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Robust optimization for a multiple-priority emergency evacuation problem under demand uncertainty

  • Ming Yang
  • Yankui Liu
  • Guoqing YangEmail author
Original Article
  • 68 Downloads

Abstract

This study examines a multiple-priority emergency evacuation optimization problem with time-dependent demand uncertainty. A multiple-priority dynamic traffic model—namely, the multiple-priority cell transmission model (MPCTM) —is developed to simulate the priority of network flows for emergency evacuation response. Moreover, a robust optimization approach is applied to formulate such an emergency evacuation response problem. The robust counterpart solutions of the proposed uncertainty model have been shown to be tractable, using the duality theorem. Finally, a real example of Ya’an earthquake emergency evacuation planning verifies the effectiveness of the proposed MPCTM.

Keywords

Emergency evacuation Multiple priority Cell transmission model Demand uncertainty Robust optimization 

1 Introduction

Today, large-scale disasters are occurring more frequently than in the past: since 1980, hundreds of natural catastrophes have occurred worldwide each year, and these disasters have claimed over two million lives and caused property losses worth 3,000 billion dollars (Wirtz et al. 2014). To ultimately mitigate the loss of human life and property, there are growing concerns about emergencyevacuation planning, which envisions the evacuation of people and materials from dangerous areas to safe locations during the initial phase of a disaster. In emergency evacuation planning, it is assumed that severely wounded people and important commodities will be moved as soon as possible. As such, they require priority treatment, which differs from general evacuation. This study focuses on a multiple-priority emergency evacuation problem.

Following large-scale disasters, the affected area generates an enormous evacuation traffic demand within a short time period; these typically comprise the wounded and various commodities. However, in reality, due to the lack of historical information, the evacuation traffic demand possesses highly dynamic uncertainty. Robust optimization method is based on a minmax formulation and produces robus solutions that are immune against data uncertainty. Furthermore, this uncertainty may not be addressed by general decisions, and this can result in reduced viability among those affected and induce serious secondary disasters. It is important to study a problem in evacuation traffic planning that is robust to uncertainty. Thus, our research adopts the robustoptimization approach to address the uncertain demand in emergency evacuation planning.

In this study, we examine a multiple-priority cell transmission model (MPCTM), based on a robust optimization approach that looks to provide a robust and computationally tractable solution to an emergency evacuation problem that features demand uncertainty. Meanwhile, the MPCTM—which provides a priority hierarchy—describes uncertain demand by using given uncertainty sets, and minimizes total transit cost in the worst-case-oriented decisions derived from the uncertainty sets. Numerical experiments relating to the Ya’an earthquake are shown to demonstrate the superior system performance of the MPCTM in emergency evacuation planning, and a robust solution outperforms a deterministic solution in coping with environmental uncertainty.

Based on this description, we highlight the main contributions of our work, as follows.
  • This study considers a multiple-priority network flows model for emergency evacuation response. The model incorporates sets of priorities into the cell transmission model (CTM), such that emergency vehicles are given higher priority for evacuation.

  • This study applies the robust optimization approach to a multiple-priority emergency evacuation problem with demand uncertainty, which belongs to a polyhedral uncertainty set. The approach demonstrates the realizations of demand uncertainty in the MPCTM.

  • This study proposes a real-world example (i.e., the Ya’an earthquake). The computational results illustrate the advantage of the proposed MPCTM and robust optimization. First, the MPCTM can minimize the transit time of the higher priority, thereby reducing the total transit cost. Second, compared to the deterministic solution, the robust solution may effectively reduce the cost. Third, the demand uncertainty level and budget level can impact the total cost, and the MPCTM can perform much better.

The remainder of this paper is organized as follows. Section 2 provides a literature review. Section 3 gives a deterministic model for the MPCTM. In Section 6, in consideration of demand uncertainty, robust optimization is applied in the MPCTM with a given uncertainty set. Section 5 presents a real example, to illustrate the features of the proposed model. Finally, Section 6 concludes the paper.

2 Literature review

This section presents a review of emergency evacuation network literature that establishes a framework for this research. The literature in this paper can be categorized into four streams: the emergency evacuation problem, the emergency evacuation problem with priority, the emergency evacuation problem in an uncertainty environment, and the application of robust optimization approach.

One stream is the research on the emergency evacuation problem. Emergency evacuation is in the response phase of disaster management, the purpose of which is to reduce the impact of disasters through timely evacuation. In an evacuation network model, the traffic flow is usually treated as a fluid for describing the traffic propagation phenomenon. The CTM (Daganzo 1993, 1995) was used to modeling the traffic propagation by many researchers. Chiu et al. (2007) proposed a multi-dimensional evacuation CTM model, which integrated the optimal evacuation destination, traffic assignment and evacuation departure schedule decisions into a unified evacuation network. Zhao et al. (2015) formulated an evacuation optimization model based on CTM which aimed to minimize the network clearance time. Hadiguna et al. (2014) built an innovative decision-support system to assess the feasibility of public facilities for evacuation after a disaster. To seek the optimal emergency evacuation routing, Liu et al. (2016) proposed a direction guidance system that guided passengers to escape from the optimal exits. Duan et al. (2016) proposed a personalized route planning system based on the Wardrop Equilibrium model for pedestrian-vehicle mixed evacuation ,which focused on the shortest clearance time. Asfaw et al. (2019) studied a wildfire emergency evacuation problems of an Indigenous community by a lack of community evacuation plan. Taneja and Bolia (2018) presented a bi-level model to optimize the evacuation strategy during mass gatherings. To minimize the quantities of unsatisfied demand, unserved wounded, and nontransferred workers, Al Theeb and Murray (2017) addressed a complex post-disaster humanitarian relief problem requiring the coordination of multiple heterogenous vehicles to facilitate three logistics operations.

The second stream is the research on the evacuation transportation planning with priority, which aims at minimizing evacuation transit time (or transit cost). Human disaster decision makers want to serve heavily injured people or other affected residents and materials evacuating as quickly as possible during the emergency response phase. Chiu and Zheng (2007) proposed a simultaneous mobilization destination, traffic assignment, and departure schedule for multi-priority groups (SMDTS-MPG) model in response to a no-notice disaster response. The objective of this model was to solve the optimal strategies for simultaneously mobilizing multiple priority groups with different types of intended destinations and priorities. Yi and Özdamar (2007) described an integrated location-distribution model that considered the priority among the injured people and all commodities, where the heavily injured people and urgent medicine held the highest priority. Parr et al. (2013) applied a transit signal priority in an urban evacuation that can save lives by reducing bus travel time. Moreover, Li et al. (2016) presented a travel itinerary problem which aimed to minimize travel cost for traveling multiple destinations. For our highlight work, we develop a novel linear programming formulation based on the CTM, named MPCTM. The model deploys the multiple priority emergency evacuation resources and achieves the minimization of both of the transit time of the higher priority resource and total transit cost.

