Equity-Driven Workload Allocation for Crowdsourced Last-Mile Delivery

Abstract

Crowdshipping, a rapidly growing approach in Last-Mile Delivery (LMD), relies on independent crowdworkers to fulfill delivery orders. Building a sustainable network of crowdshippers is crucial for the long-term success of such systems, as participation is primarily driven by fair compensation. This is especially important for workers who rely on crowdwork as their main source of income, making equitable pay not just a matter of fairness but of financial well-being. In this study, we address several key questions that gig-economy platforms concerned with fair pay may ask: How can equity be measured? What are the associated cost implications? And how can potential drawbacks be managed? Our main contribution is the development of a practical, equity-oriented framework tailored to crowdshipping within an LMD environment. Inspired by the real-world operations of several crowdshipping platforms, the framework operates in real time and is built around a bi-objective optimization model that balances equity and cost. This allows us to systematically explore trade-offs and identify the equity measures that most effectively capture this balance. We demonstrate that even a modest reduction in cost efficiency (e.g., 2.5%) can lead to substantial improvements in equity; potentially up to 65%. Our results provide actionable insights for practitioners, including guidance on selecting appropriate equity measures. We also find that the best equity outcomes occur when the crowdshipper pool is kept relatively small. Furthermore, we quantify the performance loss of high- and low-performing crowdshippers as the pool size increases, offering valuable insights for workforce planning and management. Along similar lines, we demonstrate that our framework remains effective in managing vehicle shortages in dynamic environments while achieving comparable levels of equity improvement.

Keywords

Crowdshipping Last-Mile Delivery Equitable Workload Allocation Bi-objective Optimization

1. Introduction

Last-mile delivery (LMD) consistently represents a significant expense within the logistics framework. According to CapGemini Research Institute (2019), LMD stands out as the primary cost driver in the supply chain, a fact accentuated by the exponential growth of e-commerce, which exposes the unsustainability of current delivery models. Vehicle routing in last-mile logistics is often considered non-value-added due to resource underutilization, high delivery costs, and loss of business time. In response to the growing demand for faster delivery services, several retailers are now adopting platforms allowing them to employ independent gig workers for order fulfillment. Such platforms facilitate the onboarding of gig workers based on demand dynamics, lowering operational costs compared to maintaining permanent employees, as noted by Fatehi and Wagner (2022). The employment of gig workers in the context of LMD is often referred to as crowdshipping.

It is important to distinguish crowdshipping from traditional outsourcing. While some authors (e.g., Kafle et al., 2017) describe crowdshipping as a form of outsourcing, specifically the assignment of deliveries to crowd-based workers, the term generally carries a broader meaning in the logistics literature. Traditional outsourcing typically involves delegating a set of logistics functions to a third-party provider, whose fleet performs the work under a service-level agreement (Malchow, 2023). In contrast, crowdshipping relies on independent couriers who connect directly with a platform or retailer to carry out specific delivery or pickup tasks. In a typical crowdshipping setup, a group of ad hoc drivers handles the delivery of one or more online orders to shopper locations (Alnaggar et al., 2019; Archetti et al., 2016; Arslan et al., 2019; Dahle et al., 2019; Dayarian and Savelsbergh, 2020; Macrina et al., 2017; Soto Setzke et al., 2017). Unlike conventional LMD, crowdshippers are not employees of the delivery company; they offer services occasionally, usually using spare space in their personal vehicles while en route to other destinations. As a result, crowdshippers differ in availability, vehicle capacity, and coverage area, among other factors.

While crowdshipping transforms LMD, its unique features bring operational challenges. One key issue, noted by Allon et al. (2023), is ensuring a steady and reliable supply of crowdshippers, which largely depends on their satisfaction with past participation. Achieving this requires fair and equitable workload allocation and compensation (Bai et al., 2019; Dayarian and Pazour, 2022). While fairness matters across all workforces, it is especially critical in crowdshipping, where gig workers are paid per task. Unlike salaried employees (e.g., Rea et al., 2021), crowdshippers’ earnings depend directly on workload distribution. Perceived unfairness can quickly lead to dissatisfaction and lower participation, making fair pay essential to sustain engagement and platform success. Equitable workload allocation ensures all crowdshippers have equal profit opportunities based on system conditions and individual capabilities. This supports financial stability, especially for those relying on crowd work as their main income, and guarantees outcomes nearly as favorable as efficiency-focused approaches.

Achieving workload equity involves two types of costs: operational and opportunity costs. Operational costs arise from deviations between equity-oriented and efficiency-oriented workload allocations. Opportunity costs reflect the potential loss in earnings for some crowdshippers when workload is redistributed to improve compensation for others, promoting a more equitable income distribution. Imposing operational costs on crowdshippers would reduce their total payoff, undermining the goal of encouraging participation through fairness. Thus, we focus on a setting where the company values equity, absorbs the operational costs, and propose a framework for equity-driven workload allocation. Opportunity cost is inherent in equity-focused strategies, as they redistribute opportunities by benefiting some crowdshippers while reducing pay for others compared to efficiency-based allocations. While those who benefit may be more inclined to participate, high-performing crowdshippers with greater resources may feel discouraged, even if they recognize the fairness principle. Our framework addresses this concern through effective workforce size management.

The proposed framework targets a free-market setting, specifically benefiting for-profit companies. Benefits arise when (1) the operational cost of equity remains low enough to preserve the company’s competitive edge, and (2) equity-driven outcomes favor all crowdshippers as uniformly as possible, including high performers. To achieve this, we develop a mechanism to mathematically represent crowdshippers’ workloads (the equity metric), which must be equalized. This equalization requires equity measures to evaluate the level of equality or inequality achieved. Our framework first proposes a novel equity metric capturing workload allocation across a diverse pool of drivers and adopts several established equity measures from economics. At its core is a bi-objective optimization process that balances equity and cost by approximating the nondominated (Pareto-optimal) frontier. To maintain competitiveness, the company sets an upper bound on the operational cost it is willing to incur while balancing equity and cost. Using this framework, we address three key research questions through computational analysis:

Which equity measure(s) are most reliable within the scope of our study? Since no unique equity function exists, we use multi-objective optimization to show that no established economic equity measure can be dismissed for the workload allocation problem. We then introduce a novel approach to identify the most reliable measure numerically. A measure is reliable if its nondominated frontier solutions approximate well those of other measures. Our results highlight the Coefficient of Variation as particularly promising.

How much can equity improve with varying equity budgets within a working horizon? We show that equity can improve by up to about 65% within a working horizon with no more than a 2.5% increase in cost, and up to about 75% with under a 10% cost increase.

What losses (i.e., opportunity costs) might high-performing crowdshippers face due to equity considerations? Our findings show that the framework minimizes losses for high performers. Additionally, interests align between high- and low-performing crowdshippers and equity-focused companies. Reducing the number of crowdshippers benefits all by cutting company expenses and lowering opportunity costs for both groups. Along similar lines, we also explore a complementary aspect of the problem in scenarios where courier shortages arise due to the dynamic nature of the system. We show that comparable levels of equity improvement can still be achieved under such conditions; however, this may come at the cost of increased operational expenses for the platform.

1.1. Contributions

The main contributions of this paper can be summarized as follows:

We propose a practical equity-oriented framework tailored to crowdshipping within an LMD environment. This framework seeks to mitigate the adverse effects of operational and opportunity costs while maximizing the benefits of equitable workload allocation.

We base the framework on the real-world activities of a group of crowdshipping platforms. We refer to the identified problem as the Dispatch Zone-Wave Problem (DZWP).

We propose a solution method alongside the problem formulation to provide effective strategies for addressing the DZWP. The proposed method is tailored to optimize decision-making processes, enhancing the overall effectiveness of the equity-aware framework in similar LMD contexts.

We conduct numerical experiments to evaluate the efficiency and performance of the proposed method, providing empirical evidence of its ability to handle practical-sized scenarios. A significant portion of our experiments is based on Amazon-inspired instances, which are specifically designed to address the three main research questions outlined earlier.

1.2. Paper Organization

The remainder of this paper is organized as follows: In Section 2, a brief literature review is provided. In Section 3, the theoretical foundation of the research is presented. Section 4 offers an overview of the DZWP. Section 4 provides an overview of the DZWP, including its assumptions, objective functions, and mathematical formulation. The proposed solution approach for solving the workload allocation problem is described in Section 5. Section 6 presents a solution approach to identify the most reliable equity measure(s) numerically. In Section 7, a detailed computational study is provided to address the research questions discussed above. Finally, a discussion including some concluding remarks is given in Section 8.

2. Literature Review

In this section, we review existing literature relevant to our study, organized around three key themes: (1) Optimization approaches for the LMD, (2) fairness in resource allocation, and (3) fairness in LMD-specific contexts. We emphasize summaries of strengths, limitations, and implications for each theme, highlighting how these inform our equity-driven framework for crowdshipping.

2.1. Optimization Approaches for the LMD

Exact methods aim to yield optimal solutions but require problem-specific adaptations for vehicle routing problem (VRP) variants, as seen in unified frameworks by Baldacci and Mingozzi (2009) and Pessoa et al. (2020). Applications in crowdshipping include iterative exact solving for dynamic pickups (Arslan et al., 2019) and robust optimization with queuing theory (Fatehi and Wagner, 2022). Despite these advancements, solving even moderately sized instances remains computationally intensive.

To overcome these limitations, ML has recently gained traction as either a standalone approach (Kwon et al., 2020) or combined with other methods (Morabit et al., 2021; Sobhanan et al., 2025). For example, Behrendt et al. (2023) apply ML to courier scheduling in crowdsourced deliveries. Despite their promise, ML techniques face challenges such as data dependency, long training times, and limited scalability, making them less adaptable when problem features change. As a result, heuristics have become the most widely used methods for large-scale optimization. Macrina et al. (2020) employ a variable neighborhood search for crowdshipping with intermediate depots but do not address equity considerations. Among heuristics, genetic algorithms (GAs) are particularly popular for their flexibility and speed to guide the population search towards optimality; for example, Vidal (2022) provide an open-source GA that finds near-optimal solutions to the capacitated VRP. GAs have proven effective for many VRP variants, including multi-depot (Vidal et al., 2012) and truck-drone fleets (Mahmoudinazlou and Kwon, 2024). For a comprehensive review of exact methods and heuristics applied to VRPs in freight transportation, readers can refer to Konstantakopoulos et al. (2022).

While traditional VRPs primarily focus on minimizing cost, real-world applications often involve balancing multiple objectives. For instance, Bula et al. (2019) propose a bi-objective model minimizing routing risk and cost. Recent research increasingly employs evolutionary algorithms for multi-objective VRPs (Liu et al., 2023) or adapts single-objective heuristics to incorporate route balancing (Matl et al., 2019a). Other studies optimize costs, emissions, and time using Non-dominated Sorting GA (NSGA-II) (Zhao et al., 2021) or combine heuristics with preference learning for more balanced solutions (Mesa et al., 2024; Nourmohammadi et al., 2025). Collectively, these reveal trade-offs overlooked by single-objective methods, underscoring the need for bi-objective equity-cost balancing in our framework.

2.2. Fairness in Resource Allocation

Fairness in resource allocation is a key concern across fields like healthcare, public policy, economics, and ethics. Mandell (1991) explore the cost-equity trade-off in public service delivery, unlike LMD in for-profit companies. Bertsimas et al. (2012) address efficiency-fairness trade-offs using $α$ -fairness, where $α$ guides decision-makers’ balance. For a comprehensive review of models balancing efficiency and fairness, see Xinying Chen and Hooker (2023), who note no single approach fits all contexts. Chen and Hooker (2020) propose social welfare functions combining fairness and utilitarianism in a mixed-integer program with sequential optimization prioritizing disadvantaged recipients, though such lexicographic methods may limit balanced fairness perspectives.

2.3. Fairness in LMD

For a detailed review of workload equity in vehicle routing, see Matl et al. (2018), which evaluates equity measures in bi-objective VRP models, identifying key properties and their impact on routing solutions. Matl et al. (2019b) extend this work through bi-objective optimization for route length balancing, termed the VRP with Route Balancing. Building on this, our study enhances practical relevance for crowdshipping by addressing workload equity among diverse couriers, incorporating multiple equity measures to capture varied preferences and trade-offs in crowdsourced delivery platforms.

Crowdshipping

Gig-economy delivery platforms, reliant on independent couriers, face unique challenges in equitable task allocation. Basık et al. (2018) and Li et al. (2023) explore fair task assignment in crowdshipping, focusing on fairness without route optimization. Other studies address fairness in transportation via single-objective optimization, covering scenarios like health worker allocation (McCoy and Lee, 2014), demand satisfaction with limited fulfillment (Ibarra-Rojas and Silva-Soto, 2021), and workload balancing with route consistency (Yu et al., 2024). Recently, Mahmoodian et al. (2025) proposed a vehicle routing model maximizing Nash Social Welfare of profit ratios for equitable workload distribution among crowdshippers. In contrast, our study employs a bi-objective optimization framework, balancing equity and cost to enhance fairness and operational efficiency in crowdshipping.

Meal Delivery Problem

The meal delivery problem, a key LMD scenario, emphasizes scheduling over routing due to tight delivery time windows (Agnetis et al., 2023; Cosmi et al., 2025). Operational planning prioritizes effective scheduling to ensure timely fulfillment, critical for customer satisfaction in gig-economy platforms. Martinez-Sykora et al. (2024) propose multi-objective optimization models to equitably distribute orders among food-delivery couriers, minimizing workload, travel, and waiting times, with real-world data validating significant fairness improvements over standard practices. While integrated routing and scheduling, known as the Meal Delivery Routing Problem (Reyes et al., 2018; Yildiz and Savelsbergh, 2019), is less studied, recent work by Singh et al. (2024) introduces an online heuristic algorithm incorporating fairness via a min-max objective for the $k$ -server food delivery problem. These advancements underscore the role of fairness in dynamic scheduling but address a distinct context from our crowdshipping-focused equity framework.

2.4. Research Gap

Despite advancements in LMD optimization, significant gaps remain at the intersection of efficiency and equity. Existing vehicle routing literature predominantly focuses on single-objective cost minimization, overlooking bi-objective frameworks that balance costs against workload equity for heterogeneous couriers. Moreover, broader fairness models are often ill-suited for for-profit platforms, failing to address the practical costs of implementing equity. A critical methodological void also exists, as the literature lacks both a suitable equity metric for diverse courier pools and a systematic process to validate and select the most reliable measure across a Pareto-optimal frontier. This study addresses these shortcomings by introducing a practical, bi-objective framework for the DZWP featuring a novel equity metric and a new approach for identifying the most reliable measure, thereby enhancing the operational viability and fairness of crowdshipping platforms.

