Ripple spreading algorithm: a new method for solving multi-objective shortest path problems with mixed time windows

Abstract

In emergency management, the transportation scheduling of emergency supplies and relief personnel can be regarded as the multi-objective shortest path problem with mixed time window (MOSPPMTW), which has high requirements for timeliness and effectiveness, but the current solution algorithms cannot simultaneously take into account the solution accuracy and computational speed, which is very unfavorable for emergency path decision-making. In this paper, we establish MOSPPMTW matching emergency rescue scenarios, which simultaneously enables the supplies and rescuers to arrive at the emergency scene as soon as possible in the shortest time and at the smallest cost. To solve the complete Pareto optimal surface, we present a ripple spreading algorithm (RSA), which determines the complete Pareto frontier by performing a ripple relay race to obtain the set of Pareto optimal path solutions. The proposed RSA algorithm does not require an initial solution and iterative iterations and only needs to be run once to obtain the solution set. Furthermore, we prove the optimality and time complexity of RSA and conduct multiple sets of example simulation experiments. Compared with other algorithms, RSA performs better in terms of computational speed and solution quality. The advantage is especially more obvious in the computation of large-scale problems. It is applicable to various emergency disaster relief scenarios and can meet the requirements of fast response and timeliness.

Introduction

Shortest path problem (SPP) aims to find the shortest path between source node and destination node in a network. Network optimization models can be seen in a wide range of applications, such as transportation systems, communication systems, pipeline distribution systems, fluid flow systems and neural decision systems. SPP has been solved by Dijkstra’s algorithm [1] and researchers have successively proposed many deformations of SPP [2,3,4].

Classical SPP considers only one objective in the networks while some or all of the multiple, conflicting and incommensurate objectives such as optimization of cost, profit, time, distance, risk, and quality of service may arise together in real-world applications. These types of SPPs are known as the multi-objective shortest path problem (MOSPP).

In the actual logistics and transportation process, MOSPP often needs to consider mixed time windows with a combination of hard and soft time windows at the demand node. It is urgent to complete the distribution in different time windows and plan the path reasonably. Therefore, it is important to study the multi-objective shortest path problem with mixed time windows (MOSPPMTW) to solve practical problems.

This paper is organized as follows: section "Related work" presents the relevant previous work and section Problem description and mathematical model" describes the MOSPPMTW problem. The proposed algorithm is introduced in section "Ripple spreading algorithm". Section "Theoretical analysis" analyzes the optimality and time complexity of the proposed algorithm. Section "Experimental analysis" presents the experimental results and discussion, and the conclusions are presented in section "Conclusion".

Related work

This article gives an extensive overview of multi-objective shortest path problem (MOSPP) studies by surveying lots of contributions. Since the objectives are typically conflicting, there usually does not exist a single solution that simultaneously optimizes each objective. The decision maker (DM) ideally selects his/her most-preferred solution among all Pareto optimal solutions. MOSPP is widely used in various industries and scenarios, as well as the objectives considered by DM in the existing literature are also very diverse. Therefore, according to DM's preference, this paper classifies 22 types of objectives and 3 types of solving algorithms [5], as shown below (Tables 1 and 2).

Table 1 Objectives classification in MOSPPMTW

Table 2 Search strategies in multi-objective shortest path problem

Many scholars have conducted in-depth research on MOSPP, and the research progress is as follows. Hansen [6] first introduced MOSPP and proves the intractability of the bi-objective version. M¨uller-Hannemann [7] derive conditions under which the cardinality of the Pareto-optimal frontier is polynomial. Serafini [8] shows that the decision version of the MOSPP is NP-complete. The results of Hansen [6] and Serafini [8] imply that the Pareto-optimal frontier of large MOSPP instances cannot be constructed in a reasonable amount of time. A parametric approach to the bi-objective problem is given in Mote [9] and in Sede˜no-Noda [10]. Mulmuley [11] showed that the number of breakpoints in the parametric shortest path problem can be sub-exponential. Thereafter, several researchers contributed to the study of MOSPP (Kostreva and Wiecek [12]; Papadimitriou and Yannakakis [13]; Gandibleux [14]; Chen and Nie [15]; Breugem [16]; Zhuang [17]; Nedic [18, 19]). All these studies have eventually increased the significance of MOSPP which is indispensable to the network theory.

There are various approaches to solving MOSPP. Solving algorithms are of three different types [20]: polynomial time exact algorithms (PTEAs), polynomial time approximation algorithms (PTAAs), and heuristic algorithms (HAs). A general overview of algorithms for the MOSPP is given in Skriver [21]. Mian's research on PTEAs includes Pulido [22] introduced an exact label-setting algorithm that returns the subset of Pareto optimal paths that satisfy a set of lexicographic goals, or the subset that minimizes deviation from goals if these cannot be fully satisfied. Shi [23] proposed an exact method for finding all the Pareto optimal paths for a multi-criteria constrained shortest path problem. Mian’s research on HAs includes Mandow [24] extended the A* search algorithm to find MOSPP solutions. The new algorithm, named MOA*, is a heuristic search algorithm that finds non-dominated solutions. Smith [25] proposed a MOSA algorithm. They introduced a transformation function that converts a multi-objective solution into an energy value. By comparing the energy value between two solutions, a solution can be assessed as better, equal, or worse than the alternative solution. Coello and Pulido [26] proposed micro-GA for high-quality multi-objective optimization. Deb [27] proposed NSGA-II for performing multi-objective optimization. While iterating, this algorithm seeks a different solution set by preserving two types of solutions: nondominated solutions and the most distinct solutions in the population. Bora [28] added greedy reinforcement learning (RL) to NSGA-II, which realizes parameter self-tuning and proposed a new algorithm NSGA-RL. Mian's research on PTAAs includes Tsaggouris and Zaroliagis [29] proposed an improved fully polynomial time approximation scheme (FPTAS) algorithm for solving MOSPs, which resembles the multi-objective Bellman–Ford algorithm. Horoba [30] analyzed a simple evolutionary algorithm comprising a fitness function and a mutation operation and found that it met the requirements of a fully polynomial time randomized approximation scheme. Knowles and Corne [31] proposed a multi-objective local search algorithm, named (1 + 1)-PAES algorithm. Martins and Santos [32] analyzed the labeling algorithm for the MOSPP by the generalization of the classical SPP. For an overview of MOSPP algorithms, we refer to Fu [33]; Machuca [34]; Rahimi [35]; Current [36, 37]; Coutinho-Rodrigues [38]; Climaco [39]; Wakuta [40]; Majumder [41] Bagheri [42]; Silva [43]; Yao [44]; Ajeil [45]; Gul [46]; Ren [47]. All these studies have laid a foundation and provided ideas for the further study of the MOSPP algorithm.

