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Abstract

In this article, a novel Hybrid method based on particle swarm optimization and ant colony optimization (HPA) is proposed to forecast automobile sales of China. Proposed hybrid model integrating particle swarm optimization and ant colony optimization for estimation of Chinese automobile sales (HPAE) is the first model using aforementioned meta-heuristic techniques. HPA is developed for automobile sales forecasting using highway mileage, gross domestic product, automobile ownership, and consumer price index. HPAE is developed in two forms including both Linear (HPAEL) and Quadratic (HPAEQ). Then, in order to show the accuracy of the algorithm, a comparison is made with ant colony optimization and particle swarm optimization. The results show that mean absolute errors of the HPAE model are the best among them, and better-fit solutions are provided by quadratic form (HPAEQ) due to fluctuations of the influential indicators in China.

Keywords

Particle swarm optimization ant colony optimization automobile sales forecasting hybrid meta-heuristic method

Introduction

Forecasting sales and demand is crucial for the automobile industry since it always takes a relatively long time for the development and production processes. Only with the help of accurate predictions, firms can perfect market performance, reduce profit loss, and plan manufacturing processes more efficiently. Inaccurate forecasts may give rise to several negative consequences, such as overstocking or shortage of production supplies, high costs for different workforce activities, loss of reputation for the manufacturer, and even bankruptcy. Sales forecasting^1–3 is considered as a realistic assessment of expected future demand.⁴

In the past 30 years, automobile sales forecasting has received notable attention. To improve the forecasting accuracy, many methods are usually used by researchers around the word.^5,6 Fantazzini and Toktamysova⁷ proposed new multivariate models using economic variables and Google online search data to forecast monthly automobile sales. Sa-ngasoongsong et al.⁸ presented a structural relationship identification methodology to multi-step forecast sales. Li et al.⁹ forecasted the automobile carbon emissions based on rough set theory. In addition, both data mining and the time series analysis methods have been widely used. For example, Wang et al.¹⁰ developed an adaptive network–based fuzzy inference system (ANFIS) to forecast automobile sales. Hülsmann et al.¹¹ applied data mining algorithms to model German and US automobile markets. Brühl et al.¹² proposed an automobile sales forecasting model for the German market using both time series analysis and data mining methods. Hülsmann et al.¹³ presented the general automobile sales forecasting models based on both time series analysis and data mining techniques. For automobile sales forecasting, regression is also a kind of methods which is frequently used. Lin et al.¹⁴ predicted automobile sales in China using support vector regression (SVR). Kitapcı et al.¹⁵ analyzed the effect of economic policies to automobile sales in Turkey using both multiple regression and neural network analysis. Jun and Qing¹⁶ optimized the weights and thresholds of structure of the back propagation (BP) neural network by genetic algorithm to automobile sales forecasting. A summary of automobile sales forecasting studies is given in Table 1.

Table 1.

Summary of automobile sales forecasting studies.

Method used	Author(s)
Google online search data	Fantazzini et al.
Structural relationship identification methodology	Sa-ngasoongsong et al.
Rough set theory	Li et al.
Adaptive network–based fuzzy inference system (ANFIS)	Wang et al.
Data mining algorithms	Hülsmann et al.
Time series analysis and data mining methods	Brühl et al.Hülsmann et al.
Support vector regression (SVR)	Lin et al.
Multiple regression and neural network analysis	Kitapcı et al.
BP neural network structure based on genetic algorithm	Tang et al.

BP: back propagation.

Based on the literature, most proposed models for automobile sales forecasting applied a certain structure of economic theory to econometric approaches. But that does not necessarily mean that the underlying processes of automobile sales will follow these structures. It is known that automobile sales are mainly influenced by factors such as gross domestic product (GDP), consumer price index (CPI), oil prices, gas prices, highway mileage, and automobile ownership. However, with the development of economy and the costs of automobile reduced, the price factors are no longer sensitivity factors hampering the automobile consumption in China. In addition to these macroscopic factors, many microcosmic factors affected customer behavior in the automobile market, such as aesthetics,¹⁷ form, function, and brand,¹⁸ can also influence automobile sales at segment levels but not so in determining changes of automobile sales at aggregate levels.

