Multi-objective rescue vehicle traffic optimization with dedicated link opening restriction

Zheng Liu; Zheng Liu

doi:10.48130/dts-0026-0019

In disaster emergency rescue response, rescue vehicle traffic often interferes with social vehicle traffic on roads, and rescue vehicle traffic optimization (RVTO) is always related to traffic efficiency, traffic distance, and traffic distribution (3T). Here, we develop a multi-objective combinatorial programming (MOCP) formulation of the RVTO problem, and transform MOCP-RVTO into a nonlinear weighted single-objective RVTO (NLWSO-RVTO) problem by introducing objective weight decision variables. We then design a genetic algorithm decomposition (GAD) solving approach to realize multi-objective coordination and importance analysis among the 3T. Moreover, the impact of a multi-objective dedicated rescue link opening restriction on social vehicle traffic is analyzed using Paramics. The case test on the Sioux Falls network shows that introducing traffic distance and traffic distribution into traffic efficiency can reduce the number of used rescue links, and link occupation minimization can replace traffic distance minimization to reduce used rescue links and traffic distance. There exists an optimal constant combination weight value among 3T in which traffic efficiency has the significantly greatest importance and traffic distribution has the smallest weight. 3T combinatorial optimization does not delay the quick arrival of rescue vehicles at the disaster position and can synchronously decrease the interference of dedicated rescue links on social vehicle traffic operation.

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Literature review

As is well known, RVTO is a necessary step to improve large-scale rescue traffic network operation performance in disaster situations. In current research and practical applications, RVTO is often related to evacuation vehicle traffic operation, or social vehicle traffic operation on roads, and most of them mainly focus on multiclass evacuation and rescue traffic networks. In this regard, Chiu & Zheng^[1] and Yang et al.^[2] took weight evacuation and rescue traffic travel time minimization as an objective, and developed a single-objective mathematical programming model of evacuation and rescue traffic combinatorial optimization to solve dynamic evacuation and rescue traffic assignment on the road network; Kimms & Maassen^[3] took the minimization of the number of evacuation vehicles traveling on roads as an objective, and developed a single-objective mixed integer linear programming model to discuss the combinatorial optimization problem of dynamic evacuation and rescue traffic network operation, with the aim of minimizing the weight value of evacuation cost, rescue cost, and traffic conflict; Cui et al.^[4] developed a minimum cost flow model to optimize evacuation and rescue traffic allocation on a shared road network. Recently, Li et al.^[5] aimed to minimize the total wait time for rescue of all victims and developed a single-objective mixed integer linear programming model to realize collaborative operational planning between post-disaster victim evacuation and relief distribution by considering heterogeneous rescue teams.

In addition to the above-mentioned single-objective evacuation and rescue vehicle traffic optimizations, there are also social vehicles on the rescue vehicle traffic network; however, social vehicles are usually regarded as background traffic, and rescue vehicles have traffic priority in traffic conflicts in current studies^[6−10]. Moreover, only the obstruction of social traffic to rescue traffic is considered, but the impact of rescue traffic priority on social traffic is not considered on roads. Recently, given the development of intelligent and connected vehicle traffic technology in the field of disaster emergency rescue, on the basis of the existing studies of mixed traffic flow simulation and optimization of human-driven vehicles and automated vehicles^[11−14], Liu & Pei^[15] considered micro traffic behavior differences between manual and automated driving vehicles, and studied the mixed manual-automated driving rescue vehicle traffic route coordination optimization problem by introducing automated driving vehicle penetration rate decisions and developing a single-objective nonlinear mathematical programming model.

