A Dual-Uncertainty Multi-Scenario Multi-Period Facility Location Model for Post-Disaster Humanitarian Logistics

Abstract

The frequent occurrence of natural disasters creates a symmetry-breaking scenario between pre-disaster planning and post-disaster rescue operations, such as post-disaster supply–demand mismatches for materials and the risk of potential facility failures. Thus, we propose a dual-uncertainty multi-scenario multi-period facility location allocation model for humanitarian rescue. The model employs two polyhedral uncertainty sets to represent facility failure risks and demand uncertainty at disaster points. Moreover, by constructing diverse disaster scenarios, it simulates material distribution schemes across different relief periods, enhancing its realism. Given that the model integrates three subproblems—facility location, supply–demand matching analysis, and emergency material allocation—we design a hybrid algorithm (DCSA-MA) that combines the discrete crow search algorithm (DCSA) and the material allocation (MA) method for its solution. Experimental results demonstrate that the model maintains a relatively high material satisfaction rate even under significant demand fluctuations. The number of facility failures has a direct bearing on emergency rescue effectiveness. The DCSA-MA method achieves a superior material satisfaction rate compared to other algorithms across various disaster scenarios and multiple rescue periods. Furthermore, DCSA-MA outperforms other algorithms in terms of solution quality, convergence, computational time, and stability. These findings indicate that DCSA-MA is an effective and highly stable approach.

1. Introduction

Frequent natural disasters pose a significant threat to human life, property security, and social stability [1]. The complexity of disaster response is further exacerbated by the involvement of multiple stakeholders, including governmental and non-governmental organizations, along with the uncertainty of facility failures and the acute scarcity of emergency materials [2,3,4,5,6]. These factors not only heighten the challenges of emergency response but can also lead to the failure of humanitarian relief efforts. Consequently, developing an efficient humanitarian rescue plan is imperative.

Due to various types of uncertainties associated with both facilities and disaster points, numerous scholars have focused on addressing these uncertainties. Toro-Díaz [7] developed an integer planning model and hypercube synthesis model to handle uncertainties in rescue time. Ramshani [8] proposed a two-stage non-capacity facility location model with interruption uncertainty and devised a heuristic algorithm to address it. Li [9] introduced a facility location allocation model where waste material and transport costs were considered as two factors of uncertainty. Deng [10] explored emergency facility location allocation within time windows under demand uncertainty and applied the branch and bound algorithm to solve it. Ryu [11] proposed a robust single-source capacity facility location problem with customer demand uncertainty and employed a branch pricing algorithm to solve it. Gülpınar [12] designed a stochastic facility location model for customer demand uncertainty. Li [13] examined facility location reliability in the context of facility interruption probabilities caused by natural or man-made disasters. Wen [14] developed a facility location allocation model based on customer demand uncertainty and employed a hybrid intelligent algorithm. Abounacer [15] considered a multi-objective facility location transport problem under capacity constraints at disaster points and limited quantities of multiple rescue materials with the objective of minimizing total transport time, the number of disaster points, and unmet demand for materials in disaster points. Salman [16] developed a model to simulate the spatial effects of disasters on network links, accounting for the randomness in failure and dependency. The model was designed to optimize the maximization of demand and employed the tabu search algorithm for its solution. Sun [17] proposed a robust scenario-based approach for facility interruption, safeguarding solutions from temporary medical center interruptions. Afify [18] researched two reliable facility location problems in the presence of facility interruption. Lu [19] studied the locations of capacity-constrained facilities under conditions of facility interruption, where the risk of interruption not only partially or completely reduces facility capacity but also affects customer demand patterns. Rahmani [20] portrayed the state of facility interruption in a post-earthquake scenario and developed a material chain model for humanitarian rescue. An [21] enumerated facility interruption scenarios and developed a mixed-integer nonlinear programming model, which simultaneously considered the risks of facility interruption, traffic congestion, and queuing delays to develop pre-disaster facility location plans and post-disaster evacuation plans. Zeng [22] investigated the problem of logistics center location allocation in the presence of demand uncertainty and applied a box uncertainty set to depict demand uncertainty to minimize the total cost in the worst-case scenario. Mostajabdaveh [23] developed a stochastic planning model based on scenario studies considering demand uncertainty and designed a genetic algorithm to solve it. Cheng [24] identified sources of demand uncertainty in facility location, noting that facilities face demand-side, material-side, and intermediate-side uncertainty. Balcik [25] addressed uncertainty in emergency materials and considered budget, capacity, and response time constraints by solving the problem through a scenario-based approach that maximizes the total expected demand covered by established distribution centers. Ozbay [26] used a finite set to portray scenarios with uncertain demand and gave the probability of each scenario, minimizing the expected weights of established shelters. Zokaee [27] minimized the total cost of the rescue chain in response to demand and supplier capacity uncertainty, maximizing meet in disaster points through shortage penalty costs. Elçi [28] proposed a stochastic disaster rescue network scheme by considering the uncertain factors that exist under the conditions of a post-disaster demand and transportation network, and they developed a two-stage average risk stochastic planning model with opportunity constraints. Miao et al. [29] pioneered the application of Deep Reinforcement Learning (DRL)—Attention-Dynamic Network (ADNet) to the $p_k$-median Dynamic Location Problem. Huang et al. [30] created a novel model-free demand response management scheme for industrial facilities by developing a deep reinforcement learning approach, which optimizes energy consumption schedules to reduce costs without affecting production. Teusch et al. [31] proposed a reinforcement learning-based optimization framework for strategically planning geo-fenced micro-mobility facilities (MMSFs), achieving significant improvements in parking demand.

As shown in

Table 1. Model comparative analysis.

Single Type Uncertainty	Multiple Types of Uncertainties
Rescue time [7]	Facility supply and unit transportation cost [9,10]
Disruption [8,13,15,16,17,18,19,20,21]	Demand and supplier capacity [27]
Demand [11,12,14,22,23,24,25,26]	Demand and transportation [28]

, existing studies predominantly focus on single uncertainty with limited consideration for both facility failure and demand uncertainty, which naturally cannot solve the problem of symmetry deficiency between pre-disaster planning and post-disaster rescue operations. Therefore, we design a dual-uncertainty multi-scenario multi-period facility location allocation model that incorporates facility failure and demand uncertainty. To solve this model, we design a hybrid strategy combining the discrete crow search algorithm (DCSA) with the materials allocation (MA) method, namely the DCSA-MA method. The contributions of this paper are as follows:

(1)

Considering the diversity and periodicity of disaster, our model overcomes the limitations of a single disaster scenario by incorporating various disaster scenarios to simulate emergency material allocation schemes. Moreover, the model deeply considers the periodic characteristics of disasters by evaluating facility inventories and disaster point demands across different rescue periods, enabling effective material planning. Reflecting disaster uncertainty, the model employs two polyhedral uncertainty sets to characterize facility failure and disaster point demand uncertainty.

(2)

Since the model is a mixed-integer programming problem, solutions in the facility location and supply–demand matching analysis phase are binary solutions with values of either 0 or 1. Thus, we enhance the standard crow search algorithm (CSA) with XOR/AND logic gates, swap, reverse, and mutation neighborhood operations for solution refinement, proposing a novel discrete crow search algorithm (DCSA).

(3)

Given the integer solution characteristics in the emergency material allocation phase, we design the MA method. This method efficiently and equitably allocates materials based on disaster point demand, facility inventory, distance, and facility service capacities, ensuring timely and effective emergency response.

(4)

Finally, we combine the DCSA with the MA method to solve the model. Specifically, the DCSA first solves the facility location and supply–demand matching analysis, and then the results from the DCSA are passed to the MA method for material allocation. This hybrid strategy effectively combines the strengths of both methods, enhancing overall solution efficiency and quality.

The rest of this paper is organized as follows. Section 2 performs problem description and modeling. Section 3 proposes a DCSA-MA algorithm. Section 4 conducts experimental testing and analysis. Section 5 summarizes this work.

2. Problem Description and Modeling

2.1. Problem Description

When a disaster strikes unexpectedly, the effective fulfillment of emergency material demands across N disaster points within a specific region becomes paramount. To address this problem, decision-makers select M emergency facilities to establish an efficient and adaptable emergency material deployment network denoted as $G(V,E)$, where V comprises both N disaster points and M facilities, and E signifies theses edges. Moreover, the network incorporates W disaster scenarios, each with an associated probability of occurrence. Within each scenario, decision-makers delineate P rescue periods. Each disaster point presents varying material requirements across these rescue periods. Thus, the primary optimization goal of this problem is to identify an optimal facility location allocation scheme that minimizes costs.The following assumptions are as follows:

(1)

Unmet demand at any disaster point incurs a penalty cost.

(2)

All rescue materials can be instantaneously transported from the facility to the disaster point within the current rescue period without delay.

(3)

The model does not account for mutual assistance or cooperation between different facilities.

(4)

Emergency material demand at each disaster point is uncertain.

