CROSS-DOCK SCHEDULING WITH ROUTING DECISIONS: AN INTEGRATED APPROACH

ABSTRACT

Defining delivery routes is an effective way to reduce transportation costs. In particular, a cross-dock distribution center in a retail network requires coordinated decisions on delivery routes and internal operations, leading to the cross-dock scheduling with routing decisions problem. To the best of our knowledge, no empirical study has shown the advantage of integrating these decisions. Previous works mainly focus on fresh food distribution centers with specific operational features. Based on a retail network, we address this problem by developing two mixed-integer models: one basic and another with additional routing variables that improve the problem representation. Furthermore, we assess the importance of integrating routing and cross-dock scheduling by comparing integrated and hierarchical strategies. Overall, our results indicate that integration can improve the distribution system’s scheduling efficiency under different demand patterns and dependency levels between customers and suppliers.

Keywords:
vehicle routing; cross-dock scheduling; mathematical models

1 INTRODUCTION

A Supply Chain is a logistics system that involves a series of processes from the purchase of raw materials and the manufacturing of products to customer delivery. Given the highly competitive environment and the vast number of requests, companies are interested in planning their activities to make their distribution process efficient, i.e., with minimal cost and maximum customer satisfaction level. In particular, distribution expenses impact the price of products and, thus, cost reduction contributes to more competitive prices (Lummus & Vokurka, 1999).

The cross-docking distribution strategy is highlighted in the literature as an alternative to reduce logistic costs and to achieve faster response times. There are many real successful cases reported, and the most famous example is Walmart (see, e.g., Vogt (2010); Mohtashami et al. (2015)).

This strategy consists of sending loads from suppliers to customers through cross-docks, where inbound loads are received and separated to consolidate outbound loads that are delivered to their final destinations without being stored for extended periods. Therefore, it contributes to reducing storage costs, and the number of vehicles needed in deliveries (Boysen, 2010).

The cross-docking management problem can be divided according to the level of decisions considered, which can be strategic, tactical, or operational (Agustina et al., 2010; Buijs et al., 2014). Buijs et al. (2014) analyzed the decision variables of different approaches in the literature and identified 24 problems related to the cross-dock and the cross-docking network. Among them, the most studied are the load scheduling and the vehicle routing problems. One of the research directions pointed out by the authors is the relevance of synchronizing cross-dock and network decisions to achieve a more efficient distribution process. Additionally, Yin et al. (2016) mentioned that classic cross-docking models consider the routing and scheduling problems independently, losing advantages when considered as integrated. Therefore, new studies are necessary to ensure more efficiency in the cross-docking strategy.

In addition to network integration, coordinating the cross-dock’s internal operations is essential to improve the distribution process. For example, product storage is not desirable at a cross-dock center for distributing frozen or perishable products due to the high cost and the need to maintain product quality. In this situation, temporary storage is either not allowed (see e.g., Boysen (2010)) or penalized (see e.g., Agustina et al. (2014); Ladier & Alpan (2018)). On the other hand, in most studies, storage capacity is unlimited (see, e.g., Assadi & Bagheri (2016); Li et al. (2004); Chen & Song (2009)). Another important aspect is the transportation of goods within the cross-dock, for example, the transport can be manual; e.g., workers using pallet trucks or forklifts (see e.g., Boysen (2010); Agustina et al. (2014); Shakeri et al. (2012); Assadi & Bagheri (2016)) or automatic, e.g., a conveyor network (see e.g., McWilliams (2009, 2010); McWilliams & McBride (2012)); and the time or distance to transfer loads can be decisive for the scheduling (see, e.g., Shakeri et al. (2012); Assadi & Bagheri (2016)). Furthermore, Ladier & Alpan (2016) mention different scheduling performance measures, and makespan, typically defined as the total time required to complete all operations within the cross-dock terminal, is the most considered in the literature and relevant to industry practices.

This paper presents a study on integrating cross-dock scheduling and routing problems in a distribution center based on a retail network where suppliers serve many customers (many-to-many). In short, the distribution system has a cross-dock that receives products daily from its suppliers to distribute to many stores. We consider the internal transfer time and makespan to solve the problems integrated into the distribution of non-perishable products. We developed two mixed-integer models for the problem, and we experimentally show some advantages of integrating cross-dock scheduling and routing problems.

The remainder of this paper is organized as follows. In Section 2, we present the literature related and pointed out our contributions in more detail. In Section 3, we give an overview of the problem studied and two mathematical models developed are presented in Section 4. Next, we report the computational tests in Section 5 and the final remarks and future research are discussed in Section 6.

2 RELATED LITERATURE AND OUR CONTRIBUTIONS

The integration of vehicle routing and cross-dock scheduling problems is presented in the literature as two distinct approaches. The first is the vehicle routing problem with cross-docking (VRPCD), in which the loads are collected and delivered to their destinations via cross-dock. In general, the VRPCD consists of determining the pick-up and delivery routes to meet customers’ demands by synchronizing the arrival times of the inbound loads with the departure times of the outbound loads in the cross-dock. In the VRPCD, internal operations do not deal with details. The second one addresses the vehicle routing and cross-dock scheduling problem (VRCDSP), i.e., the cross-dock scheduling decisions are defined by integrating vehicle routing decisions.

In the majority of studies about the VRPCD, the cross-docking network has one cross-dock, and the requests have predetermined pick-up and delivery points (one-to-one) and time windows (see, e.g., Wen et al. (2009); Tarantilis (2013); Morais et al. (2014); Dondo & Cerdá (2013, 2014)). In particular, Dondo & Cerdá (2014) take many docks into account, while Maknoon & Laporte (2017) study a network with multiple cross-docks with spatial and load synchronization constraints. Other studies impose simultaneous arrivals of inbound loads in the cross-dock and a maximum time to complete all the deliveries with multiple suppliers serving multiple customers (many-to-many) (see, e.g., Lee et al. (2006); Liao et al. (2010); Vahdani et al. (2012)). When a request can be delivered directly without going through the cross-dock, it is referred to as the pick-up and delivery problem with cross-docking (PDPCD). Santos et al. (2013), and Nikolopoulou et al. (2017) studied PDPCD without time windows for one-to-one case. Furthermore, Dondo et al. (2009, 2011) studied the many-to-many case with time windows and multiple products.

To the best of our knowledge, only Agustina et al. (2014) and Rahbari et al. (2019) studied the VRCDSP. The authors focused on a fresh food distribution system with a cross-dock. In Agustina et al. (2014), each customer receives their request from only one supplier, but each supplier serves many customers (one-to-many). The suppliers organize the customers’ requests in pallets and send them to the cross-dock, where they are separated to consolidate outbound loads. Each outbound load may have requests addressed to different customers, so a delivery route is necessary. In short, the decision problem involves scheduling inbound (outbound) pallets for unloading (loading) and routing the vehicle to distribute them at the minimum cost, considering time windows. The costs concern temporary storage, transportation and violation of the delivery time windows. The authors propose a mixed-integer formulation with three-index routing variables, used to perform experiments with small instances - up to 50 customers, 4 suppliers and 4 doors. For instances based on real situations - up to 300 customers, 20 suppliers and 20 doors, they proposed a formulation using clustering. In each case, the maximum time reported by CPLEX 12.3 to solve the instances was 1096.84 and 4026.84 seconds, respectively.

Rahbari et al. (2019) present an extended model of Agustina et al. (2014). The problem considers a many-to-many network (i.e., many suppliers serving many customers), heterogeneous vehicles and different processing times for the inbound and outbound loads. The objective is to maximize the freshness of the delivered products and to minimize the distribution cost. The authors show the trade-off between these two objectives. They apply robust optimization approaches to ensure the robustness of solutions against the uncertainty of the travel time and freshness-life. A many-to-many network problem is significantly more challenging to deal with than a one-to-many network problem. Consequently, the computational tests are only run to small instances - 8 customers, 3 suppliers, 2 doors. The results show it is possible to increase the freshness of the products delivered without increasing the distribution cost. Additionally, they show that the uncertain travel time of outbound vehicles has a greater impact on the deterioration of the optimal value than the uncertainty in the products’ freshness.

