Crossdocking, as a strategy to reduce lead time and enhance the efficiency of the fashion supply chain, has attracted substantial attention from both the academy and the industry. Crossdocking is a critical part of many fashion and textiles supply chains in practice because it can help to achieve many supply chain strategies such as postponement. We consider a model where there are multiple suppliers and customers in a single crossdocking center. With such a model setting, the issue concerning the coordinated routing between the inbound and outbound routes is much more complex than many traditional vehicle routing problems (VRPs). We formulate the optimal route selection problems from the suppliers to the crossdocking center and from the crossdocking center to the customers as the respective VRPs. Based on the relationships between the suppliers and the customers, we integrate the two VRP models to optimize the overall traveling time, distance, and waiting time at the crossdocking center. In addition, we propose a novel mixed 0/1 integer linear programming model by which the complexity of the problem can be reduced significantly. As demonstrated by the simulation analysis, our proposed model can be solved very efficiently by a commonly used optimization software package.
The fashion industry’s supply chain is full of uncertainty, unpredictability, and various complexities, as studied by Lo et al. [
In fact, CD is a popular logistical strategy by which packages of products are unloaded from the inbound vehicle and then are almost directly uploaded into the outbound vehicle with little or no storage in between [
However, implementing CD requires careful considerations. As Van Belle et al. [
Figure
Inbound and outbound routes merging at CDC.
It is obvious that with the CDC, even though the inbound and the outbound routes can be modeled, respectively, as two independent VRPs, a systemwide optimal performance requires the coordination between the unloading task of the inbound trucks and the uploading task of the outbound trucks, which arises a question as how to achieve this coordination. In this paper, we will devote to addressing this issue. To be specific, based on the relationships between the suppliers and the customers, we integrate the two independent VRP models to develop the strategies that can optimize the overall traveling time, distance, and waiting time at the CDC. In addition, we propose a novel mixed 0/1 integer linear programming model that can significantly reduce the complexity of the problem under study. As a matter of fact, as demonstrated by the simulation analysis, our proposed model can be solved very efficiently by a commonly used optimization software package.
The rest of this paper is organized as follows. In Section
The fashion supply chain is full of uncertainty and unpredictability. It is in general a rather complex system and requires very careful analysis in order to find rooms to improve its efficiency. As studied by Lo et al. [
To achieve agile SCM, the use of information systems is a popular measure. In a study regarding fashion retailers in the UK, Birtwistle et al. [
The previous research on CD generally focuses on the following issues:
To implement CD, the presence of CDC is critical, and its location is very influential. In fact, CDC helps coordinate suppliers and customers with distances among them. By consolidating the goods from different suppliers to a specific set of customers, the logistics costs can be reduced which helps achieve the optimal supply chain performance. In the literature, the majority of models related to this issue focus on the location problem with distribution centers. For example, Ross and Jayaraman [
In the literature, most prior studies on warehouse layout design of warehouses are closely related to the dock layout design. In fact, dock layout design problem generally includes several subproblems, such as the optimization of the quantity and location of facilities and the prediction of the frequency and strength of facility employments. Heragu et al. [
It is well known that CDCs have two typical functions: temporary storage space and consolidation of goods from trucks to trucks. However, the storage space of CDC is commonly limited. As a norm, goods should not be kept in docks for longer than 24 hours. Therefore, the main function of CDC is to consolidate goods from the inbound trucks to the outbound trucks. In order to improve the efficiency of truck usage, synchronizing the respective loading operations is a critical problem. In the literature, Soltani and Sadjadi [
Unlike many traditional warehousing “storagebased” operations, the efficiency of space and facility usage is an important criterion to assess crossdocking operations. Routing problem on loading units in dock usually focuses on these objectives. Cohen and Keren [
Transportation from suppliers to CDC and from CDC to customers requires careful planning. In fact, how to synchronize the inbound and outbound routes is a difficult problem. In the literature, this type of problem can be modeled as a coordination problem with multiple VRPs. Vahdani and Zandieh [
As reviewed above, crossdockingrelated optimization problems are widely explored. In fact, VRP is also a widely explored topic (e.g., see the reviews by Eksioglu et al. [
The scenario examined in this paper is depicted in Figure
In order to facilitate the formulation and provide solutions to the key problems, some presumptions are summarized as follows.
All trucks are waiting at CDC. When they finish the pickup or delivery tasks, they return to CDC. Therefore, all scheduled routes are close, beginning, and ending at CDC.
The capability of CDC is always adequate.
The time windows at suppliers and customers are all not considered.
The cost of VRP is measured only by traveling time.
In the planning horizon, each truck can only serve one route at a time.
In the following, we first present the indices and parameters used for the outbound VRP and for the inbound VRP. The indices and parameters can be defined in a similar way as follows.
Similarly,
The first three matrices are to represent the inbound and outbound VRPs. The fourth is used to avoid the subtours in a route. The last variable is defined to represent the measure that the outbound trucks will wait for the inbound trucks.
The objectives considered in the model can be classified into two types. The first is to optimize the inbound and outbound VRPs. The second is to minimize the waiting time of the outbound trucks for the inbound trucks.
The cost of VRP should be minimized as defined in (
The number of utilized trucks should be minimized as defined in
The waiting time of the outbound trucks for the inbound trucks should be minimized with respect to the corresponding numbers of inbound routes for all outbound routes:
With the above five objectives, the problem under study should be formulated as a multiobjective optimization model.
The first series of constraints is order to ensure that the requirements of capitalized VRP are fulfilled. We first formulate the constraints for the outbound VRP as follows, and the constraints for the inbound VRP can be formulated in a similar way:
the load must be less than the vehicle capacity, as shown in
each customer must be served by one and only one route, as defined in
if the customer
if
cost constraint is defined in
the variable of vehicle usage can be defined by the customervehicle assignment variables, which leads to the constraint defined in
the above constraints can ensure the connectivity of each route, but not avoid the subtours. Hence, the following linear constraints in (
The second series of constraints deal with the definition of
The model of coordinated VRP for CD is built upon the above indices, parameters, decision variables, objectives, and constraints. All decision variables are 0/1 integers. All analytical expressions are given in linear integer formats. Therefore, the proposed model is a multiobjective integer linear programming model.
As discussed in Section
The analysis is conducted in two steps. First, a case is demonstrated. Second, the model is solved with different parameter settings. Because the model is multiobjective, it will be transferred into a single objective model by adjusting the other objectives to be constraints. With the five objectives, the minimization of used trucks can be restricted by constraints conveniently. First, the numbers of used trucks for inbound or outbound will not be large. Second, the numbers can be adjusted by setting the upper bounds. Therefore, in the following, these two objectives are taken as constraints by adjusting
In this part, the input parameters are given for a smallscale problem. In Table
Basic parameter settings of the case demonstration.
Parameter  INN  IVN  IQ 

