Optimization of urban emergency support material distribution under major public health emergencies based on improved sparrow search algorithm

Abstract

The outbreak of major public health emergencies such as the coronavirus epidemic has put forward new requirements for urban emergency management procedures. Accuracy and effective distribution model of emergency support materials, as an effective tool to inhibit the deterioration of the public health sector, have gradually become a research hotspot. The distribution of urban emergency support devices, under the secondary supply chain structure of “material transfer center-demand point,” which may involve confusing demands, is studied to determine the actual situation of fuzzy requests under the impact of an epidemic outbreak. An optimization model of urban emergency support material distribution, based on Credibility theory, is first constructed. Then an improved sparrow search algorithm, ISSA, was designed by introducing Sobol sequence, Cauchy variation and bird swarm algorithm into the structure of the classical SSA. In addition, numerical validation and standard test set validation were carried out and the experimental results showed that the introduced improved strategy effectively improved the global search capability of the algorithm. Furthermore, simulation experiments are conducted, based on Shanghai, and the comparison with existing cutting-edge algorithms shows that the designed algorithm has stronger superiority and robustness. And the simulation results show that the designed algorithm can reduce vehicle cost by 4.83%, time cost by 13.80%, etc. compared to other algorithms. Finally, the impact of preference value on the distribution of emergency support materials is analyzed to help decision-makers to develop reasonable and effective distribution strategies according to the impact of major public health emergencies. The results of the study provide a practical reference for the solution of urban emergency support materials distribution problems.

Keywords

Emergency support material distribution vehicle routing problem fuzzy demand Credibility theory sparrow search algorithm

Introduction

As a primary means of responding to health and safety emergencies, emergency material distribution plays a crucial role in the prevention and control of major public health emergencies such as novel coronavirus. Its distribution rationality and dynamic responsiveness have become a hot spot for researchers.^1–3 Considering that the outbreak of a major health care problem is mainly concentrated in urban areas, and due to the complexity of the impact of such events, it is difficult to guarantee the effectiveness and speed of distribution referring to the traditional three-level supply chain structure of “material supply point-material transfer center-demand point.” Therefore, the current distribution of emergency supplies for urban areas is only considered the secondary supply chain structure of “material transfer center-demand point.”⁴

The urban emergency support material distribution problem (UESMDP) is still essentially a variant of the vehicle routing problem (VRP), whose main research involves a multi-center and multi-objective vehicle routing planning problem.⁵ To reflect the real distribution situation under a major public health event outbreak, researchers have extended a variety of decision factors based on the basic distribution model. Liu et al. designed a new modelling approach to solve problem of the unreasonable material distribution, in the early stages of the COVID-19 outbreak and established a dynamic distribution model to update distribution routes of vehicles.⁶ Added to that, Bodaghi et al. analyzed the emergency material distribution under multiple resource scheduling, constructed an emergency relief model involving expendable materials, and compared six heuristic algorithms to propose an effective solution for material distribution.⁷ However, the above research cannot be effectively applied to the problem of uncertainty in demand for emergency supplies; the main reason is that the patient health is not defined and the situation of urban personnel is unclear at the early stage of the outbreak of infectious major public events such as COVID-19.

Similar to the uncertainty of demand under the impact of an epidemic, there is uncertainty of demand involved in supply chain management or logistics scheduling, for which researchers have introduced fuzzy mathematics into the analysis of the problem. Based on the fuzzy mathematics these actual problems involving uncertainty can be modelled and solved, where in the job-shop scheduling problem (JSP), Li et al. investigated the fuzzy hybrid green shop scheduling problem with fuzzy processing time, and expressed the shop operation time using triangular fuzzy numbers. Then proposed a discrete artificial bee colony algorithm to solve the problem efficiently.⁸ Gao and Wang et al. then designed the improved DE algorithm and the hybrid adaptive differential evolution algorithm to efficiently solve the fuzzy JSP.^9,10 In addition, in the VRP, Zhou et al. introduced the fuzzy demand to access goods into VRP and constructed a multi-objective optimization model from the risk perspective.¹¹ Added to that, Fan et al. introduced fuzzy demand and fuzzy time window into VRP and established a multi-objective fuzzy opportunity constraint model, which fully considered the uncertainty information in real distribution tasks.¹² Yang et al. considered soft time windows based on fuzzy demand and fuzzy time windows for minimizing logistics costs.¹³ Li et al. have similarly studied the VRP with multiple time windows under fuzzy demand and developed a two-stage hybrid optimization algorithm to solve the problem.¹⁴ In a nutshell, the general distribution model can no longer meet the demand of emergency material distribution under major public health emergencies with serious social hazards such as COVID-19. For that, this paper focuses on the combination of fuzzy demand and emergency support material distribution problems to explore a realistic and feasible distribution model under the influence of COVID-19.

In terms of solving the UESMDP, researchers currently use mostly the meta-heuristic algorithms to solve the problem as it is essentially an NP-Hard problem with high complexity and difficulty to solve.¹⁵ The genetic algorithm (GA),¹⁶ particle swarm optimization (PSO),¹⁷ and bat algorithm¹⁸ have been used for the solution. In addition, as the size and the complexity of the problem increase, the general metaheuristic algorithms lack convergence accuracy and speed. So, to further improve the optimization capability of algorithm, researchers have introduced a hybrid mechanism, based on the metaheuristic algorithm.^19–21 Among them, Wang et al. designed a two-stage multi-objective evolutionary algorithm, and the simulation experimental results showed that this algorithm has a strong solving ability, but its effectiveness needs improvement.²² On the other hand, Zhang et al. combined the ant colony optimization algorithm (ACO) with artificial immunity to design an improved immune ACO, and the algorithm comparison results showed that the solution time of the new designed algorithm was shorter.²³ Based on the above research, it can be seen that the hybrid metaheuristic algorithms have better gains for solving large-scale complex optimization problems and are a feasible direction to study the solution of urban emergency material support distribution problems. The sparrow search algorithm (SSA), as a new swarm intelligence algorithm, was first proposed by Xue and Shen in 2020,²⁴ and SSA has been widely used in some areas.^25–28 Among them, Gad et al. introduced random repositioning strategy and novel local search algorithm to the traditional SSA to improve SSA to study the feature selection problem in machine learning and data mining tasks.²⁶ Xian et al. studied fuzzy time series based on SSA model for dealing with time series with trends and disturbances.²⁷ Zhang et al. improved SSA by designing a linear path strategy, neighborhood search strategy, and multi-index comprehensive evaluation method so that it can better solve the bioinspired mobile robot path planning problem.²⁸ In the above studies, the improved SSA has certain advantages in terms of optimization speed and accuracy compared with other classical population intelligence algorithms (ACO, GA, etc.). Therefore, the aim of our paper is to explore the possibility of SSA in emergency material dispatching problems and to provide a new solution for this kind of complex vehicle path planning problems.

