Delivering meals for multiple suppliers: Exclusive or sharing logistics service

Transportation Research Part E 118 (2018) 496–512

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Delivering meals for multiple suppliers: Exclusive or sharing logistics service

T

Zheng Wang School of Maritime Economics and Management, Dalian Maritime, University, Dalian, Liaoning 116025, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Meal delivery Sharing logistics service Multi-trip routing

We study a meal delivery problem faced by some logistics providers, which need to route vehicles in multiple trips to pick up meals from multiple suppliers and deliver to customers. Three types of logistics services are investigated: exclusive service and two novel sharing services. The delivery problems broadly belong to multi-trip routing with soft time windows and multiple refill locations. We develop two mainstream heuristics, iterated local search and adaptive large neighborhood search. Extensive case studies are conducted based on large-sized real-world instances. Managerial analysis shows that two sharing services generally yield much less total costs than the exclusive service.

1. Introduction Meal booking has recently become very popular internationally. During the past few years, the number of meal booking orders has skyrocketed (Visser, 2015; iiMedia, 2018). About 271,000 daily orders are placed in 2016 at the platforms of Orders2me and GrubHub (Cohen, 2016). To deliver these meals, a meal supplier usually outsources the logistics service to a third-party logistics provider (Digo, 2016). After the logistics provider accepts the orders from multiple suppliers, it needs to schedule vehicles to first pick up meals from suppliers and then deliver to customers in urban area (see Fig. 1). As a result of the bursting requests for meal delivery, many logistics providers are facing a large-scale pickup and delivery problem every day (iiMedia, 2018). The real-world data we obtained from a major logistics provider in China has up to a thousand and two hundred lunch orders and forty suppliers a day in a city. How to accomplish everyday operations in order to achieve a cost-effective vehicle routing scheme for such a large-scale problem becomes a big trouble for meal logistics providers. This paper is just focused on the meal pickup and delivery problem (MPDP) from the view of a logistics provider in a real-world setting. To handle the problem, a straightforward approach for a third-party logistics provider is to group its vehicles into subsets, each of which serves a single supplier. The approach is named as exclusive service in this paper. It is easy to manage and schedule vehicles using this service. However, every vehicle has to return to the fixed supplier for pickups, which will inevitably result in more deadheading distance from the last delivery to the supplier. This deadheading distance may be saved by another approach, named as sharing service, in which a vehicle is allowed to serve multiple suppliers and thus can visit another nearby supplier after its last delivery. To achieve more flexibility in practical operations, we present an advanced service, named as sharing+ service, in which a vehicle is allowed to visit a supplier not only after its delivery trip but also during the trip. That is, a vehicle can do new pickups before all its in-vehicle meals are delivered. Fig. 2 shows an example for the sharing and sharing+ services respectively. To the best of our knowledge, these logistics services have not been well addressed in the area of urban delivery.

E-mail address: [email protected]. https://doi.org/10.1016/j.tre.2018.09.001 Received 27 November 2017; Received in revised form 2 August 2018; Accepted 4 September 2018 1366-5545/ © 2018 Elsevier Ltd. All rights reserved.

Transportation Research Part E 118 (2018) 496–512

Z. Wang

(1) Customers make their reservations

(2) Suppliers cook and prepare meals

(3) Logistics provider collects orders

by phone Customer 1 Order details

by computer Supplier 1

Order details

Customer 2 by mobile app Supplier 2 Customer 3 Delivery

Pickup

(4) Logistics provider schedules vehicles to pick up meals from suppliers and deliver them to customers Fig. 1. The process of the ordering, preparing, and delivering of meals.

Sharing Service A

B

Orders 1 and 2 from A

Pick up meals from Supplier A

Deliver all the picked meals

Orders 3 and 4 from B

Sharing+ Service A

Pick up meals from Supplier A

Orders 1 and 2 from A

Deliver all the picked meals

Pick up meals from Supplier B

Order 1 from A Deliver one picked meal

Orders 3 and 4 from B B

Order 1 from A

Pick up meals from Supplier B

Deliver all the picked meals

Fig. 2. Examples for the sharing and sharing+ services.

From our common sense, the sharing+ service could have the least logistics cost since it has fewer restraints and is more general than the other two services. However, in the practice, the sharing+ service has more fixed cost of hiring drivers because it puts forward stricter requirements for drivers: they have to learn the road knowledge in a wider area and exactly remember the pickup and delivery sequence and the associated meals of orders. Thus the total logistics cost of the sharing+ service is not absolutely less than the other two under all the scenarios. The focus of this paper is just on solving the problem, comparing the three services, finding the best services for different scenarios, and revealing managerial insights. Our contributions are as follows. First, we present the MPDP and its three logistics services in the real-world setting. To the best of our knowledge, we are the first to study the three logistics services for meal delivery in the area of urban logistics. Second, we propose and customize a time-indexed model for the logistics services of the problem. Third, we exploit the special problem structure and design two effective heuristics to solve large-scale cases with over one thousand customers and tens of suppliers. Fourth, we conduct the extensive case studies with real-world data and perform managerial analysis. The rest of this paper is organized as follows. Section 2 reviews the related work. Section 3 formulates the problem. Section 4 presents two heuristics. Section 5 discusses the results of real-world cases. Section 6 concludes the paper and provides the future directions. 2. Related work Related studies include the delivery of perishable food products, the multi-trip vehicle routing problem, the VRPSPD, and the collaborative transportation mode. 497

