Enhancing E-Grocery-Delivery-Network Resilience with Autonomous Delivery Robots

Abstract

This paper examines the challenges associated with the efficient planning and operation of an E-grocery delivery system using Autonomous Delivery Robots (ADR) during unforeseen events. The primary objective is to minimize unfulfilled customer demands rather than focusing solely on cost reduction, considering the humanitarian aspect. To address this, a two-echelon vehicle routing problem is formulated, taking into account stochastic service times and demands. Two models, namely a deterministic model and a chance-constraint model, are employed to solve this problem. The results demonstrate that the chance-constraint model significantly reduces unmet demands compared to the deterministic model, particularly when the delivery deadline has a broad time window and the ADR/van speed ratio is moderate.

1. Introduction

Recently, Autonomous Delivery Robots (ADR) are becoming an essential component in E-grocery delivery network systems, due to the increasing need for non-contact last-mile distribution in some emerging situations, such as the COVID-19 pandemic, natural disasters etc. Their ability to navigate through urban environments, efficiently transport goods, and provide contactless delivery options has made them a vital solution. However, the successful integration of ADRs into these networks is not without its challenges.

One of the primary challenges is the inherent unpredictability of service times and demands in the delivery ecosystem. Stochastic service times, which refer to the variable time it takes to complete a delivery, and stochastic demands, which represent the fluctuating volume of orders, add layers of complexity to the system. These uncertainties can disrupt the seamless flow of deliveries, leading to potential inefficiencies and delays. Therefore, the resilient planning and operation of ADR-assisted E-grocery delivery systems considering the stochastic service time and demands have become a matter of paramount importance.

To address these challenges, researchers and practitioners are delving into innovative approaches [1,2,3]. While progress has been made in addressing the challenges of stochastic demands and service time in last-mile delivery [4,5,6], we have identified limitations in current models, including inefficiently handling the planning issues of ADR in an assisted delivery network, the inability to flexibly consolidate parcels in micro hubs amidst stochastic service times and unpredictable delivery disruptions. Thus, in this study, we developed a deterministic mixed-integer model and a chance-constraint integer model for the ADR-assisted delivery-network system and to fill these gaps. We formulated the model as a two-echelon vehicle routing problem (VRP) with stochastic service time and stochastic demands (2E-VRP-SSSD). The aim of this study was to develop a novel formulation for efficiently solving the 2E-VRP-SSSD problem, and then assess the resilience performance of the van-ADR delivery system. Thus, the contributions of this study are mainly in the following aspects:

First, we introduce a novel two-echelon van-ADR delivery network with parcel consolidation designed specifically for resilient delivery settings. Both a deterministic model and chance-constraint model were used together for the 2E-VRP-SSSD, in which the uncertainty of service time and demand were considered.

Second, we applied both of the models to the experiments to evaluate the performance of the ADR-assisted E-grocery delivery network, and tested the impact of three sensitive factors on the delivery network: the speed ratio between the ADR and van (RSAV), the service deadline (DL), and confidence level of constraints on variables. This study will potentially help transportation managers and operators to efficiently solve the related planning issues and deeply understand future resilience benefits when ADRs are adopted.

2. Literature Review

During recent years, increasing attention has been paid to the resilience of freight systems. Many studies on the topic of resilience reported in the literature cover a wide range of mode-specific problems, such as port coalitions [7], train intermodal transportation [8], airline cargo transportation [9] and urban delivery [10,11]. Since the outbreak of COVID-19, there has been fresh interest in resilient last-mile delivery, especially smart mobility-assisted delivery. For example, drones have played an important role in cargo delivery regarding disaster relief [12] and resilience [13]. Drones provide an effective solution in rural and low-density suburban areas, far-flung regions, and some areas where road accessibility is limited or in some situations where roads limit the fast delivery of goods. However, functional constraints (e.g., landing space, security, payload) may bring challenges associated with the delivery of parcels in areas of very high population density, which reduces the deployment of drones at scale in dense urban areas. Compared to drones, ADRs with built-in parcel lockers may unlock higher service levels for urban and high-density areas [14,15,16]. These ADRs would be stationary in certain areas, communicating to consumers, moving, and then making home deliveries. This kind of robot navigates on a road at a low speed and can deliver groceries with a limited weight compared with a traditional vehicle. To determine the route of a truck, schedule the drop-off locations of a robot, and plan the ADR-assisted delivery, Nils Boysen et al. modeled the delivery as a vehicle routing problem (VRP) with synchronization, where the late delivery numbers to customers were minimized [14]. However, that current model does not account for the potential role of ADR in enhancing the resiliency of a dynamic E-grocery delivery network.

To plan a resilient E-grocery network, in this study, we focused on the ADR-assisted two-echelon vehicle routing problem (2E-VRP). The 2E-VRP is attributed to the VRP with two-echelon delivery connected by micro hubs. Murray and Chu (2015) originally formulated drone-assisted last-mile delivery as a collaborative truck-and-drone routing problem [17]. Boysen et al. (2018) then extended Murry’s work by considering multiple drones launched from a single truck [18]. Afterwards, Liu. et al. (2021) described an ADR-assisted delivery network by developing a 2E-VRP with vehicle synchronization constraints [19]. These previous studies assumed that the service time of vehicles and demands are deterministic, and the objective was to minimize the operation cost or travel time.

