Journal of Applied Mathematics

Volume 2019, Article ID 2674697, 20 pages

https://doi.org/10.1155/2019/2674697

## Mixed Optimal Scheduling Model of Flexible Service System Based on Inverted Triangle

^{1}Jiyang College, Zhejiang A and F University, 77 Puyang Rd., Zhuji 311800, Zhejiang, China^{2}School of Economics and Management, Beijing Forestry University, 35 Qinghua East Rd., Haidian District, Beijing 100083, China

Correspondence should be addressed to Shipei Hu; nc.ude.naduf@420810180

Received 4 January 2019; Accepted 27 March 2019; Published 2 June 2019

Academic Editor: Frank Werner

Copyright © 2019 Shipei Hu and Yujun Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the study presented in this paper, we built a nonlinear binary integer programming model of a flexible scheduling problem for the Department of Zhejiang Provincial Local Tax Services. One difference between our model and typical ones is that whereas in the latter the number of open windows within each working day is fixed, in our model it is not. We used a variety of integer programming software in an attempt to solve our scheduling model; however, unfortunately we could not find an optimal solution. Thus, we tested all the combinations of different numbers of employees to construct the optimal solution. When we tested our model in the tax office of Lishui City, China, the average waiting time of taxpayers was less than 15 min and the employees working hours were clearly reduced. Thus, a noteworthy improvement in the quality of the service is achieved by the model.

#### 1. Introduction

Together with the change in governmental functions, the reform of the administrative approval system, and the boost of local tax informatization in Zhejiang Province, China, more service items are now provided by the local tax service windows and thus the service quantities have increased rapidly. However, service capabilities have not improved proportionately, because the number of service windows and employees is fixed, which results in a long queue of taxpayers, a decline in the service quality in the tax service halls, and damage to the external image of the local taxation bureau.

The primary cause of the queue is that the production capacity of the windows is insufficient to produce the external inventory and the windows pass the inventory cost of the production process to the taxpayers. In order to alleviate the situation of long queues, at present, the measures taken by the tax service halls management are limited mostly to increasing the input of equipment to reduce the inventory and the taxpayers waiting cost. However, the problems that accompany this simple and crude extension of improvements is that even if there is superfluous investment in various types of equipment and window employees are overequipped, a contradiction still exists between the fixed nature of production capacity and the fluctuation in service demand, which in turn may lead to an increase in the chances of periods occurring when employees are idle and in window service costs. Therefore, the determination of an appropriate arrangement of flexible working hours and window services in tax service halls is the core problem that needs to be solved.

From the perspective of management and service needs, with the aim of handling the feature of great fluctuations in service demand and so that the supply satisfies the taxpayers needs more appropriately, in this study we addressed the dynamic settings of the windows and the flexible working hours of the window employees and attempted to realize the optimal allocation of resources by using a basic queuing model and nonlinear 0-1 integer programming model. The term flexible working hours means that window employees can choose their work place and hours flexibly and autonomously and not according to the same fixed work schedule, on the premise that they have completed their regulated duties. In a flexible service system, the working time and number of window employees are dynamically adjusted by optimizing the personnel combination and implementing flexible work arrangements, according to the changing situation of the flow of taxpayers and the business volume at different time periods.

In order to provide top-quality service to the taxpayers, with the assistance of the taxation bureau of Lishui City we distributed a questionnaire to all the window employees in the city and to nearly two thousand of the taxpayers and collected almost 20000 data items. The analysis of these data items using SPSS software showed that the maximum average waiting time tolerated by taxpayers is 15 min and the average time duration for which window employees can provide high-quality services is one hour. After one hour of efficient working, the psychological and physiological state of the window employees is reflected in different levels of burnout, resulting in a decline in the quality of service and an increase in the work error rate. After the window employees practiced 15 min of stress relaxation, their quality of service returned to normal levels. Therefore, this type of working model is more flexible, can effectively include humanistic care, motivates the window employees, improves window service efficiency and quality of service, and reduces management costs. In general, it has the following advantages.