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Meal belongs to perishable food products, which may lose their value in the distribution process. A major goal of delivering food products is to keep products freshness, which is mainly achieved by three approaches in the literature: hard time constraints, penaltybased approach, and special supply chain network. Existing studies with hard time constraints include placing a time limit on vehicle routes (Tarantilis et al., 2001), imposing a constraint on the arrival of perishable foods to their destinations (Faulin, 2003), adding a restriction on the lifetime of perishable products (Devapriya et al., 2017), etc. The delivery time of perishable foods can be strictly guaranteed by hard time approach. However, if the constraint is too tight, the model may be infeasible. In addition, perishable food products are still edible when the hard constraint is violated since only part of their value may be lost. Accordingly, the penalty-based approach defines the decay value, i.e. the penalty cost, with the violation of delivery time limit in order to achieve some flexibility for real-world operations. This approach is widely applied in the delivery of perishable food products. Existing studies include a timedependent deterioration function (Hsu et al., 2007), a time-related definition for perishable food quality (Osvald and Stirn, 2008), calculating the decayed value based on time window violation (Chen et al., 2009); modeling the average freshness of food products by standardizing their remaining shelf-life (Amorim et al., 2014); and defining a decreasing function for customer satisfaction according to the elapsed time of food products (Song and Ko, 2016). Some works incorporate the time-related factors into the arcs of supply chain network, leading to special network approach. The examples consist of a multi-period supply chain network equilibrium model (Liu and Nagurney, 2012), a network-based food supply chain model (Yu and Nagurney, 2013), a closed-loop supply chain network (Hasani et al., 2012), and a sustainable supply chain network (Govindan et al., 2014). Among the three approaches, the penalty-based approach is a sensible measure for MPDP since the hard time constraint may easily result in infeasible solutions and the special supply chain network is from the view of supply chain management. A penalty-based approach is thus presented in Section 3 for MPDP. The MPDP also relates to the multi-trip vehicle routing problem (Fleischmann, 1990) since they both allow vehicles to perform multiple trips among suppliers and customers during a scheduling period. Many solution approaches are developed, which consist of multi-stage approach (Taillard et al., 1996), tabu search metaheuristic (Brãndao et al., 1998), genetic algorithms (Cattaruzza et al., 2014), constructive heuristics (Chbichib et al.,2011), adaptive guidance approaches (Battarra et al., 2009), large neighborhood search (Azi et al., 2010), pool-based metaheuristics (Wang et al., 2014), and exact algorithm (Tang et al., 2015; Hernandez et al., 2016). However, most of the existing solutions approaches consider only one supplier, i.e. the central depot, from which all the goods are picked up. While in MPDP, meals are dispersedly stored in a large number of suppliers, each of which provides meals for a subset of customers. The multiple refill locations result in much more routing choices for vehicles, which do not need to return to the fixed central supplier for pickups but can return to any suppliers after any pickups or deliveries. To model the multi-trip problem, four types of formulations are studied in the literature, as concluded by Cattaruzza et al. (2016). They are 4-index formulations that extend the 3-index vehicle flow formulation by adding a trip index, 3-index formulations without vehicle index or trip index, and 2-index formulations without vehicle index nor trip index. The 4-index formulations and the 3-index formulations without vehicle index are not suitable for MPDP because the trips in MPDP do not necessarily connect to each other from end to end and they may overlap in the sharing+ logistics service. The 3-index formulations without trip index are generally developed for the multi-trip problem without time windows. Since MPDP belongs to multi-trip routing with soft time windows and multiple refill locations, we model MPDP in Section 3 by applying the 2-index approach and adding two extra time-related variables to formulate the special requirements of soft time window and meal delivery duration. The MPDP also relates to the VRPSPD (Min, 1989) because a vehicle must perform the pickup and delivery activities simultaneously. Existing methods consist of column generation (Dumas et al., 1991), integrated heuristic based on the concepts of weak and strong feasibility (Nagy and Salhi, 2005), reactive tabu search (Wassan et al., 2008), multi-start heuristic (Subramanian et al., 2010), adaptive memory-based method (Zachariadis et al., 2010), branch-and-cut method with lazy separation (Subramanian et al., 2011), scatter search (Zhang et al., 2012), construction heuristic (Bard et al., 1998; Erdoğan and Miller-Hooks, 2012), ant colony algorithm (Paraphantakul et al., 2012), genetic algorithm (Wang and Chen, 2012) and a hybrid heuristic (Wang et al., 2013). The above existing VRPSPDs are not beyond the two mainstream VRPSPDs concluded by Berbeglia et al. (2007) and Parragh et al. (2008): one-to-one VRPSPD in which each customer has her own special pickup point, and one-to-many-to-one VRPSPD in which goods are transported from the depot to linehaul customers and from backhaul customers to the depot. However, MPDP belongs to a one-to-many VRPSPD, i.e. when a vehicle arrives at a supplier with multiple customer orders, it has multiple picking choices, each of which can pick up different orders. In addition, a given picking choice may has multiple routing sequences. Then what orders need to be picked up at a supplier and what routing sequence need to be followed are two problem aspects that should be considered together. This point has not been addressed in the VRPSPD literature. In addition, MPDP has special time-related constraints, like soft time window and meal delivery duration, which distinguish the problem from the literature and are described in Section 3. The sharing logistics services in MPDP also relate to the collaborative transportation mode, which is viewed as an attractive alternative to reduce overall routing costs by uniting vehicles of multiple shippers or carriers and reducing the empty movements (Liu et al. 2010). The related studies are focused on the collaboration of multiple entities of interest (Wang et al., 2017). Several strategies are proposed for the collaboration, including the communication of timely load information and pickup and delivery plans, the diversion capability defined as a dispatcher’s ability to divert an empty moving vehicle to serve a new request for delivery, and a longer interval in order to account for additional information on loads to be delivered, etc. (Sheu, 2007a; Zolfagharinia et al., 2017). Different from the collaborative transportation, sharing service in MPDP only represents vehicles of a logistics provider can serve orders from multiple suppliers and thus are shared by these suppliers. It is unnecessary to realize the above collaboration for MPDP since the single logistics provider in MPDP knows the load and pickup and delivery information of all its vehicles. In summary, the MPDP broadly belongs to multi-trip routing problem with soft time windows and multiple refill locations. In this problem, a fleet of vehicles starts from depot, performs multiple trips among a large number of suppliers with repeatedly refills, and finally returns to depot. Because of the problem features, the existing methods cannot be directly applied to MPDP. 498

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3. Problem and model Before presenting the three logistics services and their models, we first describe important problem aspects. As a type of perishable food products, meal has a relatively short life and usually needs to be delivered in a short duration (Hsu et al., 2007; Lisa, 2016; Song and Ko, 2016). If the in-vehicle time is longer, a meal is prone to be stale and the customer satisfaction will thus be greatly reduced although it is still edible. Therefore, a suggested delivery duration is often used in perishable food delivery in order to keep food freshness. In MPDP, the suggested delivery duration is defined for meals as from the picking time at supplier to the delivering time at customer. Any delivery duration longer than the suggested will result in penalty cost. The larger the time violation, the larger the penalty. The unit penalty cost is defined to balance food freshness and logistics cost. As a result of a maximum duration of meals, each supplier in MPDP may get multiple visits, at each of which meals of different orders are picked up. Not all the meals are picked up at the same visit because otherwise one of the meals is possibly kept in vehicle longer than the suggested delivery duration. The multiple visits result in a large number of possible pickups of a vehicle at a supplier, whose orders are averagely over two dozen in the large-scale cases we studied. Furthermore, customers usually prefer mentioning desirable receiving times instead of hard time windows. In many real-world cases, such as Meituan (Meituan, 2018) and Baidu (Waimai, 2018), suppliers only provide a set of options of preferred delivery times, not time windows, for customers to choose. Thus we model MPDP by soft time windows: preferred delivery time is window start, and window length is zero because of the modeling simplicity and effectiveness that has operational convenience from real-world settings for a large-scale of customers. Service can start earlier or later than the preferred delivery time at cost of some penalty. In fact, such soft constraint approach has been widely used for many operational problems in real-world settings (Li and Head, 2009; Taş et al., 2014). The larger the time deviation, the larger the penalty. The unit penalty cost is defined to balance customer satisfaction and logistics cost. Some assumptions are made based on the real-world setting. Unlike many variations of vehicle routing problems, meals in MPDP are small-sized in comparison with the vehicle capacity. The average meals that can be held by a vehicle are far more than the orders delivered by a vehicle trip when constrained by the maximum delivery duration. Thus the capacity limitation can be ignored. An order is not allowed to be cancelled during the delivery process and we also assume that meals are ready for pickups when vehicles arrive at suppliers who have already been informed of the vehicle schedules. 3.1. Three types of logistics services We have presented in Section 1 three types of logistics services, exclusive, sharing, and sharing+ logistics services. In the exclusive service, each vehicle only delivers the meals for one supplier; in the sharing service, each vehicle can serve another supplier after delivering all the in-vehicle meals for one supplier; in the sharing+ service, each vehicle can serve another supplier during delivering the in-vehicle meals for one supplier. Fig. 3 shows an example for each service. From the figure, it can be observed that the exclusive service uses two vehicles with the most routing cost (39) but the least deviation time (0) from customers’ preferred times;

(a) Exclusive Service C2 (8) 5 SA

3

C5 (12)

C1 (12)

4

2 6

D

7

SA: Supplier A

D: Depot

5 C4 (23)

D

Vehicle 1 (exclusive to Supplier A)

Link with travel time on it

2

SB

3

S B: Supplier B

Ci (j):Customer i with preferred delivery time j

2

C3 (25)

Vehicle 2 (exclusive to Supplier B)