Recent researches on path problems with mixed time windows are as follows, Ongcunaruk [48] used genetic algorithm (GA) to solve the transportation planning decision problem for production companies and logistics providers containing two time windows; Zhou [49] proposed an electric vehicle-routing problem (EVRP–BS–MTW) considering battery swap and mixed time windows constraints, and proposed a particle filtering and A multi-objective whale optimization algorithm (MWOA–PFLF) based on Levy flight enhancement is proposed for solving; Li [50] proposes an improved hybrid partheno-genetic algorithm (HPGA) for solving the food distribution problem in cold chain logistics with mixed time windows for the multi-depot vehicle-routing problem (MDVRPMTW); Wang [51] proposes a genetic algorithm (GA) consisting of a heuristic algorithm (ST–VNSGA) to solve a multi-objective VRP optimization model with mixed time windows and perishability (MO–VRPMTW-P).

The current research still has some shortcomings that still need to be addressed.

• (1) The existing research tends to focus on only a single type of either hard or soft time windows, which leaves a gap in understanding how different types of time windows can coexist and affect optimization.

• (2) Although polynomial time exact algorithms (PTEAs) find exact solutions to the MOSPP problem, they consume much memory and computational time, which are inappropriate for large networks.

• (3) Polynomial time approximation algorithms (PTAAs) are efficient in solving MOSPP problems but are limited by two drawbacks: they are sensitive to dynamic changes in the network and they are specific to one or few problems.

• (4) Heuristic algorithms (HAs), which is a random algorithm to find Pareto solutions of MOSPP, has great advantages in solving speed, but the quality of the solution is usually not ideal.

• (5) Most of the existing algorithms need to generate initial solution, and the solution set of MOSPP is obtained through repeated iterations, and the quality of the solution set will be affected by the initial solutions. In this paper, the proposed RSA algorithm does not need an initial solution and repeated iteration, but only needs one run to get the solution set.

In emergency management, transportation scheduling of emergency supplies and rescuers can be modeled as a MOSPPMTW problem, significantly impacting the effectiveness of rescue operations. Emergency relief has high requirements for timeliness and effectiveness: first, a dispatch path plan needs to be planned quickly; second, it is necessary to ensure that the supplies and rescuers arrive at the emergency site as soon as possible in the shortest time and at the smallest cost. Therefore, the calculation speed of the MOSPPMTW solving algorithm and the optimization level of the generated Pareto paths is particularly important during the decision-making process of the emergency.

Overall, there is a pressing need to address solution accuracy and computational speed with more advanced, efficient, and accurate optimization techniques that can manage the coexistence of different types of time windows with high precision and speed, which is suitable for various emergency disaster relief scenarios, enabling the rapid dispatch of emergency supplies and relief personnel to meet the requirements of rapid response and timeliness.

In this paper, we propose a ripple-spreading algorithm (RSA) to calculate the complete (not partial or approximated) Pareto front for multi-objective shortest path problem with mixed time windows (MOSPPMTW), seeking complete Pareto front with minimizing total time and total cost. RSA carries out a one-off ripple relay race in the route network. Nodes will generate a new ripple according to the Pareto dominance state of an incoming ripple, and the complete Pareto front will emerge at the macro level of ripple relay race, with guaranteed optimality. All complete Pareto fronts of a one-to-all MOSPPMTW can also be found in just a single run of ripple relay race. At the same time, we have also proven the optimality and time complexity of RSA while verifying its superior solution accuracy and speed through various examples with different node sizes and different network structures. RSA stands out with its robust search capabilities, fast computation speed, high solution accuracy, which is in line with the concept of "time is life" in emergency situations.

Problem description and mathematical model

Problem description

In emergency rescue scenarios, the transportation scheduling problem for emergency supplies and rescuers can be modeled as MOSPPMTW, which is a route planning problem that involves multiple objective functions and time window constraints. Here is a description of the problem and its corresponding constraints:

• (1) Problem description:

  In the emergency rescue context, given a weighted directed graph, where nodes represent locations on the map and edges represent connections between nodes, each edge has two weights (such as time and cost). In addition, there are multiple objective functions, such as minimizing travel time, minimizing cost, etc. Each node has one or more time windows, indicating the time intervals during which the node can be visited. Source node is the material storage location and the destination node is the disaster location. The goal is to find Pareto optimal paths from the source node to the destination node that minimizes total time and total cost while satisfying the time window constraints. This allows for rapid planning of dispatch path programs to ensure that emergency supplies and rescuers can arrive at the emergency site as quickly as possible, in the shortest possible time and at the lowest possible cost.

• (2) Constraints:

  Node visitation restrictions: Each node has one or more time windows, indicating the time intervals during which the node can be visited. The path planning must satisfy the time window constraints of the nodes.

  Path connectivity: The path must be a continuous sequence of nodes in the graph, from the starting point to the destination, without any disconnections or repeated nodes.

  Objective function constraints: Depending on the specific problem, the objective functions may include minimizing travel time, minimizing cost, etc. The path planning needs to consider and optimize these objective functions simultaneously.

  These are some common constraints for this type of problem. The specific problem description may involve additional constraints specific to the problem's requirements.

• (3) Assume that the following rules are satisfied:

  • (1) The location of the source node and destination node are known;

  • (2) Information about the type of time windows, cost and time of the node are known;

  • (3) The destination node has and can only be delivered by one vehicle;

  • (4) The waiting penalty factor and the late penalty factor are assumed to be the same for the soft time windows of different nodes;

  • (5) Only vehicle is considered, other transportation are not considered.

Variable definition

To model the MOSPPMTW problem in the emergency rescue context, assume that the emergency response vehicle starts from the material reserve warehouse (source node) and travels through several nodes before arriving at the disaster-stricken area (destination node). The following notation is listed in Table 3.

Table 3 Notations in MOSPPMTW and their meanings

Assume that the time window of node i is [e_i, l_i]. If node i and node j are on the same path P and serve node j immediately after serving node i, then t_j = t_i + st_i + t_ij. If t_i < e_i, then there is waiting time qt_i = e_i–t_i. Conversely, if t_i > l_i then there is delay time pt_i = l_i–t_i; Moreover, When the time window type of node i is different, the value of w_i¹ and w_i² are taken as follows:

• (1) Hard time window parameters:

  $$ w_{i}^{1} = 0,w_{i}^{2} = 0, \, i \in V. $$

  (1)

• (2) Soft time window parameters:

  $$ w_{i}^{1} = 1,w_{i}^{2} = 1, \, i \in V. $$

  (2)

  Equations (1) and (2) indicate the type of earliest service time e_i and latest service time l_i, taking into account whether node i is a hard or soft time window. When w_i¹ is 0, it means that arrival time t_i is smaller than e_i and no penalty cost is incurred. However, if w_i¹ is 1, it means that arrival time t_i is smaller than e_i incurring the penalty cost of waiting time qt_i; When w_i² is 0, it means that arrival time t_i is later than l_i and no penalty cost is incurred. However, if w_i² is 1, it means that arrival time t_i is later than e_i incurring the penalty cost of delay time pt_i.

Mathematical model

MOSPPMTW model

In emergency situations, such as disaster relief and rescue, rescue resources need to be rationally dispatched and arranged. The multi-objective shortest path problem with a mixed time window can optimize the path selection of resources to achieve the fastest response time, minimize the cost of using resources, and ensure that rescue services are provided within the time window.

