TY - JOUR
AU - Taghipourian, Mohammad Javad
AB - Abstract Construction material delivery to post-disaster reconstruction projects is challenging because of the resource and time limitations that follow a large-scale disaster. There is compelling evidence that inadequate planning jeopardises the success of a large number of post-disaster reconstruction projects. Thus, the current study proposes an integrated approach to facilitate the procurement planning of construction materials following a large-scale disaster. The proposed approach clustered the location of construction projects using a differential evolution (DE)-K-prototypes, a new partitional clustering algorithm based on DE and K-prototypes, method. Then, using a permanent matrix prioritises cluster points based on route reliability-affecting factors. The model’s objectives are to minimise the total travel time, maximise the reliability of the route, and minimise the total weighted undelivered materials to projects. In the case of distribution of material through land vehicles, the possibility of breakdowns in the vehicle is considered, allowing for the determination of vehicle breakdown under various scenarios and the minimisation of undelivered materials to projects. As a result of the uncertain character of the disaster, the demands of construction projects are fuzzy, and Jimenez’s method is used to handle it. Due to the complexity of the problem, two algorithms are proposed, a multi-objective evolutionary algorithm based on decomposition (MOEA/D) and a non-dominated sorting genetic algorithm-II (NSGA-II). The results confirm that the proposed MOEA/D has a higher accuracy while NSGA-II has a shorter computational time. By providing new theoretical perspectives on disaster recovery strategies in the construction sector, this study contributes to the growing body of knowledge about disaster recovery strategies in the sector. The findings of this study can be employed to develop an integrated planning system for the delivery of construction materials to post-disaster reconstruction projects in disaster-prone countries. Graphical Abstract Open in new tabDownload slide post-disaster reconstruction, non-dominated sorting genetic algorithm-II, post disaster, construction material, multi-objective evolutionary algorithm based on decomposition, augmented ε-constraint method Highlights The problem of construction materials delivery to post-disaster reconstruction projects is being investigated. An integrated novel optimisation model is developed. Jimenez’s method is used to address the uncertainties associated with the problem. Non-dominated sorting genetic algorithm and a multi-objective evolutionary algorithm based on decomposition are presented for the problem. 1. Introduction Almost four-fifths of the world’s cities are located in disaster-prone areas (Celentano et al., 2019). Climate change is expected to exacerbate many types of natural disasters, particularly extreme weather events, with low-income countries bearing the brunt of the impact (Yazdani et al., 2022). Disaster-related losses are more tangible in less developed regions (Dulebenets et al., 2019). Disasters have a 20-fold greater impact on the built environment in developing countries than in developed countries (Barakat, 2003), owing to widespread substandard construction, leaving many people in need of housing in countries that are already struggling with their daily economies and housing challenges (Uddin & Matin, 2021). Disaster recovery efforts prioritise the reconstruction of houses and public infrastructure (Rouhanizadeh & Kermanshachi, 2020). Effective delivery has become a significant issue when it comes to large-scale post-disaster reconstruction projects (Safapour & Kermanshachi, 2021). In the aftermath of a disaster, reconstruction must deal with the conflicting demands of the displaced population and the need for agencies to plan programmes that address both the immediate need for shelter and the long-term need for permanent housing (Bahmani & Zhang, 2021b). This longer term plan is intended to accommodate not only a socioeconomic recovery but possibly an improvement over pre-disaster conditions, as recommended by the Sendai agreement and the globally adopted building back better reconstruction guidelines (Habibi Rad et al., 2021a). Meanwhile, the spontaneous tendency of the population is to rush into restoring livelihood “back to normal,” often ending up replicating previous vulnerability by building in disaster-prone areas or using unsafe construction methods (Platt & So, 2017). As a result, this circumstance puts pressure on authorities and agencies to deliver a response that may meet the required speed while failing on socioeconomic aspects (Bahmani & Zhang, 2021a). It is important to draw the importance of providing a quick recovery in order to avoid the establishment of unsafe building structures (Habibi Rad et al., 2021b). Post-disaster reconstruction projects face challenges that go beyond standard construction issues and are prone to delivering inadequate building solutions (Siriwardhana et al., 2021). This is directly related to the need for quick reaction time in stressful situations (Mahtab et al., 2021). Although these circumstances are well known, the strategies required to overcome these difficulties appear to be less clear (Bahmani & Zhang, 2022b). Some issues have been identified as major bottlenecks in reconstruction projects, including supply chain planning challenges and resource shortages (Celentano et al., 2019). Solutions to overcome these obstacles are still being discussed (Huang et al., 2021). Bilau et al. (2017) indicate challenges associated with logistics and supplies as one of the major management issues that arise in large-scale housing reconstruction programmes. Construction materials and their supply chain in post-disaster reconstruction projects can be linked to many bottlenecks (Chang et al., 2010). After a destructive disaster, the majority of local manufacturing facilities and supply systems in manufacturing industries are likely to be damaged, causing havoc in the construction market. This also results in price fluctuations (Khodahemmati & Shahandashti, 2020). Numerous international organisations, including the IFRC and UN agencies, have also emphasised the importance of resource availability as a critical factor in optimising recovery efforts (Celentano et al., 2019). The majority of research in the field of post-disaster reconstruction is limited to certain social science perspectives or examinations of prior experiences. For example, Enshassi et al. (2017) explored factors influencing post-disaster reconstruction project management for housing provision in the Gaza Strip. Bahmani and Zhang (2021b) proposed a success evaluation framework consisting of a definition of the successful recovery measurements and how influential factors can be used to manage socio-natural disaster recovery projects. Mohammadnazari et al. (2022) presented an integrated approach based on four multi-criteria decision-making techniques: TOPSIS, ELECTRE III, VIKOR, and PROMETHEE, to aid decision makers in prioritizing post-disaster projects. Copping et al. (2022) investigated the supply networks of construction materials for disaster-relief shelters and reconstruction. A questionnaire was used to collect the data. The sample consisted of 272 displaced families from four countries (Nepal, Bangladesh, Afghanistan, and Turkey). The data were analysed using social network analysis. Charles et al. (2022) reviewed the literature on resilience factors used in post-disaster reconstruction projects and formed a framework to aid in strategic selection and application. Habibi Rad et al. (2022) proposed a conceptual framework for implementing lean construction in infrastructure recovery projects. Bahmani and Zhang (2022a) conducted a study to simplify the disaster recovery project management procedure by identifying common activities and assigning them to appropriate places based on the disaster recovery project’s timeline. However, few studies in this field have addressed the modelling approach. For example, Xu et al. (2019), Xiong et al. (2020), and Ghannad et al. (2020) proposed methods such as genetic algorithm and priority indices to determine the reconstruction order of buildings. Ghannad et al. (2021) developed a prioritisation approach for rapid and optimised post-disaster recovery that evaluates recovery priorities of damaged transportation infrastructure systems and affected regions through a multi-agent system using a reinforcement learning technique. Zokaee et al. (2021) developed a multi-resource, capacitated tour covering location-routing model to design a post-disaster reconstruction supply chain considering disruption risk during the recovery phase to minimise total costs by determining optimal locations of temporary resource distribution warehouses and optimal paths for transferring reconstruction resources to the impacted areas. Gharib et al. (2022) developed an integrated model for the distribution of post-disaster temporary shelters after a large-scale disaster. Huang et al. (2022) established a research framework for efficiently and comprehensively modelling and assessing post-disaster economic losses that can be used to analyse the full supply chain losses caused by other disasters with data availability and reliability or to simulate the full economic losses caused by an unexpected event in order to improve decision makers’ disaster management and response capacity. Diaz et al. (2022) evaluated supply, costs, and recovery times using a combination of Monte Carlo and beta models. The proposed models incorporate stochastic components into projected material, labour, and equipment flows to capture reconstruction activities’ immediate and long-term costs under highly uncertain conditions. The model could identify potential roadblocks in obtaining necessary materials and labour and prospective thresholds that could impede the housing recovery process. Alisjahbana et al. (2022) proposed a method for determining reconstruction policies and strategies, particularly for interdependent building systems like the school system. They discussed the situation of post-disaster school reconstruction, in which students are relocated to the nearest functional school if their original school was damaged. The objective of the problem was to determine the order in which damaged schools should be reconstructed by minimising the sum of the distance all students in the region have to travel until all schools in the region are reconstructed. Despite recent advancements in different phases of disaster risk management plans and strategies (Dulebenets et al., 2019; Abioye et al., 2020; Chen et al., 2022), mathematical and optimisations’ approaches have not adequately addressed the issues of post-disaster reconstructions and the challenges and obstacles faced by recovery projects (Tirkolaee et al., 2020a). As a result, this research’s primary aim is to develop an integrated optimisation model for post-disaster recovery planning to enable the most efficient use of scarce construction resources caused by natural disasters. The proposed model clusters the location of construction projects by using the proposed differential evolution (DE)-K-prototypes method and then prioritises the points of clusters by affecting factors on the route reliability using a permanent matrix. The model’s objectives are to minimise the total travel time, maximise the reliability of the route, and minimise the total weighted undelivered materials to projects. Furthermore, the possibility of the breakdown in the vehicle is considered in the model. Considering the uncertainties associated with post-disaster reconstruction, the demands for materials in each project are fuzzy. As the problem is complex, two multi-objective meta-heuristic algorithms, non-dominated sorting genetic algorithm-II (NSGA-II) and multi-objective evolutionary algorithm based on decomposition (MOEA/D), are proposed to solve the problem. 2. Contributions The review of the literature reveals that procurement issues in post-disaster reconstruction projects are still unresolved and unexplored. Many questions about formulation parameters and assumptions remain unanswered in existing studies, which can only be viewed as preliminary steps towards a more comprehensive understanding of post-disaster construction project planning. The main contributions of this study can be stated as follows: Clustering construction sites before designing the mathematical model and the formation of the routing network in order to accelerate the distribution of construction materials: Consideration of the location of construction projects with geographically common features, spatial features, and other factors in a cluster (assuming that the paths leading to those locations are impacted) can have a significant impact on accelerating and improving material distribution performance. Furthermore, the safety of material delivery operations to a construction site is critical. This study investigates the identification of common factors and features for clustering the location of construction projects in light of route disruption. Clustering is accomplished using the proposed DE-K-prototypes method in this study. In the DE-K-prototypes model, after identifying and collecting data, the factors are entered as model input, and the model considers important factors for clustering and places the most similar and relevant points in a cluster based on the type of training it has already seen. Prioritisation of construction projects based on effective factors of road safety in each cluster of construction project locations: The majority of research in the field of crisis transportation and routeing focuses on the design of routeing networks and direct distribution of materials and goods with little regard for safety and reliability. Decisions in each cluster are made using a combination of graph theory and permanent matrix theory. The priority determined by this method selects the most reliable routes for delivering materials to construction projects. Multi-mode transportation system: Considering two different modes of transportation in this study, both land and aerial vehicles, provides flexibility for the plan to serve locations with inaccessible routes. Even in the event of secondary crises in material delivery operations, such as road disruption or adverse weather, delivery operations can be completed quickly and accurately on a scheduled basis. Simultaneous consideration of routing factors such as heterogeneity of vehicles, multiple warehouses of the routing network, multi-period distribution of materials, and multi-type of construction materials: Using the same type of vehicle (homogeneous vehicle) may