Collaborative vehicle routing for equitable and effective food allocation in nonprofit settings

Abstract

Purpose

Inspired by a food bank distribution operation, this paper aims to study synchronized vehicle routing for equitable and effective food allocation. The primary goal is to improve operational efficiency while ensuring equitable and effective food distribution among the partner agencies.

Design/methodology/approach

This study introduces a multiobjective Mixed Integer Programming (MIP) model aimed at addressing the complex challenge of effectively distributing food, particularly for food banks serving vulnerable populations in low-income urban and rural areas. The optimization approach described in this paper places a significant emphasis on social and economic considerations by fairly allocating food to food bank partner agencies while minimizing routing distance and waste. To assess the performance of the approach, this paper evaluates three distinct models, focusing on key performance measures such as effectiveness, equity and efficiency. The paper conducts a comprehensive numerical analysis using randomly generated data to gain insights into the trade-offs that arise and provide valuable managerial insights for food bank managers.

Findings

The results of the analysis highlight the models that perform better in terms of equity and effectiveness. Additionally, the results show that restocking the vehicles through the concept of synchronization improves the overall quantity of food allocation to partner agencies, thereby increasing accessibility.

Research limitations/implications

This paper contributes significantly to the literature on optimization approaches in the field of humanitarian logistics.

Practical implications

This study provides food bank managers with three different models, each with a multifaceted nature of trade-offs, to better address the complex challenges of food insecurity.

Social implications

This paper contributes significantly to social responsibility by enhancing the operational efficiency of food banks, ultimately improving their ability to serve communities in need.

Originality/value

To the best of the authors’ knowledge, this paper is the first to propose and analyze this new variant of vehicle routing problems in nonprofit settings.

Citation

Bonku, R., Alkaabneh, F. and Davis, L.B. (2024), "Collaborative vehicle routing for equitable and effective food allocation in nonprofit settings", Journal of Humanitarian Logistics and Supply Chain Management, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/JHLSCM-11-2023-0113

1. Introduction

Food insecurity, or a lack of access to sufficient and nutritious food (Feeding America, 2024), affects millions of U.S. households every year. The consequences of this issue extend beyond food considerations, as it can cause lots of health problems and worsen existing conditions (Levi et al., 2023; Hussain et al., 2023). In 2022, approximately 12.8% households were food insecure (17.0 million) (Economic Research Service, 2023). The rise of food insecurity in the USA and the world at large challenges the top three Sustainable Development Goals (SDGs) to include “no poverty,” “zero hunger” and “excellent health and well-being” (United Nations, 2015). This prevailing issue of food insecurity has prompted the need for effective strategies to handle food donations while minimizing waste. The US government implements a number of public assistance measures, including the Supplemental Nutrition Assistance Program (SNAP), the Women, Infants, and Children (WIC) program and The Emergency Food Assistance Program (TEFAP) to address the issue of food insecurity (Fianu and Davis, 2018). Beyond that, nonprofit organizations such as food banks assist in alleviating the problem of food insecurity. Food banks are responsible for collecting and distributing food to the food-insecure population (Bazerghi et al., 2016). In addition to providing nutrition for millions of households, food banks play an important role in reducing global food waste; these functions align with some of the goals of the SDGs. Feeding America (FA) is the largest hunger relief charity in the USA. According to Feeding America (2024) one in six people in the USA turned to food assistance in the year 2022. By utilizing a network of over 200 food banks, 21 statewide food bank organizations, and over 60,000 partner agencies, including food pantries and meal programs, FA distributed 5.3 billion meals to tens of millions of people who were food insecure in 2023. In addition to direct relief activities, FA actively supports measures to promote equity, effective distribution, reduce food waste, and improve general food security (Feeding America, 2024). Although FA as the nationwide network, provides diverse assistance to its branches, the local food banks persistently grapple with resource inadequacies, including limited funds, vehicles and staffing, as their operations heavily depend on donors and volunteers (Reusken et al., 2023; Rivera et al., 2023).

In the postpandemic landscape, food banks are compelled to allocate a greater portion of their resources to procure additional food at prices influenced by inflation as a result of reduced surpluses available for donation and an increase in demand (Yanez and Robb, 2022) and according to food bank managers, a budgetary allocation, initially set at 1 million dollars before the pandemic, has escalated to 5 million dollars. Additionally, a shortage of drivers has arisen due to increased competition with e-commerce companies and the upward trend in wage rates (Eze, 2022). Unlike profit-oriented businesses, food banks are constrained in their ability to increase the cost of their food offerings. In response, food bank managers aim to enhance operations through innovative strategies, with the objective of reducing expenditures on nonfood-related items to allocate more funds to procure fresh food items.

Food banks have agencies located in both low-income urban areas and rural areas. Food bank vehicles are used to deliver provisions to partner agencies in these areas because of insufficient transportation resources and the absence of refrigerated vehicles for perishable item transport (Davis et al., 2014; Good news garage, 2021). However, the distribution of food to agencies in these areas comes with different challenges. For example, food banks have to travel long distances to deliver food to partner agencies in these areas due to their geographical locations. Due to inflation and the doubling of fuel costs, food banks are not able to frequently deliver food to the agencies in these areas as needed (Eze, 2022). Given financial constraints, increased demand, limited volunteers, and insufficient vehicles, how can food banks effectively deliver food to their partner agencies in these areas? This is particularly crucial, as the limited funds pose a challenge to reaching many agencies. Efficient resource administration is crucial for food banks to ensure cost-effective distribution, necessitating strategic management to optimize operations and fulfill the diverse needs of partner agencies within budgetary constraints.

It is important to note that food banks measure their operations performance based on the equity, efficiency (time/cost), and effectiveness of the food distribution across different partner agencies (Sengul Orgut and Lodree, 2023; Alkaabneh et al., 2021). Several studies in the humanitarian literature consider both de- terministic and stochastic models with the objective of maximizing these three major performance measures (Islam and Ivy, 2022; Alkaabneh et al., 2021, 2023; Davis et al., 2016; Nair et al., 2017; Eisenhandler and Tzur, 2019a, 2019b, Lien et al., 2014; Balcik et al., 2014; Sengul Orgut et al., 2017; Sengul Orgut and Lodree, 2023; Fianu and Davis, 2018). As such, our modeling focuses on increasing these essential determinants. Note that, ac- cording to Alkaabneh et al. (2023), equity and effectiveness may conflict in some real-world scenarios, which makes modeling these two measures challenging. While higher food distribution is associated with higher effectiveness, it does not necessarily guarantee fair allocation. Conversely, perfect equity, where no food is allocated, results in ineffective distribution as all food goes to waste. In general, an optimal solution to the problem we solve in this study should be both equitable and effective [1].

Synchronized vehicle routing, as a proposed strategy, has gained attention in recent literature, specifically in a profit-oriented setting (Drexl, 2012; Soares et al., 2023). The synchronized vehicle routing models are used for real-world applications such as the planning of home health care services where interdependence of services may be required (Mankowska et al., 2014; Hashemi Doulabi et al., 2020), and recently for sprayer- tanker routing in an agricultural firm (Alkaabneh, 2023). Vehicle routing problems (VRP) with synchronization (VRP-Sync.) offers the potential to improve the logistics of rural and low-income urban food bank operations by streamlining routes and coordinating vehicle movements. This approach involves organizing distribution truck routes and schedules, resulting in a well-coordinated distribution network, improved food distribution, minimized cost and reduced waste. In practice, the importance of the VRP-Sync. stems from its ability to mimic a wide range of real-world systems. Nonetheless, this approach has not been explored in a nonprofit setting.