On the basis of single-objective multiclass vehicle emergency traffic optimization, some RVTO studies are also related to the MOCP problem; for example, Chen et al.^[16] took the length of traffic routes of evacuation agents and rescue agents as two objectives to study two-way route planning for emergency rescue and emergency evacuation in a chemical industrial park; Liu et al.^[17] developed a bi-objective mixed integer linear programming formulation of maximizing the number of rescue vehicles arriving at the disaster area within the different time and minimizing the number of the affected people stranded in the emergency area to realize evacuation and rescue traffic joint optimization. To maximize the number of rescue vehicles that have arrived in the disaster position by the end of the current time period and minimize the number of evacuation vehicles that have not arrived in the outside safe area, Liu et al.^[18] developed a bi-objective evacuation and rescue traffic collaborative assignment optimization model to plan multiclass dynamic emergency traffic operation on roads. In this evacuation and rescue traffic MOCP example, every objective belongs to one specific vehicle class, and the rescue traffic optimization problem is modeled as a single-objective optimization model; thus, the classical stage-based decomposition solving approach is used, in which rescue traffic optimization is first solved without considering the evacuation process, and then evacuation traffic is optimized by reserving the optimal rescue traffic route. In addition, some studies also further focused on the impact of road time-space resource occupation scale (e.g., the number of the used intersection, the number of the used link, and the number of rescue contraflow link) of rescue vehicles on evacuation and rescue traffic optimization^[19−22], however, the impact of reserving road time-space resource occupation scale to rescue vehicles on social vehicle traffic operation also should be further focused.

Overall, current RVTO studies have achieved numerous research findings in the field of single-objective and multi-objective evacuation and rescue traffic optimization, and single-objective rescue traffic optimization considering social vehicle background traffic. However, the multi-objective rescue vehicle traffic optimization problem and the impact of different objectives on social vehicle traffic have not been fully studied to improve emergency traffic network operation performance. In this paper, we try to solve the MOCP-RVTO problem and analyze the impact of multi-objective road time-space resource reserve on social vehicle traffic operation from the perspective of dedicated rescue link opening restriction. Here, we adopt the classical link transmission model (LTM) to describe the dynamic rescue traffic network loading process and use 3T as three objectives to coordinatively optimize rescue vehicle traffic routes on the road network, and then develop the MOCP-RVTO formulation. Moreover, we introduce three weight decision variables and optimal single-objective (SO) function values to weight RVTO objectives and transform the MOCP-RVTO formulation into an NLWSO-RVTO formulation, and further design a GAD-based solving approach to optimize objective weights and rescue vehicle traffic operation in stages. And last, the impact of dedicated rescue link opening restriction on social vehicle traffic operation is analyzed on the Paramics traffic simulation network, and the case is realized on the Sioux Falls example network.

The main highlights and contributions are as follows: (1) Multi-objective coordination among 3T is focused on by MOCP-RVTO, and their importance is analyzed by multiple objective weight optimization. (2) Rescue vehicle traffic route choice characteristics are analyzed based on SO-RVTO and NLWSO-RVTO by designing a GAD-based solving approach. (3) Interference of dedicated rescue link opening restriction on social vehicle traffic is analyzed based on different RVTO objectives.

MOCP-RVTO formulation

Notations and definition

The notations and definitions in Table 2 are adopted to mathematically describe the MOCP-RVTO problem on the dynamic rescue traffic network:

Table 2. Formulation notations and definitions.

