3.1. Research Method
This study adopts second-order differential model analysis to construct the research model. This form of analysis refers to using the derivations of mathematical symbols and numerical calculations to study and represent economic or financial decision-making processes and phenomena via economic or financial decision analysis. These methods are not only simple tools for application purposes, but also contain many important theoretical concepts related to numerical methods. The optimization model based on differential equations has been applied to sustainable enterprise management (
Biswas et al. 2017), the creation of competitive advantages through sustainable environmental management and green entrepreneurship (
Skordoulis et al. 2017), the optimal control of sustainable development in bioremediation (
Nikitina et al. 2017), advertising costs (
Chalikias et al. 2016), and the sustainability of cooperative epidemiological models (
Barrios et al. 2018). Second-order differential model analysis helps organize the theoretical framework of economical or financial decision-making in a more rational, logical, and clearer manner.
Albersa et al. (
2015) proposed the use of optimization and decision models to guide sales personnel to make significant progress and explored the key concepts regarding sales response function estimation and heuristic solutions, as enterprise risk management in sustainability affects the company’s cash flow and income and reduces the company’s capital costs (
Berry-Stölzle and Xu 2018). In the assessment of risk and financial management, the most suitable training time cost allocation ratio should be invested, in order to reduce the costs or losses derived from the incidences of medical errors/disputes. Satisfying the industry’s overall maximum return through economic management is the main factor in constructing the sustainability model. When the operating managers of the chain aesthetic medicine industry of this study are making investments regarding the time of functional education for employees of two subsidiaries, according to the assessment of the company’s sustainability and financial risk management, they should use the most suitable training time cost allocation ratio to reduce the costs and losses derived from the incidence rate of medical errors/disputes. In this case, the following second-order differential equations can be used to analyze the proportion of functional training time of the two companies. The main factor for the construction of this model is to achieve the overall maximum profit through economic management under sustainability.
When faced with intense market competition, Aesthetic Medicine Group A set up two aesthetic clinics that have different business focuses, with the aim to achieve sustainable development and increase its market share and brand awareness.
Subsidiary 1 is an aesthetics medical institution with high pricing and focus on plastic surgery and microplastic surgery, which includes highly invasive medical treatments and long recovery periods. As both the appearance and posture of patients are greatly and permanently changed, Subsidiary 1 belongs to the medical industry of high risk and high unit price, and its medical professionals include licensed doctors and anesthesiologists, nursing staff, and medical beauticians. Doctors who perform plastic surgery must have many years of training in hospitals that specialize in aesthetic surgery. In addition to being qualified as a doctor, operating physicians must have specialty doctor licenses, such as a plastic surgeon license, while their nursing staff must have relevant training and experience to work in an operating room. As such functional skills require extensive training, meaning longer training time and higher cost, the investment threshold for setting up an orthopedic surgery institution is very high.
Subsidiary 2 is a microplastic surgery medical unit, which is characterized by no obvious wounds, short recovery period, and low cost and thus is easily accepted by the general public. Due to the short-term effect of microplastic surgery, frequent operation is needed to maintain the effect. Microplastic surgery is a medical industry with low investment cost, low risk, and low unit price, and its medical professionals and personnel only require a medical background and medical license. As the physicians perform micro-plastic surgery services through the training of medical equipment operation and injection techniques, there is no restriction on the licensing requirements of specialist physicians for the implementation of micro-plastic surgery; thus, skill training time is shorter. There are many training types, and training investment costs are lower.
3.2. Model Explanations
The main assumptions for the construction of this model are based on the following:
The investment of time in functional training and the incidence of medical errors or disputes of aesthetic medicine operators under sustainability are negatively correlated. In other words, the incidences of medical errors/disputes can be appropriately reduced when the optimal functional training investment of time is made. (1) The greater the optimal functional training investment of time, the more the training cost will increase. (2) Medical errors or disputes have a direct negative impact on the earnings of aesthetic medicine operators. (3) In the event of medical errors or disputes, the operators suffer direct medical losses that affect sustainability. However, the factors that affect net income of aesthetic medicine under sustainability are the time invested in functional training and training cost of the employees of Subsidiary 1 and Subsidiary 2 by the head office, where and are the ratio of time allocated to Subsidiaries 1 and 2, respectively.
When medical loss is caused by a medical error or dispute, , are the medical compensation losses of Subsidiaries 1 and 2 due to medical errors, and such medical losses also directly affect the operating income of aesthetic medicine under sustainability. and are the operating incomes of Subsidiaries 1 and 2, respectively, while and are the net incomes of Subsidiaries 1 and 2, respectively.
