This paper is concerned with the optimal decisions of blood banks in a blood logistics network (BLN) with the consideration of natural disasters. One of the biggest challenges is how to deal with unexpected disasters. Our idea is to consider the disasters as the natural consequences of interaction among multiple interdependent uncertain factors, such as the locations and the levels of disasters, the number of casualties, and the availabilities of rescue facilities, which work together to influence the rescue effects of the BLN. Thus, taking earthquakes as the example, a Bayesian Network is proposed to describe such uncertainties and interdependences and, then, we incorporate it into a dedicated two-stage multi-period stochastic programming model for the BLN. The planning stage in the model focuses on blood bank location and inventory decisions. The subsequent operational stage is composed of multiple periods, some of which may suffer disasters and initiate corresponding rescue operations. Numerical tests show that the proposed approach can be efficiently applied in blood management under the complicated disaster scenarios.
The supply of human blood relies on Blood Logistics Networks (BLNs), which are usually composed of donor points, blood banks, and relief facilities, etc. As the facilities to collect, test, inspect, store, and distribute human blood, blood banks play a fundamental role in BLNs to ensure the effective and efficient supply. Cooperating with other facilities in BLNs, blood banks should satisfy not only the daily demand of human blood but also the emergency demand caused by natural and anthropogenic disasters. Especially in recent years, frequent occurrences of disasters, such as earthquakes, hurricanes, fires, and terrorist attacks, have caused significant personnel and property losses due to its uncertain and destructive nature [
However, the blood banks related decisions are always fraught with different challenges, in which the first and most important one is the availability of the disaster information that is indispensable to make the decisions. Taking the demand on human blood as example, the daily demand may be stable. However, the emergency demand in a disaster, especially the suddenly-occurring disaster such as the earthquake in natural disasters or terrorist attack, is hard to be estimated using historical data directly [
Some researchers have adopted the integrated framework to study relief logistics network taking disasters into account. However, few works have addressed blood banks with planning stage and multiple operational stages simultaneously as well as the features of blood products. More importantly, the uncertainties of disasters are dealt with in a relatively simple way in previous BLN-related studies. Specifically, besides daily demand, blood banks should also offer emergency aid to the hospitals in case of disasters. Demand uncertainty is a crucial challenge that must be handled in disaster management [
Therefore, our study mainly deals with the location-inventory decisions on blood banks and related decisions in the BLN with the consideration of possible occurrences of disasters. We propose a two-stage multi-period Stochastic Programming (SP) model which takes the characteristics of human blood (such as multiple blood products, blood lifespan, and blood substitution) and multiple stochastic factors involved in disasters into account. In the planning stage of the model, we determine the blood bank location and the inventory levels in the selected blood bank and each hospital in the BLN. The following operational stage involves multiple periods, in which the distribution and replenishment decisions on human blood would be made periodically. When a disaster occurs in some periods, the BLN should have the ability to continuously satisfy the emergency blood transfusion requirements at the first-time treatment by its inventory. The corresponding transfusion quantity, as one of the consequences of disasters, is decided by multiple disaster-related factors, such as the magnitude of the disaster, the number of the injured, the level of injury, and the distribution of blood type among the injured. Besides, some region-related stochastic factors, such as the potential disaster points and the unavailable probability of each hospital, would decide which hospital provides continuous blood supply during a transfusion time, and, then, influence the prepositioning inventory-level decisions in this rescue hospital because there is a replenishment time due to its distance from the blood bank. Thus, we apply the Bayesian Network (BN), which is a probabilistic modelling approach in the context of data mining, to synthesize all these random factors to generate discrete scenarios which can be used to describe the possible disaster. The integrated objective is given to optimize the total cost of the strategic and operational decisions of the whole BLN during a planning horizon. A case study based on earthquakes in Sichuan Province, China, is studied and the corresponding model, which is a mixed-integer linear program, is optimally solved by IBM ILOG CPLEX to demonstrate the effectiveness of the proposed methods.
The contribution of this study lies in two aspects. Firstly, we propose a BN model to utilize the interdependences of multiple uncertain factors to generate plausible scenarios and incorporate it into a dedicated optimization model. It allows us to synthesize multiple stochastic elements in disasters and study their aggregated impacts on rescue management. To the best of our knowledge, a modelling effort for location and inventory in BLNs considering BN-based scenarios generation does not appear in existing studies. Our work is a new attempt to combine optimization method with the technique from data science to address a complicated decision-making issue in real-world. Secondly, we formulate an integrated SP model to handle with the location decision on blood banks and related inventory as well as distribution decisions in the BLN with possible disasters. The aforementioned features (e.g., coordination of location and operational activities in the whole BLN, the typical characteristics of human blood, and multiple uncertain factors considered in the BN) are considered in our model. In previous works on the BLNs, the consideration of some features of human blood (e.g., multiple blood products, blood substitution) is limited. Also, few BLN-related papers consider multiple interdependent uncertain factors, some of which are not studied as we know, such as the blood type among the injured and the level of injury. Thus, we take an early step to incorporate both multiple random factors and unique characteristics of blood into blood management considering disasters.
The remainder of this study is presented as follows. Section
In this section, the literature regarding relief logistics networks under disasters and the applications of Bayesian Networks (BNs) are unfolded as follows.