In conclusion, this paper focuses on the distribution of urban emergency support materials based on fuzzy demand. Firstly, the urban community demand, under major public health emergencies such as COVID-19, is characterized by on triangular fuzzy numbers. Thus, a heterogeneous vehicle distribution optimization model, involving multiple material transfer centers and multiple materials, is established. The distribution paths are initialized with clear values for fuzzy demands and they determine whether the vehicle will carry out material distribution through Credibility theory.^11–14 Furthermore, based on the traditional SSA, the local and the global search capabilities of the algorithm are enhanced by introducing improvement strategies such as the Cauchy variation, opposition-based learning, and adaptive strategy. The strategies also reduce the blindness of the algorithm while retaining its extensive search capability to solve the problem effectively. In addition, a simulation experiment is carried out in Shanghai as the simulation background of urban emergency support material distribution, and results were compared to existing cutting-edge algorithms. These results show that the designed algorithm has strong robustness and superiority. Finally, a sensitivity analysis is conducted for the preference value in the Credibility theory, which helps distribution decision-makers to apply strategies according to the impact of different major public health emergencies, and also provides effective reference for real urban emergency support material distribution.

To sum up, this paper will be divided as follows: In the second section, we introduce the process of establishing the optimization model for the UESMDP. Then, the traditional and improved SSA (ISSA) are provided in the third section. The numerical verification and simulations results are provided in the next section and comparison between the different methods is also established. Finally, we analyze the parameters in the model and summarize the research of the paper.

Establishment of an optimization model for UESMDP

Problem description

Under a major public health emergency, emergency supplies can effectively inhibit the development of the event. Different from the three-level supply chain structure of “material supply point-material transfer center-demand point” of the general material distribution, individuals, in urban zone, can directly transport material assistance from different areas to the affected one through the material transfer centers. Thus, in order to effectively alleviate the pressure of the transfer center and make the demand of each community in the city reasonably satisfied, this paper focuses on the optimization of the distribution under the secondary supply chain structure of “material transfer center-demand point.” Figure 1 shows the distribution diagram path.

Figure 1.

Distribution diagram.

Problem assumptions

The below points represent the problem assumption constraints of this system:

The emergency support materials distributed are all expendable, that is, therapeutic drugs, testing reagents, extermination supplies, etc.

The selection of major hospitals in the city as material transfer centers.

All kinds of tools in the material transfer center are adequate.

Each community can be served by one vehicle, and every vehicle must return to the material transfer center.

Each vehicle will depart at 8:00 AM.

Model establishment

Fuzzy demand model for emergency support materials

In the case of an epidemic outbreak, urban communities, as basic personnel management units, usually distribute emergency support materials according to communities’ needs. Because of the uncertainty of community personnel, managers cannot determine the exact demand, but can predict the lowest (most conservative) and the highest (most aggressive) demands for materials in their community. Uncertain customer demand for the dispatch center means that the dispatch center cannot accurately pre-dispatch, and therefore often leaves dispatch vehicles at full capacity when they leave the dispatch center, which not only leads to a waste of resources, but also makes it impossible to manage resources accurately, especially when supplies are limited. Thus, such fuzzy demands are likely to cause problems such as difficulty in distributing emergency supplies, poor accuracy, and low satisfaction rate of materials. Therefore, this paper uses triangular fuzzy numbers to characterize the uncertain demand for materials at communities in order to facilitate the establishment of distribution models. The demand for each emergency material a for each community i can be represented by the triangular fuzzy number $M_{i a}$ , $M_{i a} = (l_{i a}, m_{i a}, u_{i a})$ , where $l_{i a}$ denotes the lower limit of triangular fuzzy number, $m_{i a}$ represents the most probable value of the triangular fuzzy number, and $u_{i a}$ denotes the upper limit of the triangular fuzzy number.

Furthermore, after characterizing the community demand using the triangular fuzzy number, considering that the triangular fuzzy number only gives a specific range, which makes it difficult to define pre-scheduling arrangements for materials. For this reason, the expected value of the triangular fuzzy number is used for pre-scheduling, and the reliability calculation is applied to measure the service capability during the material distribution process to decide whether the vehicle continues to serve the next community. The expected value of the triangular fuzzy number of the community point material demand is calculated as shown in equation (1), and the reliability is delivered in equation (5).

E M_{i a} = \frac{1}{2} (E M_{i a 1} + E M_{i a 2})

(1)

where,

E M_{i a 1} = m_{i a} - \int_{l_{i a}}^{m_{i a}} μ_{M_{i a}} (x) d x, l_{i a} \leq x \leq m_{i a},

(2)

E M_{i a 2} = m_{i a} + \int_{m_{i a}}^{u_{i a}} μ_{M_{i a}} (x) d x, m_{i a} \leq x \leq u_{i a},

(3)

and

μ_{M_{i a}} (x) = {\begin{matrix} 0, x < l_{i a} \\ (x - l_{i a}) / (m_{i a} - l_{i a}), x \in [l_{i a}, m_{i a}] \\ (x - u_{i a}) (m_{i a} - u_{i a}), x \in [m_{i a}, u_{i a}] \\ 0, x > u_{i a} \end{matrix} .

(4)

\begin{aligned} C r {M_{j a} \leq Q_{a r}} & = C r {M_{j a} - Q_{a r} \leq 0} = C r {(l_{j a} - l q_{a r}, m_{j a} - m q_{a r}, u_{j a} - u q_{a r}) \leq 0} \\ = {\begin{matrix} 0, & i f l_{j a} \geq u q_{a r} \\ \frac{u q_{a r} - l_{j a}}{2 \times (u q_{a r} - l_{j a} + m_{j a} - m q_{a r})}, & i f l_{j a} \leq u q_{a r}, m_{j a} \geq m q_{a r} \\ \frac{u_{j a} - l q_{a r} + 2 \times (m_{j a} - m q_{a r})}{2 \times (m q_{a r} - m_{j a} + u_{j a} - l q_{a r})}, & i f m_{j a} \leq m q_{a r}, u_{j a} \geq l q_{a r} \\ 1, & i f u_{j a} \leq l q_{a r} \end{matrix}, \end{aligned}

(5)

where,

\begin{aligned} Q_{a r} & = (l q_{a r}, m q_{a r}, u q_{a r}) = Q_{a} - \sum_{i \in D} M_{i a} \\ = [(Q_{a} - \sum_{i \in D} l_{i a}) (Q_{a} - \sum_{i \in D} m_{i a}) (Q_{a} - \sum_{i \in D} u_{i a})] \end{aligned}

(6)

The

μ_{M_{i a}} (x)

in equation (4) is the membership function of

M_{i a}

, and x is independent variable of

μ_{M_{i a}} (x)

. Cr denotes the reliability when

M_{j a} \leq Q_{a r}

. Where, i and j denote arbitrarily dissimilar community points.

M_{i a}

and

M_{j a}

denote the fuzzy demand for the a^th material of communities i and j, where

M_{i a} = (l_{i a}, m_{i a}, u_{i a})

and

M_{j a} = (l_{j a}, m_{j a}, u_{j a})

are the fuzzy demand for a^th material of communities i and j, respectively. D represents the set of communities that have received the required demand by the vehicle.

Q_{a}

and

Q_{a r}

denotes, respectively, the total load and the remaining load of the a^th material for the vehicle. Since the demand of each community is an uncertain triangular fuzzy number, the remaining load of the vehicle

Q_{a r}

is likewise a triangular fuzzy number.

