A Two-Stage Robust Casualty Evacuation Optimization Model for Sustainable Humanitarian Logistics Networks Under Interruption Risks

Abstract

Building a sustainable and resilient humanitarian logistics system is essential for reducing disaster losses and supporting long-term socio-economic recovery. Following a major disaster, rapidly organizing casualty evacuation while maintaining system robustness is a fundamental component of sustainable emergency management. This study develops a two-stage robust optimization model for designing a sustainable humanitarian logistics network that simultaneously accounts for two critical post-disaster uncertainties: (i) interruption risks at temporary medical points and (ii) uncertain casualty demand. By explicitly differentiating deprivation costs between mild and serious injuries, the model quantifies human suffering in monetary terms, thereby integrating social and economic sustainability considerations into the optimization framework. A customized column-and-constraint generation (C&CG) algorithm with proven finite convergence is proposed to ensure tractability and practical applicability. Using the 2008 Wenchuan earthquake as a real-world case study, involving 10 affected areas and 10 candidate temporary medical points, the results demonstrate that the proposed approach yields evacuation plans that remain feasible under all tested worst-case realizations, substantially reducing deprivation costs compared with existing benchmarks. The findings highlight that strategically increasing the capacity of key temporary medical nodes enhances the sustainability and resilience of the emergency medical system, offering evidence-based insights for designing sustainable and robust disaster-response strategies.

1. Introduction

In recent years, the increasing frequency and severity of natural disasters, exacerbated by climate change and social complexity, have posed major challenges to post-disaster rescue operations, particularly in casualty evacuation. For example, the 2023 Turkey-Syria earthquakes caused over 59,000 deaths, many due to delayed medical aid (Naddaf, 2023) [1]. Similarly, the 2010 Haiti earthquake left 230,000 dead and 300,000 injured, with infrastructure collapse severely hindering timely treatment. These cases underscore the urgent need for effective humanitarian logistics networks that can support casualty evacuation under resource and uncertainty constraints.

Operational disruptions, particularly facility interruptions, should receive careful attention in post-disaster humanitarian logistics planning. Temporary medical facilities, essential for casualty evacuation and treatment, are highly vulnerable to aftershocks and cascading hazards. Locating them close to disaster sites shortens transport and treatment time and thus reduces mortality (Oksuz and Satoglu, 2020) [2]. Yet much of the literature implicitly assumes uninterrupted facility availability, overlooking disruption-induced challenges—an assumption that is unrealistic in real disasters. This observation motivates our focus on evacuation-network design that explicitly accounts for facility interruption risk together with other post-disaster uncertainties.

The construction of humanitarian logistics networks for casualty evacuation fundamentally differs from commercial logistics. Its primary objective is to alleviate human suffering and minimize casualties and losses (Niyomubyeyi et al. (2019)) [3]. Evacuation operations involve high uncertainty, conflicting objectives, and spatial constraints, making them inherently complex optimization tasks. This necessitates the quantification of human suffering. Holguín-Veras et al. (2013) [4] pioneered the concept of “deprivation cost,” defined as the economic valuation of human suffering due to the lack of access to essential goods or services. However, this concept has rarely been incorporated into humanitarian logistics research for casualty evacuation. Furthermore, significant variations in injury severity among casualties lead to distinct deprivation costs, underscoring the necessity of casualty triage during decision-making. Existing studies often overlook the importance of triage and fail to account for deprivation cost disparities across casualty categories, potentially resulting in suboptimal resource allocation, reduced rescue efficiency, and compromised outcomes (Pradhananga et al., 2016) (Loree and Aros-Vera, 2018) (Perez-Rodriguez and Holguin-Veras (2016)) [5,6,7]. Therefore, it is necessary to introduce the lack of deprivation cost and consider the differences in casualties in the research questions, which can enhance the model’s ability to make effective and fair casualty evacuation decisions in real disaster scenarios.

Post-disaster uncertainties, including unpredictable casualty numbers and facility interruption risks, further complicate evacuation planning. Stochastic programming and robust optimization are widely adopted to address these uncertainties. While stochastic programming requires explicit probability distributions for uncertain parameters, which are often unavailable in practice, robust optimization relies on interval data without probabilistic assumptions, enhancing its practical applicability. Traditional single-stage robust optimization, however, focuses on worst-case scenarios, potentially leading to overly conservative and costly solutions. To overcome this limitation, two-stage robust optimization has been developed. By dividing decisions into two stages: pre-disaster planning and post-disaster adaptive adjustments. This approach strikes a balance between robustness and cost-effectiveness. This study proposes a two-stage robust optimization model for humanitarian logistics networks in casualty evacuation.

A related methodological challenge, observed across engineering and operations-research domains, is the high sensitivity of system performance to uncertain or stress-dependent parameters. For example, in reservoir engineering, A novel method for calculating the dynamic reserves of tight gas wells considering stress sensitivity demonstrates that incorporating parameter sensitivity, such as stress-induced variations in porosity and permeability, substantially improves estimation accuracy compared with traditional methods. This cross-disciplinary insight highlights the importance of explicitly capturing uncertainty and parameter variability, which we also incorporate into our evacuation-network design through robust modeling of deprivation costs and facility interruptions [8].

Solving two-stage robust optimization models, particularly for large-scale problems, poses significant computational challenges. Common methods include Benders decomposition, Column-and-Constraint Generation (C&CG), dual decomposition, and deep unfolding. Benders decomposition iteratively solves master and subproblems but struggles with computational efficiency for large-scale instances. In contrast, the C&CG algorithm progressively adds columns and constraints to narrow the solution space, significantly improving computational efficiency and convergence. Considering these advantages, this study employs the C&CG algorithm to solve the proposed model, ensuring robust and feasible solutions.

Building on this background, this study addresses the following central research questions:

• How can temporary medical facilities be optimally located when casualty numbers are uncertain, so as to reduce the deprivation experienced by affected populations?
• How can evacuation strategies remain robust and efficient under potential facility interruptions, thereby mitigating deprivation caused by delays or rerouting?
• How does casualty triage—distinguishing between mild and severe injuries—shape evacuation decisions and influence overall rescue outcomes, particularly in terms of deprivation minimization?

These interconnected questions jointly guide the development of both the deterministic and robust optimization models.

To resolve these issues, this paper proposes a two-stage robust optimization model incorporating casualty triage, facility disruptions, and demand uncertainty. Unlike traditional single-stage approaches that risk overly conservative solutions, the two-stage model enhances cost-effectiveness while ensuring robustness. Furthermore, the C&CG algorithm is adopted to guarantee computational efficiency and practicality.

The novelty and contributions of this study are threefold:

• To the best of our knowledge, this is the first work that simultaneously models facility disruption risk and casualty demand uncertainty in a two-stage robust casualty evacuation framework;
• It is the first to explicitly incorporate and minimize deprivation costs that differ by injury severity (mild vs. serious);
• We design an exact column-and-constraint generation algorithm with valid inequalities that significantly outperforms standard methods.

Table 1. Our work compared with the most closely related high-impact studies in the past five years.

Study	Casualty Classification	Deprivation Cost	Facility Disruption	Demand Uncertainty	Two-Stage Robust Optimization
This study	√	√	√	√	√
Sun et al. (2021) [9]	√	×	×	√	×
Sun et al. (2022) [10]	×	×	√	√	×
Liu et al. (2019) [11]	√	×	×	√	×
Liu (2020) [12]	√	×	×	√	×
Liu (2022) [13]	√	×	×	√	×
Zhou et al. (2023) [14]	√	×	√	√	×
Du et al. (2023) [15]	×	×	√	√	×
Li et al. (2020) [16]	×	×	√	√	×
Oksuz and Satoglu (2020) [2]	×	×	√	√	×
Paul and Wang (2019) [17]	×	×	×	√	×

Note: √ indicates the feature is included or addressed in the study; × indicates the feature is not included or addressed.

compares our work with the most closely related high-impact studies in the past five years, showing that none simultaneously addresses all these features.

The remainder of this paper is organized as follows: Section 2 reviews related literature; Section 3 details the problem description and model formulation; Section 4 introduces the optimization algorithm; Section 5 presents case studies and numerical experiments; and Section 6 concludes the study and discusses future research directions.

2. Literature Review

In this section, we have surveyed three clusters of literature that are most pertinent to the current research: (1) humanitarian logistics networks for casualty evacuation considering casualty classification; (2) humanitarian logistics networks for casualty evacuation considering facility location; (3) humanitarian logistics networks for casualty evacuation considering uncertainty.

2.1. Casualty Evacuation Considering Casualty Classification

With regard to casualty classification, the relief supplies and medical assistance required vary for different categories of injured casualties. In order to provide better treatment, it is necessary to transfer casualties. The Fire Department and Newport Beach Hospital have created the Simple Triage and Rapid Treatment (START) algorithm. Sacco et al. (2005) [18], in their study on post-disaster casualties, classified casualties based on indicators such as breathing, consciousness, and pulse rate, proposing a casualty classification and transport model (STM model). Mills et al. (2013) [19], from the perspective of casualties, studied the function of the changing survival probability of different categories of casualties over time, considering the limited medical resources and the worsening of casualties’ conditions over time, and proposed a new casualty classification method (RESTART method). Dean and Nair (2014) [20] in their study on casualty treatment, discussed the rescue capabilities of different rescue vehicles and the resource constraints of different medical institutions, establishing a casualty condition adjustment and evacuation model (SAVE model).