This article presents the automobile sales forecasting model which applied hybridized particle swarm optimization and ant colony optimization for estimation of automobile sales (HPAE) and have both linear form and quadratic form. In the following section, the selection of indicators is briefly described. The basic PSO and ACO algorithms are given in section “PSO and ACO algorithms.” In section “Hybrid method based on PSO and ACO (HPA),” proposed hybrid algorithm is explained. In section “Forecasting model of automobile sales,” results obtained by automobile sales forecasting model (HPAE) and comparisons with PSO and ACO are presented. Finally, the conclusion of the study and the suggestions for future researches are given in section “Conclusion.”

Data

In this article, the sequence of annual automobile sales is the main time series, as reported in industry sources, during the period of 2000–2014. The dataset consists of automobile sales and selected indicators. As discussed in the previous section, the selection of indicators is intended to improve automobile sales forecasting. We conclude that the influential indicators must be selected to reveal a prominent relationship between automobile sales and indicators. In this article, the properties considered for the selection of indicators are as follows:

Indicators that characterize the influences of measures of traffic concerning the demand behavior in the automobile industry;

Indicators that influence the saturation degree of the automobile market;

Indicators that represent the national economy and changes in the economic cycle.

As the same time, all aspects of modeling issues in selecting the variables, such as multicollinearity, over-parameterization, and model restriction are considered. From the literature and preliminary tests as well as expert advisement, four annual influential indicators are selected: highway mileage, GDP, automobile ownership, and CPI. Table 2 summarizes the selected economic indicators for automobile sales forecasting. So, after the aforementioned selection criteria are considered, these four indicators are used to model to forecast automobile sales better. The data of automobile sales and four influential indicators between 2000 and 2014 are given in Figure 1.

Table 2.

Summary of variables.

Variable	Source	Explanation
Highway mileage	The National Bureau of Statistics	The total mileage of highway in the nation
GDP	The National Bureau of Statistics	Market value of all goods and services produced within a country
Automobile ownership	The National Bureau of Statistics	The number of resident vehicles in our country
CPI	The National Bureau of Statistics	The growth percentage of the price of the goods on the market

GDP: gross domestic product; CPI: consumer price index.

Figure 1.

The data of automobile sales and four influential indicators between 2000 and 2014.

PSO and ACO algorithms

PSO

PSO based on natural flocking and swarming behavior of insects and birds is one of the recent famous meta-heuristic techniques. The potential solutions called as particles, move through the problem space by following the current optimum particle in PSO. All particles determine the adaptive value by a fitness function to determine the location of the stand or fall at present. Each particle must be endowed with memory function. A resultant of three vectors defines the movement of each particle. One of them is a random vector known as velocity vector, and the two other vectors are created using the best locations experienced by the particles so far.^19–22

A memory of its previous best position that is called pbest is kept by each particle, and the best value is called gbest which is a global best. The velocity updating formula of particle i in the first d dimension is as follows

v_{i d}^{k} = {w v}_{i d}^{k - 1} + c_{1} r_{1} (p b e s t_{i d} - x_{i d}^{k - 1}) + c_{2} r_{2} (g b e s t_{d} - x_{i d}^{k - 1})

(1)

The position updating formula of particle i in the first d dimension

x_{i d}^{k} = x_{i d}^{k - 1} + v_{i d}^{k - 1}

(2)

where $c_{1}, c_{2}$ are acceleration constant, which determine the relative influence of the social and cognitive components; $r_{1}, r_{2}$ are two random numbers, which uniformly are distributed in the interval [0, 1].

In proposed PSO, equation (3) is used to obtain the w value. Indeed, inertia weight is an important factor that makes the algorithm convergent. The convergence is a very important target in any swarm-based technique.²³ Inertia function is obtained and used as follows

w = t - (\frac{i}{t})

(3)

where t is the maximum iteration number and i is the current iteration index. According to the above explanations and equations, the flow chart of PSO is given in Figure 2.

Figure 2.

The flow chart of particle swarm optimization.

