Dynamic Multi-trip Vehicle Routing with Unusual Time-windows for the

Pick-up of Blood Samples and Delivery of Medical Material

Nicolas Zufferey

, Byung Yun Cho

and R´emy Glardon

Geneva School of Economics and Management, GSEM - University of Geneva, Uni-Mail, 1211 Geneva 4, Switzerland

LGPP,

Ecole Polytechnique F´ed´erale Lausanne, Lausanne, Switzerland

Keywords:

Dynamic Vehicle Routing, Diversion, Time-window, Heuristics.

Abstract:

Given a ﬂeet of identical vehicles and a set of n clients to be served from a single depot, the well-known

vehicle routing problem (VRP) consists in serving each client (with a deterministic demand) once with a unique

vehicle, with the aim of minimizing the total traveled distance. In this work, the basic VRP is extended within

a medical environment, leading to MVRP (for medical VRP). Indeed, the depot is typically a laboratory for

blood analysis, and a client is assumed to be a medical location at which blood samples should be picked up by

a vehicle. In order to have efﬁcient tests at the laboratory, at most 90 minutes should elapse between the release

time of the blood sample and the delivery time at the laboratory. In addition, only a proportion of the demand

is known in advance and the travel times depend on the trafﬁc conditions. A ﬂeet of non-identical vehicle is

considered (with different speeds and capacities), and each location has to be visited anytime a blood sample

is available. Finally, medical items should be daily delivered from the laboratory to some medical locations.

A transportation cost function with three components has to be minimized. Solution methods are proposed,

which are able to account for all the speciﬁc features of the problem. The experiments highlight the beneﬁt of

considering diversion opportunities (which consists in diverting a vehicle away from its planned destinations).

1 INTRODUCTION TO THE

PROBLEM

This study is motivated by a real situation encoun-

tered in the city of Geneva (Switzerland). When the

analysis of blood samples is required at the consid-

ered hospital, the samples are sent to an external lab-

oratory (denoted LABO, which cannot be named be-

cause of a non-disclosure agreement). In order to pre-

serve the quality of the samples, it is very important

to deliver them to LABO as soon as possible. More

precisely, no more than 90 minutes should elapse be-

tween the availability of a sample and its delivery to

LABO. The pick-up of the samples at different loca-

tions is ensured by the vehicles managed by LABO.

These vehicles are continuously turning around the

city in order to collect and deliver all blood samples

(Grasas et al., 2014). In contrast with the broad exist-

ing literature on the vehicle routing problem (VRP),

the combination of the following features makes the

considered problem new. It is denoted MVRP (for

medical VRP), for which the planning horizon is a day

(from 8am to 6pm).

• Time-windows: the deliveries (blood sample and

medical items) have to be performed before their

associated deadlines (Ciavotta et al., 2009).

• Non-identical vehicles: two ﬂeet of vehicles are

available, namely cars and scooters. In the con-

sidered city, a scooter is on average slightly faster

than a car, but its capacity is lower.

• Stochastic demand: even if the static requests can

be considered before the beginning of the day,

there are stochastic requests all along the day.

• Dynamic planning: the travel times depend on the

trafﬁc conditions, and diversion (i.e., diverting a

vehicle away from its planned route) is allowed.

• Multi-trip with pick-up/delivery: a location has

to be visited when a blood sample is available.

This leads to the situation where each vehicle is

allowed to come back to depot as many times as

decided.

The objective function f to minimize contains

three types of costs (denoted f

, f

and f

), consid-

ered in a lexicographic order (i.e., a higher level ob-

jective is inﬁnitely more important than a lower level

one). Note that a lexicographic optimization is often

366

Zufferey, N., Cho, B. and Glardon, R.

Dynamic Multi-trip Vehicle Routing with Unusual Time-windows for the Pick-up of Blood Samples and Delivery of Medical Material.

DOI: 10.5220/0005733303660372

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 366-372

ISBN: 978-989-758-171-7

used in practice (Respen et al., pted; Solnon et al.,

2008; Thevenin et al., 2015; Zufferey et al., 2006).

1. Taxi cost (f

): each request which cannot be

served on-time by LABO is treated by a taxi (or

a courier service) at a high cost (involving a ﬁxed

cost as well as a variable component depending

on the traveled time).