(5)

A failed facility cannot serve any disaster point in its associated scenario.

2.2. Parameters

Before establishing the model, it is essential to define the relevant parameters and decision variables to ensure both the accuracy and operability of the model. The specific definitions are shown in

Table 2. Parameters and decision variables of the model.

Parameters and Decision Variables
(1) Sets
$\Omega =\{1,2,\dots ,W\}$: Set of disaster scenarios;
$T=\{1,2,\dots ,P\}$: Set of rescue period;
$I=\{1,2,\dots ,M\}$: Set of facilities;
$J=\{1,2,\dots ,N\}$: Set of disaster points;
(2) Parameters
$f{c}_m$: Cost of opening the facility m;
$t{c}_p$: Transport costs under rescue period p;
$p{c}_p$: Penalty cost of unmet demand at the disaster point under rescue period p;
$v_m^{wp}$: Inventory of facility m under a rescue period p in disaster scenario w;
$(f{x}_m,f{y}_m)$: Coordinates of facility m;
$(d{x}_n,d{y}_n$): Coordinates of the disaster point n;
$d_{mn}$: Distance between facility m and disaster point n;
$p{r}_w$: Probability of occurrence of disaster scenario w;
$q_n^{wp}$: Material requirements at disaster point n under a rescue period p for disaster scenario w;
${\alpha }_m^{wp}$: Material integrity rate for facility m under a rescue period p for disaster scenario w;
${\beta }_{mn}^{wp}$: Material integrity rate;
(3) Decision parameters
$y_m$: 0–1 variable, 1 if the facility m is opened, 0 otherwise;
$x_{mn}$: 0–1 variable, 1 if facility m serves disaster point n, 0 otherwise;
$z_{mn}^{wp}$: Distribution of materials.

.

2.3. MSMPUFLA Model

Based on the parameters and decision parameters defined above, we develop a dual-uncertainty multi-scenario multi-period facility location allocation model.

$$\begin{array}{c}\hfill f_1=\sum _{m\in I}f{c}_m{y}_m\end{array}$$

(1)

$$\begin{array}{c}\hfill f_2=\sum _{w\in \Omega }\sum _{p\in T}\sum _{m\in I}\sum _{n\in J}p{r}_{w}t{c}_p{d}_{mn}{z}_{mn}^{wp}\end{array}$$

(2)

$$\begin{array}{c}\hfill f_3=\sum _{w\in \Omega }\sum _{p\in T}\sum _{m\in I}\sum _{n\in J}p{r}_{w}p{c}_p(q_n^{wp}-z_{mn}^{wp})\end{array}$$

(3)

$$\begin{array}{c}\hfill F=min(f_1+f_2+f_3)\end{array}$$

(4)

$$\begin{array}{c}\hfill s.t.\sum _{m\in I}{y}_m\ge 0\end{array}$$

(5)

$$\begin{array}{c}\hfill x_{mn}\le y_m,\forall m\in I,n\in J\end{array}$$

(6)

$$\begin{array}{c}\hfill \sum _{n\in J}{x}_{mn}=1,\forall m\in I\end{array}$$

(7)

$$\begin{array}{c}\hfill z_{mn}^{wp}{x}_{mn}>0,\forall m\in I,n\in J,p\in T,w\in \Omega \end{array}$$

(8)

$$\begin{array}{c}\hfill \sum _{n\in J}{z}_{mn}^{wp}{x}_{mn}\le v_m^{wp}{\alpha }_m^{wp},\forall m\in I,p\in T,w\in \Omega \end{array}$$

(9)

$$\begin{array}{c}\hfill \sum _{m\in I}{z}_{mn}^{wp}{\beta }_{mn}^{wp}\ge q_n^{wp},\forall n\in J,p\in T,w\in \Omega \end{array}$$

(10)

$$\begin{array}{c}\hfill x_{mn}\in \{0,1\},\forall m\in I,n\in J\end{array}$$

(11)

$$\begin{array}{c}\hfill y_m\in \{0,1\},\forall m\in I\end{array}$$

(12)

$$\begin{array}{c}\hfill z_{mn}^{wp}\ge 0,\forall m\in I,\forall n\in J,p\in T,w\in \Omega \end{array}$$

(13)

where function (1) represents the facility opening cost, function (2) represents the cost of transporting materials from facilities to disaster points, function (3) is the penalty cost for unmet demand at disaster points, and function (4) aims to minimize the total cost. Constraint (5) indicates that at least one facility is open. Constraint (6) denotes that the disaster point n can be served only when the facility m is open. Constraint (7) indicates that one disaster point can only be served by one facility. Constraint (8) means that one facility can allocate materials to one disaster point only if the disaster point is served by the facility. Constraint (9) denotes that under a rescue period p of a disaster scenario w, the amount of materials allocated by the facility m to the disaster point n should not exceed the maximum inventory. Constraint (10) indicates that the amount of materials allocated by the facility m to the disaster point n should not exceed the amount needed, thus avoiding waste of materials. Constraints (11) and (12) are binary decision variables that take the value of 0 or 1. Constraint (13) denotes that the amount of materials allocated by the facility m to the disaster point n is non-negative under the rescue period p of the disaster scenario w.

2.3.1. Disaster Point Demand Uncertainty Treatment

Due to the suddenness of disasters, decision-makers cannot accurately estimate the material demand in the disaster area during the initial rescue stage, potentially leading to shortages. Thus, we employ polyhedral uncertainty sets to portray the uncertain emergency material demand $\widetilde{q_n^{wp}}$ at disaster point n during rescue period p under scenario w. Specifically, $\widetilde{q_n^{wp}}\in [q_n^{wp}-\overline{q_n^{wp}},q_n^{wp}+\overline{q_n^{wp}}]$, where $q_n^{wp}>0$ is the nominal demand and $\overline{q_n^{wp}}>0$ is the maximum offset of $q_n^{wp}$. Therefore, it incurs additional costs only if the uncertain demand fluctuates positively, so the uncertain demand fluctuation range is between $q_n^{wp}$ and $q_n^{wp}+\overline{q_n^{wp}}$, and the uncertain demand set U is defined as follows:

$$\begin{array}{c}\hfill U=\left\{\widetilde{q_n^{wp}}\left|\widetilde{q_n^{wp}}=q_n^{wp}+g_n^{wp}\overline{q_n^{wp}},\forall w\in \Omega ,p\in T,n\in J\right.\right\}\end{array}$$

(14)

$$\begin{array}{c}\hfill G=\left\{g_n^{wp}\left|\sum _{n\in J}{g}_n^{wp}\le {\Gamma }_g,g_n^{wp}\in [0,1],\forall w\in \Omega ,\forall p\in T\right.\right\}\end{array}$$

(15)

where the parameter ${\Gamma }_g$ is the uncertain budget level for the disaster point demand uncertainty set. It controls the conservative degree of constraints, reflecting the risk preference of the decision-maker. When the value of ${\Gamma }_g$ is larger, the conservative degree of decision-makers is greater. $g_n^{wp}$ in Equation (15) denotes the disturbance ratio of disaster point demand. Since Equations (3) and (10) contain the nominal demand parameter $q_n^{wp}$, we substitute it with the uncertain demand parameter $\widetilde{q_n^{wp}}$.

$$\begin{array}{c}\hfill f_3=\underset{g_n^{wp}\in G}{max}\sum _{w\in \Omega }\sum _{p\in T}\sum _{m\in I}\sum _{n\in J}p{r}_{w}p{c}_p\left(q_n^{wp}+g_n^{wp}\overline{q_n^{wp}}-z_{mn}^{wp}\right)\end{array}$$

(16)

$$\begin{array}{c}\hfill \sum _{m\in I}{z}_{mn}^{wp}{\beta }_{mn}^{wp}\ge \underset{g_n^{wp}\in G}{max}\left(q_n^{wp}+g_n^{wp}\overline{q_n^{wp}}\right),\forall w\in \Omega \end{array}$$

(17)

Model transformation:

Proposition 1.

Constraint (17) is equivalent to Equations (20)–(22).

Proof. 

For a given x, constraint (17) involves uncertain demand $\widetilde{q_n^{wp}}$ at disaster points, and it is easy to know that is equivalent to Equations (18) and (19).

$$\begin{array}{c}\hfill \sum _{m\in I}{z}_{mn}^{wp}{\beta }_{mn}^{wp}\ge q_n^{wp}+\underset{g_n^{wp}\in G}{max}{g}_n^{wp}\overline{q_n^{wp}},\forall w\in \Omega \end{array}$$

(18)

$$\begin{array}{c}\hfill \sum _{n\in J}{g}_n^{wp}\le {\Gamma }_g,0\le g_n^{wp}\le 1,\forall w\in \Omega ,p\in T\end{array}$$

(19)

Since constraint (17) contains a maximization term, we introduce dual variables $\lambda$ and ${\theta }_n^{wp}$. According to the strong duality theorem, the inner maximization problem is transformed into a minimization problem. Thus, constraint (19) is equivalent to Equations (20)–(22), completing the proof.

$$\begin{array}{c}\hfill \sum _{m\in I}{z}_{mn}^{wp}{\beta }_{mn}^{wp}\ge q_n^{wp}+\lambda {\Gamma }_g+\sum _{w\in \Omega }\sum _{p\in T}\sum _{n\in J}{\theta }_n^{wp}\end{array}$$

(20)

$$\begin{array}{c}\hfill \lambda +{\theta }_n^{wp}\ge \overline{q_n^{wp}},\forall w\in \Omega ,\forall p\in T,n\in J\end{array}$$

(21)

$$\begin{array}{c}\hfill \lambda ,{\theta }_n^{wp}\ge 0,\forall w\in \Omega ,\forall p\in T,n\in J\end{array}$$

(22)

   □

Proposition 2.

