Two-Echelon Location–Routing Problem of Perishable Products Based on the Integrated Mode of In-Store Pick-Up And Delivery

Abstract

The integrated service mode of in-store pick-up and delivery has become common in the post-epidemic period owing to the combined online and offline purchases of perishable products. This study investigates the diverse requirements of in-store pick-up and delivery customers. Then, it establishes a two-echelon location–routing model for a perishable food distribution network to minimize total cost as an objective. An adaptive large neighborhood search (ALNS) algorithm was also developed to solve the foregoing problem. To test the algorithm, instances from those of Solomon are derived. The proposed ALNS algorithm was found to achieve satisfactory performance with respect to speed and accuracy by comparing its results with those of the CPLEX software for a 12-node small-scale instance. The applicability and stability of the ALNS algorithm were further verified using different types of instances with more nodes. Different proportions of in-store pick-up and delivery customers were set, and the total cost of location–routing schemes under these proportions was compared. The results show that an integrated service type compared with the single delivery service mode and single in-store pick-up service mode can save 7.98% and 11.44% of the total cost, respectively.

Keywords

freight systems city logistics and last-mile strategies street use delivery logistics pick-up

Although e-commerce has considerably advanced in the last decade, the penetration level of online e-commerce of perishable goods is relatively low compared with other products. The COVID-19 outbreak has led to the rapid development of the e-commerce of perishable products, thus promoting online purchasing. In the post-epidemic period, some consumers have returned to offline purchasing, resulting in the integration of online and offline retailing. This integration focuses on satisfying the requirements of users with respect to in-store shopping and product pick-up as well as personalized delivery services. The integrated mode provides four different choices for customers: conventional retail shopping; online purchases requiring product pick-up; in-store purchases allowing requests for product delivery; and online purchases with a product delivery service. Based on the movement modes of perishable products, the first two options in the foregoing are referred to as “in-store pick-up” modes, and the latter two are considered as “delivery” modes. Consumers typically have higher requirements for the delivery of perishable products than for non-perishable goods. Consequently, optimizing the distribution network of perishable products and improving service quality as well as distribution efficiency while reducing costs have become urgent problems to satisfy the different requirements of customers with respect to service type.

In their research on perishable product distribution, Suraraksa and Shin ( 1 ) combined location–allocation and vehicle routing problems (VRPs) with time windows. They considered the number of distribution centers (DCs) and trucks, total travel time, total travel distance, and fairness in the distribution of work among drivers. They further designed a perishable product distribution network in Bangkok under different scenarios using a geographic information system. Zulvia et al. ( 2 ) established a multi-objective vehicle routing model for perishable products by considering time-dependent factors, including the customer time window, departure time, and peak and off-peak working days. Golestani et al. ( 3 ) studied a bi-objective green hub location problem for multiple perishable products stored at different temperatures and simultaneously distributed in the supply chain. The objectives are to minimize the total cost and maximize the delivery quality while considering the carbon emissions of the transportation system. Wang et al. ( 4 ) investigated resource sharing among multiple DCs to reduce convective transport and optimize transportation routes. They established a bi-objective optimization model of fresh product distribution with the objectives of minimizing cost and number of vehicles considering the temperature control condition. Hiassat et al. ( 5 ) conceived a location–inventory–routing model for perishable products and developed a genetic algorithm approach. Navazi et al. ( 6 ) examined the forward and reverse flows of perishable products and energy consumption of refrigeration during delivery. They developed a multi-objective model for a sustainable location–routing problem (LRP) with simultaneous pick-up and delivery that minimized network implementation costs, environmental side effects, and societal satisfaction.

The LRP is the integration of the location–allocation problem and VRP. Numerous studies, which can refer to those surveys by Nagy and Salhi ( 7 ), Prodhon and Prins ( 8 ), and Hu et al. ( 9 ), have been conducted on the LRP. The two-echelon LRP (2E-LRP) is an extension of the LRP. Research on two-echelon or multi-echelon distribution networks has gradually advanced owing to the expansion of the urban scale and traffic restrictions on trucks. Nguyen et al. ( 10 ) developed a hybrid metaheuristic algorithm to solve a 2E-LRP using learning and path relinking to reinforce the greedy stochastic adaptive search algorithm. Rahmani et al. ( 11 ) studied a 2E-LRP with multiple products, pick-up, and delivery. Darvish et al. ( 12 ) investigated a 2E-LRP with flexible network design and delivery time and then solved it using an exact method of branch and bound algorithms. Mirhedayatian et al. ( 13 ) considered synchronous flow in an intermediate facility between two echelons based on the limited capacity or waiting time of the intermediate facility. They also studied the reverse flow of the product. To study the 2E-LRP, Tirkolaee et al. ( 14 ) and Heidari et al. ( 15 ) focused on environmental factors, such as energy efficiency and CO₂ emissions.

The on-route delivery time for fresh products must be as short as possible owing to their perishable characteristics. The two-echelon distribution network has the advantage of reducing the last-mile delivery time. However, reports on the 2E-LRP of perishable products are rarely published in the literature. Govindan et al. ( 16 ) introduced a 2E-LRP with time windows for ensuring a sustainable supply chain of perishable products. They considered the costs of carbon and greenhouse gas emissions and established a mathematical model for optimizing economic and environmental objectives. Bala et al. ( 17 ) constructed a two-echelon time–space network with a given production schedule and focused on a heuristic method to solve large-scale problems. The relevant works reported in the literature are summarized in Table 1.

Table 1.

Summary of Relevant Works Reported in the Literature

Literature	Network	Fresh products	Carbon emission	Time window	Multiple products	Customer type	Pick-up and delivery
Golestani et al. ( 3 )	1E	√	√	X	√	X	X
Wang et al. ( 4 )	1E	√	X	√	√	X	X
Nguyen et al. ( 10 )	2E	X	X	X	X	X	X
Rahmani et al. ( 11 )	2E	X	X	X	√	X	√
Darvish et al. ( 12 )	2E	X	X	√	X	X	X
Mirhedayatian et al. ( 13 )	2E	X	√	√	X	X	√
Tirkolaee et al. ( 14 )	2E	X	√	X	X	X	X
Heidari et al. ( 15 )	2E	X	√	X	X	X	X
Govindan et al. ( 16 )	2E	√	√	√	X	X	X
Bala et al. ( 17 )	2E	√	X	√	X	X	X
This paper	2E	√	X	√	√	√	X

This study focuses on the background of the combined online and offline purchases of perishable products and considers the integrated service mode of in-store pick-up service, delivery, and demand for multiple products. Accordingly, a new 2E-LRP model of perishable product distribution with two service-type customers and time windows is formulated. This study provides a reference for the decision-making of perishable product distribution schemes.

The remainder of this paper is organized as follows. The second section presents a formal definition of the problem and its mathematical formulation. A two-stage adaptive large neighborhood search (ALNS) algorithm is proposed in the third section. The numerical experiments and results of the algorithm are presented in the fourth section. Finally, the conclusions are summarized in the fifth section.

