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International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
International Journal of Industrial Engineering & Production Research (2018)
June 2018, Volume 29, Number 2
pp. 147 -158
http://IJIEPR.iust.ac.ir/
A Selective Covering-Inventory-Routing Problem to the Location of
Bloodmobile to Supply Stochastic Demand of Blood
Elaheh Ghasemi & Mehdi Bashiri*
Elaheh Ghasemi, Faculty of Engineering, Department of Industrial Engineering, Yazd University
Mehdi Bashiri, Faculty of Engineering, Department of Industrial Engineering, Shahed University
KEYWORDS
ABSTRACT
Inventory-Routing;
Covering problem;
Selective Vehicle-
Routing; Mobile Blood
Collection
Supplying of blood and its products is one of the most challenging
issues in the healthcare system since blood is a extremely perishable
and vital good, and blood donation is a voluntary work. In this paper,
a two-stage stochastic selective-covering-inventory-routing (SCIR)
model was proposed to supply whole blood under uncertainty.
Herein, a set of discrete scenarios was used to display uncertainty in
stochastic parameters. Both of the fixed blood center and
bloodmobile facilities were considered in this study. It was supposed
that the number of bloodmobiles was indicated in the first stage
before knowing which scenario occurred. To verify the validation of
the presented SCIR model to supply the whole blood, the impact of
parameters variation was examined on the model outputs and cost
function using the CPLEX solver. In addition, the results of
comparison between the stochastic approach and expected value
approach were discussed.
© 2018 IUST Publication, IJIEPR. Vol. 29, No. 2, All Rights Reserved
1. Introduction1
Whole blood and its products are vital for human
lives while blood is a special product whose supply
and demand is completely random. The blood
supply chain includes the process of collection,
production, storage, and distribution blood and its
products from donor to recipient [1]. Since there is
no alternative to human blood [2], it is critical to
have a seamless process to collect and store this
scarce resource. Collection and inventory
management are two main activities in the blood
supply chain, and several studies have been
conducted in this field over the years. Figure 1
shows the general various scopes of the research
studies regarding supply chain components and
Corresponding author: Mehdi Bashiri
*
Email: bashiri. m@gmail .com
Received 24 August 2017; re vised 25 February 2018; accepted 13 March
2018
decision types (strategic, tactical, and operational),
simultaneously. In most countries around the world,
blood is mostly donated on a voluntary basis;
therefore, the management of blood collection and
blood inventory in the fixed or the mobile blood
centers is highly important for preventing shortage
in emergency situations.
Mainly, blood collection process is done in the
fixed or mobile centers. It is notable that about 80%
of all donations are made in bloodmobiles [3], and
each bloodmobile represents an investment of
$250,000 [4]. Additionally, effective use of
bloodmobiles could be helpful in increasing the
number of donations [5].
DOI: 10.22068/ijiepr.29.2.147
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148 Elaheh Ghasemi & Mehdi Bashiri A Selective Covering-Inventory-Routing Problem to
the Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
Fig. 1. Various scopes of researches in blood supply chain management
Thus, exact planning for the bloodmobile facilities
is particularly important. Determining the right
number of bloodmobiles to operate every day and
strategic deployment of bloodmobiles to various
locations to collect needed blood while minimizing
the travelled distance have been discussed as
important challenges in the literature [6]. In
addition, due to the perishable nature of blood, an
acceptable inventory policy seeks to maximize
demand satisfaction and minimize the amount of
blood units that expire [7].
According to the mentioned reasons, managing
blood collection and blood inventory with
conflicting objectives, especially under stochastic
and dynamic conditions, represent one of the
problematic issues of the blood supply chain. In this
paper, a new two-stage stochastic selective-
covering-inventory-routing (SCIR) model is
presented to supply whole blood in an uncertain
environment and manage the inventory.