impact adversely the performance of planning to distribute materials and respond quickly to the construction projects, so consider vehicles that differ in capacity, speed, fuel consumption, and so on (heterogeneous vehicles), which may provide flexibility for providing materials to different construction projects in different locations with different demands. This will solve the issue. On the other hand, the distribution network’s multi-warehouse structure can save time in serving construction projects. Consider the disruption in the vehicle operation: Emergency situations can cause a variety of disruptions, such as traffic, vehicle breakdowns, and other issues given the scale and duration of the rescue efforts. Undelivered materials to projects have resulted because some construction projects cannot receive services. Considering the uncertainty in demand for construction projects: Due to the uncertain nature of the disasters, it is reasonable to consider some effective parameters of the uncertain routeing model, such as warehouse capacity and vehicle costs for on-demand transportation. Due to the fact that demand in affected areas is high and uncertain during a disaster, the uncertainty associated with the demand parameter is considered fuzzy in this study. We will work to alleviate the distribution system’s uncertainty. Using several multi-objective methods: The epsilon-constraint method is developed to solve the proposed mathematical model in a small-size problem. However, due to the complexity of the problem, two algorithms are proposed for the large-size problems, including the NSGA-II and a MOEA/D. 3. Proposed Framework for Planning the Delivery of Construction Materials The three steps below outline the procedure for this study. Step 1 involves clustering and prioritising construction projects’ locations, while Step 2 entails providing all stages of the mathematical model. Two multi-objective meta-heuristic algorithms, including NSGA-II and MOEA/D, are then used to solve the considered model. The proposed approach for planning the delivery of construction materials to post-disaster reconstruction projects is depicted schematically in Fig. 1. Figure 1: Open in new tabDownload slide The proposed approach for planning the delivery of construction materials to post-disaster reconstruction projects. 3.1 Clustering the location of construction projects using DE-K-prototypes By clustering the location of construction projects, response strategies for sending construction materials to these areas can be accelerated. To avoid getting stuck in local optima when clustering mixed numeric and categorical data, we propose DE-K-prototypes, a new partitional clustering algorithm based on DE and K-prototypes that avoids this problem by solving global optimisation problems instead of local ones. In this study, similar to (Ji et al., 2020), a hybrid method is employed to cluster the data. 3.1.1 The K-prototypes algorithm To cluster data with both numerical and categorical attributes, Huang (1997) introduced the k-prototypes algorithm. According to this algorithm, the dataset |$\mathcal{X}$| should be partitioned into |$\mathfrak{k}$| distinct clusters by minimising a below cost function: $$\begin{equation*} E\left( {\mathcal{U},\mathcal{Z}} \right) = \mathop \sum \limits_{{\mathfrak{l}} = 1}^{\mathfrak{k}} \mathop \sum \limits_{{\mathfrak{i}} = 1}^{\mathfrak{n}} \mathfrak{u}_{{\mathfrak{il}}} \mathfrak{d} \left({\mathfrak{x}_{\mathfrak{i}}},{\mathcal{Z}_{\mathfrak{l}}}\right), \,\, 0 \le {\mathfrak{u}_{\mathfrak{il}}} \le 1, \end{equation*} $$(1) where |${\mathcal{Z}_{\mathfrak{l}}}$| denotes the prototype of the cluster |$\mathfrak{l}$|⁠; |${\mathfrak{u}_{\mathfrak{i}\mathfrak{l}}}$| denotes an element of the partition matrix |${U_{n \times k}}$|⁠; and |$d( {{x_i},{\rm{ }}{\mathcal{Z}_{\mathfrak{l}}}} )$| denotes the dissimilarity measure as follows: $$\begin{equation*} \mathfrak{d}\left( {{\mathfrak{x}_{\mathfrak{i}}},{\mathcal{Z}_{\mathfrak{l}}}} \right){\rm{ }} = \mathop \sum \limits_{{\mathfrak{j}} = 1}^{\mathfrak{m}} \mathfrak{d}\left( {{\mathfrak{x}_{{\mathfrak{i}}\mathfrak{j}}},{\mathfrak{z}_{{\mathfrak{l}}\mathfrak{j}}}} \right) . \end{equation*} $$(2) In the above formula, |$\mathfrak{d}( {{\mathfrak{x}_{\mathfrak{i}\mathfrak{j}}},{\rm{\ }}{\mathfrak{z}_{\mathfrak{l}\mathfrak{j}}}} )$| is formulated as: $$\begin{equation*} \mathfrak{d}\left( {{\mathfrak{x}_{{\mathfrak{i}}\mathfrak{j}}},{\mathfrak{z}_{{\mathfrak{l}}\mathfrak{j}}}} \right) = \left\{\begin{array}{ll} {{\left( {{\mathfrak{x}_{\mathfrak{i}\mathfrak{j}}} - {\mathfrak{z}_{\mathfrak{l}\mathfrak{j}}}} \right)}^2}, & \quad\!\!\!\! {{\rm{if\ the\ }}\mathfrak{l}{\rm{th\ attribute\ is\ numeric,}}} \\ {\mu _l} \delta \left({{\mathfrak{x}_{{\mathfrak{i}}\mathfrak{j}}}, {\mathfrak{z}_{{\mathfrak{l}}\mathfrak{j}}}} \right), & \quad\!\!\!\! {\rm{if\ the\ }}\mathfrak{l}{\rm{th\ attribute\ is\ categorical,}} \end{array} \right. \end{equation*} $$(3) where |$\delta ( {\mathfrak{p},{\rm{ }}\mathfrak{q}} ){\rm{ }} = {\rm{}}0$| if |$\mathfrak{p}$| and |$\mathfrak{q}$| have the same value; |$\delta ( {\mathfrak{p},{\rm{ }}\mathfrak{q}} ){\rm{ }} = {\rm{ }}1$| if |$\mathfrak{p}$| and |$\mathfrak{q}$| have different values; |${\mu_{\mathfrak{l}}}$| denotes the weight assigned to categorical attributes within the cluster l. If the jth attribute is numeric, |${\mathfrak{z}_{\mathfrak{l}\mathfrak{j}}}$| represents the mean of the |$\mathfrak{j}$|th numeric attribute in the cluster |$\mathfrak{l}$|⁠; if the jth attribute is categorical, |${\mathfrak{z}_{\mathfrak{l}\mathfrak{j}}}$| represents the mode of the |$\mathfrak{j}$|th categorical attribute in the cluster |$\mathfrak{l}$|⁠. The k-prototypes algorithm is implemented as follows: Generate prototypes of clusters by selecting k random data points from the dataset |$\mathcal{X}$|⁠. Each data point in dataset |$\mathcal{X}$| should be assigned to the cluster with the nearest prototype in accordance with Equation (2). After each allocation, make an update to the prototype cluster. After allocating all data objects, re-evaluate their similarity to the current prototypes. If it is discovered that the nearest prototype of a data object is in a different cluster than the current one, reallocate the data object to that cluster and update the prototypes for both clusters. If no data objects have changed clusters as a result of a full circle test of |$\mathcal{X}$|⁠, the algorithm should be terminated; otherwise, proceed to Step III. 3.1.2 The clustering algorithm based on DE and K-prototypes DE begins by generating an initial set of solutions (N) within the search space. Each individual in generation G is represented by a D-dimensional real-valued vector |$X_i^G = ( {x_{i,1}^G,{\rm{\ }}x_{i,2}^G,{\rm{\ }}x_{i,3}^G,{\rm{\ }} \ldots ,{\rm{\ }}x_{i,{\rm{\ }}D}^G} )\ $|⁠, where |$i\ = {\rm{\ }}1,2,3,{\rm{\ }} \ldots ,{\rm{\ }}N$|⁠. Following the initialisation phase, solutions are evolved and trial vectors are generated using mutation and crossover operators. Then, parents are compared to their corresponding trial vector, and the next generation’s solutions are chosen at this stage (Yazdani et al., 2021). Performing DE involves the following steps. DE’s population is made up of N individuals. DE typically begins with a randomly generated population, with the parameters of the i-th member of that population being generated as follows: $$\begin{equation*} x_{i,j}^0 = x_j^{low}{\rm{\ }} + rand{\rm{\ }} \times \left( {x_j^{up} - x_j^{low}} \right),{\rm{\ }}j\ = {\rm{\ }}1,2,3, \ldots ,D \end{equation*} $$(4) where D denotes the number of decision variables, and rand returns a uniformly distributed random number within the range [0,1]. Additionally, |$x_j^{low}$| and |$x_j^{up}$| denote the lower and upper bounds of solutions in the jth-dimensional search space, respectively. DE then uses the mutation procedure to generate mutant vectors after initialisation has taken place. In this section, “current-to-best/1:” |${\rm{\ \ }}V_i^G = X_i^G{\rm{\ }} + F.( {X_{best}^G - X_{r1}^G} ) + F.( {X_{r2}^G - X_{r3}^G} )$| is used as one of the most frequently used mutation operators. |$X_{best}^G$| is the best solution for the current generation. The difference vector’s scaling is managed using the F control parameter. The crossover operator is then used to combine the target vector with its corresponding mutant vector, resulting in a trial vector (⁠|$U_i^G$|⁠). $$\begin{equation*} {u_{i,j}^G} = \left\{ \begin{array}{ll} {v_{i,j}^G}, & \quad if\ (rand\ \lt \ Cr\ or \ j = {j_{rand}})\\ {x_{i,j}^G}, & \quad {otherwise} \end{array} \right.. \end{equation*} $$(5)|$Cr$| is the crossover rate and is between 0 and 1. Additionally, |${j_{rand}}$| is a random integer in the range [1, D]. Selecting between the target vector (⁠|$X_i^G$|⁠) and its corresponding trial vector (⁠|$U_i^G$|⁠) determines whether the vector will be retained for the next generation. The selection procedure is as follows for minimising problems: $$\begin{equation*} X_i^{G + 1} = \left\{ \begin{array}{ll} U_i^G, & \quad f\left( {U_i^G} \right) \le f\left( {X_i^G} \right) \\ X_i^G, & \quad otherwise \end{array}\right., \end{equation*} $$(6) where |$f( {X_i^G} ){\rm{\ }}$| and |$f( {U_i^G} )$| denote the objective functions of the target vector and the corresponding trial vector, respectively. Open in new tabDownload slide Table 1 contains data on the clustering of construction projects. The locations of construction projects are categorised in Table 1 into two clusters. Cluster 1 projects can be served by both land vehicles and aerial vehicles. While cluster 2 projects include locations that are inaccessible via land and should be served by air vehicles. Table 1: A numerical illustration of affected point clustering using the proposed DE-K-prototypes method. Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . 1 low Normal Very high 2739 Low 8 5 0 12 26 Low Good High 6628 Low 1 8 13 8 2 low Normal Very high 3100 Low 12 8 11 8 27 Low Good High 1300 Low 2 9 14 8 3 low Normal High 2050 Low 10 5 10 8 28 Low Good High 1400 Low 3 9 15 8 4 low Good High 450 Low 8 5 1 12 29 Low Good Medium 1500 Medium 5 10 17 8 5 low Good Very high 850 Low 9 5 3 12 30 Medium Normal Medium 700 Medium 7 11 18 8 6 low Good High 1416 Low 3 10 13 12 31 Low Normal Medium 600 Medium 7 10 17 8 7 Medium Good Medium 1200 Medium 5 13 16 8 32 Low Normal High 3000 Medium 1 7 12 8 8 Medium Normal Medium 1020 Medium 6 14 17 8 33 Low Good High 4000 Medium 6 5 10 8 9 High Bad Medium 1201 Medium 7 14 19 8 34 Low Good Medium 300 Medium 7 3 8 8 10 High Bad High 500 Medium 8 15 20 8 35 Low Good High 1400 Medium 8 3 8 8 11 High Normal Very high 900 Medium 67 70 75 12 36 Low Good High 1500 Medium 8 2 7 8 12 High Bad Very high 1316 Medium 65 72 77 12 37 Low Good High 3500 Medium 8 3 8 8 13 High Bad Very high 800 Medium 68 71 77 8 38 Medium Normal High 2000 Medium 9 3 8 8 14 Medium Normal High 500 Medium 8 13 18 8 39 Low Normal High 1000 Medium 7 2 7 8 15 Medium Good High 351 Medium 9 15 20 8 40 Low Normal Medium 1000 Medium 7 2 7 8 16 Medium Good High 450 Medium 9 11 16 8 41 Low Normal High 4000 Medium 8 7 12 8 17 Medium Normal High 500 Medium 18 16 13 12 42 High Bad Medium 50 Medium 18 4 13 8 18 low Normal High 7703 Medium 25 4 15 8 43 Low Normal High 1320 Medium 8 3 8 8 19 High Bad Very high 2584 High 75 70 65 8 44 Medium Normal High 450 Medium 8 5 8 8 20 High Bad Very high 204 High 87 82 78 8 45 Medium Bad Very high 700 High 13 8 13 8 21 High Bad Very high 1202 High 81 76 71 8 46 Medium Bad Very high 1500 High 7 5 10 8 22 High Bad Very high 202 High 80 75 70 8 47 Low Good High 500 Medium 7 4 9 8 23 High Bad Very high 302 High 75 70 65 8 48 Low Good Very high 500 Medium 8 4 12 8 24 High Bad Very high 1500 High 85 80 75 8 49 Medium Normal Medium 500 Medium 8 4 13 8 25 Medium Good High 1500 Medium 4 12 17 8 50 Low Normal High 600 Medium 8 14 16 8 Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . 1 low Normal Very high 2739 Low 8 5 0 12 26 Low Good High 6628 Low 1 8 13 8 2 low Normal Very high 3100 Low 12 8 11 8 27 Low Good High 1300 Low 2 9 14 8 3 low Normal High 2050 Low 10 5 10 8 28 Low Good High 1400 Low 3 9 15 8 4 low Good High 450 Low 8 5 1 12 29 Low Good Medium 1500 Medium 5 10 17 8 5 low Good Very high 850 Low 9 5 3 12 30 Medium Normal Medium 700 Medium 7 11 18 8 6 low Good High 1416 Low 3 10 13 12 31 Low Normal Medium 600 Medium 7 10 17 8 7 Medium Good Medium 1200 Medium 5 13 16 8 32 Low Normal High 3000 Medium 1 7 12 8 8 Medium Normal Medium 1020 Medium 6 14 17 8 33 Low Good High 4000 Medium 6 5 10 8 9 High Bad Medium 1201 Medium 7 14 19 8 34 Low Good Medium 300 Medium 7 3 8 8 10 High Bad High 500 Medium 8 15 20 8 35 Low Good High 1400 Medium 8 3 8 8 11 High Normal Very high 900 Medium 67 70 75 12 36 Low Good High 1500 Medium 8 2 7 8 12 High Bad Very high 1316 Medium 65 72 77 12 37 Low Good High 3500 Medium 8 3 8 8 13 High Bad Very high 800 Medium 68 71 77 8 38 Medium Normal High 2000 Medium 9 3 8 8 14 Medium Normal High 500 Medium 8 13 18 8 39 Low Normal High 1000 Medium 7 2 7 8 15 Medium Good High 351 Medium 9 15 20 8 40 Low Normal Medium 1000 Medium 7 2 7 8 16 Medium Good High 450 Medium 9 11 16 8 41 Low Normal High 4000 Medium 8 7 12 8 17 Medium Normal High 500 Medium 18 16 13 12 42 High Bad Medium 50 Medium 18 4 13 8 18 low Normal High 7703 Medium 25 4 15 8 43 Low Normal High 1320 Medium 8 3 8 8 19 High Bad Very high 2584 High 75 70 65 8 44 Medium Normal High 450 Medium 8 5 8 8 20 High Bad Very high 204 High 87 82 78 8 45 Medium Bad Very high 700 High 13 8 13 8 21 High Bad Very high 1202 High 81 76 71 8 46 Medium Bad Very high 1500 High 7 5 10 8 22 High Bad Very high 202 High 80 75 70 8 47 Low Good High 500 Medium 7 4 9 8 23 High Bad Very high 302 High 75 70 65 8 48 Low Good Very high 500 Medium 8 4 12 8 24 High Bad Very high 1500 High 85 80 75 8 49 Medium Normal Medium 500 Medium 8 4 13 8 25 Medium Good High 1500 Medium 4 12 17 8 50 Low Normal High 600 Medium 8 14 16 8 Open in new tab Table 1: A numerical illustration of affected point clustering using the proposed DE-K-prototypes method. Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . 1 low Normal Very high 2739 Low 8 5 0 12 26 Low Good High 6628 Low 1 8 13 8 2 low Normal Very high 3100 Low 12 8 11 8 27 Low Good High 1300 Low 2 9 14 8 3 low Normal High 2050 Low 10 5 10 8 28 Low Good High 1400 Low 3 9 15 8 4 low Good High 450 Low 8 5 1 12 29 Low Good Medium 1500 Medium 5 10 17 8 5 low Good Very high 850 Low 9 5 3 12 30 Medium Normal Medium 700 Medium 7 11 18 8 6 low Good High 1416 Low 3 10 13 12 31 Low Normal Medium 600 Medium 7 10 17 8 7 Medium Good Medium 1200 Medium 5 13 16 8 32 Low Normal High 3000 Medium 1 7 12 8 8 Medium Normal Medium 1020 Medium 6 14 17 8 33 Low Good High 4000 Medium 6 5 10 8 9 High Bad Medium 1201 Medium 7 14 19 8 34 Low Good Medium 300 Medium 7 3 8 8 10 High Bad High 500 Medium 8 15 20 8 35 Low Good High 1400 Medium 8 3 8 8 11 High Normal Very high 900 Medium 67 70 75 12 36 Low Good High 1500 Medium 8 2 7 8 12 High Bad Very high 1316 Medium 65 72 77 12 37 Low Good High 3500 Medium 8 3 8 8 13 High Bad Very high 800 Medium 68 71 77 8 38 Medium Normal High 2000 Medium 9 3 8 8 14 Medium Normal High 500 Medium 8 13 18 8 39 Low Normal High 1000 Medium 7 2 7 8 15 Medium Good High 351 Medium 9 15 20 8 40 Low Normal Medium 1000 Medium 7 2 7 8 16 Medium Good High 450 Medium 9 11 16 8 41 Low Normal High 4000 Medium 8 7 12 8 17 Medium Normal High 500 Medium 18 16 13 12 42 High Bad Medium 50 Medium 18 4 13 8 18 low Normal High 7703 Medium 25 4 15 8 43 Low Normal High 1320 Medium 8 3 8 8 19 High Bad Very high 2584 High 75 70 65 8 44 Medium Normal High 450 Medium 8 5 8 8 20 High Bad Very high 204 High 87 82 78 8 45 Medium Bad Very high 700 High 13 8 13 8 21 High Bad Very high 1202 High 81 76 71 8 46 Medium Bad Very high 1500 High 7 5 10 8 22 High Bad Very high 202 High 80 75 70 8 47 Low Good High 500 Medium 7 4 9 8 23 High Bad Very high 302 High 75 70 65 8 48 Low Good Very high 500 Medium 8 4 12 8 24 High Bad Very high 1500 High 85 80 75 8 49 Medium Normal Medium 500 Medium 8 4 13 8 25 Medium Good High 1500 Medium 4 12 17 8 50 Low Normal High 600 Medium 8 14 16 8 Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . Construction projects . Road slope . Weather conditions . The magnitude of the disaster . The density of the population . Risk on the road . Distance from depot1 (truck) Km . Distance from vehicle depot2 (truck) Km . Distance from airport (helicopter) Km . Width road (m) . 1 low Normal Very high 2739 Low 8 5 0 12 26 Low Good High 6628 Low 1 8 13 8 2 low Normal Very high 3100 Low 12 8 11 8 27 Low Good High 1300 Low 2 9 14 8 3 low Normal High 2050 Low 10 5 10 8 28 Low Good High 1400 Low 3 9 15 8 4 low Good High 450 Low 8 5 1 12 29 Low Good Medium 1500 Medium 5 10 17 8 5 low Good Very high 850 Low 9 5 3 12 30 Medium Normal Medium 700 Medium 7 11 18 8 6 low Good High 1416 Low 3 10 13 12 31 Low Normal Medium 600 Medium 7 10 17 8 7 Medium Good Medium 1200 Medium 5 13 16 8 32 Low Normal High 3000 Medium 1 7 12 8 8 Medium Normal Medium 1020 Medium 6 14 17 8 33 Low Good High 4000 Medium 6 5 10 8 9 High Bad Medium 1201 Medium 7 14 19 8 34 Low Good Medium 300 Medium 7 3 8 8 10 High Bad High 500 Medium 8 15 20 8 35 Low Good High 1400 Medium 8 3 8 8 11 High Normal Very high 900 Medium 67 70 75 12 36 Low Good High 1500 Medium 8 2 7 8 12 High Bad Very high 1316 Medium 65 72 77 12 37 Low Good High 3500 Medium 8 3 8 8 13 High Bad Very high 800 Medium 68 71 77 8 38 Medium Normal High 2000 Medium 9 3 8 8 14 Medium Normal High 500 Medium 8 13 18 8 39 Low Normal High 1000 Medium 7 2 7 8 15 Medium Good High 351 Medium 9 15 20 8 40 Low Normal Medium 1000 Medium 7 2 7 8 16 Medium Good High 450 Medium 9 11 16 8 41 Low Normal High 4000 Medium 8 7 12 8 17 Medium Normal High 500 Medium 18 16 13 12 42 High Bad Medium 50 Medium 18 4 13 8 18 low Normal High 7703 Medium 25 4 15 8 43 Low Normal High 1320 Medium 8 3 8 8 19 High Bad Very high 2584 High 75 70 65 8 44 Medium Normal High 450 Medium 8 5 8 8 20 High Bad Very high 204 High 87 82 78 8 45 Medium Bad Very high 700 High 13 8 13 8 21 High Bad Very high 1202 High 81 76 71 8 46 Medium Bad Very high 1500 High 7 5 10 8 22 High Bad Very high 202 High 80 75 70 8 47 Low Good High 500 Medium 7 4 9 8 23 High Bad Very high 302 High 75 70 65 8 48 Low Good Very high 500 Medium 8 4 12 8 24 High Bad Very high 1500 High 85 80 75 8 49 Medium Normal Medium 500 Medium 8 4 13 8 25 Medium Good High 1500 Medium 4 12 17 8 50 Low Normal High 600 Medium 8 14 16 8 Open in new tab 3.2 Reliable route selection In this section, the obtained clusters are prioritised according to factors affecting route reliability in order to determine the most reliable route for distributing construction materials. The type of road (autobahn or highway), the degree of mountainous areas, and geographical characteristics all affect the reliability of cluster 1, which are accessible through the ground and aerial vehicles. Cluster 2’s reliability is affected by the magnitude of the disaster, the regional context (urban or rural), the weather conditions, the region’s demand, and the distance between the air vehicle depot and the location of construction projects. The approach is presented in Fig. 2. Figure 2: Open in new tabDownload slide The steps of selecting the reliable routes. The graph and its dependencies are used to represent the identification of the influencing factor on the process by conducting an expert survey or using data from the literature (Geetha & Sekar, 2016). There are nodes and directed edges that make up the graph, as depicted in Fig. 3. Each node |${\mathfrak{n}_{\mathfrak{i}}}$| represents the |$\mathfrak{i}$|-th criterion for alternative selection, while the edges represent the relative importance of the criteria, with the same number of nodes and alternatives. A directed edge is drawn from |$\mathfrak{i}$| to |$\mathfrak{j}$| (⁠|${\mathfrak{e}_{\mathfrak{i}\mathfrak{j}}}$|⁠) and vice versa, if node |$\mathfrak{i}$| is more important than node |$\mathfrak{j}$| in alternative selection (Rao & Padmanabhan, 2007). Figure 3: Open in new tabDownload slide Criteria and subcriteria framework adopted from Baykasoglu (2014). The relative importance of criteria and alternative scores are defined following the first step. If the criterion is qualitative, the values of the alternatives score can be derived using a zero-to-one rating scale (Nasiri & ShisheGar, 2014), as seen in Table 2. Table 2: The scores of alternatives. Qualitative measure . Crisp score . Exceptionally low 0 Extremely low 1 Very low 2 Low 3 Below average 4 Average 5 Above average 6 High 7 Very high 8 Extremely high 9 Exceptionally high 10 Qualitative measure . Crisp score . Exceptionally low 0 Extremely low 1 Very low 2 Low 3 Below average 4 Average 5 Above average 6 High 7 Very high 8 Extremely high 9 Exceptionally high 10 Open in new tab Table 2: The scores of alternatives. Qualitative measure . Crisp score . Exceptionally low 0 Extremely low 1 Very low 2 Low 3 Below average 4 Average 5 Above average 6 High 7 Very high 8 Extremely high 9 Exceptionally high 10 Qualitative measure . Crisp score . Exceptionally low 0 Extremely low 1 Very low 2 Low 3 Below average 4 Average 5 Above average 6 High 7 Very high 8 Extremely high 9 Exceptionally high 10 Open in new tab Normalization is necessary when employing quantitative criteria. Thus, if |${\mathfrak{v}_{\mathfrak{i}}}$| and |${\mathfrak{v}_{\mathfrak{j}}}$| are criteria values of alternative |$\mathfrak{i}$| and j, respectively, |$\frac{{{\mathfrak{v}_{\mathfrak{i}}}}}{{{\mathfrak{v}_{\mathfrak{j}}}}}$| should be normalised (Mohaghar et al., 2013). Following the calculation of all the criteria values (⁠|$i\ = \ 1,\ \ldots ,\ N$|⁠) for an alternative, we create a criteria rating matrix for that alternative. $$\begin{equation*} \left[ \psi \right] = \left[ {\begin{array}{@{}*{4}{c}@{}} {{\mathcal{C}_{11}}} & \quad 0 & \quad \ldots & \quad 0\\ 0 & \quad {{\mathcal{C}_{22}}} & \quad \ldots & \quad 0\\ \ldots & \quad \ldots & \quad \ldots & \quad \ldots \\ 0 & \quad 0 & \quad \ldots & \quad {{\mathcal{C}_{\mathfrak{n}\mathfrak{n}}}} \end{array}} \right]\ . \end{equation*} $$(7) However, the relative importance of criteria (⁠|${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$|⁠) can range between zero and one. The relationship between |${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$| and |${\mathfrak{r}_{\mathfrak{j}\mathfrak{i}}}$| does not have to be compensatory. As shown in Table 3, it can be |${\mathfrak{r}_{\mathfrak{j}\mathfrak{i}}} = \frac{1}{{{\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}}}\ $|⁠. Table 3: The relative importance of criteria. Definition of the class . |${{\mathfrak{r}_{\mathfrak{ij}}}}$| . |${{\mathfrak{r}_{\mathfrak{ij}}}} = {1} - {{\mathfrak{r}_{\mathfrak{ij}}}}$| . Two criteria are equally important 0.5 0.5 One criterion is slightly more important than others 0.6 0.4 One criterion is more important than others 0.7 0.3 One criterion is very important than others 0.8 0.2 One criterion is exceptionally important than others 0.9 0.1 One criterion is the most important, other not important 1.0 0.0 Definition of the class . |${{\mathfrak{r}_{\mathfrak{ij}}}}$| . |${{\mathfrak{r}_{\mathfrak{ij}}}} = {1} - {{\mathfrak{r}_{\mathfrak{ij}}}}$| . Two criteria are equally important 0.5 0.5 One criterion is slightly more important than others 0.6 0.4 One criterion is more important than others 0.7 0.3 One criterion is very important than others 0.8 0.2 One criterion is exceptionally important than others 0.9 0.1 One criterion is the most important, other not important 1.0 0.0 Open in new tab Table 3: The relative importance of criteria. Definition of the class . |${{\mathfrak{r}_{\mathfrak{ij}}}}$| . |${{\mathfrak{r}_{\mathfrak{ij}}}} = {1} - {{\mathfrak{r}_{\mathfrak{ij}}}}$| . Two criteria are equally important 0.5 0.5 One criterion is slightly more important than others 0.6 0.4 One criterion is more important than others 0.7 0.3 One criterion is very important than others 0.8 0.2 One criterion is exceptionally important than others 0.9 0.1 One criterion is the most important, other not important 1.0 0.0 Definition of the class . |${{\mathfrak{r}_{\mathfrak{ij}}}}$| . |${{\mathfrak{r}_{\mathfrak{ij}}}} = {1} - {{\mathfrak{r}_{\mathfrak{ij}}}}$| . Two criteria are equally important 0.5 0.5 One criterion is slightly more important than others 0.6 0.4 One criterion is more important than others 0.7 0.3 One criterion is very important than others 0.8 0.2 One criterion is exceptionally important than others 0.9 0.1 One criterion is the most important, other not important 1.0 0.0 Open in new tab After this step, the relative importance matrix is defined as follows: $$\begin{equation*} \left[ \beta \right] = \left[ {\begin{array}{@{}*{4}{c}@{}} 0 & \quad {{\mathfrak{r}_{12}}} & \quad \ldots & \quad {{\mathfrak{r}_{\mathfrak{i}\mathfrak{n}}}}\\ {{\mathfrak{r}_{21}}} & \quad 0 & \quad \ldots & \quad {{\mathfrak{r}_{2\mathfrak{n}}}}\\ \ldots & \quad \ldots & \quad \ldots & \quad \ldots \\ {{\mathfrak{r}_{\mathfrak{n}1}}} & \quad \ldots & \quad \ldots & \quad 0 \end{array}} \right]\ . \end{equation*} $$(8) Following the identification of |$\beta $| and |$\psi $|⁠, the alternative evaluation matrix |$\xi $| is obtained as follows: $$\begin{equation*} \xi = \psi + \beta = \left[ {\begin{array}{@{}*{5}{c}@{}} {{\mathcal{C}_1}} & \quad {{\mathfrak{r}_{12}}} & \quad {{\mathfrak{r}_{13}}} & \quad \ldots & \quad {{\mathfrak{r}_{1\mathfrak{n}}}}\\ {{\mathfrak{r}_{21}}} & \quad {{\mathcal{C}_2}} & \quad {{\mathfrak{r}_{23}}} & \quad \ldots & \quad {{\mathfrak{r}_{2\mathfrak{n}}}}\\ {{\mathfrak{r}_{31}}} & \quad {{\mathfrak{r}_{32}}} & \quad {{\mathcal{C}_3}} & \quad \ldots & \quad {{\mathfrak{r}_{3\mathfrak{n}}}}\\ \ldots & \quad \ldots & \quad \ldots & \quad \ldots & \quad \ldots \\ {{\mathfrak{r}_{\mathfrak{n}1}}} & \quad {{\mathfrak{r}_{\mathfrak{n}2}}} & \quad {{\mathfrak{r}_{\mathfrak{n}3}}} & \quad \ldots & \quad {{\mathcal{C}_{\mathfrak{i}\mathfrak{n}}}} \end{array}} \right]\ . \end{equation*} $$(9) The permanent of matrix |$\xi $|⁠, |$per( {\xi \ } )$| provides the rating for the alternative. For every alternative |$per( {\xi \ } )$| should be calculated and arranged in a descending order (Nasiri & ShisheGar, 2014). The alternative with the highest |$per( {\xi \ } )$| value is more preferable. Below is the equation that represents the permanent value calculation function (Nasiri & ShisheGar, 2014). $$\begin{eqnarray*} {\rm{Pe}}\mathfrak{r}({\xi}) &=& \prod\limits_{{\mathfrak{i}} = 1}^{\mathcal{N}} {{\mathcal{C}_{\mathfrak{i}}}} + \mathop \sum \limits_{\mathfrak{i},\mathfrak{j},..,\mathcal{N}} ({\mathfrak{r}_{{\mathfrak{ij}}}}.{\mathfrak{r}_{{\mathfrak{ji}}}}).{\mathcal{C}_{\mathfrak{k}}}.{\mathcal{C}_{\mathfrak{l}}} \ldots {\mathcal{C}_{\mathcal{N}}}\nonumber \\ &&+ \mathop \sum \limits_{{\mathfrak{i}},\mathfrak{j},..,\mathcal{N}}({\mathfrak{r}_{{\mathfrak{{ij}}}}}.{\mathfrak{r}_{\mathfrak{jk}}}.{\mathfrak{r}_{{\mathfrak{ki}}}} + {\mathfrak{r}_{{\mathfrak{ik}}}}.{\mathfrak{r}_{{\mathfrak{kj}}}}.{\mathfrak{r}_{{\mathfrak{ji}}}}).{\mathcal{C}_{\mathfrak{l}}}.{\mathcal{C}_{\mathfrak{n}}} \ldots {\mathcal{C}_{\mathcal{N}}} \nonumber \\ && + \left\{ \mathop \sum \limits_{\mathfrak{i},\mathfrak{j},..,\mathcal{N}} ({\mathfrak{r}_{\mathfrak{ij}}}.{\mathfrak{r}_{\mathfrak{ji}}})({\mathfrak{r}_{\mathfrak{kl}}}.{\mathfrak{r}_{\mathfrak{lk}}}).{\mathcal{C}_{\mathfrak{n}}}.{\mathcal{C}_{\mathfrak{m}}} \ldots\, .{\mathcal{C}_{\mathcal{N}}}\right.\nonumber \\ && \left.+ \mathop \sum \limits_{\mathfrak{i},\mathfrak{j},..,\mathcal{N}} ({\mathfrak{r}_{\mathfrak{ij}}}.{\mathfrak{r}_{\mathfrak{kj}}}.{\mathfrak{r}_{\mathfrak{k}\ell}}.{\mathfrak{r}_{\ell\mathfrak{i}}} + {\mathfrak{r}_{\mathfrak{i}\ell }}.{\mathfrak{r}_{\ell \mathfrak{k}}}.{\mathfrak{r}_{\mathfrak{kj}}}.{\mathfrak{r}_{\mathfrak{ji}}}).{\mathcal{C}_{\mathfrak{n}}}.{\mathcal{C}_{\mathfrak{m}} \ldots {\mathcal{C}_{\mathcal{N}}}} \right\} \\ && + \left\{ \mathop \sum \limits_{\mathfrak{i},{\mathfrak{j},..,\mathcal{N}}} ({\mathfrak{r}_{{\mathfrak{ij}}}}.{{\mathfrak{r}}_{{\mathfrak{ji}}}}) \left({{\mathfrak{r}}_{{\mathfrak{k}\ell}}}.{{\mathfrak{k}}_{\ell{\rm{n}}}}.{{\mathfrak{r}}_{{\rm{n}}{\mathfrak{k}}}}\right).{{\mathcal{C}}_{\mathfrak{m}}}.{{\mathcal{C}}_{\mathfrak{o}}} \ldots {{\mathcal{C}}_{\mathcal{N}}}\nonumber\right. \\ && \left.+ \mathop \sum \limits_{{\mathfrak{i}},\mathfrak{j},..,\mathcal{N}}({{\mathfrak{r}}_{{\mathfrak{ij}}}}.{{\mathfrak{r}_{{\mathfrak{jk}}}}}.{{{\mathfrak{r}}_{{\mathfrak{k}\ell}}}}.{{\mathfrak{r}}_{\ell {\mathfrak{n}}}}.{{\mathfrak{r}}_{{\mathfrak{ni}}}}\right. \\ && \left. + {{\mathfrak{r}}_{{\mathfrak{in}}}}.{{\mathfrak{r}}_{{\mathfrak{n}\ell}}}.{{\mathfrak{r}}_{\ell{\mathfrak{k}}}}.{{\mathfrak{r}}_{{\mathfrak{kj}}}}.{{\mathfrak{r}}_{{\mathfrak{ji}}}}).{{\mathcal{C}}_{\mathfrak{m}}}.