From equation (5), we know that Cr varies between 0 and 1, and a larger value of Cr indicates that the vehicle is more able to satisfy the demand of community point j. Further, the fuzzy chance constraint is introduced, and the service preference value $ω$ is set to determine whether the vehicle continues to serve the next community. In fact, when $C r \geq ω$ , the vehicle continues to serve the next customer point; however, when Cr is smaller than ω, the vehicle returns to the transfer center. The value of $ω$ reflects the strategy of the decision maker of the allocation. A higher value of $ω$ indicates a more conservative strategy, while a lower value of $ω$ indicates a more aggressive strategy. Decision makers should choose the value of $ω$ reasonably according to the impact of a major public health emergency.

Emergency support material distribution optimization model

Under the major public health emergencies represented by COVID-19, a timely and fast emergency supplies dispatch response will greatly inhibit the deterioration of the event. Thus, it is of strong practical significance to minimize the distribution time cost as the being the main optimization objective. Based on the above analysis and the established fuzzy demand model for emergency supplies, an optimization model for emergency support materials distribution is constructed as shown below:

min O = λ \sum_{c} \sum_{t} \sum_{k} S C + (1 - λ) \sum_{c} \sum_{t} \sum_{k} T C,

(7)

where,

S C = \sum_{t} (α_{t} \times \sum_{c} \sum_{k} x_{c i k}^{t}) + β \times \sum_{a} \sum_{c} \sum_{k} Q_{a c k},

(8)

T C = w_{t} T_{k}^{t} + w_{s t} max {(T_{i}^{k} - T_{i max}), 0},

(9)

and

T_{i}^{k} = {\begin{matrix} T_{c i k}^{t}, D = \emptyset \\ T_{c i k}^{t} + \sum_{i, j \in D} T_{i j k}^{t} x_{i j k}^{t} + \sum_{i \in D} T_{h} y_{c i k}^{t} \\ T_{k}^{t}, D = H \end{matrix}, D \subset H,

(10)

T_{k}^{t} = T_{c k}^{t} v_{c k}^{t} + \sum_{i, j \in I} (T_{c i k}^{t} x_{c i k}^{t} + T_{i j k}^{t} x_{i j k}^{t}) + \sum_{i \in H} T_{h} y_{c i k}^{t},

(11)

T_{c i k}^{t} = \frac{d_{c i}}{v},

(12)

T_{i j k}^{t} = \frac{d_{i j}}{v},

(13)

x_{c i k}^{t} = {\begin{matrix} 1, indicates that the t - type vehicle k arrives \\ in the community i from the material \\ transfer center c \\ 0, otherwsie \end{matrix},

(14)

x_{i j k}^{t} = {\begin{matrix} 1, indicates that t - type vehicle k arrives \\ in the community j from the community i \\ 0, otherwise \end{matrix},

(15)

y_{c i k}^{t} = {\begin{matrix} 1, indicates t - type vehicle k \\ serves community i \\ 0, otherwise \end{matrix} .

(16)

Equation (6) is the objective function, where

λ

is a weighting factor to measure the decision maker's preference for the cost of the delivery vehicle and the cost of time. In equation (8), SC represents the distribution vehicle cost, where

Q_{a c k}

denotes the weight of the a^th material of vehicle k of the dispatch center c,

α_{t}

denotes the dispatch cost of unit t-type vehicle, and

β

denotes the dispatch cost of unit weight material. In equation (9), TC indicates the vehicle distribution time cost calculated based on the time needed by the vehicle to transfer the material from the center to the community. In addition,

T_{k}^{t}

denotes the total time for vehicle k to complete service,

T_{c k}^{t}

represents the loading time of vehicle k at transfer center c,

T_{c i k}^{t}

denotes the in-transit travel time of vehicle k from material transfer center c to community i,

T_{i j k}^{t}

denotes the in-transit travel time of vehicle k from community i to community j, and

T_{h}

denotes the service time of vehicle k at community i. Finally,

w_{t}

is the coefficient of time cost,

w_{s t}

is the coefficient of time window penalty cost, and H denotes the set of communities for which vehicle k is responsible. Equations (14–16) are 0–1 variables.

As for the constraints, they are defined as follows:

\sum_{c \in C} \sum_{t \in T} \sum_{k \in V} y_{c i k}^{t} = 1,

(17)

\sum_{i = 1} x_{i j k}^{t} = 1, j = 1, 2, \cdot \cdot \cdot, n,

(18)

\sum_{j = 1} x_{i j k}^{t} = 1, i = 1, 2, \cdot \cdot \cdot, n,

(19)

\sum_{i, j \in H} x_{i j k}^{t} \leq | H | - 1, 2 \leq | H | \leq n - 1, \forall H \subseteq I,

(20)

\sum_{i \in I} x_{c i k}^{t} = \sum_{j \in I} x_{j c k}^{t} = 1,

(21)

\sum_{i \in I} x_{i j k}^{t} = y_{c i k}^{t},

(22)

\sum_{j \in I} x_{i j k}^{t} = y_{c j k}^{t},

(23)

T_{k}^{t} \leq T_{max}^{k},

(24)

Q_{min}^{t} \leq \sum_{i \in H} \sum_{a} A M_{a i} \leq Q_{max}^{t},

(25)

C r {(Q_{a k} - \sum_{i \in D} y_{c i k}^{t} \times M_{i a}) \leq 0} \geq ω,

(26)

x_{c i k}^{t} \times d_{c i} \leq d_{max} .

(27)

Constraints (17–19) indicate that each community is served by only one vehicle. Constraint (20) indicates the elimination subloop, where H is the set of distribution communities of vehicle k. Constraint (21) shows that the vehicle returns to the transfer center. Constraints (22) and (23) indicate that vehicle k must leave community i after completing the distribution. Constraint (24) indicates that the vehicle distribution time must not exceed a certain value. Constraint (25) indicates that the total material weight of the demand point responsible for vehicle k needs to comply with the load limit. Constraint (26) is a fuzzy chance constraint, which indicates that the vehicle is required to serve at any community with a demand that does not exceed the reliability of vehicle k's service capacity which must be greater than the preset preference value

ω

. Finally, Constraint (27) represents the farthest distribution distance constraint avoiding distribution vehicles traveling to communities that are far from the material transfer centers to limit the waste of distribution resources.

Improved sparrow search algorithm

Traditional SSA

SSA was inspired by the foraging and anti-predatory behaviors of sparrows. A population of sparrows was composed of discoverers, vigilantes, and followers, and thus optimized in a continuous iterative process. The specific stages are shown below.

Foraging behavior

Discoverers in sparrow populations are responsible for searching locations with high density of food in the region, thus providing the area and direction for foraging. The discoverers with better fitness values will have priority in obtaining food. After the followers reach the foraging search range of the discoverer, the discoverer can get a larger search space. The location of the discoverer is updated as follows:

Input:

X_{i, j}

, Safety value ST

Output:

X_{i, j}^{t + 1}

R_{2} = r a n d (1)

2. if

R_{2} < S T

3. then

X_{i, j}^{t + 1} \leftarrow X_{i, j} \cdot \exp (- \frac{i}{α \cdot i t e r_{m a x}})

4. else

X_{i, j}^{t + 1} \leftarrow X_{i, j} + Q \cdot L

5. end if

Where, rand(1) indicates to generate a random number between 0 and 1, t denotes the number of current iterations, $i t e r_{m a x}$ represents the maximum number of iterations, $X_{i, j}$ shows the position of the i^th sparrow in the j^th dimension, $α$ denotes a random number that is smaller than the unit, $R_{2}$ denotes warning value that is also smaller than the unit; $S T$ denotes safety value, ranging between 0.5 and 1, Q represents a random number that follows a normal distribution, and L indicates the dimension-based unit matrix. When $R_{2} < S T$ means there are no predators around, the producer goes into broad search mode. If $R_{2} \geq S T$ , it means some sparrows have spotted the predator and all sparrows need to quickly fly to other safe areas.