(a)
(b)
Fig. 2. Illustrative example of the proposed model
level of blood to reduce wastage and holding
costs. A blood supply network is studied which
includes blood center, bloodmobile facilities,
and blood donation sites. The rest of the paper
is organized as follows: Section 2 presents the
related literature. Section 3 describes problem
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149 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering
-
Inventory
-
Routing Problem to the
Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
description, mathematical model, and
linearization scheme. Computational results of
applying the presented model on the generated
data sets are discussed in section 4. Finally, the
paper is concluded in Section 5.
2. Related Literature
Blood supply chain origin dates back to 1960s
[8]. Figure 3 illustrates the distribution of the
presented papers according to the blood supply
chain stages.
Fig. 3. Publications in blood supply chain
(adopted from [1], [9])
According to the latest review studies, Osorio
et al. [1], and Belien and Force [8], a wide
range of the researches are dedicated to the
collection and storage stages. Figures 4 and 5
indicate the trend of studies in the collection
and storage according to publication date,
respectively. Both Figures reveal the increasing
popularity of this subject after the year 2000.
In the last decade, there are several
comprehensive reviews which analyze the
related paper from various aspects in the blood
supply chain management such as Beliën and
Forcé [9], Lowalekar and Ravichandran [10],
and Osorio et al. [1]. The focus of this section
is the conducted research related to the blood
collection and blood inventory management
problems associated with our study. Although
mobile facilities are widely used for collection
of blood donations in many countries including
Iran, there are very few studies on bloodmobile
operations in the literature. Şahin et al. [11]
developed a location-allocation model for
regionalization of blood services in a
hierarchical network consisting of regional
blood centers, blood stations, and mobile
facilities. Alfonso et al. [12] addressed the
blood collection problem considering both a
fixed site and bloodmobile collection.
Fig. 4. Publications in blood collection
(adopted from [1], [9])
Fig. 5. Publications in blood inventory
management (adopted from [1], [9])
They presented the modeling and simulation of
blood collection systems in France. Blood
collection through mobile facilities has been
taken into consideration in recent years.
Ghandforoush and Sen [13] presented a
nonlinear integer programming model for
platelet production and bloodmobile scheduling
for a regional blood center to meet daily
demand. Fahiminia et al. [14] investigated the
emergency supply of blood in disasters. They
considered a supply network which includes
fixed and mobile blood center, donation sites,
and hospitals with stochastic demands.
Although they determined the inventory
constrains in their models, they did not forecast
the condition where demand exceeded capacity
of fixed and blood center (as considered in this
study). Sahinyazan et al. [5] presented a
selective vehicle routing problem suggested by
Chao et al. [15] with integrated tours. They
optimized the route of bloodmobiles which
collect blood and shuttles which transfer
collected blood by bloodmobiles to the blood
center. They only focused on the optimizing
the route of mobile facility with no
consideration about blood inventory. Based on
Sahinyazan et al. [5] study, Rabbani et al. [16]
0%
10%
20%
30%
40%
Count of publications
0%
10%
20%
30%
40%
Count of publications
(%)
0%
5%
10%
15%
20%
25%
30%
35%
Count of publications
(%)
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150 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering
-
Inventory
-
Routing Problem to
the Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
investigated the mobile blood collection system
for platelet production and optimized the
location of bloodmobiles and their routes in
two separate models. Gunpinar and Centeno [6]
proposed an integer programming approach to
the bloodmobile routing problem. They
considered uncertainty in blood potentials and
applied robust optimization. They also
determined minimum and maximum levels of
blood inventory in each period.
A wide number of papers have reported
important approaches to reducing the cost of
waste and shortage of blood products [7]. The
results of a study conducted in Transfusion
Services at Stanford University Medical Center
indicated that it was possible to reduce the loss
of blood products by 50% if supply chain tools
were implemented [17]. Although most of
integrated researches studied storage stage in
combination with distribution stage (such as
Hemmelmayr et al. [18]; Baesler et al. [7] or
production stage (such as Haijema et al. [19];
Lang and Christian [20]; Rytilä and Spens,
[21]), there is a few study in the field of
collection and storage, simultaneously.