2. Car cost (f

), formulated as the number of used

cars. Obviously, if a request is served with a

scooter, the cost is lower than if it is served by

a car.

3. Travel cost (f

), computed as the total travel time

of all the vehicles. This cost is proportional to the

fuel consumption.

Because of the difﬁculty of the problem, heuris-

tic or metaheuristic approaches have to be used (see

(Gendreau and Potvin, 2010; Zufferey, 2012a) for

general references on such solution methods). A main

contribution of this work is the design of efﬁcient so-

lutions methods able to account for all the speciﬁc

components of the MVRP. Relying on (Cho, 2015),

this study is organized as follows. The VRP and some

of its extensions, with the associated literature review,

are discussed in Section 2. The employed method-

ology is presented in Section 3. Experiments are

discussed in 4. They showed the beneﬁt of allow-

ing diversion opportunities in the proposed solution

method. Finally, a conclusion ends up the paper in

Section 5.

2 THE VRP AND SOME

EXTENSIONS

As depicted in (Zufferey,2012a), modern methods for

solving complex optimization problems are often di-

vided into exact methods and metaheuristic methods.

An exact method guarantees that an optimal solution

will be obtained in a ﬁnite amount of time. Among

the exact methods are branch-and-bound, dynamic

programming, Lagrangian relaxation based methods,

and linear and integer programming based methods

(Nemhauser and Wolsey, 1988). However, for a large

number of applications and most real-life optimiza-

tion problems, which are typically NP-hard (Garey

and Johnson, 1979), such methods need a prohibitive

amount of time to ﬁnd an optimal solution. For these

difﬁcult problems, it is preferable to quickly ﬁnd a

satisfying solution. If solution quality is not a dom-

inant concern, then a simple heuristic can be em-

ployed, while if quality occupies a more critical role,

then a more advanced metaheuristic procedure is war-

ranted. There are mainly two classes of metaheuris-

tics: local search and population based methods. The

former type of algorithm works on a single solution

(e.g., descent local search, simulated annealing, tabu

search, and variable neighborhood search), while the

latter makes a population of solutions evolve (e.g., ge-

netic algorithms, scatter search, ant colonies, adap-

tive memory algorithms). At each iteration of a lo-

cal search, a neighbor solution is generated from the

current solution by performing a modiﬁcation on the

current solution, called a move. The reader interested

in a recent book on metaheuristics is referred to (Gen-

dreau and Potvin, 2010).

As exposed in (Zufferey et al., 2015), the VRP

is one of the most popular problems in combinato-

rial optimization because of its obvious applications

in transportation. It consists in designing the route

of each of the k identical vehicles with the aim of

minimizing the total traveled distance f (or the to-

tal cost or the total travel time). All vehicles are ini-

tially in a depot, where each route starts and ends.

Each client v (with a known demand) has to be vis-

ited once by the collection of routes. The problem

is deﬁned in an undirected graph G = (V, E), where

V = {v

,...,v

} is the vertex set and E = {(v

) |

∈ V,i < j} is the edge set. Note that v

is the de-

pot and the other vertices are clients. The following

lexicographical approach is generally used: minimize

k, then the total distance f . The two most well-known

constraints associated with the VRP are: (1) capacity:

each vehicle has a limited capacity Q, thus the de-

mand of each route cannot exceed Q; (2) autonomy:

each vehicle has a limited autonomy A, thus the total

duration of each route cannot exceed A.

For survey papers on the VRP, the reader is re-

ferred to (Cordeau et al., 2005; Cordeau et al., 2002;

Cordeau and Laporte, 2004; Gendreau et al., 2002;

Golden et al., 1998; Laporte and Semet, 2002). Many

algorithms have been developed for the VRP. Among

them, there are some successful classical heuristics

such as Clarke & Wright, Two-matching, Sweep, 1-

Petal and 2-Petal, as tested in (Cordeau et al., 2002).

However, the best performance is achieved by meta-

heuristics (e.g. (Cordeau et al., 2001; Mester and

Braysy, 2007; Nagata and Braysy, 2009; Rochat and

Taillard, 1995; Toth and Vigo, 2003; Vidal et al.,

2014)). Relying on (Zufferey et al., 2015), such com-

petitive metaheuristics are discussed below.