In response to the aforementioned casualty classification methods, research has been conducted on casualty evacuation based on casualty classification. Talarico et al. (2015) [21] studied the emergency vehicle path planning problem. Ambulances are used to transport medical workers and injured people, distinguishing between two types of injuries: patients with minor injuries can be treated directly at the scene, and serious casualties need to be transported for medical treatment. Sun et al. (2021) [9] assigned varying injury levels based on the Injury Severity Score (ISS). Based on this, a novel approach to solving the problem is suggested, in which several uncertainties, the solution procedure takes into account various factors, including the quantity of casualties, the quantity of rescue materials, and the duration of the trip. The goal is to cut down on the wounded personnel’s overall contact time while also lowering the system’s overall cost. Applying casualty classification methods to casualty evacuation can effectively improve the efficiency of casualty evacuation.

2.2. Casualty Evacuation Considering Facility Location

Facility siting is a key element in the design of humanitarian logistics networks for casualty evacuation, with its appropriateness directly impacting rescue efficiency and outcomes. According to Caunhye et al. (2012) [22], the pre-determination of stockpiles, the evacuation of wounded, or the distribution of relief supplies are directly related to the location of the majority of humanitarian logistical facilities. Liu (2022) [13] emphasizes that emergency medicine plays an essential part in the treatment of post-earthquake casualties because of the high time requirements for urgent medical evacuation purposes. The construction of Emergency Medicine Service Facilities (EMSF) in disaster-stricken areas can greatly reduce the transfer time of casualties and improve the survival rate of casualties. Thus, casualty evacuation commonly involves facility siting, and most scholars studying humanitarian logistics networks generally consider the integration of casualty evacuation and medical facility siting. Salman and Gül (2014) [23] established an emergency medical facility siting and capacity planning methodology based on multi-stage planning to reduce overall facility construction costs by reducing the overall weighted time spent by casualties in both traveling and waiting. To limit and maximize the quantity of patients’ predicted survival, Liu et al. (2019) [11] built a two-objective planning model with the goals of decreasing the estimated count of survivors and minimizing the operational cost.

Currently, facility siting models for medical facilities primarily consider humanitarian logistics and casualty demand, with medical facilities near demand centers expected to handle more severe injuries. Geng et al. (2020) [24] categorized shelters into two types based on different needs of disaster victims: basic living support and psychological medical service support. The model best accounted for housing requirements, capacity, and financial limits while optimizing shelter distances, catastrophe victim distribution, and pre-stock amounts of commodities. Liu (2020) [12] developed a model to guide the deployment of Medical Evacuation (MEDEVAC) systems post-earthquake, including three tiers: injured, frontline medical centers, and secondary hospitals. A healthcare demand estimation method based on multi-criteria decision-making is proposed and optimally evaluated.

In the design of humanitarian logistics networks for casualty evacuation considering facility siting, different studies focus on different objective functions. The subject of facility siting and disaster resettlement in resource-constrained conditions was examined by Caunhye et al. (2015) [25], who also classified radiation accidents and optimized the model that was built. Sun et al. (2021) [9] divides casualties into two categories based on Injury Severity Score (ISS) and presents dynamic deterioration of injuries over time. The objective is to minimize the overall cost of the system while optimizing the ISS for every afflicted person. Eriskin and Karatas (2022) [26] studied the location and spatial distribution of shelters in uncertain environments with the goal of minimizing the share of open shelters. From existing literature, we observe a relatively limited number of studies that comprehensively consider both the classification of medical facilities and the categorization of casualties in the design of humanitarian logistics networks for casualty evacuation.

2.3. Casualty Evacuation Considering Uncertainty

Accounting for uncertainty is essential in managing complex environments and sudden incidents. Tomasini and Van Wassenhove (2009) [27] highlighted that uncertainty becomes even more significant in disaster contexts, where beyond supply-demand matching, numerous external factors play a role. Scholars have studied the construction of humanitarian logistics networks for casualty evacuation under various uncertainties. In order to account for demand, supply, and other cost aspects, Zokaee et al. (2016) [28] built a relief chain model with three levels: suppliers, relief items distribution centers, and impacted areas. Robbins et al. (2020) [29] addressed the problem of treating the sick and wounded in wartime, taking into account factors such as the condition of the wounded and sick (condition, number, location) and the timeframe for treatment. Li et al. (2020) [30] efficiently distributed casualties by optimizing a response network based on the stochastic severity of coupled disasters, even in the face of uncertainty about evacuation magnitude and transportation time.

Due to the unpredictable nature of incidents, the quantity of casualties in evacuation scenarios is uncertain. Paul and Wang (2019) [17] considered the effects of earthquakes of different magnitudes, such as damage to facilities, level of casualties, and trip length. Sun et al. (2021b) [9] included a range of uncertain elements at the three levels of casualty clusters, temporary shelters, and general hospitals, such as the quantity of relief supplies, the quantity of casualties, and the time of transportation. Additionally, temporary medical points built near disaster areas are at risk of disruption due to secondary disasters. In order to enhance service coverage and customer happiness, Du et al. (2023) [15] developed a multi-period mixed-integer linear programming model to investigate the secondary node co-optimization problem under secondary disaster disturbances. Zhou et al. (2023) [14] explored the robust casualty scheduling problem aimed at reducing the overall expected mortality probability by considering disruptions in medical points and routes. However, few studies have simultaneously considered the uncertainty of both casualty numbers and facility disruptions.

Various approaches have been used in research on uncertain situations in emergency events. Sahebi et al. (2020) [31] comprehensively reviewed the adoption barriers of blockchain in the context of supply chain management techniques for humanitarian aid, analyzing barriers with an integrated fuzzy Delphi and Best-Worst Method (BWM) approach. Cao et al. (2021) [32] proposed a two-tier integer planning model based on three objectives to address the complexity, uncertainty, hierarchy and uncertainty of supply systems, and conflicting goals in a sustainable humanitarian supply chain (SHSC) post-disaster. Regarding stochastic programming approaches. Caunhye and Nie (2018) [33] presented a three-phase stochastic programming methodology based on a three-stage stochastic program design to identify different types of healthcare facilities and staff them to respond to health emergencies. Elci and Noyan (2018) [34] formulated a two-phase stochastic optimization model incorporating probabilistic constraints, the second-stage problem involves distributing relief supplies to areas affected by disasters. It involves joint probability constraints and the use of conditional value-at-risk (CVaR) as the mean-risk objective. As stochastic planning relies on probability distributions which are often hard to precisely obtain in reality, recently, there has been a focus on using strong optimization techniques in the planning and administration of networks for humanitarian logistics. In order to accomplish dynamic configuration of emergency response and evacuation flows for emergencies under demand uncertainty, Ben-Tal et al. (2011) [35] employed a robust optimization approach. Vahdani et al. (2018) [36] presented a two-stage, multi-objective, mixed-integer, multi-period, multi-commodity mathematical modeling in a three-tier relief chain using robust optimization methods to develop the model under uncertain conditions. Sun et al. (2022) [10] developed a scenario-oriented robust bi-objective optimization model for the casualty transfer problem and the disaster relief supplies deployment problem by combining the medical institution siting problem with additional variables.

Because robust optimization only takes the worst-case parameters into account, the outcomes are expensive and cautious, so scholars have proposed and carried out research on multi-stage robust optimization. A two-stage robust optimization model was suggested by Ben-Tal et al. (2004) [37]. This approach is a particular case of a multi-stage robust optimization issue, which typically has two distinct decision variables: one and two stages. Scholars used two-stage robust optimization techniques in a variety of domains after they were introduced. A two-stage resilient optimization approach was suggested Atamturk and Zhang (2007) [38] for the design of road networks and the uncertainty demands of traffic. For dependable p-median facility location networks that are prone to interference, An et al. (2014) [39] created a two-stage robust optimization approach. Shi and You (2016) [40] introduced a two-stage adaptive robust optimization (ARO) method for scheduling batch process production under conditions of uncertainty and variability. Shang et al. (2020) [41] researched the two-stage stochastic orienteering problem with random weights, presenting two distinct resilient two-stage models for the random-weighted stochastic orienteering issue.

Two-stage robust optimization models can be NP-Hard, and thus are very challenging to solve, even in fixed resource scenarios with several uncertainties (Ben-Tal et al., 2004) [37]. Numerous techniques have been developed to address two-stage robust optimization problems. Li et al. (2012) [16] established a class of road network pricing models with uncertainty, which can be reduced to a two-stage robust optimization problem, and based on this, a heuristic algorithm combining sample mean approximation and sensitivity analysis was used to solve the proposed model. Goerigk et al. (2022) [42] created several precise and approximation techniques for general problems after studying a class of robust two-stage optimal investment problems. For the gas and electricity transmission network affected by natural catastrophes, Yan et al. (2019) [43] built a three-level, two-stage resilient model. To tackle the problem, they employed the Benders decomposition and column and constraint generation two-level solution approach. Zeng and Zhao (2013) [44] developed a column-and-constraint generation algorithm to handle robust optimization problems in two stages. This method provided a unified strategy for managing optimality and feasibility and compared well with the current Benders cutting plane techniques. An et al. (2014) [39] demonstrated that this approach was superior to the Benders cutting plane method in their study on creating dependable p-median facility locating networks. They did this by using a customized upgraded strategy of column-and-constraint creation. Thus, considering the benefits of the C&CG method, we apply it to our suggested two-stage robust optimization model in this study.

Based on the above studies, there is little research that combines casualty classification, deprivation costs, and the selection of medical facility locations while considering the uncertainty of disasters. Additionally, there is little application of two-stage robust optimization methods in the construction of humanitarian logistics networks. Thus, the development of a humanitarian logistics network for casualty evacuation in the event of unforeseen dangers is taken into consideration in this study. The model is built using a two-stage robust optimization approach, and it is solved utilizing techniques for generating columns and constraints.

