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Statistical Model Checking of Relief Supply
Location and Distribution in Natural Disaster
Management
Sohaiba Iqbal, Muhammad Usama Sardar, Faiq Khalid Lodhi, and Osman
Hasan
School of Electrical Engineering and Computer Science (SEECS)
National University of Sciences and Technology (NUST)
Islamabad, Pakistan
{sohaiba.iqbal,usama.sardar,faiq.khalid,osman.hasan}@seecs.nust.edu.pk
Abstract. Examining the efficacy of natural disaster management readi-
ness and response activities is challenging due to the involvement of many
random and uncertain components. These uncertainties can be captured
by stochastic models. The analysis of these models is carried out using
Monte Carlo simulations to judge the effectiveness of natural disaster
management solutions. However, this approach uses static estimators,
which generally rely on sampled number of events taken from the ran-
dom space. The safety-critical nature of disaster management requires
a more quantifiable analysis. In order to overcome this challenge, we
propose to use statistical model checking for relief supply location and
distribution in natural disaster management. For illustration purposes,
we use the PRISM model checker to model and analyze a real-world sce-
nario of relief supply location and distribution while considering some key
factors, like demand of medical supplies at hospitals, predestined routes
from warehouses to hospitals, capacity of warehouses and transportation
plans.
Keywords: Natural Disaster Management, PRISM Model Checker, Sta-
tistical Model Checking, Relief Supply Location and Distribution.
1 Introduction
Our lives are overwhelmed with adverse events, such as natural disasters that
occur on specific spatial and temporal scales. Some typical examples of natu-
ral disaster include earthquakes, the outbreak of diseases, volcanic eruptions,
cyclones, tornadoes, floods and bridge collapses [37]. All these sudden adverse
events may lead to disastrous consequences. For example, the Pakistani earth-
quake (2005), registered 7.6 in the Richter scale, led to the death of 75,000 peo-
ple while injuring another 106,000 [1]. Primarily, such multifaceted disasters are
caused based on the interaction of countless components, which can be broadly
categorized in three major entities, i.e., the physical environment [36]; the so-
cial and demographic characteristics of communities [35,17]; and the constructed
environment, comprising of roads, bridges and buildings [24,26].
2 S. Iqbal, et al.
To reduce the effect of such disastrous consequences a comprehensive pre-
disaster and post-disaster planning is inevitable. Numerous solutions, such as
risks reduction and disaster prevention plans [5], relief supply locations and dis-
tribution (RSLD) plans [5], evacuation and emergency response plans [32], res-
cue and relief plans [32] and reconstruction plans [21] are generally considered.
Traditional disaster management planning [40,34] considers the static and deter-
ministic location and distribution of medical supplies only. The models of such
disaster management plans are not completely adequate for capturing all the
post-disaster cases due to absence of stochastic parameters, such as the storage
capacity of relief supplies [3], location and distribution of relief supplies [20,32],
planning and routing of vehicles [29,25], vehicle destruction, efficient path selec-
tion and path destruction. Similarly, these models fail to consider the availability
of selected vehicles and paths in the case of a disaster event. In this paper, we
have considered some of these stochastic factors, i.e., path selection, path de-
struction, vehicle destruction, selection of number of vehicles and traversal time
of a specific path.
To avoid the above-mentioned limitations, stochastic modeling is considered
quite practical to capture the random and unpredictable parameters of a natural
disaster management plan [6,27]. Stochastic programming (SP) [7] is one of the
most popular stochastic modeling approaches and is used mostly in planning,
preparedness and post-disaster activities of a natural disaster event [5]. In SP,
the uncertain data is incorporated into the objective function of a mathemati-
cal stochastic program. This uncertainty is usually differentiated by probability
distribution parameters. The level of uncertainty significantly varies in the case
of natural disasters, ranging from a few possible outcomes of the data, i.e., few
scenarios, to various possible outcomes. These possible outcomes are mathemati-
cally described in terms of elements w of a set W. For example, the set of possible
demands over the next few months can be described by such a set W.
Simulations of stochastic systems (Monte Carlo methods) are well stud-
ied and have been applied extensively to analyze real-world systems for many
decades. Statistical model checking (SMC) is a similar yet distinct method for
analyzing the behavior of randomized systems. [4]. Statistical model checking is
a technique that uses the discrete-event simulation (DES) as a primary method
to generate approximate results against the probabilistic properties. In order
to carry out this activity, SMC uses the built-in discrete-event simulator of the
PRISM model checker [2]. Essentially, this is achieved by sampling i.e., gener-
ating a large number of random paths through the model, evaluating the result
of the given properties on each run, and using this information to generate an
approximately correct result. This approach is particularly useful on very large
models when normal model checking is infeasible. Traditional simulation based
analysis do not allow us to analyze formally specified properties, which are ana-
lyzed using SMC by specifying them in stochastic temporal logic. This is a key
feature of SMC as formally specified properties are generally more expressive and
reliable. Moreover, in SMC, executions of a stochastic system are first sampled,
after which statistical techniques are applied to determine whether such a prop-
Statistical Model Checking of RSLD Management 3
erty holds. While the output of sample-based methods are not always correct,
statistical inference can quantify the confidence in the result produced [4].
Simulation based analysis of all existing Stochastic Programming models is
carried out as cause and effect models. Static estimators are used in these Monte
Carlo simulation based analysis methods for computing the probability. These
estimators generally rely on a limited number of events extracted from the ran-
dom space [28]. In SMC, a limited number of events (or paths) of the model are
executed. The major advantage is that we can extract these paths directly from
a formal model on the fly. Once we have extracted the paths, we estimate the
probability by testing the property over each one of them during the verifica-
tion step. For example, we compute r/N, where r is the number of paths which
validate the property and N is the total number of extracted paths (the sample
size). The paths are linear, so the verification is performed quite quickly. Thus,
to perform SMC, we only need to extract these runs and compute the mean.
Now, the estimator we choose converges to the real probability asymptotically.
It seems necessary to get an idea of the reliability that an experimentation offers
for a certain number of runs. This reliability can be characterized, for example,
by:
–a confidence interval, with its level of confidence and width
–an asymptotic confidence interval, similar to the normal one
–a statistical test, with two types of errors
A formal model has well-defined synchronization primitives with clear seman-
tics for modeling synchronous and asynchronous communication between nodes.
Thus, model checking of the probabilistic model determines exact probabilities
and performance bounds, which can never be guaranteed by simulation [16].
Also, for a given level of reliability, it could be interesting to run the optimal
number of steps required. [28].
In statistical model checking, runs (or paths) extracted directly from the
formal model are used as events for verifying quantitative properties [22]. The
analysis can be termed as complete if all the paths are analyzed, otherwise a
probabilistic bound on the analysis error is provided, which is one of the distin-
guishing features of statistical model checking compared to simulation [18,28].
