Blood Products Inventory Pickup and Delivery Problem under Time

Windows Constraints

Imane Hssini

1

, Nadine Meskens

1

and Fouad Riane

2

1

Louvain School of Management, Catholic University of Louvain, Mons, Belgium

2

Ecole Centrale Casablanca, Casablanca, Morocco

Keywords: Blood Products, Perishable Products, Inventory Routing, Pickup and Delivery, Multi-products, Time

Windows.

Abstract: The inventory pickup and delivery problem with time windows (IPDPTW) addressed in this paper is a

variant of the well known inventory routing problem (IRP). It consists in combining the inventory

management problem and the problem of delivery and collection under the constraints of time window. In

our study, we apply this approach to model a blood products distribution system over a certain horizon. The

objective is to determine for each period of the planning horizon, the quantities of products to deliver and

collect as well as the routing to be performed by each vehicle in order to minimize the total transportation

and storage cost without allowing shortages. We present a brief review of literature related to our problem

and we provide a mathematical model that takes into account the constraint of perishability.

1 INTRODUCTION

Blood is a critical commodity for human race. The

demand for blood products is stochastic while the

supply is irregular. The management in such context

consists on matching supply and demand in an

efficient manner. This task becomes complicated

when one considers the fact that blood products are

perishable. Shortages can cause increase mortality

rates while outdates products are not accepted

(Beliën and Forcé, 2012).

An efficient manner to handle the problem of

concern is to integrate the all supply chain processes.

In this context, the Inventory Routing Problem (IRP)

method can achieve this purpose by managing

simultaneously the distribution and the storage.

Indeed, the IRP objective is to determine the optimal

distribution circuit according to the topology

adopted: from a central warehouse to a set of clients

("one to many" typology) or to a single customer

("one to one" typology), or from several warehouses

to a single client ("many to one"), or from several

warehouses to several customers (" many to many").

This circuit (or these circuits) must jointly optimize

the costs of transportation and storage without

causing any shortage. It thus consists to answer three

questions: when each customer should be supplied?

How much should we deliver to each customer

every visit? And which route(s) should be used?

The IRP problem has been addressed by different

authors in several studies. We cite among them the

work of (Coelho et al., 2014) and (Bertazzi and

Speranza, 2012). They have featured very interesting

literature reviews where they classified the IRP

according to several variants such as:

The nature of demand: deterministic or

stochastic.

The planning horizon: Finite or infinite.

The size of the fleet: A single vehicle, multiple

or unconstrained.

The nature of the fleet: Homogeneous or

Heterogeneous.

The routing: direct or multiple.

The inventory decision: nonnegative, back-

orders or lost sales.

The number and the type of products: one or

multi-products, homogeneous or heterogeneous.

Periodicity: one or multi-periods.

In addition to the variants mentioned above, the

formulation of an IRP problem requires the

definition of an objective function. Generally the

aim of this function is to minimize simultaneously

the inventory and the transportation costs. Since the

IRP problem is an extension of vehicle routing

problem (VRP), it is considered as NP-hard problem

(Coelho et al., 2014).

Hssini, I., Meskens, N. and Riane, F.

Blood Products Inventory Pickup and Delivery Problem under Time Windows Constraints.

DOI: 10.5220/0005705503490356

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 349-356

ISBN: 978-989-758-171-7

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

349

In our research, we aim at optimizing the supply

chain of blood products by ensuring the availability

of the right product in the right quantity at the right

time while reducing the inventory and transport

costs. This study will focus on the Belgian blood

supply chain.

For the French-speaking part of Belgium, the

management of the supply chain of blood is at 95%

guaranteed by the Red Cross (Croix Rouge, 2012).

This ensures the smooth running of the blood

collection process from blood donation until the

delivery of various blood products to hospitals,

passing through storage.

The blood service of the Red Cross is responsible

for the organization and the planning of blood

collection which is done in fixed collection sites and

also through mobile collection ‘bloodmobiles’.