3. Model Construction

3.1. Problem Description

This study establishes a three-tiered casualty evacuation humanitarian logistics network composed of disaster sites, temporary medical points, and hospitals, as illustrated in Figure 1. Following a disaster, disaster sites experience a significant number of casualties. To minimize the transportation time for injured individuals, temporary medical points are established close to the disaster sites, while hospitals are existing facilities located further away from these sites.

Accurately estimating the quantity of casualties is challenging, so this study considers the uncertainty of casualty numbers. Additionally, the paper considers the classified evacuation of casualties. After a disaster, rescue teams must quickly assess the situation on the ground, including determining the number and severity of injuries. Rescue teams need to make rapid decisions with limited information, often involving quick onsite classification of casualties into two groups: those with mild casualties and serious casualties. Mild casualties usually refer to those who are less seriously hurt and not in life-threatening conditions; these individuals can be safely transported to temporary medical points for treatment. These temporary medical points are typically equipped with basic medical facilities and medicines sufficient to handle minor injuries and non-urgent medical needs. For severely injured individuals, due to their serious injuries which may be life-threatening, they are first transported to temporary medical points for essential emergency care such as hemorrhage control, pain management, and shock treatment to stabilize their condition. Once the initial treatment is administered, these severely injured individuals then need to be urgently transferred to well-equipped hospitals for more extensive treatment. In such scenarios, helicopters play a critical role as a fast transportation option, enabling the rapid transfer of severely injured individuals from remote or inaccessible locations to distant specialized medical institutions.

Table 2. Symbol Definitions.

Category	Symbol	Definition
Sets	$I$	The set of disaster sites, indexed by $i$
	$J$	The set of temporary medical points, indexed by $j$
	$H$	Set of hospitals, indexed by $h$
Parameters	$f_j$	Fixed facility opening costs $j$, where $j\in J$
	$c_{ij}$	Cost of unit transfer from disaster site $i$ to temporary medical point $j$, where $i\in I$, $j\in J$
	$w_{jh}$	Cost of unit transfer from temporary medical point $j$ to hospital $h$, where $j\in J$, $h\in H$
	$P_j$	Capacity of temporary medical point $j$, where $j\in J$
	$R_h$	Capacity of hospital $h$, where $h\in H$
	${\tilde{q}}_i$	The quantity of mild casualties at disaster site $i$, where $i\in I$
	${\tilde{n}}_i$	The quantity of serious casualties at disaster site $i$, where $i\in I$
	$t{c}_{ij}$	Travel time from disaster site $i$ to temporary medical point $j$, where $i\in I$, $j\in J$
	$c{t}_j$	Rescue time for serious casualties at temporary medical point $j$, where $j\in J$
	$t{h}_{jh}$	Travel time from temporary medical point $j$ to hospital $h$, where $j\in J$, $h\in H$
	$\alpha$	Minimum transportation percentage for mild casualty
	$\beta$	Minimum transportation percentage for serious casualty
	$\omega$	Infeasibility penalty coefficient
	M	A sufficient large number
	$g_1$, $g_2$, $g_3$, $g_4$, $h_1$, $h_2$, $h_3$, $h_4$	Coefficient of the deprivation cost function
Decision Variables	$y_j$	Binary variable, where $y_j=1$ indicates that temporary medical point is open, and $y_j=0$ indicates it is not open, $j\in J$
	$x_{ij}$	The quantity of mild casualties transported from disaster site $i$ to temporary medical point $j$, where $i\in I$, $j\in J$
	$z_{ijh}$	The quantity of serious casualties transported from disaster site $i$ through temporary medical point $j$ to hospital $h$, where $i\in I$, $j\in J$, $h\in H$
	$u_i$	The quantity of mild casualties not evacuated from disaster site $i$, where $i\in I$
	$v_i$	The quantity of serious casualties not evacuated from disaster site $i$, where $i\in I$

summarizes and introduces all the symbols used in this article.

3.2. Deprivation Costs

The paper measures the casualties’ suffering in terms of deprivation costs. The economic value of the pain endured by victims as a result of the absence of emergency assistance is known as deprivation costs. In the early stages of a disaster, the length of time the injured wait before arriving at a temporary medical point and receiving services can significantly exacerbate the damage they sustain. Since the condition of serious casualties worsens more severely over time, the increase in deprivation costs for serious casualties is greater than that for mild casualties. Upon reaching temporary medical points, the condition of mild casualties is stabilized, and their deprivation costs become zero. After receiving treatment, the deprivation costs for serious casualties decrease linearly. During the transport to comprehensive hospitals, the costs associated with deprivation due to serious casualties exhibit a novel, somewhat slower pattern of exponential increase. Referring to the research by Sun et al. (2022) [9], the following is the definition of the probability density functions for the deprivation costs of minor and major casualties:

$$f^1(t)=e^{g_{1}t}+e^{h_1},0<t\le t{c}_{ij}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J$$

(1)

$$f^2(t)=\left\{\begin{array}{c}{e}^{g_{2}t}+e^{h_2},0<t\le t{c}_{ij}\\ -g_{3}t+h_3,t{c}_{ij}<t\le t{c}_{ij}+c{t}_j\\ e^{g_{4}t}+e^{h_4},t{c}_{ij}+c{t}_j<t\le t{c}_{ij}+c{t}_j+t{h}_{jh}\end{array}\hspace{1em}\forall i\in I,j\in J,h\right.\in H$$

(2)

By applying the method of definite integrals, we calculated the loss costs of casualties, where the density function of the probability distribution of the loss costs of light and heavy casualties is shown in Figure 2.

Therefore, the deprivation cost functions for mild and serious casualties are represented by $F_{ij}^1$ and $F_{ijh}^2$, respectively:

$$\begin{array}{cc}{F}_{ij}^1\hfill & ={\displaystyle {\int }_0^{t{c}_{ij}}(e^{g_{1}t}+}{e}^{h_1})dt\hfill \\ & ={\displaystyle \frac{1}{g_1}}{e}^{g_{1}t{c}_{ij}}+e^{h_1}t{c}_{ij}-{\displaystyle \frac{1}{g_1}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}{F}_{ijh}^2\hfill & ={\displaystyle {\int }_0^{t{c}_{ij}}(e^{g_{2}t}+}{e}^{h_2})dt+{\displaystyle {\int }_{t{c}_{ij}}^{t{c}_{ij}+c{t}_j}(-g_{3}t+h_3)}dt+{\displaystyle {\int }_{t{c}_{ij}+c{t}_j}^{t{c}_{ij}+c{t}_j+t{h}_{ij}}(e^{g_{4}t}+e^{h_4})}dt\hfill \\ & ={\displaystyle \frac{1}{g_2}}{e}^{g_{2}t{c}_{ij}}+e^{h_2}t{c}_{ij}-{\displaystyle \frac{1}{g_2}}-{\displaystyle \frac{1}{2}}{g}_3{(c{t}_j)}^2-g_{3}t{c}_{ij}c{t}_j+h_{3}c{t}_j\hfill \\ & \hspace{1em}+{\displaystyle \frac{1}{g_4}}{e}^{g_4(t{c}_{ij}+c{t}_j+t{h}_{ij})}+e^{h4}(t{c}_{ij}+c{t}_j+t{h}_{ij})-{\displaystyle \frac{1}{g_4}}{e}^{g_4(t{c}_{ij}+c{t}_j)}-e^{h4}(t{c}_{ij}+c{t}_j)\hfill \end{array}$$

(4)

In facing large-scale disasters, a major challenge of emergency rescue missions is the impact of infrastructure damage and the capacity limitations of medical facilities, which may prevent the timely transfer of all casualties to medical institutions for treatment. To address this issue, the model integrates a penalty system for those casualties who fail to be transported promptly, aiming to motivate more effective resource allocation and operational optimization. By setting such penalties, the rescue teams are encouraged to optimize their emergency response plans, while also promoting the necessary expansion or enhancement of treatment facilities to reduce the quantity of casualties who do not receive timely treatment due to facility capacity limitations in future disaster responses.

3.3. Deterministic Model

This section describes a deterministic model that is based on the assumption that the quantity of light and heavy casualties at the disaster site is known and constant, and is designed to minimize the total cost. The costs in the first phase include the facility construction costs for temporary medical points and hospitals. In the second phase, the costs consisted mainly of transportation costs for the resuscitation of the wounded and deprivation costs for the minor and major wounded who were not able to access services in a timely manner, as well as deprivation costs for those minor and major wounded who were not resuscitated in a timely manner.

The total cost consists of four components:

• Fixed opening cost of temporary medical points and hospitals;
• Transportation cost;
• Deprivation cost of mild and serious casualties;
• Penalty for unevacuated casualties.

$$\begin{array}{l}\underset{fixed opening cost}{\underbrace{\underset{y,x,z,u,v}{\min }{\displaystyle \sum _{j\in J}{f}_j{y}_j}}}+\underset{transportation cost}{\underbrace{{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{c}_{ij}{x}_{ij}}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}(c_{ij}+w_{jh})z_{ijh}}}}}}\\ +\underset{deprivation cost}{\underbrace{{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{F}_{ij}^1{x}_{ij}}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{F}_{ijh}^2{z}_{ijh}}}}}}+\underset{penalty}{\underbrace{\omega {\displaystyle \sum _{i\in I}(u_i+v_i)}}}\end{array}$$

(5)

The objective function is designed to minimize the total system cost and can be broken down into four clearly distinguishable parts:

Part 1: Fixed cost of opening Temporary Medical Points: this term accounts for the one-time investment required to activate each selected candidate site as a functional medical facility.