ACO

In 1997, Dorigo and Gambardella²⁴ introduced the ACO that based on the foraging behavior of real ant colonies. If a food source is found by an ant, then it carries a portion of the food to the nest. At the same time, the ant leaves some pheromone on the way back which is known as the pheromone trail. All the other ants may act in the same way due to the pheromone on the way that is the most. The amount of the pheromone deposited on the ground completely relies on the quality and quantity of the source discovered.^25–28

The quantity of the pheromone (τt) is intensified around the best objective value obtained throughout the simulation run in the continuous ACO algorithm. The location of the kth ant in the solution space is presented in the following equation

X_{t + 1}^{k} = X_{t}^{g b e s t} \pm \partial X (t = 1, 2, 3, \dots I)

(4)

where ∂X is a random vector generated from $[- α, α]$ to determine the allowed variation. The length of this jump (variation) is obtained from equation (5) at the end of $\sqrt{T}$ iterations^29,30

α_{t + 1} = 0.1 \times α_{t}

(5)

The $\pm sign$ in equation (4) determines the direction of variation from $X^{g b e s t}$ which is decided based on the ${\bar{X}}^{g b e s t}$ parameter using the objective function f(x) in the following equation

{\bar{X}}^{b e s t} = X^{g b e s t} + (X^{g b e s t} \times 0.01)

(6)

if $f ({\bar{X}}^{b e s t}) \leq f (X^{g b e s t})$ , sign (+) is used, otherwise sign (−) is used in equation (4). Equation (7) is presented to simulate the evaporation of the pheromone, and equation (8) shows the increment of the pheromone around the best objective value obtained in each iteration

τ_{t} = 0.1 \times τ_{t - 1}

(7)

τ_{t} = τ_{t - 1} + (0.01 \times f (X^{gbest}))

(8)

According to the above explanations and equations, Figure 3 shows the flow chart of ACO.

Figure 3.

The flow chart of ant colony optimization.

Hybrid method based on PSO and ACO (HPA)

HPA combines capabilities of both PSO and ACO techniques. The hybrid technique of ACO and PSO will combine the advantages of both algorithms. The two best solutions are compared and then the better one is assigned to the global best solution of system in each iteration. Both the algorithms are capable of dealing with non-convex problems such as forecasting and estimation.^31–38 PSO and ACO are motivated generating new solutions using the system solution parameters transferred to their global best. While pheromone trail is only updated around the best solution obtained in ACO, HPA are synchronous.

Originally, particles and ants are randomly distributed to the space for searching or may be set same values in HPA algorithm. While swarm and colony start showing individual behaviors, they begin to be influenced by the system global best during any iteration. Then, while equations (1)–(3) are used by the particles to close to the global minimum, equations (4)–(8) are used by the ants to reach to the global minimum. The best solutions obtained in PSO and ACO are compared during any iteration. Where pgbest and xgbest are the best solutions of swarm and colony, Table 3 shows interaction between swarm and colony.

Table 3.

Interaction between swarm and colony.

P^{g b e s t} \leq X^{g b e s t}

Then

X^{g b e s t} = P^{g b e s t}

Update the direction of ants using equation (6)

Else

P^{g b e s t} = X^{g b e s t}

Update the direction of particles using equation (2)

If the best solution of ACO is influenced by the best solution of PSO, its movement direction must be updated. Thus, the direction operator must be kept up to date during the iterations by being used equation (6). HPA has two main algorithmic parts which are initialization and solution manipulation. According to previous explanations and equations, Table 4 shows the initialization phase, and Table 5 shows solution manipulation phase of the algorithm of HPA.

Table 4.

Initialization of HPA.

Initialization

Stopping criteria is set to

n \times \sqrt{T}

Use

α_{i n i t i a l}

to generate

\partial X

vectors

For all ants in the colony

Generate the

X_{i n i t i a l}^{k}

vectors (k = 1, 2, …, m)

evaluate the location of each ant using the objective function

f (X_{i n i t i a l}^{k})

determine

X^{g b e s t}

vector by using

f (X_{i n i t i a l}^{k})

solutions

For all the particles in the swarm

Generate the

P_{i n i t i a l}^{k}

vectors (k = 1, 2, …, m)

evaluate the location of each particle by using the objective function

f (P_{i n i t i a l}^{k}) (k = 1, 2, \dots, m)

determine the self-local best solutions according to

f (P_{i n i t i a l}^{k}) (k = 1, 2, \dots, m)

{L B E S T}_{i n i t i a l}^{k} (k = 1, 2, \dots, m)

determine

P^{g b e s t}

vector by using

f (P_{i n i t i a l}^{k})

solutions

Apply the method (Table 3) to obtain the best solution of swam and colony

Determine the movement direction of each ant by using equation (6)

Update pheromone by using equations (7) and (8)

Table 5.

