A multi-tiered vehicle routing problem with global cross ... · A multi-tiered vehicle routing problem with global cross-docking A Smitha,, P Tothb, JH van Vuurena aStellenbosch Unit

A multi-tiered vehicle routing problem withglobal cross-docking

A Smitha,∗, P Tothb, JH van Vuurena

aStellenbosch Unit for Operations Research in Engineering, Department of IndustrialEngineering, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

bDEI, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

Abstract

The class of vehicle routing problems (VRPs) has been documented extensively

since its inception in 1959 with the introduction of the archetypal capacitated

vehicle routing problem (CVRP). Numerous studies have since been dedicated

to the formalisation of different variations on the CVRP that arise in more com-

plex scenarios, as well as to the establishment of suitable solution methodologies

for these variations. A new type of VRP that occurs in the pathology health-

care sector is introduced in this paper which facilitates (i) cross-docking at a

pre-specified subset of customers in the network (a feature referred to as global

cross-docking), (ii) segregation of customers (which are pathological specimen

collection clinics and specimen processing laboratories) into different tiers that

distinguish them in terms of different pathological specimen processing capabil-

ities and storage capacities, and (iii) the possibility of spill-over into subsequent

planning periods of demand for customer visitation. A mixed integer linear pro-

gramming (MILP) model for this VRP is proposed, and tested computationally

in respect of four hypothetical test instances.

Keywords: combinatorial optimisation, vehicle routing problem, integer

programming model, global cross-docking

∗Corresponding authorEmail address: [email protected] (A Smith)

Preprint submitted to Journal of LATEX Templates January 3, 2018

1. Introduction

The class of vehicle routing problems (VRPs) has enjoyed a long and colour-

ful history since its inception in 1959 by Dantzig and Ramser [11], resulting in

numerous variations on the celebrated prototype of this class, the capacitated

vehicle routing problem (CVRP). These variations have typically arisen due to

the need to accommodate practical considerations such as taking into account

operating hours of facilities, adhering to limitations in infrastructure and in-

corporating diversity into the vehicle fleet. This has led to the introduction

into the literature of widely accepted model formulations accommodating these

features, such as local cross-docking (see Wen et al. [43], and Santos et al. [35]),

multi-echelon facilities (see Dondo et al. [16], and Perboli et al. [31]) and trailer

considerations (see Chao [8], Tan et al. [38], and Drexl [17]), to name but a few.

Additional variations of the CVRP are described in Toth and Vigo [40].

In most VRP applications, a characterisation of customers or facilities vis-

ited in terms of different commodity demand needs is not applicable. In this

paper, however, we consider a variation on the VRP with time-windows that

arises in a real-life application related to the collection and delivery of patholog-

ical specimens in the transportation network of a pathology healthcare service

provider. There are different types of specimens that have to be collected from

a set of hospitals and clinics, and processed in potentially different ways at a set

of laboratories within a transportation network. The variation in pathological

specimen type may be due to the nature of the specimens themselves, such as

their purpose and processing requirements, as well as maintaining standards

associated with a specimen, or may even be due to the intended destinations

of the specimens. We segregate the available specimen processing facilities ac-

cording to their respective processing and storage capabilities into a set of tiers.

This tier allocation is nested in the sense that a facility of tier i can process

any type of specimen that can be processed at a facility of tier j if j < i, but

there exist certain commodity types which can be processed at a facility of tier

i that cannot be processed at any facility of a lower tier. Facilities of the lowest

2

tier represent customers (i.e.hospitals and clinics) at which the specimens orig-

inate and have to be collected — these facilities have no specimen processing

or storage capabilities — their only role is that they introduce new specimens

into the system. Facilities of higher tiers (i.e. laboratories) may or may not in-

troduce new specimens into the system, but their distinguishing feature is that

they all offer specimen processing capabilities or intermediate specimen storage

capabilities. All facilities, excluding facilities of the lowest tier, are assumed to

offer the same storage capabilities.

Crucially, we allow for handover of specimens at facilities in the sense that a

specimen requiring processing at a facility of a specific tier may be transported

by one vehicle to a facility of a lower tier than the required one, and then be

collected later by some other vehicle(s) which transport it to a facility of the

required tier. We refer to this type of specimen handover, which may occur at

a facility of any tier (save the lowest and the highest1), as global cross-docking2.

Another novel feature of our VRP variation is that we allow demand for spec-

imen collection to spill-over into a subsequent planning period. We essentially

assume that the time continuum may be partitioned into planning periods of

fixed length. One planning period is considered at a time, and if demand for

specimen collection occurs at a facility after the last vehicle has departed from

that facility, then this specimen is simply collected from the facility during the

following planning period (all demand for specimen collection is assumed to

be known at the beginning of the planning period). Individual specimens are

not tracked as they travel through the system, but they nevertheless all re-

quire collection at their originating customers and transportation to facilities

1Cross-docking of specimens at facilities of the highest tier is not necessary as all specimens

considered in the transportation network can be processed at facilities of the highest tier.

Cross-docking of specimens may also not occur at facilities of the lowest tier as they do not

offer any processing or storage capabilities.2As opposed to the traditional notion of cross-docking in the supply chain literature where

goods are consolidated at a dedicated cross-docking facility [20, 27], referred to here as local

cross-docking.

3

with adequate processing capabilities. This requirement is met by constructing

a model which produces a flow route (perhaps consisting of several individual

vehicle sub-routes) for specimens from any facility (except facilities of the high-

est tier) to a facility of a strictly higher tier, thereby facilitating delivery of

the specimens to facilities of the tiers required, perhaps after repeated global

cross-docking operations.

The requirements of the aforementioned problem conform to suggestions

in the so-called Maputo Declaration [44] to which a large number of countries

are signatories. The declaration suggests that the pathological specimen pro-

cessing facilities of a national health laboratory service should be segregated

into different tiers indicative of their processing capabilities (in terms of both

pathological specimen processing variation and quantity). There are four tiers

of specimen processing laboratories: A tier-one laboratory is typically referred

to as a primary laboratory where only doctors, nurses, and medical assistants

are stationed, whereas a tier-two laboratory additionally has laboratory spe-

cialists and senior technologists available. A tier-three laboratory has staff of

the same qualifications as those at a tier-two laboratory, but additionally has

equipment available to enable it to offer a complete menu of testing blood sam-

ples for HIV/AIDS, tuberculosis and malaria as well as many other diseases at

a much higher throughput. Finally, a tier-four laboratory performs the tasks

of the lower-tiered laboratories, and additionally acts as a reference laboratory

providing linkages with research laboratories, academic institutions and other

public laboratories that can provide assistance in clinical trials, the evaluation

of new technology and surveillance. The clinics where pathological specimens

originate are referred to as laboratories of tier zero as they do not offer any

processing capabilities. In rural settings, the distribution of the specimen pro-

cessing laboratories is such that for pathological specimens to reach a processing

laboratory of the required tier, global cross-docking is a necessity since it would

be impossible for a single vehicle to deliver pathological specimens originat-

ing in such settings over the long distances required to reach a suitable tier of

processing facility in view of legal maximum driving times.