3. Theoretical Foundation

In this section, we present a general theoretical framework applicable to a broad class of problems, including the specific setting of this paper. Consider an entity $G$ that must allocate $N$ indivisible goods of varying sizes among $M$ independent agents, each aiming to maximize their own utility. Each good $i \in N = {1, \dots, N}$ with size $q_{i}$ is assigned to exactly one agent. Let $S$ denote the set of all feasible allocations, and $s_{m}$ represent the goods assigned to agent $m \in M = {1, \dots, M}$ under allocation $s \in S$ . Each allocation $s$ has an associated cost $C (s)$ for $G$ , and agent utility $u (s_{m})$ , where the utility function $u (\cdot)$ is identical across agents.

Entity $G$ seeks to minimize its cost while ensuring fairness, which leads to a multi-objective optimization problem with $M + 1$ objectives: One for cost and $M$ for agent utilities. Managing an optimization problem with $M + 1$ conflicting objectives poses significant challenges, even for small $M$ values. To simplify, we consolidate the $M$ utility objectives into a single equity measure, $E M (s)$ , capturing disparity across agent utilities. By minimizing $E M (s)$ , which depends solely on $u (s_{1}), \dots, u (s_{M})$ , we convert the problem into a bi-objective optimization: minimizing cost and disparity. Solving this yields the nondominated frontier, where each Pareto-optimal solution reflects a trade-off between cost and equity. Entity $G$ can then select a solution that best balances these objectives. We later propose a selection approach tailored to for-profit settings. This framework is broadly applicable and used throughout this paper. However, reducing the $M + 1$ -dimensional objective space to two dimensions requires a careful mapping. We refer to such a transformation as a proper mapping, formally defined as follows:

Definition 1
A proper mapping reduces the full-dimensional criterion space to two dimensions such that every Pareto-optimal solution in the bi-objective space remains Pareto-optimal in the full space.

By Definition 1, proper mapping ensures that solving the bi-objective problem yields a subset of Pareto-optimal solutions in the full solution space. Without proper mapping, suboptimal solutions may arise in which improving one agent’s utility does not reduce the utility of others. This discussion highlights that not all equity measures are suitable, and those that fail to ensure proper mapping can be safely excluded. In general, both the structure of the utility function and the choice of equity measure are critical for achieving proper mapping. For instance, minimizing the Range of utilities often fails to achieve proper mapping because it focuses only on narrowing the gap between the highest and lowest utilities, while ignoring intermediate values. As a result, some agents may experience poor outcomes, and the resulting solutions may not be Pareto-optimal in the full space unless the utility function satisfies certain properties. Therefore, the use of Range as an equity measure should generally be avoided unless the utility function is explicitly designed to preserve proper mapping. In the following, we introduce a property of utility functions that, if satisfied, enables all equity measures (including the Range) to achieve proper mapping.
Theorem 1
If $u (I)$ is an increasing function of $\sum_{i \in I} q_{i}$ for all $I \subseteq N$ , then the proper mapping is guaranteed for any chosen equity measure.
Proof.
We aim to prove the assertion through a contradiction. Consider an arbitrary equity measure, and suppose $u (I)$ is an increasing function of $\sum_{i \in I} q_{i}$ , with $I \subseteq N$ , but this mapping is not a proper one. In such a scenario, there must exist a Pareto-optimal solution $s \in S$ in the bi-objective optimization problem, which is not Pareto-optimal in the full-dimensional criterion space counterpart. This suggests the existence of a feasible solution $s^{'} \in S$ that dominates $s$ in the full-dimensional criterion space. Formally, we require $C (s^{'}) \leq C (s)$ and $u (s_{m}^{'}) \geq u (s_{m})$ for all $m \in M$ , with at least one strict inequality.

Now, we consider two cases. First, suppose there exists $m \in M$ with $u (s_{m}^{'}) > u (s_{m})$ . In this case, it immediately follows that $\sum_{i \in s_{m}^{'}} q_{i} > \sum_{i \in s_{m}} q_{i}$ by assumptions. However, since we know that $\sum_{i \in N} q_{i}$ is a constant, there must exist $l \in M$ such that $\sum_{i \in s_{l}^{'}} q_{i} < \sum_{i \in s_{l}} q_{i}$ . Based on our assumptions, this implies $u (s_{l}^{'}) < u (s_{l})$ , leading to a contradiction as $s^{'}$ cannot dominate $s$ . Alternatively, if $u (s_{m}^{'}) = u (s_{m})$ for all $m \in M$ , which implies
$E M (s^{'}) = E M (s),$
then for $s^{'}$ to dominate $s$ , we must have $C (s^{'}) < C (s)$ . However, this directly contradicts the Pareto-optimality of $s$ in the bi-objective version.

Theorem 1 suggests that as long as the utility function of agents is an increasing function of the summation of item sizes allocated to them, then none of the equity measures can be considered unsuitable in theory. Therefore, the task of selecting the most appropriate equity measure(s) becomes primarily a computational challenge. Later in Section 6, we outline a computational method for this selection process. The underlying concept of our proposed approach lies in solving a bi-objective optimization problem for each equity measure. If the Pareto-optimal solutions of one equity measure closely approximate those of the others, we designate it as the most reliable equity measure. This is because by selecting that equity measure, we obtain solutions that are good across a variety of equity measures.
4. Problem Statement

In this section, we provide a detailed description of the problem under study, the DZWP. Specifically, we study the context of a company (i.e., Entity G) operating from a single depot, aiming to conduct last-mile deliveries (i.e., Indivisible Goods of varying sizes) using crowdshippers (i.e., Agents). Each day is divided into equal-length time intervals (e.g., every hour) that online shoppers can choose among for their delivery to take place. Each interval is available to the online shoppers by a fixed cutoff time, after which it cannot be selected for delivery. The group of orders received for delivery within a time interval is referred to as a “wave”. Moreover, delivery orders within a wave are further broken down based on their delivery locations. Specifically, we assume that the geographical area covered by the depot is divided into predefined delivery zones to categorize orders based on their delivery locations. The compensation offered for deliveries made at different delivery zones can vary based on their characteristics (further discussed in Section 4.2).

To streamline our discussion, we call each delivery zone within a wave a “zone-wave”, presuming the company devises a plan for each zone-wave independently. Once the orders for a given zone-wave are identified, the decision maker faces the task of securing sufficient transportation capacity to deliver the orders of each interval. As crowdshippers sign up to provide service, they choose the zone-wave they wish to serve. Subsequently, the company adopts a first-come, first-served approach for selecting crowdshippers for each zone-wave. Note that this assumption is both practical and fair. It is practical because from a crowdshipper’s perspective, once they see a company’s request for a given zone-wave in their platforms, they expect prompt accommodation rather than being put on hold for an extensive time to know if they are selected. It is fair because it prevents the company from discriminating against crowdshippers. We later, in Section 7.2.2, demonstrate that the company should prioritize selecting the minimum number of crowdshippers required to fulfill orders for each zone-wave, as if done otherwise, it could negatively affect both low- and high-performing crowdshippers. Next, the company endeavors to allocate jobs (and corresponding routes) to the selected crowdshippers by solving a variant of VRP incorporating two conflicting objectives: cost and equity.

Therefore, to address the problem of each zone-wave, as outlined in Figure 1, the company solves two optimization problems: The crowdshipper selection problem and the workload allocation problem. The former is a single-objective optimization problem aimed at determining the minimum number of crowdshippers required to fulfill all orders within a targeted zone-wave. The latter focuses on workload allocation while considering the trade-offs between cost and equity objectives, ultimately selecting a solution that achieves a desirable balance between these two objectives. The crowdshipper selection problem can be viewed as a variation of the classical bin-packing problem, dynamically solved as crowdshippers join the system and enroll for a targeted zone-wave following a first-come, first-served rule. Due to its straightforward nature, we do not explore its details in this paper; interested readers can refer to E-Companion EC.1 for the mathematical formulation and additional information. Therefore, the remainder of this paper focuses primarily on the workload allocation problem, which constitutes the main contribution of our study. We detail our proposed methodology for analyzing trade-offs in workload allocation within a targeted zone-wave and present an approach to select solutions that balance these conflicting objectives.

Figure 1.

An outline of the dispatch zone-wave problem (DZWP) optimization workflow.

In light of the above, the remainder of this section is organized as follows. We begin by outlining the key assumptions of the workload allocation problem for each zone-wave and its distinguishing features compared to existing capacitated vehicle routing variants. We then define the objective functions (cost and equity) mathematically. As shown in Figure 1, various equity measures are considered, derived from the theoretical foundations discussed in Section 3. Finally, we present a mathematical formulation for each zone-wave.

4.1. Key Assumptions and Notation

In order to formulate the workload allocation problem for each zone-wave, several key assumptions have been made. Specifically, the problem is framed as a discrete VRP in a crowdshipping context, focusing on a single decision epoch within a dynamic operational setting. Each zone contains a single depot that serves as the origin point for all deliveries within that zone. Unlike traditional VRP variants with homogeneous vehicles, our problem involves a limited fleet of vehicles with varying loading capacities.

Decision-making is considered for a specific zone-wave, where participating drivers have signed up in advance and are aware of the depot location. This means a crowdshipper can only sign up in advance for a single zone in any given wave. The fleet of vehicles is selected on a first-come, first-served basis from a queue of heterogeneous vehicles, ensuring that the minimum number of vehicles is assigned to fulfill all demands within a given zone-wave. We denote the set of crowdshippers selected for the workload allocation problem in the zone-wave under consideration by $M$ , and the vehicle capacity of each crowdshipper $m \in M$ by $Q_{m}$ . Customer demands arriving in a zone-wave are fulfilled within the same wave without any order deferment, assuming a sufficient fleet of vehicles is available to meet all demand. Later, in Section 7.2.3, we introduce methods to mitigate potential vehicle shortages and analyze their impact.

Each selected vehicle performs exactly one delivery trip within a zone-wave, and crowdshippers are free to work with other platforms outside their delivery commitments. Specifically, each crowdshipper begins their route from the depot and serves assigned customers based on prescribed routing. Unlike traditional VRP, crowdshippers are not required to return to the depot after completing their tasks. In other words, the problem is modeled as an open-route VRP, where couriers complete their routes upon delivering to the last customer. If a courier wishes to participate in a subsequent wave, they must sign up again for a zone of interest. In that case, since each zone-wave is treated independently, being selected in one wave does not guarantee selection in another. The company evaluates each zone-wave separately, with a focus on improving equity within that specific context.

Couriers are not compensated for traveling to the depot at the start of any zone-wave. Consequently, the optimization process assumes that all couriers begin each decision epoch at the depot. Additionally, crowdshippers are expected to follow the customer visit sequence recommended by the system, with all mileage costs along this route fully covered by the company. However, any additional mileage resulting from deviations will not be reimbursed. Finally, a major distinction of this study from the traditional VRP setting is that it considers not only cost minimization but also the improvement of equity. The equity objective function can be highly non-linear depending on the underlying equity measure. Let $x$ represent any feasible solution to the problem at hand. Similar to Section 3, but with a slight abuse of notation, we use $E M (x)$ to denote the equity objective function and $C (x)$ to represent the cost under solution $x$ . Note that the notational abuse arises because, in our problem, the solution $x$ does not purely reflect workload allocation; it also includes routing decisions (i.e., which arcs are traversed by each crowdshipper). In the rest of this section, to formulate the problem, we use the notation described above along with a few additional terms, which are summarized in Table 1.

Table 1.
Notation used for formulating the workload allocation problem.

Notation Description

Sets:

$N$ Set of all customers with a cardinality of $N$

$\bar{N}$ Set of all vertices, $N \cup {0}$ where $0$ is the depot

$A$ Set of all arcs

$M$ Set of all selected crowdshippers with a cardinality of $M$

Parameters:

$c_{i j}$ Cost of traversing the arc $(i, j)$

$q_{i}$ Order size of customer $i \in N$

$p_{i}$ Delivery compensation for fulfilling the order of customer $i \in N$

$Q_{m}$ Vehicle capacity of crowdshipper $m \in M$

$Q$ Maximum vehicle capacity among crowdshippers selected, that is, $max (Q_{m})_{m \in M}$

Variables:

$x_{i j m}$ Binary decision variable which equals one if crowdshipper $m \in M$ traverses the arc from $i$ to $j$ , 0 otherwise

$w_{i}$ Continuous decision variable designed for subtour elimination constraints, where $i \in N$

Notation	Description
Sets:
$N$	Set of all customers with a cardinality of $N$
$\bar{N}$	Set of all vertices, $N \cup {0}$ where $0$ is the depot
$A$	Set of all arcs
$M$	Set of all selected crowdshippers with a cardinality of $M$
Parameters:
$c_{i j}$	Cost of traversing the arc $(i, j)$
$q_{i}$	Order size of customer $i \in N$
$p_{i}$	Delivery compensation for fulfilling the order of customer $i \in N$
$Q_{m}$	Vehicle capacity of crowdshipper $m \in M$
$Q$	Maximum vehicle capacity among crowdshippers selected, that is, $max (Q_{m})_{m \in M}$
Variables:
$x_{i j m}$	Binary decision variable which equals one if crowdshipper $m \in M$ traverses the arc from $i$ to $j$ , 0 otherwise
$w_{i}$	Continuous decision variable designed for subtour elimination constraints, where $i \in N$

4.2. Cost Objective Function

The foundation of our cost objective lies in the company’s compensation policy, which consists of two components: delivery compensation and mileage compensation. Delivery compensation depends on the delivery zone and the number of deliveries. Since the total number of orders is fixed within any zone-wave, the total delivery compensation is also fixed, given by $β \sum_{i \in N} q_{i}$ , where $β \in R_{+}$ denotes the per-unit compensation, determined by factors such as demand volume, traffic, safety, and proximity to the depot. Mileage compensation is dynamic and reimburses routing costs. We assume full mileage compensation based on two axioms: (1) Crowdshippers should not benefit by deviating from assigned routes, so routes must be optimal; (2) The company alone bears the equity cost—crowdshippers should not share this burden.

To meet the first axiom, the company must assign optimal (i.e., shortest) routes and avoid reimbursing more than the actual routing cost, which would encourage manipulation. To satisfy the second, it cannot reimburse less, as this would shift equity costs to the crowdshippers. Hence, full reimbursement is required to meet both conditions. Therefore, within each zone-wave, the collective profit of crowdshippers is fixed, equal to the delivery compensation. This is akin to a fixed-size cake that must be fairly divided based on workload. As delivery compensation is constant, minimizing total routing cost becomes the company’s cost objective:

C (x) := \sum_{i \in \bar{N}} \sum_{j \in N} \sum_{m \in M} c_{i j} x_{i j m},

where

c_{i j}

is the cost of traveling from node

i

to node

j

, and

x_{i j m}

indicates whether arc

(i, j)

is used by crowdshipper

m

4.3. Equity Objective Function

The equity objective function builds on two key concepts (Matl et al., 2018; Xinying Chen and Hooker, 2023):

Equity Metric: Interpreted as the utility function of agents (i.e., crowdshippers), it represents the quantity we seek to equalize (e.g., profits, tour lengths, among others).