In spite of the fact that the deterministic 2E-VRP is considered a generalization of the VRP [20], various methods have still been developed to study the particular problems in the 2E-VRP. This is because the two sub-problems, the location-allocation problem and VRP, were thought to be NP-hard issues [21,22,23,24,25], which cannot be solved in a polynomial time. It is also challenging to optimize the realistically sized problems using purely specific methods. Considering the shortcomings of the methodology, metaheuristics approaches were chosen to optimize larger data sets, such as constructive algorithms [26,27,28], variable neighborhood-based metaheuristics [29,30,31,32], the adaptive large neighborhood search [33,34,35], tabu search [36], multi-start iterated local search [37], and tree-based search algorithm [38]. However, in reality, various parameters of the problem may be uncertain, e.g., travel times, customer demands, and/or customer presence. Although many papers consider stochastic variants of the classical single-echelon vehicle routing problem [39], limited number of papers deal with stochastic variants of the 2E-VRP [40,41,42]. Liu et al. (2017) [40], Wang et al. (2017a) [41] and Zhang et al. (2023) [42] consider stochastic customer demands and capture the uncertainty with a recourse action, in which the stochastic service time is not considered.

Addressing parameter uncertainty in optimization problems offers various approaches; the choice of the most suitable model depends on the available information regarding uncertain parameters. The chance-constrained vehicle routing problem has been a subject of research over the years [43,44]. Chance-constrained programming leverages probabilistic data and strives to determine an “optimal” solution within a probabilistic framework. This approach alleviates the need for strict adherence to absolute feasibility, given the highly improbable nature of worst-case scenarios, if they even exist.

In the context of our study, we operated under the assumption that all requisite information is readily accessible, setting aside application-specific concerns. Notably, the stochastic service time of vehicles and customer demands within the context of the ADR-assisted 2E-VRP has not been previously explored. Thus, our work extends the existing stochastic 2E-VRP problem to the realm of resilient delivery network assessment, introducing novel facets to the 2E-VRP. We propose employing chance-constrained programming as a means to address and solve this extended problem.

3. Problem Definition

3.1. Notion

In this stochastic E-grocery delivery network, we assumed that customers cannot be visited directly by vans. In order to minimize the unmet demand and the waiting time, the allocations of customers’ orders to micro hubs and micro hubs to the depots are a crucial step, which is determined according to the information of orders, including the coordination of the customers, the deadline of the orders, the weight of the orders, the coordination of the micro hubs, and the capacity of the micro hubs. We illustrate this two-echelon E-grocery network in (Figure 1), where vans connect depots ($d_1)$ to micro hubs ($s_1$,$s_2$, $s_3, \mathrm{a}\mathrm{n}\mathrm{d}$ $s_4$), and ADRs connect micro hubs to customers ($f_1$,$f_2$, $f_3$, $f_4$, $f_5$, ${\mathrm{a}\mathrm{n}\mathrm{d} f}_6$).

To mathematically formulate the two-echelon delivery network, we represented the two-echelon vehicle routing problem with the stochastic service time and demand (2E-VRP-SSSD) as an undirected graph W = (N, A) consisting of a node set N and arc set A, and a vehicle set K. Furthermore, the node set N was partitioned into four subsets, which are denoted as depots ($O$), micro hubs (S), customers ($F$), and delayed customers ($F^{\prime }$). We arranged it as follows: N = $O \cup S\cup F \cup F^{\prime }, N_1=O \cup S, N_2=S \cup F\cup F^{\prime }.$ And the arc set A was partitioned into two subsets; set $A_1=\left\{\left(i,j\right)|i\in N_1, j\in S,i\ne j\right\}$ consisted of the arcs that can be traversed by vans in the first echelon. Similarly, set $A_2=\left\{\left(i,j\right)|i\in N_2, j\in F\cup F^{\prime },i\ne j\right\}$ consisted of the arcs that can be travelled by ADRs in the second echelons. For each arc (i, j)$\in A$, let ${\rho }_{ijk}^{van}$ be the travel time of van k from i to j, (i, j) $\in A_1$, and ${\tau }_{ijk}^{ADR}$ be the travel time of the ADRs k from i to j, where $i\in N_2, j\in F$. Given that some demand j$\in F^{\prime }$ cannot be satisfied, we still assigned j a traveling time ${\tau }_{ijk}^{ADR}\ge L,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i \in N_1, j\in F’$, L is a sufficiently large value. Accordingly, the 2EVRP-SSSD was identified: A limited fleet of vans k$\in K_{van}$ with capacity $Q_{{}_{}}^{van}$ start from a depot o in $O$ in the first echelon, visit all of the assigned micro hubs, and finally return to the origin depot o with capacity $Q_o^{}$, $o\in O$. In the second echelon, a limited fleet of robots k$\in K_{ADR}$ with capacity $Q_{}^{ADR}$ start from a micro hub s in $S,$ visit all of the assigned customers, and finally return to the origin micro hub s with capacity $Q_s^{}$, $s\in S$. We made the following assumptions before the model was developed:

• Each route is served by one vehicle.
• Each trip must start and terminate at the same depot or micro hub.
• The orders assigned to a micro hub cannot be more than the maximum capacity of the micro hub.
• Without the loss of generality, travel times and preparation times are integrated into the service times.
• A limited number of vans and ADRs is available at each depot and micro hub.

Building upon the notations introduced in [45,46,47], we present the notations specific to this study.

3.2. Deterministic Model

We formulated the deterministic model of 2E-VRP-SSLD by mixed-integer programming:

Minimize ${\sum }_{f\in F}{u}_f$+ $\alpha {\sum }_{s\in S,k\in K_{van}}{T}_{sk}^{van}$+ $\beta {\sum }_{f\in F,k\in K_{ADR}}{T}_{fk}^{ADR}$.

The objective was to minimize the weighted sum of the total unmet demands and the total visit time at the depot and customer. The $\alpha$ and $\beta$ value were set to be very small to make the visit time a secondary objective compared with the unmet demand. Since we modeled a 2E-VRP in response to a resilient E-grocery delivery, the service start time at the micro hub directly impacts the arrival time at a customer. To serve the customer as early as possible for them to maintain their normal life, we used the visit time as an indicator other than the conventional objectives such as the operational time or travel time.