This paper presents an operational scenario that involves the synchronization of vehicle routing and food allocation logistics within a food bank service area. We introduce a multiobjective decision model designed to aid food banks in reducing routing time while ensuring the fair and effective distribution of donated supplies. This model synchronizes the routes of two primary types of vehicles that play pivotal roles in the food distribution process and minimizes the time of vehicle routing. Moreover, it establishes an allocation policy that maximizes both fairness and effectiveness in the distribution of food. In contrast to existing VRP-Sync. models, our model stands out for its unique objective function that minimizes routing time and maximizes the amount of food distributed while concurrently ensuring equity in food allocation among partner agencies.

The goal of our work is to optimize the allocation of food supplies to partner agencies within the food bank network by effectively coordinating truck routes to achieve maximum operational efficiency, improve equity and effectiveness, and reduce food waste. Our study addresses a complex challenge involving VRP-Sync. coupled with food allocation within nonprofit settings.

1.1 Research contributions

We present a summary of our contributions as follows:

• We formally define, model, and solve a multiobjective synchronized vehicle routing for equitable and effective food allocation in food banks (SVREEFAFB). To the best of our knowledge and according to our reviewed literature in Section 2, our paper is the first to propose and analyze this new variant of VRP in nonprofit settings.

• To analyze our proposed model approach and gain insights, we construct experiments and perform a comprehensive numerical analysis using randomly generated data with varied configurations.

• We further examine the results of three different Sync-VRP models to account for some trade-offs and flexibility in the requirements in terms of equity, effectiveness and efficiency and to offer managerial insights to food bank managers.

The rest of this paper is organized as follows: Section 2 introduces the relevant literature. The problem description, followed by illustrative example and the mathematical programming formulation, are presented in Section 3. Section 4 presents the design of the experiments. The results and discussion are presented in Section 5. Finally, conclusions, managerial implications, study limitations and future study are presented in Section 6.

2. Related literature

The operations research literature pertaining to food banks spans various topics, including network design, network allocation, donation management, collection and delivery operations and warehouse operations (Akkerman et al., 2023). Several studies within the domain of food bank literature center on decision-making processes related to the allocation of available resources to different partner organizations and determining the appropriate quantity for each recipient. Additionally, other studies explore the integration of collection and delivery operations for donations, aiming to optimize the cost-effectiveness of overall operations while upholding principles of equity. As exemplified by Islam and Ivy (2022), the authors introduce a model for assignment and distribution aiming to identify optimal assignments of counties to branches and efficient allocation of donated food to each county, allowing for a permissible deviation from perfect equity. The authors devise a mixed-integer programming model to determine the most effective allocation of demand zones (counties) to distribution centers (branches) and a fair distribution of donated food from food bank branches to these demand zones. The primary objective of their model is to minimize the overall cost associated with branch operations, the cost related to receiving and distributing food, and the costs incurred due to undistributed food, all while upholding principles of equity. Similar to our study, we integrate equity into our model, wherein we establish a specific measure for inequity. However, we introduce a multiobjective function model where we maximize both equity and effectiveness and minimize the overall routing time.

Eisenhandler and Tzur (2019a, 2019b) address the logistical challenges faced by a food bank utilizing trucks with limited capacity for regular redistribution of food from industry suppliers to welfare agencies (collecting food donations from food industry suppliers and distributing them to food relief agencies that assist people in need). Their research setting necessitates concurrent vehicle routing and resource allocation decisions with the purpose of balancing two potentially conflicting objectives: maximizing overall distribution and establishing equity in allocation. The authors introduce a novel objective function based on the Gini co- efficient method, aiming to maximize both equity and effectiveness simultaneously. We employ a similar Gini coefficient method of formulation in our model to measure inequity, except that the problem we study involves only the delivery of donations to partner agencies.

Additionally, our model introduces the unique constraints of restocking/reloading vehicles with limited capacity during distribution operations. Given that incorporating allocation decisions into cost-effective route design while ensuring fairness and effectiveness is challenging, Nair et al. (2017) propose a bi-objective formulation for the Food Rescue Allocation and Routing Problem (FRARP) which optimizes an allocation objective function and the traditional cost-driven objective in the VRP. The authors introduce two main objective functions to address fair allocation, namely, maximizing the utility of the worst-off delivery node (egalitarian or max-min) and minimizing the deviation of the utilities among delivery nodes. In our study, we employ the egalitarian model, where we maximize the minimum fillrate of the delivery node.

Lien et al. (2014) studies a single vehicle sequential resource allocation challenge for a food rescue initiative emphasizing the efficient and equitable distribution of rescued food. Their study incorporates an egalitarian welfare utility function as a metric for assessing equity. The authors make the assumption that the sequence of pickup nodes in the route remains constant and deliveries are executed subsequent to the collection of food from all the donors. Balcik et al. (2014) extends the study into a multivehicle sequential allocation problem. Our paper introduces a cost-effective multivehicle synchronized routing model for equitable and effective food allocation to partner agencies without imposing any restrictions on the sequence of visits to the agencies, allowing for greater flexibility and adaptability in the routing strategy. Other studies in the humanitarian literature have developed stochastic models with the objective of achieving equitable and effective food distribution within the context of food banks (Alkaabneh et al., 2021; Sengul Orgut et al., 2017; Fianu and Davis, 2018). For instance, Alkaabneh et al. (2021) develops an approximate dynamic programming model in which the main decision is the quantity of product to allocate to each agency. Their model implicitly integrates concave and strictly increasing equity functions. Sengul Orgut et al. (2017) designs a two-stage stochastic model with a single period that assures equitable distribution of food donations when distribution decisions are made prior to seeing capacity at receiving agencies. Fianu and Davis (2018) present a Markov decision process model to assist food banks in distributing food supplies received through random donations with the goal of optimizing distribution equitably. For a comprehensive review of recent updates in humanitarian operations research, interested readers may refer to Rivera et al. (2023).

Vehicle routing problems (VRPs) with single and multiple objective functions are modeled and solved in a number of studies in the operations research literature (Hashemi Doulabi et al., 2020; Hashemi et al., 2022; Alkaabneh et al., 2023); however, VRPs with synchronization constraints have gained attention in the most recent literature (Soares et al., 2023). In a paper by Drexl (2012), synchronization constraints in various VRPs are systematically categorized into five distinct types: task, operation, movement, load and resource synchronization. This classification implies that any major VRP involving synchronization can be classified under one of these categories. The studies by Mankowska et al. (2014), Fink et al. (2019) can be classified as task synchronization, while Meisel and Kopfer (2014), Hojabri et al. (2018) can be categorized under operation synchronization. Mankowska et al. (2014) conducts a comprehensive study on the Home Health Care Routing and Scheduling Problem (HHCRSP). In their research, the authors address a routing challenge that focuses on developing an optimal sequence of routes for home health care personnel. These personnel are tasked with visiting a specific set of patients, each requiring particular services delivered at their individual homes. The problem solved in Mankowska et al. (2014) necessitates recognizing varying qualifications among staff members for specific services and imposing synchronization demands at patient locations. Fink et al. (2019) conducts a study on the vehicle routing problem, which involves the synchronization of both workers and vehicles within the context of airport operations. This research was driven by the need to efficiently route workers to ground-handling tasks while ensuring that each task is completed within its specified time window. The study considers various decision variables, including the assignment of workers to jobs, the scheduling of workers for specific tasks, and the routing of workers.