The solution manipulation of the HPA algorithm.

For k = 1 to T

For i = 1 to

t \times \sqrt{T}

For all ants

Generate

\partial X

random vector within

[- α, α]

range

Calculate new position of each ant using equation (4)

End

Determine

X_{i}^{b e s t}

vector

f (X_{i}^{b e s t}) \leq f (X^{g b e s t})

then

X^{g b e s t} = X_{i}^{b e s t}

For all particles

Calculate new position of each particle using equations (1) and (2)

End

For all particles

Determine parameters of local best solutions

L B E S T^{k} (k = 1, 2, \dots, m)

End

Determine

p_{i}^{b e s t}

vector

f (p_{i}^{g b e s t}) \leq f (p^{g b e s t})

, then

p^{g b e s t} = p_{i}^{b e s t}

Apply method (Table 3) to obtain the best solution of swam and colony

Update pheromone by using equations (7) and (8)

End

α_{t + 1} = 0.1 \times α_{t}

End

Forecasting model of automobile sales

In this section, the model and result are introduced in detail. We first describe the establishment of the model and its performance. Finally, the comparison of the hybrid algorithm and other methods is given.

The establishment of the forecasting model

Four indicators and automobile sales of China between 2000 and 2014 are shown in Table 6. The data are collected from the National Bureau of Statistics and the China Association of Automobile Manufacturers. According to Figure 4, it can be seen that there is almost a linear relationship between four indicators and automobile sales, respectively. However, linear methods also contain many limitations. For example, some potentially useful relations are easy to be disregarded if only linear method and linear models also contain fewer coefficients. After considering these issues, the automobile sales forecasting based on four indicators was modeled using various forms in this study. Hybrid model integrating particle swarm optimization and ant colony optimization for estimation of Chinese automobile sales in linear form (HPAEL) can be expressed as follows

S_{l i n e a r} = \sum_{i = 1}^{4} α_{i} X_{i} + β

(9)

and Hybrid model integrating particle swarm optimization and ant colony optimization for estimation of Chinese automobile sales in quadratic form (HPAEQ) can be expressed as follows

\begin{matrix} S_{q u a d r a t i c} = α_{1} X_{1} + α_{2} X_{2} + α_{3} X_{3} + α_{4} X_{4} \\ + α_{5} X_{1} X_{2} + α_{6} X_{1} X_{3} + α_{7} X_{1} X_{4} \\ + α_{8} X_{2} X_{3} + α_{9} X_{2} X_{4} + α_{10} X_{3} X_{4} \\ + α_{11} X_{1}^{2} + α_{12} X_{2}^{2} + α_{13} X_{3}^{2} + α_{14} X_{4}^{2} + α_{15} \end{matrix}

(10)

Table 6.

Automobile sales, highway mileage, GDP, automobile ownership, and CPI.

Years	Automobile sales $(1 0^{4})$	Highway mileage $(1 0^{4})$	GDP $(1 0^{12})$	Automobile ownership $(1 0^{6})$	CPI (last year = 100)
2000	208.86	167.98	99.7763	160.891	100.40
2001	236.36	169.8	110.2704	180.204	100.70
2002	324.81	176.52	121.002	205.317	99.20
2003	439.08	180.98	136.5646	238.293	101.20
2004	507.11	187.07	160.7144	269.371	103.90
2005	575.82	334.52	185.8958	315.966	101.80
2006	721.6	345.7	217.6566	369.735	101.50
2007	879.15	358.37	268.0194	435.836	104.80
2008	938.05	373.02	316.7517	509.961	105.90
2009	1364.48	386.08	345.6292	628.061	99.30
2010	1806.19	400.82	408.903	780.183	103.30
2011	1850.51	410.64	484.1235	935.632	105.40
2012	1930.64	423.75	534.123	1093.309	102.60
2013	2198.41	435.62	588.0188	1267.014	102.60
2014	2349.19	446.39	636.1387	1459.811	102.00

GDP: gross domestic product; CPI: consumer price index.