The usage of a formal model to extract the paths is another feature that makes
statistical model checking superior than traditional simulation based analysis.
SMC is very helpful in scenarios where a discursive representation of the
global transition relation is required. This type of situation frequently appears
in case of cyber-physical systems [11]. The other applications of SMC include
the automated analysis of T-Cell Receptor Signaling Pathway [10] and in mixed
signal circuits where there is joint interaction between digital and analogue quan-
tities [9].
The main scope of this paper is to use statistical model checking for the
verification of RSLD in natural disaster management. We choose RSLD among
other components of natural disaster management because it is one of the most
challenging issues in the field of logistics and such operations are renowned for
their complexity. Improved planning and preparedness against natural disaster
4 S. Iqbal, et al.
can help saving lives and enabling communities to restart normal lives quickly by
reducing the sufferings of the survivors [41]. RSLD includes a number of stochas-
tic factors, such as path selection, path destruction [21], vehicle destruction [29],
selection of number of vehicles and traversal time of a specific path [25]. These
factors have a major effect on the adaptability, survivability and management of
RSLD and thus their analysis can provide very useful insights about the efficacy
of a given natural disaster management scheme.
The proposed framework for the statistical analysis of RSLD comprises of
the following main contributions:
1. The development of a formal stochastic model of RSLD incorporating its
probabilistic and non-deterministic factors. Probabilistic factors include path
selection, vehicle destruction and path destruction while non-deterministic
factors include path traversal time, disaster occurrence time and selection of
available vehicles [25].
2. The identification of a set of generic properties for the verification of the de-
veloped model, e.g., destination demand fulfillment with respect to stochastic
factors of RSLD.
3. An approach for the verification of the proposed properties by statistical
model checking for any given RSLD setting. This verification step mainly
allows us to determine the efficiency of the given RSLD by observing the
impact of above-mentioned factors on the destination demand fulfillment in
case of a natural disaster.
The proposed framework is implemented in the PRISM [22] model checking tool.
The main motivation behind this choice includes its ability to express stochastic
models as markov decision processes (MDPs). PRISM supports a wide range
of properties, such as linear temporal logic (LTL), probabilistic linear temporal
logic (PLTL) and Probabilistic Computation Tree Logic (PCTL) [22]. In order
to illustrate the effectiveness and utilization of the proposed model, we use it to
analyze a real-world scenario of RSLD implemented in Seattle, Washington [25].
2 Related Work
The pre-disaster phase include activities like disaster planning and disaster man-
agement while the post-disaster phases include activities that take into account
the stochastic considerations, such as location and distribution of supplies [32],
medical supply storage capacity, vehicle planning and routing path time calcu-
lation and selection [25]. The deterministic disaster management models, for in-
stance [34,40], lack stochastic considerations and are best suited for pre-disaster
phase instead of post disaster phase. To cater the modeling of post-disaster phase
of a natural disaster management scheme, stochastic modeling approach is con-
sidered quite triumphant as it allows incorporating its random and unpredictable
nature.
A comprehensive survey on RSLD models is presented by Ali et al. [5]. The
existing models are mainly categorized by considering some key sub-problems
of disaster management, such as relief supplies prepositioning, evacuation plan-
ning, permanent and temporary shelter location and relief supply distribution.
Statistical Model Checking of RSLD Management 5
A mixed integer multi-commodity stochastic network flow model regarding lo-
cation distribution was presented by Yi et al. [42] incorporating two key factors,
i.e. evacuation of resources and transportation of medical supplies. A two-stage
stochastic programming (SP) model was proposed by Mete et al. [25]. The first
stage activities include the ware house selection and the storage of medical sup-
plies, while the second stage incorporates the amount of medical supplies to be
delivered to hospitals and was further analyzed by a mixed-integer program-
ming (MIP) model incorporating parameters, such as vehicle assignments and
routing. Ozguven et al. [30] proposed an inventory management system while in-
corporating the uncertainty in demand after the occurrence of a disaster event.
Bozorgi-Amiri et al. [8] proposed a model regarding supply chain of relief supplies
distribution using Mixed integer non-linear programming. Duran et al. [14] de-
veloped a model regarding inventory location using MIP to analyze the response
time with respect to prepositioning of relief supplies. However, all the above-
mentioned analysis involves the usage of informal models for judging informally
specified properties, which are quite susceptible to modeling and specification
errors. Moreover, most of the existing work do not capture the randomized na-
ture of many elements in the model, like vehicle destruction [25,32,39] or path
destruction [25,39]. We overcome these limitations by using MDP to capture
the behavior of RSLD and expressing the desired characteristics as probabilistic
quantitative properties in the PRISM model checker.
Besides the above-mentioned informal models, some formal models have also
been used. Cloth et al. [12] used probabilistic model checking to check the en-
durance of a disaster management scheme. They modeled the system operations
as Continuous-Time Markov Chain that is analyzed using model checking algo-
rithms in the stochastic Petri nets to assess the system survivability. Fahadland
et. al. [15] analyzed the resilience of the disaster management schemes under
different spatiotemporal scales. They proposed an adaptive process incorporat-
ing a set of scenarios by using Petri nets. The system behavior is created and
adjusted by the model at run-time using the adaptation operator based on the
given scenarios. These existing formal methods based analysis certainly advocate
the usefulness of using formal methods in this safety-critical domain. However,
their context is not to judge the performance of RSLD, which is the main scope
of the current paper. To the best of our knowledge, this paper presents the first
formal model and a set of formally specified properties to judge the function-
ality and performance of RSLD based scheme on the statistical model checking
principles.
3 Preliminaries
This section provides a short introduction to statistical model checking and the
PRISM model checker to facilitate the understanding of the rest of the paper.
3.1 Statistical Model Checking
Statistical model checking is a computationally effective verification technique
based on selective system sampling. It tends to solve the model checking problem
by using simulation and hypothesis testing [23,33] based approaches. The main
6 S. Iqbal, et al.
idea is to simulate the given system model for a subset of its all possible execu-
tions, and then use the available samples of execution to statistically infer if the
given property [43] is satisfied or not. Statistical model checking can be used for
analyzing large models in a very effective manner. Due to the usage of hypoth-
esis testing, there is always some probability of error in the verification results.
Statistical model checkers cater for this problem by providing error bounds for
each verified property. The verification accuracy is determined by these bounds
and we can get fairly accurate results by exploring sufficiently large number of
sample paths.