The study of Meisel and Kopfer (2014) focuses on a routing issue involving the fulfillment of transportation demands using two distinct types of transportation resources: passive (used for holding the cargo that is to be shipped from pickup to delivery locations) and active (take up the passive means and carry them from one location to another) modes. The authors model the problem by synchronizing the operations of both types of transportation resources. Hojabri et al. (2018) extended the vehicle routing problem with a time window. Their study addresses a delivery and service problem featuring synchronization constraints, where the arrival of two vehicles (regular and special vehicles) at distinct customer locations is synchronized. In contrast to the above studies, our study involves tasks, operations, and load synchronization with no time window constraint.

Other major studies of VRPs with synchronization characterized by coordinating two vehicles include the pickup and delivery problem (Maknoon and Laporte, 2017; Koch et al., 2020; Masson et al., 2013), the truck and trailer routing problem (Chao, 2002), the truck and drone routing problems (Boysen et al., 2018; Dayarian et al., 2020; Boysen et al., 2018; Dienstknecht et al., 2022).

The literature reviewed above sheds light on the topics of routing and resource allocation within the operational framework of food banks, emphasizing considerations of equity, effectiveness, and route minimization. Additionally, it is noteworthy that the incorporation of synchronization in VRP has predominantly been explored within profit-oriented contexts. Notably, there is a discernible absence of studies addressing the synchronization of vehicle routing, specifically within the realm of humanitarian logistics. To the best of our knowledge, our study is the first to address such a gap in the literature. Figure 1 captures a brief summary of the related literature.

3. Formal problem description and model formulation

In this section, we present a formal description of the SVREEFAFB with the notations and key assumptions, followed by the mathematical formulation of the problem.

3.1 Problem description

Our model design is inspired by the operational framework of a food bank’s distribution operations. The SVREEFAFB can be summarized as follows: Consider a scenario where a central food bank operates both small and large trucks. The small trucks adept at navigating through communities and reaching multiple partner agencies are each laden with food supplies. These trucks embark on journeys from the depot to multiple agency locations across areas, ensuring that the requested food items are efficiently delivered. Simultaneously, the large truck loaded with food items travels from the depot to resupply the small trucks. We assume that the big truck follows a distinct route, strategically designed to intersect with the small trucks at designated rendezvous points. The primary purpose of this larger truck is to reload the supplies carried by the small trucks. Upon meeting, a seamless transfer of food items occurs, where the big truck restocks the small trucks with the necessary provisions. It is important to note that the roles of these trucks are distinct: the small trucks exclusively handle the distribution of food items to partner agencies, ensuring access. Meanwhile, the big truck serves as a resupply hub, reloading the small trucks with the food items required to serve the remaining partner agencies within the network. The synchronization of resources is carefully overseen by food bank managers, who synchronize the locations and schedules of the small trucks with those of the big trucks. This synchronization optimizes the overall distribution process, ensuring that food reaches those in need efficiently and effectively. Below is an illustrative example of our proposed approach:

3.1.1 Illustrative example of the synchronized vehicle routing for equitable and effective food allocation in food banks

We model the food distribution operation of the food bank as a graph, with nodes representing the locations of partner agencies in need of food allocation. It is imperative that each agency receives a fair share of the food supply, and these allocation operations are carried out by either food bank employees or dedicated volunteers. On a daily basis, the food bank dispatches two fully-loaded Small Trucks to visit each assigned partner agency, which may include soup kitchens, pantries, and other such organizations. Additionally, a Big Truck, also filled to capacity with food, is dispatched to serve as a reload point for the Small Trucks as they run out of supplies. It’s important to note that Big Truck is not tasked with directly delivering food to partner agencies. In the event that a small Truck exhausts its food supply before the arrival of the Big Truck for reloading, it must wait, which is not an ideal situation as it may lead to time waste. This waiting time is something we aim to minimize, as it has the potential to result in inefficiencies and potentially wasted resources.

The decisions made by food bank managers encompass various critical aspects, including determining which Small Truck should be assigned to specific nodes, as well as scheduling decisions for when and where the Big Truck should reload each Small Truck. This involves a complex synchronization of vehicle routing, scheduling, assignment and allocation decisions, all of which need to be addressed simultaneously. Additionally, there are dependencies between the Small Trucks and the Big Truck that must be taken into account to ensure smooth and efficient operations.

To illustrate the process, we refer to the example depicted in Figure 2 to demonstrate how the assignment of Small Trucks to agencies, their respective routes, food allocation quantities, reloading locations, and the Big Truck’s route are established. In this scenario, we assume that there are two available Small Trucks and a total of 13 agencies that need to be visited, as shown in Figure 2. At the outset of the food distribution operations, all three trucks are situated at the central hub known as the depot, which serves as the focal point of the food bank. As indicated in Figure 2, we assume that Small Truck 1 is designated to distribute food to agencies 9, 13, 1, 8, 12, 5 and 10, while Small Truck 2 is assigned to agencies 2, 11, 7, 4, 6 and 3. Small Truck 1 proceeds to agencies 9, 13, 1, 8 and 12 to carry out food distribution. It depletes its food supply at location 12, prompting the Big Truck to travel to location 12 for a reload. Once Small Truck 1 is replenished to full capacity, it proceeds to the next designated agency on its route to continue food delivery. Simultaneously, Small Truck 2 initiates distribution at agency 2 and later reloads at agency 4. The route of the Big Truck commences from the depot, proceeds to agency 12 for Small Truck 1’s reload, then moves to agency 4 for Small Truck 2’s reload, before returning to the depot. Similarly, upon completion of the distribution operations, all the Small Trucks return to the depot. This sequence of actions ensures a systematic and efficient distribution of food to the partner agencies, making the most of the available resources (food, staff and volunteers) and minimizing wait times.

3.2 Synchronized vehicle routing for equitable and effective food allocation in food banks

The SVREEFAFB is defined on a complete directed graph G = (N, A), where N is the set of nodes to be visited by the Small Truck in addition to the depot and A is the set of arcs connecting these nodes. The set N = {0, N_f} is comprised of the depot {0} and N_f = {1, 2, … , N} represents the nodes. For each node i ∈ N_f we denote the demand of that node as d_i and the share of the total demand as ni= di/∑j=Nfdj. The working hours considered as the planning horizon are denoted by T. We assume that agencies are prepared to accept deliveries of any quantity as long as the quantity does not exceed the agency capacity and at any time of the day during their operating hours, and that a node can only be visited once. We also assume that the network is fully connected, and hence, each pair of nodes in the network is connected via a direct arc. To each arc (i, j) ∈ A a travel time t_ij is attributed. We assume that the travel time is proportional to the distance traveled. There is a fleet of homogeneous Small Truck and a single Big Truck located at the depot.

