Inverse Optimization Method for Safety Resource Allocation and Inferring Cost Coefficient Based on a Benchmark

Abstract

Due to cost-push inflation, the trade-off between safety costs and risk prevention (safety) has become difficult worldwide. Most companies experience the difficulty of safety cost overruns and allocate safety resource inefficiently. In this paper, a forward model maximizing safety input is formulated. Because there is a wide range of variation of safety resource cost coefficient parameters, it is hard to determine safety resource cost coefficients in the forward model, to make the decisions on which types of safety resources are allocated to which potentially risky locations with what prices, and to ensure total input is as close to the benchmark as possible. Taking allocation, themes, resources, and cost coefficient parameters as new decision variables, the inverse optimization model is formulated based on a bi-level model. With consideration of quaternion decision, bi-level programming, and NP-hard problem, based on the comparison of exact penalty algorithm and an improved PSO algorithm, in which the inertia weight is adaptively changing with the number of iterations, the PSO is suitable for solving the specific inverse model. Numerical experiments demonstrated the effectiveness of the PSO algorithm, proving that it can allocate the right amount and types of safety resources with the right prices at the right places.

1. Introduction

The goal of the inverse optimization of safety resources is to reverse the resource allocation scheme and key parameters from the expected results. A production factory is subject to a wide range of regulations and standards that define acceptable levels of risk, such as OSHA, EPA, and NFPA. Noncompliance with these regulations can result in legal penalties and fines. Incorporating benchmarks is an easy way to ensure obeying the regulations and standards. A benchmark is set by those who have demonstrated excellent safety practices and whose size and risk characteristics are similar to those enterprises that adopt the benchmark management. However, detailed information on safety resource allocation plans and safety resource cost coefficient parameters for each kind of safety resources is often unknown. Moreover, the cost coefficient parameter of each safety resource has a wide range of choices. This greatly impedes the adoption of benchmark safety management.

Furthermore, the best practice of “which types of safety resources are allocated at which potentially risky locations with what prices” is a black box for any company that wants to adopt a benchmark. If this information cannot be inferred, the benchmark method cannot be adopted. Inverse optimization [1,2] involves working backward from total safety resources input value to determine the allocation plan and resource cost coefficient that would have produced the total safety input cost. This backwards approach is suitable to a situation with incomplete information about the benchmark’s best practice. Based on the above, this research proposes an inverse optimization method for inferring safety resource allocation and the cost coefficient based on a benchmark.

Safety resource allocation problems have attracted great attention in the safety management community. The following research studies are related to this study. Safety resources include human and material resources [3]. Reniers and Sörensen proposed an approach for the optimal allocation of safety resources using the knapsack problem to take aggregated cost-efficient preventive measures [4]. Rullo et al. proposed a method for ensuring security with reasonable resource and energy costs [5]. Yan et al. proposed a two-tier model to allocate budgets to airports to ensure airport safety [6]. Li et al. proposed an emergency response plan by dynamically allocating emergency resources across several scenarios [7]. Jiang et al. established a multivariate model for safety investment and accident control using grey prediction theory [8]. Lu et al. proposed a cost-effective safety investment method to ensure the best safety performance [9]. Heo et al. used budget data to realign the size and direction of the budget to reduce risk and promote effective budget operation and damage reduction within different risk groups [10]. Sato addressed the safety resource allocation decisions by combining linear programming and analytic hierarchy processes [11]. Eba et al. examined the utility of safety measures based on the return on safety investment [12]. Ma et al. gave a safety resource allocation method from the perspective of opportunity cost [13]. Roy et al. proposed a safety investment optimization framework, aimed at minimizing the risk value of potential hazards while adhering to given safety budgets. Zhang et al. explored the optimal-safety person–job matching method for major equipment [14,15]. Thus, it can be seen that the allocation of safety resources is one of the important research issues in the field of safety management. Many scholars use a forward optimization method to conduct their research.