The feasible solution space of the problem is subject to the following constraints.

3.2.1. Flow Conservation Constraints

Constraints (1) and (2) specify that the number of tours starting from a depot o must not exceed the available number of vans. Constraint (3) requires that each van k departs from a depot and returns to the same depot. Constraints (4) and (5) impose that each micro hub can be visited once. Constraint (6) guarantees that all vehicles flowing into a micro hub must flow out of it. Constraints (7) and (8) specify that the number of tours starting from a micro hub s must not exceed the available number of ADRs. Constraint (9) requires that each ADR k departs from a micro hub and returns to the same micro hub. Constraints (10) and (11) impose that each customer can be visited once. Constraint (12) guarantees that all ADRs flowing to a customer flow back from them. Constraint (13) imposes that each customer is allocated to only one micro hub.

$${\sum }_{j\in S}{\sum }_{k\in k_{van}}{xl}_{ijk}^{van}\le n \forall i\in O$$

(1)

$${\sum }_{j\in S}{\sum }_{k\in k_{van}}{xl}_{jik}^{van}\le n \forall i\in O$$

(2)

$${\sum }_{j\in S}{xl}_{ijk}^{van}={\sum }_{j\in S}{xl}_{jik}^{van}=1 \forall i\in O, k\in K_{van}$$

(3)

$${\sum }_{j\in N_1}{\sum }_{k\in K_{van}}{xl}_{ijk}^{van}=1 \forall i\in S$$

(4)

$${\sum }_{j\in N_1}{\sum }_{k\in K_{van}}{xl}_{jik}^{van}=1 \forall i\in S$$

(5)

$${\sum }_{j\in N_1}{xl}_{ijk}^{van}={\sum }_{j\in N_1}{xl}_{jik}^{van}i\in S, k\in K_{van}$$

(6)

$${\sum }_{j\in F}{\sum }_{k\in k_{ADR}}{yl}_{ijk}^{ADR}{yn}_{ij}^{ADR}\le m \forall i\in S$$

(7)

$${\sum }_{j\in F}{\sum }_{k\in k_{ADR}}{yl}_{jik}^{ADR}{yn}_{ij}^{ADR}\le m \forall i\in S$$

(8)

$${\sum }_{j\in F}{yl}_{ijk}^{ADR}={\sum }_{j\in F}{yl}_{jik}^{ADR}=1 \forall i\in S, k\in K_{ADR}$$

(9)

$${\sum }_{j\in N_2}{\sum }_{k\in K_{ADR}}{yl}_{ijk}^{ADR}=1 \forall i\in F$$

(10)

$${\sum }_{j\in N_2}{\sum }_{k\in K_{ADR}}{yl}_{jik}^{ADR}=1 \forall i\in F$$

(11)

$${\sum }_{j\in N_2}{yl}_{ijk}^{ADR}={\sum }_{j\in N_2}{yl}_{jik}^{ADR}i\in F, k\in K_{ADR}$$

(12)

$${\sum }_{s\in S}{y}_{sf}^{}\le 1 \forall f\in F$$

(13)

3.2.2. Time Constraints

This model primarily accommodates the emergency situation where late deliveries could lead to disruption to a person’s normal life. To maximize the likelihood of not delaying a person’s life necessities, groceries should be received by the customers within the specified hours without impacting their normal life. We used a hard deadline constraint of customers instead of a soft deadline. However, this was different from traditional emergency logistics with a very tight time window. For our problems, late deliveries were translated to soft deadlines, and we included a violation penalty to represent the loss due to late arrival.

In this study, each customer has a deadline of order ${tl}_f,$ which means that an ADR should arrive at the customer $f$ no later than ${tl}_f.$ If an ADR is delayed ${\delta }_{fk}^{ADR}$ at customer i, the arrival time at customer j might be delayed. In this study, we assumed that a delay happening at i would be independent from j. In other words, one delay at a customer would not result in delay at another customer. The loading time of each customer at a micro hub was denoted as ${\sigma }_f^{},f\in F.$ The speed of the vehicle type v was $u^v.$

$$T_{ik}^{van}=0 \forall i\in O,k\in K_{van}$$

(14)

$${T_{ik}^{ADR}+\tau }_{ij}^{ADR}-T_{jk}^{ADR}\le \left(1-{yl}_{ijk}^{ADR}\right)L\forall i, j\in N_2, k\in K_{ADR}$$

(15)

$$0\le T_{ik}^{ADR}\le {\sum }_{j\in N_2}{yl}_{ijk}^{ADR}L\forall i\in F, k\in K_{ADR}$$

(16)

$${0\le T}_{ik}^{ADR}-{\delta }_{ik}^{ADR}\le {tl}_f{\sum }_{j\in N_2}{yl}_{ijk}^{ADR}\forall i\in F, k\in K_{ADR}$$

(17)

$${T_{ik}^{van}+\rho }_{ij}^{van}-T_{jk}^{van}\le \left(1-{xl}_{ijk}^{van}\right)L\forall i, j\in N_1, k\in K_{van}$$

(18)

$$0\le T_{ik}^{van}\le {\sum }_{j\in N_1}{xl}_{ijk}^{van}L\forall i\in S, k\in K_{van}$$

(19)

Constraint (14) limits all vans leaving the depot at time 0. Constraint (15) restricts the time continuity based on the visiting sequence of nodes in the second echelon. Constraint (16) ensures that if an ADR does not visit a node, the visit time is zero. Constraint (17) represents that if an arrival time at a customer is after the deadline, a delay has occurred (${\delta }_{ik}^{ADR}$ = 1). Constraint (18) restricts the time continuity based on the visiting sequence of nodes in the first echelon. Constraint (19) places the limit that if the van does not visit a node, the visit time is zero.