• Adaptive Memory (AM). AM (Rochat and Tail-

lard, 1995) has been proved to be a good algo-

rithm for the VRP and introduces a very innova-

tive approach. At each generation of AM, an off-

spring solution s is built route by route from a cen-

tral memory M (which contains routes), then s is

improved with a local search, and the resulting so-

Dynamic Multi-trip Vehicle Routing with Unusual Time-windows for the Pick-up of Blood Samples and Delivery of Medical Material

367

lution is used to update M (i.e. routes of M are

replaced with routes of s).

• Uniﬁed Tabu Search (UTS). UTS (Cordeau et al.,

2001) has been proved to be a very ﬂexible al-

gorithm (easily adapted to variations of the VRP)

with competitive quality and speed. UTS relies on

a tabu search using an objective function which

dynamically penalizes the constraint violations

(the penalty component is likely to be increased

if the last iterations violate the constraints).

• Granular Tabu Search (GTS). GTS (Toth and Vigo,

2003) has proved to be a very balanced algorithm

in terms of speed and quality. It uses a tabu

search framework and relies on the use of granu-

lar neighborhoods to discard the edges that rarely

would belong to a competitive solution. GTS uses

a granularity threshold which is dynamically ad-

justed.

• Active Guided Evolution Strategies (AGES).

AGES (Mester and Braysy, 2007) has been proved

to be very efﬁcient (it is one of the best VRP

method), with a reasonable speed. AGES is a

combination of several procedures (including lo-

cal search techniques), but an important drawback

is its signiﬁcant number of parameters.

• Edge Assembly-based Memetic Algorithm

(EAMA). EAMA (Nagata and Braysy, 2009)

combines an edge-assembly crossover with

well-known local search procedures.

• Uniﬁed Solution Framework for Multi-Attribute

VRP (USFMA). USFMA (Vidal et al., 2014) is

able to tackle a wide range of VRP variants. Us-

ing a diversity management process, the proposed

method is a hybrid genetic algorithm relying on

problem-independent local search and genetic op-

erators.

Several extensions of the VRP can be found in the

literature (e.g., time-windows, pick-up and delivery,

multi-trips). As indicated in (Lorini et al., 2011; Re-

spen et al., 2014), the recent developments observed

in communication facilities have led to the consid-

eration of dynamic vehicle routing problems where

new customer requests must be inserted in the cur-

rently planned routes as soon as they occur (Gen-

dreau and Potvin, 1998; Psaraftis, 1995). A good sur-

vey about methodologies for solving different types

of dynamic vehicle routing problems can be found

in (Ichoua et al., 2000). The travel times can also

be time-dependent to account for rush hours (Fleis-

chmann et al., 2004; Horn, 2000; Ichoua et al., 2003;

Kaufman and Smith, 1993).

As detailed in (Respen et al., 2014), dynamic ve-

hicle routing, where a part of the information about

the customers to be visited is not known in advance,

is attracting a growing attention from transportation

companies (see, for example, (Gendreau and Potvin,

1998; Psaraftis, 1995)). An interesting survey on

this topic can be found in (Pillac et al., 2013), where

the problem is ﬁrst discussed, applications are re-

viewed, and solution methods are presented. In par-

ticular, tabu search led to impressive results on dif-

ferent dynamic vehicle routing problems. Nowadays,

new possibilities offered by localization devices such

as global positioning systems (GPS) can be exploited

to improve vehicle routing management. In (Potvin

et al., 2006), the authors are interested in a vehi-

cle routing problem with time windows and dynamic

travel times. The travel times include three differ-

ent components: long-term forecasts, such as those

based on long-term trends (time-dependency), short-

term forecasts, where the travel time on a link is mod-

iﬁed with a random uniform value to account for any

new information available when a vehicle is ready to

depart from its current location, and dynamic pertur-

bations, which represent any unforeseen events that

might occur while traveling on a link (e.g., accident

causing sudden congestion). A modiﬁcation to a

planned route is only possible when the vehicle is at a

customer location. That is, a planned route cannot be

reconsidered while a vehicle is traveling on a link. An

extension of this model is proposed in (Lorini et al.,

2011), where the position of each vehicle can be ob-

tained when a vehicle reaches some lateness tolerance

limit or when a new customer request occurs. Based

on this information, the planned route of each vehicle

is reconsidered, including the possibility of diversion

(i.e., redirecting a vehicle en route to its current desti-

nation). The results show that the setting of an appro-

priate lateness tolerance limit can provide substantial

improvements. In line with (Respen et al., 2014), in

the second proposed heuristic of this study, we present

a further extension by assuming that the position of

each vehicle is known at all time, thanks to accurate

GPS devices. This assumption allows the system to

react appropriately.

