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Amulti-stageemergencysuppliespre-allocationapproachforfreewayblackspots_AChinesecasestudy.pdf
RESEARCH ARTICLE
A multi-stage emergency supplies pre-
allocation approach for freeway black spots: A
Chinese case study
Siliang LuanID 1,2, Qingfang Yang1,2☯, Zhongtai JiangID
1,2☯*, Wei Wang1,2, Chao Chen3
1 School of Transportation, Jilin University, Changchun, P. R. China, 2 Jilin Research Center for Intelligent
Transportation System, Changchun, P. R. China, 3 School of Automotive Engineering, Dalian University of
Technology, Dalian, P. R. China
☯ These authors contributed equally to this work.
Abstract
This study presents a multi-stage random regret minimization (RRM) model as an emer-
gency rescue decision support system to determine the emergency resource pre-allocation
schedule for the freeway network. The proposed methodology consists of three steps: (1)
improved accident frequency approach to identify the black spots on the freeway network,
(2) stochastic programming (SP) model to determine the initial allocation plan sets, and (3)
regret-based model in the logarithmical specification to select the most minimal regret one
considering the factors of the response time, total cost and demand. The model is applied to
the case study of 2014–2016 freeway network in Shandong, China. The results show that
the random regret minimization (RRM) model can improve the full-compensation of SP
model to a certain degree. RRM in logarithmical specification performs lightly better than
random utility maximization (RUM) and RRM in the linear-additive specification in this case.
This approach emerges as a valuable tool to help decision makers to allocate resources
before traffic accident occurs, with the aim of minimizing the total regret of their decisions.
1. Introduction
Since the subject of emergency management emerged, it has already become a worldwide-notice-
able theme for natural or man-made disasters. Most of the emergency planning models are
designed to respond to irregular and unpredictable events, but ignore that road incidents have
already become the most major cause of death [1]. Every year more than 1.25 million people are
dead and approximate 50 million suffer injuries because of road safety accidents [2]. The critical
issues in traffic accidents are how to provide immediate assistance to the victims as soon as possible
and how to mitigate the effects of the incidents [3]. Emergency supply pre-allocation as the first
step of the rescue impacts the quality of the relief efforts. There is also a great need for reliable deci-
sion-support models to help evaluate and improve the performance of such systems [4]. In this
paper, a new way to set up pre-arranged emergency resources is established for decision makers.
Previous studies have developed several supply allocation models to support the decision-
making process, but there is still an understanding on how to improve the pre-allocation
PLOS ONE
PLOS ONE | https://doi.org/10.1371/journal.pone.0240372 October 8, 2020 1 / 20
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OPEN ACCESS
Citation: Luan S, Yang Q, Jiang Z, Wang W, Chen
C (2020) A multi-stage emergency supplies pre-
allocation approach for freeway black spots: A
Chinese case study. PLoS ONE 15(10): e0240372.
https://doi.org/10.1371/journal.pone.0240372
Editor: Dejan Dragan, Univerza v Mariboru,
SLOVENIA
Received: May 29, 2020
Accepted: September 25, 2020
Published: October 8, 2020
Copyright: © 2020 Luan et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: The data that support
the findings of this study are available from Jilin
University but restrictions apply to the availability of
these data, which were used under license for the
current study, and so are not publicly available. The
data is available through the Jilin Research Center
for Intelligent Transportation System, data
protection manager Ciyun Lin: [email protected].
Funding: This research was funded by China
Scholarship Council (Grant No.201906170189).
Competing interests: The authors have declared
that no competing interests exist.
process in emergency management. The design of our emergency resource pre-arrangement
strategy includes the identification of black spots and the preparedness of rescue supplies in
advance. Once the traffic incidents have occurred, the affected people’s assistance can be
launched in an orderly manner.
In this paper, we take advantage of the historical accident records that can be inferred from
the accident-prone areas. Moreover, summarizing the previous literature, we attempt to for-
mulate models that are based on stochastic programming (SP) model. SP-based model is an
appropriate tool for planning in the preparedness phase and has been successful in many appli-
cations. Notwithstanding the obvious success of the SP model for emergency management
purposes, there is much scope for the characteristics of full compensation. It assumes that one
of the attributes outperforms other attributes, which can compensate other attributes with bad
performance greatly. Therefore, we introduce the random regret minimization (RRM) model
to improve the defects of SP model. RRM’s contribution to decision support system (DSS) lies
in its semi-compensatory characteristics based on regret theory (RT), which can avoid offset-
ting an equally large decline in the performance of another attribute.
The proposed methodology is unique and has three contributions as follows:
1. The improved accident frequency method associated with freeway road network helps to
identify more risky areas. Given that the black spots on the freeway are known exactly, the
accident probability is obtained more easily according to the prediction, and the limited
supplies can be transferred to the areas where are prone to occur accidents in time. Method-
ologically, adding the method of accident identification to the spectrum of emergency res-
cue makes sense.
2. A new SP-RRM model for emergency resource pre-arrangement introduces the regret the-
ory into emergency management, which provides a different perspective to look at emer-
gency system decisions. Regret-based model intents to avoid the negative payoff and choose
the final alternative that has already taken into account the foregone alternatives, which is
adapted to emergency fields. Especially, the characteristics of semi-compensatory improve
the SP-only approaches.
3. The studies example compares SP-RRM2010(in the logarithmic specification) with the SP-
only, SP- Random Utility Maximization (RUM) and SP-RRM2008(in the linear-additive
specification) for our discussed issue. The objective to be optimized is represented in terms
of response time, allocation cost and demand.
The remainder of the paper is organized as follows. In Section 2, the relevant literature is
reviewed. Section 3 illustrates the multi-stage emergency supplies pre-allocation approach. In
Section 4, a Chinese case study for the application of our strategy is introduced. Section 5 pres-
ents conclusions and directions for further research.