Inverse optimization describes a process that is the “opposite” of traditional forward optimization. Unlike traditional forward optimization, which seeks to compute optimal decisions given objectives and constraints, inverse optimization takes decision results as inputs and determines objectives and/or constraints that make these decisions approximately or fully optimal. An inverse optimal value method is a kind of inverse optimization problem. Inverse optimal value methods aim to identify the unknown parameters of an optimization problem to achieve a given objective solution or target value as the optimal solution or the optimal value, respectively [16]. Ahmed and Guan defined the inverse optimal value model as identifying a cost vector from a set of feasible options such that the optimal objective value of the corresponding linear program is as close as possible to a desired value [17]. The methods for solving inverse optimal value problems can be categorized as accurate algorithms and intelligent algorithms. From the perspective of accurate algorithms, there are polynomial algorithms [18], alternating iteration [19], and gradient search [20], among others. From the perspective of intelligent algorithms, Li proposed an evolutionary algorithm for multi-criteria inverse optimal value problems based on the dynamic weighted aggregation method [18]. Other intelligent algorithms, such as genetic algorithm, ant colony algorithm, simulated annealing algorithm, and neural networks, have also been mentioned [21]. It should be noted that neural network algorithms are widely used to solve inverse optimization problems; however, neural network algorithms are data-driven methods, which have advantages in dealing with real-time problems. This research is a deterministic inverse optimization problem. The purpose of this research is to give a resource allocation plan with low requirements on solving speed. Moreover, only the benchmark data of the total input value of safety resources are available, so the neural network algorithm cannot deal with the problem. Chan et al. surveyed all the inverse optimization research [21]. Heuberger surveyed the inverse combinatorial optimization problem [22], and some research has paid attention to data-driven inverse optimization [23], robust inverse optimization [24], and inverse optimization applications [25,26,27]. It is worth mentioning that Wang et al. proposed an inverse optimization model for safety resource allocation, which makes the preset allocation theme to be the best theme based on the adjustment of parameters [28]; their research deals with the problem of knowing the ideal values of decision variables, and is different from this research where the ideal objective function value is known.

The existing research on the inverse optimization methodologies of safety resources provided inspiration for this research. However, quaternion problems, such as which types of safety resources should be allocated at which risky point and at what price and quantity, have not yet been solved. To provide a quaternion decision based on the total cost of a safety resource benchmark, we formulated a forward optimization model with parameters (in this model, they were initial estimate values; these were not accurate). An inverse optimization model was then formulated based on the forward model. The inverse model inferred the parameters of the forward model; the optimal input was the predetermined input of the top decision-makers. A PSO algorithm was then used to provide allocation themes and resource parameter values.

The proposed method provides a new concept for the allocation of safety resources and includes the following innovations: (1) Our quaternion decision-making method suggested quantities of the types of resources that should be allocated at which places and at what price. To the best of our knowledge, this is the first paper to address all allocation decision aspects related to the problem of the allocation of safety resources. We have provided a new method to resolve the optimization problem of safety resources, oriented by a benchmark. (2) Our research compared penalty algorithms and PSO algorithms, considering non-linearity, NP difficulty, and quaternion. Based on this comparison, a PSO algorithm with an adaptively changing inertia weight and number of iterations could provide the benchmark result. These algorithms and codes may inspire other researchers. The framework of this research is shown in Figure 1.

2. Model

2.1. Problem Analysis

Forward optimization approaches operate by acquiring a set of inputs to obtain an optimal solution based on a predefined objective function. Based on the demand points of safety resources (namely, risky points) for each system (production system, logistics system, energy system, etc.) and their subsystems, the main task of the safety management department is to develop a configuration plan for each safety resource that includes personnel, materials, equipment, and training programs. A forward model of the allocation of safety resources is a mathematical optimization model, which is used to determine the optimal allocation of safety resources to achieve the highest level of safety in a given system.

As the cost coefficient of the same type of safety resource varies, the allocation plan should simultaneously provide the quantity and cost coefficient of each type of safety resource at each safety resource demand point. Thus, an inverse optimal value model should be applied.