{{\mathcal{C}}_{\mathfrak{o}}} \ldots {{\mathcal{C}}_{\mathcal{N}}}\phantom{\mathop \sum \limits_{\mathfrak{i},{\mathfrak{j},..,\mathcal{N}}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\! \right\} + \cdots \end{eqnarray*}$$(10) 3.3 Modelling and the proposed solving method There is a specific number of post-disaster reconstruction projects in areas that have been impacted by a destructive disaster. Each project needs |$\ {\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| construction material type |$\mathfrak{m}$| in the time period |$\mathfrak{p}$|⁠. Materials should be provided for construction projects from a number of warehouses. For distributing materials from warehouses to construction sites, two different types of vehicles, including land and aerial vehicles, are used, and the number of each type is limited to |${\mathcal{N}_{\mathfrak{v} }}$| and |${\mathcal{N}_{\mathfrak{h}}}$| vehicles. All vehicles are available from the beginning of the transportation. Each vehicle can carry a limited amount of materials. The construction projects are located in two different areas; some of them are accessible through the roads, while other projects are in areas that are not accessible using the roads; therefore, aerial vehicles should be used to provide the required material for them. Vehicles move in a network represented by a graph in which nodes are locations of construction projects and depots and arcs are the routes between each pair of nodes. However, due to the impact of the disaster on the road network, travel times on network links vary depending on time, so |${\rm{\ }}{\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}$| denotes the travel time for vehicle type |$\mathfrak{v}$| from node |$\mathfrak{i}$| to |$\mathfrak{j}$| in the time period |$\mathfrak{p}$|⁠. To reach a comprehensive plan, we consider three objectives in this model, minimising the total travel time, maximising the reliability of the route, and minimising the total weighted undelivered materials to projects. See Fig.4. For the schematic presentation of the problem. Figure 4. Open in new tabDownload slide An illustration of the problem in schematic form. Real-world decision-making problems can be reduced to their essential elements by employing mathematical modelling techniques. If all the details are included in a model, it may be impossible to model real-world problems. The most useful models rely on assumptions that preserve the system’s fundamental properties while also simplifying it. Keeping models manageable and accurate in light of this fact necessitates simplifying assumptions. Mathematical models that incorporate these assumptions are highly effective, as evidenced by the following: The number of vehicles is finite, and different types of vehicles are used to deliver materials to the projects. As a result, the transport fleet of vehicles is heterogeneous, and their capacity and speed are different. The origin of all vehicles is known in advance. The vehicle’s capacity is more than the demand so that there is no interruption in service. The place of distribution of material and warehouses of vehicles is the same, and the place and number of warehouses are predetermined. Available construction material is sufficient to respond to the projects. Distribution operations are performed for several types of construction materials. The location of each project is known, and their distance from the warehouse is known. Each rescue vehicle returns to the starting point of the movement (warehouse) after the end of the operation (closed route). The distribution of construction materials is performed from several warehouses, where the warehouses for vehicles and the warehouses for storing construction materials are the same. The construction projects are located in two clusters, and all those points are prioritized based on the factors affecting reliability. A damaged vehicle cannot be repaired at an acceptable time to be able to continue its work. If the product in a particular scenario, before the car breakdown, has reached the location of construction projects, the service is complete, and shortage does not occur. After a vehicle breakdown, other vehicles should not perform their service duties and should act according to their own schedule. The demand for each project might not be fully met and may be in short supply. Construction materials are distributed over different periods. 3.3.1 The mathematical model The variables and parameters listed below are used in the development of the mathematical model. Notations, sets, and parameters   |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}{\rm{ }}}}$| Set of trucks |$\mathfrak{h} \in {\mathcal{N}_{\mathfrak{h}}}$| Set of helicopters |$\mathfrak{a} \in {\mathcal{N}_{\mathfrak{a}}}$| Set of the location of projects with a passable access road |$\mathfrak{a}^{\prime} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}$| Set of the location of projects with a damaged access road |${\mathcal{N}_{\mathfrak{a}}} \cup {\mathcal{N}_{\mathfrak{a}^{\prime}}} \subseteq {\mathcal{N}_{\hat{\mathfrak{a}}}}$| |$\mathfrak{i},\mathfrak{j} \in {\cal I},\mathcal{J}$| Set of all locations, including depots |$\mathfrak{m} \in {\cal M}$| Set of construction material types |$\mathfrak{p} \in \mathcal{P}$| Set of time periods |$\mathfrak{s} \in \mathcal{S}$| Set of time scenarios |${\mathcal{C}_{\mathfrak{v}}}$| The capacity of vehicle |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}}$| |${\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The demand of project |$\mathfrak{i}$| for construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| |${\omega _{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The value of construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| for the project |$\mathfrak{i}$| |${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$| The permanent value of node |$\mathfrak{i}$| to |$\mathfrak{j}$| based on the reliability index |${\mathcal{T}_{\mathfrak{j}\mathfrak{v}\mathfrak{p}}}$| Arrival time of vehicle |$\mathfrak{v}$| to node |$\mathfrak{i}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\delta _{\mathfrak{v}\mathfrak{p}}^{\mathfrak{s}}$| Break down scenario for vehicle |$\mathfrak{v}$|⁠, it means |$\mathfrak{v}$| in scenario|$\ \mathfrak{s}$| fails after |${\rm{\delta }}$| unit time in time |${\rm{period\ }}\mathfrak{p}$| |${\mathfrak{d}_{\mathfrak{v}}}$| The location of depot vehicle |$\mathfrak{v} \in ({\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}})$| |${\xi_{\mathfrak{p}}}$| Start time of the period |$\mathfrak{p}$| |${\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}$| Travel time for vehicle type |$\mathfrak{v}$| from node |$\mathfrak{i}$| to |$\mathfrak{j}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}{\rm{ }}}}$| Set of trucks |$\mathfrak{h} \in {\mathcal{N}_{\mathfrak{h}}}$| Set of helicopters |$\mathfrak{a} \in {\mathcal{N}_{\mathfrak{a}}}$| Set of the location of projects with a passable access road |$\mathfrak{a}^{\prime} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}$| Set of the location of projects with a damaged access road |${\mathcal{N}_{\mathfrak{a}}} \cup {\mathcal{N}_{\mathfrak{a}^{\prime}}} \subseteq {\mathcal{N}_{\hat{\mathfrak{a}}}}$| |$\mathfrak{i},\mathfrak{j} \in {\cal I},\mathcal{J}$| Set of all locations, including depots |$\mathfrak{m} \in {\cal M}$| Set of construction material types |$\mathfrak{p} \in \mathcal{P}$| Set of time periods |$\mathfrak{s} \in \mathcal{S}$| Set of time scenarios |${\mathcal{C}_{\mathfrak{v}}}$| The capacity of vehicle |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}}$| |${\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The demand of project |$\mathfrak{i}$| for construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| |${\omega _{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The value of construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| for the project |$\mathfrak{i}$| |${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$| The permanent value of node |$\mathfrak{i}$| to |$\mathfrak{j}$| based on the reliability index |${\mathcal{T}_{\mathfrak{j}\mathfrak{v}\mathfrak{p}}}$| Arrival time of vehicle |$\mathfrak{v}$| to node |$\mathfrak{i}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\delta _{\mathfrak{v}\mathfrak{p}}^{\mathfrak{s}}$| Break down scenario for vehicle |$\mathfrak{v}$|⁠, it means |$\mathfrak{v}$| in scenario|$\ \mathfrak{s}$| fails after |${\rm{\delta }}$| unit time in time |${\rm{period\ }}\mathfrak{p}$| |${\mathfrak{d}_{\mathfrak{v}}}$| The location of depot vehicle |$\mathfrak{v} \in ({\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}})$| |${\xi_{\mathfrak{p}}}$| Start time of the period |$\mathfrak{p}$| |${\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}$| Travel time for vehicle type |$\mathfrak{v}$| from node |$\mathfrak{i}$| to |$\mathfrak{j}$| in time |${\rm{period\ }}\mathfrak{p}$| Open in new tab   |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}{\rm{ }}}}$| Set of trucks |$\mathfrak{h} \in {\mathcal{N}_{\mathfrak{h}}}$| Set of helicopters |$\mathfrak{a} \in {\mathcal{N}_{\mathfrak{a}}}$| Set of the location of projects with a passable access road |$\mathfrak{a}^{\prime} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}$| Set of the location of projects with a damaged access road |${\mathcal{N}_{\mathfrak{a}}} \cup {\mathcal{N}_{\mathfrak{a}^{\prime}}} \subseteq {\mathcal{N}_{\hat{\mathfrak{a}}}}$| |$\mathfrak{i},\mathfrak{j} \in {\cal I},\mathcal{J}$| Set of all locations, including depots |$\mathfrak{m} \in {\cal M}$| Set of construction material types |$\mathfrak{p} \in \mathcal{P}$| Set of time periods |$\mathfrak{s} \in \mathcal{S}$| Set of time scenarios |${\mathcal{C}_{\mathfrak{v}}}$| The capacity of vehicle |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}}$| |${\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The demand of project |$\mathfrak{i}$| for construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| |${\omega _{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The value of construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| for the project |$\mathfrak{i}$| |${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$| The permanent value of node |$\mathfrak{i}$| to |$\mathfrak{j}$| based on the reliability index |${\mathcal{T}_{\mathfrak{j}\mathfrak{v}\mathfrak{p}}}$| Arrival time of vehicle |$\mathfrak{v}$| to node |$\mathfrak{i}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\delta _{\mathfrak{v}\mathfrak{p}}^{\mathfrak{s}}$| Break down scenario for vehicle |$\mathfrak{v}$|⁠, it means |$\mathfrak{v}$| in scenario|$\ \mathfrak{s}$| fails after |${\rm{\delta }}$| unit time in time |${\rm{period\ }}\mathfrak{p}$| |${\mathfrak{d}_{\mathfrak{v}}}$| The location of depot vehicle |$\mathfrak{v} \in ({\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}})$| |${\xi_{\mathfrak{p}}}$| Start time of the period |$\mathfrak{p}$| |${\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}$| Travel time for vehicle type |$\mathfrak{v}$| from node |$\mathfrak{i}$| to |$\mathfrak{j}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}{\rm{ }}}}$| Set of trucks |$\mathfrak{h} \in {\mathcal{N}_{\mathfrak{h}}}$| Set of helicopters |$\mathfrak{a} \in {\mathcal{N}_{\mathfrak{a}}}$| Set of the location of projects with a passable access road |$\mathfrak{a}^{\prime} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}$| Set of the location of projects with a damaged access road |${\mathcal{N}_{\mathfrak{a}}} \cup {\mathcal{N}_{\mathfrak{a}^{\prime}}} \subseteq {\mathcal{N}_{\hat{\mathfrak{a}}}}$| |$\mathfrak{i},\mathfrak{j} \in {\cal I},\mathcal{J}$| Set of all locations, including depots |$\mathfrak{m} \in {\cal M}$| Set of construction material types |$\mathfrak{p} \in \mathcal{P}$| Set of time periods |$\mathfrak{s} \in \mathcal{S}$| Set of time scenarios |${\mathcal{C}_{\mathfrak{v}}}$| The capacity of vehicle |$\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}}$| |${\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The demand of project |$\mathfrak{i}$| for construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| |${\omega _{\mathfrak{i}\mathfrak{m}\mathfrak{p}}}$| The value of construction material type |$\mathfrak{m}$| in time |${\rm{period\ }}\mathfrak{p}$| for the project |$\mathfrak{i}$| |${\mathfrak{r}_{\mathfrak{i}\mathfrak{j}}}$| The permanent value of node |$\mathfrak{i}$| to |$\mathfrak{j}$| based on the reliability index |${\mathcal{T}_{\mathfrak{j}\mathfrak{v}\mathfrak{p}}}$| Arrival time of vehicle |$\mathfrak{v}$| to node |$\mathfrak{i}$| in time |${\rm{period\ }}\mathfrak{p}$| |$\delta _{\mathfrak{v}\mathfrak{p}}^{\mathfrak{s}}$| Break down scenario for vehicle |$\mathfrak{v}$|⁠, it means |$\mathfrak{v}$| in scenario|$\ \mathfrak{s}$| fails after |${\rm{\delta }}$| unit time in time |${\rm{period\ }}\mathfrak{p}$| |${\mathfrak{d}_{\mathfrak{v}}}$| The location of depot vehicle |$\mathfrak{v} \in ({\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}})$| |${\xi_{\mathfrak{p}}}$| Start time of the period |$\mathfrak{p}$| |${\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}$| Travel time for vehicle type |$\mathfrak{v}$| from node |$\mathfrak{i}$| to |$\mathfrak{j}$| in time |${\rm{period\ }}\mathfrak{p}$| Open in new tab Decision variables $$\begin{equation*} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = \left\{ \begin{array}{ll} 1 \quad \rm{if\ vehicle}\ \mathfrak{v}\ \rm{travel\ from\ node}\ \mathfrak{i}\ \rm{to\ node}\ \mathfrak{j}\ \rm{in}\\ \quad\,\,\rm{\ time\ period}\ \mathfrak{p} \\ 0 \quad \rm{otherwise}\ \end{array}\right.\end{equation*} $$ $$\begin{equation*} \mathfrak{y}_{\mathfrak{i}\mathfrak{p}}^{\mathfrak{s}} = \left\{ \begin{array}{ll} 1 \quad {{\rm{if\ the\ material\ is\ not\ delivered\ to\ the\ project\ }}\mathfrak{i}{\rm{\ in\ }}}\\ \quad\,\,\,{{{time\ period\ }}\mathfrak{p}\ {\rm{under\ scenario\ }}\mathfrak{s}}\\ 0 \quad {{\rm{otherwise}}} \end{array}\right. \end{equation*} $$ Model formulation Mathematically, the proposed model can be expressed as follows: $$\begin{eqnarray*} && {\cal M}\mathfrak{i}\mathfrak{n}\mathop \sum \limits_{{\mathfrak{i}} \in {\cal I}} \mathop \sum \limits_{{\mathfrak{j}} \in \mathcal{J}} \mathop \sum \limits_{{\mathfrak{v}} \in \mathcal{V}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} {\mathfrak{t}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}{\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}},\nonumber \\ && {\cal M}\mathfrak{a}\mathfrak{x}\mathop \sum \limits_{{\mathfrak{i}} \in {\cal I}} \mathop \sum \limits_{{\mathfrak{j}} \in \mathcal{J}} \mathop \sum \limits_{{\mathfrak{v}} \in \mathcal{V}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} {\mathfrak{r}_{{\mathfrak{i}}\mathfrak{j}}}{\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}, \\ && {\cal M}\mathfrak{i}\mathfrak{n}\mathop \sum \limits_{{\mathfrak{s}} \in \mathcal{S}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} \mathop \sum \limits_{{\mathfrak{m}} \in {\cal M}} \mathop \sum \limits_{{\mathfrak{i}} \in {\mathcal{N}_{{\hat{\mathfrak{a}}}}}} {\mathfrak{y}_{\mathfrak{ip}}^{\mathfrak{s}}}{\tilde{\mathcal{Q}}_{{\mathfrak{imp}}}}{\omega _{{\mathfrak{imp}}}}, \end{eqnarray*} $$(11) $$\begin{equation*} \mathop \sum \limits_{{\mathfrak{i}} \in {\mathcal{N}_{\mathfrak{a}}} \cup {\mathfrak{d}_{\mathfrak{v}}}}^{s.to.