Research on the Mathematical Model for Optimal Allocation of Human Resources in the Operation and Maintenance Units of a Heavy Haul Railway

: According to the existing personnel structure, quantity, development strategy, and market demand of the Shuohuang Railway Company’s operation and maintenance project, the demand quantity of various employees of the company for the past three years is predicted, and a human resource optimization model based on existing human resources and future plans is established. Then, the optimal solutions of the two mathematical models were calculated and analyzed using LINGO software. Finally, combined with the actual situation, the optimal allocation of human resources for the operation and maintenance project of KY company was obtained. The following conclusions are obtained. (1) For the optimal allocation model of existing human resources, the maximum net proﬁt of the optimal stafﬁng model is CNY 3258000. (2) The human resources allocation cost of the minimum dismissal model is CNY 81000. (3) The human resources allocation cost of the lowest cost model is CNY 15500. The research results can effectively guide the human resource management of the operation and maintenance project of the Shuohuang Railway Company, and have important theoretical and practical signiﬁcance for further analysis of human resources model and its optimal allocation method.


Introduction
Human resource management is critical in project management and has an important impact on the competitiveness and overall development of enterprises. The value of human resources is significantly reflected in the process of allocating human resources [1][2][3][4][5].
In the development and competition of enterprises, human resource management has a great impact. Moreover, the level of human resource management ability is also directly related to the economic benefits of enterprises and projects. In order to generate increased profits for enterprises and projects, human resources should apply adjustments in the recruitment system, dismissal system, training system, and evaluations [6][7][8][9][10][11][12][13][14][15]. Human resource allocation is the management and control of enterprise human resources based on project organization, with the aim of improving efficiency and maximizing the value of human resource management. Human resource allocation should therefore aim at enhancing the effect of human resource management and ensuring a match between personnel ability and position.
The concept of human resource management is derived from traditional personnel management. The Industrial Revolution gave birth to personnel management activities. After 1970, countries all over the world began to change their management ideas. The concept of personnel management was gradually eliminated, and the concept of human resource management was gradually formed. In combing through the research conclusions of studies, it is found such research has experienced the following stages of development [15][16][17][18][19][20].
Bakke E. W. published the book Human Resources Function in 1958, which elaborated on issues related to the management of human resources. He discussed the management of human resources as a general function of management [21]. Ferguson (1960) [22] summarized the development of human resource management in the United States. Farmer (1958) [23] traced the practice of human resource management in Britain. In 1953, Hanel analyzed the impact of the political atmosphere of a country where an enterprise is located on the theory and practice of human resource management [24]. Cicourel (1958) believes that both formal and informal organizational factors within an organization have an important impact on human resource management [25].
In the mid-1970s, many scholars equated human resource management with personnel management. As Robbins described in his book Personnel, the management of Human Resources, today, personnel management aims to study the organization's human resources and how to make them serve the organization's goals more effectively [26]. Henneman, Schwab, Fossum, and Dyer used the term "Personnel/Human resource management" in their papers and equated human resource management with personnel management [27].
In the 1980s, the research on human resources and its management achieved fruitful results.
The most influential and strategic human resource management theory was put forward by Beer and others in their book Managing Human Capital in 1984. Beer and others believed that human resource management policies should be selected according to four aspects: (1) employee influence; (2) human resource flows, both within and outside the organization; (3) remuneration systems; and (4) working systems [28]. Fombrun (1984) [29] paid great attention to the importance of strategic means. That is to say, the company's strategy and human resource management's strategy are related in many aspects, such as structure, culture, and employees resources and development.
Around the definition of the concept, essence, scope, and objectives of human resource management, Walton (1985) [30] proposed a human resource management policy: clarifying common objectives, mutual influence, mutual respect, mutual reward, and mutual responsibility. Storey (1992) [31] analyzed the internal characteristics of human resource management and believed that there are 27 differences between personnel management and human resource management. He divided the 27 differences into three categories: beliefs and assumptions; strategic areas; and importance. Kamoche (1996) [32] studied the human resource management capability of enterprises, while Schuler and Huber (1993) [33] studied the purpose, behavior, and role of human resource management.