It enhances the working efficiency of the windows and reduces taxpayers waiting time. After the launch of our integrated management system for tax hall services, a performance evaluation of the window employees is also planned. Work efficiency is the main measure of performance, and the key to improving it is shortening the duration of each business transaction to a reasonable extent. The key factors influencing efficiency are the physical and mental stability of the window employees and the service capacity and level. The implementation of a flexible service system can help distinguish core from noncore work time and ensure that the window employees working in noncore time can rest sufficiently and business training. Thus, they can have sufficient energy to maintain their best working state and thereby their job performance in core work time is improved. The implementation of a flexible service system can also reduce taxpayers waiting time by allowing a sufficient number of windows to be open according to the measured flow of taxpayers and the business volume.

It alleviates the pressure on window employees and meets the requirements related to the background work. The sources of work pressure are various because of the specific characteristics of window work. The first is the fixed schedule, including the same work place and work time throughout the year. The second is the simplistic content; that is, the business operations are relatively mechanical and repetitive. The third is the uniform service standard and the enforcement of strict disciplinary standards, including the employees appearance, and the discipline related to the tax window operation process, service quality, etc. Because they face taxpayers all day, window employees remain in a state of mental pressure for a long time, which may influence their work efficiency, reduce the service level, and more seriously damage the relationship between the body that levies the tax and the taxpayer, causing the employees negative emotions such as anxiety and boredom. Therefore, it is undoubtedly beneficial to establish a comprehensive mechanism for psychological counselling and pressure relief for employees. The establishment of a flexible work system will allow the window employees an appropriate amount of time to receive psychological counselling and to decompress and rest and will effectively reduce the long-term physical fatigue and mental stress. A flexible service system will also allow the employees time to organize the taxpayers archives, record the collected information in the and send the related documents to management, as well as audit tax exemptions, effectively meeting the demands related to the background work.

It promotes the study of window service and communication and improves professional quality. Tax management refinement is improved unceasingly, while the fixed nature of window positions and the monotonous work content may affect the professional improvement of employees. Under the flexible service system, flexible working hours could allow the development of the employees professional study of window service and communication with employees in other business positions, which could convert employees more specialized talents related to tax administration to all-round talents.

It improves the management level of the tax service hall, optimizes the tax payment service, and enhances the satisfaction of taxpayers. The tax service hall is the main bridge and communication link between the two sides; that is, the tax levying body and taxpayer. It is an important window for displaying the image of the tax authorities and officials and providing solutions to tax-related issues, as well as an important platform on which to offer quality tax services. A flexible working schedule can not only solve a series of existing problems, but also embody the people-oriented, advanced service-first management idea, which will thus lead to optimization of the tax service, innovation in service methods, improvement of the management level, demonstration of the spirituality of the employees in the new era, and, to a certain extent, enhancement of the taxpayers satisfaction.

In general, the number of windows in the tax service halls in Lishui City, Zhejiang Province, is excessive relative to the demand. However, there are still a few tax service halls that cannot meet the demand, which affects the service quality. This paper presents the so-called inverted triangle flexible shift model of full- and part-time window employees to solve the problem of an insufficient number of window employees. According to the inverted triangle flexible scheduling model, full-time window employees are assigned more working hours and part-time window employees fewer working hours at the window. The length of the working hours of the window employees is from long to short, and thus the top to bottom composition of the graph is like an inverted triangle. The inverted triangle scheduling model with its combination scheduling of both full- and part-time window employees both ensures the service quality of the windows and reduces their service cost.

The rest of the paper is organized as follows. In Section 2, we review the personnel scheduling literature. In Section 3, we scientifically set the number of open windows at different time periods according to a large data analysis and queuing theory. In Section 4, we describe a flexible scheduling model that meets the needs of the tax department in Zhejiang Province. In Section 5, we establish a set of iterative algorithms to construct the optimal solution of the flexible scheduling model. In Section 6, we provide an example of flexible scheduling to verify the effectiveness and feasibility of our algorithms. In Section 7, we discuss the potential application of the scheduling model for decision-making in the Zhejiang Provincial Tax Department.

