A Layer-Based Relaxation Approach for Service Network Design

Abstract

Service-network design in transportation applications has attracted much scientific attention due to the rapid growth of online shopping. Practical service-network planning tools could help express service providers in minimizing the total cost while improving service levels. Efficient service network design is a requirement for sustainable logistical development. Express delivery has substantial negative environmental impacts, and service network design minimizes the environmental impact by reducing energy consumption costs. This paper addresses the service network design problem, which integrates a heterogeneous fleet of vehicles for vehicle dispatching in a consolidation-based time–space network to reflect the express service scenarios. Due to the NP-hard nature of this problem, we designed a layer-based relaxation algorithm to solve large-scale applications. The relaxation method relaxes and fixes the network structure on a layer-by-layer basis, and the computational experience confirms the effectiveness and efficiency of the relaxation algorithm. The solution time and quality are both improved significantly.

1. Introduction

With the rapid growth of online shopping, the demand in the logistics and transportation industry is continuously increasing. The number of parcels shipped in the United States reached a record 21.5 billion in 2021. Express delivery plays a vital role in our daily life, and the express service market is highly competitive [1]. Express service providers must improve the reliability of their delivery services while minimizing total costs [2]. The increasing number of parcels has become a challenge for the express delivery service system. Developing decision-making tools for service providers is necessary to improve this situation and stand out from the competition, since manual decisions for large-scale network planning have become intractable. To mitigate the energy crisis and carbon emissions, effectively addressing service network design issues can also contribute to environmental and energy sustainability. This work develops an integer programming model for the express service network design (SND) problem and proposes a layer-based relaxation method for large-scale applications.

In an express service network, parcels are sorted through a sequence of sorting centres before arriving at their destinations, as shown in Figure 1. Optimal planning can be achieved by solving the consolidation-based SND problem [3], which determines the services and the delivery paths. SND designates decision-making tools to schedule the activities and assets of a transportation system, aiming to satisfy customers with a high-quality standard. In SND, carriers’ most severe challenge is how to reduce costs while continuously improving service quality. The cost is mainly composed of the cost of transportation fuel. Therefore, while increasing the loading rate, choosing the optimal route and vehicle model minimizes the total cost while reducing carbon emissions and air pollution. Carriers must efficiently manage limited transportation assets that they own [4,5]. Researchers proposed models with solution methodologies that explicitly examine resource management issues in the context of SND [6,7]. In particular, design-balance constraints are introduced in SND models for resource management. Design-balance constraints guarantee the mass balance of transportation assets entering and leaving every time–space node [8]. Researchers explicitly considered the resource-availability issue in an uncertain situation [9]. SND is particularly relevant in a consolidation-based delivery system, an umbrella term for service providers that group and deliver parcels for customers within the exact vehicle, aiming for a balance between high service standards for customers and economy-of-scale-based costs [10]. There is still a lack of efficient and practical algorithms for solving large-scale SND problems with heterogeneous vehicles for express delivery service. As the express delivery business grows, SND plays a vital role in planning time-critical express processes.

This paper addresses the consolidation-based SND problem with a heterogeneous fleet of vehicles in which vehicle repositioning could happen within multiple service cycles. The contributions are threefold: First, we propose a mixed-integer programming model for the variant of the SND problem, and the formulation is arc-based, with which it is easier to model the consolidation-based operations. Second, a layer-based relaxation algorithm is introduced to solve large-scale applications; the computational experiments demonstrate that the proposed approach efficiently solves the problem with high-quality solutions. Third, the numerical results confirm that the total costs could be reduced significantly for service providers by introducing a heterogeneous fleet of vehicles.

The rest of this work is as follows. In Section 2, a literature review on SND problems is elaborated. The problem and the model formulation are presented in Section 3. Section 4 provides the framework of the solution methodology, and Section 5 includes the numerical experiments. Conclusions are provided in Section 6.

