# Discrete Robustness Optimization on Emergency Transportation Network Based on Prospect Theory

Authors: Zheng-kun Zhang; Chang-feng Zhu; Qing-rong Wang; Jia-shan Yuan
Journal: Journal of Advanced Transportation
 (2019)

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## Abstract

This paper focuses on the discrete robustness optimization of emergency transportation network with the consideration of timeliness and decision behavior of decision-maker under the limited rationality. Based on a situation that the nearer to disaster area, the higher probability of time delay, prospect theory is specially introduced to reflect the subjective decision behavior of decision-maker. Then, a discrete robustness optimization model is proposed with the purpose of the better timeliness and robustness. The model is based on the emergency transportation network with multistorage centers and multidisaster points. In order to obtain the optimal solution, an improved genetic algorithm is designed by introducing a bidirectional search strategy based on a newfangled path cluster to obtain specific paths that connect each storage centers and each disaster points. Finally, a case study is exhibited to demonstrate the reasonability of the model, theory, and algorithm. The result shows that the path cluster with the better timeliness and robustness can be well obtained by using the prospect theory and improved genetic algorithm. The analysis especially reveals that the robustness is correspondent to the risk aversion in prospect theory.


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## Body

## 1. Introduction

As mentioned at the Copenhagen conference in 2009, the frequency and strength of natural disaster are trending higher than ever because of the damage on ecosystem getting serious. According to the disaster generation mechanism, natural disaster can be generally divided into geological disaster, meteorological disaster, environmental pollution disaster, fire disaster, marine disaster, and biological disaster [1]. Usually, those disasters have serious destruction on social and economic development in different degree. Especially, for some destructive disasters, such as earthquake, hail, and flood, they may cause unthinkable outcomes and leave us no time to respond. In most instances, destructive disaster usually leads to secondary disaster, which may enhance the destruction and incur some additional difficulties to rescuing. However, no matter how serious the situation is, some rescue decisions and measures must be taken immediately.In order to reduce losses and improve rescue effectiveness, emergency response and guarantee system is scientifically constructed during the normal condition. Roughly, emergency response and guarantee system includes policies and regulations, institutions and organization, emergency prearranged planning, and emergency planning system. Obviously, emergency planning system is the precondition to enact emergency prearranged planning, and, also, it is supported by policies and regulations. Therefore, emergency planning system plays a fundamental and essential role in emergency response and guarantee system. The main content of emergency planning system is shown in Figure1.Figure 1
The main content of emergency planning system.As shown in Figure1, transportation network planning of emergency materials (TNPES) is a vital step in emergency planning system. The significance of TNPES is to prearrange suitable paths from existing transportation network, and emergency materials will be dispatched immediately to the disaster area through those paths, once unexpected disaster takes place. Unquestionably, if the suitable path is not preplanned, emergency materials may be delayed, which may have a bad effect on rescuing and lead to additional loss.Obviously, the problem of TNPES belongs to the category of decision-making under uncertainty and risk. Supported by the policies and regulations, the selected path will be specialized for transporting emergency materials [2]. Without the interruption of other traffic during the emergency period, the transportation of emergency materials can be guaranteed in a great degree. However, the interruption caused by other uncertainty factors, like secondary disaster, cannot be predicted and averted. Especially, if the unexpected interruption is serious enough, the preplanned path may lose the value of use. Therefore, how to select out a suitable path from existing transportation network under complicated and uncertain environment is a difficulty and hot topic.


## 2. Literature Reviews and Discussions

According to the significance of TNPES, related specialists and scholars have made excellent researches from different viewpoints.


### 2.1. Emergency Materials Chain (ESC)

Based on the substance of ESC, the problem of facility location and vehicle routing is effectively researched by literatures [3, 4]. The more comprehensive reviews about this problem are neatened and analyzed in [5, 6]. In recent years, research on management and response is a new and hot topic about ESC. With respect to the management, some literatures provide the further study, such as [7–9]. Thereinto, Othman et al. [7] concentrate on the study of decision support system for resources scheduling. Gaonkar and Viswanadham [8] incline to the framework of management. The risk management of ESC especially is lucubrated by [9]. For the response, Rawls and Turnquist [10] reveal that the response can be improved by scientifically arranging and planning the locations and demands of emergency materials. Based on the case of Nepal earthquakes in 2015, the internal response mechanism of ESC is analyzed and identified by [11]. Besides, some studies focusing on strategy and assessment about this problem, such as scientific and technical rescue work [12], integrated effect caused by disaster are analyzed in [13].The research about ESC belongs to the field of emergency planning system. Those researches provide the foundation of TNPES to further study this problem. Beyond doubt, TNPES must obey operation mechanism of ESC. The excellent framework, management, and planning especially should be referenced during the research of TNPES.


### 2.2. Emergency Transportation Network (ETN)

Research on optimization of ETN is another main content in the field of emergency planning system. Generally, those researches can be roughly divided as uncertainty research, indicators evaluation, and scheduling path planningFor uncertainty of ETN, vulnerability is lucubrated in [14–16]. Thereinto, Sheu, and Pan [14] construct three subnetworks to presupport the operation of ETN. According to the property of ETN, postdisaster vulnerability of ETN is analyzed by [15]. Diversely, vulnerability based spatial spreading degradations in road network is systematically revealed by [16]. In addition, Berger and Lo [17], Mišković et al. [18], and Coco et al. [19] pay their attentions on robustness of ETN. Based on the deficiency of estimating optimality in most researches, Berger and Lo [17] formulate a mixed-interlinear programming model by considering robust bound. Mišković et al. [18] specially focus on the search strategy, and Coco et al. [19] focus on the formulation and algorithm. Together with the uncertainty research, related indicators of ETN are also elaborately evaluated in [20–22]. Obviously, those researches are meaningful and give us insight to further study ETN.Regarding researches on scheduling and planning, Alem et al. [23] concentrate on the postdisaster relief problem based emergency network. In addition, Duque et al. [24] study the problem of optimal route by repairing the rural road network. The study implies that multipath is necessary to emergency materials dispatch scheduling. Based on the timeliness, Bahaabadi et al. [25] present a model with multiobjective to find an optimal path from road network, in which the time-dependent stochastic travel time is taken into account. Inversely, Nikoo et al. [2] propose a three-objective interprogramming model to seek the path with best robustness by sacrificing the timeliness. Regarding the algorithm, genetic algorithm is frequently used in related researches, such as [26–28].The common attitudes held by those literatures are that timeliness is vital to emergency rescue, and stochastic factors caused by unexpected disturbance make the environment of ETN more complex and uncertain. The conclusions especially resulting from the researches can improve the tolerance and sensitivity of ETN to complex environment. However, the decision behavior under uncertainty and risk is not considered in those researches.


### 2.3. Researches from Other Viewpoints

In recent years, more and more scholars focus on the study of emergency problems from other viewpoints. Thereinto, researches like Kahneman and Tversky [29], Geroliminis et al. [27], and Córdova and Vásquez [30] optimize ETN from the viewpoint of gridding unit. With the consideration of dependencies of emergency transportation network, supernetwork theory is used to describe information transaction in [31]. Besides, motivated by a fact that different travelers usually have different path-choice preferences, a multiobjective path-finding model with time-dependent stochastic travel time is proposed in [32], and the similar theory is also involved in [25]. Slightly different, focus on the emergency path planning, decision tree is used in [33]. One of newfangled contributions in [33] is that human factors are analyzed meticulously. Moreover, based on the scenario of postdisaster, planning of rescue supporting route is also lucubrated in [34].Generally speaking, research on optimization of ETN, especially path planning, belongs to the category of decision-making. Those literatures give us some new insights to analyze the problem. Above all, the introduction of decision preference and scenario planning makes the research closer to the real world.


### 2.4. Literature Discussion and Contributions in This Paper

#### 2.4.1. Literature Discussion

According to the classified analysis of aforementioned researches, the problem of ETN is a main body of ESC, and emergency path planning is a detailed one for ETN. Without a doubt, existing researches have provided some significant references. However, some shortcomings are still conspicuous.For ETN, the complex and uncertainty of environment are considered, and optimization measures are also used to improve the tolerance and sensitivity of ETN to the complex environment, such as [14, 16, 18, 19]. But, the uncertainty scale and degree of ETN in related researches are uncontrollable. Therefore, the probability given stochastically does not reflect the disturbance mechanism in real world. For emergency path planning, performance metrics, especially the timeliness, are adopted by related literatures, like [2, 17, 27]. However, the time delay is given at random, which may lead to misconception for planner during making the decision. Moreover, some researches, like [25, 32], take the preference of route-choice into consideration, but the decision risk is not attached adequately. Although this shortcoming is made up by [33], the insufficient is still left over, because the decision behavior is analyzed just under the rational condition, which obviously violates the decision behavior under limited rationality when decision-maker faces an uncertainty situation.


#### 2.4.2. Contributions in This Paper

Based on the aforementioned analysis, this paper focuses on the discrete robustness optimization of ETN with the consideration of transportation timeliness and decision behavior of decision-maker under limited rationality. The main contributions about this paper are the following:(a) With the consideration of uncertainty of secondary disaster that may lead to the time delay, prospect theory is introduced, based on the required arrival time of emergency materials, to reflect the subjective decision of decision-maker under limited rationality.(b) A discrete robustness optimization model is proposed based on an ETN with multistorage center and multidisaster point. The purpose of this model is to find several paths with the better timeliness and better robustness to transport emergency materials.(c) An improved genetic algorithm is specially designed to obtain the optimal solution. Technically, bidirectional search strategy is designed based on a newfangled path cluster that integrates the specific paths to connecting each storage centers and disaster points.The subsequent structure of the paper is arranged as follows. Section3 is the description of discrete optimization model of ETN, and prospect theory is also introduced in this section. Section 4 is the algorithm design. A series of rationalities are analyzed based on a case study in Section 5. Section 6 is some conclusions.


## 2.1. Emergency Materials Chain (ESC)

Based on the substance of ESC, the problem of facility location and vehicle routing is effectively researched by literatures [3, 4]. The more comprehensive reviews about this problem are neatened and analyzed in [5, 6]. In recent years, research on management and response is a new and hot topic about ESC. With respect to the management, some literatures provide the further study, such as [7–9]. Thereinto, Othman et al. [7] concentrate on the study of decision support system for resources scheduling. Gaonkar and Viswanadham [8] incline to the framework of management. The risk management of ESC especially is lucubrated by [9]. For the response, Rawls and Turnquist [10] reveal that the response can be improved by scientifically arranging and planning the locations and demands of emergency materials. Based on the case of Nepal earthquakes in 2015, the internal response mechanism of ESC is analyzed and identified by [11]. Besides, some studies focusing on strategy and assessment about this problem, such as scientific and technical rescue work [12], integrated effect caused by disaster are analyzed in [13].The research about ESC belongs to the field of emergency planning system. Those researches provide the foundation of TNPES to further study this problem. Beyond doubt, TNPES must obey operation mechanism of ESC. The excellent framework, management, and planning especially should be referenced during the research of TNPES.


## 2.2. Emergency Transportation Network (ETN)

Research on optimization of ETN is another main content in the field of emergency planning system. Generally, those researches can be roughly divided as uncertainty research, indicators evaluation, and scheduling path planningFor uncertainty of ETN, vulnerability is lucubrated in [14–16]. Thereinto, Sheu, and Pan [14] construct three subnetworks to presupport the operation of ETN. According to the property of ETN, postdisaster vulnerability of ETN is analyzed by [15]. Diversely, vulnerability based spatial spreading degradations in road network is systematically revealed by [16]. In addition, Berger and Lo [17], Mišković et al. [18], and Coco et al. [19] pay their attentions on robustness of ETN. Based on the deficiency of estimating optimality in most researches, Berger and Lo [17] formulate a mixed-interlinear programming model by considering robust bound. Mišković et al. [18] specially focus on the search strategy, and Coco et al. [19] focus on the formulation and algorithm. Together with the uncertainty research, related indicators of ETN are also elaborately evaluated in [20–22]. Obviously, those researches are meaningful and give us insight to further study ETN.Regarding researches on scheduling and planning, Alem et al. [23] concentrate on the postdisaster relief problem based emergency network. In addition, Duque et al. [24] study the problem of optimal route by repairing the rural road network. The study implies that multipath is necessary to emergency materials dispatch scheduling. Based on the timeliness, Bahaabadi et al. [25] present a model with multiobjective to find an optimal path from road network, in which the time-dependent stochastic travel time is taken into account. Inversely, Nikoo et al. [2] propose a three-objective interprogramming model to seek the path with best robustness by sacrificing the timeliness. Regarding the algorithm, genetic algorithm is frequently used in related researches, such as [26–28].The common attitudes held by those literatures are that timeliness is vital to emergency rescue, and stochastic factors caused by unexpected disturbance make the environment of ETN more complex and uncertain. The conclusions especially resulting from the researches can improve the tolerance and sensitivity of ETN to complex environment. However, the decision behavior under uncertainty and risk is not considered in those researches.


