Nonlinear strategic human resource management based on organisational mathematical model

Hao Guo

Dziennik i szczegóły wydania

Applied Mathematics and Nonlinear Sciences

Informacje o czasopiśmie

Format: Czasopismo
eISSN: 2444-8656
Pierwsze wydanie: 01 Jan 2016
Częstotliwość wydawania: 2 razy w roku
Języki: Angielski

Human resource management organisation's goal must be achieved by regulating employee behaviour through a restraint system. We conduct quantitative research on employee behaviour under organisational constraints, which is an essential means to ensure the organisation's healthy and stable development. The employees themselves adapt to the organisational goals, obey the organisation's constraint system and allocate their resources so that there is less research on optimising their behaviour. This article briefly reviews the research and application of economic theory, organisational behaviour, human resource management and other aspects of optimising employee resource allocation behaviour. The thesis defines organisational resources and divides them into three types. Based on the constraint system, we turn the organisation's control, requirements or incentives on employee behaviour into above or below employee operating constraints; we believe that the constraint system has the characteristics of discrete and hanging river effects, and we have obtained an evaluation. Two algorithms of the organisational constraint system. The article discusses employee resources and behaviours. At the same time, it attempts to construct a quantitative model based on the employee's multi-objective, nonlinear, continuous and discrete constraints, on the allocation of resources and behaviour optimisation and on the employee's adaptation to the organisational constraints. Under the premise of optimisation, several situations in which the model solves the optimal solution are studied. The research found that this article can fill the gaps in employees’ self-adaptation to organisational goals and behaviour optimisation quantitative models.

Keywords

employee behaviour
human resource management
nonlinear mathematical model
goal planning
organisational resources
optimisation of the constraint system

MSC 2010

34A34

Artykuł

Introduction

The organisation regulates the behaviour of employees through a restraint system to achieve organisational goals. The employees adapt to the organisation's goals, obey the organisation's constraint system and allocate their resources to optimise their behaviour, and in this manner the best results can be obtained. Research in this area can see the penetration of economics research and be found in the category of individual behaviour optimisation in organisational behaviour and personal career planning in human resource management. Of course, all these are inseparable from the strong support of psychology, and the attempt of mathematical model description is also indispensable [1]. From the perspective of the research and application of individual behaviour optimisation in the field of human resource management, most of the issues involved are considered from the perspective of managers, and employees are passive; there are more qualitative researches and fewer combined qualitative and quantitative studies; further, there is more business management, and less proactive career planning by employees.

This article delves into the concepts of organisational resources, constraint system, employee behaviour, employee resources, etc., and gives corresponding functional expressions to the discussion's factors. From the perspective of individual employees, taking into account the continuity of employee behaviour and the constraint system's discreteness, we tried to construct a multi-objective nonlinear employee under organisational constraints, and at the same time, a resource allocation optimisation model with continuous and discrete constraints [2]. And on the premise of maximising the benefits of employees, we researched the optimisation of the model.

Method

The organisation's constraint system refers to an orderly set of discrete points constructed under the existing organisational resources and according to the needs of business processes to ensure the effective operation of the organisation and to regulate the continuous behaviour of employees, which is composed of requirements, control, incentives and other indicators.

2.1

Constraint system function

For organisations, the restraint system is mainly used by managers to regulate employee behaviour under existing organisational resources so that employee behaviour can achieve a particular effect. The restraint system is divided into requirements, control and incentives for all employees, and the behaviour of employees in specific positions is based on the standards of the system [3]. Above, below, and the above operation and the above operation's constant control can be transformed into above and below requirements or incentives. We assume that through the organisation's constraint system, there is a total of n₂(n₃), which requires all employees to run on it, and the functional groups of its operating effects are, respectively: (1) $G_{s_{k}} = G_{s_{k}} (X), k = 1, 2, \dots, n_{2} (G_{s_{k}}^{'} = G_{s_{k}}^{'} (X), k = 1, 2, \dots, n_{3})$ {G_{{s_k}}} = {G_{{s_k}}}(X),\;k = 1,2, \cdots ,{n_2}(G_{{s_k}}^\prime = G_{{s_k}}^\prime(X),\;k = 1,2, \cdots ,{n_3}) We suppose that through the organisation's constraint system, there are a total of $n_{2}^{'} (n_{3}^{'})$ n_2^\prime(n_3^\prime) that require employees of a specific position to operate on it, and the operating effect function groups are, respectively: (2) $T_{s_{k}} = T_{s_{k}} (X), k = 1, 2, \dots, n_{2}^{'} (T_{s_{k}}^{'} = T_{s_{k}}^{'} (X), k = 1, 2, \dots, n_{3}^{'})$ {T_{{s_k}}} = {T_{{s_k}}}(X),\;k = 1,2, \cdots ,n_2^\prime(T_{{s_k}}^\prime = T_{{s_k}}^\prime(X),\;k = 1,2, \cdots ,n_3^\prime)