2. Literature review
Our literature review consists of two sub-sections. The first sub-section reviews the recent
studies that are emergency management related to resource allocation. The second sub-section
reviews past researches that are tightly connected to the theory and methodology used in this
paper.
2.1 Emergency resource allocation
Much of the effort in emergency supplies allocation focuses on natural disaster management.
Haghani and Oh presented a formulation to solve a large-scale multi-commodity, multi-
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modal network problem with time windows in disaster relief management [5]. Fiedrich et al.
minimized the total facilities after strong earthquakes and used the detailed descriptions of the
operational areas and of the available resources to calculate the model for different disasters
[6]. Sherali et al. considered comprehensively the aftermath of a natural disaster, terrorist
attack, and so on, and stated a tight linear programming relaxation [7]. Dodo et al. put forward
a linear program to minimize the overall risk of earthquakes and to avoid losses in the future
earthquake [8]. Minciardi et al. presented two main phases to manage natural hazards: a pre-
event phase is to reallocate resources closer to sites; the during event-phase is to manage the
available resources in real-time. This approach is useful for the risk management of natural
hazards [9]. Mete and Zabinsky proposed a stochastic optimization to store and distribute
medical supplies in preparation for disasters [10]. In the paper by Rawls and Turnquist, a two-
stage stochastic mixed integer program was presented to provide response strategies for hurri-
canes or other natural disasters, which was solved by Lagrangian L-shaped method (LLSM)
[11]. In the paper by Wex et al., a decision support model in natural disaster management that
minimizes the sum of completion time of incidents was developed and was solved by a heuris-
tics algorithm that was selected among Monte Carlo-based heuristic, the joint application of 8
construction heuristics and 5 improvement heuristics [12].
Increased attention in the literature has been focused on allocating resources by Stochastic
Program (SP) model. Primary concerns are often the development of the allocation of medical
resources such as ambulances to the emergency occurrence areas [13–16]. Baker et al. pro-
posed an integer, non-linear mathematical programming model to allocate emergency medical
service ambulances to meet the government-mandated response-time criterion and also reflect
the criterion for budget and workload [17]. In addition, Ahmed formulated a 0–1 integer lin-
ear programming problem to allocate resources to maximize the total maintenance effective-
ness for highway maintenance management [18]. Yin presented a min-max bi-level
programming model to allocate tow trucks among patrol beats to contribute to the traffic inci-
dent management systems on the freeway [19]. In 2008, Yin extended his original model and
proposed a mixed-integer nonlinear programming model to minimize the expected loss of dif-
ferent incidents [20]. Ozbay et al. introduced the concept of quality of service during a poten-
tial incident and put forward an SP model with probabilistic constraints to respond the
incidents and allocate resources [21]. In the paper by Garrido et al. [22], a spatio-temporal sto-
chastic process model for the logistics of a flood emergency can help the decision makers well
to deliver enough supplies to satisfy the demands before and after a flood occurs. Feng et al.
[23] optimized the hospital emergency departments(Eds) system in Taiwan and used multi-
objective stochastic mathematical model for the limited medical resources allocation. This
mathematical model was solved by an integrated non-dominated sorting genetic algorithm
(NSGA) II.
2.2 Regret theory and regret-based model
In contributing to the continued discussion on emergency management in transportation
research, and following this generally accepted methodological approach, in this paper, we
introduce the notion of regret theory and apply RRM model instead of only SP-based model in
DSS. It is widely assumed that regret is based on a kind of emotion that individuals do not sat-
isfy their expectation, that is, the chosen alternative performs worse than the non-chosen one.
Hence, Bell utilized this choice behavior to develop the regret theory to provide assistance to
make a good decision with regard to tradeoff and lotteries in order to avoid regret in 1982
[24]. In search of discrete choice models, the concept of regret has recently attracted the atten-
tion of several scholars in transportation research [25–40]. Regret-based models assume that
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individuals minimize regret rather than maximize utility when they face several choice
alternatives.
Chorus et al. introduced RRM model into transportation research [25]. The original regret-
based model is in a linear function of attribute differences between the considered alternative
and the best of foregone alternatives. In 2010, Chorus proposed a regret-based discrete choice-
model in logarithmic specification instead of the discontinuous function [26]. This new model
not only considered all foregone alternative, but also can use standard discrete-choice software
packages. Later, G-RRM model replaced the fixed constant of one in the attribute level regret
function of the RRM2010 model by regret-weight variable [33]. Cranenburgh et al. developed
the classical RRM model and allowed a shape parameter μ to be estimated [35].
This fundamental work has led to an upsurge of applications of regret-based models in dif-
ferent fields. In term of the transportation research, it contains departure times [41], travel
mode [42], route choices [28, 36, 43] and freight transport [32]. These literatures performed in
this present study indicates that these applications focus on the individual choice from the
user’s point of view and contribute to the user’s service system. However, few of the applica-
tions are in the context of emergency resource allocation. This issue belongs to the decision
support system, and the decision makers are policymakers that have the duty to select the opti-
mal determinate alternative to help people who may suffer the accidents.
A total of fifteen literature in Table 1 summarizes their choice type, data type, and model.
Two papers do not use statistical methods, only analyzing the specification methodologically.
3. Mathematical formulation
One of the topics in the development of emergency management concerns the analysis of traf-
fic accident data. A fundamental property of the traffic accident data is its enormousness and
complexity, which is hard to infer accident-prone areas. To start with, introducing the
improved accident frequency method gives the operators the way to identify the black spots on
the freeway. Then we attempt to address an SP-RRM model to allocate emergency supplies
before the accidents occur. In our model, multiple potential scenarios with various demand for
supplies are allowed, and the response time, the total cost and the demand are assumed to be
the major factors to make decisions.