The third stream is the research on the emergency evacuation problem with uncertainty which is characterized by random and fuzzy variables. The theoretical researches and applications mentioned above are deterministic demand. However, in real-world evacuation network, the demand is mostly highly uncertain. So what should the network system do when more demand is realized than used for prediction? In order to reduce the unnecessary cost, several researches take into account the well-known tendency of realized demand uncertainty as documented in the following. In stochastic environment, Sumalee et al. (2011) and Zhong et al. (2013) extended the stochastic cell transmission model to simulate traffic flow on networks with stochastic demand and supply. Do Chung et al. (2012) provided a chance-constrained programming approach for a joint chance-constrained cell transmission model based SO-DTA where uncertain demand was only partial distributional information known. Ghasemi et al. (2019) presented an uncertain multi-objective multi-commodity multi-period multi-vehicle location-allocation mixed-integer mathematical programing model, where the uncertainty was modeled using a scenario-based probability approach. Wang et al. (2016) investigated a stochastic optimization model to generate the evacuation plans based on different evaluation criteria. To address stochastic demand, Levin and Boyles (2016) proposed a cell transmission model for dynamic lane reversal with a Markov decision process formulation. Ukkusuri et al. (2017) proposed a stochastic linear mixed-integer programming model that considers three key areas of emergency logistics: facility and stock pre-positioning, evacuation planning and relief vehicle planning. The objective of the model is to minimize the total cost of opening distribution centre, transporting evacuees and shipping relief supplies. In fuzzy environment, Zheng and Ling (2013) considered three correlated fuzzy ranking criteria in an emergency transportation planning of disaster relief supply chain. Yang et al. (2015) used multi-objective biogeography-based optimization algorithm to solve supply chain network design with uncertain transportation cost and uncertain customer demand, which were characterized by continuous possibility distributions. To evaluate preparedness and response ability, Celik and Gumus (2016) proposed a hybrid approach based on interval type-2 fuzzy sets, which aimed to minimize the losses and number of affected people.

The last stream is the research on the robust optimization approach for dealing with optimization problems with the underlying uncertain data. Robust optimization is a specific methodology that may be outperformed with an unknown or partial known probability distributions, which more likely true in real network. First of all, Soyster (1973) introduced the idea of robust optimization. He considered a linear optimization problem that the solution was feasible for a given convex set, however, the solution was too conservative. After that, a prosperous development of robust optimization was put forward by Bertsimas and Sim (2003) and Ben-Tal et al. (2009) and so on. The approach could handle a variety of optimization problems, such as linear programming (LP), conic-quadratic programming (CQP) and semi-definite programming (SDP). The robust optimization approach has been applied in various fields, such as inventory management (See and Sim 2010), supply chain network (Bai and Liu 2016), and project portfolio (Liu and Liu 2017). For our related emergency evacuation network, Karoonsoontawong and Waller (2007) presented a robust optimization model for the dynamic traffic assignment based continuous network design problem, which accounted for a bilevel objective and long-term origin-destination demand uncertainty. The model also embedded the Daganzo’s cell transmission model. Ben-Tal and Do Chung (2011) applied an affinely adjustable robust counterpart approach for dynamically assigning emergency response and evacuation traffic flow planning with time dependent demand uncertainty. Rezaei-Malek et al. (2016) used a scenario-based robust stochastic approach for a disaster relief logistics network with perishable commodities. In order to address the hospital evacuation problem under uncertainty, Rabbani et al. (2016) developed a bi-objective programming model using a robust possibilistic programming approach. In this paper, we apply robust optimization approach for MPCTM with demand uncertainty.

For our paper differing from the aforementioned literature, no work has been done in multiple priority dynamic traffic assignment based CTM formulation with uncertain demand. In this paper, our effort is given to study the MPCTM based on a robust optimization method. Then we verify the performance of the MPCTM in a real-world evacuation transportation network.