An inverse optimization is defined as follows: max $\mathbf{P}(\mathbf{x},\mathbf{c})$, $\mathbf{x}\in \mathbf{X}$ is a forward optimization problem, where $\mathbf{X}$ is a feasible domain, and c is a parameter vector. With a known $\widehat{\mathbf{x}}\in \mathbf{X}$ and a vector c, it is necessary to question the value of the parameter inferring to find out whether there is a parameter vector of c, $\widehat{\mathbf{c}}$. This makes the benchmark value equal to the problem of max $\mathbf{P}(\mathbf{x},\widehat{\mathbf{c}})$ optimal value. A description of the scenarios for the inverse optimization of safety resources is shown in Figure 2.

2.2. Assumptions

We made the following assumptions in our research: (1) There was a positive linear correlation between safety performance and the total amount of safety investment. (2) There were diminishing returns inputted in safety beyond a certain point. The input of the benchmark was the certain point. (3) The safety resource allocation plan of the benchmark was unknown. (4) The total investment in the safety resources of the benchmark was known. (5) Each safety resource price had a large range.

2.3. Notations

The definitions of the model symbols and their meanings are shown in

Table 1. Mathematical symbols and meanings.

Symbol	Meaning
$m$	Number of systems
$n$	Number of subsystems
$p$	Quantity of equipment resources
$q$	Quantity of training resources
$u$	Quantity of material resources
$v$	Quantity of human resources
$g_1$	Upper limit of equipment resources
$g_2$	Upper limit of training resources
$g_3$	Upper limit of material resources
$g_4$	Lower limit of material resources
$g_5$	Upper limit of human resources
$g_6$	Lower limit of human resources
$x_{ijk}$	Quantity of the k-th equipment resource in the j-th subsystem of the i-th system
$t_{ijr}$	Time of the r-th resource in the j-th subsystem of the i-th system
$x_{ijs}^{\prime }$	Quantity of the s-th resource in the j-th subsystem of the i-th system
$t_{ij\mu }^{\prime }$	Time of the μ-th resource in the j-th subsystem of the i-th system
$a_{ijk}$	Cost coefficient of the k-th resource in the j-th subsystem of the i-th system
$b_{ijr}$	Cost coefficient of the r-th resource in the j-th subsystem of the i-th system
$c_{ijs}$	Cost coefficient of the s-th resource in the j-th subsystem of the i-th system
$d_{ij\mu }$	Cost coefficient of the μ-th resource in the j-th subsystem of the i-th system
$e_i$	Upper limit of the total investment of the i-th system
$\gamma$	Coefficient of converting the total input in safety resources into safety output
$\alpha$	Proportion of human resources and non-human resources
$f^{*}$	Total input of safety resources of the benchmark

.

2.4. Forward Optimization Model of Safety Resource allocation

The forward optimization model is as follows:

$$\underset{x,t,x^{\prime },t^{\prime }}{\max }\gamma ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})$$

(1)

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}\le g_1$$

(2)

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}\le g_2$$

(3)

$${\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}\le e_i$$

(4)

$$g_4\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}\le g_3$$

(5)

$$g_6\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\le g_5$$

(6)

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\ge \alpha ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})$$

(7)

$$x_{ijk\_\min }\le x_{ijk}\le x_{ijk\_\max }$$

(8)

$$t_{ijr\_\min }\le t_{ijr}\le t_{ijr\_\max }$$

(9)

$$x_{ijs\_\min }^{\prime }\le x_{ijs}\le x_{ijs\_\max }^{\prime }$$

(10)

$$t_{ij\mu \_\min }^{\prime }\le t_{ij\mu }\le t_{ij\mu \_\max }^{\prime }$$

(11)

In the forward optimization model, Formula (1) indicates that the greater the safety performance, the better. The decision variables refer to the input amounts of various resources at each risky point. Constraints (2), (3), (5), and (6) are the resources available to supply domains from the perspectives of each type of resource. Constraint (4) indicates that each system has an upper limit for the total input of safety resources. Constraint (7) is the combination ratio of the requirements of human resources to non-human resources. Constraints (8) to (11) provide the upper- and lower-value limits of each decision variable.

2.5. Inverse Optimization Model

An inverse optimization model was formulated, as demonstrated in Figure 3.

In the inverse optimization model (12), the upper-level model pursued the total input as close to the benchmark as possible, with coefficient parameters of $a,b,c,d$. The lower-level model was the original model.