Recently, frequent-occurring disasters drive researchers to study relief logistics networks with the aim of mitigating the impacts of disasters. In a disaster, since two stages exist naturally, i.e., pre/postdisaster, most of the existing studies formulate two-stage models to study the related issues of relief logistics networks. Barbarosoǧlu and Arda’s work [
Besides, only a few authors have incorporated the interdependences between random disaster-related factors within humanitarian operations. For example, Verma and Gaukler [
Moreover, it is worth noting that the studies above only focus on general relief goods. Due to the importance of human blood in disaster rescue, more and more studies focus on emergency BLN design, taking the special structure of blood supply network as well as distinctive features of blood products into account. For example, Kochan et al. [
SP is extensively applied to the design of emergency logistics networks, including BLNs. However, in most previous studies, the stochastic scenario which is indispensable in SP is assumed to be known. Obviously, such assumption is not always realistic, especially in the case of disasters, due to their unpredictability. This motivates more researchers to utilize Robust Optimization (RO), which only requires the worst-case information instead of entire distribution, to study related problems. For example, Jabbarzadeh et al. [
To clearly demonstrate the existing BLN research under the setting of disasters, Tables
Characteristics of related literature in disaster management
Reference | Model features | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Blood | Perishability | Blood substitution | Product | Stochastic factors | ||||||||
Single | Multiple | Demand | Disaster point | Rank of disaster | Severity |
Distribution of blood type | Facility failure | Interdependence | ||||
Barbarosoǧlu and Arda 2004 | √ | √ | ||||||||||
Mete and Zabinsky 2010 | √ | √ | ||||||||||
Jabbarzadeh et al. 2014 | √ | √ | √ | √ | ||||||||
Gunpinar and Centeno 2015 | √ | √ | √ | √ | √ | |||||||
Tofighi et al. 2015 | √ | √ | ||||||||||
An et al. 2015 | √ | √ | ||||||||||
Verma and Gaukler 2015 | √ | √ | √ | √ | √ | |||||||
Zahiri et al. 2015 | √ | √ | √ | √ | ||||||||
Shishebori and Babadi 2015 | √ | √ | √ | |||||||||
Alem et al. 2016 | √ | √ | √ | |||||||||
Kochan et al. 2016 | √ | √ | √ | √ | ||||||||
Paul and MacDonald 2016 | √ | √ | √ | √ | √ | |||||||
Rezaei-Malek et al. 2016 [ |
√ | √ | √ | |||||||||
Fereiduni and Shahanaghi 2016 | √ | √ | √ | √ | √ | |||||||
Mohamadi and Yaghoubi 2017 | √ | √ | √ | |||||||||
Fahimnia et al. 2017 | √ | √ | √ | |||||||||
Ramezanian and Behboodi 2017 | √ | √ | √ | |||||||||
Haghi et al. 2017 | √ | √ | √ | |||||||||
Salehi et al. 2017 | √ | √ | √ | √ | √ | |||||||
Khalilpourazari and Khamseh 2017 | √ | √ | √ | |||||||||
Samani et al., 2018 | √ | √ | √ | √ | ||||||||
Samani and Hosseini-Motlagh 2018 | √ | √ | √ | √ | ||||||||
Rahmani 2018 | √ | √ | √ | |||||||||
Kamyabniya et al. 2018 | √ | √ | √ | √ | ||||||||
Eskandari-Khanghahi et al. 2018 [ |
√ | √ | √ | √ | ||||||||
This paper | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ | √ |
Decisions and methods of related literature in disaster management
Reference | Predisaster | Postdisaster | Single |
Multiple |
Decisions | Methods | ||||
---|---|---|---|---|---|---|---|---|---|---|
Location | Prepositioning | Inventory |
Distribution flow | SP | RO | |||||
Barbarosoǧlu and Arda 2004 | √ | √ | √ | √ | √ | √ | ||||
Mete and Zabinsky 2010 | √ | √ | √ | √ | √ | √ | √ | |||
Jabbarzadeh et al. 2014 | √ | √ | √ | √ | √ | √ | √ | |||
Gunpinar and Centeno 2015 | √ | √ | √ | √ | √ | √ | ||||
Tofighi et al. 2015 | √ | √ | √ | √ | √ | √ | √ | |||
An et al. 2015 | √ | √ | √ | √ | √ | √ | ||||
Verma and Gaukler 2015 | √ | √ | √ | √ | √ | √ | ||||
Zahiri et al. 2015 | √ | √ | √ | √ | √ | |||||
Shishebori and Babadi 2015 | √ | √ | √ | √ | √ | √ | ||||
Alem et al. 2016 | √ | √ | √ | √ | √ | √ | √ | |||
Kochan et al. 2016 | √ | √ | √ | √ | √ | √ | √ | |||
Paul and MacDonald 2016 | √ | √ | √ | √ | √ | √ | √ | |||
Rezaei-Malek et al. 2016 | √ | √ | √ | √ | √ | √ | √ | √ | ||
Fereiduni and Shahanaghi 2016 | √ | √ | √ | √ | √ | √ | √ | |||
Mohamadi and Yaghoubi 2017 | √ | √ | √ | √ | √ | √ | ||||
Fahimnia et al. 2017 | √ | √ | √ | √ | √ | √ | √ | |||
Ramezanian and Behboodi 2017 | √ | √ | √ | √ | √ | |||||
Haghi et al. 2017 | √ | √ | √ | √ | √ | √ | √ | |||
Salehi et al. 2017 | √ | √ | √ | √ | √ | √ | √ | √ | √ | |
Khalilpourazari and Khamseh 2017 | √ | √ | √ | √ | √ | √ | √ | |||
Samani et al. 2018 | √ | √ | √ | √ | √ | √ | ||||
Samani and Hosseini-Motlagh 2018 | √ | √ | √ | √ | √ | √ | √ | |||
Rahmani 2018 | √ | √ | √ | √ | √ | √ | ||||
Kamyabniya et al. 2018 | √ | √ | √ | √ | √ | |||||
Eskandari-Khanghahi et al. 2018 | √ | √ | √ | √ | √ | √ | ||||
This paper | √ | √ | √ | √ | √ | √ | √ | √ |
In summary, the rescue effects of BLNs are significantly influenced by the consequences of disasters, which embody in multiple aspects, including the emergency demand, facility failure, and the place where the disaster occurs. In previous studies, such consequences are treated in a relatively simple way. Specifically, most works only consider the random demand. Also, SP-based studies usually assume there is a known discrete distribution about the uncertain demand, etc. RO, which does not require a probabilistic distribution, only considers extreme scenarios and thus may lead to conservative decisions. Recently, only few BLN-related works, such as Fereiduni and Shahanaghi [
In recent years, BNs have become a popular method for extracting knowledge from data in complex and uncertain systems, which encode the joint probability distribution of a set of random variables by making conditional dependence assumptions [
These works indicate that BNs can be used to describe complicated disaster scenarios and improve the quality of decisions. However, it is worth noting that different types of disasters involve distinctive stochastic factors and their relationships are also diverse. Fortunately, for some common natural disasters, such as earthquakes and hurricanes, there are lots of studies in the context of disasters research which can provide solid foundation for our BN modelling. Moreover, with the consideration of more stochastic factors, we need to consider more decisions of BLNs, which means a dedicated model is required. Thus, in the next section, we describe the problem we studied, give the corresponding mathematical formulation and then, propose our BN for the case of earthquake.