Table 1. Summary of feature used in selected papers.
Lead Author (Year) Choice type Data type Model
Kaplan & Prato (2012) [29] Travel route RP RRM2010
Chorus (2012a) [27] RUM, RRM2008
Chorus (2012b) [28] Travel route SP RRM2010
Chorus et al. (2013) [30] Travel route SP RUM, RRM2010, contextual concavity model
Hensher et al. (2013) [31] Travel mode SP RUM, RRM2010
Chorus (2014) [33] Travel route SP G-RRM (logarithmic)
Boeri et al. (2014) [32] Transportation mode, time, cost, punctuality of the transport SP RUM, RRM2010
Hess et al. (2014) [34] Information acquisition SP RUM, RRM
An et al. (2015) [42] Travel mode SP Hybrid model
Cranenburgh et al. (2015) [35] Shopping location SP μRRM (logarithm), PRRM (logarithm)
Rasouli & Harry (2017) [37] Parking fee SP RRMax, RRSum, RRlog
Li & Huang (2017) [36] Travel route SP RUM, RRM2010
Chorus and Cranenburgh (2018) [38] Ten datasets SP RUM, RRM2008, RRM2010, G-RRM, μRRM
Rasouli & Harry (2018a) [39] RRM2008, RRM2010
Rasouli & Harry (2018b) [40] Shopping destination, travel mode SP Regret-rejoice model
https://doi.org/10.1371/journal.pone.0240372.t001
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3.1 The identification of black spots on the freeway
The identification of black spots contributes to the safety and the quality of service of the road
network. The results of road accident statistics stress the need for more systematic mechanisms
for accident analysis and prediction. The main goal of the identification of black spots is not
only to find out the high-risk areas for potential accidents, but also can guarantee the next
work concerned about resource arrangement.
Inherently, freeway networks constitute complex dynamic systems impacted by various
uncertain factors. Black spot is not just “a point”, it can represent a point, a road section or an
area [44]. In this paper, we stipulate that the black spot is a certain road section and put for-
ward the improved accident frequency method to identify the black spots on the freeway.
Given the critical number of accidents CN as the identification criterion, the number of acci-
dents in a certain section is larger than the critical value, so it is considered as the prone acci-
dent occurrence location.
We define λ as the average number of accidents in unit sections.
l ¼
X mi
n ð1Þ
Where mi is the number of accidents on the road section i and n is the total number of the
unit road section.
We define the critical number of accidents as CN, which is regarded as the accident thresh-
old for the black spots. Assume the Eq (2) obey Poisson distribution and u is the confidence
level. If the confidence level u(1−α)/2 is 95%, the critical number of accidents CN is:
CN ¼ lþ uð1� aÞ=2 � ffiffiffi l p
; i ¼ 1; 2; . . .; n ð2Þ
If the actual number of traffic accidents per year is larger than the critical number of acci-
dents CN, that road section seems to be the black spot. However, this road segmentation
method results in considerable errors as it may divide one whole accident-prone into two sec-
tions. Hence, the original fixed segmentation method needs to be modified by the cut-sectional
technology to improve the defect of missing some black spots. The principle of this non-fixed
length accident representation of black spots is to move some unit sections with a great num-
ber of accidents to a suitable area or integrate into one area on the basis of the distribution of
their locations, in case of ignoring the potential black spots.
3.2 A SP-RRM model for pre-allocating emergency supplies at the black
spots
Usually, the government decision-makers and scholars studying for emergency management
prefer to use stochastic programming (SP) model for emergency resource allocation. SP model
is a linear-additive specification with the maximum or minimum objective function. SP-only
based models may potentially capture a valid choice mechanism in some decision contexts
because SP model has the characteristics of full-compensation, that is, the attributes with better
performance compensate for the other attributes with worse performance. For that very rea-
son, many scholars tried to use different weights or penal coefficients to improve the signifi-
cance of some factors. Indeed, this is an effective way to some extent, but we might doubt the
process of calibration of the parameters. The parameters are obtained by referring to existing
literatures or relying on personal experiences. If experts as the decision-makers express their
preference based on their knowledge and empirical studies, the estimated results of the weight
of considerable factors are likely to be more convincing. Our specific model not only provides
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a new way of thinking to make related emergency decisions, but improves the process of the
calibration of the parameters.
Regret-based models have been introduced in several fields such as route choices, travel
information acquisition choices, parking lot choices, shopping location choices and so on.
Most researches related to regret-based models have in common that the decision makers are
individuals with different personal attributes. Decision makers prefer to choose a regretless
plan rather than a maximum utility schedule in emergency management.
Original RRM model is proposed by Chorus in 2008 [25] and it is called RRM2008 in the
next contexts. The main idea of the RRM model is the compassion of different attributes of
every alternative and to select the alternative with the minimal regret value. This method com-
prehensively considers every attribute’s performance and balance the overall situations. It can
be expressed as:
φxðxi; xjÞ ¼ maxf0;bx � ðxj � xiÞg ð3Þ
φyðyi; yjÞ ¼ maxf0;by � ðyj � yiÞg ð4Þ
φzðzi; zjÞ ¼ maxf0;bz � ðzj � ziÞg ð5Þ
Where φ is the linear regret function for altribute x,y,z for alternative i,j. Let β represent the
estimated parameter of alternative x,y,z. And the regret associated with alternative i when com-
pared to j is equal to:
Rij ¼ φxðxi; xjÞ þ φyðyi; yjÞ þ φzðzi; zjÞ ð6Þ
The regret of alternative i,j,k is:
Ri ¼ maxfRij;Rikg ð7Þ
Rj ¼ maxfRji;Rjkg ð8Þ
Rk ¼ maxfRki;Rkjg ð9Þ
Finally, we select the minimal regret among Ri,Rj,Rk.