When $R_{2} \geq S T$ , the discoverer moved randomly to the vicinity of the current position according to a normal distribution. When $R_{2} < S T$ , the discoverer's search space gradually shifted from the range from (0,1) to (0.0.4). And as can be seen from Figure 2, as the value of ST increases, the sparrow's search space tends to move towards the space where R₂ < ST and the global search for superiority is weaker. When ST = 1, the population reaches a minimum search space at the end of the algorithm and tends to fall into a local optimum.

Figure 2.

The effect of ST taking on the SSA search space.

After a follower joins a finder's foraging area, some of the followers will monitor the finder. If the finder gets a better foraging area (i.e. if the discoverer finds a solution space that gives a better value for the objective function, the follower will move with the discoverer towards the better solution space), the followers will immediately leave their current position to grab the new foraging position of the finder. If the follower succeeds, it will immediately receive food. If the grab fails, the follower will leave the current area due to its low fitness value and go to another spotter's area. The location of the follower is updated as follows:

Input:

X_{i, j}

X_{w o r s t}

, Number of iterations i

Output:

X_{i, j}^{t + 1}

R_{2} \leftarrow r a n d (1), Q \leftarrow r a n d (1)

A^{+} \leftarrow A^{T} (A A^{T})^{- 1}

2. if

i > \frac{n}{2}

3. then

X_{i, j}^{t + 1} \leftarrow Q \cdot \exp (\frac{X_{w o r s t} - X_{i, j}^{t}}{i^{2}})

4. else

X_{i, j}^{t + 1} \leftarrow X_{P}^{t + 1} + | X_{i, j} - X_{P}^{t + 1} | \cdot A^{+} \cdot L

5. end if

Where $X_{p}$ denotes the optimal position occupied by the discoverer, $X_{w o r s t}$ represents the global worst position, and A denotes a matrix with dimension-based elements assigned to 1 or −1.

Anti-predatory process

The sparrow population will also be predated by the corresponding natural predators when they are foraging, so after their appearance, the sparrow population will develop a sense of danger; thus, the sparrow population at the edge will go to the low-risk location area, and the ones in the middle will go to random location areas. The sparrow population locations for the anti-predation process are updated as follows:

Input:

X_{i, j}

X_{w o r s t}

Output:

X_{i, j}^{t + 1}

1. if

f_{i} > f_{g}

3. then

X_{i, j}^{t + 1} \leftarrow X_{b e s t}^{t} + β \cdot | X_{i, j}^{t} - X_{b e s t}^{t} |

4. else

X_{i, j}^{t + 1} \leftarrow X_{i, j}^{t} + K \cdot [\frac{| X_{i, j}^{t} - X_{b e s t}^{t} |}{(f_{i} - f_{w}) + ε}]

5. end if

where $X_{b e s t}^{t}$ denotes the current global optimal value, $β$ represents the satisfaction of the step control parameter, K is a variable varying between −1 and 1, $f_{i}$ denotes the adaptation value of the current individual sparrow, $f_{g}$ and $f_{w}$ are, respectively, the current global best and worst fitness values and, finally, $ε$ is a constant.

Improvement strategies

The SSA algorithm has a strong global search capability due to its rich and characteristic search mechanism, and it can get better optimization accuracy compared to the general population intelligence algorithm.²⁴ However, in solving the complex VRP problem, studied in this paper, the algorithm suffers from premature development and easily falls into local optimality, as most swarm intelligence algorithms do. Therefore, it cannot be directly used to solve the UESMDP problem. Therefore, based on the search mechanism of SSA algorithm and developed model, the following improvement strategies were introduced to enhance the local and global search capability of the algorithm.

Initialize the population position based on the Sobol sequence

In order to improve the global search ability, a deterministic low-difference sequence is used instead of a pseudo-random sequence, also known as the Quasi-Monte Carlo (QMC) method. The sampling direction selected by the QMC method is more reasonable, and the sampling points are more evenly filled into the multi-dimensional hypercube unit, which has higher efficiency and better uniformity when dealing with probability problems. In this paper, the Sobol sequence is used to initialize the population:

Input: Optimal solution range

x_{n} \in [x_{m i n}, x_{m a x}]

Sobol sequence to generate random numbers

K_{n} \in [0, 1]

Output: Initial location of the population

x_{n} \leftarrow x_{m i n} + K_{n} \cdot (x_{m a x} - x_{m i n})

Cauchy variation

The Cauchy distribution function has a smaller peak at the origin but a longer distribution at both ends. The use of the Cauchy variation can generate larger perturbations in the vicinity of the current variation of the sparrow individual, thus making the Cauchy distribution function wider and easier to jump out of the local optimum. Incorporating the Cauchy operator, the effect of the variation at both ends of this distribution function is fully used to optimize the calculation of the global optimal individual, and to adjust the performance of the algorithm to reach the global optimum. After the optimal solution $X_{b e s t}^{t}$ is obtained, the formula for performing the Cauchy variation is as follows:

X_{n e w b e s t}^{t} = X_{b e s t}^{t} + X_{b e s t}^{t} \cdot C a u c h y (0, 1) .

(28)

where Cauchy(0,1) is a standard Cauchy sequence.

Combining bird swarm algorithm to update population flight strategy

In the SSA, the larger the early warning value, the easier it is for sparrows to perceive the existence of natural enemies and move to other areas. When $R_{2} < S T$ , the search space of individual discoverers is decreasing and the population diversity is also declining. Compared with the bird swarm algorithm (BSA), the search strategy of SSA is prone to fall into local optimum.

Figure 3 is the schematic diagram of search strategy of SSA, and Figure 4 is the schematic diagram of search strategy of BSA. The equations in the vertical coordinates of the two figures are the way how algorithms generate a new population. It can be seen that as the number of iterations increases, the search space of SSA gradually decreases while the search space of BSA remains largely unchanged. Although the SSA search strategy allows the algorithm to converge faster, the reduced search space means that the search results will be worse. The search strategy using BSA clearly helps the algorithm to obtain better search results on a global scale. Hence, we introduce the search strategy of BSA into the structure of SSA to improve the SSA's global search capability, and the update method of finder location is as follows:

Figure 3.

Schematic diagram of search strategy of SSA.

Figure 4.

Schematic diagram of search strategy of BSA.

Input:

X_{i, j}

, Safety value ST

Output:

X_{i, j}^{t + 1}

R_{2} = r a n d (1)

2. if

R_{2} < S T

3. then

X_{i, j}^{t + 1} \leftarrow X_{i, j} + X_{i, j} \cdot r a n d n (0, 1)

4. else

X_{i, j}^{t + 1} \leftarrow X_{i, j} + Q \cdot L

5. end if

Where, randn(0,1) indicates to generate a random number with a variance of 0–1 and a mean of 0 between 0 and 1.