3.2.3. Synchronization Constraints

Constraint (20) shows that the micro hub is operationally balanced. Constraint (21) guarantees time continuity based on the visiting sequence of nodes between two echelons.

$${\sum }_{i\in F}{yl}_{ijk}^{ADR}={\sum }_{i\in N_1}{xl}_{ijk\prime }^{van}=y_j\forall j\in S, k\in K_{ADR},k\prime \in K_{van}$$

(20)

$${T_{sk}^{van}+\tau }_{sf}^{ADR}+{\delta }_f{y}_{sf}{\sum }_{f\in F}-T_{fk}^{ADR}\le \left(1-{yl}_{fsk}^{ADR}\right)L\forall s\in \mathrm{S},f\in F$$

(21)

3.2.4. Service Constraints

Constraint (22) ensures that the deadline can only be violated when ${yn}_{ik}^{ADR}$ is equal to zero. Constraints (23) and (24) define that only a physical vehicle travels on an arc; the service variable can only be true. Constraints (25) and (26) establish the connection between the service flows and the grocery flows.

$${\delta }_{ik}^{ADR}\le (1-{yn}_{ik}^{ADR})L \forall i\in F, k\in K_{ADR}$$

(22)

$${yn}_{ik}^{ADR}\le {\sum }_{j\in N_2}{yl}_{ijk}^{ADR} \forall i\in F, k\in K_{ADR}$$

(23)

$${xn}_{ik}^{van}\le {\sum }_{j\in N_1}{xl}_{ijk}^{van} \forall i\in F, k\in K_{van}$$

(24)

$${yn}_{ik}^{ADR}L\ge {\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}\forall i\in F, k\in K_{ADR}$$

(25)

$${xn}_{ik}^{van}L\ge {\sum }_{j\in N_1}{\mu }_{jik}^{van}-{\sum }_{j\in N_1}{\mu }_{ijk}^{van}\forall i\in S, k\in K_{van}$$

(26)

3.2.5. Capacity Constraints

Constraint (27) restricts the requirement of the balanced grocery flow for a customer. Constraints (28) and (29) restrict the flow of grocery to not exceed the vehicle capacity. Constraint (30) guarantees that the grocery packed at the micro hub does not exceed the micro-hub capacity.

$${\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}+u_i-{\zeta }_i\ge 0 \forall i\in \mathrm{F}$$

(27)

$${\mu }_{jik}^{ADR}\le {yl}_{ijk}^{ADR}{Q}_{{}_{}}^{ADR} \forall i, j\in N_2, k\in K_{ADR}$$

(28)

$${\mu }_{jik}^{van}\le {xl}_{ijk}^{van}{Q}_{{}_{}}^{van} \forall i, j\in N_1, k\in K_{van}$$

(29)

$${\sum }_{f\in F}{y}_{sf}^{}{\zeta }_f\le w_s{y}_s^{} \forall s\in S$$

(30)

3.2.6. Battery-Life Limit Constraints

Constraint (31) defines that each tour of ADR in the second echelon should be less than the maximum operating life of a battery. In addition, the right side of the inequality ensures the reserved safety buffer for unforeseen circumstances during one navigation.

$${\sum }_{j\in F}WT{u}^{ADR}{yl}_{ijk}^{ADR}\le {\beta }^{range}-{\sigma }^{range}\forall i\in S, k\in K_{ADR}$$

(31)

3.2.7. Definition of Variables

The definition of the variables are listed in Formulas (32)–(40).

$${xl}_{ijk}^{van},{xn}_{sk}^{van}\in \left\{\mathrm{0,1}\right\}\forall i,j\in N_1,s\in S, k\in K_{van}$$

(32)

$${yl}_{ijk}^{ADR}{,yn}_{fk}^{ADR},y_{sf}^{}{,y}_s^{}\in \left\{\mathrm{0,1}\right\}\forall i,j\in N_2,s\in S,f\in F,k\in K_{ADR}$$

(33)

$$y_{sf}^{}{,y}_s^{}\in \left\{\mathrm{0,1}\right\}\forall o\in O,f\in F$$

(34)

$${\mu }_{ijk}^{van}\ge 0,\forall i,j\in N_1,k\in K_{van}$$

(35)

$$U_f^{}\ge 0,\forall f\in F$$

(36)

$$T_{sk}^{van}\ge 0,\forall s\in S,k\in K_{van}$$

(37)

$$T_{fk}^{ADR}\ge 0,\forall f\in F,k\in K_{ADR}$$

(38)

$${\delta }_{sk}^{van}\ge 0,\forall s\in S,k\in K_{van}$$

(39)

$${\delta }_{fk}^{ADR}\ge 0,\forall f\in F,k\in K_{ADR}$$

(40)

3.3. Chance-Constraint Model

Our approach stemmed from the need to address the challenges posed by a 2E-VRP designed for resilient E-grocery delivery. In this context, the starting time of service at a micro hub directly influences the arrival time at a customer’s location. To ensure the earliest possible service to maintain a regular customer experience, we utilized the visit time as a key performance indicator, distinct from conventional metrics like the operational time or travel time. This strategic emphasis on visit time underscores its significance in achieving our overarching goal. By integrating chance-constrained programming into our approach, we can effectively manage the uncertainty associated with travel times and customer demand, ensuring that a delivery network remains resilient and capable of consistently meeting customer expectations, even in the face of unpredictable factors.