} \ {\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = \mathop \sum \limits_{{\mathfrak{i}} \in {\mathcal{N}_{\mathfrak{a}}} \cup {\mathfrak{d}_{\mathfrak{v}}}} {\mathfrak{x}_{{\mathfrak{v}}\mathfrak{j}\mathfrak{i}\mathfrak{p}}}\qquad \forall \mathfrak{j} \in {\mathcal{N}_{\mathfrak{a}}},\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(12) $$\begin{equation*} \mathop \sum \limits_{\mathfrak{i} \in {\mathcal{N}_{\hat{\mathfrak{a}}}} \cup {\mathfrak{d}_{\mathfrak{h}}}} \ {\mathfrak{x}_{\mathfrak{h}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = \mathop \sum \limits_{\mathfrak{i} \in {\mathcal{N}_{\hat{\mathfrak{a}}}} \cup {\mathfrak{d}_{\mathfrak{h}}}} {\mathfrak{x}_{\mathfrak{h}\mathfrak{j}\mathfrak{i}\mathfrak{p}}}\qquad \forall \,\,\mathfrak{j} \in {\mathcal{N}_{\hat{\mathfrak{a}}}},\mathfrak{h} \in {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(13) $$\begin{equation*}\ \mathop \sum \limits_{\mathfrak{i} \in {\mathfrak{d}_{\mathfrak{v}}}} \mathop \sum \limits_{\mathfrak{j} \in {\mathcal{N}_{\hat{\mathfrak{a}}}}} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = 1\qquad \qquad \qquad \forall \,\,\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(14) $$\begin{equation*}\ \mathop \sum \limits_{\mathfrak{i} \in {\mathcal{N}_{\hat{\mathfrak{a}}}}} \mathop \sum \limits_{\mathfrak{j} \in {\mathfrak{d}_{\mathfrak{v}}}} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = 1\qquad \qquad \qquad \forall \,\,\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(15) $$\begin{equation*} \mathop \sum \limits_{\mathfrak{i} \in {\mathfrak{d}_{\mathfrak{v}}}} \mathop \sum \limits_{\mathfrak{j} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} + \mathop \sum \limits_{\mathfrak{i} \in {\mathcal{N}_{\mathfrak{a}^{\prime}}}} \ \mathop \sum \limits_{\mathfrak{j} \in {\mathfrak{d}_{\mathfrak{v}}}} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} = 0\qquad \forall \,\,\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(16) $$\begin{equation*} \mathop \sum \limits_{\mathfrak{i} \in {\mathfrak{d}_{\mathfrak{v}}}} \mathop \sum \limits_{\mathfrak{j} \in {\mathcal{N}_{\hat{\mathfrak{a}}}}} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}{\tilde{\mathcal{Q}}_{\mathfrak{i}\mathfrak{m}\mathfrak{p}}} \le {\mathcal{C}_{\mathfrak{v}}}\,\,\qquad \qquad \forall \,\,\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(17) $$\begin{equation*} \ {\mathcal{T}_{\mathfrak{j}\mathfrak{v}\mathfrak{p}}} = \mathop \sum \limits_{\mathfrak{i} \in {\mathfrak{d}_{\mathfrak{v}}}} {\mathfrak{t}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}{\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}\ + {\xi _{\mathfrak{p}}}\qquad\!\!\! \forall \mathfrak{j} \in {\mathcal{N}_{\hat{\mathfrak{a}}}},\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*} $$(18) $$\begin{equation*} {\cal M}\left( {1 - \mathfrak{y}_{\mathfrak{i}\mathfrak{p}}^{\mathfrak{s}}} \right) \ge \left( {{\mathcal{T}_{\mathfrak{i}\mathfrak{v}\mathfrak{p}}} - \delta _{\mathfrak{v}\mathfrak{p}}^{\mathfrak{s}}} \right)\quad\quad\!\! \forall \mathfrak{i} \in {\mathcal{N}_{\hat{\mathfrak{a}}}},\mathfrak{s} \in \mathcal{S},\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P} \end{equation*}$$(19) $$\begin{equation*} {\mathfrak{x}_{\mathfrak{v}\mathfrak{i}\mathfrak{j}\mathfrak{p}}} \in \left\{ {0,1} \right\}\qquad \qquad \forall \mathfrak{v},\mathfrak{i},\mathfrak{j},\mathfrak{p} \end{equation*} $$(20) $$\begin{equation*} {\mathcal{T}_{\mathfrak{i}\mathfrak{v}\mathfrak{p}}} \ge 0\qquad \qquad \forall \mathfrak{v},\mathfrak{i},\mathfrak{p} \end{equation*} $$(21) $$\begin{equation*} \mathfrak{y}_{\mathfrak{i}\mathfrak{p}}^{\mathfrak{s}} \in \left\{ {0,1} \right\}\qquad \qquad \forall \mathfrak{i},\mathfrak{p},\mathfrak{s}. \end{equation*} $$(22) The first objective function minimises the total travel time, the second objective function maximises the reliability of routes, and the third objective function minimises the total weighted undelivered materials to projects. Constraint (2) ensures that each truck leaves a project after entering and servicing it. Constraint (3) ensures that each helicopter leaves a project after entering and servicing it. Constraints (4) and (5) ensure that after servicing any nodes, all vehicles (i.e. truck and helicopter) must return to the start point, successfully closing the route. Constraint (6) asserts that trucks cannot serve projects with impassable roads. Constraints (7) refer to the vehicle capacity limitations. Constraint (8) calculates the vehicle’s arrival time at construction sites. Constraint (9) indicates whether or not the shortage exists in the given scenarios. Constraints (10) to (12) define the type of the variables. Pareto-optimal or non-dominated solutions are explored in multi-objective problems rather than single optimal solutions. If a solution cannot be improved in one objective function without degrading its performance in at least one other objective, it is Pareto optimal (Mavrotas, 2009). As a result of these Pareto optimal solutions, the Pareto front of the problem is constructed, from which decision makers can choose the final preferred compromise solution. Several methods for multi-objective problems have been proposed in the literature. In this study, the augmented |${\boldsymbol {\varepsilon }}$|-constraint method (AUGMECON), similar to Esmaili et al. (2011), transforms the multi-objective problem into a single-objective problem. Due to the uncertainty in a disaster situation and becoming closer to reality, the received demand for construction projects is fuzzy numbers in the proposed model. A symmetrical triangle distribution is considered for indicating the fuzzy parameter due to its applicability and ease of calculation. Because of its high efficiency, the Jimenez method is used to transform the current model into its corresponding deterministic model (Jiménez et al., 2007). The following equations represent changes in the objective function and constraints: $$\begin{eqnarray*} && {\cal M}\mathfrak{i}\mathfrak{n}\mathop \sum \limits_{{\mathfrak{i}} \in {\cal I}} \mathop \sum \limits_{{\mathfrak{j}} \in \mathcal{J}} \mathop \sum \limits_{{\mathfrak{v}} \in \mathcal{V}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} {\mathfrak{t}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}{\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}},\nonumber \\ && {\cal M}\mathfrak{a}\mathfrak{x}\mathop \sum \limits_{{\mathfrak{i}} \in {\cal I}} \mathop \sum \limits_{{\mathfrak{j}} \in \mathcal{J}} \mathop \sum \limits_{{\mathfrak{v}} \in \mathcal{V}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} {\mathfrak{r}_{{\mathfrak{i}}\mathfrak{j}}}{\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}, \\ && {\cal M}\mathfrak{i}\mathfrak{n}\mathop \sum \limits_{{\mathfrak{s}} \in \mathcal{S}} \mathop \sum \limits_{{\mathfrak{p}} \in \mathcal{P}} \mathop \sum \limits_{{\mathfrak{m}} \in {\cal M}} \mathop \sum \limits_{{\mathfrak{i}} \in {\mathcal{N}_{\hat{\mathfrak{a}}}}} \mathfrak{y}_{{\mathfrak{i}}\mathfrak{p}}^{\mathfrak{s}}\left( {\frac{{\mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^{1} + \mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^{2} + \mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^{3}}}{4}} \right){\omega _{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}}, \end{eqnarray*} $$(23) $$\begin{eqnarray*} && {\rm{Subject\ to:}} \nonumber \\ && {\rm{Other\ constraints\ in\ the\ model}}. \end{eqnarray*} $$(24) $$\begin{equation*} \mathop \sum \limits_{{\mathfrak{i}} \in {\mathfrak{d}_{\mathfrak{v}}}} \mathop \sum \limits_{{\mathfrak{j}} \in {\mathcal{N}_{\hat{\mathfrak{a}}}}} {\mathfrak{x}_{{\mathfrak{v}}\mathfrak{i}\mathfrak{j}\mathfrak{p}}}\left[ {a.\left( {\frac{{\mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^1 + \mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^2}}{2}}\right.} \right)\\ \quad+ \left({\left. {1 - a} \right)\left( {\frac{{\mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^2 + \mathcal{Q}_{{\mathfrak{i}}\mathfrak{m}\mathfrak{p}}^3}}{2}} \right)} \right] \le {\mathcal{C}_{\mathfrak{v}}}\ \ \ \ \,\,\ \ \forall \,\,\mathfrak{v} \in {\mathcal{N}_{\mathfrak{v}}} \cup {\mathcal{N}_{\mathfrak{h}}},\mathfrak{p} \in \mathcal{P}. \end{equation*} $$(25) 4. Multi-Objective Meta-heuristics Algorithms When a problem has multiple objectives, it is difficult to find the best solution (Tirkolaee et al., 2022; Yuan et al., 2022). Many real-world optimisation problems are made more difficult by the presence of multiple objectives (Pasha et al., 2022). Meta-heuristic algorithms can be used to solve these problems (Fathollahi-Fard et al., 2021b). Meta-heuristic algorithms have some advantages (Dulebenets, 2021). Any problem that can be expressed as a function-optimisation problem can be solved with these methods (Fathollahi-Fard et al., 2021b); these methods tend to be more straightforward to understand and implement (Theophilus et al., 2021), and they can handle more complex problems with ease (Pasha et al., 2020). In this study, two widely used multi-objective meta-heuristic algorithms, NSGA-II and MOEA/D, are used to solve the problem. The pseudocode and algorithm properties are discussed in the following section. 4.1 NSGA-II NSGA-II, which was introduced for the first time by Deb et al. (2002), is one of the most widely applicable and well-proposed genetic algorithm (GA)-based algorithms for solving multi-objective optimisation problems. NSGA-II begins by randomly generating a population of size |${A_\xi }$| (Babaeinesami et al., 2022). The objective values of a population are evaluated using an objective function. The population is then ranked according to the non-domination sorting procedure in order to generate Pareto fronts. Each individual in the population under consideration is assigned a rank equal to its non-domination level, with the first front containing individuals with the lowest rank, the second front containing individuals with the second rank, and so on (Tirkolaee et al., 2020b). The following step calculates the crowding distance between members on each front using a linear distance criterion. Due to the fact that a binary tournament selection operator based on a crowded-comparison operator is used, both the rank and crowding distance of each member of the population must be calculated. Two members of the population are initially selected using this selection operator. The member with the greater crowding distance is then chosen if their ranks are equal. Otherwise, the lower ranking member is chosen. Following that, a new population of offspring with a size of n is created using the selection, crossover, and mutation operators to create a population composed of the current population and the new population of size (⁠|${A_\xi }$| + n). In this study, one point crossover has been used. Finally, using the sorting procedure, a population with the exact size of |${A_\xi }$| is obtained. The solutions are sorted twice in this procedure: first by their crowding distances in descending order, and then by their ranks in ascending order (Tirkolaee et al., 2019). By repeating the preceding steps in order, the new population is used to generate the next generation of offspring. This procedure is repeated until the termination condition is satisfied. After implementing NSGA-II, a set of non-dominated Pareto-optimal solutions is obtained, as all solutions are optimal in terms of multi-objective optimisation. Algorithm 2 presents the pseudocode of NSGA-II. Open in new tabDownload slide 4.2 MOEA/D The MOEA/D algorithm for the multi-objective problems is introduced by Zhang and Li (2007). MOEA/D uses a set of uniformly distributed weight vectors |$W\ = \ \{ {{w^1},\ \ldots ,\ {w^\mu }} \}$| to decompose a multi-objective optimisation problem with M objectives into single-objective subproblems. At the start of the search, all members of the population are generated randomly in the search space S. An index list |${B^i} = \ \{ {{i_1},\ \ldots ,\ {i_T}} \}$| is initialised for each subproblem index |$i \in \{ {1,\ \ldots ,\ \mu } \}$|⁠, which is used for mating and replacement selections: |${B^i}$| is composed of the indices of the T weight vectors that are closest to |${w^i}$| in the weight vector space, where T denotes the neighbourhood size (Fathollahi-Fard et al., 2021a). Following initialisation, the following steps are repeated for each subproblem |$i \in \{ {1,\ \ldots ,\ \mu } \}$| until the search criteria for termination are met. The parent indices k and l are randomly chosen from |${B^i}$| for each i. Then, by combining |${x^k}$| and |${x^l}$|⁠, a child |${u^i}$| is generated. If necessary, a mutation opera-tor is applied to the child |${u^i}$|⁠. After generating |${u^i}$|⁠, a pre-defined scalarising function g is used to perform the replacement selection. On the basis of g, the individual |${x^j}$| is compared to the child |${u^i}$| for each |$j \in {B^i}$|⁠. If |${u^i}$| is superior to |${x^j}$| in terms of their weight vector |${w^j}$| scalarising function values, |${x^j}$| is replaced by |${u^i}$|⁠. Open in new tabDownload slide Since the replacement of individuals is based on their scalarising function values, g plays a crucial role in MOEA/D. Although there are a number of scalarising functions as reviewed in (Pescador-Rojas et al., 2017), in this paper, we follow the approach of (Wang et al., 2015). 4.3 Solution representation The solution representation should be simple to decode in order to minimise the algorithm’s computational cost. As a result, the solution representation illustrated in Fig. 5 is used in this study. A cube is considered in this type of answer representation, with the dimensions of the solution being construction projects (J), vehicles (V), and time period (t). In the solution depicted in Fig. 5, five construction projects, three vehicles, and three periods are assumed. Each cell in this solution representation is filled with a value between 0 and 1. Figure 5: Open in new tabDownload slide A schematic sample of the solution representation. To determine how vehicles are used to service construction projects during each period, the maximum number in each column and row is determined and marked. The columns’ maximum values are indicated in blue, while the rows’ maximum values are indicated in red. According to the priority winch established in the previous step, in each period, construction projects are assigned to a different vehicle, and the sequence of visits to each project by a particular vehicle is determined by the cell values. For instance, in the first period, vehicle one goes to project 5, and then goes to project 3. Figure 6 shows the answer for each of the three periods separately. During the first period, vehicle 1 is assigned to projects 3 and 5, vehicle 2 is assigned to project 2, and vehicle 3 is assigned to projects 1 and 4. Then we find that the construction materials are distributed in the following order: Vehicle 1 is routed to project 5 first, then to project 3. Vehicle 2 only serves project 2, while vehicle 3 first serves project 4 and then project 1. Vehicle 1 is assigned to projects 3 and 5, vehicle 2 is assigned to project 4, and vehicle 3 is assigned to projects 1 and 2. Then we find that the construction materials are distributed in the following order: Vehicle 1 goes to project 5 and then to project 3. Vehicle 2 only serves project 4, whereas vehicle 3 first serves project 2 and then project 1. During the third period, vehicle 1 is assigned to projects 2 and 3, vehicle 2 is assigned to project 4, and vehicle 3 is assigned to projects 1 and 5. Then we see that the construction materials are distributed in the following order: Vehicle 1 goes to project 2 and then to project 3. Vehicle 2 only serves project 4, while vehicle 3 first serves project 5 and then project 1. Figure 6: Open in new tabDownload slide Decoding the solution representation. 