With the deepening of human resource management research, Legee (1989) [34] linked culture with human resource management; Legee believes that the real focus of human resource management is organizational culture as well as its change and management. Culture is closely connected with organizational strategy and structure and also has an impact on employee recruitment, selection, evaluation, training, and reward. Moorhead and Griffin (1992) [35] believe that organizational culture is a set of values that helps people in organizations understand which behaviors are acceptable and which are unacceptable. These values often circulate among employees in stories or other symbolic ways. Ouchi (1981) [36] attributed Japan's economic success in the early 1980s to the combination of the American and Japanese models in his Z theory, specifically the emphasis on people, long-term employment guarantee, being based on value sharing and collective decisionmaking, the participation of "small groups" and the social pressure on performance, high trust in the final judgment of managers, and the nonspecialized career promotion path. The final result is mutual obligation; that is, employees respond to the employer's commitments to them with a sense of responsibility to the organization. Peters and Waterman (1982) [37] focused on the relationship between American organizational culture and performance when analyzing various successful management models.   [38][39][40][41][42] analyzed the factors influencing land-leasing based on mathematical models.
The quantitative analysis and optimal allocation model of human resources has been recognized by increasing numbers of researchers and project managers. At the macro level, the value of human resources has begun to be reflected in the national strategic level, and China has successively implemented numerous human resources development policies. At the micro level, human resource planning, development, training, and management have become the basic content of project organization and management, and the effect of human resource management is more ideal.
It is of great significance to carry out a quantitative analysis of human resources and research into human resource optimization allocation to improve the enterprise efficiency and competitiveness of the Shuohuang Railway Company's operation and maintenance project in the past three years based on the following rationales: (1) Helping the project improve the effect of human resource management and improving the comprehensive competitiveness of the enterprise. The current project is facing fierce competition and a more complex competition pattern. It is impossible for enterprises to realize long-term competitive advantage based on one kind of competitiveness. Following its emergence, the project's competitive advantage will be constantly imitated and then gradually disappear. Competitive advantage formed on the basis of core human resources, however, is unique and therefore difficult for other enterprises to copy and surpass. Therefore, the project needs to establish the mechanisms for the right human resources management; (2) Achieving the profit maximization effect, based on cost minimization. First, in the process of staff allocation, we should consider the ability and experience of employees to ensure that the results after distribution can support the realization of the personal value of employees. Under the control of human resources, employees can communicate in a timely manner and flexibly adjust their work, thus effectively reducing the operating cost; (3) Improving the skills and quality of employees to promote the technological renewal of enterprises. Circulation of information together with improving the technical ability of employees can be realized through the exchange of information between employees in the project. This is very beneficial for the mining of staff skills and abilities, which can help the enterprise to continuously develop.
The method based on mathematical models can characterize the quantitative distribution of all kinds of human resources and thus fulfill the objective and serve as an effective decision-making basis for enterprise managers.
The main goal of this paper is to create the maximum net profit according to the existing human resources. Then, according to the human resource planning in the next three years, the optimal allocation scheme of the minimum dismissal model and the minimum cost model is obtained.
Therefore, this paper analyzes the optimal allocation of human resources under the existing human resources conditions by establishing an optimal allocation model based on the existing human resources and the dynamic optimization model based on the future plans of human resources. Furthermore, the optimal allocation of human resources is considered under conditions of either minimum dismissal (when the Shuohuang Railway Company's operation and maintenance project benefit is high) or lowest cost (when the Shuohuang Railway Company's operation and maintenance project benefit is low) and can effectively improve the utilization efficiency, transportation efficiency, and profit rate through the Shuohuang Railway Company's human resources.
Because there are many uncertainties in the future transportation conditions, the possible changes of transportation elasticity [43,44] and external costs [45,46] are not considered in this paper.
The main methods adopted in this paper are: (1) Literature induction Collect, analyze, and study relevant literature on the optimal allocation of human resources at home and abroad; carefully sort out, analyze, summarize, and select the theory and method of optimal allocation of human resources related to the paper to achieve the purpose of investigation and research; (2) Questionnaire survey method Through the questionnaire survey method, collect and sort out the planning data of the transportation project of the Shuohuang Railway Company in terms of post establishment, recruitment, dismissal, training, etc., as the material of this quantitative study; (3) Mathematical modeling method According to the existing human resources and future plans of the transportation project of the Shuohuang Railway Company, a mathematical model for the optimal allocation of existing human resources in the four bridge and tunnel operation and maintenance units of the transportation project of the Shuohuang Railway Company, as well as the minimum dismissal and costs in the next three years, is established. Then, Lingo software is used to analyze and calculate the optimal solution of the two mathematical models, and the optimal allocation of human resources in the transportation project of the Shuohuang Railway Company is obtained based on the actual situation.

