Properties and Bounds for the Single-vehicle Capacitated Routing

Problem with Time-dependent Travel Times and Multiple Trips

T. Adamo

, G. Ghiani

, P. Greco and E. Guerriero

Dipartimento di Ingegneria per l’Innovazione, Universit

a del Salento, Via Monteroni, Lecce, Italy

Keywords:

Time-dependent Routing, Multi-trip, Capacitated Vehicle Routing Problem, Path Ranking Invariance.

Abstract:

This paper deals with a problem where the same vehicle performs several routes to serve a set of customers

and arc traversal times vary along the planning horizon. The relationship with its time-invariant counterpart is

investigated and a procedure to compute lower and upper bounds on the optimal solution value is developed.

Computational results on instances, based on the Paris (France) road graph, show the effectiveness of this

approach.

1 INTRODUCTION

In this work, we consider a variant of the capaci-

tated vehicle routing problem, where the same vehi-

cle can perform several routes during its workday and

the travel times depends on the departure times. This

problem arises in e-groceries, where customers can

order goods through the internet and have them de-

livered at home. In particular in the home delivery

of perishable goods, like foods, routes are of short

duration and must be combined to form a complete

workday. Indeed in urban areas, physical street struc-

tures essentially allow only small-sized vehicles for

delivery. This leads to routes shorter than the work-

day. At the time of writing, this type of problem

is becoming more and more important: Coronavirus

disease (COVID-19) pandemic has rapidly acceler-

ated the shift toward online grocery shopping and the

digitization of stores. The idea is that, once people

will get used to order online their groceries, they will

stick with this habit even after the pandemic. A com-

prehensive review on the variants of multi-trip ve-

hicle routing problem can be found in (Cattaruzza

et al., 2016). In urban areas, taking into account time-

dependencies of travel times helps to capture conges-

tion phenomena and improve route design and logis-

tics costs. Some existing researches have studied ve-

hicle routing problems under time-dependent settings

(Gendreau et al., 2015). To the best of our knowl-

https://orcid.org/0000-0002-9505-5869

https://orcid.org/0000-0002-5243-1799

https://orcid.org/0000-0002-8959-5017

edge, only in (Sun et al., 2018), (Pan et al., 2020)

and (Karoonsoontawong et al., 2020) it has been con-

sidered both time-dependent travel time and multiple

use of vehicles together. In (Sun et al., 2018) the au-

thors report about a tabu search heuristic that can ef-

ﬁciently handle different types of constraints includ-

ing time windows and multiple uses of vehicles. The

authors take into consideration the time-dependent

travel times between different customers in order to

satisfy time windows constraints, and also minimize

the total scheduling time of all vehicles. They adopted

a piece-wise linear travel speed model which leads

to a quadratic travel time function, characterized by

complicated calculations of travel times. In (Pan

et al., 2020) and (Karoonsoontawong et al., 2020)

the time-dependent setting is modeled by the widely

used piecewise linear travel time function paradigm

(Ichoua et al., 2003) only in order to satisfy time win-

dows constraints, whilst arc costs are assumed to be

constant.

In this paper, we investigate some properties of

the time dependent capacitated single-vehicle rout-

ing problem with multiple trips (TD-CSVRPMT).

In particular we investigated the relationship be-

tween TD-CSVRPMT and its time-independent coun-

terpart. We exploit some results recently provided

by (Adamo et al., 2020), where the authors stud-

ied a fundamental property of time-dependent graphs

called path ranking invariance. A time-dependent

graph is path ranking invariant if the ordering of its

paths w.r.t. travel duration is not dependent on the

start travel time. (Adamo et al., 2020) proved that

this property can be exploited to solve a large class

Adamo, T., Ghiani, G., Greco, P. and Guerriero, E.

Properties and Bounds for the Single-vehicle Capacitated Routing Problem with Time-dependent Travel Times and Multiple Trips.