4

The specimen collection and processing system with global cross-docking and

demand spill-over to subsequent planning periods described above is modelled

in this paper as a tri-objective VRP which may form the basis of a decision

support system capable of assisting tiered-facility services in respect of cost-

effective planning, routing and scheduling of a fleet of homogeneous vehicles

dedicated to specimen collection. The mathematical model formulation builds

on a combination of various well-known variants of the celebrated CVRP in the

literature, but exhibits various novel features, as outlined above. An acceptable

trade-off between the three objectives is pursued in the model, namely minimi-

sation of the total time required to transport specimens, minimisation of the

difference between the longest and shortest travel times associated with vehicles

(i.e., balancing of driver workload) and, finally, minimisation of the number of

vehicles required to implement the specimen collection routing schedule.

The paper is organised as follows. Section 2 is devoted to a brief review

of various VRPs from the literature that are related to the problem considered

here. After carefully noting the assumptions underlying our novel VRP in §3,

we proceed to cast the problem as a mixed integer linear programming (MILP)

model in §4, and then validate the model logic in §5 by implementing the model

in the commercial MILP solver CPLEX and applying it to four small, hypo-

thetical problem instances. The paper closes in §6 with a brief summary and

a suggestion with respect to possible follow-up future work. The main goal of

the paper is to introduce a new rich variation of the CVRP, possibly having

real-world applications3 other than pathological specimen transportation while

3Although specifically modelled for a specimen collection and delivery transportation net-

work, a postal service collection and consolidation network may potentially also benefit from

a VRP of the type considered in this paper. In this case, the segregation of facilities may

refer to the extent to which mail sorting takes place in each sorting centre within the system.

There may, for example, be local, provincial, national, and international mail sorting centres

in the system, giving rise to four tiers of mail sorting facilities. Letters destined to be sent

abroad may then conceivably experience repeated global cross-docking operations — first at a

local sorting centre, then at a provincial sorting centre and finally at a national sorting centre

5

also formalising the requirements of the specimen collection and delivery trans-

portation network of a pathology service provider as described above, and to

propose a MILP model taking into account the objectives and constraints of

the problem considered. Instead of focusing on algorithmic and computational

performance aspects associated with solving large instances of the pathological

specimen transportation VRP described above, our main concern in this paper

is to establish a verified MILP model of the problem.

2. Literature review

The problem considered in this paper, which will be described in further

detail in Section 3, belongs to the family of the so-called rich VRPs, since it

represents a real-world generalisation of the classical variations of the CVRP

mentioned in the introduction, and of those briefly reviewed in this section. For

a more extensive review of these problems, see also Irnich et al. [21].

In the basic version of the Pickup and Delivery Problem (PDP), each trans-

portation request consists of the transportation of a commodity between two

locations: one where the commodity is picked up (the origin), and a correspond-

ing location where the commodity is delivered (the destination). It is generally

required that each transportation request is served by a single vehicle, which

first visits the origin and then the destination. The commodities to be trans-

ported may represent goods, people, or any other type of commodity (mail,

parcels, etc.). The PDP for the transportation of goods has been considered,

among many others, in the surveys by Savelsbergh and Sol [36], Desaulniers et

al. [12], and Battarra et al. [3], while the recent survey by Doerner and Salazar-

Gonzalez [15] concerns the transportation of people (this version of the PDP is

also called the Dial-a-Ride Problem (DARP).

The classical Vehicle Routing Problem with Time Windows (VRPTW) is

the extension of the CVRP in which each customer is associated with a time

before finally being consolidated at an international sorting centre.

6

interval (called a time window) and a service time. It is required that servicing

a customer must start within the associated time window, and that the vehicle

must stop at the customer location for a time period equal to the associated

service time. In addition, in case of arrival at the location of a customer before

the beginning of the associated time window, the vehicle is allowed to wait until

servicing may start. The VRPTW has been considered, among others, in the

surveys by Kolen et al. [25], Desrochers et al. [14], Braysy and Gendreau [4, 5],

Kallehauge [24], and Desaulniers et al. [13].

When the available vehicles are homogeneous, but must start and end their

routes at different depots, the corresponding variation of the CVRP is called the

Multi Depot VRP (MDVRP). Although each available vehicle could potentially

have its own specific starting and ending locations, the vehicles are generally

grouped and assigned to a limited number of depots in the classical MDVRP.

The MDVRP was introduced by Renaud et al. [33]. A recent review on the

MDVRP may be found in Vidal et al. [42].

In many variations of the CVRP, a feasible solution is represented by a

set of routes such that each single route satisfies the corresponding intra-route

(or local) constraints, and all the transportation requests are partitioned in an

appropriate manner. In these cases, each route depends on the other routes

only in respect of the partitioning of the transportation requests. There are,

however, also important variations on the CVRP where the feasibility of a so-

lution depends on inter-route (or global) constraints as well, i.e., on how the

routes are related to each other. A typical example is the class of the so-called

inter-route resource constraints, which arises when the vehicles used compete

for globally limited resources (such as a limited number of docks at a depot,

or a limited processing capacity for the commodities arriving at a destination

location). Variations on the CVRP dealing with inter-route constraints have

been considered by, among others, Hempsch and Irnich [19], and Rieck and

Zimmermann [34].

There are variations of the CVRP that consider more than one level of

the distribution network, referred to in the literature as multi-echelon VRPs,

7

with city logistics and multi-modal transportation systems among the most

cited examples of such a network. Two-echelon VRPs, introduced by Jacob-

sen and Madsen [22], consider transportation networks in which the goods are

available from different origins and have to be delivered to the respective des-

tinations moving mandatorily through intermediate facilities. Models, exact

algorithms and metaheuristics for the two-echelon VRP have been proposed in

[16, 31, 30, 1, 6]. See Cuda et al. [10] for a survey on two-echelon routing prob-

lems. The TVRPGC presented in this paper differs from multi-echelon VRPs

in that consolidation at intermediary facilities is not compulsory as pathological

specimens may be delivered directly to capable facilities.