## 2.3. Researches from Other Viewpoints

In recent years, more and more scholars focus on the study of emergency problems from other viewpoints. Thereinto, researches like Kahneman and Tversky [29], Geroliminis et al. [27], and Córdova and Vásquez [30] optimize ETN from the viewpoint of gridding unit. With the consideration of dependencies of emergency transportation network, supernetwork theory is used to describe information transaction in [31]. Besides, motivated by a fact that different travelers usually have different path-choice preferences, a multiobjective path-finding model with time-dependent stochastic travel time is proposed in [32], and the similar theory is also involved in [25]. Slightly different, focus on the emergency path planning, decision tree is used in [33]. One of newfangled contributions in [33] is that human factors are analyzed meticulously. Moreover, based on the scenario of postdisaster, planning of rescue supporting route is also lucubrated in [34].Generally speaking, research on optimization of ETN, especially path planning, belongs to the category of decision-making. Those literatures give us some new insights to analyze the problem. Above all, the introduction of decision preference and scenario planning makes the research closer to the real world.


## 2.4. Literature Discussion and Contributions in This Paper

### 2.4.1. Literature Discussion

According to the classified analysis of aforementioned researches, the problem of ETN is a main body of ESC, and emergency path planning is a detailed one for ETN. Without a doubt, existing researches have provided some significant references. However, some shortcomings are still conspicuous.For ETN, the complex and uncertainty of environment are considered, and optimization measures are also used to improve the tolerance and sensitivity of ETN to the complex environment, such as [14, 16, 18, 19]. But, the uncertainty scale and degree of ETN in related researches are uncontrollable. Therefore, the probability given stochastically does not reflect the disturbance mechanism in real world. For emergency path planning, performance metrics, especially the timeliness, are adopted by related literatures, like [2, 17, 27]. However, the time delay is given at random, which may lead to misconception for planner during making the decision. Moreover, some researches, like [25, 32], take the preference of route-choice into consideration, but the decision risk is not attached adequately. Although this shortcoming is made up by [33], the insufficient is still left over, because the decision behavior is analyzed just under the rational condition, which obviously violates the decision behavior under limited rationality when decision-maker faces an uncertainty situation.


### 2.4.2. Contributions in This Paper

Based on the aforementioned analysis, this paper focuses on the discrete robustness optimization of ETN with the consideration of transportation timeliness and decision behavior of decision-maker under limited rationality. The main contributions about this paper are the following:(a) With the consideration of uncertainty of secondary disaster that may lead to the time delay, prospect theory is introduced, based on the required arrival time of emergency materials, to reflect the subjective decision of decision-maker under limited rationality.(b) A discrete robustness optimization model is proposed based on an ETN with multistorage center and multidisaster point. The purpose of this model is to find several paths with the better timeliness and better robustness to transport emergency materials.(c) An improved genetic algorithm is specially designed to obtain the optimal solution. Technically, bidirectional search strategy is designed based on a newfangled path cluster that integrates the specific paths to connecting each storage centers and disaster points.The subsequent structure of the paper is arranged as follows. Section3 is the description of discrete optimization model of ETN, and prospect theory is also introduced in this section. Section 4 is the algorithm design. A series of rationalities are analyzed based on a case study in Section 5. Section 6 is some conclusions.


## 2.4.1. Literature Discussion

According to the classified analysis of aforementioned researches, the problem of ETN is a main body of ESC, and emergency path planning is a detailed one for ETN. Without a doubt, existing researches have provided some significant references. However, some shortcomings are still conspicuous.For ETN, the complex and uncertainty of environment are considered, and optimization measures are also used to improve the tolerance and sensitivity of ETN to the complex environment, such as [14, 16, 18, 19]. But, the uncertainty scale and degree of ETN in related researches are uncontrollable. Therefore, the probability given stochastically does not reflect the disturbance mechanism in real world. For emergency path planning, performance metrics, especially the timeliness, are adopted by related literatures, like [2, 17, 27]. However, the time delay is given at random, which may lead to misconception for planner during making the decision. Moreover, some researches, like [25, 32], take the preference of route-choice into consideration, but the decision risk is not attached adequately. Although this shortcoming is made up by [33], the insufficient is still left over, because the decision behavior is analyzed just under the rational condition, which obviously violates the decision behavior under limited rationality when decision-maker faces an uncertainty situation.


## 2.4.2. Contributions in This Paper

Based on the aforementioned analysis, this paper focuses on the discrete robustness optimization of ETN with the consideration of transportation timeliness and decision behavior of decision-maker under limited rationality. The main contributions about this paper are the following:(a) With the consideration of uncertainty of secondary disaster that may lead to the time delay, prospect theory is introduced, based on the required arrival time of emergency materials, to reflect the subjective decision of decision-maker under limited rationality.(b) A discrete robustness optimization model is proposed based on an ETN with multistorage center and multidisaster point. The purpose of this model is to find several paths with the better timeliness and better robustness to transport emergency materials.(c) An improved genetic algorithm is specially designed to obtain the optimal solution. Technically, bidirectional search strategy is designed based on a newfangled path cluster that integrates the specific paths to connecting each storage centers and disaster points.The subsequent structure of the paper is arranged as follows. Section3 is the description of discrete optimization model of ETN, and prospect theory is also introduced in this section. Section 4 is the algorithm design. A series of rationalities are analyzed based on a case study in Section 5. Section 6 is some conclusions.


## 3. Optimization Model of ETN

### 3.1. Problem Description and Analysis

With contrast to the traditional transportation network, ETN belongs to a rescue system that emergency materials are transported from storage centers to the disaster points. In addition, the timeliness is more important than economical efficiency, and the environment is also more complex. Without loss of generality, we make an appointment that ETN is a connected network with multistorage centers and multidisaster points. Then, letN=V,E be an ETN, where V and E are the sets of point vi and oriented edge ei,j, respectively. Besides,V s = v s ∣ v s ∈ V, set of storage center points.V c = v c ∣ v c ∈ V, set of crossover points.V d = v d ∣ v d ∈ V, set of disaster points.Especially,Vs∩Vc=⌀, Vc∩Vd=⌀, Vs∩Vd=⌀, and Vs∪Vc∪Vd=V. Let xi,j be a 0-1 binary variable to describe ETN.(1)xi,j=1,ifei,j∈E0,OhtersBased on the description, know that at least one path is valid for disaster pointvd∈Vd to obtain emergency materials from storage center vs∈Vs. As a preliminary vision with a hopeful attitude, let Ps,d be a path that is selected to transport emergency materials from vs to vd. Then, it can be described by a binary decision variable xi,js,d that(2)xi,js,d=1,ifei,j∈ps,d0,OhtersObviously,xi,js,d resulted from the planning finally determined. As having analyzed previously, timeliness is the most prominent factor to emergency materials because of the emergency rescue period within 72 hours, after which the chance of survival population may decrease dramatically [35]. Because of the importance of timeliness to emergency materials, we use timeliness as a main factor for decision-maker to make decision during the path preplanning.With respect to the transportation time, there are nominal transportation timeti,j on ei,j and potential transportation time t~i,js,d on ei,j∈ps,d. The relationship between ti,j and t~i,js,d is that(3)t~i,js,d=ti,j+Δti,js,dwhere Δti,js,d is the potential time delay on ei,j∈ps,d.Obviously, for an experienced decision-maker,t~i,js,d is more reasonable than ti,j during the decision-making. In fact, ti,j is easy to be obtained according to the design criterion of ei,j or empirical value. However, Δti,js,d is nearly impossible to be precaptured and premastered precisely, because the reason of time delay is over tens of thousands, and it is nearly impossible to take all of them into consideration. In order to facilitate the research, Δti,js,d is given stochastically in some related researches, like [36]. However it may not well reflect the real situation.As we all known, emergency events, like macroquake, usually lead to the secondary disasters, majority of which may provoke traffic congestion during the emergency materials transportation. Therefore, we can make a feasible assumption thatΔti,js,d is mainly caused by secondary disaster, like earthquake, landslide, or debris flow. Based on the assumption, we can intuitively image that the higher probability of those secondary disasters on the edge ei,j, the more potential to time delay. With respect to the probability of secondary disaster, some simulation models are proposed by related researches, like [37]. However, the decision-maker, despite capturing the probability of secondary disasters, is still under limited rationality of when and where the time delay potentially will take place. In addition, the scale of time delay still cannot be captured. Therefore, how to predict the time delay on each ei,j is extremely important for a decision-maker during the decision-making.