DOI: 10.5220/0010322500820087

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 82-87

ISBN: 978-989-758-485-5

of time-dependent routing problems including the

Time-Dependent Travelling Salesman Problem and

the Time-Dependent Rural Postman Problem. We ex-

tended these results to the TD-CSVRPMT by proving

that the optimal solution of a time-independent ca-

pacitated vehicle routing problem (CVRP) provides

both a lower bound and an upper bound for the orig-

inal TD-CSVRPMT. The paper is organized as fol-

lows. Section 2 introduces the notation used through-

out the paper. Section 3 presents a procedure to com-

pute lower and upper bounds on the optimal solution

value. Section 4 is devoted to computational exper-

iments. Finally some conclusions follow in Section

2 NOTATION AND PROBLEM

DEFINITION

The problem considered is deﬁned on a time-

dependent directed complete graph G := (V ∪

{0},A,τ,q

,Q ) , where V = {1,. .. ,n} is the set of

customers, vertex 0 is the depot and A := {(i, j) : i ∈

V, j ∈ V }

{(0,i) : i ∈ V }

{(i,0) : i ∈ V } is the set of

arcs. We have a single vehicle of capacity Q deliver-

ing goods from the depot to the set of customer nodes

V . The vehicle workday corresponds to a route made

up of a set of R trips, where each trip starts and ends

at the depot (some of these trips might be empty).

We assume, without loss of generality, that the trips

are served in the order 1, 2,. .. ,R. Let denote with

[0,T ] the time interval associated to a single work-

ing day. We denote with τ : A × R

→ R a function

that associates to each arc (i, j) ∈ A and starting time

t ∈ [0,+∞) the traversal time when a vehicle leaves

the vertex i at time t. Without loss of generality we

suppose that the travel time functions are constant in

the long run, that is τ(i, j,t) := τ(i, j,T ) with t ≥ T .

For the sake of notational simplicity, we use τ

i j

(t)

to designate τ(i, j,t). We suppose that traversal time

i j

(t) satisfy the ﬁrst-in-ﬁrst-out (FIFO) property, i.e.,

leaving the vertex i later implies arriving later at ver-

tex j. Each customer i ∈ N is characterized by a de-

mand q

, which is deterministic, known in advance

and cannot be split.

For any given path p

:= (i

,. .. ,i

), the cor-

responding duration z(p

,t) can be computed recur-

sively as:

z(p

,t) := z(p

k−1

,t) + τ

k−1

(z(p

k−1

,t)), (1)

with the initialization z(p

,t) := 0. The TD-

CSVRPMT aims to determine the optimal multi-trip

route on G := (V ∪ {0},A, τ,q

,Q ) used by a single

vehicle, based at the depot, to serve the set of cus-

tomers. Only the capacity restriction for the vehicle

is imposed, and the objective is to minimize the total

travel time needed to serve all the customers when the

vehicle leaves the depot at a time instant t = 0, that is:

min

p∈P

z(p,0).

where P denotes the set of paths feasible for TD-

CSVRPMT. It is worth noting that the time indepen-

dent counterpart of the TD-CSVRPMT is the classi-

cal CVRP. Indeed, in the classical CVRP routes do

not need to correspond to vehicles. In other words,

any feasible solution of the CVRP may be used to

model a real-world situation where a single vehicle

will perform all routes in sequence. Algorithms de-

veloped for the CVRP are not able to consider time-

varying travel times without essential structural mod-

iﬁcations. Nevertheless, we observe that the absence

of time constraints implies that time-varying travel

times have an impact on the ranking of solutions of

the TD-CSVRPMT, but they do not pose any difﬁ-

culty for feasibility check of solutions. In particular,

one can assert that there always exists a time-invariant

(dummy) cost function c : A → R

such that a least

duration route of TD-CSVRPMT is also a least cost

solution of the time-invariant CVRP, deﬁned on the

time-invariant graph G

= (V ∪ {0},A,c,q

,Q ).

Deﬁnition 2.1 (Valid Cost Function). A time-

invariant cost function c : A → R

is valid for the

TD-CSVRPMT deﬁned on G = (V ∪ {0},A,τ, q

,Q ),

if the least duration solution p

∗

= min

p∈P

z(p,0) corre-

sponds to a least cost solution of the time-invariant

CVRP deﬁned on G

= (V ∪ {0},A,c,q

,Q ).