2. Literature Review

SND arises in the transportation sector when there is a requirement to determine cost-effective routes and schedules to meet customer demand with a specified service level. SND methodologies are extensively used in many fields, including telecommunication, logistics, transportation, and manufacturing systems [11,12,13]. In transportation systems, SND was successfully applied to maritime transportation [14,15], airport transportation [16], express shipment [17,18,19], rail transportation [20,21,22], air traffic flow management [23,24], and multimodal transportation systems [25,26,27]. In addition, different network structures were addressed in the literature, such as multimodal networks [28], long-haul and local transportation structures [29], autonomous fleets [30], and heterogeneous fleets [31]. Furthermore, SND methodologies have attracted much scientific research attention in developing green logistics and green supply-chain systems: for example, the green closed-loop supply chain network design for ventilators during the COVID-19 epidemic [32], the multitier supply network design methodology in reducing cost and carbon emissions [33], the intermodal green p-hub median problem [34], an approach to evaluating the coal transportation chain to reduce carbon emissions and pollution [35], reducing the negative environmental impact in freight transportation by understanding service buyers [36], effective green transit network design methodologies under urban expansion [37,38], an electrical transportation network design method to reduce congestion and carbon emissions [39,40], and a collaborative fleet routing methodology for sustainable seaport transportation [41].

Recently, SND with resource management has attracted much scientific attention [42], especially regarding the application of design-balanced constraints in SND to improve the utilization of transportation assets [43,44,45]. The design-balanced constraints guarantee that the fleet of vehicles are used cyclically to keep a high utilization under a fixed fleet size. Four models for the SND with asset management were introduced, and the numerical results demonstrated that the arc-based model formulation is computationally better than the others [46]. This paper offers an arc-based SND formulation with design-balance constraints, since the express service network is consolidation-based. The arc-based formulation allows for consolidation operations at each node. An early express SND model was developed in 2002 [47]. The authors studied SND application for a next-day air network in a multimodal express package-delivery problem; however, it was not a consolidation-based network. There is still a lack of academic research on express service networks based on heterogeneous vehicles.

There are many heuristic approaches for solving SND problems [48,49]. Two common heuristic approaches frequently appearing in the literature are local search [50,51], and slope scaling [52]. However, the solution quality of these methods cannot be guaranteed in large-scale applications. So far, none of these heuristics has been able to solve our problem directly, because our problem has incorporated a heterogeneous fleet and allowed for the vehicle to continuously serve multiple service cycles. For the exact algorithm, a branch-and-price algorithm is proposed to effectively solve the cycle-based formulation [53]. Since the vehicle does not necessarily return within the service period, this exact solution method is not suitable for our problem. In this work, we propose an efficient algorithm based on the large-scale application of a layer-based relaxation algorithm. The computational experiments demonstrate that the solution’s quality and the algorithm’s efficiency can significantly improve.

3. Problem Formulation

In SND models, nodes represent terminals, and arcs correspond to services that can be executed by service providers. Assets are generally vehicles that are employed to transport a commodity with origin–destination (O–D). Considering the time–space network and a set of O–D commodities, service providers must offer transportation services to meet all demands at a lower total cost while ensuring service quality for a long cycle.

This work assumes that services must be repeated within the planning horizon. That is, services operate repetitively and cyclically. Let’s consider that the planning horizon $\mathcal{T}$ is a set of discrete time periods, e.g., $\mathcal{T}=\{1,2,3,...,t_{max}\}$. Services can be scheduled with these periods. The vehicles that execute services would travel through a circular path in the time-space network. The network is depicted in Figure 2.

Let $\mathcal{P}$ be the set of physical terminals in the transportation network where service providers can perform consolidation-based operations. Each node corresponds to a tuple in the time-space network, including a terminal and a time index. Demand is a total volume to be moved between its origin and destination terminals within a certain time limit. Let $k=(o,d)$ be a commodity with source and destination. $\mathcal{K}$ denotes the set of all commodities. Each commodity k has a total volume $w_k$ that must be transported from its origin o to its destination d.

We modeled a carrier’s operations over a time–space network, $\mathcal{G}=(\mathcal{N},\mathcal{A})$, where $\mathcal{N}$ is the set of time–space nodes, and $\mathcal{A}$ is the time–space arc set. Each time–space node $i\in \mathcal{N}$ includes a terminal $p\in \mathcal{P}$ with a time period $t\in \mathcal{T}$. Each arc represents a service between the two time–space nodes. Arc set $\mathcal{A}$ contains service, waiting, and rotation arcs. Let $\mathcal{S}$, $\mathcal{G}$, and $\mathcal{R}$ represent service, waiting, and rotation arcs, respectively. The waiting arcs are the horizontal arcs that connect nodes with consecutive time periods of an identical terminal from $(p,t)$ to $(p,t+1)$, as depicted in Figure 2. Blue arcs are service arcs, and oblique arcs in Figure 2 are rotation arcs. One service arc denotes service operation at a certain point in time between two terminals. Each time-space arc $(i,j)$ has a service time $t_{ij}$, a fixed traveling cost $c_{ij}$, and a volume flow cost $h_{ij}^k$.