Equity Measure: A mathematical function (e.g., range, standard deviation, among others) used to evaluate the level of achieved equity. It acts as the criterion for assessing fairness.

Together, these define the equity objective function to be optimized. However, no universal form exists for this function. In our setting, the challenge lies in selecting both the equity metric and measure. This difficulty arises from heterogeneous crowdshipper characteristics, such as varying vehicle capacities. Although profits may seem like a natural equity metric, since crowdshippers aim to maximize earnings, ignoring their investments (i.e., contributed resources) risks bias and misleading conclusions. Crowdshippers behave as rational investors, expecting higher returns with greater input. Hence, the equity metric should reflect their level of investment. In our model, crowdshippers provide two resources: Time and vehicle capacity. Because zone-waves are analyzed independently and mileage costs are reimbursed, time is effectively standardized across crowdshippers within a zone-wave. Thus, the only remaining differentiating resource is vehicle capacity, which directly affects their profit and must be accounted for in the equity metric. With this consideration, we next introduce our proposed equity metric or utility function for each crowdshipper, termed “adjusted profits.” Specifically, we define

N_{m} (x) := {j \in N : \exists i \in \bar{N} such that x_{i j m} = 1}

as the set of customers to be serviced by crowdshipper

m \in M

under solution

x

. Utilizing this notation, the adjusted profit (i.e., the utility function) for crowdshipper

m \in M

under solution

x

, denoted by

u_{m} (x)

, is expressed as:

u_{m} (x) = (\frac{\sum_{i \in N_{m} (x)} p_{i}}{Q_{m}}) (\frac{\sum_{i \in N_{m} (x)} q_{i}}{Q_{m}}) .

(1)

Note that the numerator of the first term in the equation represents the total profit gained by crowdshipper

m

, while the numerator of the second term corresponds to the capacity of the vehicle used for delivery purposes. The denominator in both terms reflects the total initial capacity of the vehicle. With this in mind, Equation (1) captures the crowdshipper’s preference to maximize profits by fully utilizing their vehicle capacity, while also prioritizing the delivery of small yet highly profitable items. The first term reflects the former preference, while the second term encapsulates the latter. It is also important to note that, based on our discussion in Section 4.2, since mileage costs are fully covered by the company, we have:

\sum_{i \in N_{m} (x)} p_{i} = β \sum_{i \in N_{m} (x)} q_{i} .

Consequently, an additional feature of the proposed equity metric is that in Equation (1), both the numerator and denominator are consistent in form, that is, both are quadratic expressions:

u_{m} (x) = β {(\frac{\sum_{i \in N_{m} (x)} q_{i}}{Q_{m}})}^{2},

and squaring inherently makes the equity objective function sensitive to outliers, akin to the concept of mean squared errors in statistical methods. Additionally, the quadratic form gives rise to the the following important observation.

Observation 1

The proposed utility function for crowdshipper $m \in M$ is an increasing function of $\sum_{i \in N_{m} (x)} q_{i}$ .

Now that we have established the equity metric, the primary question arises: What form should the equity measure take? The following result can be directly proved from Theorem 1 and Observation 1.

Proposition 1

Any equity measure offers proper mapping when the proposed utility function is employed for the crowdshippers.

The aforementioned proposition indicates that any equity measure available in the literature is theoretically suitable for the context of our research. Therefore, we examine several equity measures in this study, treating the selection of the most reliable measures as a key research question. In Section 6, we will present an approach to numerically compare and select the most reliable equity measure(s) for the problem under study. Specifically, within the context of our research, we examine five well-known equity measures, that is, range, mean absolute deviation (MAD), standard deviation, coefficient of variation, and Gini coefficient. While there are other measures, we do not include them in this study as they perform similarly to those we have already considered. For instance, Min-Max, a popular method, shares similarities with the range. We choose to use range since our research emphasizes distributional justice over the condition of the worst-paid-off crowdshipper. Building upon this foundation, we define

E M (x) \in {Range (x), MAD (x), SD (x), CV (x), GINI (x)},

and the mathematical representations of these equity measures are presented in E-Companion EC.2.

4.4. Mathematical Formulation

With the objective functions defined, we can now present the complete formulation of the workload allocation problem. Based on the notation outlined in Table 1, the workload allocation problem for any zone-wave can be formulated as follows:

\begin{aligned} min & {C (x), E M (x)} \end{aligned}

(2a)

\begin{aligned} s.t. & \sum_{i \in \bar{N}} \sum_{m \in M} x_{i j m} = 1 & \forall j \in N \end{aligned}

(2b)

\begin{aligned} \sum_{i \in \bar{N}} x_{i j m} = \sum_{i \in \bar{N}} x_{j i m} & \forall j \in \bar{N}; m \in M \end{aligned}

(2c)

\begin{aligned} \sum_{j \in N} x_{0 j m} = 1 & \forall m \in M \end{aligned}

(2d)

\begin{aligned} \sum_{m \in M} x_{i i m} = 0 & \forall i \in \bar{N} \end{aligned}

(2e)

\begin{aligned} q_{i} \leq w_{i} \leq \sum_{m \in M} Q_{m} \sum_{j \in \bar{N}} x_{j i m} & \forall i \in N \end{aligned}

(2f)

\begin{aligned} w_{i} - w_{j} + Q \sum_{m \in M} x_{i j m} \leq Q - q_{j} & \forall i, j \in N; i \neq j \end{aligned}

(2g)

\begin{aligned} w_{i} \in R & \forall i \in N \end{aligned}

(2h)

\begin{aligned} x_{i j m} \in {0, 1} & \forall i, j \in \bar{N}; m \in M \end{aligned}

(2i)

The first objective function aims to minimize the total routing cost incurred by all crowdshippers, and the second objective function retains its abstraction, allowing for the integration of any equity measure to evaluate the disparity in adjusted profits. Constraint (2b) ensures the fulfillment of all customer demands. It is important to note that while constraints (2c) enforce flow conservation at each node, the first objective function solely minimizes traversal costs for active routes. Hence, the cost of returning from the last served location to the depot, although considered in the constraint set, is omitted from the objective function. Crowdshippers depart from the depot according to constraints (2d), and self-visits are prohibited by constraints (2e). Subtour elimination constraints (2f)–(2g), also known as Miller-Tucker-Zemlin constraints (Miller et al., 1960), ensure the connectivity of crowdshipper routes based on heterogeneous load capacities. Variables $w_{i}$ are assigned real values as per (2h), yet their range is bounded by constraints (2f). Finally, the integrality conditions of routing decision variables are enforced by constraints (2i). It is important to note that we have not explicitly introduced capacity constraints as they are inherently captured by constraints (2f)–(2g). This is because each customer must be satisfied by a crowdshipper, implying that customer $i \in N$ must be in the sequence of customers to be serviced by a crowdshipper. Thus, variable $w_{i}$ can be interpreted as the capacity occupied by customer $i$ and its preceding customers in that crowdshipper’s route.

5. Proposed Solution Approach

In this section, we present our proposed approach for solving this problem. The proposed approach consists of two components: A bi-objective optimization process aimed at generating the nondominated frontier, and a practical, generic method for selecting a solution from this frontier that balances cost and equity in alignment with the company’s needs.

5.1. Biobjective Optimization Process

Solving the workload allocation problem (2) is challenging. This is evident as even classical VRPs with a single objective pose considerable difficulty. In our research, we confront a variant of the VRP that is notably more complex due to its bi-objective nature, particularly when one objective is nonlinear (partly due to the nonlinear nature of the adjusted profits). Consequently, employing exact multi-objective optimization methods for solving the workload allocation problem becomes impractical. Therefore, we propose a custom-built heuristic approach tailored to address the problem, irrespective of the equity measure employed. Our proposed method quickly approximates the nondominated frontier for any instance of the workload allocation problem. It combines the core principles of the well-known NSGA-II(Deb et al., 2002) with the concepts of Multi-Directional Local Search (MDLS; Tricoire, 2012), offering an effective solution approach for handling the complexities of the workload allocation problem.

We refer to our proposed solution method as $HybridNsgaII$ . At the heart of our proposed method is NSGA-II, known for its computational efficiency in approximating optimal trade-offs between conflicting objectives by ranking and selecting solutions based on their dominance relationship (Gullotta et al., 2021; Jozefowiez et al., 2005; Srivastava et al., 2021; Zhou et al., 2018). Specifically, during each iteration, NSGA-II evaluates the rank of each candidate solution using a non-dominated sorting technique. This technique categorizes the solutions into different approximations of the nondominated frontiers based on their dominance relationship with others. This implies that the first approximation has rank one and is better than the second, and the second is better than the third one, and so forth. To generate a new population of offspring solutions, NSGA-II then performs selection, crossover, and mutation operations. Additionally, to ensure well-distributed objective function values or diversity in the population of solutions, NSGA-II uses a crowding distance metric.

To effectively solve the workload allocation problem, we either customize some of the main operations or add some new operations to NSGA-II. Customized operations include:

Solution Representation: We employ a sequence-based solution representation for $N$ customers and $M$ crowdshippers by a two-part chromosome: A giant tour gene of length $N$ specifying the order of customer visits, and a breakpoint gene of length $M - 1$ indicating where the giant tour is split into crowdshipper routes. The breakpoint gene contains increasing positive integers less than $N$ , ensuring each crowdshipper serves at least one customer.

Initialization: The initial population is generated using both targeted and random methods. Targeted methods such as Clarke-Wright savings, sweep, and nearest neighbor heuristics generate good initial solutions and accelerate convergence. Some solutions are constructed by heuristically balancing the equity metric among vehicles, while random permutations of customers and breakpoints are incorporated to maintain sufficient population diversity.

GA Operations: Parent selection, a critical step in the GA, is implemented using binary tournament selection based on fitness scores. To generate offspring, we employ effective crossover operations specifically designed for sequence-based chromosome representations. Additionally, mutations are introduced with a predetermined probability and tournament selection.

Newly integrated operations include:

Local Search: We attempt to refine the set of best solutions by employing MDLS at each iteration. This process prioritizes cost and equity separately, leveraging Large Neighborhood Search and workload reassignment strategies to achieve targeted improvements.

Infeasible Population Management: Due to vehicle capacity constraints, some solutions generated during the search may be infeasible. Instead of discarding these solutions, we either attempt to repair them or preserve them in a separate population. This diversifies the total population and aids in escaping local optima. To ensure proper integration into the selection process, the objective values of infeasible solutions are penalized based on the extent of capacity violations, weighted by an adaptive penalty factor.

Diversity Score: Our method enhances population diversity using a diversity score and alternately focuses on cost, equity, and solution rank throughout the search process to ensure a balanced exploration of the solution space.

Additional information about $HybridNsgaII$ is available in E-Companion EC.3 for interested readers. $HybridNsgaII$ serves as a key tool for addressing the research questions in this study. Therefore, in our computational analysis, we prioritize demonstrating its reliability before utilizing it to tackle the research questions. To show its reliability, we borrow two main measures from the literature of multi-objective optimization:

Cardinality: The Cardinality, sometimes referred to as ‘Card.’ in this study, shows the number of approximate nondominated points found. We would like to highlight that, unlike exact methods, in heuristic solution approaches, the number of approximate nondominated points does not necessarily increase over time, as it is possible for a point to be found in one iteration that dominates a subset of the points in the previous iteration. However, one can hope that the Cardinality measure gets stabilized over time.

Hypervolume (HV): This is a widely used performance indicator in multi-objective optimization (Pal and Charkhgard, 2019), and we sometimes refer to it as ‘HV’ in this study. It is a single value reflecting the area of the criterion space that is dominated by the approximate nondominated frontier found and is trapped between the approximated nondominated frontier and a given reference point. Larger values for the HV indicate better approximations. To ensure this premise, it is critical for the reference point to be selected carefully and remain fixed when comparing multiple approximations. In the case of our study, because we are minimizing both objective functions, the correct selection of the reference point means each component of the reference point should be larger than or equal to the same component among all approximate nondominated points across all comparisons.

5.2. Balancing Objectives: Equity and Cost

As previously noted, the equity and cost objective functions are two antithetical objectives, making it unlikely for a solution to simultaneously optimize both. Consequently, in tackling such bi-objective optimization problems, the primary aim for the company is to identify the trade-offs between the objectives, determining what is known as the “nondominated frontier.” With the nondominated frontier calculated, the company faces the task of selecting a nondominated point (and its corresponding Pareto-optimal solution) that effectively balances the equity and cost objectives from its perspective. This selection process is closely tied to the specific problem at hand and is heavily influenced by the application’s context and underlying research philosophy. In our research context, the guiding philosophy centers on favoring an equity-driven workload allocation, that is, balancing adjusted profits, within a free-market environment, with a focus on benefiting for-profit companies. Benefits are attained when (1) the cost of equity remains sufficiently low to preserve the company’s competitive edge, and (2) the outcomes driven by equity are uniformly favorable for all crowdshippers to the highest degree feasible. In essence, the cost of equity borne by the company should not yield undesirable consequences, resulting in uniformly unfavorable outcomes for all crowdshippers.

The latter condition is addressed by the proposed compensation policy outlined earlier in Section 4.2, which guarantees that crowdshippers will not bear the burden of equity costs. The former condition is pivotal in balancing the cost and equity objective functions. This means that the company can only afford to sacrifice a small fraction of its least-achievable costs to achieve a higher workload allocation for crowdshippers. Consequently, the company selects a point from the nondominated frontier with a capped percentage increase in costs, denoted by $γ$ (e.g., 5%), and then chooses the closest feasible nondominated point to it—effectively selecting the nondominated point that maximizes the equity objective function while adhering to the cost cap. As an aside, we sometimes refer to $γ$ as the equity budget of the company in this paper.