If we are given a cost function valid for an in-

stance of the TD-CSVRPMT deﬁned on a time-

dependent G = (V ∪ {0},A, τ,q

,Q ) , then we can de-

termine the least duration solution p

∗

by exploiting

algorithms developed for CVRP. For this purpose we

introduce a property of time-dependent graphs called

path ranking invariance.

Deﬁnition 2.2 (Path Ranking Invariance). A time-

dependent graph G is path ranking invariant, if the

path dominance rule holds true for any pair of paths

and p

of G, it results that:

z(p

,t) ≥ z(p

,t) ∀t ≥ 0.

Since travel time function are constant in the

long run, if a time-dependent graph G = (V ∪

{0},A, τ,q

,Q ) is path ranking invariant then a valid

cost function is c(i, j) = τ

i j

(T ). In the following sec-

tion we exploit the path ranking invariance property

in order to devise a procedure to compute lower and

upper bounds for the TD-CSVRPMT.

Properties and Bounds for the Single-vehicle Capacitated Routing Problem with Time-dependent Travel Times and Multiple Trips

3 PROPERTIES AND BOUNDS

Given a time-dependent graph G = (V ∪

{0},A, τ,q

,Q ), we deﬁne an auxiliary path ranking

invariant graph G

= (V ∪ {0}, A,τ, q

,Q ) where τ

i j

(t)

is a lower approximation of the original traversal time

i j

(t), that is:

i j

(t) ≤ τ

i j

(t),

with (i, j) ∈ A and t ∈ [0, T ]. We suppose that the

traversal time function is generated by the travel time

model proposed in (Ichoua et al., 2003) (IGP model

for short), in which each arc (i, j) ∈ A is character-

ized by a constant stepwise speed function v

i j

(t) and a

length L

i j

. We suppose that the horizon is partitioned

into H subintervals [T

h+1

] (h = 0,. ..,H − 1), with

= 0 and T

= T . We assume that all arcs of the

auxiliary graph share a common speed function, such

that

i j

(t) = v

with t ∈ [T

h+1

] and h = 0,.. ., H − 1. According

to the IGP model, given a start time t the travel time

value τ

i j

(t) is computed by the following iterative

procedure.

Algorithm 1: Computing the travel time τ

i j

(t).

k ← h : t

≤ t ≤ t

h+1

d ← L

i j

;

← t + d/v

;

while t

> T

k+1

d ← d − v

k+1

−t);

t ← T

k+1

;

← t + d/v

k+1

;

k ← k + 1

return t

−t

In the IGP model the speed of a vehicle is not a

constant over the entire length of arc (i, j) ∈ A but

it changes when the boundary between two consecu-

tive time periods is crossed. The relationship between

the input parameters and the output value of the IGP

model can be expressed in a compact fashion as fol-

lows:

i j

t+τ

i j

(t)

v(µ)dµ. (2)

We denote with z(p

,t) the traversal time of a path p

at time instant t on the time-dependent graph G, that

z(p

,t) = z(p

k−1

,t) + τ

k−1

(z(p

k−1

,t)), (3)

with the initialization z(p

,t) = 0.

Proposition 3.1. The time dependent graph G = (V ∪

{0},A, τ,q

,Q ) is path ranking invariant.

Proof. We observe that from (2) it follows that given

a path p we have that:

∑

(i, j)∈p

i j

t+z(p,t)

v(µ)dµ,

where the notation (i, j) ∈ p means that the arc (i, j) ∈

A is traversed by the path p. This implies that if a path

is shorter that a path p

then p

is also quicker that

for any start time t ∈ [0,T ]:

∑

(i, j)∈p

i j

≤

∑

(i, j)∈p

i j

⇔ z(p

,t) ≤ z(p

,t),

which proves the thesis.

In order to determine the IGP parameters we

follow the two steps procedure proposed in (Adamo

et al., 2020).

Step 1 - Determining the Potential Speed Breakpoints.