Upper-level model:

$$\begin{array}{c}\underset{a,b,c,d}{\min }{({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}-f^{*})}^2\\ a_1\le a\le a_2\\ b_1\le b\le b_2\\ c_1\le c\le c_2\\ d_1\le d\le d_2\end{array}$$

Lower-level model:

$$\begin{array}{c}\underset{x,t,x^{\prime },t^{\prime }}{\max }\gamma ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}\le g_1\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}\le g_2\\ {\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}\le e_i\\ g_4\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}\le g_3\\ g_6\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\le g_5\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\ge \alpha ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})\\ x_{ijk\_\min }\le x_{ijk}\le x_{ijk\_\max }\\ t_{ijr\_\min }\le t_{ijr}\le t_{ijr\_\max }\\ x_{ijs\_\min }^{\prime }\le x_{ijs}^{\prime }\le x_{ijs\_\max }^{\prime }\\ t_{ij\mu \_\min }^{\prime }\le t_{ij\mu }^{\prime }\le t_{ij\mu \_\max }^{\prime }\end{array}$$

(12)

3. Algorithm

3.1. Penalty Algorithm

The inverse optimization model is difficult to solve for the following reasons. First, it involves two levels of optimization that are interconnected, making it more complex than single-level optimization problems. Second, the problem has multiple local optima; therefore, the problem can have multiple feasible solutions. Thus, obtaining global optima becomes more complex. Third, the bi-level programming model is non-linear and NP-hard [17], making it more difficult to obtain an optimal solution in a reasonable amount of time. We solved the model-based KKT condition using the penalty function method. The Kuhn–Tucker optimality condition was used to transform the non-linear bi-level optimization problem into a regular non-linear programming problem for the low-level problem. The complementarity and slackness conditions of the low-level problem were applied to the high-level problem through a penalty function [19]. The inverse model was then transferred as the model (13) and directly solved using the MATLAB solver.

$$\begin{array}{c}\underset{x,x^{\prime },t,t^{\prime },a,b,c,d,u}{\min }{({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}-f^{*})}^2\\ a_1\le a\le a_2,b_1\le b\le b_2,c_1\le c\le c_2,d_1\le d\le d_2\\ \begin{array}{l}(u_1+\gamma ){\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}}}}+(u_2+\gamma ){\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}}}}+\gamma {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}}}}+\gamma {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{u}_{2+i}{a}_{ijk}}}}\\ +{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{u}_{2+i}{b}_{ijr}}}}+(u_{11}-u_{10}){\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}})}}+(u_{13}-u_{12}){\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }}}}\\ +u_{14}((\alpha ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }}}}))-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}}}}-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }}}})=0\end{array}\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}\le g_1\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}\le g_2\\ {\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}\le e_i\\ g_4\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}\le g_3\\ g_6\le {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\le g_5\\ {\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}\ge \alpha ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})\\ u_1({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}-g_1)=0\\ u_2({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}-g_2)=0\\ u_{2+i}({\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}}-e_i)=0\\ u_{10}({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}-g_3)=0\\ u_{11}(g_4-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}})=0\\ u_{12}({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}}-g_5)=0\\ u_{13}(g_6-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})=0\\ u_{14}(\alpha ({\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p{a}_{ijk}{x}_{ijk}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{r=1}^q{b}_{ijr}{t}_{ijr}}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{s=1}^u{c}_{ijs}{x}_{ijs}^{\prime }}}}-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{\mu =1}^v{d}_{ij\mu }{t}_{ij\mu }^{\prime }}}})=0\\ x_{ijk\_\min }\le x_{ijk}\le x_{ijk\_\max },t_{ijr\_\min }\le t_{ijr}\le t_{ijr\_\max },x_{ijs\_\min }^{\prime }\le x_{ijs}^{\prime }\le x_{ijs\_\max }^{\prime },t_{ij\mu \_\min }^{\prime }\le t_{ij\mu }^{\prime }\le t_{ij\mu \_\max }^{\prime },u\ge 0\end{array}$$

(13)

3.2. PSO Algorithm

A particle swarm algorithm has a strong advantage when solving multi-variable, non-linear, and large-scale optimization functions. The inertia weight of the PSO algorithm is determined by adaptively changing with the number of iterations. This is conducive to a quicker search for the global optimal value at the initial stage, obtaining a greater number of optimal solutions. With the increase in the number of iterations, the inertia weight decreases, and the local search ability is enhanced, which is conducive to a local search for other solutions close to the potential optimal solution.

