two literature approaches exhibit promising computational tractability,

they heavily exploit the structures of uncertainty sets and hence can not be generalized to many

uncertainty sets commonly used in the RO literature (e.g., ellipsoidal sets). Therefore, it is of great

necessity to develop a new approach that can work for solving robust VRPs under various, general

types of uncertainty sets. [22] and [2] proposed the cutting-plane idea to enforce robust feasibility

and embedded it into a branch-and-cut framework for addressing robust VRPs with uncertainty

5

in demands and travel times, respectively, while the authors of [20] adopted a similar approach

for solving the distributionally robust chance-constrained vehicle routing problem. These works

have demonstrated the versatility of cutting planes in handling several classes of uncertainty sets.

Recently, the work of [15] integrated the cutting-plane idea into a BPC framework for solving the

chanced-constrained vehicle routing problem with stochastic demands. This motivates us to develop

a generic approach of combining cutting-plane techniques with the deterministic BPC algorithm,

so as to address robust VRPs under a variety of uncertainty sets.

The distinct contributions of our work can be summarized as follows.

• We propose a novel BPC algorithm to address robust VRPs under demand and travel time

uncertainty. Our algorithm embeds cutting-plane techniques into the deterministic BPC

framework. In particular, we utilize deterministic pricing routines to generate partially ro-

bust feasible routes, and then dynamically separate robust rounded capacity inequalities and

infeasible path elimination constraints to guarantee the immunity of a routing design against

infeasibility due to variability in demands and travel times, respectively. We demonstrate

that separating these inequalities can be done efficiently for all classes of uncertainty sets

that have been introduced in the literature to-date.

• We provide an overview of existing methods from the literature that have demonstrated

success in extending BPC algorithms for the solution of robust VRPs. We then compare these

literature approaches with our own approach in terms of (i) which uncertain parameters and

uncertainty sets they can accommodate, (ii) the time complexity of their underlying pricing

subproblems, and (iii) the tightness of their LP relaxations, demonstrating the applicability

and limitations of each approach.

• We derive tight upper bounds on the number of non-dominated extreme points of the cardinality-

constrained and factor model uncertainty sets. Their relatively small value justify the tractabil-

ity of tailored approaches for these sets that implicitly enumerate these points.

• We evaluate our algorithm on instances of robust VRPs with uncertainty in customer de-

mands and vehicle travel times, both separately and jointly, over five classes of uncertainty

sets, namely cardinality-constrained sets, budget sets, factor models, ellipsoids, and discrete

sets. Our computational studies illustrate that the proposed approach is versatile and compet-

6

itive against current state-of-the-art approaches. In particular, for previously open literature

instances, we are able to certifiably close the gap in 56 cases and improve upon 8 best-known

solutions.

The remainder of the paper is organized as follows. In Section 2, we give a formal problem

definition, while in Section 3, we present the various uncertainty sets we utilize in this study. In

Section 4, we study several polyhedral uncertainty sets from a geometric perspective, reducing

them to equivalent discrete sets. A brief overview of the BPC algorithm for solving deterministic

VRPs is given in Section 5. In Section 6, we summarize the existing approaches from the literature

and propose a new one for adapting BPC algorithms to address robust VRPs. Section 7 presents

computational results on the BPC algorithm’s performance. Finally, we conclude our work in

Section 8.

2 Problem Definition

The robust vehicle routing problem with time windows (RVRPTW) is defined on a directed graph

G = (V,A), where V := Vc ∪ 0, n + 1 denotes the set of nodes, which is composed of a set

of customers Vc := 1, 2, . . . , n, the origin depot 0 and the destination depot n + 1, while A :=

(i, j) ∈ V × V : i 6= j, i 6= n+ 1, j 6= 0 \ (0, n + 1) is the set of arcs. We consider a fleet of K

identical vehicles of capacity Q ∈ R>0, initially located at the origin depot. Every used vehicle

can only leave the origin depot after time 0 and has to return to the destination depot by time

H. Customer i ∈ Vc has a demand qi ∈ R>0 that has to be delivered during a given time window

indicated by [ei, `i]. A vehicle is allowed to wait, if it arrives at customer i before ei, while arriving

after `i is prohibited. For convenience, each depot node i ∈ 0, n + 1 is also associated with a

time window such that ei = 0 and `i = H. Service at customer i ∈ Vc must start during the

corresponding time window and takes time of si ∈ R≥0. Let cij ∈ R≥0 and tij ∈ R≥0 represent

the cost and time, respectively, for a vehicle to traverse arc (i, j) ∈ A. In this work, we consider

the demand vector q and/or the travel time vector t as uncertain. They may independently take

any value from postulated uncertainty sets Q and T , respectively. A feasible vehicle route, r =

(0, v1, v2, . . . , vp, n+ 1), starts from the origin depot and ends at the destination depot such that:

(C1) each customer is visited at most once (i.e., an elementary route);

7

(C2) the vehicle capacity constraint is respected under any customer demand realization q ∈ Q,

that is,p∑i=1

qvi ≤ Q for all q ∈ Q; and

(C3) the time window constraints are satisfied under any travel time realization t ∈ T , that is,

ai(t) ≤ `i for all i ∈ v1, v2, . . . , vp, n + 1 and for all t ∈ T , where ai(t) denotes the earliest

service start time at node i under scenario t.

We refer to (C2) and (C3) as the robust capacity feasibility and the robust time window feasibility

conditions, respectively. The objective is to determine the cost-effective feasible routes for vehicles

to traverse, such that each customer is visited exactly once and no more than K vehicles are used.

The above defined problem can be reduced to the well-studied deterministic vehicle routing

problem with time windows (VRPTW), if Q and T are both singleton sets. Additionally, it an be

reduced to the robust capacitated vehicle routing problem (RCVRP), if the time window constraints

(C3) are neglected.

3 Uncertainty Sets

In this section, we review uncertainty sets that are commonly used in the RO literature to model

demand and travel time uncertainty in VRP applications. In particular, we consider five demand

uncertainty sets and two travel time time uncertainty sets. These sets can be categorized into five

groups: cardinality-constraint sets, budget sets, factor models, ellipsoidal sets and discrete sets.

3.1 Cardinality-Constrained Sets

The first set we consider for modeling demand uncertainty is the cardinality-constrained set, as

shown in Equation (1).

QG :=

q ∈ Rn : qi = q0

i + qi ξi ∀i ∈ 1, 2, . . . , n ,n∑i=1

ξi ≤ Γq, ξ ∈ [0, 1]n

(1)

Here, q0 ∈ Rn>0, q ∈ Rn≥0 and Γq ∈ [0, n] are parameters that need to be specified by the modeler.

This uncertainty set stipulates that each customer demand qi can deviate upwards by up to qi from

its nominal value, q0i , and that the sum of the relative deviations of all customers is bounded from

above by Γq. In particular, at most bΓqc of these demands may achieve their maximum deviations.

This uncertainty set was originally proposed in [11] and is also often referred to as a “budgeted” or

8

“gamma” uncertainty set. This set has been used widely to model customer demand uncertainty

in the VRP literature [29, 33, 38, 48].

The cardinality-constrained set is also often used to model travel time uncertainty, as shown in

Equation (2).

TG :=

t ∈ R|V |×|V | : tij = t0ij + tij ξij ∀(i, j) ∈ A,∑

(i,j)∈A

ξij ≤ Γt, ξij ∈ [0, 1]|V |×|V |

(2)

Here, t0ij ∈ R≥0 and tij ∈ R≥0, for all arcs (i, j) ∈ A, as well as Γt ∈ [0, |A|] are parameters that need

to be specified by the modeler. This set stipulates that the time tij for traversing arc (i, j) ∈ A can

deviate upwards by up to tij from its nominal value, t0ij , and that the sum of relative deviations of

all arcs is bounded from above by Γt. In particular, at most bΓtc of these travel times may achieve

their maximum deviations. This set has also been widely used for VRP aplications [3, 29, 33].

3.2 Budget Sets

Consider the demand uncertainty set of the following form (3).

QB :=

q ∈ Rn : qi≤ q ≤ qi ∀i ∈ 1, 2, . . . , n ,

∑i∈B`

qi ≤ b` ∀` ∈ 1, 2, . . . , L

(3)

Here, q ∈ Rn>0, q ∈ Rn>0, B` ⊆ Vc and b` ∈ R>0 for ` ∈ 1, 2, . . . , L are parameters chosen by the

modeler such that (i) the customer subsets B` are pairwise disjoint, i.e., B` ∩B`′ = ∅ for all ` 6= `′;

(ii) b` ≤∑

i∈B`qi for ` ∈ 1, 2, . . . , L; and (iii) QB 6= ∅. The uncertainty set (3) stipulates that

a customer demand qi can deviate within an interval [qi, qi] and that the total demand in every

customer set B` is bounded from above by b`. This uncertainty set was originally proposed in the

work of [22] to model demand uncertainty and was subsequently also used in [21, 38, 48].

3.3 Factor Model Sets

Consider the demand uncertainty set of the following form (4).

QF :=

q ∈ Rn : q = q0 + Ψξ, −βF ≤F∑f=1

ξf ≤ +βF, ξ ∈ [−1, 1]F

(4)

Here, q0 ∈ Rn≥0, F ∈ N,Ψ ∈ Rn×F≥0 and β ∈ [0, 1] are parameters that need to be specified by the

modeler. The uncertainty set (4) stipulates that the customer demand vector q results from adding

9

a disturbance of Ψξ to the nominal demand vector q0. This disturbance is a linear combination

of independent factors, ξ1, ξ2, . . . , ξF , which reside in an F -dimensional polytope. This set was

originally proposed in the work of [22] to model demand uncertainty, and was later also considered

in [21, 48].

3.4 Ellipsoidal Sets

Consider the demand uncertainty set of the following form (5).

QE :=q ∈ Rn :

(q − q0

)>Σ−1

(q − q0

)≤ 1. (5)

In the above set, q0 ∈ Rn≥0 and Σ ∈ Sn++ are the parameters that need to be specified by the modeler;

here, we use Sn++ to represent the set of positive definite matrices. The uncertainty set (5) stipulates

that the customer demand vector q can only attain values in an ellipsoid centered at the nominal

demand vector q0. This set was originally proposed by [48] to model customer demand uncertainty

in the context of routing a heterogeneous fleet. We emphasize that QE is not a polyhedral set,

representing a feasible region dictated by nonlinear constraints.

3.5 Discrete Sets

Consider the demand uncertainty set of the following form (6).

QD := conv(qd : d = 1, 2, . . . , Dq

), (6)

where conv(·) denotes the convex hull of a finite set of points. Here, q1, q2, . . . , qDq ∈ Rn≥0 are a

total of Dq ∈ N modeler-specified scenarios for the uncertain customer demands. The uncertainty

set (6) stipulates that the customer demand vector may realize to any value corresponding to a

convex combination of the specified demand scenarios. This set was originally proposed in the work

of [48] to model uncertain customer demands in the context of routing a heterogeneous fleet.

We can also define a discrete set for the travel time vector t, as shown in (7).

TD := conv(td : d = 1, 2, . . . , Dt

). (7)

Here, t1, t2, . . . , tDt ∈ R|A|≥0 are Dt ∈ N distinct scenarios of the uncertain travel times that need to

be specified by the modeler. The uncertainty set (7) was originally proposed in [45].

10

4 Polyhedral Studies

In this section, we study the extreme points of polyhedral sets QG and QF and then reduce each

set to an equivalent discrete set of the form QD. We do not consider the polyhedral set QB because

the resulting discrete set can be of exponential size. Before proceeding, we first present some

observations and a definition.

Observation 1. If the demand uncertainty set Q is compact and convex, then we can equivalently

replace Q by Ext(Q) in condition (C2), where Ext(Q) denotes the set of extreme points of Q.

Proof. Clearly, Ext(Q) ⊆ Q. We only need to show that, for any route r = (0, v1, v2, . . . , vp, n+ 1)

such thatp∑i=1

qvi ≤ Q for all q ∈ Ext(Q), it also holds thatp∑i=1

q′vi ≤ Q for all q′ ∈ Q. This

is true, owing to the fact that any point q′ ∈ Q can be expressed as a convex combination of

points q ∈ Ext(Q), and hence, the former inequalities collectively imply any inequality of the latter

form.