This study assumes that training cost allocation ratio and time of the optimal aesthetic medical professional functional training under sustainability also directly affects the professional judgment of the aesthetic medical professional, which can reduce medical errors or disputes and thus reduce corporate risk. Assuming that the professionals’ training knowledge in the aesthetic medicine industry is insufficient, and results in litigation caused by medical errors or high dispute rates, if reported through the media and passed on by customers, such negative exposure will inevitably affect revenues. In other words, the company will fail to reach the final goal of net income , which affects sustainable development.
The assumption that medical professionals are given more training time can reduce medical errors and disputes
and avoid litigation, which is relatively helpful to the net incomes
and
of the two subsidiary companies under the sustainability of the aesthetic medicine industry. On the contrary, if the aesthetic medical professionals of Subsidiaries 1 and 2 often cause medical errors and disputes
, which result in litigation and compensation due to inadequate training time, in turn, this will inevitably affect the company’s reputation, leading to a decline in performance, or even making the company go out of business or undergo bankruptcy. The interactive parameters between the two are ultimately significantly related to the net income
of the aesthetic medicine industry under sustainability. The framework of optimal decision making for aesthetic medical functional training time is shown in
Figure 1.
This model is based on the aforementioned assumptions, and the relevant parameter symbols are explained in
Table 1.
The relevant functions and parameter symbols are assumed as follows: 1. Assumption of Subsidiaries 1 and 2 aesthetic medicine functional training investment time cost
, the cost–benefit function:
Cost and benefit relationship between the time cost invested in aesthetic medicine functional training allocation by Subsidiary 1, and the time invested in aesthetic medicine functional training : , if , and (b1 > a certain negative number)
Assume:
where the parameters meet the conditions of
,
,
,
and satisfy the first-order differential greater than 0. Thus,
To satisfy the second-order differential greater than 0, the cost–benefit of Subsidiary 1′s aesthetic medicine functional training investment of time at this stage is calculated:
The cost–benefit relationship between the time cost invested in aesthetic medicine functional training allocation by Subsidiary 2 and the time invested in aesthetic medicine functional training : if and .
Assume:
where the parameters meet the conditions of
,
,
,
, satisfy the first-order differential greater than 0. Thus,
To satisfy that the second-order differential is less than 0 and calculate the cost–benefit of Subsidiary 2’s optimal aesthetic medical functional training investment of time at this stage:
Assumption of the cost–benefit relationship function of the operating income
of Subsidiaries 1 and 2 and their functional training allocation investment time
:
The cost–benefit relationship between operating income
of the aesthetic medicine of Subsidiary 1 and its time invested in the allocation of functional training
:
Assume:
where the parameters meet the conditions of
,
,
and satisfy the first-order differential greater than 0. Thus,
To satisfy that the second-order differential is greater than 0,
and
; the cost–benefit relationship between the time invested in the allocation of functional training
of Subsidiary 1 and its operating income
is calculated:
The cost–benefit relationship between operating income
of the aesthetic medicine of Subsidiary 2 and the time invested in the allocation of functional training
:
Assume:
where the parameters meet the conditions of
,
and
, and satisfy the first-order differential greater than 0. Thus,
To satisfy that the second-order differential is less than 0,
and
,
, the cost–benefit relationship between the optimal time invested in the allocation of functional training
of Subsidiary 2 and its operating income
is calculated:
Assumption of the cost–benefit relationship between the cost of medical losses due to medical errors
of Subsidiaries 1 and 2 and their time invested in the allocation of functional training
:
Assumption of the cost–benefit relationship between the cost of medical losses due to medical errors
of Subsidiary 1 and its time invested in the allocation of functional training
:
Assume:
where the parameter satisfies the condition of
,
,
satisfy the first-order differential less than 0. Thus,
To satisfy that the second-order differential is less than 0,
, the cost–benefit relationship between the cost of medical losses due to medical errors
of Subsidiary 1 and its time invested in the allocation of functional training
is calculated:
Assumption of the cost–benefit relationship between the cost of medical losses due to medical errors
of Subsidiary 2 and its time invested in the allocation of functional training
:
Assume:
where the parameter satisfies the condition of
,
,
satisfy the first-order differential less than 0. Thus,
To satisfy the second-order differential is less than 0,
, the cost–benefit relationship between the cost of medical losses due to medical errors
of Subsidiary 2 and its time invested in the allocation of functional training
is calculated:
The objective function is to satisfy the following instructions:
, and given
,
The cost–benefit of optimal time invested in the allocation of functional training
meets the maximum benefit
:
That is, to determine the optimal time invested in the allocation of functional training while satisfying .
Extremum first-order condition necessary condition:
; therefore,
In other words, the optimal time invested in the allocation of functional training
to satisfy
Extremum second-order sufficient condition