#### 2. Literature Review

In the research field of queuing and flexible scheduling problems, Deutsch et al. [1] presented a successful application of queuing theory to the scheduling of a large bank. Targeting cost optimization, Hammond et al. [2] set up a spreadsheet model based on queuing theory to achieve the required number of service personnel in a bank and verified the effectiveness of the proposed model through simulations. Jones et al. [3] analysed the relationship between the actual and acceptable queue waiting time and validated it in an empirical study. So et al. [4] established a queuing theory model to verify the ability of the dynamic adjustment service function to reduce the length of the service line. Nosek et al. [5] claimed that queuing theory can be used to evaluate the employees working arrangements, working environment and productivity, and the customers waiting time and waiting environment. They applied queuing theory and customer satisfaction to the field of pharmaceutical research and showed that, if managers use queuing theory correctly to make the appropriate decisions, then customers, employees, and managers will all be satisfied. On the basis of queuing theory, Wang et al. [6] presented a fast channel model to improve the efficiency of the bank queuing system. By modifying a greedy algorithm and then using MATLAB to execute the numerical simulation of a multiple optimization model, they obtained the experimental results that a system that includes a fast channel can reduce customers average waiting time in both the regular and fast channel queue. Using the queuing model, Ogunwale et al. [7] conducted a comparative research study on the waiting time of customers in two banks and provided corresponding suggestions.

Moondra [8] set up a linear programming model of bank employee scheduling. Kra-Jewski et al. [9] described a scheduling system for check-decoding personnel and its implementation in large banks. They then assessed its effect on cost savings and other functions of the system. Ernst et al. [10] identified the volatility of customer demands in a day as the source of the main difficulties in bank employee scheduling and that handling this volatility depends on appropriate arrangements for full-time employees and part-time employees. Mabert et al. [11] proposed two types of heuristic scheduling methods to meet the volatility of customer demands by means of using part-time employees. Both methods were aimed to minimize the number of windows and the window employee transfers between branches; the validity of the method was verified by using the actual data of a bank. In particular, Burns et al. [12] introduced an approach for solving multiple-shift manpower scheduling problems by means of an algorithm that constructs an optimal schedule for a large and common class of scheduling problems.

#### 3. Forecast to Meet the Needed Number of Open Windows

##### 3.1. Prediction of the Flow of Customers in the Tax Service Hall

According to an analysis of the historical data provided by several tax service halls of the local taxation bureau of Lishui City, the number of taxpayers obeys Poisson distribution, and in each month the numbers of taxpayers at the beginning, middle, and end of the month may be different. The business in the tax service hall can be divided into two seasons: slack and busy. In addition, there are two types of visitor-flow-rate data of taxpayers: one is called the busy time type and the other the idle time type. The first type occurs usually at the beginning and end of a month and the second in the middle of a month. The average number of taxpayers arriving at the tax service hall at different times (busy and idle) is divided into busy season and slack season separately, where different time refers to, for example, 08:30–09:30, 09:30–10:30, etc. Meanwhile, we can predict the average total number of taxpayers arriving at the tax service hall at different times (busy and idle) in different seasons, separately, according to the autoregressive moving average (ARMA) model.

##### 3.2. Calculation of the Minimum Required Window Number in Different Time Periods According to Queuing Theory

The analysis of all the data collected from the tax service halls showed that the entire service process from the customers arrival to departure obeys the standard queuing model of . The predicted value of the average arrival rate of the taxpayer per hour can be calculated using the historical data and the average service number per hour can be calculated by dividing the total number of services with the total taxpayers service time.

In the standard queuing theory, represents the number of open windows in the tax service hall, the log-run per hour average arrival rate of the taxpayers, the daily average service rate at the windows and the number of the taxpayers in the queue. Then, denotes the service intensity. According to the queuing theory, the average waiting time [13] until receiving service isAccording to the complaints of the taxpayers and the data from the responses to the questionnaire, the average waiting time for the taxpayers should not be more than 15 min. The local tax bureau in Lishui City has a requirement, that is, hr. In addition, on the basis of quality of service assurance, the number of open windows should be as small as possible. Because is a positive integer and it is difficult to compute its analytical solution, the calculation is performed according to the following steps.

*Step 1. *Let , based on the queuing process to achieve the steady state.

*Step 2. *Use formula (1) to compute . If , then the algorithm is terminated and is the minimum number of open windows required to satisfy the demands; if , then go to Step 3.