Current Human Resource Allocation Overview
This section takes the four bridge and tunnel operation and maintenance units of the Shuohuang Railway Company as an example to optimize human resource allocation. There are 328 managers, senior workers, intermediate workers, and junior workers. According to the quantitative analysis in the above sections, the average salary level distribution of all types of personnel is shown in Table 1. The four bridge and tunnel operation and maintenance units A, B, C and D of the Shuohuang Railway Company are located in Yuanping, Dongye, Xiaojue, and Suning, respectively. Due to the different economic conditions and freight volume in the four regions, the contribution of different types of personnel to the average transportation benefits of each bridge and tunnel operation and maintenance unit differs, as shown in Table 2.

Model Hypothesis
(1) All types of personnel are paid a fixed salary every month; (2) The four bridge and tunnel operation and maintenance units are working every day, and there is no suspension of the bridge and tunnel operation and maintenance units; (3) No additional temporary recruiters; (4) The same person is not considered as two bridge and tunnel operation and maintenance units at the same time; (5) The influence of various factors on the bridge and tunnel operation and maintenance unit and staff is excluded; (6) This paper does not consider many of the uncertainties of future transportation conditions, including possible changes in transportation elasticity and external costs.

Model Construction
(1) Definitions A, B, C, D: the operation and maintenance unit of each bridge and tunnel; x ij : the number of the i-th type personnel required for the j-th bridge and tunnel operation and maintenance unit; c ij : the average transportation benefit of the i-th type personnel assigned to the j-th bridge and tunnel operation and maintenance unit; R: the direct income of four bridge and tunnel operation and maintenance units; Q: the total wages paid to 328 personnel per month; W: the direct transportation benefits, W = R − Q; W 1 : the direct income of the fixed part; W 2 : the direct income from the assignment; ρ ij : the net profit generated by the i-th type personnel as the j-j-th bridge and tunnel operation and maintenance unit to the company; k i : the salary paid by the four bridge and tunnel operation and maintenance units to the i-th type personnel.
(2) Model preparation First, we make a rough estimate of the model. Suppose ρ ij represents the net profit brought to the company when the i-th type person assigned to the j-th bridge and tunnel operation and maintenance unit. k i represents the monthly salary the company gives to its personnel. c ij represents the average transportation benefit created by the i-th type personnel as the j-th bridge and tunnel operation and maintenance unit. Then: From this, we come to the values of ρ ij shown in Table 3. (3) Integer programming model Through careful analysis of the problem, it can be described through the integer programming model: i = 1, 2, 3, 4 refers to the personnel type, namely 1-management personnel, 2-senior worker, 3-intermediate worker, 4-junior worker; j = 1, 2, 3, 4 is the type of bridge and tunnel operation and maintenance unit, where 1-bridge and tunnel operation and maintenance unit A, 2-bridge and tunnel operation and maintenance unit B, 3-bridge and tunnel operation and maintenance unit C, 4-bridge and tunnel operation and maintenance unit D; x ij (i = 1, 2, 3, 4; j = 1, 2, 3, 4) represents the number of i-th class personnel required by the j-th bridge and tunnel operation and maintenance unit; c ij represents the benefit standard of i-th class personnel required by the j-th bridge and tunnel operation and maintenance unit; Q is the total monthly wage; R is the total monthly income of the four bridge and tunnel operation and maintenance units. The net profit created by the company is W = R − Q. The total salary is Q = 8000 × 16 + 6000 × 64 + 5000 × 156 + 4000 × 92 = CNY 1660000.
The integer programming model with maximum benefit can be established as follows: According to the overall transportation benefits of each bridge and tunnel operation and maintenance unit of the Shuohuang Railway Company's operation and maintenance project, the structural restrictions of various personnel in each bridge and tunnel operation and maintenance unit can be estimated, as shown in Table 4.