3 Multiple-priority cell transmission model for emergency evacuation

In this section, we present the emergency evacuation planning model. The model is used to plan the immediate and urgent movement of affected people away from the occurrence of a disaster in circumstances with limited time and roads. Let the affected areas be the origin, and the safe areas be the destination; the evacuation roads link them. Vehicles transporting injured people represent the generated demands. We divide, on an average basis, the evacuation roads and planning horizon into several segments and periods, respectively. The size of each segment derives from the distance a vehicle drives in a unit period when moving at a free-flow speed. Due to there being finite space, the segment limits the maximum capacity and maximum flow capacity of vehicles in a unit period. Based on the properties of the proposed model, we employ the CTM (Daganzo 1993, 1995), which simplifies the segments into cells. It provides a relatively simple method by which to optimize a dynamic evacuation network, by connecting the nodes with arcs. The notation A = [aij], an adjacency matrix, is defined as follows.
$$ \begin{array}{@{}rcl@{}} a_{ij}=\left\{ \begin{aligned} &1, &&\text{cell~ i~ is~ connected~ to~ cell~ j}\\ &0, &&\text{else}. \end{aligned} \right. \end{array} $$
This is used to represent the cell connectivity in the CTM. According to the above design, vehicles can travel only to adjacent cells in a unit of time, and all the traffic flows through the network, from source cells to the sink cells.
In considering the emergency evacuation problem while assuming both heavily and moderately injured people in the affected areas, this study proposes a multiple-priority evacuation mode; thus, the evacuation planning network consists of different types of demands, and each corresponds to one priority mode. The priority may occur at any node while exceeding the maximum flow capacity. Therefore, based on the general CTM, we introduce the reformulation version, which is called the MPCTM. This model includes different vehicle types, which are assigned to different priorities. The greater the emergency, the more likely vehicles will be given higher priority. We can then generalize the concept of multiple-priority evacuation in the network. A transit cost coefficient \({\rho _{l}^{t}}\) is proposed, whereby a higher-priority vehicle type is assumed to have a higher cost when vehicles cannot arrive at the sink cells at the end of the time horizon. Therefore, let \({\rho _{1}^{t}}, {\rho _{2}^{t}},..., {\rho _{l}^{t}}\) be the partition costs of different vehicle types, such that the evacuation vehicles of \({\rho _{1}^{t}}\) have the highest, the vehicles of \({\rho _{2}^{t}}\) have the second highest, and so on. Based on the notations listed in Table 1, the deterministic linear programming MPCTM formulation can be presented as follows.
$$ \begin{array}{@{}rcl@{}} \min \qquad &\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\rho_l^tc_{li}^tx_{li}^t\quad(DLP-MPCTM1) \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} \text{s. t.}\qquad &x_{li}^t-x_{li}^{t-1}-\sum\limits_{k\in C}a_{ki}y_{lki}^{t-1}+\sum\limits_{i\in C}a_{ij}y_{lij}^{t-1}=d_{li}^{t-1},\\ & \quad \quad \quad \quad \quad \quad \quad \forall l\in L,~i\in C,~t\in \Im \end{array} $$
(2)
$$ \begin{array}{@{}rcl@{}} &\sum\limits_{l\in L}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^t+\delta_i^tx_{li}^t\right)\leq \delta_i^tN_i^t,\quad \forall i\in C,~t\in \Im \end{array} $$
(3)
$$ \begin{array}{@{}rcl@{}} &\sum\limits_{l\in L}\sum\limits_{k\in C}a_{ki}y_{lki}^t\leq Q_i^t, \quad \forall i\in C,~t\in \Im \end{array} $$
(4)
$$ \begin{array}{@{}rcl@{}} &\sum\limits_{j\in C}a_{ij}y_{lij}^t-x_{li}^t\leq0,\quad \forall l\in L,~i\in C,~t\in \Im \end{array} $$
(5)
$$ \begin{array}{@{}rcl@{}} &\sum\limits_{l\in L}\sum\limits_{j\in C}a_{ij}y_{lij}^t\leq Q_i^t, \quad \forall i\in C,~ t\in \Im \end{array} $$
(6)
$$ \begin{array}{@{}rcl@{}} &x_{li}^0=\hat{x}_{li},\quad \forall l\in L,~i\in C \end{array} $$
(7)
$$ \begin{array}{@{}rcl@{}} &y_{lij}^0=0,\quad \forall l\in L,~(i,j)\in C\times C \end{array} $$
(8)
$$ \begin{array}{@{}rcl@{}} &x_{li}^0\geq0, \quad \forall l\in L,~i\in C \end{array} $$
(9)
$$ \begin{array}{@{}rcl@{}} &y_{lij}^t\geq0, \quad \forall l\in L,~(i,j)\in C\times C,~t\in \Im. \end{array} $$
(10)
The objective function (1) of this model is to minimize the total evacuation cost, i.e., to maximum reduce the cost of evacuating the wounded by vehicles of different priorities. Constraint (2) regulates the flow conservation of each vehicle type in cell i at time t, i.e., the change of vehicle quantity is determined by traffic flow and demand at each node and in each time period. Only source cells generate demand \(d_{li}^{t}\). Constraints (3) and (4) indicate that the bounds of the total inflow are limited by the remaining capacity and the inflow capacity of the cell. In other words, the total inflow into a cell is bounded by the minimum of the remaining space and inflow capacity. Constraint (5) means that the total output flow of each vehicle type from a cell is restricted by the occupancy of the current vehicle type; additionally, Constraint (6) expresses the total outflow capacity of the cell, which is similar to Constraint (4). The remaining constraints from Eqs. 7 to 10 set initial conditions and nonnegative flow conditions. It is assumed that the capacities of the source and sink are infinite. Equation 9 is a redundant constraint, since \({\sum }_{j\in C}a_{ij}y_{lij}^{t}-x_{li}^{t}\leq 0\) and \({\sum }_{j\in C}a_{ij}y_{lij}^{t}\geq 0\). Then \(x_{li}^{t}\geq 0\) can be eliminated.
Table 1

Notations

Symbol

Description

I

set of discrete time intervals

C

set of cells

CR

set of source cells

CS

set of sink cells

A

adjacency matrix representing transportation network connectivity

L

set of priority

\({\rho _{l}^{t}}\)

transit cost coefficient

\({Q_{i}^{t}}\)

maximum number of vehicles that can flow into or out of cell i during time interval t

\({N_{i}^{t}}\)

maximum number of vehicles in cell i at time interval t

\({\delta _{i}^{t}}\)

traffic flow parameter in cell i at time interval t

\(d_{li}^{t}\)

demand of priority l in cell i at time interval t

\(x_{li}^{t}\)

number of vehicles of priority l in cell i at time interval t

\(y_{lij}^{t}\)

number of vehicles of priority l moving from cell i to cell j at time interval t

According to (7) and (8), formulation (2) can be further integrated in the following form:
$$ \begin{array}{@{}rcl@{}} x_{li}^{t}=\hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\right). \end{array} $$
(11)
Substituting formulation (11) into the DLPMPCTM1, we can obtain the following equivalent formulation:
$$ \begin{array}{@{}rcl@{}} \min &&z\quad(DLP-MPCTM2)\\ {\mathrm s. t.} &&\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\rho_l^tc_{li}^t\left( \hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\right)\right)\leq z\\ &&\sum\limits_{l\in L}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^t+\delta_i^t\left( \hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\right)\right)\right)\leq \delta_i^tN_i^t,\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \forall i\in C,~t\in \Im\\ &&\sum\limits_{j\in C}a_{ij}y_{lij}^t-\left( \hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\right)\right)\leq0,\\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\forall l\in L,~i\in C,~t\in \Im\\ &&\sum\limits_{l\in L}\sum\limits_{k\in C}a_{ki}y_{lki}^t\leq Q_i^t, \quad \forall i\in C, ~t\in \Im\\ &&\sum\limits_{l\in L}\sum\limits_{j\in C}a_{ij}y_{lij}^t\leq Q_i^t, \quad \forall i\in C, ~t\in \Im\\ &&x_{li}^0=\hat{x}_{li},\quad \forall l\in L,~i\in C\\ &&y_{lij}^0=0,\quad \forall l\in L,~(i,j)\in C\times C\\ &&x_{li}^0\geq0, \quad \forall l\in L,~i\in C\\ &&y_{lij}^t\geq0, \quad \forall l\in L,~(i,j)\in C\times C,~t\in \Im. \end{array} $$
(12)
A penalty cost parameter \(c_{li}^t\) is given to punish, so that vehicles cannot arrive at the destination at the end of time horizon T. That is to say,
$$ \begin{array}{@{}rcl@{}} c_{li}^t=\left\{ \begin{array}{ll} &1, \quad~~ l\in L,~i\in C\backslash C_S,~t\neq T\\ &M, \quad l\in L,~i\in C\backslash C_S,~t= T, \end{array} \right. \end{array} $$
where M is a number sufficiently positive and large to represent the penalty cost. By using this penalty cost parameter, the total evacuation cost consists of a travel cost and a penalty cost. In emergency evacuation networks, vehicles do not reach the destination during the planning horizon, and this will lead to the potential loss of life and property. The penalty cost parameter can play an important role in minimizing the quantity of vehicles remaining in the evacuation network.