6. Identifying the Most Reliable Equity Measure(s)

In this section, we explore the process of identifying the most reliable equity measures for solving the workload allocation problem within a given zone-wave. As per Proposition 1, all equity measures considered in this study offer proper mappings. Therefore, selecting the most reliable measure(s) is primarily an empirical question, best addressed through computational analysis. Thus, we propose a method for this purpose in this section. The approach proposed here conducts a comprehensive analysis over the entire set of approximate Pareto-optimal solutions found using each measure. Later in Section 7.1.3 and E-Companion EC.5, we highlight other alternative techniques that rely on analyzing a subset of Pareto-optimal solutions found using each measure. However, such approaches raise some major questions that are not trivial how they can be addressed. We posit that a measure identified as the most reliable using full-set analysis is likely to perform well under subset-based analysis but not necessarily vice versa. In Section 7.1.3 and E-Companion EC.5, we provide numerical evidence supporting this claim. With this in mind, we next present our proposed selection approach.

Our proposed approach starts with generating approximate nondominated frontiers for each equity measure using $HybridNsgaII$ for each instance. Given that we are investigating five equity measures in this study, we produce five distinct approximate nondominated frontiers for each instance. The proposed approach involves mapping the approximate Pareto-optimal solutions identified under each equity measure into the bi-objective space of every other equity measure. For each pair of equity measures, this mapping can be expressed as follows:

\begin{aligned} Map ((Cost, θ^{I}) \to (Cost, θ^{I I})), \\ \forall θ^{I}, θ^{I I} \in Θ := {Range, MAD, SD, CV, GINI} . \end{aligned}

The mapping process is straightforward. For instance, let $θ^{I}$ be Range and $θ^{I I}$ be MAD. This indicates a desire to map the approximate Pareto-optimal solutions obtained by $HybridNsgaII$ for Cost and Range into the space of Cost and MAD. To achieve this, we simply compute the values of Cost and MAD for each solution. While the Cost values remain constant as in the Cost and Range space, this mapping generates several feasible points in the Cost and MAD space. However, some of these feasible points may dominate others in this new space, necessitating the removal of dominated points. Through a removal process, an approximate nondominated frontier is derived for the Cost and MAD space using the solutions originally found for the Cost and Range space.

Considering the exploration of five distinct equity measures in $Θ$ , for every equity measure $θ^{I I} \in Θ$ , we obtain five different approximated nondominated frontiers through this mapping procedure, namely

\begin{aligned} Approximation (θ^{I}, θ^{I I}) : Map ((Cost, θ^{I}) & \to (Cost, θ^{I I})), \\ \forall θ^{I} \in Θ . \end{aligned}

Note that among these approximations, $Approximation (θ^{I I}, θ^{I I})$ , where the approximate Pareto-optimal solutions obtained by $θ^{I I}$ is re-mapped into the space of $θ^{I I}$ , is expected to be the best approximation of the nondominated frontier for the space of Cost and $θ^{I I}$ . This is because such an approximation implies that $HybridNsgaII$ has directly generated an approximate nondominated frontier for the space of Cost and $θ^{I I}$ , rendering the mapping redundant. However, such certainty is not guaranteed given that $HybridNsgaII$ is a heuristic approach. Hence, we regard $Approximation (θ^{I I}, θ^{I I})$ as a benchmark for assessing the quality of all $Approximation (θ^{I}, θ^{I I})$ , $\forall θ^{I} \in Θ ∖ θ^{I I}$ . By comparing the HVs of all five approximations to that of $Approximation (θ^{I I}, θ^{I I})$ , we can compute the HV gaps. Naturally, the HV gap will be 0% for $Approximation (θ^{I I}, θ^{I I})$ . However, for each $θ^{I} \in Θ ∖ θ^{I I}$ , the HV gap corresponding to $Approximation (θ^{I}, θ^{I I})$ may vary positively or negatively. Smaller HV gaps across all approximations denote better performance.

After computing all five different approximations for each equity measure, we can now readily identify the most reliable measure(s). An equity measure earns this distinction if it not only produces the best-known approximation for itself but also comes close to achieving the same level of accuracy for other measures through mapping. We gauge this performance using the HV gap. Clearly, determining the best measure through this method requires a comprehensive computational study across various instances to assess the average performance of each equity measure and its corresponding mappings.

7. Computational Study

In this section, we conduct an extensive computational study using large-sized instances to not only showcase the performance of $HybridNsgaII$ but also to address the main research questions delineated in Section 1. All simulation and optimization tools necessary for conducting the experiments in this paper are implemented and executed in Julia 1.11.5. The experiments are carried out on a MacBook Pro featuring an Apple M1 Chip with 16GB RAM, running on macOS Sonoma 14.1.2. To ensure the reliability of the results reported by $HybridNsgaII$ , we solve each instance 10 times and consider the average performance across these runs as the representative outcome for that instance. Consequently, when we state that a particular result is averaged over 10 instances for $HybridNsgaII$ , it implies that a total of 100 independent runs of $HybridNsgaII$ were conducted to compute that value.

We conduct two series of experiments, each based on a distinct set of instances. The first series, reported in Section 7.1, focuses on operations at a single zone-wave level, exclusively addressing the workload allocation problem without focusing on the process of crowdshipper selection, and is designed to answer research questions of a theoretical and computational nature. In contrast, the second series of experiments, reported in Section 7.2, investigates the end-to-end operations of a company over a typical working horizon. These experiments involve multiple zones and waves, jointly addressing both the crowdshipper selection and workload allocation problems on a recurring basis. Therefore, the second part aims to answer research questions grounded in real-world operational settings.

Specifically, the first series of experiments is designed primarily to highlight the performance of the proposed workload allocation approach and to identify the most reliable equity measures. To ensure the generality of the results, we focus on a relatively large set of randomly generated instances, as this phase centers on pure performance analysis through computational experiments. Specifically, 14 different size classes of randomly generated instances are considered, with 10 instances in each size class, yielding a diverse experimental set. In contrast, the second part of the experiments mainly addresses real-world operational questions introduced in the Introduction of this paper, and demonstrates how the courier selection and workload allocation model operates in a dynamic setting where orders arrive over time. For this series of experiments, we utilize the 2021 Amazon Last Mile Routing Research Challenge dataset (Merchán et al., 2024) to simulate a planning horizon consisting of 100 waves of customer orders spanning five geographically distinct zones in the Los Angeles, California region. Each wave constitutes a synthetic instance derived from realistic data, capturing the spatial and demand characteristics typical of LMD operations, in which 20% of drivers are high-capacity and 80% are low-capacity. Further details about the instance generation and adoption process are provided in E-Companion EC.4.

7.1. Part I: Evaluation of the Algorithmic Performance

The first series of experiments consists of several components: Analyzing the overall performance of the proposed heuristic, validating its effectiveness, and ultimately identifying the most reliable equity measure(s) based on the proposed approach. Identifying the most reliable measure is particularly important, as it will be fixed and consistently used in the second series of experiments.

7.1.1. Overall Performance Analysis

In this subsection, we focus on assessing the overall performance of the $HybridNsgaII$ across various equity measures and its evolution over time. To accomplish this, we generated 80 random instances using our random instance generator across 8 classes of different sizes, each containing up to 100 nodes and 20 crowdshippers. The solution times of $HybridNsgaII$ under different equity measures are detailed in Table 2, with the values in each row representing averages across 10 instances. We observe from the table that the proposed approach can generate an approximate nondominated frontier within a matter of seconds even for large-sized instances. Unsurprisingly, we observe a relationship between the solution time of $HybridNsgaII$ and the complexity of the chosen equity measure. The equity measure with the least complexity, Range, exhibits an average solution time of 8.88 seconds, while the most complex measure, GINI, necessitates 16.91 seconds on average to solve an instance.

Table 2.
Average run time of $HybridNsgaII$ across different equity measures.

Average CPU time (s)

$N$ $M$ Range MAD SD CV GINI

25 3 2.31 2.72 2.90 2.27 2.27

25 5 2.67 2.47 2.61 3.37 3.16

50 5 4.55 5.00 5.52 5.72 5.21

50 10 6.57 7.93 8.12 8.04 8.79

75 8 9.33 12.36 12.78 12.73 13.88

75 15 11.54 20.04 21.12 23.95 23.68

100 10 12.83 24.16 25.21 24.06 28.83

100 20 21.20 45.49 45.48 45.90 49.48

Average 8.88 15.02 15.47 15.76 16.91

		Average CPU time (s)
25	3	2.31	2.72	2.90	2.27	2.27
25	5	2.67	2.47	2.61	3.37	3.16
50	5	4.55	5.00	5.52	5.72	5.21
50	10	6.57	7.93	8.12	8.04	8.79
75	8	9.33	12.36	12.78	12.73	13.88
75	15	11.54	20.04	21.12	23.95	23.68
100	10	12.83	24.16	25.21	24.06	28.83
100	20	21.20	45.49	45.48	45.90	49.48
Average	8.88	15.02	15.47	15.76	16.91

MAD: mean absolute deviation; CV: coefficient of variation; SD: standard deviation; NSGA: non-dominated sorting genetic algorithm.

The progression of $HybridNsgaII$ in terms of both HV and Cardinality at each iteration is illustrated in Figures 2 and 3. Specifically, Figure 2 focuses solely on the GINI measure, depicting the progress of two instances simultaneously. Both instances consists of $N = 100$ deliveries, with one having $M = 10$ crowdshippers and the other $M = 20$ . The former instance is further depicted in Figure 3 across all equity measures considered. From these figures, it is evident that almost after 400 iterations, HV has nearly converged, indicating that high-quality approximations have been generated within only 10% of the total runtime of the $HybridNsgaII$ . This implies that early termination can still result in high-quality approximations by $HybridNsgaII$ , which can be useful for solving instances significantly larger than those used in this study.

Figure 2.

Progression of $HybridNsgaII$ results until termination under GINI for two instances with $N = 100$ . (a) Hypervolume, (b) Cardinality. NSGA: non-dominated sorting genetic algorithm.

Figure 3.

Progression of $HybridNsgaII$ results under various equity measures for an instance with $N = 100$ and $M = 10$ . (a) Hypervolume, (b) Cardinality. NSGA: non-dominated sorting genetic algorithm.

Regarding Cardinality, natural fluctuations are observed, indicating continuous improvement of its best-known approximate nondominated frontier over time, facilitated by various enhancement techniques such as dynamic weight adjustments and periodic repopulation of the solution set. It is noteworthy that a decrease in the nondominated frontier can occur when a newly found feasible point dominates a subset of points in the best-known approximate nondominated frontier, resulting in a reduction in Cardinality as dominated points are removed. Furthermore, an interesting but expected observation is that more complex equity measures lead to larger Cardinality values. This implies that in practice, companies may struggle to enhance equity when employing simpler measures like Range. The challenge lies in selecting a point from the approximate nondominated frontier that balances equity without significantly impacting costs. Due to the limited options available, attributed to the small Cardinality under these simpler equity measures, companies may find it difficult to achieve substantial improvements.

7.1.2. Validating the Proposed Heuristic Approach

In this subsection, our goal is to present numerical evidence demonstrating the high-quality performance of our proposed heuristic, that is, $HybridNsgaII$ , in solving the workload allocation problem. This demonstration inherently supports the claim that $HybridNsgaII$ is a suitable tool for addressing our research questions. We provide two types of analysis to showcase the performance of $HybridNsgaII$ : bi-objective analysis and single-objective analysis. The former focuses on evaluating the nondominated frontier as a whole, while the latter examines the quality of the endpoints of the nondominated frontier —specifically, solutions that are purely equity-driven or purely cost-driven.

For the bi-objective analysis, we compare the performance of the proposed heuristics with an off-the-shelf exact solver. In light of the complex nature of the VRP variation we are examining, exact bi-objective optimization solvers face significant challenges in solving instances unless the equity measure chosen is Range from our list of options. This preference stems from the inherently linear nature of Range, coupled with the fact that existing off-the-shelf multi-objective optimization tools are primarily designed for integer linear programs. Consequently, our comparison with an exact off-the-shelf solver is focused solely on Range. However, it is worth noting that despite the Range’s linear nature, the range of adjusted profits is non-linear due to the underlying non-linearity of adjusted profits (refer to Equation (1)). To address this, we have decided to employ the range of vehicle utilization values (specifically, $\frac{\sum_{i \in N_{m} (x)} q_{i}}{Q_{m}}$ ) for our comparison. For this purpose, we have employed the Julia package MultiObjectiveAlgorithms, leveraging the $ϵ$ -constraint method and Gurobi Optimizer 11.0.

We did not impose any time limit for the Exact Solver. Consequently, six instances took up to 75 minutes to be solved using the exact solver, while the rest were solved within 30 minutes. The results of our comparison across various instance sizes are presented in Table 3, with each row displaying averages from 10 randomly generated instances, totaling 60 instances overall. In the final column of the table, we showcase the gap between the HV reported by our proposed approach and the exact method for the fixed reference point of $(700, 1)$ .

Table 3.
Comparing $HybridNsgaII$ to an exact solver using range as the equity measure.

Exact solver $HybridNsgaII$

Time Time HV gap

$N$ $M$ Card. HV (s) Card. HV (s) (%)

8 2 4.8 475.32 2.05 4.8 475.32 0.70 0.00

8 3 8.7 468.12 4.26 8.7 468.12 0.99 0.00

10 2 7.7 462.42 23.79 7.7 462.42 0.90 0.00

10 3 10.8 452.03 84.29 10.4 451.80 1.37 0.05

12 2 6.3 439.78 969.98 6.2 439.46 1.14 0.07

12 3 14.6 423.71 1,625.90 13.4 423.39 1.95 0.07

		Exact solver	$HybridNsgaII$
8	2	4.8	475.32	2.05	4.8	475.32	0.70	0.00
8	3	8.7	468.12	4.26	8.7	468.12	0.99	0.00
10	2	7.7	462.42	23.79	7.7	462.42	0.90	0.00
10	3	10.8	452.03	84.29	10.4	451.80	1.37	0.05
12	2	6.3	439.78	969.98	6.2	439.46	1.14	0.07
12	3	14.6	423.71	1,625.90	13.4	423.39	1.95	0.07

NSGA: non-dominated sorting genetic algorithm; HV: hypervolume.

Overall, our findings underscore the efficacy of our proposed approach, which significantly outpaces the exact solver in terms of speed. Notably, for networks of size 8, our approach successfully generated all nondominated points. This trend continues for networks of size 10 with 2 crowdshippers. Furthermore, our observation from the table reveals that even for relatively larger-sized instances, the proposed approach consistently finds almost all nondominated points.

Next, we focus on the single-objective analysis. We conduct two benchmarking experiments: pure cost-driven performance comparison and pure equity-driven performance comparison. For the former, we solve the single-objective cost-minimization model using Gurobi and compare the best solution reported by Gurobi within a time limit of $3, 600$ seconds against the minimum-cost point on the nondominated frontier produced by $HybridNsgaII$ . Table 4 reports averages over 10 instances. In this table, “TL” indicates that the time limit was reached and optimality could not be proven for any of the instances corresponding to that row. Out of the 80 instances summarized in Table 4, Gurobi successfully reported solutions for 61 instances. Among the 19 failed instances, one belongs to the set $(N = 75, M = 15)$ , eight belong to $(N = 100, M = 10)$ , and ten belong to $(N = 100, M = 20)$ . In the table, due to the existence of failed instances, the cost values reported by Gurobi and the corresponding gap values are averaged only over the subset of successful instances, that is, those for which Gurobi produced a solution.