(b) Sharing Service 4 1 C2 (8)

C5 (12)

C1 (12)

2

C5 (12) 2

SB

C2 (8)

7

5 SA

1

(c) Sharing+ Service C1 (12)

2

SB

5 3

10

D

SA

C4 (23) 7 C3 (25)

3

D

C4 (23) 7

2

Vehicle 1 (shared by Supplier A and B)

C3 (25)

Vehicle 1 (shared by Supplier A and B)

Fig. 3. Examples for the exclusive and sharing logistics services. 499

2

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(b) A plane graph with three copies of supplier B for different times

(a) A space‐time diagram with rectangles for states and arrows for transitions between states Time

C5 (12)

D

2

2

SB (10) 25

C3

23

3

C4

18

SB

14

SB

SB (14) 4

D

SB (18) Wait at SB 5

Wait at SB

7 C4 (23)

12

2

C5

10

C3 (25)

SB

D: Depot SB: Supplier B

Location D

SB

C6

C5

(t): Visiting time t Ci: Customer i

C3

Fig. 4. Illustrations of the detailed schedule of vehicle 2 in Fig. 3(a).

the two sharing services employs only one vehicle with positive deviation times but less routing costs. All the three services show their own advantages respectively and no any service can dominate the other two in the example. Then, what factors contribute to the decreasing of the logistics cost for each service, and which is the best service for a certain range of situations, are the questions that need to be answered. These answers will be revealed after modeling and solving the MPDP. 3.2. Modeling An important aspect in MPDP is the multiple visits of each supplier during the operational horizon. Each visit has a different arrival time, which makes the problem challenging since it is impossible to designate a single variable for the arrival time at the supplier. The mutual exclusion constraints proposed by Hernandez et al. (2016) is a significant contribution in handling the “multiple visits”. However, it cannot be applied to MPDP because MPDP has multiple suppliers and the mutual exclusion constraints cannot ensure the incoming vehicles are consistent with the outgoing vehicles in visiting times and sequences for each supplier. To model the multiple visits and the possible waiting time for vehicles at suppliers, we propose a time-indexed model following the idea of the time-indexed graph presented by Macedo et al. (2011). This can be explained by the schedule of vehicle 2 in Fig. 3(a). The vehicle visits supplier B twice and waits at B during its second visit. The detailed schedule of the vehicle is shown in Fig. 4, where supplier B is copied to three vertices with same location but different visiting times. A state of a vehicle at a supplier depends on not only the location but also the visiting time. Then each supplier can be copied to multiple vertices indexed by different times. In this way, the multiple visits at a supplier can be distinguished by the time-indexed vertices of the supplier, and the waiting time of a vehicle at a supplier can be figured out by the time gap of two connected time-indexed vertices of the supplier. Overall, this discretization modeling approach is applicable to MPDP. Based on the time-indexed modeling idea, we formulate MPDP into a mixed integer programming model. The notations used in the modeling are shown in Table 1. The three logistics services have the same vertex set V = {ds} ∪ UI ∪ Uc ∪ {dt } but different edge set E , which is concluded in Table 2. The MPDP for all the three logistics services is formulated as the following. It is worth mentioning that formulae (1)–(12) are applicable to all the three logistics services and formulae (13) and (14) are special for the sharing+ service. The same (1)–(12) are defined on different edge set E for each logistics service concluded in Table 2.

min z = ρ

∑ (i, j ) ∈ E Ev

x i, j t i, j + σ

∑ (i, j ) ∈ E : i = d

x i, j + τ1

∑ i ∈ UC

pi1 + τ2

∑ i ∈ UC

s. t. 500

pi2 + τ3

∑ i ∈ UC

pi3

(1)

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Table 1 Notations used in modeling. Name

Type

Description

V E S Is

Set

UI

the vertex set the edge set defined in Table 2 the set of all the suppliers the set of time-indexed suppliers where s ∈ S and each item is associated with the same location as s but different visiting time the set of ⋃s ∈ S Is representing all the time-indexed suppliers

Cs UC

the set of customers who order meals from supplier s (s ∈ S ) the set of ⋃s ∈ S Cs representing all the customers the depot (Let ds and dt be the starting and terminating depot respectively; they are the same depot but represented by different symbols for modeling convenience) the visiting time wherei ∈ Is the preferred delivery time of customeri ∈ UC the supplier from which customer i (∈UC ) orders her meal the suggested delivery duration of meal the travel time of an edge(i, j ) ∈ E the routing cost per time unit the fixed cost of employing a vehicle and a driver penalty cost per time unit earlier than customers' preferred times penalty cost per time unit later than customers' preferred times penalty cost per time unit deviated from the suggested delivery duration xi, j = 1if the edge (i, j ) ∈ E is traveled by a vehicle and xi, j = 0 otherwise

Parameter

d

ti Di si P ti, j Parameters used in the objective function

ρ σ τ1 τ2 τ3 x i, j

Variable

Ai Ti, s pi1

the arrival time of a vehicle, i.e. the visiting time, at vertexi (i ∈ UI ∪ UC ) the accumulative travel time at vertex i (i ∈ UC ∪ UI ) since last visit to suppliers (s ∈ S ) the tardiness penalty if Ai ≤ Di (i ∈ UC )

pi2

the tardiness penalty if Ai > Di (i ∈ UC )

pi3

the tardiness penalty ifTi, si ≥ P (i ∈ UC )

Table 2 The connections in the edge set E for each logistics service. Connection Type From From From From

Exclusive Service

depot to supplier supplier to customer customer to customer customer to supplier

Connections Connections Connections Connections wheres ∈ S Connections

From supplier to supplier

Sharing Service

Sharing+ Service

ds

from to anyi ∈ UI Connections from any i ∈ UI to anyc ∈ UC from any i ∈ Is to any c ∈ Cs wheres ∈ S from any c ∈ Cs to c'∈ Cs where s ∈ S andc ≠ c' Connections from any c ∈ UC to c '∈ UC (c ≠ c') Connections from any c ∈ UC to anyi ∈ UI from any c ∈ Cs to any i ∈ Is from any i ∈ Is to j ∈ Is , (i ≠ j ) , satisfying ti < t j that indicates a waiting time of t j−ti at suppliers (∈S )

(We name the set of the connections between two time-indexed suppliers in the same set Is (s ∈ S ) as Ev since they are virtual edges that are “traveled” in no fuel consumption cost but only in waiting time.) Connections from any i ∈ Is to any j ∈ Is' satisfying s ≠ s' andti, j = t j−ti Connections from any c ∈ ⋃s ∈ S Cs todt

From customer to depot

∑

x i, j =

(i, j ) ∈ E

∑

x j, k = 1 ∀ j ∈ UC (2)

∑

x i, j =

(i, j ) ∈ E : i = d

∑

∑ (j, k ) ∈ E

x i, j =

(i, j ) ∈ E

x i', j'

(i', j' ) ∈ E : j' = d

∑

x j, k

(3)

∀ j ∈ UI (4)

(j, k ) ∈ E

x i, j (Ai + ti, j ) ≤ x i, j Aj

∀ (i, j ) ∈ E : i, j ∈ UC ∪ UI

(5)

Ai = ti ∀ i ∈ UI

(6)

Ti, s = 0 ∀ i ∈ Is , s ∈ S

(7)

x i, j (Ti, s + ti, j ) = x i, j Tj, s ∀ (i, j ) ∈ E : i, j ∈ UC ∪ UI , j ∉ Is , s ∈ S

(8)

pi1

(9)

≥ Di−Ai

∀ i ∈ UC 501

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pi2 ≥ Ai −Di ∀ i ∈ UC

(10)

pi3

≥ Ti, si−P ∀ i ∈ UC

(11)

pi1 ,

pi2 ,

(12)

pi3

≥ 0 ∀ i ∈ UC

Formula (1) is the objective function. Constraint (2) ensures that each customer is served exactly once. Constraint (3) guarantees that all the vehicles departing from depot should return to depot. Constraint (4) is conservation flow for suppliers. Constraint (5) imposes the time continuity for visiting two neighbour vertices. Constraint (6) indicates that for a time-indexed supplier, its starting time is equal to its time point. Constraint (7) guarantees the accumulative travel time as 0 when a trip starts from a time-index supplier. Constraint (8) ensures that the accumulative travel time since the last supplier is increased by the travel time along an edge. Constraints (9)–(11) are tardiness constraints for calculating for each customer the early arrival time, late arrival time, and the overtime for in-vehicle meals in a vehicle. Constraint (12) are nonnegative constraints for the three tardiness penalty variables.