The parameters ${\tau }_{ij}^{ADR}$ in constraints (15) (21) represent the stochastic travel time of an ADR, ${ \rho }_{ij}^{van}$ in constraint (18) represents the stochastic travel time of the van, and ${\zeta }_f$ in constraint (27) represents the uncertain demand in our model. If we ignore the random characteristics and represent the parameters by their mean value ${\mu }_{{ \tau }_{ij}^{ADR}},{ \mu }_{{ \rho }_{ij}^{van}},$ $\mathrm{a}\mathrm{n}\mathrm{d} {\mu }_{{\zeta }_f},$ the stochastic 2E-VRP is simplified as a deterministic 2E-VRP. In our study, we assumed that ${ \tau }_{ij}^{ADR},$ ${ \rho }_{ij}^{van}$ and ${\zeta }_f$ were unknown at the time of planning but followed some known probability distributions independently and uniformly. For a given distribution of these parameters, we used chance-constrained programming (CCP) to rewrite constraints (15) (18) (21) (27), where ${\epsilon }_{\rho }^{van}, { \epsilon }_{\tau }^{ADR} {,\epsilon }_f$ represented the confidence level defining the arrival-time constraint of the van, the ADR, and the unmet demand constraint of customers, respectively:

$$\mathrm{P}\left\{\left(\tau |{T_{ik}^{ADR}+\tau }_{ij}^{ADR}-T_{jk}^{ADR}\le \left(1-{yl}_{ijk}^{ADR}\right)L\right)\right\} \ge 1- {\epsilon }_{\tau }^{ADR} \forall i, j\in N_2, k\in K_{ADR}$$

(41)

$$\mathrm{P}\left\{\left(\rho |{T_{ik}^{van}+\rho }_{ij}^{van}-T_{jk}^{van}\le \left(1-{xl}_{ijk}^{van}\right)L\right)\right\}\ge 1- {\epsilon }_{\rho }^{van} \forall i, j\in N_1, k\in K_{van}$$

(42)

$$P\left\{\left(\tau |{T_{sk}^{van}+\tau }_{sf}^{ADR}+{\delta }_f{y}_{sf}-T_{fk}^{ADR}\le \left(1-{yl}_{fsk}^{ADR}\right)L\right)\right\}\ge 1-{\epsilon }_{\tau }^{ADR}\forall s\in \mathrm{S},f\in F$$

(43)

$$\mathrm{P}\left\{\left(\zeta |{\sum }_{k\in K_{van}}{\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}+u_i-{\zeta }_i\ge 0\right)\right\} \ge 1-{\epsilon }_f \forall i\in \mathrm{F}$$

(44)

Constraint (41)–(44) are the chance-constraint model, which replaces the deterministic constraint. To simplify the representation of the parameters, in this paragraph, we use short notation $\tau ,$ $\rho$ and $\zeta$ to substitute ${\tau }_{ij}^{ADR}$, ${\rho }_{ij}^{van}$ and ${\zeta }_f$, respectively. Then, we can define $\tau ,$ $\rho$, and $\zeta$ to follow a lognormal distribution with their means ${\mu }_{\tau }^{},{\mu }_{\rho }^{},$ ${\mu }_{\zeta }$, respectively, and with their standard deviations ${\sigma }_{\tau }^{},{\sigma }_{\rho }^{},{\sigma }_{\zeta }$, respectively. The logarithm log($\tau$), log($\rho$), and log($\zeta$) are normally distributed as normal (${\mu }_{\tau }^{\prime },{{\sigma }_{\tau }^{\prime }),(\mu }_{\rho }^{\prime },{\sigma }_{\rho }^{\prime }),$ $({\mu }_{\zeta }^{\prime },$ ${\sigma }_{\zeta }^{\prime }$), where ${\mu }_{}^{\prime }=log{\mu }_{}^{}-\frac{1}{2}{{\sigma }_{}^{\prime }}^2,$ ${{\sigma }_{}^{\prime }}^2=\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{{\mu }_{}^2+{\sigma }_{}^2}{{\mu }_{}^2}\right)$. We let ${\kappa }_T^{ADR},$ ${\kappa }_T^{van},\mathrm{and} {\kappa }_F$ represent the Z value for the normal distribution corresponding to ${\epsilon }_{\tau }^{ADR},$ ${\epsilon }_{\rho }^{van},$ $\mathrm{a}\mathrm{n}\mathrm{d} {\epsilon }_f$. Therefore, constraints (41)–(44) can be rewritten as

$$T_{ik}^{ADR}+e^{{\mu }_{{ \tau }_{ij}^{ADR}}^{\prime }+{ \sigma }_{{\tau }_{ij}^{ADR}}^{\prime }{\kappa }_T^{ADR}}-T_{jk}^{ADR}\le \left(1-{yl}_{ijk}^{ADR}\right)L \forall i, j\in N_2, k\in K_{ADR}$$

(45)

$$T_{ik}^{van}+e^{{\mu }_{{\rho }_{ij}^{van}}^{\prime }+{ \sigma }_{{\rho }_{ij}^{van}}^{\prime }{\kappa }_T^{van }}-T_{jk}^{van}\le \left(1-{xl}_{ijk}^{van}\right)L \forall i, j\in N_1, k\in K_{van}$$

(46)

$$T_{sk}^{van}+e^{{\mu }_{{ \tau }_{ij}^{ADR}}^{\prime }+{ \sigma }_{{\tau }_{ij}^{ADR}}^{\prime }{\kappa }_T^{ADR}}+{\delta }_f{y}_{sf}-T_{fk}^{ADR}\le \left(1-{yl}_{fsk}^{ADR}\right)L \forall s\in \mathrm{S},f\in F$$

(47)

$${\sum }_{k\in K_{van}}{\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}+u_i-e^{{\mu }_{{\zeta }_i}^{\prime }+{ \sigma }_{{\zeta }_i}^{\prime }{\kappa }_F}\ge 0 \forall i\in F$$

(48)