Table 4.

Pure cost-driven performance comparison: $HybridNsgaII$ vs. Gurobi.

		Average cost value		Gap	Time
$N$	$M$	$HybridNsgaII$	Gurobi optimizer	(%)	(s)
25	3	369.7	366.8	0.77	2.3
25	5	391.4	389.4	0.54	1.1
50	5	552.0	536.6	2.87	40.9
50	10	635.3	611.9	3.82	1,971.2
75	8	717.1	684.3	4.79	2,498.5
75	15	847.9	803.4	5.54	3,180.0
100	10	901.4	843.0	6.93	TL
100	20	1,164.0	–	–	TL

NSGA: Non-dominated sorting genetic algorithm; TL: time limit.

Note that as shown in the previous subsection, our proposed approach can approximate the entire nondominated frontier within seconds, rather than focusing solely on a single least-cost solution. In contrast, the Gurobi focuses only on pure cost-driven solutions, and it requires a significant amount of time. In fact, as shown in Table 4, Gurobi Optimizer is even unable to provide a feasible solution within a one-hour time limit for large instances. Consequently, in terms of computational time, the superiority of the proposed approach is evident. Regarding solution quality, the performance of the proposed heuristic is also reasonable. Even for large datasets $(N = 100, M = 10)$ , the performance gap between the pure cost-driven solution of $HybridNsgaII$ and Gurobi remains below $7 %$ , and across all other instances, it is approximately $3 %$ .

In light of the above, in terms of pure cost-driven solutions, the performance of the proposed approach is competitive and within an acceptable range. However, a natural follow-up hypothetical question arises: If we had access to a black box that could instantly provide the solution that Gurobi reports after one hour, could such information be used to warm-start $HybridNsgaII$ and substantially improve the Pareto-optimal approximation compared to the standard $HybridNsgaII$ runs? The answer to this question demonstrates the potential benefit of feeding the initial population of $HybridNsgaII$ with high-quality single-objective solutions. To answer this question, we conducted an experiment and reported the results in Table 5, which includes only the classes of instances where Gurobi was successful in generating solutions for all instances. The table shows a slight improvement in terms of the average HV gap, which remains under $2 %$ for the largest class of instances and under $0.8 %$ on average. These results indicate that although some improvement may be achieved by feeding the initial population of $HybridNsgaII$ with high-quality single-objective solutions, such improvements are marginal. Therefore, only if high-quality, single-objective solutions are readily available at a low cost would it be worthwhile to use them. These results also highlight the stability of the proposed heuristic technique and its overall approximation quality.

Table 5.

Performance improvement by seeding $HybridNsgaII$ with the pure cost-driven solution reported by Gurobi.

		Average HV
$N$	$D$	$HybridNsgaII$	$HybridNsgaII$ + Gurobi Solution	Gap (%)
25	3	5,624.3	5,628.5	0.07
25	5	5,592.6	5,600.5	0.14
50	5	5,434.5	5,453.7	0.35
50	10	5,313.1	5,365.4	0.99
75	8	5,235.0	5,297.6	1.20
75	15	5,066.8	5,165.6	1.95

NSGA: non-dominated sorting genetic algorithm; HV: hypervolume.

Next, we conduct a similar experiment, this time by focusing on the pure equity-driven performance evaluation, specifically analyzing the opposite end of the nondominated frontier produced by our approach. For this purpose, we employ SCIP as a nonlinear mixed-integer optimizer, since all the equity measures under consideration are highly nonlinear. As discussed earlier, even the Range measure is nonlinear due to the nature of adjusted profits. In practice, we observed that among the equity measures, only Range and MAD can be handled robustly by SCIP (i.e., without encountering numerical issues), owing to their relatively lower degree of nonlinearity. Therefore, in this section, we report results exclusively for these two measures. Even for Range and MAD, however, the solution time remains significant, even for instances with fewer than 12 nodes. As a result, we restrict our analysis to such small instances. To further assist SCIP, we warm-start the solver using the pure equity-driven solutions obtained by $HybridNsgaII$ (located at the equity end of the nondominated frontier produced by $HybridNsgaII$ ). Additionally, we increase the time limit to five hours to enhance the likelihood of reaching optimal solutions. If the solver does not terminate within the time limit, we report “TL”. The results are summarized in Table 6, where all reported values are averages over 10 instances. From the table, we observe that in all cases, the final solution reported by SCIP is identical to the feasible solution provided as a warm-start, with the quality gap between SCIP and $HybridNsgaII$ consistently reported as zero. Notably, for instances with 8 nodes, SCIP was able to certify optimality within the time limit. These results highlight the efficacy of the proposed heuristic in generating high-quality, equity-driven solutions. Finally, since SCIP was unable to improve upon the warm-start solution, there is no need to consider a similar follow-up hypothetical question as raised for the pure cost-driven case. The answer is evident: Even if we had access to a black box capable of instantly providing SCIP’s five-hour solution, using it to warm-start $HybridNsgaII$ would not improve the Pareto-optimal approximation beyond what is achieved by the standard $HybridNsgaII$ .

Table 6.

Pure equity-driven performance comparison: $HybridNsgaII$ vs. SCIP.

		Average range value				Average MAD value
				Gap	Time			Gap	Time
$N$	$M$	$HybridNsgaII$	SCIP	(%)	(s)	$HybridNsgaII$	SCIP	(%)	(s)
8	2	0.0038	0.0038	0.0	7,071.7	0.0019	0.0019	0.0	3,685.1
8	3	0.0069	0.0069	0.0	7,067.7	0.0027	0.0027	0.0	3,490.3
10	2	0.0005	0.0005	0.0	TL	0.0002	0.0002	0.0	TL
10	3	0.0045	0.0045	0.0	TL	0.0017	0.0017	0.0	TL
12	2	0.0001	0.0001	0.0	TL	0.0001	0.0001	0.0	TL
12	3	0.0011	0.0011	0.0	TL	0.0004	0.0004	0.0	TL

NSGA: non-dominated sorting genetic algorithm; MAD: mean absolute deviation; TL: time limit.

7.1.3. The Most Reliable Equity Measures

In this subsection, we embark on exploring one of our key research questions: Which equity measure(s) prove most reliable within the scope of our study? To achieve this goal, we utilize the selection method introduced in Section 6. This method identifies a measure as most reliable when it can generate a set of Pareto-optimal solutions using $HybridNsgaII$ that remain robust across all measures. As discussed in Section 6, alternatives to our proposed approach may involve analyzing a subset of Pareto-optimal solutions rather than the entire set holistically. However, such approaches encounter challenges as determining the appropriate subset selection method is not straightforward. Additionally, we posit that a method identified as most reliable through holistic analysis is likely to maintain its efficacy relative to subset-based approaches, but the reverse may not hold true. This implies that selections made using our proposed approach are expected to possess an additional layer of robustness against subset-based approaches. For those interested, E-Companion EC.5 provides numerical evidence and further discussions supporting this claim, focusing on a special case of subset-based approaches when the subset size is one.

Given these considerations, we revisit the 80 instances outlined in Table 2 from the preceding subsection. These instances are evenly distributed across four classes, categorized by the number of customers in their networks, denoted as $N \in {25, 50, 75, 100}$ . Utilizing the mapping techniques detailed in Section 6, we attempt to identify the most reliable measure(s). Table 7 provides a comprehensive summary of the HV gap for each observed instance class in the mapping experiments. Each row reports approximate Pareto-optimal solutions obtained using a specific equity function, reevaluated against the remaining functions investigated in this study. To maximize the reliability of our findings, we first solve each instance 10 times using $HybridNsgaII$ for every equity measure. The best approximation, based on HV, is then selected from these 10 runs for each equity and will be used for subsequent mapping processes across various criterion spaces for that instance. Furthermore, to ensure consistent comparison of the HV gap across diverse problem sizes, each experiment class uses a fixed reference point. This point is determined by the maximum observed cost within the respective class.

Table 7.
Mapping results: Hypervolume gaps.

(a) Class $N = 25$

Map HV gap (%) using

From $∖$ To Range MAD SD CV GINI Average

Range 0.00 0.19 −0.65 0.07 −0.23 −0.12

MAD −0.03 0.00 −0.30 −0.09 −0.22 −0.13

SD 0.93 1.19 0.00 1.07 0.64 0.77

CV 0.04 0.08 −0.33 0.00 −0.20 −0.08

GINI 0.27 0.42 −0.39 0.31 0.00 0.12

(b) Class $N = 50$

Map HV gap (%) using

From $∖$ To Range MAD SD CV GINI Average

Range 0.00 0.99 −0.83 0.32 0.60 0.22

MAD −0.61 0.00 −1.46 0.60 −0.28 −0.35

SD 1.53 2.71 0.00 2.06 1.92 1.64

CV −0.15 −0.61 −1.02 0.00 0.27 −0.30

GINI −0.29 0.39 −1.37 −0.21 0.00 −0.30

(c) Class $N = 75$

Map HV gap (%) using Average

From $∖$ To Range MAD SD CV GINI Average

Range 0.00 0.30 0.47 0.51 0.36 0.33

MAD −0.19 0.00 −0.33 0.23 −0.22 −0.10

SD 0.32 0.88 0.00 1.07 0.47 0.55

CV −0.56 −0.22 −0.65 0.00 −0.44 −0.37

GINI −0.10 0.30 −0.28 0.50 0.00 0.08

(d) Class $N = 100$

Map HV gap (%) using

From $∖$ To Range MAD SD CV GINI Average

Range 0.00 1.13 1.54 0.98 0.65 0.86

MAD −0.90 0.00 0.10 −0.11 0.71 −0.04

SD −0.79 0.53 0.00 0.38 −0.42 −0.06

CV −1.12 0.15 −0.17 0.00 −0.70 −0.37

GINI −0.28 −0.86 0.56 0.73 0.00 0.03

(a) Class $N = 25$
Range	0.00	0.19	−0.65	0.07	−0.23	−0.12
MAD	−0.03	0.00	−0.30	−0.09	−0.22	−0.13
SD	0.93	1.19	0.00	1.07	0.64	0.77
CV	0.04	0.08	−0.33	0.00	−0.20	−0.08
GINI	0.27	0.42	−0.39	0.31	0.00	0.12
(b) Class $N = 50$
Map	HV gap (%) using
From $∖$ To	Range	MAD	SD	CV	GINI	Average
Range	0.00	0.99	−0.83	0.32	0.60	0.22
MAD	−0.61	0.00	−1.46	0.60	−0.28	−0.35
SD	1.53	2.71	0.00	2.06	1.92	1.64
CV	−0.15	−0.61	−1.02	0.00	0.27	−0.30
GINI	−0.29	0.39	−1.37	−0.21	0.00	−0.30
(c) Class $N = 75$
Map	HV gap (%) using	Average
From $∖$ To	Range	MAD	SD	CV	GINI	Average
Range	0.00	0.30	0.47	0.51	0.36	0.33
MAD	−0.19	0.00	−0.33	0.23	−0.22	−0.10
SD	0.32	0.88	0.00	1.07	0.47	0.55
CV	−0.56	−0.22	−0.65	0.00	−0.44	−0.37
GINI	−0.10	0.30	−0.28	0.50	0.00	0.08
(d) Class $N = 100$
Map	HV gap (%) using
From $∖$ To	Range	MAD	SD	CV	GINI	Average
Range	0.00	1.13	1.54	0.98	0.65	0.86
MAD	−0.90	0.00	0.10	−0.11	0.71	−0.04
SD	−0.79	0.53	0.00	0.38	−0.42	−0.06
CV	−1.12	0.15	−0.17	0.00	−0.70	−0.37
GINI	−0.28	−0.86	0.56	0.73	0.00	0.03

MAD: mean absolute deviation; SD: standard deviation; CV: coefficient of variation; HV: hypervolume.

Note that a negative HV gap in Table 7 indicates that the approximation found through mapping is better than the original approximation for that criterion space. This can partly be attributed to the fact that $HybridNsgaII$ is a heuristic approach; however, by running each instance 10 times, we aim to minimize its impact. Overall, we observe from the table that the average HV gap through mapping is up to about 2% better or worse than the original approximation on average. The mapping results indicate that while MAD and GINI exhibit good performance for medium-sized instances, CV emerges as the most reliable equity measure, consistently demonstrating the best performance across all instance classes—particularly in the larger ones. Additionally, we observe that although Range is computationally efficient, it performs significantly worse than the other measures due to its limited ability to capture variations in adjusted profits. Based on these observations, we select CV as the equity measure to be used in Part II of the computational study.

7.2. Part II: Investigating Real-World Operational Scenarios

The second series of experiments focuses on three key practical research questions rooted in real-world operations, which involve multiple zones and span over a sequence of multiple waves, referred to as the decision horizon. Specifically, we use a case study inspired by the publicly available Amazon routes, involving 5 zones and a decision horizon of 100 waves. For each zone-wave, a list of available vehicles along with their sign-up times is provided. From this list, the minimum number of drivers needed to meet the demand is selected by solving the crowdshipper selection problem discussed in E-Companion EC.1. Next, we examine the extent to which equity can be improved under varying equity budgets by the end of the decision horizon (Section 7.2.1). Then, we investigate the potential losses that crowdshippers may incur as a result of incorporating equity considerations (Section 7.2.2). This analysis reveals that selecting the minimum number of drivers required to satisfy demand minimizes such losses, while deviations from this baseline can have negative impacts in terms of opportunity cost for some drivers. Finally, we explore the expected consequences of insufficient driver availability in dynamic environments (Section 7.2.3). While the first two research questions assume sufficient driver availability to meet demand in every zone-wave, the third question relaxes this assumption and studies the impact of that. In such cases, demand satisfaction may require either postponing excess demand to future waves with an associated penalty cost or offering additional incentives to attract more drivers in each zone-wave. We investigate both scenarios in detail.