### 3.2. Time Delay Prediction Based on Prospect Theory

As having researched by [37], let us turn now to the situation that the nearer to the disaster area, the higher probability the secondary disasters will occur, which also means the higher probability the time delay will be. In order to facilitate the research and without loss of generality, we make an appointment that the secondary disaster just indicates earthquake, landslide, or debris flow. Xu and Li [37] reveal that the occurrence number of secondary disaster at point vi obeys the following.(4)fri,d=-Alg⁡ri,d+Bwhere ri,d is the distance from vi to disaster point vd. A and B are the parameters that can be simulated by matching the variance of number of secondary disasters.In order to get the total number of secondary disasters onei,j, ei,j can be roughly divided into τ small sections. For edge ei,j, let mi,jd be the potential number of predicted secondary disasters caused by disaster point vd; then, mi,jd is approximately equal to (5)mi,jd=∑k=1τdi,jkfrk,dwhere di,jk is the length of thek-th section and rk,d is the middle distance from thek-th section to vd.According to (5), the probability pi,j of predicted secondary disasters on each edge ei,j can be further conjectured by (6)pi,j=∑vd∈Vdmi,jd∑vd∈Vd∑vp∈V∑vq∈Vmp,qd·xp,qEquations (5) and (6) show that pi,j is just an approximate value that reflects the probability of predicted secondary disasters. However, the time delay, which is mainly caused by those predicted secondary disasters, still cannot be predicted precisely by decision-maker because of limited rationality. In fact, in the behavior theory, prospect theory reveals that no matter how precise the probability is, the decision result is finally made based on the subjective reflection of probability, not the probability directly. Just as the weight function in prospect theory [29], it reflects that decision-maker usually overweight the smaller probability and underestimates the higher one. Therefore, we can use the weight function to mimic the decision behavior of decision-maker under limited rationality. That is we have the following.When decision-maker faces gains,(7)ω+pi,j=pi,jγpi,jγ+1-pi,jγ1/γWhen decision-maker faces losses,(8)ω-pi,j=pi,jδpi,jδ+1-pi,jδ1/δwhere γ and δ are the parameters of decision weight.With respect to the gains and losses, they are based on the reference point, which is the neutral attitude of decision-maker. As behavior theory showed, decision-maker usually focuses on the deviation between impersonal outcome and subjective outcome, not the impersonal outcome itself [29]. Based on this reason, we now give a reasonable explanation of gains and losses of time delay on each edge ei,j∈ps,d.During the emergency materials path planning, the value ofΔti,js,d cannot be predicted. However, in order to without losing the rescue value of emergency materials, each category of emergency materials has its own required arrival time, and it is also the latest arrival time, beyond which the value may decrease abruptly. Just like a scarce medicine, if the medicine from manufacturer arrives at the hospital beyond the required arrival time, it will lose its value in that the patient has died. Therefore, let each category of emergency materials i have its required arrival time Ti. Based on preplanning, decision-maker pins his hope on the prediction that the expected transportation time T from storage center to disaster point is without beyond the required arrival time Ti. That is,(9)T=min⁡T1,T2,T3,…,TnIn other words, for the decision-maker with positive attitude, the maximal tolerated time delayt-s,d on ps,d is expected as (10)t-s,d=T-ts,dwhere ts,d is the nominal transportation time on ps,d; that is,(11)ts,d=∑vi∑vjti,j·xi,js,dLett-0s,d be the neutral maximal tolerant time delay per hour on path ps,d. Then, t-0s,d can be formulated by (12)t-0s,d=t-s,dts,dObviously,t-0s,d is average time delay per hour on path ps,d, which can be regarded as the neutral reference point or impersonal outcome under complete rationality on each edge ei,j∈ps,d. As mentioned that the nearer to the disaster point, the higher probability the time delay on each ei,j∈ps,d, therefore, if we let t-i,js,d be the potential time delay on ei,j∈ps,d, then, t-i,js,d can be formulated by (13)t-i,js,d=t-s,d·hi,js,dti,jwhere(14)hi,js,d=pi,j∑vi∑vjpi,j·xi,js,dThen,t-i,js,d can be regarded as the potential reference point or subjective outcome under the limited rationality on ei,j. Obviously, if t-0s,d>t-i,js,d, the result means gains for decision-maker or, otherwise, means losses. In fact, value function in prospect theory reveals that decision-maker usually holds the attitude of risk aversion in positive domain and risk seeking in negative domain, especially that the attitude is more sensitive to losses than gains. The affection caused by this attitude can be described as the following value function.(15)gti,js,d=t-0s,d-t-i,js,dα,ift0s,d≥ti,js,d-λt-i,js,d-t-0s,dβ,ifti,js,d>t0s,dwhere α is the parameter of risk aversion. β is the parameter of risk seeking. λ is the parameter of sensitive to loss.Then, we can give subjective reflection between neutral tine delay and predicted time delay on edgeei,j∈ps,d. The prospect value of each edge ei,j can be obtained by (16)Vti,js,d=gti,js,d·ω+pi,j,ifgti,js,d≥0gti,js,d·ω-pi,j,ifgti,js,d<0Based on the descriptions and analysis above, we now turn back to (3), and the predicted transportation time can be replaced by (17).(17)t^i,js,d=ti,j+Δt-i,js,d(18)Δt-i,js,d=ti,j·t-0s,d-r·Vti,js,dwhere r is the weight factor of prospect valve in time delay, and it can be stipulated as(19)r=ts,dt-s,dThere is no doubt thatΔt-i,js,d is not the genuine time delay on edge ei,j∈ps,d, but the predicted one after subjective reflection of decision-maker under limited rationality. Similarly, t^i,js,d is the predicted transportation time with the consideration of time delay under limited rationality.


### 3.3. Discrete Robustness Optimization Model

The transportation time on each edgeei,j is analyzed explicitly in Section 3.2. The introduction of predicted time delay especially based on prospect theory can give an expression to the subjective reflection of decision-maker under limited rationality, by which the result is more reasonable for decision behavior in real word. However, as shown in (17), the timeliness is apparently yielded with the consideration of predicted time delay, for which the selected path is more adaptable to its complex environment. In fact, it is not necessary to take all of the time delay into consideration, because it is nearly impossible that all the edges ei,j∈ps,d are interrupted. Therefore, during the decision-making, this weakness can be avoided by controlling the number of interrupted edges and the corresponding optimal decision will be made.Therefore, letU be a controllable uncertainty set of edges, where U⊆E. The edge especially in set U can be selected from ETN with probability Pi,j to mimic the disturbance caused by the secondary disaster in real world. Let U be the number of edges in U, and Γ is an integer to denote the uncertainty controllable variable to the ETN. Then, the objective function based on the decision variable xi,js,d can be given as follows.(20)minZ=∑vs∈Vs∑vd∈Vd∑ei,j∈Eti,j·xi,js,d+maxU∣U⊆E,U≤Γ⁡∑ei,j∈UΔt^i,js,d·xi,js,d(21)S.T.∑vd∈Vd∑vj∈Vxs,js,d-∑vi∈Vxi,ss,d=Vd,∀vs(22)min∑vs∈Vs∑vi∈Vxi,ds,d-∑vj∈Vxd,js,d=Vs,∀vd(23)min∑vs∈Vs∑vd∈Vd∑vi∈Vxi,cs,d-∑vj∈Vxc,js,d=0,∀vc(24)minxi,js,d≤xi,j,∀s,d,i,jEvidently, the purpose of objective function is to obtain several paths with the best robustness and better timeliness. Hence, the robustness and adaption of each path to complexity environment are improved. In order to solve the problem easily, the objective function, especially the robustness function, must be transformed into an equivalent linear function. According to the property of the objective function and the transformation rule proposed by [38], let Γs,d be an integer to denote uncertainty controllable variable of edge ei,j on the ps,d. Obviously, Γs,d resulted based on Γ, and both of them are accordant. Then, the equivalent transformation of the objective function can be given as follows.(25)minZ=∑vs∈Vs∑vd∈Vd∑ei,j∈Eti,j·xi,js,d+zs,dΓs,d+∑ei,j∈Uqi,js,d(26)S.T.∑vd∈Vd∑vj∈Vxs,js,d-∑vi∈Vxi,ss,d=Vd,∀vs(27)min∑vs∈Vs∑vi∈Vxi,ds,d-∑vj∈Vxd,js,d=Vs,∀vd(28)min∑vs∈Vs∑vd∈Vd∑vi∈Vxi,cs,d-∑vj∈Vxc,js,d=0,∀vc(29)minzs,d+qi,js,d≥Δt^i,js,d·xi,js,d,ei,j∈U,∀s,d(30)minzs,d≥0,∀s,d(31)minqi,js,d≥0,∀s,d(32)minxi,js,d≤xi,j,∀s,d,i,j


## 3.1. Problem Description and Analysis

With contrast to the traditional transportation network, ETN belongs to a rescue system that emergency materials are transported from storage centers to the disaster points. In addition, the timeliness is more important than economical efficiency, and the environment is also more complex. Without loss of generality, we make an appointment that ETN is a connected network with multistorage centers and multidisaster points. Then, letN=V,E be an ETN, where V and E are the sets of point vi and oriented edge ei,j, respectively. Besides,V s = v s ∣ v s ∈ V, set of storage center points.V c = v c ∣ v c ∈ V, set of crossover points.V d = v d ∣ v d ∈ V, set of disaster points.Especially,Vs∩Vc=⌀, Vc∩Vd=⌀, Vs∩Vd=⌀, and Vs∪Vc∪Vd=V. Let xi,j be a 0-1 binary variable to describe ETN.(1)xi,j=1,ifei,j∈E0,OhtersBased on the description, know that at least one path is valid for disaster pointvd∈Vd to obtain emergency materials from storage center vs∈Vs. As a preliminary vision with a hopeful attitude, let Ps,d be a path that is selected to transport emergency materials from vs to vd. Then, it can be described by a binary decision variable xi,js,d that(2)xi,js,d=1,ifei,j∈ps,d0,OhtersObviously,xi,js,d resulted from the planning finally determined. As having analyzed previously, timeliness is the most prominent factor to emergency materials because of the emergency rescue period within 72 hours, after which the chance of survival population may decrease dramatically [35]. Because of the importance of timeliness to emergency materials, we use timeliness as a main factor for decision-maker to make decision during the path preplanning.With respect to the transportation time, there are nominal transportation timeti,j on ei,j and potential transportation time t~i,js,d on ei,j∈ps,d. The relationship between ti,j and t~i,js,d is that(3)t~i,js,d=ti,j+Δti,js,dwhere Δti,js,d is the potential time delay on ei,j∈ps,d.Obviously, for an experienced decision-maker,t~i,js,d is more reasonable than ti,j during the decision-making. In fact, ti,j is easy to be obtained according to the design criterion of ei,j or empirical value. However, Δti,js,d is nearly impossible to be precaptured and premastered precisely, because the reason of time delay is over tens of thousands, and it is nearly impossible to take all of them into consideration. In order to facilitate the research, Δti,js,d is given stochastically in some related researches, like [36]. However it may not well reflect the real situation.As we all known, emergency events, like macroquake, usually lead to the secondary disasters, majority of which may provoke traffic congestion during the emergency materials transportation. Therefore, we can make a feasible assumption thatΔti,js,d is mainly caused by secondary disaster, like earthquake, landslide, or debris flow. Based on the assumption, we can intuitively image that the higher probability of those secondary disasters on the edge ei,j, the more potential to time delay. With respect to the probability of secondary disaster, some simulation models are proposed by related researches, like [37]. However, the decision-maker, despite capturing the probability of secondary disasters, is still under limited rationality of when and where the time delay potentially will take place. In addition, the scale of time delay still cannot be captured. Therefore, how to predict the time delay on each ei,j is extremely important for a decision-maker during the decision-making.


## 3.2. Time Delay Prediction Based on Prospect Theory

As having researched by [37], let us turn now to the situation that the nearer to the disaster area, the higher probability the secondary disasters will occur, which also means the higher probability the time delay will be. In order to facilitate the research and without loss of generality, we make an appointment that the secondary disaster just indicates earthquake, landslide, or debris flow. Xu and Li [37] reveal that the occurrence number of secondary disaster at point vi obeys the following.(4)fri,d=-Alg⁡ri,d+Bwhere ri,d is the distance from vi to disaster point vd. A and B are the parameters that can be simulated by matching the variance of number of secondary disasters.In order to get the total number of secondary disasters onei,j, ei,j can be roughly divided into τ small sections. For edge ei,j, let mi,jd be the potential number of predicted secondary disasters caused by disaster point vd; then, mi,jd is approximately equal to (5)mi,jd=∑k=1τdi,jkfrk,dwhere di,jk is the length of thek-th section and rk,d is the middle distance from thek-th section to vd.According to (5), the probability pi,j of predicted secondary disasters on each edge ei,j can be further conjectured by (6)pi,j=∑vd∈Vdmi,jd∑vd∈Vd∑vp∈V∑vq∈Vmp,qd·xp,qEquations (5) and (6) show that pi,j is just an approximate value that reflects the probability of predicted secondary disasters. However, the time delay, which is mainly caused by those predicted secondary disasters, still cannot be predicted precisely by decision-maker because of limited rationality. In fact, in the behavior theory, prospect theory reveals that no matter how precise the probability is, the decision result is finally made based on the subjective reflection of probability, not the probability directly. Just as the weight function in prospect theory [29], it reflects that decision-maker usually overweight the smaller probability and underestimates the higher one. Therefore, we can use the weight function to mimic the decision behavior of decision-maker under limited rationality. That is we have the following.When decision-maker faces gains,(7)ω+pi,j=pi,jγpi,jγ+1-pi,jγ1/γWhen decision-maker faces losses,(8)ω-pi,j=pi,jδpi,jδ+1-pi,jδ1/δwhere γ and δ are the parameters of decision weight.With respect to the gains and losses, they are based on the reference point, which is the neutral attitude of decision-maker. As behavior theory showed, decision-maker usually focuses on the deviation between impersonal outcome and subjective outcome, not the impersonal outcome itself [29]. Based on this reason, we now give a reasonable explanation of gains and losses of time delay on each edge ei,j∈ps,d.During the emergency materials path planning, the value ofΔti,js,d cannot be predicted. However, in order to without losing the rescue value of emergency materials, each category of emergency materials has its own required arrival time, and it is also the latest arrival time, beyond which the value may decrease abruptly. Just like a scarce medicine, if the medicine from manufacturer arrives at the hospital beyond the required arrival time, it will lose its value in that the patient has died. Therefore, let each category of emergency materials i have its required arrival time Ti. Based on preplanning, decision-maker pins his hope on the prediction that the expected transportation time T from storage center to disaster point is without beyond the required arrival time Ti. That is,(9)T=min⁡T1,T2,T3,…,TnIn other words, for the decision-maker with positive attitude, the maximal tolerated time delayt-s,d on ps,d is expected as (10)t-s,d=T-ts,dwhere ts,d is the nominal transportation time on ps,d; that is,(11)ts,d=∑vi∑vjti,j·xi,js,dLett-0s,d be the neutral maximal tolerant time delay per hour on path ps,d. Then, t-0s,d can be formulated by (12)t-0s,d=t-s,dts,dObviously,t-0s,d is average time delay per hour on path ps,d, which can be regarded as the neutral reference point or impersonal outcome under complete rationality on each edge ei,j∈ps,d. As mentioned that the nearer to the disaster point, the higher probability the time delay on each ei,j∈ps,d, therefore, if we let t-i,js,d be the potential time delay on ei,j∈ps,d, then, t-i,js,d can be formulated by (13)t-i,js,d=t-s,d·hi,js,dti,jwhere(14)hi,js,d=pi,j∑vi∑vjpi,j·xi,js,dThen,t-i,js,d can be regarded as the potential reference point or subjective outcome under the limited rationality on ei,j. Obviously, if t-0s,d>t-i,js,d, the result means gains for decision-maker or, otherwise, means losses. In fact, value function in prospect theory reveals that decision-maker usually holds the attitude of risk aversion in positive domain and risk seeking in negative domain, especially that the attitude is more sensitive to losses than gains. The affection caused by this attitude can be described as the following value function.(15)gti,js,d=t-0s,d-t-i,js,dα,ift0s,d≥ti,js,d-λt-i,js,d-t-0s,dβ,ifti,js,d>t0s,dwhere α is the parameter of risk aversion. β is the parameter of risk seeking. λ is the parameter of sensitive to loss.Then, we can give subjective reflection between neutral tine delay and predicted time delay on edgeei,j∈ps,d. The prospect value of each edge ei,j can be obtained by (16)Vti,js,d=gti,js,d·ω+pi,j,ifgti,js,d≥0gti,js,d·ω-pi,j,ifgti,js,d<0Based on the descriptions and analysis above, we now turn back to (3), and the predicted transportation time can be replaced by (17).(17)t^i,js,d=ti,j+Δt-i,js,d(18)Δt-i,js,d=ti,j·t-0s,d-r·Vti,js,dwhere r is the weight factor of prospect valve in time delay, and it can be stipulated as(19)r=ts,dt-s,dThere is no doubt thatΔt-i,js,d is not the genuine time delay on edge ei,j∈ps,d, but the predicted one after subjective reflection of decision-maker under limited rationality. Similarly, t^i,js,d is the predicted transportation time with the consideration of time delay under limited rationality.


