Nonlinear strategic human resource management based on organisational mathematical model

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) Gsk=Gsk(X), k=1,2,⋯,n2(Gsk′=Gsk′(X), k=1,2,⋯,n3) {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 n2′(n3′) 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) Tsk=Tsk(X), k=1,2,⋯,n2′(Tsk′=Tsk′(X), k=1,2,⋯,n3′) {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)

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.

Fig. 1

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) Fi=Fi(X), i=1,2,⋯,n1 {F_i} = {F_i}(X),\;i = 1,2, \cdots ,{n_1}

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) OptYj1=Yj1(X′), j1=1,2,3,⋯,j1 Opt{Y_{j1}} = {Y_{j1}}\left( {{X^\prime}} \right),\quad {j_1} = 1,2,3, \cdots ,{j_1} (12) s.t{Yj2=Yj2(X′)≥0,j2=1,2,3,⋯,j2Yj3′=Yj3′(X′)≥0,j3=1,2,3,⋯,j3Gsk=Gsk(X)≥0,k=1,2,⋯,n2Gsk′=Gsk′(X)≤0,k=1,2,⋯,n3Tsk=Tsk(X)≥0,k=1,2,⋯,n2′Tsk′=Tsk′(X)≤0,k=1,2,⋯,n3′X′≥0 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 j2+j3+n2+n3+n2′+n3′ {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.

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].

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) OptYj1=Yj1(X′), j1=1,2,3,⋯,j1 Opt{Y_{j1}} = {Y_{j1}}\left( {{X^\prime}} \right),\quad {j_1} = 1,2,3, \cdots ,{j_1} (14) s.t{Yj2=Yj2(X′)≥0,j2=1,2,3,⋯,j2 (a)Yj3′=Yj3(X′)≥0,j3=1,2,3,⋯,j3 (b)X′≥0(g) 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.

When max{Gsk′(X*)}≤minYj1′(X*)(min{Gsk(X*)}≥maxYj1′(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{Tsk′(X*)}≤minYj1′(X*)(min{Tsk(X*)}≥maxYj1′(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].