7.2.1. Quantifying Potential Equity Improvement

In this subsection, we explore a key research question: How much can equity improve with varying equity budgets by the end of a decision horizon? The goal is to demonstrate that equity can be significantly enhanced across various zones even with only a small increase in the operational cost for the company in different zones. To explore the question, we consider different equity budgets for the company, ranging from $γ \in {2.5 %, 5 %, 7.5 %, 10 %}$ , and conduct a full simulation of a long term decision horizon. Following our assumptions, within each zone-wave, we select the minimum necessary crowdshippers from available crowdshippers based on the first-come, first-served principle. Furthermore, our method of selecting a point from the approximate nondominated frontier precisely follows the approach outlined in Section 5.2. Specifically, we choose the point that maximizes equity for the equity budget under consideration. An illustrative example of the selection process can be found in Figure 4, where we present the set of approximate Pareto-optimal points for a specific zone-wave instance with 90 orders. The shaded region within the plot delineates the company’s equity cost budget, allowing for a 2.5% deviation from the least cost. To strike a balance between workload allocation and cost, we select the point found with the best equity value within this constrained region, as highlighted in the figure.

Figure 4.

Approximate pareto-optimal frontier obtained for a specific zone-wave.

Table 8 shows the average improvement for each zone in where the numbers in each row represent averages across all 100 waves. In this table, $M_{L}$ and $M_{H}$ denote the average number of low-performance (i.e., low-capacity) and high-performance (i.e., high-capacity) vehicles, respectively. Additionally, $\bar{u}$ represents the average adjusted profit across all vehicles.

Table 8.

Summary of results from simulating the entire decision horizon.

									Improvement in Coefficient
						Least-cost solution			of Variation (CV) (%)
Zone	$β$	$N$	$M_{L}$	$M_{H}$	Card.	Cost	$\bar{u}$	CV	$γ = 2.5 %$	$γ = 5 %$	$γ = 7.5 %$	$γ = 10 %$
1	0.7	99.8	11.9	2.5	55.4	735.5	0.65	0.0267	54.77	58.82	62.88	67.23
2	0.8	100.5	9.2	2.3	59.2	249.5	0.72	0.0463	64.65	68.76	71.50	74.25
3	0.9	99.5	12.7	2.9	63.3	466.8	0.83	0.0295	50.72	55.86	60.82	64.48
4	0.9	100.0	10.1	2.6	64.3	376.7	0.82	0.0414	63.13	66.49	71.52	75.50
5	1.0	98.6	10.8	2.8	59.5	574.5	0.92	0.0368	47.61	54.98	61.39	65.59
Average		99.7	10.9	2.6	60.3	480.6	0.79	0.0361	56.18	60.98	65.62	69.41

Table 8 demonstrates that even a minor deviation of 2.5% from the least-cost solution can significantly enhance equity in workload allocation. Across the five zones, distinguished by the geographical segmentation and $β$ values, we observe an average equity improvement of 56.17% when allowing for this slight cost increase. For example, in Zone 1, a 2.5% deviation from the least-cost solution permits the selection of a solution with a route cost averaging 753.89, compared to the average least cost of 735.5. Similarly, with an equity budget of 5%, the average equity improvement is observed as 58.82% compared to the least-cost solution. This highlights that even with a modest equity budget, the company can foster a more equitable workload distribution, incentivizing participation from crowdshippers and aiding a sustainable business model. The results show that the compensation per-unit of delivery, $β$ , has no observable effect on equity; instead, as expected, it impacts $\bar{u}$ , that is, the couriers’ average adjusted profits.

Figure 5.

Average trends in equity, cost increase ratio, and adjusted profits when $γ = 2.5 %$ . (a) Zone 1, (b) Zone 2, (c) Zone 3, (d) Zone 4 and (e) Zone 5.

7.2.2. Quantifying the Loss of Crowdshippers

In this subsection, we explore a key research question: How much potential loss (i.e., opportunity cost) might a crowdshipper experience due to the company’s equity considerations? Here, gains (or losses) refer to the disparity between the adjusted profits under an equity-aware solution and those under the least-cost solution. To provide a more comprehensive view, we offer insights about both sides of the equation in this section: low-performing and high-performing crowdshippers. The response to the research question depends on the number of crowdshippers within the system. Our proposed approach, by construction, targets the minimum number of crowdshippers (denoted as $M$ ) employed in a first-come-first-served framework, when solving the crowdshipper selection problem. It is anticipated that deviating from this minimum count and admitting more crowdshippers will increase the loss of crowdshippers. Thus, we examine the research question across different scenarios, contingent upon the crowdshipper count. Specifically, we increase $M$ by 50% and 100%, that is, $1.5 M$ and $2 M$ . Echoing the preceding subsection, we also explore various equity budgets for the company, spanning $γ \in {2.5 %, 5 %, 7.5 %, 10 %}$ .

Table 9 summarizes the results of the zone-wave experiments for varying numbers of crowdshippers, with each row reporting averages computed over 20 waves. Note that this represents 20% of the total number of waves in the decision horizon, which is expected to provide a good approximation for the entire decision horizon, as there is no transfer of information between waves. The main reason for limiting the analysis to 20 waves is the computational burden. Recall that each instance is solved 10 times using $HybridNsgaII$ , and we consider multiple scenarios for the number of crowdshippers (see the horizontal breakpoints in Figure 5) across 5 zones. Even with only 20 waves, the total number of runs becomes substantial, requiring significant computational effort. In the table, ${\bar{u}}_{L}$ and ${\bar{u}}_{H}$ represent the average adjusted profits for low-performing and high-performing crowdshippers, respectively. Within each zone, results are categorized by the equity budgets discussed previously. We observe the following five key managerial insights:

Insight I: Expanding the number of crowdshippers can initially reduce the company’s operational costs, as all selected crowdshippers must be utilized on a first-come, first-served basis. However, beyond a certain point, further increases in the number of crowdshippers lead to higher operational costs. For example, in Zone 1, a 50% increase in crowdshippers lowers operational costs, but a 100% increase results in higher costs compared to the baseline.

Insight II: Even if the company hires more crowdshippers than necessary, it can still notably improve equity across all zones with a small equity budget. This improvement is evident in the shifts in the CV values for each crowdshipper size.

Insight III: While equity gains remain notable even with a limited equity budget across different workforce sizes, the disparities between the average adjusted profits of high-performing and low-performing crowdshippers exhibit minimal variation. In essence, ${\bar{u}}_{L}$ and ${\bar{u}}_{H}$ exhibit comparable magnitudes across different budget allocations within each zone and the same workforce size. Thus, according to the definition of CV, equity gains predominantly arise from reductions in standard deviations.

Insight IV: The equity deteriorates with an increase in the number of crowdshippers in general. This is apparent from the rising trend in the CV values across each row. According to the definition of CV, this suggests that as the number of crowdshippers increases, the average adjusted profits decrease significantly, to an extent where even a decrease in standard deviation cannot compensate for it.

Insight V: When the minimum number of crowdshippers is employed, changing the equity budget has negligible impact on the average opportunity cost, as reflected in the adjusted profits of both high-performing and low-performing crowdshippers.

Insight VI: Finally and interestingly, selecting more crowdshippers than needed leads to higher opportunity costs for both low-performing and high-performing crowdshippers. This is evident in the reductions incurred in the adjusted profits of both high-performing and low-performing crowdshippers as the number of crowdshippers increases.

Table 9.
Comparing the impacts of different numbers of crowdshippers.

Number of crowdshippers selected

$M$ $1.5 M$ $2 M$

Zone Equity budget (%) Cost CV ${\bar{u}}_{L}$ ${\bar{u}}_{H}$ Cost CV ${\bar{u}}_{L}$ ${\bar{u}}_{H}$ Cost CV ${\bar{u}}_{L}$ ${\bar{u}}_{H}$

1 0.0 726.0 0.0259 0.6527 0.5876 710.6 0.3330 0.2850 0.2697 862.7 0.5062 0.1934 0.1403

2.5 732.4 0.0131 0.6525 0.5872 718.2 0.1642 0.2787 0.2665 872.0 0.2962 0.1795 0.1459

5.0 740.9 0.0115 0.6523 0.5878 730.2 0.1537 0.2784 0.2645 885.5 0.2509 0.1739 0.1458

7.5 755.5 0.0068 0.6533 0.5862 748.6 0.1186 0.2768 0.2622 914.3 0.2085 0.1680 0.1488

10.0 780.8 0.0059 0.6534 0.5859 766.2 0.0997 0.2776 0.2587 932.0 0.1786 0.1652 0.1502

2 0.0 235.1 0.0483 0.7152 0.7181 243.1 0.3846 0.3320 0.3213 305.2 0.4543 0.1992 0.1785

2.5 237.5 0.0174 0.7169 0.7159 246.0 0.2568 0.3369 0.3080 309.2 0.3048 0.1973 0.1714

5.0 240.3 0.0137 0.7165 0.7155 249.9 0.2348 0.3314 0.3119 315.0 0.2652 0.1964 0.1686

7.5 242.8 0.0121 0.7168 0.7142 255.3 0.2143 0.3313 0.3081 323.9 0.1714 0.1862 0.1715

10.0 244.8 0.0112 0.7168 0.7139 261.3 0.1715 0.3129 0.3074 329.6 0.1532 0.1856 0.1720

3 0.0 502.3 0.0266 0.8349 0.8380 368.2 0.3783 0.3790 0.3585 441.8 0.5976 0.2406 0.2358

2.5 508.9 0.0131 0.8357 0.8359 373.0 0.2322 0.3687 0.3500 448.5 0.3576 0.2224 0.2126

5.0 518.1 0.0118 0.8362 0.8345 381.7 0.1888 0.3597 0.3538 458.5 0.2936 0.2141 0.2111

7.5 527.3 0.0105 0.8361 0.8346 391.7 0.1598 0.3593 0.3540 468.0 0.2660 0.2116 0.2110

10.0 539.6 0.0086 0.8352 0.8362 398.1 0.1462 0.3610 0.3487 481.1 0.2503 0.2114 0.2042

4 0.0 381.7 0.0563 0.8090 0.8119 384.6 0.3260 0.3815 0.3732 481.3 0.5031 0.2391 0.2359

2.5 386.2 0.0166 0.8105 0.8077 390.5 0.1873 0.3732 0.3667 490.0 0.3049 0.2316 0.2162

5.0 389.1 0.0152 0.8104 0.8078 394.2 0.1692 0.3738 0.3644 497.8 0.2334 0.2253 0.2101

7.5 393.7 0.0138 0.8104 0.8072 403.2 0.1378 0.3719 0.3628 512.3 0.1845 0.2211 0.2132

10.0 399.5 0.0132 0.8103 0.8072 417.3 0.0888 0.3701 0.3670 521.7 0.1552 0.2212 0.2083

5 0.0 613.5 0.0268 0.9406 0.9545 550.9 0.3496 0.4431 0.4527 652.3 0.5551 0.2823 0.2835

2.5 617.8 0.0119 0.9420 0.9506 558.4 0.2481 0.4493 0.4089 663.4 0.3690 0.2665 0.2774

5.0 629.5 0.0085 0.9433 0.9480 568.1 0.2117 0.4386 0.4173 679.7 0.3417 0.2652 0.2728

7.5 641.4 0.0070 0.9438 0.9472 584.0 0.1631 0.4380 0.4082 693.6 0.2872 0.2575 0.2703

10.0 650.2 0.0066 0.9440 0.9466 599.4 0.1291 0.4323 0.4041 713.5 0.2413 0.2526 0.2626

		Number of crowdshippers selected
1	0.0	726.0	0.0259	0.6527	0.5876	710.6	0.3330	0.2850	0.2697	862.7	0.5062	0.1934	0.1403
	2.5	732.4	0.0131	0.6525	0.5872	718.2	0.1642	0.2787	0.2665	872.0	0.2962	0.1795	0.1459
	5.0	740.9	0.0115	0.6523	0.5878	730.2	0.1537	0.2784	0.2645	885.5	0.2509	0.1739	0.1458
	7.5	755.5	0.0068	0.6533	0.5862	748.6	0.1186	0.2768	0.2622	914.3	0.2085	0.1680	0.1488
	10.0	780.8	0.0059	0.6534	0.5859	766.2	0.0997	0.2776	0.2587	932.0	0.1786	0.1652	0.1502
2	0.0	235.1	0.0483	0.7152	0.7181	243.1	0.3846	0.3320	0.3213	305.2	0.4543	0.1992	0.1785
	2.5	237.5	0.0174	0.7169	0.7159	246.0	0.2568	0.3369	0.3080	309.2	0.3048	0.1973	0.1714
	5.0	240.3	0.0137	0.7165	0.7155	249.9	0.2348	0.3314	0.3119	315.0	0.2652	0.1964	0.1686
	7.5	242.8	0.0121	0.7168	0.7142	255.3	0.2143	0.3313	0.3081	323.9	0.1714	0.1862	0.1715
	10.0	244.8	0.0112	0.7168	0.7139	261.3	0.1715	0.3129	0.3074	329.6	0.1532	0.1856	0.1720
3	0.0	502.3	0.0266	0.8349	0.8380	368.2	0.3783	0.3790	0.3585	441.8	0.5976	0.2406	0.2358
	2.5	508.9	0.0131	0.8357	0.8359	373.0	0.2322	0.3687	0.3500	448.5	0.3576	0.2224	0.2126
	5.0	518.1	0.0118	0.8362	0.8345	381.7	0.1888	0.3597	0.3538	458.5	0.2936	0.2141	0.2111
	7.5	527.3	0.0105	0.8361	0.8346	391.7	0.1598	0.3593	0.3540	468.0	0.2660	0.2116	0.2110
	10.0	539.6	0.0086	0.8352	0.8362	398.1	0.1462	0.3610	0.3487	481.1	0.2503	0.2114	0.2042
4	0.0	381.7	0.0563	0.8090	0.8119	384.6	0.3260	0.3815	0.3732	481.3	0.5031	0.2391	0.2359
	2.5	386.2	0.0166	0.8105	0.8077	390.5	0.1873	0.3732	0.3667	490.0	0.3049	0.2316	0.2162
	5.0	389.1	0.0152	0.8104	0.8078	394.2	0.1692	0.3738	0.3644	497.8	0.2334	0.2253	0.2101
	7.5	393.7	0.0138	0.8104	0.8072	403.2	0.1378	0.3719	0.3628	512.3	0.1845	0.2211	0.2132
	10.0	399.5	0.0132	0.8103	0.8072	417.3	0.0888	0.3701	0.3670	521.7	0.1552	0.2212	0.2083
5	0.0	613.5	0.0268	0.9406	0.9545	550.9	0.3496	0.4431	0.4527	652.3	0.5551	0.2823	0.2835
	2.5	617.8	0.0119	0.9420	0.9506	558.4	0.2481	0.4493	0.4089	663.4	0.3690	0.2665	0.2774
	5.0	629.5	0.0085	0.9433	0.9480	568.1	0.2117	0.4386	0.4173	679.7	0.3417	0.2652	0.2728
	7.5	641.4	0.0070	0.9438	0.9472	584.0	0.1631	0.4380	0.4082	693.6	0.2872	0.2575	0.2703
	10.0	650.2	0.0066	0.9440	0.9466	599.4	0.1291	0.4323	0.4041	713.5	0.2413	0.2526	0.2626

To demonstrate the broader applicability of our findings beyond the $1.5 M$ and $2 M$ thresholds, Figure 5 showcases the average values of CV, ${\bar{u}}_{L}$ , ${\bar{u}}_{H}$ , and the Cost Increase Ratio (CIR) for all four zones across various workforce sizes, with an equity budget set at 2.5%. The CIR quantifies the percentage increase or decrease in costs compared to hiring the minimum number of crowdshippers. For instance, a CIR of 0.1 indicates a 10% increase, while -0.1 implies a 10% decrease in costs. Overall, Figure 5 reaffirms that our key findings extend beyond the $1.5 M$ and $2 M$ thresholds. Consequently, the managerial imperative highlighted in this section is to prioritize selecting the minimum number of crowdshippers.