2.2

The characteristics of the constraint system function

2.2.1

The organisational constraint system is discrete

The organisational constraint system requires employees to achieve particular effects of the function group, and its geometric figure is an ordered set of discrete points in a dimensional space. For example, in G_s1 = G_s1(X), if in the X +αX′(0 ≤ α ≤ 1) line segment formed by point X, X′, the function is meaningful only at the endpoint X, X′, then in X + αX′(0 ≤ α ≤ 1), the function g_s1 has only two end values: (3) $G_{s_{1}} = G_{s_{1}} (X), G_{s_{1}} = G_{s_{1}} (X^{'})$ {G_{{s_1}}} = {G_{{s_1}}}(X),\;{G_{{s_1}}} = {G_{{s_1}}}({X^\prime})

2.2.2

The constraint system has a hanging river effect

As employees become familiar with the restraint system, employees gradually become ‘smart’, and thus the system no longer possesses sufficiently the capability to restrain employees’ behaviour; with the organisation's development, technical issues are becoming more and more complex, and human coordination is becoming more and more difficult. To make the organisation's operation more effective, it is necessary to produce new management regulations continuously [4]. The more perfect the regulations, the larger the constraint system will be, and it is expected to eventually become an inverted pyramid. The bottom of the pyramid is a specific matter that needs to be dealt with during the organisation's operation. At the same time, the top is a constraint system composed of various management rules and regulations. The top of the pyramid has a trend of continuous expansion. As reflected in the organisational constraint system, the discreteness of these abstract functions of operational effects is weakened and gradually becomes ‘more continuous’.

2.3

Organisational resources

In a narrow sense, organisational resources are all quantifiable, tangible materials that can support organisational activities to achieve specific goals.

For example: for a productive organisation, front-line production workers, engineering and technical personnel, management personnel, capital, production equipment, workshops, buildings, raw materials, administrative office equipment, etc., are organisational resources. Figure 1 shows the system dynamics model between organisational resources and management efficiency.

System dynamics model between organisational resources and management efficiency

Organisational resources can be divided into three categories according to their usage. The first category is resources allocated to specific employees; the second category is organisational resources that some specific employees cannot use; and the third category is outside-organisational resources. Organisations where employees can only use the first type of resources are called closed organisational systems; organisations where employees can use all types of resources except for the third type are called internally open organisational systems; and organisations that can use all three types of resources are called open organisational systems.

The thesis has m kinds of organisational resources, and its controllable variable is X = (x₁,x₂,⋯ ,x_m)^T; then the function group of employee n₁ project objective can be set as follows: (4) $F_{i} = F_{i} (X), i = 1, 2, \dots, n_{1}$ {F_i} = {F_i}(X),\;i = 1,2, \cdots ,{n_1}

2.4

Employee resources and behaviour

2.4.1

Staff Resources

Employee resources include employees’ resources and resources allocated by the organisation. Own resources include factors such as the time that employees can invest, some simple self-owned tools, the professional knowledge accumulated by employees in long-term work, etc.; organisational rationing resources are resources allocated to employees in a specific position to enable them to complete tasks. Figure 2 shows the system dynamics model of employee resources and behaviour.