Consider a region where a government plans to launch a central blood bank to satisfy both the daily demand and the uncertain demand caused by possible future disasters. Hence, the location-inventory problem is proposed in a three-layer BLN composed by donor points, a central blood bank, and hospitals, taking uncertain disasters and the characteristics of human blood into account. After the blood bank is located, the donor points collect blood from donors and provide blood to the prespecified blood bank, which is responsible for distributing blood to hospitals. The operational stage is divided into multiple periods (time interval of each period is
The problem consists of four different decisions: (1) the central blood bank location; (2) inventory level in the prespecified blood bank and each hospital at the beginning of each operational period; (3) the distribution flow from donor points to the prespecified blood bank at each operational period; and (4) the distribution flow from the prespecified blood bank to each hospital at each operational period. The decisions are made in two stages. Taking the BLN composed by three donor points, two central blood bank candidates and two hospitals as the example, the planning stage includes the strategic decisions on the selection of a central blood bank from two candidates and the inventory level of both the selected blood bank and each hospital. Given these strategic decisions, the activities in one period of the operational stage under the daily or disaster setting are given respectively in Figure
The illustration of activities in one period of the operational stage.
The daily operations of the whole BLN are displayed in Figure
More specifically, the inventory status of each facility in Figure
The status of inventory level in three-layers BLN in the operational stage.
As shown in the right part in Figure
Figure
In summary, the BLN itself should have the ability to satisfy the first-time blood demand. The consumed inventory in the BLN can be recovered before the next period. Regarding the blood demand of subsequent treatments on the injured, it also can be satisfied by other neighbor regional BLNs or extra blood donations, which is not discussed in our study.
The main assumptions of the above problem are listed as follows: A fixed construction cost for each blood bank candidate is available. The holding cost of in-transit inventory from each donor point to the prespecified blood bank and the prespecified blood bank to each hospital will be included in the inventory cost of the blood bank and corresponding hospital, respectively. The inventory holding cost is proportional to the inventory duration and quantity. The transportation cost is proportional to the distance and distribution flow. In each period, at most one disaster occurs. At least one hospital is available during the disaster period, and the unavailable hospital can recover in the next period.
The last two assumptions are reasonable in our study. First, the disasters that require emergency blood rescue usually would not occur frequently. Moreover, a hospital may be unavailable because of surrounding road damage or the damage to the hospital itself in a disaster. However, even all hospitals are unavailable, in order to implement the relief, the government would restore the most slightly damaged one or build a temporary rescue facility nearby some hospital as soon as possible. Hence, it is reasonable to assume there is at least one available hospital. Furthermore, some hospitals that are seriously damaged may not be restored soon; however, the daily demand in the area covered by the damaged hospital still exists and may be guided to a nearby substituting facility in a short time. Thus, the similar daily operational activities still exist. Therefore, we give the last assumption.
Sets, parameters, and variables used are given in Tables
Sets and indices.
Symbol | Description |
---|---|
|
Set of donor points |
|
Set of blood bank candidates |
|
Set of hospitals |
|
Set of blood products (plasma, platelets, et al.) |
|
Set of blood types (A, B, AB, O, et al.) |
|
Set of disaster scenarios |
|
Set of whole scenarios (union of |
|
Indices to donor points, and |
|
Indices to blood bank candidates, and |
|
Indices to hospitals, and |
|
Indices to blood products, and |
|
Indices to blood types, and |
|
Indices to scenarios, and |
Model parameters.
Symbol | Description | |
---|---|---|
Deterministic parameters | Φ | The number of periods in planning horizon |
|
Time interval of each period | |
|
Unit transportation fee (RMB/ km•U, U is the abbreviation of blood Units. 1 Unit equals 200 ml) | |
|
The velocity of transportation (km/h) | |
|
Demand service level (confidence level) | |
|
Inventory holding cost per unit of blood bank | |
|
Fixed construction cost of blood bank | |
|
Inventory holding cost per unit of hospital | |
|
Transportation time from donor point | |
|
Transportation time from blood bank | |
|
Distance from potential disaster point | |
|
Lifespan of blood product | |
|
Supply capacity of blood product | |
|
Daily demand for blood product | |
|
1 if blood type |
|
∆ |
The continuous blood transfusion time in the first-time treatment (h) | |
|
||
Stochastic parameters |
|
Probability of scenario |
|
Transportation time from the actual disaster point to hospital | |
|
1 if hospital | |
|
1 if the injured are sent to hospital | |
|
Emergency demand for blood product |
Decision variables.