Observation 1 simply implies that the demand uncertainty set Q can be reduced to the set of

its extreme points, Ext(Q). Note that, if Q is a polytope, Ext(Q) is a finite set. Furthermore, if Q

has a favorable structure, one may be able to precisely characterize Ext (Q).

Definition 1. A demand vector q ∈ Q is said to dominate another vector q′ ∈ Q \ q ifp∑i=1

qvi ≥p∑i=1

q′vi, for every route r := (0, v1, v2, . . . , vp, n+ 1) satisfying (C1), (C2) and (C3). Conversely,

a vector q′ is said to be non-dominated if there is no q ∈ Q \ q′ for which this condition holds.

The immediate implication from Definition 1 is that, if the vehicle capacity constraint along

a route r is respected under demand scenario q, then it is also respected under any scenarios

dominated by q. It is obvious that q ≥ q′ (element-wise) is a sufficient condition for this dominance

relationship to hold.

Observation 2. If the demand uncertainty set Q is compact and convex, then we can equivalently

replace Q by Ext(Q) in condition (C2), where Ext(Q) ⊆ Ext(Q) denotes the set of non-dominated

points of Ext(Q).

Proof. This deduces readily by combining Observation 1 with Definition 1.

11

One may also replace the travel time uncertainty set T by the set of its extreme points, Ext (T ),

and define a similar dominance relationship. In the following, we derive upper bounds on∣∣Ext(QG)

∣∣and

∣∣Ext(QF )∣∣, noting that a similar result could also be obtained for the case of set TG. We use

the following well-known result on extreme points of knapsack polytopes.

Lemma 1 (Proposition 10 in [23]). Suppose a ∈ Rm>0 and b ∈ R satisfy ai ≤ b, for i ∈ 1, 2, . . . ,m,

and a1 +a2 + . . .+am > b. Then, x is an extreme point of P :=x ∈ [0, 1]m : a>x ≤ b

if and only

if there exist index sets I0, I1 ⊆ 1, 2, . . . ,m such that I0 ∩ I1 = ∅, |I0 ∪ I1| ≥ m − 1,∑i∈I1

ai ≤ b,

aj > b−∑i∈I1

ai for j ∈ 1, 2, . . . ,m \ (I0 ∪ I1), and

xj =

0, if j ∈ I0,

1, if j ∈ I1,(b−

∑i∈I1

ai

)/aj if j /∈ I0 ∪ I1.

Proposition 1 (Non-dominated points of cardinality-constrained sets).

∣∣Ext(QG)∣∣ ≤

(nΓq

)if Γq ∈ N,

(n− bΓqc)(nbΓqc)

otherwise.

(8)

Proof. For the cardinality-constrained set QG, the demand vector q results from an affine trans-

formation of ξ ∈ ΞG; that is, q = q0 + diag (q1, . . . , qn) ξ, where diag (·) denotes a diagonal matrix,

and ΞG is the following polyhedral set:

ΞG :=

ξ ∈ [0, 1]n :

n∑i=1

ξi ≤ Γq

.

Therefore, each ξ ∈ Ext(ΞG) maps to an extreme point q = q0 +diag (q1, . . . , qn) ξ such that q ∈

Ext(QG). If Γq = n, QG is a hyper-box; hence, Ext(QG) becomes a singleton (i.e.,∣∣Ext(QG)

∣∣ = 1)

and the stated bound holds. Otherwise, the set Ext(ΞG) can be characterized by applying Lemma 1

with m = n, ai = 1, b = Γq, and P = ΞG. |Ext(ΞG)| is then precisely the number of index sets

I0, I1 ⊆ 1, 2, . . . , n satisfying the conditions of the Lemma. Among these, we only need to count

those which satisfy |I1| = bbc = bΓqc since other choices would result in extreme points of ΞG that

map to dominated points of QG. Indeed, if I1 defining some ξ ∈ Ext(ΞG) satisfies∣∣I1

∣∣ < bΓqc, then

the extreme point ξ∗ ∈ Ext(ΞG) defined by I∗1 = I1 ∪ j for some arbitrary j ∈ 1, . . . , n \ I1

12

satisfies ξ∗j = 1 > ξj ; that is, ξ∗ ≥ ξ leading to a dominant point q∗ ≥ q since all qi are non-negative.

If Γq is integral, then by the same reasoning, we only need to count those index sets I0, I1 such that

|I0|+|I1| = n; otherwise, we must have |I0|+|I1| = n−1 with exactly one fractional component that

must be chosen among n− bΓqc components. This leads to the statement of the proposition.

Proposition 2 (Non-dominated points of factor model sets).

∣∣Ext(QF )∣∣ ≤

(F

(F+βF )/2

)if βF ∈ N and F + βF is an even number,

F−bβF c2

(F

(F+bβF c)/2)

if βF /∈ N and F + bβF c is an even number,

F−bβF c+12

(F

(F+bβF c−1)/2

)if F + bβF c is an odd number.

(9)

Proof. For the factor model set QF , the demand vector q results from an affine transformation of

ξ ∈ ΞF ; that is, q = q0 −Ψe + 2Ψξ, where e ∈ RF is a vector of ones, and ΞF is the polytope:

ΞF :=

ξ ∈ [0, 1]F : (F − βF )/2 ≤F∑f=1

ξf ≤ (F + βF )/2

.

Therefore, each ξ ∈ Ext(ΞF ) maps to an extreme point q = q0−Ψe+2Ψξ such that q ∈ Ext(QF ). To

characterize Ext(ΞF ), note that for β = 1, ΞF is a hyper-box; hence, Ext(QF ) becomes a singleton

(i.e.,∣∣Ext(QF )

∣∣ = 1) and the stated bound holds. Otherwise, for any extreme point ξ ∈ Ext(ΞF ):

(i) exactly one of the inequalities bounding∑F

f=1 ξf is active at ξ, and (ii) ξj < 1 for at least one

j ∈ 1, . . . , F. Therefore, for any extreme point ξ ∈ Ext(ΞF ) at which the lower inequality is

active,∑F

f=1 ξf = (F − βF )/2, we can construct ξ∗ ∈ ΞF such that the upper inequality is active,

and for which ξ∗j > ξj ; that is, ξ∗ ≥ ξ leading to a corresponding dominant point q∗ ≥ q since

Ψ ∈ Rn≥0.

In other words, we can follow precisely the same reasoning as in Proposition 1, to characterize

Ext(ΞF ) by applying Lemma 1 with m = F , ai = 1, b = (F + βF )/2, and P = ξ ∈ [0, 1]F :∑Ff=1 ξf ≤ (F + βF )/2. This leads to the following bound, analogous to (8):

∣∣Ext(QF )∣∣ ≤

(

F(F+βF )/2

)if (F + βF )/2 ∈ N,

(F − b(F + βF )/2c)(

Fb(F+βF )/2c

)otherwise.

Note that (F + βF )/2 ∈ N if and only if βF ∈ N and F + βF is an even number, leading to the

first case of (9). Otherwise, if (F + βF )/2 /∈ N, then note that b(F + βF )/2c = b(F + bβF c)/2c.

13

Furthermore, if F + bβF c is an even number, then the latter becomes (F + bβF c)/2, while we also

have that (F −b(F +βF )/2c) = (F −bβF c)/2, leading to the second case of (9); if F + bβF c is an

odd number, then b(F +bβF c)/2c = (F +bβF c−1)/2 and (F −b(F +βF )/2c) = (F −bβF c+1)/2,

leading to the third case of (9).

These results imply that, in theory, QG and QF can be reduced to equivalent discrete sets of

the form QD. In the case of QG, Proposition 1 indicates that the size of the resulting set grows

polynomially with the number of customers n (Γq being fixed). Nevertheless, this size can still

become impractical even for moderate values of n that may be encountered in practice. In contrast,

Proposition 2 indicates that the equivalent discrete set for QF has a size that is independent of n,

and it depends only on the number of factors F which is typically much smaller than n. In such

cases, the equivalent set QD could be of manageable size and one might be able to convert a robust

VRP with the demand uncertainty set QF to the one with an equivalent discrete set QD.

5 Branch-Price-and-Cut

Before we present the solution approaches to the RVRPTW, we will first discuss the use of BPC

for solving its deterministic counterpart, the VRPTW. We remark that, over recent years, the

BPC method has been gradually accepted as the most efficient exact approach for solving the

VRPTW and its variants. In this section, we only highlight the most important ingredients of

the BPC algorithm, referring readers to [35, 37] for many of the details behind the state-of-the-art

implementations.

5.1 Set-Partitioning Model

Let R denote the set of feasible routes for a VRPTW, and let cr denote the cost of traversing route

r ∈ R by each of K homogeneous vehicles. Let the parameter δir denote the number of times

customer i ∈ Vc is covered in route r ∈ R. Let λr be a binary variable indicating whether route

r ∈ R is selected or not in the optimal solution. The VRPTW can be formulated as the following

14

set-partitioning model (10)–(13).

minλr

∑r∈R

crλr (10)

s.t.∑r∈R

δirλr = 1 ∀i ∈ Vc (11)

∑r∈R

λr ≤ K (12)

λr ∈ 0, 1 ∀r ∈ R (13)

The objective function (10) calls for minimizing the cumulative cost of all selected routes. The

degree constraints (11) guarantee that every customer is served exactly once, while the fleet size

constraint (12) enforces that no more than the available vehicles are used. Finally, constraints (13)

simply enforce the binarity of the route selection variables.

It is well-known that, without sacrificing optimality, we may relax the feasible space of the

above set-partitioning model by including in R non-elementary vehicle routes (i.e., by relaxing

condition C1). In particular, we replaceR with the set of so-called ng-routes that are not necessarily

elementary [8]. In the remainder of this paper, we use R to denote the set of ng-feasible routes.

We emphasize that both capacity feasibility and time window feasibility are ensured implicitly by

including only feasible routes in the set R.

Since there exist exponentially many feasible routes, the formulation (10)–(13) is a mixed-integer

linear programming model with a huge number of binary variables. To address this issue, the LP

relaxations at each node of the branch-and-bound tree are tackled via column generation [32].

Furthermore, valid inequalities can be dynamically separated and added to strengthen the LP

relaxations, yielding a BPC algorithm for addressing the set-partitioning model. In the following

subsection, we discuss the specifics of such an algorithm.

5.2 Algorithm Overview

In the BPC algorithm, we first replace the binarity constraints (13) in the set-partitioning model by

non-negativity constraints and obtain the LP relaxation, which is usually referred to as the master

problem. However, as generating all of the ng-feasible routes to explicitly define the master problem

is obviously impractical, we work with a restricted master problem (RMP) defined by a subset of

ng-routes R ⊆ R, and we resort to column generation. More specifically, after optimizing the RMP,

15

columns with negative reduced costs are appended to R and the resulting RMP is reoptimized.1

This procedure iterates until no such columns exist. In that case, column generation has converged

and the master problem has achieved its optimality. If the solution to the master problem is

fractional, we then proceed by adding valid inequalities or by branching. In particular, rounded

capacity inequalities [28] and limited-memory subset row cuts [26, 34, 35] are often considered to

tighten the LP relaxations. In terms of branching, doing so on the number of used vehicles (left-

hand side of 12) is usually prioritized over branching on edges [17, 42].

5.3 Pricing Subproblems

After the RMP is solved, we check whether it is necessary to enlarge R by including some ne-

glected ng-feasible routes that may potentially improve the RMP’s objective value. This entails

identifying columns with negative reduced costs, which is achieved by solving a pricing subproblem.