Further, the follower in BSA joins the discoverer with a certain probability, which not only ensures the convergence of the algorithm, but also enables the population to jump out of the local optimum and enhances the global search ability of the algorithm. Hence, the position update equation of improved SSA based on BSA is as follows:

Input:

X_{i, j}

X_{w o r s t}

Output:

X_{i, j}^{t + 1}

R_{2} = r a n d (1), Q = r a n d (1)

2. if

i > \frac{n}{2}

3. then

X_{i, j}^{t + 1} \leftarrow Q \cdot \exp (\frac{X_{w o r s t} - X_{i, j}^{t}}{i^{2}})

4. else

X_{i, j}^{t + 1} \leftarrow X_{P}^{t + 1} + | X_{i, j} - X_{P}^{t + 1} | \cdot F L \cdot r a n d n (0, 1)

5. end if

In summary, the total pseudo-code of the designed algorithm is shown in Table 1.

Table 1.

Pseudo-code of ISSA algorithm.

Algorithm: ISSA
Input: Maximum number of iterations G , population number pop, proportion of discoverers PD, proportion of warners SD , alarm value, warning value ST
Output: Optimal value fitness , Optimal location $X_{b e s t}$
1:	Initialize population location based on Sobol sequence
2:	while $(i < G)$
3:	Dividing populations into discovers and followers
4:	for $j \leftarrow 1$ to PDNumber do
5:	$R_{2} = rand (1)$
6:	if $R_{2} < ST$
7:	then $X_{i, j}^{t + 1} \leftarrow X_{i, j} + X_{i, j} \cdot randn (0, 1)$
8:	else $X_{i, j}^{t + 1} \leftarrow X_{i, j} + Q \cdot L$
9:	end if
10:	end for
11:	for $j \leftarrow P D N u m b e r + 1$ to pop do
12:	if $i gt \frac{n}{2}$
13:	then $X_{i, j}^{t + 1} \leftarrow Q \cdot \exp (\frac{X_{worst} - X_{i, j}^{t}}{i^{2}})$
14:	else $X_{i, j}^{t + 1} \leftarrow X_{P}^{t + 1} + \| X_{i, j} - X_{P}^{t + 1} \| \cdot FL \cdot randn (0, 1)$
15:	end if
16:	$e n d f o r$
17:	for $j \leftarrow 1$ to SDNumber do
18:	if $f_{i} > f_{g}$
19:	then $X_{i, j}^{t + 1} \leftarrow X_{best}^{t} + β \cdot \| X_{i, j}^{t} - X_{best}^{t} \|$
20:	else $X_{i, j}^{t + 1} \leftarrow X_{i, j}^{t} + K \cdot [\frac{\| X_{i, j}^{t} - X_{best}^{t} \|}{(f_{i} - f_{w}) + ε}]$
21:	end if
22:	end for
23:	Cauchy Variant Optimization Global Optimal
24:	update $X_{b e s t}$ and fitness
25:	$i \leftarrow i + 1$
26:	end while
27:	return $X_{b e s t}$ and fitness

Numerical verification

In order to guarantee the timely and effective distribution of emergency materials and to form a means of containment for major public health emergencies in time, the distribution of urban emergency support materials needs to make distribution decisions for a large number of demand points in a short period of time. Therefore, there are high requirements for the robustness of the solution algorithm under high latitude problems and the superiority of the solution. To this end, different types of test functions, at different latitudes, are selected for numerical validation, and PSO, GA, and the traditional SSA algorithm are selected for comparison with the designed ISSA algorithm, considering that PSO and GA have been used in the initial research of emergency material distribution. To ensure the validity of the validation results, the mean and standard deviation of each algorithm was taken after 30 runs on each test function. The specific test function information and validation results are shown in Appendix (Table 1) and the iteration graphs are shown in Figure 5.

Figure 5.

Numerical validation iterative curve diagram. (A) Schwefel's Problem 1.2 Function. (B) Six-Hump Camel-Back Function. (C) Quartic Function. (D) Shekel's Function.

From Table 1 in Appendix, it is clear that the designed ISSA algorithm and the PSO, GA, and SSA algorithms are basically optimized on the low-latitude function $f_{4}$ with the optimal value of the function. Added to that, on the high-latitude single-peak function $f_{1}$ , high-latitude multi-peak function $f_{2}$ and noise function $f_{3}$ , the designed ISSA algorithm has the smallest optimization result among the four algorithms, which tends to 0 and is closest to the optimal value of the function. The standard deviation of the designed ISSA algorithm is minimum and close to 0 in 30 runs for the four test functions, thus it has a good robustness. From Figure 5, the iteration curve shows that the designed ISSA algorithm has the smallest number of iterations on the test functions $f_{1}$ and $f_{4}$ which can quickly jump out of the local optimum and has a strong global search capability, but the iteration speed on the test functions $f_{3}$ and $f_{4}$ is better than that of PSO and GA algorithms, and the difference with ISSA algorithm is not obvious. In summary, the designed ISSA algorithm has better optimization effect and robustness than the existing PSO, GA, and SSA algorithms on both high-latitude single-peak and multi-peak functions and it can effectively solve high-latitude complex problems such as urban emergency support material distribution optimization problems.

When assessing the performance of metaheuristic algorithms, it is not sufficient to make judgements based on mean and standard deviation alone. To demonstrate whether the results obtained by the proposed algorithm are statistically significantly different from other algorithms, the ISSA and PSO, GA and SSA algorithms are subjected to the Wilcoxon Rank Sum Test in turn. The p-value in the Wilcoxon Rank Sum Test is 0.05. If the statistical value p < 0.05, then there is a significant difference between the results of the two algorithms, and vice versa, it means that there is no significant difference. We conducted 100 Wilcoxon Rank Sum Tests of ISSA with the other three algorithms and averaged the test values as shown in Table 2.

Table 2.

Statistical test results.

p-value Function	p1(ISSA&PSO)	p2(ISSA&GA)	p3 (ISSA&SSA)
f1	7.066 × 10⁻¹⁸	7.066 × 10⁻¹⁸	7.987 × 10⁻¹⁹
f2	7.066 × 10⁻¹⁸	1.669 × 10⁻¹³	7.066 × 10⁻¹⁸
f3	4.944 × 10⁻⁰⁶	7.066 × 10⁻¹⁸	7.066 × 10⁻¹⁸
f4	2.555 × 10⁻⁰⁶	4.104 × 10⁻⁰⁶	1.396 × 10⁻⁰⁴

From Table 2 it can be seen that the p-value of the designed ISSA algorithm compared to PSO, GA, and the classical SSA on the four tested functions is less than 0.05, indicating a significant advantage of ISSA's optimization capability over the above algorithms.

Simulation experiments

Standard case test

To verify the optimization of ISSA under different sized networks, three standard test sets (Oliver30, eil76, and eil101) in Travelling Salesman Problem were chosen as the locations of the test community points. And the scheduling centers’ locations were random locations within the network. The fuzzy demand for all test community points is a random value and there is no limit to the provisioning capacity of the dispatch centers. In addition, the PSO and simulated annealing (SA) algorithms were selected for cross-sectional comparison and the results of the runs are shown below. Figure 6 shows the results of the runs in Oliver30. Figure 7 shows the results of the runs in eil76. Figure 8 shows the results of the runs in eil101.

Figure 6.

Results of ISSA, PSO, and SA runs under Oliver30. (A) Algorithm iteration curve. (B) Vehicle route map for c1. (C) Vehicle route map for c2. (D) Vehicle route map for c3.