The robust location problem (23) is equivalent to the mixed integer optimization problem shown in (24)–(26).

$$\begin{array}{c}\hfill f_3=\sum _{w\in \Omega }\sum _{p\in T}\sum _{m\in I}\sum _{n\in J}p{r}_{w}p{c}_p(q_n^{wp}-z_{mn}^{wp})x_{mn}+\eta \end{array}$$

(23)

$$\begin{array}{c}\hfill \mathrm{s}.\mathrm{t}.\left(5\right)-\left(13\right)\end{array}$$

(24)

$$\begin{array}{c}\hfill \pi +{\gamma }_n^{wp}\ge p{r}_{w}p{c}_p\overline{q_n^{wp}}{x}_{mn},\forall w\in \Omega ,p\in T,n\in J\end{array}$$

(25)

$$\begin{array}{c}\hfill t,\pi ,{\gamma }_n^{wp}\ge 0,\forall w\in \Omega ,p\in T,n\in J\end{array}$$

(26)

Proof. 

Since Equation (18) contains the maximization problem over uncertainty set G, it can be reformulated as Equations (27) and (28).

$$\begin{array}{cc}\hfill \underset{g\in G}{max}\pi {\Gamma }_g+\sum _{w\in \Omega }\sum _{p\in T}\sum _{n\in J}{\gamma }_n^{wp}& \hfill \end{array}$$

(27)

$$\begin{array}{c}\hfill \sum _{n\in J}{g}_n^{wp}\le {\Gamma }_g,0\le g_n^{wp}\le 1,\forall w\in \Omega ,p\in T,n\in J\end{array}$$

(28)

According to the duality theorem, we introduce dual variables $\pi$ and ${\gamma }_n^{wp}$ to derive the dual problem seen in (29)–(31). In addition, $\lambda$ and $\pi$ represent the marginal cost of uncertainty. They quantify the incremental penalty cost per unit increase in the uncertainty budget ${\Gamma }_g$. Higher values indicate greater cost sensitivity to demand fluctuations. ${\theta }_n^{wp}$ and ${\gamma }_n^{wp}$ reflect the criticality of disaster points. They measure the cost impact of unmet demand at specific locations (n) during rescue periods (p) in scenarios (w).

$$\begin{array}{c}\hfill min\sum _{w\in \Omega }\sum _{p\in T}\sum _{m\in I}\sum _{n\in J}p{r}_{w}p{c}_p{g}_n^{wp}\overline{q_n^{wp}}{x}_{mn}\end{array}$$

(29)

$$\begin{array}{c}\hfill \pi +{\gamma }_n^{wp}\ge p{r}_{w}p{c}_p\overline{q_n^{wp}}{x}_{mn},\forall w\in \Omega ,p\in T,n\in J\end{array}$$

(30)

$$\begin{array}{c}\hfill \pi ,{\gamma }_n^{wp}\ge 0,\forall w\in \Omega ,p\in T,n\in J\end{array}$$

(31)

Consequently, the inner maximization problem is equivalent to the minimization problem (25)–(28) through the introduction of an auxiliary variable. This completes the proof of proposition 2.    □

2.3.2. Facility Failure Treatment

In the existing literature studies, most of them assume that the probability of facility failure is known. However, disasters are characterized by suddenness, and it is impossible to determine the probability of facility failure. In order to better describe facility failure uncertainty, we introduce a binary state variable $o_m\in \{0,1\}$ for each facility m. The facility failure uncertainty set is formally defined by Equation (32).

$$\begin{array}{c}\hfill O_k=\left\{o\in {\{0,1\}}^{\left|I\right|}:\sum _{m\in I}{o}_m\le k\right\}\end{array}$$

(32)

$$\begin{array}{c}\hfill o_m\in \{0,1\},\forall m\in I\end{array}$$

(33)

where $o_m$ is used to describe the state of facility m, $o_m=1$ signifies facility m has failed and $o_m=0$ signifies facility m is operational. Parameter k denotes the number of failed facilities. To account for potential failures, we employ the failed facility m in constraint (6).

$$x_{mn}^w\le y_m(1-o_m),\forall m\in I,n\in J$$

(34)

where Equation (34) indicates that disaster point n can be served by facility m only if facility m is open and not failed.

3. Algorithm Analysis and Design

Since our model is a mixed-integer programming model, it is not easy to find the optimal emergency material allocation scheme. The solution space will increase exponentially with the scale of the problem, which is an NP-hard problem [32]. Therefore, we develop a DCSA-MA method to solve the model.

3.1. Basic CSA Algorithm

The CSA is an evolutionary algorithm [33], which has stronger optimization ability than traditional optimization algorithms such as PSO and GA [34,35,36]. In addition, CSA is successfully applied to image segmentation [37], manufacturing processes [38], and task scheduling [39]. The CSA is derived from the hiding and stealing behaviors of crows. The features of CSA include the following: (1) crows can memorize the locations of hidden food; (2) crows will follow each other and steal food; (3) crows will try their best to protect food from being stolen by a probability.

Assuming that there exists a D-dimensional search space with K crows, the position of crow i at the t-th iteration represents a candidate solution $po{s}_i^t(i=1,2,\dots ,K$$t=1,2,\dots ,Maxiter)$, where $Maxiter$ is the maximum number of iterations. In addition, each crow can memorize the position of its hiding food. The hiding food position of crow i at the t-th iteration can be denoted as $R_i^t$, which is the best position that crow i has obtained so far. Assume that at iteration t, crow j wants to visit its hiding position $R_j^t$. At this iteration, crow i decides to follow crow j to approach its hiding food position.

(1)

If crow j does not realize that crow i is following it, then crow i will steal the food from crow j. The new position of crow i can be calculated from Equation (35).

$$\begin{array}{c}\hfill po{s}_i^{t+1}=po{s}_i^t+r_i\times F\times (R_j^t-po{s}_i^t)\end{array}$$

(35)

where $r_i$ is a random number between [0, 1] and F is the flight length of crow i.

(2)

If crow j realizes that crow i is following it, crow j will fly to a random position to protect its food. Thus, the position update strategy of crow i is calculated by Equation (36).

$$\begin{array}{c}\hfill po{s}_i^{t+1}=\left\{\begin{array}{c}po{s}_i^t+r_i\times F\times (R_j^t-po{s}_i^t),\mathrm{if}\mathrm{rand}\left(\right)\ge AP\hfill \\ po{s}_{\mathrm{random}},\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

(36)

where $po{s}_{random}$ is the random position and $AP$ is the probability that crow j will notice being tracked.

When crow i uses Equation (36) for a position update, the fitness function is utilized to evaluate the new position of crow i. If the new position $po{s}_i^{t+1}$ is better than $R_i^t$, then $R_i^{t+1}$ is updated as Equation (37).

$$\begin{array}{c}\hfill R_i^{t+1}=\left\{\begin{array}{c}po{s}_i^{t+1},\mathrm{if}fit\left(po{s}_i^{t+1}\right)\mathrm{is}\mathrm{better}\mathrm{than}fit\left(R_i^t\right)\hfill \\ R_i^t,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

(37)

where $R_i^{t+1}$ denotes the hiding food position of crow i at the $(t+1)$-th iteration and $fit(·)$ represents a fitness function. The pseudo-code of CSA is described in Algorithm 1.    

Algorithm 1: The pseudo-code of CSA

3.2. DCSA Algorithm

Our model is a discrete optimization problem in the facility location and supply–demand matching stages, which involves 0–1 decision variables. To address these complexities, we design a hybrid encoding scheme, discrete evolutionary mechanism, and neighborhood operation to further optimize the quality of feasible solutions.

3.2.1. Hybrid Encoding

Our model has three subproblems: facility location, supply–demand matching, and material allocation problems. For a facility location problem, we employ a single-layer binary encoding method of dimension I, where a value of 1 indicates facility m is open, and 0 indicates it is closed. For a supply–demand matching problem, we use a multi-layer binary matrix, which is represented by an $I\times J$ binary matrix. In this matrix, the value at the m-th row and n-th column indicates whether facility m provides service to disaster point n. For a material allocation problem, we employ a multi-layer integer encoding approach, which is represented by an $I\times J$ dimensional matrix. In this matrix, the value located at the m-th row and n-th column of the matrix indicates the emergency material allocated from facilities to disaster points during a specific rescue period in a disaster scenario.