The idea of the algorithm is that the particle moves in the solution space to track the individual extreme value and the population extreme value to update the position. The particle is required to calculate the fitness value every time it updates the position. It updates the individual extreme value and the population extreme value by comparing the fitness value. In this algorithm, the particle updates its velocity using Equation (14) and its position using Equation (15),

$$V_{\mathrm{id}}^{k+1}={\omega }^k{V}_{\mathrm{id}}^k+c_1{r}_1(P_{\mathrm{id}}^k-X_{\mathrm{id}}^k)+c_2{r}_2(P_{gd}^k-X_{id}^k)$$

(14)

$$X_{id}^{k+1}=X_{id}^k+V_{id}^{k+1}$$

(15)

where w^k is the inertia weight; D represents that the search space is D-dimensional, such that $d=1,2,\dots \dots ,D$; $i=1,2,\dots \dots ;n$ represents that there are n particles; $k$ is the current iteration number; $c_1$ and $c_2$ are the acceleration factors in the PSO algorithm; $r_1$ and $r_2$ are random numbers in the interval (0,1); $P_i$ is the position of the individual extreme value; $X_i$ is the position of the particle; and $P_g$ is the position of the group extreme value.

The inertia weight varied according to the number of iterations and

$${\omega }^k={\omega }_{\max }-\frac{({\omega }_{\max }-{\omega }_{\min })\times k}{K}$$

(16)

the particle swarm algorithm could be used to jointly optimize the upper and lower layers to solve the problem.

The steps used to solve the PSO algorithm were as follows:

Step 1: We entered the basic parameters, including the particle swarm algorithm acceleration factor, maximum termination generation, population size, maximum and minimum particle position, and maximum and minimum particle velocity.

Step 2: We initialized the population. According to the upper and lower limits of the decision variables of the upper-layer and lower-layer models, the initial population and the initial velocity of the particles were generated. The initial population was a matrix of random values generated by a rand function between the upper and lower limits of the variables.

Step 3: We calculated the fitness and discovered the optimal particle. Taking the objective function as the fitness function, the smaller the upper fitness value, the better the particle. The larger the lower fitness value, the better the particle. The initial population was included in the fitness function to calculate the fitness value. According to the obtained fitness value, the individual extreme value and the group extreme value as well as the individual best fitness and the group best fitness were discovered.

Step 4: We updated the weight, particle speed, and position. We obtained a weight value according to a formula and obtained a new subgeneration population.

Step 5: Boundary processing. According to the upper and lower limits of the decision variable, if the updated particle position was smaller than the lower-limit value or greater than the upper-limit value, the rand function was used to regenerate the value between the upper and lower limits.

Step 6: We judged whether the number of iterations was greater than the maximum termination generation. If yes, we proceeded to the next step. Otherwise, we returned to Step 4.

Step 7: We outputted the fitness value of the optimal individual and the optimal position of the particle and the iteration ended.

4. Numerical Analysis

4.1. Background and Parameter Setting

S Chemical Industry Park covers almost 30 square kilometers. There are 7 systems, such as raw material transportation system, production system, energy system, wastewater treatment system, environmental protection system, environmental monitoring system, and material storage system. Each system is divided into 9 subsystems. For example, the production system contains distillation, crystallization, adsorption, membrane processes, absorption, stripping, extraction, separation, and other subsystems. There are 16 kinds of equipment resources, 12 kinds of training resources, 28 kinds of material resources, and 19 kinds of human resources. $\mathsf{\alpha }\mathrm{i}\mathrm{s}$ 60%. The upper and lower limits of the price are the upper and lower limits of the bargaining range, which are 0.9 to 1.1 times the reference price, and the lower limit of the lower decision variable is the minimum quantity demanded, and the upper limit is 2 times the minimum quantity demanded. Other parameters can be seen in

Table 2. The upper and lower limits of each resource input.