7.2.3. Impact of Vehicle Shortages in Dynamic Environments

In this section, we focus on another key research question: How does vehicle shortage affect the company’s cost and the equity among drivers over the entire decision horizon? To fulfill demand under limited vehicle availability, we consider two scenarios: (1) Postponing excess demand to future waves with an associated penalty cost (referred to as the Postponement (PP) scenario), and (2) offering additional incentives to attract more drivers (referred to as the Additional Vehicles (AVs) scenario). To enable a fair comparison across scenarios, we assume that the final wave in the decision horizon does not experience any shortage, that is, all remaining demands are met with sufficient driver availability.

To set up the experiments for both scenarios, we first identify the so-called “at-risk” orders, that is, demand received at a given zone-wave that, due to a shortage of vehicles, must be handled according to one of the two scenarios described earlier. In our experiments, to create different levels of vehicle shortage, for each zone $z \in {1, \dots, 5}$ , the at-risk orders are considered to be the last $5 % \times z$ orders of each wave (according to the placement times of the orders). This varying percentage reflects the variability in the number of orders affected by vehicle shortage. Note that at-risk orders are served in the same wave under the AV scenario by recruiting extra drivers. In contrast, under the PP scenario, they are deferred to the next wave. In this case, such orders are given the highest priority in the following wave, as they have remained in the system longer.

We first focus on the PP scenario. We assume that postponing each order from one wave to the immediate next wave incurs a fixed, user-defined cost parameter, denoted by $λ$ . Let $N_{defer}$ represent the average number of deferred orders over all 100 waves in the decision horizon for a given zone. In this case, the average PP cost per wave for a given zone is $N_{defer} \times λ$ . Under the PP scenario, deferred orders are selected in the reverse order of their placements. As a result, the value of $λ$ does not influence the decision-making process itself but only affects the total operational cost, which includes both routing and PP costs. That is, as $λ$ increases, the total cost increases accordingly, but the solution structure remains unchanged. This behavior is illustrated in Table 10, which reports the average least cost per zone under the PP scenario.

Table 10.
Least cost under the postponement (PP) scenario.

Average cost

Zone $N_{defer}$ $λ = 0$ $λ = 2$ $λ = 4$ $λ = 6$ $λ = 8$ $λ = 10$

1 5.0 654.6 664.6 674.6 684.6 694.6 704.6

2 10.1 214.6 234.8 255.0 275.2 295.4 315.6

3 14.9 368.0 397.8 427.6 457.4 487.2 517.0

4 20.0 301.7 341.7 381.7 421.7 461.7 501.7

5 24.7 476.1 525.5 574.9 624.3 673.7 723.1

		Average cost
1	5.0	654.6	664.6	674.6	684.6	694.6	704.6
2	10.1	214.6	234.8	255.0	275.2	295.4	315.6
3	14.9	368.0	397.8	427.6	457.4	487.2	517.0
4	20.0	301.7	341.7	381.7	421.7	461.7	501.7
5	24.7	476.1	525.5	574.9	624.3	673.7	723.1

Table 11 further summarizes the results of simulating the entire decision horizon with $λ = 4$ , where the values in each row represent averages over 100 waves. It is important to note that the results in Table 11 are not directly comparable to those in Table 8, which represents a setting with no vehicle shortages. The PP scenario results in an entirely different nondominated frontier due to inherent differences in order composition and vehicle availability across waves. Nevertheless, one observation is that, even under a shortage of drivers, the magnitude of equity (i.e., CV) improvement across zone-waves is comparable to the case without vehicle shortages. However, total cost outcomes vary significantly depending on the value of $λ$ . Overall, while the PP scenario may be necessary when more direct remedies are infeasible, it can inherently erode customer trust and should be used with caution.

Table 11.

Summary of results under the PP scenario with $λ = 4$ .

				Least-cost solution			Improvement in CV (%)
Zone	$M_{L}$	$M_{H}$	Card.	Cost	$\bar{u}$	CV	$γ = 2.5 %$	$γ = 5 %$	$γ = 7.5 %$	$γ = 10 %$
1	12.0	2.5	67.2	674.6	0.65	0.0349	46.33	51.62	55.97	61.49
2	9.2	2.2	58.5	255.0	0.73	0.0470	57.29	62.68	64.81	67.47
3	12.8	2.8	66.6	427.6	0.84	0.0309	40.45	45.81	49.60	54.02
4	10.3	2.5	66.1	381.7	0.81	0.0502	50.49	54.45	59.41	66.25
5	10.9	2.7	59.0	574.9	0.93	0.0385	44.34	50.23	55.85	59.67
Average	11.0	2.5	63.5	462.8	0.79	0.0403	47.78	52.96	57.13	61.78

PP: postponement; CV: coefficient of variation.

Next, we focus on the AV scenario, where extra drivers are incentivized to participate through premium payments. For comparability, we assume the sequence of driver arrivals remains the same as in the no-shortage case. However, a smaller subset of the earliest arrivals agree to participate under the original terms, while the remaining drivers will join only if offered higher per-delivery compensation (a premium). To determine which drivers should receive the premium, we proceed as follows. For each zone-wave, we first identify the minimum number of vehicles required to meet demand under a first-come, first-served policy, assuming no orders are at risk. Next, for the same zone-wave, we find the minimum number of vehicles needed to serve only the non-at-risk orders under the same policy. The difference between these two vehicle sets corresponds to those requiring the premium.

For the AV scenario experiment, we consider three premium levels offered to additional crowdshippers: 10%, 20%, and 30% increases in per-delivery compensation. In this setting, the total cost depends not only on routing costs but also on which deliveries are assigned to the additional vehicles, since a bonus is paid for those deliveries. To solve this problem using our proposed method, which is designed to minimize routing costs, we convert the per-delivery bonus into an equivalent travel cost. Specifically, for additional vehicles, the route length is augmented by the bonuses associated with the deliveries assigned to that route. Thus, unlike the PP scenario, changes in payment can also affect workload allocation, as the adjusted profits of drivers depend on the offered premium.

Table 12 presents the least-cost solutions under each of the three bonus levels. The “Total Cost” in this table captures the total operational cost to the company, which includes both the standard routing cost and the bonus compensation paid only for deliveries made by the additional crowdshippers. In the table, all reported values are averages over 100 waves per row. Notably, the least-cost solutions found by the proposed heuristic approach in Table 12 are comparable to those reported in Table 8, which assumes no vehicle shortages and no bonuses. This comparability holds because the set of orders and vehicles is fixed across zone-waves; the only variation is that a subset of crowdshippers receives a per-delivery bonus. As anticipated, the total cost is higher under the AV scenario due to the inclusion of the compensation bonuses, particularly in zones with a high number of at-risk orders (e.g., Zone 5). Furthermore, a clear pattern emerges within this scenario: as the bonus value increases, the total cost consistently rises, a trend readily observable across Table 12.

Table 12.

Least-cost solutions under the AV scenario.

				$+ 10 %$ Bonus				$+ 20 %$ Bonus				$+ 30 %$ Bonus
Zone	$M_{L}$	$M_{H}$	AV	Bonus	Total cost	$\bar{u}$	CV	Bonus	Total cost	$\bar{u}$	CV	Bonus	Total cost	$\bar{u}$	CV
1	11.9	2.5	1.2	10.3	751.2	0.66	0.0399	19.8	769.2	0.66	0.0462	29.0	770.2	0.66	0.0555
2	9.2	2.3	1.6	14.4	264.2	0.73	0.0693	27.3	279.3	0.73	0.0887	40.3	298.0	0.73	0.0936
3	12.7	2.9	2.9	28.6	495.9	0.85	0.0520	55.6	524.2	0.85	0.0695	82.0	557.9	0.85	0.0797
4	10.1	2.6	2.9	29.3	410.3	0.84	0.0667	56.8	445.6	0.84	0.0791	84.1	474.9	0.84	0.0888
5	10.8	2.8	3.8	43.5	622.3	0.95	0.0526	85.7	675.6	0.95	0.0755	126.7	725.3	0.95	0.0898
Average	10.9	2.6	2.5	25.2	508.8	0.81	0.0561	49.0	538.8	0.81	0.0718	72.4	565.3	0.81	0.0815

AV: additional vehicle; CV: coefficient of variation.

As for the impact of bonuses on CV, we observe that the bonus increases profit inequality among crowdshippers compared to both the No-Shortage (Table 8) and PP (Table 11) scenarios. This occurs because a higher bonus directly increases the adjusted profits for a few additional drivers hired to satisfy the zone-wave order fulfillment requirement without order deferment. This, in turn, makes achieving an equitable workload allocation more challenging. Notably, despite the different bonus levels offered, the algorithm is able to maintain the same average adjusted profit, $\bar{u}$ , by reallocating workloads. However, this comes at the cost of greater disparity among drivers: As the bonus level increases, the standard deviation of adjusted profits rises and, consequently, the CVs increase.

In summary, the AV scenario experiment shows an increasing trend in both total cost and CV, as observed across all zones for each 10% increase in the bonus. Finally, we note that the equity improvements observed in the AV scenario remain substantial (37%–79%) across various equity budgets, consistent with the trends observed in other experiments presented in this paper. For complete details across equity budgets, interested readers are referred to the E-Companion EC.6.

8. Final Remarks

This study focused on three key questions faced by crowdshipping-based LMD platforms aiming to ensure fair pay and long-term sustainability: (1) How can equity be effectively measured? (2) How can cost benefits be evaluated? (3) What are the potential losses of employing additional couriers compared to the minimum required? The first question concerns selecting an appropriate equity measure from the vast literature in statistics and economics. The second underscores the need for LMD platforms (especially for-profit ones) to evaluate equity alongside cost, requiring tailored tools to assess both dimensions. The third addresses the potential drawbacks of promoting equity, particularly the notion of opportunity cost, which reflects how redistributing opportunities to promote fairness may unintentionally reduce profit-making opportunity for high-performing crowdshippers, thus discouraging participation.

To address these questions, we proposed a practical equity-oriented framework inspired by real-world crowdshipping operations. It consists of: (1) a fair crowdshipper selection process based on first-come, first-served principles, and (2) a bi-objective optimization approach for workload allocation and routing that balances cost and equity. This optimization approach is a heuristic, named $HybridNsgaII$ , and was validated through extensive computational analysis.

For the first question, we established theoretical foundations using Pareto-optimality concepts and proposed a systematic method for identifying most reliable equity measures. The method is simple: If Pareto-optimal solutions under one equity measure perform well across all measures, that measure is deemed reliable. Applied with $HybridNsgaII$ , our results identify CV as the most reliable measure overall, with MAD and GINI as strong alternatives. Range, while less robust, requires less computational efforts. To answer the second question, we explored cost–equity tradeoffs across different equity budget levels on Amazon-inspired instances. A key insight emerged: Even a small budget increase can yield significant equity gains. Specifically, a 2.5% increase in the equity budget can improve equity by about 65%, while an additional 7.5% increase yields only 10% more, suggesting a logarithmic-like relationship between budget and benefit. Regarding the third question, our analysis reveals that minimizing the number of crowdshippers is in the interest of all parties—low- and high-performing drivers and equity-driven firms alike. Excess crowdshippers raise both operational and opportunity costs and reduce equity outcomes regardless of budget level. Hence, careful control of courier volume is vital for achieving both fairness and efficiency. Additionally, we explored a complementary aspect of the problem in scenarios where courier shortages arise due to the dynamic nature of the system. We showed that comparable levels of equity improvement can still be achieved under such conditions; however, this may come at the expense of higher operational costs for the platform or potentially a loss of customer trust.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261425875 - Supplemental material for Equity-Driven Workload Allocation for Crowdsourced Last-Mile Delivery

Supplemental material, sj-pdf-1-pao-10.1177_10591478261425875 for Equity-Driven Workload Allocation for Crowdsourced Last-Mile Delivery by Abhay Sobhanan, Hadi Charkhgard and Iman Dayarian in Production and Operations Management

Footnotes

Acknowledgment

The authors thank the senior editor and the two anonymous reviewers for their valuable comments on an earlier version of this manuscript. The first author also gratefully acknowledges that part of this research was conducted during his doctoral program at the University of South Florida.

Declaration of Conflicting Interests

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

Funding

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

ORCID iDs

Abhay Sobhanan

Iman Dayarian

Supplemental Material

Supplemental material for this article is available online (doi: ).

How to cite this article

Sobhanan A, Charkhgard H and Dayarian I (2026) Equity-Driven Workload Allocation for Crowdsourced Last-Mile Delivery. Production and Operations Management XX(XX): 1–24.

References

Agnetis

Cosmi

Nicosia

, et al. (2023) Two is better than one? order aggregation in a meal delivery scheduling problem. Computers & Industrial Engineering 183: 109514.

Allon

Cohen

Sinchaisri

(2023) The impact of behavioral and economic drivers on gig economy workers. Manufacturing & Service Operations Management 25(4): 1376–1393.

Alnaggar

Gzara

Bookbinder

(2019) Crowdsourced delivery: A review of platforms and academic literature. Omega 98: 102139.

Archetti

Savelsbergh

Speranza

(2016) The vehicle routing problem with occasional drivers. European Journal of Operational Research 254(2): 472–480.

Arslan

Agatz

Kroon

, et al. (2019) Crowdsourced delivery—a dynamic pickup and delivery problem with ad hoc drivers. Transportation Science 53(1): 222–235.

Bai

Tang

, et al. (2019) Coordinating supply and demand on an on-demand service platform with impatient customers. Manufacturing & Service Operations Management 21(3): 556–570.

Baldacci

Mingozzi

(2009) A unified exact method for solving different classes of vehicle routing problems. Mathematical Programming 120: 347–380.