Model Solving
LINGO 11 software (developed by Lindo System Inc. located in Chicago, IL, USA) is used to write and solve the program for the human resources optimal allocation model. The results of the program operation are shown in Figure 1.

Model Solving
LINGO 11 software (developed by Lindo System Inc. located in Chicago, IL, USA) is used to write and solve the program for the human resources optimal allocation model. The results of the program operation are shown in Figure 1.
The optimal personnel structure allocation model of four bridge and tunnel operation and maintenance units of the Shuohuang Railway Company's operation and maintenance project is solved by LINGO software, as shown in Table 5. The maximum net profit of the optimal personnel structure allocation model is CNY 3,258,000.
The variables according to their sensitivity from large to small are as follows: Through the sensitivity test of the model shown in Table 5, it is found that the optimal allocation model of human resources is reasonable, the benefits to transportation are the largest, and the restrictions of personnel structure are fully met.  The optimal personnel structure allocation model of four bridge and tunnel operation and maintenance units of the Shuohuang Railway Company's operation and maintenance project is solved by LINGO software, as shown in Table 5. The maximum net profit of the optimal personnel structure allocation model is CNY 3258000. The variables according to their sensitivity from large to small are as follows: Through the sensitivity test of the model shown in Table 5, it is found that the optimal allocation model of human resources is reasonable, the benefits to transportation are the largest, and the restrictions of personnel structure are fully met.

Human Resources Planning
According to the existing personnel structure, quantity, development strategy, and market demand for the operation and maintenance project of the Shuohuang Railway Company, the demand quantity of various employees of the company in the past three years can be predicted, as shown in Table 6. As the number of senior workers in the operation and maintenance project of the Shuohuang Railway Company is small and remains unchanged, it is only necessary to make adjustments to the intermediate workers and junior workers of the Shuohuang Railway Company through recruitment, retraining, demotion, dismissal, additional recruitment, and recruitment of non-staff personnel, so as to establish the optimal dynamic adjustment scheme of the personnel structure to meet the requirements of the Shuohuang Railway Company. Due to the automatic dismissal of workers or for other reasons, the operation and maintenance project of the Shuohuang Railway Company has the problem of natural attrition. Table 7 shows the natural dismissal rate of the bridge and tunnel operation and maintenance unit of the operation and maintenance project of the Shuohuang Railway Company. As the recruitment cycle is one year, the existing workers have worked for more than one year. (3) Lower level Incompetent workers can be demoted. Although the company does not need to pay fees, 40% of the workers who are demoted will be dismissed; (4) Dismissal The dismissal cost of senior workers is CNY 1000/person, the dismissal cost of intermediate workers is CNY 800/person, and the dismissal cost of junior workers is CNY 500/person; Each type of personnel of the whole company can employ five more than needed to cope with the rapid development environment of the logistics market. The additional cost is CNY 3000/person/month for junior workers, CNY 4000/person/month for intermediate workers, and CNY 6000/person/month for senior workers; (6) Non-staff personnel Junior workers, intermediate workers, and senior workers can each have no more than 50 non-staff personnel to complete the production task of half a person. In this way, the company pays CNY 3000/person/month.