Assuming that in a specific period p under a disaster scenario w, there are five facilities and five disaster points, which are denoted as $I=5$ and $J=5$. As illustrated in Figure 1, the encoding method for the facility location problem allows for the opening of two facilities, satisfying the constraint of at least one open facility. In the supply–demand matching encoding, only facilities 2 and 4 provide services to disaster points. This encoding approach ensures that disaster points can only receive service when the facility is open. Furthermore, the supply–demand matching encoding also satisfies Equation (7). Finally, the emergency material allocation is coded by integers with each disaster point served by a single facility.

3.2.2. Discrete Evolutionary Mechanism

In the many discrete evolutionary algorithms, transfer functions [40,41,42] are employed to solve discrete optimization problems. García [43] stated that the transfer function tends to miss some solutions in the search space, which makes the algorithm solve poorly. Recently, scholars have turned to logic gates (e.g., AND, OR, and XOR) for discretizing continuous evolutionary algorithms due to the inputs and outputs of these logic gates being binary values [44,45,46]. If we consider the OR logic gate, the output of this gate is 1 with a probability of 0.75 and 0 with a probability of 0.25. If we consider the AND logic gate, the output of this gate is 1 with a probability of 0.25 and 0 with a probability of 0.75. In the XOR logic gate, the probabilities are equal, providing a fair solution [47].

In order to provide a robust searching ability for the CSA, we utilize both the AND and XOR logic gates to modify Equation (37). The AND logic gate outputs 1 only when both inputs are 1, which makes it relatively conservative in the search process and maintains the current solution state. In contrast, the XOR logic gate outputs 1 when its inputs differ, promoting search diversity and aiding escape from local optima. By integrating these logic gates, the CSA can strike a balance between conservatism and diversity, as formalized in Equation (38).

$$\begin{array}{c}\hfill po{s}_i^{t+1}=R_i^t\oplus \left(rb\&\left(R_j^t\oplus R_i^t\right)\right)\end{array}$$

(38)

where $rb$ is a random number equal to 0 or 1, ⊕ is the XOR logic gate, and & is the AND logic gate. In this case, we use the optimal hidden food positions of crow i and crow j to perform logical operations, which makes the CSA more focused on local details and discovers the locally optimal solutions in the solution space.

3.2.3. Neighborhood Operation

For our model, when $rand\left(\right)\ge AP$, the CSA randomly generates facility location and supply–demand matching schemes to update the position of the crow. In this case, the random position update method suffers from blindness, which means that the CSA has a small probability of finding the optimal position. To address this problem, we introduce neighborhood operations based on the best hiding food position $R_i^t$ of the crow i, thereby generating new neighborhood solutions. The CSA uses the information from $R_i^t$ during the position update process to find structural and characteristic features of superior solutions, which enhances search efficiency and mitigates blind exploration. Neighborhood operations include swap, reverse, and mutation. By performing these operations on the facility location scheme and material–demand matching scheme, it helps the algorithm avoid blindness in the search process and generate diverse neighborhood solutions. The neighborhood operation is illustrated in Figure 2.

As shown in Figure 2, when $rand\left(\right)\ge AP$, we note the $R_i^t$ as $Parent$. Subsequently, the $Parent$ randomly selects two indexes $i_1$ and $i_2$ and then swaps their values to generate $Child1$. In addition, $Child1$ randomly selects indices $i_3$ and $i_4$ and then reverses the values of these indices to generate $Child2$. If $rand\left(\right)<0.5$, $Child2$ randomly selects some positions for mutation, resulting in the generation of a new solution, which is referred to as $Child3$. Consequently, the position update formula for the DCSA is expressed in Equation (39).

$$\begin{array}{c}\hfill po{s}_i^{t+1}=\left\{\begin{array}{c}{B}_i^t\oplus \left(rb\&\left(B_j^t\oplus B_i^t\right)\right),r_i\ge AP\hfill \\ Child2,r_i<AP\hfill \\ Child3,r_i<AP\mathrm{and}\mathrm{rand}\left(\right)<0.5\hfill \end{array}\right.\end{array}$$

(39)

3.3. Design of Emergency Material Allocation Algorithm

In response to natural disasters or emergencies, an efficient material allocation strategy is critical for promptly meeting affected areas’ needs. This section presents a material allocation algorithm (MA). The MA method integrates multidimensional information including the demand at the disaster point, the inventory of the facility, the distance between the facility and the disaster point, and the service capacity of the facility. The specific steps of the MA method are listed below.

(1)

Initialization data

First, there is initialization of the demand at the disaster point, the inventory of the facility, and the distance between the facility and the disaster point. The facility location, the supply–demand matching scheme, and the initial emergency material allocation schemes of the DCSA are transmitted to the MA algorithm.

(2)

Facility service capacity assessment

For rescue period p in disaster scenario w, we integrate a supply–demand matching scheme with the inventory of the facility to calculate its service capacity $servic{e}_m$, i.e., $servic{e}_m={\sum }_{n\in J}{x}_{mn}·\left(v_m^{wp}-q_n^{wp}\right),\forall w\in \Omega ,p\in T,m\in I$.

(3)

Emergency material allocation strategy

For rescue period p in disaster scenario w, if the $servic{e}_m<0$, it means that the inventory of facility m is insufficient. Based on the supply–demand matching scheme, the facility m replenishes the disaster points of its service in ascending order according to the distance, i.e., emergency materials demands from closer disaster points are met first. The sorted list of disaster points is traversed to assess the demand $q_n^{wp}$ of disaster point n and the inventory $v_m^{wp}$ of facility m. If $q_n^{wp}\le v_m^{wp}$, then $z_{mn}^{wp}=q_n^{wp}$, and we update the inventory of facility m, i.e., $v_m^{wp}=v_m^{wp}-z_{mn}^{wp}$. Otherwise, $z_{mn}^{wp}=v_m^{wp}$ and $v_m^{wp}=0$.

For the rescue period p in disaster scenario w, if the $servic{e}_m\ge 0$, it means that the inventory of facility m is sufficient. In order to prevent the waste of materials, we implement redundancy treatment. Based on the material–demand matching scheme, the facility m replenishes the disaster points of its service in ascending order according to the distance, i.e., emergency materials demands from closer disaster points are met first. Furthermore, the relationship between $z_{mn}^{wp}$, the remaining inventory of facility $las{t}_m^{wp}$, and $q_n^{wp}$ can be assessed by analyzing the initial allocation of resources. Here, $las{t}_m^{wp}=v_m^{wp}-z_{mn}^{wp}$. If $z_{mn}^{wp}\ge q_n^{wp}$, it indicates that the emergency materials allocated from facility m to disaster point n are excessive. Thus, we employ an overspending treatment strategy to avoid emergency material wastage: specifically, the overspending of emergency materials $tmp$ allocated from facility m to disaster point n, i.e., $tmp=z_{mn}^{wp}-q_n^{wp}$. Subsequently, we have $z_{mn}^{wp}=q_n^{wp}$ and $las{t}_m^{wp}=las{t}_m^{wp}+tmp$. If $z_{mn}^{wp}<q_n^{wp}$, it indicates that facility m allocates fewer emergency materials to disaster point n. In order to meet the emergency material demand of disaster point n, we implement a shortage treatment strategy. Consequently, the emergency material shortage amount $tmp1$ of disaster point n is equal to $q_n^{wp}-z_{mn}^{wp}$. Furthermore, we will address the restocking of disaster point n. If $tmp1<las{t}_m^{wp}$, then $z_{mn}^{wp}=z_{mn}^{wp}+tmp1$ and $las{t}_m^{wp}=las{t}_m^{wp}-z_{mn}^{wp}$. If $tmp1>las{t}_m^{wp}$, then $z_{mn}^{wp}=z_{mn}^{wp}+las{t}_m^{wp}$ and $las{t}_m^{wp}=0$. The execution process of the MA algorithm is illustrated in Algorithm 2.

3.4. DCSA-MA Method for Solving the Model

In this section, we integrate the DCSA and MA method into a hybrid DCSA-MA algorithm, which consists of an inner optimization and an outer optimization. The DCSA is first employed in the inner layer for facility location and supply–demand matching analysis. Subsequently, the optimized results from the DCSA are passed to the outer layer MA for material allocation. Algorithm 3 presents the pseudo-code for the DCSA-MA method. The proposed DCSA fundamentally differs from the traditional CSA and its variants [48] in three aspects: (1) CSA variants use transfer functions (such as S-shaped/V-shaped) for binary conversion, while the DCSA employs XOR/AND logic gates to preserve solution feasibility and avoid premature convergence. (2) Unlike GA (mutation/crossover), the DCSA integrates swap, reverse, and mutation directly into the position update, enhancing local search capability. (3) DCSA-MA decouples facility location (discrete optimization) and material allocation (integer programming).   