Symbol | Description | |
---|---|---|
The |
|
1 if blood bank candidate |
|
Maximum inventory of blood product | |
|
Emergency stock of blood product | |
|
Emergency stock of blood product | |
|
||
The |
|
Supply quantity of blood product |
|
Distribution flow of blood product | |
|
Distribution flow of blood product |
|
|
Supply quantity of blood product | |
|
Distribution flow of blood product | |
|
Distribution flow of blood product |
The mutual substitution among common blood types (set
|
|
||||||||
---|---|---|---|---|---|---|---|---|---|
|
O+ | O– | A+ | A– | B+ | B– | AB+ | AB– | |
|
O+ | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
O– | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
A+ | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | |
A– | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | |
B+ | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | |
B– | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | |
AB+ | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |
AB– | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
The objective of the two-stage multiperiod SP model aims at minimizing the total cost of the BLN in both the planning stage and operational stage, which can be divided into three terms: (1) the construction cost in the planning stage, (2) the operational cost of the daily period, and (3) the operational cost of the disaster period in the operational stage.
In our study, the scenario set in each operational period is denoted as
Notice that the first two cost items in formula (
Moreover, the constraints include the following three groups.
Constraint (
Constraints (
Constraints (
Moreover, constraints (
Finally, constraints (
In this section, based on earthquakes, we propose the corresponding BN model to generate the scenario set
Variables (nodes) and states in the BN.
Variables (Nodes) | States | Description | |
---|---|---|---|
Set | Name | ||
|
Potential |
Point 1 | Consider several potential epicenters and each with corresponding earthquake occurrence probabilities. |
… | |||
Point |
|||
|
|||
|
Level of earthquake | Level 1 | Earthquake can be divided into different levels, and each of them has the corresponding probabilities. |
… | |||
Level |
|||
|
|||
|
Level of injury | Seriously | The level of injury of each injured person is subject to the two-point distribution. Each level of injury has corresponding probability and demand of blood. |
Slight | |||
|
|||
|
Distribution of blood type | Distribution 1 | The distribution of different blood types in the injured may be uncertain. |
… | |||
Distribution |
|||
|
|||
|
States of hospitals | Unavailable | Use unavailable probability, which is affected by the random events in |
Available |
Thus, the proposed BN, as shown in Figure
Proposed BN for the earthquake scenarios.
Next, their inter-relationship can be quantified as follows.
Then, with
At first, Wyss and Trendafiloski [
Second, for each injured, the demand for blood products
The event that an earthquake occurs at potential epicenter
According to Peng [
As we mentioned before, the blood prepositioned in the BLN needs to satisfy the blood demand of the transfusion time
When the hospital
Finally, one disaster scenario
All disaster scenarios constitute disaster scenario set
In this section, we implement simulation study based on earthquakes in Sichuan Province, China. In order to evaluate the impacts of the BN model, we would test and compare simulation cases under different scenario sets and parameters, such as the probability of disasters and the unit inventory holding cost.
We would introduce our simulation background based on the Longmenshan earthquake zone in Sichuan, China.
In the operational stage, each period
The random epicenters, donor points, blood bank candidates, and hospitals are shown in the Figure
The geographic picture of blood bank supply chain.
Firstly, their parameters are elaborated below.
(1) Potential epicenters set
(2) Let the blood bank candidates set
The parameters of the blood bank candidate
Blood bank | Fixed cost | Inventory holding |
---|---|---|
candidates |
|
cost |
Chengdu | 1.2 | 0.012 |
Yibin | 1.1 | 0.01 |
Nanchong | 1.08 | 0.008 |
Yaan | 1.05 | 0.006 |
Deyang | 1.12 | 0.007 |
(3) Considering that the number of blood donors is positively correlated with the population density, top four cities in population density in Sichuan Province are designated as main blood donation points. Thus, by World Population Network [
(4) Set
Secondly, the parameters of blood products are given as follows.