Problem Formulations

Problem Description

In a certain region, the distribution system of perishable products includes DCs, stores, delivery customers, and in-store pick-up customers. Consider a perishable product distribution network, $G = (N, A)$ , where $N = {N_{D} \cup N_{S} \cup N_{C}}$ is the set of all vertices containing the set of candidate DCs, $N_{D}$ , and the set of candidate stores, $N_{S}$ ( ${N_{S}}^{'}$ is the copy of $N_{S}$ ). Moreover, the network includes the set of customers, $N_{C} = N_{C}^{1} \cup N_{C}^{2}$ , which can be divided into in-store pick-up customers, $N_{C}^{1}$ , and delivery customers, $N_{C}^{2}$ . The network further includes $A = A_{1} \cup A_{2}$ , which denotes the set of arcs (where $A_{1} = {(i, j) \in A | i, j \notin N_{C}}$ in the first-echelon network that can be traversed by vehicle set $K$ , and $A_{2}$ is the set of arcs in the second-echelon network traversed by vehicle set $V$ ). The set of perishable product types is represented by $P$ . Each customer, $i \in N_{C}$ , has a demand, $q c_{ip}$ , for a product type, $p \in P$ ; moreover, each store, $i \in N_{S}$ , has a demand, $q s_{ip}$ . Each customer has a time window, $[e_{i}, l_{i}]$ , in which service starts; each also has service time ( $w_{i}$ ). To satisfy each customer’s demand and provide accessible stores for the customers of in-store pick-up, this study determines the location and number of open DCs and stores. Furthermore, it plans the optimal vehicle routes of the first-level distribution network and second-level delivery network while minimizing the total cost and matching in-store pick-up customers with appropriate stores. An overview of the 2E-LRP of perishable products considered in this study is shown in Figure 1.

Figure 1.

Two-echelon location–routing problem of perishable products.

Assumptions

(1) Because of the refrigerating system in DCs during first-echelon transportation and in-store shopping, the decay of products is assumed to be insensitive to time until they leave the stores of the second echelon. The initial freshness of all perishable products is set to $θ_{0}$ when they leave the stores (by delivery vehicles or in-store pick-up).

(2) The locations of candidate DCs, stores, and customers are known. Moreover, the capacity of DCs and stores is limited.

(3) The first-echelon and second-echelon trips are served by identical large refrigerated vehicles and delivery vehicles with small capacities, respectively. In addition, the vehicles are driven at a constant speed.

(4) No delivery failures or secondary deliveries occur.

(5) A customer is only visited once by a delivering electric vehicle or picks up a product at a store once, that is, the demand cannot be split.

(6) The distance of a delivery trip is shorter than the driving distance restrictions of vehicles.

(7) The start time of second-echelon trips for each vehicle is recorded as zero; the distribution time of the first-echelon network is excluded.

(8) The first level (from DCs to stores) delivers products without breaking a case apart. The second level serves customers who order perishable products in a low-intensity and high-frequency manner. Therefore, delivery from DCs to customers is not allowed.

Notations

The definitions of variables and constants are summarized in Table 2.

Table 2.

Definition of Notations

Parameters	Description
$F_{d}$ ( $F_{s}$ )	Fixed cost to open DC; $d \in N_{D}$ (store $s \in N_{S}$ )
$Q_{d}$ ( $Q_{s}$ )	Capacity of DC; $d \in N_{D}$ (store, $s \in N_{S}$ )
$F_{k}$ ( $F_{v}$ )	Fixed cost of dispatching first-echelon vehicle; $k \in K$ (second-echelon vehicle, $v \in V$ )
$C_{k}$ ( $C_{v}$ )	Unit transportation cost of first-echelon vehicle, $k \in K$ (second-echelon vehicle, $v \in V$ )
$Q_{k}$ ( $Q_{v}$ )	Capacity of first-echelon vehicle, $k \in K$ (second-echelon vehicle, $v \in V$ )
$M_{K}$ ( $M_{V}$ )	Maximum number of vehicles used by DCs (stores)
$\| S \|$	Number of open stores
$\| C \|$	Number of delivery customers
$v_{a}$	Average travel speed of second-echelon vehicles
$v_{b}$	Average walking speed of in-store pick-up customers
$d_{ij}$	Distance between vertices $i$ and $j$ , $(i, j) \in A$
$t_{ij}$	Travel time between vertices $i$ and $j$ in second-echelon network, $(i, j) \in A_{2}$
$s t_{i}$	Walking time of in-store pick-up customer, $i \in N_{C}^{1}$ , returning from store
$α$	Early arrival penalty cost
$β$	Late arrival penalty cost
$q c_{ip}$	Demand for product, $p \in P$ , of customer, $i \in N_{C}$
$q s_{ip}$	Demand for product, $p \in P$ , of store, $i \in N_{S}$
$G_{p}$	Unit value of product, $p \in P$
$f_{p}$	Unit cooling cost of product, $p \in P$
$D_{i}$	Farthest acceptable distance for customer ( $i \in N_{C}^{1}$ ) to visit
$φ_{i}$	Optimal acceptable distance for customer ( $i \in N_{C}^{1}$ ) to visit
$R$	Offline service scope of stores
$μ$	Distance penalty cost coefficient
$δ$	Labor cost for each in-store pick-up customer
$M$	Maximum constant number
Variables	Description
$u_{ik} / u_{iv}$	Auxiliary variables for eliminating sub-tours
$s_{iv}$	Arrival time of second-echelon vehicle, $v \in V$ , at customer, $i \in N_{C}^{2}$
$r_{i}$	Binary variable: 1 if candidate DC, $i \in N_{D}$ , is opened; otherwise, 0
$z_{i}$	Binary variable: 1 if candidate store, $i \in N_{S}$ , is opened; otherwise, 0
$x_{ijk}$	Binary variable: 1 if vehicle, $k \in K$ , travels by $(i, j) \in A_{1}$ ; otherwise, 0
$y_{ijv}$	Binary variable: 1 if vehicle, $v \in V$ , travels by $(i, j) \in A_{2}$ ; otherwise, 0
$η_{ij}$	Binary variable: 1 if store, $j \in N_{S}$ , is served by DC, $i \in N_{D}$ ; otherwise, 0
$λ_{ij}$	Binary variable: 1 if customer, $j \in N_{C}$ , is served by store, $i \in N_{S}$ ; otherwise, 0

Note: DC = distribution center.

Mathematical Formulation

Objective Function

This study considers the relevant costs of the distribution network layout of enterprises and builds a model to minimize the total costs in the distribution network of fresh products. The total costs include the fixed costs for opening and operating the selected DCs and stores ( $Z_{1}$ ); fixed costs of dispatching a first-echelon refrigeration vehicle and a second-echelon delivery vehicle ( $Z_{2}$ ); delivery costs in the first-echelon and second-echelon networks ( $Z_{3}$ ); decay costs ( $Z_{4}$ ); cooling cost ( $Z_{5}$ ); time penalty cost ( $Z_{6}$ ); distance penalty cost ( $Z_{7}$ ); and labor service cost for in-store pick-up customers ( $Z_{8}$ ). The objective function is expressed as in Equation 1:

min Z = Z_{1} + Z_{2} + Z_{3} + Z_{4} + Z_{5} + Z_{6} + Z_{7} + Z_{8} .