Algorithm 2: The pseudo-code of MA

Algorithm 3: The pseudo-code of DCSA-MA

1 Input: Parameters of our model, such as W, T, M, N, and other variables (seen in Table 2)    Output: Emergency material distribution scheme and the best fitness value  2 Initialization parameters of DCSA-MA, including K, $Maxiter$, and $AP$  3 Hybrid encoding of K crows  4 Initialize the hiding food position of K crows  5 Use Equations (5)–(7) to constrain each crow and prevent them from crossing the boundary  6 Use Equations (1)–(4) to calculate the fitness value of each crow and find the best crow;

3.5. Time Complexity Analysis of the DCSA-MA Algorithm

The computational complexity of evolutionary optimization algorithms refers to the number of times the algorithm performs arithmetic operations such as addition, subtraction, multiplication, and division. Let the population size be K and the maximum iterations be $Maxiter$. Additionally, the dimensionality of our model is related to the number of disaster scenarios W, the number of periods P, the number of facilities M, and the number of disaster points N. The time complexity of the DCSA is as follows: the time complexity for evaluating the fitness of each crow is $O\left(M\right)+2\times O(W\times P\times M\times N)=O(W\times P\times M\times N)$. The time complexity for hybrid encoding is $O(K\times M)+O(K\times M\times N)+O(K\times W\times P\times M\times N)=O(K\times W\times P\times M\times N)$. The time complexity of the discrete evolutionary mechanism is $O(K\times M)+O(K\times M\times N)=O(K\times M\times N)$. The time complexity of neighborhood operations is $O(K\times M\times N)$. The time complexity for the MA method is $O(K\times W\times P\times M\times N)$. Therefore, the time complexity of DCSA-MA is $O(K\times W\times P\times M\times N)+Maxiter\times \left(O\right(K\times M\times N)+O(K\times M\times N)+O(K\times W\times P\times W\times N\left)\right)$, and the overall time complexity is $O(K\times W\times P\times M\times N)+Maxiter\times O(K\times W\times P\times M\times N)$. It can be seen that the highest sub-term of the overall time complexity is the product of the dimension of our model and the population size K. In fact, the population size K is a smaller value, so the DCSA-MA algorithm has a lower time complexity.

4. Numerical Experiments and Analysis

This section performs experiments on a PC configured with an Intel(R) Core(TM) i7-12700K CPU @ 3.61GHz and 32GB of RAM, and the programming software is Matlab R2020a. Section 4.1 introduces the dataset. Section 4.2 and Section 4.3 conduct sensitivity analysis of our model. Section 4.4 presents a comparative analysis of algorithmic performance.

4.1. Dataset

This study employs the logistics network dataset provided by Tsinghua University as the basis for experimental validation. The dataset can be downloaded from the following link: http://jzw.ie.tsinghua.edu.cn/info/1219/1158.htm (accessed on 4 May 2025). The logistics network dataset gives the 10 disaster scenarios (seen in

Table 3. Disaster level and occurrence probability.

Disaster Scenario	Disaster Level	Probability of Occurrence
1	extra major disaster (level I)	0.1
2	extra major disaster (level II)	0.122
3	extra major disaster (level III)	0.026
4	major disaster (level I)	0.011
5	major disaster (level II)	0.09
6	major disaster (level III)	0.12
7	larger disaster (level I)	0.098
8	larger disaster (level II)	0.17
9	general disaster (level I)	0.234
10	general disaster (level II)	0.029

), the open cost of each facility, the division of each disaster scenario into 12 rescue periods, and the unit transportation cost of materials for different rescue periods. Meanwhile, the dataset contains 48 disaster points, eight facilities, the coordinates of the facilities and disaster points, and the emergency material demands of the disaster points.

4.2. Model Sensitivity Analysis Under Demand Fluctuations

This section evaluates the performance of the model under different demand conditions by adjusting the uncertain budget level $\Gamma$. The uncertain budget level $\Gamma$ is set to 10, 30, 50, 70, and 90, respectively. We set $\overline{q_n^{wp}}=\phi ·q_n^{wp}$, where $\phi$ is the perturbation ratio, which takes on values of 0.2, 0.4, 0.6, 0.8, and 1. Figure 3 and Figure 4 depict the emergency rescue cost and the meet rate at the disaster point under uncertain demand conditions, respectively.

As shown in Figure 3, the emergency rescue cost shows a gradual upward trend with consistent growth rates as $\Gamma$ increases from 10 to 90. This result indicates a positive correlation between $\Gamma$ and cost; i.e., the higher the $\Gamma$, the greater the cost required. Concurrently, for a fixed $\Gamma$, the greater the $\phi$ value, the faster the cost value increases. Specifically, as $\phi$ increases from 0.2 to 1, the cost value escalates. Figure 4 illustrates that the meet rate of emergency materials at the disaster point declines with increasing $\Gamma$. At a constant $\Gamma$, the material meet rate decreases with the increase of $\phi$, indicating that larger $\phi$ values correspond to more severe positive demand fluctuations beyond projected levels.

4.3. Model Sensitivity Analysis Under Facility Failures

This section validates the impact of the number of facility failures on the emergency rescue cost and the meet rate at the disaster point by adjusting the k. The experimental results are presented in Figure 5.

As shown in Figure 5, the emergency rescue cost exhibits an upward trend with the increase in k. This trend is attributed to the fact that as the number of facility failures increases, it becomes increasingly difficult to meet the material demands of numerous disaster points, resulting in a higher penalty cost. Concurrently, the meet rate at the disaster point shows a significant downward trend with the increase in k. This illustrates that there is a positive correlation between the number of facility failures and the meet rate at the disaster point. Therefore, facility failure not only has a direct impact on the effectiveness of emergency rescue but also highlights the importance of maintaining the stability and reliability of facilities in emergency management.

4.4. Comparative Analysis of Algorithm Performance

In this section, the effectiveness of DCSA-MA is verified from three aspects: the meet rate, convergence, and stability.

4.4.1. Analysis of Meet Rate at Disaster Points

To validate the effectiveness of the DCSA-MA algorithm, this section evaluates the meet rate results computed by various algorithms across 12 rescue periods in 10 disaster scenarios (Scenario 1 to Scenario 10). Specifically, Scenarios 1 to 3 represent extra major disaster, Scenarios 4 to 6 denote major disaster, Scenarios 7 to 8 mean larger disaster, and Scenarios 9 to 10 represent general disaster. In addition, we combine the genetic algorithm (GA) [49] and variants (S1, S2, S3, S4, V1, V2, V3, and V4) of the binary crow search algorithm with time-varying flight length (BCSA-TVFL) [48] with the MA algorithm, respectively. These combined algorithms are named GA-MA, S1-MA, S2-MA, S3-MA, S4-MA, V1-MA, V2-MA, V3-MA, and V4-MA, respectively. The experimental results are presented in

Table 4. The emergency material meet rates at disaster points across 12 rescue periods in Scenario 1∼Scenario 3.