In this paper, we approach the VRCDSP in a retail network inspired by a real situation. In this context, many suppliers send different types of products to fulfill multiple requests, which are organized in pallets within the cross-dock and consolidated into outbound loads (many-to-many). Handling some products may be complex; therefore, we take into account the time required to transfer products between the unloading and loading docks in the cross-dock, here referred to as internal transfer time. The internal transfer time and unloading order can affect the release time of the outbound loads and, consequently, the delivery time. On the other hand, the unloading and loading schedules can also impact the internal workload required (Van Belle et al., 2012).

Concerning the integration of routing and cross-dock scheduling problems, Agustina et al. (2014) and Rahbari et al. (2019) have the most similar studies to ours. However, our context differs from theirs, with significant differences in modeling and problem complexity. In Table 1, we summarize the main differences among the three problems. First, in our case and Rahbari et al. (2019), the requests are composed of several types of products sent by different suppliers (many-to-many); therefore, they need to be consolidated into pallets in the cross-dock. In Agustina et al. (2014), the request consolidation does not exist once they treat the one-to-many case as explained before. In this paper, the products have different handling times, which affect the processing (inbound/outbound) and internal transfer times. For Agustina et al. (2014), the unloading and loading times are the same for all loads and do not consider internal transfer time. Rahbari et al. (2019) consider only different processing times. Another difference among them is the delivery vehicles. In Agustina et al. (2014) and Rahbari et al. (2019), the vehicles are homogeneous and heterogeneous, respectively, and the number of vehicles is fixed a priori in both. In our case, they are homogeneous, and the vehicle number is the problem decision. Finally, Agustina et al. (2014) and Rahbari et al. (2019) deal with perishable products, making both the temporary storage time within the cross-dock and the storage costs relevant. In our problem, products are not perishable, i.e., they can wait for loading without being damaged, and, consequently, there are no storage costs. According to Van Belle et al. (2012), in this situation, the makespan (the total time to complete the loading of all requests) can represent the internal workload required. However, in this study, we do not explicitly address workforce requirements and consider makespan only as the operational time of the cross-dock.

Table 1
Differences between the VRCDSPs in the literature and this paper.

Aspect	Agustina et al. (2014 )	Rahbari et al. (2019 )	Problem studied
Type of network	one-to-many	many-to-many	many-to-many
Processing times (inbound/outbound)	equal	different	different
Internal transfer time	not considered	not considered	different
Number of delivery	known a priori	known a priori	decision
vehicles	(homogeneous)	(heterogeneous)	(homogeneous)
Temporary storage time per costumer	minimized (per customer)	minimized (per customer)	not considered
Makespan	not considered	not considered	minimized

Our main contributions are: (i) we develop a mixed-integer formulation for the VRCDSP inspired by a typical retail network, considering the internal transfer time and the makespan using two-index routing variables; and (ii) we present an alternative formulation with a better linear relaxation. Even with the differences pointed out, our formulations allow us to deal with instances larger than those treated by Agustina et al. (2014); and (iii) we present a comparative analysis of three other approaches to solve the problem to highlight the importance of solving the integrated cross-docking scheduling and vehicle routing problems. In the first step, we solve the cross-dock scheduling problem, assuming a single route for each customer (direct delivery). The two others are hierarchical approaches. In one approach, we begin by obtaining the cross-dock schedule before solving the vehicle routing problem. In the other, we determine the delivery routes by solving the vehicle routing problem before defining the cross-dock scheduling of the inbound and outbound loads. The computational results confirm that integrating the problems may benefit a cross-dock network, and this study should consequently motivate further research on the VRCDSP.

3 PROBLEM OVERVIEW

We consider a distribution system with a cross-dock that receives loads with one or more types of products to compose a set of customer requests. In Figure 1, we show an example of a set of inbound loads and a set of customer requests. Each request can require products from many inbound loads (suppliers), and each one can serve many customers, which defines a many-to-many system. The loads received are unloaded at the inbound docks, and their goods are organized into pallets according to customer demands to consolidate the output loads collected by delivery vehicles at the outbound docks. The number of pallets required to organize each customer demand is known and lower than or equal to the vehicles’ capacity and is delivered either directly or by routes. The number of pallets is based on the total volume demanded and the volume that each pallet can support. The system’s work schedule includes decision-making for unloading, consolidating and loading activities as well as delivery routes, considering the time to transfer the products within the cross-dock and the delivery time windows for each customer. The transfer time depends on the handling complexity of different product types and the unloading and loading positions.

Figure 1
Representation of an instance of the distribution problem.

In Figure 2, we illustrate an example of the distribution system studied. From left to right, we have representations of the load set received from the suppliers, the cross-dock, the homogeneous vehicle set used for the distribution and the customer set, respectively. The number of loads received exceeds the number of inbound docks. Moreover, the number of delivery vehicles is higher than the number of outbound docks. The distances between inbound and outbound docks are different. Thus, the time to transfer the products within the cross-dock depends on the unloading and loading positions. Apart from this, the transfer of some products takes longer than others due to their handling complexity (e.g., some products are heavier or require more care). In the loading and routing parts, the continuous lines exemplify delivery routes. Therefore, the scheduling distribution system needs to define the dock and the sequence for the unloading and loading operations without preemption (i.e., when a load processing starts, it cannot be interrupted). Besides these decisions, we need to know how to consolidate the output loads considering the customers served by the same vehicle. Note that the number of output loads defines the number of vehicles.

Figure 2
Representation of an example of the distribution system studied.

Internally, at the cross-dock, the received products are placed in an area close to the inbound docks. The products are immediately separated and transferred to the outbound docks according to the consolidation and loading position. The unloading times for each inbound load are known and different. The consolidation area in front of the outbound docks is sufficient for temporarily storing consolidated loads and incurs no additional costs. When all the products of an outbound load are in the consolidation area, loading can begin. This instant is defined as the release time of the outbound load. The loading time is known and depends on the number of pallets.

The objective is to obtain a schedule for the cross-dock and a distribution plan that minimizes the costs. The total cost includes the operational cost associated with the cross-dock makespan, the transportation cost, and the violation cost when a request is not delivered within its time window.

When we analyze the whole distribution system from an operational perspective, we can notice the interrelated activities, as illustrated in Figure 3. The delivery time depends on how the operations are carried out in the cross-dock, especially when the output loads are available. Decisions to unload the received loads can reflect directly when the output loads are available to be collected (release time). Similarly, the availability of the output loads and their loading order reflect on the time the vehicles can leave the cross-dock. Furthermore, the vehicle’s departure time affects each customer’s delivery time, depending on the route. Reversely, when the routes are established, they determine the consolidation of the outbound loads and the loading order according to its priorities and, consequently, the unloading order. This fact reinforces the importance of consider integrated decisions. Besides, it is known from the literature of vehicle routing and cross-docking that the vehicle routing and cross-dock scheduling problems are NP-hard. Hence, the integration of the two problems is also NP-hard.

Figure 3
Interdependence relationship among the decisions for the system scheduling.

4 MATHEMATICAL FORMULATION

In this section, we present a mixed-integer model for the VRCDSP studied. We formulate the assignment/sequencing decisions in the cross-dock based on Assadi & Bagheri (2016) and assume the following assumptions.

1. All inbound loads are available at the beginning of the planning horizon.

2. The unloading (loading) of each inbound (outbound) load is processed in only one inbound (outbound) dock without preemption.