(1)

The costs considered in this model are described and calculated as follows:

Z_{1} = \sum_{i \in N_{D}} F_{d} r_{i} + \sum_{i \in N_{S}} F_{s} z_{i},

(2)

Z_{2} = \sum_{i \in N_{D}} \sum_{j \in N_{S}} \sum_{k \in K} F_{k} x_{ijk} + \sum_{i \in N_{S}} \sum_{j \in N_{C}^{2}} \sum_{v \in V} F_{v} y_{ijv},

(3)

Z_{3} = \sum_{(i, j) \in A_{1}} \sum_{k \in K} C_{k} d_{ij} x_{ijk} + \sum_{(i, j) \in A_{2}} \sum_{v \in V} C_{v} d_{ij} y_{ijv} .

(4)

Because fresh products are perishable, their quality gradually deteriorates after leaving a store. Therefore, cooling measures are required during second-echelon deliveries. Because different refrigeration methods with different costs affect the degree of decay of perishable products during the delivery process, a fresh-keeping cost coefficient of fresh products is introduced. The freshness of a product is affected by the dual factors of time and product preservation costs. Therefore, when the perishable product, $p \in P$ , is delivered to a customer, $i \in N_{C}^{2}$ , the use of a freshness function (i.e., Equation 5) is proposed:

θ_{ip} = {\begin{matrix} θ_{0} \cdot {θ_{0}}^{(σ_{p} - f (ε_{p})) s_{iv}}, \forall i \in N_{C}^{2} \\ θ_{0} \cdot {θ_{0}}^{(σ_{p} - f (ε_{p})) s t_{i}}, \forall i \in N_{C}^{1} \end{matrix},

(5)

where $θ_{0}$ is the initial freshness of fresh products; $σ_{p}$ is the decay coefficient; $f (ε_{p})$ is the fresh-keeping cost coefficient of product $p$ ; and $f (ε_{p}) = 1 - e^{- ε_{p}}$ . Thus, the decay cost is expressed by Equation 6:

\begin{matrix} Z_{4} = \sum_{i : (i, j) \in A_{2}} \sum_{j \in N_{C}^{2}} \sum_{v \in V} \sum_{p \in P} y_{ijv} G_{P} q c_{jp} (θ_{0} - θ_{jp}) \\ + \sum_{i \in N_{S}} \sum_{j \in N_{C}^{1}} \sum_{p \in P} λ_{ij} G_{P} q c_{jp} (θ_{0} - θ_{jp}) . \end{matrix}

(6)

Because delivery vehicles do not have refrigeration functions, refrigeration is required to reduce the decay of perishable products during delivery, resulting in refrigeration costs. The additional cooling cost incurred by opening the door during the distribution process is ignored. The cooling cost is calculated using Equation 7:

Z_{5} = \sum_{v \in V} \sum_{i : (i, j) \in A_{2}} \sum_{j \in N_{C}^{2}} \sum_{p \in P} f_{p} q c_{jp} s_{jv} y_{ijv} .

(7)

Because of the perishability of products and delivery time mismatch, customers typically have specific time window requirements for delivery. This study uses soft time windows to analyze the penalty cost of violating time windows. The penalty function of the soft time window is given by Equation 8. Based on this equation, the time penalty cost equation, Equation 9, is derived:

F (s_{iv}) = {\begin{matrix} \begin{matrix} α (e_{i} - s_{iv}), & s_{iv} < e_{i} \end{matrix} \\ \begin{matrix} 0, & e_{i} \leq s_{iv} \leq l_{i} \end{matrix} \\ \begin{matrix} β (s_{iv} - l_{i}), & l_{i} < s_{iv} \end{matrix} \end{matrix},

(8)

Z_{6} = \sum_{j : (i, j) \in A_{2}} \sum_{i \in N_{C}^{2}} \sum_{v \in V} y_{jiv} F (s_{iv}) .

(9)

Customers located close to perishable product stores may choose the in-store pick-up service; therefore, the pick-up distance is a significant factor to consider while locating perishable product stores. In addition, stores provide a certain range of offline services. The willingness of customers to visit a store beyond this range is virtually zero. The distance penalty is defined as a piecewise function, as given by Equation 10. Thus, the distance penalty cost can be derived as in Equation 11:

F (d_{ij}) = {\begin{matrix} 0 d_{ij} < φ_{i} \\ 1 - (\frac{D_{i} - d_{ij}}{D_{i} - φ_{i}}) φ_{i} < d_{ij} < D_{i} \\ 1 + (d_{ij} - D_{i}) D_{i} < d_{ij} < R \\ M d_{ij} > R \end{matrix},

(10)

Z_{7} = \sum_{i \in N_{S}} \sum_{j \in N_{C}^{1}} μ λ_{ij} F (d_{ij}) .

(11)

The last term, $Z_{8}$ , is the labor service cost for in-store pick-up customers. Stores provide labor services (such as reception, guidance, and in-store cleaning) for in-store pick-up customers, resulting in the customer service labor cost given by Equation 12:

Z_{8} = \sum_{i \in N_{S}} \sum_{j \in N_{C}^{1}} δ λ_{ij} .

(12)

Constraints

The constraints for the first-echelon distribution network are expressed by Equations 13–23:

\sum_{i \in N_{D} \cup N_{S}} \sum_{k \in K} x_{ijk} = z_{j}, \forall j \in N_{S},

(13)

\sum_{i \in N_{D} \cup N_{S}} x_{ijk} = \sum_{i \in N_{D} \cup N_{S}} x_{jik}, \forall j \in N_{D} \cup N_{S}, \forall k \in K,

(14)

\sum_{i \in N_{D}} \sum_{j \in N_{D}} x_{ijk} = 0, \forall k \in K,

(15)

\sum_{i \in N_{D}} \sum_{j \in N_{S}} x_{ijk} \leq 1, \forall k \in K,

(16)

\sum_{i \in N_{S}} x_{dik} + \sum_{i \in N_{D} \cup N_{S}} x_{isk} \leq 1 + η_{ds}, \forall d \in N_{D}, s \in N_{S}, k \in K,

(17)

\sum_{i \in N_{S}} \sum_{j \in N_{S} \cup N_{D}} \sum_{p \in P} x_{ijk} q s_{ip} \leq Q_{k}, \forall k \in K,

(18)

\sum_{j \in N_{S}} \sum_{p \in P} η_{ij} q s_{jp} \leq Q_{d}, \forall i \in N_{D},

(19)

\sum_{j \in N_{S}} η_{ij} \geq r_{i}, \forall i \in N_{D},

(20)

η_{ij} \leq r_{i}, \forall j \in N_{S}, \forall i \in N_{D},

(21)

\sum_{j \in N_{S}} \sum_{k \in K} x_{ijk} \leq M_{K}, \forall i \in N_{D},

(22)

u_{ik} - u_{jk} + | S | * x_{ijk} \leq | S | - 1, \forall i, j \in N_{S}, i \neq j, v \in K .

(23)

The constraint given by Equation 13 indicates that each open fresh product store is served only once. Equation 14 represents the flow balance constraint of each node in the first-level distribution network. Equation 15 forbids paths among the DCs. Equation 16 guarantees that each vehicle in the first-echelon distribution network serves one route at most. Equation 17 indicates that a route between a store and DC exists only when the store is allocated to the DC to provide services. The capacity of refrigerated vehicles in the first-echelon network is limited by Equation 18. Equation 19 indicates that the demand of stores does not exceed the capacity of the DC. Equations 20 and 21 ensure that the store is allocated to the DC only when the DC is open. The number of vehicles available at each DC is limited by Equation 22. Equation 23 eliminates sub-tours in the first-echelon distribution network.