In our context, the pricing subproblem can be modeled as a shortest path problem with resource

constraints (SPPRC) [39]. The SPPRC is defined on a directed graph G = (V,A). We associate

demand qi, capacity Q and a time window [ei, `i] with vertex i ∈ V . Note that qi = 0, if vertex i

denotes a depot. Associated with each arc (i, j) ∈ A are a travel time tij and a cost cij . This cost is

obtained by properly modifying cij , in order to account for the contribution from current dual values

to constraints (11) and (12). While arc (i, j) ∈ A is traversed by a path, time resource (si + tij) and

capacity resource qj are consumed, and the paths should be constructed such that the accumulated

consumptions of each and every resource do not exceed their corresponding limits. The goal of the

SPPRC is to determine the minimum-cost path among all such paths that start from the vertex 0

and end at the vertex n+ 1. It is well-known that the SPPRC is weakly NP-hard [16]. The most

successful solution approach is the labeling algorithm, a dynamic programming method that has a

pseudo-polynomial time complexity [39]. The labeling algorithm works as follows. We associate an

ng-feasible partial path P with a label L(P ) := (pred(P ), c(P ), v(P ),Π(P ), d(P ), a(P )) that stores

a pointer to its predecessor label, reduced cost, end vertex, a set of forbidden vertices, total vehicle

load, and earliest time to start servicing. We initialize the labeling algorithm by storing the first

label (null, 0, 0, ∅, 0, 0) into a pool. Choosing a label L(P ) from the pool, we attempt to extend it

to vertex j ∈ V \ Π(P ) with (v(P ), j) ∈ A, in order to generate a new label P ′. To achieve this

1In our context, columns denote ng-feasible routes, and the two terms are used interchangeably.

16

extension, we use the following procedure (14)–(19).

pred(P ′)← L(P ), (14)

c(P ′)← c(P ) + cv(P )j , (15)

v(P ′)← j, (16)

Π(P ′)← Π(P ) ∩NG(j) ∪ j , (17)

d(P ′)← d(P ) + qj , (18)

a(P ′)← maxej , a(P ) + sv(P ) + tv(P )j

, (19)

where NG(j) ⊆ Vc denote the ng-set for node j. Usually, the ng-set is chosen to be the set of

nearest neighbors as suggested by [8]. Before storing label L(P ′) to the pool, we check whether this

is a feasible extension in terms of respecting the resource consumption constraints; that is, whether

d(P ′) ≤ Q and a(P ′) ≤ `j . If that is the case, we obtain a new ng-feasible label and store it in the

pool; otherwise, we proceed with other applicable label extensions. When no more label extensions

can be made, we collect all paths that end at vertex n+ 1 and return those with negative reduced

costs.

To accelerate the labeling algorithm, it is crucial to utilize dominance relationships to avoid

non-interesting label extensions. To that end, before storing a new ng-feasible label, we first check

whether it dominates (or is dominated by) existing labels stored in the pool. We say that a label

L(P1) dominates another label L(P2) if, for any feasible path extended from L(P2), we can always

find a feasible path extension from L(P1) and this extended path is no more costly. Sufficient

conditions for this are given by (20)–(24).

c(P1) ≤ c(P2), (20)

v(P1) = v(P2), (21)

Π(P1) ⊆ Π(P2), (22)

d(P1) ≤ d(P2), (23)

a(P1) ≤ a(P2). (24)

A newly generated ng-feasible label, L(P ), is saved only if it is not dominated. Furthermore, when

L(P ) is saved, any existing labels that are dominated by it are removed from the pool. Since

checking for a dominance relationship can be computationally expensive, it is important to not

17

perform this too aggressively. Usually, labels of proximity in resource consumption levels are saved

into a bucket and dominance is checked only among labels residing in the same (or neighboring)

buckets [35, 43].

As long as the pricing subproblem is solved exactly, one can obtain a Lagrangian dual bound [32].

When the primal-dual gap becomes small, it is advisable to apply a route enumeration step [7, 13],

in which one may identify all elementary columns with reduced costs less than the primal-dual gap,

since only these columns may contribute to a solution better than the incumbent. This step works in

a similar fashion as the aforementioned labeling algorithm, with only a couple of notable exceptions:

(i) the extension rule (17) is updated to Π(P ′)← Π(P ) ∪ j, so as to produce elementary routes;

(ii) sufficient conditions (20) and (22) are replaced by (25) and (26), respectively.

c(P1) ≤ c(P2), (25)

Π(P1) = Π(P2), (26)

where c(P ) denote the actual monetary cost for traversing path P .

In order to further expedite the labeling algorithm, various advanced pricing techniques, such

as heuristic pricing [18], bidirectional labeling [40], and variable fixing [25], among others, have

also been proposed. Furthermore, if strengthening inequalities are incorporated to the RMP, then

proper modifications have to be made when solving the SPPRC [35]. Finally, we remark that

various techniques, including primal heuristics [44], stabilized column generation [36], dynamic ng-

set [41], and strong branching [43] have also been considered to improve overall performance. We

refer readers to [37] for details on these techniques.

6 Incorporating Uncertainty in Branch-Price-and-Cut

In this section, we first review two existing approaches in the literature for adapting the BPC

algorithm to solve robust VRPs under uncertainty. Both of these approaches can be categorized as

robust pricing approaches, aiming to ensure the robust feasibility of routing designs in the pricing

subproblems. We then introduce the intrisically different robust cutting-plane approach, which

seeks robust feasibility via dynamically enforcing necessary constraints in the master problems.

Notably, our new approach can tackle VRPs under all types of uncertainty sets we discussed in

Section 3. For notational convenience, let R denote the set of robust ng-feasible routes with respect

18

to any realization (q, t) ∈ Q × T ; that is, for any route r ∈ R, the robust feasibility conditions

(C2) and (C3) are satisfied.

6.1 Robust Pricing Approach

The robust pricing approach is based on the set-partitioning model (10)–(13) but replaces the set of

deterministic ng-feasible routes R with that of the robust ng-feasible routes R. The BPC algorithm

remains largely the same, the only exception being that the resulting pricing subproblem becomes a

robust shortest path problem with resource constraints (RSPPRC), in order to dynamically introduce

robust ng-feasible routes. Thus, the robust feasibility of the optimal routes is ensured when these

routes were first generated via pricing subproblems. We remark that the RSPPRC under resource

uncertainty is strongly NP-hard for arbitrary uncertainty sets [4]. In the following, however,

we review two literature methods that have demonstrated success in solving the RSPPRC under

specific uncertainty sets from Section 3.

6.1.1 Direct Pricing

We first focus on cardinality-constrained uncertainty sets QG and TG. To tackle the RVRPTW

in which the uncertain demand vector q and uncertain travel time vector t fall into respective

cardinality-constrained sets, QG and TG, the authors of [33] proposed a modified labeling algorithm

to directly solve the resulting RSPPRC, leading us to refer to this approach as direct pricing. We

remark that the approach of [33] considered only the case of integral Γq and Γt, but below we

present it in the context of the more general case. Their proposed labeling procedure is similar to

the deterministic counterpart we discussed in Section 5.3 but with the following three modifications:

(i) for a given path P , the labeling notation is redefined as L(P ) :=(pred(P ), c(P ), v(P ),

Π(P ), d0(P ), d1(P ), . . ., ddΓqe(P ), a0(P ), a1(P ), . . ., adΓte(P )), where the first four labels

retain the same meanings as before, while for the remaining ones we have: dγ(P ) (where

γ ∈ 0, 1, . . . , dΓqe − 1) denotes the maximum total vehicle load along path P when up

to γ customer demands attain their maximum values; ddΓqe (P ) denotes the maximum total

vehicle load when the cumulative relative deviation does not exceed Γq; aγ(P ) (where γ ∈0, 1, . . . , dΓte − 1

) denotes the worst-case earliest time to start servicing node i, considering

that up to γ travel times attain their maximum values; and adΓte (P ) denotes the worst-case

19

earliest time when the cumulative relative deviation does not exceed Γt;

(ii) when extending a path P to some node j, creating a new path P ′, the extension procedures

(18)–(19) are replaced by (27)–(32),

d0(P ′)← d0(P ) + q0j , (27)

dγ(P ′)← max dγ(P ), dγ−1(P ) + qj+ q0j ∀γ ∈ 1, 2, ..., dΓqe − 1 , (28)

ddΓqe(P′)← max

ddΓqe(P ), ddΓqe−1(P ) + (Γq − dΓqe+ 1) qj

+ q0

j , (29)

a0(P ′)← maxej , a0(P ) + sv(P ) + t0v(P )j

, (30)

aγ(P ′)← maxej ,max

aγ(P ), aγ−1(P ) + tv(P )j

+ sv(P ) + t0v(P )j

∀γ ∈

1, 2, ..., dΓte − 1

,

(31)

adΓte(P′)← max

ej ,max

adΓte(P ), adΓte−1(P ) +

(Γt − dΓte+ 1

)tv(P )j

+ sv(P ) + t0v(P )j

,

(32)

and hence, the resulting label P ′ can be accepted as a feasible extension only if ddΓqe(P′) ≤ Q

and adΓte(P′) ≤ `j ; and

(iii) sufficient conditions (23) and (24) for checking dominance should be respectively replaced by

conditions (33) and (34),

dγ(P1) ≤ dγ(P2) ∀γ ∈ 0, 1, . . . , dΓqe, (33)

aγ(P1) ≤ aγ(P2) ∀γ ∈ 0, 1, . . . , dΓte. (34)

The direct pricing method could effectively solve the RSPPRC due to the fact that, for demand

uncertainty set QG and/or travel time uncertainty set TG, one can efficiently compute the maximum

vehicle load and/or worst-case earliest service start time along a path by introducing(dΓqe+ dΓte

)extra relevant resources into a deterministic SPPRC and keeping track of their consumption. In

particular, every path-extending operation can be achieved in O(dΓqe+ dΓte

)time. The immediate

implication from the dominance rules (33) and (34) is that, when compared with the deterministic

case, path P1 is “less likely” to dominate path P2 due to more restrictive sufficient conditions. As

a result, more labels will be kept and processed in the modified labeling algorithm, on average,

causing an increase in its time complexity.

20

We also remark that the direct pricing idea can be readily applied when the demand and/or

travel time vector are assumed to attain realizations from the discrete sets QD × TD. Correspond-

ingly, the pricing subproblem entails the enforcement of(Dq +Dt

)resource constraints. Similarly,

the path-extending operation could be achieved in O(Dq +Dt

)time. Meanwhile, as also pointed

out above, one would expect a time complexity increase of the modified labeling algorithm. This

implies that, for a discrete set QD × TD that consists of a large number of scenarios (in either the

demand or travel time side), solving the corresponding RSPPRC via a direct pricing method will

become prohibitive.

6.1.2 Transformed Pricing

In this section, we consider that the travel time vector t is constant (i.e., set T is a singleton) and

that the customer demand vector q can take any value from a non-empty knapsack set, QK , given

by

QK :=

q ∈

[q, q]

:∑i∈Vc

a`iqi ≤ b`,∀` ∈ 1, 2, . . . , L

, (35)

where a`i ≥ 0, for i ∈ Vc, and ` ∈ 1, 2, . . . , L. Note that this knapsack set is a more general

uncertainty set and can be reduced to (i) QG, when q = q0, q = q0 + q, L = 1, a1i = 1/qi, and

b1 = Γq +∑

i∈Vc q0i /qi; (ii) QB, when a`i = 1i∈B`.

The work of [38] focused on how to solve a robust VRP with knapsack uncertainty of the

form (35) via the BPC approach. For this setting, the authors presented an important proposition,

which we reframe below to aid readers understand how the robust feasibility of the generated routes

is ensured in the transformed pricing approach.

Proposition 3 (Equivalent to Theorem 1 in [38]). If Q := QK and T is a singleton, then solving

the RSPPRC is equivalent to solving at mostL∑=0

(L`

)(nL−`)

deterministic SPPRCs.

Proof. For an elementary route r ∈ R, the robust capacity feasibility condition (C2) is satisfied,

as shown by (36).

maxq∈QK

∑i∈Vc

δirqi ≤ Q (36)

Considering the definition of set QK and replacing q by q+η, where 0 ≤ η ≤ q−q, we can represent

21

the left-hand side of inequality (36) as the LP problem (37).

maxη≥0

∑i∈Vc

δir

(qi+ ηi

)s.t. ηi ≤ qi − qi ∀i ∈ Vc∑

i∈Vc

a`iηi ≤ b` −∑i∈Vc

a`iqi ∀` ∈ 1, 2, . . . , L

(37)

Since QK is a non-empty and bounded set, strong duality holds for problem (37). Introducing dual

variables z ∈ Rn≥0 and ϑ ∈ RL≥0, we obtain its dual as the LP problem (38).

minz≥0,ϑ≥0

∑i∈Vc

δirqi +∑i∈Vc

(qi − qi

)zi +

L∑`=1

(b` −

∑i∈Vc

a`iqi

)ϑ`

s.t. zi +L∑`=1

a`iϑ` ≥ δir ∀i ∈ Vc

(38)

Clearly, for the minimizer (z∗, ϑ∗) of the above problem, we have that

z∗i = max

0, δir −

L∑`=1

a`iϑ∗`

= max

0, 1−

L∑`=1

a`iϑ∗`

δir, (39)

where the last equality holds because δir ∈ 0, 1 and a`i ≥ 0.