The set of Small Truck is denoted as K, each with a capacity of Q^s. On the other hand, the Big Truck has a capacity of Q^s and Q^b is greater than or equal to Q^s. M is a large number. A Big Truck can only replenish the Small Truck at a designated stop after the Small Truck finishes its deliveries at that stop. This means that the Big Truck can only transfer resources to the Small Truck when it is not in motion and is stationary at a specific node. Realistically, the reloading time should be dependent on the quantity; however, we assume that the time taken by a Big Truck to reload a Small Truck remains constant and denote the reloading time as ζ, which represents the duration in units of time needed to carry out a reloading operation. We assume there exists a set of identical Small Trucks. Given the set of nodes, our objective is to determine:

• a routing plan for the Small Truck that dictates its movement;

• an allocation of resources specifying the quantity to be allocated to each node on the route; and

• identify the reloading locations, i.e., the nodes where the Small Truck is replenished with resources by the Big Truck.

In our prior discussion, we clarified that our central objective is to increase the quantity of food allocated to each agency. In practical terms, this entails the effective distribution of the maximum amount of food to fulfill the demands of diverse agencies, all while ensuring fairness. This entails ensuring that no agency is put at a disadvantage. Equity here is defined by maintaining similar filling rates or service levels across all nodes. The extent of inequity is controlled by a parameter known as α which can take values ranging from 0 to 1. When α is set to 0, it signifies perfect equity, meaning all agencies receive the same quantity of allocation. Conversely, when α is set to 1, it represents perfect inequity, where only one node receives the entire allocation, leaving the others with nothing.

The decision variables are defined as follows:

• xijk: a binary routing variable equal to 1 if arc (i, j) ∈ A is traversed by Small Truck k ∈ K and 0 otherwise.

• y_ij: a binary routing variable equal to 1 if arc (i, j) ∈ A is traversed by the Big Truck and 0 otherwise.

• δ_i: a binary variable equal to 1 if reloading of Small Truck takes place at node i ∈ N_f after the Small Truck delivers food items at node i and 0 otherwise.

• θ_i: a continuous positive variable denoting the reloading start time of Small Truck at node i ∈ N_f.

• aik: a continuous positive variable denoting the arrival time of Small Truck k ∈ K at node i ∈ N_f.

• mik: a continuous positive variable denoting the waiting time of Small Truck k ∈ K at node i ∈ N_f.

• qik: a continuous positive variable denoting the quantity of the food items delivered/duration of service by Small Truck k ∈ K at node i ∈ N_f.

• w_i: a continuous positive variable denoting the arrival time of the Big Truck at node i ∈ N_f.

• h_i: a continuous positive variable denoting the remaining quantity of food items in the Big Truck upon arrival to node i ∈ N.

• lik: a continuous positive variable denoting the remaining quantity of food items in Small Truck k ∈ K upon arrival to node i ∈ N_f.

• v_i: a continuous positive variable denoting the quantity of food items reloaded to a Small Truck k ∈ K when node i ∈ N_f is a reloading node.

• f_i: a continuous positive variable denoting the fill rate of node i ∈ N_f where 0 ≤ f_i ≤ 1.

• ∈_ij an auxiliary decision variable used to linearize the |ηjfidi – ηifjdj| in the equity term of the objective function.

The mathematical formulation is as follows:

(1) max∑i∈Nf fidi−∑i,j∈Nf:i≠jεij

(2) min∑i,j∈Nf:i≠j ∑k∈Ktijxijk+∑i,j∈Nf:i≠jtijyij+∑k∈K ∑i∈Nmik

s.t.

(3) ϵij≥ηjfidi−ηifjdj ∀k∈K,i,j∈Nfi≠j,

(4) ϵij≥ηifjdj−ηjfidi ∀ k∈K,i,j∈Nfi≠j,

(5) ∑k∈K ∑i∈N:i≠jxijk=1 ∀j∈Nf, 

(6) ∑i∈N:i≠j xijk=∑i∈N:i≠jxjik   ∀j∈N,k∈K,

(7) ∑i∈N:i≠jyij=∑i∈N:i≠j yji   ∀j∈N,

(8) ∑j∈Nf x0jk=1  ∀k∈K,

(9) ∑j∈Nfxj0k=1       ∀k∈K, 

(10) δi=∑j∈N:j≠iyji   ∀i∈Nf,

(11) ∑i∈Nfy0i=∑i∈Nfyi0=1,

(12) aik+qik≤θi   ∀i∈Nf,k∈K,

(13) ajk≥aik+qik+tijxijk+ζδi−M(1−xijk)   ∀i,j∈Nf:i≠j,k∈K,

(14) ajk≤aik+qik+tijxijk+ζδi+M(1−xijk), ∀i,j∈Nf:i≠j,k∈K,

(15) aik≤t0ix0ik+M(1−x0ik) ∀i∈Nf,k∈K,

(16) aik≥t0ix0ik ∀i∈Nf,k∈K,

(17) θi≥t0iy0i ∀i∈Nf,

(18) aik+qik+ti0xi0k≤T     ∀i∈Nf,k∈K,

(19) θi+(tij+ζ)yij−M(1−yij)≤wj  ∀i∈Nf, j∈N:i≠j,

(20) θi−(aik+qik)−M(1−δi)≤mik≥0       ∀i∈Nf,k∈K,

(21) wi≤θi      ∀i∈Nf,

(22) qik≤di=∑j∈N:j≠ixjik  ∀i∈Nf,k∈K,

(23) ljk≤lik−qik+Qs(1−xijk)    ∀i,j∈Nf:i≠j,k∈K,

(24) ljk≥lik−qik−Qs(1−xijk)    ∀i,j∈Nf:i≠j,k∈K,

(25) vi≤−lik+qik+Qs+Qs(1−δi)     ∀i∈Nf,k∈K,

(26) vi≤Qsδi    ∀i∈Nf,k∈K,

(27) hj≤hi−vi+M(1−yij)    ∀i,j∈Nf:i≠j,

(28) l0k=Qs,

(29) h0=Qb, 

(30) ljk≤Qs    ∀j∈Nf,

(31) hj≥Qb−M(1−y0j)    ∀j∈Nf,

(32) fi≤∑k∈Kqikdi    ∀i∈Nf,

(33) fi≤1   ∀i∈Nf,

(34) yij,xijk,δi∈{0,1},θik,aik,vi,wi,qik,ϵij≥0  ∀i,j,k.

The objective function (1) maximizes effectiveness and equity concurrently. Effectiveness is computed as the sum of the allocated quantity. Constraints (3) and (4) linearize the equity part of the objective function. The objective function (2) aims to minimize the total traveled time for both the Small Trucks and the Big Truck. This involves minimizing overall travel time and waiting time. Note that both objectives have equal weight. These two objectives are subject to the following constraints below:

Vehicle routing constraints (5)–(11): Constraint (5) ensures that each node is visited exactly once by the Small Truck. Constraint (6) establishes the inflow and outflow conservation: Small Truck entering a delivery node must also exit it. Similarly, constraint (7) ensures the inflow and outflow constraints for the Big Truck. Constraints (8) and (9) establish that a Small Truck can only leave and return to the depot once. Constraint (10) ensures that a node can be visited by the Big Truck if and only if it functions as a reloading node. Constraint (11) establishes that there is only one Big Truck.