4 Robust Optimization Formulation of the Multiple-Priority Cell Transmission Model

In this section, we apply the MPCTM to study an emergency evacuation flow network; in this way, we can depict the dynamic evacuation traffic flow. In the deterministic CTM, the demand quantity is known; however, in real emergency events, it is very difficult to accurately forecast real demand information. A small perturbation of uncertain demand information may lead to severe suboptimality or even infeasibility. To overcome evacuation demand information uncertainty, we explore the robust optimization approach (Bertsimas and Sim 2003) to convert a deterministic linear programming model into a robust counterpart model.

It is assumed that the demand quantity is unknown and belongs to a prescribed uncertainty set—that is to say, the box uncertainty set
$$ U_{box}=\left[\tilde{d}_{li}^{t^{\prime}}(1-\theta), \tilde{d}_{li}^{t^{\prime}}(1+\theta)\right], $$
where 𝜃 is the uncertainty level and \(\tilde {d}_{li}^{t^{\prime }}\) is the nominal demand 1 for type l in cell i at time \(t^{\prime }\).
In a realistic emergency event, there is only a very small probability that the demand of evacuation traffic flow used to achieve maximum flow values will be simultaneously generated in all the time intervals. Hence, for every evacuation demand cell i, we introduce a parameter Γli—which is not necessarily an integer—that adjusts the optimality and robustness of the demand perturbation. The following is a polyhedral uncertain data set:
$$ \begin{array}{@{}rcl@{}} U_{pol}&= & \bigg\{d_{li}^{t^{\prime}}:\tilde{d}_{li}^{t^{\prime}}(1-\theta)\leq d_{li}^{t^{\prime}}\leq \tilde{d}_{li}^{t^{\prime}}(1+\theta),\\ && \qquad \qquad \qquad {\sum}_{{t^{\prime}}=0}^{t-1}\left|\frac{d_{li}^{t^{\prime}}-\tilde{d}_{li}^{t^{\prime}}}{\tilde{d}_{li}^{t^{\prime}}\theta}\right|\leq {\Gamma}_{li}\bigg\} . \end{array} $$

According to the polyhedral set Upol, the robust counterpart with uncertain demand data is equivalent to the following formulation:

$$ \begin{array}{@{}rcl@{}} \min &&z\quad(RC-MPCTM) \end{array} $$
(13)
$$ \begin{array}{@{}rcl@{}} {\mathrm s. t.} &&{\sum}_{l\in L}{\sum}_{t\in\Im}{\sum}_{i\in C\backslash C_S}\rho_l^tc_{li}^t\bigg(\hat{x}_{li}+{\sum}_{{t^{\prime}}=0}^{t-1}\bigg({\sum}_{k\in C}a_{ki}y_{lki}^{t^{\prime}}\\ && \qquad \qquad -{\sum}_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\bigg)\bigg)\leq z,\quad \forall d_{li}^{t^{\prime}}\in U_{pol} \end{array} $$
(14)
$$ \begin{array}{@{}rcl@{}} &&{\sum}_{l\in L}\Bigg({\sum}_{k\in C}a_{ki}y_{lki}^t+\delta_i^t\bigg(\hat{x}_{li}+{\sum}_{{t^{\prime}}=0}^{t-1}\bigg({\sum}_{k\in C}a_{ki}y_{lki}^{t^{\prime}}\\ && -{\sum}_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+d_{li}^{t^{\prime}}\bigg)\bigg)\Bigg)\leq \delta_i^tN_i^t, \forall i\in C, t\in \Im ,~d_{li}^{t^{\prime}}\in U_{pol} \end{array} $$
(15)
$$ \begin{array}{@{}rcl@{}} &&{\sum}_{j\in C}a_{ij}y_{lij}^t-\Bigg(\hat{x}_{li}+{\sum}_{{t^{\prime}}=0}^{t-1}\bigg({\sum}_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-{\sum}_{j\in C}a_{ij}y_{lij}^{t^{\prime}}\\ && \qquad +d_{li}^{t^{\prime}}\bigg)\Bigg)\leq0, \forall l\in L, i\in C, ~t\in \Im ,~d_{li}^{t^{\prime}}\in U_{pol} \end{array} $$
(16)
$$ \begin{array}{@{}rcl@{}} &&{\sum}_{l\in L}{\sum}_{k\in C}a_{ki}y_{lki}^t\leq Q_i^t, \quad \forall i\in C, ~t\in \Im \end{array} $$
(17)
$$ \begin{array}{@{}rcl@{}} &&{\sum}_{l\in L}{\sum}_{j\in C}a_{ij}y_{lij}^t\leq Q_i^t, \quad \forall i\in C, ~t\in \Im \end{array} $$
(18)
$$ \begin{array}{@{}rcl@{}} &&x_{li}^0=\hat{x}_{li},\quad \forall l\in L,~i\in C \end{array} $$
(19)
$$ \begin{array}{@{}rcl@{}} &&y_{lij}^0=0,\quad \forall l\in L,~(i,j)\in C\times C \end{array} $$
(20)
$$ \begin{array}{@{}rcl@{}} &&x_{li}^0\geq0, \quad \forall l\in L,~i\in C \end{array} $$
(21)
$$ \begin{array}{@{}rcl@{}} &&y_{lij}^t\geq0, \quad \forall l\in L,~(i,j)\in C\times C,~t\in \Im. \end{array} $$
(22)
The above robust counterpart formulation is an intractable semi-infinite programming problem. A model variant that can be reformulated as a tractable optimization problem is presented in Theorem 1.

Theorem 1

Given the polyhedral uncertainty set Upolfor evacuation demand, and due to the duality theorem, the robust counterpart problem is equivalent to the following tractable deterministic formulation. Note thatrandq are dual variables.