Cross-docking has been applied in industry since the 1980s, however, has

only recently attracted attention from academia with more the 85% of papers

published from 2004 onwards [41]. The two key points of cross-docking are,

typically, simultaneous arrival and consolidation. If all vehicles do not arrive

simultaneously. some vehicles have to wait and therefore the core issue is to

synchronise the arrival of vehicles at cross-docking facilities. The cross-docking

facilities, typically, do not offer any processing or storage capabilities and are

known a priori. Several applications of cross-docking exist in the supply chain

management literature [16, 27, 26]. Models, exact algorithms and metaheuristics

for the VRP with cross-docking have been proposed in [9, 29, 39, 32, 18] and

[28]. Recent reviews of VRPs with cross-docking may be found in Buijs et al. [7]

and Van Belle et al. [41]. The global cross-docking component within the VRP

presented in this paper differs from the typical cross-docking components in

that the cross-docking facilities offer both storage and processing capabilities,

and in addition, the facilities which act as consolidation centres are not known

a priori, but must be decided by considering the objectives and the constraints

of the VRP considered in this paper.

Most of the existing variations on the CVRP involve the optimisation of a

single objective (i.e., minimisation of the global distance travelled by the vehicles

used), or of hierarchical objectives (e.g., first minimisation of the number of

vehicles used, and then minimisation of the global distance). Other variations

8

on the CVRP reside within the realm of multi-objective optimisation, where the

aim is to find an acceptable compromise between the optimisation of several

conflicting objectives (e.g., global distance, completion time, or the balancing

of the routes). See Jozefowiez et al. [23] for a survey on the aforementioned

variations on the CVRP.

3. Model assumptions

In the mathematical model proposed in this paper, certain assumptions are

required in order to arrive at a mathematical description of tiered-facility routing

operations described in §1. These assumptions, introduced in order to simplify

the mathematical model, are, however, still reflective of the real-life operations

of tiered-facility networks in which global cross-docking occurs, as agreed upon

in conjunction with a senior decision maker at a large pathology healthcare

service provider in South Africa [2], and are as follows:

1. The nature of the facilities. The transportation network consists of cus-

tomers, consolidation points, and facilities of varying specimen processing

and storage capabilities, which are collectively referred to as facilities.

Specimens introduced into the network of facilities exhibit varying pro-

cessing requirements, which are in certain cases only satisfiable by some

subset of facilities. Therefore, the facilities are segregated into a collec-

tion of tiers according to the specimen processing capabilities that they

offer, with a higher tier suggestive of superior processing capabilities. The

tiers are ordered in such a manner that the lowest-tier facilities only re-

quire specimen collection, the highest-tier facilities only offer processing

capabilities, and all the other facilities both require specimen collection

and offer processing capabilities as these facilities are all able to process

certain specimens, but may require specimens to be transported to more

capable facilities for processing. As mentioned in §1, the various tier levels

of facilities are assumed to exhibit nested specimen processing capabilities

in the sense that a facility of tier i can perform all the types of specimen

9

processing (and more) than a facility of tier j if i > j. Facilities of the

lowest and highest tiers furthermore do not offer any storage or consol-

idation capabilities. All other tiers of facilities, however, offer the same

storage or consolidation capabilities.

2. The nature of the vehicles. It is assumed that a fleet of homogeneous ve-

hicles is available for specimen collection. The capacities of the vehicles

are assumed to be sufficiently large to handle any demand requirements.

This is usually a realistic assumption in the case of pathological speci-

men transportation, becase these commodities typically exhibit negligible

volume and weight. A capacity constraint may nevertheless easily be in-

cluded in the model formulation, if required. Each vehicle may perform

at most one route.

3. Home depot allocation. It is assumed that each vehicle has a fixed home

depot which may be located at any of the facilities within the network. All

vehicles must begin and end their routes at their respective home depots.

4. Multiple visits and global cross-docking. The lowest-tier facilities must

be visited by exactly one vehicle during the planning period. The other

facilities may be visited by more than one vehicle during the planning

period, although any specific vehicle may visit any facility at most once

during the planning period. In particular, a specimen may be delivered

to a facility by a vehicle, and then later be collected by a different vehicle

for further transportation in the network.

5. Service times. The service time of a facility by a vehicle is limited to the

loading and/or unloading of specimens at the facility and does not include

the processing times of the specimens. The facilities in the transportation

network are not assumed to be operational for twenty four hours a day.

Therefore, there is a need for collection and delivery of specimens by ve-

hicles within certain time windows that reflect the operational hours of

each facility.

6. Rolling demand horizon. It is assumed that demand for specimen col-

lection occurs on a continual basis at all but the highest-tiered facilities,

10

regardless of the time within the planning period. Unmet demand from

the previous planning period may therefore be brought forward to the

current planning period. This allows for a vehicle to deliver specimens

to and collect specimens from the same facility without having to wait at

the facility for all demand to have realised there. Demand for specimen

collection that occurs at a facility after the last vehicle has departed from

the facility may be satisfied during the following planning period.

7. Facility visitation sequence. For feasibility of a route, it is required that

every facility (except the highest-tiered facilities) should be visited by at

least one vehicle that also visits a higher-tier facility at a later stage within

the planning period or should participate with another vehicle in cross-

docking at a consolidation facility such that the specimens of the facility

reach a strictly higher-tiered facility (this allows a facility of a tier different

from the lowest and the highest tiers to be visited by a vehicle that later

visits a facility of the same tier, if this facility is visited by another vehicle

visiting a higher tier facility at a later stage).

8. Specimen destinations. Individual specimen collection and transportation

is not tracked explicitly in the model formulation as numerous types of

specimens may be collected and an even larger number of possible types

of specimen processing may be required by these specimens. The only

constraint is that a specimen should eventually be delivered to a facility

capable of processing it (perhaps over the course of several successive

planning periods).

9. Specimen expiration. The possible deterioration of the quality of a spec-

imen over time is limited to the time it takes for the specimen to be

collected from a facility of the lowest tier and transported to the first

facility that has the appropriate processing, storage or consolidation ca-

pabilities (i.e. specimen deterioration occurs only as a result of being in

transit prior to the first facility of a tier greater than zero). It is there-

fore assumed that once a specimen has been delivered to a facility (of tier

greater than the lowest tier), the specimen is either processed there or

11

stored in such a manner that its expiration window remains unaffected

during storage (i.e. in a vacuum or at a low temperature) or future trans-

portation (i.e. repackaged in such a manner so as to retain the specimen’s

integrity).