3. The transshipment time for products depends on the distance between the receiving and shipping docks and the product volume. The transfer of products from the inbound area to the outbound area starts when the inbound load (with the products) is completely unloaded.

4. The capacity of the temporary storage is unlimited.

5. The number of pallets to organize the products demanded by each customer is lower than or equal to the delivery vehicle’s capacity. Each customer request is delivered by only one vehicle.

6. The delivery vehicles are homogeneous in capacity and speed.

7. The loading of each outbound load starts after all its requests are ready.

8. Each outbound load (delivery vehicle) may leave the cross-dock after all its requests are loaded.

9. Each delivery route starts and ends at the cross-dock.

10. All delivery vehicles must return to the cross-dock before the end of the planning horizon.

To formulate the problem, we do not use an index explicitly for the vehicle. Each vehicle collects one outbound load for which it is necessary to define a route to deliver its requests. Additionally, each outbound load (delivery vehicle) is identified by the first customer visited and binary variables indicate the route each customer takes part in (more details are given when we explain the constraints). Thus, the number of outbound loads determines the number of vehicles required for distribution. Also, we use binary two-index variables and the indexes 0 and N+1 to represent the cross-dock in the routing. In Tables 2 and 3, we define the following notation to write the mathematical models.

Table 2
Notation for the modeling: sets and parameters.

Sets:
ℒ	set of L inbound loads (ℒ={1, ..., L}),
𝒫	set of P types of product (𝒫={1, ..., P}),
𝒩	set of N customers (𝒩={1, ..., N}),
ℱ	set of F inbound docks (ℱ={1, ..., F}),
ℋ	set of H outbound docks (ℋ={1, ..., H}).
Parameters:
pt l	processing time of inbound load l,
$r_l^p$	number of product type p brought into inbound load l,
θp	volume of one product of type p,
vol	volume of one pallet,
$t{r}_{fh}^p$	time needed to transfer one product type p from inbound dock f to the consolidating area in front of outbound dock h,
lp	time needed to load one pallet, i.e., to transfer one pallet from the consolidating area to the delivery vehicle,
ch	changeover time of vehicles in one outbound dock,
cap	capacity (in pallets) of each delivery vehicle,
$d_i^p$	number of products type p demanded by customer i,
q i	number of pallets needed to transport the request of customer i,
co	cost per unit of time to keep the cross-dock running,
ttij (tc ij )	travel time (travel cost) between customers i and j,
ce (cl)	earliness (tardiness) penalty cost per unit of time,
st i	service time to unload the pallets of customer i (proportional to the number of pallets of each customer),
[a i , b i ]	time window to deliver the request of customer i,
T max	maximum time to complete all the activities,
M	a big number (we took M=T max ).

Table 3
Notation for the modeling: variables.

Variables:
$z_{lf}^{in}$	takes the value of 1, if inbound load l is assigned to inbound dock f to be processed, and 0, otherwise,
$w_{lm}^{in}$	takes the value of 1, if inbound loads l and m are assigned to the same inbound dock and l is processed before m, and 0, otherwise,
$z_{ih}^{out}$	takes the value of 1, if the delivery vehicle, whose route begins with customer i, collects its load in outbound dock h, and 0, otherwise,
$w_{ij}^{out}$	takes the value of 1, if the delivery vehicles whose route begins with customer i and j are assigned to the same outbound dock and the loading of the first precedes the second, and 0, otherwise,
y li	takes the value of 1, if at least one type of product is transferred from inbound load l to the outbound load collected by the delivery vehicle whose route begins with customer i, and 0, otherwise,
${\rho }_{lfih}^p$	number of products p transferred from inbound load l received in inbound dock f for the outbound load collected by the vehicle whose route begins with customer i and loaded in outbound dock h,
${\pi }_i^p$	number of products p in the delivery vehicle when it arrives at customer i,
np i	number of pallets in the delivery vehicle when it arrives at customer i,
ut l	time when the processing of inbound load l is completed,
rt i	release time to load the outbound pallets collected by the delivery vehicle whose route begins with customer i,
dt i	time of the delivery vehicle whose route begins with customer i leaves the cross-dock,
dt max	time the last delivery vehicle leaves the cross-dock (equivalent to cross-dock makespan),
x ij	takes the value of 1, if a delivery vehicle travels from customer i to customer j, and 0, otherwise,
t i	arrival time of a delivery vehicle at customer i,
eti (lt i )	delivery earliness (tardiness) at customer i.

The mathematical formulation (F1) for the problem studied has the Objective Function (1) subject to the sets of constraints (2)-(30) written as follows.

$min\sum _{i\in \mathcal{N}\cup \left\{0\right\}}\sum _{\underset{:j\ne i}{j\in \mathcal{N}\cup \left\{N+1\right\}}}t{c}_{ij}{x}_{ij}+\sum _{i\in \mathcal{N}}cee{t}_i+\sum _{i\in \mathcal{N}}cll{t}_i+cod{t}^{max}.$ (1)

The Objective Function (1) minimizes not only the sum of costs with the routing but also the violation of the time window and the cross-dock operational time (makespan).

Subject to:

//Inbound assignment - sequencing

$\sum _{f\in \mathcal{F}}{z}_{lf}^{in}=1,\forall l\in \mathcal{L},$ (2)

$w_{lm}^{in}+w_{ml}^{in}\ge z_{lf}^{in}+z_{mf}^{in}-1,\forall l,m\in \mathcal{L}:l<m,f\in \mathcal{F},$ (3)

$u{t}_m\ge u{t}_l+p{t}_m-M\left(1-w_{lm}^{in}\right),\forall l,m\in \mathcal{L}:l\ne m,$ (4)

$u{t}_l\ge p{t}_l,\forall l\in \mathcal{L}.$ (5)

The inbound assignment/sequencing decisions are modeled by constraints (2)-(5). Constraints (2) ensure that each inbound load is assigned to only one inbound dock. Constraints (3) and (4) determine the precedence relation among all the inbound loads assigned to the same dock, i.e., if $z_{lf}^{in}=1$ and $z_{mf}^{in}=1$, then $w_{lm}^{in}=1$ or $w_{ml}^{in}=1$. The processing completion time (ut _l ) of each inbound load is defined by constraints (4) and (5) according to the unloading sequence.

//Outbound assignment - sequencing

$\sum _{h\in \mathcal{H}}{z}_{ih}^{out}=x_{0i},\forall i\in \mathcal{N},$ (6)

$w_{ij}^{out}+w_{ji}^{out}\ge z_{ih}^{out}+z_{jh}^{out}-1,\forall i,j\in \mathcal{N}\left(i<j\right),h\in \mathcal{H},$ (7)

$d{t}_j\ge d{t}_i+lpn{p}_j+ch-M\left(1-w_{ij}^{out}\right),\forall i,j\in \mathcal{N}\left(i\ne j\right),$ (8)

$d{t}_i\ge r{t}_i+lpn{p}_i-M\left(1-x_{0i}\right),\forall i\in \mathcal{N},$ (9)

$d{t}^{max}\ge d{t}_i,\forall i\in \mathcal{N}.$ (10)

Constraints (6)-(10) model the outbound assignment/sequencing decisions. As already mentioned, each outbound load (vehicle) is associated with a delivery route and is indicated by the first customer i visited. Figure 4 illustrates the cross-dock internal process with two outbound loads, whose delivery routes begin with customers 3 and 7. In this example, x ₀₃ and x ₀₇ take the value of 1. When x _0i takes the value of 1, constraints (6) ensure outbound load i is assigned to one of the outbound docks and the sequence of loading is defined by constraints (7) and (8), similarly to (3) and (4). Furthermore, the departure time (dt _i ) must respect the release time (rt _i ) and the loading time, which is modeled by constraints (9). As the makespan (dt ^max ) of the cross-dock is part of the objective function, constraints (10) impose it to be equal to the completion time of the last loading.