Using the expression (38), we can eliminate variables z from problem (38) and obtain the

equivalent problem (40).

minϑ≥0

∑i∈Vc

(qi+(qi − qi

)max

0, 1−

L∑`=1

a`iϑ`

)δir +

L∑`=1

(b` −

∑i∈Vc

a`iqi

)ϑ`

(40)

Let f(ϑ; r) denote the objective function of problem (40). A key observation is that f(ϑ; r) is a con-

vex piecewise linear function along the domain RL≥0, while since by construction b`−∑

i∈Vc a`iqi ≥ 0,

we have that limϑ`→+∞

f(ϑ; r) = +∞ for all ` ∈ 1, 2, . . . L; thus, f(ϑ; r) achieves its minimum at

some breaking point ϑ′ ∈ RL≥0. Let Θ ⊆ RL≥0 denote the set of breaking points, each of which can

be identified as a solution to a subsystem of L linearly independent equations among the following

L+ n equations:

ϑ` = 0 ∀` ∈ 1, 2, . . . , L ,

1−L∑`=1

a`iϑ` = 0 ∀i ∈ Vc.

Note that |Θ| ≤L∑=0

(L`

)(nL−`).

22

For a ϑ ∈ Θ, let Rϑ denote the set of routes such that, for any route r ∈ Rϑ, we have (i) ng-

feasibility is satisfied;2 (ii) the time window constraints are respected along this route; (iii) f(ϑ; r) ≤

Q.3 The robust feasibility of a route r ∈ R implies that f(ϑ∗; r) ≤ Q for some ϑ∗ ∈ Θ. As a

result, we have that r ∈ Rϑ∗ , and thus, R ⊆⋃ϑ∈Θ

Rϑ. Conversely, given a route r ∈⋃ϑ∈Θ

Rϑ, let

ϑ∗ := argminϑ∈Θ

f(ϑ; r). Then, ϑ∗ is the the optimal solution of the linear program (40). As a result,

we have that r ∈ R, and thus,⋃ϑ∈Θ

Rϑ ⊆ R. Therefore, we conclude that R =⋃ϑ∈Θ

Rϑ, leading to

at most |Θ| deterministic SPPRCs that have to be solved.

Proposition 3 indicates that solving an RSPPRC can be transformed into solving polynomially

many deterministic SPPRCs.4 Hence, we refer to this approach as transformed pricing. At this

point, we provide the following remarks:

1. The resulting SPPRC has roughly the same time complexity as the one we discussed in

Section 5.3, since condition (iii) for the definition of Rϑ is enforced exactly like a capacity

constraint.

2. The number of transformed SPPRCs increases polynomially, but not mildly, with the number

of customers n. In particular, if the demand vector q is supported on a general knapsack

support QK , solving the RSPPRC via the transformed approach will still be impractical for

relatively large n values.

3. For QK := QG, we have |Θ| ≤ n,5 while for QK := QB, we have |Θ| ≤ 2L. This indicates that

the RSPPRC under uncertainty sets of the form QG or QB is weakly NP-hard and that one

can efficiently solve the relevant RSPPRC via the transformed pricing method, as succesfully

done in [31] and [38].

2Relaxing the elementary condition (C1) is again permitted due to the degree constraints (11).3f(ϑ; r) ≤ Q can be properly mapped as a capacity constraint in the resulting SPPRC defined by ϑ. More

specifically, a demand of

(qi

+(qi − q

i

)max

0, 1−

L∑=1

a`iϑ`

)is assigned to customer i ∈ Vc and the vehicle

capacity becomes Q−L∑=1

(b` −

∑i∈Vc

a`iqi

)ϑ`.

4The number of deterministic SPPRCs that have to be solved might be reduced if, for example, one can deduce

that Rϑ′= ∅ for some ϑ′ ∈ Θ. Readers are referred to [38] for details.

5This bound can be further tightened to⌈(n− Γq) /2

⌉+ 1, when Γq is integral (see [30]).

23

6.2 Robust Cutting-Plane Approach

Another avenue to ensure robust feasibility of routing designs is through cutting planes. Let Q ⊆ Q

and T ⊆ T denote finite sets, and let R denote the set of ng-feasible routes with respect to any

realization (q, t) ∈ Q × T ; that is, for route r = 0, v1, v2, . . . , vp, n+ 1 ∈ R:

p∑i=1

qvi ≤ Q for all q ∈ Q, and

ai(t) ≤ `i for all i ∈ v1, v2, . . . , vp, n+ 1 and for all t ∈ T ,

where ai(t) denotes the earliest service start time at node i under scenario t. Based on this

definition, we observe the following: (i) R ⊆ R, that is, all robust feasible routes are included

within R; (ii) since capacity feasibility and time window feasibility are only ensured against a

subset of anticipated scenarios, robust feasibility conditions (C2) and (C3) are not guaranteed

yet. In order to forbid the selection of routes r ∈ R \ R in the optimal solution, we rely on

the enforcement of necessary constraints in the master problem. In particular, we utilize robust

rounded capacity inequalities (robust RCI) and infeasible path elimination constraints (IPEC) to

ensure robust capacity feasibility and robust time window feasibility, respectively. Since the number

of necessary constraints is exponentially many, these constraints are separated and introduced

dynamically at every node of the branch-and-price process. Hence, we refer to this as a robust

cutting-plane approach. We emphasize that adding these necessary constraints to an RMP will

not complicate the solution of pricing subproblems, because their corresponding dual values can be

properly accommodated into the modified arc cost in the SPPRC.

To summarize, our BPC algorithm starts with a subset of vehicle routes (not necessarily robust

feasible) and iteratively introduces some neglected routes via solving the deterministic SPPRC

under finitely many resource constraints, each corresponding to an element from Q or T . When

column generation converges, IPEC and robust RCI are separated and added to the RMP to prohibit

solutions that are not robust feasible. Next, we will discuss two key aspects of our cutting-plane

approach, namely how to construct sets Q and T , as well as how to separate IPEC and robust RCI.

24

6.2.1 Constructing Q and T

Since both Q and T are compact and convex sets, we can replace them by Ext (Q) and Ext (T ),

respectively.6 Thus, our goal is to construct Q and T such that Q ⊆ Ext (Q) and T ⊆ Ext (T ),

respectively. When Q and T contain more scenarios, routes generated in pricing subproblems are

“more likely” to be robust feasible, and consequently, less effort will be made to separate nec-

essary constraints. However, pricing subproblems correspondingly become computationally more

expensive, since the time complexity of an SPPRC generally increases with the number of resource

constraints that are enforced. Next, we encapsulate a proper number of selected demand and travel

time scenarios (denoted by NSq and NSt) into Q and T , respectively, so as to balance the efforts of

solving pricing subproblems and separating necessary constraints.

• QG. We use the k-means clustering method to partition the customer set Vc into NSq clusters

based on their coordinates. We can now focus on a given cluster and generate an ordered list of

customers such that they are sorted by their worst-case demands in descending order. In case

the cluster contains fewer than dΓqe elements, we augment the list by inserting (in sorted

order) customers from other clusters, starting with those exhibiting the largest worst-case

demand in their cluster, and continuing with those exhibiting the second largest worst-case

demand, and so on, until the list contains a total of dΓqe elements. Once the final ordered

list is obtained, we assign ξi = 1 to the first bΓqc customers in this sorted list, ξi = Γq − bΓqc

to its last customer (only when Γq not an integer), and ξi = 0 to all other customers, yielding

a demand scenario q∗. Clearly, q∗ ∈ Ext (QE). By repeating this procedure focusing on a

different cluster each time, we can obtain up to NSq different extreme demand scenarios. Let

Q be the set of these scenarios.

• QB. We sort customers from each partitioned set B` by their worst-case demands in de-

scending order. We fix the demands for higher-ranking and lower-ranking customers at their

upper and lower bounds, respectively, and allow exactly one customer demand to take a value

within its bounds, such that the the budget constraint is active. This yields a scenario q∗.

One can easily show that q∗ ∈ Ext (QB). Taking the reverse direction, we fix the demands for

6One may be only interested in Q ⊆ Ext (Q)[T ⊆ Ext (T )

]. However, given an extreme point q ∈ Ext (Q)

[t ∈ Ext (T )], it is generally not convenient to determine whether q [t] is dominated or not by any other scenario from

Ext (Q) [Ext (T )].

25

lower-ranking and higher-ranking customers at their upper and lower bounds, respectively.

Considering all possible combinations among different customer subsets B`, we have in total

2L scenarios. We sort them, in descending order, by the number of customer demands that

have achieved their upper bounds. Let Q be the set of the first NSq scenarios.

• QF . In the proof of Proposition 2, we have identified the set Ext (QF ). We sort the elements

of Ext (QF ) in descending order according to their cumulative customer demands. Let Q be

the set of the first NSq scenarios.

• QE . We consider the maximization of a weighted sum of customer demands qi that is subject

to q ∈ QE ∩q ∈ Rn : q ≥ q0

. For this, we choose 1/q0

i to be the weight corresponding to

each qi. Optimizing this problem will yield an extreme demand vector q∗ that dominates q0.

To obtain additional extreme scenarios, the randomly shuffled list

1/q0j

j∈Vc

is used as the

ordered list of weights and the resulting problem is re-optimized. In total, we generate NSq

scenarios. Let Q be the set of those.

• QD. We first eliminate from QD any scenarios that are either dominated or are not extreme

points (e.g., by checking each one via solving a linear program) and then sort the remaining

scenarios by the total customer demands in descending order. Let Q be the set of the first

NSq scenarios. Clearly, Q ⊆ Ext (QD).

• TG. We begin by ranking all feasible arcs (i, j) ∈ A based on the following ordered criteria:

(i) whether the arc is connecting to the depot, i.e., being an arc of the type (0, j); (ii) the

traversal cost cij . Then, we assign ξij = 1 to the first bΓtc arcs, ξij = Γt − bΓc to the next

one in the list, and ξij = 0, to any remaining arcs, yielding a travel time scenario t∗. Clearly,

t∗ ∈ Ext (TG). Replacing the depot by some customer node in criterion (i) above, we may

repeat the same procedure to obtain a new extreme travel time scenario. In total, we generate

NSt scenarios. Let T be the set of those.

• TD. We first eliminate from TD any scenarios that are either dominated7 or are not extreme

points and then sort the remaining scenarios by the total arc traversal time in descending

order. Let T be a set of the first NSt scenarios. Clearly, T ⊆ Ext (TD).

7We use the fact that t dominates t′, if t ≥ t′.

26

We can now empirically adjust the time complexity of pricing subproblems by simply controlling

NSq and NSt, the number of resource constraints that have to be enforced in the pricing step.

6.2.2 Robust Rounded Capacity Inequalities

For notational convenience, we introduce arc-based variables xij , for each (i, j) ∈ A, and relate it

to route-based variables λr, as shown by (41).

xij =∑r∈R

τijrλr ∀(i, j) ∈ A (41)

To ensure the robust feasibility of selected routes against demand uncertainty, the authors of [22]

and [48] proposed to enforce robust RCI (42) in a branch-and-cut framework.

∑i/∈S

∑j∈S

xij ≥

⌈1

Qmaxq∈Q

∑i∈S

qi

⌉∀S ⊆ Vc (42)

In this work, we adopt the same idea but enforce it in the context of BPC. We now focus on the

RCI separation routine. Let xij be the arc variable values corresponding to the optimal routes

of an RMP solution. When all xij values are integral, one can simply loop through all vehicle

routes and check whether robust capacity feasibility is satisfied or not. In contrast, the exact

separation of robust RCI at a fractional solution is an NP-complete problem, since separating its

deterministic counterpart is itself NP-complete [6]. Hence, solving the separation problem exactly

is computationally prohibitive, and we resort to heuristic methods.