System dynamics model of employee resources and behaviour

We suppose that in addition to the rationed resources in the organisation resources, there are m types of employees [5]. If they have their available resources expressed by X_m+1, X_m+2, ⋯, X_m, the employee resources can be set as the following: (5) $X^{'} = {(x_{1}, \dots, x_{m}, \dots, x_{m + 1}, \dots, x_{m^{'}})}^{T}$ {X^\prime} = {\left( {{x_1}, \cdots ,{x_m}, \cdots ,{x_{m + 1}}, \cdots ,{x_{{m^\prime}}}} \right)^T} The function of staff allocation resources to reach the goal of j₁ can be set as follows: (6) $Y_{J_{1}} = Y_{J_{1}} (X^{'}), j_{1} = 1, 2, 3, \dots, j_{1}$ {Y_{{J_1}}} = {Y_{{J_1}}}({X^\prime}),\;{j_1} = 1,2,3, \cdots ,{j_1} By configuring rationed resources and own resources, the functions for employees to achieve the effect of the item j₂, j₃ can be set to the following: (7) $Y_{j_{2}} = Y_{j_{2}} (X^{'}), j_{2} = 1, 2, 3, \dots, j_{2} Here X^{'} = {(X_{1}, \dots, X_{M}, 0, \dots, 0)}^{T}$ {Y_{{j_2}}} = {Y_{{j_2}}}({X^\prime}),\;{j_2} = 1,2,3, \cdots ,{j_2}\quad {\rm{Here}}\;{X^\prime} = {\left( {{X_1}, \cdots ,{X_M},0, \cdots ,0} \right)^T} (8) $Y_{j_{3}} = Y_{j_{3}} (X^{'}), j_{3} = 1, 2, 3, \dots, j_{3} Here X^{'} = {(0, \dots, 0 X_{m + 1}, \dots, X_{m^{'}})}^{T}$ {Y_{{j_3}}} = {Y_{{j_3}}}({X^\prime}),\;{j_3} = 1,2,3, \cdots ,{j_3}\quad {\rm{Here}}\;{X^\prime} = {\left( {0, \cdots ,0{X_{m + 1}}, \cdots ,{X_{{m^\prime}}}} \right)^T}

2.4.2

Behaviour of employees

Some scholars believe that the causes of employee behaviour in an organisation are very complicated. What is the reason for determining it? There are very complicated researches in psychology. An employee's behaviour is related to economic status, quality, physical state, emotion, family members, society and organisation. Employees’ normal behaviour refers to the behaviour that employees exhibit that meets the requirements of managers and conforms to the restraint system, which is conducive to the realisation of organisational goals [6]. The behaviour of employees is continuous and continuous. Abnormal behaviour refers to employees’ behaviour deviating from the organisation's goals, violating the requirements of the restraint system, and harming the interests of the organisation. It can be said that the continuity of employee behaviour, the discreteness of the restraint system, the hysteresis of the influence of the restraint system on employee behaviour and the hanging effect of the restraint system are the profound foundations for employees to produce abnormal behaviours.

Under the organisational constraints, based on the premise of employees’ best interests, the following ‘smart principles’ can be set for employees’ behaviour. ‘Smart Principle.’ If the constraints of the organisation require employees to operate above a particular effect, the employees must choose to operate on the minimum requirement to achieve this effect; if the constraints of the organisation require employees to operate under a particular effect, the employees must choose to operate under a particular effect. Run-on the highest requirements to achieve this effect.

Results

The restraint system's effect on employee behaviour can be evaluated by the total benefits obtained by the organisation to restrain employee behaviour. Two evaluation methods are discussed here, pre-algorithm and post-algorithm. The algorithm before the event has errors, and the algorithm after the event is more accurate [7]. Now we give an example. For convenience, the discussion is only on the two-dimensional plane, and the upward curve of employee behaviour and restraint system requirements for employees are, respectively, Y = Y (x), G₁ = G₁ ≥ 0, as shown in Figure 3.