In 2010, Chorus illustrated that the previous regret-based model (RRM2010 for short) has
two limitations [26]: first, the previous model is anticipated with respect to only the best of
foregone alternatives, but the new model assumes that regret is experienced about each fore-
gone alternative that performs well. Second, the logarithmic specification’s likelihood function
is smooth, unlike the non-smooth characteristics of previous model. The new model’s estima-
tion doesn’t rely on handwritten code anymore. Therefore, he formulated a new model specifi-
cation, and it is:
RRi ¼ Ri þ εi ¼ X
j6¼i
X
m
ln½1þ expðbmðxjm � ximÞÞ� þ εi ð10Þ
To map regret into choice probabilities, both model specifications assumed that regret is a
stochastic variable. Its error terms are independently Gumbel distributed, and the probability
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of the choice models is:
Pi ¼ expð� RiÞX
j¼1...J
expð� RjÞ ð11Þ
Where i,j are alternatives and i,j2J. Pi is the selected probability of the alternative i. The fitting surface of logarithmic function is shown in S1 Fig. ln[1+exp(βm(xjm−xim))] is a
monotonically increasing function of βm and (xjm−xim). The regret value at the attribute level
depends on the importance of the attributes and the difference between the two alternatives on
the comparison of the properties. The estimated parameter βm signifies the importance of the
attribute m. When the chosen alternative i outperforms unselected alternative j, i.e.,
(xjm−xim)<0, the regret value is closer to 0. On the contrary, when the performance of the
selected alternative i is not as good as the unselected scheme j, the larger the value of (xjm−xim)
is, the greater the regret value is.
In line with these findings, we decide to use the RRM model in logarithmical specification
for the emergency rescue discipline. All alternative plans are obtained from the SP model with
the constraint of the probability of traffic accidents on the freeway network. RRM model is
applied to get the optimum resources arrangement strategy, in response to different bad
weather scenarios.
3.2.1 Assumptions. We make the following assumptions about this model:
1. This model only considers resource allocation on hypothetical traffic accidents under dif-
ferent weather. Natural disasters such as earthquakes, hurricanes, floods, etc. are not
considered.
2. We assume that the supply locations are connected to the demand locations, their con-
nected roads are not damaged and the travel time can be predicted.
3. In this model, the demand location of the freeway network is the accident black spot (the
location at which the high probability of accident occurs). The black spot is simplified to
the particle point of the road segment.
3.2.2 Notations of parameters and variables. To formulate our model before, we need to
establish the model’s parameters and variables in Table 2.
We introduced the concept of scenario ξ during the modeling, which illustrates the future
possible conditions [45]. Since the distinct characteristics of incidents cause different demands
for resources and travel time, it is necessary to discuss emergency response schemes according
to the specific emergency type. The scenario-based applications have grown extensively into
the uncertainty emergency issues such as the railway infrastructure maintenance [46], the
flood emergency logistics preparation [47] and highway transportation industry [48]. In our
paper, the approach is based on the Bayesian Networks. Since the approach of Bayesian Net-
works has been discussed in detail in a number of previous papers, we don’t present deeply
how to classify the scenarios [49].
As different bad weather usually leads to traffic congestion, we introduce this congestion
delay index that is used widely to evaluate the current traffic situation in China, and we apply
this index to modify the velocity of rescue vehicles. In China, different cities have their defini-
tion and standard for the congestion delay index. In order to avoid ambiguity, applying the
definition of congestion delay index proposed by Autonavi company as in our model then
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gives the following calculation form:
Congestion delay index ¼ Peak travel time
Free flowðnon� congestionÞ travel time
This parameter is obtained by analyzing historical records and current floating cars feed-
back, and makes some sense to modify the travel time from supply location i to demand point
j. The formulation of transportation time is:
tijðxÞ ¼ dijðxÞ � lij
vðxÞ ð12Þ
Table 2. Notification.
Indices and index sets
I Set of all the supply locations i2I J Set of all the black spots j2J parameters
pj(ξ) Accident probability of black spots j under the scenario ξ t0ij Emergency resources transportation time from supply location i to black spot j �t�i The average processing time after receiving the alarm at the supply location i
ai The maximum stock capacity of the supply location i A Number of total available emergency supplies
B The maximum budget for the inventory and procurement in the system.
r The minimal number of emergency resources that can handle a minor accident independently
dj(ξ) Demand for emergency supplies in location j under the scenario ξ Gi Service level of supply locations i. Grade 1 is the top of them, i.e., the supply location has the best rescue
service. Grade 3 is the bottom of them, i.e., the supply location has the worst rescue ability.
Gi ¼
3 Grade 1
2 Grade 2
1 Grade 3
8 >><
>>:
Cs The inventory cost of emergency supplies
Cb The procurement cost of emergency supplies
C The attribute of total cost, includes inventory and procurement costs
T The attribute of response time. The response time includes reaction time after receiving the alarm and travel time from zone i to zone j.
D The attribute of demand for emergency supplies
β The estimated parameter of the attributes.
lij The shortest distance from zone i to zone j. v(ξ) The average transportation velocity under the scenario ξ. tij(ξ) Transportation time from zone i to zone j under the scenario ξ. δij(ξ) Congestion delay index from zone i to zone j under the scenario ξ. Variables
θij(ξ) 1, if black spot j is covered by supply location i; 0, otherwise.
xij(ξ) The number of emergency supplies, sent to location j, originated from a supplier in location i under the
scenario ξ. ε An independent and identically distributed (i.i.d.) error term.