The work of [22] proposed a tabu search algorithm to identify violated robust RCI whenQ := QB

or Q := QF . The authors of [48] extended this to every demand uncertainty set presented in

Section 3, and here we adopt the separation routines from that work. In particular, the separation

procedure starts with a randomly selected customer set S ⊆ Vc and then iteratively perturbs this set

through a sequence of operations in which individual customers are added or removed. For this, we

maintain tabu lists of customers that have recently been added or removed to avoid cycling and to

escape local optima. At each iteration, the separation algorithm greedily chooses a customer whose

inclusion or removal maximizes the slack of the robust RCI (42). Computing this slack requires the

computation of the right-hand side, which in turn, requires the efficient evaluation of the worst-

case demand over the current candidate set of customers S, i.e., maxq∈Q

∑i∈S

qi. Generally speaking,

maximizing a linear objective over a convex set has a polynomial-time complexity [10]. Fortunately,

27

for those convex uncertainty sets of our interest, this problem can be optimized analytically in

roughly linear time by exploiting the structure of the given uncertainty set. Furthermore, since

the set S results from perturbing another set S′ by adding or removing only a single customer,

we can apply quick incremental or decremental updates and obtain the worst-case demand for S′

from its counterpart for S, which is more efficient than computing it from scratch. For the exact

closed-form expressions, time and storage complexities, as well for implementation details of how

to compute the worst-case demand of a given customer set S under each of the demand uncertainty

sets presented in Section 3, the readers are referred to [48].

We remark that, when the travel time vector t is deterministic, i.e., T is a singleton, sufficient

conditions (21) and (23)–(26) for claiming dominance of a path P1 over path P2 during route

enumeration are still valid.8 Their certification comes from the fact that, if P1 dominates P2, then

both paths cover exactly the same customer set (condition 26), and hence, the vehicle loads are

equal under any demand realization q ∈ Q. After route enumeration, only those routes that are

robust feasible are kept for consideration.

6.2.3 Infeasible Path Elimination Constraints

In applications of routing with time windows, infeasible path elimination constraints [27] are often

introduced to forbid routes along which time window conditions are violated. Let a sequence of

vertices P = (v1, v2, . . . , vp) denote an elementary path, and let AP ⊆ A denote the set of arcs

that are traversed along path P . Note that path P does not necessarily start or end at a depot.

Furthermore, let P denote the set of paths that are deemed infeasible with respect to time window

constraints. One can simply forbid these paths via enforcing IPEC given by (43).

∑(i,j)∈AP

xij ≤ |AP | − 1 ∀P ∈ P (43)

It is well-known that, due to the degree constraints (11), one can further lift IPEC to tournament

inequalities [5]. In particular, we replace the subscript of the summation with (i, j) ∈ tr.cl.(P ),

where tr.cl.(P ) denotes the transitive closure of path P . This so-called tournament form of the

inequality is stronger than the version presented above, hence we always do so in the remainder

8This was also pointed out in [37], in which capacity inequalities were enforced as necessary constraints when

solving the VRPTW via a BPC algorithm.

28

of our paper. To tackle the RVRPTW, the work of [2] applied IPEC within the branch-and-cut

framework to forbid those routes along which time window constraints are violated. In this work,

we adopt the same idea, enforcing them in the BPC framework.

In regards to IPEC separation, we note that it is trivial at any integral solution, while at a

fractional solution, it can be shown that there are polynomially many paths P for which inequal-

ities (43) are violated [5]. These paths can be easily detected by a simple enumeration procedure

for the case of an uncertainty set of the form TG. In particular, the algorithm starts with choosing

i ∈ V as a root node and then runs a depth-first search for extending paths. If a violated tour-

nament inequality is found for some path P ′, we then check whether time window feasibility is

respected (i.e., P ′ ∈ P) under any travel time realization t ∈ TG. This can be achieved in O(dΓte

)time, using proper data structures similar to the ones in Section 6.1.1. In our implementation, we

use the enumeration procedure to identify all violated infeasible paths and only forbid those if they

are minimal infeasible.9

One can simply extend the above separation routine to the case of a discrete support TD. Using

proper data structures, checking the time window feasibility along a path takes O(Dt) time. We

emphasize that, if travel times are variable, sufficient conditions (21) and (23)–(26) for claiming

dominance are no longer valid,10 since they cannot suffice the dominance relationship between P1

and P2 for some realization t ∈ T \ T . Thus, route enumeration is turned off in this case.

6.3 Approach Comparison

We have discussed two distinct approaches for adapting the BPC algorithm to solve robust VRPs

under demand and travel time uncertainty. The robust pricing approach guarantees robust fea-

sibility of routing designs in pricing subproblems, while the robust cutting-plane approach seeks

for robust feasibility via dynamically enforcing necessary constraints in master problems. To solve

the resulting pricing subproblem in the former, we have synopsized two existing methods from

the literature, the direct pricing method and the transformed pricing method. To apply cutting

planes dynamically in the latter, we have discussed efficient separation routines. In this section, we

9An infeasible path P = (v1, v2, . . . , vp) is said to be minimal infeasible if the truncated subpaths defined by

AP \ (v1, v2) and AP \ (vp−1, vp) are feasible [27].10Our previous work [49] also pointed this out in a similar situation where path inequalities were dynamically

enforced as necessary constraints in the context of solving the VRPTW variant via a BPC algorithm.

29

compare these two approaches in a number of important aspects.

• Uncertainty sources. Both approaches can deal with robust VRPs with uncertainty in

demands and travel times. We note that for both uncertainty sources, ensuring robust feasi-

bility of a routing design can be decomposed into ensuring the same for every single route.

Stated differently, if every vehicle route in the returned solution is robust feasible, so is the

whole routing design. However, for other types of uncertainty sources, such as the customer

order uncertainty we mentioned in Section 1, one may not be able to encapsulate the robust

feasibility condition into the pricing subproblems and instead has to enforce it explicitly in

the master problems (e.g., see [47]). In such a case, the robust cutting-plane approach might

become the only choice.

• Uncertainty sets. Compared with the robust pricing approach, the robust cutting-plane

approach is applicable to more general types of uncertainty sets. We synopsize in Table 1 the

applicability of each BPC algorithm to different types of uncertainty sets. A check mark “3”

indicates that the specific approach has been demonstrated to be applicable in the context of a

given uncertainty set, while a cross mark “7” denotes a case where this approach is not fit. To

the best of our knowledge, our robust cutting-plane approach can handle all popular classes of

uncertainty sets listed in Table 1, while the robust pricing approach has limited applicability.

Specifically, the transformed pricing method only works for two types of demand uncertainty

sets, while the direct pricing method is limited to three types of demand uncertainty sets. The

work of [15] has showed that the RSPPRC with an ellipsoidal set QE for demand uncertainty

is strongly NP-hard, and hence, the robust pricing approach is not suitable. As we have

pointed out, the direct pricing method will quickly become computationally prohibitive in

practice when the parameters Γq and Γt in QG and TG, or the number of scenarios Dq and

Dt in QD and TD, increase.

• Time complexity of pricing subproblems. The robust pricing approach always has to

solve instances of the RSPPRC as the pricing subproblems via either the direct or transformed

method. The former solves the RSPPRC directly in each pricing iteration, while the latter

transforms the RSPPRC into polynomially many deterministic SPPRCs. In either case, the

RSPPRC has a larger time complexity than the deterministic SPPRC. In contrast, the robust

30

Table 1. Applicability of different BPC algorithms under different uncertainty sets.

ApproachQ T

QG QB QF QE QD TG TD

Direct pricing 3 3 7 3 3 3

Transformed pricing 3 3 7

Robust cutting-plane 3 3 3 3 3 3 3

cutting-plane approach always solves a deterministic SPPRC of a controlled time complexity.

Specifically, one can choose a proper number of demand and/or travel time scenarios to define

the SPPRC so as to generate partially robust feasible routes.

• Tightness of LP relaxations. One can also apply robust RCI and IPEC as strengthening

constraints in the robust pricing approach. From this perspective, both approaches can be

viewed as using exactly the same formulation (i.e., the backbone set-partitioning model (10)–

(13) augmented with robust RCI and IPEC) noting that they are based on different route

sets, namely R and R. Given that R ⊆ R, the LP relaxation in the robust pricing approach

is always stronger. This implies that the robust pricing approach might solve a robust VRP

more effectively on the condition that the RSPPRC could be solved efficiently.

To summarize, our robust cutting-plane algorithm only entails the solution of deterministic

pricing subproblems to generate vehicle routes and relies on the enforcement of necessary constraints

in master problems to ensure robust feasibility of routing designs. Compared with the robust

pricing algorithm, our proposed approach has two distinctive advantages. Firstly, it is more widely

applicable, as it can readily deal with parameter variability related to customer demands and/or

vehicle travel times under any known type of uncertainty set, as shown in Table 1. Secondly, it is

more versatile, allowing us to proactively balance the effort between generating robust routes and

separating robustifying cuts. In particular, one is allowed to consider only a manageable number

of scenarios in the pricing step, adjusting its computational burden. This feature becomes crucial

when the resulting RSPPRC is strongly NP-hard (e.g., in the case of set QE), when the robust

pricing method is not practically viable.

31

7 Computational Studies

In this section, we test our proposed BPC algorithm on RCVRP and RVRPTW benchmark in-

stances and compare it against the existing methods from the literature. Our algorithm relies

on a deterministic BPC engine to generate vehicle routes and a cut generation routine to enforce

necessary constraints. In particular, we utilize VRPSolver 0.3 [37] as our BPC engine via its Julia

interface, in which all subordinate linear and mixed-integer linear programs were solved using the

IBM ILOG CPLEX Optimizer 12.9.0. The separation routines for IPEC and robust RCI were

implemented in C++ and compiled into a C library for use in Julia. The VRPSolver provides a

callback function for users to separate and add necessary constraints. The experiments were run

on an Intel Xeon E5-2689 v4 server running at 3.10 GHz with 128 GB of available RAM that was

shared among 10 copies of the algorithm running in parallel on the server. Each instance was solved

by one copy of the algorithm using a single thread. We compare our proposed algorithm against

the transformed pricing approach from [38] for solving RCVRP instances, as well as against the

direct pricing approach from [33] for solving RVRPTW instances. The authors of [38] ran their

experiments on an Intel Core i7-3770 3.40 GHz system, while [33] did so on an Intel Xeon E5-

2680 2.70 GHz system. According to https://www.cpubenchmark.net/singleThread.html, our

machine runs approximately 2.1 times faster than the former and 1.5 times faster than the latter.

7.1 Computational Results on RCVRP Instances

In this section, we evaluate our proposed algorithm on solving RCVRP instances under demand

uncertainty. We adapt the classic CVRP benchmark instances for generating RCVRP instances in

which the customer demands are supported on the various demand uncertainty sets from Section 3.

We consider five classes of CVRP instances, namely A, B, E, F, M, and P. These benchmark

instances are available at http://vrp.galgos.inf.puc-rio.br/index.php/en/. To be consistent

with the literature, when Q := QB or Q := QF , we consider 26 instances from class A, 23 instances

from B, 11 instances from E, 3 instances from F, 3 instances from M, and 24 instances from P, as [22]

did, while when Q := QG, we consider one extra instance from class A and two extra instances

from E, discarding one instance from P and two large-size instances from F and M, as [38] did. For

the cases of Q := QE and Q := QD, since no RCVRP benchmarks were available in the literature,

we considered the same instances from [22]. For each uncertainty set, there are 90 benchmarks

32

in total, with the number of customers ranging from 13 to 150. Following the convention in the

literature, every entry of the travel cost matrix is calculated from coordinates and then rounded

to the nearest integer. The customer demands specified in the original datasets are regarded to

be the nominal values q0. As [22] did, we partition the customer set Vc into four geographic

quadrants, NE, NW, SW, and SE, based on the customer coordinates reported in the datasets.

For each deterministic CVRP instance, we construct the following five types of uncertainty sets, as

described in the following.

(a) Cardinality-constrained set (originally proposed in [29])

QG :=

q ∈ Rn : qi = q0

i + αq0i ξi, ξi ∈ [0, 1] ∀i ∈ Vc,

∑i∈Vc

ξi ≤ Γq

.