Statistical model checking offers various advantages compared to other clas-
sical analysis methods. Firstly, the model only requires the system’s sample
executions, which makes statistical model checking a very flexible analysis tech-
nique and it can be applied to analyze a wide range of systems, including infinite
state systems. Secondly, the approach can be used to analyze a wide range of
probabilistic properties [23], such as (PLTL). The main difference between the
Monte Carlo method and statistical model checking is that in the Monte Carlo
method, the computation of an approximate probability with a statistic esti-
mator is based on a limited number of events from the random space while, in
statistical model checking, the limited number of events are runs (or paths) of a
formally specified model. The major advantage is that these paths are extracted
directly from a complete and rigorous model and the probability is estimated by
testing the property over each of the extracted path. Moreover, the verification
time for the properties is very short as the paths are linear [28].
3.2 PRISM Model Checker
PRISM [22] is a widely used probabilistic model checker for formal modeling
and analysis of schemes or systems that exhibit probabilistic or random behav-
ior.The probabilistic behavior of systems is basically captured based on the Re-
active Modules formalism [2]. PRISM supports modern algorithms and symbolic
data structures by the agency of Binary Decision Diagrams (BDDs) and Multi-
Terminal Binary Decision Diagrams (MTBDDs) along with statistical model
checking using its discrete events based simulation engine. The verification of
Markov processes, i.e., continuous-time Markov chains (CTMC), discrete-time
Markov chains (DTMC), Markov decision processes (MDP), and probabilistic
timed automata (PTA) is also a distinguished feature of this tool. The com-
ponents of the given distributed system are modeled as modules, which can be
synchronous or asynchronous in nature incorporating variables and commands.
The possible states of modules are described by variables while its behavior is
described by commands. The variables in PRISM can be declared (globally, lo-
cally) using both integer and Boolean data-types. In PRISM, the state space
can be explored automatically and manually. In the case of manual state space
exploration, a user can define the number of steps and time for simulation. Dif-
ferent simulation paths are generated that can be backtracked and stored in a
text file. In PRISM, the verification of Markov processes is performed by the
conformance of formulated properties. Properties in PRISM are expressed using
Linear Temporal Logic (LTL), Probabilistic Linear Temporal Logic (PLTL) and
Statistical Model Checking of RSLD Management 7
Probabilistic Computation Tree Logic (PCTL). PRISM automatically verifies
these properties against the associated model and the results can also be plotted
and logged [22]. For performing statistical model checking, PRISM supports four
different methods i.e., Confidence Interval (CI), Asymptotic Confidence Inter-
val (ACI), Approximate Probabilistic Model Checking (APMC )and Sequential
Probability Ratio Test (SPRT).
The parameters of CI are Confidence (alpha), Number of samples (N) and
Width (w). On the basis of the number of samples and given confidence level,
this method provides confidence interval for the approximate value generated
for a property P= ?. Let X be the actual result of the property P= ? and Y
be the approximation generated then actual result lies in the confidence interval
[Y-w,Y+w]100 (1-alpha)%. ACI has the same functionality as CI except that it
uses Normal Distribution for approximation when finding the confidence inter-
val. The APMC method provides a probabilistic guarantee on the accuracy of
the approximate value generated for a P=? property. SPRT is more appropriate
for bounded properties, such as P<=p[..] and P>=p[..]. It is based on accep-
tance sampling techniques. It uses Wald’s sequential probability ratio test, which
generates a series of samples, determining when an answer with sufficiently high
confidence can be given on runtime. We used the CI method in this paper as this
is the default method and is considered more effective in performing statistical
model checking [28].
4 Proposed RSLD Model
The main objective is to develop a formal model of RSLD in natural disaster
management that can be specialized to analyze any real-world scenario of RSLD.
For this purpose, we have identified three major components, as depicted in Fig.
1.
1. Source: Represents a facility, where medical supplies are stored and trans-
portation vehicles are kept.
2. Destination: Represents a facility, where the demand of a medical supply
arises.
3. Paths: Represents route information between sources and destinations.
These components are used as inputs to the proposed RSLD model incorporating
the key stochastic parameters as exhibited in Fig. 1. The detail of the above-
mentioned inputs is described below:
4.1 Source
The source location and storage capacity for emergency supplies is the most im-
portant part of the disaster readiness process [25]. The nature of sources depends
on the type of disaster, i.e., earthquake, floods, volcanic eruption, windstorms
etc. For example, a warehouse can be a source of commodities for earthquake
based disaster. Sources should be capable of satisfying the needs of one or more
destinations depending upon their location, capacity and their distance from the
destination. In addition to relief supplies, sources also have transportation vehi-
cles to fulfill the destination demand. While deciding the transportation modes,
8 S. Iqbal, et al.
Fig. 1: Proposed RSLD Model
the two main factors are considered: the requirements (Urgency, distance to the
destination and other conditions, such as transportation routes, weather, etc.)
and feasible forms of transport (available means, such as trucks, boats, planes
etc., cost in terms of time, transmission capacities, etc.) [38,13].
In our proposed RSLD model, we assume that sources are always available
and operational to meet the required needs. This assumption is made to eval-
uate the quality of the underlying disaster management plan as, in the case of
unavailability of sources, the required demand can never be fulfilled. Our model
also incorporates vehicle destruction as this event can happen with a relatively
high probability during a disaster. To the best of our knowledge, this feature is
not yet incorporated in any other RSLD model.
4.2 Destination
The destination is defined as the facility that generates the demand to receive
relief supplies from the sources. The main goal of the logistics chain in relief
operations is delivering the aid to the affected people [31]. Destinations can be
different with respect to the disaster type, e.g., hospitals or temporary medical
centers are commonly used destination points in case of earthquakes and floods.
The destination should be properly identified as its role is equally important in all
of the three stages of natural disaster management - pre-disaster prevention and
planning, disaster situation management and post-disaster phases of resolution
and return to normality [21,31]. The successful and timely delivery of the relief
supplies to their required destinations is the only way to lessen the impact of
natural disaster in a scenario where the severity and frequency of natural disaster
are rising alarmingly [31]. In our proposed RSLD model, the destinations are
Statistical Model Checking of RSLD Management 9
categorized into primary and secondary destinations. Primary destination is the
one closest to the source and secondary destination is the next to the closest
source. On the basis of the shortest distance from the source to the destination,
we are assuming the range of the primary destinations from 1 to 5 while the
secondary destinations from 1 to 2.
4.3 Paths
The transportation of supplies from the source to the destination is a vital part
of the logistic chain [38]. The late delivery of supplies can adversely affect the
performance of RSLD. Therefore, the main challenge is to ensure secure trans-
portation and timely delivery of supplies by choosing appropriate transporta-
tion modes (roads, waterways, air) [37,13]. In general, humanitarian operations
largely use the road and air transport. However, other modes e.g., water can
efficiently support distribution activities in both the strategy of delivery and
logistical support to the process. Our proposed RSLD model selects the shortest
path among all the available paths. Paths or routes selected for transportation
of relief supplies are also vulnerable to damage as a consequence of a natural
disaster. To overcome this vulnerability, our model also incorporates path de-
struction.