RR(ξ) The total regret value under the scenario ξ. RT(ξ) The regret of the attribute T under the scenario ξ. RC(ξ) The regret of the attribute C under the scenario ξ. RD(ξ) The regret of the attribute D under the scenario ξ.
https://doi.org/10.1371/journal.pone.0240372.t002
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3.2.3 The optimization formulation. With the parameters and variables previously
defined, we now present SP model to obtain initial plans.
min X
i2I
X
j2J
yijðxÞpjðxÞtijðxÞxijðxÞ ð13Þ
Subject to
X
j2J
xijðxÞ � ai; 8i 2 I ð14Þ
X
j2J
xijðxÞ � r; 8i 2 I ð15Þ
X
i2I
xijðxÞ � dj; 8j 2 J ð16Þ
yi ¼ 0 tijðxÞ > t0ij � �ti 1 tijðxÞ � t0ij � �ti
; 8i 2 I ð17Þ
(
xijðxÞ � 0; and integer 8i 2 I ð18Þ
The objective function of this stage (13) incorporates the total response time and accident
probability of black spots in order to provide limited emergency supplies to the demand loca-
tions. The goal of the objective function is to minimize the response time and the number of
resources. Constraint (14) ensures that the storage of emergency supplies in the supply loca-
tion cannot exceed the capacity. Constraint (15) ensures that the number of emergency sup-
plies in every supply location can handle an accident independently. Constraint (16)
guarantees that the emergency supplies that may be distributed to zone j cannot exceed the
actual demand. Constraint (17) states that if the black spot j is within the coverage of supply
location i, θij = 1; otherwise, θij = 0. Constraint (18) illustrates the non-negative nature and the
integer requirement of xij(ξ). Recall that the regret-based models contribute to the development of the decision support
system. We propose the RRM model in the lograthimic specification to select the optimum
alternative.
min RRðxÞ ð19Þ
RRðxÞ ¼ RTðxÞ þ RCðxÞ þ RDðxÞ þ ε ð20Þ
RTðxÞ ¼ X
others6¼choice
lnð1þ expðbTðTothers � TchoiceÞÞÞ ð21Þ
RCðxÞ ¼ X
others6¼choice
lnð1þ expðbCðCothers � CchoiceÞÞÞ ð22Þ
RDðxÞ ¼ X
others6¼choice
lnð1þ expðbDðDothers � DchoiceÞÞÞ ð23Þ
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T ¼ X
i2I
X
j2J
tijðxÞxijðxÞ ð24Þ
C ¼ X
i2I
X
j2J
ðCs þ CbÞGixijðxÞ ð25Þ
D ¼ X
j2J
ðdj � xijðxÞÞ ð26Þ
Subject to:
X
j2J
xijðxÞ � ai; 8i 2 I ð27Þ
X
i2I
X
j2J
xijðxÞ � A ð28Þ
X
j2J
xijðxÞ � r; 8i 2 I ð29Þ
X
i2I
xijðxÞ � dj; 8j 2 J ð30Þ
X
i2I
X
j2J
ðCs þ CbÞGixijðxÞ � B ð31Þ
xijðxÞ � 0; and integer 8i 2 I ð32Þ
The objective function (19) of the final stage is to select the allocation plan with the minimal
regret value and formula (20) is the linear function to calculate the total regret value of an alter-
native plan. Formula (21)–(23) respectively presents the regret value of the attribute of
response time, cost and demand under the scenario ξ by comparing every alternative. Eq (24)
indicates the meaning of the attribute of response time. Eq (25) states that the total costs
include the storage cost and acquisition cost of resources. Eq (26) defines that the attribute of
demand equals the difference between actual demands and actual emergency supplies. If the
actual dispatches don’t meet the need for demands at black spots, the regret value will increase.
The limitations on the capacities of warehouses and the maximum number of emergency
resources are represented by (27) and (28), respectively. Constraint (29) ensures that the num-
ber of emergency supplies in every supply location can handle an accident independently.
Constraint (30) guarantees that the emergency supplies that may be distributed to zone j can-
not exceed the actual demand. The maximum number of budgets B are taken into account by
constraint (31). Constraint (32) illustrates the non-negative nature and the integer require-
ment of xij(ξ). Since RT(ξ),RC(ξ),RD(ξ) have different dimensions, it is necessary to use the unified dimen-
sion. We carry out non-dimensionalization of (Tothers−Tchoice), (Cothers−Cchoice) and (Wothers− Wchoice). This process is also called formatting the parameters. We use standardization method
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to convert them to unified dimension data. Standardized formula is as follows:
yi ¼ xi � �x
s ð33Þ
Where �x ¼ 1
n
Xn
i¼1
xi and s ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
n� 1
Xn
i¼1
ðxi � �xÞ2 s
.
4. A case study
In this section, we illustrate the newly developed methodological insights and present an exam-
ple to demonstrate the validity of our SP-RRM model. The data we used contains the three-
year (2014–2016) traffic incidents on a rainy day in three Chinese cities, i.e., Linyi, Weifang
and Rizhao where they occupied the highest traffic accident rate in Shandong province. These
data are analyzed once before [50].
4.1 The identification of black spots on the freeway
We analyze a recent three-year accident portion of Shandong province and find that Linyi has
the largest number of accidents. As Rizhao and Weifang are near Linyi according to the geo-
graphic location, we take these three cities as our study areas to identify the black spots. Linyi has
four freeways, i.e., G2, G20, G22, and G25; Weifang embodies G18, G25 and G20; Rizhao
includes G15, G1511 and G25. We take G2 as an example to present the process of the identifica-
tion of black spots on the freeway. G2 has a length of 1261.99 kilometers from K540+195 to K709
+800 and is across six provinces in all. In this paper, we only discuss G2 in the Linyi section.
4.1.1 The critical value of accidents. We determine 1km as a unit length and divide G2
in the Linyi section into several units. And there are 170 road sections in total. The total num-
ber of accidents for every unit road section is 864.