This set stipulates that each customer’s demand can deviate upward from the nominal value

by at most α · 100%. As [38] did, we choose α = 0.3, Γq =⌊0.75n/K

⌋and modify the vehicle

capacities to be Q =⌊0.3Cmax + 0.7Cmin

⌋, where Cmax and Cmin are values available in the

appendix of [38].

(b) Budget sets (originally proposed in [22])

QB :=

q ∈

[(1− α)q0, (1 + α)q0

]:∑i∈Ω

qi ≤ (1 + αβ)∑i∈Ω

q0i ∀Ω ∈ NE,NW,SW,SE

.

This set stipulates that each customer’s demand can deviate from the nominal value by at

most α ·100%, but the cumulative demand of each quadrant may not exceed its nominal value

by αβ · 100%. As [22] did, we chose α = 0.1 and β = 0.5, while also increasing the vehicle

capacities by 20%.

(c) Factor models (originally proposed in [22])

QF :=

q ∈ Rn : q = q0 + Ψξ, ξ ∈ [−1, 1]4 ,

∣∣∣∣∣∣4∑

f=1

ξf

∣∣∣∣∣∣ ≤ 4β

.

Here, the disturbance term Ψξ is a linear combination of 4 factors, with each entry of the

matrix Ψ depending on the relative proximity of the corresponding customer to corresponding

quadrant’s centroid. More specifically, let Ψif = αq0i ψif/

∑4f ′=1 ψif ′ , where ψif denotes the

inverse distance between customer i and the centroid of quadrant f ∈ 1, 2, 3, 4. As [22] did,

we chose α = 0.1 and β = 0.5, while also increasing the vehicle capacities by 20%.

33

Using the reduction procedure described in the proof of Proposition 2, one can represent

Ext (QF ) as a discrete set with only 4 demand scenarios. In particular, these correspond to

(1, 1, 1, 0), (1, 1, 0, 1), (1, 0, 1, 1), (0, 1, 1, 1) ∈ Ext(ΞF ). Since the number of these scenarios is

small, we consider Q := Ext (QF ), which leads all routes generated from pricing subproblems

to be robust feasible, i.e., R = R. We remark that, in this case, our approach can be viewed

as an application of the direct pricing approach strengthened via cutting planes that are valid

in the robust setting.

(d) Ellipsoidal sets (originally proposed in [48])

QE :=q ∈ Rn :

(q − q0

)>Σ−1

(q − q0

)≤ 1.

Here, Σ = (1 − β)ΨΨ> + βdiag(αq0

1, αq02, . . . , αq

0n

), where Ψ is the same matrix as the one

defined above for set QF , while diag (·) denotes a diagonal matrix. We chose α = 0.1 and

β = 0.5, while increasing the vehicle capacities by 10%.

(e) Discrete sets (originally proposed in [48])

QD := conv(q0∪qd : d = 1, 2, . . . ,nint (βn)

).

Here, nint (βn) denotes the nearest integer to βn. The points qd are generated as uniform sam-

ples of the[(1− α)q0, (1 + α)q0

]hyper-rectangle. We choose α = 0.1, β = 0.2 and increase

the vehicle capacity by 10%.

To be consistent with the literature [21, 22, 38], the number of used vehicles is fixed to be the

fleet size K for the cases of sets QG, QB and QF , while the number of used vehicles should not

exceed K for the cases of sets QE and QD. In other words, we enforce equality for constraint (12) in

the former cases and a “less than” inequality in the latter. In what follows, we present synopsized

computational results of our proposed BPC algorithm for addressing RCVRP instances. The

detailed results are presented in the Appendix.

7.1.1 Set QG

For a fair comparison, we obtain from [38] the heuristic solution cost to each benchmark and use

it as an initial upper bound in our BPC algorithm. In our experiments, NSq was chosen to be

34

the fleet size K. We imposed a time limit of 2 hours for each instance and report in Table 2 the

consolidated results of our robust cutting-plane algorithm, comparing those against the reported

performance of the transformed pricing method from [38] (PPSV21). The column “Class” reports

the instance class, while the column “# inst.” reports the number of instances in this class. In

columns “# opt.”, “Avg. t (sec)” and “Avg. gap (%)”, we respectively report the number of

instances that were solved to optimality, the geometric mean solution time (rounded to the nearest

integer) for those solved instances, and the average residual gap for those instances for which the

algorithm was terminated due to the time limit but with valid lower and upper bounds identified.

Out of 90 RCVRP instances, our robust cutting-plane approach solved 37 of them to optimality

within an average solution time of 79 seconds, resulting in an average residual gap of 4.26% in the

remaining ones; the transformed pricing approach performed significantly better than our approach,

solving to optimality all but three instances. We remark that the superior performance of PPSV21

was expected in this case, since the robust pricing approach can transform the RSPPRC into a

small number (around 20 on average) of deterministic SPPRCs and thus efficiently generate robust

feasible routes in pricing subproblems.

Table 2. Computational results for our BPC algorithm on RCVRP instances under uncertainty set QG and

comparison with literature.

Class # inst.PPSV21 This work

# opt.Avg. t

(sec)

Avg. gap

(%)# opt.

Avg. t

(sec)

Avg. gap

(%)

A 27 27 19 – 11 262 4.79

B 23 20 70 1.65 5 278 4.81

E 13 13 28 – 7 11 4.03

F 2 2 223 – 2 101 –

M 2 2 108 – 0 – 2.44

P 23 23 11 – 12 53 3.05

All 90 87 26 1.65 37 79 4.26

35

7.1.2 Set QB

For a fair comparison, we obtain from [38] the heuristic solution cost to each benchmark and use

it as an initial upper bound in our BPC algorithm. In our experiments, NSq is chosen to be 2.

We imposed a time limit of 2 hours for each instance and report in Table 3 the consolidated

results of our robust cutting-plane algorithm, comparing those against the reported performance

of the transformed pricing method from [38] (PPSV21). Out of 90 RCVRP instances, our robust

cutting-plane approach solved 80 of them to optimality within an average solution time of 36

seconds, resulting in an average residual gap of 1.96% in the remaining ones; the transformed

pricing approach performed slightly better, solving 89 instance to optimality. Compared with the

branch-and-cut algorithm from [21, 22], our BPC algorithm solved 37 more instances to optimality,

which results from the fact that the route-based formulation (e.g., the set-partitioning model) has

a tighter LP relaxation than the arc-based formulation used in those previous studies.

Table 3. Computational results for our BPC algorithm on RCVRP instances under uncertainty set QB and

comparison with literature.

Class # inst.PPSV21 This work

# opt.Avg. t

(sec)

Avg. gap

(%)# opt.

Avg. t

(sec)

Avg. gap

(%)

A 26 26 3 – 23 50 1.48

B 23 23 6 – 20 75 2.06

E 11 11 11 – 10 38 0.76

F 3 2 833 0.89 2 36 4.81

M 3 3 154 – 1 32 1.72

P 24 24 1 – 24 14 –

All 90 89 5 0.89 80 36 1.96

7.1.3 Sets QF ,QE and QD

When Q := QF , we obtain from [21] the heuristic solution cost to each instance and use it as an

initial upper bound in the BPC algorithm, while for the cases of Q := QE and Q := QD, we run the

heuristics code from [48] with a time limit of 2 seconds and utilize the returned heuristic solution

36

value as an initial upper bound. For each instance, we chose NSq to be 4, 1 and 6 when Q := QF ,

QE and QD, respectively. We imposed a time limit of 2 hours for each instance and report the

consolidated computational results in Table 4. We remark that we do not present any comparisons

in this case, as no literature BPC approaches had been previously applied on these datasets.

Among the three sets, our robust cutting-plane algorithm performs the best under the factor

model set, QF , solving 85 out of 90 instances to optimality. This can be attributed to the fact that

every generated route in this case is guaranteed to be robust feasible, since we enforce Q := Ext (QF )

when defining pricing subproblems. As a consequence, the LP relaxation of the set-partitioning

model is tighter and our algorithm could solve RCVRP instances more efficiently in this case.

As expected, the BPC framework outperforms the branch-and-cut implementation from [21, 22],

solving 33 more instances to optimality. Our proposed algorithm performs roughly the same on

solving RCVRP instances when Q := QE and Q := QD, solving 76 and 72 instances, while resulting

in average residual gaps 3.69% and 3.35%, respectively.

Table 4. Computational results for our BPC algorithm on RCVRP instances under uncertainty sets QF ,

QE and QD.

Class # inst.QF QE QD

# opt.Avg. t

(sec)

Avg.

gap (%)# opt.

Avg. t

(sec)

Avg.

gap (%)# opt.

Avg. t

(sec)

Avg.

gap (%)

A 26 26 11 – 23 53 3.23 23 71 3.01

B 23 21 19 1.58 16 46 2.97 15 52 3.70

E 11 11 21 – 9 39 6.22 9 38 3.00

F 3 2 68 1.47 2 18 5.29 2 74 5.16

M 3 1 1,573 1.46 2 1,508 3.44 0 – 3.14

P 24 24 6 – 24 11 – 23 20 1.01

All 90 85 13 1.51 76 32 3.69 72 41 3.35

Through the above experiments on RCVRP instances, we can make the following observations:

1. The approach of embedding robust RCI into a deterministic BPC algorithm to address the

RCVRP is versatile and efficient for various types of demand uncertainty sets, including sets

QE and QD, for which existing approaches based on robust pricing are not applicable.

37

2. The transformed pricing approach demonstrated superior performance for RCVRP instances

when demands are realized from sets of the form QG and QB.

3. Embedding robust RCI into a BPC framework yields significantly better performance than

the case of branch-and-cut.

7.2 Computational Results on RVRPTW Instances

We now evaluate our proposed algorithm on solving RVRPTW instances under demand and/or

travel time uncertainty. We follow the same procedure from [33] to adapt Solomon datasets [46]

for generating RVRPTW instances, in which the customer demands and vehicle travel times are

considered uncertain and are postulated to realize from cardinality-constrained sets QG and TG,

respectively. The Solomon datasets include 100 customers and are classified according to the

spatial distribution of the latter into classes “C1” and “C2”, which correspond to clustered distri-

butions, classes “R1” and “R2”, which correspond to random distributions, as well as classes “RC1”

and “RC2”, which correspond to mixed distributions. In addition, datasets from C2, R2, and RC2

have wider time windows and larger vehicle capacities than their “1” counterparts. The Solomon

datasets are available at http://neo.lcc.uma.es/vrp/. We apply the convention that travel times

and costs are calculated from coordinates and then truncated with one decimal place. The customer

demands specified in the original datasets, as well as the calculated travel times, are regarded to

be the nominal values q0 and t0, respectively. The maximum allowable deviations in cardinality-

constrained sets (1) and (2) are qi = trunc(αq × q0

i

)and tij = 0.1× trunc

(αt × 10× t0ij

), respec-

tively. Thus, each customer demand can deviate upward from its nominal value by about αq ·100%,

while each arc traversal time can deviate upward from its nominal value by about αt ·100%. In our

experiments, we chose αq = αt = 0.1, and Γq = Γt = 5, as used in [33].

For each RVRPTW instance, we consider three cases: (i) only demand uncertainty, (ii) only

travel time uncertainty, and (iii) both demand and travel time uncertainty. Since we did not have at

hand sophisticated heuristics for solving the RVRPTW, we chose to use as the initial upper bound

the best known value of each instance, modified upwards by an offset of 0.1. Correspondingly,

our BPC algorithm always has to locate by itself a feasible solution with a value better than the

initial upper bound provided. As [34] pointed out, for VRPTW instances from classes C2, R2, and

RC2, the capacity constraints are not really binding. Therefore, to ease the solution of the pricing

38

subproblem, we do not include these constraints (i.e., Q := ∅) but enforce them in master problems

through capacity inequalities whenever needed. Also, as already mentioned, route enumeration has

to be deactivated whenever travel time uncertainty applies. In what follows, we present synopsized

computational results for RVRPTW instances. The detailed results are presented in the Appendix.

7.2.1 Set QG

In these experiments, NSq was chosen to be 10. We imposed a time limit of 1 hour for each instance

and report in Table 5 the consolidated results of our proposed robust cutting-plane algorithm, com-

paring those against the reported performance of the direct pricing approach from [33] (MMVAGM19).