In this paper, the MOSPPMTW model is developed by considering the timeliness, economy, and mixed time windows of nodes. Based on the above-mentioned notation and variable descriptions, the objective function and constraints of the model are defined as follows:

$$ \min Z = \left[ {f_{1} \left( P \right),f_{2} \left( P \right)} \right]^{T} , $$

(3)

$$ f_{1} \left( P \right) = \sum\limits_{i \ne j} {t_{ij} x_{ij} } , $$

(4)

$$ f_{2} \left( P \right) = \sum\limits_{i \ne j} {c_{ij} x_{ij} } + x_{ij} w_{j}^{1} \mu_{1} \sum\limits_{j = 1}^{N} {\max \left( {e_{j} - t_{j} ,0} \right)} + x_{ij} w_{j}^{2} \mu_{2} \sum\limits_{j = 1}^{N} {\max \left( {t_{j} - l_{j} ,0} \right)} , $$

(5)

$$ \sum\limits_{i \ne j} {x_{ij} - } \sum\limits_{i \ne j} {x_{ji} } = \left\{ {\begin{array}{*{20}l} {1{ ,}\quad i = 1} \\ { - 1{,}\quad i = n} \\ {0{,} \quad i \ne 1,n} \\ \end{array} } \right., $$

(6)

$$ \sum\limits_{i,j \in S} {x_{ij} \le \left| S \right|} - 1, $$

(7)

$$ t_{ij} = \frac{{d_{ij} }}{{v_{ij} }}, $$

(8)

$$ x_{ij} = \left\{ {0,1} \right\}, $$

(9)

$$ \left( {1{ - }w_{i}^{2} } \right)t_{i} \le \left( {1{ - }w_{i}^{2} } \right)l_{i} , \, \forall i \in V, $$

(10)

$$ \left( {1{ - }w_{i}^{1} } \right)e_{i} \le \left( {1{ - }w_{i}^{2} } \right)\left( {t_{i} + qt_{i} } \right) \le \left( {1{ - }w_{i}^{2} } \right)l_{i} , \, \forall i \in V, $$

(11)

$$ t_{i} + (1{ - }w_{i}^{1} )qt_{i} + st_{i} + w_{i}^{2} pt_{i} + t_{ij} + (1 - x_{ij} )T \le t_{j} , \, \forall i,j \in V,i \ne j, $$

(12)

$$ t_{1} = qt_{1} = st_{1} = pt_{1} = 0, $$

(13)

$$ qt_{i} = \max \left\{ {0,e_{i} - t_{i} } \right\}, \, i \in V, $$

(14)

$$ pt_{i} = \max \left\{ {0,t_{i} - l_{i} } \right\}, \, i \in V, $$

(15)

$$ t_{i} ,st_{i} ,qt_{i} ,pt_{i} \ge 0, \, i \in V. $$

(16)

Equation (3) represents the two minimization objectives; Eq. (4) represents the total time minimization; Eq. (5) represents the total cost minimization, including transportation costs, penalty costs of waiting time qt_i and delay time pt_i; Eq. (6) represents the entry and exit relations of the nodes along path P, at the source node 1 without and no entry, the value is 1; at the destination node n with in and no out, the value is – 1; In all other cases, the value is zero regardless of whether node i is on the path; Eq. (7) indicates that the path P contains no any loops; Eq. (8) indicates the time it takes to pass between nodes; Eq. (9) indicates the value of x_ij, x_ij = 1 indicates that the path P includes edge ij, otherwise it is 0; Eq. (10) indicates that the arrival time is not allowed to exceed the latest service time of the node; Eq. (11) indicates that the start service time of the node must be between the earliest start service time and between the earliest start service time and the latest start service time; Eq. (12) indicates the time relationship between precursor and successor nodes in the case of soft and hard different time windows. Here, T is a sufficiently large integer. The appropriate range of parameter T needs to consider the duration of the time window and carry out sensitivity analysis; Eq. (13) indicates the time parameter setting at the source node; Eq. (14) indicates the formula of waiting time; Eq. (15) indicates the formula of delay time; Eq. (16) indicates that the time parameters are all non-negative.

Pareto optimal path

The MOSPPMTW model, as described above, requires the simultaneous optimization of two conflicting objectives in finding the path with the shortest time while minimizing costs. This optimization problem does not have a unique solution, but rather a set of optimal solutions consisting of multiple Pareto optimal paths, with the following definition:

• (1) Pareto optimal paths: paths that cannot be improved in any further objective function without weakening at least one other objective function;

• (2) Pareto solution set: a set of Pareto optimal paths with solutions that are mutually non-dominant;

• (3) Pareto frontier: the set consisting of the vectors of objective values corresponding to each solution of the Pareto solution set;

• (4) Pareto front surface: a curve or surface formed by connecting the objective values corresponding to the Pareto optimal paths. For a two-objective problem, the Pareto front surface is a line, while for multiple objectives, it is a hypersurface.

The Pareto optimal path P^* of the MOSPPMTW problem has the following characteristics: there does not exist any path P that makes:

$$ f_{k} \left( P \right) \le f_{k} \left( {P^{*} } \right), \, k = 1,...,K, $$

(17)

$$ f_{{k{^\prime} }} \left( P \right) < f_{{k^{^\prime} }} \left( {P^{*} } \right), \, k{^\prime} = \left[ {1,...,K} \right]. $$

(18)

That is, the path P^* is not dominated by any other path P.

The process of solving the MOSPPMTW problem involves finding the Pareto optimal paths. Without additional subjective preference information, all Pareto-optimal paths are considered equally good, since the vectors cannot be completely ordered. Therefore, the most important thing is to find as accurate and as many Pareto-optimal paths as possible.

Ripple spreading algorithm

Basic steps

Ripple spreading algorithm [52,53,54,55,56,57,58] is a path optimization approach that leverages independent ripple relays competing with each other in the road network. Basic RSA is a method for solving single-objective path optimization problems. The following relevant definitions are used:

• (1) Ripples: ripples are the waves that spread from one node to neighboring nodes;

• (2) Ripple spreading speed: the speed at which ripples spread in the network;

• (3) Update time: the cycle time of ripple spreading;

• (4) Ripple radius: the length of the ripples as it spreads in concentric circles;

• (5) Ripple relay: the order of ripples passing through the nodes are connected at the source node and destination node to form a path.

In this paper, based on the characteristics of the MOSPPMYW problem, we apply an improved RSA that can directly complete the Pareto frontiers, and the main improvements are as follows:

• (1) Judgment mechanism for non-dominated solutions at nodes: non-dominated ripples are used based on the original ripple concept. By judging whether the ripple that activates a node is a non-dominated ripple or not, the non-dominated solution at the node is screened to ensure that the ripple that activates the subsequent node is a non-dominated ripple generated by the non-dominated solution of the preceding node.

• (2) Redefining node behavior and termination conditions: it can be guaranteed that the set of Pareto solutions on a node is available when Paul no longer generates new ripples.

• (3) The formula for calculating the ripple spreading speed has been modified from a fixed value to one based on the ripple spreading condition and network structure at each moment t, which ensures that at least one ripple reaches the neighboring node at each moment t.