Parameter	${\mathit{g}}_1$	${\mathit{g}}_2$	${\mathit{g}}_3$	${\mathit{g}}_4$	${\mathit{g}}_5$	${\mathit{g}}_6$
Value	2,000,000	1,200,000	4,000,000	3,000,000	1,800,000	1,500,000

and

Table 3. The upper limit of the total cost for each system.

Parameter	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$	${\mathit{e}}_7$
Value	500,000	500,000	500,000	500,000	600,000	600,000	600,000

.

4.2. Calculation Process

The forward optimization was solved using a computer with the following basic configuration: Intel (R) Core (TM) i7-1065G7 CPU @ 1.30 GHz, 16 GB memory (15.7 GB available) running Microsoft Windows 10 Home Edition, using MATLAB R2016b configured with CPLE 12.10. The basic computer configuration used in solving the inverse optimization model was an Intel (R) Core (TM) i7-1065G7 CPU @ 1.30 GHz, 16 GB memory (15.7 GB available) running Microsoft Windows 10 Home Edition and using MATLAB R2020b.

We solved the instance using an actuate algorithm and a PSO algorithm. A numerical analysis showed that the exact algorithm could only solve small-scale problems, such as problems with only four locations. In practice, the number of locations of safety resources needed in chemical industry parks is much larger than that. The particle swarm optimization parameters were as follows: maximum inertia weight $w_{\max }$ = 0.9; minimum inertia weight $w_{\min }$ = 0.4; acceleration constants $c_1$ = 1.5 and $c_2$ = 2; particle velocity set in the range [−1, 1]; and a population size of 20. The maximum number of iterations was 100.

4.3. Result of the Forward Optimization Model

The forward optimization problem was solved using CPLEX. The results of the forward optimization model based on the original parameters are shown in the following figures. The maximum safety input of the forward optimization was nine million. The running time of the forward optimization was 3.5374 s.

Shown in Figure 4a is the maintenance material configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the k-th maintenance material, and the input quantity of the material is represented by color. For example, x (1,1,1) = 20; that is, the input quantity of the first maintenance material in the first subsystem of the first system was 20, and so on. Figure 4b is the maintenance person-hour configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the r-th maintenance of human resources, and the person-hour input is represented by color. For example, t (1,1,1) = 40; that is, the person-hour input of the first repair and maintenance of human resources in the first subsystem of the first system was 40, and so on. Figure 4c is the configuration diagram of the direct safety guarantee materials. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, and the z-axis represents the s-th direct safety guarantee material. The quantity of material input is represented by color. For example, x’ (1,1,1) = 1; that is, the input amount of the first direct security material in the first subsystem of the first system is 1, and so on. Figure 4d is the direct security labor person-hour configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the u-th direct security human resource, and the person-hour input is represented by color. For example, t’ (1,1,1) = 8; that is, the person-hour input of the first direct security human resource in the first subsystem of the first system was 8, and so on.

4.4. Result of the Inverse Optimization Model

As shown in Figure 5, after about 30 iterations, the algorithm results tended to be stable, and the running time was 1.806268 s. The ordinate was the fitness value, which was the square of the difference. The optimal value of the result calculated by inverse optimization was F = 8,499,771.1; that is, the input value was 8,499,771.1. The difference from the expected result was 0.052 × 10⁶, which was 52,395.21. The safety resource input value obtained by the KKT condition was equal to the assigned value; that is, 8.5 × 106. The running time was 1442.292392 s. The result from the particle swarm optimization algorithm was close to the exact solution and the calculation time was much faster.

Shown in Figure 6a is a maintenance material configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the k-th maintenance material, and the input quantity of the material is represented by color. For example, x (1,1,1) = 23; that is, the input quantity of the first maintenance material in the first subsystem of the first system was 23, and so on. Figure 6b is the maintenance person-hour configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the r-th maintenance human resource, and the person-hour input is represented by a color. For example, t (1,1,1) = 41; that is, in the first subsystem of the first system, the person-hour input of the first repair and maintenance human resource was 41, and so on. Figure 6c is the configuration diagram of direct safety guarantee materials. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, and the z-axis represents the s-th direct safety guarantee material. The input quantity of materials is represented by color. For example, x’ (1,1,1) = 1; that is, the input amount of the first direct security material in the first subsystem of the first system was 1, and so on. Figure 6d is the direct security labor person-hour configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the u-th direct security human resource, and the person-hour input is represented by color. For example, t’ (1,1,1) = 11; that is, the person-hour input of the first direct security human resource in the first subsystem of the first system was 11, and so on.