We assumed that the stochastic parameters ${\tau }_{ij}^{ADR}$, ${\rho }_{ij}^{van}$ and ${\zeta }_f$ are only known to belong to a given uncertainty set $\pi$. The robust optimization requires that the solutions satisfy the constraints with stochastic parameters in the set $\pi$. Thus, constraints (45)–(48) can be expressed as

$${T_{ik}^{ADR}+\tau }_{ij}^{ADR}-T_{jk}^{ADR}\le \left(1-{yl}_{ijk}^{ADR}\right)L \forall {\tau }_{ij}^{ADR}\in \pi ,i, j\in N_2, k\in K_{ADR}$$

(49)

$${T_{ik}^{van}+\rho }_{ij}^{van}-T_{jk}^{van}\le \left(1-{xl}_{ijk}^{van}\right)L \forall {\rho }_{ij}^{van}\in \pi , i, j\in N_1, k\in K_{van}$$

(50)

$${T_{sk}^{van}+\tau }_{sf}^{ADR}+{\delta }_f{y}_{sf}-T_{fk}^{ADR}\le \left(1-{yl}_{fsk}^{ADR}\right)L \forall {\tau }_{ij}^{ADR}\in \pi ,s\in S,f\in F$$

(51)

$${\sum }_{k\in K_{van}}{\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}+u_i-{\zeta }_i\ge 0 \forall {\zeta }_i\in \pi ,i\in F$$

(52)

All the stochastic parameters ${\tau }_{ij}^{ADR}\in \pi ,$ ${\rho }_{ij}^{van}\in \pi ,$ ${\zeta }_i\in \pi$ in constraints (49)–(52) can be substituted by the maximum possible uncertainty parameters ${{{\tau }_{\mathit{ij}}^{\mathit{ADR}}}^{\mathrm{*}}=\mathrm{m}\mathrm{a}\mathrm{x}\tau }_{ij}^{ADR}\in \pi ,$ ${{\rho }_{ij}^{van}}^{\mathrm{*}}=$ max ${\rho }_{ij}^{van}\in \pi ,$ $\mathrm{a}\mathrm{n}\mathrm{d} {{\zeta }_i}^{\mathrm{*}}=\mathrm{m}\mathrm{a}\mathrm{x}{\zeta }_i\in \pi$. Considering the maximum uncertainty parameters, constraints (49)–(52) can be rewritten as

$$T_{ik}^{ADR}+{{\tau }_{ij}^{ADR}}^{*}-T_{jk}^{ADR}\le \left(1-{yl}_{ijk}^{ADR}\right)L \forall i, j\in N_2, k\in K_{ADR}$$

(53)

$${{T_{ik}^{van}+\rho }_{ij}^{van}}^{*}-T_{jk}^{van}\le \left(1-{xl}_{ijk}^{van}\right)L \forall i, j\in N_1, k\in K_{van}$$

(54)

$$T_{sk}^{van}+{{\tau }_{ij}^{ADR}}^{*}+{\delta }_f{y}_{sf}-T_{fk}^{ADR}\le \left(1-{yl}_{fsk}^{ADR}\right)L \forall s\in \mathrm{S},f\in F$$

(55)

$${\sum }_{k\in K_{van}}{\sum }_{j\in N_2}{\mu }_{jik}^{ADR}-{\sum }_{k\in K_{ADR}}{\sum }_{j\in N_2}{\mu }_{ijk}^{ADR}+u_i-{ {\zeta }_i}^{*}\ge 0 \forall i\in F$$

(56)

4. Experimentation

In this section, we examine both the deterministic model and the chance-constraint model for 2E-VRP-SSSD across various scenarios, taking into account influential factors such as the ratio of travel speed between the ADRs and vans (RSAV), and the customer service deadline (DL). The procedure of our experimentation is illustrated in (Figure 2). We firstly described the datasets and parameter setting for our experiments, followed by an assessment of the impact of sensitive factors ${\epsilon }_{\rho }^{van}, {\epsilon }_{\tau }^{ADR}{,\epsilon }_f$ on our optimization model. Finally, we delved into a comprehensive discussion of the results obtained from our experiments.

4.1. Data Sets and Parameters

In our study, we evaluated the performance of the proposed deterministic model and the chance-constraint modal through virtual simulation experiments, with randomly generated instances with different scales (

Table 1. Characteristics of the data sets.

	Data Sets
No. of depots (O), micro hubs (S), customers (F) and vehicles $K_{van}, K_{ADR}$
Small datasets Depot = 1, micro hub = 1, customers = 10	Medium datasets Depot = 1, micro hub = 2, customers = 50	Large datasets Depot = 1, micro hub = 4 Customers = 100
Parameters	Deterministic model/Chance-constraint model
Coordination (Small)	(20, 20)
Coordination (Medium)	(100, 100)
Coordination (Large)	(200, 200)
n, m (Small-scale)	2, 2
n, m (Medium-scale)	4, 10
n, m (Large-scale)	10, 20
The average speed $u_{}^{van},u_{}^{ADR }$	U (1, 6), U (1, 6)
Handling capacity of micro hub	300
Service deadline of customer ${tl}_f$	${ \mu }_{{\tau }_{ijk}^{ADR}}+$U (2, 6) × WT + 20
Loading time at a node ${\delta }_f$	U (0.1, 0.5) × WT
Safety buffer of the battery ${\alpha }^{range}$	U (1.5, 2.5) × WT
Maximum operating range of the battery ${\beta }^{range}$	${\tau }_{ijk}^{ADR}+U$ (1.5, 2.5) × WT
Parameters	Deterministic model	Chance-constraint model
Demand of customer ${\zeta }_i$	Lognormal (${ \mu }_{{\zeta }_i}$, $\frac{1}{5}{ \mu }_{{\zeta }_i}$)	Lognormal (${ \mu }_{{\zeta }_i}$, $\frac{1}{5}{ \mu }_{{\zeta }_i}$) considering ${\epsilon }_{\rho }^{van}, {\epsilon }_{\tau }^{ADR}{,\epsilon }_f$
Travel time of van ${\rho }_{ijk}^{van}$	Lognormal (${ \mu }_{{\rho }_{ijk}^{van}}$, $\frac{UB-{ \mu }_{{\rho }_{ijk}^{van}}}{100}{ \mu }_{{\rho }_{ijk}^{van}}$) *	Lognormal (${ \mu }_{{\rho }_{ijk}^{van}}$, $\frac{UB-{ \mu }_{{\rho }_{ijk}^{van}}}{100}{ \mu }_{{\rho }_{ijk}^{van}}$) *considering ${\epsilon }_{\rho }^{van}, {\epsilon }_{\tau }^{ADR}{,\epsilon }_f$
Travel time of ADR ${\tau }_{ijk}^{ADR}$	Lognormal (${\tau }_{ijk}^{ADR}$, $\frac{UB-{\tau }_{ijk}^{ADR}}{100}{\tau }_{ijk}^{ADR}$)	Lognormal (${\tau }_{ijk}^{ADR}$, $\frac{UB-{\tau }_{ijk}^{ADR}}{100}{\tau }_{ijk}^{ADR}$) considering ${\epsilon }_{\rho }^{van}, {\epsilon }_{\tau }^{ADR}{,\epsilon }_f$
${\epsilon }_{\rho }^{van}, {\epsilon }_{\tau }^{ADR}{,\epsilon }_f$	none	1.56