Equations 24–42 are the constraints in the second-level distribution network:

\sum_{i \in N_{s} \cup {N_{s}}^{'}} λ_{ij} = 1, \forall j \in N_{c},

(24)

\sum_{i \in {N_{s}}^{'}} λ_{ij} = 1, \forall j \in N_{C}^{1},

(25)

\sum_{i \in N_{s}} λ_{ij} = 1, \forall j \in N_{C}^{2},

(26)

\sum_{i : (i, j) \in A_{2}} \sum_{v \in V} y_{ijv} = 1, \forall j \in N_{C}^{2},

(27)

\sum_{i : (i, j) \in A_{2}} y_{ijv} = \sum_{i : (j, i) \in A_{2}} y_{jiv}, \forall j \in N_{C}^{2}, v \in V,

(28)

\sum_{i \in N_{C}^{2}} y_{ijv} = \sum_{i \in N_{C}^{2}} y_{jiv}, \forall j \in N_{S}, v \in V,

(29)

\sum_{i \in N_{S}} \sum_{j \in N_{S}} y_{ijv} = 0, \forall v \in V,

(30)

\sum_{i \in N_{S}} \sum_{j \in N_{C}^{2}} y_{ijv} \leq 1, \forall v \in V,

(31)

\sum_{i : (i, j) \in A_{2}} y_{siv} + \sum_{i : (i, j) \in A_{2}} y_{icv} \leq 1 + λ_{sc}, \forall s \in N_{S}, c \in N_{C}^{2}, v \in V,

(32)

\sum_{i \in N_{C}^{2}} \sum_{j : (i, j) \in A_{2}} \sum_{p \in P} y_{ijv} q c_{ip} \leq Q_{v}, \forall v \in V,

(33)

\sum_{j \in N_{C}} λ_{ij} q c_{jp} \leq q s_{ip} z_{i}, \forall i \in N_{S}, p \in P,

(34)

\sum_{j \in N_{C}} \sum_{p \in P} λ_{ij} q c_{jp} \leq z_{i} Q_{s}, \forall i \in N_{S},

(35)

\sum_{j \in N_{C}} λ_{ij} \geq z_{i}, \forall i \in N_{S},

(36)

λ_{ij} \leq z_{i}, \forall i \in N_{S}, j \in N_{C},

(37)

\sum_{j \in N_{C}^{2}} \sum_{k \in K} y_{ijk} \leq M_{V}, \forall i \in N_{S},

(38)

u_{iv} - u_{jv} + | C | * y_{ijv} \leq | C | - 1, \forall i, j \in N_{C}^{2}, i \neq j, v \in V,

(39)

s_{iv} + w_{i} + t_{ij} - s_{jv} \leq (1 - x_{ijv}) M, \forall i, j \in N_{C}^{2} \cup N_{S}, v \in V,

(40)

t_{ij} = x_{ijv} \frac{d_{ij}}{v_{a}}, \forall i, j \in A_{2}, v \in V,

(41)

s t_{j} = λ_{ij} \frac{d_{ij}}{v_{b}}, \forall i \in N_{S}, j \in N_{C}^{1} .

(42)

Equations 24–26 indicate that each customer is served once and only by the store within reach. Equation 27 shows that each delivery customer is served exactly once by a second-echelon vehicle, and Equations 28 and 29 indicate the flow balance in customers and stores, respectively. Equation 30 forbids paths among perishable product stores. Equation 31 ensures that each vehicle in the second-echelon delivery network serves one route at most. Equation 32 indicates that a route exists between a customer and store if the customer is served by the store. The capacity of the electric vehicles in the second-echelon network is limited by Equation 33. Equations 34 and 35 indicate that the total demand of all customers does not exceed the demand and storage capacity of the perishable product stores. Equations 36 and 37 state that customers are allocated to selected open stores. Equation 38 limits the number of vehicles available for each store. Equation 39 eliminates the sub-tours in the second-echelon delivery network. Equation 40 represents the continuity of vehicle travel time. Equation 41 demonstrates the travel time of a delivery vehicle between two vertices. Equation 42 represents the return time of the in-store pick-up customer to the store.

Equations 43 and 44 and 45–48 indicate the non-negative and 0–1 variables, respectively:

u_{ik} \geq 0, \forall i \in N_{S}, k \in K,

(43)

u_{iv} \geq 0, s_{iv} \geq 0, \forall i \in N_{C}^{2}, v \in V,

(44)

r_{i}, η_{ij} \in {0, 1}, \forall i \in N_{D}, j \in N_{S},

(45)

z_{i}, λ_{ij} \in {0, 1}, \forall i \in N_{S}, j \in N_{C},

(46)

x_{ijk} \in {0, 1}, \forall (i, j) \in A_{1}, k \in K,

(47)

y_{ijv} \in {0, 1}, \forall (i, j) \in A_{2}, v \in V .

(48)

Adaptive Large Neighborhood Heuristic for the Two-Echelon Location–Routing Problem

The 2E-LRP for perishable product distribution is an extension of the LRP, which is a non-deterministic polynomial time-hard problem. The idea of the large neighborhood search (LNS) algorithm is to improve the current solution by expanding the search space using the destruction and repair operators of the initial solution. The ALNS algorithm adds a mechanism and selects an operator with a large weight (i.e., an operator that significantly improves the solution with each iteration). The weight of each operator is updated based on the performance of optimization after a certain number of iterations to improve the solution efficiency and convergence speed. A flowchart of the ALNS algorithm developed in this study is shown in Appendix Figure 1.

Initial Solution

Step 1: Among the candidate stores, select $IN$ stores randomly according to the total customer demand and store capacity: $IN \geq \sum_{i \in N_{C}} \sum_{p \in P} q c_{ip} / Q_{s}$ .

Step 2: Randomly allocate customers (delivery and in-store pick-up customers) to open stores in order. Check whether the demands of the allocated customers exceed the store capacity. If this is the case, the current customer is allocated to another open store until all customers are allocated.

Step 3: Randomly select a delivery customer, $i \in N_{C}^{2}$ , and store, $s \in N_{S}$ , to which the customer is allocated. Then, establish a route for $(s, i, s)$ .

Step 4: Calculate the sum of the time window deviation and distance between $i \in N_{C}^{2}$ and other delivery customers, $j \in N_{C}^{2}$ , allocated to the store.

Step 5: Select the customer with the minimum summation value in Step 4, and predetermine whether the vehicle capacity is exceeded after inserting the delivery to the customer into the route established in Step 3. If the vehicle capacity is not exceeded, accommodate customer delivery; otherwise, construct a new delivery route between the delivery to the customer and corresponding store.

Step 6: Repeat Steps 3–5 until all delivery customers are arranged along a route.

The foregoing initial solution generation method is used to generate the initial solution of the second-echelon delivery network. Because the structure of the first-echelon distribution network is similar to that of the second-echelon network, the initial solution of the first-echelon distribution network is generated using similar steps based on the store location and demand output from the result of the second-echelon delivery network.