5. Computational Results This section discusses the model proposed and the proposed solution approach for improving the distribution of post-disaster construction supplies after a large-scale disaster. First, in Section 5.1, the performance of the model and proposed algorithms for small-size problems are evaluated. The study’s algorithms were implemented in MATLAB 2018b, and the mathematical model was coded in GAMS 25.0.2 and solved using the CPLEX solver on an Intel Core i7 @ 2.5 GHz PC with 8 GB of RAM running Microsoft Windows 10. Section 5.2 explains the evaluating metrics and the procedure for generating the instances of the problem. Section 5.3 includes numerical examples and analyses that illustrate how the proposed algorithms perform under various large-size real-world conditions. Section 5.4 conducts the statistical test to evaluate the performance of the proposed method in more detail—finally, section 5.5 details parameter sensitivity analyses. 5.1 Analysing the performance of the model and algorithms in small-size problems There are ten problems of varied dimensions to test the performance of the algorithms and the model proposed. The outputs of meta-heuristics are compared to the GAMS results achieved using AUGMECON. As presented in Table 4, the proposed meta-heuristics’ objective function values (OFV) and errors are reported for each problem compared to the AUGMECON. The maximum run time for meta-heuristic algorithms has been set to 3600 s. In this paper, the error rate of each algorithm in comparison to the AUGMECON is determined using the following equations for each objective function of the mathematical model (Yazdani et al., 2017): $$\begin{equation*} {\rm{Ga}}{{\rm{p}}_{O{F_{\mathfrak{i}}}}} = \frac{{O{F_{\mathfrak{i}}}_{Metahurestic} - O{F_{\mathfrak{i}}}_{{\rm{AUGMECON}}}}}{{O{F_{\mathfrak{i}}}_{AUGMECON}}}{\rm{ }} \times 100,\!\!\!\qquad \mathfrak{i} = 1,2,3 \end{equation*} $$(26) where |$O{F_{\mathfrak{i}}}_{Metahurestic}$| is the best results obtained by the meta-heuristic for objective |$\mathfrak{i}$|⁠, while |$O{F_{\mathfrak{i}}}_{{\rm{AUGMECON}}}$| is the obtained result for objective |$\mathfrak{i}$| using the AUGMECON. Table 4: Results obtained by AUGMECON, NSGA-II, and MOEA/D. (Problem no., depots, affected areas, time period, product no.) . AUGMECON . NSGA-II . Gap (%) . MOEA/D . Gap (%) . Best OF (1,2,3) . CPU time . Best OF (1,2,3) . CPU time . OF (1,2,3) . Best OF (1,2,3) . CPU time . OF (1,2,3) . (1,2,5,3,2) (74.7,8.4,43) 284 (74.7, 8.4,43) 426 (0.000,0.000,0.000) (74.7,8.4,43) 472 (0.000,0.000,0.000) (2,3,10,3,2) (82.3,13.2,37) 573 (83,11.3,39) 661 (0.009,0.144,0.054) (82.5,12.6,38) 682 (0.002,0.045,0.027) (3,4,9,3,2) (80.9,14,58) 825 (84.1,12.6,63) 837 (0.040,0.100,0.086) (84.1,13.1,62) 819 (0.040,0.064,0.069) (4,3,12,4,3) (101.4,18.8,79) 1135 (107.9,18.1,82) 1023 (0.064,0.037,0.038) (107.9,17.6,82) 963 (0.064,0.064,0.038) (5,5,14,4,3) (107.3,10.3,86) 1637 (110.4,10.1,87) 1251 (0.029,0.019,0.012) (110.4,9,89) 1104 (0.029,0.126,0.035) (6,6,15,5,3) (128.1,17.4,123) 2563 (135.6,15.7,127) 1386 (0.059,0.098,0.033) (135.6,16.3,129) 1284 (0.059,0.063,0.049) (7,7,17,5,4) (136.8,14.4,109) 3048 (141.9,13.2,124) 1503 (0.037,0.083,0.138) (141.9,13.9,114) 1403 (0.037,0.035,0.046) (8,8,18,6,4) (-,-,-) - (174.3,18.3,149) 1783 (-,-,-) (174.3,17.9,151) 1583 (-,-,-) (9,9,19,6,5) (-,-,-) - (186.3,16,156) 1980 (-,-,-) (186.3,18.3,149) 1792 (-,-,-) (10,10,21,8,5) (-,-,-) - (219.6,14.9,173) 2194 (-,-,-) (219.6,14.8,172) 1941 (-,-,-) Average (203.3,27.57,76.43) 284 (131.78,13.86,104.3) (3.38,6.88,5.14)% (131.73,14.19,102.9) (3.30,5.68,3.76)% (Problem no., depots, affected areas, time period, product no.) . AUGMECON . NSGA-II . Gap (%) . MOEA/D . Gap (%) . Best OF (1,2,3) . CPU time . Best OF (1,2,3) . CPU time . OF (1,2,3) . Best OF (1,2,3) . CPU time . OF (1,2,3) . (1,2,5,3,2) (74.7,8.4,43) 284 (74.7, 8.4,43) 426 (0.000,0.000,0.000) (74.7,8.4,43) 472 (0.000,0.000,0.000) (2,3,10,3,2) (82.3,13.2,37) 573 (83,11.3,39) 661 (0.009,0.144,0.054) (82.5,12.6,38) 682 (0.002,0.045,0.027) (3,4,9,3,2) (80.9,14,58) 825 (84.1,12.6,63) 837 (0.040,0.100,0.086) (84.1,13.1,62) 819 (0.040,0.064,0.069) (4,3,12,4,3) (101.4,18.8,79) 1135 (107.9,18.1,82) 1023 (0.064,0.037,0.038) (107.9,17.6,82) 963 (0.064,0.064,0.038) (5,5,14,4,3) (107.3,10.3,86) 1637 (110.4,10.1,87) 1251 (0.029,0.019,0.012) (110.4,9,89) 1104 (0.029,0.126,0.035) (6,6,15,5,3) (128.1,17.4,123) 2563 (135.6,15.7,127) 1386 (0.059,0.098,0.033) (135.6,16.3,129) 1284 (0.059,0.063,0.049) (7,7,17,5,4) (136.8,14.4,109) 3048 (141.9,13.2,124) 1503 (0.037,0.083,0.138) (141.9,13.9,114) 1403 (0.037,0.035,0.046) (8,8,18,6,4) (-,-,-) - (174.3,18.3,149) 1783 (-,-,-) (174.3,17.9,151) 1583 (-,-,-) (9,9,19,6,5) (-,-,-) - (186.3,16,156) 1980 (-,-,-) (186.3,18.3,149) 1792 (-,-,-) (10,10,21,8,5) (-,-,-) - (219.6,14.9,173) 2194 (-,-,-) (219.6,14.8,172) 1941 (-,-,-) Average (203.3,27.57,76.43) 284 (131.78,13.86,104.3) (3.38,6.88,5.14)% (131.73,14.19,102.9) (3.30,5.68,3.76)% Open in new tab Table 4: Results obtained by AUGMECON, NSGA-II, and MOEA/D. (Problem no., depots, affected areas, time period, product no.) . AUGMECON . NSGA-II . Gap (%) . MOEA/D . Gap (%) . Best OF (1,2,3) . CPU time . Best OF (1,2,3) . CPU time . OF (1,2,3) . Best OF (1,2,3) . CPU time . OF (1,2,3) . (1,2,5,3,2) (74.7,8.4,43) 284 (74.7, 8.4,43) 426 (0.000,0.000,0.000) (74.7,8.4,43) 472 (0.000,0.000,0.000) (2,3,10,3,2) (82.3,13.2,37) 573 (83,11.3,39) 661 (0.009,0.144,0.054) (82.5,12.6,38) 682 (0.002,0.045,0.027) (3,4,9,3,2) (80.9,14,58) 825 (84.1,12.6,63) 837 (0.040,0.100,0.086) (84.1,13.1,62) 819 (0.040,0.064,0.069) (4,3,12,4,3) (101.4,18.8,79) 1135 (107.9,18.1,82) 1023 (0.064,0.037,0.038) (107.9,17.6,82) 963 (0.064,0.064,0.038) (5,5,14,4,3) (107.3,10.3,86) 1637 (110.4,10.1,87) 1251 (0.029,0.019,0.012) (110.4,9,89) 1104 (0.029,0.126,0.035) (6,6,15,5,3) (128.1,17.4,123) 2563 (135.6,15.7,127) 1386 (0.059,0.098,0.033) (135.6,16.3,129) 1284 (0.059,0.063,0.049) (7,7,17,5,4) (136.8,14.4,109) 3048 (141.9,13.2,124) 1503 (0.037,0.083,0.138) (141.9,13.9,114) 1403 (0.037,0.035,0.046) (8,8,18,6,4) (-,-,-) - (174.3,18.3,149) 1783 (-,-,-) (174.3,17.9,151) 1583 (-,-,-) (9,9,19,6,5) (-,-,-) - (186.3,16,156) 1980 (-,-,-) (186.3,18.3,149) 1792 (-,-,-) (10,10,21,8,5) (-,-,-) - (219.6,14.9,173) 2194 (-,-,-) (219.6,14.8,172) 1941 (-,-,-) Average (203.3,27.57,76.43) 284 (131.78,13.86,104.3) (3.38,6.88,5.14)% (131.73,14.19,102.9) (3.30,5.68,3.76)% (Problem no., depots, affected areas, time period, product no.) . AUGMECON . NSGA-II . Gap (%) . MOEA/D . Gap (%) . Best OF (1,2,3) . CPU time . Best OF (1,2,3) . CPU time . OF (1,2,3) . Best OF (1,2,3) . CPU time . OF (1,2,3) . (1,2,5,3,2) (74.7,8.4,43) 284 (74.7, 8.4,43) 426 (0.000,0.000,0.000) (74.7,8.4,43) 472 (0.000,0.000,0.000) (2,3,10,3,2) (82.3,13.2,37) 573 (83,11.3,39) 661 (0.009,0.144,0.054) (82.5,12.6,38) 682 (0.002,0.045,0.027) (3,4,9,3,2) (80.9,14,58) 825 (84.1,12.6,63) 837 (0.040,0.100,0.086) (84.1,13.1,62) 819 (0.040,0.064,0.069) (4,3,12,4,3) (101.4,18.8,79) 1135 (107.9,18.1,82) 1023 (0.064,0.037,0.038) (107.9,17.6,82) 963 (0.064,0.064,0.038) (5,5,14,4,3) (107.3,10.3,86) 1637 (110.4,10.1,87) 1251 (0.029,0.019,0.012) (110.4,9,89) 1104 (0.029,0.126,0.035) (6,6,15,5,3) (128.1,17.4,123) 2563 (135.6,15.7,127) 1386 (0.059,0.098,0.033) (135.6,16.3,129) 1284 (0.059,0.063,0.049) (7,7,17,5,4) (136.8,14.4,109) 3048 (141.9,13.2,124) 1503 (0.037,0.083,0.138) (141.9,13.9,114) 1403 (0.037,0.035,0.046) (8,8,18,6,4) (-,-,-) - (174.3,18.3,149) 1783 (-,-,-) (174.3,17.9,151) 1583 (-,-,-) (9,9,19,6,5) (-,-,-) - (186.3,16,156) 1980 (-,-,-) (186.3,18.3,149) 1792 (-,-,-) (10,10,21,8,5) (-,-,-) - (219.6,14.9,173) 2194 (-,-,-) (219.6,14.8,172) 1941 (-,-,-) Average (203.3,27.57,76.43) 284 (131.78,13.86,104.3) (3.38,6.88,5.14)% (131.73,14.19,102.9) (3.30,5.68,3.76)% Open in new tab The NSGA-II and AUGMECON values differ by 3.38, 6.88, and 5.14% for the first to the third objective functions, respectively. The MOEA/D and AUGMECON have average differences of 3.30, 5.68, and 3.76%, respectively, for the first, second, and third objective functions. The fact that the average gaps of three objective functions are slight leads us to conclude that the meta-heuristic approaches proposed are effective. Figure 7 shows the CPU time needed to solve each meta-heuristic algorithm compared to the AUGMECON. The AUGMECON’s solving time grows exponentially in proportion to the problem’s size, and for problems 9 and 10, the method was not able to solve the problem. In contrast, meta-heuristic algorithms work much faster. Figure 7: Open in new tabDownload slide CPU time different approaches. 5.2 Metrics used for evaluation In this work, multiple criteria were utilised to assess the performance of various algorithms in multi-objective optimisation problems where problem solutions incorporate the optimal Pareto front. An algorithm’s performance may be evaluated quantitatively by counting the number of Pareto solutions it generates. The larger this number, the better the algorithm. The distance index may also be used to figure out how far apart two solutions are. The better it is if this index has a lower value. The index of distance is determined as follows (Schott, 1995): $$\begin{equation*} SM = \frac{{\mathop \sum \nolimits_{\mathfrak{i} = 1}^{\mathcal{N} - 1} \left| {\bar{\mathfrak{d}} - {\mathfrak{d}_{\mathfrak{i}}}} \right|}}{{\left( {\mathcal{N} - 1} \right)\mathfrak{d}}}, \end{equation*} $$(27) where d is the average of |$dis$|⁠, N is the total number of Pareto solutions, and |${d_i}$| is the distance between two successive solutions in the optimum front obtained by each method. The variety index is another way to measure the diversity of possible solutions, and its larger values are more suitable. Calculating this index is as follows (Zitzler & Thiele, 1999): $$\begin{equation*} DM\ = \sqrt {{{\left( {\frac{{max{f_{1\mathfrak{i}}} - min{f_{1\mathfrak{i}}}}}{{f_{1,total}^{max} - f_{1,total}^{min}}}} \right)}^2} + {{\left( {\frac{{max{f_{2\mathfrak{i}}} - min{f_{2\mathfrak{i}}}}}{{f_{2,total}^{max} - f_{2,total}^{min}}}} \right)}^2}}. \ \end{equation*} $$(28) In total, 12 problems of varied sizes were created, the performance of two meta-heuristic methods was compared, and Pareto-optimal solutions were generated for each. Tables 5 and 6 list the created sample problems, failure times, fuzzy demands, and the values of evaluation metrics for methods. Table 5: Values for generating sample problems. Problem no. . Projects with accessible roads . Projects with damaged roads . Vehicle depot . Truck . Helicopter . Types of materials . Periods . 1 7 5 6 3 2 2 2 2 10 7 6 3 2 3 2 3 15 9 7 4 2 4 2 4 16 7 7 4 2 3 3 5 18 8 7 5 3 4 3 6 20 8 8 6 3 5 3 7 24 10 8 6 3 4 4 8 25 11 9 5 4 5 4 9 30 15 9 7 4 6 4 10 33 16 9 7 5 4 5 11 35 20 10 9 5 5 5 12 40 25 10 10 5 6 5 Problem no. . Projects with accessible roads . Projects with damaged roads . Vehicle depot . Truck . Helicopter . Types of materials . Periods . 1 7 5 6 3 2 2 2 2 10 7 6 3 2 3 2 3 15 9 7 4 2 4 2 4 16 7 7 4 2 3 3 5 18 8 7 5 3 4 3 6 20 8 8 6 3 5 3 7 24 10 8 6 3 4 4 8 25 11 9 5 4 5 4 9 30 15 9 7 4 6 4 10 33 16 9 7 5 4 5 11 35 20 10 9 5 5 5 12 40 25 10 10 5 6 5 Open in new tab Table 5: Values for generating sample problems. Problem no. . Projects with accessible roads . Projects with damaged roads . Vehicle depot . Truck . Helicopter . Types of materials . Periods . 1 7 5 6 3 2 2 2 2 10 7 6 3 2 3 2 3 15 9 7 4 2 4 2 4 16 7 7 4 2 3 3 5 18 8 7 5 3 4 3 6 20 8 8 6 3 5 3 7 24 10 8 6 3 4 4 8 25 11 9 5 4 5 4 9 30 15 9 7 4 6 4 10 33 16 9 7 5 4 5 11 35 20 10 9 5 5 5 12 40 25 10 10 5 6 5 Problem no. . Projects with accessible roads . Projects with damaged roads . Vehicle depot . Truck . Helicopter . Types of materials . Periods . 1 7 5 6 3 2 2 2 2 10 7 6 3 2 3 2 3 15 9 7 4 2 4 2 4 16 7 7 4 2 3 3 5 18 8 7 5 3 4 3 6 20 8 8 6 3 5 3 7 24 10 8 6 3 4 4 8 25 11 9 5 4 5 4 9 30 15 9 7 4 6 4 10 33 16 9 7 5 4 5 11 35 20 10 9 5 5 5 12 40 25 10 10 5 6 5 Open in new tab Table 6: The failure rates of vehicles and the fuzzy demand in various scenarios. |$\tilde{\mathcal{Q}} = ( {{\mathcal{Q}_1},{\mathcal{Q}_2},{\mathcal{Q}_3}} )$| . Vehicle 5 . Vehicle 4 . Vehicle 3 . Vehicle 2 . Vehicle 1 . Scenario . (20,46,75) 10 16 13 7 10 1 (25,50,80) 10 9 14 4 20 2 (30,53,95) 16 8 11 16 10 3 (35,60,100) 10 20 7 10 10 4 (45,70,120) 8 10 9 10 15 5 |$\tilde{\mathcal{Q}} = ( {{\mathcal{Q}_1},{\mathcal{Q}_2},{\mathcal{Q}_3}} )$| . Vehicle 5 . Vehicle 4 . Vehicle 3 . Vehicle 2 . Vehicle 1 . Scenario . (20,46,75) 10 16 13 7 10 1 (25,50,80) 10 9 14 4 20 2 (30,53,95) 16 8 11 16 10 3 (35,60,100) 10 20 7 10 10 4 (45,70,120) 8 10 9 10 15 5 Open in new tab Table 6: The failure rates of vehicles and the fuzzy demand in various scenarios. |$\tilde{\mathcal{Q}} = ( {{\mathcal{Q}_1},{\mathcal{Q}_2},{\mathcal{Q}_3}} )$| . Vehicle 5 . Vehicle 4 . Vehicle 3 . Vehicle 2 . Vehicle 1 . Scenario . (20,46,75) 10 16 13 7 10 1 (25,50,80) 10 9 14 4 20 2 (30,53,95) 16 8 11 16 10 3 (35,60,100) 10 20 7 10 10 4 (45,70,120) 8 10 9 10 15 5 |$\tilde{\mathcal{Q}} = ( {{\mathcal{Q}_1},{\mathcal{Q}_2},{\mathcal{Q}_3}} )$| . Vehicle 5 . Vehicle 4 . Vehicle 3 . Vehicle 2 . Vehicle 1 . Scenario . (20,46,75) 10 16 13 7 10 1 (25,50,80) 10 9 14 4 20 2 (30,53,95) 16 8 11 16 10 3 (35,60,100) 10 20 7 10 10 4 (45,70,120) 8 10 9 10 15 5 Open in new tab The number of ground vehicles ranges between three and ten, while the number of air vehicles ranges between two and five. There is a single warehouse for air vehicles in all situations. The capacity and transportation time parameters are produced using the uniform distributions between 50–60 and 8–20, respectively. The demand is represented by a triangle fuzzy number with the structure |$\mathcal{Q}\ = {\rm{\ }}( {{\mathcal{Q}_1},{\rm{\ }}{\mathcal{Q}_2},{\rm{\ }}{\mathcal{Q}_3}} )$|⁠, which generates using uniform distributions: |${\mathcal{Q}_1}$| from [20, 45], |${\mathcal{Q}_2}$| from [46, 70], and |${\mathcal{Q}_3}$| from [71, 140]. Additionally, the travel time of vehicles used to distribute goods is considered to be generated