//Internal transfer (consolidation)

$\sum _{i\in \mathcal{N}}\sum _{h\in \mathcal{H}}{\rho }_{lfih}^p=r_l^p{z}_{lf}^{in},\forall p\in \mathcal{P},l\in \mathcal{L},f\in \mathcal{F},$ (11)

$\sum _{l\in \mathcal{L}}\sum _{f\in \mathcal{F}}\sum _{h\in \mathcal{H}}{\rho }_{lfih}^p\ge {\pi }_i^p-\left(\sum _{l\in \mathcal{L}}{r}_l^p\right)\left(1-x_{0i}\right),\forall p\in \mathcal{P},i\in \mathcal{N},$ (12)

$\sum _{p\in \mathcal{P}}\sum _{l\in \mathcal{L}}\sum _{f\in \mathcal{F}}{\rho }_{lfih}^p\le \left(\sum _{p\in \mathcal{P}}\sum _{l\in \mathcal{L}}{r}_l^p\right)z_{ih}^{out},\forall i\in \mathcal{N},h\in \mathcal{H},$ (13)

$\sum _{p\in \mathcal{P}}\sum _{f\in \mathcal{F}}\sum _{h\in \mathcal{K}}{\rho }_{lfih}^p\le \left(\sum _{p\in \mathcal{P}}{r}_l^p\right)y_{li},\forall l\in \mathcal{L},i\in \mathcal{N},$ (14)

$r{t}_i\ge u{t}_l+\sum _{p\in \mathcal{P}}\sum _{f\in \mathcal{F}}\sum _{h\in \mathcal{K}}t{r}_{fh}^p{\rho }_{lfih}^p-M\left(1-y_{li}\right),\forall l\in \mathcal{L},i\in \mathcal{N}.$ (15)

Figure 4
Instant view of the internal cross-dock process.

The set of constraints (11)-(15) links unloading times to release times, taking the internal transfer times, $t{r}_{fh}^p$, into account according to the number of products transferred, ${\rho }_{lfih}^p$. Constraints (11), (12) and (13) ensure that the total number of products of each type received from the inbound loads is enough to meet the number of products needed to compose the requests consolidated in the same outbound load (i.e., the number of products demanded by customers on the route associated, ${\pi }_i^p$) considering the unloading and loading positions. Constraints (14) indicate the transfers from inbound to outbound loads forcing variable y _li to take the value of 1. One outbound load is collected by a delivery vehicle if all the products needed are available to be loaded, i.e., the requests are consolidated and ready to be loaded at the outbound docks. This condition is imposed by constraints (15), which consider the internal transfer time according to the products’ unloading and loading positions. For example, in the instant view illustrated in Figure 4, we can observe outbound load 3 is ready to leave the cross-dock. Differently, the loading of the outbound load 7 has not started because some products are still missing, i.e., their inbound loads are still being processed.

//Routing

$\sum _{\underset{\overline{)i\ne j}}{i\in \mathcal{N}\cup \left\{0\right\}}}{x}_{ij}=1,\forall j\in \mathcal{N},$ (16)

$\sum _{\underset{\overline{)i\ne j}}{j\in \mathcal{N}\cup \left\{N+1\right\}}}{x}_{ij}=1,\forall i\in \mathcal{N},$ (17)

$n{p}_j\ge q_j+n{p}_i-cap\left(1-x_{ji}\right),\forall i,j\in \mathcal{N}\left(j\ne i\right),$ (18)

$q_i\le n{p}_i\le cap,\forall i\in \mathcal{N},$ (19)

${\pi }_j^p\ge d_j^p+{\pi }_i^p-\left(\sum _{l\in \mathcal{L}}{r}_l^p\right)\left(1-x_{ji}\right),\forall p\in \mathcal{P},\forall i,j\in \mathcal{N}\left(j\ne i\right),$ (20)

${\pi }_i^p\ge d_i^p,\forall p\in \mathcal{P},\forall i\in \mathcal{N},$ (21)

$t_i\ge d{t}_i+t{t}_{0i}-M\left(1-x_{0i}\right),\forall i\in \mathcal{N},$ (22)

$t_j\ge t_i+s{t}_i+t{t}_{ij}-M\left(1-x_{ij}\right),\forall i\in \mathcal{N},j\in \mathcal{N}\cup \left\{N+1\right\}\left(i\ne j\right),$ (23)

$a_i-e{t}_i\le t_i\le b_i+l{t}_i,\forall i\in \mathcal{N},$ (24)

$t_{N+1}\le T^{max}.$ (25)

In the routing part, constraints (16) and (17) guarantee each customer is visited exactly once. Constraints (18) determine the number of pallets in the delivery vehicle when it arrives at each customer j (np _j is the number of pallets of customer j plus the number of pallets of the next customers on the route). The lower and upper bounds for np _i , by constraints (19), define the number of pallets when the vehicle arrives at its last customer and impose the vehicle capacity, respectively. Although np _i is modeled as a continuous variable, fractional pallet values do not arise in practice, since it is determined by routing decisions and integer customer demands. For the consolidation of outbound loads, it is necessary to know the number of products of each type to meet the demands of the customers on each route. Constraints (20) and (21) define this number $\left({\pi }_i^p\right)$, similarly to constraints (18) and (19). The succession of times according to the delivery order is determined by constraints (22) and (23). Constraints (24) define the time windows allowing earliness and tardiness to each delivery. Finally, constraint (25) imposes a maximum time for the return of the delivery vehicles.

//Domain of variables

$z_{lf}^{in},w_{lm}^{in}\in \left\{0,1\right\},u{t}_l\ge 0,\forall l,m\in \mathcal{L}:l\ne m,\forall f\in \mathcal{F},$ (26)

$z_{ih}^{out},w_{ij}^{out}\in \left\{0,1\right\},d{t}^{max},d{t}_i,r{t}_i\ge 0,\forall i,j\in \mathcal{N}:i\ne j,\forall h\in \mathcal{H},$ (27)

$y_{li}\in \left\{0,1\right\},{\pi }_i^p,{\rho }_{lfih}^p\ge 0,\forall l\in \mathcal{L},\forall f\in \mathcal{F},\forall i\in \mathcal{N},\forall h\in \mathcal{H},\forall p\in \mathcal{P},\forall i\in \mathcal{N},\forall h\in \mathcal{H},$ (28)

$x_{ij}\in \left\{0,1\right\},t_{N+1},t_i,e{t}_i,l{t}_i,n{p}_i\ge 0,\forall i,j\in \mathcal{N}:i\ne j,$ (29)

$x_{0i},x_{iN+1}\in \left\{0,1\right\},\forall i\in \mathcal{N},$ (30)

Finally, constraints (26)-(30) define the domain of the variables.

Remark 1: In case the different types of cost are the same per time unit and are proportional to the respective times, the model may be written equivalently with the substitution of objective function (1) by (31).

$min\sum _{i\in \mathcal{N}\cup \left\{0\right\}}\sum _{\overline{)j\in \mathcal{N}\cup \left\{N+1\right\}}i\ne j}t{t}_{ij}{x}_{ij}+\sum _{i\in \mathcal{N}}e{t}_i+\sum _{i\in \mathcal{N}}l{t}_i+d{t}^{max}.$ (31)

Remark 2: In case the minimization of the total internal transfer times is relevant, it is possible to include it by adding sum (32) to objective function (31). The total internal transfer times may represent the workload needed to perform the internal activities.

$\sum _{p\in \mathcal{P}}\sum _{l\in \mathcal{L}}\sum _{f\in \mathcal{F}}\sum _{i\in \mathcal{N}}\sum _{h\in \mathcal{H}}t{r}_{fh}^p{\rho }_{lfih}^p.$ (32)

Remark 3: If there is a limited number of delivery vehicles, it is necessary to include constraint Σ_i∈𝒩 x _0i ≤K, in which K is the number of delivery vehicles available.