## 3.3. Discrete Robustness Optimization Model

The transportation time on each edgeei,j is analyzed explicitly in Section 3.2. The introduction of predicted time delay especially based on prospect theory can give an expression to the subjective reflection of decision-maker under limited rationality, by which the result is more reasonable for decision behavior in real word. However, as shown in (17), the timeliness is apparently yielded with the consideration of predicted time delay, for which the selected path is more adaptable to its complex environment. In fact, it is not necessary to take all of the time delay into consideration, because it is nearly impossible that all the edges ei,j∈ps,d are interrupted. Therefore, during the decision-making, this weakness can be avoided by controlling the number of interrupted edges and the corresponding optimal decision will be made.Therefore, letU be a controllable uncertainty set of edges, where U⊆E. The edge especially in set U can be selected from ETN with probability Pi,j to mimic the disturbance caused by the secondary disaster in real world. Let U be the number of edges in U, and Γ is an integer to denote the uncertainty controllable variable to the ETN. Then, the objective function based on the decision variable xi,js,d can be given as follows.(20)minZ=∑vs∈Vs∑vd∈Vd∑ei,j∈Eti,j·xi,js,d+maxU∣U⊆E,U≤Γ⁡∑ei,j∈UΔt^i,js,d·xi,js,d(21)S.T.∑vd∈Vd∑vj∈Vxs,js,d-∑vi∈Vxi,ss,d=Vd,∀vs(22)min∑vs∈Vs∑vi∈Vxi,ds,d-∑vj∈Vxd,js,d=Vs,∀vd(23)min∑vs∈Vs∑vd∈Vd∑vi∈Vxi,cs,d-∑vj∈Vxc,js,d=0,∀vc(24)minxi,js,d≤xi,j,∀s,d,i,jEvidently, the purpose of objective function is to obtain several paths with the best robustness and better timeliness. Hence, the robustness and adaption of each path to complexity environment are improved. In order to solve the problem easily, the objective function, especially the robustness function, must be transformed into an equivalent linear function. According to the property of the objective function and the transformation rule proposed by [38], let Γs,d be an integer to denote uncertainty controllable variable of edge ei,j on the ps,d. Obviously, Γs,d resulted based on Γ, and both of them are accordant. Then, the equivalent transformation of the objective function can be given as follows.(25)minZ=∑vs∈Vs∑vd∈Vd∑ei,j∈Eti,j·xi,js,d+zs,dΓs,d+∑ei,j∈Uqi,js,d(26)S.T.∑vd∈Vd∑vj∈Vxs,js,d-∑vi∈Vxi,ss,d=Vd,∀vs(27)min∑vs∈Vs∑vi∈Vxi,ds,d-∑vj∈Vxd,js,d=Vs,∀vd(28)min∑vs∈Vs∑vd∈Vd∑vi∈Vxi,cs,d-∑vj∈Vxc,js,d=0,∀vc(29)minzs,d+qi,js,d≥Δt^i,js,d·xi,js,d,ei,j∈U,∀s,d(30)minzs,d≥0,∀s,d(31)minqi,js,d≥0,∀s,d(32)minxi,js,d≤xi,j,∀s,d,i,j


## 4. Algorithm Design

### 4.1. Problem Analysis

The problem of obtaining an optimal path from start point to terminal point in a network belongs to the shortest path problem, which is also an NP problem. In addition, the obvious differences in this paper are as follows.( 1 ) The predicted time delay Δt^i,js,d is calculated based on the ts,d on ps,d, not on the edge ei,j directly. That is, only the path is confirmed and can Δt^i,js,d be loaded on the ei,j to reflect the limited rationality of decision-maker.( 2 ) The ETN in this paper is a multisource and multiterminal network. Therefore, at least Vs×Vd paths are needed to realize the emergency materials transportation from vs to vd.( 3 ) Based on the objective function, the optimal solution with the smallest objective value entails that the path in optimal solution must be the shortest path from vs to vd. Otherwise, it will be replaced by the better one.From the analysis, when the scale of ENT is bigger enough, the optimal solution can hardly be obtained by traditional classical algorithm because of the parallelism and complexity.


### 4.2. Improved Genetic Algorithm

Genetic algorithm (GA) is proposed based on the evolutionary theory and genetic mechanism. With contrast to other intelligence algorithms, GA has advantage in solving the problem with property of parallelism because of its better robustness, flexibility, and, especially, the adaption without constraints from the problem [39, 40]. In GA, crossover operator, mutation operator, and selection operator are the cores of search strategy of the algorithm. Generally, majority of existing researches use GA to search path from network with single-source and single-terminal. Therefore, the GA is improved in this paper to search the optimal solution from ETN with multistorage centers and multiterminal points. Besides, the search strategy is also improved.


#### 4.2.1. Chromosome

Chromosome is coded as the role that combining the problem solution domain with search domain that GA can deal with. Since there are several paths in a path cluster, we present a notion of chromosome cluster. Then, chromosome in cluster represents path.For chromosome, the loca of gene represents the starting point of edgeei,j, and the gene represents the end point of ei,j. If ei,j∉ps,d, the corresponding gene is appointed as -1. Especially, if vj∈Vd, the corresponding gene is appointed as 0.Based on decision variablexi,js,d, if the chromosome represents the path ps,d, and the value of gene i is j, xi,js,d=1, otherwise, xi,js,d=0. A description of chromosome is given in Figure 2.Figure 2
A description of chromosome.As there are several paths obtained as the optimal solution to realize the emergency materials transportation fromvs to vd, chromosome cluster can be introduced, and it can be recorded in a two-dimensional matrix. Specifically, the column subscript of this matrix has same means of loca in chromosome. The rower of matrix is consistent with the subsequence of obtained chromosome. Now, we give an example to illustrate the chromosome cluster as Figure 3 shown.Figure 3
An example to illustrate chromosome cluster.Obviously, in order to ensure the constraint that each disaster point can obtain emergency materials from each storage center, the chromosomes in the optimal cluster isVs×Vd at least.


#### 4.2.2. Search Strategy

In terms of the objective function, chromosome can be generated with the search mechanism that the edge with the smaller nominal timeti,j has the higher probability to be selected. The advantage of this searching mechanism can guarantee the convergence of the algorithm and avoid the result falling into local optimum. However, for monodirectional search, the ergodicity of each storage center or disaster point cannot be satisfied. In order to make up this weakness, we improve the strategy with bidirectional search, respectively, named father search with forward search strategy and mother search with backward search strategy.(  1) Father Search Strategy. Father search direction is consistent with the direction of edge. We appoint different vs as starting point for each search. Once all vs are traversed, the father search will be over, and the chromosomes are recorded in a chromosome cluster, which can be called father chromosome cluster. As the stochastic search mechanism, the terminal point of each search is uncertainty. The flow chart of father search is shown in Figure 4.Figure 4
Flow chart of father search.(  2) Mother Search Strategy. The direction of mother search is the reverse of edge direction. For each search, vd is appointed as starting point, and the total search time is Vd. The search will be over when the search meets terminal point vs. As the selected path is converse to the real path, therefore, it needs to be reverted before coding as a chromosome. This chromosome can also be recorded in a cluster and called mother chromosome cluster. The flow chart of mother search is shown in Figure 5.Figure 5
Flow chart of mother search.Obviously, after each father search and mother search, all the storage centers and disaster points are traversed. The proportionality and availability of solution resulting from two searches can also be guaranteed.


#### 4.2.3. Crossover Operator

Crossover operator is proposed to find a better offspring during the interaction. Since the path in the father cluster is optimal in terms of starting point of each edgeei,j and for the mother cluster, the optimal one is in terms of the ending point of each ei,j; therefore, we are incline to find offspring with optimal property in terms of both staring point and ending point. In order to make it possible, we give the following description.(  1) Crossover Point. We appoint a gene (except disaster point), which is contained both in father chromosome and in mother chromosome, as a crossover point. Especially, if there is no crossover point, we grant those parent chromosomes are irreproducible. If there are several crossover points in parent chromosomes, we can select one stochastically as a real crossover point.(  2) Crossover Mechanism. For each chromosome in father cluster, it will be taken out to match each chromosome in mother cluster. Once matched successfully, they will take part in the crossover with the mechanism that the crossover point together with the subsequent genes replaced each other both in father chromosome and in mother chromosome. This process can be denoted as ⊗.Based on the description of crossover point and crossover mechanism, an example of crossover operator is shown in Figure6.Figure 6
An example of crossover operator.After the crossover operator, all the chromosomes in father cluster, mother cluster, and offspring cluster will become a new population.


#### 4.2.4. Selection Operator

Selection operator is introduced to find a chromosome with the best evaluation indicator from the existing population. In this paper, we give a stable optimal cluster to contain the exclusive chromosome. Obviously, the scale of this cluster isVs×Vd. For each selection, all the chromosomes in population will be compared with the corresponding one in the optimal cluster. Once a better one is found, the corresponding one will be replaced. The terminal condition of selection operator is when each chromosome in optimal cluster is compared. Based on the objective function, the evaluation indicator of each chromosome can be given as (33)Fs,d=min∑ei,j∈Eti,j·xi,js,d+zs,dΓs,d+∑ei,j∈Uqi,js,dNow, we give the max cycle timeTe as the terminal condition of the algorithm. According to the description above, the flow chart of improved genetic algorithm is given as Figure 7.Figure 7
Flow chart of improved genetic algorithm.An improved genetic algorithm is designed in this section. The main improvement is that a bidirectional search strategy is proposed based on the ETN with multisource and multiterminal. Besides, chromosome cluster is also newfangled one. Additional illustration is that mutation operator is omitted in that the weakness of falling into local optimum is avoided by introduction of father search and mother search.