Destroy Operators

The destroy phase mainly consists of removing $ϕ$ customers from the current solution and adding them to a removal list, $L_{d}$ . The size of $ϕ$ affects the degree of destruction of the solution. It affects the efficiency of the algorithm, regardless of whether it is large or small. Generally, the size of $ϕ$ is $ϕ = [| N_{c} | / 8; | N_{c} | / 4]$ . The eight removal operators used in the proposed algorithm are as follows.

(1) Closure of open DCs/stores.

For the second-echelon delivery network, one of the open stores is randomly selected and then closed. All customers currently served by the store and corresponding delivery routes are deleted and placed in the removal list, $L_{d}$ , as shown in Figure 2. This operation destroys some of the allocation status of customers by changing the status of the store (open or closed). For the first-echelon distribution network, the foregoing operations are performed on open DCs and the stores they serve.

Figure 2.

Closure of an open store.

(2) Reopening of DCs/stores

A closed store in the second-echelon network is randomly selected and then reopened. Moreover, the $ϕ$ customers closest to it are removed from the current route and placed in the removal list, $L_{d}$ , as shown in Figure 3. For the first-echelon distribution network, the foregoing operations are performed on the DC and the stores allocated to it.

Figure 3.

Opening of a closed store.

(3) Random customer removal

For the second-echelon delivery network, randomly remove $ϕ$ delivery customers from the current route and put them into the removal list, $L_{d}$ . The operator is simple to use and can aid in increasing the diversity of solution space. For the first-level network, this operation involves the random removal of stores.

(4) Customer removal

The operator for customer removal was first proposed and used by Shaw et al (18). Their idea was to remove nodes with high correlation simultaneously such that subsequent insertion operations can improve the diversity of solutions. The correlation between delivery customers $i$ and $j$ is $R (i, j)$ :

R (i, j) = ρ_{1} d_{ij} + ρ_{2} (| e_{i} - e_{j} | + | l_{i} - l_{j} |) + ρ_{3} (| q_{i} - q_{j} |) + ρ_{4},

(48)

where $q_{i} = \sum_{p \in P} q c_{ip}$ ; $ρ_{1}, ρ_{2}, and ρ_{3}$ are the weights of the distance, time window difference, and demand difference, respectively; $ρ_{4}$ denotes whether they are on the same route (0 if they are; otherwise, 1). The smaller the value of $R (i, j)$ , the higher the correlation between delivery customers $i$ and $j$ . Delivery customer $i$ is randomly removed and then added to the removal list, $L_{d}$ . The correlation between each unremoved customer and removed customer, $i$ , is calculated. The delivery customers, $ϕ - 1$ , are removed according to the order of relevance from low to high. No time window exists when this operation is performed for stores in the first-echelon network.

(5) Worst cost removal

The time penalty cost, transportation cost, and cargo damage cost of a route may change after delivery customers are removed from the route. This operator implements calculations for each delivery customer, $i$ ; changes the objective function value (OFV) after removing the customer; and removes the customer with the largest change. Then, the repair operator is used to select a better location. After removing delivery customer $i$ , the difference among the OFVs corresponding to the current solution, $s_{current}$ , is as follows:

Δ f (i, s_{current}) = f (s_{current}) - f_{- i} (s_{current}),

(48)

where $f (s_{current})$ is the OFV corresponding to the current solution ( $s_{current}$ ), and $f_{- i} (s_{current})$ is the corresponding OFV after removing customer $i$ from the current solution. Sort $Δ f (i, s_{current})$ in descending order, remove the top $ϕ$ delivery customers, and place them in the removal list, $L_{d}$ .

(6) Worst time removal

Delivery customers have certain requirements with respect to delivery time windows; accordingly, a time penalty cost is set. For each customer, $i$ , this operator calculates the deviation of the vehicle arrival time from the expected service time window and then removes the customer with the largest deviation. The concept is to prevent long waiting times or delayed service start times. The difference between the vehicle arrival time and customer time window is set as follows:

Δ T_{i} = min {\frac{α}{β} | s_{iv} - e_{i} |, | s_{iv} - l_{i} |} .

(48)

If the vehicle arrives within the time window expected by the delivery customer, $Δ T_{i} = 0$ is used. Customers are sorted in the descending order of $Δ T_{i}$ . The first $ϕ$ customers are removed and then added to the removal list, $L_{d}$ .

(7) Removal of in-store pick-up customers

Because no time window constraint or path connection exists for in-store pick-up customers, the roulette principle is applied to select customers for removal until $ϕ$ in-store pick-up customers are removed.

(8) Random route removal

This operator removes the entire route from the solution. A route randomly selected from the set of routes in the solution is to be removed. If the number of customers on this route is less than $ϕ$ , the operator repeatedly selects another route to remove until no less than $ϕ$ customers are eliminated. Similar to random customer removal, this operator increases the randomness of the algorithm and reduces the likelihood of falling into local optima. This operator is also applied to the routes in the first-echelon distribution network.

Repair Operators

After the destruction operator removes $ϕ$ nodes from the current solution, the use of the repair operator is necessary to reinsert the deleted nodes in the removal list, $L_{d}$ , into the current solution to form a new neighborhood solution. Before performing the repair operation, ensure that DCs and stores are open; otherwise, the DC or store must be reopened. This section presents two insertion operators used in the ALNS algorithm.

(1) Greedy insertion

This operator repeatedly inserts a removed node in the best possible position of a route, causing a minimal increase in the OFV. At each inserted iteration, a deleted node, $i$ , is selected from the removal list, $L_{d}$ . The increased value of the objective function is calculated after the node is inserted into each feasible position. Then, the position with a minimum increase in OFV is selected and inserted until the removal list, $L_{d}$ , is empty.

(2) Random insertion.

The removed node, $i \in L_{d}$ , is randomly inserted into feasible positions until the removal list, $L_{d}$ , is empty. The performance of the randomly inserted solution is relatively inadequate; however, the complexity is low and can increase the diversity and randomness of the solution; consequently, falling into the local optimum is avoided.

Adaptive Mechanism

The proposed ALNS algorithm uses a roulette wheel mechanism to select operators. Initially, all removal and insertion operators have equal probabilities; they are set to 1/8 and 1/2, respectively. The operator scores are marked according to the performance of the operator in the last $Ren$ iterations (stage). Subsequently, the operator weights are updated.

If the operator finds the current global optimal solution in the latest $Ren$ iterations, the score of the operator is increased by $s c_{1}$ . If the neighborhood solution found is not optimal but better than the current solution, the score is increased by $s c_{2}$ . If the neighborhood solution found is worse than the current solution but is acceptable, the score is increased by $s c_{3}$ . Then, the score of the operator is calculated as $π_{dr}^{sg} = π_{dr}^{sg - 1} + {\begin{matrix} s c_{1}, s_{ne} < s_{best} \\ s c_{2}, s_{best} < s_{ne} < s_{current} \\ s c_{3}, s_{current} < s_{ne}, accept \end{matrix}$ , where $π_{dr}^{sg - 1}$ is the operator score of the last iteration stage, and $π_{dr}^{sg}$ is the operator score of the current stage (i.e., $s c_{1} > s c_{2} > s c_{3}$ ). The weight of operators in each stage is updated as $ω_{dr}^{sg} = (1 - τ) ω_{dr}^{sg - 1} + τ π_{dr}^{sg - 1} / m_{dr}$ , where $ω_{dr}^{sg}$ is the operator weight of the current iteration stage; $ω_{dr}^{sg - 1}$ is the operator weight of the last iteration stage; $m_{dr}$ is the number of times $ω_{dr}^{sg - 1}$ is used during the last $Ren$ iterations; and $τ$ is a weight control parameter. Therefore, the probability that the operator finds the solution by roulette selection is $ω_{dr}^{i} / \sum_{i = 0}^{sg} ω_{dr}^{i}$ .