(1) The multiproduct set is denoted as
(2) By “The Minister of Health of the People’s Republic of China [
(3) Table
The maximum supply quantity (
Donor point |
Plasma | Red blood cells | Platelets | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | AB | O | A | B | AB | O | A | B | AB | O | |
Chengdu | 435 | 377 | 103 | 569 | 274 | 238 | 65 | 359 | 122 | 106 | 29 | 160 |
Neijiang | 279 | 242 | 66 | 366 | 176 | 153 | 42 | 231 | 79 | 68 | 19 | 103 |
Zigong | 305 | 264 | 72 | 399 | 192 | 167 | 46 | 252 | 86 | 74 | 21 | 112 |
Leshan | 112 | 97 | 27 | 146 | 71 | 61 | 17 | 92 | 32 | 28 | 8 | 41 |
(4) By Wang et al. [
The daily blood demand (
Blood product |
Blood type |
Hospitals |
|||
---|---|---|---|---|---|
WCH | BPH | WPH | MPH | ||
Plasma | A | 3.050 | 0.213 | 0.081 | 1.191 |
B | 2.644 | 0.184 | 0.070 | 1.032 | |
AB | 0.718 | 0.050 | 0.019 | 0.280 | |
O | 3.007 | 0.281 | 0.107 | 1.562 | |
|
|||||
Red blood cells | A | 1.932 | 0.137 | 0.052 | 0.755 |
B | 1.675 | 0.118 | 0.045 | 0.655 | |
AB | 0.455 | 0.032 | 0.012 | 0.178 | |
O | 2.532 | 0.179 | 0.068 | 0.989 | |
|
|||||
Platelets | A | 0.829 | 0.058 | 0.022 | 0.324 |
B | 0.719 | 0.050 | 0.019 | 0.281 | |
AB | 0.195 | 0.013 | 0.005 | 0.076 | |
O | 1.087 | 0.076 | 0.029 | 0.425 |
Furthermore, the parameters of transportation are given as follows:
(1) The transportation velocity
(2) The transportation time from donor points to blood bank candidates and from blood bank candidates to hospitals and the distance from potential epicenters to hospitals, which can be obtained from Google Maps, are given in Tables
The transportation time (
Blood bank candidate |
Donor point |
|||
---|---|---|---|---|
Chengdu | Neijiang | Zigong | Leshan | |
Chengdu | 0.15 | 2.5 | 3 | 2 |
Yibin | 3.2 | 1.5 | 1 | 2 |
Nanchong | 3 | 2.2 | 2.7 | 3.5 |
Yaan | 1.8 | 2.8 | 2.8 | 1.5 |
Deyang | 1.5 | 2.7 | 3 | 2.8 |
The transportation time (
Hospitals |
Blood bank candidate |
||||
---|---|---|---|---|---|
Chengdu | Yibin | Nanchong | Yaan | Deyang | |
WCH | 0.15 | 3.2 | 2.8 | 2 | 1.5 |
BPH | 3 | 5.2 | 4.1 | 3.7 | 1.7 |
WPH | 2.3 | 4.5 | 3.8 | 3 | 2.5 |
MPH | 1.8 | 4 | 2.4 | 2.5 | 0.9 |
The distance (
Epicenter |
Hospital |
|||
---|---|---|---|---|
WCH | BPH | WPH | MPH | |
Wenchuan | 147 | 130 | 5 | 173 |
Beichuan | 140 | 5 | 232 | 53.4 |
Maoxian | 187 | 89.1 | 41.9 | 212 |
Lushan | 170 | 305 | 261 | 241 |
Pingwu | 273 | 124 | 368 | 211 |
Based on above parameters, we show how to construct scenario sets by the proposed BN model. Notice the final scenario set may vary due to the states and possibilities of the random factors. Hence, we only give one scenario set here. Other kinds of scenario sets can be given in similar way.
(1) For
The probability of single epicenter and no disaster.
Epicenter | ||||||
---|---|---|---|---|---|---|
Wenchuan | Beichuan | Maoxian | Lushan | Pingwu | Probability | |
No earthquake | 0.903803 | |||||
|
||||||
Single-point earthquake | √ | 0.027953 | ||||
√ | 0.023174 | |||||
√ | 0.018445 | |||||
√ | 0.013763 | |||||
√ | 0.009129 | |||||
|
||||||
Multipoint earthquake | 25–5–1 cases | 0.003733 | ||||
|
||||||
|
1 |
Next, since at most one epicenter at each period is assumed, the normalized results are displayed in Table
The normalization of probability.
Epicenter | ||||||
---|---|---|---|---|---|---|
Wenchuan | Beichuan | Maoxian | Lushan | Pingwu | Probability | |
No earthquake | 0.907 | |||||
|
||||||
Single-point earthquake | √ | 0.028 | ||||
√ | 0.023 | |||||
√ | 0.019 | |||||
√ | 0.014 | |||||
√ | 0.009 | |||||
|
||||||
|
1 |
(2) Let
The levels of the earthquake and set
Rank | Magnitude | The level of the earthquake | Set |
---|---|---|---|
I | ≥7.0 | Especially large earthquake |
|
II | 6.5~7.0 | Large earthquake |
|
III | 6.0~6.5 | Relatively large earthquake |
|
IV | 5.0~6.0 | General earthquake |
|
(3) For each blood product,
(4) Two demand levels, seriously and slightly, are assumed because the blood demand varies from person to person by injury level difference. Two random proportions,
The quantity of two-level demand for each blood product (per injured).
Blood product | Rank | Quantity(U/h) |
---|---|---|
Plasma | High | 0.225 |
Low | 0.15 | |
Red blood cells | High | 1.136 |
Low | 0.522 | |
Platelets | High | 0.182 |
Low | 0.094 |
(5) Utilize formulas (
The conditional unavailable probability of hospitals (
Epicenter (Set |
The level of the earthquake (Set | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Rank IV |
Rank III (6.5) | Rank II |
Rank I |
|||||||||||||
WCH | BPH | WPH | MPH | WCH | BPH | WPH | MPH | WCH | BPH | WPH | MPH | WCH | BPH | WPH | MPH | |
Wenchuan | 0 | 0 | 0.909 | 0 | 0 | 0 | 0.95 | 0 | 0.065 | 0.173 | 0.968 | 0 | 0.513 | 0.57 | 0.983 | 0.427 |
Beichuan | 0 | 0.909 | 0 | 0.029 | 0 | 0.95 | 0 | 0.464 | 0.11 | 0.968 | 0 | 0.66 | 0.537 | 0.983 | 0.232 | 0.823 |
Maoxian | 0 | 0 | 0.238 | 0 | 0 | 0.105 | 0.579 | 0 | 0 | 0.433 | 0.734 | 0 | 0.381 | 0.705 | 0.861 | 0.298 |
Lushan | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.437 | 0 | 0.136 | 0.202 |
Pingwu | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.211 | 0.978 | 0 | 0.096 | 0.589 | 0 | 0.301 |
Till now, Figure
The scenario tree for one epicenter in the dataset 1200_1.