Remark 4: It is possible to calculate a lower bound for variable dt _i . Let ${\overline{\mathcal{L}}}_i$ be $\left\{\overline{)l\in \mathcal{L}:\exists p\in \mathcal{P}}{r}_l^p>0\wedge d_i^p>0\right\}$, i.e., the set of all inbound loads with at least one product type required by customer i; ${\alpha }_i^{*}=\underset{l\in {\overline{\mathcal{L}}}_i}{argmax}\left\{p{t}_l\right\}$, which is the largest processing time among the inbound loads that have any product required by customer i; and ${\tau }_i^{*}=\underset{f\in \mathcal{F},h\in \mathcal{H}}{mmin}\left\{\sum _{p\in \mathcal{P}:r_{{\alpha }_i^{*}}^p>0}t{r}_{fh}^p{d}_i^p\right\}$, i.e., the minimum time to transfer the products required by customer i to inbound load ${\alpha }_i^{*}$ from input to output. Supposing the request of customer i may be delivered directly, the lower bound for the departure time dt _i of the delivery vehicle is the processing time of inbound load ${\alpha }_i^{*}$ plus the transfer time and the loading time, as written in inequality (33).

$d{t}_i\ge p{t}_{{\alpha }_i^{*}}+{\tau }_i^{*}{q}_{i}lt.$ (33)

4.1 An alternative formulation with route indicative variables

The efficiency of exact methods for solving mixed-integer problems can be affected by the formulation. Theoretically, formulations with a better linear relaxation can improve the performance of exact methods. We developed an alternative formulation by including binary variables to indicate the route each customer takes part of (v _ij takes 1, if customer i is on the route starting with customer j, and 0, otherwise). With these variables, we eliminated some constraints and included or rewrote others. We used the new variables to write the number of pallets and products demanded on each route; thus, variables nq _i and ${\pi }_i^p$ and constraints (18)-(21) were eliminated. The alternative formulation (F2) is written as follows.

$min\sum _{i\in \mathcal{N}\cup \left\{0\right\}}\sum _{\underset{\overline{)i\ne j}}{j\in \mathcal{N}\cup \left\{N+1\right\}}}t{c}_{ij}{x}_{ij}+\sum _{i\in \mathcal{N}}cee{t}_i+\sum _{i\in \mathcal{N}}cll{t}_i+cod{t}^{max}.$

Subject to:

//Inbound assignment - sequencing

(2)-(5),

//Outbound assignment - sequencing

(6), (7), (10),

$d{t}_j\ge d{t}_i+lp\left(q_i{x}_{0i}+\sum _{\overline{)n\in N}n\ne i}{q}_n{v}_{ni}\right)+ch-M\left(1-w_{ij}^{out}\right),\forall i,j\in \mathcal{N}\left(i\ne j\right),$ (34)

$d{t}_i\ge r{t}_i+lp\left(q_i{x}_{0i}+\sum _{\overline{)j\in N}j\ne i}{q}_j{v}_{ji}\right)-M\left(1-x_{0i}\right),\forall i\in \mathcal{N},$ (35)

//Internal transfer (consolidation)

(11), (13) - (15),

$\sum _{l\in \mathcal{L}}\sum _{f\in \mathcal{F}}\sum _{h\in \mathcal{H}}{\rho }_{lfih}^p=d_i^p{x}_{0i}+\sum _{\overline{)j\in N}j\ne i}{d}_j^p{v}_{ji},\forall p\in \mathcal{P},i\in \mathcal{N},$ (36)

//Routing

(16), (17), (22) - (25),

$v_{ij}\ge x_{0j}-\left(1-x_{ji}\right),\forall i,j\in \mathcal{N}\left(i\ne j\right),$ (37)

$v_{ij}\ge v_{nj}-\left(1-x_{in}\right),\forall i,j,n\in \mathcal{N}\left(i\ne j,i\ne l,j\ne l\right),$ (38)

$\sum _{\overline{)j\in \mathcal{N}}j\ne i}{v}_{ij}+x_{0i}=1,\forall i\in \mathcal{N},$ (39)

$\sum _{\overline{)i\in \mathcal{N}}j\ne i}{q}_i{v}_{ij}+q_j{x}_{0j}\le cap{x}_{0j},\forall j\in \mathcal{N}.$ (40)

In constraints (34) and (35), variables q _i and ${\pi }_i^p$ are replaced by equivalent expressions written with the new variables. Constraints (36) ensure the transfers of products needed to meet the customers’ demands are consolidated in the same outbound load. Constraints (37) and (38) guarantee that indicative variables v _ij take values 1 if customer i is in a route in which j is the first customer. Constraints (39) ensure that each customer is assigned to only one route if it is not the first one visited. Constraints (40) impose the vehicle capacity and prevent a customer from being assigned to a non-existent route. The primary difference between the two models proposed lies in the modeling of route capacity.

5 COMPUTATIONAL EXPERIMENTS

The computational experiments are presented in three phases. In the first (Subsection 5.2), we analyze the performance of formulations F1 and F2. In the second phase (Subsection 5.3), we present a brief comparison between the VRCDSP studied and the problem of Agustina et al. (2014). In the third (Subsection 5.4), we evaluate the relevance of integrating the vehicle routing and cross-dock scheduling problems to obtain a schedule for the distribution system, compared to hierarchical strategies.

The instances are solved by the Gurobi Solver version 8.0 with default settings and a two-hour time limit on an Intel Core i7-2600 2.40 GHz computer with 15.6 GB of RAM (Linux 16.04 LTS).

5.1 Set of instances

To the best of our knowledge, there is nothing in the literature that deals specifically with the problem studied here. First, we adapt Solomon’s Instances (R101 - R112) (Solomon, 1987) for the problem studied. We select the first N customers in instances of Solomon (1987). For each instance, we maintain the distances between two customers and between a customer and a depot (cross-dock). Due to the operational time required for unloading, consolidation, and loading activities at the cross-dock, we add the average inbound processing time per inbound dock to every time window of the original instances. Consequently, the time window of each customer i ([a _i , b _i ]) and the maximum time T ^max are updated as follows:

$\tau =⌈\frac{{\Sigma }_{l\in \mathcal{L}}p{t}_l}{F}⌉,\left[a_i,b_i\right]=\left[a_i^o+\tau ,b_i^o+\tau \right]and{T}^{max}=T^{ma{x}^o}+\tau ,$

where $a_i^o$, $b_i^o$ e $T^{ma{x}^o}$ are values from the original instances.

We preserve the variation of Solomon’s instances concerning time windows. In the original instances, some of them R101-R104, R105-R108 and R109-R112, the time window is equal to 10, 30 and 65 units of time, respectively. Also, in each case, the number of customers whose time windows are equal to their respective intervals is decreasing. For example, in case of 25 customers, R101, R102, R103 and R104 have, respectively, 25, 18, 12 and 8 customers with time windows equal to 10, and the remaining customers have higher time windows.

In Table 4, we define other parameters. The vehicle capacity is 16 pallets (cap=16) and the maximum volume a pallet supports is 1.2 (vol=1.2) - values based on real data.

Table 4
Parameter values regarding the number of customers.

