This paper addresses a special truck routing optimization problem in openpit mines based on the minimization of timevarying transport energy consumption. A mixedinteger programming model is formulated to clearly describe the engineering problem, and a series of constraints are deduced to strengthen the model. To ensure that the model has timevarying characteristics, a method to estimate timevarying parameters is proposed by using pattern recognition and trend surface estimation. This timevarying resistance coefficient is mainly used to describe the process of road damage caused by frequent rolling of heavy trucks on the road surface. At the same time, in order to make the truck routing converge to the optimal energy consumption solution quickly, some definitions and properties are then provided based on stochastic theory, and a strategy to improve the computational efficiency of the model is proposed using these properties. Finally, an improved genetic algorithm is designed to solve the model. The results of experiments show that the proposed algorithm is effective and efficient.
Truck’s routing along openpit mines is an important link connecting all process flows and external transportation tasks for mineral and waste, which is one of the key factors that will directly influence the cost of transportation systems [
The transport routing optimization problem in openpit mines, which ultimately aims to optimize global transportation cost under specific physical and economic conditions, is a kind of multiobjective combinatorial optimization problem. It has attracted the interest of scholars in the fields of logistics planning and truck dispatching for openpit mines owing to its effect on reducing transportation costs. At present, methods used to optimize routes for openpit mines can be divided into two categories. One of them is to try to use the static equivalent distance as a weight or selection criterion to search for the optimal route. For example, White and Olson [
In light of the above considerations, the problem of finding the shortest path in terms of timevarying transport energy consumption under several constraints is examined in this paper. First, we provide a brief overview of the characteristics of the road transportation network in openpit mines and establish a mixedinteger programming model to describe the state of global timevarying transport. Second, to ensure that the model has timevarying characteristics, we propose a method to calculate resistance in different road segments. Finally, to improve the computational efficiency of the timevarying model, we propose several optimization strategies based on stochastic theory and improve the genetic algorithm to solve the mixedinteger programming model.
The remainder of this paper is organized as follows: In Section
In an openpit mine with trucks as the main transport equipment, there are usually several routing nodes in the road transport system as shown in Figure
3D model of openpit mine (the main body of the figure is a threedimensional (3D) model of an openpit mine, and the left side shows a partial stripping transportation system of the north slope; the right side is a satellite image mainly used to illustrate the spatial relationship between the road and the bench).
In the problem scenario that we consider, it is generally believed that the energy consumption of trucks is dominated by the work needed to overcome driving resistance. Thus, if optimization is based on the minimum global timevarying energy consumed for transport, we must first quantitatively analyze the fluctuation in driving resistance in different states. In general, for mining trucks, this resistance fluctuation effect can be divided into three parts [
To define and formulate this problem explicitly, the following parameters are introduced first:
To represent routing node decisions, we introduced the following decision variables:
Based on the above notations, the openpit transport routing optimization problem can be formulated as the following mixedinteger programming model:
According to resistance characteristics described in the literature [
The objective function (
Through formula (
Tests of the coefficient of rolling resistance [
Maximum driving speeds for various road types.
Road types  Maximum driving speed (km/h)  

Loading truck  Unloading truck  
+8%  0  −8%  +8%  0  −8%  
Transport artery  35  40  40  45  50  50 
Transport semimain  30  35  35  40  45  45 
Temporary track  25  30  30  35  40  40 
Turn back road  15  20  15  20  25  20 
In the process of establishing the type of surface pattern recognition algorithm, the most critical operation involves classifying pavement types according to the differences in the coefficient of rolling resistance to establish a variety of labels of pavement type. Therefore, we first need to consider how to effectively create tags for the road surface.
For this reason, we first classify periodic damage to the road, which is divided into eight types of road surfaces in Table
Rolling resistance for various types of road surfaces (Zhahanaoer OpenPit Mine).
Various types of road surfaces  Rolling resistance 