Every day, the Red Cross distribution service

organizes shuttles to pick up the blood bags

collected at blood centers and route them to the Red

Cross blood central service (central and unique

geographical location) where they will be

decomposed, qualified biologically and then stored

in the form of three perishable products called blood

products (plasma, red cells, platelets) before being

distributed. Theses shuttles carry also the bags of

plasma and platelets collected by Apheresis in order

to be qualified biologically and then stored before

being distributed. The Apheresis is a blood sampling

technique for taking a single blood component

(plasma, platelets, red blood cells) using a cell

separator

Thus, during an apheresis donation, the

required component is collected in a sample bag, and

other components are returned to the donor.

Due to the heterogeneity of these products in

terms of shelf life and storage temperature, the

conditions of storage and distribution must be

appropriate to each product. The blood service uses

the same vehicle (with limited capacity) pick up the

collected products (Whole blood, non qualified

plasma and platelets) from the blood centers and at

the same time to deliver the blood products to

hospitals (figure 1).

The hospitals served by the Red Cross are

divided into four regions (clusters), each hospital has

a time window during which it should be visited and

has a frequency of visits that is based on its

consumption.

The plasma is transported separately from other

products, as it is distributed in a frozen state and

under a temperature of -25 ° C. Therefore, a specific

vehicle should be used to transport this product. Our

study will then cover only the distribution of

platelets and red blood cells.

Figure 1: The route for Pickup and Delivery of blood

products.

The stochastic nature of the demand, either in the

number of patients who need blood products or in

number of blood units required by each patient, may

imply shortages which are not authorized because

blood products are vital.

To improve the performance of blood products

distribution process, it is necessary to optimize both

the inventory management problem and the vehicle

routing problem with pickup and delivery and time

windows (VRPPDTW). This problem is referred to

as an inventory pickup and delivery problem with

time windows (IPDPTW).

Furthermore, in the VRPPDTW, the vehicles

must ensure during the same tour a dual service:

pickup and delivery of products to customers during

specific periods of time called time windows. This

temporal constraint can be divided into two types:

tight time windows and wide time windows.

In the case of the tight time windows, the

customer cannot be visited outside the time window

i.e. if a vehicle arrives to customer location before

the start of the time window, then it must wait until

the beginning of the time window to deliver, and if it

arrives late (after the end of the time window), it

cannot serve the customer. While in the case of the

wide windows, the time windows cannot be

respected: the customer can be visited outside the

time window; however, the penalties can be applied

to the supplier. In this paper, we study the IPDPTW

problem as part of a two stage blood product supply

chain (a single supplier with multiple hospitals and

blood centers). In reality, the demand of blood

products is stochastic; however, to simplify our

model we assume that the demand is deterministic.

Referring to the classification of the IRP problem

presented by (Coelho et al., 2014), the problem of

concern is a deterministic, multi-period, multi-

product, multi-vehicle, multi-routing problem over

finite planning horizon.

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

350

The objective function is the minimization of the

inventory and the transportation costs. This paper is

organized as follows. We start reviewing related

studies encountered in the literature. We then

suggest a mathematical formulation for the problem

that we solved to optimality for small size instances.

We finally analyze the relevant computational results.

2 RELATED LITERATURE

During the last two decades, the literature on the IRP

problem has increased exponentially. Despite the

abundant literature on the IRP problem, we did not

find any papers that address the problem of

Inventory Pickup and Delivery Problem with Time

Windows for Perishable Products (IPDPTWPP). To

this end, we have introduced in this state of the art

the articles dealing the IRP problem applied to

perishable products since blood products share this

relevant characteristic.

Indeed, the IRP problem researches related to

perishable products are very rare. Of these, we

include those performed by (Federgruen et al.,

1986) ; (Coelho and Laporte, 2014) ; (Jia et al.,

2014) ; (Chen and Lin, 2009) ; (Hemmelmayr et al.,

2010) ; (Hemmelmayr et al., 2009); (Rusdiansyah

and Tsao, 2005) ; (Zanoni and Zavanella, 2007) ;

(Le et al., 2013) ; (Niakan and Rahimi, 2015) ;

(Mirzaei and Seifi, 2015) ; (Diabat et al., 2014);

(Soysal et al., 2015); (Al Shamsi et al., 2014);

(Kande et al., 2014).