In the ripple spreading process, certain conditions need to be satisfied. The node behavior and termination conditions are designed according to the MOSPPMTW problem:

• (1) Pareto non-dominated ripples (PNDRs): a ripple at a node is not dominated by any other ripple that arrived before that node;

• (2) Pareto optimal paths: the sequence of nodes recorded by some Pareto non-dominated ripple relay racing from the source node through several nodes to the destination node;

• (3) Conditions for ripple generation at a node: if a ripple that has just arrived at a node is not dominated by any previous PNDRs at that node, then that ripple becomes a new PNDR at that node and triggers node to generate new ripples;

• (4) The termination condition for Ripple Relay: no new ripples are created.

• (5) The ripple will disappear when it reaches all neighboring nodes of a node;

• (6) The continuous spreading of all the ripples that exist, triggers more ripples at nodes further out in space;

• (7) When the termination condition is satisfied, the ripple relay ends and the Pareto solution set is produced.

The relevant variables are defined as follows: node 1 denotes the source node, node n denotes the destination node, 1 ∈ V, n ∈ V; i denotes any node, i ∈ V; E(r_w) = i denotes node i generates ripple r_w; T(r₂) = r₁ indicates that ripple r₁ generates ripple r₂; When S_R(r_w) = 0, ripple r_w is in dead state. Conversely, S_R (r_w) = 1 indicates ripple r is in active state. The following notation is listed in Table 4.

Table 4 Notations in the RSA and their meanings

In this paper, ripple spreading is based on the first link value as the length, that is, k = 1. The formulae for the parameters associated with the RSA algorithm for solving the complete Pareto front for MOSPPMTW are as follows:

• (1) Ripple spreading speed

  Assuming that the ripple spreading speed between node i and its neighbor node j at moment t is denoted by v, the corresponding equations are expressed in (19) and (20):

  $$ v_{{r_{w} }}^{t} = \min \left[ {C_{1} (i,j) - R_{1}^{t} \left( {r_{w} } \right)} \right], $$

  (19)
  $$ v^{t} = \mathop {\min }\limits_{{}} v_{{r_{w} }}^{t} . $$

  (20)

• (2) Update ripple spreading radius

  During a ripple relay race, the ripples update the ripple radius once per unit time. Specifically, the ripple radius at moment t + 1 is determined by multiplying the ripple radius of the same ripple at moment t by the sum of the ripple spreading speed and the unit time. All ripples have an initial radius of 0, and each spreads to reach exactly one of its neighboring nodes without overtraveling. The ripple radius of ripple r_w from node i spreading to its neighboring node j and the initial radius of newly generated ripples are shown in Eqs. (21) and (22):

  $$ R_{1}^{t + 1} \left( {r_{w} } \right) = R_{1}^{t} \left( {r_{w} } \right) + v^{t} , $$

  (21)
  $$ R_{1}^{{}} (r_{w} ) = 0. $$

  (22)

• (3) Update PNDRs and ripple state

  If ripple r_w arrives at node j, first judge whether f₁(r_w, j) satisfies the time window of node j, second compare the arriving ripple r_w, with the current PNDRs on node j to and judge whether r_w is a PNDRs on node j; If both are satisfied, then node j generates a new ripple r_w+1, updated set Ω_PNDR (j) is shown in Eqs. (23) and (24):

  $$ R_{1}^{t} (r_{w} ) = C_{1} (E(r_{w} ),j), $$

  (23)
  $$ \Omega_{{{\text{PNDR}}}} (j) = \Omega_{{{\text{PNDR}}}} (j) + \{ r_{w} \} . $$

  (24)

The MOSPPMTW problem from source node 1 to destination node n is solved using RSA to obtain the complete Pareto front, as shown in Fig. 1.

Fig. 1

Flow chart of ripple spreading algorithm

The main steps are described as follows: Step 1 generates a network containing N nodes and M edges, given the source node 1 and the destination node n; Step 2 sets initial ripple spreading speed; Step 3 initializes the parameters and generates the initial ripples at the source node 1; Step 4 determines whether the termination condition is satisfied and no more ripples are generated, if yes, go to step 10, otherwise go to step 5; Step 5 updates the ripple spreading speed and ripple spreading radius; Step 6 judges whether satisfies the time window; Step 7 judges whether satisfies PNDR; Step 8 indicates generating new ripple; Step 9 represents the switching of the ripple state, that is, the ripple that has reached its all neighboring nodes enters a dead state and no more spread; Step 10 indicates outputting Pareto optimal path.

Operating principle

As an illustration, consider a four-node graph, where the edge values denote time and cost as example. In this graph, node 0 and node 2 are hard time window nodes, with respective intervals of [0,5] and [1, 6], while node 1 and node 3 are soft time window nodes with respective intervals of [1, 9] and [3, 7]. The waiting and delay penalty cost coefficients for the soft time windows are 0.5. Assuming that the source node is node 0 and the destination node is node 3, we need to find the Pareto optimal paths from node 0 to node 3. The time objective is the ripple length for spreading. The moment t is updated once per unit time. Ripple spreading speed is determined to be 1 according to the spreading condition at moment t and the network structure. The initial node network is shown in Fig. 2.

Fig. 2

Node topology diagram

The operational schematic of RSA to solve the one-to-one MOSPPMTW problem is shown in Fig. 3.

Fig. 3

Schematic diagram of algorithm operation

At the moment t = 0, the source node 0 becomes active and generates an initial ripple r₁.

During the moment from t = 0 to t = 1, the initial ripple r₁ spreads to the neighboring nodes 1 and 2. Ripple r₁ first reaches node 1 and its target value is (1,3) with time value 1 within the soft time window [1, 9] of node 1. Therefore, ripple r₁ becomes the first PNDR on node 1 and triggers a new ripple r₂ on node 1.

During the moment from t = 1 to t = 2, ripple r₂ spreads to neighboring nodes 2 and 3; meanwhile, ripple r₁ reaches node 2 with a target value (2,5) and time value 2 is within the hard time window [1, 6] of node 2. Therefore, ripple r₁ triggers a new ripple r₃ on node 2, becoming the first PNDR on node 2.

During the moment from t = 2 to t = 3, ripple r₁ has reached all its neighboring nodes, so r₁ disappears and source node 0 is in the dead state; Meanwhile, ripple r₂ continues to spread and reaches node 2. Its time value of 4 is within the hard time window [1, 6] of node 2. The objective function value of r₂ is (4,4), which is then compared with the previous objective function value (2,5) of ripple r₁ when it reaches node 2. Since r₂ is not r₁ dominates, r₂ triggers a new ripple r₄ on node 2, becoming the second PNDR on node 2.

During the moment from t = 3 to t = 4, ripple r₂ and r₃ reach node 3 simultaneously with time values both of 4 within the soft time window [3, 7] of node 3, and then compare the objective function value (4,8) of r₂ when it arrives at node 3. The objective function value (4,8) of r₃ when it arrives at node 3. Since r₃ is not dominated by r₂, and r₃ is not dominated by r₂, ripple r₂ and ripple r₃ simultaneously become PNDRs on termination node 3, making paths 0-1-3 and 0-2-3 both Pareto optimal paths. In addition, ripple r₃ reaches node 1 and the time value of ripple r₃ is 4, which is within the soft time window [1, 9] of node 1. Then, the objective function value (4,6) of r₃ when it arrives at node 1 is compared with the objective function value (1,3) of r₁ when it arrives at node 1. As r₃ is dominated by r₁, ripple r₃ is not a PNDR on node 1 and cannot trigger a new ripple.