Shown in Figure 7a is the price configuration diagram of repair and maintenance materials. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the k-th repair and maintenance materials, and the unit price of materials is represented by color. For example, a (1,1,1) = 178.2; that is, the unit price of the first repair and maintenance material resource in the first subsystem of the first system was 178.2, and so on. Figure 7b is the price configuration diagram of repair and maintenance labor hours. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the r-th repair and maintenance human resources, and the unit price of labor hours is represented by color. For example, b (1,1,1) = 23.9; that is, the unit price of the first repair and maintenance person-hour in the first subsystem of the first system was 23.9, and so on. Figure 7c is the price configuration diagram of direct security guarantee materials. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the s-th direct security guarantee material, and the unit price of the material is represented by color. For example, c (1,1,1) = 79.6; that is, the unit price of the first direct security material in the first subsystem of the first system was 79.6, and so on. Figure 7d is the direct security labor person-hour price configuration diagram. The x-axis represents the i-th system, the y-axis represents the j-th subsystem, the z-axis represents the u-th direct security human resource, and the person-hour unit price is represented by a color. For example, d (1,1,1) = 42.2; that is, the unit price of the labor hour of the first direct security human resource in the first subsystem of the first system was 42.2, and so on.

Finally, a comparison was conducted. The results are listed in

Table 4. Comparison of forward and inverse optimization results.

	Forward Optimization	Inverse Optimization
Total safety resources input	9,000,000	8,499,771.1
Total input time of training and human resources (hour)	80,391.9	80,819.2
Sum of the cost coefficient of time	72,669	67,149.9
Total input of non-human resources	36,208.9	37,052.0
Sum of unit prices of materials	455,266	425,020.2
Difference from the benchmark	500,000	228.9

.

4.5. Analysis of Results

(1) Regarding the input of the total safety resources, based on the original parameters, the optimal value obtained by the forward optimization model was F = 900 × 10⁴. This exceeded the benchmark of 850 × 10⁴. Based on the inverse optimization method, the input value was 8,499,771.1, which was much closer to the benchmark value.

(2) As shown in Figure 4, the error rate of the best value of the PSO was acceptable from the comparison of the results between random instances and actual instances. As the size of the instances increased, an exact algorithm was too slow to provide accurate solutions. The solving speed of the PSO was acceptable. Therefore, we concluded that the PSO could be used to solve the quaternion decision of the inverse optimization problem.

5. Conclusions

Most companies bear the discomfort of safety cost overruns and safety resource allocation, which is inefficient. Our research adopted a benchmark to solve the problem. The benchmark was a company that could minimize the consumption of human resources, financial resources, and material resources under the premise of ensuring safety. However, the allocation plans of safety resources are usually confidential. To infer a detailed plan of the benchmark, we adopted a quaternion decision of which types of safety resources were allocated at which point and at what quantity and price. This provided a key breakthrough point based on the target of benchmark safety. An optimal model of security and limited safety resource constraints were established. Once the security had reached a specific high point, the input of additional resources did not generate any effects to improve safety. Therefore, we applied the reverse-thinking of academic ideas and the inverse optimization method, converting the optimization model of maximizing safety to the inverse optimization model. Driven by the benchmark and by focusing on the inverse optimization model and algorithms for the bi-level programming of the quaternion decision, we achieved the modeling and solved the process of mechanism analysis → optimization modeling → dual conversion for inverse optimization → calculation by computer. We then used S Chemical Industry Park as a case study to apply the research.

This research is novel in solving quaternion decisions, changing the concept of safety management to guarantee safety by minimizing the consumption of resources. It not only provides new ideas and tools to solve the problem of the allocation of safety resources, but also promotes and strengthens the research into reverse engineering.

One future direction for the development of this research is the application of data-driven inverse optimization methods to consider real-time quaternion decisions.