Moreover, notice
Characteristics of all datasets.
The dimensionality |
Dataset’s name | Random factors | |
---|---|---|---|
|
|
||
| |
1200_1 | ( |
( |
1200_2 | ( |
( | |
1200_3 | ( |
( | |
1200_4 | ( |
( | |
1200_5 | ( |
( | |
1200_6 | ( |
( | |
|
|||
| |
3600_1 | ( |
( |
3600_2 | ( |
( | |
|
|||
| |
4500_1 | ( |
( |
Assigning each dataset to the SP model which can be optimally solved by IBM ILOG CPLEX 12.6.3, the corresponding recourse problem (RP) solution can be obtained. Taking the dataset 1200_1 as the example, in its RP solution, Deyang is selected as the blood bank. The planning and daily operational cost (first four lines in formula (
Specifically, Table
The decisions of maximum inventory and emergency stock of the blood bank.
The prespecified blood bank: Deyang | |||||||
---|---|---|---|---|---|---|---|
Maximum |
Plasma | A | 4092.21 | Emergency stock |
Plasma | A | 3379.45 |
B | 2460.53 | B | 1821.26 | ||||
AB | 766.14 | AB | 598.29 | ||||
O | 6154.89 | O | 5220.76 | ||||
Red blood cells | A | 4914.84 | Red blood cells | A | 4428.12 | ||
B | 3090.94 | B | 2701.67 | ||||
AB | 204.17 | AB | 98.43 | ||||
O | 9402.44 | O | 8721.99 | ||||
Platelets | A | 2842.62 | Platelets | A | 2633.79 | ||
B | 1212.25 | B | 1029.25 | ||||
AB | 596.31 | AB | 549.32 | ||||
O | 4543.67 | O | 4269.8 |
The distribution flow (
Blood product |
Blood type |
Hospital |
|||
---|---|---|---|---|---|
WCH | BPH | WPH | MPH | ||
Plasma | A | 461.65 | 36.14 | 13.81 | 201.16 |
B | 448.15 | 31.22 | 12.11 | 147.79 | |
AB | 108.83 | 8.49 | 3.24 | 47.29 | |
O | 577.87 | 47.69 | 18.24 | 290.33 | |
|
|||||
Red blood cells | A | 327.47 | 23.24 | 8.87 | 127.52 |
B | 242.03 | 20.02 | 7.67 | 119.55 | |
AB | 77.12 | 5.43 | 2.05 | 21.14 | |
O | 471.43 | 30.38 | 11.59 | 167.04 | |
|
|||||
Platelets | A | 140.51 | 9.84 | 3.75 | 54.72 |
B | 121.87 | 9.57 | 4.09 | 47.46 | |
AB | 33.05 | 1.11 | 0.00 | 12.84 | |
O | 184.25 | 12.90 | 4.94 | 71.78 |
The emergency stock (
Blood product |
Blood type |
Hospital |
|||
---|---|---|---|---|---|
WCH | BPH | WPH | MPH | ||
Plasma | A | 0 | 2451.57 | 1604.92 | 15.13 |
B | 0 | 1932.33 | 1265.26 | 11.92 | |
AB | 0 | 596.29 | 391.02 | 3.68 | |
O | 0 | 2674.08 | 1750.46 | 16.5 | |
|
|||||
Red blood cells | A | 0 | 3547.9 | 2333.35 | 21.9 |
B | 0 | 2799.65 | 1842.68 | 17.28 | |
AB | 0 | 870.96 | 576.4 | 5.37 | |
O | 0 | 870.96 | 2543.47 | 23.87 | |
|
|||||
Platelets | A | 0 | 1886.98 | 1236.37 | 11.64 |
B | 0 | 1487.63 | 975.02 | 9.18 | |
AB | 0 | 459.76 | 302.01 | 2.83 | |
O | 0 | 2058.09 | 1348.35 | 12.70 |
Notice there is no prepositioning emergency stock in WCH as shown in Table
Similarly, other datasets can also be optimally calculated and Deyang is always selected as final blood bank, while the inventory results would be different. Specifically, the emergency stock in both blood bank and hospitals of each dataset is shown in Figure
Total emergency stock in blood bank and all hospitals of different datasets.
Moreover, for the datasets with the same
In summary, the decision on blood bank location is consistent, but the different random variables would significantly influence the inventory decisions. Specifically, with the increase of the dimensionality of scenario set, more disaster scenarios, although their possibilities are getting smaller, would be taken into account. Thus, to handle these extreme situations, larger inventory would be prepositioned which makes the solution more conservative.
This section further implements sensitivity analysis by changing (1) the disaster probability, i.e., P(
After calculation, it is found that, for each dataset, the decisions of blood bank location and emergency stock remain unchanged under the different disaster probabilities. And these two decisions are in the first stage, i.e., the planning stage, of two-stage SP which are what the decision-maker mainly care about. The reason is that P(
Since the decisions on location as well as emergency stock under different P(
Daily transportation cost and rescue cost under varying disaster probabilities (million).
The disaster probability change ratio | 0.2 | 0.25 | 0.5 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|
Daily transportation cost | ||||||||
All datasets | 34.80 | 34.62 | 33.80 | 32.20 | 29.33 | 26.88 | 24.64 | 22.69 |
|
||||||||
Rescue cost | ||||||||
1200_1~1200_3 | 1.14 | 1.44 | 2.74 | 5.32 | 9.96 | 13.87 | 17.38 | 20.51 |
1200_4~1200_6 | 1.21 | 1.52 | 2.90 | 5.63 | 10.56 | 14.69 | 18.41 | 21.73 |
3600_1 | 1.18 | 1.46 | 2.77 | 5.39 | 10.10 | 14.07 | 17.63 | 20.80 |
3600_2 | 1.20 | 1.48 | 2.83 | 5.50 | 10.30 | 14.34 | 17.97 | 21.20 |
4500_1 | 1.20 | 1.49 | 2.82 | 5.43 | 10.18 | 14.18 | 17.76 | 20.96 |
Total cost under varying disaster probabilities.