4. Mathematical model formulation

This section contains a detailed description of the sets of constraints and

planning objectives required to translate the transportation of pathological spec-

imens within a tiered-facility network, as introduced briefly in §1 and elaborated

upon in §3, into a formal MILP model. After defining the model parameters

and variables in §4.1 and §4.2, respectively, the model objectives are formulated

mathematically in §4.3. The focus then shifts in §4.4 to the formulation of the

model constraints.

4.1. Model parameters

Suppose there are f + 1 different tiers of facilities in the system, and that

each facility tier (save the lowest) is associated with specific specimen processing

capabilities. Suppose, furthermore, that indices are assigned to these facility

tiers in such a manner that a facility of tier d > 1 possesses a superset of the

processing capabilities of a facility of tier e for any e ∈ {1, . . . , d−1}, but that all

facilities of the same tier have identical processing capabilities. As mentioned in

§3, the customers at which specimens originate for collection and the processing

facilities, which may also exhibit demand for specimen collection, are together

referred to as facilities. An indexing convention is, however, followed where all

customers exhibiting no processing capabilities are referred to as facilities of

tier zero, while all processing facilities of tier d ∈ {1, . . . , f} are referred to as

facilities of tier d. Let Fd denote the set of all facilities of tier d ∈ {0, 1, . . . , f},

and define F = ∪fd=0Fd as the set of all the facilities. Any facility in F0 therefore

has no specimen processing capability, but only exhibits demand for specimens

to be collected there. Any facility in Ff , on the other hand, only processes

12

specimens, and exhibits no demand for the collection of such specimens. Finally,

any facility in F \ (F0 ∪ Ff ) may or may not exhibit demand for specimen

collection as a result of cross-docking operations there and also offers certain

processing capabilities. Facility i ∈ F furthermore has an associated vehicle

arrival capacity γi (i.e. a limit on the number of vehicle arrivals the facility can

accommodate during the planning period), a required service time of si time

units and a service time window [ai, gi] during which vehicles have access to the

facility.

Let V represent the set of homogeneous vehicles that constitute the specimen

collection fleet. As mentioned in §3, it is assumed that this set of vehicles is

sufficiently large to facilitate feasible specimen collection routing and scheduling

at a 100% service level. The homogeneity of the fleet implies that all vehicles

have the same autonomy level µ (the maximum allowable route duration of a

vehicle, measured in units of expected travel time) and that any two vehicles

are expected to traverse a given road link in the same amount of time. Denote

the subset of facilities acting as home depots for vehicles by D and denote the

home depot of vehicle k ∈ V within this set by bk. As is customary in the VRP

literature, each home depot bk is associated with a virtual, identical copy of the

depot, denoted by b+k , in order to be able to distinguish between the departure

time of a vehicle from its home depot and the later arrival time of the vehicle

when returning to its home depot. In particular, bk represents the home depot

of vehicle k ∈ V when it departs from the depot, while b+k represents the same

home depot when the vehicle returns to the depot upon completion of its route.

The departure time T ′bkk of vehicle k ∈ V from the depot bk is known a priori.

The set of all specimens that have to be collected is partitioned into f distinct

types, indexed by the set S = {1, . . . , f}, according to the convention that a

specimen of type c ∈ S can be processed at any facility in ∪fd=cFd. Each

specimen of type c ∈ S is assumed to have an associated expiration time τc

which is an upper bound on the time the specimen may be in transit before it

is delivered to a facility in ∪fd=cFd.

Let G = (F , E) be a complete directed weighted graph with vertex set F and

13

arc set E representing all possible road network connections between facilities

in F , where the weight of an arc (i, j) ∈ E is the expected travel time tij of a

vehicle traversing the arc from facility i ∈ F to facility j ∈ F . It is assumed

that the triangle inequality is upheld.

The planning period is limited to a schedule of fixed length, implemented

(possibly in slightly altered form) along a rolling horizon. A subset of facilities

in F \ Ff may perhaps not exhibit demand for specimen collection within the

planning period under consideration, due to demographic variability and fluc-

tuating demand. Let the binary parameter αic therefore assume the value 1 if

specimens of type c ∈ S have to be collected from facility i ∈ F \ Ff , or the

value 0 otherwise.

Finally, let N denote a set of global event numbers associated with the ve-

hicle routing schedule over the planning period. The elements of this set induce

a global ordering of vehicle arrivals over time at the various facilities in the

spirit of Dondo et al. [16] (who considered the special case of local cross-docking

in supply chain management). In their application, the arrival of each vehicle

at a pre-specified local cross-docking facility was associated with a unique in-

teger value in such a manner that a later arrival of any vehicle at the facility

was associated with a larger integer value. These values were employed in a

two-echelon VRP as to reflect real-life distribution problems in which several

vehicles may stop at the same manufacturing site or warehouse to accomplish

pickup or delivery operations. In this kind of application, a vehicle may be visit-

ing a source node several times during the same tour, and product requirements

at some destination may be satisfied through various partial shipments using

more than one vehicle. Therefore, a sequence of operations may be performed

at every location and a vehicle stop is no longer characterised by just the visited

node. Dondo et al. [16] overcame this obstacle by including an ordered set of

event numbers in their model. In our application, we also adopt the practice

of assigning the arrival of each vehicle a unique integer value. Our application,

however, differs from that of Dondo et al. [16] in that we consider the arrival

times of all vehicles at all the facilities in the network as opposed to at a specific

14

cross-docking facility only. The integer values included in the set N are rep-

resentative of the global arrival sequence of vehicles at all destination facilities

of the network. This sequence facilitates monitoring of the global cross-docking

and tier-visitation of vehicles.

4.2. Model variables

In the model formulation, decision and auxiliary variables are required to

keep track of the movement of vehicles and their service allocation to facilities.

In order to facilitate the orchestration of global cross-docking operations, a

global ordering is assigned to the arrivals of all vehicles in the routing schedule,

as described above. The auxiliary variables

ynik =

1, if the arrival of vehicle k ∈ V at facility i ∈ F is global

event n ∈ N during the current planning period,

0, otherwise

achieve this purpose in conjunction with the auxiliary variables

zijkn =

1, if the arrival of vehicle k ∈ V at facility i ∈ F \ (F0 ∪ Ff )

is global event n ∈ N , following which vehicle k also visits

facility j ∈ F` (with j 6= i) at some later stage, where facilities i and j

are of the same tier `,

0, otherwise.