Let {t

i jk

,k = 0,...,K

i j

− 1} be the set of breakpoints

of the travel time function τ

i j

(t) and let Γ

i j

(t) be

the arrival time function, i.e. Γ

i j

(t) = t + τ

i j

(t),

with (i, j) ∈ A. In the ﬁrst phase, we determine

a set Ω = {T

,. .. ,T

} of speed breakpoints as

Ω =

(i, j)∈A

Ω

i j

, where each Ω

i j

is an ordered set

determined by means of an iterative procedure

(Algorithm 2) composed of a main while loop in

which each travel time breakpoint t

i jk

is added to Ω

i j

Moreover, for each t

i jk

1. Ω

i j

is iteratively enriched by the arrival time

i j

i jk

) associated to a starting time equal to t

i jk

by the arrival time Γ

i j

(Γ

i j

i jk

)) associated to a

starting time equal to Γ

i j

i jk

), etc, until no speed

breakpoint less than or equal to t

i j,K

i j

−1

can be

generated;

2. ﬁnally, Ω

i j

is iteratively enriched by the starting

time Γ

−1

i j

i jk

) associated to an arrival time equal

to t

i jk

, by the starting time Γ

−1

i j

(Γ

−1

i j

i jk

)) asso-

ciated to an arrival time equal to Γ

−1

i j

i jk

), etc,

until no speed breakpoint greater than or equal to

i j0

= 0 can be generated.

Step 2 - Determining the Speed Levels and the Length

of the IGP Model.

We start by observing that τ is a lower approximation

of τ, if the following relationships holds true for each

arc (i, j) ∈ A and time instant t ∈ [0, T ]:

t+τ

i j

(t)

v(µ)dµ ≥

t+τ

i j

(t)

v(µ)dµ = L

i j

. (4)

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

Algorithm 2 : Determine a set of speed breakpoints Ω

i j

given the set of time breakpoints {t

i j0

,.. . ,t

i jK

i j

−1

Ω

i j

for all t ∈ {t

,. .. ,t

i jK

i j

−1

} do

if t /∈ Ω

i j

then

Ω

i j

← t

while (t

≤ t

i jK

i j

−1

) ∧ (Γ

i j

) /∈ Ω

i j

) do

Ω

i j

← Γ

i j

)

← Γ

i j

)

← t

while (t

≥ Γ

i j

) ∧ (Γ

−1

i j

) /∈ Ω

i j

) do

Ω

i j

← Γ

−1

i j

)

← Γ

−1

i j

)

return Ω

i j

Theorem 3.1. (Adamo et al., 2020) Given two time-

dependent graphs G = (V,A,τ) and G = (V,A,τ), if

the relationships (4) holds true for any arc (i, j) ∈ A

and time instant t ∈ Ω, then the traversal time func-

tion τ is a lower approximation of the original travel

time function τ.

Let a

i jkh

= min(T

h+1

− T

,max(0, Γ

i j

i jk

) − T

))

if k ≤ h, 0 otherwise, with (i, j) ∈ A, h = 0,...,H − 1,

k = 0, .. ., |Ω

i j

| − 1. Since v(t) is a constant stepwise

function the relationship (4) can be expressed by the

following linear equality:

|Ω|−1

∑

h=0

i jkh

× v

− s

i jk

= L

i j

, (5)

where s

i jk

denotes the surplus of the right-hand-

side of (4) with respect to L

i j

, with (i, j) ∈ A, h =

0,. .. ,H − 1, k = 0,. .. ,|Ω

i j

| − 1. We observe that

the maximum ﬁtting deviation between the original

travel time function τ

i j

(t) and its lower approxima-

tion τ

i j

(t)), depends on the quantity

i j

= max

∈Ω

i j

i jk

− min

∈Ω

i j

i jk

with (i, j) ∈ A.

Remark 3.2. If ζ

i j

is equal to zero, then the travel

time function τ

i j

(t) is a perfect ﬁt, for all arcs

(i, j) ∈ A. In this case, the original graph G = (V ∪

{0},A, τ,q

,Q ) is path ranking invariant and the opti-

mal solution of TD-CSVRPMT can be determined by

solving a classical CVRP on G

(V ∪ {0}, A,c, q

,Q )

where c

i j

= τ

i j

(T ), with (i, j) ∈ A .

In all other cases (i.e. ζ

i j

> 0 for some (i, j) ∈ A)

the value of ζ =

∑

(i, j)∈A

i j

represents a measurement

of the distance from this special case.

