We study the tractor and semitrailer routing problem (TSRP), a variant of the vehicle routing problem (VRP). In the TSRP model for this paper, vehicles are dispatched on a trailerflow network where there is only one main depot, and all tractors originate and terminate in the main depot. Two types of decisions are involved: the number of tractors and the route of each tractor. Heuristic algorithms have seen widespread application to various extensions of the VRP. However, this approach has not been applied to the TSRP. We propose a heuristic algorithm to solve the TSRP. The proposed heuristic algorithm first constructs the initial route set by the limitation of a driver’s onduty time. The candidate routes in the initial set are then filtered by a twophase approach. The computational study shows that our algorithm is feasible for the TSRP. Moreover, the algorithm takes relatively little time to obtain satisfactory solutions. The results suggest that our heuristic algorithm is competitive in solving the TSRP.
In this paper, we consider the tractor and semitrailer routing problem (TSRP), a variant of the vehicle routing problem (VRP). The VRP is one of the most significant problems in the fields of transportation, distribution, and logistics. The basic VRP consists of some geographically dispersed customers, each requiring a certain weight of goods to be delivered (or picked up). A fleet of identical vehicles dispatched from a depot is used to deliver the goods, and the vehicles must terminate at the depot. Each vehicle can carry a limited weight and only one vehicle is allowed to visit each customer. It is assumed that some parameters (e.g., customer demands and travel times) are known with certainty. The solution of the problem consists of finding a set of routes that satisfy the freight demand at minimal total cost. In practice, additional operational requirements and restrictions, as in the case of the truck and trailer routing problem (TTRP), may be imposed on the VRP [
The basic types of vehicles. Note: In practice, many vehicle types are used in road freight transportation. This figure only lists four basic types. A large number of other types can be derived from the four basic types by the number of axles, tires and the combination style. Enterprises in most of the countries in the world employ various types.
The VRP and its various extensions have long been one of the most studied combinatorial optimization problems due to the problem’s complexity and extensive applications in practice [
Gerdessen [
Villegas et al. [
Research to date has considered most types of road vehicles, especially trucks and truck and trailer combinations. However, there has been little research on the types of tractor and semitrailer combinations. Hall and Sabnani et al. [
We aim to propose a heuristic for the TSRP. This aim is based on the practical knowledge that tractor and semitrailer combinations are popular in some countries, particularly China. The remainder of this paper is organized as follows. Section
Although there is little literature devoted to the definition and solution of the TSRP in the fields of transportation or logistics, plenty of research has been done on the TTRP, providing important references for the TSRP. In the TTRP, a heterogeneous fleet composed of
Different types of vehicle routes in the TTRP.
The vehicle types in the TSRP are different from those in the TTRP. The TTRP focuses on trucks and trailers, both of which can carry cargo. The TSRP involves
The TSRP can be formally defined on a directed graph
There are various tractordriving modes on graph
In practice, there are many depots on the freight transportation network of an enterprise. Depots have different functions and sizes. In the TSRP, we classify these depots into two types: main depots and customer depots. The flow of freight between any two depots is usually uneven. In our method, we abstract the freight transportation network onto a graph (denoted by
Because the freight flows among various depots are unequal, we select a freight flow network denoted by
An example of selecting unitflow networks. Note: Black points denote depots. Lines denote the distribution of freight flows. Numbers near lines denote freight flow volume (unit: one semitrailer).
The TSRP model in this paper uses the unitflow network. Vertex 0 represents the main depot where some loaded semitrailers are waiting to be delivered to customers. The vertices in
Vehicle routes in the TSRP model.
The TSRP model consists of determining the number of tractors to be used and the route of each tractor so that the variable costs and service level are balanced, while each route starts and ends at the main depot. Variable costs are reduced by decreasing the overlap distance of tractors running alone. The service level is based on the percentage of customer demand that is satisfied.
Drivers are assigned to ensure flexible running of tractors. A driver’s onduty hours per day are determined by the legal onduty time and driver dispatching mode. Onduty hours consist of driving hours plus temporary rest time or residence time in depots. The residence time at the main depot (denoted by
The elements of the initial solution set are tractor routes. To give the form of a route, we suggest the following procedure. The number of drivers assigned to each tractor is
We suggest the steps below to construct elements of the initial solution set.
Transform the distance matrix into a running time matrix. Use
Search the running time matrix. Let
Compare the routes with freight flow demand. Every route in Step
The more elements (i.e., tractor routes) in the initial solution set, the more choices for freight enterprises. We classify tractor routes into certain types, according to the number of customers on a route. There is one customer passed by in the 1st type, two customers passed by in the 2nd type, and so on. When there are many customers on a route, the tractor can serve more freight demand. When more temporary rest time is consumed at customer points, the effective running hours of the tractor are reduced. Routes of the same type generally have some “overlap arcs” in which only one tractor pulls a semitrailer, and others run alone. We suggest reducing the total distance of “overlap arcs.”
The first step is for the same type. An overlap arc
The second step is for different types. A “tractor route—overlap arc” matrix (
In the first phase, a transitional solution set that contains such elements as the
In the second phase, we propose the “fillandcut” approach to attain a satisfactory solution to the TSRP.
Construct a zero matrix
All route segments in the transitional solution set actually have corresponding elements in matrix
In some cases, the number of nonzero elements of
We abstract the transportation network on an N
The tractor running time between two depots (hours).
From  To  