It follows that |N | ≤ |F0|+ |V||F \ F0|. The assignment decision variables

rikn =

1, if global event n ∈ N involves the assignment of vehicle k ∈ V to

visit facility i ∈ F \ Ff and this vehicle later visits a

facility of a higher tier than that of facility i,

0, otherwise

15

are used in a disjunctive fashion to enforce appropriate facility visitation se-

quences. The flow decision variables

xijk =

1, 1 if vehicle k ∈ V travels directly from facility i ∈ F to j ∈ F ,

0, otherwise

monitor the movement of vehicle k ∈ V, while the non-negative, real auxiliary

variables Tik denote the time at which vehicle k ∈ V arrives at facility i ∈ F ,

with Tik assuming the value zero for all i ∈ F if vehicle k is not used. Finally,

consider the conditions

Tjk > Tik + si, i ∈ F0, j ∈ F \ F0, k ∈ V, (1)

Tjk ≤ Tik + si + minc∈S:αic=1

{τc}, i ∈ F0, j ∈ F \ F0, k ∈ V, (2)

and define the binary variables

δijk =

0, If (1)–(2) are imposed for facilities i ∈ F0 and j ∈ F \ F0 when

vehicle k ∈ V visits both facilities i and j, or if i and j are not

visited by the same vehicle k,

1, otherwise,

which are employed to enforce that the specimen expiration time window is

respected.

4.3. Model objectives

Following the discussion in §1, the aim of the model proposed in this paper is

to pursue an acceptable trade-off between the realisation of three objectives. The

first of these objectives is to minimise the expected global travel time4 associated

with the transportation of all specimens from the various original specimen

4The decision not to minimise the distance travelled by vehicles stems from possibly very

rural locations of some of the facilities. The potentially poor quality of roads leading to these

remote facilities in a developing context often brings about considerable deviations in the

expected travel time per unit distance.

16

collection facilities to appropriate facilities where they are to be processed or

stored. This objective may be formulated mathematically as

minimise∑i∈F

∑j∈F

∑k∈V

tijxijk. (3)

The second objective is to balance the workload of the delivery vehicles in terms

of their total service travel times, that is to

minimise maxk∈V

(Tb+k k− T ′bkk). (4)

The final objective is to

minimise∑k∈V

∑j∈F

xbkjk, (5)

which is equivalent to minimising the number of vehicles required for specimen

collection at a service level of 100% by reducing the number of vehicles departing

from their home depots.

4.4. Model constraints

The model includes numerous constraints reflecting the various specimen

transportation requirements outlined in §3. The first such constraint states

that every vehicle must initially depart from and eventually return to its home

depot at the end of its route, as required by Assumption 3 of §3. This constraint

is enforced by requiring that∑j∈F

xbkjk ≤ 1, k ∈ V

and that ∑j∈F

xjb+k k=∑j∈F

xbkjk, k ∈ V.

The constraint set ∑i∈F

xijk ≤∑`∈F

xbk`k, j ∈ F , k ∈ V

17

ensures that any vehicle k ∈ V visits a facility j ∈ F at most once during the

planning period according to Assumption 4. The flow conservation constraint

set ∑i∈F

xijk −∑`∈F

xj`k = 0, j ∈ F \ {bk, b+k }, k ∈ V

states that if any vehicle k ∈ V arrives at facility j, then the same vehicle must

traverse an arc departing from facility j, for all j ∈ F \ {bk, bk+}. Since not all

facilities i ∈ F \ Ff necessarily exhibit demand for specimen collection during

the planning period, the constraint set∑j∈F

∑k∈V

xijk ≥ αi, i ∈ F \ Ff

ensures that at least one vehicle k ∈ V should visit facility i ∈ F \ Ff if there

is actually demand for specimen collection at facility i, where

αi =

1, if∑c∈S αic ≥ 1

0, otherwise.

The constraint set

Tik + si + tij − Tjk ≤ (1− xijk)M, i ∈ F , j ∈ F , k ∈ V

is included to monitor the arrival time of vehicle k ∈ V at each vertex along its

route. This constraint set ensures, if vehicle k ∈ V travels from facility i ∈ F

to facility j ∈ F , that the time instant at which it starts to service facility j

is bounded from below by the time instant at which it started servicing facility

i together with the combined service time duration at facility i and the time

required to travel from facility i to facility j. Here M is a large positive number.

The services provided by the processing facilities are furthermore not typically

twenty four hour operations, but should be rendered within acceptable time

windows associated with each facility according to Assumption 5. Since there

is a possibility that not all vehicles k ∈ V may be used, the constraint set

T ′bkk + tbkj −M(1− xbkjk) ≤ Tjk, j ∈ F , k ∈ V

18

defines the arrival time of vehicle k ∈ V at the first facility j ∈ F visited by

vehicle k, where M is again a large positive number. If vehicle k is not used,

the values of Tik should be equal to zero for all i ∈ F . The constraint set

ai∑j∈F

xjik ≤ Tik ≤ gi∑j∈F

xjik, i ∈ F , k ∈ V

states that vehicle k may not arrive at a facility i ∈ F outside of its associated

time window and enforces the requirement mentioned above that if vehicle k ∈ V

does not visit facility i ∈ F , the value of Tik is equal to zero. The constraint set

Tb+k k− T ′bkk ≤ µ, k ∈ V

ensures that vehicle k ∈ V does not undertake a route that is expected to

take longer to complete than the allowable time autonomy level assigned to the

vehicle. Apart from the multiple problem objectives, an aspect of the novelty

of the VRP formulated here is elucidated in the next constraint sets. Each

specimen of type c ∈ S has a certain time window associated with it during

which the specimen remains viable. As discussed in Assumption 8, the specific

requirements of each individual specimen and its intended purpose is not traced

explicitly. Instead, a more abstract approach is taken by imposing the constraint

sets

Tjk > Tik + si −Mδijk −M

(2−

∑`∈F

xi`k −∑`∈F

x`jk

), i ∈ F0, j ∈ F \ F0,

k ∈ V,

and

Tjk ≤ Tik + si + minc∈S:αic=1

{τc}+Mδijk +M

(2−

∑`∈F

x`ik −∑`∈F

x`jk

),

i ∈ F0, j ∈ F \ F0, k ∈ V,

which require a specimen to be delivered to a facility that is able to process

or store it in such a manner that its integrity is not affected (see Assumption

9). Here M is again a large positive number. The above constraint sets require

an appropriate linking of the variable δijk with the flow variables such that it

19

assumes the value of 0 if facilities i ∈ F0 and j ∈ F \ F0 are not visited by

the same vehicle k ∈ V (while the variable δijk can assume the values 0 or 1 if

facilities i and j are visited by the same vehicle k). This linking is achieved by