3 METHODOLOGY

In this section, additional information is ﬁrst given,

and the proposed heuristics are then presented. Be-

cause the planning horizonis a day (from 8am to 6pm)

and the concerned territory is a rather small city, the

autonomy constraint (i.e., the refueling of vehicles)

can be ignored, because fuel can be provided to the

vehicles before or after the planning horizon.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

368

3.1 Additional Assumptions

The graph G is built based on the Geneva network,

which allows to deduce the distance matrix D. Two

types S (for scooters) and C (for cars) of vehicles are

considered. Each vehicle has its own speed coefﬁ-

cient ρ (in km/h), and it is assumed that ρ

= 1.1· ρ

where ρ

(resp. ρ

) is the speed coefﬁcient for scoot-

ers (resp. cars). Indeed, a scooter generally moves

faster than a car within a city environment. The stan-

dard travel time

between two locations i and j is

computed as D

/ρ (where ρ ∈ {ρ

,ρ

}). A coefﬁ-

cient T is used to model the trafﬁc situation: T = 1 in

standard conditions, T > 1 if there is trafﬁc conges-

tion (e.g., beginning and end of the day), and T < 1

if we have low trafﬁc conditions (e.g., second part of

the morning, ﬁrst part of the afternoon). The actual

travel time t

between locations i and j is computed

· T. Based on the situation of the city of Geneva,

two coefﬁcients are considered: low trafﬁc condition

between 10am and 3pm (with T = 0.5) and high traf-

ﬁc condition for the remaining part of the planning

horizon (with T = 1.25).

Two types of request are considered: (1) the blood

samples represent 80% of the demand; (2) freight

(i.e., the delivery of medical material from the depot)

constitutes 20% of the demand. The previous requests

are all static (i.e., their release times are known before

the planning horizon), whereas 40% of the blood re-

quests are dynamic (i.e., they appear randomly during

the day). Each request j is associated with: a unique

location, a volume q

, a release time r

(at the cus-

tomer location), and a deadline d

(at the LABO de-

pot). In other words, a time window [r

] is asso-

ciated with each request j, where in contrast with the

classical VRP literature, d

is not a due date (or time)

at location j, but a deadline at the depot. Note that

formally, a due date can be exceeded but penalized,

whereas a deadline cannot be exceeded. For each re-

quest j, its release time r

is generated based on a

uniform distribution during the whole day (but not

within the 90 last minutes of the day for the blood

request). The deadline d

of any blood sample is as-

sumed to be 90 minutes after its release time (i.e.,

= r

+ 90), whereas the deadline for the freight is

randomly generated with a uniform distribution in in-

terval min(r

+ 60;d

), where d

is the closing time

of LABO (i.e., 6pm).

3.2 Heuristics

Because of the dynamic nature of MVRP, quick

reactions have to be taken during the day. For

this reason, sophisticated metaheuristics cannot be

employed, thus a straightforward but fast solution

method is designed.

The proposed heuristic relies on a ﬁxed ﬂeet F =

(S,C), where S is the set of scooters and C the set

of cars. In a ﬁrst phase, before the beginning of the

planning horizon, a static solution is generated with a

greedy insertion procedure GR, followed by a descent

local search method DLS based on the well-known

CROSS-exchanges (as in (Lorini et al., 2011)). GR

starts from scratch and at each step, it inserts a re-

quest j to a route R (of a scooter or a car) in order

to minimize the augmentation of f

(ties are broken

with f

, and then it tries to balance the request load

over all the vehicles), while satisfying the capacity

and the deadline constraints. If it is not possible to

ﬁnd a feasible insertion, a taxi is used for request j

(i.e., the value of f

augments). Because the static

solution will be signiﬁcantly modiﬁed with the occur-

rence of random events (e.g., dynamic travel times,

stochastic requests), there is no need to use more ad-

vanced methods to generate it.