Relation diagram of restraint system and employee behaviour

Beforehand algorithm. After the organisational constraint system is given, the employee's behaviour has not yet occurred [8]. Therefore, by constraining the employee's behaviour, the organisation's total benefits can be divided into two parts: discrete points and non-discrete points. When employees happen to meet the constraint requirements, the discrete point benefit is (9) $S_{1} = \sum_{i = 0}^{4} G_{1} (x_{1})$ {S_1} = \sum\limits_{i = 0}^4 {G_1}({x_1}) We set the benefit outside the discrete point as S_x (the benefit between two discrete points) and S_x is a significant amount that cannot be calculated. To solve this problem, the area enclosed by the broken line can be used instead. Therefore, employees strictly follow the requirements of the upward curve in the restraint system, and the estimated total benefit obtained by the organisation is the area enclosed by the broken line from x₀ to x₄ and the x-axis.

After the fact algorithm. Since the employee's behaviour has occurred after the constraint system is given, the exact value of the total benefit obtained by the organisation is given by: (10) $S = \int_{x_{0}}^{x_{4}} Y (x) dx$ S = \int\limits_{{x_0}}^{{x_4}} Y(x)dx

Discussion

The general planning model is very closely related to real problems. For example, in the natural environment, people have ideals. Since the expectations are unlimited and the environment is limited, it is necessary to use specific methods to find the combination of existing resources to achieve the best expectations [9]. The general abstract function programming model is as follows:

Objectives and Evaluation Criteria Group – V = f (x₁,y_j,ξ_k)

Constraint group – g(x₁,y_j,ξ_k) ≥ 0

Among them, x₁ is a controllable variable, y_j is a known parameter and ξ_k is a random factor.

4.1

Abstract function model

The organisation, constraint system, employee behaviour, etc., discussed in this article are abstract, and so only abstract models can be established. The model based on the optimisation of employee resource allocation behaviour under the constraint system is as follows: (11) $Opt Y_{j 1} = Y_{j 1} (X^{'}), j_{1} = 1, 2, 3, \dots, j_{1}$ Opt{Y_{j1}} = {Y_{j1}}\left( {{X^\prime}} \right),\quad {j_1} = 1,2,3, \cdots ,{j_1} (12) $s . t {\begin{array}{l} Y_{j 2} = Y_{j 2} (X^{'}) \geq 0, & j_{2} = 1, 2, 3, \dots, j_{2} \\ Y_{j 3}^{'} = Y_{j 3}^{'} (X^{'}) \geq 0, & j_{3} = 1, 2, 3, \dots, j_{3} \\ G_{s_{k}} = G_{s_{k}} (X) \geq 0, & k = 1, 2, \dots, n_{2} \\ G_{s_{k}}^{'} = G_{s_{k}}^{'} (X) \leq 0, & k = 1, 2, \dots, n_{3} \\ T_{s_{k}} = T_{s_{k}} (X) \geq 0, & k = 1, 2, \dots, n_{2}^{'} \\ T_{s_{k}}^{'} = T_{s_{k}}^{'} (X) \leq 0, & k = 1, 2, \dots, n_{3}^{'} \\ X^{'} \geq 0 \end{array}$ s.t\left\{ {\matrix{ {{Y_{j2}} = {Y_{j2}}\left( {{X^\prime}} \right) \ge 0,} \hfill & {{j_2} = 1,2,3, \cdots ,{j_2}} \hfill \cr {Y_{j3}^\prime = Y_{j3}^\prime\left( {{X^\prime}} \right) \ge 0,} \hfill & {{j_3} = 1,2,3, \cdots ,{j_3}} \hfill \cr {{G_{{s_k}}} = {G_{{s_k}}}\left( X \right) \ge 0,} \hfill & {k = 1,2, \cdots ,{n_2}} \hfill \cr {G_{{s_k}}^\prime = G_{{s_k}}^\prime\left( X \right) \le 0,} \hfill & {k = 1,2, \cdots ,{n_3}} \hfill \cr {{T_{{s_k}}} = {T_{{s_k}}}\left( X \right) \ge 0,} \hfill & {k = 1,2, \cdots ,n_2^\prime} \hfill \cr {T_{{s_k}}^\prime = T_{{s_k}}^\prime\left( X \right) \le 0,} \hfill & {k = 1,2, \cdots ,n_3^\prime} \hfill \cr {{X^\prime} \ge 0} \hfill & {} \hfill \cr } } \right. In Model 1, the decision variable has m′, the objective function has j₁ and the constraint condition is $j_{2} + j_{3} + n_{2} + n_{3} + n_{2}^{'} + n_{3}^{'}$ {j_2} + {j_3} + {n_2} + {n_3} + n_2^\prime + n_3^\prime , an optimisation model of multi-objective nonlinear programming. In particular, it needs to be explained that there are both hypersurfaces in dimensional space and discrete point sets in m-dimensional space in the constraint condition group. Among them, the ‘≥ , ≤ ’ in (c), (e), (e), (f) is understood by running up and down [10]. The optimisation of the model is discussed below.