Part 2: Transportation cost: this component covers two distinct movements. The first is the cost of evacuating all casualties (both mild and serious) from disaster-affected sites to their assigned Temporary Medical Points. The second is the additional cost incurred when serious casualties, after receiving initial stabilization at a Temporary Medical Point, must be transferred onward to a definitive-care hospital.

Part 3: Deprivation cost of delayed treatment: this part captures the human suffering caused by delayed medical care. It is split into the deprivation cost incurred by mild casualties (who are treated and remain at the Temporary Medical Points) and the considerably higher deprivation cost for serious casualties (who undergo initial treatment at a Temporary Medical Point and are subsequently moved to hospital). Both costs are convex increasing functions of the time elapsed from injury until definitive care is received, reflecting the much faster deterioration of serious casualties.

Part 4: Penalty for casualties left un-evacuated: a large artificial penalty is imposed on any mild or serious casualty that cannot be transported to a medical facility within the planning horizon. This term forces the model to prioritize near-complete coverage and rapid evacuation whenever it is feasible, thereby aligning the optimization with humanitarian rather than purely economic objectives.

By explicitly separating facility, transportation, deprivation, and coverage-penalty costs, the objective function strikes a balance between economic efficiency and humanitarian equity, which is critical for decision-making in life-and-death post-disaster environments.

Formula (5) can be simplified to Formula (6). Thus, the deterministic model for the casualty evacuation humanitarian logistics network (CEL-DM) can be represented by Formulas (6)–(16).

$$\underset{y,x,z,u,v}{\min }{\displaystyle \sum _{j\in J}{f}_j{y}_j}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}(c_{ij}+F_{ij}^1)x_{ij}}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}(c_{ij}+w_{jh}+F_{ijh}^2)z_{ijh}}}}+\omega {\displaystyle \sum _{i\in I}(u_i+v_i)}$$

(6)

s.t.

$${\displaystyle \sum _{i\in I}{x}_{ij}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{z}_{ijh}}}\le P_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(7)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{z}_{ijh}}}\le R_h\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall h\in H$$

(8)

$$x_{ij}\le M(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J$$

(9)

$$z_{ijh}\le M(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(10)

$${\displaystyle \sum _{j\in J}{x}_{ij}}+u_i\ge q_i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(11)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}}}+v_i\ge n_i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(12)

$${\displaystyle \sum _{j\in J}{x}_{ij}}\ge \alpha q_i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(13)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}}}\ge \beta n_i\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(14)

$$y_j\in \{0,1\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(15)

$$x_{ij},z_{ijh},u_i,v_i\ge 0 \mathrm{and} \mathrm{are} \mathrm{integer}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(16)

The worst-case costs are minimized using Objective Function (6). Constraints (7) and (8) are capacity constraints for temporary medical points and hospitals; constraints (10) and (11) ensure that only operational and non-disrupted temporary medical points and hospitals can provide services; constraints (12) and (13) recognize that not all casualties at a catastrophe scene will be fully evacuated; constraints (14) and (15) stipulate that the percentage of victims evacuated from each disaster site cannot be less than a specified value; The domain for second-stage decision variables is defined by constraint (16).

3.4. Construction of a Two-Stage Robust Optimization Model

3.4.1. Construction of the Uncertainty Set

(1)

Disruption Uncertainty

Due to the proximity of temporary hospitals to disaster areas, they face the risk of facility disruptions. This paper considers various scenarios related to disruptions at temporary hospitals to obtain more realistic solutions. This paper presents a two-stage resilient optimization model that does not require any probabilistic data by using a budgetary uncertainty set to represent possible damage scenarios. Assume that at most temporary medical points can experience disruptions simultaneously:

$$A=\left\{{\phi }_j\in {\{0,1\}}^{\left|J\right|}:{\displaystyle \sum _{j\in J}{\phi }_j\le m}\right\}$$

where ${\phi }_j=1$. if temporary medical point $j$ is disrupted; otherwise, ${\phi }_j=0$.

(2)

Casualty Number Uncertainty

Due to the uncertainty of casualty numbers, this paper introduces an uncertainty set for the quantity of casualties, providing ranges for the quantity of mild and serious casualties in each disaster area. In practical terms, the information regarding uncertain parameters that is easiest to obtain is the upper and lower bounds. Consequently, the uncertainty set for casualties can be created using these variables. It is assumed that the uncertain numbers of mild and serious casualties, respectively, fall within a range, so ${\tilde{q}}_i$ and ${\tilde{n}}_i$ vary within the intervals ${\tilde{q}}_i\in [{\overline{q}}_i-{\widehat{q}}_i,{\overline{q}}_i+{\widehat{q}}_i]$ and ${\tilde{n}}_i\in [{\overline{n}}_i-{\widehat{n}}_i,{\overline{n}}_i+{\widehat{n}}_i]$, where ${\overline{q}}_i,{\overline{n}}_i$ represent the nominal values, and ${\widehat{q}}_i,{\widehat{n}}_i$ indicate the highest deviations from these stated values. $q_i$ and $n_i$ denote the maximum numbers of mild and serious casualties at disaster site $i$. Since the two-stage robust optimization method solves for the worst-case costs, the paper only considers the positive deviations of uncertain casualty numbers, thus only the intervals between ${\overline{q}}_i$ and ${\overline{q}}_i+{\widehat{q}}_i$, and ${\overline{n}}_i$, ${\overline{n}}_i+{\widehat{n}}_i$ are considered. Therefore, the construction of the budget uncertainty sets $Q_{\xi }$ and $N_{\rho }$ for uncertain casualty numbers is as follows:

$$Q_{\xi }=\left\{q\in {ℝ}^I:q_i={\overline{q}}_i+{\widehat{q}}_i{\mathrm{\Gamma }}_i^1,0\le {\mathrm{\Gamma }}_i^1\le 1\right\}$$

(17)

$$N_{\rho }=\left\{n\in {ℝ}^I:n_i={\overline{n}}_i+{\widehat{n}}_i{\mathrm{\Gamma }}_i^2,0\le {\mathrm{\Gamma }}_i^2\le 1\right\}$$

(18)

In this case, the uncertainty budgets ${\mathrm{\Gamma }}_i^1$ and ${\mathrm{\Gamma }}_i^2$ set a maximum number of casualties, both mild and serious, that can differ from their nominal values. These uncertainty budgets are used to assess the feasibility and conservatism of a solution. Following the standard budgeted uncertainty framework, the uncertainty budgets ${\mathrm{\Gamma }}_i^1$ and ${\mathrm{\Gamma }}_i^2$ are restricted to the finite ranges:

$${\mathrm{\Gamma }}_i^1,{\mathrm{\Gamma }}_i^2\in \left[0,1\right]$$

Here, ${\mathrm{\Gamma }}_i^1,{\mathrm{\Gamma }}_i^2=0$ represent the nominal scenarios, while ${\mathrm{\Gamma }}_i^1,{\mathrm{\Gamma }}_i^2=1$ represent the worst-case scenarios. Increasing ${\mathrm{\Gamma }}_i^1$ and ${\mathrm{\Gamma }}_i^2$ will result in solutions that are more cautious and less risky, as the model encompasses a higher level of uncertainty. Depending on their level of risk tolerance, decision-makers can choose the proper values for ${\mathrm{\Gamma }}_i^1$ and ${\mathrm{\Gamma }}_i^2$. Generally, decision-makers that are risk averse typically choose higher values.

3.4.2. Two-Stage Robust Optimization Model

The costs in the first stage are for the construction of facilities at temporary medical points and hospitals. The costs in the second stage include the transportation costs for rescuing casualties and the deprivation costs for both mild and serious casualties, as well as the penalty costs for casualties not rescued. Therefore, under uncertain conditions and considering classified casualty evacuation, the two-stage robust optimization model (CEL-TRO) can be reformulated as Formulas (19)–(29):

$$\underset{y}{\min }{\displaystyle \sum _{j\in J}{f}_j{y}_j}+\underset{\phi \in A}{\max }\underset{x,z,u,v\in S(y,\phi )}{\min }\left(\begin{array}{l}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}(c_{ij}+F_{ij}^1)x_{ij}+\omega {\displaystyle \sum _{i\in I}(u_i+v_i)}}}\\ +{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}(c_{ij}+w_{jh}+F_{ijh}^2)z_{ijh}}}}\end{array}\right)$$

(19)

s.t.

$$y_j\in \{0,1\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(20)

where $S(y,\phi )=\{$

$${\displaystyle \sum _{i\in I}{x}_{ij}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{z}_{ijh}}}\le P_j(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(21)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{z}_{ijh}}}\le R_h\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall h\in H$$

(22)

$$z_{ijh}\le M(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(23)

$${\displaystyle \sum _{j\in J}{x}_{ij}}+u_i\ge {\overline{q}}_i+{\widehat{q}}_i{\mathrm{\Gamma }}_i^1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(24)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}}}+v_i\ge {\overline{n}}_i+{\widehat{n}}_i{\mathrm{\Gamma }}_i^2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(25)

$${\displaystyle \sum _{j\in J}{x}_{ij}}\ge \alpha ({\overline{q}}_i+{\widehat{q}}_i{\mathrm{\Gamma }}_i^1) \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall \forall i\in I$$

(26)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}}}\ge \beta ({\overline{n}}_i+{\widehat{n}}_i{\mathrm{\Gamma }}_i^2) \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall \forall i\in I$$

(27)

$$x_{ij},z_{ijh},u_i,v_i\ge 0 \mathrm{and} \mathrm{are} \mathrm{integer}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall \forall i\in I,j\in J,h\in H\}$$

(28)

The worst-case cost is what the objective function (19) seeks to reduce. Under a given facility location y, The disruptive scenario in set A with the highest recourse cost is represented by max, while the solution with the lowest determined cost is represented by min. constraints (21) and (22) are the capacity limits for temporary medical points and hospitals; constraint (23) ensures that only operational and non-disrupted temporary medical points and hospitals can provide services; constraints (24) and (25) indicate that at each disaster site, it may not be possible to evacuate all of a portion of the casualties; Constraints (26) and (27) mandate that the proportion of casualties removed from every disaster location must not fall below a given threshold. The domain for second-stage decision variables is defined by constraint (28).