Value  8  4  40 
Parameter  ONN  OVN  OQ 
Value  8  4  40 
Cost matrix of the inbound VRP.
0  1  2  3  4  5  6  7  8  

0  0  3  10  3  6  9  9  10  4 
1  3  0  10  7  7  10  5  5  1 
2  10  10  0  2  7  6  3  2  6 
3  3  7  2  0  6  2  7  4  8 
4  6  7  7  6  0  9  1  9  10 
5  9  10  6  2  9  0  10  1  3 
6  9  5  3  7  1  10  0  8  10 
7  10  5  2  4  9  1  8  0  5 
8  4  1  6  8  10  3  10  5  0 
Cost matrix of the outbound VRP.
0  1  2  3  4  5  6  7  8  

0  0  3  5  1  10  9  1  7  10 
1  3  0  7  5  1  4  2  9  3 
2  5  7  0  3  9  4  9  9  6 
3  1  5  3  0  8  1  2  3  1 
4  10  1  9  8  0  2  4  6  2 
5  9  4  4  1  2  0  6  5  7 
6  1  2  9  2  4  6  0  8  4 
7  7  9  9  3  6  5  8  0  1 
8  10  3  6  1  2  7  4  1  0 
Supply matrix between suppliers and customers.
0  1  2  3  4  5  6  7  8  

0  0  0  0  0  0  0  0  0  0 
1  0  5  8  0  0  0  0  0  0 
2  0  0  4  8  0  0  0  0  0 
3  0  0  0  6  8  0  0  0  0 
4  0  0  0  0  6  4  0  0  0 
5  0  0  0  0  0  5  8  0  0 
6  0  0  0  0  0  0  5  6  0 
7  0  0  0  0  0  0  0  7  0 
8  0  8  0  0  0  0  0  0  4 
By aggregating the three objectives by a weight vector
XpressIVE produces optimal solutions corresponding to different settings of the weight vector. The weight vectors and objective values are shown in Table
Optimal objective values of the demonstrative case.
Case 