imposing the constraint set

δijk ≤

(∑`∈F

x`ik +∑`∈F

x`jk

)/2, i ∈ F0, j ∈ F \ F0, k ∈ V,

in conjunction with the constraint set∑j∈F\F0

δijk ≤∑

j∈F\F0

∑`∈F

x`jk − 1, i ∈ F0, k ∈ V

which ensures that for each facility i ∈ F , conditions (1) and (2)are imposed

for at least one facility j ∈ F \ F0 visited by the same vehicle k ∈ V that also

visits facility i. In addition, the constraint set ensures that each facility i ∈ F0

is visited by exactly one vehicle. Accordingly, we require that∑j∈F

∑k∈V

xijk = 1, i ∈ F0.

Every facility tier has an associated processing capability in respect of speci-

mens, as described in Assumption 2. As the model does not, however, track

individual specimen processing requirements, the more practical approach, de-

scribed in Assumption 8, is adopted, whereby the number of vehicles arriving at

a facility is limited in order to prevent processing bottlenecks. The constraint

set ∑k∈V

∑i∈F

xijk ≤ γj , j ∈ F \ F0

requires that the number of vehicles arriving at facility j ∈ F\F0 should not ex-

ceed the arrival capacity of the facility over the scheduling window. The novelty

of the VRP considered here is further showcased by the remaining constraint

sets, which all contribute to controlling the sequencing of vehicle arrivals at

facilities so as to facilitate global cross-docking. The constraint set∑i∈F

∑k∈V

ynik ≤ 1, n ∈ N

20

ensures that the arrival of each vehicle at every facility i ∈ F is assigned at most

one global event index n ∈ N , with every facility actually exhibiting specimen

collection demand being assigned a unique global event index by prescribing the

constraint set ∑n∈N

∑k∈V

ynik ≥ αi, i ∈ F \ Ff .

It is required that the global event indices assigned to vehicle arrivals should

reflect the order of their arrival sequence in global time. The constraint set

Tj` − Tik ≥M(ynik + ymj` − 2), i, j ∈ F , k, ` ∈ V, m, n ∈ N : m > n

achieves this requirement by ensuring that Tj` ≥ Tik if ynik = 1 and ymj` = 1.

Here M is again a sufficiently large positive number. For every facility i ∈

F \Ff there must be some vehicle k ∈ V visiting a higher-tiered facility at some

time after having visited facility i, as explained in Assumptions 7 and 8. The

disjunctive constraint sets∑k∈V

∑n∈N

rikn = 1, i ∈ F0

and ∑k∈V

∑n∈N

rikn +∑

j∈F`\{i}

zijkn

≥ 1, i ∈ F`, ` ∈ {1, . . . , f − 1}

enforce this requirement. These constraint sets ensure that for each facility i of

tier ` < f there exists a vehicle k ∈ V visiting the facility with a corresponding

event number n ∈ N such that rikn = 1 (indicating that vehicle k later visits

some facility of a tier higher than `) or (if i ∈ F` with ` ∈ {1, . . . , f − 1})

zijkn = 1 for some facility j of tier `, provided that j 6= i (indicating that

vehicle k later visits facility j), in accordance with Assumption 7. The second

disjunctive constraint set may, however, allow for the situation where a specimen

is transported by several vehicles to facilities of the same tier without eventually

reaching a facility of a higher tier. In order to avoid this kind of situation, the

constraint set∑k∈V

∑n∈N

zijkn +∑k∈V

∑n∈N

zjikn ≤ 1, i, j ∈ F`, ` ∈ {1, . . . , f − 1}

21

is introduced. The above constraint set still allows for multiple vehicles to visit

the same facility, but zijkn may only assume the value of 1 for one of the routes

in which the facility is visited by a vehicle that visits a facility of the same tier

at a later stage. The linking constraint set

pynik +∑m∈Nm>n

∑j∈∪f

`=c+1F`

ymjk ≥ (p+ 1)rikn, i ∈ Fc, c ∈ {0, . . . , f − 1},

k ∈ V, n ∈ N

furthermore ensures that the variable rikn may only assume a value of 1 if

vehicle k ∈ V actually visits facility i ∈ Fc and at some later stage also visits

facility j of a tier higher than c, where p denotes the number of vertices in the

transportation graph G. The constraint set∑j∈F

xjik =∑n∈N

ynik, i ∈ F , k ∈ V

ensures that an event n ∈ N cannot be assigned to the arrival of a vehicle

k ∈ V at a facility i ∈ F , unless vehicle k actually visits facility i. The powerful

disjunctive constraint sets above depend on the values of auxiliary variables

rikn. The linking constraint set∑n∈N

rikn ≤∑n∈N

ynik, i ∈ F \ Ff , k ∈ V

enforces the correct assignment of values to these binary variables. The global

cross-docking component of the model allows for facilities of the same tier to have

their specimens consolidated at any facility of that tier within the transportation

network. The constraint set

ynik +∑m∈Nm>n

ymjk ≥ 2zijkn, i, j ∈ F`, ` ∈ {1, . . . , f − 1}, n ∈ N , k ∈ V

ensures that the auxiliary variable zijkn only assumes the value 1 if a vehicle

visits facility i ∈ F` (with ` 6= 0, f) and then at a later time also visits facility

j ∈ F`, allowing for consolidation of specimens of both facilities at facility j, to

be collected by a possibly different vehicle k ∈ V for transportation to a higher-

tiered facility. Finally, the computational burden associated with satisfying

22

the aforementioned constraints may be lowered by introducing the symmetry-

breaking constraint set∑i∈F

∑j∈F

xijk ≥∑i∈F

∑j∈F

xijk+1, k ∈ {1, . . . , |V| − 1}.

This constraint set ensures that the number of facilities visited by vehicle k ∈ V

is not smaller than the number of facilities visited by vehicle k + 1.