The column names have the same meanings as before, adding an additional column “# no LB” to

denote the number of instances for which no lower bounds were reported in the literature paper.

Out of 56 RCVRP instances, our robust cutting-plane approach solved 51 of them to optimal-

ity within an average solution time of 42 seconds, resulting in an average residual gap of 2.92%

in the remaining ones. In comparison, the direct pricing approach was only able to solve 34 in-

stances, noting that it did not idenify any valid lower bound for 7 of the unsolved ones. These

results indicate that solving pricing subproblems exactly via the direct pricing approach is compu-

tationally prohibitive for these instances, confirming our earlier hypothesis that that solving the

resulting RSPPRC subproblems directly comes with increased time complexity and might become

intractable. Compared with the direct pricing method, our robust cutting-plane algorithm also

performs better in terms of both the average solution time and the average residual gap.

7.2.2 Set TG

In these experiments, NSt was chosen to be 1. We imposed a time limit of 1 hour for each instance

and report in Table 6 the consolidated results of our proposed robust cutting-plane algorithm, com-

paring those against the reported performance of the direct pricing approach from [33] (MMVAGM19).

Out of 56 RCVRP instances, our robust cutting-plane approach solved 35 of them to optimality

within an average solution time of less than 181 seconds, resulting in an average residual gap of

2.05% in the remaining ones. In comparison, the direct pricing approach was able to solve a total of

40 instances, noting that it did not idenify any valid lower bound for 7 of the unsolved ones. Upon

closer inspection of these results, we note that the direct pricing approach solved more instances

39

Table 5. Computational results for our BPC algorithm on RVRPTW instances under uncertainty set QG

and comparison with literature.

Class # inst.MMVAGM19 This work

# opt.# no

LB

Avg. t

(sec)

Avg.

gap† (%)# opt.

Avg. t

(sec)

Avg. gap

(%)

C1 9 3 0 881 9.94 5 158 2.32

R1 12 9 0 62 2.30 12 18 –

RC1 8 6 0 94 4.67 8 62 –

C2 8 7 1 196 – 8 18 –

R2 11 4 4 598 1.28 10 96 5.36

RC2 8 5 2 283 1.62 8 37 –

All 56 34 7 175 5.44 51 42 2.92

†Not including “no LB” instances.

from classes C1, R1 and RC1, while our algorithm performed better for instances from classes C2,

R2 and RC2. This could be explained by the fact that instances from the “1” classes have tighter

time windows, and hence, there only exists a relatively small number of robust feasible routes in the

RSPPRC, such that the direct pricing method turns out to be computationally cheap. In contrast,

instances from the “2” classes have wider time windows, and hence, small perturbations on the

travel time vector will not deprive most routes of their feasibility, such that solving the SPPRC

defined by T rather than the RSPPRC defined by T to generate routes bears less burden.

7.2.3 Set QG × TG

In these experiments, NSq and NSt were chosen to be 10 and 1, respectively. We imposed a time

limit of 1 hour for each instance and report in Table 7 the consolidated results of our proposed

robust cutting-plane algorithm, comparing those against the reported performance of the direct

pricing approach from [33] (MMVAGM19). Out of 56 RCVRP instances, our robust cutting-plane

approach solved 30 of them to optimality within an average solution time of less than 287 seconds,

resulting in an average residual gap of 2.35% in the remaining ones. In comparison, the direct

pricing approach was able to solve a total of 32 instances, noting that it did not idenify any valid

40

Table 6. Computational results for our BPC algorithm on RVRPTW instances under uncertainty set TGand comparison with literature.

Class # inst.MMVAGM19 This work

# opt.# no

LB

Avg. t

(sec)

Avg.

gap† (%)# opt.

Avg. t

(sec)

Avg. gap

(%)

C1 9 8 0 154 2.55 8 67 0.34

R1 12 10 0 156 2.37 3 57 1.29

RC1 8 7 0 148 5.16 0 – 2.04

C2 8 6 1 796 1.29 8 1,093 –

R2 11 4 4 482 2.24 8 165 4.95

RC2 8 5 2 241 1.44 8 134 –

All 56 40 7 233 2.43 35 181 2.05

†Not including “no LB” instances.

lower bound for 7 of the unsolved ones. Again, our robust cutting-plane algorithm performs better

for instances with wider time windows, while the direct pricing method did better for instances

with tighter time windows.

Through the above experiments on RVRPTW instances, we can make the following observations:

1. Enforcing IPEC as necessary constraints in a BPC framework is a viable strategy to address

travel time uncertainty, irrespectively of whether demand uncertainty also applies or not.

2. The direct pricing approach should be preferred when the time windows are tight, while the

robust cutting-plane approach should be preferred when the time windows are wide.

8 Conclusions

This work focused on robust vehicle routing problems, where customer demands and vehicle travel

times were assumed to be random variables that can take any values from their respective uncer-

tainty sets. To that end, we considered five popular classes of uncertainty sets, namely cardinality-

constrained sets, budget sets, factor models, ellipsoidal sets and discrete sets, and we explored

different avenues for how BPC algorithms can be employed to address this setting. More specif-

41

Table 7. Computational results for our BPC algorithm on RVRPTW instances under uncertainty set

QG × TG and comparison with literature.

Class # inst.MMVAGM19 This work

# opt.# no

LB

Avg. t

(sec)

Avg.

gap†# opt.

Avg. t

(sec)

Avg. gap

(%)

C1 9 3 0 1,860 9.73 3 969 2.02

R1 12 10 0 180 2.31 3 74 1.29

RC1 8 5 0 126 4.38 0 – 2.13

C2 8 5 1 850 1.65 8 1,076 –

R2 11 4 4 621 4.39 8 173 6.75

RC2 8 5 2 343 1.44 8 133 –

All 56 32 7 349 5.53 30 287 2.35

†Not including “no LB” instances.

ically, we synposized the literature approaches and argued how they fall into the robust pricing

family of approaches, in which the full robust feasibility of routes is ensured during their generation

in pricing subproblems. We then proposed a fundamentally different BPC algorithm that combines

robust cutting-plane techniques with the advances currently present in deterministic BPC solvers.

In particular, our proposed approach calls for the utilization of a deterministic pricing engine to

generate partially robust feasible routes and enforces IPEC and robust RCI as necessary constraints

to complete the guarantee of robust feasibility of the routing designs. We also conducted extensive

computational studies using RCVRP and RVRPTW instances under all classes of uncertainty sets

mentioned above. Our computational results demonstrated the versatility, flexibility, and efficiency

of our robust cutting-plane approach, while it elucidated settings (e.g., specific unceratinty sets

and/or specific data regimes) in which the new robust cutting-plance approach should be preferred

over the existing robust pricing approaches.

Acknowledgments

The authors gratefully acknowledge financial support from the Center for Advanced Process Decision-

making at Carnegie Mellon University, as well as support for Akang Wang from the James C.

42

Meade Graduate Fellowship and the H. William and Ruth Hamilton Prengle Graduate Fellowship

at Carnegie Mellon University. We also gratefully acknowledge the VRPSolver developers for their

generosity to provide academic use of the software.

43

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49

Appendix

In this Appendix, we present the detailed computational results for RCVRP and RVRPTW in-

stances in Tables 8–11 and Tables 12–13, respectively. As a reference, we also present the com-

putational results for solving deterministic CVRP and VRPTW instances (with the optimal value

provided as an upper bound for each instance) via the VRPSolver. In each table, the column

“Instance” lists the instance name, the column “Opt [UB]” reports the corresponding optimal ob-

jective value, while the column “t (sec) [LB]” provides the time to solve the instance to optimality;

if an instance could not be solved within the allotted time limit, these columns report (in brackets)

the best upper and lower bounds found within this time limit. A total of 56 instances that had

not been solved to optimality in the literature but were solved in this work for the first time are

indicated with an asterisk (*), while a total of 8 new best-known solutions found are indicated in

boldface.

50

Tab

le8.

Det

ail

edre

sult

sfo

rou

rB

PC

alg

ori

thm

on

RC

VR

Pin

stan

ces

A.

Inst

ance

q0QG

QB

QF

QE

QD

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

A-n

32-k

578

42

857

119

748

274

82

755

475

55

A-n

33-k

566

12

675

3064

213

631

265

22

652

14

A-n

33-k

674

22

758

215

717

371

02

733

273

03

A-n

34-k

577

85

776

4,22

671

53

702

274

756

743

252

A-n

36-k

579

93

[823

][8

06]

755

376

68

783

1376

94

A-n

37-k

566

92

706

292

650

464

83

667

566

528

A-n

37-k

694

94

[948

][9

05]

892

1489

26

907

290

76

A-n

38-k

573

04

714

2270

411

693

370

93

709

5

A-n

39-k

582

24

[818

][7

89]

777

4877

25

806

3280

362

A-n

39-k

683

13

850

230

787

278

62

813

1380

914

A-n

44-k

693

72

930

5990

930

289

22

928

3191

916

A-n

45-k

694

42

918

278

896

889

14

923

1192

110

A-n

45-k

71,

146

5[1

,163

][1

,121

]–

––

––

––

–

A-n

46-k

791

42

988

3,35

388

831

883

790

613

790

255

A-n

48-k

71,

073

3[1

,129

][1

,067

]1,

033

293

1,03

3*41

1,06

01,

014

1,04

219

4

A-n

53-k

71,

010

5[1

,019

][9

83]

974

287

967

6598

719

984

11

A-n

54-k

71,

167

8[1

,169

][1

,110

]1,

106

738

1,09

7*42

1,14

53,

358

1,14

41,

228

A-n

55-k

91,

073

21,

107

1,49

41,

030

391,

007

41,

055

391,

055

157

A-n

60-k

91,

354

10

[1,4

08]

[1,3

11]

[1,2

80]

[1,2

62]

1,26

4*42

1,29

215

01,

290

345

A-n

61-k

91,

034

7[1

,022

][9

84]

983

419

974*

191,

010

133

1,00

311

8

A-n

62-k

81,

288

16

[1,3

39]

[1,2

60]

[1,2

17]

[1,1

97]

1,20

1*67

1,24

53,

128

1,23

26,

058

A-n

63-k

91,

616

17

[1,6

20]

[1,5

01]

1,50

54,

754

1,49

8*19

31,

571

3,34

3[1

,575

][1

,533

]

A-n

63-k

101,3

1410

[1,3

48]

[1,2

81]

1,23

358

61,

222*

191,

257

482

1,24

91,

611

A-n

64-k

91,

401

18

[1,4

17]

[1,3

38]

1,32

51,

017

1,31

4*97

[1,3

85]

[1,3

43]

[1,3

80]

[1,3

35]

A-n

65-k

91,

174

6[1

,184

][1

,120

]1,

106

188

1,09

45

1,16

41,

582

1,14

43,

572

A-n

69-k

91,

159

10

[1,1

77]

[1,1

36]

1,10

995

31,

096*

18[1

,149

][1

,114

]1,

122

5,29

5

A-n

80-k

101,7

6316

[1,8

03]

[1,7

02]

[1,6

62]

[1,6

39]

1,6

44

*1,

253

[1,7

43]

[1,6

80]

[1,7

15]

[1,6

62]

#op

t.27

1123

2623

23

51

Tab

le9.

Det

ail

edre

sult

sfo

rou

rB

PC

alg

ori

thm

on

RC

VR

Pin

stan

ces

B.