During the moment from t = 4 to t = 5, ripple r₂ has reached all its neighboring nodes and disappears accordingly, causing node 1 to become the dead state. Meanwhile, ripple r₄ reaches node 3 and the time value of ripple r₄ is 5, within the soft time window [3, 7] of node 3. The objective function values of r₄ when it reaches node 3 is (5,7), which is compared to the objective function values of previous ripples r₂, r₃ when they reach node 3, which are (4,8) and (4,8). As r₄ is not dominated by the existing PNDRs ripples r₂ and r₃ at node 3, r₄ becomes the PNDR on destination node 3 and path 0-1-2-3 is a Pareto optimal path.

After moment t = 5, ripple r₄ reaches the destination node 3 and ripple r₄ disappears, ending the ripple relay with no more active ripples.

In summary, there are three Pareto optimal paths from the source node 0 to the destination node 3. The first Pareto optimal path is 0-1-3 with the objective value [4, 8]; the second Pareto optimal path is 0-2-3 with the objective value [4, 8], and the third Pareto optimal path is 0-1-2-3 with the objective value [5, 7] (Fig. 4).

Fig. 4

Pareto optimal path diagram

Theoretical analysis

Algorithm optimality

This section aims to prove the optimality of RSA for solving the MOSPPMTW problem, and to that end, it presents five theorems to support the verification process. Theorem 1 establishes that PNDRs exclude all irrelevant ripples on the Pareto-optimal paths from the source node to the destination node at intermediate nodes, which significantly improves the computational efficiency. In addition, all ripples triggered at the destination node of the path correspond to the Pareto-optimal paths, which is a necessary condition for optimality. Theorem 2 ensures that when the ripple relay race ends, each Pareto-optimal path connecting the source node and destination node would have triggered the corresponding ripple at the destination node, which is a sufficient condition for optimality [52,53,54,55,56,57,58]. The relevant theorem is as follows:

Similar to the dynamic programming algorithm, RSA to find the optimality of the MOSPPMTW can consider the optimal substructure, that is, the optimal solution of the MOSPPMTW can be constructed by the optimal solution of the subproblem.

When designing the RSA algorithm, it is required that the objective value of the ripples arriving at a node must be the non-dominated solution, that is, this ripple is a PNDR to motivate this node to generate new ripples for spreading propagation, which guarantees the optimality of arbitrary subproblems.

Theorem 1

MOSPPMTW satisfies the optimal substructure.

Proof

Suppose Pareto optimal path P from source node 1 to destination node n, denoted as P(1, n), now considering any node i in this path, denoted the path as P(1, i, n). We want to prove that path P(1, i) and P(i, n) are the Pareto optimal paths from source node 1 to node i and from node i to destination node n, respectively.

Using the counterfactual, if path P(1, i) is not the Pareto optimal path, there must exist another path P'(1, i) that dominates P(1, i).

We have Eqs. (25) and (26):

$$ P\left( {1,i,n} \right) = P\left( {1,i} \right) + P\left( {i,n} \right), $$

(25)

$$ P^{\prime}\left( {1,i,n} \right) = P^{\prime}\left( {1,i} \right) + P\left( {i,n} \right). $$

(26)

It can be deduced that P'(1,i,n) dominates P(1,i,n). This contradicts the original hypothesis. The same reasoning leads to the fact that P(i,n) is also the Pareto optimal path.

Therefore, we show that in the RSA algorithm, if we have found the Pareto optimal path from 1 to i, then the path P(1, i) and P(i, n) must also be the Pareto optimal path, that is, the subproblem optimality property holds. Combined with the optimal substructure property, we can conclude that the RSA algorithm to find the MOSPPMTW is Pareto optimal.

Theorem 2

Suppose a path P^* is a Pareto-optimal path connecting the source node 1 and the destination node n. Then, when the ripple relay ends, there must exist a triggered ripple at the destination node n. Moreover, the path of the node through which the ripple relay passes is this path P^*.

Proof

Assume that the path P^* is a path containing a total of N_* nodes, from the source node 1 through the N_ith node i, eventually to the destination node n. There exist Eqs. (25)–(29):

$$ P^{*} (1)\,{ = }\,1, $$

(27)

$$ P^{*} (N_{i} )\, = \,i, $$

(28)

$$ P^{*} (N_{*} )\, = \,n. $$

(29)

If the path P^* is a Pareto optimal path, according to Theorem 1, the ripple in the path P^* from the source node to any node is the PNDR of this node, that is, the ripple of the path P^*(1), …, P^* (N_i -1) arriving at P^*(N_i) and the ripple of the path P^*(N_i), …, P^*(N_* -1) arriving at P^*(N_*). Therefore, there must exist a triggered PNDR ripple at the destination node n. The node through which the ripple passes is the path P^*.

Theorem 3

During a ripple relay race, the jth PNDR that reaches the finish line and passes through the mixed time window is the jth Pareto optimal path.

Proof

The comparison between time windows and PNDRs only acts as a filter for ripples that satisfy the time window constraint and non-dominated ripples. It does not affect the ripple spreading optimization principle. Therefore, referring to the mathematical analysis of the single-objective shortest path given in the literature, it can be straightforwardly deduced that the jth ripple relay that reaches the destination node n in the ripple relay determines the jth Pareto-optimal path in the mixed time window network.

Theorem 4

When K = 2 and the ripple length is diffused with the kth = 1st objective value, the first PNDR that reaches the destination node and passes through the mixed time window is the Pareto-optimal path with the smallest cumulative value of the kth = 1st objective function among all paths. Similarly, the last PNDR that reaches the destination node and passes through the time window is the Pareto-optimal path with the smallest cumulative value of the kth = 2nd objective function among all paths.