*Step 3. *Let , and then return to Step 2.

Because of the volatility of the number of taxpayers arriving at the tax service hall in different time periods and seasons, the average arrival rate also varies. In addition, the service rate differs because of the varying business skills of the window employees. The minimum number of open windows that can satisfy the demands in different time periods and seasons can be calculated using the above calculation steps.

#### 4. Staff Scheduling Model

##### 4.1. Parameters and Decision Variables

The opening time of the windows of the tax service hall in Lishui City is 08:30–17:00, divided into the morning shift from 08:30 to 12:00, the noon shift from 12:00 to 14:00, and the afternoon shift from 14:00 to 17:00. To provide high-quality service, each window employee should rest for 15 min after working continuously for 45 min or 1 hr. Let 15 min be one time period; then, the open window time in any one working day can be divided into 27 time periods. 08:30–08:45 is recorded as the first time period and 08:45–09:00 as the second and so on. Then, 11:45–12:00 is recorded as the 14th, 12:00–14:00 as the 15th (consisting of eight times 15 min), and 16:45–17:00 as the 27th time period.

In the tax service hall, there is only one full-time window employee on duty at noon and this window employee does not work from 11:30 to 12:00. Since each employee’s working time cannot exceed eight hours in a working day, each employee works for only two of the three shifts in every working day: morning, noon, and afternoon. The difference in the working time of any two full-time window employees is as small as possible in a scheduling cycle. According to the inverted triangle scheduling model, each full-time window employee needs to work at least three and at most four consecutive time periods. Each full-time window employee can be allowed to work for two consecutive time periods only immediately before 12:00 or immediately before 17:00. If a window employee in the morning or afternoon has rested for two consecutive time periods, then he/she will not work again in the morning or afternoon at the window.

The parameters and decision variables of the flexible scheduling model are defined as follows. Let denote the total days of the scheduling period (usually one month is considered a cycle) and let and denote the total number of full-time window employees and part-time window employees, respectively. Let denote the minimum number of open windows required to meet the taxpayers service demand for time period on day (calculated by the formula (1)); =1, 5, 9, 13, 16, 20, and 24 correspond to 08:30, 09:30, 10:30, 11:30, 14:00, 15:00, and 16:00, respectively, and , , , , , , , . We define also the following decision variables:

##### 4.2. Objective Function

In order to meet the requirement that the taxpayers average waiting time does not exceed 15 min, the number of employees is the minimum for the optimal objective; that is, the objective function is

The first constraint (4) assures that each full-time window employee works for at least three consecutive time periods (45 min) after he/she has started to work. Constraint (5) assures that each full-time window employee works for at most four consecutive time periods (60 min). Constraint (6) assures that if each full-time window employee has rested for two consecutive time periods in the morning/afternoon, then he/she no longer works at the window in the morning/afternoon. Constraint (7) assures that if a full-time window employee is on duty from 12:00-14:00, then he/she does not work from 11:30 to 12:00; thus, he/she has time to eat lunch and prepare for the noon duty. Constraint (8) assures that each full-time window employee in every working day works only two of the three shifts, morning, noon and afternoon; otherwise, his/her working time is more than eight hours (in violation of labour law). A rule of the tax service hall in Lishui City is that if a full-time window employee is on duty at noon, he/she must also work at the window in the morning. The advantage of the rule is that it makes it convenient for an employee on this duty to take leave, make a business trip, or organize documents in the background in the afternoon. Constraint (9) assures that if a full-time window employee is on duty at noon, then he/she does not work at the window in the afternoon. Constraint (10) assures that each full-time employee is allowed to work for two consecutive time periods before 12:00/17:00. Constraint (11) assures that each part-time window employee has to work for at least two consecutive time periods after he/she has started to work. Constraint (12) assures that if each part-time window employee is on duty for the second time at the window, then he/she has to work for at least two consecutive time periods. Constraint (13) assures that each part-time window employee works for at most four consecutive time periods. Constraint (14) assures that if each part-time window employee has rested for two consecutive time periods in the morning/afternoon, then he/she no longer works at the window in the morning/afternoon. Constraint (15) assures that in the last period of the morning/afternoon shift each part-time window employee is allowed to work only for two consecutive working time periods. Constraint (16) assures that there is a sufficient number of employees to cover the demand for each time period in any working day. In the following algorithm, our optimal solution takes the equality in constraint (16); i.e., under the condition that is the minimum, we obtain the optimal solution. Constraint (17) assures that there is only one full-time window employee on duty at noon on any day . Constraint (18) assures that the difference in the working hours of any two full-time window employees in one cycle is less than (nonnegative integer constant) time periods. In order to take fairness among all full-time window employees into account, we usually calculate the best constant (the difference in the working time of any two full-time window employees is less than or equal to 15 min in one cycle).