Inst

ance

q0QG

QB

QF

QE

QD

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

B-n

31-k

5672

369

45,

842

651

265

12

604

260

24

B-n

34-k

5788

678

91,

565

768

1,09

574

810

782

264

769

151

B-n

35-k

5955

5[9

86]

[952

]88

34

883

3[9

46]

[921

]92

125

4

B-n

38-k

6805

482

333

729

472

93

705

6968

83

B-n

39-k

5549

456

176

532

2352

917

534

953

442

B-n

41-k

6829

3[8

38]

[797

]79

645

679

126

800

480

110

B-n

43-k

6742

8[7

79]

[734

]68

116

768

034

683

268

322

B-n

44-k

7909

5[9

43]

[912

]83

510

835

1085

69

855

95

B-n

45-k

5751

7[7

39]

[717

]70

142

168

013

708

1670

237

B-n

45-k

6678

6[6

68]

[631

]66

011

657

967

025

668

2,27

1

B-n

50-k

7741

1075

874

679

369

98

732

1,02

171

788

B-n

50-k

81,

312

17[1

,330

][1

,286

][1

,224

][1

,208

][1

,217

][1

,202

][1

,251

][1

,215

][1

,226

][1

,206

]

B-n

51-k

71,

032

15[1

,027

][9

44]

961

238

928

6[9

99]

[988

][9

96]

[973

]

B-n

52-k

7747

4[7

75]

[750

]67

530

670

1466

711

662

15

B-n

56-k

7707

4[7

40]

[730

]62

38

623

2761

48

612

11

B-n

57-k

71,

153

12[1

,132

][1

,054

]1,

055

778

1,05

2*62

[1,1

36]

[1,0

94]

[1,1

26]

[1,0

71]

B-n

57-k

91,

598

5[1

,656

][1

,594

]1,

540

221,

539

22[1

,520

][1

,488

][1

,490

][1

,474

]

B-n

63-k

101,

496

8[1

,588

][1

,471

]1,

407

6,80

91,

405*

392

[1,5

07]

[1,4

59]

[1,4

97]

[1,4

31]

B-n

64-k

9861

6[8

65]

[839

]80

366

803

3080

519

804

41

B-n

66-k

91,

316

21[1

,319

][1

,209

][1

,251

][1

,217

]1,

210

331,

269

2,57

1[1

,266

][1

,215

]

B-n

67-k

101,

032

11

[1,0

86]

[1,0

48]

1,00

77,

030

1,00

1*24

81,

012

1,12

91,

001

2,65

8

B-n

68-k

91,

272

40[1

,298

][1

,247

][1

,205

][1

,179

][1

,197

][1

,174

][1

,255

][1

,190

][1

,245

][1

,167

]

B-n

78-k

101,

221

10

[1,2

61]

[1,1

61]

1,13

11,

697

1,13

0*81

1,16

26,

256

[1,2

00]

[1,1

40]

#op

t.23

520

2116

15

52

Tab

le10.

Det

aile

dre

sult

sfo

rou

rB

PC

alg

ori

thm

on

RC

VR

Pin

stan

ces

E,

Fan

dM

.

Inst

ance

q0QG

QB

QF

QE

QD

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

E-n

13-

k4

247

127

72

––

––

––

––

E-n

22-

k4

375

137

31

373

237

31

373

137

32

E-n

23-

k3

569

257

02

563

254

42

564

256

92

E-n

30-

k3

534

1049

56

475

349

27

495

349

54

E-n

31-

k7

379

237

911

––

––

––

––

E-n

33-

k4

835

283

634

381

412

581

442

828

5682

19

E-n

51-

k5

521

251

910

151

628

516

2451

95

518

27

E-n

76-

k7

682

27[6

99]

[681

]66

190

661*

283

665

294

661

76

E-n

76-

k8

735

19[7

36]

[715

]70

931

470

0*26

721

511

714

199

E-n

76-k

10

830

15[8

30]

[796

]79

61,

580

782*

1080

920

280

741

0

E-n

76-k

14

1,02

17

[1,0

22]

[963

]95

241

952*

2998

73,

198

982

7,08

2

E-n

101-

k8

815

61[8

26]

[802

][7

89]

[783

]78

3*13

9[8

58]

[790

][8

08]

[784

]

E-n

101-k

14

1,0

6731

[1,1

21]

[1,0

54]

1,01

186

1,00

9*77

[1,0

85]

[1,0

36]

[1,0

57]

[1,0

25]

F-n

45-k

472

410

736

6571

837

714

121

721

972

171

F-n

72-k

423

724

236

156

232

3523

238

235

3523

477

F-n

135

-k7

1,16

23,0

58–

–[1

,122

][1

,068

][1

,086

][1

,070

][1

,171

][1

,109

][1

,162

][1

,102

]

M-n

101

-k10

820

4[9

18]

[907

]80

932

804

1,57

381

158

5[8

27]

[795

]

M-n

121-k

71,

034

44[1

,030

][9

92]

[994

][9

80]

[987

][9

80]

1,00

43,

892

[999

][9

84]

M-n

151

-k12

1,01

5119

––

[987

][9

67]

[991

][9

69]

[1,0

18]

[983

][1

,010

][9

69]

#op

t.19

913

1413

11

53

Tab

le11.

Det

ail

edre

sult

sfo

rou

rB

PC

alg

ori

thm

on

RC

VR

Pin

stan

ces

P.

Inst

ance

q0QG

QB

QF

QE

QD

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

Op

t

[UB

]

t(s

ec)

[LB

]

P-n

16-

k8

450

1–

–43

91

439

145

02

448

2

P-n

19-

k2

212

219

51

195

219

51

195

119

51

P-n

20-

k2

216

220

82

208

120

81

209

120

91

P-n

21-

k2

211

120

82

208

220

81

211

221

11

P-n

22-

k2

216

221

32

213

221

32

216

221

52

P-n

22-

k8

603

160

12

537

155

71

592

258

72

P-n

23-

k8

529

152

74

504

250

3*1

524

252

42

P-n

40-

k5

458

246

814

447

344

72

456

345

513

P-n

45-

k5

510

351

238

501

4149

43

503

350

39

P-n

50-

k7

554

356

325

853

99

537*

454

640

541

114

P-n

50-

k8

631

4[6

14]

[591

]59

26

588*

660

55

605

13

P-n

50-k

10

696

269

51,

093

656

265

6*2

676

1067

07

P-n

51-k

10

741

273

615

470

729

698*

272

23

721

10

P-n

55-

k7

568

458

32,

083

549

5154

414

550

8954

211

P-n

55-

k8

588

9[6

24]

[604

]57

212

656

8*4

570

2457

068

2

P-n

55-k

10

694

3[7

18]

[691

]67

018

565

7*8

669

866

951

P-n

55-k

15

989

2[9

45]

[908

]88

94

877*

293

011

923

5

P-n

60-k

10

744

2[7

55]

[727

]71

223

705*

672

626

726

246

P-n

60-k

15

968

2[1

,020

][9

68]

931

1191

6*3

950

994

950

P-n

65-k

10

792

4[8

09]

[780

]76

510

576

1*13

781

1,14

676

611

1

P-n

70-k

10

827

10[8

24]

[794

]78

520

783*

680

938

780

11,

038

P-n

76-

k4

593

28[5

90]

[585

]59

038

459

032

159

038

[595

][5

89]

P-n

76-

k5

627

33[6

21]

[612

]61

697

061

5*1,

462

621

5061

621

9

P-n

101-

k4

681

144

[681

][6

77]

673

852

673

6,34

767

377

673

3,93

2

#op

t.24

1224

2424

23

54

Table 12. Detailed results for our BPC algorithm on RVRPTW instances (C1, R1 and RC1)

Instance

q0×t0

QG ×t0

q0× TG QG × TG

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

C101 827.3 2 981.3 15 848.0 5 994.5 366

C102 827.3 4 975.2* 280 846.2 207 976.6* 3,248

C103 826.3 9 [972.6] [950.8] 838.8 161 [974.5] [952.1]

C104 822.9 13 [953.3] [934.4] [834.0] [831.2] [961.2] [933.7]

C105 827.3 3 981.3 218 848.0 27 [984.2] [977.7]

C106 827.3 3 981.3 45 848.0 132 984.1 765

C107 827.3 3 981.3* 2,423 842.8 18 [982.9] [969.7]

C108 827.3 4 [974.7] [948.9] 842.8 136 [973.9] [949.0]

C109 827.3 5 [966.1] [943.0] 842.8 315 [966.1] [943.0]

R101 1,637.7 2 1,637.7 3 1,692.1 9 1,692.1 19

R102 1,466.6 2 1,466.6 3 1,505.3 17 1,505.3 16

R103 1,208.7 4 1,208.7 6 1,235.3 1,230 1,235.3 1,363

R104 971.5 27 975.8* 165 [999.5] [981.6] [999.5] [982.9]

R105 1,355.3 3 1,355.3 4 [1,391.7] [1,368.9] [1,391.7] [1,365.9]

R106 1,234.6 5 1,234.6 7 [1,265.5] [1,244.0] [1,265.5] [1,244.0]

R107 1,064.6 13 1,064.6 18 [1,081.3] [1,074.4] [1,082.3] [1,071.4]

R108 932.1 41 938.6* 124 [951.8] [934.9] [953.7] [938.0]

R109 1,146.9 13 1,147.2 19 [1,169.2] [1,161.9] [1,169.4] [1,166.0]

R110 1,068.0 12 1,068.0 15 [1,097.0] [1,076.8] [1,097.0] [1,076.7]

R111 1,048.7 33 1,048.7 44 [1,071.0] [1,065.8] [1,070.9] [1,064.5]

R112 948.6 62 950.9* 164 [961.4] [950.3] [961.3] [951.5]

RC101 1,619.8 4 1,619.8 6 [1,674.8] [1,651.4] [1,677.5] [1,651.1]

RC102 1,457.4 20 1,472.7 32 [1,500.9] [1,476.2] [1,512.6] [1,493.5]

RC103 1,258.0 22 1,264.6 32 [1,331.0] [1,265.7] [1,341.3] [1,273.5]

RC104 1,132.3 48 1,156.7* 2,371 [1,154.4] [1,138.6] [1,179.7] [1,155.6]

RC105 1,513.7 16 1,513.7 16 [1,563.0] [1,545.2] [1,565.4] [1,546.2]

RC106 1,372.7 35 1,388.8 69 [1,400.4] [1,377.8] [1,413.0] [1,391.7]

RC107 1,207.8 14 1,236.7 63 [1,244.0] [1,214.8] [1,279.7] [1,242.4]

RC108 1,114.2 51 1,155.0* 228 [1,141.4] [1,119.8] [1,161.2] [1,143.9]

# opt. 29 25 11 6

55

Table 13. Detailed results for our BPC algorithm on RVRPTW instances (C2, R2 and RC2)

Instance

q0×t0

QG ×t0

q0× TG QG × TG

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

Opt

[UB]

t (sec)

[LB]

C201 589.1 9 589.1 10 605.4 488 605.4 557

C202 589.1 15 589.1 20 605.2 1,990 605.2* 1,832

C203 588.7 35 588.7 24 597.7* 1,130 597.7* 1,069

C204 588.1 40 588.1* 43 594.0* 536 594.0* 510

C205 586.4 14 586.4 12 598.9 965 598.9 910

C206 586.0 14 586.0 16 598.9 1,414 598.9 1,688

C207 585.8 20 585.8 17 598.3 1,074 598.3 1,097

C208 585.8 14 585.8 15 598.3 2,369 598.3 1,923

R201 1,143.2 11 1,143.2 12 1,143.2 11 1,143.2 12

R202 1,029.6 80 1,029.6 93 1,032.6 174 1,032.6 204

R203 870.8 87 870.8 70 873.3 113 873.3 120

R204 731.3 195 731.3* 180 731.3* 203 731.3* 219

R205 949.8 87 949.8 86 949.8 406 949.8 365

R206 875.9 137 875.9* 149 875.9* 137 875.9* 149

R207 794.0 96 794.0* 114 794.1* 149 794.1* 151

R208 701.0 3,296 [737.4] [697.9] [741.7] [698.4] [772.2] [698.3]

R209 854.8 129 854.8* 134 858.4* 1,460 858.4* 1,513

R210 900.5 168 900.5* 159 [931.7] [907.6] [913.2] [906.1]

R211 746.7 146 746.7* 149 [799.1] [747.8] [830.2] [747.9]

RC201 1,261.8 8 1,261.8 10 1,263.0 21 1,263.0 21

RC202 1,092.3 21 1,092.3 15 1,095.6 37 1,095.6 29

RC203 923.7 28 923.7 28 933.0 387 933.0 524

RC204 783.5 66 783.5* 67 787.5* 520 787.5* 491

RC205 1,154.0 17 1,154.0 15 1,154.0 36 1,154.0 33

RC206 1,051.1 55 1,051.1 53 1,051.1 92 1,051.1 95

RC207 962.9 83 962.9* 112 963.3* 375 963.3* 341

RC208 776.1 80 776.1* 140 778.4* 558 778.4* 580

# opt. 27 26 24 24

56