$$ \begin{array}{@{}rcl@{}} \min \quad &&z\quad (RC_{pol}-MPCTM)\\ {\mathrm s. t.} \quad &&\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\Bigg(\rho_l^tc_{li}^t\bigg(\hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\bigg(\sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}} -\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}} \\ && \ \ \ +\tilde{d}_{li}^{t^{\prime}}\bigg)\bigg)+\sum\limits_{{t^{\prime}}=0}^{t-1}r_{1li}^{t^{\prime}}+{\Gamma}_{li}q_{1li}\Bigg)\leq z\\ & &r_{1li}^{t^{\prime}}+q_{1li}\geq \rho_l^tc_{li}^t\tilde{d}_{li}^{t^{\prime}}\theta, \qquad \forall l\in L, ~i\in C, ~t\in \Im, {t^{\prime}}=\{0,...,t-1\}\\ &&r_{1li}^{t^{\prime}},q_{1li}\geq 0, \qquad \forall l\in L, ~i\in C, ~t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&\sum\limits_{l\in L}\Bigg(\sum\limits_{k\in C}a_{ki}y_{lki}^t+\delta_i^t\bigg(\hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\bigg(\sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}- \sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}+\tilde{d}_{li}^{t^{\prime}}\bigg)\bigg) \\ && \ \ +\sum\limits_{{t^{\prime}}=0}^{t-1}r_{2li}^{t^{\prime}}+{\Gamma}_{li}q_{2li}\Bigg)\leq \delta_i^tN_i^t, \forall i\in C, ~t\in \Im\\ &&r_{2li}^{t^{\prime}}+q_{2li}\geq \delta_i^t\tilde{d}_{li}^{t^{\prime}}\theta, \qquad \forall l\in L, ~i\in C,~ t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&r_{2li}^{t^{\prime}},q_{2li}\geq0, \qquad \forall l\in L, ~i\in C, ~t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&\sum\limits_{j\in C}a_{ij}y_{lij}^t-\Bigg(\hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\bigg(\sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}} +\tilde{d}_{li}^{t^{\prime}}\bigg)\Bigg) \\ &&\qquad \qquad +\sum\limits_{{t^{\prime}}=0}^{t-1}r_{3li}^{t^{\prime}}+{\Gamma}_{li}q_{3li}\leq0, \qquad \forall l\in L, ~i\in C, ~t\in \Im\\ &&r_{3li}^{t^{\prime}}+q_{3li}\leq \tilde{d}_{li}^{t^{\prime}}\theta, \qquad \forall l\in L, ~i\in C, ~t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&r_{3li}^{t^{\prime}},q_{3li}\geq0, \qquad \forall l\in L, ~i\in C, ~t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&\sum\limits_{l\in L}\sum\limits_{k\in C}a_{ki}y_{lki}^t\leq Q_i^t, \qquad \forall ~i\in C, ~t\in \Im\\ &&\sum\limits_{l\in L}\sum\limits_{i\in C}a_{ij}y_{lij}^t\leq Q_i^t, \qquad \forall ~i\in C, ~t\in \Im\\ &&y_{lij}^0=0, \qquad \forall l\in L,~(i,j)\in C\times C\\ &&y_{lij}^t\geq0, \qquad \forall l\in L,~(i,j)\in C\times C,~t\in \Im. \end{array} $$

For the constraint maximum protected criterion, the Constraint (1) affected by the uncertain demand is equivalent to

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\rho_l^tc_{li}^t\left( \hat{x}_{li}+\sum\limits_{{t^{\prime}}=0}^{t-1}\left( \sum\limits_{k\in C}a_{ki}y_{lki}^{t^{\prime}}-\sum\limits_{j\in C}a_{ij}y_{lij}^{t^{\prime}}\right)\right) \\&& \qquad \qquad \qquad +\underset{d_{li}^{t^{\prime}}}{\max}\left( \sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\sum\limits_{{t^{\prime}}=0}^{t-1}\rho_l^tc_{li}^td_{li}^{t^{\prime}}\right)\leq z, \end{array} $$
where
$$ \begin{array}{@{}rcl@{}} \ && d_{li}^{t^{\prime}}\in U_{pol}=\Bigg\{d_{li}^{t^{\prime}}:\tilde{d}_{li}^{t^{\prime}}(1-\theta)\leq d_{li}^{t^{\prime}}\leq \tilde{d}_{li}^{t^{\prime}}(1+\theta),\\ && \qquad \qquad \qquad \sum\limits_{{t^{\prime}}=0}^{t-1}\left|\frac{d_{li}^{t^{\prime}}-\tilde{d}_{li}^{t^{\prime}}}{\tilde{d}_{li}^{t^{\prime}}\theta}\right|\leq {\Gamma}_{li}\Bigg\}. \end{array} $$
Let \(v_{li}^{t^{\prime }}\) be defined as \(v_{li}^{t^{\prime }}=\left |\frac {d_{li}^{t^{\prime }}-\tilde {d}_{li}^{t^{\prime }}}{\tilde {d}_{li}^{t^{\prime }}\theta }\right |\), which takes values in interval [0,1], and right-hand side vector \(b=z-\sum \limits _{l\in L}\sum \limits _{t\in \Im }\sum \limits _{i\in C\backslash C_S}\rho _l^tc_{li}^t\left (\hat {x}_{li}+\sum \limits _{{t^{\prime }}=0}^{t-1}\left (\sum \limits _{k\in C}a_{ki}y_{lki}^{t^{\prime }}-\sum \limits _{j\in C}a_{ij}y_{lij}^{t^{\prime }}\right )\right )\). Then, the uncertain set Upol can be reformulated as \( U^{\prime }_{pol}=\left \{v_{li}^{t^{\prime }}:0\leq v_{li}^{t^{\prime }}\leq 1, \sum \limits _{{t^{\prime }}=0}^{t-1}v_{li}^{t^{\prime }}\leq {\Gamma }_{li}\right \} \). We consider the following linear optimization problem (P),
$$ \begin{array}{@{}rcl@{}} \max &&\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\sum\limits_{{t^{\prime}}=0}^{t-1}\rho_l^tc_{li}^t\tilde{d}_{li}^{t^{\prime}}\\ && +\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\sum\limits_{{t^{\prime}}=0}^{t-1}\rho_l^tc_{li}^t\tilde{ d}_{li}^{t^{\prime}}\theta v_{li}^{t^{\prime}}\leq b\quad (P)\\ {\mathrm s. t.} &&0\leq v_{li}^{t^{\prime}}\leq1, \forall l\in L, ~i\in C, ~{t^{\prime}}=\{0,...,t-1\}\\ &&\sum\limits_{{t^{\prime}}=0}^{t-1}v_{li}^{t^{\prime}}\leq{\Gamma}_{li}, \forall l\!\in\! L, ~i\!\in\! C, ~t\!\in\! \Im, ~{t^{\prime}}=\{0,...,t-1\}.\\ \end{array} $$
Based on the duality theorem, we can derive an equivalent linear optimization problem of uncertain Constraint (D),
$$ \begin{array}{@{}rcl@{}} \min &&\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\sum\limits_{{t^{\prime}}=0}^{t-1}\rho_l^tc_{li}^t\tilde{d}_{li}^{t^{\prime}}+\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}{\Gamma}_{li}q_{li}\\ &&+\sum\limits_{l\in L}\sum\limits_{t\in\Im}\sum\limits_{i\in C\backslash C_S}\sum\limits_{{t^{\prime}}=0}^{t-1}r_{li}^{t^{\prime}}\leq b\quad (D)\\ {\mathrm s. t.} &&r_{li}^{t^{\prime}}+q_{li}\geq \rho_l^tc_{li}^t\tilde{d}_{li}^{t^{\prime}}\theta, \\ &&\qquad\qquad \forall l\in L, ~i\in C, ~t\in \Im, ~{t^{\prime}}=\{0,...,t-1\}\\ &&q_{li}, r_{li}^{t^{\prime}}, \rho_l^t\geq0, \forall l\in L, ~i\in C, ~{t^{\prime}}=\{0,...,t-1\}, \end{array} $$
where \(r_{li}^{t^{\prime }}\) and qli are dual variables.