Nahmias [22] studied the inventory ordering
policies for perishables including blood bank
management. A review paper was reported in
Prastacos [23] on the theory and practice of
blood inventory management. Baesler et al. [7]
created a discrete event simulation model to
analyze and propose inventory policies in a
blood center. Gunpinar and Centeno [6]
focused on reducing wastages and shortages of
red blood cells and platelet components of
whole blood units at a hospital.
Table 1 illustrates some of the characteristics of
related studies according to collection,
inventory, and general characteristics. The last
row in table 1 indicates the present study. As
Table 1 shows, most of researches in the
collection stage are focused on the planning of
mobile and fixed blood collection facilities;
however, the combination of inventory
management with a collection stage has been
considered in few studies.
The aim of this study is to present a new SCIR
model to whole blood collection and its
inventory management under uncertainty. To
the best of author’s knowledge, a modeling
effort for blood supply similar to the present
study is non-existent.
This paper contributes to the area of blood
supply in the following ways:
A new selective covering inventory-routing
problem is designed for blood supply for
the first time in the literature.
This study considers that the blood demand
and blood donation in the fixed or mobile
facilities is stochastic.
The inventory level constraints and blood
campaigns are considered when the
inventory levels in blood center are critical
in order to manage the shortage in each
period.
A two-stage stochastic programming
approach is presented for blood SCIR
problem.
3. The Proposed SCIR Model
3-1. Problem description
A new SCIR model is designed for whole
blood collection and its inventory management
in a blood center during the planning horizon,
simultaneously. Periodic stochastic demands
could be supplied through collected blood by
blood center or bloodmobiles. The
bloodmobiles start their tours at the beginning
of the planning horizon from blood centers and
do not need to return to the blood center at the
end of each period. In this study, it is supposed
that a vehicle transfers collected blood by each
bloodmobile to the blood center. The stop
points for bloodmobiles are selected based on
the level of stochastic blood donation in
candidate location and also the amount of
stochastic blood potential in donation sites in
its coverage distance. Each bloodmobile visits
only one location in each period and collects
blood only from donation sites located within
its coverage radius. An illustration of the
proposed model is shown in Figure 2. Figure 2a
depicts the blood center, donation sites, and
potential stops of the bloodmobiles and their
coverage areas; figure 2b shows the covering
tours of three bloodmobiles as an illustrative
example. In the proposed model, a policy is
considered for inventory management of whole
blood under the rules of Iranian.
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151 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering-Inventory-Routing Problem to the
Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
Tab. 1. Summery of the related literature in blood collection and inventory management
Authors
Year
Collection
characteristics
Inventory
characteristics
General characteristics
Collect
ion
facility
Model
structure
Short
age
Invento
ry level
Quality of
information
Stocha
stic
parame
ter
Stochastic
programm
ing
approach
Soluti
on
approa
ch
Fixed
Mobile
Location
Allocation
Routing
Other
Yes
No
Yes
No
Deterministic
Stochastic
Other
Demand
Donation
Other
Two- stage
Other
Exact
heuristic
Şahin et al.
[11]
200
7 * * *
*
* * *
Ghandforo
ush and
Sen [13]
201
0 * *
*
* * * *
Alfonso et
al. [12]
201
2 * * *
*
* * * *
Fahiminia
et al. [14]
201
4 * * *
*
* * * *
* *
Sahinyazan
et al. [5]
201
5 *
* * * *
Gunpinar
and
Centeno [6]
201
6 *
* * * *
*
Rabbani et
al. [16]
201
7 * *
* * * *
*
This study - * * *
*
* * * *
*
* *
Blood Transfusion Organization (IBTO). Based
on these rules, at least three days of safety stock
must be carried, and if the inventory level in each
day is less than five days of demand, a special
campaign for extra donations begins. To avoid
blood spoilage and reduce the cost of inventory
holding, it is supposed that blood center may
transfer the extra blood inventory to the other
blood centers regarding their requirements if and
only if its inventory level is more than seven days
of demand. To the best of our knowledge, such
modeling with these inventory rules
considerations in combination of collection
process has not been considered in the literatures.