	15 customers	25 customers	40 Customers
Suppliers (L)	5	10	15
Inbound docks (F)	3	4	6
Outbound docks (H)	3	4	6
Products (P)	10	15	25

The remainder of the data is generated randomly in two scenarios with three groups each, as summarized in Table 5. In Scenario 1, the number of pallets (Q _i ) needed to organize each request is between 25% and 100% of the vehicle capacity and, in Scenario 2, this number is between 12.5% and 50%. Consequently, we may have more customers in a route in the second case. For each scenario, we generate three groups of instances based on the number of product types (η_i ) required by each customer. This number is obtained randomly between one and 20%, 40% and 100% of the total product types (P). Matrixes $\left(I_i^p\right)\in {\left\{0,1\right\}}^{\mathcal{P}\times \mathcal{N}}$ and $\left(J_l^p\right)\in {\left\{0,1\right\}}^{\mathcal{P}\times \mathcal{L}}$ indicate what types of product are demanded by each customer and sent by each supplier, respectively. $\left(I_i^p\right)$ is randomly defined, respects the number of product types required by each customer and ensures that every product is required by at least one customer. $\left(J_l^p\right)$ is randomly obtained by ensuring that each product is sent by only one supplier and that each supplier sends at least one product type and no more than three. We also consider the volume of each product type (θ^p ) randomly - between 1% and 10% of the pallet capacity.

Table 5
Data generated randomly.

qi ∈ℤ	number of pallets needed to organize the products
	Scenario 1: U [0.25cap, cap] - Scenario 2: U [0.125cap, 0.5cap]
ηi ∈ℤ	Number of product types required by customer i
	Group A: U [1, 0.2P] - Group B: U [1, 0.4P] - Group C: U [1, P]
$I_i^p\in \left\{0,1\right\}$	takes 1, if product of type p is required by customer i, and 0, otherwise
$J_l^p\in \left\{0,1\right\}$	takes 1, if product p is sent by supplier l, and 0, otherwise
θp ∈ℝ	volume of one unit of product of type p
	U [0.01vol, 0.1vol]

As presented in Table 6, part of the data is calculated with the values generated randomly and the parameters. First, for each customer i, the total volume of each product type required $\left(v_i^p\right)$ is the floor of the value obtained by equally dividing the volume that corresponds to the number of pallets by the number of product types. The number of each product type $\left(d_i^p\right)$ required by each customer is the floor of the division of $v_i^p$ by θ^p . With the product demands of each customer, we calculate the number of each product type in each inbound load $\left(r_l^p\right)$ and its processing time (pt _l ), considering that the processing time of one pallet is 5 units of time. The distance between two docks (dist _fh ) is calculated according to their positions and, after, we define the transfer times of each product type $\left(t{r}_{fh}^p\right)$, which depends on the transfer speed, proportionally to its volume in comparison to the pallet volume.

Table 6
Data calculated.

$v_i^p$	total volume approximated of product of type p required by customer i	$v_i^p=I_i^p⌊\frac{q_{i}vol}{{\Sigma }_{p\in \mathcal{P}}{I}_i^p}⌋$
$d_i^p$	number of products p demanded by customer i	$d_i^p=⌊\frac{v_i^p}{{\theta }^p}⌋$
$r_l^p$	number of products p in inbound load l	$r_l^p=J_l^p\sum _{i\in \mathcal{N}}{d}_i^p$
pt l	time to process inbound load l	$p{t}_l=\sum _{p\in \mathcal{P}}\left(5⌈\frac{r_l^p{\theta }^p}{vol}⌉\right)$
dist fh	distance between inbound dock f and outbound dock h	distfh =10+2|f-h|
$t{r}_{fh}^p$	time to transfer one unit of product p according to docks f and h	$t{r}_{fh}^p=0.4\frac{{\theta }^p}{vol}dis{t}_{fh}$

Finally, we define ch=l p=2 and st _i =2q _i . Assuming the proportionality between costs and time, we consider the minimization of total travel (distance) time plus the earliness and tardiness times and the total cross-dock operational time (makespan). Each instance is referred to in the following way: Solomon’s name - scenario - group. For example, instance R111-2B refers to the instance generated by Solomon’s instance R111, considering Scenario 2 and Group B.

5.2 F1 and F2 efficiency

We evaluated the two formulations developed (F1 and F2) using instances with 15 and 25 customers. First, we solved the linear relaxation of all instances with and without the lower bounds for variables dt _i , as defined in Section 4. On average, the lower bounds increase the objective function value of linear relaxation by about 85% for F1 and 51% for F2. Next, we present a comparison of the linear relaxation for both formulations with lower bounds for variables dt _i defined by inequalities (33).

Figure 5 shows the objective function values of the linear relaxation using each formulation. F2 provides the highest values. For instances with 15 customers, on average, the values provided by F2 are approximately 28% (Scenario 1) and 16% (Scenario 2) higher than the values provided by F1. The highest and lowest differences are respectively 33.6% and 20.6% in Scenario 1 and 21.9% and 12.7% in Scenario 2. Also, F2 provides higher values for 25 customer instances with differences of 60% (Scenario 1) and 30% (Scenario 2) on average, and the highest and lowest differences are respectively 72.6% and 49.6% in Scenario 1 and 25.8% and 12.7% in Scenario 2. As expected, F2, which has a smaller number of disjunctive constraints, observed in Subsection 4.1, presented higher objective function values of the linear relaxation.

Figure 5
Linear Relaxation objective function values for F1 and F2.

To compare the two formulations’ performance solving the problem, we present results to 15 and 25 customers. In the case of 15 customers, an optimal solution is obtained for 52 instances out of 72. In Figure 6, we show the cumulative number of instances solved optimally by each formulation. We can observe F2 has a small advantage over F1 presenting optimal solutions for three more instances, when the time limit is reached.

Figure 6
Number of optimal solutions over the time by F1 and F2 - 15 customers.

In the case of 25 customers, just instance R101-1A has its optimal solution found and the time spent by F2 is lower than half the time spent by F1. We compare the lower and upper bounds obtained by both formulations in two hours, as illustrated in Figure 7. Concerning the upper bounds, the values provided by the formulations in both scenarios are very close; there are few cases in which F2 obtains lower values than F1. As can be seen, we get higher lower bounds with F2, especially in Scenario 2. As a result, F2 presents solutions with the lowest gaps for most of the instances.

Figure 7
Lower and upper bounds obtained in two hours by F1 and F2 - 25 customers.

In general, the results show a small advantage for F2. Thus, for the next discussions, we consider the results obtained with this formulation.

5.3 VRCDSP and literature comparison

As already mentioned, Agustina et al. (2014) and Rahbari et al. (2019) have the most similar studies to ours. The authors deal with the VRCDSP in the context of perishable food distribution, which implies specific operational features in the cross-dock, as previously mentioned. As the problems are different and Rahbari et al. (2019) extended Agustina’s model to a bi-objective approach, we adapt the problem studied by Agustina et al. (2014) and ours to make a comparison.

In the formulation of Agustina et al. (2014), we do not consider storage time. Consequently, we eliminate storage costs and redundant constraints. In our formulation (F2), we set the processing times and costs, as defined in Agustina et al. (2014), and fix the number of delivery vehicles (K). We also ignore the internal transfer, cross-dock operational and service times. To establish a one-to-many distribution, we assume each customer request has just one product. This way, we obtain equivalent formulations.

In Agustina et al. (2014), instances are not available. We generate eight sets of instances based on their data. The number of inbound/outbound docks is 2, the number of inbound loads (L), the number of delivery vehicles (K), the number of requests/customers (N), and the processing times and costs are defined as in Agustina et al. (2014). We generated the demands randomly between 2 and 8 pallets. Each test set has five instances adapted from instances R201-R205 (Solomon, 1987). These instances are used because they allow many customers to be serviced by the same vehicle, as in Agustina et al. (2014).

We solve the instances by using the Gurobi solver with a 2-hour time limit. In Table 7, we show, for each instance set, the number of optimal solutions obtained (OS), the average run time (T) in seconds, and the average gap (Gap) for both formulations. In general, our model achieved better results than Agustina et al. (2014) (33 optimal solutions versus 29). In particular, for the instances in the 4/4/40 instance set, our proposal is significantly better. These results indicate the performance of Agustina et al. (2014)’s formulation would be improved if it were reformulated with two-index variables.