The model formulation is as follows.

$$\begin{array}{c}\hfill \mathrm{minimize}\sum _{v\in \mathcal{V}}\sum _{(i,j)\in {\mathcal{A}}_v}{c}_{ij}^v{x}_{ij}^v+\sum _{k\in \mathcal{K}}\sum _{(i,j)\in \mathcal{S}}{h}_{ij}^k{y}_{ij}^k\end{array}$$

(1)

subject to:

$$\begin{array}{cc}\hfill & \sum _{j\in \mathcal{N}:(i,j)\in {\mathcal{A}}_v}{x}_{ij}^v-\sum _{j\in \mathcal{N}:(j,i)\in {\mathcal{A}}_v}{x}_{ji}^v=0\forall v\in \mathcal{V},\forall i\in \mathcal{N},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k:i=k.o}{y}_{ij}^k=w_k\forall k\in \mathcal{K},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k:j=k.d}{y}_{ij}^k=w_k\forall k\in \mathcal{K},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k}{y}_{ij}^k-\sum _{(j,i)\in {\mathcal{A}}_k}{y}_{ji}^k=0\forall k\in \mathcal{K},\forall j\in \mathcal{N}:j\ne k.o,j\ne k.d,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \sum _{v\in \mathcal{V}}{x}_{ij}^v{u}^v-\sum _{k\in \mathcal{K}}{y}_{ij}^k\ge 0\forall (i,j)\in \mathcal{S},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & x_{ij}^v\in {\mathbb{Z}}_{+}\forall v\in \mathcal{V},\forall (i,j)\in {\mathcal{A}}_v,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & y_{ij}^k\in {\mathbb{R}}_{+}\forall k\in \mathcal{K},\forall (i,j)\in \mathcal{S}.\hfill \end{array}$$

(8)

The objective function is to minimize the combination of vehicle operating and commodity transportation costs. Constraints (2) are the mass balance constraints for vehicle routing. Constraints (3), (4) and (5) ensure the mass balance for the commodity flow. The supply and demand at the origin and destination node for commodity k are $d_k$, and inflow equals outflow on other time–space nodes. Constraints (6) guarantee that the total volumes of all commodities on arc $(i,j)$ cannot exceed the total vehicle capacity. Lastly, decision variable restrictions appear in Constraints (7) and (8).

4. Solution Approach

The SND problem is NP-hard [43]; in this section, we designed a layer-based relaxation algorithm to efficiently solve large-scale instances. First, we separated the time–space nodes into layer-based sets according to the time index, as shown in Figure 3.

Then, we chose the service arcs outgoing from nodes in one layer into an arc layer set. In Figure 4, the arcs in red are in Layers 1 and 2, respectively. In the solution process of the algorithm, in each iteration, we selected an arc layer and released integer decision variables $x_{ij}^v$ for arcs $(i,j)$ in other arc layers. We denoted the problem as the layer-based linear programming relaxation of SND (LLPRSND). LLRPSND can be formulated as follows.

Notations:

${\mathcal{A}}_l\subseteq \mathcal{S}$ Arc set of layer l.

Formulation:

$$\begin{array}{c}\hfill \mathrm{minimize}\sum _{v\in \mathcal{V}}\sum _{(i,j)\in {\mathcal{A}}_v}{c}_{ij}^v{x}_{ij}^v+\sum _{k\in \mathcal{K}}\sum _{(i,j)\in \mathcal{S}}{h}_{ij}^k{y}_{ij}^k\end{array}$$

subject to:

$$\begin{array}{cc}\hfill & \sum _{j\in \mathcal{N}:(i,j)\in {\mathcal{A}}_v}{x}_{ij}^v-\sum _{j\in \mathcal{N}:(j,i)\in {\mathcal{A}}_v}{x}_{ji}^v=0\forall v\in \mathcal{V},\forall i\in \mathcal{N},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k:i=k.o}{y}_{ij}^k=w_k\forall k\in \mathcal{K},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k:j=k.d}{y}_{ij}^k=w_k\forall k\in \mathcal{K},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \sum _{(i,j)\in {\mathcal{A}}_k}{y}_{ij}^k=\sum _{(j,i)\in {\mathcal{A}}_k}{y}_{ji}^k\forall k\in \mathcal{K},\forall j\in \mathcal{N}:j\ne k.o,j\ne k.d,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \sum _{v\in \mathcal{V}}{x}_{ij}^v{u}^v-\sum _{k\in \mathcal{K}}{y}_{ij}^k\ge 0\forall (i,j)\in \mathcal{S},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & x_{ij}^v\in \left\{\begin{array}{cc}\hfill & {\mathbb{Z}}_{+}\forall v\in \mathcal{V},(i,j)\in {\mathcal{A}}_l,\hfill \\ \hfill & {\mathbb{R}}_{+}\forall v\in \mathcal{V},(i,j)\in {\mathcal{A}}_v\backslash {\mathcal{A}}_l,\hfill \end{array}\right.\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & y_{ij}^k\in {\mathbb{R}}_{+}\forall k\in \mathcal{K},(i,j)\in \mathcal{S}\hfill \end{array}$$

(15)

Then, we solved LLPRSND from Layer 1 to Layer $t_{max}$, and with each iteration, we deleted arc $(i,j)$ in layer ${\mathcal{A}}_l$ if $x_{ij}^v$ was 0, and fixed the backbone arcs of that layer. Then, we solved the SND model in the backbone network.

Figure 5 depicts the working mechanism of the layer-based relaxation algorithm with a flow chart. Solving LLRPSND at each layer deletes service arcs with a fleet size that does not meet the constraints, and the network shrinks, fixing the backbone of each layer’s service arcs. Then, the SND was solved on the reconstructed network to obtain the optimal solution.

5. Numerical Study

In this section, we assess the computational performance of the layer-based relaxation algorithm. The algorithm was coded in Python, integrating Gurobi v9.1.1 as the optimization solver. The computational tests were performed using a workstation with 32 processors, 2.9 GHz, and 256 GB with the Windows 10 operating system.

This work generated two types of instances with simulated data on the basis of an express service company: medium- and large-scale instances. The medium instances were 10 terminals with 10–20 time periods, and the large-scale instances were 21–30 terminals with 15 periods. Three vehicle models were considered in this case. The dimensionality of all instances is demonstrated in

Table 1. Instance dimensionality.

Instance	$\left|\mathcal{P}\right|$	$\left|\mathcal{T}\right|$	$\left|\mathcal{S}\right|$	$\left|\mathcal{G}\right|$	$\left|\mathcal{R}\right|$	$\left|\mathcal{K}\right|$
1	10	10	900	100	10	90
2	10	11	990	110	10	90
3	10	12	1080	120	10	90
4	10	13	1170	130	10	90
5	10	14	1260	140	10	90
6	10	15	1350	150	10	90
7	10	16	1440	160	10	90
8	10	17	1530	170	10	90
9	10	18	1620	180	10	90
10	10	19	1710	190	10	90
11	10	20	1800	200	10	90
12	21	15	6300	315	21	420
13	22	15	6930	330	22	462
14	23	15	7590	345	23	506
15	24	15	8280	360	24	552
16	25	15	9000	375	25	600
17	26	15	9750	390	26	650
18	27	15	10,530	405	27	702
19	28	15	11,340	420	28	756
20	29	15	12,180	435	29	812
21	30	15	13,050	450	30	870

.

Columns 2–3 show the number of terminals and the time periods. Columns 4–7 present the cardinality of the service-arc, waiting-arc, rotation-arc, and commodity sets. We refer to Instances 1–11 as medium, and 12–21 as large. The large-scale instances are capable of reflecting the practical situations of the express service system.

For medium-scale instances, the time limit was 1000 s, and we set the MIP gap as 5%; if the feasible solution reached any bound, the solver stopped. For large-scale instances, the time limit was 3600 s, and we set 10% as the MIP gap.

5.1. Numerical Results for Medium-Scale Instances

Table 2. Numerical results for medium-scale instances with one vehicle model.