4.2

The impact of organisational openness on the optimal solution of the model

In a closed organisational system, the optimal solution of Model 1 is theoretically the optimal solution obtained by solving the multi-objective nonlinear programming optimisation model. In reality, there will be a phenomenon that the actual optimal solution under Model 1 is far greater than its theoretical optimal solution [11]. This phenomenon can be explained to a certain extent by the open and internal open organisation system introduced in the previous section. We know that in a closed organisational system, resources are conserved, while in the open and internally open organisational systems, since employees can deliberately occupy resources inside and outside the organisation based on their subjectivity, the feasible domain of Model 1 is enlarged, resulting in the upper limit of the objective function value becoming enlarged or the lower limit becoming reduced [12].

4.3

Discussion on the optimisation of the model

To discuss the optimal solution of Model 1, we do not consider the (c), (d), (e), (f) constraint groups in Model 1 and get Model 2. (13) $Opt Y_{j 1} = Y_{j 1} (X^{'}), j_{1} = 1, 2, 3, \dots, j_{1}$ Opt{Y_{j1}} = {Y_{j1}}\left( {{X^\prime}} \right),\quad {j_1} = 1,2,3, \cdots ,{j_1} (14) $s . t {\begin{array}{l} Y_{j 2} = Y_{j 2} (X^{'}) \geq 0, & j_{2} = 1, 2, 3, \dots, j_{2} & (a) \\ Y_{j 3}^{'} = Y_{j 3} (X^{'}) \geq 0, & j_{3} = 1, 2, 3, \dots, j_{3} & (b) \\ X^{'} \geq 0 & (g) \end{array}$ s.t\left\{ {\matrix{ {{Y_{j2}} = {Y_{j2}}\left( {{X^\prime}} \right) \ge 0,} \hfill & {{j_2} = 1,2,3, \cdots ,{j_2}}\hfill &{ (a)} \hfill \cr {Y_{j3}^\prime = {Y_{j3}}\left( {{X^\prime}} \right) \ge 0,} \hfill & {{j_3} = 1,2,3, \cdots ,{j_3}}\hfill &{(b)} \hfill \cr {{X^\prime} \ge 0} \hfill & {} \hfill &{(g)} \hfill \cr } } \right. In Model 1, suppose the feasible region of constraint condition group (a)∩(b) is DY ∩Y, the definition domain of constraint condition group (c) ∩ (d) ∩ (e) ∩ (f) is D_{G∩G∩′T∩T′}, and the definition domain of upstream constraint condition group (c) ∩ (e) is D_G∩T. The definition domain of downstream constraint condition group (d) ∩ (f) is D_G∩′T′. Since Model 1 is based on multi-objective nonlinear programming, and some constraints are discrete point sets, it is not easy to solve it directly. It can be converted to solve Model 2 first [see Xu Guanghui (1999) for Model 2]; and thereafter Model 1 can be solved. We suppose that Model 2 has an optimal solution X^* under a particular target order requirement [13]. When employees act according to the ‘smart principle’, there is the following theorem to find the optimal solution of Model 1 under the same goal order. The proof is omitted.

Theorem 1

Suppose Model 2 has an optimal solution X*, if X* ∈ D_Y∩Y′ and X* ∉ D_{G∩G∩′T∩T′}, then Model 1 still has an optimal solution X*.