Proof

Assume that the spreading speed is v^t at each moment t, and the path P₁ is the path traversed by the 1st PNDR ripple relay that reaches the destination node n. It takes T₁ time units and the cumulative value of the objective function is (a,b); Eq. (30) can be obtained:

$$ f_{k = 1} (P_{1} ) = \sum\limits_{t = 1}^{{T_{1} }} {v^{t} } = a. $$

(30)

We can use the converse method to assume that Theorem 4 is false, meaning there exists another Pareto-optimal path with a better cumulative value of the kth = 1st objective function, but requires more time to reach the destination node n. Let call this path be P₂, which is the path traversed by a certain PNDR ripple relay that reaches the destination node n and requires T₂ time units, with a cumulative value of the objective function (c, d). There exists Eq. (31) as follows:

$$ f_{k = 1} (P_{2} ) = \sum\limits_{t = 1}^{{T_{2} }} {v^{t} } = c. $$

(31)

For paths P₁ and P₂ to be Pareto optimal paths, Eqs. (17)–(18) must be satisfied. This means that the PNDR ripples of path P₁ reach the destination node n first and P₂ is not dominated by P₁. Thus, we can derive Eqs. (32)–(35):

$$ c < a, $$

(32)

$$ d > b, $$

(33)

$$ T_{2} > T_{1} , $$

(34)

$$ f_{k = 1} (P_{2} ) - f_{k = 1} (P_{1} ) = c - a < 0. $$

(35)

However, if Theorem 4 is false, then there exists, Eq. (36), which leads to a contradiction:

$$ f_{k = 1} (P_{2} ) - f_{k = 1} (P_{1} ) = \sum\limits_{{t = T_{1} + 1}}^{{T_{2} }} {v^{t} } > 0. $$

(36)

Because the ripple spreading speed v^t is positive numbers greater than 0, it follows that Eq. (36) should be greater than 0, which obviously contradicts, Eq. (35), which shows that the cumulative value of the objective function for P₁ is less than or equal to that of P₂. Therefore, our assumption that Theorem 4 is false must be incorrect, and the original statement holds true.

Similarly, assuming that path P₃ is the path traversed by the last PNDR ripple that reaches the destination node n. It takes T₃ time units and the cumulative value of the objective function is (A, B). If Theorem 4 is incorrect, then there must exist another path P₄ that takes less time and has a better cumulative value of the objective function for the kth = 2nd to reach the destination node n. Let us define path P₄ as the path traversed by the penultimate PNDR ripple that reaches the destination node n. It takes T₄ time units and has a cumulative objective function value of (C, D). There exists Eqs. (37)–(41):

$$ f_{k = 2} (P_{3} ) = \sum\limits_{t = 1}^{{T_{3} }} {v^{t} } = B, $$

(37)

$$ f_{k = 2} (P_{4} ) = \sum\limits_{t = 1}^{{T_{4} }} {v^{t} } = D, $$

(38)

$$ A > C, $$

(39)

$$ B > D, $$

(40)

$$ T_{3} > T_{4} . $$

(41)

We can conclude that the PNDR ripple of path P₃ is dominated by the PNDR ripple of P₄, which contradicts Eqs. (17) and (18). Therefore, path P₃ cannot be the Pareto optimal path of destination node n, which is contrary to our previous assumption. As a result, Theorem 4 must hold.

Time complexity

To demonstrate the algorithmic merit of RSA for solving the MOSPPMTW problem, this section analyzes mathematically the time complexity:

Theorem 5

Consider a road network with N nodes, M edges and a given source node. Assume that the complete Pareto front of each of the (N-1) nodes have an average of N_P Pareto points. Then, the computational complexity of the corresponding RSA is approximately O (K × N_N × N_P²) for the one-to-one MOSPPMTW problem with K objective functions.

Proof

In the ripple relay race, each node (except the starting node) has N_P PNDRs on average, triggering N_P ripples. RSA is divided into three basic computational steps: (1) increasing the ripples radius by ripple spreading speed; (2) determining whether ripple reach its neighboring nodes; (3) determining whether the arriving ripples are dominated by the PNDRs already present on the node.

Assuming that each node has an average of M/N edges and that it takes N_ATU time units for a ripple to pass through a path. Then, it takes on average (N_ATU + K × N_P) calculations before that ripple disappears. Since the source node will only produce one ripple, while the other (N-1) nodes will produce N_P ripples on average, a total of [(N-1) × N_P] × (N_ATU + K × N_P) × (M/N) calculations are needed when all the ripples disappear, that is, when the ripple relay race is over. Typically, N_P >  > K and K × N_P >  > N_ATU. Thus, the time complexity of RSA is approximately O (K × M × N_P²).

Experimental analysis

Randomized network experiments

To verify the optimality of RSA under different node sizes and network structures, this experiment uses the number of nodes N_N = [10, 50, 100], the average number of edges N_L = [2, 4, 6], time and cost randomly generated in the range of [10, 100], the optimization objective number K = 2, the mixed time window randomly generated in the range of [0, 300], the waiting time penalty factor μ₁ = 0.5 and delay time penalty factor μ₂ = 1.

We randomly generated node networks and example settings are shown in Table 5 and Fig. 5. It can be categorized into the following different types of emergency response network topologies. When the average number of edges and nodes is small, it can be regarded as a sparse topology, with relatively few connections between nodes, resulting in a limited number of direct communication paths, which is consistent with an emergency response road network that is in a suburb; when the average number of edges and nodes is medium, it belongs to a moderately dense connection that involves both the urban and suburb transportation road networks; when the average number of edges and nodes are large, it can be regarded as a dense topology, with a large number of connections between nodes, and a more complex urban road network, with a wide variety of intersections and multiple routes for emergency response vehicles. The testing was conducted uniformly using Python programming language, and the testing environment of all algorithms was a personal computer with an Intel(R) Core (TM) i5-8250 UCPU@1.60 GHz.

Table 5 Example of randomized network

Fig. 5

Different emergency response node networks

Combining RSA, NSGA-II [59, 60], MOEA/D [61, 62], Labeling Algorithms (LA) [63] and Simulated Annealing (SA) [64], Tabu Search (TS) [65] are solved for the above example. Each algorithm is applied to three different scenarios, namely, starting at node 0 and ending at node 9, node 49 and node 99, respectively. Each scenario runs 30 times. According to the experimental results in reference [59,60,61,62,63,64,65], and on this basis, we determined the better parameter configuration of the algorithm by comparing its performance under different parameter settings. The algorithm parameters are set as shown in Table 6.

Table 6 Parameter setting table

In this section, we propose two metrics to evaluate the performance of the above algorithm: N_Pareto indicates the number of Pareto points found by the algorithm, the more Pareto points, the better the performance; R_Pareto indicates the ratio of the number of complete Pareto fronts found to the number of experiments, the higher the value of R_Pareto, the more complete the Pareto fronts. In addition, we consider the running time, solution accuracy and stability of the algorithm, and the comparison tables of the algorithm results are shown in Tables 8, 9, 10, 11, 12 and 13.

The indicators, including Generational Distance (GD), Hypervolume (HV), Spacing, Spread, Coverage, and Error Ratio (ER), were selected based on their established relevance and effectiveness in assessing the performance of multi-objective optimization algorithms (Table 7).

Table 7 Multi-objective evaluation indicators

GD and HV are widely used to evaluate the proximity and diversity of the obtained Pareto fronts, providing insights into the algorithm’s convergence and coverage of the true Pareto front. Spacing and Spread measure the dispersion of solutions along the Pareto front, indicating the algorithm’s ability to explore the entire Pareto front and maintain a well-distributed set of solutions. Coverage assesses the closeness of the generated Pareto front to the true Pareto front, reflecting the algorithm's accuracy in approximating the optimal solutions.

In addition, we employed the Error Ratio (ER) to gauge the deviation of our proposed RSA’s solutions from the ideal Pareto front, offering a comprehensive view of the algorithm's performance in terms of solution quality.

Using these evaluation indicators, we aimed to comprehensively assess the effectiveness and efficiency of our proposed RSA in solving the MOSPPMTW problem. Each indicator provides a specific aspect of the algorithm's performance, enabling a well-rounded evaluation.