Table 7
Comparison of the two adapted formulations.

Instance set	Agustina et al. (2014 )			Our formulation
L/K/N	OS	T(s)	Gap(%)	OS	T(s)	Gap(%)
2/2/10	5	<1	0.00	5	<1	0.00
2/2/20	5	34	0.00	5	25	0.00
2/2/30	4	2980	1.11	5	2055	0.00
2/2/40	0	7200	17.61	0	7200	11.62
4/4/10	5	<1	0.00	5	<1	0.00
4/4/20	5	297	0.00	5	35	0.00
4/4/30	5	2494	0.00	5	469	0.00
4/4/40	0	7200	9.38	3	4650	3.46
Average	3.63	2527	3.51	4.13	1804	1.89

5.4 Integrated strategy relevance

In this section, we compare the Integrated Strategy (IS) with three strategies frequently used in practice to obtain a schedule for the distribution system. The first one accepts only direct deliveries to the customers (Direct Delivery Strategy - DD). In the other two, we hierarchically solve the cross-dock scheduling and the routing problems. In the Cross-Dock Scheduling/Routing Strategy (CDR), we obtain the cross-dock schedule before defining delivery routes. In the Routing/Cross-Dock Scheduling Strategy (RCD), we first define the delivery routes and then solve the cross-dock scheduling problem respecting the outbound loads associated with previously established routes. For each strategy, we solve the instances generated for the problem studied (Subsection 5.1) by using F2 with minor adaptations, as follows, and with one-hour time limit for each problem (routing and cross-dock scheduling).

• DD: one route for each customer is fixed by routing variables (x _0i =x _{i, N+1} =1) and the redundant constraints concerning routing problems are eliminated.

• CDR: first, we solve the cross-dock scheduling problem with the Direct Delivery Strategy. Then, we fixed the decisions related to the inbound load scheduling, the internal transfers, and outbound docks for each customer. In the next step, we solve the routing problem for each outbound dock and allow delivery routes for customers to be served in the same outbound dock.

• RCD: first, we solve the routing problem by forcing the departure time to be higher than τ (defined in Subsection 5.1) to simulate the cross-dock operation time. Second, we solve the cross-dock scheduling problem, in which each outbound load is associated with a route defined by a solution of the routing problem.

5.4.1 Results for instances with 15 customers

As previously mentioned, we have optimal solutions for 52 out of 72 instances given by the IS (F2). With the other strategies, we obtained optimal solutions for the subproblems in almost all instances. We only have eight solutions provided by the DD strategy with gaps lower than 2% and, consequently, the same happens with the CDR strategy when the cross-dock scheduling is solved first. The non-optimality of these solutions does not influence our conclusions since the IS solutions are significantly better than those provided by the DD and CDR.

Figure 8 shows the objective function values according to the solutions given by each strategy. As expected, the IS strategy obtained the best solution for most instances. Even in Scenario 1 and for customers with high demands, the DD strategy provides the worst solution, reinforcing the importance of routing in this distribution system. Regarding the objective function, we note the two best strategies are the IS and RCD. However, the RCD strategy cannot get feasible solutions for R110-1A, R111-1B, R101-2B, and R109-2B since the routes defined force infeasible scheduling, i.e., for some outbound loads, the deliveries are not possible before timeout. Besides, for nine instances, the RCD strategy provides the worst solutions.

Figure 8
Objective function values obtained by strategies - 15 customers.

We analyzed the performance of the IS over the RCD strategy. Table 8 summarizes the following information on the IS: the number of instances (NI^B ) in which it performs better, including the cases for which the RCD does not provide a feasible solution. We calculate the percentage difference (D) of the best solutions (BS), D=100×(BS^RCD -BS^IS )/BS^RCD , which represents how much better the IS performs compared to the RCD, and we report the average (AD), the lowest (LD) and the major (MD) values of the percentage differences. We also count the number of instances (OS) for which an optimal solution is obtained by the IS, the average gap (Gap) only considering the instances that are not solved optimally and have a feasible solution by the RCD. As it can be observed, for most of the instances, the solutions given by the IS are better, and the gaps show the difference may still increase. Furthermore, it is possible to achieve differences greater than 20%, representing a significant cost reduction for the distribution system.

Table 8
The performance of the IS compared with the RCD - 15 Customers.

Scenario	Group	NIB	AD (%)	LD (%)	MD (%)	OS	Gap (%)
1	A	10	4.58	0.03	22.79	10	0.00
	B	11	4.11	0.01	9.98	10	2.29
	C	11	5.72	1.22	15.66	8	6.56
2	A	11	9.48	1.86	18.43	10	10.01
	B	11	11.12	3.40	24.59	6	11.28
	C	11	8.34	1.34	20.33	6	8.73
Average			7.23	1.31	18.63		6.58

To compare the schedules obtained by the IS and RCD strategies and evaluate the effects of the integration decision, we expand the objective function. For both scenarios, we have very similar conclusions; however, they are more evident in Scenario 2. Due to this, we choose Scenario 2 to be illustrated here. In Figure 9, we have the cross-dock operation time, routing time and tardiness time for each instance, according to the solution given by each strategy. As the solutions do not present earliness time, we do not include this information in Figure 9. In general, the difference between the objective function values given by both strategies is mainly due to the difference between routing and tardiness times. According to the results, when we solve the routing problem first, we define delivery routes with a lower total time, about 9.4% on average. However, we have high tardiness when we use these routes to obtain a schedule for the system. This happens since the routing decisions do not consider the time to consolidate the outbound loads in the cross-dock. The schedules provided by the IS present a tardiness time about 87.7% lower on average. Also, note that the cross-dock operational time in solutions provided by the RCD is either the same or slightly higher than the ones provided by the IS (approximately 4.2% on average).

Figure 9
Times according to IS and RCD (Scenario 2).

By analyzing these costs individually, the advantage of integrated decisions is more evident. In our distribution system, the availability of the requests and, consequently, the outbound loads directly affects the departure time, i.e., when a route begins. Therefore, it does not make sense to define the routes without coordinating with the operational times in the cross-dock.

5.4.2 Results for instances with 25 and 40 customers

As the problems increase in size, almost all instances with 25 customers have non-zero gap solutions. In the case of hierarchical strategies, at least one of the subproblems is not solved optimally for most of the instances. We compare the four strategies and verify that the IS and RCD provide the best solutions, and none of them is predominantly better. Even with the possible improvements concerning the gaps, the solutions obtained with the CDR and DD strategies are not competitive with the IS and RCD strategies regarding solutions with a lower objective function value.

In Table 9, we summarize information to compare the performance of the IS over RCD. The table is similar to Table 8; however, we add the number of instances (NI^W ) in which the IS performs worse than the RCD and the number of non-optimal solutions between parenthesis next to the average gap. In the case of the RCD strategy, we show gaps for each subproblem solved.

Table 9
The performance of IS compared with RCD (25 customers).