Theorem 2

Suppose Model 2 has an optimal solution X^*, if X^* ∈ D_Y∩Y′ and X^* ∉ D_{G∩G∩′T∩T′} are not necessarily the optimal solution of Model 1, the optimal solution of Model 1 needs to be discussed separately.

Theorem 3

If Model 1 is a maximisation (minimisation) problem, and Model 2 has an optimal solution X^*, if X^* ∈ D_Y∩Y′, X^* ∈ D_{G∩G∩T∩T′} and X^* ∈ D_G∩T(X^* ∈ D_G∩T′), when X^* is the only point with a running effect function value, then X^* is still the optimal solution of Model 1.

Theorem 4

If Model 1 is a maximisation (minimisation) problem, and Model 2 has an optimal solution X^*, if X^* ∈ D_Y∩Y′, X^* ∈ D_{G∩G∩T∩T′}, and X^* ∈ D_G∩′T′(X^* ∈ D_G∩T), when X^* is not the only point with operating effect function value, then the optimal solution of Model 1 needs to be discussed separately.

Theorem 5

If Model 1 is a minimisation problem, and Model 2 has an optimal solution X^*, if X^* ∈ D_Y∩′Y′ X^* ∈ D_G∩G∩T′ and X^* ∈ D_G∩T′ (X^* ∈ D_G∩T), meanwhile X^* is not the only point where $G_{s_{k}}^{'} (X)$ G_{{s_k}}^\prime(X) , $T_{s_{k}}^{'} (X) (G_{s_{k}} (X) T_{s_{k}} (X))$ T_{{s_k}}^\prime(X)\left( {{G_{{s_k}}}(X){T_{{s_k}}}(X)} \right) has a running effect function value.

When $max {G_{s_{k}}^{'} (X^{*})} \leq min Y_{j 1}^{'} (X^{*}) (min {G_{s_{k}} (X^{*})} \geq max Y_{j 1}^{'} (X^{*}))$ \max \left\{ {G_{{s_k}}^\prime\left( {{X^*}} \right)} \right\} \le \min Y_{j1}^\prime\left( {{X^*}} \right)\left( {\min \left\{ {{G_{{s_k}}}\left( {{X^*}} \right)} \right\} \ge \max Y_{j1}^\prime\left( {{X^*}} \right)} \right) and at the same time $max {T_{s_{k}}^{'} (X^{*})} \leq min Y_{j 1}^{'} (X^{*}) (min {T_{s_{k}} (X^{*})} \geq max Y_{j 1}^{'} (X^{*}))$ \max \left\{ {T_{{s_k}}^\prime\left( {{X^*}} \right)} \right\} \le \min Y_{j1}^\prime\left( {{X^*}} \right)\left( {\min \left\{ {{T_{{s_k}}}\left( {{X^*}} \right)} \right\} \ge \max Y_{j1}^\prime\left( {{X^*}} \right)} \right) , then X^* is still the optimal solution of Model 1; otherwise, Model 1 has no optimal solution, only a sub-optimal solution. Steps to solve a satisfactory solution of Model 1 are the following:

Theorem 6 selects the sub-optimal solution X^* ≠ X^* of Model 2. If it satisfies the condition that there is no optimal solution in Theorem 5, comparison can be made with Theorem 1, 2, 3, 4 and 5; if X^* satisfies Theorem 1, 2, 3 and 4 (the optimal solution in the theorem is changed to the sub-optimal solution), then X^* is the suboptimal solution of Model 1; otherwise, another sub-optimal solution of Model 2 is selected, and the above steps are performed, and so on until the sub-optimal solution of Model 1 is obtained as a solution [14].

Conclusion

This article presents an abstract mathematical model for optimising employee resource allocation and discusses the model's optimisation. The model considers the continuity of employee behaviour and the discrete nature of the constraint system. The hysteresis of the restraint system's influence on employee behaviour, the hanging river effect of the restraint system and the organisation's openness caused the variation of the model and the optimal solution, which were not discussed. Additionally, from the perspective of employees adapting to organisational goals, there are still many issues in employee resource optimisation worthy of in-depth study.

Ilustracje i tabele

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