Ti, s = −M ∀ i ∈ UI Is , s ∈ S

(13)

Ti, si ≥ 0 ∀ i ∈ UC

(14)

For the sharing+ service, we add the constraints (13) and (14) to its model to exclude the possibility that a supplier si is not visited before visiting customer i . In (13), M is a large number. If customer i is visited before its supplier si , Ti, si will be negative and thus constraint (14) will be violated. The other two services do not necessarily have the constraint because it has been ensured by the connections defined in their edge sets. From the model, the problem broadly belongs to the multi-trip routing problem with soft time windows and multiple refill locations. Furthermore, the problem contains some nonlinear components in (5) and (8), which are linearized in (15) and (16) respectively. Because the MPDP extends the general vehicle routing problem by considering multiple suppliers and multiple visits at suppliers, it is clearly NP-hard. It is thus not practical to apply any exact methods to real-world cases that can consists of tens of suppliers and thousands of customers. We then present two heuristics in Section 4, and test their effectiveness in Section 5.

Ai + ti, j−Aj ≤ (1−x i, j ) M ∀ (i, j ) ∈ E : i, j ∈ UC ∪ UI

(15)

Ti, s + ti, j−Tj, s ≤ (1−x i, j ) M ∀ (i, j ) ∈ E : i, j ∈ UC ∪ UI , j ∉ Is , s ∈ S

(16)

4. Algorithms As mentioned in Section 3, the MPDP belongs to a type of complicated vehicle routing problem with special features. We thus develop two efficient heuristics in this section to solve large-scale real-world problems. The heuristics are an iterated local search in Section 4.2 and an adaptive large neighborhood search in Section 4.3. Both the heuristics start from an initial solution generated by the construction heuristic introduced in Section 4.1. 4.1. Construction heuristic A typical vehicle route can be classified into three types of trips: initial deadheading trip, middle service trips, and final deadheading trip. The initial trip and the final trip are denoted as {ds , i} and {c, dt }(i ∈ UI , c ∈ UC ) respectively. A middle service trip is defined as a vehicle path starting from a supplier to pick up some orders and then visiting the customers sequentially. Thus a service trip can be denoted as {i, c1, ⋯, cn} (i ∈ Is , cj ∈ Cs, s ∈ S ) . The service trips contribute to the major portion of a solution to the MPDP. To make an initial solution, we first construct middle service trips and then link the trips to become vehicle routes. (1) Construction of middle service trips. Common trip construction approaches are to keep clustering customers until a hard constraint is reached, such as vehicle capacity or maximum travel distance. However, only some soft constraints are used for the MPDP to handle its flexibility in the real-world setting. Among the soft constraints, the suggested delivery duration is a good candidate because it follows the spirit of multi-trip routing. We thus use it as a break point to stop including more customers into a trip since otherwise penalty costs have to be incurred. A good solution often clusters the customers together if they have a same supplier, similar preferred delivery times, and nearby delivery locations. Thus, a spatial-temporal based clustering method is developed to construct service trips. As illustrated in Fig. 5, we divide the whole delivery time, i.e. the temporal axis, into multiple periods, each of which is represented by an oval. Within the spatial area of each oval, the customers are grouped together if their preferred delivery times are in the temporal period. We then apply the principle of sweep to scan the customers clockwise in each oval until all the customers in the subset have been assigned. Three points in the construction procedure are worth mentioning. First, the oval number is set as [L/ P ] + 1, where L is the whole delivery time from the vehicle starting time to the latest customer’s preferred delivery time. Second, when a swept customer is added to a trip, only the insertion with the minimum cost will be made. Third, the evaluation of an inserted trip depends on two components in the objective function, the routing cost and the penalty cost of the deviation from customers’ preferred times. However, the penalty 502

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Temporal axis Vertical axis

Vertical axis

Horizontal axis

Horizontal axis

Customer

Spatial based clustering

Spatial-temporal based clustering

Fig. 5. Illustration of the spatial-temporal based clustering method.

of a trip will change with its starting time, which is unknown since its former trips are not determined yet. Then we use the starting time with the minimal deviation to approximate the penalty. (2) Linking of middle service trips. The three logistics services have different constraints in linking middle service trips for vehicle routes. In the exclusive service, only the service trips that have a same supplier can be linked end to end. In the sharing service, any service trips can be linked end to end. In the sharing+ service, any service trips can be linked by overlapping any parts of them. We postpone the overlapping to the guided neighborhood search of the two heuristics because a good overlapping needs to be made by local optimization. By referencing the linking principle in the classical savings heuristic (Clarke and Wright, 1964), we develop a greedy trip linking heuristic to find promising vehicle routes. The heuristic first assign each service trip to a vehicle, and then iteratively merges the two vehicle routes with the maximum cost decrease until no routes can be merged for less cost. An example in Fig. 6 is used to illustrate the construction process. The suggested delivery duration is set as 13, resulting in two time periods, [0,13] and (13,24]. Customers 1, 2, and 5 are in the first period while the other customers are in the second. Then customers of each supplier and each period are swept and assigned to three service trips. These trips are further assigned to vehicles by greedy trip linking heuristic. Afterwards, the two routes of supplier B are linked together because they are the best linking with maximum cost decrease. Finally, the construction stops when no other routes can be linked with cost decrease.

4.2. An iterated local search heuristic In Section 4.1, we use the suggested delivery duration to decompose the whole time horizon into a number of smaller time horizons and apply the construction algorithms on each smaller horizon. Such decomposition and construction well correspond to the cost structure that imposes penalty if the trip duration is beyond the suggested delivery duration. However, the suggested delivery duration is defined as a soft constraint that can be violated at a penalty; it is thus very possible that the durations of some trips in the optimal solution are slightly beyond the suggested delivery duration, after routing and fixed costs are considered. Based on this observation, we present an iterated local search, whose major principle is to make random disturbation to the suggested delivery duration and provide different initial solutions for improvement, thereby increasing the possibility of escaping from local minimum and exploring the solution space extensively. The iterated local search has been used successfully in many combinatorial optimization problems, such as the traveling salesman problem (Lin and Kernighan, 1973) and vehicle routing problem (Ibaraki et al., 2005). We realize the heuristic by first initializing a solution based on a randomized suggested delivery duration and then improving the solution by local search. Our preliminary experiments show that a good range is between 85% and 115% of the suggested delivery duration. The heuristic is run for 10 iterations and finally returns the best found solution.