Among the researchers interested in the IRP

problem with a single perishable product we quote:

(Coelho and Laporte, 2014) ; (Hemmelmayr et al.,

2009); (Rusdiansyah and Tsao, 2005) ; (Diabat et al.,

2014); (Soysal et al., 2015); (Jia et al., 2014). Each

of them analyzed the problem in a different way

according to the used variants (number and nature of

product, nature of demand, size of fleet...) or the

proposed solution. For example, (Rusdiansyah and

Tsao, 2005) ; (Diabat et al., 2014); (Jia et al.,

2014); (Mirzaei and Seifi, 2015) have studied the

IRP problem in the case of a deterministic demand,

others such as (Soysal et al., 2015); (Coelho and

Laporte, 2014) are interested in the case of a

stochastic demand. In these articles, the authors

have used several homogeneous vehicles to transport

products; they targeted the minimization of costs

(transport, storage and shortages) as the main

objective function. In the case of multiple perishable

products, (Al Shamsi et al., 2014); (Zanoni and

Zavanella, 2007) ; (Le et al., 2013) studied the IRP

with deterministic demand. In addition to the

multiplicity of products, the stochasticity of demand

was considered by researchers as (Federgruen et al.,

1986) ; (Chen and Lin, 2009); (Hemmelmayr et al.,

2010) and (Niakan and Rahimi, 2015).

Blood products have been the subject of some

works such as those of (Federgruen et al., 1986) ;

(Hemmelmayr et al., 2010) ; (Hemmelmayr et al.,

2009). Indeed, (Hemmelmayr et al., 2009) studied

the impact of the adoption of the VMI policy by the

Austrian Red Cross blood service. The objective of

this study was to minimize the distribution cost.

They have developed a flexible vehicle routing

system to deliver one blood product with

deterministic demand. This study has been extended

by (Hemmelmayr et al., 2010) to cover the case of

several blood products with a stochastic demand.

Certainly the optimization of inventory

management and distribution of blood products is

complex given the perishable nature of these

products and their heterogeneity. In fact, each blood

product has its own shelf life, storage temperature,

temperature to be maintained during transport,

storage conditions: in the case of blood platelets

(must be stored under continuous agitation), so each

product should be stored separately from other

products and transported separately in insulated

containers. All these constraints further complicate

the management of the distribution process and

storage of blood products.

According to this state of art we noticed that the

articles on the IRP problem applied to perishable

products are few in number especially in the case of

blood products (three papers), and, the only paper

that has searched the almost similar characteristics to

ours is that of (Niakan and Rahimi, 2015). However,

our research is distinguished by taking into account

constraints such as the respect time windows and

treatment of problems of delivery and pickup at the

same time. To our knowledge no research has

focused on optimizing inventory management

problems of perishable products and vehicle routing

problems with pickup and delivery with time

window simultaneously. In the following, we

present the mathematical model.

3 PROBLEM DESCRIPTION

In our problem, a single warehouse receives from

the laboratory during each period three

heterogeneous perishable products (red cells, plasma

and platelets) (indexed ∈ each with a quantity

N

rt

. Each product has an age a

rt

at period t. In this

study, we will be limited to the case of distribution

Blood Products Inventory Pickup and Delivery Problem under Time Windows Constraints

351

of platelets and red blood cells seen as plasma are

distributed separately from the other blood

components. The warehouse uses a fleet of

heterogeneous vehicles V (indexed byv ∈ V), with

a charge capacity Cap

v

of insulated containers K and

a travel cost per km of CT

v

to distribute these

products to a set of geographically dispersed

hospitals denoted H (indexed by i or j Є H) and at

the same time to collect the bags of whole blood and

plasma and platelets (indexed r’ Є P’) collected in

blood centers and which are not yet qualified

biologically. The quantity of qualified products is

Z

r’t

. In the following we present the parameters, the

sets and the decision variables that are used:

Table 1: The used sets.

H

The set of hospitals

H

+

The set of hospitals plus the warehouse (0 for

warehouse, 1 and higher for hospitals).