Generally, in this paper, the following decisions
are determined:
The stopping location of bloodmobiles among
candidate locations;
The covering tour of bloodmobiles;
The quantity of collected blood by blood
center and bloodmobiles;
The quantity of collected blood by blood
campaign;
The quantity of transferred blood to the other
centers.
3-2. Mathematical model
Let graph G (V, E) be a geographical network
where V is the node of considered network and E
represents roads between nodes. In this study,
0V I L . 0 is the blood center; I= {1, ...,
n} is the set of candidate locations for
bloodmobiles; L= {1, …, m} is the set of
potential donor points that must be covered.
Note that blood from donors is collected in both
of location indexed by {0}, I, and L; however,
bloodmobile facilities could settle only in special
places such as universities or industrial
complexes where indexed by I and donors move
blood center or bloodmobile location based on
their distances to donation (i.e., based on or
coverage distance of each blood collection
facility: fixed or mobile). Assumption, indices,
parameters, and decision variables as well as
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152 Elaheh Ghasemi & Mehdi Bashiri A Selective Covering-Inventory-Routing Problem to
the Location of Bloodmobile to Supply Stochastic . . .
International Journal of I ndustrial Engineering & Production Research, June 2018, Vol. 29, No. 2
mathematical formulation of the proposed model
are described in the following.
Assumptions:
The framework under study considers the
following assumptions:
We use a homogenous fleet of bloodmobiles
and do not consider the required capacity for
them.
The inventory level of blood in the blood
center is checked at the end of each period.
The shortage is not allowed in each period.
The demand level of whole blood and donated
blood (the number of people who are willing
to donate blood) in each period is considered
stochastic.
The operation cost of collecting blood in
bloodmobile and blood center is not
considered in the proposed model.
The indices, parameters, and decision variables to
the mathematical model formulation are describes
as below.
Indices:
I: Set of candidate locations for bloodmobile
indexed by i
L: Set of potential blood donation sites indexed by
l
T: set of time period indexed by t
S: set of scenario indexed by s
Parameters:
h: Inventory holding cost
: Cost of call for extra donation
: Cost of transferring blood unit to the other center
ij
c: Distance between bloodmobile locations i and j
li
r: Distance between bloodmobile location i and donation sites l
s
t
d: Demand of blood center in period t under scenario s
: Percentage of collected units discarded due to a disease detected after testing
s
l
b: Blood potential of blood donation site l under scenario s.
s
j
b: Blood potential of blood donation in bloodmobile location j under scenario s.
: Coverage distance
fb : Fixed cost of establishing a bloodmobile facility
M: A very large number
s
: Probability of scenario s occurrence
Decision variables:
s
t
I: Inventory level of the whole blood at the end of period t under scenario s.
s
ijt
x
: Equals 1 if bloodmobile k be sent to j after i in period t under scenario s; 0 otherwise.
y: Number of bloodmobiles.
s
t
Q: Total unit of received blood to the blood center in period t under scenario s.
s
t
BQ : Total unit of collected blood from donors in the blood center in period t under scenario s.
s
t
CQ : Total unit of received blood to the blood center by bloodmobiles in period t under scenario s.
's
t
Q: Total unit of received blood through campaign of blood extra donation to the blood center in
period t under scenario s.
s
t
RQ : Amount unit of extra blood sent to the other blood centers in period t under scenario s.
's
t
q: Quantity of collected blood through campaign in period t under scenario s.
s
t
q: Quantity of collected blood by bloodmobiles in period t under scenario s.
s
t
qq : Quantity of collected blood units in the blood center in period t under scenario s.