5 Modeling RSLD in PRISM
We selected MDP to develop the formal model for RSLD because it allows us to
capture both probabilistic and non-deterministic factors of the proposed RSLD
model. Probabilistic factors include path selection, vehicle destruction and path
destruction while non-deterministic factors include path traversal time, occur-
rence time of disaster and selection of available vehicles. This model can be used
to analyze the proposed properties by varying different RSLD parameters in the
PRISM model checker.
The inputs of the proposed model are described in Algorithm 1 where, S,
Dand Prepresents the total number of sources, destinations and the intercon-
necting paths. srepresents a specific source, drepresents a specific destination
and prepresents a specific path with values ranging 1to S,1to Dand 1to P
respectively. The primary and secondary destinations of a source are also inte-
grated into the model by developing separate modules. The basic functionality of
modules remains the same and is illustrated in Algorithm 1. To handle multiple
sources and destinations the model also incorporates the concept of module re-
naming which allows duplication of existing modules. The working of our RSLD
model is divided into two phases, i.e, at the pre-disaster and post-disaster phase.
5.1 Pre-disaster Phase
The pre-disaster phase includes the coverage of events that can be taken up be-
fore the occurrence of a disaster. In the proposed model, we use the minimum
distance from the sources as the foremost criteria for destination selection [25].
The original map of the affected area is used to calculate the distance from each
source to the destination, which is then used to determine the primary destina-
tion, i.e., the nearest destination for a source, and the secondary destination.
10 S. Iqbal, et al.
Algorithm 1 : Source Destination Module
Input:
S Dd;Set of primary destinations for each source s[D1...Dd], D1 represents destination 1 where
destination range is from 1 to d
S ExDd; Set of secondary destinations for each source s[ExD1..ExDd], where E xD1 represents
alternative destination 1
T D : [1 : 3]; Disaster time (1, 2 and 3 represents rush, working and nonworking hours, respec-
tively) and P(D)=1/3
T ime p s Dd : [1 : P]; Time of each path p from source sto destination d
W w work ing : [1 : K]; Time coefficients for working hours
W w nonwork ing : [1 : K]; Time coefficients for non-working hours
W w rush : [1 : K]; Time coefficients for rush hours
where
s: [1 : S]; Number of sources
d: [1 : D]; Number of destinations
p: [1 : P]; Number of paths
Vs: [0..V ]; No of Vehicles in source s.
Cs: [0..C]; Source total capacity.
Cv: [0..v]; Vehicle transportation capacity .
Dd: [0..R]; Destination Demand or Requirement.
P pathdes, [0.1...0.9]; probability of path destruction
P vehdes, [0.1...0.9]; probability of vehicle destruction
veh s Dd; No of vehicles assigned.
avg speed; Average speed of vehicles.
dist s Dd; distance of destination dfrom source s
Xs, Y s; X and Y coordinates of source s
Xd, Y d; X and Y coordinates of destination d
Path Time Calculation:
1: if Disaster = 1 then
2: 1/3:(T D = 1) + 1/3:(T D = 2) + 1/3:(T D = 3);
3: T D = (1|2|3)−> dist s D d = (pow((Xd −Xs),2) + pow((Y d −Y s),2),1/2)));
where dist s Dd is computed from speed formula;
4: T D = (1|2|3) →(T ime 1s D d =dist s Dd/avg speed);
5: T D = (1|2|3) →(T ime p s Dd =T ime 1s D d +random value p);
6: T D = 1 →T ime p s Dd =T ime p s D d ∗W w working;
7: T D = 2 →T ime p s Dd =T ime p s D d ∗W w nonworking;
8: T D = 3 →T ime p s Dd =T ime p s D d ∗W w rush;
Path Destruction and Path Selection:
9: P pathdes : (T ime p s Dd =Time p s Dd ∗0) + (1 −P pathdes) : (T ime p s Dd =
T ime p s Dd ∗1);
10: Rs d =min(T ime 1s Dd......T ime p s Dd);
where Rs d is Route form source sto destination d;
11: if Rs d != 0 then
12: path selected;
13: end if
Vehicle Selection:
14: if total(num primary destinations)>= 1 then
15: veh s Dd =Vs/num;
where num is number of primary destinations of source s
16: else
17: V eh s Dd = (1/C v)∗Dd;
18: end if
Vehicle Destruction:
From Source stowards the Destination d;
19: (S d =s)&Vs! = 0&save v Dd!=0→
P vehdes/save v Dd : (found v Dd =save v IDd −1) + .. +Pv ehdes/save v Dd :
(found v Dd = 0) + 1 −P v ehdes : (found v Dd =save v Dd)
S d =srepresents source sfor destinations d,
where save v D d is number of vehicles transported with goods on the selected path,
found v Dd is number of non-destructed or safe vehicles;
20: if (found v Dd = 0|found v Dd = 1|..f ound v Dd =save v D d)then
21: RD d =Dd+ (save v Dd −found v Dd)∗Cv;
22: Vs=Vs+found v Dd;
where RD d is the remaining demand of destination d;
23: end if
From Destination dtowards the Source s;
24: same as line number 19
25: if (found v Dd = 0|found v Dd = 1|..f ound v Dd =save v D d)then
26: Vs=Vs+found v Dd;
27: end if
28: if (RD d > 0)&(Cs! = 0&Vs! = 0) then
29: go to line number 9;
30: else
31: if (RD d > 0)&(Cs= 0|Vs= 0)&(S E xDd =S Dd)then;
32: W h h =w;
secondary source selected and goto line number 9;
33: else
34: if (RD h > 0)&(Cw= 0|Vw= 0) then
secondary source capacity is consumed or vehicles are utilized;
35: Destination demand is not fulfilled
36: end if
37: Destination demand is fulfilled
38: end if
39: end if
40: end if
Statistical Model Checking of RSLD Management 11
5.2 Post-disaster Phase
Post-disaster phase includes the coverage of events that can be taken up after
the occurrence of a disaster. The disaster event, based on its occurrence time is
categorized into rush hours (R), working hours (W) and non-working hours (N)
[25]. The occurrence time of disaster T D is selected non-deterministically in our
model as shown in Path Time Calculation part of Algorithm 1.
The traversal time of all the paths from the source to the destination is calcu-
lated. The modeling of this step in PRISM is performed by calculating the nor-
mal traversal time by using the formula T ime 1s Dd =dist s Dd/avg speed,
where avg speed is the average speed of the vehicle and dist s Dd is the min-
imum distance from the source to the destination. Similarly, the traversal time of
other paths are calculated by addition of a random value in the above computed
traversal time and is also dependent on the occurrence time of a disaster event.
The next step is to check the path availability based on the damages caused by
a disaster event, e.g., landsliding etc. This activity is handled in Path Destruction
and Path Selection part of Algorithm 1. If the path with minimum traversal
time is available then it is selected, otherwise the next available shortest path is
chosen.