Notations	Definitions
	Sets
$ L $	Set of links that constituting the rescue vehicle traffic network
$ L' $	Set of links where rescue vehicles enter the disaster position from the road network, $ L'\subset L $
$ L'' $	Set of links where rescue vehicles enter the road network from the rescue station, $ L''\subset L $
$ S $	Set of rescue stations
$ \Gamma _{i}^{+} $	Set of links that connect with link i in its downstream direction,$ i\in L,\Gamma _{i}^{+}\subset L $
$ \Gamma _{i}^{-} $	Set of links that connect with link i in its upstream direction,$ i\in L,\Gamma _{i}^{-}\subset L $
$ T $	Set of integer time periods that describe the dynamic rescue vehicle traffic network loading process
	Indices
$ i,j $	Index of any link on the road network, $ i,j\in L $
$ t,\tau $	Index of any time period, $ t,\tau \in T $
$ s $	Index of any rescue station, $ s\in S $
$ o $	Index of the disaster position
	Parameters
$ \alpha ,\delta $	Linear interpolation coefficient of cumulative traffic volume in non-integer time periods
$ {d}_{s} $	Rescue traffic demand that represents the number of rescue vehicles called from rescue station$ s $, $ s\in S $
$ {l}_{i} $	Length of link i, $ i\in L $
$ {n}_{i} $	Number of lanes of link i, $ i\in L $
$ {Q}_{i} $	Road capacity, that is, the maximum number of rescue vehicles that can pass every lane of link i within any time period t, $ t\in T,i\in L $
$ \rho _{jam}^{(i)} $	Jam traffic density, that is, the maximum number of rescue vehicles that can be accommodated on every lane of link i, $ i\in L $
$ {\tau }_{i} $	Length of traffic free-flow time periods from upstream end to downstream on link i, $ i\in L $
$ {\iota }_{i} $	Length of backward traffic congestion shockwave propagation time periods from downstream end to upstream on link i, $ i\in L $
	Variables
$ U_{t}^{(i)} $	Cumulative number of rescue vehicles entering link i by the end of current time period t, $ t\in T,i\in L $
$ V_{t}^{(i)} $	Cumulative number of rescue vehicles leaving link i by the end of current time period t, $ t\in T,i\in L $
$ q_{t}^{(i,j)} $	Number of rescue vehicles entering link j from link i within current time period t, $ t\in T,i\in L,j\in \Gamma _{i}^{+} $
$ q_{t}^{(i,o)} $	Number of rescue vehicles arriving in the disaster position o from link i within current time period t, $ t\in T,i\in L' $
$ q_{t}^{(s,i)} $	Number of rescue vehicles entering link i from rescue station s within current time period t, $ t\in T,i\in L'',s\in S $
$ x_{t}^{(i)} $	Number of rescue vehicles on link i in the beginning of current time period t, $ t\in T,i\in L $

Multi-objective combinational modelling

We consider a collaborative optimization among traffic efficiency, traffic distance, and traffic distribution: traffic efficiency is modelled as Z1 to maximize the number of rescue vehicles arriving at the disaster position by the end of any time period; traffic distance is modelled as Z2 to minimize the travel distance of rescue vehicles from the rescue station to the disaster position; and traffic distribution is modelled as Z3 to minimize the number of links used by rescue vehicles.

$ {\mathrm{Max}}\;\; \text{Z1 =}{\sum }_{t\in T}{\sum}_{\tau =1}^{t}{\sum }_{i\in L'}q_{\tau }^{(i,o)} $

$ {\mathrm{Min}}\;\; \text{Z2}={\sum }_{i\in L}U_{|T|}^{(i)}{l}_{i} $

$ {\mathrm{Min}}\;\; \text{Z3}={\sum }_{i\in L}U_{|T|}^{(i)} $

Here, Z1 is equal to $ \sum_{t=1}^{|T|}(|T|-t+1){\sum }_{i\in L'}q_{t}^{(i,o)} $ in which the time period is the weight of the number of rescue vehicles arriving at the disaster position; obviously, the earlier arrival of rescue vehicles can obtain a larger Z1 value; Z2 is obtained by multiplying the total number of rescue vehicles passing different links by the length of corresponding links, in which the length of links is the weight to minimize traffic distance from the rescue station to the disaster position. In addition, the total number of rescue vehicles passing all links is summed by Z3; obviously, if more links are used, the corresponding number of rescue vehicles will be summed more times, and traffic distance will not be minimized. Therefore, Z3 can optimize the distribution of rescue vehicles on fewer links.

LTM-based dynamic rescue traffic network loading

Based on the classical LTM theory^[23,24], we develop Eqs (1)–(4) to realize the dynamic loading process of large-scale rescue vehicle traffic on the road network by the relaxed LTM^[17]:

$ V_{t}^{(i)}\leq U_{t-{\tau }_{i}}^{(i)}\;\;t\in T,i\in L $

(1)

$ V_{t}^{(i)}-V_{t-1}^{(i)}\leq {n}_{i}{Q}_{i}\;\;t\in T,i\in L $

(2)

$ U_{t}^{(i)}\leq V_{t-{\iota }_{i}}^{(i)}+{n}_{i}{l}_{i}\rho _{jam}^{(i)}\;\;t\in T,i\in L $

(3)

$ U_{t}^{(i)}-U_{t-1}^{(i)}\leq {n}_{i}{Q}_{i}\;\;t\in T,i\in L $

(4)

here, Eq. (1) describes the cumulative number of rescue vehicles that leave link i by the end of time period t is less than or equal to the cumulative number of rescue vehicles that enter link i by the end of time period $ t-{\tau }_{i} $ considering the restriction of traffic free-flow time period; Eqs (2) and (4) require the number of rescue vehicles that leave and enter link i within time period t is not more the value of road capacity; Eq. (3) restricts the number of rescue vehicles that can enter link i in any time period t does not exceed its remaining space.