* UB is the upper bound value, which is greater than the longest arc in the graph.

): small (within 10 nodes), medium (50 nodes), and large (100 nodes). For each scale of datasets, customers with randomly generated mean demands are located in a specific square (20, 20), (100, 100), (200, 200). Instances generated for each scale of datasets are distinguished by the distribution of customers and micro-hubs. Each instance includes depots (O), micro hubs (S), customers (F), and vehicles ($K_{van}, K_{ADR}$), representing the generated network. We performed simulations on five scenarios for each instance and calculated the average value as the result. Another factor we tested was the impact of the confidence level defining the arrival time and demand constraint (${\epsilon }_{\rho }^{van},{\epsilon }_{\tau }^{ADR}{,\epsilon }_f).$

In our experiment, the parameters of demand and travel time were set using data from randomly generated instances. We set the parameters of demand and travel time as their mean values. The choice of the mean values as the parameters was made because they are commonly used as central values in statistical modeling and provide a starting point for our analysis. In the deterministic model, the demand and travel time followed a lognormal distribution, with the same $\mu$ as was used in the chance-constraint model. The choice of a lognormal distribution to model the demand and travel time is often based on the characteristics of the data and the assumption that various factors combine multiplicatively, or that the data exhibit right-skewed behavior. Also, when there is no dominant influence on demand and travel time, and many random factors contribute, a lognormal distribution can be a reasonable approximation. In this study, $\sigma$ was set proportional to the $\mu$ of demand (20% of the $\mu$) and inversely proportional to the $\mu$ of travel time ($\sigma =\frac{UB-{ \mu }_{{\rho }_{ijk}^{van}}}{100}{ \mu }_{{\rho }_{ijk}^{van}}$; UB is the upper bound for the transformation). For the chance-constraint model, the confidence level was set to 88%; thus, ${\epsilon }_{\rho }^{van},{\epsilon }_{\tau }^{ADR}{,\mathrm{a}\mathrm{n}\mathrm{d} \epsilon }_f$ were set to 1.56.

4.2. Comparison of Deterministic Model and Chance-Constraint Model

In the stochastic two-echelon delivery network, the route decision is sensitive to both the travel speed of the vehicles and the deadline requests. We simulated the produced routes under a combination impact of the two parameters in terms of the unmet demand: the travel speed between the ADR and the van, and the service deadline of the customer. We varied the levels of the two parameters and observed the produced results by both the deterministic model and chance-constraint model. Thus, we divided the ratio of the travel speed between the ADR and the van (RSAV) into six levels: 60%, 80%, 100%, 120%, 150% and 200%. The base speed for the van was 10 miles/h (The average speed of a van is lower than the speed traveled in a normal delivery network). The deadline (DL) was set to 40%, 60%, 80%, 100% and 120% of the route length. The base route length was five times the average length of all the links in the network. There were, in total, 30 cases generated in our experiments, which were identified by the combination of two factors with different levels (6 types ∗ 5 types). To summarize, in this experiment, we experimented with 12 instances, and for each instance we performed 30 cases with various speed ratios and deadlines. Hence, the simulation results were obtained through 360 instances. These instances were tested using both the deterministic model and chance-constraint model under the impact of RSAV and DL (as shown in

Table 2. Percentage of unmet demand produced by RSAV and DL (%).

	DDL 200 (40%)	DDL 300 (60%)	DDL 400 (80%)	DDL 500 (100%)	DDL 600 (120%)	CDL 200 (40%)	CDL 300 (60%)	CDL 400 (80%)	CDL 500 (100%)	CDL 600 (120%)
RSAV60 (60%)	18.91	17.27	16.83	14.26	9.78	18.45	16.34	16.19	14.33	11.09
RSAV80 (80%)	14.48	12.36	12.11	9.67	6.05	13.03	11.82	11.47	10.04	7.88
RSAV100 (100%)	11.22	10.56	9.86	8.08	4.40	10.92	10.20	9.53	9.01	3.85
RSAV120 (120%)	8.44	6.89	5.72	4.01	0.00	8.21	6.31	5.06	4.70	0.00
RSAV150 (150%)	6.78	6.06	4.23	2.59	0.00	5.15	5.52	4.14	2.67	0.00
RSA200 (200%)	4.58	4.11	3.27	1.95	0.00	4.00	3.90	2.99	2.04	0.00

DDL: deterministic deadline; CDL: chance-constraint deadline.