Constraints (15) and (16) have forms similar to that in (14), corresponding to a similar dual form. Therefore, based on the strong duality property, the dual linear optimization problem RCpol has an equal optimal solution with RC that becomes tractable.

The model RCpolMPCTM becomes a deterministic linear programming problem with finite constraints; then, we can process it with some classical algorithms.

5 Numerical Experiments

In this section, to illustrate the performance of the proposed model and its potential achievements, we propose a real-world instance based on the Ya’an earthquake. Moreover, a robust solution can be chosen to guarantee MPCTM optimality. The linear programming problem is solved using CPLEX 12.6 on a personal computer with AMD 1.80 GHz CPU and 4 GB RAM, under Windows 7.

5.1 Problem Instance Design

Ya’an, Sichuan is located in the seismic area of China; it was there that a 7.0-magnitude earthquake occurred on April 20, 2013. As such, this is a typical study case that, when put into practice, may minimize loss through significant emergency evacuation planning. The evacuation routes from Ya’an are shown in bold in Fig. 1. The characteristics of the problem instances are described below.
Fig. 1

Ya’an evacuation routes

The evacuation routes form a single-origin multi-destination network that has seven nodes and nine connectors, including one quake-stricken node and three safe nodes. The network is equivalent to evacuating the wounded from the origin node to the three destination nodes (i.e., there is a one-way transmission network). During evacuation, due to there being varying road conditions, the average vehicle speed is assumed to be 60 mph. The cell length is set to be 10 miles; then, all vehicles can move from one cell to the next cell within 10 minutes. By dividing the connector length by the cell length, a cell network as shown in Fig. 2 represents the simplified transmission network of Fig. 1. Overall, during a planning horizon of 5 hours, the problem transforms into a network with 40 cells, 42 links, and 30 time intervals.
Fig. 2

CTM evacuation network

Furthermore, for this network, it is necessary to consider that there will be congestion near the epicenter, and that this may have a serious impact on disaster relief. Thus, traffic control for parts of cells can transit more emergency vehicles during the relief efforts. More specifically, under traffic control, the flow capacity \({Q_{i}^{t}}\) of the cells is designed to be 60 vehicles per time interval (10 mins), which is twice the flow rate of the other cells. In certain cases, aftershocks may trigger landslides, hampering transit; in such cases, the flow capacity of cell 8 is designed to vanish from time 15. Besides, the origin and destination cells are assigned as infinite, so that there will be no congestion. Next, the capacity \({N_{i}^{t}}\) is determined by the cell length, such that there are 10 vehicles per mile. Note that the destination cell 15 is equipped with a more advanced medical level so that the cell has a sufficient capacity that can staff all the evacuation wounded. Related to the cell 15, the other two destination cells (26 and 33) possess poor medical levels, and therefore their capacities are assigned as finite. The initial flow condition \(x_{li}^{0}\) and traffic flow parameter \({\delta _{i}^{t}}\) are assumed to be 0 and unity, respectively. The penalty cost (M) is 100. All the characteristics of the cells are summarized in Tables 2 and 3.
Table 2

Time-invariant cell properties

 

Cell

              
 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

\({N_{i}^{t}}\)

\(\infty \)

100

100

100

100

100

100

100

100

100

100

100

100

100

\(\infty \)

\({Q_{i}^{t}}\)

\(\infty \)

60

60

60

60

60

60

 

30

30

30

30

30

30

\(\infty \)

\(\hat {x}_{i}^{t}\)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

\({N_{i}^{t}}\)

100

100

100

100

100

100

100

100

100

100

120

100

100

100

100

\({Q_{i}^{t}}\)

60

60

60

60

60

60

30

30

30

30

\(\infty \)

60

60

30

30

\(\hat {x}_{i}^{t}\)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

31

32

33

34

35

36

37

38

39

40

\({N_{i}^{t}}\)

100

100

150

100

100

100

100

100

100

100

\({Q_{i}^{t}}\)

30

30

\(\infty \)

60

60

30

30

30

30

30

\(\hat {x}_{i}^{t}\)

0

0

0

0

0

0

0

0

0

0

Table 3

Time-dependent cell properties

 

Time

              
 

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

\({Q_{8}^{t}}\)

30

30

30

30

30

30

30

30

30

30

30

30

30

30

0

 

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

\({Q_{8}^{t}}\)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

The time-dependent demands are defined to be generated at the source cell, between time 0 and time 3. The nominal demand is set to be 500 vehicles, which evacuates two levels of injuries—namely, the heavily and moderately wounded. These correspond to two modes of sets within the network. The sets are assigned with a distinguishing transit cost coefficient, according to the emergency evacuation level. Under the priority mode, the vehicles of priority set 1 are assumed to have a transit cost coefficient \({\rho _{1}^{t}}\)= 2, to guarantee that the heavily wounded are given a higher priority. By default, the other vehicles in priority set 2 have the transit cost coefficient \({\rho _{2}^{t}}\)= 1. Moreover, in nonpriority mode, there is no consideration of the priority hierarchy among the vehicles. Then the nonpriority 1 and 2 have equality in terms of the transit cost coefficient (i.e., \({\rho _{1}^{t}}\)=\({\rho _{2}^{t}}\)= 1). Table 4 lists the setting of the priority mode of time-dependent demands and transit cost coefficients. The total transit cost equals the transit cost coefficients multiplied by time; consequently, the object of the problem is to minimize the total transit cost by minimizing the transit time of the higher-priority vehicles within the network.
Table 4

Time-dependent demand and transit cost coefficient for two priority sets

  

Time

              
  

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Priority set 1

\(d_{11}^{t}\)

35

20

10

5

0

0

0

0

0

0

0

0

0

0

0

 

\({\rho _{1}^{t}}\)

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Priority set 2

\(d_{21}^{t}\)

200

120

80

30

0

0

0

0

0

0

0

0

0

0

0

 

\({\rho _{2}^{t}}\)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

  

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Priority set 1

\(d_{11}^{t}\)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

\({\rho _{1}^{t}}\)

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Priority set 2

\(d_{21}^{t}\)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

\({\rho _{2}^{t}}\)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

In this instance, the MPCTM with 40 nodes and 42 time intervals has 123,203 constraints and 106,643 variables; this was solved using CPLEX 12.6.

5.2 Experimental results

In this section, we summarize the experimental results for the network in terms of transit time and transit cost, corresponding to priority and nonpriority cases, respectively.