## 4.1. Problem Analysis

The problem of obtaining an optimal path from start point to terminal point in a network belongs to the shortest path problem, which is also an NP problem. In addition, the obvious differences in this paper are as follows.( 1 ) The predicted time delay Δt^i,js,d is calculated based on the ts,d on ps,d, not on the edge ei,j directly. That is, only the path is confirmed and can Δt^i,js,d be loaded on the ei,j to reflect the limited rationality of decision-maker.( 2 ) The ETN in this paper is a multisource and multiterminal network. Therefore, at least Vs×Vd paths are needed to realize the emergency materials transportation from vs to vd.( 3 ) Based on the objective function, the optimal solution with the smallest objective value entails that the path in optimal solution must be the shortest path from vs to vd. Otherwise, it will be replaced by the better one.From the analysis, when the scale of ENT is bigger enough, the optimal solution can hardly be obtained by traditional classical algorithm because of the parallelism and complexity.


## 4.2. Improved Genetic Algorithm

Genetic algorithm (GA) is proposed based on the evolutionary theory and genetic mechanism. With contrast to other intelligence algorithms, GA has advantage in solving the problem with property of parallelism because of its better robustness, flexibility, and, especially, the adaption without constraints from the problem [39, 40]. In GA, crossover operator, mutation operator, and selection operator are the cores of search strategy of the algorithm. Generally, majority of existing researches use GA to search path from network with single-source and single-terminal. Therefore, the GA is improved in this paper to search the optimal solution from ETN with multistorage centers and multiterminal points. Besides, the search strategy is also improved.


### 4.2.1. Chromosome

Chromosome is coded as the role that combining the problem solution domain with search domain that GA can deal with. Since there are several paths in a path cluster, we present a notion of chromosome cluster. Then, chromosome in cluster represents path.For chromosome, the loca of gene represents the starting point of edgeei,j, and the gene represents the end point of ei,j. If ei,j∉ps,d, the corresponding gene is appointed as -1. Especially, if vj∈Vd, the corresponding gene is appointed as 0.Based on decision variablexi,js,d, if the chromosome represents the path ps,d, and the value of gene i is j, xi,js,d=1, otherwise, xi,js,d=0. A description of chromosome is given in Figure 2.Figure 2
A description of chromosome.As there are several paths obtained as the optimal solution to realize the emergency materials transportation fromvs to vd, chromosome cluster can be introduced, and it can be recorded in a two-dimensional matrix. Specifically, the column subscript of this matrix has same means of loca in chromosome. The rower of matrix is consistent with the subsequence of obtained chromosome. Now, we give an example to illustrate the chromosome cluster as Figure 3 shown.Figure 3
An example to illustrate chromosome cluster.Obviously, in order to ensure the constraint that each disaster point can obtain emergency materials from each storage center, the chromosomes in the optimal cluster isVs×Vd at least.


### 4.2.2. Search Strategy

In terms of the objective function, chromosome can be generated with the search mechanism that the edge with the smaller nominal timeti,j has the higher probability to be selected. The advantage of this searching mechanism can guarantee the convergence of the algorithm and avoid the result falling into local optimum. However, for monodirectional search, the ergodicity of each storage center or disaster point cannot be satisfied. In order to make up this weakness, we improve the strategy with bidirectional search, respectively, named father search with forward search strategy and mother search with backward search strategy.(  1) Father Search Strategy. Father search direction is consistent with the direction of edge. We appoint different vs as starting point for each search. Once all vs are traversed, the father search will be over, and the chromosomes are recorded in a chromosome cluster, which can be called father chromosome cluster. As the stochastic search mechanism, the terminal point of each search is uncertainty. The flow chart of father search is shown in Figure 4.Figure 4
Flow chart of father search.(  2) Mother Search Strategy. The direction of mother search is the reverse of edge direction. For each search, vd is appointed as starting point, and the total search time is Vd. The search will be over when the search meets terminal point vs. As the selected path is converse to the real path, therefore, it needs to be reverted before coding as a chromosome. This chromosome can also be recorded in a cluster and called mother chromosome cluster. The flow chart of mother search is shown in Figure 5.Figure 5
Flow chart of mother search.Obviously, after each father search and mother search, all the storage centers and disaster points are traversed. The proportionality and availability of solution resulting from two searches can also be guaranteed.


### 4.2.3. Crossover Operator

Crossover operator is proposed to find a better offspring during the interaction. Since the path in the father cluster is optimal in terms of starting point of each edgeei,j and for the mother cluster, the optimal one is in terms of the ending point of each ei,j; therefore, we are incline to find offspring with optimal property in terms of both staring point and ending point. In order to make it possible, we give the following description.(  1) Crossover Point. We appoint a gene (except disaster point), which is contained both in father chromosome and in mother chromosome, as a crossover point. Especially, if there is no crossover point, we grant those parent chromosomes are irreproducible. If there are several crossover points in parent chromosomes, we can select one stochastically as a real crossover point.(  2) Crossover Mechanism. For each chromosome in father cluster, it will be taken out to match each chromosome in mother cluster. Once matched successfully, they will take part in the crossover with the mechanism that the crossover point together with the subsequent genes replaced each other both in father chromosome and in mother chromosome. This process can be denoted as ⊗.Based on the description of crossover point and crossover mechanism, an example of crossover operator is shown in Figure6.Figure 6
An example of crossover operator.After the crossover operator, all the chromosomes in father cluster, mother cluster, and offspring cluster will become a new population.


### 4.2.4. Selection Operator

Selection operator is introduced to find a chromosome with the best evaluation indicator from the existing population. In this paper, we give a stable optimal cluster to contain the exclusive chromosome. Obviously, the scale of this cluster isVs×Vd. For each selection, all the chromosomes in population will be compared with the corresponding one in the optimal cluster. Once a better one is found, the corresponding one will be replaced. The terminal condition of selection operator is when each chromosome in optimal cluster is compared. Based on the objective function, the evaluation indicator of each chromosome can be given as (33)Fs,d=min∑ei,j∈Eti,j·xi,js,d+zs,dΓs,d+∑ei,j∈Uqi,js,dNow, we give the max cycle timeTe as the terminal condition of the algorithm. According to the description above, the flow chart of improved genetic algorithm is given as Figure 7.Figure 7
Flow chart of improved genetic algorithm.An improved genetic algorithm is designed in this section. The main improvement is that a bidirectional search strategy is proposed based on the ETN with multisource and multiterminal. Besides, chromosome cluster is also newfangled one. Additional illustration is that mutation operator is omitted in that the weakness of falling into local optimum is avoided by introduction of father search and mother search.


## 4.2.1. Chromosome

Chromosome is coded as the role that combining the problem solution domain with search domain that GA can deal with. Since there are several paths in a path cluster, we present a notion of chromosome cluster. Then, chromosome in cluster represents path.For chromosome, the loca of gene represents the starting point of edgeei,j, and the gene represents the end point of ei,j. If ei,j∉ps,d, the corresponding gene is appointed as -1. Especially, if vj∈Vd, the corresponding gene is appointed as 0.Based on decision variablexi,js,d, if the chromosome represents the path ps,d, and the value of gene i is j, xi,js,d=1, otherwise, xi,js,d=0. A description of chromosome is given in Figure 2.Figure 2
A description of chromosome.As there are several paths obtained as the optimal solution to realize the emergency materials transportation fromvs to vd, chromosome cluster can be introduced, and it can be recorded in a two-dimensional matrix. Specifically, the column subscript of this matrix has same means of loca in chromosome. The rower of matrix is consistent with the subsequence of obtained chromosome. Now, we give an example to illustrate the chromosome cluster as Figure 3 shown.Figure 3
An example to illustrate chromosome cluster.Obviously, in order to ensure the constraint that each disaster point can obtain emergency materials from each storage center, the chromosomes in the optimal cluster isVs×Vd at least.


## 4.2.2. Search Strategy

In terms of the objective function, chromosome can be generated with the search mechanism that the edge with the smaller nominal timeti,j has the higher probability to be selected. The advantage of this searching mechanism can guarantee the convergence of the algorithm and avoid the result falling into local optimum. However, for monodirectional search, the ergodicity of each storage center or disaster point cannot be satisfied. In order to make up this weakness, we improve the strategy with bidirectional search, respectively, named father search with forward search strategy and mother search with backward search strategy.(  1) Father Search Strategy. Father search direction is consistent with the direction of edge. We appoint different vs as starting point for each search. Once all vs are traversed, the father search will be over, and the chromosomes are recorded in a chromosome cluster, which can be called father chromosome cluster. As the stochastic search mechanism, the terminal point of each search is uncertainty. The flow chart of father search is shown in Figure 4.Figure 4
Flow chart of father search.(  2) Mother Search Strategy. The direction of mother search is the reverse of edge direction. For each search, vd is appointed as starting point, and the total search time is Vd. The search will be over when the search meets terminal point vs. As the selected path is converse to the real path, therefore, it needs to be reverted before coding as a chromosome. This chromosome can also be recorded in a cluster and called mother chromosome cluster. The flow chart of mother search is shown in Figure 5.Figure 5
Flow chart of mother search.Obviously, after each father search and mother search, all the storage centers and disaster points are traversed. The proportionality and availability of solution resulting from two searches can also be guaranteed.


## 4.2.3. Crossover Operator

Crossover operator is proposed to find a better offspring during the interaction. Since the path in the father cluster is optimal in terms of starting point of each edgeei,j and for the mother cluster, the optimal one is in terms of the ending point of each ei,j; therefore, we are incline to find offspring with optimal property in terms of both staring point and ending point. In order to make it possible, we give the following description.(  1) Crossover Point. We appoint a gene (except disaster point), which is contained both in father chromosome and in mother chromosome, as a crossover point. Especially, if there is no crossover point, we grant those parent chromosomes are irreproducible. If there are several crossover points in parent chromosomes, we can select one stochastically as a real crossover point.(  2) Crossover Mechanism. For each chromosome in father cluster, it will be taken out to match each chromosome in mother cluster. Once matched successfully, they will take part in the crossover with the mechanism that the crossover point together with the subsequent genes replaced each other both in father chromosome and in mother chromosome. This process can be denoted as ⊗.Based on the description of crossover point and crossover mechanism, an example of crossover operator is shown in Figure6.Figure 6
An example of crossover operator.After the crossover operator, all the chromosomes in father cluster, mother cluster, and offspring cluster will become a new population.


## 4.2.4. Selection Operator

Selection operator is introduced to find a chromosome with the best evaluation indicator from the existing population. In this paper, we give a stable optimal cluster to contain the exclusive chromosome. Obviously, the scale of this cluster isVs×Vd. For each selection, all the chromosomes in population will be compared with the corresponding one in the optimal cluster. Once a better one is found, the corresponding one will be replaced. The terminal condition of selection operator is when each chromosome in optimal cluster is compared. Based on the objective function, the evaluation indicator of each chromosome can be given as (33)Fs,d=min∑ei,j∈Eti,j·xi,js,d+zs,dΓs,d+∑ei,j∈Uqi,js,dNow, we give the max cycle timeTe as the terminal condition of the algorithm. According to the description above, the flow chart of improved genetic algorithm is given as Figure 7.Figure 7
Flow chart of improved genetic algorithm.An improved genetic algorithm is designed in this section. The main improvement is that a bidirectional search strategy is proposed based on the ETN with multisource and multiterminal. Besides, chromosome cluster is also newfangled one. Additional illustration is that mutation operator is omitted in that the weakness of falling into local optimum is avoided by introduction of father search and mother search.