Figure 4.

The relationship between four factors and automobile sales.

HPAE model optimizes coefficients $(α_{i}, β)$ of the four indicators $(X_{i}, (i = 1, 2, 3, 4))$ , which are highway mileage, GDP, automobile ownership, and CPI, respectively.

PSO algorithm uses fitness function as reference, optimizing prediction model, and finding the optimal estimation.^39–43 The following fitness function is established by minimizing the mean absolute error between the observed and estimated values⁴⁴

f = \frac{1}{N} \sum_{i = 1}^{N} | S_{A c t} (i) - S_{E s t} (i) |

(11)

where $S_{A c t}$ and $S_{E s t}$ are the actual and predicted automobile sales, respectively; i is the number of observations; and N is the sample size.

The performance of the forecasting model

HPAE models (HPAEL and HPAEQ) are developed to the future automobile sales forecasting based on highway mileage, GDP, automobile ownership, and CPI (Table 6). The HPAE model is coded with MATLAB. Setting the optimal parameters of PSO and ACO is also important. The values of c1 and c2 are set to be 2 according to Shi and Eberhart⁴⁵ who performed an extensive study for finding proper values. Hence, these fixed values are also used in this study. And the other parameters are considered as follows: number of particles (m): 20; number of ants (n): 40; inertia weight (w): 1; number of maximum iteration (I): 100,000. The data (2000–2009) are used to determine the coefficients of HPAE models. Following HPAE (linear and quadratic) equations are obtained for automobile sales forecasting. In the linear form, coefficients obtained are given below

\begin{matrix} S_{l i n e a r} = 0.031015 X_{1} + 4.27102 X_{2} \\ - 0.12801 X_{3} + 1.40062 X_{4} - 324.644 \end{matrix}

(12)

f_{l i n e a r} = 54.49

In the quadratic form, coefficients obtained are given below

\begin{matrix} S_{q u a d r a t i c} = 17.98229 X_{1} - 25.1556 X_{2} - 0.56399 X_{3} \\ - 9.08862 X_{4} + \\ 0.064811 X_{1} X_{2} - 0.00943 X_{1} X_{3} \\ - 0.20868 X_{1} X_{4} + 0.08177 X_{2} X_{3} \\ + 0.252951 X_{2} X_{4} + 0.035843 X_{3} X_{4} \\ - 0.01082 X_{1}^{2} - 0.1319 X_{2}^{2} \\ - 0.01588 X_{3}^{2} + 0.073538 X_{4}^{2} + 238.338 \end{matrix}

(13)

f_{q u a d r a t i c} = 11.73

where $X_{1}$ is highway mileage, $X_{2}$ is GDP, $X_{3}$ is automobile ownership, $X_{4}$ is CPI, and f is mean absolute error. The data (2010–2014) are used to validate the models. Table 7 shows relative errors between estimated and observed data. According to Table 7, proposed HPAE approach for automobile sales forecasting in China is successful and robust. It can be seen that the better-fit forecast can be provided by quadratic form than the linear form because of the fluctuations of the economic indicators in China.

Table 7.

Automobile sale estimation of proposed models between 2010 and 2014.

Years	Observed sale	Estimated sale		Relative errors (%)
Years	Observed sale	Linear	Quadratic	Linear	Quadratic
2010	1806.19	1479.03	1803.16	−18	0.17
2011	1850.51	1783.65	1849.19	3.61	0.07
2012	1930.64	1973.50	1958.77	−2.21	−1.46
2013	2198.41	2182.82	2172.39	0.71	1.18
2014	2349.19	2362.15	2355.28	−0.55	−0.26

The comparison of the proposed hybrid algorithm and other methods

When 5-year data are considered (2010–2014), proposed approach finds less relative error than the single method in both linear form and quadratic forms. Tables 8 and 9 give coefficients and forecasting relative errors of each method in linear form and quadratic forms, respectively. And the comparisons between proposed HPAE model, PSO, and ACO are shown in Figures 5 and 6. It exposes that proposed HPAE model is providing better-fit estimation than ACO and PSO whether in linear form or quadratic form.