1  2  3  4  5  6  7  8  9  10  11  12  13  

0  5.5  4.0  4.5  1.5  1.0  1.0  4.0  3.0  4.5  1.0  3.5  3.5  3.0 
1  5.5  0  1.5  1.0  4.0  4.5  4.5  2.5  2.5  3.0  5.5  4.0  5.0  6.5 
2  4.0  1.5  0  2.5  2.5  3.0  3.0  4.0  3.0  4.5  4.0  3.5  3.5  5.0 
3  4.5  1.0  2.5  0  3.0  3.5  3.5  1.5  1.5  2.0  4.5  3.0  4.0  5.5 
4  1.5  4.0  2.5  3.0  0  0.5  0.5  4.5  3.5  5.0  1.5  4.0  4.0  3.5 
5  1.0  4.5  3.0  3.5  0.5  0  1.0  5.0  4.0  5.5  2.0  4.5  4.5  4.0 
6  1.0  4.5  3.0  3.5  0.5  1.0  0  4.0  3.0  4.5  1.0  3.5  3.5  3.0 
7  4.0  2.5  4.0  1.5  4.5  5.0  4.0  0  1.0  0.5  4.0  2.5  3.5  5.0 
8  3.0  2.5  3.0  1.5  3.5  4.0  3.0  1.0  0  1.5  3.0  1.5  2.5  4.0 
9  4.5  3.0  4.5  2.0  5.0  5.5  4.5  0.5  1.5  0  3.5  2.0  3.0  4.5 
10  1.0  5.5  4.0  4.5  1.5  2.0  1.0  4.0  3.0  3.5  0  2.5  2.5  2.0 
11  3.5  4.0  3.5  3.0  4.0  4.5  3.5  2.5  1.5  2.0  2.5  0  1.0  2.5 
12  3.5  5.0  3.5  4.0  4.0  4.5  3.5  3.5  2.5  3.0  2.5  1.0  0  1.5 
13  3.0  6.5  5.0  5.5  3.5  4.0  3.0  5.0  4.0  4.5  2.0  2.5  1.5  0 
The freight flow between two depots (one semitrailer).
From  To  