Synchronization constraints (12)–(21): Constraint (12) sets the reloading start time to be the arrival time of Small Truck k to node i plus the delivery duration of the Small Truck. Note that this constraint ensures that the reloading of the Small Truck is performed after service completion. Constraints (13) and (14) determine the arrival time (and hence the delivery start time) of node j that is visited by Small Truck k after serving node i. Constraints (15)–(16) set the delivery start time of the first node visited by a Small Truck right after departing from the depot. These constraints also suggest that the time it takes to travel from the depot to the initial node is included within the timeframe for planning. Likewise, constraint (17) determines the arrival time of the first node visited by the Big Truck. Constraint (18) ensures that each Small Truck completes their tasks and gets back to the depot within the maximum allowable time. Constraint (19) computes the arrival time of the Big Truck at the next node after performing a reloading operation at the previous node. Constraint (21) ensures that reloading only starts at node i after the Big Truck arrives at node i. Constraint (20) sets the waiting time of Small Truck.

Vehicle capacity constraints (22)–(31): Constraint (22) ensures that the quantity of food items delivered at node i does not exceed the maximum allowable quantity or demand. Constraints (23) and (24) track the load level of Small Truck k upon arriving at node j after visiting node i and also account for the event of reloading at node i prior to arriving at node j. Constraint (25) computes the quantity of reload for a Small Truck at node i based on the excess food items after delivery at node i. Constraint (26) sets the appropriate value of the reloading quantity to be at most Q^s in case there is a reloading at node i and 0 otherwise. Constraint (27) tracks the level of food resources in the Big Truck. Constraints (28) and (29) ensure that the Small Truck and the Big Truck depart the depot fully loaded. Constraint (30) ensures that the Small Truck does not hold more food items than its capacity. Constraint (31) ensures that a Big Truck leaves the depot fully loaded. Constraint (32) calculates the filling rate for each node i. Constraint (33) ensures that the allocation at node i does not exceed demand. Finally, constraint (34) specifies the feasible ranges of the decision variables.

In our study, we employ similar objective functions from the humanitarian resource allocation models presented in the studies by Eisenhandler and Tzur (2019a, 2019b) and Alkaabneh et al. (2023). The researchers in their study create a novel objective function that advances the operational objectives of food banks. This objective function encompasses both effectiveness and fairness. This function combines the measure of effectiveness, defined as the total allocation provided to all agencies, with an equity metric. The equity metric is derived by subtracting the Gini coefficient of the allocated distribution from 1. The Gini coefficient, widely used to measure inequity in social welfare, is an important tool in economics. It serves to quantify the distribution of various values, such as income or wealth, within a given population. This coefficient operates by assessing inequality through a comparison between the area beneath the perfect equity line and the Lorenz curve, relative to the total area under the perfect equity line. The result is a numerical measurement ranging between zero and one, which serves as an indicator of the degree of inequity in the distribution. A value of 0 signifies perfect equity, where everyone possesses an equal share, while a value of 1 represents extreme inequity, where a single entity possesses everything, leaving others with nothing. Moreover, this Gini coefficient can be effectively applied to the allocation of resources in food banks, allowing for an objective assessment of how resources are distributed among various agencies. It provides a quantitative means to gauge the level of fairness in the allocation process, thus aiding decision-makers in addressing potential disparities and promoting more equitable resource distribution. Interested readers may refer to Anand (1978) and Mandell (1991). In our study, we calculated the Gini coefficient as follows:

(35) G=∑i∈Nf∑j∈Nf| ηjfidi−ηifjdj |∑i∈Nffidi

Since the Gini coefficient is a measure of inequity, to measure equity in our objective function, we subtract the Gini coefficient from 1. Thus, following the above description, the function becomes (36) and simplified to (37):

(36) ∑i∈Nffidi(1−∑i∈Nf∑j∈Nf| ηjfidi−ηifjdj |∑i∈Nffidi)

(37) ∑i∈Nffidi−∑i∈Nf∑j∈Nf| ηjfidi−ηifjdj |

The absolute value |η_jf_id_i − η_if_jd_j| of (37) is linearized using the variable ϵ_ij by introducing constraints (3) and (4), thus providing MIP models (1) – (34) that may be solved using general MIP model solver.

3.3 Social objective

The objective function (1) in the social context is geared toward ensuring that every affiliate within the food bank network receives an equitable portion of donated food resources. Furthermore, it underscores the crucial importance of acknowledging and valuing the pivotal societal role played by these food banks in mitigating food insecurity challenges. As demonstrated within the mathematical framework, the initial segment of the objective function (1) takes into account the aggregate volume of food distributed by the food bank to its partner agencies. The subsequent element is dedicated to minimizing disparities in distribution, aiming to achieve a balanced and fair allocation process. Recall that constraints (3) and (4) are strategically implemented to linearize the equity component of the model. This approach to modeling reflects the model’s commitment to promoting perfect equity within the food distribution network. By using quantitative measures of food allocation and equity, the social objective function works cohesively to foster an environment of equal opportunity and support for all partner agencies, ultimately advancing the goal of reducing food insecurity in the community.

We formulate two additional distinct social objective functions to address different perspectives, modifying the original Model1, which maximizes both effectiveness and equity simultaneously and minimizes total routing time.

In Model2, the emphasis is solely on maximizing equity, specifically by maximizing the minimum filling ratio while minimizing the overall routing time. This concept we adapt was introduced by Lien et al. (2014) and Balcik et al. (2014). To implement this, we replace the objective function (1) with the new objective function (38), and add the constraint (39) to the model:

(38) max    Φ 

(39) fi≥Φ    ∀i∈Nf

With Model3 we replace the objective function (1) with (40) and we set equity as a constraint. Model3 aims at maximizing efficiency which is measured as the total quantity of food allocated to all the nodes while also minimizing the overall routing time:

(40) max∑i∈Nffidi

(41) |fi−fj|≤α       ∀i,j∈Nf,

Constraint (41) ensures that the deviation between the maximum filling rate and the minimum filling rate does not exceed the inequality threshold α. Note that in both Model2 and Model3, the objective function 2 remains unchanged.

3.3.1 Economic objective

The primary goal of the economic objective function is to identify the most efficient routes with the least total travel time. The first and second segments of the objective function [equation (2)] account for the total travel distance/time covered by both the Small Trucks and the Big Truck. This is accomplished by seeking to minimize the metric of travel time. The third component represents the value/cost of the waiting time created by the operation of Small Trucks. By prioritizing the reduction of travel time, the economic objective function aims to streamline operational efficiency. This strategy not only cuts down on fuel costs but also maximizes the productive hours of the trucks.