Scenarios	Periods	GA-MA	S1-MA	S2-MA	S3-MA	S4-MA	V1-MA	V2-MA	V3-MA	V4-MA	DCSA-MA
Scenario 1	P1	95.31%	97.19%	97.21%	99.16%	97.85%	97.57%	97.87%	97.77%	97.74%	98.75%
	P2	98.45%	98.66%	98.58%	98.87%	99.49%	99.46%	99.93%	99.39%	99.06%	99.96%
	P3	99.47%	99.75%	99.96%	97.69%	99.93%	99.79%	99.98%	99.99%	99.74%	100.00%
	P4	93.69%	97.97%	98.99%	99.00%	98.01%	98.83%	97.02%	98.48%	98.06%	97.77%
	P5	97.54%	99.52%	99.77%	99.39%	99.57%	99.81%	99.57%	99.73%	99.71%	99.72%
	P6	99.78%	99.90%	100.00%	99.41%	99.99%	100.00%	99.97%	100.00%	99.99%	100.00%
	P7	88.25%	89.90%	90.00%	96.25%	91.85%	92.38%	89.58%	90.84%	90.24%	95.39%
	P8	99.47%	99.95%	99.85%	97.90%	99.98%	100.00%	99.91%	99.93%	99.84%	100.00%
	P9	99.97%	100.00%	100.00%	99.39%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P10	98.96%	100.00%	99.27%	99.66%	100.00%	100.00%	100.00%	99.92%	100.00%	100.00%
	P11	99.99%	100.00%	99.89%	99.16%	100.00%	100.00%	99.99%	100.00%	100.00%	100.00%
	P12	97.59%	99.95%	99.53%	98.87%	99.01%	99.93%	99.40%	100.00%	99.29%	99.74%
	Mean	97.37%	98.57%	98.59%	98.73%	98.81%	98.98%	98.60%	98.84%	98.64%	99.28%
Scenario 2	P1	87.65%	91.26%	89.54%	95.68%	88.85%	92.11%	90.59%	91.60%	90.64%	94.24%
	P2	90.14%	94.20%	92.51%	95.43%	91.78%	93.23%	96.51%	94.69%	94.44%	97.58%
	P3	93.83%	97.65%	97.67%	92.42%	96.49%	98.49%	99.15%	98.60%	98.23%	99.18%
	P4	86.71%	93.87%	92.00%	96.59%	92.98%	95.25%	92.48%	93.34%	93.27%	95.39%
	P5	92.40%	96.38%	94.91%	96.21%	96.56%	97.72%	95.01%	95.89%	95.13%	97.32%
	P6	92.29%	95.20%	97.64%	96.66%	97.90%	99.01%	98.19%	97.87%	98.61%	99.02%
	P7	73.76%	73.87%	74.14%	88.48%	75.88%	79.68%	79.53%	78.64%	71.98%	83.31%
	P8	94.14%	97.05%	98.19%	92.79%	95.78%	96.61%	96.59%	97.28%	95.88%	98.35%
	P9	97.11%	99.91%	99.38%	96.41%	99.78%	99.98%	99.25%	99.61%	99.15%	100.00%
	P10	94.61%	98.96%	95.83%	97.70%	97.84%	99.49%	99.27%	98.48%	99.06%	98.92%
	P11	98.17%	99.97%	99.17%	95.68%	99.93%	99.98%	98.81%	99.86%	98.95%	99.66%
	P12	90.88%	97.47%	93.41%	95.43%	94.58%	97.75%	95.16%	96.26%	94.58%	96.86%
	Mean	90.97%	94.65%	93.70%	94.96%	94.03%	95.78%	95.05%	95.18%	94.16%	96.65%
Scenario 3	P1	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P2	100.00%	100.00%	100.00%	99.51%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P3	100.00%	100.00%	100.00%	99.84%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P4	99.37%	99.85%	100.00%	100.00%	99.45%	100.00%	99.70%	99.97%	99.87%	99.94%
	P5	100.00%	100.00%	100.00%	99.89%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P6	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P7	98.84%	100.00%	100.00%	99.33%	100.00%	99.99%	100.00%	100.00%	100.00%	100.00%
	P8	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P9	99.29%	99.12%	99.53%	99.79%	99.38%	99.48%	99.21%	98.58%	99.15%	100.00%
	P10	100.00%	100.00%	100.00%	99.94%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P11	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P12	100.00%	100.00%	100.00%	99.51%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	Mean	99.79%	99.91%	99.96%	99.82%	99.90%	99.96%	99.91%	99.88%	99.92%	100.00%

,

Table 5. The emergency material meet rates at disaster points across 12 rescue periods in Scenario 4∼Scenario 6.

Scenarios	Periods	GA-MA	S1-MA	S2-MA	S3-MA	S4-MA	V1-MA	V2-MA	V3-MA	V4-MA	DCSA-MA
Scenario 4	P1	89.39%	87.43%	93.33%	99.40%	93.13%	90.84%	89.90%	91.57%	88.42%	93.39%
	P2	93.78%	93.08%	96.15%	91.31%	96.19%	94.13%	93.95%	94.71%	92.39%	96.29%
	P3	95.74%	94.08%	96.96%	95.83%	98.26%	96.35%	96.24%	96.46%	95.08%	98.15%
	P4	88.69%	89.38%	94.61%	99.37%	95.86%	94.28%	91.81%	94.95%	92.02%	93.93%
	P5	96.24%	95.06%	97.50%	96.04%	98.40%	97.45%	96.93%	97.70%	95.74%	98.52%
	P6	98.73%	99.43%	99.23%	98.73%	99.43%	99.66%	99.20%	99.52%	97.81%	99.73%
	P7	90.84%	89.98%	93.18%	91.54%	96.22%	94.07%	91.54%	94.46%	91.64%	97.07%
	P8	99.79%	99.76%	99.66%	98.18%	99.56%	99.70%	99.59%	99.75%	99.55%	100.00%
	P9	100.00%	100.00%	100.00%	95.35%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P10	100.00%	100.00%	99.99%	97.47%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P11	100.00%	100.00%	100.00%	99.40%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P12	99.95%	100.00%	100.00%	91.31%	100.00%	100.00%	100.00%	100.00%	99.99%	100.00%
	Mean	96.10%	95.68%	97.55%	96.16%	98.09%	97.21%	96.60%	97.43%	96.05%	98.09%
Scenario 5	P1	95.31%	97.19%	97.21%	95.55%	97.85%	97.57%	97.87%	97.77%	97.74%	98.75%
	P2	98.45%	98.66%	98.58%	97.55%	99.49%	99.46%	99.93%	99.39%	99.06%	99.96%
	P3	99.47%	99.75%	99.96%	94.02%	99.93%	99.79%	99.98%	99.99%	99.74%	100.00%
	P4	93.69%	97.97%	98.99%	97.04%	98.01%	98.83%	97.02%	98.48%	98.06%	97.77%
	P5	97.54%	99.52%	99.77%	98.03%	99.57%	99.81%	99.57%	99.73%	99.71%	99.72%
	P6	99.78%	99.90%	100.00%	96.21%	99.99%	100.00%	99.97%	100.00%	99.99%	100.00%
	P7	88.25%	89.90%	90.00%	93.81%	91.85%	92.38%	89.58%	90.84%	90.24%	95.39%
	P8	99.47%	99.95%	99.85%	96.89%	99.98%	100.00%	99.91%	99.93%	99.84%	100.00%
	P9	74.12%	68.51%	69.06%	98.80%	65.42%	69.24%	73.30%	70.68%	69.19%	77.74%
	P10	90.82%	93.03%	88.52%	96.55%	88.01%	93.32%	92.81%	90.63%	90.91%	96.87%
	P11	99.99%	100.00%	99.89%	95.55%	100.00%	100.00%	99.99%	100.00%	100.00%	100.00%
	P12	97.59%	99.95%	99.53%	97.55%	98.90%	99.93%	99.40%	100.00%	99.29%	99.74%
	Mean	94.54%	95.36%	95.11%	96.46%	94.92%	95.86%	95.78%	95.62%	95.31%	97.16%
Scenario 6	P1	83.18%	83.51%	80.39%	94.62%	78.04%	80.43%	84.80%	83.16%	84.84%	89.51%
	P2	86.98%	89.04%	83.86%	97.52%	83.56%	85.44%	91.30%	89.82%	89.57%	92.68%
	P3	96.63%	95.21%	96.83%	97.51%	96.58%	97.66%	98.97%	96.90%	97.95%	98.58%
	P4	90.93%	96.49%	95.48%	97.31%	92.69%	97.15%	96.56%	96.10%	97.07%	97.68%
	P5	97.44%	99.49%	99.75%	98.00%	96.74%	99.80%	99.53%	99.66%	98.55%	99.68%
	P6	99.76%	99.54%	100.00%	96.75%	99.99%	100.00%	99.97%	100.00%	99.98%	100.00%
	P7	86.91%	88.91%	87.32%	92.42%	90.04%	91.50%	90.71%	92.08%	87.35%	94.80%
	P8	99.41%	99.94%	99.84%	96.42%	99.97%	99.99%	99.88%	99.92%	99.82%	100.00%
	P9	99.96%	100.00%	100.00%	98.05%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P10	98.92%	99.99%	99.24%	96.82%	99.99%	100.00%	100.00%	99.92%	99.99%	100.00%
	P11	98.92%	100.00%	99.88%	94.62%	100.00%	100.00%	99.99%	100.00%	100.00%	100.00%
	P12	97.46%	99.94%	99.26%	97.52%	98.90%	99.92%	98.58%	99.63%	98.54%	99.71%
	Mean	94.71%	96.01%	95.15%	96.46%	94.71%	95.99%	96.69%	96.43%	96.14%	97.72%

and

Table 6. The emergency material meet rates at disaster points across 12 rescue periods in Scenario 7∼Scenario 10.