).

We found that the when the deadlines were tight, for example, 40% to 80% of the route length, the outcome of deterministic model outperformed the chance-constraint model, since the route generated by the deterministic model can mostly arrive on time to serve the demand requested in the beginning. In the situation where the values of the RSAV were small, for example, RSAV60 (60%)-DDL500/CDL500 (100%), and RSAV80 (80%)-DDL500/CDL500 (100%), the deterministic model and chance-constraint model yielded a higher percentage of unmet demand (Figure 3). We suspect that this is due to the low speed of ADRs and vans.

4.3. The Impact of Confidence Level ${\epsilon }_{\rho }^{van},{\epsilon }_{\tau }^{ADR}{,and \epsilon }_f$

In this section, we tested the impact of the confidence level on the route decisions, where the deadline and the ratio between the ADR and van was fixed. For example, for one specific network (one instance), we fixed the speed ratio at 100% and the deadline at 80%. In the van–ADR delivery system, we assumed that the ADR was much more reliable than the van, considering the lack of a truck driver and other uncertainties of van deliveries in the network. Thus, we ran the simulation under the chance-constraint model with different values of the confidence level only for ${ \epsilon }_{\rho }^{van}{,\epsilon }_f$, and fixed the ${\epsilon }_{\tau }^{ADR}.$ The percentage of unmet demand is shown in Figure 4. Since the service time and demand of the deterministic model follow lognormal distribution, the mean value of the deterministic route was bigger than 0.5 rather than exactly 0.5 (normal distribution). The minimal unmet demand of the deterministic model is achieved when ${ \epsilon }_{\rho }^{van}=0.75{,\epsilon }_f=0.75$ (marked with orange square). Then, we can generate different routes by changing the confidence level ${ \epsilon }_{\rho }^{van}{,\epsilon }_f$ for the chance-constraint model. We used the same value of confidence level for both ${ \epsilon }_{\rho }^{van}{,\epsilon }_f$. For all the instances that we tested, the minimal percentage of unmet demand corresponding to the confidence level was 1.56, which was also the number we used before in the experimentation. If the confidence level is too big, then all the routes are planned to overpass the deadline (move towards the right end of the confidence level). If the confidence level is too small, then all the routes will be too conservative (move towards the left end of the confidence level).

5. Discussion

One of the central observations from the study is the sensitivity of route decisions in a stochastic two-echelon delivery network to both the RSAV and DL. This sensitivity is critical to understanding how the delivery network performs under various conditions. The study systematically varied the RSAV and DL to observe their combined impact on the percentage of unmet demand. As expected, the results showed that when deadlines were tight, the deterministic model outperformed the chance-constraint model. This is because the deterministic model prioritizes on-time delivery, resulting in a lower percentage of unmet demand. Conversely, the chance-constraint model, designed to manage uncertainties, occasionally led to a higher percentage of unmet demand, particularly when the values of RSAV were small. This sensitivity analysis highlights the trade-off between risk and reliability in logistics operations. Tighter deadlines improve reliability but can lead to higher costs or unmet demand, especially when faced with slower vehicle speeds.

This study also explored the impact of confidence levels ${\epsilon }_{\rho }^{van},{\epsilon }_{\tau }^{ADR}{,\mathrm{a}\mathrm{n}\mathrm{d} \epsilon }_f$ on route decisions while keeping the RSAV and DL fixed. The study assumed that ADRs are more reliable than vans due to factors like the lack of truck drivers and uncertainties associated with van deliveries. Under the chance-constraint model, different values of confidence levels were tested, leading to varying percentages of unmet demand. The results show that balancing risk and reliability is crucial. Overly conservative confidence levels result in low unmet demand but may lead to inefficiencies. Conversely, overly optimistic confidence levels can result in missed deadlines and higher unmet demand.

These findings have practical implications for enterprise risk management within E-grocery delivery operations. It underscores the importance of understanding how different factors, such as travel speed, deadlines, and confidence levels, interact and influence the reliability and risk profiles of delivery networks. For enterprises, this study suggests the need to strike a balance between risk and reliability when designing delivery routes and operations. Tight deadlines may improve customer satisfaction but increase the risk of unmet demand, especially in scenarios with slower vehicle speeds. Additionally, the choice of confidence levels in chance-constraint models should be carefully considered to optimize both efficiency and risk mitigation.

In summary, the chance-constraint programming model of 2E-VRP-SSSD, as explored in this study, offers valuable insights into the complexities of logistics and risk management. The findings underscore the importance of adopting a nuanced approach that carefully balances risk, reliability, and resilience to optimize logistics operations in an uncertain and dynamic environment.

6. Conclusions and Future Work

In this study, we investigated the resilience of an ADR-assisted E-grocery delivery network in which the service time and demand are stochastic. We set the unmet demand instead of cost as the objective. The problem was formulated as a mixed-integer programming problem by both deterministic model and chance-constraint model. Unique features of this problem were embedded in our model, such as a limited number of vehicles, the battery life of the ADRs, synchronization of two echelons, and service-time requirement. Experimentations were implemented to compare the effectiveness of the deterministic model and chance-constraint model under different deadlines and speed ratios between the ADR and the van. Furthermore, the confidence level defining the arrival time and demand constraint was also demonstrated by the experimentations. We concluded that the chance-constraint model is more effective in providing optimal routes that better cover the overall demand, using a moderate speed ratio between the ADR and van and a loose deadline. A balanced confidence level could produce a lower percentage of unmet demand. Future studies in this area will involve developing a tighter lower bound, which might improve the effectiveness of the model, and also facilitate the elimination of the infeasible solutions to reduce the calculation time.