To proceed further, the transportation resources are chosen, i.e., the number
of vehicles meant for transporting relief supplies from the source to the destina-
tion. The RSLD model keeps an update regarding the status of vehicles while
sending medical supplies on a selected path from source to destination and vice
versa. On detection of a vehicle destruction, the model recomputes the num-
ber of vehicles and the destination demand. Upon demand fulfillment, the main
objective of the model is considered to be accomplished, otherwise, the next
alternate source for the destination is selected based on the minimum distance
criteria. This set of activities continues until the destination demand is fulfilled.
As shown in Vehicle Selection part of Algorithm 1, the vehicle selection activity
is performed according to the number of primary destinations, i.e., the avail-
able vehicles are distributed equally among all primary destinations. In case of
single primary destination, the available vehicles are assigned according to the
destination demand. The selected vehicles veh s Dd carrying relief supplies are
transported to the destination accordingly. Afterwards, the vehicle destruction
along-with demand fulfillment of the destination is checked. Based on above out-
put, the remaining demand of destinations RD d, remaining capacity of sources
RC s, and remaining vehicles RV s are computed. In case of vehicle destruction,
the remaining demand of the destination RD d and remaining vehicles RV s are
updated accordingly, as shown in the Vehicle Destruction part of Algorithm 1.
If no vehicle destruction occurs, then only the total number of available vehicles
V s are updated. Similarly, vehicle destruction from a destination to a source
is monitored and the total number of available vehicles V s are updated. After-
wards, if the remaining demand of the destination RD d is greater than 0 and the
remaining source capacity RC s or remaining number of vehicles are consumed
then destination dselects the secondary source. Otherwise, the destination se-
12 S. Iqbal, et al.
lects the same source for demand fulfillment. If the remaining demand RD d
becomes 0, we conclude that the destination demand is fulfilled.
After modeling RSLD in PRISM, The next step is to verify the desired prop-
erties and thus analyze the model using statistical model checking.
6 Analyzing RSLD using Statistical Model Checking
The analysis process of the RSLD using statistical model checking principles is
depicted in Fig. 2.
The first step is to develop the RSLD model while identifying its sources,
destinations, and paths, described as RSLD Modeling Parameters in Fig. 2.
After modeling, a set of functional properties based on the working description
of the given RSLD model is identified. The functional properties ensure the
precise and accurate working of the model. These properties are represented by
quantitative operators in PRISM. The RSLD performance parameters, i.e., the
probability of path destruction, the number of destinations, expected time and
vehicle destruction are used to analyze the impact on the demand fulfillment
of the destination. The sub-attributes of vehicle destruction include a number
of lost vehicles and the probability of vehicle destruction. The RSLD modeling
parameters and performance parameters are translated into the MDP model.
PRISM, based on the LTL properties formally verifies the MDP model and
provides the quantitative information. This quantitative information can play a
vital role in developing effective RSLD model.
The transportation of relief supplies from source to a destination requires
available paths and vehicles and it depends on a highly vulnerable transporta-
tion system thus making it a very critical parameter in the analysis. Demand
fulfillment is analyzed with respect to a varying number of destinations instead
of a number of sources to study the capability of a source to fulfill the demand
of multiple destinations.
We now provide a set of properties to formally analyze the functionality
of RSLD with respect to sources and destinations. For instance, we evaluated
the probability of eventually fulfilling the destination demand by expressing the
following property:
P=?[F RD d = 0 & var d n = 1] (1)
where RD d represents the total demand of destination dwith an associated flag
var d n. The destination dvaries from 1 to Dand ncorresponds to the num-
ber of primary destination of source s. This property evaluates the probability
of demand fulfillment of a destination dwith respect to a source. Property 1
corresponds to the demand fulfillment of destination dwhen RD d becomes 0
and var d n updates its status to 1. It can also be used for the verification of
demand fulfillment of primary and secondary destinations.
It is quite important in the case of disasters to meet the destination de-
mand within a certain time. This makes the expected time a vital parameter
Statistical Model Checking of RSLD Management 13
Fig. 2: Proposed Analysis Framework
in assessing the performance of RSLD model. Keeping it in view, we have pro-
posed another property which evaluates the probability of destination demand
fulfillment with respect to the expected time.
P=?[F RD d = 0 & var d n = 1 &
estimatedtime Dd <=expectedtime Dd](2)
Property 2 corresponds to the demand fulfillment of destination within expected
time when RD d is set to 0 and var d n is set to 1 and the estimated time of
demand fulfillment estimatedtime Dd is less than or equal to the expected
time expectedtime Dd. Expected time is computed by taking into account the
best and the worst case scenarios of the given RSLD model and the estimated
time is the approximate time computed by the number of steps involved in the
destination demand fulfillment in PRISM. The best case scenario occurs when
the source has the maximum capacity for destination demand fulfillment with no
path destruction and vehicle destruction. The worst case scenario occurs when
the source has only a single path to reach its destination and the path has a
maximum traversal time with an unavoidable probability of vehicle destruction.
14 S. Iqbal, et al.
7 Analyzing RSLD for Seattle
For illustration purposes, we applied the proposed RSLD model to analyze the
RSLD in Seattle [25] as shown in Fig. 3. To specialize the proposed generic
model to the given RSLD scenario, the following translations are considered.
Warehouses and Hospitals correspond to sources and destinations while only
roads and trucks are considered as transportation modes and medium. Seattle
Fig. 3: Seattle Map with Hospitals and Warehouses
has five warehouses represented by square boxes and ten hospitals represented by
circles, as depicted in Fig. 3. Warehouse 1has five primary hospitals, i.e., 3,4,
8,9and 10 and a secondary hospital i.e., 1as shown in Fig. 3. Fig. 4 represents
a more detailed and scaled down version of the same scenario incorporating
only warehouse 1with associated hospitals for better understanding. In Fig.
4a, warehouse 1has multiple available paths to connect with a hospital. The
traversal time for four different available paths between warehouse 1and hospital
4are labeled as T1,T2,T3 and T4. After disaster, as shown in Fig. 4a, the
path traversal times T1,T2,T3 and T4 are changed, based on the occurrence
time of the disaster, i.e., working, rush and non-working hours. The updated
values of path traversal time are T1+∆T,T2+∆T,T3+∆Tand T4+∆Twhere ∆T
represents the change in time. The value of ∆Tis unique for each path. Similarly,
the post-disaster traversal times against all other hospitals are computed.