using a uniform distribution. We have a scenario matrix for each problem, in which the rows represent scenarios, the columns represent vehicles, and the numbers in the table represent the failure time of vehicles. That is highlighted in the table as |$\delta _{24}^3 = {\rm{\ }}16$|⁠, indicating that vehicle 2, in period 4, in scenario three, breaks down after 16 h, resulting in undelivered materials to projects. The parameters of the algorithms presented in Table 7 are given after tuning the parameters in the proposed algorithms using the response surface methodology (RSM) method. Table 7: Tunned parameters. Algorithms . Parameters . Small-size problems . Large-size problems . NSGA-II Maximum number of iterations 73 205 Population size 51 192 Crossover rate 0/39 0/34 Mutation rate 0/16 0/19 Mu 0/07 0/8 MOEA/D Maximum number of iterations 97 186 Population size 38 149 nArchive 29 101 Size of neighbourhood 13 20 Lambda 0/21 0/18 Algorithms . Parameters . Small-size problems . Large-size problems . NSGA-II Maximum number of iterations 73 205 Population size 51 192 Crossover rate 0/39 0/34 Mutation rate 0/16 0/19 Mu 0/07 0/8 MOEA/D Maximum number of iterations 97 186 Population size 38 149 nArchive 29 101 Size of neighbourhood 13 20 Lambda 0/21 0/18 Open in new tab Table 7: Tunned parameters. Algorithms . Parameters . Small-size problems . Large-size problems . NSGA-II Maximum number of iterations 73 205 Population size 51 192 Crossover rate 0/39 0/34 Mutation rate 0/16 0/19 Mu 0/07 0/8 MOEA/D Maximum number of iterations 97 186 Population size 38 149 nArchive 29 101 Size of neighbourhood 13 20 Lambda 0/21 0/18 Algorithms . Parameters . Small-size problems . Large-size problems . NSGA-II Maximum number of iterations 73 205 Population size 51 192 Crossover rate 0/39 0/34 Mutation rate 0/16 0/19 Mu 0/07 0/8 MOEA/D Maximum number of iterations 97 186 Population size 38 149 nArchive 29 101 Size of neighbourhood 13 20 Lambda 0/21 0/18 Open in new tab 5.3 Analysing the performance of the model and algorithms in small-size problems In Table 8, all the problems generated in Table 5 are solved separately by the algorithm, and the values of the standard indicators used are reported. To better evaluate the algorithms’ performances, each indicator is shown schematically in Figs. 8 to 10. Figure 8: Open in new tabDownload slide Comparison between NSGA-II and MOEA/D according to the spacing metric. Figure 9: Open in new tabDownload slide Comparison between NSGA-II and MOEA/D according to the diversity metric. Figure 10: Open in new tabDownload slide Comparison between NSGA-II and MOEA/D according to the number of Pareto solutions. Table 8: Obtained values for different instances by NSGA-II and MOEA/D. Problem no. . MOEA/D . NSGA-II . . SM . DM . NOPS . CPU time . SM . DM . NOPS . CPU time . 1 7.939933 7.939933 6 615 0.4396193 7.479914 7 652 2 3.18014 3.18014 5 763 0.7331419 7.15122 5 782 3 5.655873 5.655873 6 1043 0.4650282 7.922465 9 1023 4 4.878148 4.878148 9 1245 0.5922232 6.916572 7 1202 5 3.805205 3.805205 6 1684 0.8037516 7.637064 7 1502 6 7.288054 7.288054 7 1943 1.0505771 4.179673 5 1732 7 7.508725 7.508725 8 2692 0.9946069 5.532396 11 2319 8 3.031869 3.031869 9 3042 0.416409 5.124544 6 2548 9 6.712907 6.712907 6 4174 0.6263944 3.038738 7 3657 10 6.978169 6.978169 8 6031 0.8053852 3.411685 9 5662 11 4.955259 4.955259 8 7453 0.4990383 8.496907 7 6650 12 7.859942 7.859942 5 10554 0.4833295 5.1066 8 8636 Average 5.816185 5.816185 6.916667 3436.583 0.6591254 5.999815 7.333333 3030.4167 Problem no. . MOEA/D . NSGA-II . . SM . DM . NOPS . CPU time . SM . DM . NOPS . CPU time . 1 7.939933 7.939933 6 615 0.4396193 7.479914 7 652 2 3.18014 3.18014 5 763 0.7331419 7.15122 5 782 3 5.655873 5.655873 6 1043 0.4650282 7.922465 9 1023 4 4.878148 4.878148 9 1245 0.5922232 6.916572 7 1202 5 3.805205 3.805205 6 1684 0.8037516 7.637064 7 1502 6 7.288054 7.288054 7 1943 1.0505771 4.179673 5 1732 7 7.508725 7.508725 8 2692 0.9946069 5.532396 11 2319 8 3.031869 3.031869 9 3042 0.416409 5.124544 6 2548 9 6.712907 6.712907 6 4174 0.6263944 3.038738 7 3657 10 6.978169 6.978169 8 6031 0.8053852 3.411685 9 5662 11 4.955259 4.955259 8 7453 0.4990383 8.496907 7 6650 12 7.859942 7.859942 5 10554 0.4833295 5.1066 8 8636 Average 5.816185 5.816185 6.916667 3436.583 0.6591254 5.999815 7.333333 3030.4167 Open in new tab Table 8: Obtained values for different instances by NSGA-II and MOEA/D. Problem no. . MOEA/D . NSGA-II . . SM . DM . NOPS . CPU time . SM . DM . NOPS . CPU time . 1 7.939933 7.939933 6 615 0.4396193 7.479914 7 652 2 3.18014 3.18014 5 763 0.7331419 7.15122 5 782 3 5.655873 5.655873 6 1043 0.4650282 7.922465 9 1023 4 4.878148 4.878148 9 1245 0.5922232 6.916572 7 1202 5 3.805205 3.805205 6 1684 0.8037516 7.637064 7 1502 6 7.288054 7.288054 7 1943 1.0505771 4.179673 5 1732 7 7.508725 7.508725 8 2692 0.9946069 5.532396 11 2319 8 3.031869 3.031869 9 3042 0.416409 5.124544 6 2548 9 6.712907 6.712907 6 4174 0.6263944 3.038738 7 3657 10 6.978169 6.978169 8 6031 0.8053852 3.411685 9 5662 11 4.955259 4.955259 8 7453 0.4990383 8.496907 7 6650 12 7.859942 7.859942 5 10554 0.4833295 5.1066 8 8636 Average 5.816185 5.816185 6.916667 3436.583 0.6591254 5.999815 7.333333 3030.4167 Problem no. . MOEA/D . NSGA-II . . SM . DM . NOPS . CPU time . SM . DM . NOPS . CPU time . 1 7.939933 7.939933 6 615 0.4396193 7.479914 7 652 2 3.18014 3.18014 5 763 0.7331419 7.15122 5 782 3 5.655873 5.655873 6 1043 0.4650282 7.922465 9 1023 4 4.878148 4.878148 9 1245 0.5922232 6.916572 7 1202 5 3.805205 3.805205 6 1684 0.8037516 7.637064 7 1502 6 7.288054 7.288054 7 1943 1.0505771 4.179673 5 1732 7 7.508725 7.508725 8 2692 0.9946069 5.532396 11 2319 8 3.031869 3.031869 9 3042 0.416409 5.124544 6 2548 9 6.712907 6.712907 6 4174 0.6263944 3.038738 7 3657 10 6.978169 6.978169 8 6031 0.8053852 3.411685 9 5662 11 4.955259 4.955259 8 7453 0.4990383 8.496907 7 6650 12 7.859942 7.859942 5 10554 0.4833295 5.1066 8 8636 Average 5.816185 5.816185 6.916667 3436.583 0.6591254 5.999815 7.333333 3030.4167 Open in new tab In order to gain a better understanding of how well the meta-heuristic algorithms perform, the following computations are shown to illustrate the results of comparison metrics. With the spacing metric, we can see that the MOEA/D outperforms the NSGA-II when compared to the other meta-heuristic algorithms in Fig. 8. Comparing solution times shows that the MOEA/D finds better solutions faster, resulting in better overall performance. (Fig. 11). Figure 11: Open in new tabDownload slide Comparison between NSGA-II and MOEA/D according to CPU time. The results of the diversity metric shown in Fig. 9 demonstrate that there is no particular trend for the algorithms. The number of Pareto solutions determined by the MOEA/D is depicted in Fig. 10, demonstrating its superior performance. 5.4 Statistical test For each comparison metric, a number of statistical tests were conducted to see if there was any statistically significant difference between the two methods. The results of the independent t-test to compare the proposed algorithms are provided in Tables 9 and 10. Table 9: Results of statistical hypothesis test to compare the performance of algorithms (independent t-test). . Levene’s test for equality of variances . t-Test for equality of means . . F . Sig. . t . df . Sig. (2-tailed) . Mean difference . Std. error difference . 95% Confidence interval of the difference . . . . . . . . . Lower . Upper . SM Equal variances assumed 0.069 0.795 1.094 21 0.286 0.10257 0.09375 −0.09239 0.29753 Equal variances not assumed 1.091 20.502 0.288 0.10257 0.09403 −0.09325 0.29840 DM Equal variances assumed 0.087 0.770 −0.498 21 0.623 −0.37670 0.75596 −1.94881 1.19541 Equal variances not assumed −0.499 20.950 0.623 −0.37670 0.75450 −1.94600 1.19261 NOPS Equal variances assumed 22.911 0.000 −3.841 21 0.001 −3023.41667 787.21206 −4660.51377 −1386.31956 Equal variances not assumed −4.019 11.000 0.002 −3023.41667 752.20719 −4679.01339 −1367.81994 RUN Equal variances assumed 22.930 0.000 4.108 21 0.001 3685.75758 897.27824 1819.76532 5551.74983 Equal variances not assumed 3.924 10.000 0.003 3685.75758 939.21166 1593.06374 5778.45141 . Levene’s test for equality of variances . t-Test for equality of means . . F . Sig. . t . df . Sig. (2-tailed) . Mean difference . Std. error difference . 95% Confidence interval of the difference . . . . . . . . . Lower . Upper . SM Equal variances assumed 0.069 0.795 1.094 21 0.286 0.10257 0.09375 −0.09239 0.29753 Equal variances not assumed 1.091 20.502 0.288 0.10257 0.09403 −0.09325 0.29840 DM Equal variances assumed 0.087 0.770 −0.498 21 0.623 −0.37670 0.75596 −1.94881 1.19541 Equal variances not assumed −0.499 20.950 0.623 −0.37670 0.75450 −1.94600 1.19261 NOPS Equal variances assumed 22.911 0.000 −3.841 21 0.001 −3023.41667 787.21206 −4660.51377 −1386.31956 Equal variances not assumed −4.019 11.000 0.002 −3023.41667 752.20719 −4679.01339 −1367.81994 RUN Equal variances assumed 22.930 0.000 4.108 21 0.001 3685.75758 897.27824 1819.76532 5551.74983 Equal variances not assumed 3.924 10.000 0.003 3685.75758 939.21166 1593.06374 5778.45141 Open in new tab Table 9: Results of statistical hypothesis test to compare the performance of algorithms (independent t-test). . Levene’s test for equality of variances . t-Test for equality of means . . F . Sig. . t . df . Sig. (2-tailed) . Mean difference . Std. error difference . 95% Confidence interval of the difference . . . . . . . . . Lower . Upper . SM Equal variances assumed 0.069 0.795 1.094 21 0.286 0.10257 0.09375 −0.09239 0.29753 Equal variances not assumed 1.091 20.502 0.288 0.10257 0.09403 −0.09325 0.29840 DM Equal variances assumed 0.087 0.770 −0.498 21 0.623 −0.37670 0.75596 −1.94881 1.19541 Equal variances not assumed −0.499 20.950 0.623 −0.37670 0.75450 −1.94600 1.19261 NOPS Equal variances assumed 22.911 0.000 −3.841 21 0.001 −3023.41667 787.21206 −4660.51377 −1386.31956 Equal variances not assumed −4.019 11.000 0.002 −3023.41667 752.20719 −4679.01339 −1367.81994 RUN Equal variances assumed 22.930 0.000 4.108 21 0.001 3685.75758 897.27824 1819.76532 5551.74983 Equal variances not assumed 3.924 10.000 0.003 3685.75758 939.21166 1593.06374 5778.45141 . Levene’s test for equality of variances . t-Test for equality of means . . F . Sig. . t . df . Sig. (2-tailed) . Mean difference . Std. error difference . 95% Confidence interval of the difference . . . . . . . . . Lower . Upper . SM Equal variances assumed 0.069 0.795 1.094 21 0.286 0.10257 0.09375 −0.09239 0.29753 Equal variances not assumed 1.091 20.502 0.288 0.10257 0.09403 −0.09325 0.29840 DM Equal variances assumed 0.087 0.770 −0.498 21 0.623 −0.37670 0.75596 −1.94881 1.19541 Equal variances not assumed −0.499 20.950 0.623 −0.37670 0.75450 −1.94600 1.19261 NOPS Equal variances assumed 22.911 0.000 −3.841 21 0.001 −3023.41667 787.21206 −4660.51377 −1386.31956 Equal variances not assumed −4.019 11.000 0.002 −3023.41667 752.20719 −4679.01339 −1367.81994 RUN Equal variances assumed 22.930 0.000 4.108 21 0.001 3685.75758 897.27824 1819.76532 5551.74983 Equal variances not assumed 3.924 10.000 0.003 3685.75758 939.21166 1593.06374 5778.45141 Open in new tab Table 10: Statistical characteristics of the three criteria for both algorithms (independent t-test). . Method . N . Mean . Std. deviation . Std. error mean . SM NSGA-II 11 0.7617 0.23212 0.06999 MOEA/D 12 0.6591 0.21751 0.06279 DM NSGA-II 11 5.6231 1.77043 0.53381 MOEA/D 12 5.9998 1.84714 0.53322 NOPS NSGA-II 11 7.0000 1.48324 0.44721 MOEA/D 12 3030.4167 2605.72169 752.20706 RUN NSGA-II 11 3693.0909 3115.01225 939.21153 MOEA/D 12 7.3333 1.72328 0.49747 . Method . N . Mean . Std. deviation . Std. error mean . SM NSGA-II 11 0.7617 0.23212 0.06999 MOEA/D 12 0.6591 0.21751 0.06279 DM NSGA-II 11 5.6231 1.77043 0.53381 MOEA/D 12 5.9998 1.84714 0.53322 NOPS NSGA-II 11 7.0000 1.48324 0.44721 MOEA/D 12 3030.4167 2605.72169 752.20706 RUN NSGA-II 11 3693.0909 3115.01225 939.21153 MOEA/D 12 7.3333 1.72328 0.49747 Open in new tab Table 10: Statistical characteristics of the three criteria for both algorithms (independent t-test). . Method . N . Mean . Std. deviation . Std. error mean . SM NSGA-II 11 0.7617 0.23212 0.06999 MOEA/D 12 0.6591 0.21751 0.06279 DM NSGA-II 11 5.6231 1.77043 0.53381 MOEA/D 12 5.9998 1.84714 0.53322 NOPS NSGA-II 11 7.0000 1.48324 0.44721 MOEA/D 12 3030.4167 2605.72169 752.20706 RUN NSGA-II 11 3693.0909 3115.01225 939.21153 MOEA/D 12 7.3333 1.72328 0.49747 . Method . N . Mean . Std. deviation . Std. error mean . SM NSGA-II 11 0.7617 0.23212 0.06999 MOEA/D 12 0.6591 0.21751 0.06279 DM NSGA-II 11 5.6231 1.77043 0.53381 MOEA/D 12 5.9998 1.84714 0.53322 NOPS NSGA-II 11 7.0000 1.48324 0.44721 MOEA/D 12 3030.4167 2605.72169 752.20706 RUN NSGA-II 11 3693.0909 3115.01225 939.21153 MOEA/D 12 7.3333 1.72328 0.49747 Open in new tab According to the results reported in Table 9, it can be seen that the significant level of Levene’s test for equality of variances for SM, DM, NOPS, and Run indices are 0.795, 0.770, 0, and 0. The resulting p-values of Levene's test for SM and DM are greater than significance level 0.05, indicating that the null hypothesis of equal variances is not rejected, and no difference between the variances is concluded. While the P-values obtained from Levene’s test for NOPS and Run are less than the significance level of 0.05. As a result, the null hypothesis of equal variances is rejected, and it is concluded that the variances differ. Based on the significant level of the independent t-test, the obtained values for DM and SM are 0.623 and 0.286, which are both greater than 5%. As a result, the mean of the two samples is not significantly different, and the ability of both algorithms in the SM, DM criteria is not significantly different. In contrast, the obtained values for NOPS and Run are 0.002 and 0.003, respectively, which are less than 5%, so the assumption of the equal mean of the two samples is rejected. In NOPS criteria, the algorithm with the highest value is preferred. As a result, the MOEA/D algorithm outperforms the NSGA-II algorithm. In the Run criteria, however, the algorithm with the shortest run time is preferred. The NSGA-II method’s average value (3693.0909) is lower than the MOEA/D method’s (7.3333). As a result, the NSGA-II algorithm outperforms the MOEA/D algorithm. 