Initial State

C1 C2

SA

C5

SB

C2

C2 C4

C4

SB

C3

SB D

C5

SA

C5

C1

C1

SB

D

Initial solution

Constructed service trips

SA

C3

Trip Construction

Trip Linking

Fig. 6. An example of solution construction. 503

C4

D C3

D: Depot SA: Supplier A S B: Supplier B Ci (j): Customer i with preferred time j

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To improve an initial solution, we design special local search operators for MPDP to exploit the solution space intensively. Local search has been widely used for solving the vehicle routing problem (Bräysy and Gendreau, 2005) and consists of inter-route operations (Ibaraki et al., 2005; Lin, 1965), intra-route relocations (Or, 1976; Hu et al., 2015), and integrated operations (Sheu, 2007b; Li, 2013; Wang et al., 2015). Different from the general vehicle routing problem, each vehicle route of the MPDP can be interpreted as two-levels: (1) each route is composed of a sequence of service trips; (2) each service trip serves a supplier and a sequence of customers. In order to exploit this special structure, we develop a local search method with three types of operations: at the customer level, at the trip level, and at the integrated level of customers and trips. In the customer level, we apply 2-opt (Lin, 1965) and or-opt operators (Or, 1976) to improve the sequence of customers in a service trip. In the trip level, we use the following two local search operations: (1) relocating a service trip from a vehicle route to another route, and (2) exchanging a service trip in a vehicle route with a service trip in another vehicle route. We also develop an integrated operation for both customers and trips. Consider a situation in which the total cost may be reduced if a customer can be deleted and inserted into another trip, and the inserted service trip may be further relocated to another vehicle. Neither the trip-level operators nor the customer-level operators can achieve such an improvement. We then develop two integrated operators, relocation first-assignment second and exchange first-assignment second. The former operation first moves a customer from its service trip to another trip which is then assigned to the vehicle with the minimum total cost. The latter operation first exchanges two customers of two service trips which are then assigned to the vehicles with the minimum total cost. Both the “first” operations cannot be made on two service trips that start from different suppliers. After any local search operation, the feasibility of the obtained solution will be checked for the three logistics services. Only the feasible solution with less total cost will be accepted. The procedure stops until the maximum iterations. Fig. 7 illustrates the operation of relocation first-assignment second by applying it to the initial solution obtained from Section 4.1 for the sharing service. As shown in the figure, customer 5 is first relocated to the other service trip of the same vehicle and then the changed trip is assigned to the other vehicle. Thus an improved solution, which not only uses one vehicle less but also save over 20% routing cost, is obtained. 4.3. An adaptive large neighborhood search heuristic We also develop an adaptive large neighborhood search, which is another mainstream heuristic for vehicle routing (Shaw, 1998; Pisinger and Ropke, 2010; Masson et al., 2013). The heuristic contains the following steps. Step 1: Remove a large number of elements, say 20% of the customers in our case studies, from the current solution using removal heuristics. Step 2: Insert these removed elements back with a series of heuristic rules. Step 3: Update the current solution using a simulated annealing acceptance criterion (Masson et al., 2013); and repeat the above two steps until the maximum iterations are reached. In order to achieve good performance, we develop special removal and insertion heuristics that exploits the special structure of the problem. 4.3.1. Removal heuristics We first develop some common removal heuristics for vehicle routing, such as the worst removal, random removal, and related removal (Pisinger and Ropke, 2010; Masson et al., 2013). In the related removal, the relatedness of two customers in MPDP depends on the travel times between their pickup locations and between their delivery locations, as well as the difference between their preferred delivery times. Formally, the relatedness of customer i and j , is defined in (17), where α , β , and γ are the weights. Any pair of customers that have a small relatedness are prone to being removed.

Ri, j = αti, j + βt si, sj + γ |Di−Dj | i, j ∈ Uc

(17)

Note that a major difficulty in solving our problem is to determine what customers need to be picked together from a supplier for a delivery trip. For the real-world cases studied in Section 5, a supplier has over two dozen customers on average. Thus there are a large number of options in picking up their orders. A good solution is likely to be found by exploring different picking decisions. For example, the initial solution in Fig. 7 picks up twice at supplier B; if we merge the two pickups into one, better solutions can be found Initial solution

Intermediate solution

C5 C1

C1 SB

C2

SB

C2

D SA

Improved solution

C5 Relocation First

C2

Assignment Second

D C4

D

SA

C4

D

C3

SA

Assignment Second

C4 C3

D: Depot SA: Supplier A S B: Supplier B Ci (j): Customer i with preferred time j

Fig. 7. An example of relocation first-assignment second for the sharing service solution. 504

SB

D

C3

Relocation First

C5

C1

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with much saving in routing cost. Such a picking exploration requires removing all the customers of a supplier from a solution before repairing them. However, the existing removal heuristics cannot ensure that all the customers of a supplier are removed, which makes it difficult to examine different combinations of order picking within the scope of a supplier. Based on this observation, we develop a supplier-oriented removal heuristic, which first chooses a supplier randomly and removes all the associated customers. If the associated customers consist of less than the given percentage 20%, choose another supplier randomly and repeat the removal process until enough customers are removed. If the customers of a chosen supplier are more than the given percentage, remove part of its customers randomly. A supplier itself needs to be removed if all the associated customers are removed. 4.3.2. Insertion heuristics To insert the removed customers, we develop the best insertion and regret-k insertion methods (Masson et al., 2013). The value of k is selected between two and four according to our experiments. In MPDP, order pickup occurs before delivery; a solution is then infeasible if an inserted customer is visited before her supplier. Thus, in order to obtain a feasible insertion, a customer needs to be inserted after her supplier that is just inserted or already exists in the route. Generally, there are a number of optional positions for insertion in a route. However, only some of the positions are likely optimal. We design the following three rules to choose a certain range of insertion positions. Rules 1 and 2 consider inserting a customer with her supplier, while rule 3 considers inserting only customers without suppliers. Rule 1: Do not insert supplier s into a position between the same s and its subsequent customer j (j ∈ Cs ) because otherwise supplier s would be visited twice without its customers in between. Rule 2: After inserting supplier s , potential positions of inserting its customer i (i ∈ Cs ) are after the inserted s and before any subsequent s in the route because otherwise the insertion of s is unnecessary. Rule 3: Potential positions of inserting customer i (i ∈ Cs ) are after any visit of her supplier s in the route. These three rules are illustrated by an example in Fig. 8. There are five positions to insert customer 1 whose supplier is A. Based on rule 1, position 2 is not an option. Based on rule 2, if supplier A is inserted at position 3, the options of inserting customer 1 are positions 3, 4, and 5; there are respectively 1, 3, 2, and 1 potential position(s) for inserting customer 1 if supplier A is inserted at the positions 1, 3, 4, and 5. Based on rule 3, potential positions of customer 1 include positions 2, 3, 4, 5. Based on the three rules, there are totally 11 insertion candidates, among which the best position is 5 for inserting customer 1 herself. However, without using the rules, up to 30 possible insertions need to be examined. The computing time is significantly saved to repair a solution in real-world case studies. At each iteration, the best insertion method chooses a customer with the lowest insertion cost. The regret-k insertion method k inserts customer i∗ (and her supplier) at her best position so that i∗ = argmax( ∑ j = 2 Δfi j −Δfi1 ) , where UR is the set of removed cusi ∈ UR

tomers and Δfi j designates the insertion cost of customer i (and her supplier) in the j th best position of a solution. The methods stop when all the removed customers are inserted. An example of the supplier removal method and the best insertion method is illustrated in Fig. 9. Customers 3, 4, and 5 of supplier B are first removed from the initial solution and then these customers are inserted respectively at the best positions. Finally, an improved solution to the sharing+ service is obtained. 5. Case studies In this section, we first describe real-world data and estimate the relevant parameters used in case studies. We then compare our algorithms with the MIP solver of CPLEX and the classical insertion (Solomon, 1987). Finally, we present managerial insights of the three logistics services, and conduct sensitivity analysis. 5.1. Case description and parameter estimation We obtained real-world data from a major logistics provider in China. The data contains nine-day lunch orders in Dalian metropolitan area. Everyday data is considered as a case so that we have nine cases. Each case has different numbers of suppliers and orders. Each order has different delivery requirements, including the delivery location, the picking supplier, and the preferred delivery time. And a supplier in each case has different orders. Numbers of orders and suppliers are detailed in Table 3. The options of the preferred delivery time ranges from 11:15 to 14:00 with 15 min interval between each. SA

D Position 1

C2 (8) Position 2

SB

Position 3

C5 (12) Position 4

D

Position 5

Possible positions for inserting a customer of supplier A

Fig. 8. A route ready for the insertion of a customer of supplier A.