Instances	Original Solution			Algorithm Solution
	Obj.	Solution Time (sec.)	MIP Gap	Obj.	Solution Time (sec.)	MIP Gap
1	175,275	25.88	4.173%	170,855	10.04	3.303%
2	157,731	34.90	4.972%	150,347	7.18	3.283%
3	151,760	53.83	4.681%	150,394	15.50	3.002%
4	161,311	60.39	4.694%	157,942	14.46	3.366%
5	156,563	76.14	3.836%	154,833	20.82	3.491%
6	151,567	129.19	3.925%	148,417	11.32	3.577%
7	154,718	145.04	4.806%	150,892	12.32	3.054%
8	158,865	130.25	4.884%	156,135	10.51	3.881%
9	160,215	321.72	4.914%	158,469	39.81	2.746%
10	162,947	333.88	4.616%	160,157	107.93	3.869%
11	162,691	671.70	4.711%	160,976	307.25	3.615%

,

Table 3. Computational results for medium-scale instances with two vehicle models.

Instances	Original Solution			Algorithm Solution
	Obj.	Solution Time (sec.)	MIP Gap	Obj.	Solution Time (sec.)	MIP Gap
1	89,585	25.98	4.782%	85,110	13.60	2.60%
2	80,207	41.20	4.770%	76,777	26.59	3.999%
3	77,059	42.15	3.287%	74,910	26.62	2.212%
4	79,810	68.81	3.023%	78,523	31.71	2.468%
5	81,507	76.15	3.132%	76,586	31.25	2.235%
6	79,351	81.76	4.017%	74,671	31.62	2.003%
7	77,068	108.02	4.436%	75,145	43.06	2.529%
8	79,626	117.29	3.325%	76,527	49.54	1.631%
9	83,030	178.93	3.031%	80,964	85.53	1.753%
10	83,077	200.93	4.006%	79,779	130.26	1.855%
11	79,818	244.00	3.760%	76,958	136.35	2.308%

and

Table 4. Computational results for medium-scale instances with three vehicle models.

Instances	Original Solution			Algorithm Solution
	Obj.	Solution Time (sec.)	MIP Gap	Obj.	Solution Time (sec.)	MIP Gap
1	44,624	32.36	3.960%	43,279	27.07	2.897%
2	40,309	91.03	4.255%	38,267	41.53	3.292%
3	40,041	58.67	4.613%	38,775	46.00	2.938%
4	41,934	80.46	4.886%	40,555	61.07	4.326%
5	41,576	145.22	3.621%	38,969	74.07	3.303%
6	40,035	138.57	4.174%	38,344	78.21	3.342%
7	40,314	178.66	4.338%	38,481	125.19	4.214%
8	41,185	159.04	4.147%	39,288	121.50	4.191%
9	42,615	306.53	3.796%	41,354	213.47	1.977%
10	42,660	320.70	3.981%	40,989	299.21	2.879%
11	41,755	397.73	4.657%	39,450	350.16	4.621%

reports the numerical results of medium-scale instances with three types of vehicle models. To show the difficulty of solving SND instances, we first employed Gurobi to solve the original model for each instance. In these tables, Columns 2 and 3 show the objective value and the running time for obtaining the best feasible integer solutions after termination. Column 4 gives each instance’s mixed integer programming (MIP) gap. Columns 5–7 are numerical results obtained with the layer-based relaxation algorithm. Another advantage of the proposed approach is that it took substantially less computing time to achieve a better solution. The computational results demonstrate that, regarding these 11 medium-scale instances, the objective value received by the proposed algorithm was considerably better than that solved directly by the commercial solver. The average computational time for obtaining the best integer solutions was about 552 s, which was only 27.92% of that solved by Gurobi. The numerical results show that the computational time was reduced, while the solution quality was improved significantly. All these results confirm the effectiveness and efficiency of the layer-based relaxation algorithm.

The objective value comparison for the medium-scale instances is depicted in Figure 6, and M indicates the number of vehicle models. The objective value obtained by the proposed algorithm was better in the same setting. Another trend is that the objective value decreased with the number of vehicle models because a heterogeneous fleet increases the flexibility of cost reduction. The cost reduction by introducing more vehicle models reached about 26.67%. Figure 7 demonstrates the running time comparison between the original and proposed algorithms. The time needed by the layer-based relaxation algorithm was remarkably shorter.

5.2. Computational Experience for Large-Scale Instances

The numerical results for large-scale instances with heterogeneous vehicle models are shown in

Table 5. Numerical results for large-scale instances.