's
lt
p: Equals 1 if the donor located in site l refers to blood center for blood donation in period t under
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153 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering-Inventory-Routing Problem to the
Location of Bloodmobile to Supply Stochastic . . .
International Journal of I ndustrial Engineering & Production Research, June 2018, Vol. 29, No. 2
scenario s.
s
ljt
p
: Equals 1 if the donor located in site l refers to bloodmobile k located in site j for blood donation in
period t under scenario s.
The objective function and constraints for the proposed model are as in (1)-(23):
1 1
0 0
min fb* *( ( * h ' *
* * ))
S T
s s
s t t
s t
n n
s s
t ij ijt
i j
z y I Q
RQ c x
(1)
Subject to:
1 1 1
' ,
s s s s s s
t t t t t t
I I Q Q d RQ s t
(2)
' (1 ) ' ,
s s
t t
Q q s t
(3)
' max(0,5 ) ,
s s s
t t t
q d I s t (4)
,
s s s
t t t
Q CQ BQ s t (5)
(1 ) ,
s s
t t
CQ qq s t
(6)
1
* ' ,
l
m
s s s
t lt
l
qq b p s t
(7)
(1 ) ,
s s
t t
BQ q s t
(8)
1 0 1
( * * * ) ,
n n m
s s s s s s
t j ijt l ljt ijt
j i l
q b x b p x s t
(9)
3 ,
s s
t t
I d s t (10)
max ( 7 ), 0 ,
s s s
t t t
RQ I d s t (11)
* , , ,
s
lj ljt
r p s t l j
(12)
0* ' , ,
s
l lt
r p s t l
(13)
1
' 1 , ,
n
s s
ljt lt
j
p p s t l
(14)
1
0 0
, 1, 0
n n
s s
ijt jit
i i
x x s t T j
(15)
0 1
1 , 0
n T
s
ijt
i t
x s j
(16)
0 1
1 , 0
n T
s
jit
j t
x s i
(17)
0 1
0
1 1
n n
s s
i
i T
i i
x x s
(18)
1
1 0
0
n n
s
ij
i j
x s
(19)
0 0
,
n n
s
ijt
i j
x y s t
(20)
, , , , ' , , ' 0 ,
s s s s s s s
t t t t t t t
I Q BQ CQ Q RQ q s t (21)
, , ' 0,1 ,
s s s
ijt ljt l t
x p p s t (22)
, , ,
s s
t t
y q qq s t
(23)
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154 Elaheh Ghasemi & Mehdi Bashiri A Selective Covering-Inventory-Routing Problem to
the Location of Bloodmobile to Supply Stochastic . . .
International Journal of I ndustrial Engineering & Production Research, June 2018, Vol. 29, No. 2
Equation (1) minimizes the total cost function,
which includes establishing bloodmobiles,
holding cost, cost of blood collection campaign,
and transportation cost. Constraint (2) shows the
blood inventory level in period t under scenario s.
This level is equal to the inventory level in period
t-1, plus total unit of blood collected by blood
center and bloodmobiles in period t under
scenario s, plus total unit of received blood
through campaign of blood extra donation to the
blood center in period t-1 under scenario s, minus
the stochastic demand in period t, minus amount
unit of extra blood sent to the other blood centers
in period t-1 under scenario s (the inventory level
in the end of each period is checked and, after
that, the amount of sending extra blood or
supplying shortage blood is determined).
Constraint (3) indicates the total units of received
blood through blood campaign to the blood center
in period t under scenario s, regarding reduction
coefficient β. Constraint (4) indicates that the
campaign for blood collection must be done when
the blood inventory level is less than five times
the demand. Constraint (5) computes the total
unit of received blood to the blood center by itself
or through bloodmobiles in period t under
scenario s. Constraint (6) defines the total unit of
received blood to the blood center by all of
bloodmobiles in period t under scenario s.