Considering the error between the approximate Pareto front X and the true Pareto front Y of the algorithm. In this paper, the reference point in HV index selects the maximum value of each target dimension as the coordinates of the reference point, and the selected reference point will be located at the boundary of the target space, containing as many real Pareto frontier sets as possible. The threshold value ϵ = 0.3 of ER, which < ϵ means that Pareto within the specified error range is better than.

Based on Tables 8 and 9, we can conclude the following:

• (1) In all the examples, the metrics are the same for all the algorithms except the LA algorithm, indicating that the solutions of the RSA algorithm are very close to the true Pareto front and are well-distributed in the target space.

• (2) In Example (10,2), the LA algorithm obtains a good Spacing value but sacrifices GD, HV, Spread, Coverage and ER. This suggests that the LA algorithm has an unreasonable coverage of the Pareto frontier, which may lead to a lack of diversity in solutions.

Table 8 Comparison table of algorithms results for small-scale node networks

Table 9 Comparison table of algorithm metrics of small-scale node networks

Based on Tables 10 and 11, we can conclude the following:

• (1) In Example (50,2), the RSA, NSGA-II, MOEA/D, SA and TS algorithms perform equally well in all aspects, but the LA algorithm has only one solution set, with no non-dominated solutions found [0, 5, 45, 10, 49], leading to the obtained high GD value (50.9755), and the other metrics are unsatisfactory.

• (2) In Example (50,4): the RSA、SA and TS algorithms again show the best performance in terms of GD, HV, Spacing, Coverage and ER to find high-quality solutions efficiently. Among the other algorithms, the GD values of NSGA-II, MOEA/D and LA are too high (11.04654, 22.09308 and 11.6812, respectively), and the Spread value of LA is better than that of the RSA algorithm, but not as good as the other values.

• (3) In Example (50,6), all algorithms perform equally well, effectively finding diverse and well-distributed solutions.

Table 10 Comparison table of algorithm metrics of medium-scale node networks

Table 11 Comparison table of algorithms results for medium-scale node networks

Based on Tables 12 and 13, we can conclude the following:

• (1) In Examples (100,2) and (100,4), all algorithms perform equally well, effectively finding diverse and well-distributed solutions.

• (2) In Example (100,6), RSA’s advantages over the other algorithms in terms of GD, HV, Spacing, Coverage and ER are more pronounced, finding complete Pareto solutions. In the Spacing metric, SA and RSA have the same value (19.0804) and in the Spread metric, LA’s value 0.4418) is better than RSA's value (0.4037) but other values of LA are not as good as RSA.

Table 12 Comparison table of algorithms results for large-scale node networks

Table 13 Comparison table of algorithm metrics of large-scale node networks

By analyzing the data in Tables 8, 9, 10, 11, 12 and 13, the following conclusions can be drawn:

• (1) From the comparison of Pareto optimal path solution sets, it is clear that RSA outperforms other algorithms in obtaining Pareto solution sets for all instances. The LA, which reduces multiple objectives into multiple single objectives for solving, cannot provide a complete Pareto frontier for some instances. In addition, NSGA-II and MOEA/D are random search algorithms, failing to find the full Pareto frontier in a few instances. As highlighted in Tables 11 and 12, a considerable part of the solutions obtained are usually not Pareto optimal, which shows that RSA has better optimality finding accuracy;

• (2) N_Pareto and R_Pareto are effective metrics for evaluating and can reflect the algorithm of ability and stability to find the Pareto solution set. RSA can both find all the Pareto points consistently across all runs, with an R_Pareto of 1, which indicates excellent solving stability, but other methods solve the problem with general results.

• (3) From the running time comparison, it can be seen that RSA has significantly lower than other algorithms, and its running time increase is not significant with the increase of node size, indicating its advantage in solution speed. While the running time of LA is slightly better than that of the RSA, its solution quality is not high and it lacks an advantage in the solving multi-objective problem;

• (4) Node size has no effect on RSA, and the complete Pareto frontier can still be found. The larger N and N_L are, the more obvious the advantage of RSA in solution accuracy and stability, with the solution effect of other algorithms will become worse. In large-scale instances (100,6), RSA running time in approximately 0.618% of NSGA-II, 1.161% of MOEA/D, 0.38% of SA, 2.3.0% of TS;

• (5) Most of the algorithms find the true Pareto frontier for medium and small-scale instances, LA algorithm performs the worst in (10,2) and (50,2), instances, NSGA-II and MOEA/D have too large GD in (50,4); In large-scale instances (100,6), all algorithms except RSA algorithm do not find the true Pareto frontier.

In summary, RSA fully traverses the search solution space, and the search optimization mechanism better balances the breadth and depth of the algorithm. This avoids the algorithm from entering a blind and disorderly search state, reducing the computation time and ensuring the solution accuracy, resulting in a better search optimization effect. Therefore, considering the algorithm running time, solution accuracy and solution stability, RSA is more advantageous, especially in large-scale instances.

Benchmark network experiments

The benchmark network experiments were performed on San Francisco Bay Area (BAY), Colorado (COL), and New York city (NY) road networks [66]. All of these road networks are huge as evident by the number of nodes and edges in each road network, as shown in Table 14. Each edge in the road network is assigned two weights: travelling time and edge length. The test case was a source and a destination node randomly selected from the road network. Each experiment consisted of ten trials of the test case. The benchmark network testing was also conducted uniformly using Python programming language, and the testing environment was a personal computer with an Intel(R) Core (TM)i5-8250UCPU @1.60 GHz.

Table 14 Example of benchmark network

From Table 15, we can observe the following results:

Table 15 Comparison table of algorithm metrics of benchmark network

For each dataset (BAY, COL, NY), the RSA algorithm has a value of 0 on the GD metric, with a larger value of HV and a smaller value of Spacing, which indicates that the solution set of RSA is completely overlapped with the real Pareto frontier, and the uniformity and dispersion of the solution set are better. The value of 1 for Coverage indicates that the solutions in the solution set almost completely cover the real Pareto frontier, which is a very good performance. RSA is able to explore the Pareto frontier better and get a high-quality solution set, which is advantageous compared with other algorithms.

In the ER index, the value of RSA is 0, which means that the execution time of RSA is equal to itself, indicating that RSA has high execution efficiency in these experiments, and the solution set obtained is very close to the real Pareto frontier, which is superior to other algorithms.

As far as the NY dataset is concerned, the RSA has a higher Spacing value compared to the TS and the solution set is not uniform enough. Compared to the LA, RSA has a lower Spread value and the RSA solution set covers a smaller portion of the true Pareto front. Although RSA is at a disadvantage, the gap between the metrics is small, and the RSA still has a significant advantage when considering other data sets and metrics.

Taken together, RSA shows high exploration performance and low execution time on these test datasets, which makes it a powerful multi-objective optimization algorithm.

Sensitivity analysis

This section consists of three parts. The first part focuses on adjusting the time window type, while the second part focuses on adjusting the time window-related parameters. Finally, the third part adjusts the RSA parameters to explore their effects on the total time, total cost, and Pareto solution set for a network of nodes with N = 100 and N_L = 6 for comparison.