Table 5 lists the experimental results regarding transit time with priority. In this tabulation, the entries in the total column show the total transit time; additionally, pr.sets refers to the respective transit time of two priority sets of vehicles, where the upper one represents the priority set 1 and the lower one is the priority set 2. The tabulation presents the change in transit time with three different uncertainty levels (i.e., 𝜃 = 0.1, 0.2, and 0.3) and four different budget levels (i.e., Γ = 1, 2, 3, and 4). Then, let \(u_{prl}^{\theta {\Gamma }}\) be the transit time of the l vehicle type in the conditions 𝜃 and Γ with priority (e.g., \(u_{pr1}^{0.11}=803.5\)) and when in the DLP model, \({\Gamma }=0(u_{prl}^{\theta 0})\).

The uncertainty level 𝜃 is a protection parameter. The larger 𝜃 is, the longer the transit time will be, relative to the outlined value. Moreover, the budget level Γ is a conservative degree parameter. As the budget level increases, the RCpol entries will increase accordingly. When Γ = 4—that is, RCpol = RCbox—the robust strategy gives the best worst-case objective value that will guarantee 100% immunization against perturbations. In line with various risk attitudes, the decision-maker can trade off between robustness and optimality, based on the budget level. Note that the DLP transit time is equivalent to its robust solution with the zero uncertainty level (𝜃 = 0).

In addition, Table 6 summarizes the experimental results regarding transit time with nonpriority. It differs from Table 5, in that in the case of nonpriority, different vehicle types can arrive at the destination in random order. The entries in column npr.sets show the expected transit times of two types of vehicles with the same perturbation uncertainty levels and budget levels. Additionally, let \(u_{nprl}^{\theta {\Gamma }}\) be the transit time of the l vehicle type with nonpriority (e.g., \(u_{npr1}^{0.11}=1181.25\)), where \({\Gamma }=0(u_{nprl}^{\theta 0})\) when in the DLP model.
Table 5

Transit time with priority (\(u_{prl}^{\theta {\Gamma }}\))

𝜃

DLP

RCpol

   

Γ = 1

Γ = 2

Γ = 3

Γ = 4(RCbox)

 

total

pr.sets

total

pr.sets

total

pr.sets

total

pr.sets

total

pr.sets

0.1

7145.0

770.0

7699.0

803.5

8030.0

830.5

8229.5

843.0

8292.5

847.0

  

6375.0

 

6895.5

 

7199.5

 

7386.5

 

7445.5

0.2

7145.0

770.0

8263.0

838.0

8930.0

891.0

9347.0

916.0

9480.0

924.0

  

6375.0

 

7425.0

 

8039.0

 

8431.0

 

8556.0

0.3

7145.0

770.0

8827.5

874.5

9875.0

951.5

10537.0

989.0

10757.5

1001.0

  

6375.0

 

7953.0

 

8923.5

 

9548.0

 

9756.5

Table 6

Transit time with nonpriority (\(u_{nprl}^{\theta {\Gamma }}\))

𝜃

DLP

RCpol

   

Γ = 1

Γ = 2

Γ = 3

Γ = 4(RCbox)

 

total

npr.sets

total

npr.sets

total

npr.sets

total

npr.sets

total

npr.sets

0.1

7145.00

1077.50

7699.00

1181.25

8030.00

1242.25

8229.50

1269.25

8292.50

1278.25

  

6067.50

 

6617.75

 

6787.75

 

6960.25

 

7014.25

0.2

7145.00

1077.50

8263.00

1273.50

8930.00

1360.50

9347.00

1420.50

9480.00

1440.00

  

6067.50

 

6989.50

 

7569.50

 

7926.50

 

8040.00

0.3

7145.00

1077.50

8827.50

1341.00

9875.00

1505.25

10537.00

1596.50

10757.50

1631.75

  

6067.50

 

7486.50

 

8369.75

 

8940.50

 

9125.75

In comparing Tables 5 and 6, one can see that the sum of the transit time of the two priority sets remains unchanged over the two different cases. The reason is that the vehicles of priority sets 1 and 2 can travel to a destination along the same route and in different orders. However, the actual travel time of different priority sets will decrease when the transit cost coefficient increases, as the MPCTM may bestow upon emergency vehicles a higher priority.

Figure 3 shows the transit time characteristics of two different cases with three different uncertainty levels. As shown in Fig. 3, three priority sets 1 of Fig. 3a (i.e., \(u_{pr1}^{0.1{\Gamma }}\), \(u_{pr1}^{0.2{\Gamma }}\), and \(u_{pr1}^{0.3{\Gamma }}\)) are narrower than the corresponding nonpriority sets 1 in Fig. 3b (i.e., \(u_{npr1}^{0.1{\Gamma }}\), \(u_{npr1}^{0.2{\Gamma }}\), and \(u_{npr1}^{0.3{\Gamma }}\)); this indicates the shorter transit time (e.g., \(u_{pr1}^{0.1{\Gamma }}<u_{npr1}^{0.1{\Gamma }}\)). Nevertheless, in both modes, the overall length of the total transit time under each uncertainty level remains unchanged (e.g., \(u_{pr1}^{0.1{\Gamma }}+u_{pr2}^{0.1{\Gamma }}=u_{npr1}^{0.1{\Gamma }}+u_{npr1}^{0.2{\Gamma }}\)).
Fig. 3

Transit time with priority and nonpriority

The computational results regarding transit cost with priority and nonpriority are shown in Tables 7 and 8, respectively. In line with the results in Tables 5 and 6, we denote the objective cost function value \(z_{pr}^{\theta {\Gamma }}\) and \(z_{npr}^{\theta {\Gamma }}\), which can be concluded by the product of transit cost coefficient and respective transit time, such as \(z_{pr}^{\theta {\Gamma }}=\sum \limits _{l\in L}{\rho _{l}^{t}}u_{prl}^{\theta {\Gamma }}\) and \(z_{npr}^{\theta {\Gamma }}=\sum \limits _{l\in L}{\rho _{l}^{t}}u_{nprl}^{\theta {\Gamma }}\). We see that the total transit cost is always increasing or decreasing with respect to transit time. Moreover, the total transit cost with nonpriority is generally higher than that with priority.
Table 7

Total cost with priority (\(z_{pr}^{\theta {\Gamma }}\))

𝜃

DLP

RCpol

  

Γ = 1

Γ = 2

Γ = 3

Γ = 4(RCbox)

0.1

7915.0

8502.5

8860.5

9072.5

9139.5

0.2

7915.0

9101.0

9821.0

10263.0

10404.0

0.3

7915.0

9702.0

10826.5

11526.0

11758.5

Table 8

Total cost with nonpriority (\(z_{npr}^{\theta {\Gamma }}\))

𝜃

DLP

RCpol

  

Γ = 1

Γ = 2

Γ = 3

Γ = 4(RCbox)

0.1

8222.50

8880.25

9272.25

9498.75

9570.75

0.2

8222.50

9536.50

10290.50

10767.50

10920.00

0.3

8222.50

10168.50

11380.25

12133.50

12389.25

The total transit costs of \(z_{pr}^{\theta {\Gamma }}\) and \(z_{npr}^{\theta {\Gamma }}\) are plotted in Fig. 4: they increase with increases in budget Γ and uncertainty level 𝜃. In addition, under any circumstance, the total travel cost with priority is lower than that with nonpriority.
Fig. 4

Transit cost with priority and nonpriority

5.3 Results analysis

To illustrate the performance of our proposed MPCTM and robust optimization approach under uncertainty, we present the analytical results in two parts. First, we compare the transit cost with priority to that with nonpriority. Second, we compare the robust transit cost to the deterministic transit cost.