Scenarios	Periods	GA-MA	S1-MA	S2-MA	S3-MA	S4-MA	V1-MA	V2-MA	V3-MA	V4-MA	DCSA-MA
Scenario 7	P1	94.82%	96.03%	96.72%	90.91%	96.70%	97.26%	96.05%	97.32%	97.95%	98.92%
	P2	97.56%	98.81%	98.69%	95.59%	99.59%	99.52%	99.97%	99.47%	99.17%	99.88%
	P3	99.42%	99.84%	99.98%	87.00%	99.94%	99.82%	99.98%	100.00%	99.79%	100.00%
	P4	94.22%	98.09%	99.07%	93.59%	97.83%	98.94%	97.17%	97.49%	98.24%	97.82%
	P5	97.85%	99.61%	99.82%	96.24%	99.42%	99.84%	99.59%	99.78%	99.78%	99.80%
	P6	99.83%	99.95%	100.00%	90.03%	99.99%	100.00%	99.98%	100.00%	99.99%	100.00%
	P7	67.93%	62.96%	64.64%	90.12%	62.89%	65.08%	67.43%	69.24%	63.61%	73.08%
	P8	82.00%	81.37%	80.16%	94.92%	76.67%	81.84%	81.94%	80.91%	78.15%	85.84%
	P9	84.74%	83.78%	81.65%	97.44%	78.89%	84.89%	83.98%	83.12%	82.14%	88.22%
	P10	86.84%	91.96%	83.45%	92.92%	81.28%	87.85%	88.73%	86.25%	86.23%	93.18%
	P11	90.68%	88.35%	87.65%	90.91%	86.49%	90.40%	89.72%	89.95%	89.06%	91.48%
	P12	85.28%	86.03%	83.15%	95.59%	82.33%	90.16%	85.77%	87.49%	84.87%	91.19%
	Mean	90.10%	90.57%	89.58%	92.94%	88.50%	91.30%	90.86%	90.92%	89.92%	93.28%
Scenario 8	P1	93.60%	94.54%	96.28%	94.05%	96.81%	96.83%	97.82%	97.00%	96.78%	98.67%
	P2	97.81%	98.36%	98.41%	93.56%	99.28%	99.44%	99.73%	99.17%	97.31%	99.94%
	P3	99.43%	99.71%	99.94%	90.71%	99.92%	99.77%	99.98%	99.99%	98.18%	100.00%
	P4	53.43%	47.46%	50.28%	92.62%	44.52%	50.73%	54.00%	52.20%	49.09%	59.99%
	P5	81.13%	81.16%	80.55%	95.25%	73.81%	77.57%	82.28%	78.23%	80.00%	85.19%
	P6	99.56%	99.57%	100.00%	92.22%	99.22%	100.00%	99.97%	100.00%	99.98%	100.00%
	P7	86.66%	88.49%	88.85%	90.64%	91.40%	91.58%	88.71%	88.84%	87.76%	94.06%
	P8	98.39%	99.94%	98.96%	91.84%	99.97%	99.99%	99.07%	99.92%	99.51%	100.00%
	P9	99.95%	100.00%	100.00%	95.39%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P10	98.91%	99.99%	99.23%	93.39%	99.99%	100.00%	100.00%	99.80%	99.99%	100.00%
	P11	99.98%	100.00%	99.88%	94.05%	100.00%	100.00%	99.99%	100.00%	96.72%	100.00%
	P12	97.34%	99.94%	98.99%	93.56%	99.19%	99.92%	99.36%	100.00%	98.37%	99.70%
	Mean	92.18%	92.43%	92.61%	93.11%	92.01%	92.99%	93.41%	92.93%	91.97%	94.80%
Scenario 9	P1	97.64%	98.22%	98.30%	92.64%	98.26%	98.77%	98.44%	98.70%	98.85%	99.44%
	P2	99.62%	98.37%	99.52%	96.34%	99.90%	99.82%	100.00%	99.81%	99.67%	100.00%
	P3	99.34%	99.70%	100.00%	95.46%	99.99%	99.98%	100.00%	100.00%	99.94%	100.00%
	P4	95.08%	98.57%	99.13%	93.16%	98.41%	99.22%	96.86%	99.06%	98.72%	97.96%
	P5	99.17%	99.30%	99.98%	95.62%	99.80%	99.96%	100.00%	99.95%	99.98%	100.00%
	P6	74.54%	72.92%	68.82%	93.06%	64.00%	68.19%	77.88%	71.47%	73.20%	77.33%
	P7	67.20%	66.86%	62.85%	88.85%	57.48%	64.18%	66.74%	63.62%	64.91%	74.04%
	P8	93.97%	94.19%	92.46%	93.00%	88.33%	92.35%	96.82%	92.62%	94.73%	97.14%
	P9	98.51%	98.56%	96.50%	95.71%	93.71%	99.45%	99.94%	98.11%	99.27%	100.00%
	P10	99.28%	99.77%	98.15%	93.10%	98.62%	100.00%	100.00%	99.44%	99.83%	100.00%
	P11	100.00%	100.00%	100.00%	92.64%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P12	99.36%	100.00%	99.89%	96.34%	99.64%	100.00%	99.79%	100.00%	99.92%	99.99%
	Mean	93.64%	93.87%	92.97%	93.83%	91.51%	93.49%	94.71%	93.57%	94.09%	95.49%
Scenario 10	P1	67.96%	66.61%	68.31%	90.18%	73.20%	66.25%	69.21%	70.44%	69.41%	75.84%
	P2	71.37%	71.84%	72.03%	80.72%	76.74%	69.06%	73.86%	75.60%	74.03%	79.36%
	P3	81.03%	78.26%	79.07%	88.14%	85.28%	81.22%	79.62%	81.06%	81.90%	85.02%
	P4	73.38%	73.35%	73.61%	94.25%	77.59%	72.77%	74.57%	75.24%	76.55%	77.71%
	P5	84.13%	80.90%	82.65%	89.58%	86.81%	83.15%	82.12%	83.52%	83.52%	87.34%
	P6	90.71%	89.43%	90.96%	91.88%	92.71%	92.13%	91.87%	92.27%	90.72%	93.94%
	P7	78.60%	75.65%	78.91%	81.13%	83.41%	78.69%	77.43%	80.60%	78.59%	87.14%
	P8	95.46%	93.08%	96.92%	91.57%	98.99%	96.33%	96.39%	96.47%	95.00%	97.55%
	P9	99.65%	99.70%	99.72%	88.36%	99.93%	99.78%	99.55%	99.35%	99.48%	100.00%
	P10	99.68%	100.00%	99.79%	90.84%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P11	100.00%	100.00%%	100.00%	90.18%	100.00%	100.00%	100.00%	100.00%	100.00%	100.00%
	P12	99.37%	100.00%	99.89%	80.72%	99.98%	100.00%	100.00%	100.00%	99.92%	99.99%
	Mean	86.78%	85.74%	86.82%	88.13%	89.55%	86.62%	87.05%	87.88%	87.43%	90.32%

.

As demonstrated in Table 4, within the extra major disaster shown in Scenarios 1 to 3, the DCSA-MA algorithm outperforms the other nine algorithms in emergency material meet rates. Notably, in Scenario 1’s P1 period, Scenario 2’s P9 period, and Scenario 3’s P9 period, the DCSA-MA algorithm achieves a 100% meet rate. Furthermore, the average emergency material meet rate of DCSA-MA in Scenario 3 approaches 100%, significantly surpassing the other nine algorithms. These results demonstrate the high effectiveness and stability of DCSA-MA across various disaster scenarios, highlighting its robust capability for handling extreme complexities. As shown in Table 5, for major disaster (i.e., Scenarios 4 to 6), DCSA-MA not only exceeds other algorithms with a 97.72% emergency material meet rate but also achieves 100% during Scenario 4’s P8 period, Scenario 5’s P3 period, and Scenario 6’s P8 period, significantly outperforming all nine alternatives.

As shown in Table 6, for larger disaster scenarios (Scenarios 7 to 8) and general disaster scenarios (Scenarios 9 to 10), although the emergency material meet rate of the DCSA-MA algorithm slightly decreases, its overall performance remains significantly better than other algorithms. DCSA-MA maintains the highest average meet rate across Scenarios 7–10, which is attributable to its robust algorithmic architecture that continues to excel despite reduced problem complexity as disaster severity decreases. Other algorithms exhibit greater performance fluctuations due to inherent limitations.

In summary, the DCSA-MA algorithm demonstrates exceptional overall performance, exhibiting superior effectiveness and stability across disaster scenarios and rescue periods. Its significantly higher average meet rate and consistent performance contrast with the fluctuations and deficiencies observed in other algorithms, establishing DCSA-MA’s distinct advantage for disaster scenario material allocation and providing robust decision support for actual rescue operations.

4.4.2. Analysis of Algorithm Convergence and Stability

To validate the convergence and stability of the DCSA-MA algorithm, this section presents a comparative analysis across 10 different algorithms. Figure 6 illustrates the convergence curves, while Figure 7 employs a box plot to depict the stability of each method. The box plot provides detailed information about each method, including the maximum value, minimum value, second quartile, median, third quartile, and outliers.

From Figure 6, it is evident that the GA-MA algorithm exhibits significant fluctuations during convergence, indicating that the GA-MA algorithm is prone to local optimum. In the S-series algorithms, S1-MA demonstrates the highest convergence accuracy but suffers from a slower convergence speed. Conversely, S2-MA converges fastest at the expense of accuracy. Within the V-series algorithms, V2-MA achieves the highest convergence accuracy but experiences significant fluctuations during the convergence process. Overall, the S-series algorithms outperform the V-series in terms of convergence accuracy and speed. Notably, DCSA-MA stabilizes around the 400th iteration and surpasses all other algorithms in both convergence speed and accuracy.