5.5 Sensitivity analysis The sensitivity analysis is taken into account in solving problem 6. Tables 8 and 9 show the results of changing demand and vehicle capacity at the same time, as well as the mean failure time of the vehicles. As shown in Table 11, there will be no change in OFV or route selection if we simultaneously reduce demand and increase vehicle capacity. When we simultaneously increase material demand while decreasing vehicle capacity, we will see that transportation times and undelivered materials to projects (shortage due to delays and failures) will increase, as will the objective function, with the exception of one case that will remain constant. Obviously, with increasing demand and decreasing vehicle capacity, more vehicles will be required to service, increasing transportation time (objective function 1) and the likelihood of vehicle failure. As a result, the level of shortage (objective function 3) will rise. On the other hand, the likelihood of taking less reliable routes will rise. If demand remains constant while reducing vehicle capacity, we effectively reduce overcapacity, and the routeing will not change again. However, all three objective functions will be altered if demand remains constant and vehicle capacity is reduced from 0.9 to 0.8. For instance, vehicle one previously had to serve three projects due to its reduced capacity (0.8), but it will no longer be able to do so due to its reduced capacity (0.8), so this task will be assigned to vehicle 2. When machine 2 wishes to travel through the arc, the transport time (objective function 1) increases and becomes non-optimal. Second, in various scenarios, the delays in meeting demand (objective function 3) increase. Third, the previously chosen arc had higher reliability, which was reduced from 10.3 to 10.1 by making this change and by lowering the capacity to 0.7. Now, if we assume the vehicle’s capacity is fixed and reduce the demand, the route will not change, nor will the objective functions 1 (transport time) and 2 (reliability), but objective function 3 (undelivered materials to projects) will decrease because we reduced the demand. As a result, all of the demands will be met. However, if we increase the demand, the vehicle will be unable to meet this level of demand due to constant capacity and increasing demand. As a result, transportation time (objective 1) will increase while reliability (objective 2) will decrease. As a result, the duration of the delay (undelivered materials to projects) (objective 3 function) will increase. Table 11: Sensitivity analysis of vehicle capacity and demand variations at the same time. Capacity . Demand . . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . 0.7 107 10.3 72.9 107 10.3 80.5 113.7 10.1 84.9 120.1 10.1 90.6 127.4 9.7 96.1 134.2 9.2 104.2 147.3 8.9 116.9 0.8 107 10.3 72.9 107 10.3 78.1 107 10.3 83.6 111.6 10.1 88.1 119.4 9.9 93.9 126 9.4 99.9 134.8 9.1 110.8 0.9 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 112.7 10.1 91.3 119.8 10.1 97.2 128.9 9.9 106.5 1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 114.5 10.3 95.2 121.3 10.1 103.5 1.1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.8 115.1 10.1 98.1 1.2 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 1.3 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 Capacity . Demand . . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . 0.7 107 10.3 72.9 107 10.3 80.5 113.7 10.1 84.9 120.1 10.1 90.6 127.4 9.7 96.1 134.2 9.2 104.2 147.3 8.9 116.9 0.8 107 10.3 72.9 107 10.3 78.1 107 10.3 83.6 111.6 10.1 88.1 119.4 9.9 93.9 126 9.4 99.9 134.8 9.1 110.8 0.9 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 112.7 10.1 91.3 119.8 10.1 97.2 128.9 9.9 106.5 1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 114.5 10.3 95.2 121.3 10.1 103.5 1.1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.8 115.1 10.1 98.1 1.2 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 1.3 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 Open in new tab Table 11: Sensitivity analysis of vehicle capacity and demand variations at the same time. Capacity . Demand . . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . 0.7 107 10.3 72.9 107 10.3 80.5 113.7 10.1 84.9 120.1 10.1 90.6 127.4 9.7 96.1 134.2 9.2 104.2 147.3 8.9 116.9 0.8 107 10.3 72.9 107 10.3 78.1 107 10.3 83.6 111.6 10.1 88.1 119.4 9.9 93.9 126 9.4 99.9 134.8 9.1 110.8 0.9 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 112.7 10.1 91.3 119.8 10.1 97.2 128.9 9.9 106.5 1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 114.5 10.3 95.2 121.3 10.1 103.5 1.1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.8 115.1 10.1 98.1 1.2 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 1.3 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 Capacity . Demand . . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . OF1 . OF2 . OF3 . 0.7 107 10.3 72.9 107 10.3 80.5 113.7 10.1 84.9 120.1 10.1 90.6 127.4 9.7 96.1 134.2 9.2 104.2 147.3 8.9 116.9 0.8 107 10.3 72.9 107 10.3 78.1 107 10.3 83.6 111.6 10.1 88.1 119.4 9.9 93.9 126 9.4 99.9 134.8 9.1 110.8 0.9 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 112.7 10.1 91.3 119.8 10.1 97.2 128.9 9.9 106.5 1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 114.5 10.3 95.2 121.3 10.1 103.5 1.1 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.8 115.1 10.1 98.1 1.2 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 1.3 107 10.3 72.9 107 10.3 78.1 107 10.3 81.3 107 10.3 86 107 10.3 89.4 107 10.3 93.1 107 10.3 96.4 Open in new tab Table 12 shows the sensitivity analysis results based on changes in the average vehicle failure time. Figures 11–19 show the results of a sensitivity analysis of simultaneous changes in demand and vehicle capacity. [Note: The amount of change in demand and capacity is shown as (⁠|$\Delta Dem,{\rm{\ }}\Delta Cap)$|⁠.] Figure 12. Open in new tabDownload slide Sensitive analysis when the capacity is constant and demand increases. Figure 13. Open in new tabDownload slide Sensitive analysis when the demand is constant and capacity increases. Figure 14. Open in new tabDownload slide Sensitive analysis when the capacity is constant and demand decreases. Figure 15. Open in new tabDownload slide Sensitive analysis when the demand is constant and capacity decreases. Figure 16. Open in new tabDownload slide Sensitive analysis when the capacity increases and demand increases. Figure 17. Open in new tabDownload slide Sensitive analysis when the capacity decreases and demand decrease. Figure 18. Open in new tabDownload slide Sensitive analysis when the capacity decreases and demand increases. Figure 19. Open in new tabDownload slide Sensitive analysis when the capacity increase and demand decreases. Table 12: Sensitivity analysis of vehicle failure time variations. Mean of vehicle’s failure time . −0.7 . −0.6 . −0.5 . −0.4 . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . 0.4 . 0.5 . 0.6 . 0.7 . OFV1 126.8 126.6 125.1 121.0 120.2 109.5 107.5 107.0 104.3 98.7 96.3 94.2 91.2 91.2 91.2 OFV2 9.4 9.4 9.4 9.8 10.1 10.1 10.3 10.3 10.3 10.5 10.5 10.7 10.7 10.7 10.7 OFV3 103.4 103.4 103.3 101.8 99.1 95.1 91.2 86.0 81.7 78.7 75.1 73.9 73.9 73.8 73.8 Mean of vehicle’s failure time . −0.7 . −0.6 . −0.5 . −0.4 . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . 0.4 . 0.5 . 0.6 . 0.7 . OFV1 126.8 126.6 125.1 121.0 120.2 109.5 107.5 107.0 104.3 98.7 96.3 94.2 91.2 91.2 91.2 OFV2 9.4 9.4 9.4 9.8 10.1 10.1 10.3 10.3 10.3 10.5 10.5 10.7 10.7 10.7 10.7 OFV3 103.4 103.4 103.3 101.8 99.1 95.1 91.2 86.0 81.7 78.7 75.1 73.9 73.9 73.8 73.8 Open in new tab Table 12: Sensitivity analysis of vehicle failure time variations. Mean of vehicle’s failure time . −0.7 . −0.6 . −0.5 . −0.4 . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . 0.4 . 0.5 . 0.6 . 0.7 . OFV1 126.8 126.6 125.1 121.0 120.2 109.5 107.5 107.0 104.3 98.7 96.3 94.2 91.2 91.2 91.2 OFV2 9.4 9.4 9.4 9.8 10.1 10.1 10.3 10.3 10.3 10.5 10.5 10.7 10.7 10.7 10.7 OFV3 103.4 103.4 103.3 101.8 99.1 95.1 91.2 86.0 81.7 78.7 75.1 73.9 73.9 73.8 73.8 Mean of vehicle’s failure time . −0.7 . −0.6 . −0.5 . −0.4 . −0.3 . −0.2 . −0.1 . 0 . 0.1 . 0.2 . 0.3 . 0.4 . 0.5 . 0.6 . 0.7 . OFV1 126.8 126.6 125.1 121.0 120.2 109.5 107.5 107.0 104.3 98.7 96.3 94.2 91.2 91.2 91.2 OFV2 9.4 9.4 9.4 9.8 10.1 10.1 10.3 10.3 10.3 10.5 10.5 10.7 10.7 10.7 10.7 OFV3 103.4 103.4 103.3 101.8 99.1 95.1 91.2 86.0 81.7 78.7 75.1 73.9 73.9 73.8 73.8 Open in new tab According to Table 12, we can expect the parameters related to vehicle breakdown times to decrease as the average vehicle breakdown time increases. They perform better by increasing the breakdown time from 1 to 1.1 and because the vehicle has a longer breakdown time. It attempts to select routes that were previously damaged in those vehicles and had an unmet demand that it did not select. When we reduce the average vehicle breakdown time, the model tries to increase the vehicle breakdown time, but the vehicle breakdown rate is very high. Figures 20 and 21 depict the sensitivity analysis as a result of changes in average vehicle breakdown time. [Note: The amount of change in demand and capacity is show as (⁠|$\Delta Dem,{\rm{\ }}\Delta Cap)$|⁠.] Figure 20. Open in new tabDownload slide Sensitive analysis when the average failure time of vehicle decrease. Figure 21. Open in new tabDownload slide Sensitive analysis when the average failure time of vehicle increases. 6. Conclusions and Future Works Reconstruction projects following a destructive disaster can be challenging due to time and resources constraints. A large number of post-disaster reconstruction efforts are compromised by inadequate planning. Consequently, this paper proposed an integrated model for delivering materials to the location of reconstruction projects following a large-scale disaster. Clustering the location of construction projects through the use of the proposed DE-K-prototypes method and designing the most reliable route for trucks through the use of a decision-making approach and graph theory (Permanent matrix) were two suggested approaches for designing a reliable system for delivering construction materials to the reconstruction projects. Consideration of concurrent disruptions in distribution operations is a necessity that is frequently overlooked, resulting in deficiency and delay in the process of distributing construction materials. Thus, in order to avoid disruptions in the route of trucks, serving and distribution to the construction projects were prioritised (in terms of the factors affecting route reliability), and, on the other hand, a strategy was adopted in which, in order to minimise the total weighted undelivered materials to projects, trucks were scheduled under various scenarios so that, in the event of a disruption, the distribution system and also the serenity are not damaged. We attempted to minimise both total travel time and shortage in this research by proposing a clustered and disrupted vehicle routeing model. As a result, a model was developed in which disparate vehicles from multiple depots initiated relief operations on the ground and in the air and delivered construction material to the projects. The optimal solution was found using the AUGMECON in a small size. Considering the complexity of the problem, two algorithms were proposed, including an NSGA-II and a MOEA/D, to solve the problems in large-size instances. The mean difference between the objective values of the NSGA-II and the epsilon-constraint method was 3.38 for the first objective function (transport time), 6.88 for the second objective function (reliability), and the same difference for the third objective function (the total weighted undelivered materials to projects) is 5.14%. We found that the average gaps of the three objective functions with meta-heuristic algorithms are very small. As the size of the problem increases, the time to solve the epsilon-constraint method increases exponentially to the extent that it is unable to respond from problem 8 onwards, while the execution time of meta-heuristic algorithms increases with a slight incline and is capable even with relatively slight slopes. NSGA-II and MOEA/D have the same capability in terms of SM and DM, but MOEA/D has a better performance in terms of NOPS than NSGA-II. And NSGA-II has better performance in terms of CPU time than MOEA/D. Providing decision makers with a set of reliable solutions to help them make decisions about the delivery of construction materials to post-disaster reconstruction projects is one of the most important management insights that can be considered in this research. With the help of this study’s advanced approach, planners will be able to find the best solution to the problem. The results of the sensitivity analysis of the model show that if the demand of the projects decreases and at the same time the capacity of the vehicles increases or remains constant, almost no change will occur in all three objective functions and the optimal routes will remain unchanged. The capacity of the vehicle will remain constant (decrease). In most of the time, vehicles distribute several types of materials in a few periods, in which the probability of using less reliable routes will increase, and as a result, due to the slow process and the possibility of equipment failure, the undelivered materials to projects will increase. If we increase the demand for construction projects and at the same time reduce the capacity of vehicles, the duration of operation will be longer and vehicles will provide assistance on less reliable routes, and as a result, the undelivered materials to projects will increase. Managers and decision makers should specify potential warehouse locations in more secure areas to avoid the use of expensive transportation modes or less reliable access roads, in light of the project’s sensitivity to construction material. Additionally, managers and decision makers are encouraged to outsource certain tasks to reputable contractors. Many studies in disaster risk management have their own set of limitations. This study is no exception. Generally, there is no reliable database to define the values of the model’s parameters; thus, in this study, some parameters were obtained after consulting with experts and others were generated randomly. Furthermore, uncertainty was addressed in the problem only for a subset of parameters, whereas in the real world, other parameters are associated with uncertainty. Additionally, while many multi-objective meta-heuristic algorithms exist in the literature that may perform better, only NSGA-II and MOEA/D were used. Finally, some future research directions are possible. To begin, nondeterministic values can be investigated for other model parameters. Additionally, employing other multi-objective metahurestic algorithms or even developing some exact or semi-exact methods such as benders decomposition algorithm can be investigated in the future. 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TI - Developing an integrated model for planning the delivery of construction materials to post-disaster reconstruction projects
JF - Journal of Computational Design and Engineering
DO - 10.1093/jcde/qwac042
DA - 2022-06-27
UR - https://www.deepdyve.com/lp/oxford-university-press/developing-an-integrated-model-for-planning-the-delivery-of-0nAHc78ZyY
SP - 1135
EP - 1156
VL - 9
IS - 3
DP - DeepDyve
ER -