First, the priority value is compared to the nonpriority value in a deterministic scenario. Then, the cost decrease degree CDD𝜃 relative to the priority value and the nonpriority value is calculated as 𝜃 is varied from 0.1 to 0.3 in intervals of 0.1. Thus, the cost decrease degree CDD𝜃 is calculated as follows:
$$ CDD^{\theta}=\frac{(z_{npr}^{\theta0}-z_{pr}^{\theta0})}{z_{npr}^{\theta0}}\times 100<percent>. $$
(23)
In the deterministic scenario, the CDD𝜃 remains unchanged for all uncertainty levels. This is because the future demand information is not affected by uncertainty. From the entries in column DLP of Table 9, the CDD𝜃 represents a 3.740% improvement. Therefore, the evacuation network with priority becomes more attractive.
Table 9

Cost decrease degree (CDD)

𝜃

DLP

RCpol

  

Γ = 1

Γ = 2

Γ = 3

Γ = 4(RCbox)

0.1

3.740%

4.254%

4.441%

4.487%

4.506%

0.2

3.740%

4.567%

4.562%

4.685%

4.725%

0.3

3.740%

4.588%

4.866%

5.007%

5.091%

Second, a similar setting is used to compare the robust transit cost to the deterministic transit cost. Then, the cost decrease degree CDD𝜃Γ relative to the robust transit cost and the deterministic transit cost is calculated as 𝜃 is varied from 0.1 to 0.3 in intervals of 0.1; Γ is varied from 1 to 4 in intervals of 1. The cost decrease degree CDD𝜃Γ is calculated as follows:
$$ CDD^{\theta{\Gamma}}=\frac{(z_{npr}^{\theta{\Gamma}}-z_{pr}^{\theta{\Gamma}})}{z_{npr}^{\theta{\Gamma}}}\times 100<percent>. $$
(24)
The entries in column RCpol of Table 9 show that a 4.254%–5.091% increase in improvement exceeds the deterministic transit cost by the robust transit cost, under a varying uncertainty level and budget level. Then, let the uncertainty level 𝜃 be fixed; the CDD𝜃Γ concluded from the robust transit cost is higher than that of the corresponding deterministic transit cost—that is to say, with the demand uncertainty, the robust solution can stipulate reducing the total transit cost of a larger proportion. More specifically, the robust solution performs better at a higher budget level. However, there can be a more conservative solution when increasing the budget level Γ. By undertaking sensitivity analysis of Γ, decision-makers can learn about their preferences regarding the number of appropriate vehicles to be dispatched for a variety of uncertain environments. Similarly, we fix the budget level Γ, and the improvement in CDD𝜃Γ shows that the robust solution can be more effective at high uncertainty levels.
Further, Fig. 5 shows the values of the cost decrease degree CDD𝜃Γ for all demand scenarios, with various uncertainty levels 𝜃 and budget levels Γ. That figure shows that higher values are associated with higher uncertainty—that is, under uncertain demand, the MPCTM guarantees a more stable and robust solution.
Fig. 5

The cost decrease degree of zpr versus znpr

In summary, the objective function derived by implementing a robust network design solution with uncertain demand generated by a given uncertainty set shows sound performance in the MPCTM. This may be significant, as the decrease in transit cost with priority corresponds to the loss of human life and property during a disaster. Meanwhile, one can employ the polyhedral uncertainty set to obtain a robust solution, which will afford a more realistic choice in emergency evacuation planning.

The multiple-priority emergency evacuation model can not only accurately simulate the real-time road conditions of each section, but also reflect the dynamic characteristics of the evacuation process. At the same time, it can also reflect the impact of congestion and the difference of crowd priority in the actual dispersion process, so as to provide some reference for the formulation of optimal management decision-making scheme.

6 Conclusions

This study applies the robust optimization approach with the cell transmission model -based multiple-priority emergency evacuation model under demand uncertainty. The multiple-priority cell transmission model (MPCTM) develops an efficient emergency evacuation flows network, which deploys higher-priority flows over lower-priority ones. When the roadway’s flow capacity is restricted, higher-priority vehicles will always be first to get through. We also studied a variant of the MPCTM that considers demand uncertainty, and which belongs to a polyhedral set and uses the robust optimization approach to manage it. Integrating with uncertainty renders the original model computationally intractable, since it is a semi-infinite program; additionally, by using the duality theorem, it can be reformulated as an equivalent and tractable optimization problem. By using real-world numerical results, we find that the MPCTM can provide a higher-quality solution than the model with nonpriority. Furthermore, a robust solution improves feasibility compared to a deterministic solution under uncertainty level 𝜃. In addition, the budget level Γ can be developed to reduce the solution conservativeness, in line with the decision preference.

Finally, we would like to suggest future research directions. Other uncertainty sets—including an ellipsoidal uncertainty set or a norm uncertainty set—should be developed in the MPCTM to derive a more optimized solution. Other effective algorithms (such as fuzzy optimization) should be employed with the model when there is more information about the uncertain data.

Footnotes

  1. 1.

    The nominal demand means the perturbation value of uncertain demand in constraint (2) is zero, that is to say, the demand is a deterministic value. In this paper, nominal model is the same as deterministic model.

Notes

Acknowledgments

This work are supported by the National Natural Science Foundation of China (Grant No. 61773150 and Grant No. 71801077), the Top-notch talents of Heibei province (Grant No. 702800118009), Humanities Social Sciences Research Project of Colleges and Universities in Hebei Province (SD171005) and Social Science Foundation of Hebei Province (HB17GL012).

Compliance with Ethical Standards

Conflict of interests

The authors declare that there is no conflict of interest regarding the publication of this article.

Human and animals participants

This article does not contain any studies with human participants or animals performed by any of the authors.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei UniversityBaodingChina
  2. 2.School of ManagementHebei UniversityBaodingChina

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