At first, for each disaster probability, the daily transportation cost of all datasets is identical. It is because their daily demand and the probability of normal scenario
Firstly, Figure
Blood bank location of each dataset under varying UIH costs.
Moreover, it is found that the emergency stock prepositioned in the whole BLN, which is decided by the BN, remains unchanged under the different UIH costs. But the inventory in each facility may vary due the different location results. It also means the planning capacity of the blood bank is influenced by the UIH cost.
In summary, different from to the BN-related parameters which only affect the inventory decisions, the UIH cost would influence the location decision. Thus, it is necessary to measure the UIH cost exactly and analyze the location decision’s sensitivity with respect to such cost-related parameters.
To prove the effectiveness of the proposed two-stage SP model, in this section, we give the corresponding so-called Expected Value (EV) Model, in which the random variable is replaced by the expected value, and compare the results of two models.
At first, we reformulate the two-stage SP in Section
In the EV model, in addition to the original constraints (
Let
Taking the dataset 1200_1 in Section
At first, according to the distance from epicenters to hospitals in Table
The one-to-one match between potential epicenters and hospitals.
Potential epicenters | Hospitals |
---|---|
Wenchuan | WPH |
Beichuan | BPH |
Maoxian | WPH |
Lushan | WCH |
Pingwu | BPH |
According to the probabilities of four magnitudes, formulas (
The proportions of two-level demand and four blood types are both turned into corresponding expected value. According to the proportion of two-level demand, blood type distribution and the corresponding probability stated in Section
The average blood demand (U) of each epicenter.
Blood product | Blood type | Epicenters | ||||
---|---|---|---|---|---|---|
Wenchuan | Beichuan | Maoxian | Lushan | Pingwu | ||
Plasma | A | 282.28 | 645.10 | 291.22 | 775.02 | 317.68 |
B | 217.14 | 496.23 | 224.01 | 596.17 | 244.37 | |
AB | 65.14 | 148.87 | 67.20 | 178.85 | 73.31 | |
O | 304.00 | 694.72 | 313.62 | 834.63 | 342.12 | |
|
||||||
Red blood cells | A | 1180.22 | 2697.16 | 1217.58 | 3240.33 | 1328.23 |
B | 907.86 | 2074.74 | 936.60 | 2492.56 | 1021.71 | |
AB | 272.36 | 622.42 | 280.98 | 747.77 | 306.51 | |
O | 1271.01 | 2904.64 | 1311.24 | 3489.59 | 1430.40 | |
|
||||||
Platelets | A | 199.88 | 456.79 | 206.21 | 548.78 | 224.95 |
B | 153.75 | 351.37 | 158.62 | 422.14 | 173.04 | |
AB | 46.13 | 105.41 | 47.59 | 126.64 | 51.91 | |
O | 215.26 | 491.92 | 222.07 | 590.99 | 242.25 |
Multiplying the expected blood demand (see Table
The daily demands and expected emergency demands caused by the earthquake in hospital
Blood product | Blood type | Hospitals | ||||
---|---|---|---|---|---|---|
WCH | BPH | WPH | MPH | |||
Daily demands |
Plasma | A | 3.050 | 0.213 | 0.081 | 1.191 |
B | 2.644 | 0.184 | 0.070 | 1.032 | ||
AB | 0.718 | 0.050 | 0.019 | 0.280 | ||
O | 3.007 | 0.281 | 0.107 | 1.562 | ||
Red blood |
A | 1.932 | 0.137 | 0.052 | 0.755 | |
B | 1.675 | 0.118 | 0.045 | 0.655 | ||
AB | 0.455 | 0.032 | 0.012 | 0.178 | ||
O | 2.532 | 0.179 | 0.068 | 0.989 | ||
Platelets | A | 0.829 | 0.058 | 0.022 | 0.324 | |
B | 0.719 | 0.050 | 0.019 | 0.281 | ||
AB | 0.195 | 0.013 | 0.005 | 0.076 | ||
O | 1.087 | 0.076 | 0.029 | 0.425 | ||
|
||||||
Expected emergency demands |
Plasma | A | 10.850 | 17.696 | 13.437 | 0 |
B | 8.346 | 13.613 | 10.336 | 0 | ||
AB | 2.504 | 4.084 | 3.101 | 0 | ||
O | 11.685 | 19.058 | 14.471 | 0 | ||
Red blood |
A | 45.365 | 73.989 | 56.180 | 0 | |
B | 34.896 | 56.914 | 43.216 | 0 | ||
AB | 10.469 | 17.074 | 12.965 | 0 | ||
O | 48.854 | 79.680 | 60.502 | 0 | ||
Platelets | A | 7.683 | 12.531 | 9.515 | 0 | |
B | 5.910 | 9.639 | 7.319 | 0 | ||
AB | 1.773 | 2.892 | 2.196 | 0 | ||
O | 8.274 | 13.494 | 10.246 | 0 |
The parameters of the EV models under other datasets can be obtained in the same way.