In Eqs (1)–(4), the cumulative inflow and outflow numbers of rescue vehicles that enter and leave link i also need to be obtained by the end of the non-integer time period. Here, based on the linear interpolation method, we introduce the interpolation coefficient α to calculate the cumulative inflow number of rescue vehicles in Eq. (5), and introduce the interpolation coefficient δ to calculate the cumulative outflow number of rescue vehicles in Eq. (6).

$ U_{t+\alpha }^{(i)}=(1-\alpha )U_{t}^{(i)}+\alpha U_{t+1}^{(i)}t\in T,i\in L,\alpha \in [0,1] $

(5)

$ V_{t+\delta }^{(i)}=\left\{\begin{aligned} &(1-\delta )V_{t}^{(i)}t=\left\lfloor {\tau }_{i}\right\rfloor ,&&0\leq \delta \leq {\tau }_{i}-\left\lfloor {\tau }_{i}\right\rfloor &\\ &\dfrac{\delta -({\tau }_{i}-\left\lfloor {\tau }_{i}\right\rfloor )}{1-({\tau }_{i}-\left\lfloor {\tau }_{i}\right\rfloor )}V_{t+1}^{(i)} &&t=\left\lfloor {\tau }_{i}\right\rfloor ,{\tau }_{i}-\left\lfloor {\tau }_{i}\right\rfloor \leq \delta \leq 1&\\ &(1-\delta )V_{t}^{(i)}+\delta V_{t+1}^{(i)} &&t\in T \{\left\lfloor {\tau }_{i}\right\rfloor \},0\leq \delta \leq 1& \end{aligned}\right. i\in L $

(6)

In addition, Eq. (7) is adopted to optimize the number of rescue vehicles that enter link i from its upstream adjacent links and the rescue station within time period t; Eq. (8) is adopted to optimize the number of rescue vehicles that enter its downstream adjacent links and arrive in the disaster position from link i within time period t.

$ U_{t}^{(i)}-U_{t-1}^{(i)}={\sum }_{j\in {{\Gamma }^-}}q_{t}^{(j,i)}+{\sum }_{s\in S}q_{t}^{(s,i)}t\in T,i\in L $

(7)

$ V_{t}^{(i)}-V_{t-1}^{(i)}={\sum }_{j\in {{\Gamma }^+}}q_{t}^{(i,j)}+q_{t}^{(i,o)}t\in T,i\in L $

(8)

In the traffic conservation equation of Eq. (9), the number of rescue vehicles on link i in the beginning of time period t + 1 should be equal to the number of rescue vehicles on link i in the beginning of time period t, plus the number of rescue vehicles that enter link i from its upstream adjacent links and the rescue station within time period t, and then minus the number of rescue vehicles that enter its downstream adjacent links and arrive in the disaster position from link i within time period t.

$ x_{t+1}^{(i)}=x_{t}^{(i)}+\left({\sum }_{j\in {{\Gamma }^-}}q_{t}^{(j,i)}+{\sum }_{s\in S}q_{t}^{(s,i)}\right)-\left({\sum }_{j\in {{\Gamma }^+}}q_{t}^{(i,j)}+q_{t}^{(i,o)}\right)t\in T,i\in L $

(9)

When rescue vehicles travel on roads, they originate from the outside rescue stations and finally arrive at the disaster position. Therefore, the dynamic network loading of rescue vehicles should obey traffic generation and attraction balance, that is, all called rescue vehicles should enter the disaster position from the rescue station. Here, traffic generation of rescue vehicles from the rescue station can be mathematically expressed as Eq. (10), and traffic attraction of rescue vehicles in the disaster position can be mathematically expressed as Eq. (11):

$ {d}_{s}={\sum }_{i\in {{L}^{''}}}{\sum }_{t\in T}q_{t}^{(s,i)}s\in S $

(10)