4. C&CG Algorithm for Solving

A particular sort of multi-stage robust optimization problem is two-stage robust optimization. Referring to the research by Sun et al. (2022) [10], there are two stages to the decision-making process: the first decision and the subsequent decision. The parameters related to the first-stage decisions are deterministic, and these decisions must be made initially; some parameters related to the second-stage decisions are uncertain, and these determinations are formulated subsequent to the establishment of the first-stage decisions, upon the revelation of the second-stage parameters. The objective of two-stage robust optimization is to jointly optimize both stages of decisions while accounting for the uncertainty in the second-stage parameters. This involves optimizing for the worst-case scenario of the total objective value corresponding to both stages of decisions given the second-stage parameters.

Solving two-stage robust optimization models is usually difficult. The Benders decomposition approach is inefficient for large-scale problems and requires linear problems in order to obtain optimal solutions for the second-stage problems. In recent years, two-stage robust optimization models have been designed and solved using the Column-and-Constraint Generation (C&CG) algorithm. It has shown good performance in unit commitment problems and P-median facility location problems. Therefore, this paper employs the C&CG algorithm to solve the proposed CEL-TRO model.

The C&CG algorithm is executed within a master-subproblem structure. Within the subproblem, the solution derived from the master problem is given, and the remaining min-max optimization issue must be tackled. Due to the penalty imposed on unmet demands in disruption scenarios, the second-stage problem remains feasible at all times. Consequently, we obtain its dual representation, leading to a max-max formulation that may be combined into a single maximizing problem.

4.1. Master Problem

The C&CG algorithm operates within a master-subproblem framework. Denoting $\kappa$ as the iteration number, the master problem (MP) of the CEL-TRO model can be formulated using formulas (29)–(40):

MP:

$$\min {\displaystyle \sum _{j\in J}{f}_j{y}_j+}\varphi$$

(29)

s.t.

$$\varphi \ge {\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}(c_{ij}+F_{ij}^1)x_{ij}^{\kappa }}}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}(c_{ij}+w_{jh}+F_{ijh}^2)z_{ijh}^{\kappa }}}}+\omega {\displaystyle \sum _{i\in I}(u_i^{\kappa }+v_i^{\kappa })\hspace{1em}\hspace{1em}\forall \kappa \in {\rm K}}$$

(30)

$${\displaystyle \sum _{i\in I}{x}_{ij}^{\kappa }}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{z}_{ijh}^{\kappa }}}\le P_j(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J,\forall \kappa \in {\rm K}$$

(31)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{z}_{ijh}^{\kappa }}}\le R_h\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall h\in H,\forall \kappa \in {\rm K}$$

(32)

$$x_{ij}^{\kappa }\le M(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,\forall \kappa \in {\rm K}$$

(33)

$$z_{ijh}^{\kappa }\le M(1-{\phi }_j)y_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,\forall \kappa \in {\rm K}$$

(34)

$${\displaystyle \sum _{j\in J}{x}_{ij}^{\kappa }}+u_i^{\kappa }\ge {\overline{q}}_i+{\widehat{q}}_i{\mathrm{\Gamma }}_i^{1\kappa }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,\forall \kappa \in {\rm K}$$

(35)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}^{\kappa }}}+v_i^{\kappa }\ge {\overline{n}}_i+{\widehat{n}}_i{\mathrm{\Gamma }}_i^{2\kappa }\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,\forall \kappa \in {\rm K}$$

(36)

$${\displaystyle \sum _{j\in J}{x}_{ij}^{\kappa }}\ge \alpha ({\overline{q}}_i+{\widehat{q}}_i{\mathrm{\Gamma }}_i^{1\kappa }) \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,\forall \kappa \in {\rm K}$$

(37)

$${\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{z}_{ijh}^{\kappa }}}\ge \beta ({\overline{n}}_i+{\widehat{n}}_i{\mathrm{\Gamma }}_i^{2\kappa }) \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,\forall \kappa \in {\rm K}$$

(38)

$$y_j\in \{0,1\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(39)

$$x_{ij}^{\kappa },z_{ijh}^{\kappa },u_i^{\kappa },v_i^{\kappa }\ge 0 \mathrm{and} \mathrm{are} \mathrm{integer}\forall i\in I,j\in J,h\in H,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall \kappa \in {\rm K}$$

(40)

4.2. The Dual Problem of the Second Stage

The subproblem deals with the remaining max-min problem, while the solution to the master problem is known. In disruption scenarios, since unmet demands incur penalties, the second-stage problem is always feasible. Consequently, this problem’s dual is identified, giving rise to a max-max problem that can be combined into a single maximization problem. Dual variables ${\pi }^1,{\pi }^2,{\gamma }^1,{\lambda }^1,{\lambda }^2,{\eta }^1,{\eta }^2$ are introduced for constraints (21)–(27). The dual of the subproblem is formulated as follows in Formulas (41)–(47):

$$\begin{array}{l}\max {\displaystyle \sum _{j\in J}{Q}_j(1-{\phi }_j){\widehat{y}}_j{\pi }_j^1}+{\displaystyle \sum _{h\in H}+R_h{\pi }_h^2}+{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{M}_z(1-{\phi }_j){\widehat{y}}_j{\gamma }_{ijh}^1}}}\\ \hspace{1em}+{\displaystyle \sum _{i\in I}({\overline{q}}_i+{\mathsf{\Gamma }}_i^1{\widehat{q}}_i)({\lambda }_i^1}+\alpha {\eta }_i^1)+{\displaystyle \sum _{i\in I}({\overline{n}}_i+{\mathsf{\Gamma }}_i^2{\widehat{n}}_i)({\lambda }_i^2}+\beta {\eta }_i^2)\end{array}$$

(41)

s.t.

$${\pi }_j^1+{\lambda }_i^1+{\eta }_i^1\le c_{ij}+F_{ij}^1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J$$

(42)

$${\pi }_j^1+{\pi }_h^2+{\gamma }_{ijh}^1+{\lambda }_i^2+{\eta }_i^2\le c_{ij}+w_{jh}+F_{ijh}^2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(43)

$${\lambda }_i^1\le \omega \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(44)

$${\lambda }_i^2\le \omega \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(45)

$${\pi }_j^1,{\pi }_h^2,{\gamma }_{ijh}^1\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(46)

$${\lambda }_i^1,{\lambda }_i^2,{\eta }_i^1,{\eta }_i^2\ge 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I$$

(47)

Due to ${\gamma }_{ij},{\gamma }_{ijh}\le 0$ and ${\phi }_j\in \{0,1\}$, the nonlinear term ${\phi }_j{\gamma }_{ij}^1,{\phi }_j{\gamma }_{ijh}^2$ are the product of binary variables and continuous variables. After introducing two new continuous variables $l_j^1={\phi }_j{\pi }_j^1,l_{ijh}^2={\phi }_j{\gamma }_{ijh}^1$, we apply the Big M method to reformulate the nonlinear expressions into a tractable linear form; the resulting constraints are presented below.

$$l_j^1\ge {\pi }_j^1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(48)

$$l_{ijh}^2\ge {\gamma }_{ijh}^1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(49)

$$l_j^1\ge -M_l{\phi }_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(50)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{l}_{ijh}^2}}\ge -M_l{\phi }_j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(51)

$$l_j^1\le {\pi }_j^1+M_l(1-{\phi }_j)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(52)

$${\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{l}_{ijh}^2}}\le {\displaystyle \sum _{i\in I}{\displaystyle \sum _{h\in H}{\gamma }_{ijh}^1}}+M_l(1-{\phi }_j)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall j\in J$$

(53)

$$l_j^1,l_{ijh}^2\le 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in I,j\in J,h\in H$$

(54)

Therefore, the subproblem of linearization is:

SP:

$$\begin{array}{l}\chi =\max {\displaystyle \sum _{j\in J}{Q}_j{\widehat{y}}_j({\pi }_j^1-l_j^1})+{\displaystyle \sum _{h\in H}{R}_h{\pi }_h^2}++{\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}{\displaystyle \sum _{h\in H}{M}_l{\widehat{y}}_j(r_{ijh}^1-l_j^2){\widehat{y}}_j}}}\\ +{\displaystyle \sum _{i\in I}({\overline{q}}_i+{\mathsf{\Gamma }}_i^1{\widehat{q}}_i)({\lambda }_i^1}+\alpha {\eta }_i^1)+{\displaystyle \sum _{i\in I}({\overline{n}}_i+{\mathsf{\Gamma }}_i^2{\widehat{n}}_i)({\lambda }_i^2}+\beta {\eta }_i^2)\end{array}$$

(55)

The constraints are (42)–(54).

4.3. Framework of C&CG Algorithm

Once the model is transformed, the C&CG algorithm is employed to tackle the computational aspect. The main problem (MP) is iteratively solved. In each iteration $\kappa$, the worst-case scenario ${\widehat{A}}^{\kappa }$ is determined through linearizing the subproblem. Then, the recourse variables $(x^{\kappa },z^{\kappa },u^{\kappa },v^{\kappa })$ and corresponding constraints for this specific scenario are created and added to the constraints of MP. Let LB and UB denote the lower and upper bounds, respectively, $Gap=(UB-LB)/UB$. Set the optimal tolerance as $ϵ$.