4. Experimental design

In this section, we conduct an in-depth numerical analysis to gain valuable managerial insights from our study. Given that we address VRP-Sync. for the first time in humanitarian literature, there are no benchmark instances to use for analysis. Thus, we generate test instances with n = 10, 15, 20, 25, 30, 35, 40, and 50 agencies. In these Euclidean instances, the travel time is the same as the distance. We assume the cost of travel is proportional to the travel time. The agencies are randomly located in the instances. Additional parameter values embedded in our model include ζ set at 30 units of time, inequity measure α set to 0.05 and 0.1, Q^s established at 120 units of food resources, and Q^b set at [120, 150, 200] corresponding to (L1 = Low, L2 = Medium, L3 = High) supply levels, respectively. The demand d_i of each agency is set within the range of 20–50 units. This carefully designed experimental setup allows for a thorough evaluation and validation of the performance of our proposed optimization approach, ensuring its effectiveness under conditions closely aligned with real-world operational scenarios.

We coded our mathematical model and generated instances in Python using Jupiter Notebook as an interface, and we use Gurobi 9.1.1 optimization software as the MIP solver. All the experiments were con- ducted on a computer with Intel(R) Core(TM) i7-8700 CPU @ 3.20 GHz processor with 16.0 GB RAM. All analysis of data were performed using R Studio statistical software version 4.3.1.

4.1 Baseline

In this section, we establish a base model BModel to understand the current state of the food bank distribution operation. We simulate a food bank distribution operation without the implementation of the proposed multiobjective model. This base model aims to represent the standard operational approach followed by food banks for equitable food allocation in low-income and remote areas serving vulnerable populations. The BModel relies on methods of routing and food allocation without synchronization (non-Sync-VRP) with the objective of maximizing the total quantity of food allocated. The current model we propose places specific emphasis on minimizing routing time and maximizing equity and the total quantity of food allocated simultaneously. The results from the base model serve as a valuable reference point with which all other three Sync-VRP models (Model1, Model2, and Model3) are compared. Note that the key performance measures we assess are waste, effectiveness, equity, and efficiency (the total routing time). We present a pseudo code of the BModel in the Appendix. The test instances can be found in the Mendeley data set repository at: https://data.mendeley.com/datasets/pdz4cys2r7/1

5. Results and discussion

In this section, we provide the results that were obtained by solving the MILP models introduced in Section 3.2 using the solver Gurobi Optimizer.

5.1 Comparison of models

In this section of our manuscript, we provide an extensive analysis that entails the comparison of the three multiobjective MIP models developed, as detailed in Section 3.2. We also compare all three Sync-VRP models to the base model (non-Sync-VRP). This comparative analysis aims to offer a comprehensive understanding of the performance variations among the models under different configurations of the key parameters α and varying levels of total supply. Through this exploration, we seek to unveil insights into performance of the models and contribute to the optimization of food distribution in food banks.

5.1.1 Inequity

The assessment of inequity in food distribution within food banks, as highlighted in our analysis, emerges as a critical metric for gauging the fairness of the distribution process. The measurement of inequity not only serves as a vital indicator but also plays a pivotal role in evaluating how effectively food resources are allocated among various agencies within the food bank network. Figure 3 visually depicts the inequity measure across different supply levels for the four distinct models. In this context, inequity is calculated as the difference between the node with the highest filling ratio and the node with the lowest filling ratio. We observe that Model2 showed the lowest inequity measure, characterized by zero inequity. This is reasonable because Model2 minimizes the difference between the maximum and minimum filling ratios. Bmodel and Model3 yield inequity values between 0.05 and 0.10 as specified by the inequity parameters. Model1 exhibits relatively positive performance, with inequity being approximately less than 0.035. Note that in Model1, the threshold of the inequity is not explicitly defined. It is interesting to see that the inequity measure of Model1 is not consistently distributed across the different supply levels, suggesting that an increase in supply does not necessarily lead to a proportional improvement in equitable distribution. This observation underscores the complexity of distribution dynamics under Model1.

Our analysis challenges the notion of setting a fixed inequity, as demonstrated by Model1, which consistently produces lower inequity even with high levels of supply compared to BModel and Model3. This suggests that a more flexible approach, as adopted by Model1, is more effective in achieving equitable food distribution. An intriguing observation arises when comparing the behavior of inequity in Model1 with that of other models, both Sync-VRP and non-Sync-VRP. Unlike the other models, where mean inequity remains constant as supply levels increase, the behavior of inequity in Model1 is nonmonotone. This suggests that maintaining equity may be increasingly challenging as supply levels rise. This finding highlights the complexity inherent in achieving equitable food allocation, particularly in scenarios of abundance of supplies.

5.1.2 Effectiveness

Maximizing the distribution of food resources is a crucial aspect of the humanitarian response supply chain. Strategic food allocation by food banks enables them to support a larger number of individuals and families experiencing food insecurity. Effectively distributing as much food as possible in a food bank setting holds paramount importance for addressing the pressing challenges of hunger and food insecurity. In this section of the paper, we analyze the effectiveness of the four models. Figure 4 provides a visual representation of the average fill rates across various α values and total supply levels for the four model types. The key observations indicate that BModel consistently produces the lowest fill rates, whereas Model1, Model3 and Model2 exhibit higher fill rate levels, following in descending order, respectively. This implies that Model2 is less effective in distributing food compared to Model1 and Model3. Note that this outcome aligns with our expectations, as the objective function of both Model1 and Model3 maximizes the total quantity of food distributed while Model2 maximizes equity among the agencies. Furthermore, in the BModel, the capacity of the truck is constant throughout without restocking, thus the number of agency visits remains constrained by the limited capacity of the truck, regardless of the available supply at the food bank. The findings highlight the significance of maximizing the total quantity of food allocation and restocking vehicles through the synchronization of the routes during distribution to increase overall food accessibility.

5.1.3 Efficiency

Efficient routing in food distribution is critical for the effective delivery of food to agencies in low-income and rural areas, offering cost savings and operational improvements. Figure 5 presents a graphical representation of the average time covered across various α values and supply levels for each of the four models. The visualization allows us to compare the transportation efficiency of the three models to BModel based on different parameters. Upon analysis, we note that the BModel results in the highest total distance traveled, with Model1 and Model3 following in descending order. Conversely, Model2 yields the lowest total time among the models. This pattern holds true across diverse α values and supply levels, indicating a consistent trend in the performance of these models.

The findings regarding average times align with our expectations, underscoring that optimizing routing in models such as Model1, Model2, and Model3 proves more efficient in achieving reduced travel times compared to BModel, which solely prioritizes maximizing overall food distribution. Additionally, it is noteworthy that in Model1, Model2 and Model3, the total time traveled increases as the total supply rises, irrespective of the α value. This correlation is logically explained by the fact that a higher supply necessitates visits to more agencies for food distribution, thus increasing accessibility, leading to an extension of the overall route. In our study, we show in Figure 5 that all the three Sync-VRP models are efficient, highlighting the effect of synchronization and minimizing the overall routing time.