In Fig. 4b, warehouse 1checks the availability of all active paths and the
paths having the shortest traversal time are selected for transporting relief sup-
plies to the respective hospitals. It also depicts a scenario of vehicle selection
and its destruction from warehouse to hospital (WH→H) and from hospital to
warehouse (H→WH). Considering WC is the warehouse capacity, VC is the vehicle
capacity, D-EX1 is the demand of secondary Hospital and D-3,D-4,D-8,D-9,
D-10 are the demands of the corresponding primary hospitals. Vehicles avail-
able at warehouses are assigned to all primary hospitals and relief supplies are
transported on the pre-selected paths. During transportation we also perform
Statistical Model Checking of RSLD Management 15
Warehouse 1
Hospital 8
Hospital 4
Hospital 9
Hospital 1
Hospital 10
Hospital 9
Hospital 3
(a) Path Selection after disaster
D - 4
D - 8
WC
Hospital 4
Warehouse 1
Hospital 9
Hospital 8
Vehicles Capacity = VC
D – EX1
D -
9
D - 3
D - 10
Hospital 9
Hospital 3
Hospital 1
Hospital 10
(b) Vehicle Selection and Destruction
Fig. 4: Case Study Scenario
the vehicle destruction check to update the status of vehicles along with the
demand of hospitals.
To verify the demand fulfillment property of any primary hospital, e.g., Hos-
pital 3of Warehouse 1, we modify the generic Property 1 as follows:
P=?[F rem H3 = 0 & var ID 1 = 1] (3)
where rem H3 is the remaining demand of Hospital 3 and var ID 1 is its associ-
ated flag. The same property is used for the verification of demand fulfillment of
other primary and secondary hospitals of Warehouse 1. Warehouse 1, after fulfill-
ing the demand of its primary hospitals will satisfy the demand of its secondary
hospital upon availability of its capacity. Similarly, we used Property 1, to verify
the demand fulfillment of primary and secondary hospitals of warehouses i.e.,
2,3,4,5 as shown in Fig. 3. In order to verify the accumulative demand fulfillment
of primary hospitals of warehouse 1, we modify property 1 as follows:
P=? F[rem H3 = 0 & var ID 1 = 1 & rem H4 = 0 &
var ID 2 = 1 & rem H8 = 0 & var ID 3 = 1 & rem H9 = 0 &
var ID 4 = 1 & rem H10 = 0 & var ID 5 = 1]
(4)
where rem H4,rem H8,rem H9 and rem H10 are the remaining demands of
hospital 4,8,9 and 10 and var ID 2,var ID 3,var ID 4, and var ID 5 are
their associated flags. The proposed model can be used to obtain interesting
insights about the given RSLD by varying the probability of vehicle and path
destruction, number of hospitals and vehicles. We used Version 4.3 of the PRISM
model checker along with CentOS Linux (6.5), storage capacity (22TB) and
total memory (1.312TB). The system has 34 computing nodes, 272 processing
cores and a peak performance of 132 Teraflops. We also used default maximum
path length (10000), confidence (0.01) which is 99% confidence level, simulation
16 S. Iqbal, et al.
method CI and the number of samples (100000) in statistical model checking
for the verification of the properties. The sample size (100000) is reasonably fine
since very low probabilistic bound is observed against property verification.
In order to analyze the RSLD, we computed the probability of hospital de-
mand fulfillment against path and vehicle destruction probability by using the
above-mentioned property 1. For example, Figs. 5a and 5b depict the demand
fulfillment probability of primary hospitals 3, 4, 8, 9 and 10 of warehouse 1 for
the scenario consisting of one warehouse and six hospitals (five primary and
one secondary hospital) against the probability of vehicle destruction and the
probability of path destruction, respectively.
The graph in Fig. 5a exhibits a decreasing trend in hospital demand ful-
fillment probability with respect to an increase in the probability of vehicle
destruction assuming no path destruction.
We have made a generalized formal model of RSLD in natural disaster man-
agement which can analyze any real-world scenario. For verification and analysis
purpose, we chose to use the RSLD of Seattle as a case study. The corresponding
map is shown in Fig. 3 [25]. It has five warehouses and ten hospitals. We altered
the demand of the hospitals to analyze the impact of vehicle destruction on hos-
pital demand fulfillment. If a hospital has a demand such that the number of
available vehicles are not sufficient for demand fulfillment, the chance of vehicle
destruction is more as compared to the case in which the hospital demand is
low because we are observing two way vehicle destruction. It is important to
note that the proposed model is generic so we can cater for more hospitals with
different demands as well but we have verified this particular scenario because
of the considered case of Seatle RSLD. [25].
Moreover, we also change the hospital demand, i.e., Hospital 3 gets a higher
demand compared to Hospital 10 to analyze its impact on the demand fulfillment
probability. We can thus deduce that with an increase in hospital demand, the
impact of vehicle destruction increases. Similarly, for the same scenario, the
probability of hospital demand fulfillment decreases with an increase in the path
destruction probability assuming no vehicle destruction as shown in Fig. 5b.
0.4
0.6
0.8
1
Hospital Demand Fulfillment
Hospital 3
Hospital 4
Hospital 8
Hospital 9
Hospital 10
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Hospital Demand Fulfillment
Probability of Vehicle Destruction
(a)
0.4
0.6
0.8
1
Hospital Demand Fulfillment
Hospital 3
Hospital 4
Hospital 8
Hospital 9
Hospital 10
0
0.2
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Hospital Demand Fulfillment
Probability of Path Destruction
(b)
Fig. 5: Demand Fulfillment of Hospitals Vs Probability of Vehicle and Path
Destruction
Statistical Model Checking of RSLD Management 17
0.6
0.8
1
Hospital Demand Fulfillment
Exp Time 15
Exp Time 20
Exp Time 25
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Hospital Demand Fulfillment
Probability of Path Destruction
(a)
0.6
0.8
1
Hospital Demand Fulfillment
Exp Time 40
Exp Time 42
Exp Time 45
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Hospital Demand Fulfillment
Probability of Vehicle Destruction
(b)
Fig. 6: Demand Fulfillment of a Hospital Vs Probability of Path and Vehicle
Destruction within the Expected Time
We also analyzed the hospital demand probability within expected time for
the same scenario. For illustration of the effect of expected time on demand ful-
fillment, the demand fulfillment of a single hospital, i.e., Hospital 3of warehouse
1is plotted against the probability of path destruction and probability of vehicle
destruction by varying expected time. The required time or the expected time
is the time stated by respective hospitals to meet their corresponding demands.
The values of expected time are selected based on the best and worst cases of
hospital demand fulfillment time. In Fig. 6a, the demand fulfillment of a hospital
within an expected time of 15,20 and 25 units is observed while varying the
probability of path destruction. The slope of the curve for the expected time
equal to 15 is steeper compared to the curve having an expected time equal to
20. Similarly, the slope of the curve for the expected time equal to 20 is steeper
as compared to the curve having expected time equal to 25. This trend depicts
that with a decrease in expected time of demand fulfillment, the impact of path
destruction increases. This trend enables a disaster manager to route its vehicle
on potentially safer paths in case of having strict constraints on the expected
time and route remaining vehicles on alternate paths to efficiently fulfill the
demand.