$ {\sum }_{i\in L'}{\sum }_{t\in T}q_{t}^{(i,o)}={\sum }_{s\in S}{\sum }_{i\in {{L}^{''}}}{\sum }_{t\in T}q_{t}^{(s,i)} $

(11)

Last, Eqs (12) and (14) are presented to describe that there are no rescue vehicles on roads before the disaster occurs. Equation (13) describes that no rescue vehicles leave the links under the restriction of their traffic-free-flow time period. Equation (15) is the non-negative constraint:

$ U_{0}^{(i)}=0\;\;i\in L $

(12)

$ V_{t}^{(i)}=0\;\;t\in \{0,1,\cdots ,\left\lfloor {\tau }_{i}\right\rfloor \},i\in L $

(13)

$ x_{0}^{(i)}=0\;\;i\in L $

(14)

$ x_{t}^{(i)}\geq 0,q_{t}^{(i,j)}\geq 0,q_{t}^{(i,o)}\geq 0,q_{t}^{(s,i)}\geq 0\;\;i\in L,j\in \Gamma _{i}^+,s\in S $

(15)

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Objective index	Route index
Objective index	A	B	C
Traffic efficiency	L	M	H
Traffic distance	L	M	H
Traffic distribution	L	M	H
H, M, and L represent the high, medium, and low levels, respectively.

Link index	Node series	Lanes	Length (m)	Link index	Node index	Lanes	Length (m)	Link index	Node index	Lanes	Length (m)
1	1→2	4	900	27	10→11	2	300	53	17→19	1	200
2	1→3	4	200	28	10→15	2	400	54	18→7	4	200
3	2→1	2	900	29	10→16	1	300	55	18→16	3	300
4	2→6	1	200	30	10→17	1	360	56	18→20	4	855
5	3→1	2	200	31	11→4	1	400	57	19→15	3	300
6	3→4	3	300	32	11→10	2	300	58	19→17	1	200
7	3→12	4	400	33	11→12	1	300	59	19→20	1	400
8	4→3	3	300	34	11→14	1	400	60	20→18	4	855
9	4→5	3	300	35	12→3	4	400	61	20→19	1	400
10	4→11	1	400	36	12→11	1	300	62	20→21	1	300
11	5→4	3	300	37	12→13	4	800	63	20→22	1	360
12	5→6	1	300	38	13→12	4	800	64	21→20	1	300
13	5→9	2	200	39	13→24	1	300	65	21→22	1	200
14	6→2	1	200	40	14→11	1	400	66	21→24	1	300
15	6→5	1	300	41	15→14	1	300	67	22→15	2	200
16	6→8	1	200	42	14→23	1	200	68	22→20	1	360
17	7→8	1	300	43	15→10	2	400	69	22→21	1	200
18	7→18	4	200	44	14→15	1	300	70	23→22	1	300
19	8→6	1	200	45	15→19	3	300	71	23→14	1	200
20	8→7	1	300	46	15→22	2	200	72	22→23	1	300
21	8→9	1	300	47	16→8	1	200	73	23→24	1	200
22	8→16	1	200	48	16→10	1	300	74	24→13	1	300
23	9→5	2	200	49	16→17	1	200	75	24→21	1	300
24	9→8	1	300	50	16→18	3	300	76	24→23	1	200
25	9→10	2	200	51	17→10	1	360
26	10→9	2	200	52	17→16	1	200

Weight	d_s = 30	d_s = 60	d_s = 90	d_s = 120	d_s = 150	d_s = 180	d_s = 210	d_s = 240
a₁	0.9918	0.9918	0.9918	0.9918	0.9897	0.9918	0.9918	0.9918
a₂	0.0050	0.0050	0.0050	0.0050	0.0070	0.0050	0.0050	0.0050
a₃	0.0032	0.0032	0.0032	0.0032	0.0033	0.0032	0.0032	0.0032

Objectives	Combination weights (a₁, a₂, a₃)
Objectives	(0.9918, 0.0050, 0.0032)	(0.9897, 0.0070, 0.0033)
Z1	4,179.75	4,179.75
Z2	262,500	262,500
Z3	807	807
\|L\|	22	22
Z	−0.97838	−0.98298

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Multi-objective rescue vehicle traffic optimization with dedicated link opening restriction

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