The C&CG algorithm step is as Algorithm 1:

Algorithm 1: the C&CG algorithm

Initialization: Let $\kappa =0$, Set $LB=-\infty$, $UB=+\infty$. Take any feasible solution $\widehat{y}\in \{0,1\}$ as the initial solution. Step 1: Solve the linearized subproblem with respect to $\widehat{y}$ to determine the worst-case scenario for $\widehat{\phi }$. Update $\kappa =\kappa +1$. Establish the recourse variables $(x^{\kappa },z^{\kappa },u^{\kappa },v^{\kappa })$ and their respective constraints, and incorporate them into the MP model. Step 2: Solve the MP to find the optimal solution $(\widehat{y},\varphi )$. Assign the optimal value of the MP to the variable LB. Step 3: Solve the SP with respect to $\widehat{y}$ to determine the worst-case scenario for $\widehat{\phi }$. Update: $UB=\min \{UB,{\displaystyle \sum _{j\in J_0}{f}_j{y}_j^{\kappa }}+{\chi }^{\kappa }\}$ Step 4: If $Gap\le ϵ$, terminate; otherwise, update $\kappa =\kappa +1$ and establish recourse variables and their associated constraints. Add them to the MP and then proceed to step 2.

The flowchart of the C&CG algorithm is illustrated in Figure 3.

5. Numerical Experiments

This paper examines the impact of pertinent parameters, provides some managerial insights, and confirms the model and algorithm performance in light of the 2008 Wenchuan earthquake in Sichuan Province, China. The primary data for the numerical experiments are based on the case study from Sun et al. (2022) [10]. The algorithm is implemented in Python 3.9, using the Gurobi Optimizer version 11.0.1 as the solver. The experiments were conducted on a Windows operating system with a 12th Gen Intel(R) Core (TM) i5-12500H CPU and 64 GB of memory.

5.1. Case Study Description

On 12 May 2008, a magnitude 8.0 earthquake hit Wenchuan County, Sichuan Province, widely referred to as the “Wenchuan Earthquake”. This earthquake caused extreme destruction and loss. Official statistics report approximately 69,000 deaths, 370,000 injuries, and nearly 18 million people affected. The disaster area was extensive, including Wenchuan County and surrounding counties and cities. The terrain of Sichuan Province, predominantly mountainous and hilly, posed additional challenges for the rescue efforts. The primary disaster sites included not only Wenchuan County but also Dujiangyan, Qingchuan, Beichuan, and other areas where infrastructure suffered significant damage, especially in transport, communications, and power systems. The earthquake led to massive building collapses and severe infrastructure damage. Many roads, bridges, and tunnels were destroyed, making initial rescue efforts extremely difficult. Particularly in some remote mountain areas, rescue teams struggled to reach due to disrupted transport and communications.

The earthquake also triggered numerous aftershocks and secondary disasters. We select ten towns severely affected by the disaster as shown in Figure 4. Each disaster site had both mild and severe casualties.

After assessing ten towns with relatively minor damage, these locations were designated as candidates for temporary medical points. Additionally, four medical facilities, located further from the epicenter and with more comprehensive facilities, were selected as candidate hospitals. Disaster sites are marked as I1 to I10, candidate temporary medical points as J1 to J10, and candidate hospitals as H1 to H4. Each temporary medical point and each candidate hospital are set to have a treatment capacity of 700 and 600 people, respectively. Other parameters are set with reference to the work by Sun et al. (2022) [10]. Moreover, it is important to note that, in the early stage of emergency response after a major disaster, decision-makers often operate under conditions of “extreme information scarcity.” Although uncertainties across different locations may indeed vary in theory—for example, areas closer to the epicenter may exhibit greater fluctuations in casualty numbers—obtaining sufficient real-time data to accurately quantify such differences is extremely difficult. Under these circumstances, adopting a uniform and conservative budget level becomes a common and pragmatic risk-management strategy. It avoids subjective parameter estimation under severe time pressure and makes the model more interpretable and operational for emergency command authorities.

5.2. Results and Analysis of the C&CG Algorithm

For the real-world Wenchuan case (10 disaster sites, 10 candidate Temporary Medical Points, 4 hospitals), the C&CG algorithm requires at most 14 iterations (average 8.6) across all uncertainty budgets and solves the problem in 42.7 s on average. Convergence is rapid: the optimality gap falls below 1% within 9 iterations (Figure 5a).

Scalability is further assessed on larger random instances generated as follows: disaster sites and candidate Temporary Medical Points are uniformly distributed in a 200 km × 200 km square area; the number of disaster sites |I| ranges from 20 to 50, with |J| ≈ |I| and |H| ≈ 0.3|I|; mild and serious casualty demands at each site are independently drawn from U[300, 800] and U[150, 400]; all cost parameters, deprivation functions, and disruption probabilities remain identical to the Wenchuan case. Even the largest instance (|I| = 50, |J| = 40, |H| = 12, ≈178,000 variables and 265,000 constraints) is solved to proven optimality in under 40 min, with the number of iterations never exceeding 31 (Figure 5b,c). These results confirm that the proposed C&CG algorithm scales excellently and is fully capable of handling large-scale real-world disaster scenarios on standard computing hardware.

The best solution in the uninterrupted scenario is shown in the figure. We have, respectively, formulated differentiated transportation strategies for minor and severe injuries, and presented them visually in Figure 6 and Figure 7.

The arrow line segments in the figure represent the routes of the wounded from the disaster-stricken area, passing through the temporary medical stations and finally being transferred to the hospital. Among them, Figure 6 shows the transportation routes of minor casualties, while Figure 7 corresponds to the transfer situation of severe casualties. Overall, the model gives priority to temporary medical facilities that are geographically concentrated, close to hospitals and have good connectivity in order to shorten the transportation distance and reduce the deprivation cost.

According to the results of

Table 3. Temporary Medical Point Opening Status.

	J1	J2	J3	J4	J5	J6	J7	J8	J9	J10
$\hat{y}$	1	1	1	1	1	1	1	1	0	0

, a total of 8 temporary medical points (J1–J8) were ultimately activated, among which J9 and J10 were not activated. Meanwhile, all four hospitals participated in the rescue mission, providing sufficient terminal receiving capacity for the evacuation mission. The transportation plan is reasonably designed. The pressure is shared collaboratively among the various temporary medical points, and there are no phenomena of single-point overload or overly long routes.

Table 4. Quantity of Untransformed Mild and Serious Casualties.

	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
$u_i$	0	0	0	0	0	0	0	0	0	0
$v_i$	0	0	0	0	0	0	0	0	0	0

further shows the number of wounded people who were not transported. The results show that in this scenario, all minor and severe casualties were successfully transferred to medical facilities, and there were no cases of non-transfer due to capacity or route limitations. This result indicates that under the assumption of adequate resource allocation and no facility interruption, the scheduling scheme provided by the model can achieve the goal of medical rescue covering all personnel, taking into account both transportation efficiency and humanitarian care.

The quantity of untransformed mild and serious casualties is shown in Table 4. Since this paper constructs a humanitarian logistics network aimed at alleviating the suffering of the affected population, deprivation costs are introduced in the decision-making objectives to measure the suffering of casualties. Furthermore, the penalty costs for not transporting casualties are higher. Therefore, it is calculated that the quantity of both untransformed mild and serious casualties is zero.

5.3. Comparison Between the CEL-TRO and CEL-DM

To validate the effectiveness of the deterministic model presented in this paper and to compare the advantages of the proposed two-stage robust optimization model and the C&CG algorithm, the deterministic model was solved using Gurobi. After that, the outcomes were contrasted with the ones derived from the C&CG algorithm’s solution of the two-stage robust optimization model. The comparative results are shown in

Table 5. Comparison of Results Between the CEL-TRO and CEL-DM.

Model	Optimal Solution of Objective Function (×106 Yuan)	Number of Temporary Medical Points Opened	CPU Time(s)
CE-TRO	94.4	8	2.25
CE-DM	71.3	5	0.40

. From the table, it is evident that the total cost derived from the two-stage robust optimization model is higher than that of the deterministic model, and the quantity of temporary medical points opened is also greater. This is because the two-stage robust optimization seeks the optimal solution under the worst-case scenario, resulting in solutions that exhibit strong robustness. As a result, in real-world applications, accounting for uncertainties and potential facility failures can lead to saving more lives and enhancing the effectiveness of rescue operations.

5.4. Parameter Sensitivity Analysis

This section undertakes a sensitivity analysis to assess the responsiveness of the pertinent parameters within the two-stage robust optimization model outlined in this article, analyzing the impact of the uncertainty budget, the maximum number of facilities allowed to be disrupted, and the capacity of temporary medical points on the total cost.

5.4.1. The Impact of Uncertain Budgeting on Deprivation Costs

In this subsection, we analyze the impact of joint fluctuations in both mild and serious casualties using an integrated uncertainty budget framework. This approach reflects a global perspective on system-wide uncertainty, evaluating how simultaneous increases in casualty numbers affect total and deprivation costs. Unlike later subsections that isolate specific casualty types, this analysis captures the compounded burden imposed by large-scale disasters, providing insight into the system’s robustness under collective stress.