B

c

The set of blood centers

HB

c

The set of blood centers plus hospitals (= H+B

c

)

HB

c

+

The set HB

c

plus the warehouse (= H

+

+ B

c

)

P

The set of blood products (delivered products)

P’

The set of collected products

Table 2: The used parameters.

c

Consumption rate of each product r whose age a

rt

(at

period t) at each hospital during each period time t

C

ir

The capacity of storage at each hospital i for each

product r

Cc

r’

The capacity of storage at the warehouse for each

collected product collected r’

I

The inventory level of the product r whose age a

rt

(at pe-

riod t) at each location j Є H

+

by the beginning of period t

IC

0

r’t

The initial inventory of collected products r’ at the

warehouse at each period

N

rt

The quantity produced of products r at period t

Z

r’t

The quantity of qualified products r’ at period t

Ic

r’t

The final inventory of collected product r’ at the

warehouse in period t

Cap

v

The capacity of vehicle v

a

rt

The age of product r at period t (in days)

s

lr

The shelf life of product r (in days)

k

r

The insulated container of delivered product r

k

r’

The insulated container of collected product r’

Dist

ij

The distance between locations i,j Є HB

c

+

(in Km)

φ

ij

The travel time between locations i,j Є HB

c

+

(in hours)

b

i

start of time window for location i Є HB

c

e

i

end of time window for location i Є HB

c

CS

rjt

The holding cost of product r at location j Є H

+

at

period t

CT

v

Travel cost per Km

τ

t

Total working hours per driver per each period t

M

A big number

Table 3: The decision variables.

Q

The quantity of product r whose age a

rt

(at period t)

delivered to hospital i Є H by vehicle v Є V in

period t∈T

CP

The quantity of product r’ collected from blood

center z Є B

c

by vehicle v Є V in period t∈T

O

rjt

The amount of outdated product r at location j Є H

+

during the period t

I

rjt

sl

r

+1

The inventory level of product r whose age sl

r

+1 at

location j Є H

+

by the end of period t

I

The inventory level of product r whose age a

rt

(at

period t) at location j Є H

+

by the end of period t

x

A binary variable set to 1 if location j is visited

immediately after location i by vehicle v at each

period t, 0 otherwise

q

Total quantity of product r whose age a

rt

(at period

t) transported in insulated container k

r

in vehicle v

from a location i∈ HB

c

+

to location j∈ HB

c

+

at

period t

qc

′

Total quantity of product r’ transported in insulated

container k

r’

in vehicle v from a location i∈ HB

c

+

(HB

c

+

= H

+

+ B

c

) to location j∈ HB

c

+

at period t

y

A binary variable set to 1 if vehicle v visits location

i Є HB

c

at period t, 0 otherwise

s

The arriving time of vehicle v at hospital i at period

t

Some assumptions are made in this study:

Each tour must begin and end at the warehouse

and each hospital and blood center must be

served by a single vehicle.

The storage capacity at each hospital cannot be

exceeded.

Given that the products concerned are perishable

products, we assume that each product r is

delivered only if its age a

rt

is less than or equal to

its shelf life sl

r

, thus any outdated product is no

longer appearing in the inventory and it cannot

be used to satisfy the demand.

Also we assume that no stock-out is allowed.

Since each blood product has its own conditions

of conservation, each product is transported

in insulated container k

r

Є K separately from

other products. We assume that all insulated

containers k

r

and k

r’

have the same size.

The model is:

Min CT

v

*Dist

ij

*x

ijt

v

v t

HB

c

+

j=i+1

HB

c

+

-1

i=0

+

CS

rjt

*I

rjt

a

rt

r ta

rt

≤ sl

r

H

j

(1)

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

352

Subject to:

x

ijt

v

i Є HB

c

+

i ≠ j

≤1 ∀ j Є HB

c

+

, v Є V , t Є T

(2)

y

it

v

vЄ V

≤1 ∀ i Є HB

c

, t Є T

(3)

x

ijt

v

i ЄHB

c

+

i ≠ j

- x

jmt

v

m ЄHB

c

+

m ≠ j

=0

∀ j Є HB

c

, v Є V , t Є T

(4)

q

ijrk

r

t

a

rt

v

i ЄH

+

i ≠ j

- q

jmrk

r

t

a

rt

v

=

m ЄH

+

m ≠ j

Q

jr t

a

rt

v

∀jЄH,∀k

ЄK,

∀a

sl

, v Є V , r Є P, t Є T

(5)

qc

ji r'k

r'

t

v

iЄHB

c

i≠j

- qc

mjr'k

r'

t

v

=

mЄHB

c

m≠j

CP

r'jt

v

∀jЄB

,∀

k

Є

K

,

v

ЄV,r

ЄP

,tЄT

(6)

qc

′

′

′

Є

q

Є

Cap

∗x

∀i,jЄHB

,rЄP,r

ЄP

,

v

ЄV,tЄT

(7)

b

∗y

s

e

∗y

∀iЄHB

,

v

ЄV,tЄT

(8)

s

φ

s

M1x

∀i,jЄHB

,ij,

v

ЄV,tЄT

(9)

x

ijt

v

HB

c

+

j

HB

c

+

-1

i

* φ

ij

≤ τ

t

∀ vЄ V ,

t

Є T

(10)