Basık

Gedik

Ferhatosmanoğlu

, et al. (2018) Fair task allocation in crowdsourced delivery. IEEE Transactions on Services Computing 14(4): 1040–1053.

Behrendt

Savelsbergh

Wang

(2023) A prescriptive machine learning method for courier scheduling on crowdsourced delivery platforms. Transportation Science 57(4): 889–907.

10.

Bertsimas

Farias

Trichakis

(2012) On the efficiency-fairness trade-off. Management Science 58(12): 2234–2250.

11.

Bula

Afsar

González

, et al. (2019) Bi-objective vehicle routing problem for hazardous materials transportation. Journal of Cleaner Production 206: 976–986.

12.

CapGemini Research Institute (2019) The last-mile delivery challenge. Technical report, CapGemini Research Institute.

13.

Chen

Hooker

(2020) Balancing fairness and efficiency in an optimization model. arXiv preprint arXiv:2006.05963.

14.

Clarke

Wright

(1964) Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12(4): 568–581.

15.

Cosmi

Oriolo

Piccialli

, et al. (2025) Courier assignment in meal delivery via integer programming: A case study in rome. Omega 133: 103237.

16.

Dahle

Andersson

Christiansen

, et al. (2019) The pickup and delivery problem with time windows and occasional drivers. Computers & Operations Research 109: 122–133.

17.

Dayarian

Pazour

(2022) Crowdsourced order-fulfillment policies using in-store customers. Production and Operations Management 31(11): 4075–4094.

18.

Dayarian

Savelsbergh

(2020) Crowdshipping and same-day delivery: Employing in-store customers to deliver online orders. Production and Operations Management 29(9): 2153–2174.

19.

Deb

Pratap

Agarwal

, et al. (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2): 182–197.

20.

Dixon

Weiner

Mitchell-Olds

, et al. (1987) Bootstrapping the gini coefficient of inequality. Ecology 68(5): 1548–1551.

21.

Fatehi

Wagner

(2022) Crowdsourcing last-mile deliveries. Manufacturing & Service Operations Management 24(2): 791–809.

22.

Gullotta

Campisano

Creaco

, et al. (2021) A simplified methodology for optimal location and setting of valves to improve equity in intermittent water distribution systems. Water Resources Management 35: 4477–4494.

23.

Ibarra-Rojas

Silva-Soto

(2021) Vehicle routing problem considering equity of demand satisfaction. Optimization Letters 15(6): 2275–2297.

24.

Jozefowiez

Semet

Talbi

(2005) Enhancements of NSGA II and its application to the vehicle routing problem with route balancing. In: International Conference on Artificial Evolution (Evolution Artificielle), pp.131–142. Springer.

25.

Kafle

Zou

Lin

(2017) Design and modeling of a crowdsource-enabled system for urban parcel relay and delivery. Transportation Research Part B: Methodological 99: 62–82.

26.

Konstantakopoulos

Gayialis

Kechagias

(2022) Vehicle routing problem and related algorithms for logistics distribution: A literature review and classification. Operational Research 22(3): 2033–2062.

27.

Kwon

Choo

Kim

, et al. (2020) Pomo: Policy optimization with multiple optima for reinforcement learning. Advances in Neural Information Processing Systems 33: 21188–21198.

28.

Larranaga

Kuijpers

CMH

Murga

, et al. (1999) Genetic algorithms for the travelling salesman problem: A review of representations and operators. Artificial Intelligence Review 13: 129–170.

29.

Mei

, et al. (2023) Fairness-guaranteed task assignment for crowdsourced mobility services. IEEE Transactions on Mobile Computing 23(5): 5385–5400.

30.

Liu

Gui

, et al. (2023) Heuristics for vehicle routing problem: A survey and recent advances. arXiv preprint arXiv:2303.04147.

31.

Macrina

Pugliese

LDP

Guerriero

, et al. (2017) The vehicle routing problem with occasional drivers and time windows. In: International conference on optimization and decision science, pp.577–587. Springer.

32.

Macrina

Pugliese

LDP

Guerriero

, et al. (2020) Crowd-shipping with time windows and transshipment nodes. Computers & Operations Research 113: 104806.

33.

Mahmoodian

Charkhgard

Dayarian

(2025) Equitable workload allocation in vehicle routing problem with heterogeneous drivers. Production and Operations Management 34(7): 1875–1900.

34.

Mahmoudinazlou

Kwon

(2024) A hybrid genetic algorithm with type-aware chromosomes for traveling salesman problems with drone. European Journal of Operational Research 318(3): 719–739.

35.

Malchow

(2023) What is the difference between outsourcing and crowdsourcing? https://shorturl.at/w5MAd (accessed 12 May 2025).

36.

Mandell

(1991) Modelling effectiveness-equity trade-offs in public service delivery systems. Management Science 37(4): 467–482.

37.

Martinez-Sykora

McLeod

Cherrett

, et al. (2024) Exploring fairness in food delivery routing and scheduling problems. Expert Systems with Applications 240: 122488.

38.

Matl

Hartl

Vidal

(2018) Workload equity in vehicle routing problems: A survey and analysis. Transportation Science 52(2): 239–260.

39.

Matl

Hartl

Vidal

(2019a) Leveraging single-objective heuristics to solve bi-objective problems: Heuristic box splitting and its application to vehicle routing. Networks 73(4): 382–400.

40.

Matl

Hartl

Vidal

(2019b) Workload equity in vehicle routing: The impact of alternative workload resources. Computers & Operations Research 110: 116–129.

41.

McCoy

Lee

(2014) Using fairness models to improve equity in health delivery fleet management. Production and Operations Management 23(6): 965–977.

42.

Merchán

Arora

Pachon

, et al. (2024) 2021 amazon last mile routing research challenge: Data set. Transportation Science 58(1): 8–11.

43.

Mesa

Montoya

Ramos-Pollán

, et al. (2024) A bi-objective approach to last-mile delivery routing considering driver preferences. arXiv preprint arXiv:2405.16051.

44.

Miller

Tucker

Zemlin

(1960) Integer programming formulation of traveling salesman problems. Journal of the ACM (JACM) 7(4): 326–329.

45.

Morabit

Desaulniers

Lodi

(2021) Machine-learning–based column selection for column generation. Transportation Science 55(4): 815–831.

46.

Mueller

(2024) Haversine vs. OSRM: Distance and cost experiments on a vehicle routing problem (VRP). NextMV blog. https://shorturl.at/QXd7C (accessed 5 July 2025).

47.

Nourmohammadi

Rey

, et al. (2025) A data-driven preference learning approach for multi-objective vehicle routing problems in last-mile delivery. Transportation Research Part C: Emerging Technologies 174: 105101.

48.

Pal

Charkhgard

(2019) A feasibility pump and local search based heuristic for bi-objective pure integer linear programming. INFORMS Journal on Computing 31(1): 115–133.

49.

Pessoa

Sadykov

Uchoa

, et al. (2020) A generic exact solver for vehicle routing and related problems. Mathematical Programming 183: 483–523.

50.

Rea

Froehle

Masterson

, et al. (2021) Unequal but fair: Incorporating distributive justice in operational allocation models. Production and Operations Management 30(7): 2304–2320.

51.

Reyes

Erera

Savelsbergh

, et al. (2018) The meal delivery routing problem. Optimization Online 6571: 2018.

52.

Shaw

(1998) Using constraint programming and local search methods to solve vehicle routing problems. In: International conference on principles and practice of constraint programming, pp.417–431. Springer.

53.

Singh

Kumar

Chakraborty

(2024) Towards fairness in online service with k servers and its application on fair food delivery. In: Proceedings of the AAAI conference on artificial intelligence, Vol. 38, pp.19929–19936.

54.

Sobhanan

Park

, et al. (2025) Genetic algorithms with neural cost predictor for solving hierarchical vehicle routing problems. Transportation Science 59(2): 322–339.

55.

Soto Setzke

Pflügler

Schreieck

, et al. (2017) Matching drivers and transportation requests in crowdsourced delivery systems.

56.

Srivastava

Singh

Mallipeddi

(2021) NSGA-II with objective-specific variation operators for multiobjective vehicle routing problem with time windows. Expert Systems with Applications 176: 114779.

57.

Tricoire

(2012) Multi-directional local search. Computers & Operations Research 39(12): 3089–3101.

58.

Vidal

(2022) Hybrid genetic search for the CVRP: Open-source implementation and SWAP* neighborhood. Computers & Operations Research 140: 105643.

59.

Vidal

Crainic

Gendreau

, et al. (2012) A hybrid genetic algorithm for multidepot and periodic vehicle routing problems. Operations Research 60(3): 611–624.

60.

Xinying Chen

Hooker

(2023) A guide to formulating fairness in an optimization model. Annals of Operations Research 326(1): 581–619.

61.

Yildiz

Savelsbergh

(2019) Provably high-quality solutions for the meal delivery routing problem. Transportation Science 53(5): 1372–1388.

62.

(2024) The consistent vehicle routing problem considering driver equity and flexible route consistency. Computers & Industrial Engineering 187: 109803.

63.

Zhao

Liu

Guo

, et al. (2021) Bi-objective optimization for vehicle routing problems with a mixed fleet of conventional and electric vehicles and soft time windows. Journal of Advanced Transportation 2021(1): 9086229.

64.

Zhou

Geng

Jiang

, et al. (2018) Multi-objective capacity allocation of hospital wards combining revenue and equity. Omega 81: 220–233.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.19 MB

		Exact solver			$HybridNsgaII$
				Time			Time	HV gap
$N$	$M$	Card.	HV	(s)	Card.	HV	(s)	(%)
8	2	4.8	475.32	2.05	4.8	475.32	0.70	0.00
8	3	8.7	468.12	4.26	8.7	468.12	0.99	0.00
10	2	7.7	462.42	23.79	7.7	462.42	0.90	0.00
10	3	10.8	452.03	84.29	10.4	451.80	1.37	0.05
12	2	6.3	439.78	969.98	6.2	439.46	1.14	0.07
12	3	14.6	423.71	1,625.90	13.4	423.39	1.95	0.07

		Number of crowdshippers selected
		$M$				$1.5 M$				$2 M$
Zone	Equity budget (%)	Cost	CV	${\bar{u}}_{L}$	${\bar{u}}_{H}$	Cost	CV	${\bar{u}}_{L}$	${\bar{u}}_{H}$	Cost	CV	${\bar{u}}_{L}$	${\bar{u}}_{H}$
1	0.0	726.0	0.0259	0.6527	0.5876	710.6	0.3330	0.2850	0.2697	862.7	0.5062	0.1934	0.1403
	2.5	732.4	0.0131	0.6525	0.5872	718.2	0.1642	0.2787	0.2665	872.0	0.2962	0.1795	0.1459
	5.0	740.9	0.0115	0.6523	0.5878	730.2	0.1537	0.2784	0.2645	885.5	0.2509	0.1739	0.1458
	7.5	755.5	0.0068	0.6533	0.5862	748.6	0.1186	0.2768	0.2622	914.3	0.2085	0.1680	0.1488
	10.0	780.8	0.0059	0.6534	0.5859	766.2	0.0997	0.2776	0.2587	932.0	0.1786	0.1652	0.1502
2	0.0	235.1	0.0483	0.7152	0.7181	243.1	0.3846	0.3320	0.3213	305.2	0.4543	0.1992	0.1785
	2.5	237.5	0.0174	0.7169	0.7159	246.0	0.2568	0.3369	0.3080	309.2	0.3048	0.1973	0.1714
	5.0	240.3	0.0137	0.7165	0.7155	249.9	0.2348	0.3314	0.3119	315.0	0.2652	0.1964	0.1686
	7.5	242.8	0.0121	0.7168	0.7142	255.3	0.2143	0.3313	0.3081	323.9	0.1714	0.1862	0.1715
	10.0	244.8	0.0112	0.7168	0.7139	261.3	0.1715	0.3129	0.3074	329.6	0.1532	0.1856	0.1720
3	0.0	502.3	0.0266	0.8349	0.8380	368.2	0.3783	0.3790	0.3585	441.8	0.5976	0.2406	0.2358
	2.5	508.9	0.0131	0.8357	0.8359	373.0	0.2322	0.3687	0.3500	448.5	0.3576	0.2224	0.2126
	5.0	518.1	0.0118	0.8362	0.8345	381.7	0.1888	0.3597	0.3538	458.5	0.2936	0.2141	0.2111
	7.5	527.3	0.0105	0.8361	0.8346	391.7	0.1598	0.3593	0.3540	468.0	0.2660	0.2116	0.2110
	10.0	539.6	0.0086	0.8352	0.8362	398.1	0.1462	0.3610	0.3487	481.1	0.2503	0.2114	0.2042
4	0.0	381.7	0.0563	0.8090	0.8119	384.6	0.3260	0.3815	0.3732	481.3	0.5031	0.2391	0.2359
	2.5	386.2	0.0166	0.8105	0.8077	390.5	0.1873	0.3732	0.3667	490.0	0.3049	0.2316	0.2162
	5.0	389.1	0.0152	0.8104	0.8078	394.2	0.1692	0.3738	0.3644	497.8	0.2334	0.2253	0.2101
	7.5	393.7	0.0138	0.8104	0.8072	403.2	0.1378	0.3719	0.3628	512.3	0.1845	0.2211	0.2132
	10.0	399.5	0.0132	0.8103	0.8072	417.3	0.0888	0.3701	0.3670	521.7	0.1552	0.2212	0.2083
5	0.0	613.5	0.0268	0.9406	0.9545	550.9	0.3496	0.4431	0.4527	652.3	0.5551	0.2823	0.2835
	2.5	617.8	0.0119	0.9420	0.9506	558.4	0.2481	0.4493	0.4089	663.4	0.3690	0.2665	0.2774
	5.0	629.5	0.0085	0.9433	0.9480	568.1	0.2117	0.4386	0.4173	679.7	0.3417	0.2652	0.2728
	7.5	641.4	0.0070	0.9438	0.9472	584.0	0.1631	0.4380	0.4082	693.6	0.2872	0.2575	0.2703
	10.0	650.2	0.0066	0.9440	0.9466	599.4	0.1291	0.4323	0.4041	713.5	0.2413	0.2526	0.2626

		Average cost
Zone	$N_{defer}$	$λ = 0$	$λ = 2$	$λ = 4$	$λ = 6$	$λ = 8$	$λ = 10$
1	5.0	654.6	664.6	674.6	684.6	694.6	704.6
2	10.1	214.6	234.8	255.0	275.2	295.4	315.6
3	14.9	368.0	397.8	427.6	457.4	487.2	517.0
4	20.0	301.7	341.7	381.7	421.7	461.7	501.7
5	24.7	476.1	525.5	574.9	624.3	673.7	723.1