Figure 7.

Results of ISSA, PSO, and SA runs under eil76. (A) Algorithm iteration curve. (B) Vehicle route map for c1. (C) Vehicle route map for c2. (D) Vehicle route map for c3. (A) Algorithm iteration curve. (B) Vehicle route map for c1. (C) Vehicle route map for c2. (D) Vehicle route map for c3.

Figure 8.

Results of ISSA, PSO, and SA runs under eil101. (A) Algorithm iteration curve. (B) Vehicle route map for c1. (C) Vehicle route map for c2. (D) Vehicle route map for c3.

In Figures 6 to 8, the blue dot is the location of the dispatch center and the red circle is the location of the community point. The results in Figures 6 to 8 and Table 3 show that ISSA has faster and more accurate convergence than PSO and SA on three different sizes of networks, Oliver30, eil76, and eil101. On Oliver30, ISSA improved by 1.14% and 4.52% compared to PSO and SA respectively, on eil76, ISSA improved by 5.08% and 2.51% compared to PSO and SA, respectively. And on eil101, ISSA improved by 3.86% and 2.55% compared to PSO and SA, respectively. ISSA has the fastest convergence speed on Oliver30 and eil101. While on eil76, although the convergence speed is not dominant, it can be seen that ISSA has a better global search ability in the later stages, and is able to go beyond the local optimum to find a better solution, which again supports the improvement in the global search ability of the introduced improvement strategy.

Table 3.

Results of specific runs with different test sets.

Value Algorithm	Objective function value
Value Algorithm	Oliver30	eil76	eil101
PSO	3695	7815	9985
SA	3826	7609	9851
ISSA	3653	7418	9600

Data preparation

To verify the model and the algorithm established in this paper, Shanghai Pudong district is used as the background of real distribution tasks. Three hospitals are selected as emergency material transfer centers and 30 communities are collected as demand points. Among them, the fuzzy demands of the 30 community points are shown in Appendix (Table 2). The names and numbers of the three selected hospitals are shown in Table 4. The expected values of the various types of material demands after defuzzification used to generate the initialized distribution path are shown in Table 5. The distance matrices of the three hospitals and 30 communities collected are shown in Table 6. Finally, the parameters of different models of vehicles are shown in Table 7. Figure 9 shows a diagram of the real locations of the 30 community sites and the three dispatch centers selected.

Figure 9.

Visualization of the location of 30 community points and 3 dispatch centers.

Table 4.

Name and number of the selected hospital.

Name	Number
Shuguang Hospital East Hospital, Shanghai University of Traditional Chinese Medicine	c1
Shanghai Pudong Hospital	c2
Shanghai Children's Medical Center	c3

Table 5.

Expected value of demand for various types of materials in 30 communities.

Community	a1	a2	a3	Community	a1	a2	a3
1	1.41	1.11	1.47	16	1.47	0.00	0.00
2	0.73	1.26	0.00	17	1.03	1.28	1.28
3	3	1.28	0.57	18	0.00	0.43	3.01
4	0.43	1.42	1.06	19	0.66	1.16	1.42
5	1.43	0.35	1.47	20	1.86	1.42	1.06
6	1.03	0.00	2.41	21	1.47	1.43	0.43
7	1.35	1.41	0.66	22	1.11	0.35	0.35
8	2.35	1.61	1.47	23	1.47	1.47	0.00
9	2.16	0.73	0.35	24	0.73	0.00	1.35
10	0.73	0.43	1.28	25	1.34	0.66	0.66
11	0.57	1.16	1.47	26	0.00	1.35	1.41
12	0.35	0.35	1.35	27	3.01	1.28	1.28
13	1.41	1.11	1.41	28	1.28	0.35	2.35
14	0.57	1.47	1.47	29	1.16	0.73	0.35
15	1.42	0.73	1.34	30	1.47	1.11	2.16

Table 6.

Distance matrix of the simulation experiment (km).

	c1	c2	c3	…	28	29	30
c1	0	19.69	7.12	…	15.51	10.67	21.59
c2	19.69	0	26.17	…	9.60	28.87	5.91
c3	7.12	26.17	0	…	22.60	10.20	27.20
1	9.86	16.22	16.61	…	7.95	14.23	20.54
2	21.94	2.19	0.29	…	11.11	30.97	6.00
3	3.80	16.99	10.80	…	11.83	12.01	19.65
…	…	…	…	…	…	…	…
28	15.51	9.60	22.60	…	0	21.92	15.01
29	10.67	28.87	10.20	…	21.92	0	31.85
30	21.59	5.91	27.20	…	15.01	31.85	0

Table 7.

Parameters of different models of vehicles.

Vehicle type T	Lower load limit (t)	Upper load limit (t)
t1	11	25
t2	14	25
t3	13	23

Simulation experiment

Since the demands of each community are fuzzy values that cannot be directly applied to the algorithm, the actual demand needs to be determined first. The upper and lower limits of the fuzzy demand of each community are randomly selected, using a random function, and rounded to be treated as the actual demand of each community, and the specific values are shown in Table 8.

Table 8.

Actual demands of the communities in the simulation experiment (t).

Community	a1	a2	a3	Community	a1	a2	a3
1	1.73	1.40	1.87	16	1.99	0	0
2	0.86	1.82	0	17	0.68	1.41	1.00
3	3.96	1.09	0.73	18	0	0.77	2.02
4	0.53	1.60	0.77	19	0.83	1.27	1.63
5	1.61	0.56	1.78	20	1.09	1.65	0.56
6	0.82	0	2.09	21	1.96	1.72	0.57
7	1.80	1.51	0.90	22	1.04	0.23	0.43
8	2.61	1.12	1.26	23	1.84	1.96	0.00
9	2.11	0.74	0.23	24	0.55	0	1.76
10	0.66	0.50	1.43	25	1.54	0.58	0.95
11	0.59	1.11	1.86	26	0	1.89	1.23
12	0.56	0.37	1.41	27	3.51	1.85	1.40
13	1.74	1.04	1.22	28	1.30	0.31	2.36
14	0.52	1.56	1.98	29	1.45	0.68	0.29
15	1.78	0.54	1.01	30	1.68	0.96	2.19

Simulation experiments are carried out based on the above relevant data. The values of the relevant parameters in the designed ISSA algorithm are shown in Table 9. The parameters in the optimized model distribution are taken as follows: $α = 1.8$ , $β = 3$ , $v = 50 km / h$ , $w_{t} = 0.5$ , $w_{s t} = 0.3.$ The credibility of each type of material at each distribution community point is calculated after generating the initialized path results using the expected values to iterate the path results in accordance with the actual demand. The distribution can be completed when the credibility is higher than the preferred value, and vice versa. Further, to verify the effectiveness of the designed algorithm, the initial preference value $ω$ is 0.1. In order to measure the impact of different weightings $λ$ on the optimization results, weights $λ$ of 0.3, 0.5, 0.7, and 0.9 were selected for testing. Figure 10 shows the iterative graphs of the five algorithms PSO, SA, GA, SSA, and ISSA when $λ = 0.3$ . Figure 11 shows the iterative graphs of the five algorithms PSO, SA, GA, SSA, and ISSA when $λ = 0.5$ . Figure 12 shows the iterative graphs of the five algorithms PSO, SA, GA, SSA, and ISSA when $λ = 0.7$ . Figure 13 shows the iterative graphs of the five algorithms PSO, SA, GA, SSA, and ISSA when $λ = 0.9$ . Table 10 shows average results of the five algorithms PSO, SA, GA, SSA, and ISSA for different $λ$ . Table 11 shows specific results for any one of the five algorithms PSO, SA, GA, SSA, ISSA when $λ = 0.5$ . And the results of a specific scheduling, generated by the ISSA algorithm, are shown in Table 12, and the calculation results of the specific confidence in this scheduling are shown in Table 13. Finally, Figure 14 shows a diagram of scheduling results planning based on Table 12. The software used in this paper is MATLAB R2020a, the operating system is Windows 10, the computer memory is 32G, and the CPU is Intel i7-10870H.