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Removed solution (Customers of supplier B are removed)

Intermediate solution (Customer 5 and supplier B are inserted) C5

C1

C5

C1

C2

C2

Improved solution (Customers 4 and 3 are inserted respectively) C1 C5

SB

C2

SB

C4 SA

D

SA

C3

D

SA

C4

D C3

Insert customers 4 and 3 respectively

Insert customer 5 with supplier B

D: Depot SA: Supplier A S B: Supplier B Ci (j): Customer i with preferred time j

Fig. 9. An example of removing the customers of supplier B and inserting them for the sharing+ service.

Table 3 Data used in the case studies. Date Case Case Case Case Case Case Case Case Case

1 2 3 4 5 6 7 8 9

Nov. Nov. Nov. Nov. Nov. Nov. Nov. Nov. Nov.

22, 23, 24, 25, 26, 27, 28, 29, 30,

2014 2014 2014 2014 2014 2014 2014 2014 2014

Orders

Suppliers

794 660 920 1162 1219 681 734 595 665

35 32 39 41 42 30 31 27 28

Since the addresses of customers and suppliers are known, their longitudes and latitudes can be determined by mapping their addresses to the points on a digital map. The mapping is achieved by Baidu API. To visualize the distribution of the depot, customers, and suppliers in our data, we plot their locations by a thermodynamic diagram on a Baidu map, as seen in Fig. 10. To calculate the distance of a pair of vertices, we determined the shortest travel distance based on the longitudes and latitudes of the pair of vertices. Accordingly, the travel time between a pair of vertices can be then estimated using its travel distance and an average travel speed, which was set as 40 km per hour because of the off-peak hours during the lunch period. Fuel consumption cost per km was set as 0.48 RMB because fuel consumption is usually 0.08 L per km for a delivery vehicle and oil cost stabilizes at around 6 RMB per liter oil in China. Therefore, parameter ρ was set to 0.32 RMB per minute. The settings of parameters are concluded in Table 4. Many settings are in accordance with real operation cost of the logistics provider. 5.2. Computational studies To testify the performance of the developed heuristics, we first compare the iterated local search (ILS) with the MIP solver of CPLEX 12.7 and then compare the ILS and the large neighborhood search (LNS) with the adapted insertion algorithm. We use 1 min as the time unit to discretize the suppliers. Since CPLEX cannot solve real-world sized instances, we generate a number of small instances with 2 or 3 suppliers and tens of customers for the sharing service. The computational results are presented in Table 5. It can be observed that it takes CPLEX nearly 1.5 h to solve an instance with 10 customers and 2 suppliers. CPLEX often fails to find an optimal solution for the problem with 20 customers after 5 h run. For the instances with less than 20 customers we tested, the ILS generates the solutions as good as CPLEX in no more than two seconds. For the instances with 50 and 100 customers, the ILS is obviously better than CPLEX. The poor performance of CPLEX is not a surprise since the linearized model has nearly 5000 constraints and over 3100 variables for the problem with only 20 customers and 3 suppliers in the sharing service. In order to testify the performance of our heuristics, we adjusted the Solomon Insertion heuristic 1 (Solomon, 1987) by setting the parameters of μ , α1, α2 , and λ to 1, 0.5, 0.5, and 1 respectively and completing a service trip construction when the suggested delivery duration has to be violated. The service trips constructed by the insertion heuristic are further merged and improved by our trip linking method (Section 4.1) and the classical local search techniques of 2-opt, or-opt, and trip relocation and exchanging (Section 4.2), so that a solution can be obtained for comparison. Table 6 shows the comparison. In Table 6, column 1 is case ID, and column 2 specifies the algorithm. Columns 3–5 present service trip number per vehicle (Trip/Veh.), total cost in RMB, and CPU time in seconds for the exclusive service. Columns 6–8 and 9–11 present the corresponding results for the two sharing services respectively.

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Supplier

Dense

Depot Customer density Sparse

Fig. 10. A thermodynamic diagram of the distribution of depot, customers and suppliers. Table 4 Values of parameters. P : 40 min ρ : 0.32 RMB/min τ1, τ2 : 0.1 RMB τ3 : 1 RMB The maximum iterations of the iterated local search: 2000 The maximum iterations of the large neighborhood search: 8000

σ1: 150 RMB α : 0.3

σ2 : 180 RMB β : 0.3

σ3 : 210 RMB γ : 0.4

Table 5 Computational comparison between CPLEX and ILS for the sharing service. Suppliers

Customers

Time Span

Total Cost

CPU (s)

Cplex

ILS

Cplex

ILS

2 2 2 2 2

5 10 20 50 100

11:00–11:30

298.37 496.16 617.85 (936.72) (1637.18)

298.37 496.16 617.85 930.59 1621.45

3 5164 15,275 =5 h =5 h

<1 <1 2 17 49

3 3 3 3 3

5 10 20 50 100

11:00–11:30

314.20 (521.74) (635.45) (1004.79) (1692.95)

314.20 521.74 635.45 996.54 1667.24

7 =5 h =5 h =5 h =5 h

<1 <1 2 22 53

2 2 2 2 2

5 10 20 50 100

11:00–11:40

273.91 (500.48) (634.73) (1013.68) (1714.36)

273.91 500.48 630.91 998.52 1687.69

24 =5 h =5 h =5 h =5 h

<1 <1 2 34 63

*The results in parentheses are the feasible solutions obtained by CPLEX in 5 h. 507

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Table 6 Computational results of Insertion, ILS, and LNS. Case

Algorithm

Exclusive Service

Sharing Service

Sharing+ Service

Trip/Veh.

Total Cost

CPU (s)

Trip/Veh.

Total Cost

CPU (s)

Trip/Veh.