## 5. Case Study

An ETN with 19 points is introduced in this section, wherev1, v2, and v3 are the storage centers, v18 and v19 are the disaster points, and others are the crossover points. The ETN in case study is shown in Figure 8.Figure 8
The ETN in case study.Based on the ETN shown in Figure8, values of related parameters are presented in Table 1.Table 1
Values of related parameters.
No. e i,j d i,j (km) t i,j (h) Distance tov18 (km) Distance tov19 (km) r i,18 r j,18 r i,19 r j,19 1 (1,4) 34.42 0.95 145.33 113.12 154.63 119.35 2 (1,5) 47.27 1.21 145.33 117.03 154.63 114.71 3 (1,7) 51.04 1.52 145.33 96.63 154.63 112.28 4 (2,4) 33.23 0.78 150.23 113.12 146.18 119.35 5 (2,5) 32.78 0.84 150.23 117.03 146.18 114.71 6 (3,4) 41.86 1.04 140.54 113.12 133.74 119.35 7 (3,5) 25.19 0.62 140.54 117.03 133.74 114.71 8 (3,6) 35.13 1.12 140.54 113.79 133.74 102.39 9 (4,7) 28 0.63 113.12 96.63 119.35 112.28 10 (4,8) 25.14 0.75 113.12 94.73 119.35 95.13 11 (4,10) 50.09 1.21 113.12 64.48 119.35 81.61 12 (5,6) 25.36 0.67 117.03 113.79 114.71 102.39 13 (5,8) 23.58 0.65 117.03 94.73 114.71 95.13 14 (5,9) 41.05 1.05 117.03 88.58 114.71 76.47 15 (6,8) 33.38 0.73 113.79 94.73 102.39 95.13 16 (6,9) 26.92 0.84 113.79 88.58 102.39 76.47 17 (6,11) 49.1 1.36 113.79 65.12 102.39 65.23 18 (7,8) 32.12 0.79 96.63 94.73 112.28 95.13 19 (7,10) 31.66 0.81 96.63 64.48 112.28 81.61 20 (7,11) 50.12 1.23 96.63 65.12 112.28 65.23 21 (7,14) 59.35 1.77 96.63 37.56 112.28 67.67 22 (8,9) 31.78 0.71 94.73 88.58 95.13 76.47 23 (8,10) 33.72 0.97 94.73 64.48 95.13 81.61 24 (8,11) 32.22 0.85 94.73 65.12 95.13 65.23 25 (9,11) 27.55 0.63 88.58 65.12 76.47 65.23 26 (9,12) 29.92 0.86 88.58 65.32 76.47 46.48 27 (9,15) 48.27 1.09 88.58 42.06 76.47 34.67 28 (10,11) 26.76 0.58 64.48 65.12 81.61 65.23 29 (10,13) 25.66 0.74 64.48 42.7 81.61 57.08 30 (10,14) 29.97 0.90 64.48 37.56 81.61 67.67 31 (11,12) 31.25 0.87 65.12 65.32 65.23 46.48 32 (11,13) 23.26 0.6 65.12 42.7 65.23 57.08 33 (11,15) 30.96 0.69 65.12 42.06 65.23 34.67 34 (12,15) 25.36 0.77 65.32 42.06 46.48 34.67 35 (12,17) 30.05 0.9 65.32 46.43 46.48 18.97 36 (13,14) 19.73 0.45 42.7 37.56 57.08 67.67 37 (13,15) 24.43 0.59 42.7 42.06 57.08 34.67 38 (13,16) 28.86 0.74 42.7 24.61 57.08 32.99 39 (13,18) 44.21 0.98 42.7 2 57.08 45.7 40 (14,16) 34.67 0.77 37.56 24.61 67.67 32.99 41 (14,18) 38.18 1.06 37.56 2 67.67 45.7 42 (15,16) 17.43 0.48 42.06 24.61 34.67 32.99 43 (15,17) 20.13 0.52 42.06 46.43 34.67 18.97 44 (16,17) 25.39 0.60 24.61 46.43 32.99 18.97 45 (16,18) 25.1 0.56 24.61 2 32.99 45.7 46 (16,19) 32.68 0.88 24.61 45.14 32.99 2 47 (17,19) 18.03 0.45 46.43 45.14 18.97 2According to the values in Table1, uncertainty probability pi,j can be obtained. Based on serious of matching, when A=8.23 and B=20.38 in (4) and τ=1000 in (5), the uncertainty probability of each edge under controllable variable Γ=47 is given as Figure 9.Figure 9
The uncertainty probability of each edge under controllable variableΓ=47.From Figure9, we can see that the probability becomes different along with the edge gets close to corresponding disaster point. However, the situation that the nearer it is to disaster point, the higher the probability of time delay is not reflected vividly, the reason is that the result takes the length of each edge into consideration additionally. If we eliminate this factor, the probability of unit length on each edge is shown as in Figure 10.Figure 10
Probability of unit length on each edge.Obviously, Figure10 can well reflect the phenomenon we appointed. Especially, for three conditions in Figure 10, the value of the edge that is nearest to the disaster point is beyond triple to the farthest one. In addition, both in Figure 9 and in Figure 10, the synthesis probability is caused by disaster points v18 and v19 commonly, and the result is also in accordance with the real word.


### 5.1. Optimal Solutions

Based on the essential information given above, we acquiescently appoint that the minimum expected arrive timeT=8 in (9) and let the max cycle of algorithm Te=250. When the C++ runs for 0.274s, the path cluster with the best predicted transportation time can be obtained. The results of the best path cluster under Γ=47 are shown in Table 2.Table 2
The results of the best path cluster underΓ=47.
No. Path Details Nominal time Time delay Predicted time 1 P 1,18 1→4→10→13→18 3.88 4.12 8 2 P 1,19 1→4→7→8→9→12→17→19 5.29 2.85 8.14 3 P 2,18 2→4→10→13→18 3.71 4.29 8 4 P 2,19 2→5→9→15→17→19 3.95 4.05 8 5 P 3,18 3→5→8→10→13→18 3.96 4.04 8 6 P 3,19 3→5→8→11→15→17→19 3.78 4.22 8 Note: time/hour.As the result in Table2, the nominal time, time delay, and expected time of the optimal path cluster are 24.57, 23.57, and 48.14 hours, respectively. The path (10→13→18) especially is commonly shared byP1,18,P2,18, andP3,18; the main reason is that it has the better timeliness and robustness, and it is significant to disaster point p18. The same reason can also go for path (15→17→19) toP2,19 andP3,19. In addition, the predicted time ofP1,19 is beyond the required time, which means that this path has the higher probability to time delay, and the decision-maker should pay much attention to this path during the emergency materials transportation.In terms of the optimal path cluster, the convergence procedure based on predicted time, time delay, and nominal time are shown in Figures11 and 12, respectively.Figure 11
The convergence procedure based on predicted time.Figure 12
The convergence procedure based on time delay and nominal time.According to Figures11 and 12, the predicted time in Figure 11 tends to be stable when iteration is at 104. However, the nominal time and time delay in Figure 12 is stable when iteration is at 144. This means that a better path cluster, in which the increment of time delay is exactly equal to the decrement of nominal time, is found when iteration is at 144. Generally, the predicted time is the comprehensive reflection of time delay and nominal time. In addition, in order to meet the required arrival time, the convergence procedures in Figures 11 and 12 show that the path with the better robustness has the better timeliness in the condition of all the edges under the uncertainty; that is, Γ=47. In order to analyze the convergence of each path in cluster, the convergence procedure of each path based predicted transportation time is shown in Figure 13.Figure 13
The convergence procedure of each path based on predicted transportation time.From Figure13, we can see the convergence procedures of each path are different, but the trends are strictly consistent with the convergence procedure in Figure 11. However, a peculiar phenomenon is that the predicted transportation time ofP1,19 is finally beyond the required arrival time, which is also mentioned in Table 1. Further analysis shows that the reason is that all the paths from v1 to v19 are much great and comprehensively influenced; consequently, both the timeliness and robustness have to be yield as much as possible.As mentioned in Section4, the main improvement of genetic algorithm is that a bidirectional search strategy is proposed according to the ETN with multisource and multiterminal. In some of the other researches, the network is usually transformed as the one with single-source and single-terminal. In order to verify the effectiveness of the improved genetic algorithm in this paper, we transform the ETN as the one with single-source and single-terminal and solve it by the traditional genetic algorithm. Then, the comparison of convergence procedure based on two genetic algorithms is shown in Figure 14.Figure 14
The comparison of convergence procedure based on two genetic algorithms.According to the comparison in Figure14, the convergence of improved genetic algorithm tends to table when iteration is at 104; however, the table convergence of traditional genetic algorithm is at 184. Besides, the results of improved genetic algorithm are much more obvious than traditional genetic algorithm before the iteration at 30, which reflects that the bidirectional search strategy has higher search ability. Combined with analysis above, the results demonstrate the improved genetic algorithm is more reasonable and efficient to solve the path optimization problem in the network with multisource and multiterminal.


### 5.2. Influence Analysis of Subjective Decision

#### 5.2.1. Influence Analysis Based Network Viewpoint

The advantage of discrete robustness optimization model is that the uncertainty scale is controllable. From the viewpoint of network,Γ is an important parameter to reflect the optimism or pessimism attitude of decision-maker. In order to analyze the influence caused by attitude of decision-maker, the value of optimal path cluster under different Γ are shown in Figure 15.Figure 15
The value of optimal path cluster under differentΓ.Obviously, with the uncertainty scale up, the predicted time tends to increase. It reflects that the attitude of decision-maker becomes more and more pessimistic. Generally, the attitude of decision-maker is insensitive whenΓ≤8, and the attitude is in the hesitation when 8<Γ≤22. Especially, Γ>22, attitude becomes pessimistic. Next, we take p2,18 and p2,19 as examples for further analysis. The related information of e p1,18 and p1,19 is shown in Table 3.Table 3
The related information ofp2,18 and p2,19.
Γ Path Details Nominal time Time delay Predicted time 0 P 2,18 2→4→10→13→18 3.71 0 3.71 P 2,19 2→5→9→15→17→19 3.95 0 3.95  7 P 2,18 2→5→8→11→13→18 3.92 0 3.92 P 2,19 2→5→9→12→17→19 4.10 0 4.10  17 P 2,18 2→5→8→10→13→16→18 4.50 0.76 5.26 P 2,19 2→5→8→10→13→15→17→19 4.76 0.25 5.01  27 P 2,18 2→4→10→13→14→18 4.24 0.97 5.21 P 2,19 2→4→10→13→15→17→19 4.29 1.11 5.40  37 P 2,18 2→4→8→11→13→18 3.96 1.87 5.83 P 2,19 2→4→8→11→13→16→19 4.60 1.27 5.87  47 P 2,18 2→4→10→13→18 3.71 4.29 8 P 2,19 2→5→9→15→17→19 3.95 4.05 8 Note:time/hour.The result in Table3 shows that the optimal paths are different with Γ taking different values. It entails that the paths in optimal cluster are amended to adapt the circumstance under different Γ. Especially, when Γ takes the value in median parts of the uncertainty scale, the optimal paths have the poorest timeliness of nominal time. Strangely, when Γ=47, the optimal path is some as to the optimal path under Γ=0. This phenomenon gives us a conclusion that the optimal solution under medium scale of uncertainty is the most difficult scenario for decision-maker to make decision, and the final result usually has higher sacrifice of the timeliness.