Table 8.

Comparisons of coefficients and mean absolute error in linear form.

Coefficients	HPAEL	ACO	PSO
$α_{1}$	0.031015	0.046101	0.052015
$α_{2}$	4.27102	4.19152	4.27152
$α_{3}$	−0.12801	−0.0999	−0.10801
$α_{4}$	1.40062	1.436636	1.447626
$α_{5}$	−324.644	−340.903	−324.644
Mean absolute error	54.49	55.898	56.27

ACO: ant colony optimization; PSO: particle swarm optimization.

Table 9.

Comparisons of coefficients and relative errors in quadratic form.

Coefficients	HPAEQ	ACO	PSO
$α_{1}$	17.98229	18.08329	18.00604
$α_{2}$	−25.1556	−24.9866	−25.0799
$α_{3}$	−0.56399	−0.57299	−0.57379
$α_{4}$	−9.08862	−9.14963	−9.14859
$α_{5}$	0.064811	0.063999	0.063879
$α_{6}$	−0.00943	−0.009399	−0.009387
$α_{7}$	−0.20868	−0.20799	−0.20799
$α_{8}$	0.08177	0.081769	0.08177
$α_{9}$	0.252951	0.25199	0.25194
$α_{10}$	0.035843	0.03584	0.03584
$α_{11}$	−0.01082	−0.01083	−0.01083
$α_{12}$	−0.1319	−0.1312	−0.1312
$α_{13}$	−0.01588	−0.01599	−0.01588
$α_{14}$	0.073538	0.073449	0.07345
$α_{15}$	238.338	229.447	230.985
Mean absolute error	11.73	14.47	23.35

ACO: ant colony optimization; PSO: particle swarm optimization.

Figure 5.

Comparisons of automobile sales in linear form.

Figure 6.

Comparisons of automobile sales in quadratic form.

Conclusion

Accurate automobile sales forecasting is critical in the automobile industry. With the evolution of intelligent technology, meta-heuristic techniques on automobile sales forecasting have been popular methods to forecast better. And it can lead to a higher quality result with a faster convergence profile by the hybridization of the two algorithms to optimize the forecasting model. So, automobile sales forecasting with a hybrid model based on PSO and ACO is adopted in this article. There are a lot of factors affecting automobile sales of China; among the factors, four key indicators considered here are highway mileage, GDP, automobile ownership, and CPI. In order to meet the fluctuations of automobile sales, there are two forms (linear and quadratic) of the HPAE model developed. In order to show the availability and advantages of proposed approach, 15-year data (2000–2014) are collected, in which 10-year data (2000–2009) are used to model, and the 5-year data (2010–2014) are calculated to validate the accuracy of results. Three optimization methods are compared to automobile sales forecasting in the years 2000–2014 using the two forms of the HPAE. The methodology of hybridization is extensively explained, and the results show its capability in providing a robust forecasting model. The empirical results indicate that the accuracy of the proposed approach to the automobile sales forecasting of China is acceptable and better than PSO and ACO.

Although two kinds of model have been provided to the automobile sales forecasting in this study, researching the combination of different meta-heuristic methods also will be necessary in the future. In addition, the number of economic indicators of the model proposed is limited, so the suggestion for next work is to take more economic indicators into consideration to the automobile sales forecasting.

Footnotes

Academic Editor: Tao Feng

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by Humanity and Social Science Youth foundation of Ministry of Education of China 13YJCZH042 and Specialized Research Fund for the Doctoral Program of Higher Education of China 20120041120018.

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A hybrid optimization approach to forecast automobile sales of China

Abstract

Keywords

Introduction

Data

PSO and ACO algorithms

PSO

ACO

Hybrid method based on PSO and ACO (HPA)

Forecasting model of automobile sales

The establishment of the forecasting model

The performance of the forecasting model

The comparison of the proposed hybrid algorithm and other methods

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References