#### 5. The Algorithm

We decompose the original scheduling model into several submodels and use the descending dimension method to construct the optimal solution of the flexible scheduling model. Before examining the algorithm of the flexible scheduling model, we first repeat the scheduling rules and introduce some scheduling variables. The series of time periods are defined as the vertical direction and the number of windows of each time period is defined as the horizontal direction. When each full-time window employee has started to work, he/she has to work for at least three but not more than four consecutive time periods and then rest for one time period. Each window employee working at the end of the morning/afternoon is allowed to work for two consecutive time periods. When a part-time window employee has started to work, he/she has to work for at least two consecutive time periods. In any fixed working day, there are two variables for any time period. The first variable is the state variable of the time period, which has two values: if the state variable takes 1, it shows that the window employee works at the window and if the state variable takes 0, it shows that the window employee does not work at the window. The second variable is the flag variable of the time period. If the flag variable takes 0, it shows that the working state of the window employee is uncertain and if the flag variable takes 1, it shows that the working state of the window employee is certain. If the flag variable takes 3, it shows that a part-time window employee is working his/her first shift, and there are two time periods from the current state variable to the determined state. (If the flag variable value is 3, it shows that the part-time window employee has worked for two consecutive time periods, and thus, the state of the second time period of the current period is determined.) If the flag variable takes 2, it shows that the state variable has one time period to the determined state. If the flag variable takes 10, it shows that the state variable of the time period previous to the current time period takes 0; the window employee has finished working for consecutive working time periods and has rested for one time period, that is, the first flag variable of the new cycle takes 10 (the state variable of the current time period is uncertain). If the flag variable takes 11, it shows that the state variables of the current time period and the subsequent time period can be assigned only 0. (If the state variables of the two consecutive time periods take 0, the value of the corresponding second flag variable is set to 11; i.e., when the flag variable is set to 11, the window employee no longer works for the remaining time periods of the morning/afternoon.) If the flag variable takes 13, it shows that the current state variable to the determined state has at most two time periods, and if 0 is assigned to the current state variable, then the corresponding flag variable is set to 11, and the state variable of the time period subsequent to the current time period is set to 0 and the corresponding flag variable is set to 11; i.e., the window employee no longer works for the remaining time period of the morning/afternoon, and therefore, the state variable of the time period previous to the current time period is the last working state in the morning/afternoon; if the flag variable takes 13 and 1 is assigned to the current state variable, then the values of the state variable and the flag variable of the time period subsequent to the current time period are set to 0 and 2, respectively. If the flag variable takes 14, it shows that the state variable of the current time period does not participate in the horizontal cycle permutation of the combinatorial tuple consisting of 0 and 1. Each full-time window employee’s symbol variable is set to green and each part-time window employee’s symbol variable is set to red. In order to improve the efficiency of iterative return, we usually set the values of the state variable and the flag variable only of the subsequent two consecutive time periods in the current time period (except the value of flag variable is 14).

*Step 1 (initialize variables). *The initial state variable and the initial flag variable of each time period of each window employee are all set to 0.

*Step 2 (the vertical iterations for flexible scheduling algorithm in the morning). *The rule of the iterative scheme is as follows. We first examine the vertical (time periods 1-11) iterative schemes of each full-time window employee. The iterative schemes in Figures 1 and 2 are suitable for time periods 1-11. The digits at the top of all the boxes indicate the values of the state variables of the corresponding time periods, respectively, and the digits at the bottom of all the boxes represent the values of the flag variables of the corresponding time periods, respectively.