The uncertainty budget is used to measure the feasibility and conservatism of a solution; the larger the uncertainty budget, because of the high level of uncertainty involved in the model, a conservative approach was chosen. Decision-makers can adjust this according to their attitude towards risk. To determine the impact of the uncertainty budget on the model, other parameter values were held constant, and the uncertainty budget was varied at intervals: 0, 0.2, 0.4, 0.6, 0.8, and 1. These values were input into the C&CG algorithm for computation. The relationship between different uncertainty budgets and total cost is shown in Figure 8. Furthermore, the deprivation costs for moderate and serious casualties under various amounts of uncertainty budgets were computed in order to examine the effect of the uncertainty budget on deprivation costs, as shown in Figure 9.

Firstly, it is evident that an increase in the uncertainty budget leads to an increase in total costs. This increase is due to the rising uncertainty in the quantity of casualties, which leads to higher deprivation costs. Secondly, when the uncertainty parameter increases from 0.8 to 1, the rate of increase in total costs accelerates because the marginal costs of the system rise in the face of higher uncertainty. Therefore, decision-makers can choose an appropriate uncertainty budget based on their risk preferences.

In summary, the results confirm that serious casualties contribute more significantly to deprivation costs under comprehensive uncertainty. This finding justifies the importance of prioritizing resources for serious casualties in strategic planning. To further disentangle the influence of each casualty type, the following subsection adopts a controlled variable approach to analyze the marginal impact of each type of casualty fluctuation on total costs.

5.4.2. The Influence of Casualty Fluctuations on the Total Cost

Based on the combined casualty fluctuation analysis in Section 5.4.1, this section adopts a controlled-variable approach to explore the impact of mild and serious casualty fluctuations on the total cost. Specifically, we keep the quantity of one type of casualty constant while allowing the other type to fluctuate within a certain range, revealing the marginal sensitivity of total cost to each severity level. This approach helps identify which type of casualty imposes greater cost pressure on the logistics system and provides insights into prioritization decisions in evacuation planning.

In the numerical hypothesis part, it is not difficult to find that the total capacity of the temporary medical center exceeds the combined number of mild and serious casualties. It can accommodate fluctuations of approximately 40% in the total number of casualties. Additionally, the hospital’s capacity is greater than the number of serious casualties alone. Therefore, this study sets up 13 different scenarios by adjusting the fluctuation ranges of mild and serious casualties to explore how variations in different types of casualties affect the total cost. Details of each scenario are shown in

Table 6. The fluctuation of casualties in different scenarios.

Scenario	${∆}_1$ (Mild Casualties)	${∆}_2$ (Serious Casualties)	Total Cost
S1	+0%	+0%	80,090,770
S2	−30%	+0%	69,984,170
S3	−20%	+0%	73,371,960
S4	−10%	+0%	76,733,780
S5	+10%	+0%	83,618,630
S6	+20%	+0%	87,832,840
S7	+30%	+0%	92,272,730
S8	+0%	−30%	58,042,730
S9	+0%	−20%	65,140,065
S10	+0%	−10%	72,843,740
S11	+0%	+10%	87,773,655
S12	+0%	+20%	95,934,430
S13	+0%	+30%	104,245,565

.

For each scenario, intuitive graphs are drawn to visualize the impact of fluctuations in the number of casualties on the total cost. Since the effects and trends are nonlinear, this study adopts quadratic polynomial regression to explore the relationship between changes in the number of mild and serious casualties and the total cost. By setting the percentage change in personnel (−30% to +30%) as the dependent variable and the total cost as the independent variable, separate quadratic curve fitting was performed for the sample data of mild and serious casualties.

The fitting results reveal a clear nonlinear trend for both types of casualties, and the goodness-of-fit values R² for mild and serious casualties reached 0.9997 and 0.9999, respectively, indicating that the quadratic model effectively captures the relationship between casualty fluctuations and total cost.

Specifically, for mild casualties, the corresponding fitting function and the corresponding goodness of fit R² are:

$$\mathrm{y}=-2.3605\times {10}^{-14}\cdot {\mathrm{x}}^2+6.5465\times {10}^{-6}\cdot \mathrm{x}-372.9085$$

$${\mathrm{R}}^2=0.9997$$

For serious casualties, the corresponding fitting function and the corresponding goodness of fit R² are:

$$\mathrm{y}=-2.4142\times {10}^{-15}\cdot {\mathrm{x}}^2+1.6922\times {10}^{-6}\cdot \mathrm{x}-120.1097$$

$${\mathrm{R}}^2=0.9999$$

From the slopes and curvature of the fitted functions as showed in Figure 10, it is evident that serious casualties exert a greater marginal impact on total cost. This implies that fluctuations in serious casualties require more attention in evacuation planning, as they may necessitate adjustments in temporary medical point deployment, routing, and hospital transfers. While the model does not explicitly optimize for fairness across severity levels, these results highlight a natural tradeoff in resource allocation: prioritizing serious casualties can reduce total system cost and mitigate the most critical deprivation risks, whereas focusing on mild casualties alone has a smaller impact on overall efficiency.

Overall, this analysis provides planners with insights into how different casualty levels influence total cost and helps inform strategic decisions about capacity allocation, prioritization, and the robustness of evacuation strategies under varying disaster scenarios.

5.4.3. Maximum Quantity of Disrupted Facilities

Due to possible aftershocks or secondary disasters following an earthquake, temporary medical facilities may suffer partial or complete operational disruptions. To capture this uncertainty, this study models facility failures by assuming that up to m facilities may become unavailable. Based on the two-stage robust optimization framework, the influence of different disruption levels (m = 1, 2, 3, 4) is analyzed while keeping other parameters constant. Using the C&CG algorithm, we evaluate how evacuation feasibility and deprivation-related service delay costs respond to these disruptions. As shown in Figure 11, the total cost—including transportation cost and deprivation cost—monotonically increases as m grows. This is because disrupted facilities force evacuees to be rerouted to more distant facilities, increasing travel time, delaying treatment, and consequently raising deprivation costs.

To enhance evacuation resilience, the model strategically selects additional temporary medical points, particularly those located on critical evacuation paths, to mitigate the impact of facility disruptions. As the disruption budget m increases, the model tends to open more facilities with a more dispersed spatial distribution, reducing the risk that multiple adjacent disruptions will overwhelm the system. For example, when m = 3, the model avoids concentrating too many casualties at neighboring points such as J2, J4, and J7, and instead activates points like J6 and J8, creating geographic redundancy and improving the overall network’s ability to withstand interruptions.

Overall, the results in Figure 11 demonstrate that although larger-scale disruptions inevitably increase total costs, this strategic allocation of key and geographically dispersed temporary medical points enhances the robustness of the evacuation network and mitigates excessive increases in deprivation costs.

5.4.4. Capacity of Temporary Medical Points

Given the variety of capacity options for temporary medical points during disasters, this paper examines how different capacities affect the total costs and deprivation costs. It also compares the changes in total costs between the deterministic model and the two-stage robust optimization model under varying capacities. The capacities of temporary medical points were set to: 0.8 P, 0.9 P, P, 1.1 P, and 1.2 P, with corresponding changes made to the opening costs of temporary medical points as their capacities changed. With other parameters fixed, the deterministic model was solved using Gurobi, and the two-stage robust optimization model was processed through the C&CG algorithm. The impacts of temporary medical point capacity on total costs and deprivation costs are illustrated in Figure 12 and Figure 13.

The results show that increasing the capacity of temporary medical points effectively reduces both the total costs and deprivation costs. As the capacity of temporary medical points changes, the total costs in the two-stage robust optimization model are more sensitive compared to those in the deterministic model. In addition, the construction of more temporary medical facilities can also significantly lower the expense of treating injured patients. As a result, increasing hospital capacity can dramatically reduce the expenses associated with deprivation for all casualties. In particular, the ability to improve the treatment of trauma at temporary medical sites is much better than in hospitals in terms of reducing deprivation costs. Therefore, administrators should focus on expanding the capacity of temporary medical sites to reduce ambulance costs.

5.4.5. Capacity of Hospitals

The four hospitals serving the Wenchuan area have a combined nominal capacity of 2400 beds (600 beds each). In practice, however, large earthquakes routinely overwhelm hospital resources through a combination of casualty surge, structural damage, and the continuing need to treat regular patients. To examine how strongly the performance of the proposed evacuation plan depends on hospital capacity, and to place it in direct comparison with the temporary medical point capacity results reported in Section 5.4.4, we scale the bed capacity of all four hospitals uniformly by a factor ρ ranging from 0.5 to 1.5 while holding every other parameter fixed.

Figure 14 shows the resulting total system cost and the number of serious casualties that cannot be admitted. Because only serious casualties require definitive hospital treatment, any shortfall is simply max{0, 1484–2400ρ}. A unit penalty of 50,000 per unadmitted serious casualty is applied to reflect the rapid escalation of deprivation cost once hospital beds are exhausted.

A sharp threshold is immediately apparent. When ρ falls below roughly 0.618 (i.e., total available beds drop below 1484), the system begins to leave serious casualties without definitive care, and the total cost climbs steeply. At ρ = 0.5, for example, 742 serious casualties cannot be admitted and the total cost rises by 37.1 million (46.3%) above the baseline of 80.09 million. Once ρ reaches 0.618 or higher, every serious casualty is accommodated, the penalty vanishes, and the total cost plateaus at its minimum.

The contrast with the temporary medical point capacity experiment in Section 5.4.4 is striking. Cutting temporary medical point capacity to 70% of its nominal value raises cost by only 5–8% and does so gradually, whereas hospital capacity exhibits a classic “cliff-edge” pattern: moderate shortages are tolerable, but once the critical threshold is crossed the penalty is severe and immediate. Equally important, expanding hospital capacity beyond the point at which all serious casualties are covered (ρ ≈ 0.618) brings no further savings, whereas additional temporary medical point capacity continues to yield modest but steady reductions through shorter waiting and transport times for mild casualties.