I

rjt

0a

rt

a

r

≤ sl

r

+ Q

jrt

a

rt

v

a

r

≤ sl

r

≤C

jr

∀ jЄ H ,∀ vЄ V,

r

Є P ,

t

Є T

(11)

IC

r't

0

+ CP

r'jt

v

≤Cc

r’

∀

jЄB

,

∀

v

ЄV,r

ЄP

,tЄT

(12)

O

rjt

= I

rjt

sl

r

+1

∀ j Є H

+

,rЄ P , tЄ T

(13)

I

rjt

0a

rt

=I

rjt-1

a

rt-1

- O

rjt-1

∀

j Є H

+

, a

rt

≤ sl

r

,rЄ P , tЄ T

(14)

I

rjt

a

rt

=I

rjt-1

a

rt-1

+ Q

jrt

a

rt

-c

rjt

a

rt

∀

jЄ H,

∀

vЄ V, a

rt

≤ sl

r

, rЄ P ,tЄ T

(15)

I

r0t

a

rt

=I

r0t-1

a

rt-1

+ N

rt

- Q

jrt

a

rt

v

jЄ H

∀

v Є V, a

rt

≤ sl

r

,

r

Є P ,

t

Є T

(16)

Ic

r't

=Ic

r't-1

+ CP

r'jt

v

-Z

r't

∀

j Є B

c

,

∀

v Є V, r'Є P',

t

Є T

(17)

x

ijt

v

HB

c

+

i

= y

jt

v

∀

j Є HB

c

,i ≠ j ,

∀

v Є V,

t

Є T

(18)

y

it

v

, x

ijt

v

Є

0,1

∀ i , j Є HB

c

+

, i ≠ j , ∀ vЄ V, tЄ T

(19)

CP

r'zt

v

; Q

jrt

a

rt

v

; I

rjt

a

rt

; I

rjt

0a

rt

≥ 0

∀ i,jЄ H

+

, i≠j , ∀ vЄ V, ∀a

rt

≤ sl

r

,

r Є P, r

'

Є P

'

, z ЄB

c

, tЄ T

(20)

Objective function of the proposed model is defined

by (1), it includes the total transportation cost and

the total inventory holding cost at the end of each

period. Constraints (2) ensure that each location

(hospital or blood center) is visited at most once in

period t. Constraints (3) define that each location

(hospital or blood center) can be visited by one

vehicle maximum per period. Constraints (4) ensure

the continuity of a tour, so that if a vehicle arrives at

a location (hospital or blood center), it must leave

after it has served it to a next location or to the

warehouse. Constraints (5) determine the quantity of

delivered product r Є P to a hospital and eliminate

sub-tours. Constraints (6) determine the quantity of

collected product r’ Є P’ from a blood center and

also eliminate sub-tours. Constraints (7) ensure that

the quantity transported is less than or equal to the

Blood Products Inventory Pickup and Delivery Problem under Time Windows Constraints

353

vehicle’s capacity. Vehicles arriving time at location

(hospital or blood center) must be in given time

windows for each location (hospital or blood center),

constraints (8) ensure that these time windows are

respected for each location (hospital or blood center)

and constraints (9) define the time arriving.

Constraints (10) ensure that the total travel time of a

vehicle should not exceed the planned total working

hours in period. Constraints (11) ensure the respect

of storage capacity at each hospital. Constraints (12)

ensure the respect of storage capacity of collected

product at the warehouse. Constraints (13) define the

amount of outdated product. Constraints (14) ensure

that the inventory at the beginning of a period is

equal to the inventory at the end of the previous

period minus the quantity of outdated products

during this period. Constraints (15), (16) and (17)

define the inventory conservation conditions for the

warehouse and the hospitals. Constraints (18)

indicate that a vehicle cannot be used to serve any

hospital or blood center unless it is selected.