5. A worked example

The logic of the model of §4 is verified in this section by implementing it in

a commercially available MILP solver within the context of small, hypothetical

problem instances. A worked example based on a hypothetical instance with

seven facilities is first described in detail. Computational results on three larger

hypothetical instances (with up to ten facilities) are reported later. The aim of

the worked example is not to evaluate experimentally the computational perfor-

mance of the proposed MILP model (which could be substantially improved by

applying effective preprocessing procedures to decrease the number of variables

and constraints while also incorporating more efficient branch-and-bound pro-

tocols), but to show its capability to deal with the novel global cross-docking

properties and the peculiar constraints of the model proposed in §4.

In the first hypothetical instance there are seven facilities of three different

tiers, and so f = 2 in this case. The first of these facilities, listed in Table 1,

is the depot. Facilities 2, 5 and 6 are hospitals or clinics where pathological

samples originate. These collection stations have no blood analysis capabilities,

and so they are classified as facilities of tier zero. Facilities 3 and 4 are hospitals

where blood sample analysis laboratories of tier one are located, while Facility

7 is a tier-two laboratory.

The travel times between these facilities are shown in Table 2, and were

calculated as the corresponding Euclidean distances (rounded up) between the

facilities.

This instance was constructed in a manner to highlight the concept of global

cross-docking and hence some of the model parameters of §4.1 which do not

23

Table 1: Seven facilities in a small, hypothetical test problem instance of a tiered-facility

network.

Facility Number Facility Type X-coordinate Y-coordinate

1 Depot 190 190

2 0 230 210

3 1 220 260

4 1 110 230

5 0 150 270

6 0 50 180

7 2 10 0

Table 2: Travel times (in minutes) between the respective facilities.

Facility 1 2 3 4 5 6 7

1 — 45 77 90 90 141 262

2 45 — 51 122 100 183 305

3 77 51 — 115 71 188 335

4 90 122 115 — 57 79 251

5 90 100 71 57 — 135 305

6 141 183 188 79 135 — 185

7 262 305 335 251 305 185 —

affect cross-docking, such as the imposition of time windows and the adherence

to arrival capacities of facilities, were set to generally unconstraining values,

so as to reduce the complexity of finding an initial feasible solution. Thus,

the values ai = 0, gi = 5 000 (expressed in minutes) were specified for every

facility i ∈ F , together with a specimen expiration time limit of 250 minutes.

A maximum driver autonomy value of 740 minutes was also imposed in order

to prohibit a single vehicle from servicing all the facilities. Finally, the arrival

capacities of facilities were specified as γi = 1 for all facilities i ∈ F0, γj = 2 for

all facilities j ∈ F1 and γ` = 3 for all facilities ` ∈ F2.

24

A complete enumeration of all feasible routes was performed, implemented

in Wolfram’s Mathematica [45], in order to generate the true Pareto front for

the hypothetical problem instance, with a view to validate the logic of the model

in the cases where either k = 2 or k = 3 delivery vehicles are employed. This

enumeration process consisted of seven phases:

Phase 1. A nonempty subset of the set of facilities was selected for visitation

by a delivery vehicle. Since the depot (Facility 1) necessarily has to be

included in the visitation set, this resulted in∑6i=1

(6i

)= 26 − 1 = 63

possible facility visitation subsets for any single vehicle.

Phase 2. The facility visitation subsets identified during Phase 1 were then

combined in order to form an assignment of facilities to be visited by each

vehicle in the fleet. This led to 3 969 (for k = 2) and 250 047 (for k = 3)

facility-to-vehicle assignment alternatives, respectively.

Phase 3. From the set of facility-to-vehicle assignment alternatives, all those

alternatives in which not all facilities are visited, were removed. This

reduced the set of facility-to-vehicle assignment alternatives to a total of

727 (for k = 2) and 115 464 (for k = 3) alternatives, respectively.

Phase 4. All alternatives in which the vehicle arrival capacities at facilities

are exceeded, were removed next. Accordingly, all alternatives in which a

facility of tier 0 appears more than once and all alternatives in which a

facility of tier 1 appears more than twice were removed from consideration.

This led to 214 (for k = 2) and 6 159 (for k = 3) remaining facility-to-

vehicle assignment alternatives, respectively.

Phase 5. The orders in which facilities are visited by each vehicle were taken

into account by permuting (in all possible ways) the non-depot facilities

in each of the facility-to-vehicle visitation sets within the alternatives that

remained after the filtering process of Phase 4, ensuring that the depot

(Facility 1) remains in the first and last position of each permutation. This

25

resulted in 54 288 (for k = 2) and 370 800 (for k = 3) potential vehicle

routing combinations, respectively.

Phase 6. Infeasible vehicle routing combinations were next removed from those

combinations identified in Phase 5. These infeasibilities occurred due to

violations of the requirement that each facility of tiers 0 and 1 must be vis-

ited by a vehicle that visits a strictly higher-tiered facility or participates

in cross-docking such that all pathological specimens are eventually able

to reach a strictly higher-tiered facility. This resulted in 13 104 (for k = 2)

and 72 662 (for k = 3) feasible vehicle routing combinations, respectively.

Phase 7. For each of the vehicle routing combinations that remained after the

filtering process of Phase 6, (1) the total travel time and (2) the maxi-

mum driver autonomy were recorded. All vehicle routing combinations

that were dominated in terms of both these objectives were then filtered

out, and combinations that violated the individual vehicle autonomy spec-

ification (740 minutes per vehicle) were also removed, yielding only three

(for k = 2) and two (for k = 3) Pareto-optimal vehicle routing combina-

tions, as depicted in objective function space in Figure 1.

Although it violates the driver autonomy bound of 740 mins, the objective

function values of the optimal solution single-vehicle TSP are also included for

reference purposes in Figure 1.

The six numbered solutions in Figure 1 are depicted in solution space in

Figure 2. Among these solutions, the concept of global cross-docking is clearly

illustrated in Solutions 2 and 5.

The mathematical model of §4 was also implemented in CPLEX 12.5 (on an

i7-4 770 processor running at 3.40 GHz within a Windows 7 operating system)

in respect of the problem instance described above in an attempt to validate

the logic of the mathematical formulation. In order to accommodate the pur-

suit of trade-offs between minimising the total travel time and balancing the

driver workload in a solution, the number of vehicles utilised was fixed first as

k = 2 and then as k = 3. Since CPLEX 12.5 can only handle single-objective

26

MILPs, we decided to focus our CPLEX search on replicating Solutions 2 and

6, respectively. This allows for single-objective consideration, as the number of

vehicles may be fixed, as described above, after which the non-relevant model

objective may simply be disregarded.