We determine the auxiliary graph G by determin-

ing the lower approximation that minimize the value

of ζ. For this purpose we formulate the linear pro-

gram (6)-(14), where s

i j

and s

i j

model, respectively,

the minimum and maximum value of the surplus vari-

able s

i jk

, with (i, j) ∈ A and k = 0,.. .,|Ω

i j

|− 1. A so-

lution of such linear programming model represents

the parameters of a constant piecewise function y(t)

and the constant values x

i j

, with (i, j) ∈ A. The con-

tinuous variable y

represents the value of y(t) during

the h −th time interval, that is:

y(t) = y

with t ∈ [t

h+1

] and h = 0,...,|Ω| − 1. The set of

feasible solutions of the linear program (6)-(14) repre-

sents the IGP input parameters for generating a family

of lower approximations of the travel time function τ.

∗

:= min

∑

(i, j)∈A

i j

(6)

s.t.

|Ω|−1

∑

h=0

i jkh

· y

− s

i jk

= x

i j

(7)

, k = 0,...,|Ω

i j

| − 1

(i, j) ∈ A

i j

≥ s

i j

− s

i j

(i, j) ∈ A (8)

i j

≤ s

i jk

k = 0,...,|Ω

i j

| − 1, (i, j) ∈ A (9)

i j

≥ s

i jk

k = 0,...,|Ω

i j

| − 1, (i, j) ∈ A (10)

≥ ρ h = 0, .. ., |Ω| − 1 (11)

i jk

≥ 0 k = 0,...,|Ω

i j

| − 1, (i, j) ∈ A (12)

i j

≥ 0 (i, j) ∈ A (13)

i j

≥ 0 (i, j) ∈ A (14)

The objective function (6) states that the opti-

mization model aims to determine a constant step-

wise function y

∗

(t), such that it is minimized the total

maximum ﬁtting deviation between the original travel

time function τ and its lower approximation τ. Con-

straints (7) state the relationship between y(t), x

i j

and

i jk

at time instant t

i jk

∈ Ω

i j

. Constraints (8) state

the relationship between the objective function and

the range value of ζ

i j

, modeled as the difference be-

tween s

i j

and s

i j

. Constraints (9) and (10) state the

relationship between s

i j

, s

i j

and the continuous vari-

ables s

i jk

. In order to cut off the trivial (pointless)

solution y(t) = 0 for t ≥ 0, constraints (11) state that

the constant stepwise linear function y(t) has to be

Properties and Bounds for the Single-vehicle Capacitated Routing Problem with Time-dependent Travel Times and Multiple Trips

greater or equal than the input parameter ρ > 0. Con-

straints (12), (13) and (14) provide the non-negative

conditions of the remaining decision variables.

Let y

∗

(t) and x

∗

denote, respectively, the step

function and the x’s values associated with the opti-

mal solution of the the linear program (6)-(14). The

lower approximation τ

i j

(t) is generated by the IGP

model with the following input parameters:

v(t) = y

∗

(t), L

i j

= x

∗

i j

with (i, j) ∈ A and t ∈ [0, T ].

Summing up the proposed lower bounding proce-

dure consists of three main steps.

• STEP 1. Solve linear program (6)-(14). Set the

travel speed function v(t) equal to y

∗

(t). Similarly

we set the L

i j

to x

∗

i j

for each (i, j) ∈ A.

• STEP 2. Determine the solution p

∗

as the least

cost solution of the following time-independent

CVRP:

min

p∈P

∑

(i, j)∈p

i j

• STEP 3. Compute the lower bound z

∗

by evaluat-

ing p

∗

w.r.t. τ obtained as output of the IGP model

with input parameters set according the optimal

solution of the linear program (6)-(14) determined

at STEP 1, that is:

∗

= z(p

∗

,0)

We ﬁnally observe that since the path p

∗

belongs

to the set of feasible solutions P, we also generate a

parameterized family of upper bound z obtained by

evaluating p

∗

w.r.t. the original travel time function τ

z := z(p

∗

,0).