(1) Very hard, smooth roadway or dirt surface, no penetration or flexing  0.010∼0.018 
(2) Hard, smooth, stabilized surfaced roadway without penetration under load, watered, maintained  0.018∼0.020 
(3) Firm, smooth, rolling roadway with dirt or light surfacing, flexing surfacing, flexing slightly, maintained slightly, maintained fairly regularly, watered  0.020∼0.030 
(4) Dirt roadway, rutted or flexing under load, little maintenance, no water, 25 mm tire penetration or flexing  0.030∼0.045 
(5) Dirt roadway, rutted or flexing under load, little maintenance, no water, 50 mm tire penetration or flexing  0.045∼0.060 
(6) Rutted dirt roadway, soft under travel, no maintenance, no stabilization, 100 mm tire penetration or flexing  0.060∼0.075 
(7) Rutted dirt roadway, soft under travel, no maintenance, no stabilization, 200 mm tire penetration and flexing  0.075∼0.1 
(8) Very soft, muddy, rutted roadway, 300 mm tire penetration, no flexing  0.1∼0.14 
The multilabel classification algorithm is supported by training data, and thus another task of algorithm design is to collect them. In this section, we used the Zhahanaoer OpenPit Mine as an example to show the format of the training data in Figure
The partial training data.
The core step of the parameter estimation algorithm is to classify the training data using pattern tags formed by the type of surface. As this kind of algorithm [
For the multilabel classification scenario considered in this paper, the first hypothesis is that the training sample is
The above SVM technology is only a binaryclass method that could find the type of goal in a variety of types of road surfaces. Therefore, to ensure that the algorithm has the capacity for multilabel classification based on the binary class, a decision tree SVM was introduced to carry out multilabel classification. The basic idea of the method is to set up an SVM model at each level of the decision tree and select one of the subclasses at each level. The above operation is repeated until all subclasses have been chosen. Figure
Fourclassification logic.
The goal of the multilabel pattern classification algorithm is to find the type of road surface estimated by using the index parameters at any given time. The main task here is to use grouping data to estimate the coefficient of rolling resistance on the basis of the type of road surface. The method of estimation used here is not unique, but we find that data for the Zha mine are suitable for the quadratic trend surface analysis method in this paper through the detection of goodness of fit and significance. We thus only introduce the basic principle of quadratic trend surface analysis in the next section. The trend surface model can be expressed as follows:
We now need only use the least squares’ model to fit the parameters of the model to construct an available trend surface. The least squares’ model can be written as below, and the trend surface parameter estimation process is shown in Figure
The trend surface estimation of the quadratic polynomial.
In Section
In the network graph
It is noteworthy that the once the timevarying transport energy consumption is considered a stochastic variable, we can construct its probability density function and redefine the concept of the global timevarying shortest route as follows:
In the road network graph
It is noteworthy that even though the concept of the global shortest route is redefined in Definition 2, probability density estimation for each route remained difficult. Therefore, to simplify the calculation further, we summarize the following two optimization properties by using statistics:
For any two stochastic variables
Property 1 demonstrates that when two stochastic variables are compared, the probability of the one with a smaller mathematical expectation is greater than that of the other. To demonstrate the practical correctness of this property, the proof of is given as shown in formulae (
Suppose that random variables
Its equivalent form is as follows:
We introduce another random variable
Therefore, the proposition can be rewritten as
From the definition of a random variable, the expectation of
According to the scope of the domain, the integral is rewritten as follows:
According to the known conditions of Property 1, we give the sign criteria as follows:
And, the initial conditions are given by formula (
Therefore, the following absolute value relationship can be obtained:
Therefore, the probability events
Because the full probability event of
The following formula can be derived from formulas (
We thus get the following:
provided only a prior condition for the comparison between stochastic variables. To further illustrate the practical effectiveness of this property in the optimization of the algorithm, by combining with a realistic optimization scenario, the second property of the optimization problem is provided.
When searching for the optimal route along routes
Based on the theory of optimization given in Section
Assuming there were
Thus, the traditional openpit truck routing optimization problem could be transformed into the event probability problem shown as follows:
Finally, considering the computational complexity of the mass transport network, the expectation of these stochastic variables is used in an algorithm of adaptive values according to Properties 1 and 2, and an improved genetic algorithm is used to improve the efficiency of route searching.
The openpit truck route optimization problem is a kind of TSP (traveling salesman problem), which is a typical problem of combinatorial optimization. The outstanding feature of this kind of combinatorial optimization problem is that the search space increased sharply with an increase in the scale of the problem. Thus, in the practical openpit route optimization process, the traditional method of enumeration struggles to obtain the optimal solution. To ensure that the algorithm has better global optimization ability and adaptability, a genetic algorithm [
Step A21: calculate the Euclidean distance between nodes segment
Step A22: calculate the number of degrees of the output of each node
Step A23: add the source node to the position of the first gene and search for the adjacent node with the minimum value
Step A24: judge the outdegree of the node. If
Step A25: loop Step 2 until the desired population size is reached.
Step A41: select
Step A42: calculate the fitness of the individual in the remaining population
Step A43: using the roulette method, divide the probability internally. Create random numbers
Step A44: according to the formula
Crossoperation.
Mutation operation.
To avoid the phenomenon of premature convergence of the algorithm in the initial iteration, the algorithm redesigns the crossover and mutation parameters to ensure that they could be adaptively adjusted according to the fluctuation of fitness.
Adaptive adjustment of the crossover probability. The crossover operator is important to ensure that the algorithm produced new offspring, and the probability of the crossover operation
Adaptive adjustment of mutation probability. In the entire genetic algorithm, gene mutation is a key operation to control the diversity of the population. Its iterative nature is often the opposite of the crossover operation, and its characteristic is that if the mutation rate is too high in early iterations, it will lead to the algorithm getting caught during blindly searching and further lose the genetic characteristics of excellent individuals. Conversely, if the mutation rate is too low in later iterations, it is easy for the algorithm to fall into a local optimal solution, which makes it difficult to be found the global optimal solution. Therefore, to avoid the above problems, we design the following iterative relationship concerning mutation probability:
To test the performance of the mathematic model and the proposed algorithm with improvement strategies, we wrote all relevant programs under in C# .NET to solve for route optimization in the Zhahanaoer Openpit Mine. The experimental platform and parameters are as follows:
Experimental platform: all experiments were performed on a computer with a 2.9 GHz Intel Core I5 CPU, 8 GB of RAM, and Win 7 X64 operating system.
GA parameters: population size,
Other parameters of the calculation: the Zha mine uses two types of trucks: 1200 HP and 2000 HP. The capacity of the trucks was 60 m^{3} and 90 m^{3}, respectively. The slope is
To intuitively express the effect of the optimization of the algorithm, we select 5 sets of test cases and calculate the energy consumption by generations. These five groups of examples are the transportation routings from mining and stripping sites to unloading sites of Zhahanaoer Mine in 2018. The energy consumption relation in the process of searching the optimal solution for each group of routings is as shown in Figure
Energy consumption by generations.
From Figure
To further demonstrate the effectiveness and advantages of parameters optimization, we use all test cases to build simulation experiments, and the results of a comparison are shown in Figure
Energy consumption by generations (parameters not optimized).
Figure
To further demonstrate the superiority of the algorithm in solving the problem of route optimization, this section reports a comparative analysis of the optimization results obtained by the improved algorithm and the traditional algorithm. The data are shown in Table
Data for comparison of the effect of algorithmic optimization.
Case  Load status  Contrast algorithm  Node  Average speed (Km/h)  Haul distance (Km)  Transport energy (KJ)  Execution time (s) 