800 900 1000 1100 1200

600

650

700

750

Total travel time (min)

Dri

ver

auto

nom

y(m

in)

Three vehiclesTwo vehiclesOne Vehicle

1

2

3

4

5

6

Figure 1: True Pareto fronts for the hypothetical test problem instance consisting of the seven

facilities of Table 1 in the cases of using one, two and three vehicles, respectively.

Accordingly, the number of vehicles was fixed at two and objective (4) above

was removed from consideration in order to replicate Solution 2. The values of

the non-zero decision variables returned by CPLEX in this case are shown in

Table 3. The facility index 8 in the tables refers to the virtual copy of the depot

(Facility 1).

The total travel time of the two vehicles in Solution 2 is 899.53 minutes,

while the times spent by vehicles 1 and 2 are, respectively, 171.87 and 727.66

minutes, giving a maximum driver autonomy value of 727.66 minutes.

27

Similarly, the number of vehicles was fixed at three and objective (3) above

was removed from consideration in order to replicate Solution 6. The non-zero

decision variables returned by CPLEX 12.5 in this case are shown in Table 4.

The total travel time of the three vehicles in solution 6 is 1222.79 minutes, while

the times spent by vehicles 1, 2 and 3 are, respectively, 468.50, 601.97 and 152.32

minutes, giving a maximum driver autonomy of 601.97 minutes.

Table 3: Non-zero decision variables returned by CPLEX 12.5 for Solution 2 depicted in Figure

2, obtained when the number of vehicles was set to two and model objective (4) was removed

from consideration.

Decision variable Value

xijk x131 = 1 x351 = 1 x541 = 1 x461 = 1 x671 = 1

x781 = 1 x122 = 1 x232 = 1 x382 = 1

ynik y122 = 1 y231 = 1 y332 = 1 y451 = 1 y541 = 1

y661 = 1 y771 = 1

Tik T22 = 45 T31 = 77 T32 = 96 T51 = 148 T82 = 172

T41 = 205 T61 = 284 T71 = 469 T81 = 728

rikn r221 = 1 r312 = 1 r415 = 1 r514 = 1 r616 = 1

δijk δ531 = 1 δ571 = 1 δ631 = 1 δ641 = 1

zijkn z3412 = 1

The solutions represented in Tables 3 and 4 are exactly those depicted in

Figures 2(b) and 2(f), respectively. The computation times required by CPLEX

to reach these solutions are listed in Table 5.

The numerical experiment described above was repeated for eight, nine and

ten facilities, respectively. The data for these three instances are available online

[37]. Objective (4) was removed from consideration and the number of vehicles

was fixed at three for all instances. The driver autonomies were set at 550 min-

utes, 410 minutes and 600 minutes, respectively. The combinatorial explosion

associated with solving these problem instances is elucidated in Table 6. The

CPLEX implementation was allocated a computational budget of 200 000 sec-

28

(a) Solution 1 (b) Solution 2

(c) Solution 3 (d) Solution 4

(e) Solution 5 (f) Solution 6

Figure 2: The numbered solutions reported in objective function space in Figure 1 are depicted

here in solution space.

29

Table 4: Non-zero decision variables returned by CPLEX 12.5 for Solution 6 depicted in Figure

2, obtained when the number of vehicles was set to three and model objective (3) was removed

from consideration.

Decision variable Value

xijk x121 = 1 x231 = 1 x351 = 1 x561 = 1 x641 = 1

x481 = 1 x142 = 1 x472 = 1 x782 = 1 x133 = 1

x383 = 1

ynik y121 = 1 y233 = 1 y342 = 1 y431 = 1 y551 = 1

y661 = 1 y772 = 1 y841 = 1

Tik T21 = 45 T33 = 77 T42 = 90 T31 = 96 T83 = 153

T51 = 167 T61 = 302 T72 = 341 T41 = 381 T81 = 469

T82 = 602

rikn r211 = 1 r515 = 1 r616 = 1 r424 = 1

δijk δ241 = 1 δ531 = 1 δ631 = 1

zijkn z3414 = 1

Table 5: Computational times (expressed in seconds) required by CPLEX 12.5 to generate

the solutions in Figures 2(b) and 2(f) on an i7-4770 processor running at 3.40 GHz with a

working memory limit of 6GB within the Windows 7 operating system.

Solution 2 6

Time to find initial feasible solution 651.35 s 456.25 s

Time to find an optimal solution 2 046.39 s 2 106.25 s

Time to prove optimality 10 755.46 s 17 855.81 s

onds and was not able to prove optimality for the instance with ten customers

within the allotted time.

6. Conclusion

A new type of VRP was introduced in this paper. It is an extension of the

celebrated CVRP in which pathological specimens have to be collected from a

30

Table 6: Computational times (expressed in seconds) required by CPLEX 12.5 to generate

solutions for hypothetical problem instances of eight, nine and ten facilities on an i7-4770

processor running at 3.40 GHz with a working memory limit of 6GB within the Windows 7

operating system and a time limit of 200 000 seconds.

Number of Facilities 8 9 10

Time to find initial feasible solution 457.79 s 2 665.15 s 26 433.48 s

Time to find an optimal solution 2 046.39 s 13 183.58 s 155 813.09 s *

Time to prove optimality 11 192.28 s 118 233.02 s —

* Similarly, a complete enumeration was performed to confirm optimality of the

solution returned

number of facilities and which facilitates global cross-docking (i.e. cross-docking

that can occur at any vertex within a subset of vertices). The model also

provides for the segregation of intermediate facilities into a variety of tiers,

arranged according to unique specimen processing capabilities and allows for

the possibility of the spill-over of unmet demand for specimen collection into a

next planning period. A MILP formulation was proposed for finding the optimal

solution to the problem considered.

It is evident from Tables 5 and 6 that the computational time required to

solve the mathematical model exactly is extremely high. This is to be expected

in view of the model versatility and complexity. This type of complexity clearly

calls for the design of approximate solution methodologies in order to facilitate

application of the model of §4 to real-world problem instances5 of pathology

healthcare service providers within South Africa. The mathematical model pre-

sented in this paper serves the purpose of formalising the constraints of specimen

transportation by a pathology healthcare service provider in accordance with the

Maputo Declaration [44] and to introduce a new rich VRP into the literature.

5There are, for example, 377 facilities within the Western Cape provincial portion of the

transportation network of a large pathology healthcare service provider in South Africa. The

same organisation has more than 6 000 facilities nationwide.

31

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