Based on above parameters, IBM ILOG CPLEX 12.6.3 is used to optimally solve the corresponding EV model. In this EV solution, the central blood bank is located in Chengdu, and the total cost is 4.86 × 107(RMB). The maximum inventory and emergency stock of Expected Value Model are shown in Table
The decisions of maximum inventory and emergency stock of the blood bank in the EV solution.
The prespecified central blood bank: Chengdu | |||||||
---|---|---|---|---|---|---|---|
Maximum inventory |
Plasma | A | 807.29 | Emergency stock |
Plasma | A | 42 |
B | 695.67 | B | 33 | ||||
AB | 189.74 | AB | 10 | ||||
O | 882.33 | O | 46 | ||||
Red blood cells | A | 660.87 | Red blood cells | A | 176 | ||
B | 555.74 | B | 136 | ||||
AB | 154.74 | AB | 41 | ||||
O | 824.9 | O | 190 | ||||
Platelets | A | 237.79 | Platelets | A | 30 | ||
B | 203.26 | B | 23 | ||||
AB | 55.63 | AB | 7 | ||||
O | 304.89 | O | 33 |
By comparing the results of the RP and EV solution, the EV solution performs better in terms of the total cost. However, the EV model, which is under the deterministic setting, only takes the expected demand into account. It is necessary to validate its performance under the disaster scenarios [
Thus, for each dataset as shown in Table
At first, the FP results under different disaster probabilities are displayed in Table
The FP of the EV solutions of each dataset under varying disaster probabilities.
The disaster probability change ratio | 1200 | 3600 | 4500_1 | ||||||
---|---|---|---|---|---|---|---|---|---|
1200_1 | 1200_2 | 1200_3 | 1200_4 | 1200_5 | 1200_6 | 3600_1 | 3600_2 | ||
0.2 | 83.92% | 84.30% | 84.30% | 83.92% | 83.92% | 83.92% | 83.80% | 83.80% | 83.95% |
0.25 | 84.08% | 84.08% | 84.08% | 84.08% | 84.08% | 84.08% | 84.06% | 84.06% | 84.22% |
0.5 | 83.93% | 83.93% | 83.93% | 83.93% | 83.93% | 83.93% | 83.95% | 83.95% | 83.97% |
1 | 83.98% | 83.98% | 83.98% | 64.80% | 64.80% | 77.43% | 78.61% | 77.39% | 77.45% |
2 | 64.80% | 64.80% | 64.80% | 64.80% | 64.80% | 64.80% | 64.83% | 64.83% | 64.80% |
3 | 64.96% | 64.96% | 64.96% | 64.96% | 64.96% | 64.96% | 64.96% | 64.96% | 64.97% |
4 | 65.30% | 65.30% | 65.30% | 65.30% | 65.30% | 65.30% | 65.31% | 65.31% | 65.31% |
5 | 65.30% | 65.30% | 65.30% | 65.30% | 65.30% | 65.30% | 65.29% | 65.29% | 65.30% |
Furthermore, as to the UIH cost, its change would not influence the EV model’s decisions on inventory because they are actually decided by the expected disaster demand. Hence, the FP results would not vary with the UIH cost and their values are same with the case of the disaster probability change ratio 1 as shown in Table
In summary, compared with the EV solution, the RP solution can ensure the supply while reducing cost as possible by taking all possible disaster scenarios into account. Hence, the SP model which can generate the RP solution has the significant value for the blood management in the setting of disasters.
Human blood is a crucial and precious relief good. In this paper, we proposed a BN-based two-stage SP model to study the BLN planning problem, taking the suddenly-occurring disasters into account. The planning decisions in the BLN and the long-term operational decisions under the setting of daily and disaster scenarios are optimized globally by the two-stage SP model. The stochastic scenarios used in the SP model is generated by the proposed BN model in which multiple disaster-related factors that contribute the emergency blood demand are considered in a systematic way. Based on earthquakes in Sichuan Province, China, the simulations and related sensitivity analysis are implemented. Several observations and corresponding implications can be obtained from the results.
First, the BNs are able to describe the disaster scenarios and thus, work with the SP model to generate the effective solutions. Results indicate all these solutions significantly outperform the corresponding EV solutions in terms of the effects of rescue. Moreover, the scenario set with different dimensionalities can give the consistent location results. Hence, it is important and necessary to put efforts on the modelling of the BNs.
Furthermore, the larger scenario set can cover more detailed disaster scenarios, but it does not mean the bigger dimensionality is better. Because it not only requires more modelling efforts but also may cause conservative solutions, such as more prepositioning inventory. Considering the preciousness and perishability of human blood, over-stock also should be avoided.
At last, the parameters, including the random factors involving the BNs and the deterministic ones, would influence the final decisions in different ways. For example, the total quantity of the emergency stock is only influenced by the some of the BN-related factors, while the decision on blood bank location is decided by the cost-related parameters, such as the unit inventory holding cost. Hence, by the utilization of the BNs, we can reveal the specific relationship between the random factors and the final decisions. For example, if the decision-maker cares about the planning of the total inventory, we should focus on the estimation of the severity of the injured, instead of the distribution of blood types, the disaster probability, and the unit inventory cost, etc. Thus, we can identify the specific efforts that should be made in terms of different planning purposes. This is the main benefit of the BNs.
There are some future directions as follows. At first, the BN model we proposed here only fits earthquakes. The applications of the BNs in the setting of other disasters still leave lots of opportunities. Moreover, we only consider the uncertainty of demand in this paper. But, when the earthquake occurs, blood shortage may be satisfied by urgent blood donations partially which are also random. And the more serious the disaster is, the more relief goods, including blood, the government and the people donate. Alfonso et al. [
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by National Natural Science Foundation of P. R. China (Grant Nos. 71572033 and 71832001).