Scenario	Group	NIB	NIW	AD (%)	LD (%)	MD (%)	Gap (%) - IS	Gap (%) - RCD
								Routing	CD
1	A	6	6	2.74	0.13	14.14	2.26 (11)	0.00	1.81 (7)
	B	6	5	0.79	0.04	1.71	5.59 (12)	0.00	3.16 (11)
	C	7	5	2.47	0.02	6.27	7.34 (12)	0.00	6.19 (12)
2	A	7	5	3.69	0.43	8.87	14.07 (12)	8.91 (8)	0.00
	B	5	7	1.90	0.25	3.64	19.22 (12)	9.47 (9)	2.01 (1)
	C	5	6	2.20	0.31	6.39	20.80 (12)	7.11 (9)	1.96 (8)
Average				2.30	0.20	6.84	11.55	4.25	2.52

As it can be noticed, the two strategies present the same performance concerning the best solution (50% each). Only for R112-2C, both strategies provided solutions with the same objective function value. However, the IS provides better quality solutions (2.30% on average). It is also important to highlight that the gaps presented with the IS are higher than the sums of the gaps when we solved the two subproblems (Routing+CD). We can also observe that the number of instances that are not solved optimally with the IS is high, indicating a great chance of improvement in the IS strategy’s performance. Besides, in the RCD strategy, the gaps reflect the difficulty to solve the subproblems. In Scenario 1, the routes may have four customers at most and, in Scenario 2, they may have eight at most. In the cross-dock scheduling problem, we have more outbound loads considered in Scenario 1 than in Scenario 2. Consequently, the routing problem’s combinatorial structure is more extensive in Scenario 2 than in Scenario 1. And, the inverse happens with the cross-dock scheduling problem.

For 40 customers, the IS and RCD strategies provide the best solutions. As can be observed in Table 10, the IS strategy did not maintain the same performance observed in the previous case. We have feasible solutions for 63 out of 72 instances given by the IS (F2), and we obtain feasible solutions for almost all cases with the RCD strategy, except for instance R108-2A, in which the routes obtained by solving the routing problem first were infeasible because, for some outbound loads, the deliveries could not be completed before the time limit. Since Table 10 reports the advantage of IS over RCD, we additionally evaluate the advantage of RCD over IS. In Scenario 1, both strategies are competitive; RCD is, on average, 2.9% better than the IS. However, in Scenario 2, the difference is more significant, with RCD outperforming the IS by 12.7% on average.

Table 10
The performance of IS compared with RCD (40 customers).

Scenario	Group	NIB	NIW	AD (%)	LD (%)	MD (%)	Gap (%) - IS	Gap (%) - RCD
								Routing	CD
1	A	2	10	0.19	0.08	0.31	6.79	0.00	5.92
	B	2	10	1.00	0.49	1.52	11.50	0.00	9.35
	C	3	9	3.54	0.45	8.87	19.69	0.00	15.99
2	A	1	11	2.09	2.09	2.09	35.55	23.22	7.59
	B	0	12	-	-	-	39.49	22.82	8.22
	C	0	12	-	-	-	42.76	23.20	12.87
Average				1.70	0.87	3.20	25.96	11.54	9.99

As expected, dealing with the routing and cross-docking problems separately is easier than solving them in an integrated way. However, for instances with 40 customers, the computational difficulty of solving the IS strategy prevented a completely fair comparison between them. Although the RCD strategy appears to perform better, this behavior should not be generalized, since solving the routing problem first does not guarantee feasible routes for the overall system schedule, whereas the IS explicitly accounts for these constraints through the integrated optimization process. Consequently, there is a need to develop more efficient solution methods, particularly for the integrated problem, in order to enable a more comprehensive comparison on larger instances. Although the instance data are not real, it is worth noting that the retail network on which this study is based currently has 44 customers; thus, instances with 40 customers represent a number of customers close to the real situation.

5.4.3 Solution time: IS versus RCD

Table 11 shows each strategy’s average run time (IS and RCD) to solve the instances. We separated the values according to the number of customers, Scenarios 1 and 2, and Groups A, B and C. In the RCD case, we have the time spent to solve the routing and the cross-dock scheduling problems (CD), with a time-limit of one hour each and their sum (Total).

Table 11
Average run time in seconds to obtain the best solutions according to the IS and RCD strategies.

	Scenario	Group	IS	RCD
				Routing	CD	Total
15 Customers	1	A	259	<1	3	3
		B	1460	<1	3	3
		C	3824	<1	137	137
	2	A	3163	14	<1	14
		B	4582	11	<1	11
		C	5108	11	<1	11
25 Customers	1	A	6670	<1	2525	2525
		B	7200	<1	3600	3600
		C	7200	<1	3600	3600
	2	A	7200	3019	396	3415
		B	7200	3094	915	4009
		C	7200	3210	3182	6392
40 Customers	1	A	7200	1	3600	3601
		B	7200	1	3600	3601
		C	7200	1	3600	3601
	2	A	7200	3600	3294	6894
		B	7200	3600	3600	7200
		C	7200	3600	3600	7200

In general, the RCD provided quicker solutions than the IS. It is essential to highlight that the RCD solution, regardless of whether the routing solution is optimal or not, does not guarantee feasible routes for the system schedule, as observed in some instances. In instances with 25 and 40 customers, mainly in Scenario 1, the run times of RCD increase significantly, being close or equal to the IS run times. As already identified, it is advantageous for the system schedule when we solve the integrated routing and cross-dock scheduling problems. However, the IS performance presented in Table 11 has high run times. Thus, we can point out the development of methods to solve the integrated problems as future research.

6 CONCLUSIONS AND FUTURE RESEARCH

In this paper, we study the vehicle routing and cross-dock scheduling problem (VRCDSP) in a retailer network context. The problem involves scheduling cross-dock activities, which include simultaneous unloading, consolidation, and loading, as well as defining delivery routes.

We consider that customers’ requests are organized within the cross-dock, which receives products from many suppliers (many-to-many). Furthermore, the transfer times from inbound to outbound docks are also considered.

Although the literature highlights that integrating decisions in a distribution system may contribute to a more efficient schedule, to the best of our knowledge, only Agustina et al. (2014) and Rahbari et al. (2019) have studied this integrated problem. However, their approach is specific to fresh food distribution centers with some particularities as indicated. Unlike the three-index routing variables used by Agustina et al. (2014) and extended by Rahbari et al. (2019), we present a model based on two-index routing variables. In addition, we proposed an improved formulation (F2). Besides, computational experiments show F2 outperforms the results obtained by Agustina et al. (2014).

The benefits of solving the integrated routing and cross-dock scheduling problems are evident. We obtain better quality solutions when compared to those obtained by solving the problems separately. We analyzed three possibilities. When we solved: i) the cross-dock scheduling problem with direct deliveries; and ii) the cross-dock scheduling, and after the routing problems, we obtained the worst solutions with high costs. However, when we solved the vehicle routing first and the cross-dock scheduling problem after, the solutions present lower transportation costs but higher scheduling costs (tardiness time). Even so, they are worse than the solutions obtained by solving the integrated problems. Furthermore, when we do not consider the integrated problems, we can obtain non-optimal solutions or infeasible routes for the distribution system.

Especially for the range of instances with 25 customers, we do not obtain optimal solutions. Due to this, the gains from integrating decisions may be more meaningful, especially because gaps reveal more possibilities of finding better solutions with the IS. For instances with 40 customers, this is more evident.

As future research, the computational tests reveal the need to develop faster methods to solve the VRCDSP for instances with a large number of customers. It is also possible to incorporate the order information to unload the products sent by suppliers at the cross-dock. Depending on the context, it may be part of the decisions, and it may be interesting to investigate how this impacts the schedule for the distribution system.

Data Availability

Datasets related to this article are available upon request to the corresponding author.

Acknowledgements

The authors sincerely thank Professor Vinicius Amaral Armentano, in memorian, for his valuable contributions during the development of this study.

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• Funding
  This research was supported by São Paulo Research Foundation (FAPESP) [grant numbers 2016/20504-1, 2017/15029-5 and 2013/07375-0] and National Council for Scientific and Technological Development (CNPq) [grant number 309161/2022-3].

• Editor responsible for the review
  Editor-in-Chief: Antônio Augusto Chaves.

Publication Dates

• Publication in this collection
  28 Aug 2026
• Date of issue
  2026

History

• Received
  13 Dec 2025
• Accepted
  23 June 2026