N1  1  IGA  29  37  1.39 


Dijkstra/PSO  30 

776826.4  7.07/4.72  
N2  1  IGA  31  34  1.17 


Dijkstra/PSO  27 

704826.5  8.77/3.04  
N3  1  IGA  43  36  1.68 

3.97 
Dijkstra/PSO  32 

914732.3  11.87/ 

N4  1  IGA  59  36  1.93 


Dijkstra/PSO  30 

967439.5  15.02/5.35  
N5  1  IGA  74  27  1.94 


Dijkstra/PSO  22 

979261.8  21.17/7.79 
From Table
This study solved the problem of truck route optimization in openpit mines under the influence of a fluctuation in resistance. A mixedinteger programming model was formulated, and timevarying parameters were estimated and used to simulate the characteristics of the timevarying rolling resistance in different segments. To solve the problem of computational complexity caused by the addition of timevarying characteristics, some effective improvement strategies based on stochastic theory were proposed, and an improved genetic algorithm was used to solve the model. Finally, the validity of the algorithm was verified through numerical experiments.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors have no conflicts of interest to declare regarding the publication of this paper.
This work was supported by NSFC (Project 51304104); Liaoning Province Ordinary University Academic Leaders Fund (Project LJYL038); and Liaoning Research Center of Coal Resources Safe Mining and Clean Utilization, School of Mining, Liaoning Technical University (Project TU15KF07).