5.1.4 Waste

Food waste is a global issue, with 33% of edible food discarded at different stages of the supply chain, according to estimates from the UN Food and Agriculture Organization (FAO) and the UN Environment Program (UNEP) (UNEP, 2021; The Global Food Banking Network, 2021). Measuring food waste in distribution operations is essential for food bank managers to enhance efficiency and address the environmental impact of their operations. Quantifying and analyzing food waste provides insights into areas where improvements can be made, such as optimizing allocation practices. This data-driven approach enables food banks to operate more effectively and align their efforts with SDGs. Figure 6 provides a visual representation of the variation in food waste percentages across different levels of total supply for the three distinct models compared to the base model. The percentage of food waste is computed by taking the difference between the total supply designated for distribution and the overall quantity allocated, divided by the total available supply for distribution. We note that BModel produces the highest percentage of food waste, followed by Model2, while Model1 and Model3 record the lowest food waste. This aligns with our expectations, as Model2 prioritizes maintaining a consistent filling ratio across demand nodes, even if it results in food waste. In contrast, both Model1 and Model3 adopt an allocation strategy that emphasizes maximizing the total quantity allocated. Interestingly, similar to Model1 and Model3, the BModel maximizes the total quantity of food allocated as well. However, BModel does not consider restocking the truck during food distribution. This results in a higher percentage of the available supply being wasted. While the Model2 sacrifices optimization for equity, the Model1 seeks a delicate balance between waste reduction and maximizing a fair and effective food distribution. Furthermore, the higher percentage of waste generated by BModel and Model2 (refer to Figure 6) aligns with the lower fill rates seen in Figure 4 showcasing the inherent ineffectiveness as evidenced by the elevated waste percentages demonstrated in Figure 6. The results derived from our analysis, particularly the lower food waste percentages exhibited by Model1 and Model3, highlights the effectiveness of these models as seen in Figure 4.

5.1.5 Volunteer workload balance

Ensuring workload equity is essential for maintaining a balanced distribution of tasks among volunteer drivers or distribution personnel, fostering a positive and inclusive working environment. The workload ratio serves as a quantifiable metric that delineates the correlation between the most demanding workload (pertaining to a route sequence with a high number of visits to agencies) and the least demanding workload (associated with a route sequence featuring a lower number of visits to agencies). Computed by dividing the heavy workload by the least workload, this ratio yields a numerical value index of the relative magnitude of the heavier workload to the least workload. A workload ratio exceeding 1 signal that the heavy workload is notably more substantial than the lowest workload. Figure 7 illustrates the volunteer driver workload associated with the Small Trucks during distribution across all three model types. We notice that the workload distribution among drivers is not balanced for all three Sync-VRP models. Notably, Model2 demonstrated a more balanced workload with a consistent ratio across the different levels of supply compared to Model1. Overall, we noticed the worst balance in Model3. Particularly showing an increase in workload ratio as the levels of supply increase.

5.2 Managerial implications

The significance of our analysis of the proposed Sync-VRP models emphasizes the substantial variations in performance that can emerge based on the selection of one model over another and provides guidance to decision-makers on the trade-off and the model to select depending on the aim of the food bank. Table 1 presents a comprehensive overview of the performance of various Sync-VRP models investigated in our study. When selecting a model, food bank managers must carefully consider the trade-offs inherent in each option. For instance, while Model1 demonstrates low inequity, low waste, and high effectiveness, it may compromise efficiency compared to other models. Conversely, Model3 shows moderate efficiency but exhibits higher in- equity. Food bank managers can leverage these findings to align model selection with their operational goals and constraints, taking into account factors such as resource availability and community needs.

In recent years, various research papers have emphasized the need to add equitable factors into human- Italian distribution decision-making approaches (Balcik et al., 2014; Sengul Orgut and Lodree, 2023). Food bank operations rely heavily on equitable distribution since these institutions are required to disclose their distribution equity levels to organizations such as FA (Feeding America, 2024). This emphasis originates from FA’s instruction to distribute food allotments according to the extent of food insecurity in each county. The method of allocating resources based on the proportion of food insecure population or demand in each county is commonly referred to as perfect equity. This approach, widely adopted by food banks in current practice, reveals a notable trend. Counties with larger capacity-to-demand ratios tend to receive an oversupply of re- sources, while those with smaller ratios tend to be undersupplied (Feeding America, 2024). This suggests that the current allocation strategies may inadvertently exacerbate disparities in resource distribution, favoring areas with higher demand capacity and potentially neglecting regions with greater need. Some studies suggest that food banks allow strategic deviations from perfect fairness or establish an allowed inequity buffer in their initial distribution strategy. However, this approach of determining a permitted inequity margin may not be useful to the food bank’s operations in a long planning period (Sengul Orgut and Lodree, 2023). Our work emphasizes the significance of flexible techniques in food allocation operations, as shown in 5.1.1. Our findings encourage food bank managers to go beyond setting rigid inequality boundaries and instead examine innovative approaches, such as those described in this study, that account for the factors driving distribution dynamics. This adaptable and flexible equitable food distribution enables food bank managers to respond quickly to changing circumstances. This aligns with concepts of supply chain systems resilience under complex conditions. Our study provides a tool for food bank managers to constantly analyze and adjust distribution strategies to guarantee fair access to resources across all populations.

Our results in Section 5.1.2 offer practical guidance to food bank managers in selecting models that align with their desired maximum distribution goals. These managers often set high-level objectives concerning equity and effectiveness to steer their distribution strategies. Insights from effectiveness and equity distribution models, as depicted in Sections 5.1.2 and 5.1.1, shed light on the different distribution approaches and highlight common ground. With this, food bank managers can balance the goals of efficiently distributing resources to address immediate needs while also addressing disparities in access and resource allocation across different communities with careful consideration and strategic planning.

In the functioning of food banks, reducing operating expenses is equally important as ensuring equitable and effective resource distribution. Because food banks rely significantly on donations to fund their operations, there is a continuous need to improve procedures to remain sustainable (Hasnain et al., 2021). While striving for fairness and effectiveness in resource distribution, it is critical to identify efficient techniques that allow food banks to maximize their impact with limited resources. This includes simplifying processes, lowering overhead expenses, and looking for inventive ways to stretch every dollar farther. We introduce the distinctive concept of route synchronization, a new variant of the VRP that is frequently ap- plied in profit-driven settings. Our results show that route synchronization improves accessibility to food resources by increasing food distribution efficiency. Even food banks with staffing limitations or volunteer shortages can accomplish their distribution objectives with this optimization strategy. Our sync-VRP model strategy, for example, may achieve the same goal with only three drivers if a regular distribution requires four drivers, proving its efficacy in optimizing resources and overcoming logistical restrictions. Insight from this study provides understanding of the performance differences across models, allowing managers to make educated decisions within budgetary constraints.

5.3 Environmental, social and governance implications

In this section, we provide the environmental, social and governance (ESG) implications of our study.

ESG includes a comprehensive set of standards utilized to evaluate an organization’s holistic impact. The environmental component examines the organization’s preservation of the environment, covering issues like pollution, waste management, greenhouse gas emissions, and climate change. Social considerations evaluate the organization’s impact on people, societies, and cultures by examining aspects such as human rights, diversity, inclusion, and ethical supply chain management. Governance looks at the policies that control how an organization manages its operations and safeguards the interests of its shareholders (Krantz and Jonker, 2024). When taken as a whole, these three factors offer a strong framework for assessing the overall sustainability and ethical performance of an organization.