Acceptance

When a new neighborhood solution is found by the destruction and repair operations of the LNS algorithm, the neighborhood solution is accepted in any case if the OFV of the new solution is smaller than that of the current solution, that is, $f (s_{ne}) < f (s_{current})$ . If the OFV of the neighborhood solution is extremely low, it is accepted based on the Metropolis criterion, which is typically used in simulated annealing. The acceptance probability is $p = e^{- \frac{f (s_{current}) - f (s_{ne})}{T}}$ , where $T$ denotes the cooling temperature. The initial temperature and cooling coefficient are set as $T_{0}$ and $\partial$ , respectively. Then, $T = \partial {T_{0}}^{gen}$ , where $gen$ is the number of iterations.

Numerical Experiments and Analysis of Results

All numerical experiments in this study are performed on a laptop with 8 GB of RAM and an Intel (R) Core (TM) i5-8265u processor with a Windows 10 operating system. The ALNS algorithm is implemented using MATLAB R2017b.

Verification of the Model and Proposed ALNS Algorithm

Instances generated by Solomon ( 19 ) are widely used in the research on VRPs, LRPs, and their extensions. Solomon analyzed networks with three different node distributions: random, clustered, and semi-clustered. Subsequently, three different sets of instances are generated, named respectively R, C, and RC. The node distribution configurations are shown in Figure 4.

Figure 4.

Node distribution configuration: (a) random (R), (b) clustered (C), and (c) semi-clustered (RC).

No standard instances of a 2E-LRP with time windows considering different customer service types are reported in the literature. In this study, an instance with 12 nodes (two candidate DCs, three candidate stores, four delivery customers, and three in-store pick-up customers) is derived from Solomon instance R101. This instance is solved using CPLEX and the proposed ALNS algorithm to verify the effectiveness of the model and algorithm. Based on the empirical values of the parameters reported in the literature and several pre-experiments, the parameters of the algorithm and 2E-LRP perishable product network are as summarized in Table 3.

Table 3.

Parameter Settings

Parameter	Value		Parameter	Value
$ϕ$	$ϕ = [\frac{\| N_{c} \|}{8}, \frac{\| N_{c} \|}{4}]$ ( $\| N_{c} \| = 25 / 50 / 100$ )		$s c_{1}$	2
$ρ_{1}$	4		$s c_{2}$	1
$ρ_{2}$	3		$s c_{3}$	0.5
$ρ_{3}$	2		$T_{0}$	100
$τ$	0.7		$\partial$	0.99
$Ren$	1		$GenMax$	200/400/800/1000
Description	Value		Description	Value
Opening cost of DCs, $F_{d}$ (RMB)	300		Capacity of DCs, $Q_{d}$ (kg)	1500
Opening cost of stores, $F_{s}$ (RMB)	120		Capacity of stores, $Q_{s}$ (kg)	100/500
Capacity of first-echelon vehicles, $Q_{k}$ (kg)	500		Fixed cost of first-echelon vehicles, $F_{k}$ (RMB)	100
Capacity of second-echelon vehicles, $Q_{v}$ (kg)	40/80		Fixed cost of second-echelon vehicles, $F_{v}$ (RMB)	30
Unit transportation cost of first echelon, $C_{k}$ (RMB/km)	10		Early arrival penalty coefficient, $α$	10
Unit transportation cost of second echelon, $C_{v}$ (RMB/km)	3		Late arrival penalty coefficient, $β$	20
Travel speed of second-echelon vehicles, $v_{a}$ (km/h)	15		Distance penalty coefficient, $μ$	10
Best in-store pick-up distance for customer (km)	0.5		Acceptable farthest in-store pick-up distance for customer (km)	1.0
Travel speed of in-store pick-up customer, $v_{b}$ (km/h)	5		Delivery range of stores, $R$ (km)	3
Customer service cost, $δ$ (RMB)	1		Maximum number of vehicles in DCs, $M_{K}$	3
Initial freshness, $θ_{0}$	0.98		Maximum number of vehicles in stores, $M_{V}$	3
Unit value of product, $G_{p}$	Normal temperature	1.5	Unit cooling cost, $f_{p}$ (RMB/kg)	Normal temperature	0.1
	Cold storage	2.5		Cold storage	0.3
	Freezing	5		Freezing	0.5
Freshness sensitivity coefficient, $σ_{p}$	Normal temperature	2.0	Fresh-keeping cost coefficient, $ε_{p}$	Normal temperature	0.2
	Cold storage	2.2		Cold storage	1.8
	Freezing	2.5		Freezing	3.2

Note: DC = distribution center.

The results and location–routing scheme of the 12-node network are summarized in Table 4 and shown Figure 5, respectively. As indicated in Table 4, the ALNS algorithm and CPLEX obtained the same optimal solution, OFV, location, allocation, and routing scheme, indicating the high accuracy of the proposed ALNS algorithm. With respect to running time, the ALNS algorithm consumed 9.1275 s (highlighted in bold) for 200 iterations, whereas CPLEX ran for 1692 s. The proposed algorithm saves approximately 99% of the time consumed by CPLEX. Moreover, it can solve large-scale instances.

Table 4.

Comparison Between the Solution Results of CPLEX and the Proposed Adaptive Large Neighborhood Search (ALNS) Algorithm

	CPLEX	ALNS
OFV	801.865	801.865
CPU time	1692 s	9.1275 s
Location scheme	Open DC:1, open store: 3,4	Open DC:1, open store: 3,4
Routing scheme	Route of first-echelon vehicle: 1-4-3-1; Routes of first-echelon vehicle: 3-9-11-3 and 4-10-12-4 Route of in-store pick-up customers: customer 6 to store 3; customers 7 and 8 to store 4	Route of first-echelon vehicle: 1-4-3-1 Routes of first-echelon vehicle: 3-9-11-3, 4-10-12-4 Route of in-store pick-up customers: customer 6 to store 3; customers 7 and 8 to store 4

Note: OFV = objective function value; DC = distribution center.

Figure 5.

Location–routing results of a 12-node network.

Algorithm Performance Analysis

A set of instances (C101, R101, and RC101) is selected from the Solomon instances and then modified to verify the stability and applicability of the ALNS algorithm further. By employing a method reported in the literature by Grangier et al. ( 20 ), additional locations of candidate DCs and stores that are within the customer distribution range of the selected Solomon instances are added. To test the proposed ALNS algorithm in small-scale, medium-scale, and large-scale instances, three sets of instances are generated. These include a set of small-scale instances with 31 nodes (2 candidate DCs, 4 candidate stores, and 25 customers, simplified as #2-4-25); a set of medium-scale instances with 58 nodes (#2-6-50); and a set of large-scale instances with 111 nodes (#3-8-100). Instances can be found at https://kdocs.cn/l/cabQm4QhHZqV. The customers are classified into two groups: delivery and in-store pick-up customers. The proportion of in-store pick-up customers is set as one-third of the total number of customers. Each instance is solved 10 times using the ALNS algorithm. The best optimal solution and average solution obtained among the 10 runs are recorded as $bes t^{ALNS}$ and $Av e^{ALNS}$ , respectively. The gap between the optimal and average solutions is presented using ${GAP}_{ave}^{ALNS}$ , and $t_{ave}^{ALNS}$ is the average running time for 10 runs. The results are summarized in Table 5.