Human Resource Model Analysis
(1) Minimum dismissal model When the Shuohuang Railway Company has high transportation efficiency, the dismissal of employees needs to be subsidized to pay to ensure company humanization. If too many employees are dismissed, this may impact on the company's economy. The first model is unilaterally considered from the number of personnel along with a variety of factors, and the company also adopts methods for training, recruitment, demotion, and non-staff personnel. In the establishment of the mathematical model, all of the factors involved are taken into account as limiting conditions in the optimization data. Demotion will result in the dismissal of 40% of workers, excluding natural dismissal. In summary, the company's requirements for the number of people are gradually increasing, the demand for senior workers and intermediate workers is gradually decreasing, and the demand for junior workers is gradually increasing.
(2) Lowest cost model When the transportation efficiency of the Shuohuang Railway Company is poor, personnel changes are bound to occur due to economic factors. Whether it is through dismissal or non-staff personnel and training, economic expenditure will be incurred. Obviously, under a reasonable scope and arrangement, the establishment of the mathematical model and linear optimization can find the best scheme.

Model Hypothesis
(1) Assuming that the number of workers per year is just the estimated number, recruitment will occur in strict accordance with demand. The maximum number of recruits is the sum of the estimated workforce demand of the current year and the number of overstaffed workers; (2) Assuming that the cost is considered, only recruitment, training, demotion, and dismissal are involved. The salary and workload of all types of work are not taken into account; (3) Demotion does not only consider demotion; that is, senior workers are demoted to intermediate workers, intermediate workers are demoted to junior workers, and senior workers can also be demoted to junior workers; (4) The processes of staff reduction, recruitment, dismissal, retraining, and establishment of non-staff personnel are carried out in a short time at the beginning of each year, and recruitment, demotion, dismissal, and retraining are carried out after natural staff reduction; (5) After adjustment at the beginning of each year, the workforce demand of the Shuohuang Railway Company's operation and maintenance project can be met, and there will be no changes to personnel in the next year; (6) When calculating the workload, a non-staff member is equal to half a regular worker.
When calculating recruitment, retraining, demotion, dismissal, and overemployment dismissal, a non-staff person is calculated as an independent person. Non-staff personnel are the result of the company's transfer of various types of work according to the needs of personnel transfer; (7) All types of workers (senior workers, intermediate workers, and junior workers) have the same opportunities to be dismissed, demoted, promoted, and trained; (8) This paper does not consider many of the uncertainties of future transportation conditions, including possible changes in transportation elasticity and external costs.

Model Building
(1) Definition X ij : j-th type workers employed in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, and 3 represents senior workers). XC ij : j-th type workers dismissed in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, and 3 represents senior workers). XD ij : recruitment of j-th type workers among non-staff personnel in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, 3 represents senior workers). XE ij : additional j-th type workers recruited in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, 3 represents senior workers). XZ ij : j-th type workers recruited in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, 3 represents senior workers).
XP ij : j-th type workers trained in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, 3 represents senior workers). X J ij : j-th type workers demoted in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents junior workers, 2 represents intermediate workers, 3 represents senior workers). XH ij : j-th type workers promoted after demotion in the i-th year (i represents the number of years from 0 to 3; j indicates the type of worker, where 1 represents intermediate workers recovered after demotion, 2 represents senior workers that were promoted to intermediate level after being demoted, 3 represents senior workers recovered after being demoted to intermediate level, and 4 represents senior workers recovered after being demoted to primary level). Workers who have been reinstated do not need additional training.
In the second year, the number of junior workers recruited is XZ 11 , the number of intermediate workers recruited is XZ 12 , the number of senior workers recruited is XZ 13 .
In the third year, the number of junior workers recruited is XZ 11 , the number of intermediate workers recruited is XZ 12 , the number of senior workers recruited is XZ 13 .