In order to obtain a full solution, a discrete event

simulator (e.g., (Silver and Zufferey, 2005)) is needed

to generate the random events occurring all along the

day. The used time bucket is a minute. Anytime a

request appears, it is greedily assigned to a route (as

in GR), and DLS is then directly performed. There is

no need to discuss the computing time of the proposed

overall method, because the insertion of a new request

only requires a small fraction of a second.

In the basic version H1 of the heuristic, each in-

sertion can only be performed after the current desti-

nation of each vehicle, whereas in the enhanced ver-

sion H2 of the heuristic, it is allowed to divert a vehi-

cle away from its current destination. Of course, H2

is only possible if there is an information system al-

lowing an efﬁcient communication between the vehi-

cles and the dispatching ofﬁce, as described in (Lorini

et al., 2011). Giving an opportunity to divert a vehi-

cle away from its initially planned route is a signiﬁ-

cant advantage: it gives more ﬂexibility to the plan-

ner. Note that the insertion procedure (involving GR

and DLS) works with expected travel times, which are

different from the actual (i.e., simulated) travel times.

In such a context, the use of diversion actions is more

than relevant.

Let s

⋆

be the best solution encountered during the

search process, and let f

⋆

= ( f

⋆

, f

⋆

, f

⋆

) be its asso-

ciated simulated values. We have now all the ingre-

dients to formulate a generalizable approach in Al-

gorithm 1, where the stopping condition can be the

non-reduction of f

⋆

at the end of the main loop.

Dynamic Multi-trip Vehicle Routing with Unusual Time-windows for the Pick-up of Blood Samples and Delivery of Medical Material

369

Algorithm 1: General solution method for MVRP.

Initialization

1. Choose an initial ﬂeet F = (S,C) of vehicles (e.g.,

the one associated with the involved company).

2. Set f

⋆

= ( f

⋆

, f

⋆

, f

⋆

) = (+∞,+∞, +∞).

While a stopping condition is not met, do

1. Set s

as the empty solution (i.e., it does not con-

tain any request).

2. Before the planning horizon, generate a static so-

lution s

while considering only the known static

requests. For this purpose, GR and DLS are se-

quentially used to insert the requests one by one.

3. Use the discrete event simulator to extend s

as a

full solution. Anytime a request appears, insert it

greedily to the current solution s

(i.e., to a route)

with GR, followed by DLS. In the variant H1 of

the method, each insertion can only be performed

after the current destination of a vehicle, whereas

in the variant H2, diversion is allowed.

4. Update s

⋆

: if f (s

) < f

⋆

(under the lexicographic

approach), set s

⋆

= s

and f

⋆

= f(s

5. Modify the ﬂeet F = (S,C) by augmenting or re-

ducing S or C (but not both) by one unit (it is for-

bidden to consider again an already investigated

ﬂeet).

Return s

⋆

with value f

⋆

= ( f

⋆

, f

⋆

, f

⋆

4 EXPERIMENTS

In addition to the above provided information, the fol-

lowing data is also assumed to be given.

• Instance size: 300 static requests, 200 dynamic

requests, 25 material requests, n = 20 locations.

• The distances (in km) between the medical loca-

tions belong to [5,30].

• LABO ﬂeet: 15 cars and 10 scooters, car capacity

= 900 (liters), scooter capacity = 60 (liters).

• Volume q

(integer) of a blood sample request:

uniformly generated in interval [1,10] (liters).

• Volume q

of a medical material request: uni-

formly generated in set {10,20,30,40,50, 60}

(liters).

• For each location, a blood request j is generated

every t minutes, where t is an integer uniformly

generated in interval [23, 33] (if j is static)

or in interval [40,50] (if j is dynamic).

• The car base speed (i.e., ρ

) is ﬁxed to 17km/h

(based on the provided practical information).