1  2  3  4  5  6  7  8  9  10  11  12  13  



1  1  1  1  1  1  1  1  1  1  1  1 
1  1  0  1  0  1  1  0  0  0  0  0  0  0  1 
2  1  1  0  0  0  0  0  0  1  0  1  1  0  0 
3  1  0  0  0  1  0  0  0  0  0  1  0  1  1 
4  1  0  0  1  0  0  0  0  1  1  0  0  0  1 
5  1  0  0  1  1  0  0  0  0  0  1  0  0  1 
6  1  0  0  0  1  1  0  0  0  0  1  1  0  0 
7  1  0  0  0  1  0  0  0  0  0  1  1  1  0 
8  1  0  1  0  0  0  1  0  0  1  0  1  0  0 
9  1  0  1  0  1  0  1  0  0  0  0  1  0  0 
10  1  1  0  0  0  1  0  1  1  0  0  0  0  0 
11  1  0  1  0  1  0  0  0  1  0  0  0  0  1 
12  1  0  1  0  0  1  0  0  0  0  0  1  0  1 
13  1  0  0  0  1  0  1  0  1  1  0  0  0  0 
According to some enterprise experience, a driver’s onduty time is 8.5 hours per day. One or two drivers are assigned to a tractor. A tractor with two drivers can work consecutively for no more than 17 hours in a 24consecutivehour period. The temporary rest time in customer depots is 0.5 hour, and the residence time in
The satisfactory solution of the TSRP achieved by the heuristic algorithm.
Types of the routes  Form of route  Working time needed by routes (hours) 

A tractor with two drivers. Two different customer depots are passed by. 

17 
A tractor with two drivers. Three different customer depots are passed by. 

17.5 
A tractor with two drivers. Four different customer depots are passed by. 

17 
A tractor with two drivers. Five different customer depots are passed by. 

17.5 
A tractor with two drivers. Six different customer depots are passed by. 

17 
A tractor with two drivers. Two different customer depots are passed by. 

17.5 
It is feasible to propose exact algorithms (e.g., integer programming) for the TSRP when the initial solution set is constructed. We proposed a 01 integer programming for the No. 1 example and attained the exact solution. The exact solution can satisfy 84 percent of all transportation demand. Sixteen tractors and thirtytwo drivers are needed during a 24consecutivehour period. In 10 percent of the total running time, tractors run alone. We implemented the proposed heuristic algorithm using Matlab and the 01 integer programming with QS. Although the exact algorithm can attain a slightly better solution, it requires more calculating time. For the No. 1 example, the solving time using the heuristic algorithm was approximately 80 seconds while that for the exact algorithm was approximately 2000 seconds. We suggest that the heuristic algorithm has an advantage for solving the TSRP.
We have repeated the generation of random arrays over 50 times to obtain some typical computational networks. The heuristic algorithm was employed on these networks. We ran the experiments of this section on a computer with an AMD Athlon(tm) X2 DualCore QL65 running at 2.10 GHz under Windows 7 ultimate (32 bits) with 2 GB of RAM. Table
The results of TSRP experiments achieved by the heuristic algorithm.
Transportation network  On the solution  

Number of nodes  Number of freight flows  Fraction of demand satisfied (%)  Number of tractors  Average time of tractors running alone (hours)  Calculation time (seconds) 
10  54  65  10  0.9  3 
11  60  85  18  1.9  5 
12  66  89  17  1.8  26 
13  72  92  22  2.1  45 
14  78  89  23  1.6  78 
15  84  71  18  1.8  47 
17  96  89  28  1.7  59 
18  102  81  25  1.6  86 
19  108  86  32  1.7  114 
21  120  74  28  1.6  198 
22  126  71  28  1.4  234 
23  132  86  37  1.8  310 
In this paper, we proposed a TSRP model and suggested a heuristic algorithm to solve it. The TSRP concentrated on a unitflow network, and all tractors originated and terminated at a main depot. Unlike most approaches to the TTRP or VRP, the heuristic algorithm for the TSRP did not regard the number of vehicles as a precondition. Therefore, the solution to the TSRP was able to balance the variable costs and service level by altering the vehicle number. The main characteristics of the heuristic algorithm are the initial solution set constructed by the limitation of driver onduty time and the combination of a twophase filtration on candidate routes. The computational study shows that our algorithm is feasible and effective for the TSRP. Although some exact algorithms for the TSRP are feasible after the initial solution set is constructed, the heuristic algorithm is efficient because it takes relatively less time to obtain satisfactory solutions. Future research may try to extend the TSRP to include more practical considerations, such as time window constraints. Other efficient heuristics for the TSRP may also be proposed. In this regard, the benchmark instances generated in this study may serve as a testbed for future research to test the efficiency of specific algorithms for TSRP.
This work was partially funded by the Science and Technology Plan of Transportation of Shandong Province (2009R58) and the Fundamental Research Funds for the Central Universities (YWF1002059). This support is gratefully acknowledged.