Number of workers trained:
In the first year, the number of junior workers trained is XP 11 , the number of intermediate workers trained is XP 12 .
In the second year, the number of junior workers trained is XP 21 , the number of intermediate workers trained is XP 22 .
In the third year, the number of junior workers trained is XP 31 , the number of intermediate workers trained is XP 32 .
The training cost is P = ∑(800 × XP i1 + 1000 × XP i2 ) (i = 1,2,3). 4 Number of workers demoted: X J ij (i represents the number of years from 0 to 3; j indicates the type of worker represented, where 1 represents intermediate workers demoted to junior workers, 2 represents senior workers demoted to junior workers, 3 represents senior workers demoted to intermediate workers). 1, 2, 3). 5 The number of demoted workers who are promoted again: Note: promotion is only for workers who were demoted in the previous year. 6 Number of workers dismissed: The dismissal fee is C = ∑(500 × C i1 + 800 × C i2 + 1000 × C i3 ), (i = 1,2,3). 7 Additional recruitment of all types of worker per year: ∑ XE ij ≤ 4.
The expenses for recruiting non-staff are D = ∑(3000 × XD i ).
(3) Establish objective function: The decision variables are integrated, and the minimum dismissal model and cost minimum model are established for further analysis.

1.
Objective function of minimum dismissal: MI N 2.
Objective function of minimum cost model:

Model Solving
The program of the human resource optimization configuration model is compiled and solved using LINGO software.
(1) Minimum dismissal model The number of dismissals, recruitments, trainings, demotions, and promotions of all types of personnel in the past three years is calculated according to rounding out and rounding, as shown in Tables 8-10. The minimum number of dismissed personnel is 0, and the cost of human resource allocation is CNY 81000.    (2) Lowest cost model The number of dismissals, recruitments, trainings, demotions, and promotions after demotion of various personnel in the past three years are rounded, as shown in Tables 11-13. The cost of human resources allocation is CNY 15500.

Conclusions
In this study, the demand quantity of various employees of the company in the past three years is predicted according to the existing personnel structure, quantity, development strategy, and market demand of the Shuohuang Railway Company's operation and maintenance project, and a human resource optimization model based on existing human resources and future plans is established. The following conclusions are obtained: (1) Optimal allocation model of existing human resources The optimal personnel structure allocation model of bridge and tunnel operation and maintenance unit of the Shuohuang Railway Company's operation and maintenance project is solved using LINGO software, as shown in Table 14. The optimal allocation model of human resources is reasonable, the benefits to transportation are the largest, and the restrictions of personnel structure are fully met. The maximum net profit of the optimal staffing model is CNY 1506000. In the third year, senior workers were demoted to 20 junior workers; (vi) Scheme for promotion and employment of workers after demotion: three workers will be restored after senior workers are demoted to junior workers in the second year. In the third year, 4 senior workers were reinstated after being demoted to intermediate workers, and 43 senior workers were reinstated after being demoted to primary workers.
(3) Optimal allocation scheme of the lowest cost model The human resources allocation cost of the lowest cost model is CNY 15500.
(i) Dismissal plan: dismiss one junior worker whose working ability is not competent for the post requirements in the first year. There is no need to dismiss any category of worker in the second and third years; (ii) Recruitment Scheme: recruit six senior workers in the first year. Recruit four senior workers in the second year. In the third year, there is no need to recruit any type of worker; (iii) Staff training program: a total of 12 intermediate workers will be trained in the first year. Three intermediate workers will be trained in the second year. In the third year, there is no need to train any type of worker; (iv) Additional recruitment scheme: there is no need to recruit any additional category of worker; (v) Scheme for demotion of workers: in the first year, intermediate workers are demoted to six junior workers. In the second year, intermediate workers were demoted to 18 junior workers. In the third year, intermediate workers were demoted to 9 junior workers and senior workers to 30 junior workers; (vi) Scheme for promotion and employment of employees after demotion: 12 employees will be restored after senior workers are demoted to junior workers in the second year. In the third year, 11 intermediate workers were reinstated after being demoted to primary workers, and 64 senior workers were reinstated after being demoted to primary workers.

Discussion
The main goal of this paper is to create the maximum net profit according to the existing human resources. Then, according to the human resource planning in the next three years, the optimal allocation scheme of the minimum dismissal model and the minimum cost model is obtained.
The research results can effectively guide the human resource management of the operation and maintenance project of the Shuohuang Railway Company, and have important theoretical and practical significance for further analysis of the human resources model and its optimal allocation method.
Because there are many uncertainties in the future transportation conditions, the possible changes in transportation elasticity and external costs are not considered in this paper. The authors will consider these uncertainties in future studies.