Constraints (19) and (20) require that inventory

levels at hospitals and warehouse, quantity of

products delivered to hospitals, and quantity of

products collected from blood centers are non-

negative and define the binary nature of decision

variables.

4 COMPUTATIONAL RESULTS

In order to validate the proposed mathematical

model we tested it on small size fictitious instances.

We have randomly generated seven test instances

with the following parameters: one warehouse

delivers two products to a set of hospitals, varying

from 3 to 14 and picks up three products from a set

of blood centers

,

varying from 2 to 4, two

heterogeneous vehicles are used. The capacity of

each vehicle is [Cap

v1

=60, Cap

v2

=140]; Hospital

consumption by product is integer value randomly

generated within the interval [1, 90]. We consider a

planning horizon of three days (T=3). The travel

distance between locations i and j is given in

kilometer. The inventory level at location i Є H

+

at

the beginning of planning horizon is randomly

generated as an integer between [7, 65]; the

inventory level of each collected product at the

warehouse at the beginning of planning horizon is

randomly generated as an integer between [2, 40];

the maximum working hours is set to 8 hours per

day (b

i

=0 min, e

i

=480 min); The big number in

constraints is set to M=100, the travel cost is 1.2

euros per km and the storage cost at each location

i Є H

+

is [r

1

=20, r

2

=10] in euro. The IPDPTWPP

was tested by the Cplex 12.5 on the Intel(R)

Core(TM) i5-2450M CPU 2.50 GHz with 4 GB

RAM.

The results for the seven test instances are

presented in Table 4. The table indicates the

objective function values, the transportation cost, the

storage cost and the CPU computing time. Given the

small size of the first four instances, we have chosen

to present the delivery routes, the quantities of

products (

1

,

2

) to be delivered and of products

[

1

′

,

2

′

,

3

′

] to be collected during the fifth instance

(figure 2).

t=1

t=2

t=3

Figure 2: Delivery routes in each day for the instance5.

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

354

Table 4: Results for the seven test instances.

Inst Nb of

blood

centers

Nb of

hospitals

Storage

cost

(in €)

Transport

cost

(in €)

Total

cost

(in €)

CPU

time

(in

sec)

1 2 3 9190 1944 11134 2.90

2 2 5 8860 2693 11553 2.37

3 2 7 9080 4490 13570 12.06

4 4 3 9080 2208 11288 3.24

5 4 5 7570 2952 10522 14.26

6 4 7 9470 5054 14524

3

8.41

7 4 14 17470 5611 23081 900

The obtained results show that by increasing the

number of hospitals, the distribution cost increases.

This can be explained by the fact that by increasing

the number of the served hospitals, the distance

traveled during the deliveries increases, and

consequently the related cost also increases. As for

the change in the storage cost, it is considered as a

consequence of the variation in the final level of

stock in the warehouse and in the hospitals. This is

due to the variation in the level of consumption from

one period to another. Also, it should be noted that

for very small size instances, the model presented in

Section 3 can be solved by using Cplex solver in

very small CPU time.

However, the instance 7 cannot be solved in

maximum allowed CPU time of 900 sec. This

implies that the resolution of the presented model is

affected by any change in instance input parameters.

Additionally, CPU time for solving the small

instances shows that the instances of realistic sizes

cannot be solved to optimality in a reasonable time.

5 CONCLUSIONS

In this paper, we have modeled the optimization

problem of the blood products supply chain as an

IPDPTWPP problem with the objective of

determining the quantities to be delivered and

collected and fixing the optimal routes while

respecting the constraints of storage and

transportation related to the perishable nature of the

products as well as the time windows during which

each location must be visited. The studied products

are heterogeneous and perishable, each with its own

characteristics in terms of shelf life and storage

conditions. Hence, the need to respect some

constraints in the storage and distribution such as the

separation of products and delivery in specific

insulated containers. Also, no shortage will be

permitted because of the criticality of these products.

We conducted an experimental analysis of the model

on seven instances of small size. Since the IRP

problem is NP-difficult, we are now looking for a

heuristic approach to solve this problem for real life

instances of realistic sizes. Also we plan include

additional constraints such as the possibility to visit

hospitals more than once per period and the

stochastic nature of the demand.

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