As shown in Figure 7, the S3-MA, V3-MA, and V4-MA algorithms exhibit the presence of outliers, indicating instability in these three methods. In contrast, DCSA-MA shows no outliers and has the smallest standard deviation, demonstrating superior stability and optimization performance.

4.4.3. Wilcoxon Rank Sum Test of Algorithm

To comprehensively evaluate the performance of DCSA-MA across various disaster scenarios, this section validates its superiority compared to other algorithms. We employ the Wilcoxon rank sum test statistic method for significance analysis. By comparing results from 30 independent runs, this statistical method effectively determines whether significant performance differences exist between DCSA-MA and other algorithms. We establish the following hypotheses:

(1)

Original Hypothesis (H0): No significant performance difference exists between DCSA-MA and other algorithms.

(2)

Alternative Hypothesis (H1): A significant performance difference exists between DCSA-MA and other algorithms.

The resulting p-value determines if a significant difference exists. If the p-value is less than 0.05, we reject H0 and accept H1, indicating a significant difference between the two algorithms. If the p-value exceeds 0.05, we accept H0; i.e., the performance of the two algorithms are similar.

Table 7. The p-value of the rank sum test comparing DCSA-MA with other algorithms in different disaster scenarios.

Scenarios	GA-MA	S1-MA	S2-MA	S3-MA	S4-MA	V1-MA	V2-MA	V3-MA	V4-MA
Scenario 1	3.34 $\times {10}^{-11}$	2.20 $\times {10}^{-7}$	2.03 $\times {10}^{-9}$	1.20 $\times {10}^{-10}$	1.09 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	6.06 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	9.91 $\times {10}^{-11}$
Scenario 2	3.02 $\times {10}^{-11}$	2.67 $\times {10}^{-9}$	9.92 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	4.08 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	3.34 $\times {10}^{-11}$	3.02 $\times {10}^{-11}$	4.08 $\times {10}^{-11}$
Scenario 3	6.78 $\times {10}^{-7}$	2.81 $\times {10}^{-2}$	9.62 $\times {\mathbf{10}}^{-\mathbf{2}}$	7.36 $\times {\mathbf{10}}^{-\mathbf{2}}$	3.50 $\times {10}^{-3}$	4.10 $\times {10}^{-4}$	2.68 $\times {10}^{-4}$	2.60 $\times {10}^{-4}$	1.20 $\times {10}^{-3}$
Scenario 4	2.44 $\times {10}^{-9}$	1.08 $\times {10}^{-2}$	3.99 $\times {10}^{-4}$	6.36 $\times {10}^{-5}$	7.04 $\times {10}^{-7}$	1.17 $\times {10}^{-5}$	3.59 $\times {10}^{-5}$	5.19 $\times {10}^{-7}$	1.11 $\times {10}^{-6}$
Scenario 5	4.50 $\times {10}^{-11}$	1.03 $\times {10}^{-2}$	4.74 $\times {10}^{-6}$	1.41 $\times {10}^{-9}$	3.20 $\times {10}^{-9}$	2.87 $\times {10}^{-10}$	1.07 $\times {10}^{-9}$	5.57 $\times {10}^{-10}$	2.44 $\times {10}^{-9}$
Scenario 6	3.02 $\times {10}^{-11}$	4.42 $\times {10}^{-6}$	1.01 $\times {10}^{-8}$	6.72 $\times {10}^{-10}$	1.96 $\times {10}^{-10}$	5.49 $\times {10}^{-11}$	4.62 $\times {10}^{-10}$	6.07 $\times {10}^{-11}$	1.46 $\times {10}^{-10}$
Scenario 7	1.41 $\times {10}^{-9}$	1.05 $\times {\mathbf{10}}^{-\mathbf{1}}$	1.68 $\times {10}^{-4}$	5.09 $\times {10}^{-8}$	5.09 $\times {10}^{-8}$	2.92 $\times {10}^{-9}$	1.85 $\times {10}^{-8}$	7.69 $\times {10}^{-8}$	3.08 $\times {10}^{-8}$
Scenario 8	3.02 $\times {10}^{-11}$	3.03 $\times {10}^{-3}$	2.78 $\times {10}^{-7}$	5.49 $\times {10}^{-11}$	5.57 $\times {10}^{-10}$	3.02 $\times {10}^{-11}$	1.21 $\times {10}^{-10}$	1.09 $\times {10}^{-10}$	2.67 $\times {10}^{-9}$
Scenario 9	7.39 $\times {10}^{-11}$	4.22 $\times {10}^{-4}$	1.17 $\times {10}^{-4}$	2.13 $\times {10}^{-5}$	1.03 $\times {10}^{-6}$	4.62 $\times {10}^{-10}$	7.04 $\times {10}^{-7}$	3.20 $\times {10}^{-9}$	2.39 $\times {10}^{-8}$
Scenario 10	7.12 $\times {10}^{-9}$	2.38 $\times {10}^{-3}$	1.06 $\times {10}^{-3}$	4.31 $\times {10}^{-8}$	1.75 $\times {10}^{-5}$	1.73 $\times {10}^{-6}$	2.49 $\times {10}^{-6}$	3.96 $\times {10}^{-8}$	3.59 $\times {10}^{-5}$
$+/\approx$	10 / 0	9 / 1	9 / 1	9 / 1	10 / 0	10 / 0	10 / 0	10 / 0	10 / 0

presents the Wilcoxon rank sum test p-value comparing DCSA-MA compared to other algorithms across various disaster scenarios. Symbols + and ≈ indicate instances where DCSA-MA outperforms others or demonstrates similar performance, respectively.

As shown in Table 7, across all 10 disaster scenarios, the p-values for comparisons between DCSA-MA and GA-MA, V1-MA, V2-MA, V3-MA, and V4-MA are consistently below 0.05. Theses results indicate that DCSA-MA is significantly superior to these algorithms in these scenarios. For comparison with S1-MA, S2-MA, S3-MA, and S4-MA, DCSA-MA demonstrates significant advantages in most scenarios while showing comparable performance in a limited number of cases. Overall, DCSA-MA significantly outperforms other algorithms in most disaster scenarios and has superior optimization performance. Although its performance is comparable to that of some algorithms in a few scenarios, this does not affect the overall superiority of DCSA-MA. Therefore, DCSA-MA is a reliable and effective optimization algorithm, which is applicable to solving optimization problems in various disaster scenarios.

5. Conclusions

The study is aiming to solve the symmetry breaking problem between pre-disaster planning and post-disaster rescue operations. Firstly, we account for the diversity and periodicity characteristics of disaster, establishing a deterministic multi-scenario multi-period facility location model. Subsequently, to address the fluctuations in material demand and facility failure, we incorporate a polyhedral uncertainty set into the deterministic model, thereby constructing a dual-uncertainty multi-scenario multi-period facility location model that aims to minimize the costs associated with facility opening, emergency material transportation, and penalties. To solve this model, we design a DCSA-MA algorithm. In the first stage, the DCSA-MA employs hybrid encoding, discrete evolutionary mechanism, and neighborhood operation strategies to determine optimal facility locations and supply–demand matching schemes. The second stage then performs the actual material allocation.

This paper uses a logistics network dataset to verify the effectiveness of our model and the DCSA-MA method. When the demand fluctuation is small, its impact on the material satisfaction rate of our model is relatively low. Although the material satisfaction rate decreases under high demand fluctuation, our model still maintains a relatively high level. Secondly, an increase in the number of facility failures leads to a lower material satisfaction rate at disaster points, verifying the direct impact of facility failure on emergency rescue effectiveness and underscoring the critical importance of maintaining facility stability and reliability in emergency management. Furthermore, the DCSA-MA method outperforms other algorithms in terms of material satisfaction rate across various disaster scenarios and different rescue periods. The DCSA-MA method also demonstrates superior performance not only in solution quality but also in convergence speed and stability, validating the efficacy of the DCSA-MA approach. Therefore, the models and methods presented in this study not only provide novel theoretical insights into the emergency resource allocation problem but also offer valuable decision support for emergency management departments. In practical applications, the model can dynamically update multi-scenario parameters using real-time disaster data and leverage the DCSA-MA method to rapidly generate cost-effective and robust multi-period material allocation schemes. This provides automated decision support for multi-stage resource scheduling in response to sudden disasters such as earthquakes and floods. Future research will focus on the following aspects:

(1)

Construct a multi-objective optimization model. On the basis of the existing cost minimization, we plan to introduce more realistic factors to build a multi-objective optimization model, such as fairness, rescue time, and other realistic factors. To balance the potential conflicts between these objectives, we will employ two strategies: First, we transform multiple objectives into a single objective using a weighting approach. Second, multi-objective evolutionary algorithms such as NSGA-II are used to explore a set of Pareto-optimal solutions to provide a diverse set of options to the decision-maker.

(2)

Algorithm innovation and computational efficiency improvement. We will explore cutting-edge heuristics and exact algorithms to obtain more efficient and accurate model solutions. In addition, we will parallelize existing algorithms and use multi-core processors or multi-node computing clusters to speed up the computational efficiency.