ivrpc  ovrpc  izs  izs 

obj  Is it optimal?  Computation time/s 











2 

3  3  76  66  8  26.4  Y  0.5 
3 

4  4  58  47  10  25.9  Y  0.4 
4 

4  3  72  53  9  32.9  Y  0.5 
5 

4  4  58  61  10  40.3  Y  0.4 
6 

4  4  53  48  11  37.4  Y  0.5 
7 

4  4  63  61  11  51.2  Y  0.5 
8 

4  4  65  66  9  60.2  Y  0.4 
9 

4  4  57  51  9  22.8  Y  0.7 
10 

3  4  55  41  9  24.6  Y  0.6 
11 

3  4  61  44  10  30.4  Y  0.4 
12 

4  4  53  64  9  39.8  Y  0.4 
13 

4  4  57  62  11  45.7  Y  0.5 
14 

3  4  58  49  10  43  Y  0.6 
15 

4  4  65  58  9  54.5  Y  0.6 
16 

3  3  80  50  9  34.4  Y  0.5 
17 

3  3  67  53  9  35.2  Y  0.7 










19 

4  4  75  47  10  44.3  Y  0.5 
20 

4  3  64  47  8  44.3  Y  0.4 
21 

4  4  61  69  10  60.7  Y  0.4 
22 

4  4  64  48  11  35.9  Y  0.5 
23 

4  4  57  51  9  36.6  Y  0.4 
24 

4  3  69  50  7  44.7  Y  0.4 
25 

4  4  60  42  10  42.8  Y  0.5 
26 

3  3  69  77  7  66.8  Y  0.4 
27 

3  3  62  44  9  39  Y  0.4 
28 

4  4  69  45  11  46.8  Y  0.4 
29 

3  4  74  73  8  60.5  Y  0.5 
30 

4  4  67  56  11  57  Y  0.4 
31 

4  4  69  41  10  48.5  Y  0.4 
32 

4  3  70  50  10  54  Y  0.4 
33 

4  4  72  44  9  57.3  Y  0.4 
34 

3  3  71  73  8  58.6  Y  0.4 
35 

3  4  61  73  8  58.1  Y  0.4 
36 

4  4  80  57  11  70.8  Y  0.5 
VRP plans of the “case 1”.
VRP plans of the “case 18”.
The model deals with two VRPs and the coordination between them. Therefore, one may believe that the corresponding optimization problem cannot be solved with an acceptably good performance because VRP is itself a notorious problem in terms of computational efficiency in mathematical programming. However, in the application of crossdocking, the two VRPs only deal with littlescale or mediumscale problems. The scheduled trucks, passed nodes, and node numbers of routes are all limited. Therefore, it is probable to obtain a promising performance to solve the problem. In order to reveal whether the model can be applied in realworld CDCs, a series of cases generated randomly are solved, and the computation times are recorded. For each setting with the involved inbound and outbound nodes and trucks, the test is repeated for 30 times with different cost matrices and supply matrices. The average computational time is recorded in Table
Performance evaluation of different problem scales.
No.  INN  IVN  ONN  OVN  Time (s)  Is it optimal? 

1  2  1  2  1  0.1  Y 
2  4  2  4  2  0.1  Y 
3  6  3  6  3  0.3  Y 
4  8  4  8  4  0.5  Y 
5  10  5  10  5  6.4  Y 
6  12  6  12  6  20.6  Y 
7  14  7  14  7  73.2  Y 
8  16  8  16  8  175.4  Y 
9  18  9  18  9  312.4  Y 
10  20  10  20  10  911.2  Y 
11  22  11  22  11  1862.9  Y 
12  24  12  24  12  3982.7  Y 
Logistical efficiency plays an increasingly important role in shortening the lead time for an agile fashion supply chain. Crossdocking approach can help to increase the coordination efficiency and reduce the transshipment time and operations cost. This is especially important in the quick responsebased fashion supply chain with multiproduct and smallbatch distribution. In crossdocking operations, a crossdocking center separates the suppliers and customers into two groups, which can be modeled as two VRP models, respectively. However, it is obvious that a systemwide optimal operations performance in the crossdocking center requires coordinating these two models so as to obtain the systemwide optimal strategy. In this paper, we have proposed a truck routing model based on two coordinated VRP models with respect to the commonly observed crossdocking center operations in the fashion supply chain. A 0/1 integer linear programming model has been built for solving it. Furthermore, our numerical analysis illustrates that our proposed optimization problem, with a realistic problem size, can be solved by commonly available optimization software with a very promising performance. However, notice that the VRP is itself an NPhard problem, and our proposed model focuses on coordinating two VRP models and hence is much more complex in computation. Thus, when the scale of served suppliers and customers is increased or the number of nodes in routes is increased, it will be very difficult to obtain the optimal solution within a few seconds or minutes. Therefore, like many prior studies on VRP in the literature, heuristicbased approaches will play an important role to tackle the problem under such situations. In addition, this study does not take into account other practical elements, such as stochastic nature of various model parameters. Considering that some of these elements will make the problem more complex and challenging, we leave them as the topics to be pursued in future research.
The paper was supported partially by the National Natural Science Foundation of China (71101088, 71101028, and 71171129), the National Social Science Foundation of China (11&ZD169), the Shanghai Municipal Natural Science Foundation (10ZR1413200, 10190502500, 11510501900, and 12ZR1412800), the China Postdoctoral Science Foundation (2011M500077 and 2012T50442), the Science Foundation of Ministry of Education of China (10YJC630087), the Doctoral Fund of Ministry of Education of China (20113121120002), the Program for Excellent Talents and the Program for Innovative Research Team in UIBE. The authors claim that none of them has any financial relation with the commercial identities mentioned in the paper that might lead to conflict of interests.