Figure 10.

Iteration curves for PSO, SA, GA, SSA, and ISSA when $λ = 0.3$ .

Figure 11.

Iteration curves for PSO, SA, GA, SSA, and ISSA when $λ = 0.5$ .

Figure 12.

Iteration curves for PSO, SA, GA, SSA, and ISSA when $λ = 0.7$ .

Figure 13.

Iteration curves for PSO, SA, GA, SSA, and ISSA when $λ = 0.9$ .

Figure 14.

Scheduling results planning schematic.

Table 9.

Parameters of ISSA algorithm.

Parameter	Meaning	Value
$i t e r_{m a x}$	Number of iterations	200
N	Population size	60
lm	Control position change parameter	2
ST	Early warning value	0.6
PD	Proportion of discoverers	0.7
SD	Proportion of early warner	0.2
$θ$	Adjustment parameters	0.05

Table 10.

Average results of the five algorithms PSO, SA, GA, SSA, and ISSA for different $λ$ .

Value Algorithm	Objective function value
Value Algorithm	$λ = 0.9$	$λ = 0.7$	$λ = 0.5$	$λ = 0.3$
PSO	3185	2913	2792	2547
GA	3217	3061	2749	2493
SA	3155	2984	2673	2317
SSA	3204	2956	2719	2425
ISSA	3033	2803	2556	2290

Table 11.

Specific results for any one of the five algorithms PSO, SA, GA, SSA, ISSA when $λ = 0.5$ .

Algorithm Index	PSO	GA	SA	SSA	ISSA
Vehicle cost SC	3153.54	3139.92	3009.32	2969.40	2917.90
Time cost TC	2495.70	2354.42	2420.04	2360.36	2074.16
Total time (h)	32.58	31.57	32.14	31.82	29.07
Total distance (km)	482.85	457.23	470.72	440.18	422.08
Objective function	2824.62	2747.17	2714.68	2684.68	2573.27

Table 12.

Specific distribution results generated by ISSA algorithm.

No.	Center	Route	Vehicle type	Departure time	Return time
1	c1	c1→1→21→12→c1	t1	8:00	11:38
2	c1	c1→24→28→30→c1	t1	8:00	11:23
3	c2	c1→3→25→23→c1	t3	8:00	10:09
4		c1→6→5→29→c1	t3	8:00	11:05
5		c2→19→7→c2	t1	8:00	11:26
6	c3	c2→10→22→11→4→c2	t2	8:00	11:23
7		c3→14→17→18→15→c3	t1	8:00	11:50
8		c3→2→9→16→c3	t2	8:00	10:52
9		c3→20→26→c3	t2	8:00	9:30
10		c3→8→13→27→c3	t3	8:00	10:59

Table 13.

Confidence calculation results generated by ISSA algorithm.

Community	a1	a2	a3	Community	a1	a2	a3
1	1	1	1	16	1	1	1
2	1	1	1	17	1	1	1
3	1	1	1	18	1	1	1
4	1	1	1	19	1	1	1
5	1	1	1	20	1	1	1
6	1	1	0.256	21	1	1	1
7	1	1	0.250	22	1	1	1
8	1	1	1	23	1	1	1
9	1	1	1	24	1	1	1
10	1	1	0.500	25	1	1	1
11	1	1	1	26	1	1	1
12	1	1	1	27	1	1	1
13	1	1	1	28	1	1	1
14	1	1	1	29	1	1	1
15	1	1	1	30	1	1	1

Analysis of results

From Figures 10 to 13, together with Table 10, it can be seen that the designed ISSA algorithm is the best among the five algorithms for different weights, and as the weight λ decreases, the objective function value also gradually decreases, indicating that SC has a greater influence on the objective function than TC. As can be seen from Table 10, when λ = 0.9, the designed ISSA algorithm reduces the objective function value by 5.01%, 6.07%, 4.02%, and 5.64%, respectively, compared to PSO, GA, SA, and SSA algorithms, with an average reduction of 5.18%; when λ = 0.7, it reduces the objective function value by 3.92%, 9.20%, 6.46%, and 5.46%. When λ = 0.5, the reductions were 9.23%, 7.55%, 4.58%, and 6.38%, respectively, with an average reduction of 6.93%; when λ = 0.3, the reductions were 11.22%, 8.86%, 1.18%, and 5.90%, respectively, with an average reduction of 6.79%. This shows the robustness and optimization of the ISSA algorithm.

As ISSA has the best optimization capability when λ = 0.5, the results at this point were chosen for analysis. Combining the specific values in Table 11, it can be seen that:

- the distribution vehicle cost SC, generated by ISSA algorithm, is reduced by 7.47%, 7.07%, 3.04%, and 1.73%, respectively, compared to PSO, GA, SA, and SSA, with an average reduction of 4.83%;

- the distribution time cost TC, generated by ISSA algorithm, is reduced by 16.89%, 11.90%, 14.29%, and 12.13%, respectively, compared to PSO, GA, SA, and SSA, with an average reduction of 13.80%;

- the total distribution time, generated by ISSA algorithm, is reduced by 10.77%, 7.92%, 9.55%, and 8.64%, respectively, compared to PSO, GA, SA, and SSA, with an average reduction of 9.22%;

- the total distribution distance, generated by the ISSA algorithm, is reduced by 12.59%, 7.69%, 10.33%, and 4.11%, respectively, compared to PSO, GA, SA, and SSA, with an average reduction of 8.68%;

- the total distribution cost, generated by ISSA algorithm, is reduced by 8.90%, 6.33%, 5.21%, and 4.15%, respectively, compared to PSO, GA, SA, and SSA, with an average reduction of 6.15%.

In summary, the designed ISSA algorithm can be superior to the existing algorithms in terms of distribution vehicle cost SC, distribution time cost TC, total distribution time, and total distribution distance, and can effectively solve the urban emergency support material distribution optimization problem.

From the specific scheduling results in Table 12, transfer centers c1, c2, and c3 are responsible for 2, 3, and 5 distribution tasks, respectively, for a total of 10 distribution tasks. Among them, a maximum of four deliveries are affected per vehicles. As for the return time, the earliest return to the transfer center is 9:30, the latest return to the transfer center is 11:50, and the average time spent is about 3 h.