Instances	M	Original Solution			Algorithm Solution
		Obj.	Solution Time (sec.)	MIP Gap	Obj.	Solution Time (sec.)	MIP Gap
12	1	184,367	288.83	6.253%	181,814	23.87	3.290%
13	1	227,056	402.56	6.284%	223,123	56.80	3.619%
14	1	291,441	766.21	6.372%	286,697	87.40	3.447%
15	1	311,050	1345.20	6.322%	311,025	131.05	3.340%
16	1	366,682	2101.25	6.534%	366,567	152.66	3.609%
17	1	400,584	2342.15	6.307%	400,445	201.22	3.403%
18	1	473,864	2494.13	6.737%	473,788	234.01	3.779%
19	1	549,456	2652.48	6.381%	549,202	353.00	3.465%
20	1	N.A.	3600.00	N.A.	601,265	486.34	3.513%
21	1	N.A.	3600.00	N.A.	852,916	615.20	3.705%
12	2	95,161	161.46	5.571%	93,911	53.95	4.967%
13	2	115,937	841.43	6.189%	112,248	270.95	2.381%
14	2	151,734	1040.30	6.752%	141,291	635.94	1.971%
15	2	158,888	1405.70	6.850%	153,644	983.96	2.049%
16	2	190,525	2615.10	8.687%	189,835	1399.00	6.760%
17	2	N.A.	3600.00	N.A.	198,638	2178.78	2.494%
18	2	N.A.	3600.00	N.A.	246,125	2382.01	7.239%
19	2	N.A.	3600.00	N.A.	282,696	2528.29	7.868%
20	2	N.A.	3600.00	N.A.	292,659	2868.56	7.986%
21	2	N.A.	3600.00	N.A.	555,522	3023.69	8.625%
12	3	50,232	222.07	4.746%	47,062	85.88	4.422%
13	3	60,020	1003.77	4.997%	57,452	125.58	4.153%
14	3	74,895	1885.41	6.063%	72,156	611.13	3.470%
15	3	80,417	2236.50	6.952%	78,070	717.15	3.547%
16	3	93,859	2880.06	7.767%	92,379	892.33	3.918%
17	3	N.A.	3600.00	N.A.	105,432	1277.73	7.534%
18	3	N.A.	3600.00	N.A.	120,914	2555.93	4.879%
19	3	N.A.	3600.00	N.A.	137,701	2874.57	9.361%
20	3	N.A.	3600.00	N.A.	155,457	3059.14	7.296%
21	3	N.A.	3600.00	N.A.	157,694	3210.25	5.534%

. For these instances, the number of O–D commodities and service arcs exponentially increased, which was computationally expensive.

The computational results show the complexity and challenge of handling large-scale SND instances. Within the computing time limit, the original algorithm could find feasible solutions for all medium instances within the MIP gap requirement; however, only 18 out of 30 cases were solved with feasible solutions for large-scale instances. The layer-based relaxation algorithm could efficiently solve all instances and find better solutions, as depicted in Figure 8. The average computing times for our algorithm were 233, 1632, and 1540 s. The computational times of primal formulation were several times longer, as shown in Figure 9. The computational results demonstrate that our approach is more efficient.

6. Conclusions

We studied one variant of the SND problem for express services in this work and proposed a mixed-integer optimization model. The model jointly considers vehicle dispatching, heterogeneous fleet sizing, and commodity flow. A distinguishing feature of the model is that there is no restriction of each vehicle having to return in one cycle, a realistic rule that prevents existing algorithms from directly solving the problem. Due to the problem’s NP-hard nature and the lack of efficient algorithms, we propose a layer-based relaxation algorithm for solving large-scale instances. The computational results indicate that our algorithm outperforms solving directly by a state-of-the-art commercial solver on computational time and solution quality. We also found that the heterogeneous fleets could reduce the total cost.

As the demand for transportation continues to grow, the design algorithms for service networks are becoming increasingly important. Solving service network design problems with high quality can not only improve the competitiveness of service providers, but also help in reducing energy consumption and carbon emissions in terms of sustainable development. Although the algorithm is very efficient, it is heuristic. Theoretical proofs do not guarantee the quality of the solution. Therefore, future research focuses on developing efficient and precise algorithms to solve the proposed formula, i.e., a branch-cut-and-price framework for large fleet sizes. The branch-and-bound process is NP-hard and determines the overall efficiency of the entire branch-and-cut-and-price algorithm. To enhance the algorithm, we employed reinforcement learning techniques to improve the efficiency of the branch-and-bound process. The next research direction is to search for valid inequalities via mathematical structure-property to obtain a tight model formulation. The valid inequalities narrow the feasible region and facilitate finding the optimal solution. In future research on sustainability, we will also try to explore the modelling and solution methods of electric vehicles in express service network design.