#### 5.2.2. Influence Analysis Based Path Viewpoint

During the decision of optimal path cluster, prospect theory is introduced to reflect the subjective decision of decision-maker under limited rationality. Actually, the influence is embodied in the time delay. In order to analyze the influence caused by of subjective decision under limited rationality, the time delay of differentP1,18 based different viewpoints are listed in Tables 4, 5, and 6, respectively.Table 4
The time delay ofP1,18 with the better robustness and the poorer timeliness.
Items P 1,18(1→7→10→13→18) ∑ e i , j (1,7) (7,10) (10,13) (13,18) — t i , j (h) 1.520 0.810 0.740 0.980 4.050 h i , j 0.195 0.172 0.181 0.452 1.000 DND 1.483 0.790 0.722 0.956 3.950 DPD 0.770 0.678 0.717 1.786 3.950 DSD 1.282 0.756 0.719 1.763 4.520 Note: DND is the time delay based neutral decision under complete rationality. DPD is the time delay based probabilistic decision under complete rationality. DSD is the time delay based subjective decision under limited rationality.hi,j is the relative probability in (14).Table 5
The time delay ofP1,18 with the better timeliness and the poorer robustness.
Items P 1,18(1→4→10→13→18) ∑ e i , j (1,4) (4,10) (10,13) (13,18) — t i , j (h) 0.950 1.210 0.740 0.980 3.880 h i , j 0.122 0.254 0.179 0.445 1.000 DND 1.009 1.285 0.786 1.041 4.120 DPD 0.503 1.046 0.736 1.835 4.120 DSD 1.009 1.285 0.786 1.041 4.120Table 6
The time delay ofP1,18 with the poorer timeliness and the poorer robustness.
Items P 1,18(1→7→11→15→16→18) ∑ e i , j (1,7) (7,11) (11,15) (15,16) (16,18) — t i , j (h) 1.520 1.230 0.690 0.480 0.560 4.480 h i , j 0.164 0.237 0.204 0.142 0.252 1.000 DND 1.194 0.966 0.542 0.377 0.440 3.520 DPD 0.577 0.836 0.718 0.501 0.888 3.520 DSD 1.031 0.918 0.663 0.446 0.746 3.804As Tables4 and 6 showed, for each ei,j, DSD is smaller than DND but bigger than DPD, when hi,j takes smaller values. It illustrates that decision-maker holds risk aversion attitudes when he faces the situation with smaller probability of time delay. On the contrary, DSD is bigger than DND but smaller than DPD, when hi,j takes bigger values; that is, the attitude is risk seeking when he faces the higher probability of time delay. From the viewpoint of each path, decision-maker holds attitude of risk aversion because of ∑DSD beyond both ∑DPD and ∑DND. Obviously, the more obvious the attitude of risk aversion holds, the better the robustness is.DSD especially is all equal to DND in Table5, but it does not violate the conclusion obtained above. The reason is mainly caused by the weight factor of prospect valve in time delay in equation (19). Further analysis shows that the tolerant time delay in Table 4 is so plenty that subjective reflection turns invalid. In fact, it is coincident with decision behavior in the real world.


#### 5.2.3. Parameters Analysis in Prospect Theory

The subjective reflection of decision-maker is mainly influenced by the value of parameters in prospect theory. Based on time delay, this section focuses on the analysis of influence caused by the value of parameters in prospect theory. The influence of time delay caused byα and β is shown in Figure 16.Figure 16
The influence of time delay caused byα and β.α and β are the parameters in value function. From Figure 16 we can see that the bigger the value that α takes, the higher the time delay is. In other words, the decision attitude turns more risk aversion. On the contrary, the attitude turns more risk seeking when β turns bigger. Similarly, the influence of time delay caused by γ and δ is shown in Figure 17.Figure 17
The influence of time delay caused byγ and δ.Compared with value function, Figure17 shows that parameters in weight function have higher influence on time delay. Especially, when γ takes values in 0.4,0.8, the attitude of risk seeking is obvious, and the risk aversion is obvious when δ takes values in 0.6,1. Generally, δ has higher influence than γ.In addition, prospect theory presents that bothα and γ have comprehensive influence on gains, and so do β and δ on losses. Therefore, the influence of time delay caused by α and γ is shown in Figure 18.Figure 18
The influence of time delay caused byα and γ.Obviously, time delay is gradually increased with the value ofα up. When α<0.8, the variation of γ has greater influence on time delay. However, when α≥0.8, the influence is unconspicuous. Generally, the influence caused by variation of γ is greater than that of α. In other words, subjective reflection is more sensitive to weight function than value function. Similarly, the influence of time delay caused by β and δ is shown in Figure 19.Figure 19
The influence of time delay caused byβ and δ.As shown in Figure19, the influence on time delay caused by variation of β is obvious when δ≥0.4, and the influence caused by δ is obvious when β≤0.7. Similar to the reflection in Figure 18, Figure 19 also shows that the time delay is much more sensitive to weight function than value function. Combining with Figures 18 and 19, we can make conclusion that weight function has greater influence on subjective reflection than value function in this paper.


## 5.1. Optimal Solutions

Based on the essential information given above, we acquiescently appoint that the minimum expected arrive timeT=8 in (9) and let the max cycle of algorithm Te=250. When the C++ runs for 0.274s, the path cluster with the best predicted transportation time can be obtained. The results of the best path cluster under Γ=47 are shown in Table 2.Table 2
The results of the best path cluster underΓ=47.
No. Path Details Nominal time Time delay Predicted time 1 P 1,18 1→4→10→13→18 3.88 4.12 8 2 P 1,19 1→4→7→8→9→12→17→19 5.29 2.85 8.14 3 P 2,18 2→4→10→13→18 3.71 4.29 8 4 P 2,19 2→5→9→15→17→19 3.95 4.05 8 5 P 3,18 3→5→8→10→13→18 3.96 4.04 8 6 P 3,19 3→5→8→11→15→17→19 3.78 4.22 8 Note: time/hour.As the result in Table2, the nominal time, time delay, and expected time of the optimal path cluster are 24.57, 23.57, and 48.14 hours, respectively. The path (10→13→18) especially is commonly shared byP1,18,P2,18, andP3,18; the main reason is that it has the better timeliness and robustness, and it is significant to disaster point p18. The same reason can also go for path (15→17→19) toP2,19 andP3,19. In addition, the predicted time ofP1,19 is beyond the required time, which means that this path has the higher probability to time delay, and the decision-maker should pay much attention to this path during the emergency materials transportation.In terms of the optimal path cluster, the convergence procedure based on predicted time, time delay, and nominal time are shown in Figures11 and 12, respectively.Figure 11
The convergence procedure based on predicted time.Figure 12
The convergence procedure based on time delay and nominal time.According to Figures11 and 12, the predicted time in Figure 11 tends to be stable when iteration is at 104. However, the nominal time and time delay in Figure 12 is stable when iteration is at 144. This means that a better path cluster, in which the increment of time delay is exactly equal to the decrement of nominal time, is found when iteration is at 144. Generally, the predicted time is the comprehensive reflection of time delay and nominal time. In addition, in order to meet the required arrival time, the convergence procedures in Figures 11 and 12 show that the path with the better robustness has the better timeliness in the condition of all the edges under the uncertainty; that is, Γ=47. In order to analyze the convergence of each path in cluster, the convergence procedure of each path based predicted transportation time is shown in Figure 13.Figure 13
The convergence procedure of each path based on predicted transportation time.From Figure13, we can see the convergence procedures of each path are different, but the trends are strictly consistent with the convergence procedure in Figure 11. However, a peculiar phenomenon is that the predicted transportation time ofP1,19 is finally beyond the required arrival time, which is also mentioned in Table 1. Further analysis shows that the reason is that all the paths from v1 to v19 are much great and comprehensively influenced; consequently, both the timeliness and robustness have to be yield as much as possible.As mentioned in Section4, the main improvement of genetic algorithm is that a bidirectional search strategy is proposed according to the ETN with multisource and multiterminal. In some of the other researches, the network is usually transformed as the one with single-source and single-terminal. In order to verify the effectiveness of the improved genetic algorithm in this paper, we transform the ETN as the one with single-source and single-terminal and solve it by the traditional genetic algorithm. Then, the comparison of convergence procedure based on two genetic algorithms is shown in Figure 14.Figure 14
The comparison of convergence procedure based on two genetic algorithms.According to the comparison in Figure14, the convergence of improved genetic algorithm tends to table when iteration is at 104; however, the table convergence of traditional genetic algorithm is at 184. Besides, the results of improved genetic algorithm are much more obvious than traditional genetic algorithm before the iteration at 30, which reflects that the bidirectional search strategy has higher search ability. Combined with analysis above, the results demonstrate the improved genetic algorithm is more reasonable and efficient to solve the path optimization problem in the network with multisource and multiterminal.


## 5.2. Influence Analysis of Subjective Decision

### 5.2.1. Influence Analysis Based Network Viewpoint

The advantage of discrete robustness optimization model is that the uncertainty scale is controllable. From the viewpoint of network,Γ is an important parameter to reflect the optimism or pessimism attitude of decision-maker. In order to analyze the influence caused by attitude of decision-maker, the value of optimal path cluster under different Γ are shown in Figure 15.Figure 15
The value of optimal path cluster under differentΓ.Obviously, with the uncertainty scale up, the predicted time tends to increase. It reflects that the attitude of decision-maker becomes more and more pessimistic. Generally, the attitude of decision-maker is insensitive whenΓ≤8, and the attitude is in the hesitation when 8<Γ≤22. Especially, Γ>22, attitude becomes pessimistic. Next, we take p2,18 and p2,19 as examples for further analysis. The related information of e p1,18 and p1,19 is shown in Table 3.Table 3
The related information ofp2,18 and p2,19.
Γ Path Details Nominal time Time delay Predicted time 0 P 2,18 2→4→10→13→18 3.71 0 3.71 P 2,19 2→5→9→15→17→19 3.95 0 3.95  7 P 2,18 2→5→8→11→13→18 3.92 0 3.92 P 2,19 2→5→9→12→17→19 4.10 0 4.10  17 P 2,18 2→5→8→10→13→16→18 4.50 0.76 5.26 P 2,19 2→5→8→10→13→15→17→19 4.76 0.25 5.01  27 P 2,18 2→4→10→13→14→18 4.24 0.97 5.21 P 2,19 2→4→10→13→15→17→19 4.29 1.11 5.40  37 P 2,18 2→4→8→11→13→18 3.96 1.87 5.83 P 2,19 2→4→8→11→13→16→19 4.60 1.27 5.87  47 P 2,18 2→4→10→13→18 3.71 4.29 8 P 2,19 2→5→9→15→17→19 3.95 4.05 8 Note:time/hour.The result in Table3 shows that the optimal paths are different with Γ taking different values. It entails that the paths in optimal cluster are amended to adapt the circumstance under different Γ. Especially, when Γ takes the value in median parts of the uncertainty scale, the optimal paths have the poorest timeliness of nominal time. Strangely, when Γ=47, the optimal path is some as to the optimal path under Γ=0. This phenomenon gives us a conclusion that the optimal solution under medium scale of uncertainty is the most difficult scenario for decision-maker to make decision, and the final result usually has higher sacrifice of the timeliness.


### 5.2.2. Influence Analysis Based Path Viewpoint

During the decision of optimal path cluster, prospect theory is introduced to reflect the subjective decision of decision-maker under limited rationality. Actually, the influence is embodied in the time delay. In order to analyze the influence caused by of subjective decision under limited rationality, the time delay of differentP1,18 based different viewpoints are listed in Tables 4, 5, and 6, respectively.Table 4
The time delay ofP1,18 with the better robustness and the poorer timeliness.
Items P 1,18(1→7→10→13→18) ∑ e i , j (1,7) (7,10) (10,13) (13,18) — t i , j (h) 1.520 0.810 0.740 0.980 4.050 h i , j 0.195 0.172 0.181 0.452 1.000 DND 1.483 0.790 0.722 0.956 3.950 DPD 0.770 0.678 0.717 1.786 3.950 DSD 1.282 0.756 0.719 1.763 4.520 Note: DND is the time delay based neutral decision under complete rationality. DPD is the time delay based probabilistic decision under complete rationality. DSD is the time delay based subjective decision under limited rationality.hi,j is the relative probability in (14).Table 5
The time delay ofP1,18 with the better timeliness and the poorer robustness.
Items P 1,18(1→4→10→13→18) ∑ e i , j (1,4) (4,10) (10,13) (13,18) — t i , j (h) 0.950 1.210 0.740 0.980 3.880 h i , j 0.122 0.254 0.179 0.445 1.000 DND 1.009 1.285 0.786 1.041 4.120 DPD 0.503 1.046 0.736 1.835 4.120 DSD 1.009 1.285 0.786 1.041 4.120Table 6
The time delay ofP1,18 with the poorer timeliness and the poorer robustness.
Items P 1,18(1→7→11→15→16→18) ∑ e i , j (1,7) (7,11) (11,15) (15,16) (16,18) — t i , j (h) 1.520 1.230 0.690 0.480 0.560 4.480 h i , j 0.164 0.237 0.204 0.142 0.252 1.000 DND 1.194 0.966 0.542 0.377 0.440 3.520 DPD 0.577 0.836 0.718 0.501 0.888 3.520 DSD 1.031 0.918 0.663 0.446 0.746 3.804As Tables4 and 6 showed, for each ei,j, DSD is smaller than DND but bigger than DPD, when hi,j takes smaller values. It illustrates that decision-maker holds risk aversion attitudes when he faces the situation with smaller probability of time delay. On the contrary, DSD is bigger than DND but smaller than DPD, when hi,j takes bigger values; that is, the attitude is risk seeking when he faces the higher probability of time delay. From the viewpoint of each path, decision-maker holds attitude of risk aversion because of ∑DSD beyond both ∑DPD and ∑DND. Obviously, the more obvious the attitude of risk aversion holds, the better the robustness is.DSD especially is all equal to DND in Table5, but it does not violate the conclusion obtained above. The reason is mainly caused by the weight factor of prospect valve in time delay in equation (19). Further analysis shows that the tolerant time delay in Table 4 is so plenty that subjective reflection turns invalid. In fact, it is coincident with decision behavior in the real world.