Total Cost

CPU (s)

Case 1

Insertion ILS LNS

1.00 2.91 2.97

9746.51 8725.45 8429.78

317 738 804

1.25 3.64 3.75

9122.84 7162.17 6869.22

441 935 963

1.35 4.14 4.05

8950.03 6547.98 6402.17

476 1066 1088

Case 2

Insertion ILS LNS

1.00 2.84 2.81

8444.53 7429.41 7243.43

287 548 570

1.28 3.77 3.82

7937.79 6379.09 5978.42

394 872 959

1.39 4.10 4.15

7834.39 5822.40 5695.70

418 1003 1103

Case 3

Insertion ILS LNS

1.02 2.82 2.87

11048.83 9517.43 9003.69

624 1119 1175

1.34 3.84 3.92

9835.67 7843.56 7508.59

770 1271 1384

1.44 4.04 4.09

8869.65 7169.57 7027.06

816 1614 1464

Case 4

Insertion ILS LNS

1.00 3.00 3.05

11845.29 10178.23 10115.04

935 1424 1367

1.14 3.84 3.88

11081.44 8865.14 8625.17

1040 1782 1746

1.21 4.03 4.07

10241.93 8144.06 7987.41

1054 2049 2165

Case 5

Insertion ILS LNS

1.00 2.88 2.90

12986.35 10851.31 10556.54

1603 2140 2289

1.17 4.44 4.47

11563.56 8524.36 8264.19

1627 2219 2332

1.24 4.86 4.90

10882.97 7772.72 7716.16

1654 2407 2364

Case 6

Insertion ILS LNS

1.03 2.97 3.00

8460.51 7692.9 7724.56

315 526 501

1.11 3.70 3.70

7730.43 6519.66 6245.59

435 942 972

1.20 3.95 3.90

7672.34 5939.43 5925.91

474 1008 1018

Case 7

Insertion ILS LNS

1.03 2.84 2.90

8711.31 7793.51 7344.99

385 578 584

1.15 3.54 3.58

8067.78 6704.25 6423.88

416 871 879

1.24 3.82 3.86

7807.83 6031.98 5948.35

453 1097 1064

Case 8

Insertion ILS LNS

1.00 2.96 3.00

7774.5 6947.28 6929.04

195 404 368

1.08 3.67 3.57

7240.14 5890.23 5830.97

250 497 510

1.17 4.00 3.89

7169.20 5351.41 5352.79

310 522 638

Case 9

Insertion ILS LNS

1.00 2.93 2.96

8129.25 7231.6 7069.93

262 425 408

1.17 3.90 3.85

7644.37 5811.45 5590.09

295 918 920

1.27 4.28 4.28

7414.80 5182.67 5120.27

322 1092 1196

Avg.

Insertion 1.01 ILS 2.91 LNS 2.94 ILS / Insertion LNS / Insertion

9683.01 8485.24 8268.56 87.63% 85.39%

547.00 878.00 896.22

1.19 3.82 3.84

8913.78 7077.77 6815.12 79.40% 76.46%

629.78 1145.22 1185.00

1.28 4.14 4.13

8538.13 6440.25 6352.87 75.43% 74.41%

664.14 1317.68 1344.28

From Table 6, we can observe that the total costs from our heuristics are consistently lower than the cost from the insertion algorithm for all the cases. For the three services, the average total costs from the ILS cover only 87.63%, 79.40%, and 75.43% of those from the insertion algorithm respectively, and the related coverages of the LNS are 85.39%, 76.46%, and 74.41% respectively, which indicates a significant improvement over the classical algorithm. It shows that our heuristics are competitive and the solution frameworks and some special operators we developed, e.g. the local search and customer removal and insertion heuristics, are effective for solving the MPDP. In addition, average service trips per vehicle of the insertion algorithm are very close to 1 no matter what type of logistics services, while those from our algorithms are much more than 1. We observed from the detailed routes of the insertion algorithm that almost all their service trips have so large travel times that a vehicle almost has no more time to accomplish another service trip after it completes the current service trip. On the contrary, our algorithm splits longer service trips into multiple shorter ones, each of which falls in different time periods. Vehicles are then able to complete more than one service trip at expense of slight deviation from customers’ preferred delivery times. Having shorter service trips can provide more possibilities for optimization in trip linking. The good performance is due to considering the vicinity from both spatial side and temporal side. Our clustering principle is confirmed as effective. As a result, short service trips can be obtained, and hence the total costs for the three services by our algorithms are less than those by the insertion algorithm. Table 6 also shows that our two heuristics can solve the large-scale real-world cases in half an hour averagely. The two algorithms are comparable in the computation time. While in the aspect of total costs, the LNS is slightly better than the ILS despite one exception on case 8. We found the special supplier removal heuristic in the LNS contributes to its better results significantly. Among the total iterations of updating the best found solution by LNS on case 1, the supplier removal method accounts for 34.67%, far beyond other removal methods. We observe that the computation times of the two sharing services are longer than that of the exclusive service. The longer computation of the sharing services is expected since additional connections between the suppliers are allowed, resulting in much 508

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Table 7 Results of the ILS for the three logistics services. Case

Service Type

Overall Result

Normalized Result

Veh.

Routing Cost

Total Cost

Trip/Veh.

Deviation/Order (m)

Total Cost Percentage (Sharing/Exclusive)

Case 1

Exclusive Service Sharing Service Sharing+ Service

35 25 22

1214.05 1054.78 931.49

8725.45 7162.17 6547.98

2.91 3.64 4.14

23.93 18.35 11.32

– 82.08% 75.04%

Case 2

Exclusive Service Sharing Service Sharing+ Service

32 22 20

979.12 891.20 766.44

7429.41 6379.09 5822.40

2.85 3.77 4.10

20.13 19.72 9.42

– 85.86% 78.37%

Case 3

Exclusive Service Sharing Service Sharing+ Service

39 25 23

1334.06 1200.63 1001.58

9517.43 7843.56 7169.57

2.82 3.84 4.04

21.33 21.31 11.83

– 82.41% 75.33%

Case 4

Exclusive Service Sharing Service Sharing+ Service

41 32 30

1414.64 1286.35 1106.62

10178.23 8865.14 8144.06

3.00 3.84 4.03

19.07 14.29 6.11

– 87.10% 80.01%

Case 5

Exclusive Service Sharing Service Sharing+ Service

42 32 29

1917.41 1506.50 1050.39

10851.31 8524.36 7772.72

2.88 4.44 4.86

18.15 9.49 4.89

– 78.56% 71.63%

Case 6

Exclusive Service Sharing Service Sharing+ Service

30 23 21

1157.72 958.79 971.91

7692.90 6519.66 5939.43

2.97 3.70 3.95

25.40 19.95 7.37

– 84.75% 77.21%

Case 7

Exclusive Service Sharing Service Sharing+ Service

31 24 22

1239.27 1140.55 942.28

7793.51 6704.25 6031.98

2.84 3.54 3.82

21.76 14.30 5.22

– 86.02% 77.40%

Case 8

Exclusive Service Sharing Service Sharing+ Service

27 21 19

1308.01 1037.96 925.69

6947.28 5890.23 5351.41

2.96 3.67 4.00

22.11 15.16 6.32

– 84.78% 77.03%

Case 9

Exclusive Service Sharing Service Sharing+ Service

28 20 18

1157.46 973.13 914.48

7231.60 5811.45 5182.67

2.93 3.90 4.28

23.79 16.28 6.12

– 80.36% 71.67%

Avg.