Environmental: Food banks contribute significantly to environmental conservation by reducing food waste, which increases greenhouse gas emissions. Millions of pounds of donated food are saved by food banks each year from ending up in landfills, where organic waste like food may release the dangerous greenhouse gas methane (Greater Pittsburgh Community Food Bank, 2024). The Environmental Protection Agency (EPA) states that landfills are the third-largest source of methane emissions in the US. Food banks prevent billions of tons of food waste from entering the environment, which lowers greenhouse gas emissions (The Global Food Banking Network, 2021). Our study, which maximizes food distribution, has significant environmental implications. To reduce greenhouse gas emissions overall, our proposed Sync-VRP models decrease overall waste and transportation time, as seen in Sections 5.1.2 and 5.1.4. Reducing vehicle emissions and fuel use through effective transportation strategies decreases air pollution and slows down the degradation of the environment. Managers of food banks may reduce the carbon footprint of their entire operations by implementing the approach we propose. Furthermore, this research supports a more environmentally responsible manner of food delivery, which helps to promote global sustainability initiatives by preventing landfill waste via efficient food distribution.

Social: Food banks’ capacity to embrace diversity, equity, and inclusion is critical to their efficiency in serving their communities, in addition to maintaining resilient systems of operation (Hamilton et al., 2024). Several humanitarian studies present models on equity, and our approach follows suit. In this study, we con- sider equity not only when allocating food to partner agencies but also when ensuring an appropriate task division. Food bank managers can provide equal access to necessary food supplies by improving distribution operations through equitable allocation and taking into account elements like workload equity. This improves community resilience by supporting vulnerable individuals and promotes social cohesion by addressing dis- parities in food access. Additionally, this study encourages social responsibility and community participation by placing a higher priority on volunteer satisfaction and retention through workload fairness, strengthening the connections between volunteers, food banks, and the communities they serve. This all-encompassing approach places a strong emphasis on the role that advocacy, fairness, and accessibility play in reducing food insecurity and building a more inclusive society.

Governance: Food banks cater to the needs of a varied demographic by implementing various distribution programs tailored to specific groups such as children, seniors, immigrants, refugees, and individuals facing transportation challenges (Feeding America, 2024). Their priority lies in delivering food in a manner that is fair, while also striving to minimize food waste, given the common occurrence where demand surpasses available supply. In the management of their distribution processes, food banks collaborate closely with the community to advocate for local and national policy changes, aiming to drive impactful change in society. Our research provides valuable insights into the policymaking processes concerning food distribution and initiatives to enhance food security. By shedding light on the various Sync-VRP models, policymakers acquire the understanding needed to make informed decisions regarding food allocation and support for food banks. Additionally, our study underscores the importance of adopting flexible strategies and approaches to food allocation, thereby informing the development of policies that prioritize efficiency, equity, and sustainability within the food distribution system. Furthermore, our findings may be used to inform policymakers, allowing for the establishment of measures that will improve food distribution and access while also addressing problems like food waste and food insecurity at the local and national levels. In addition, this study offers practical solutions to promote some of the main goals of the SDGs, including eliminating hunger, reducing inequality, and eliminating poverty.

6. Conclusion

In conclusion, our research focuses on optimizing food bank distribution operations to achieve fair, effective, and efficient food allocation within food banks, ultimately enhancing overall distribution processes. We address a multiobjective synchronized vehicle routing problem, introducing three distinct mathematical models that prioritize key performance measures such as effectiveness, efficiency, and equity. Notably, our findings highlight that Model1 excels in achieving high efficiency, increased effectiveness, minimal waste, and reduced inequity. In contrast, while Model3 demonstrates commendable attributes with low waste and improved efficiency, it records the highest level of inequity among the three Sync-VRP models. Model2 exhibits a moderate level of waste and efficiency, lower effectiveness, and zero inequity. The trade-off between equity and waste reduction is evident in Model2. Although Model2 may enhance equity in food distribution, the observed increase in waste indicates a potential ineffectiveness in food allocation. This requires a careful balance between achieving fair distribution practices and minimizing food waste within the operational framework of food banks utilizing Model2. Decision-makers should consider this delicate equilibrium when choosing Model2 for implementation, aiming for solutions that optimize both equity and waste reduction, ultimately leading to a more sustainable and effective food distribution system. Overall, we show that the non-Sync-VRP model produces the worst results compared to the Sync-VRP models.

By studying Sync-VRP models, we draw attention to the inherent trade-offs that decision-makers have to take into account when deciding how best to allocate resources and serve food banks. Our research emphasizes how important efficiency, equity, and sustainability are when designing policies for food distribution, underscoring the necessity for adaptable systems. Our study contributes to the SDGs by providing insights that can inform policymaking and interventions aimed at addressing diverse community needs.

6.1 Recommendation

All three Sync-VRP models have unique characteristics that may benefit decision-makers one way or another. Nonetheless, we recommend Model1 for decision-makers. Model1’s deliberate trade-offs and distinctive features, encompassing maximizing food distribution, equity, waste reduction, and minimizing total time traveled, offer a unique and balanced approach for decision-makers. This model presents a valuable tool for decision-makers aiming to optimize food allocation, especially for vulnerable populations dependent on food assistance, considering the current demand surge, decrease in donations, inadequate transportation resources, and limited funds. The insights gained from Model1 will enable decision makers to make informed decisions and craft adaptive strategies that consider their objectives and operational constraints.

6.2 Study limitation and future work

The study limitations are as follows:

• The assumptions of deterministic demand and constant supply levels do not reflect the stochastic real-world settings. In practice, demand for food assistance may fluctuate owing to a variety of circumstances, including economic situations, seasonal variations, and unanticipated occurrences such as natural disasters. Furthermore, supply levels may change due to donor patterns and food availability.

• While numerical analyses using randomly simulated data provide valuable insights into model performance and behavior under controlled conditions, they may not fully capture the complexities of real-world operational environments. Without validation against actual operational data from food banks, there remains uncertainty regarding the applicability and effectiveness of the proposed models in practical settings.

The current synchronized routing and allocation problem represents a complex challenge, and addressing large-scale instances poses intractable complexities. To tackle this, future studies aim to devise high-quality metaheuristics leveraging advanced algorithmic schemes to efficiently solve the proposed models. Additionally, our research envisions incorporating the dimension of picking up donations using the distribution vehicles during operations. Given the random nature of donations to food banks, future investigations will explore stochastic synchronized pickup and distribution operations within a nonprofit setting. These enhancements will contribute to a more comprehensive understanding and practical application of optimized distribution strategies for food banks, ensuring adaptability to real-world challenges and promoting the efficiency of nonprofit operations.

Figures

Figure 1

Summary of related literature

Figure 2

Proposed solution approach illustration

Figure 3

Comparison of mean inequity by total supply

Figure 4

Comparison of the mean effectiveness under different α value by total supply

Figure 5

Comparison of the mean total distance traveled

Figure 6

Comparison of the mean percentage food waste by total supply levels

Figure 7

Comparison of workload

Table 1

Summary of model performance

	Model type
Performance measure	Bmodel	Model1	Model2	Model3
Inequity	High	Low	Least	High
Effectiveness	Least	High	Low	Medium
Efficiency	Least	Low	High	Medium
Waste	High	Low	Medium	Least

Notes:

Least < Low < Medium < High, Model Type

Source: Table created by authors

Note

1 Equity refers to achieving equal service levels for all partner agencies. Conversely, effectiveness here refers to the allocation of food items to the fullest extent possible, in other words, distributing as much food as possible to the agencies.

Appendix