As can be seen from Table 13, 27 communities had a distribution confidence level of one for all three types of supplies, indicating that the needs of these communities were fully met. The remaining seven communities have seven distribution reliabilities above the preference value but not one, indicating that the distribution vehicles are allowed to distribute the material, but the demand for this type of material cannot be fully satisfied, and the overall material satisfaction rate is 96.67%. In summary, it can be seen that the ISSA algorithm, designed in this paper, benefits from the introduction of improved strategies such as the Cauchy variational operator and the opposition-based learning strategy, which can better reduce the overall total cost of distribution compared with the general algorithm and provide a boost to realize the efficient dispatching of emergency supplies.

Analysis of the model

In order to help distributors to develop reasonable decision-making distribution strategies, this paper analyzes the effects of preference value setting on the number of distribution vehicles, distribution time cost, and total cost. Based on the simulation experimental environment, only the ISSA algorithm delivers an optimized distribution where the preference values are taken from 0 to 1, and the average value is taken after 50 times optimization using ISSA algorithm at each preference value. The influence curves of preference values on the number of dispatched vehicles, time cost, and total cost are obtained as shown in Figures 15 to 17, respectively. In addition, the results of time cost and total cost per vehicle trip are shown in Table 14.

Figure 15.

Effect of preference value on the number of vehicles.

Figure 16.

Effect of preference value on time cost TC.

Figure 17.

Effect of preference value on total cost.

Table 14.

Relationship between preference values and average cost.

Preference value	Vehicle number	Average vehicle cost	Average time cost
0.1	9	224.000	208.900
0.2	10	228.200	216.400
0.3	10	215.455	200.455
0.4	11	219.182	204.364
0.5	11	220.273	211.455
0.6	11	216.828	209.727
0.7	11	204.081	196.917
0.8	12	216.083	209.500
0.9	13	202.615	193.923
1.0	14	197.786	187.357

From Figures 15 to 17, it can be seen that when the preference value is increasing, the values of the distribution vehicles, the time cost, and the total cost, in general, are also increasing. This is because, in order for distribution trucks to finish the distribution with the same demand at the community point, the requirements to do so grow more onerous as the value of preference increases. Therefore, more vehicles are needed for distribution, which leads to the increase of the total distribution time cost and the total cost. The time cost decreases when the value of $ω$ is 0.6, and the total cost decreases when the value of $ω$ is 0.5 and 0.6. This is due to the fact that ISSA is, by nature, a population-based metaheuristic algorithm with a randomized search mechanism that leads to uncertainty in its optimization results.

After discarding these outliers, Table 14 shows that the minimum average time cost is 187.357 when the value of $ω$ is 1, and the minimum average total cost is 197.786 when the value of $ω$ is 1. This is because the use of more vehicles inevitably means that the average cost per vehicle will be lower, given the abundance of vehicle resources. In summary, it can be seen that the decision maker can set a larger preference value to ensure that the demand of each community is satisfied as much as possible when the number of distribution vehicles is unlimited. Added to that, when the outbreak of a major public health emergency affects the distribution resources, the decision maker can set a smaller preference value to guarantee that the minimum demand of as many communities as possible is met. In summary, decision makers need to choose preference values according to their objectives in order to make the best use of distribution resources.

Conclusion

In this paper, we study the problem of distribution of urban emergency support materials under major public health emergencies. Considering the fuzzy demand in a such emergency case as the coronavirus epidemic, it is introduced into the distribution optimization model, and the distribution credibility is used to decide whether the distribution vehicle continues to make the distribution. Based on this event, a distribution optimization model, with a minimum total cost as the optimization objective, is constructed for urban emergency support supplies. Added to that, an ISSA, introducing the Cauchy variational operator, the opposition-based learning strategy and the adaptive strategy, is designed for solving the model. The comparison with existing cutting-edge algorithms shows that the improved strategies can make the algorithm more robust in solving complex high-latitude problems, can effectively enhance the ability of the algorithm to jump out of the local optimum and can improve the global search capability.

In addition, this paper analyzes the influence of preference values on urban emergency support material distribution and provides reasonable and effective decision-making strategies for taking actions. The urban emergency support material distribution optimization model, constructed in this paper, is effectively applied to the UESMDP, and the model can be further extended to urban cold chain distribution, medical material distribution, and other fields involving material vehicle distribution, providing effective research value and reference direction for the VRP problem. It is worth noting that the model developed in this paper is mainly used for emergency response under major public health emergencies such as the new coronavirus epidemic and it is idealized for road traffic conditions. In addition to SSA, other intelligent algorithms such as monarch butterfly optimization,²⁹ earthworm optimization algorithm,³⁰ elephant herding optimization,³¹ moth search algorithm,³² Slime mould algorithm,³³ hunger games search,³⁴ Runge Kutta optimizer,³⁵ colony predation algorithm,³⁶ and Harris hawks optimization³⁷ have been applied to complex optimization problems such as engineering optimization, resource allocation, and vehicle planning. Therefore, in future research, we will further explore the feasibility of other intelligent algorithms in solving emergency material dispatching problems involving fuzzy demands and the distribution task planning involving time-varying speed and complex traffic environment will also be further considered.

Footnotes

Acknowledgements

This work was supported in part by the Science and Technology Project of China Southern Power Grid Co., Ltd. (YNKJXM20220174).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Nan Pan

Jianing Cao

Jingcheng Zhang

Yuhang Han

Author biographies

Nan Pan is an associate professor. He received his PhD in mechanical engineering. His research interests include swarm intelligence and deep learning.

Xiao Ye is a master's degree candidate in Transportation. Her research interests include intelligent transportation.

Jianing Cao is a bachelor's degree candidate in mechanical engineering. His area of research is swarm intelligence and machine learning.

Jingcheng Zhang is a bachelor's degree candidate in data science and big data technology. His research interests include swarm intelligence, data mining, and in-depth learning.

Yuhang Han is a bachelor's degree candidate in mechanical engineering. His main research direction is swarm intelligence and deep learning.

Zhaolei He holds a bachelor's degree in engineering. His research area is electric energy metering, storage logistics, and intelligent sensing.

Appendix

Table 2.

Test function information and verification results.

Test function				Result
Symbols	Name	Range of values	Optimal solution	Algorithm	Mean value	Standard deviation
$f_{1}$ (30 latitude)	Schwefel's Problem 1.2 Function	$[- 100, 100]$	0	PSO	348.4255	214.249
				GA	2.030 × 10⁻⁹	6.419 × 10⁻⁹
				SSA	6.065 × 10⁻⁵³	1.918 × 10⁻⁵²
				ISSA	0	0
$f_{2}$ (30 latitude)	Six-Hump Camel-Back Function	$[- 500, 500]$	−12569.5	PSO	−1.0316	2.318 × 10⁻⁵
				GA	−1.0316	6.6038 × 10⁻⁷
				SSA	−1.0316	4.2168 × 10⁻⁸
				ISSA	−1.0316	1.1253 × 10⁻⁸
$f_{3}$ (30 latitude)	Quartic Function	$[- 1.28, 1.28]$	0	PSO	0.052	0.028
				GA	1.843 × 10⁻³	1.234
				SSA	1.204 × 10⁻³	7.092 × 10⁻⁴
				ISSA	2.999 × 10⁻⁴	1.492 × 10⁻⁴
$f_{4}$ (7 latitude)	Shekel's Function	$[0, 10]$	−10.536	PSO	−10.027	0.933
				GA	−8.904	2.998
				SSA	−9.333	2.237
				ISSA	−10.377	0.037

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