### 5.2.3. Parameters Analysis in Prospect Theory

The subjective reflection of decision-maker is mainly influenced by the value of parameters in prospect theory. Based on time delay, this section focuses on the analysis of influence caused by the value of parameters in prospect theory. The influence of time delay caused byα and β is shown in Figure 16.Figure 16
The influence of time delay caused byα and β.α and β are the parameters in value function. From Figure 16 we can see that the bigger the value that α takes, the higher the time delay is. In other words, the decision attitude turns more risk aversion. On the contrary, the attitude turns more risk seeking when β turns bigger. Similarly, the influence of time delay caused by γ and δ is shown in Figure 17.Figure 17
The influence of time delay caused byγ and δ.Compared with value function, Figure17 shows that parameters in weight function have higher influence on time delay. Especially, when γ takes values in 0.4,0.8, the attitude of risk seeking is obvious, and the risk aversion is obvious when δ takes values in 0.6,1. Generally, δ has higher influence than γ.In addition, prospect theory presents that bothα and γ have comprehensive influence on gains, and so do β and δ on losses. Therefore, the influence of time delay caused by α and γ is shown in Figure 18.Figure 18
The influence of time delay caused byα and γ.Obviously, time delay is gradually increased with the value ofα up. When α<0.8, the variation of γ has greater influence on time delay. However, when α≥0.8, the influence is unconspicuous. Generally, the influence caused by variation of γ is greater than that of α. In other words, subjective reflection is more sensitive to weight function than value function. Similarly, the influence of time delay caused by β and δ is shown in Figure 19.Figure 19
The influence of time delay caused byβ and δ.As shown in Figure19, the influence on time delay caused by variation of β is obvious when δ≥0.4, and the influence caused by δ is obvious when β≤0.7. Similar to the reflection in Figure 18, Figure 19 also shows that the time delay is much more sensitive to weight function than value function. Combining with Figures 18 and 19, we can make conclusion that weight function has greater influence on subjective reflection than value function in this paper.


## 5.2.1. Influence Analysis Based Network Viewpoint

The advantage of discrete robustness optimization model is that the uncertainty scale is controllable. From the viewpoint of network,Γ is an important parameter to reflect the optimism or pessimism attitude of decision-maker. In order to analyze the influence caused by attitude of decision-maker, the value of optimal path cluster under different Γ are shown in Figure 15.Figure 15
The value of optimal path cluster under differentΓ.Obviously, with the uncertainty scale up, the predicted time tends to increase. It reflects that the attitude of decision-maker becomes more and more pessimistic. Generally, the attitude of decision-maker is insensitive whenΓ≤8, and the attitude is in the hesitation when 8<Γ≤22. Especially, Γ>22, attitude becomes pessimistic. Next, we take p2,18 and p2,19 as examples for further analysis. The related information of e p1,18 and p1,19 is shown in Table 3.Table 3
The related information ofp2,18 and p2,19.
Γ Path Details Nominal time Time delay Predicted time 0 P 2,18 2→4→10→13→18 3.71 0 3.71 P 2,19 2→5→9→15→17→19 3.95 0 3.95  7 P 2,18 2→5→8→11→13→18 3.92 0 3.92 P 2,19 2→5→9→12→17→19 4.10 0 4.10  17 P 2,18 2→5→8→10→13→16→18 4.50 0.76 5.26 P 2,19 2→5→8→10→13→15→17→19 4.76 0.25 5.01  27 P 2,18 2→4→10→13→14→18 4.24 0.97 5.21 P 2,19 2→4→10→13→15→17→19 4.29 1.11 5.40  37 P 2,18 2→4→8→11→13→18 3.96 1.87 5.83 P 2,19 2→4→8→11→13→16→19 4.60 1.27 5.87  47 P 2,18 2→4→10→13→18 3.71 4.29 8 P 2,19 2→5→9→15→17→19 3.95 4.05 8 Note:time/hour.The result in Table3 shows that the optimal paths are different with Γ taking different values. It entails that the paths in optimal cluster are amended to adapt the circumstance under different Γ. Especially, when Γ takes the value in median parts of the uncertainty scale, the optimal paths have the poorest timeliness of nominal time. Strangely, when Γ=47, the optimal path is some as to the optimal path under Γ=0. This phenomenon gives us a conclusion that the optimal solution under medium scale of uncertainty is the most difficult scenario for decision-maker to make decision, and the final result usually has higher sacrifice of the timeliness.


## 5.2.2. Influence Analysis Based Path Viewpoint

During the decision of optimal path cluster, prospect theory is introduced to reflect the subjective decision of decision-maker under limited rationality. Actually, the influence is embodied in the time delay. In order to analyze the influence caused by of subjective decision under limited rationality, the time delay of differentP1,18 based different viewpoints are listed in Tables 4, 5, and 6, respectively.Table 4
The time delay ofP1,18 with the better robustness and the poorer timeliness.
Items P 1,18(1→7→10→13→18) ∑ e i , j (1,7) (7,10) (10,13) (13,18) — t i , j (h) 1.520 0.810 0.740 0.980 4.050 h i , j 0.195 0.172 0.181 0.452 1.000 DND 1.483 0.790 0.722 0.956 3.950 DPD 0.770 0.678 0.717 1.786 3.950 DSD 1.282 0.756 0.719 1.763 4.520 Note: DND is the time delay based neutral decision under complete rationality. DPD is the time delay based probabilistic decision under complete rationality. DSD is the time delay based subjective decision under limited rationality.hi,j is the relative probability in (14).Table 5
The time delay ofP1,18 with the better timeliness and the poorer robustness.
Items P 1,18(1→4→10→13→18) ∑ e i , j (1,4) (4,10) (10,13) (13,18) — t i , j (h) 0.950 1.210 0.740 0.980 3.880 h i , j 0.122 0.254 0.179 0.445 1.000 DND 1.009 1.285 0.786 1.041 4.120 DPD 0.503 1.046 0.736 1.835 4.120 DSD 1.009 1.285 0.786 1.041 4.120Table 6
The time delay ofP1,18 with the poorer timeliness and the poorer robustness.
Items P 1,18(1→7→11→15→16→18) ∑ e i , j (1,7) (7,11) (11,15) (15,16) (16,18) — t i , j (h) 1.520 1.230 0.690 0.480 0.560 4.480 h i , j 0.164 0.237 0.204 0.142 0.252 1.000 DND 1.194 0.966 0.542 0.377 0.440 3.520 DPD 0.577 0.836 0.718 0.501 0.888 3.520 DSD 1.031 0.918 0.663 0.446 0.746 3.804As Tables4 and 6 showed, for each ei,j, DSD is smaller than DND but bigger than DPD, when hi,j takes smaller values. It illustrates that decision-maker holds risk aversion attitudes when he faces the situation with smaller probability of time delay. On the contrary, DSD is bigger than DND but smaller than DPD, when hi,j takes bigger values; that is, the attitude is risk seeking when he faces the higher probability of time delay. From the viewpoint of each path, decision-maker holds attitude of risk aversion because of ∑DSD beyond both ∑DPD and ∑DND. Obviously, the more obvious the attitude of risk aversion holds, the better the robustness is.DSD especially is all equal to DND in Table5, but it does not violate the conclusion obtained above. The reason is mainly caused by the weight factor of prospect valve in time delay in equation (19). Further analysis shows that the tolerant time delay in Table 4 is so plenty that subjective reflection turns invalid. In fact, it is coincident with decision behavior in the real world.


## 5.2.3. Parameters Analysis in Prospect Theory

The subjective reflection of decision-maker is mainly influenced by the value of parameters in prospect theory. Based on time delay, this section focuses on the analysis of influence caused by the value of parameters in prospect theory. The influence of time delay caused byα and β is shown in Figure 16.Figure 16
The influence of time delay caused byα and β.α and β are the parameters in value function. From Figure 16 we can see that the bigger the value that α takes, the higher the time delay is. In other words, the decision attitude turns more risk aversion. On the contrary, the attitude turns more risk seeking when β turns bigger. Similarly, the influence of time delay caused by γ and δ is shown in Figure 17.Figure 17
The influence of time delay caused byγ and δ.Compared with value function, Figure17 shows that parameters in weight function have higher influence on time delay. Especially, when γ takes values in 0.4,0.8, the attitude of risk seeking is obvious, and the risk aversion is obvious when δ takes values in 0.6,1. Generally, δ has higher influence than γ.In addition, prospect theory presents that bothα and γ have comprehensive influence on gains, and so do β and δ on losses. Therefore, the influence of time delay caused by α and γ is shown in Figure 18.Figure 18
The influence of time delay caused byα and γ.Obviously, time delay is gradually increased with the value ofα up. When α<0.8, the variation of γ has greater influence on time delay. However, when α≥0.8, the influence is unconspicuous. Generally, the influence caused by variation of γ is greater than that of α. In other words, subjective reflection is more sensitive to weight function than value function. Similarly, the influence of time delay caused by β and δ is shown in Figure 19.Figure 19
The influence of time delay caused byβ and δ.As shown in Figure19, the influence on time delay caused by variation of β is obvious when δ≥0.4, and the influence caused by δ is obvious when β≤0.7. Similar to the reflection in Figure 18, Figure 19 also shows that the time delay is much more sensitive to weight function than value function. Combining with Figures 18 and 19, we can make conclusion that weight function has greater influence on subjective reflection than value function in this paper.


## 6. Conclusions

This paper focuses on the discrete robustness optimization of emergency logistic network based on a situation that the nearer to the disaster area, the higher probability the time delay. Thereinto, prospect theory is especially introduced to describe the subjective reflection of decision-maker under limited rationality. In addition, an improved genetic algorithm is also designed based on the optimization model to obtain the optimal path cluster with the better timeliness and robustness from ETN with multistorage centers and multidisaster points. According to the analysis of case study, some conclusions can be summarized as follows.( 1 ) With the required arrival time, prospect theory can well describe the subjective reflection of decision-maker under limited rationality. Generally, decision-maker who faces a path with more nominal transportation time and less tolerant delay time holds an attitude of risk seeking, which can be described as a proverb that what is hopeless and bold is often successful. On the contrary, the attitude will turn to risk aversion when decision-maker faces a path with less nominal transportation time and more tolerant delay time, which also can be described as a proverb that caution is the mother of security.( 2 ) Based on the realistic situation that the nearer to the disaster area, the higher the probability of time delay, decision-maker who is under limited rationality, holds the attitude of risk aversion when he faces an edge with lower probability of time delay, and the attitude turns risk seeking when he faces an edge with higher probability.( 3 ) The advantage of discrete robustness optimization in this paper is that the uncertainty scale is controllable, by which the uncertainty environment of ETN can be vividly mimicked. Robustness especially in the model is correspondent to the attitude of risk aversion in prospect theory, and the optimal solution under different scale of uncertainty means the better risk aversion attitude of decision-maker with consideration of timeliness.In addition, genetic algorithm has better advantage in obtaining path cluster from ETN with multistorage centers and multidisaster points. The shortage in this paper is that reference point is only just based on the timeliness. In fact, the optimization resulting from multireference point, such as capacity of each edge and path, is more practically significance. Besides, group decision can also be used in this paper, by which universality of optimal solution is more general. Those will be the next step to further study.



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*Source: 2728073-2019-03-04.xml*