The average number of accidents λ on a unit is:
l ¼
X mi
n ¼
864
170 � 5
If the confidence level is 95%, the critical number of accidents CN is:
CN ¼ lþ uð1� aÞ=2 � ffiffiffi l p ¼ lþ 1:96
ffiffiffi l p � 10
4.1.2 Initial identification process of black spots. By comparing the actual accidents and
the critical value, we identified 27 black spots initially in Table 3.
4.1.3 The modification of the initial approach. Although most of the black spots have
been identified initially, the rough results may ignore some hidden black spots owing to the
fixed segmentation method. Thus, it is valuable to use cut section technology to modify the ini-
tial results. The approach mainly modifies two conditions in S2 Fig. The first condition is
regarding the black spots that have been identified in the previous step. As can be seen, the
K707-K708 is regarded as a black spot, for the number of accidents exceeds the critical number
of accidents CN. However, the road sections around K707-K708 also have accidents and the
total number of the accident is high. Hence, in light of the accident distribution of the adjacent
road sections, we integrated them into a new road section and stated point A as the black spot
for this new road section K704-K709. In terms of condition 2, the number of accidents of road
section K570-K571 and K571-K572 is less than the critical number of accidents CN and they
do not seem as the black spots. However, it is unreasonable to ignore the potential black spot,
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since the total number of accident between K570 and K572 is larger than the critical number
of accidents CN. Therefore, K570-K572 is modified by the cut section technology as a new
black spot.
Based on the modified approach, the final results are obtained as in Table 4.
We identified 14 black spots on the G2 finally. The rest of freeway also applied the same
way to identify the black spots.
4.1.4 The calculation of accident probability of black spots. After considering the actual
situation of every freeway, i.e., the traffic flow and the total number of accidents on every free-
way, we calculate the probability of accident at black spot that is just a relative value. The
method is expressed as follows.
pi ¼ wtf � wa � niX
i
ni
ð34Þ
Table 3. The initial result of the identification of black spots.
Original Pile No Final Pile No The amount of accident Original Pile No Final Pile No The amount of accident
K542 K543 12 K581 K582 10
K545 K546 11 K599 K600 10
K546 K547 11 K600 K601 10
K547 K548 10 K604 K605 11
K551 K552 16 K609 K610 10
K553 K554 11 K613 K614 10
K555 K556 11 K615 K616 10
K556 K557 10 K616 K617 10
K563 K564 10 K617 K618 11
K564 K565 11 K639 K640 10
K567 K568 10 K642 K643 11
K577 K578 10 K649 K650 10
K578 K579 10 K681 K682 10
K580 K581 13 K707 K708 10
https://doi.org/10.1371/journal.pone.0240372.t003
Table 4. The final result of modification.
Black spots j The number of accidents
Original pile No Final pile No
K542 K543 12
K545 K548 32
K551 K557 48
K561 K568 39
K570 K572 16
K575 K585 20
K588 K591 60
K599 K605 41
K609 K618 51
K622 K627 15
K639 K643 25
K647 K650 18
K681 K682 10
K704 K709 19
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Where pi is the accident probability of black spot i and ni is the number of accident of black
spot i. wtf denotes the weight of traffic flow, and wa is the weight of accident and expresses by
wa ¼ mjX
j
mj
, where mj is the total number of accidents of highway j.
In S3 Fig, the proportion of accidents on every freeway is counted. The final results of acci-
dent probability of black spots are shown in Table 5.
4.2 SP-RRM model for emergency supplies pre-allocation
4.2.1 SP model for initial alternative plans. Usually, different types of supplies need to
be pre-positioned in storage warehouses for different scenarios. In the example application in
Section 4, we only consider one type of commodity(wreckers) on rainy days, but in general,
the list of commodities might be several kinds of supplies. This case study focusing on pre-
paredness for traffic accident threats in three cities serves to illustrate the SP model. To high-
light the validity of our methodology, the case study is small enough to also be solved by a
computer software package but detailed enough to be of interest as an illustration.
The natural scenario is considered, that is, the rainfall intensity is less than 2.5 mm�h−1. The
average speed is 82.4 km/h and the average congestion index is 1.62. In the given case, 40
wreckers are available and 18 warehouses (road administration brigades and road administra-
tion squadrons) can store wreckers. The grade of road administration brigades is 1 and the
grade of road administration squadrons is 2. The emergency manual in Shandong province
stipulates that the maximum rescue time is 30 minutes and the average process time is 5
Table 5. The results of accident probability.
NO Name Original Pile No Final Pile No Accident Probability
1 G15 K704 K707 0.012234
2 K712 K715 0.015293
3 K742 K744 0.006117
4 G18 K363 K364 0.00534
..
. .. . ..
. .. .
10 K443 K445 0.00534
11 G2 K542 K543 0.006662
..
. .. . ..
. .. .
24 K704 K709 0.011104
25 G20 K101 K102 0.010531
..
. .. . ..
. .. .
36 K213 K216 0.038614
37 G22 K101 K103 0.005706
..
. .. . ..
. .. .
43 K159 K160 0.005706
44 G25 K1406 K1408 0.005464
..
. .. . ..
. .. .
61 K1604 K1606 0.010927
62 G1511 K16 K18 0.009786
..
. .. . ..
. .. .
72 K156 K158 0.009786
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minutes. According to the time constraint, the major element θij can judge whether the supply
location i covers the black spot j. Genetic algorithm is one of the widely useful heuristic algo-
rithms and it usually solves the SP model. By changing the number of iterations, initial scales
of the population, etc., of the genetic algorithm, the alternative plans are obtained in Table 6.
The values in Table 6 illustrate the number of rescue vehicles for black spots when this area
occurs in the same condition. We coded our models in MATLAB2014a and obtained 9 plans
as our alternatives in the second stage.