An important issue is to determine the ﬂeet F =

(S,C) of vehicles. One can deduce that if F is too

small (resp. C too large, S too large), f

will be too

high (resp. f

too high, f

too high). Initially, F is

deﬁned as the current situation encountered by LABO

(i.e., 15 cars and 10 scooters). For each ﬂeet F, 10

runs of the heuristic (either H1 or H2) on 10 scenarios

are performed, and average costs (i.e., over 100 exper-

iments) are computed for each component ( f

, f

and

). Then, other ﬂeets are tested by adding a single

vehicle (either a scooter or a car), until bad results

(i.e., solutions with high costs) are obtained. It was

ﬁrst observed that the initial ﬂeet is signiﬁcantly un-

derstaffed. As a goal of LABO consists in assigning at

most 10% of the total number of requests to the taxis,

then a ﬂeet of 18 scooters and 12 cars was found to be

the most efﬁcient. We have performed various sensi-

tivity analysis, which are summarized as follows.

• With the initial ﬂeet of vehicles, if ρ

increases

from 17km/h to 27km/h, the number of taxi re-

quests averagely decreases from 19.5% to 7.3%.

The most sensitive interval for ρ

is [19,22], cor-

responding to taxi requests in [18.8%, 11.2%].

• With 15 cars, the average number of requests as-

signed to taxis decreases from 84 to 62 if the num-

ber of scooters increases from 10 to 24. The most

sensitive interval for the number of scooters is

[10,15], corresponding to taxi requests in [84, 66].

• With 18 scooters, the average number of requests

assigned to taxis decreases from 79 to 62 if the

number of cars increases from 8 to 18. The most

sensitive interval for the number of cars is [8, 12],

corresponding to taxi requests in [79, 64].

We have observed the beneﬁt of the proposed in-

gredients introduced in the heuristics. Firstly, the use

of the CROSS-exchanges (Taillard et al., 1997) leads

to a reduction of the total travel time (i.e., f

) by

roughly 15%. More precisely, the total travel time

averagely decreases from 114.8 hours to 101.9 hours.

Secondly, considering T ∈ {0.5, 1.25} for two distinct

time periods (versus T = 1 for the whole day) leads

to an augmentation of 22% on the number of assigned

requests to F. It means that no attention should be

paid to standard trafﬁc conditions (i.e., with T = 1)

when designing and tuning a solution method, be-

cause standard trafﬁc conditions are far away from

real conditions. Thirdly, the use of diversion oppor-

tunities has an important impact, as H2 is able to

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

370

roughly assign the double of dynamic requests when

compared to H1.

5 CONCLUSION

In this paper, we have introduced MVRP, a new prob-

lem relying on the well-known vehicle routing prob-

lem. In MVRP, blood samples have to be picked-up

(when available) at some medical locations and then

delivered on-time (in order to preserve the quality of

the blood samples) to the depot, which is a labora-

tory denoted LABO. The planning horizon is a day.

Two ﬂeets of vehicles are managed by LABO: cars

and scooters. If LABO is not able to assign a request

to one of its vehicle, it can call an external taxi to

treat the request (but at a higher cost). For LABO, the

involved transportation functions to minimize are the

taxi costs, the number of employed cars, and the total

traveled distance of its vehicles. Because of the dy-

namic nature of the problem (indeed, the demand is

stochastic and the travel times depend on the trafﬁc),

a quick solution method has to be employed.

The performanceof a solution method can be eval-

uated according to several criteria (Zufferey, 2012a).

The most relevant criteria are presented below.

• Quality: value of the obtained results, according

to a given objective function.

• Speed: time needed to get good results.

• Robustness: sensitivity to variations in problem

characteristics and data quality.

• Ease of adaptation of the method to a problem,

because, as mentioned in (Woolsey and Swanson,

1975), ”people would rather live with a problem

they cannot solve than accept a solution they can-

not understand”.

• Possibility to incorporate properties of the prob-

lem. It is admitted that an efﬁcient metaheuris-

tic should incorporate knowledge from the con-

sidered problem (Grefenstette, 1987).

The second heuristic, able to divert away a vehicle

from its current destination, seems to perform well

according to all the above criteria. Future works

might include: the consideration of maintenance con-

straints with an extended planning horizon (Hertz

et al., 2009), the use of other constructive algorithms

with a learning process (Zufferey, 2012b), and the de-

velopment of exact methods (e.g., based on linear pro-

gramming) to benchmark the heuristics on determin-

istic cases.

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