Exclusive Service Sharing Service Sharing+ Service

33.89 24.89 22.67

1302.41 1116.66 956.76

8485.24 7077.77 6440.25

2.91 3.82 4.14

21.74 16.54 7.62

– 83.55% 75.97%

larger search space. In comparison with the insertion algorithms, our heuristics require longer computation because more complicated operations are involved. However, even for the large case with over one thousand customers, the computational time is < 40 min on a very common computer with Intel I7 2.69 GHz processor and 8 GB RAM. 5.3. Managerial analysis for the logistics services In this section, we conduct managerial analysis for the three logistics services. Table 7 presents the results of the services. In Table 7, column 1 is case ID, and column 2 is the service type. Columns 3–5 present overall results, including the number of vehicles (Veh.), routing cost in RMB, and total cost in RMB. Columns 6–8 present normalized results, including trips per vehicle (Trip/Veh.), deviation from customers’ preferred delivery times per order in minutes (Deviation/Order), and total cost percentage as the total cost of any sharing service over the total cost of the exclusive service. We use the ILS to make the managerial analysis since the LNS has very similar results. Table 7 shows that the routing costs and the total costs of the two sharing services are less than those of the exclusive service respectively for each case, and the sharing+ service spends less cost than the sharing service on average. The average total cost percentages are 83.55% and 75.97% for sharing and sharing+ services, reducing over 16% and 24% total cost respectively. Furthermore, the average vehicles are 33.89, 24.89, and 22.67 for exclusive, sharing, and sharing+ services respectively, which indicates the two sharing services require over 26% and 33% vehicles less on average. The savings in vehicles are due to the fact that vehicles in the sharing services can undertake more deliveries than in the exclusive service. The vehicle savings are also confirmed by the trips per vehicle, as in column 6. The average trips per vehicle of the two sharing services are 3.82 and 4.14, much more than that of the exclusive service, 2.91. The large vehicle savings in the sharing services are due to the operational innovation. Vehicles in the sharing services do not need to return to starting supplier and thus can serve a closer supplier and shorten the deadheading distance. The saved travel distance and time help a vehicle to undertake more deliveries. Table 7 shows that the sharing+ service saves over 7% more than the sharing service in the total cost. Note that in the sharing service, all the picked meals need to be delivered before any new pickup; such a constraint does not exist in the sharing+ service. 509

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(a) Sensi ty analysis for the sharing service with the exclusive service as the basis.

(b) Sensi ty analysis for the sharing+ service with the sharing service as the basis.

Fig. 11. Illustration of the sensitivity analysis.

This important operational innovation in the sharing+ service leads to more flexibility in routing vehicles, which is the major reason of having less total cost. We also observe that the deviation times per order of the sharing services are smaller than those of the exclusive service. The average deviation times per order are 16.54 and 7.62 min for the sharing services respectively, much less than that of the exclusive service. The better results of the sharing services are not a surprise. As mentioned above, the vehicles are much more flexible in the sharing services, thereby improving on-time delivery quality. 5.4. Sensitivity analysis The total cost of the MPDP consists of routing cost, penalty cost from delivery time deviation and fixed cost. The first two types of costs in the solution have relatively clear definitions and are generally not impacted by the logistics service type. While the fixed cost is mainly composed of the costs of hiring vehicles and drivers. Note that the driver hiring costs are generally different in the three logistics services: the cost of sharing+ service is higher than that of sharing service, whose driver hiring cost is further higher than that of exclusive service. This is because the drivers need to know more service areas in the sharing services and it is more complicated for a driver to exactly remember the pickup and delivery sequence and the associated products of orders in the sharing+ service. Thus, a driver generally prefers the exclusive service rather than the two sharing services given the same hiring wage. The fixed costs of employing a vehicle and a driver, i.e. σ1, σ2 , and σ3, are set as 150, 180 and 210 RMB for the three services respectively in the previous experiments. However, the driver hiring cost is related to human behaviors and the degrees of their familiarity with the traffic in a certain area. Such a cost varies with different areas, different traffic conditions, different groups of drivers, or even different standards of living, and it is thus subject to more uncertainties. Therefore, we conduct sensitivity analysis using a set of varied unit fixed costs, from 150 to 250 RMB, for the two sharing services. The analysis is done on one of the cases (Case 1) by our ILS since the other cases and the LNS have very similar results. The sensitivity analysis results are shown in Fig. 11 (a) and (b), where x-axes are σ2 , and σ3 for the two sharing services respectively, and y-axes are the percentages of three types of costs from the two sharing services with the corresponding costs from the exclusive service and the sharing service as the bases respectively. For the comparison purpose, both the basic σ1, and σ2 are set as 150 RMB in Fig. 11 (a) and (b). From Fig. 11 (a) and (b), when σ1, σ2 , and σ3 are all set as 150, the total cost of the sharing service covers about 74% of that of the exclusive service, and the total cost of the sharing+ service is about 85% of that of the sharing service and thus about 63% (=85% × 74%) of that of the exclusive service. The two sharing services bring a lot of cost savings due to their operational innovations. From Fig. 11 (a), it can be clearly seen that the fixed cost is in an increasing trend with the increasing of σ2 . Although there are fewer vehicles used in the sharing service, the larger unit fixed cost results in the larger overall fixed cost. However, the total cost percentage does not exceed 100% in Fig. 11 (a) despite its increasing trend with the fixed cost. This is because the percentages of the sum of routing cost and penalty cost are far less than 100%, making the total cost of the sharing service less than that of the exclusive service in a wide range of σ2 . As mentioned before, the operations across the suppliers bring much more routing flexibility, thus the routing cost and penalty cost can be significantly reduced. In Fig. 11 (b), the overall trends of the three types of costs are similar to those in Fig. 11 (a) respectively. However, different from Fig. 11 (a) where the percentage of the total cost is below 100%, the percentage of the total cost is above 100% when σ3 is larger than 510

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220. Thus, 220 is a good line for decision makers to choose a right sharing service to the case. When the practical unit fixed cost is larger, the sharing service is the better option. Otherwise, the sharing+ service is recommended. In summary, the sensitivity analysis shows that the sharing services have significant advantages over the exclusive service in a broad range of fixed cost uncertainty. Due to the operational innovation, the sharing services are the better options. The special guideline of the unit fixed cost can be obtained by our algorithms for the decision makers to choose a right sharing service in practice. 6. Conclusions In this paper, we study a large-scale meal pickup and delivery problem that consists of thousands of orders, tens of food suppliers, multiple visits of each supplier, and delivery time and meal freshness requirements. The problem broadly belongs to the multi-trip routing problem with soft time windows and multiple refill locations. We propose three logistics services: in the exclusive service, each vehicle only delivers the meals for one supplier; in the sharing service, each vehicle can serve another supplier after finishing the delivery for the current supplier; in the sharing+ service, each vehicle can serve multiple suppliers simultaneously. All the three services lead to NP-hard problems. We propose a time-indexed model to formulate the problems. The MIP solver of CPLEX can solve very small-sized instances due to problem complexity, in particular a large number of suppliers and multiple visits. We then develop two mainstream heuristics, an iterated local search and an adaptive large neighborhood search, both of which start from a construction heuristic. The construction heuristic generates an initial solution with short service trips that provide more possibilities for optimization than long trips. The iterated local search improves a solution by the special operators designed the problem structure with both trip and customer levels. The adaptive large neighborhood search is beneficial to handling the one-to-many pickup/ delivery problems by special supplier removal heuristic and insertion rules. We compare the heuristics with the MIP solver of CPLEX and the adapted insertion algorithm since there is no existing heuristic for the problem, and examine the performance of the two heuristics on nine real-world cases with over one thousand customers and tens of suppliers. The computation shows our iterated local search can obtain an optimal solution for small-sized instances in no more than two seconds, while CPLEX needs to spend over 4 h solving the instances. The two heuristics have a significant improvement over the adapted insertion algorithm. In comparison with the iterated local search, the large neighborhood search improves the solution by about 2% on average. The computation of the three logistics services reveals that, under a typical cost setting, the sharing and sharing+ services save beyond 16% and 24% total cost over the exclusive service respectively. The saving is mainly due to the important operational innovation that reduces deadheading travel distances and makes the delivery closer to the preferred receiving time. Extensive sensitivity analyses show the sharing service has great advantage over the exclusive service when the unit fixed cost of the sharing service ranges from 150 to 250; and the sharing+ service is better than the sharing service given the unit fixed cost below 220, which is a certain guideline that can be provided by our heuristics for the decision makers to choose a right service in practice. Future research can go in several directions. An important topic is to study and develop a decision support system based on geographical information system for logistics providers. 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