Constraint (7) calculates the quantity of collected
blood units in the blood center in period t under
scenario s. Constraint (8) is the total unit of
collected blood from donors in the blood center
in period t under scenario s regarding reduction
coefficient β. Constraint (9) determines the
quantity of collected blood by bloodmobiles in
period t under scenario s. Constraint (10)
guarantees that, at least, three periods (days) of
safety stock are considered in blood center.
Constraint (11) makes sure that blood transfer to
the other centers is possible if only and if the
blood inventory level in blood center be more
than seven times the demand in each period under
each scenario. Constraints (12) and (13) ensure
that bloodmobiles and blood center only accept
donors within their coverage distance. Constraint
(14) imposes that each donor site is served by
only blood center or a bloodmobile. Constraint
(15) specifies that if there is a bloodmobile
coming to node j in period t, there should be also
an outgoing one from node j in period t+1.
Constraints (16) and (17) force that each potential
location of bloodmobile under each scenario will
be visited utmost one in planning period.
Constraint (18) determines that each
bloodmobile, which starts its tour from center,
must go back to it in the end of its tour.
Constraint (19) prevents tours starting from any
site other than the blood center. Constraint (20)
ensures that the number of used bloodmobiles
does not exceed the number of established
bloodmobiles. In this model, it is supposed that
variable y determines the number of requirement
bloodmobiles independent of each period and
each scenario. It is noteworthy that Constraints
(3), (16), and (17) altogether prevent sub-tours of
bloodmobiles. Constraints (21)-(23) define the
eligible domains of the decisions variables.
3-3. Model linearization
The proposed mathematical model is nonlinear in
the present form because of constraints (4), (9),
and (11). The linearization scheme is based on
the method introduced in [24] to linearize
constraint (4), and new binary variables are
introduced as s
t
s
tlvlv 2,1 ; constraint (4) should be
replaced by constraints (24)-(30).
* 1 ,
s s
t t
q M lv t s
(24)
* 1 ,
s s
t t
q M lv t s
(25)
5 * 2 ,
s s s s
t t t t
q d I M lv t s
(26)
5 * 2 ,
s s s s
t t t t
q d I M lv t s
(27)
5 * 1 2 ,
s s s
t t t
d I M lv t s
(28)
5 * 1 2 ,
s s s
t t t
d I M lv t s
(29)
1 2 1 ,
s s
t t
lv lv t s
(30)
In addition, variable s
ijlt
lv3 is used instead of
multiplication of two binary variables s
ljt
s
ijt px , in
constraint (9); therefore, this constraint is
replaced by equation (31), and additional
constraints (32)-(35) should be added to the
model.
1 0 1
( * * lv 3 ) ,
n n m
s s s s s
t j ijt l ijlt
j i l
q b x b
s t
(31)
3 , , , ,
s s
ijlt ljt
lv p t s i j l (32)
3 , , , ,
s s
ijlt ijt
lv x t s i j l (33)
3 1 , , , ,
s s s
ijlt ijt ljt
lv x p t s i j l
(34)
3 0 , , , ,
s
ijlt
lv t s i j l (35)
Finally, to linearize the proposed model, binary
variables s
t
s
tlvlv 5,4 are defined and (11) should
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155 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering-Inventory-Routing Problem to the
Location of Bloodmobile to Supply Stochastic . . .
International Journal of I ndustrial Engineering & Production Research, June 2018, Vol. 29, No. 2
be replaced with the following additional
constraints.