Time windows type

To examine the impact of time window type, nodes are assigned either a hard or soft time window type, respectively, and compared with the mixed time window approach used in section "Randomized network experiments". The time cost value, time window range, waiting and delay time penalty factors μ₁ and μ₂ and other parameters are kept the same as in section "Randomized network experiments". Each case is run several times and the comparison results are shown in Table 16.

Table 16 Comparison table of results for different time window types

Based on the analysis of the data presented in Table 9, the following conclusions are drawn:

• (1) By comparing the Pareto solution sets, it is evident that RSA can find more N_Pareto under the mixed time window; Some paths are removed under the hard time window to obtain new Pareto optimal paths, resulting in an increase in total cost, from 179 to 234; Meanwhile, no waiting is required under the soft time window, but a penalty cost is incurred, resulting in a decrease in total time from 201 to 110, and an increase in total cost increases from 179 to 244.5. In addition, the operational efficiency remains stable.

• (2) The running time of RSA under different time window types shows minimal difference. However, the solution space of the mixed time window is larger, which increases computational complexity and also provides more viable path solution sets, which ultimately increases the feasibility and computational efficiency of path planning while preserving the flexibility of time window. Therefore, it can be concluded that the mixed time window is better suited to actual logistics distribution requirements.

Time windows parameter

Parameters related to the time window primarily include the time window width (Table 17), the size of the time window values (Table 18) and the penalty cost factors (Table 19). All other parameters remain unchanged. The comparison results are shown in the following table.

Table 17 Comparison table of results for different time window widths

Table 18 Comparison table of numerical results for different time windows

Table 19 Comparison table of the results of different penalty coefficients

Analysis of the data in Table 17 leads to the following conclusions:

• (1) Due to waiting, rejection and penalty mechanisms, differences in the Pareto optimal paths and N_Pareto are evident at different time window widths. However, R_Pareto consistently equals 1, indicating complete identification of the Pareto frontier.

• (2) Path [0, 78, 96, 99] appears most frequently under different time window widths, as highlighted in red font. The total time for this path is 129 with widths of [0,200], [0,250], [0,450], and [0,500], while the total cost varies among these scenarios. Similarly, the Pareto paths [0, 95, 80, 23, 99], [0, 43, 52, 99] and [0, 43, 97, 99] also appear multiple times.

• (3) It can be inferred that the Pareto solution set for this example in the subsequent different time window widths contains the aforementioned paths, with inconsistent total time and total cost. In most cases, the optimal path solution sets involve nodes 0, 78, 96 and 99.

After analyzing the data presented in Table 18, the following conclusions can be drawn:

As the range of values of the mixed time window increases, the waiting mechanism of certain nodes causes the total time of the Pareto solution set to increase substantially, while the total cost appears to decrease slightly. For example, the total time for path [0, 78, 96, 99] increases from 189 to 206, while the total cost decreases from 283.5 to 255.5. Similarly, the total time for path [0, 95, 80, 23, 99] increases from 201 to 371, 395, but the total cost remains relatively unchanged. Finally, the total time for path [0, 95, 16, 52, 99] increases from 304 to 410, while the total cost remains the same at 144.

Analysis of the data in Table 19 leads to the following conclusions:

• (1) The penalty factors μ₁ and μ₂ mainly affect the total cost of Pareto solution set containing soft time window nodes. When the factors are small, the solution set remains relatively unchanged. Conversely, when the factors increase, new Pareto optimal paths are generated, such as the newly generated path [0, 38, 27, 12, 99] when μ₂ = 5.

• (2) When μ₁ is kept constant, the total cost of paths [0, 76, 41, 12, 99], [0, 38, 33, 12, 99] and [0, 78, 96, 99] remains relatively unchanged, while the total cost of path [0, 95, 80, 23, 99] increases. On the other hand, when μ₂ is kept constant, the total cost of paths [0, 38, 33, 12, 99] and [0, 95, 80, 23, 99] remains constant, while the total cost of paths [0, 76, 41, 12, 99] and [0, 78, 96, 99] increases.

Algorithm parameters

The main parameters of RSA are ripple spreading speed v^t at moment t. According to Eqs. (19) and (20), we can determine the optimal value of v^t for the node network at each moment. To explore the effect of different values on the results of the algorithm, the comparison results of different combinations of parameters set artificially are shown in Table 20.

Table 20 Comparison table of results of different algorithm parameters

Analysis of the data in Table 20 leads to the following conclusions:

• (1) For parameters v^t of 0.1, 0.5, 1 and 10, the Pareto solution set is the same. The solution set changes when v^t is set to 50, where the total cost of path [0, 11, 9, 96, 99] increases, and new path such as [0, 38, 27, 12, 99] appear. For parameters v^t of 100, 200 and 500, there are no solutions available. From existing literature [67], it is known that the ripple spreading speed v^t must be less than the minimum node link min(d_ij), which can avoid the overtravel problem and ensure the optimality of RSA.

• (2) The computational speed and solution accuracy of RSA depend on the ripple spreading speed v^t. As the value of v^t increases, the running time decreases slightly, but the magnitude is small. When the value of v^t is large, the larger the ripple radius of each update, the fewer the number of updates, which is beneficial to the local search and reduces the algorithm running time but may be accompanied by overtravel problems and reduction in solution accuracy. On the other hand, when the value of v^t is small, the smaller the ripple radius of the update, the more the number of updates, which is beneficial to the global search, but it needs to repeat the spreading process multiple times, which increases the algorithm running time and reduces the computational efficiency.

In conclusion, as v^t increases, the solution accuracy of RSA tends to decrease, while the running time decreases only slightly. This finding suggests that determining the ideal range of v^t has strong practical significance and research value for RSA when attempting to solve large-scale problems.

Conclusion

Previous studies have largely focused on single types of time windows in multi-objective shortest path problems, overlooking the coexistence of multiple time window types. Furthermore, existing multi-objective algorithms have faced challenges in striking the right balance between solution accuracy and computational speed. To address these limitations, we proposed Ripple Spreading Algorithm (RSA) specifically tailored to solve the Multi-Objective Shortest Path Problem with Mixed Time Windows (MOSPPMTW). Notably, the proposed RSA algorithm stands out as it can efficiently calculate the complete Pareto front with just one ripple relay race. In addition, the optimality and time complexity of the RSA are theoretically proven with rigorous theoretical analysis, and its effectiveness and feasibility are verified through several illustrative examples.

Looking ahead, we aim to further optimize the RSA's performance by incorporating additional factors. These factors will include path reliability, path threatened degree, node priority, and node satisfaction, contributing to a more practical and effective multi-objective problem-solving method that better aligns with real-world logistics requirements. Moreover, we plan to extend the time window to encompass soft-hard and hard-soft time windows, catering to a broader range of scenarios.

To expand the applicability of the algorithm, we plan to conduct more extensive numerical experiments and parameter studies on RSA for solving other complex optimization problems. This exploration will broaden the practical applications of the algorithm and uncover its potential across diverse domains.