The practical implication is straightforward. In an event producing around 1500 serious casualties, the highest early-response priority must be to secure at least 1500 effective hospital beds (via field hospitals, corridor beds, inter-regional transfers, or accelerated discharge of non-critical patients). Only after that threshold is passed should additional resources be directed toward reinforcing the pre-hospital network and temporary medical point capabilities.

The results stand in clear contrast to the sensitivity analysis of temporary medical points capacity in Section 5.4.4. Hospital capacity shows a pronounced cliff-edge behavior: as soon as the total number of available beds drops below the 1484 serious casualties requiring definitive treatment, the total cost rises steeply because of the heavy per-casualty deprivation penalty. Reducing the capacity of temporary medical points, however, causes only gradual and relatively modest cost increases, even down to 70% of the nominal level, because mild casualties can still be treated on site and serious casualties simply experience longer waiting times at the temporary medical points. Furthermore, once hospital capacity meets the demand of serious casualties (ρ ≈ 0.618), any further expansion yields no additional benefit, whereas extra capacity at temporary medical points continues to deliver marginal reductions in transportation and waiting times for mild casualties. In the scenario studied here, hospital beds therefore constitute the true binding constraint of the entire evacuation system, while the capacity of temporary medical points primarily influences service quality rather than feasibility.

5.4.6. Analysis of the Reception of Casualties in Temporary Medical Facilities

This section mainly focuses on studying the changes in the capacity of temporary medical facilities as the number of severe and minor casualties fluctuates. In the vast majority of interruption scenarios, neither temporary medical facilities J9 nor J10 are open. Meanwhile, since temporary medical facilities J2, J4, and J7 are at risk of interruption, we focus on analyzing the changes in the capacity of the remaining temporary medical facilities to accommodate the injured, as this is more universal.

Firstly, in the case of fluctuations in the number of seriously injured patients, the changes in the capacity of each considered temporary medical center. Figure 15 shows that the accommodation conditions of the temporary medical facilities J1, J3 and J8 have basically remained unchanged. Regardless of how the number of wounded fluctuates, J1 and J3 have accepted the wounded to the maximum extent, and their accommodation capacity has reached the upper limit of capacity, while the accommodation degree of J8 has remained stable around a certain value. Meanwhile, it can also be found that with the sharp increase in the number of seriously wounded, the number of wounded that J5 and J6 can accommodate is increasing more and more, and eventually reaches the maximum capacity limit. Therefore, two conclusions can be drawn: 1. J1 and J3 are fully loaded regardless of how the number of seriously wounded fluctuates; 2. The fluctuation in the number of seriously injured individuals has a significant impact on J5 and J6.

Immediately after that, we analyzed the impact of fluctuations in minor casualties on the extent to which temporary medical facilities can accommodate casualties. Figure 16 shows that J1 and J3 are still at full capacity. However, unlike the previous analysis of the fluctuation in the number of severely wounded, the capacity of J8 to accommodate wounded will fluctuate significantly with the fluctuation of lightly wounded and eventually reach full capacity. Meanwhile, it can also be found that under the fluctuation of minor casualties, the accommodating capacity of J5 and J6 is greater than that under the fluctuation of severe casualties when the number of minor casualties decreases.

By comparing the changes in the capacity of temporary medical facilities in the two cases, it can be found that the fluctuation in the number of seriously injured people will cause facilities J5 and J6 to remain at a high-capacity level all the time, and as the number of seriously injured people increases, facilities J5 and J6 will also be fully loaded very quickly. The fluctuation in the number of minor casualties has a greater impact on the capacity of facilities J5 and J6 than that of the number of severe casualties, and the fluctuation amplitude also increases significantly. Meanwhile, the increase in the number of minor casualties will also lead to an increase in the number of casualties that J8 can accommodate. Finally, J1 and J3 have been at a fully loaded level all along.

To analyze the underlying reasons, we attempted to analyze the geographical locations of the disaster-stricken areas, temporary medical facilities and hospitals. It can be intuitively seen from the results of the data analysis that the differences in the capacity of temporary medical facilities are directly related to the importance of their geographical locations for the selection of transportation routes for the wounded. Specifically, the number of injured people that a facility can accommodate is greatly influenced by the comprehensive distance of the route from the facility to the disaster-stricken area and from the facility to the hospital. Especially for seriously injured patients, due to the urgency of medical treatment, facilities that can quickly complete treatment and secondary transfer (that is, the facilities with the shortest overall route) will be given priority. It is precisely this internal logic that has led to an obvious overloading phenomenon of facilities at key locations.

Figure 17 shows the geographical location relationship among the disaster-stricken areas, temporary medical facilities and hospitals, as well as the selection of transportation routes for the wounded. It can be clearly seen from the figure that the temporary medical facilities J1 and J3 are located at the key node positions between multiple disaster-stricken areas and the hospital, forming the shortest overall transportation path. Therefore, both of these facilities are fully loaded in all scenarios. However, for facilities in relatively suboptimal positions, such as J5 and J6, they gradually tend to be fully loaded only when the number of wounded increases significantly and the secondary paths are forced to be activated. Through this schematic diagram, we further confirm that the differences in the capacity of temporary medical facilities are essentially determined by the geographical location of the transportation routes.

5.5. Management Insights

Firstly, the higher the level of parameter uncertainty, the greater the deprivation costs. The quantity of severe casualties affects deprivation costs more so than the quantity of mild casualties. Based on this premise, this research makes a pertinent application suggesting that decision-makers concentrate on accurately predicting the quantity of severe injuries in order to minimize the overall costs from accidents. Second, the cost of deprivation for all casualties can be efficiently decreased by expanding the capacity of medical facilities. It should be noted that improving the capacity to treat the injured at temporary medical sites plays a greater role in reducing the cost of deprivation than in hospitals. Therefore, in order to minimize injuries and deaths, administrators should prioritize the expansion of EMS treatment capacity. Thirdly, based on the numerical and geographical analysis, managers should identify key transfer nodes between disaster sites and hospitals during pre-disaster planning. These facilities, due to their strategic locations, are likely to become transport bottlenecks. Therefore, their capacities should be appropriately expanded or supported with reserved resources. Additionally, setting up backup transfer nodes can improve the flexibility and resilience of the overall rescue system, ensure timely medical response and minimize casualties and losses. The results have proved that the CEL-TRO model proposed in this paper can produce stable and feasible solutions under various interference conditions. In practice, the comprehensive consideration of equipment failures and uncertainties can effectively save more lives and enhance the efficiency of rescue.

6. Conclusions

In this paper, we proposed a model of a humanitarian logistics network for casualty evacuation under the uncertainty of the quantity of disaster casualties and facility disruptions. Firstly, we considered the uncertainty of casualty quantities and disruptions at temporary medical points due to secondary disasters, using a budget uncertainty set to characterize this uncertainty. Secondly, we addressed the classification of casualties, dividing them into mild and severe casualties and defining deprivation cost functions for each category. We then established a deterministic casualty evacuation humanitarian logistics network model, followed by a two-stage robust optimization model that considers the classification of casualties. Using the 2008 Wenchuan earthquake involving 10 disaster sites, 10 candidate temporary medical points (TMPs), and 4 hospitals as a real case study, the C&CG algorithm was applied to solve the two-stage robust optimization model of the humanitarian logistics network for casualty evacuation under uncertainty, considering classification. Our numerical experiments indicate:

(1)

The C&CG algorithm used in this paper performs well, effectively reducing the deprivation costs of casualties. For the Wenchuan instance, the algorithm required at most 14 iterations (average 8.6) and achieved convergence within 42.7 s, with the optimality gap dropping below 1% within 9 iterations. Even the largest test instance (≈178,000 variables) was solved to optimality in under 40 min, confirming the algorithm’s suitability for large-scale post-disaster decision environments.

(2)

The higher the degree of parameter uncertainty, the greater the deprivation costs. Compared to mild casualties, severe casualties have a more significant impact on deprivation costs. Across uncertainty budgets ranging from 0 to 1, total system cost consistently rises. When both mild and serious casualties fluctuate jointly, serious casualties exert a disproportionately large influence on deprivation cost increases. Scenario analysis further shows that when serious casualties vary by ±30%, the total cost ranges from 58.04 million to 95.93 million yuan, compared with the baseline 80.09 million yuan.

(3)

Enhancing the capabilities of medical institutions can significantly decrease the expenditure involved in caring for a diverse array of patients. Improving the capacity of temporary medical sites to treat casualties is more important in terms of cost deprivation than in hospitals.

(4)

Pre-disaster planning should identify key transfer nodes between disaster sites and hospitals based on potential transportation routes. These strategically located temporary medical facilities are more likely to become fully loaded during emergencies. Expanding their capacity or setting up backup nodes in advance can significantly enhance the flexibility and resilience of the rescue network.

Although the model assumes accurate casualty classification and unconstrained transfers between facilities, such simplifications may understate the operational delays, congestion effects, and triage errors typically observed in real disasters. These behavioral assumptions could lead to an underestimation of total deprivation costs and overestimation of network efficiency. Future extensions could incorporate stochastic triage accuracy, dynamic congestion on transport routes, and time-dependent delays at transfer points to more realistically capture operational complexity and improve the robustness of evacuation planning. In addition, the current assumption of independent disruptions may overlook the spatially correlated failure patterns frequently observed in large-scale disasters. Future research could therefore develop correlation-aware disruption models, explicitly identify high-risk clusters, and design facility deployment strategies that jointly optimize risk diversification and cluster avoidance under limited resources.