Table 5.

Results of Different Instances Solved by the Proposed Adaptive Large Neighborhood Search (ALNS) Algorithm

Instances	$bes t^{ALNS}$	$Av e^{ALNS}$	$t_{ave}^{ALNS}$	${GAP}_{ave}^{ALNS}$ (%)
C1(2-4-25)	1055.07	1061.15	34.61	0.64
R1(2-4-25)	1207.98	1212.00	35.39	0.37
RC1(2-4-25)	1326.59	1327.68	43.86	0.12
Average	1196.55	1200.28	37.95	0.38
C1(2-6-50)	1756.86	1771.16	248.22	0.90
R1(2-6-50)	1583.91	1598.38	231.42	1.01
RC1(2-6-50)	2188.29	2193.35	321.40	0.26
Average	1843.02	1854.30	267.01	0.72
C1(2-6-100)	3566.39	3610.10	1128.97	1.36
R1(2-6-100)	2522.89	2579.45	1089.34	2.49
RC1(2-6-100)	3325.41	3342.44	1025.26	0.57
Average	3138.23	3177.33	1081.19	1.47

Table 5 indicates that although the running time of the ALNS algorithm increases with the number of nodes in the instance, a large-scale instance with 111 nodes can be solved within 1200 s. The gap between the optimal and average solutions slightly increases with scale, reaching a maximum of 2.49% because of the larger solution space. In addition, the gap among the RC instances is extremely small and does not exceed 1%. The foregoing analysis indicates that the proposed ALNS algorithm exhibits satisfactory performance with respect to speed and stability in solving the 2E-LRP of perishable products developed in this study.

Sensitivity Analysis

Under the integrated service mode of in-store pick-up and delivery, the impact of different proportions of customers of different service types on the total cost of the 2E-LRP schemes is analyzed. The proportions of in-store pick-up customers to the number of delivery customers are set; these are 0:1, 1/3:2/3, 1/2:1/2, 2/3:1/3, and 1:0, where 0:1 indicates that all customers are delivery customers, whereas 1:0 indicates that all customers are in-store pick-up customers. By considering medium-scale instances as examples, the total cost, number of dispatched second-echelon vehicles, and number of open stores are calculated under five different proportions of the two types of customers. As summarized in Table 6, the foregoing is computed for instances C1(2-6-50), R1(2-6-50), and RC1(2-6-50) to analyze the sensitivity of the solutions to changes in the proportion of customers.

Table 6.

Analysis of Results of Different Customer Proportions

Proportions	C1(2-6-50)			R1(2-6-50)			RC1(2-6-50)
Proportions	Total cost	No. of vehicles	No. of stores	Total cost	No. of vehicles	No. of stores	Total cost	No. of vehicles	No. of stores
0:1	2057.10	12	4	1732.22	9	3	2253.56	9	3
1/3:2/3	1756.86	8	3	1583.91	6	2	2188.29	9	4
1/2:1/2	1768.11	6	3	1650.26	5	2	2157.33	7	4
2/3:1/3	1671.44	4	3	1688.84	4	2	2216.24	5	4
1:0	1802.11	0	5	1914.68	0	4	2561.57	0	5

Figure 6 shows the OFVs of different instances with different customer proportions. The figure indicates that the total cost of considering integrated service types (delivery and in-store pick-up) is lower than the single service type of either providing delivery only or in-store pick-up only. For all sets of instances, when considering integrated delivery and in-store pick-up customers, the total cost is reduced by 7.98% and 11.44% on average compared with complete delivery customers (customer proportion 0:1) and complete in-store pick-up customers (customer proportion 1:0), respectively. The reasons for the foregoing results are as follows. For a network of complete delivery customers, the total transportation distance and number of vehicles used increase. In addition, this network may generate more time penalty costs compared with the integrated service types of customers owing to the time window requirements of delivery customers. For a network of complete in-store pick-up customers, the distance penalty costs increase because of the long distances from stores to customers. Furthermore, more stores are opened to increase coverage and avoid excessive distance penalty costs. In-store customer service costs also increase as customers visit stores, resulting in an increase in total cost.

Figure 6.

Objective function value with different customer proportions.

Table 6 indicates that the proportion of customers generating the minimum total cost varies for different node distribution networks. For instance, C1(2-6-50) has the lowest total cost when the customer proportion is 2/3:1/3, saving 18.74% and 6.47% of the total costs compared with complete delivery customers and complete in-store pick-up customers, respectively. In series C, the customers are clustered and close to candidate stores. Consequently, a higher proportion of in-store pick-up customers saves more on the cost of the delivery system than complete delivery customers. In series R, the total cost is the lowest when the customer proportion is 1/3:2/3, saving 8.56% and 17.28% of the total costs compared with complete delivery customers and complete in-store pick-up customers, respectively. In series RC, the best customer proportion is 1/2:1/2, saving 4.27% and 15.78% of the total costs compared with complete delivery customers and complete in-store pick-up customers, respectively. In an instance in series RC, the customers are clustered but distant from candidate stores; therefore, a large distance penalty cost is possibly incurred. Therefore, the savings in the cost of a delivery system with integrated customers is lower than a system with complete delivery customers but higher than a system with complete in-store pick-up customers.

Furthermore, a detailed analysis of the total costs based on C/R/RC1(2-6-50) instances is implemented; the eight costs considered are shown in Figure 7. The long-term investment for DCs and stores and the fixed cost for dispatching vehicles are expected to be the highest. The delivery cost is high and can be further optimized. Interestingly, the distance penalty for in-store pick-up customers has a high proportion. This is because a relatively high distance penalty coefficient (10) and a strict acceptable longest walking distance (1 km) are set such that the necessity for customers with heavy bags to walk long distances is avoided.

Figure 7.

Total cost structure.

Figure 8 shows that the total cost of the second echelon is virtually double that of the first echelon for all sets of instances. This is because the activities in the second echelon become complicated because of high frequency, small batches, and strict requirements. Figure 9 shows the proportion of costs incurred by delivery and in-store pick-up customers. For instance, in C1(2-6-50), the cost for delivery customers accounts for approximately 70%, whereas the costs for delivery and in-store pick-up customers are divided into two for the other two sets of instances. This result is probably caused by the distribution characteristics of the nodes.

Figure 8.

Total costs for two echelons.

Figure 9.

Proportion of cost for the two service types.