4.2.2 RRM model for the selection of the optimum plan. This stage is based on a choice
experiment constructed to analyze expert choice of resource allocation plan. The experiment
described the allocation plan in terms of three generic attributes—response time, total cost and
demand for the black spots. Each attribute was categorized into four levels. The choice set was
blocked into 9 choice sets based on the results of the SP model. The experiment, comple-
mented with questions about the importance of these three attributes, was administered via
Web-based questionnaires. This study only recruited respondents among a group of people
who have studied the emergency management or taken part in related work. The target sample
size was 146 respondents. Data collection started on July 10, 2018. The target sample size was
achieved on July 19, 2018. 119 valid questionnaires were obtained.
Table 7 reports the frequency distributions of the selected socio-demographic characteris-
tics. It shows that 58.9% of the sample is male, implying 41.1% is female. Age was classified
into four categories. The percentages for these categories are respectively 22.3, 36.7, 30.1 and
10.9%. Table 7 also shows that the respondents receive high education in the sample.
In this case, the original regret model and utility-based model were compared with the loga-
rithmic specification to figure out the differences between these models based on different
decision rules in Table 8 (See S1 Appendix for the SP model, original regret model and RUM
model). For the reason of clear distinction, we use the same software, PandaPython [51, 52], to
estimate the parameters.
Table 8 shows the estimation results. The rows show model performance 213indicators and
estimates for the three parameters: response time, total costs and demand. The columns show
the model types: SP-RUM, SP-RRM2008 and SP-RRM2010.
In terms of model fit, the overall goodness of fit of the RRM2010 model empirically outper-
forms the standard RUM model and classical RRM model. Log-likelihood is improved by
Table 6. The initial alternative plans by SP model.
Scheme No
Black spot
1 2 3 4 5 6 7 8 9
1 0 1 1 1 0 0 2 0 0
2 0 0 2 2 0 0 1 0 1
3 0 0 0 0 1 2 0 0 0
4 0 0 0 0 0 0 0 0 0
..
. .. . ..
. .. . ..
. .. . ..
. .. . ..
. .. .
65 1 1 2 0 0 0 0 0 1
66 1 2 1 2 0 1 2 1 1
67 1 0 1 1 1 1 1 1 1
68 1 1 2 1 0 1 2 0 0
69 1 0 0 0 1 0 0 0 0
70 1 2 1 1 1 1 1 1 0
71 2 0 0 0 0 0 0 0 1
72 0 1 0 1 0 1 1 1 2
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about 10 LL points and 12 LL points as compared to respectively RUM and RRM2008. A statis-
tical test also shows that higher Adj.Rho-square for RRM2010 model means a good overall fit-
ness and a lower Adj.Rho-square for RUM and RRM2008 model illustrates poor fitness.
In a sequel, we turn to the parameter estimates. As expected, the related parameters (i.e.
response time, total costs and demand) of the utility-based model and regret-based model are
all significant at 90% or 95% level. The positive sign implies that decision makers more high-
light the importance of time and demand and the negative sign implies that the total costs are
not as important as the time and demand under emergency context. We can apply this param-
eter estimation to calculate the final alternative results.
Since the SP model is in the mainstream for emergency management, we compare the SP-
only model with SP-RRM model and analyze their differences. The results are shown in
Table 8. Meanwhile, we focus on empirical studies that report comparisons between RRM and
RUM. Therefore, we convert SP model to the utility-based model which added the estimated
parameter β and observe the differences between SP-RUM and SP-RRM model in Table 8.
The overview of the optimal plan ranking based on the different decision rules is shown in
Table 9.
Table 10 reports the optimal plan ranking for the results of SP-only, SP-RUM, and SP-RRM
model. As shown in Table 10, Plan2 is the optimal scheme for the SP-only model, but it is the
worst one in the SP-RUM and SP-RRM model. As can be seen in Table 9, the attribute of
demand extremely compensates the other attributes in terms of SP model. However, the results
Table 7. Frequency distribution of socio-demographic characteristics.
Socio-demographic variables Percentage (%)
Gender Male 58.9
Female 41.1
Age 20–30 22.3
30–40 36.7
40–50 30.1
>50 10.9
Education Undergraduate 17.9
Master 55.4
Doctor 26.7
Whether participated in related decision-making Yes 66.1
No 33.9
https://doi.org/10.1371/journal.pone.0240372.t007
Table 8. Estimation results for different models with the PandaPython.
Parameters SP-RUM (T-test) SP-RRM2008 (T-test) SP-RRM2010 (T-test)
Response time 0.186(3.424)��� -0.127(-8.010)��� -0.033(7.240)���
Total costs -0.100(6.785)��� -0.001(-5.019)��� -0.017(-5.170)���
Demand 1.020(5.511)��� 0.031(1.740)�� 0.012(1.770)��
Null log likelihood -261.468 -261.468 -261.468
Final log likelihood -223.8414 -225.471 -213.906
Rho-square 0.132 0.126 0.145
Adjuested rho-square 0.132 0.126 0.145
Note:
��: robust t-value <0.09
���: robust t-value<0.095.
https://doi.org/10.1371/journal.pone.0240372.t008
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for the RUM and RRM models could be modified by the estimated parameter β. In light of the
estimation results, obviously, the attribute of demand is the least consideration among three
attributes. The estimated parameters of the SP-RUM and SP-RRM model getting from the
experts’ investigations is necessary to assist the decision-maker to balance every important fac-
tor. Plan8 is the optimal scheme for the SP-RUM and SP-RRM, which illustrates that SP-RUM
and SP-RRM in the emergency context are substantial in final results. Therefore, we choose
Plan8 as the final decision.