4 COMPUTATIONAL RESULTS

The algorithms have been implemented in Java and

run on a Linux machine clocked at 2.8 GHz and

equipped with 16GB of RAM. We used IBM ILOG

CPLEX 12.10 as a black-box solver to ﬁnd the solu-

tion of the linear program (6)-(14), and VRPSolver

from (Pessoa et al., 2020) as exact solver for the

Asymmetric CVRP. We imposed a time limit of 3600

seconds for both stages.

We have generated 7 classes of test instances each

containing 10 individual instances, based on the Paris

(France) road graph (Ghiani et al., 2020), with |V | =

20, 30, 40, 50, 60, 70 and 80 nodes, respectively. We

assigned a demand q

∈ {6, 8,10, 12} (i ∈ V ) to each

customer. Therefore, customers can be partitioned in

a family of 4 subsets sharing the same demand, i.e.

V =

c∈{6,8,10,12}

. The number of daily trip R is

chosen to be 3, 4, or 5. Moreover we set the value of ρ

equal to 1/ min

h=0,...,H−1

h+1

−T

). Table 1 summarizes

customers demands distribution and vehicles capacity

Q according to the number of nodes. Q

∑

i∈V

is the

total demand.

Table 1: Test instances.

|V | |V

| |V

| Q

20 5 5 5 4 168 66

30 8 7 7 7 262 97

40 10 10 10 9 348 126

50 13 12 12 12 442 157

60 15 15 15 14 528 186

70 18 17 17 17 622 217

80 20 20 20 19 708 246

The results are reported in Table 2. The headings

are as follows:

• T IME

: average computing time for the STEP 1

in seconds;

• ζ

∗

: average objective value determined at STEP

• OPT : number of instances solved to optimality in

STEP 2 by VRPSolver out of 10 ;

• T IME

: average computing time for the VRP-

Solver in seconds;

• GAP: average optimality gap

z − z

∗

(%).

Table 2: Computational results.

STEP 1 STEP 2

|V | R T IME

∗

OPT T IME

GAP

5 7.4 0.138 10 1.2 1.19

4 6.7 0.117 10 1.3 1.03

3 5.9 0.112 10 1.3 1.03

5 22.3 0.214 10 1.6 1.93

4 20.7 0.172 10 1.5 1.39

3 20.1 0.148 10 1.8 1.20

5 50.7 0.374 10 10.7 4.87

4 49.8 0.343 10 3.7 4.42

3 48.2 0.307 10 4.6 3.81

5 160.5 0.537 10 7.5 8.33

4 164.6 0.514 10 8.7 8.05

3 147.3 0.499 10 18.5 7.86

5 427.8 0.596 10 52.8 8.90

4 452.6 0.586 10 57.1 8.99

3 363.2 0.576 10 63.8 8.98

5 988.8 0.619 9 340.4 7.91

4 988.8 0.619 10 580.6 8.98

3 988.8 0.619 9 244.2 8.08

5 1749.8 0.628 10 223.6 8.73

4 1749.8 0.628 10 226.8 8.81

3 1749.8 0.628 10 405.5 8.90

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

The procedure does not exceeds the time limits

for 208 out of 210 instances: in particular the algo-

rithm fails to determine a lower bound for two test

cases. Therefore, with reference to columns T IME

and GAP, each row is the average across instances

solved to optimality by VRPSolver. The overall aver-

age time to solve the linear program (6)-(14) (STEP

1) is about 484 seconds with an average ζ

∗

equal

to 0.427, while the overall average time required by

VRPSolver (STEP 2) is about 108 seconds obtaining

an average GAP of 5.88%. We underline that as ζ

∗

grows, the GAP also increases. In particular when ζ

∗

raises up to 0.5, GAP doubles its value.

5 CONCLUSIONS

This paper has introduced a procedure to compute

lower and upper bounds of the optimal solution value

of the time dependent capacitated single-vehicle rout-

ing problem with multiple trips. For the special case

where the graph is path ranking invariant, we have

shown that the upper bound computed in this way pro-

vides an optimal solution. Future work will focus on

embedding the lower and upper bounding procedure

introduced in this paper in an enumerative search al-

gorithm.

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Properties and Bounds for the Single-vehicle Capacitated Routing Problem with Time-dependent Travel Times and Multiple Trips