Conclusions

Based on the combination of online and offline purchases of perishable products in the post-epidemic period, the integrated service mode of in-store pick-up service and delivery is considered to study the 2E-LRP of perishable product distribution networks. A two-echelon location–routing model with the objective of minimizing the total cost of a perishable product distribution network is established. An ALNS algorithm is developed to solve this problem, and instances from those of Solomon are derived to test the algorithm. A 12-node small-scale instance is solved by CPLEX and the proposed ALNS algorithm. A comparison of results shows the satisfactory performance of the proposed ALNS algorithm with respect to speed and accuracy. The results of the instances with different scales obtained by the proposed ALNS algorithm indicate the applicability and satisfactory stability of this algorithm. Different proportions of in-store pick-up and delivery customers are set and then the total costs of location–routing schemes under different proportions of the two service-type customers are compared. The results show that an integrated service mode compared with single delivery and single in-store pick-up service mode can save the total retail cost by 4.21 RMBs per customer and 5.77 RMBs per customer, respectively, in instances with 50 customers. With the foregoing, enterprises of perishable products may establish a fresh product distribution network by reasonably determining the location and number of stores as well as choosing the ideal proportion of in-store pick-up and delivery customers based on customer distribution such that a variety of services can be offered. Furthermore, the proportion of customers contributing to the minimum total cost varies for different node distribution networks. This conclusion provides a reference for the layout of the distribution network of perishable product enterprises.

This study has limitations because of the need to impose various assumptions. To start, some nonlinearities, such as dynamic customer demands, seasonality, or geographic concentration, are not part of this exercise. The foregoing can be considered as challenges that must be overcome by future modeling efforts so as to achieve practical goals. In addition, a bottom-up solution was adopted for the 2E-LRP, but the algorithm for solving this problem should be further explored by considering the mutual iterative influence between the two echelons.

Supplemental Material

sj-docx-1-trr-10.1177_03611981231218008 – Supplemental material for Two-Echelon Location–Routing Problem of Perishable Products Based on the Integrated Mode of In-Store Pick-Up And Delivery

Supplemental material, sj-docx-1-trr-10.1177_03611981231218008 for Two-Echelon Location–Routing Problem of Perishable Products Based on the Integrated Mode of In-Store Pick-Up And Delivery by Xiqiong Chen, Yanni Jiu and Dawei Hu in Transportation Research Record

Footnotes

Acknowledgements

We would like to thank the editors and referees for their valuable and constructive comments.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: D. Hu, X. Chen; data collection: Y. Jiu; analysis and interpretation of results: Y. Jiu, X. Chen; draft manuscript preparation: X. Chen, Y. Jiu. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the Natural Science Foundation of China (72274024) and the Natural Science Foundation of Shaanxi Province (2021JZ-20 and 2022JQ-728). This support is greatly acknowledged.

Data Accessibility Statement

Data can be found at .

Supplemental Material

Supplemental material for this article is available online.

References

Suraraksa

Shin

K. S.

Urban Transportation Network Design for Fresh Fruit and Vegetables Using GIS–The Case of Bangkok. Applied Sciences, Vol. 9, No. 23, 2019, p. 5048.

Zulvia

F. E.

Kuo

R. J.

Nugroho

D. Y.

A Many-Objective Gradient Evolution Algorithm for Solving a Green Vehicle Routing Problem with Time Windows and Time Dependency for Perishable Products. Journal of Cleaner Production, Vol. 242, 2020, p. 118428.

Golestani

Moosavirad

S. H.

Asadi

Biglari

A Multi-Objective Green Hub Location Problem with Multi Item-Multi Temperature Joint Distribution for Perishable Products in Cold Supply Chain. Sustainable Production and Consumption, Vol. 27, 2021, pp. 1183–1194.

Wang

Zhang

Liu

M.-z.

Optimization of Fresh Goods Multi-Center Vehicle Routing Problem Based on Resource Sharing and Temperature Control. Chinese Journal of Management Science, Vol. 30, No. 11, 2022, pp. 272–285.

Hiassat

Diabat

Rahwan

A Genetic Algorithm Approach for Location-Inventory-Routing Problem with Perishable Products. Journal of Manufacturing Systems, Vol. 42, 2017, pp. 93–103.

Navazi

Sedaghat

Tavakkoli-Moghaddam

A New Sustainable Location-Routing Problem with Simultaneous Pickup and Delivery by Two-Compartment Vehicles for a Perishable Product Considering Circular Economy. IFAC-PapersOnLine, Vol. 52, No. 13, 2019, pp. 790–795.

Nagy

Salhi

Location-Routing: Issues, Models and Methods. European Journal of Operational Research, Vol. 177, No. 2, 2007, pp. 649–672.

Prodhon

Prins

A Survey of Recent Research on Location-Routing Problems. European Journal of Operational Research, Vol. 238, No. 1, 2014, pp. 1–17.

D. W.

Chen

X. Q.

Gao

Review on Location-Routing Problem. Journal of Traffic and Transportation Engineering, Vol. 18, No. 1, 2018, pp. 111–129.

10.

Nguyen

V. P.

Prins

Prodhon

Solving the Two-Echelon Location Routing Problem by a GRASP Reinforced by a Learning Process and Path Relinking. European Journal of Operational Research, Vol. 216, No. 1, 2012, pp. 113–126.

11.

Rahmani

Cherif-Khettaf

W. R.

Oulamara

A Local Search Approach for the Two–Echelon Multi-Products Location–Routing Problem with Pickup and Delivery. IFAC-PapersOnLine, Vol. 48, No. 3, 2015, pp. 193–199.

12.

Darvish

Archetti

Coelho

L. C.

Speranza

M. G.

Flexible Two-Echelon Location Routing Problem. European Journal of Operational Research, Vol. 277, No. 3, 2019, pp. 1124–1136.

13.

Mirhedayatian

S. M.

Crainic

T. G.

Guajardo

Wallace

S. W.

A Two-Echelon Location-Routing Problem with Synchronization. Journal of the Operational Research Society, Vol. 72, No. 1, 2021, pp. 145–160.

14.

Tirkolaee

E. B.

Goli

Mardani

A Novel Two-Echelon Hierarchical Location-Allocation-Routing Optimization for Green Energy-Efficient Logistics Systems. Annals of Operations Research, Vol. 324, 2023, pp. 795–823.

15.

Heidari

Imani

D. M.

Khalilzadeh

Sarbazvatan

Green Two-Echelon Closed and Open Location-Routing Problem: Application of NSGA-II and MOGWO Metaheuristic Approaches. Environment, Development and Sustainability, Vol. 25, No. 9, 2023, pp. 9163–9199.

16.

Govindan

Jafarian

Khodaverdi

Devika

Two-Echelon Multiple-Vehicle Location–Routing Problem with Time Windows for Optimization of Sustainable Supply Chain Network of Perishable Food. International Journal of Production Economics, Vol. 152, 2014, pp. 9–28.

17.

Bala

Brcanov

Gvozdenovic

Two-Echelon Location Routing Synchronized with Production Schedules and Time Windows. Central European Journal of Operations Research, Vol. 25, No. 3, 2017, pp. 525–543.

18.

Shaw

Using Constraint Programming and Local Search Methods to Solve Vehicle Routing Problems. Proc., International conference on principles and practice of constraint programming, Springer, Berlin, Heidelberg, 1998, pp. 417–431.

19.

Solomon

Algorithms for the Vehicle Routing and Scheduling Problem with Time Window Constarints. Operations Research, Vol. 35, No. 2, 1987, pp. 254–265.

20.

Grangier

Gendreau

Lehuédé

Rousseau

L. M.

An Adaptive Large Neighborhood Search for the Two-Echelon Multiple-Trip Vehicle Routing Problem with Satellite Synchronization. European Journal of Operational Research, Vol. 254, No. 1, 2016, pp. 80–91.

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