* 4 ,
s s
t t
RQ M lv t s (36)
* 4 ,
s s
t t
RQ M lv t s (37)
7 * 5 ,
s s s s
t t t t
RQ I d M lv t s
(38)
7 * 5 ,
s s s s
t t t t
RQ I d M lv t s
(39)
7 * 1 5 ,
s s s
t t t
I d M lv t s
(40)
7 * 1 5 ,
s s s
t t t
I d M lv t s
(41)
4 5 1 ,
s s
t t
lv lv t s (42)
4. Computational Results
Since the problem described in this paper has not
been studied before, no benchmark instances are
available in the literature. We therefore generate
a set of three numerical examples with small and
medium sizes according to the real situation in
blood center in Isfahan Blood Transfusion
Center. Characteristics of the generated datasets
are shown in Table 2. In the present study, after
linearizing the model, the GAMS (23.5)– with
CPLEX solver is used for optimization. Table 3
shows the numerical results obtained by using
GAMS for all of data sets at different reduction
percentage (β). In Table 3, “Absolute gap”
column represents the difference between MIP
solution and the best solution obtained using
GAMS; also, the model runtime is given in the
last column.
Number of determined bloodmobiles, Average
quantity of collected blood by bloodmobiles,
blood center, and blood campaign under each
scenario are presented for three datasets and
varying β. The value of β between is changed 0.1
and 0.9 in increments of 0.1. By increasing β, we
observe that the number of bloodmobiles as well
as cost function increase in three datasets.
At a constant level of bloodmobiles in each
dataset, first, quantity of collected blood by blood
center increases; after that, the level of collected
blood by bloodmobiles increases.
Tab. 2. Characteristics of the generated datasets
characteristics Data set
1
Data set
2 Data set 3
Number of candidate
bloodmobile
locations
4 7 9
Number of blood
donation sites 10 15 20
Time period horizon 3 3 4
Number of scenario 2 4 5
Finally, if the amounts of periodic demands exceed
capacity of blood center and blood mobiles, the
needed blood will be collected by blood campaign
and high costs. The results indicate that the
proposed SCIR model produces the expected output
correctly. For some instances in which the absolute
gap is large, a solution approach to solving them
would be more useful. In this study, the benefit of
the two- stage stochastic programming approach is
examined by comparing its performance against
that of an expected value approach. In this paper,
we used stochastic parameters under discrete
scenarios for demand and amount of potential blood
in donation sites and bloodmobiles locations in each
period. There are several measures to evaluate the
benefit of scenario-based approaches. Value of the
Stochastic Solution (VSS) introduced by Birge [25]
is one of the common criteria to investigate the
benefits of the two-stage stochastic programming
approach.
VSS can be formulated as follows:
EEV HN
VSS z z (43)
where zEEV and zHN indicate the objective values
under expected value and stochastic programming
approaches, respectively. zHN is the optimal
objective value presented in section 3. To compute
zEEV, the presented model is solved by replacing the
values of the stochastic parameters with their
expected values. The solution obtained from
solving this model provides the optimal number of
bloodmobiles (y) for the expected value approach.
In the final step, initial model is solved based on
each scenario, separately while the number of
bloodmobiles is fixed to the optimal number of
bloodmobiles (y) obtained in the previous step.
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156 Elaheh Ghasemi & Mehdi Bashiri A Selective Covering-Inventory-Routing Problem to
the Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
Based on the computed results, VSS for data set 1
for all of β values is equal to 0, since there are only
2 scenarios in data set 1 and occurrence of scenarios
is equal; there is no difference between stochastic
and expected value approaches. Figures (6) and (7)
indicate the trend of VSS over a range of β for
datasets 1 and 2, respectively. The results show that
trend of VSS completely depends on value of β and
is not regular regarding increasing β. In dataset 1,
for β values more than 0.7, not only the stochastic
programming approach is not useful, but also VSS
value is negative. Note that, in higher level of β, the
majority of blood demand supply by blood
campaigns; therefore, considering stochastic value
for blood potential in donation sites could not
improve cost function. No difference observed
between stochastic or deterministic approaches
when β equals 0.1 or 0.2. In the range of 0.3 to 0.7
for β values, the proposed stochastic programming
approach has a higher priority than the expected
value approach. The obtained results show that
trend of VSS for dataset 2 follows a similar pattern
to those of dataset 1 with some of the differences.