5. Conclusion and future work
Pre-allocation of emergency supplies can be an effective mechanism for improving response to
traffic accidents. We have developed a multi-stage SP-RRM model whose solution provides a
pre-allocation strategy for the storage and distribution of emergency supplies under different
scenarios. The improved accident frequency method is proposed to identify black spots on the
freeway, which increases the accuracy and efficiency for the next work. The emergency sup-
plies pre-allocation approach is a combination of SP model with the constraint of the response
time and accident probability and RRM model in the logarithmical specification considering
the response time, total cost and demand for black spots.
A case study addressing black spots on the freeways in the three cities of China illustrates
both the improved accident frequency and a new SP-RRM model in a practical context. In the
case study, after identifying 72 black spots on the freeway in Linyi, Rizhao and Weifang from
the historical data in 2014–2016 and calculating their accident probability, we obtained 9 ini-
tial plans as our alternatives by SP model and selected the optimum one by RRM model. More-
over, this case study has also allowed experiments to assess the performance of RUM model,
RRM2008 model and RRM2010 model. By analyzing the results of the case, we are confident
that a multi-stage SP-RRM model can be used as a large-scale resource pre-allocating planning
tool.
This paper contributes to analyzing the emergency resources allocation choice based on
utility or regret decision rules. The regret model was used to account for the negative
Table 9. The results of SP-only, SP-RUM and SP-RRM model.
Plan No T C D SP SP-RUM RT RC RD SP-RRM
1 3.497 1.436 3.207 2.140 3.779 6.173 6.124 6.224 18.521
2 3.365 2.205 0.723 0.293 1.145 6.192 6.183 6.354 18.730
3 2.085 1.949 2.793 0.826 3.044 6.385 6.164 6.246 18.795
4 3.349 3.231 3.621 4.201 3.997 6.195 6.263 6.203 18.660
5 1.988 3.744 3.621 3.352 3.693 6.400 6.303 6.203 18.906
6 2.439 4.513 1.965 2.917 2.011 6.332 6.364 6.289 18.984
7 2.208 2.462 4.035 2.705 4.283 6.367 6.203 6.181 18.751
8 5.336 2.718 3.621 5.675 4.417 5.903 6.223 6.203 18.329
9 3.183 4.000 2.793 3.976 0.195 6.220 6.323 6.246 18.789
Note: T, C and D are the dimensional values. The results of RRM is based on the logarithmic specification.
https://doi.org/10.1371/journal.pone.0240372.t009
Table 10. The optimal plan ranking for SP-only, SP-RUM and SP-RRM model from small to large.
SP-only (from small to large) 2 3 1 6 7 5 9 4 8
SP-RUM (from large to small) 8 7 4 1 5 9 3 6 2
SP-RRM (from small to large) 8 1 4 2 7 9 3 5 6
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psychology, addressing the semi-compensatory issues. In this study, we broke the traditional
decision-making method (only SP model) and proposed a RRM model based on the results of
the SP models. Meanwhile, in order to improve the defect of the unreasonable weight of impor-
tant influencing factors in the related literature, we investigated 146 scholars and experts who
have studied the emergency management or have taken part in related work. The traditional
utility model, regret-based model in the liner-addictive specification and regret-based model
in the logarithmic specification are all estimated, respectively. Compared with RUM and
RRM2008, an advantage of the RRM2010 was found. It is found that the experts are willing to
sacrifice inventory and procurement costs in exchange for safety benefits. However, for the final
optimal choice, there is no difference between RUM and RRM models. The findings will pro-
vide a new way of thinking from different angles and perspectives for the decision support sys-
tem and help policy-makers to develop better allocation plans in the emergency context.
Several avenues present themselves as direction for further work. Firstly, one of the prob-
lems faced is data available for the casualties of traffic accidents. Due to the confidentiality of
accident data, the identification of black spots we proposed didn’t consider the casualties,
which impacts the accuracy of the results. In the future study, we will take account into the nat-
ural geography locations and the number of casualties on this issue. Secondly, a generalization
that treats the multi-commodity flow problem under different weather is certainly worth
developing. Although our case highlights the process of allocating one type of supplies on
rainy days, our methodology is capable of providing robust preparedness plans for many types
of circumstances. Hence, the cases with regard to the pre-allocation of many types of emer-
gency supplies under different scenarios will be proposed to increase the range of potential
applicability for the model. Moreover, this paper only illustrates the application of the regret-
based model in the field of emergency management, but doesn’t certify RRM models is better
than RUM model, or RRM in the linear-additional specification is less than RRM in logarith-
mical specification, because we are based on a single dataset and based on one scenario. In the
future, we will compare with different decision rules based on several databases theoretically
and empirically. In addition, the limitations of comparison is our method is a time-consuming
task. It relies on the preference and experiences of the experts and often ignores the actual con-
ditions of the specific areas. In future work, we may improve our method and be better applied
in the engineering project.
Supporting information
S1 Fig. The attribute layer of regret function based on βm and (xjm−xim).
(PDF)
S2 Fig. Sketch of cut section technology.
(PDF)
S3 Fig. The proportion of accident on every freeway.
(PDF)
S1 Appendix.
(DOCX)
S1 File. Part of the accident data.
(XLSX)
S2 File. Part of the result of questionnaire.
(XLSX)
PLOS ONE A multi-stage emergency supplies pre-allocation approach for freeway black spots: a Chinese case study
PLOS ONE | https://doi.org/10.1371/journal.pone.0240372 October 8, 2020 17 / 20
Author Contributions
Conceptualization: Siliang Luan.
Data curation: Zhongtai Jiang.
Formal analysis: Chao Chen.
Investigation: Wei Wang.
Methodology: Siliang Luan.
Project administration: Qingfang Yang.
Supervision: Qingfang Yang, Wei Wang.
Validation: Zhongtai Jiang.
Writing – original draft: Siliang Luan.
Writing – review & editing: Qingfang Yang, Zhongtai Jiang.
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