Fig. 6b shows the demand fulfillment of a hospital within an expected time of
40,42 and 45 units based on the probability of vehicle destruction. So basically
we wanted to have three integer values within the interval of 40-45. The middle
value could be 42 or 43 and we chose 42. As anticipated, the expected time of
hospital demand fulfillment in case of vehicle destruction is higher as compared
to path destruction. Reason being, path destruction and path selection activity
is performed prior to routing of vehicles as depicted in Algorithm, 1 thus max-
imizing the chance of demand fulfillment. All the curves exhibit a decreasing
trend in hospital demand fulfillment with respect to an increase in the proba-
bility of vehicle destruction assuming safe paths. This experiment indicates that
a curve with a larger expected time has more probability of demand fulfillment
and a larger value of slope as compared to a curve with a smaller expected time.
This trend enables a disaster manager to route the mechanically fit vehicles (to
18 S. Iqbal, et al.
avoid vehicle destruction) to hospitals having larger expected time and route the
remaining vehicles to hospitals having smaller expected times to efficiently fulfill
the demand.
0.4
0.6
0.8
1
Hospital Demand Fulfillment
1 Hospital
2 Hospitals
3 Hospitals
4 Hospitals
5 Hospitals
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Hospital Demand Fulfillment
Probability of Vehicle Destruction
(a)
0.4
0.6
0.8
1
Hospital Demand Fulfillment
1 lost vehicle
2 lost vehicles
3 lost vehicles
4 lost vehicles
5 lost vehicles
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Hospital Demand Fulfillment
Probability of Vehicle Destruction
(b)
Fig. 7: Demand Fulfillment of Hospital Vs Probability of Vehicle Destruction
with respect to the number of Hospitals and number of Vehicles Lost
We analyzed the effect of varying number of hospitals on probability of vehi-
cle destruction and hospital demand fulfillment as shown in Fig. 7a. It is worth
mentioning here that the hospital demand fulfillment is independent of a spe-
cific number i.e., hospital 3, 4, 8, 9 and 10. The same impact is observed for
each case. Fig. 7a, represents the demand fulfillment of Hospital 3which is the
first primary hospital of Warehouse 1. It exhibits a decreasing trend in hospital
demand fulfillment with respect to an increase in the probability of vehicle de-
struction for different number of served hospitals i.e., 1 to 5. It is evident from
the graph that the demand fulfillment of the hospital decreases with an increase
in the number of serving hospitals for a specific value of probability of vehicle
destruction, because of resource sharing, i.e., capacity and number of vehicles of
a warehouse. Similarly, we analyzed the effect of varying number of lost vehicles
on probability of vehicle destruction and hospital demand fulfillment as shown
in Fig. 7b. It shows a decreasing trend in hospital demand fulfillment with re-
spect to an increase in the number of lost vehicles and probability of vehicle
destruction. We also analyzed the accumulative effect of the probability of vehi-
cle and path destruction on the probability of demand fulfillment as depicted in
Fig. 8. This is the probability of the event when both the path and vehicles are
destroyed simultaneously. This event can happen even though its occurrence is
rare compared to the ones where only the path or only the vehicles are destroyed.
As expected, the slope of the curve is steeper as compared to the one given in
Fig. 5. In other words, we can say that as we increase the number of parameters
in the transportation system, i.e., path and vehicle destruction, the impact gets
inevitable and even more critical in nature. The probability of error or the proba-
bilistic bound for each verified property is approximately 0.0003168941551804785
based on a 99.0% confidence level, indicates the quality of our results.
Statistical Model Checking of RSLD Management 19
Probability of Hospital
Demand Fulfillment
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2 0.3 0.4 0.5 0.60.7
0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Fig. 8: Demand Fulfillment of Hospital Vs Accumulative Probability of Vehicle
and Path Destruction
We have considered a number of random factors associated with RSLD in
the analysis including path selection, path destruction, vehicle selection, vehicle
destruction and traversal time of a specific path. To the best of our knowledge,
all these factors have not been tackled simultaneously in any existing model of
RSLD. The development of a formal stochastic model of RSLD enabled us to
verify interesting properties regarding demand fulfillment of hospitals in terms
of the probability of vehicle destruction, the probability of path destruction, the
probability of vehicle destruction within expected time, the probability of path
destruction within expected time, the probability of vehicle destruction with
varying number of hospitals, the probability of vehicle destruction with varying
number of lost vehicles and the probability of vehicle and path destruction. These
analysis results facilitate the disaster managers to adopt a proactive approach
in fulfilling the demand of hospitals in exigency scenarios.
8 Conclusions
The main contribution of this paper includes the development of a formal model
of RSLD in natural disaster management that can be specialized to analyze
any real-world scenario of RSLD. In modeling RSLD, a number of stochastic
factors are considered, such as path selection, path destruction, vehicle selec-
tion, vehicle destruction and traversal time of a specific path. To the best of our
knowledge, these stochastic factors are not implemented simultaneously in any
existing RSLD model. One distinguishing feature of developing such a model is
its basis on a formal semantic of systems as it allows the disaster manager to
reason about very complex behavioral properties of the system using statistical
model checking. For example, we analyzed the number of vehicles required to
fulfill the hospital demand within some expected time with respect to the accu-
mulative probability of path and vehicle destruction, hospital demand fulfillment
20 S. Iqbal, et al.
with respect to the expected time etc. In order to illustrate the usefulness of the
proposed framework, we used it to analyze a real-world RSLD scenario based in
the Seattle area. The analysis results demonstrate the effectiveness of the pro-
posed framework, which can be further extended to formally model and analyze
other domains of natural disaster management as well, e.g., evacuation shelters
and evacuation planning.
References
1. 25 worst natural disasters ever recorded. http://list25.com/
25-worst- natural-disasters- recorded/ (2013)
2. Prism website. http://www.prismmodelchecker.org/ (2016)
3. Afshar, A., Haghani, A.: Modeling integrated supply chain logistics in real-time
large-scale disaster relief operations. Socio-Economic Planning Sciences 46(4), 327–
338 (2012)
4. Agha, G., Palmskog, K.: A survey of statistical model checking. ACM Transactions
on Modeling and Computer Simulation (TOMACS) 28(1), 6 (2018)
5. Ali, I., Tanchoco, J.M., Lee, S.: Evacuation shelter location problem for emergency
planning: Review. In: Industrial and Systems Engineering Research. pp. 1–10. Y.