Generally, the stochastic programming approach
does not show a clear preference over an expected
value approach and depends directly on β value in
both datasets.
Fig. 6. VSS results for data set 2
Fig. 7. VSS results for data set 3
Tab. 3. The impact of varying reduction percentage (β) on cost function and optimal number of blood
facilities, collected blood by bloodmobiles, centers, and campaign
Data set
Β
Cost function (MIP solution)
Number of used blood mobile
Average quantity of collected blood
by bloodmobiles in each scenario
Average quantity of collected blood
by blood center in each scenario
Average quantity of collected blood
by blood campaign in each
scenario
Absolute gap
Solution time (S)
Data set 1
0.9 112811 2 193 49 98.8 0 13.949
0.8 77142 2 193 49 74.5 312 13.594
0.7 53184 2 193 49 49.5 117.93 12.187
0.6 29128 2 193 49 26.2 17.997 11.646
0.5 7636 2 188 49 3 0.15 12.355
0.4 3184 2 156 49 0 106.4 13.758
0.3 3184.5 2 127 49 0 172.58 12.517
0.2 3182.5 2 105 49 0 232.36 12.841
0.1 3138.5 1 88 49 0 216.3 12.769
Data set 2
0.9 109120.750 3 264.75 47 106.725 7490.1 39.373
0.8 70958.500 3 264.75 47 68.15 1470 26.124
0.7 38580.750 3 264.75 47 35.475 1971.5 24.046
0.6 13806.000 3 249.25 47 10.5 0 18.742
0.5 4623.250 3 208.5 47 1.25 0.01 18.247
0.4 3387.750 3 186.25 47 0 98.24 21.418
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157 Elaheh Ghasemi & Mehdi Bashiri
A Selective Covering-Inventory-Routing Problem to the
Location of Bloodmobile to Supply Stochastic . . .
International Journal of Industrial Engineering & Production Research, June 2018, Vol. 29, No. 2
0.3 3387.000 3 137.75 47 0 167.75 19.409
0.2 3335.25 2 114.5 47 0 0 19.633
0.1 3341.5 2 96.75 47 0 0.19181 21.352
Data set 3
0.9 109856.6 4 329 47 110.8 10.96 450.132
0.8 61833.6 4 344 46 57.68 3544.2 178.532
0.7 22518.8 4 334 46 18 0 95.611
0.6 7028.8 4 273.8 46 2.4 287.6 27.867
0.5 4620 4 218 46 0 46.777 74.966
0.4 4564 3 174 46 0 18.204 46.930
0.3 4566.8 3 143 46 0 8.5832 25.580
0.2 4513.6 2 125 46 0 239.25 24.874
0.1 4517.8 2 101 46 0 31.043 41.294
5. Conclusion
The timely supply of safe blood is a
challenging issue in the healthcare problems.
This work presented a selective-covering-
inventory-routing (SCIR) model for supply
stochastic demand of whole blood under
uncertainty conditions. In the presented two-
stage stochastic approach, the number of the
needed bloodmobiles was determined in the
first stage before knowing information about
occurrence of the scenarios and other decisions
were made in the second stage. This proposed
model considered inventory management rules
of Iranian Blood Transfusion Organization
(IBTO) as well as the bloodmobile routing
problem. Three randomized datasets were
generated and sensitivity analysis was done
based on model parameters.
Numerical results showed that the small
instances of the problem could be solved to
optimality using GAMS in reasonable amount
of time; however, a solution approach to
solving the big instances was needed. Results
showed the collected blood by blood center,
bloodmobiles and campaign increase,
respectively, when demand and reduction
percentage of blood increases. In addition, the
benefit of stochastic value approach versus
expected value approach by using VSS criteria
was investigated. Results showed that the
stochastic programming approach does not
show a clear preference over an expected value
approach and depends directly on the β value in
each dataset.
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