Guan and H. Liao, eds. (2014)
6. Barbarosoglu, G., Arda, Y.: A two-stage stochastic programming framework for
transportation planning in disaster response. Journal of the operational research
society 55(1), 43–53 (2004)
7. Birge, J.R., Louveaux, F.: Introduction to stochastic programming pp. 421–477
(2011)
8. Bozorgi-Amiri, A., Jabalameli, M.S., Alinaghian, M., Heydari, M.: A modified par-
ticle swarm optimization for disaster relief logistics under uncertain environment.
Advanced Manufacturing Technology 60(1), 357–371 (2012)
9. Clarke, E., Donz´e, A., Legay, A.: Statistical model checking of mixed-analog cir-
cuits with an application to a third order δ-σmodulator. In: Haifa Verification
Conference. pp. 149–163. Springer (2008)
10. Clarke, E.M., Faeder, J.R., Langmead, C.J., Harris, L.A., Jha, S.K., Legay, A.:
Statistical model checking in biolab: Applications to the automated analysis of
t-cell receptor signaling pathway. In: International Conference on Computational
Methods in Systems Biology. pp. 231–250. Springer (2008)
11. Clarke, E.M., Zuliani, P.: Statistical model checking for cyber-physical systems. In:
International Symposium on Automated Technology for Verification and Analysis.
pp. 1–12. Springer (2011)
12. Cloth, L., Haverkort, B.R.: Model checking for survivability! In: Quantitative Eval-
uation of Systems. pp. 145–154. IEEE (2005)
13. da Costa, S.R.A., Campos, V.B.G., de Mello Bandeira, R.A.: Supply chains in hu-
manitarian operations: cases and analysis. Procedia-Social and Behavioral Sciences
54, 598–607 (2012)
14. Duran, S., Gutierrez, M.A., Keskinocak, P.: Pre-positioning of emergency items
for care international. Interfaces 41(3), 223–237 (2011)
15. Fahland, D., Woith, H.: Towards process models for disaster response. In: Business
Process Management. pp. 254–265. Springer (2008)
16. Fehnker, A., Gao, P.: Formal verification and simulation for performance analy-
sis for probabilistic broadcast protocols. In: International Conference on Ad-Hoc
Networks and Wireless. pp. 128–141. Springer (2006)
Statistical Model Checking of RSLD Management 21
17. Frankenberg, E., Laurito, M., Thomas, D.: The demography of disasters. In: Pre-
pared for the International Encyclopedia of the Social and Behavioral Sciences.
vol. 3, p. 12. Citeseer (2014)
18. H´erault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic
model checking. In: Verification, Model Checking, and Abstract Interpretation. pp.
73–84. Springer (2004)
19. Holmes, D.: What is stochastic programming (2001), http://users.iems.
northwestern.edu/˜jrbirge/html/dholmes/StoProIntro.html
20. Knott, R.: The logistics of bulk relief supplies. Disasters 11(2), 113–115 (1987)
21. Kovacs, G., Spens, K.M.: Relief supply chain management for disasters: humani-
tarian aid and emergency logistics. Information Science Reference Hershey (2012)
22. Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: Verification of probabilistic
real-time systems. In: Computer Aided Verification. LNCS, vol. 6806, pp. 585–591.
Springer (2011)
23. Legay, A., Delahaye, B., Bensalem, S.: Statistical model checking: An overview. In:
Runtime Verification. pp. 122–135. Springer (2010)
24. Lindell, M.K., Prater, C.S.: Assessing community impacts of natural disasters.
Natural hazards review 4(4), 176–185 (2003)
25. Mete, H.O., Zabinsky, Z.B.: Stochastic optimization of medical supply location and
distribution in disaster management. Production Economics 126(1), 76–84 (2010)
26. Mileti, D.: Disasters by design: A reassessment of natural hazards in the United
States. Joseph Henry Press (1999)
27. Najafi, M., Eshghi, K., Dullaert, W.: A multi-objective robust optimization model
for logistics planning in the earthquake response phase. Transportation Research
Part E: Logistics and Transportation Review 49(1), 217–249 (2013)
28. Nimal, V.: Statistical approaches for probabilistic model checking. MSc mini-
project dissertation, Oxford University Computing Laboratory (2010)
29. ¨
Ozdamar, L., Ekinci, E., K¨u¸c¨ukyazici, B.: Emergency logistics planning in natural
disasters. Annals of operations research 129(1), 217–245 (2004)
30. Ozguven, E., Ozbay, K.: Case study-based evaluation of stochastic multicommodity
emergency inventory management model. Transportation Research Record: Jour-
nal of the Transportation Research Board (2283), 12–24 (2012)
31. Preparedness, P.A.H.O.E., Program, D.R.C., of Emergency, W.H.O.D., Action, H.:
Humanitarian supply management and logistics in the health sector. Pan American
Health Organization (2001)
32. Rawls, C.G., Turnquist, M.A.: Pre-positioning of emergency supplies for disaster
response. Transportation research part B: Methodological 44(4), 521–534 (2010)
33. Reijsbergen, D., de Boer, P.T., Scheinhardt, W., Haverkort, B.: On hypothesis
testing for statistical model checking. International journal on software tools for
technology transfer 17(4), 377–395 (2015)
34. Rottkemper, B., Fischer, K., Blecken, A., Danne, C.: Inventory relocation for over-
lapping disaster settings in humanitarian operations. Operations Research (OR)
spectrum 33(3), 721–749 (2011)
35. Schultz, J., Elliott, J.R.: Natural disasters and local demographic change in the
united states. Population and Environment 34(3), 293–312 (2013)
36. Seneviratne, S.I., Nicholls, N., et al.: Changes in climate extremes and their impacts
on the natural physical environment. Managing the risks of extreme events and
disasters to advance climate change adaptation pp. 109–230 (2012)
37. Simonovic, S.P.: Systems approach to management of disasters–a missed opportu-
nity. Integrated Disaster Risk Management 5(2), 70–81 (2015)
22 S. Iqbal, et al.
38. Timoleon, C.: The logistics chain of emergency supplies in disasters (2012)
39. Tzeng, G.H., C.H.H.: Multi-objective optimal planning for designing relief delivery
systems pp. 673–686 (2007)
40. Widener, M.J., Horner, M.W.: A hierarchical approach to modeling hurricane dis-
aster relief goods distribution. Transport Geography 19(4), 821–828 (2011)
41. Wisetjindawat, W., Ito, H., Fujita, M., Eizo, H.: Planning disaster relief operations.
Procedia-Social and Behavioral Sciences 125, 412–421 (2014)
42. Yi, W., ¨
Ozdamar, L.: A dynamic logistics coordination model for evacuation and
support in disaster response activities. European Journal of Operational Research
179(3), 1177–1193 (2007)
43. Younes, H.L., Simmons, R.G.: Probabilistic verification of discrete event systems
using acceptance sampling. In: Computer Aided Verification. pp. 223–235. Springer
(2002)
