Advances in Fuzzy Systems

Volume 2018, Article ID 4756520, 11 pages

https://doi.org/10.1155/2018/4756520

## Parameter Estimation and Sensitivity Analysis of an Optimal Control Model for Capital Asset Management

^{1}Department of Mathematics, P.M.B 373, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria^{2}Department of Mathematics, P.M.B 1515, University of Ilorin, Ilorin, Nigeria

Correspondence should be addressed to Tolulope Latunde; gn.ude.eyouf@ednutal.epolulot

Received 23 May 2018; Accepted 16 July 2018; Published 12 August 2018

Academic Editor: Katsuhiro Honda

Copyright © 2018 Tolulope Latunde and Olabode Matthias Bamigbola. 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

Optimal control is a very significant field of modern control theory which has been applied in many areas like medicine, science, and finance. This work is based on realization of asset values as a benefit of asset management where a capital asset management problem is modelled and expressed mathematically from the perspective of an investor whose income is generated by return and capital gains on investments with price and return on assets assumed to satisfy uncertainty process. This results in an optimal control model based on uncertainty theory which relates two or more parameters that measures the condition or state of individual’s investments. These parameters enable us to know the condition of risk involved in asset management and how to maintain and manage the assets in order to maximize expected present value of the utility of asset and minimize the risk involved to aid capital investment decision-making. Parameter sensitivity analysis is an approach given to a model so as to define significance of the factors related to the model where the whole parameter space is fully described. However, the model is applied to a real-life problem of capital asset management to deal with debt crisis of a nation’s economy and the sensitivity analysis to determine the effects of the input factors on the model is investigated such that relative significance and sensitivity of each parameter on the model results are presented using parameter estimations. Finally the optimal control decision policy is obtained and discussed.

#### 1. Introduction

Capital assets are significant pieces of property such as plants and machinery, land and buildings, vehicles, and estates. For businesses, they have useful life usually longer than a year. Risky capital assets are the capital assets that carry a degree of risk, that is, assets that have a significant degree of price volatility, depreciation, hazards, inflation, or liability.

The Institute of Asset Management (IAM) explains that asset management which is the planned action of an organization to acknowledge value from assets is no more new; people and organizations have been managing assets for a large number of year. What has changed, however, is the cumulative recognition of what good asset management involves, the optimizing of costs, risk performance, resources and benefits over a given time, and likewise considering that risks are inherent in all decision-making. The typical priorities of asset management are keeping stakeholders happy, substantial returns on investment compliance and sustainability, systems performance, cost and risk optimization, and efficiency and effectiveness of an asset’s life cycle.

In dealing with real-life or physical problems, mathematical modelling is always of great advantage because of its power to predict system behaviour and a clear insight of the important inputs and outputs. Mathematical models are of various forms such as deterministic, stochastic, fuzzy, and uncertain forms, Mazur [1].

However, there is a need to model problems arising from the successful management of capital assets using the following essential methodology: identifying the needs of customers, regulators, or investors; designing or formulating the model; utilizing and maintaining the model; and managing residual liabilities. Thus, following the methodology, an uncertain optimal control model of capital assets is developed to tackle some problems that arise in the optimization of capital assets such that the expected net worth is maximized and capital growth is attained while risk is minimized. Here, maximizing the expected net worth of the investment is considered as the objective of the optimal control while the present net worth is considered as the constraint which is expressed as asset-liability problem.

In most practical problems, we are interested not only in the optimal solution of the control problem but also in how the solution changes when the parameters of the problem change. The change in the parameters may be discrete or continuous. The study of the effect of discrete parameter changes on the optimal solution is called the sensitivity analysis while that of the continuous changes is termed the parametric programming, Rao [2].

Parameter sensitivity analysis is an essential method for examining mathematical models of a real-life problem. A detailed parameter sensitivity analysis gives a broad set of predictions that show how changes in a model parameter affect relevant model outputs. A parameter is a characteristic, a measurable factor, a feature that can help in defining a particular system. Therefore, the effects of changes in the parameters of a model are determined by solving the model and comparing the results with respect to changes made with parameters in the model’s configuration space.

From existing works, it is noticed that Merton [3], Zhu [4], and Deng and Zhu [5] examined the selection risk-free asset and risk asset together in the formulation of the models. Meanwhile, Stein [6] examined the selection of risky assets only with stochastic optimal control approach without considering depreciation and taxation as input factors. Consequently, in this research work, uncertainty theory is utilized in the model formulation whereby the selection of risky assets only is examined; thus, depreciation and taxation are considered as input factors in the formulation of the model. Furthermore, the relative significance of each parameter of the model is determined in order to describe the ability to move the model around the assets’ entire configuration space with minimum execution time.

#### 2. Literature Review

Real-life problems are often modelled as mathematical expressions which sometimes include parameters. Examples abound in electrical engineering, finance, economics, and medical sciences, to name a few. A model is defined as a simplified representation of certain aspects of real-life system. A mathematical model is a model created using mathematical concepts such as functions and equations. When mathematical models are created, it is assumed that there is a movement from real world into the theoretical world of mathematical concepts, where the model is built. The model is then manipulated using mathematical or computer aided techniques. Finally the real world is reentered, which is then translated into useful solution to the real problem which implies that the start and end are in the real world, Edward and Hamson [7]. Thus, several researchers have worked on the applications of mathematical modelling into solving real-life situations such as in mathematical modelling, a useful tool for STI control policy, Ashleigh et al. [8], mathematical modelling application to corrosion in a petroleum industry, Oyelami and Asere [9], dynamic modelling of a wind turbine with doubly fed induction generator, Poller and Achilles [10], and a mathematical theory to model complex socioeconomical systems by functional subsystems representation, Giulia et al. [11]. As such, much emphasis has been on the operational aspect rather than the economical aspect of mathematical modelling. However, several researchers have also worked on the sensitivity analysis of the modelling parameters used in various types of models. Rosa and Torres [12] carried out a sensitivity and cost-effectiveness analysis on periodic epidemic model, Papageorgiou et al. [13] carried out sensitivity analysis for optimal control problems which is governed by nonlinear evolution inclusion, Fordor et al. (2010) worked on parameter sensitivity analysis of a synchronous generator, Zivarian (2002) also worked on sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics, Kathirgamanathan and Neitzart [14] worked on optimal control parameter estimation in Aluminium extrusion for given product characteristics, Guo et al. [15] worked on the performance evaluation and parameter sensitivity of energy-harvesting shock absorbers on different vehicles, Christopher and Fathalla [16] worked on sensitivity analysis of parameters in modelling with delay-differential equations, Burns et al. [17] worked on sensitivity analysis and parameter estimation for a model Chlamydia Trachomatis infection, and Bastidas et al. [18] worked on parameter sensitivity and uncertain analysis for a storm surge and wave model to mention but a few. The main concern in mathematical modelling is to establish relationships between factors but this also almost invariably involves parameters. Parameters are quantified factors that have constant values for a particular problem but can change from problem to problem, Edward and Hamson [7]. In general, a model can be of some use in predicting general behaviour in a descriptive fashion. To use the model in a practical way, we must solve for the numerical values of the parameters from given data. The use of data to obtain parameter values relevant to a particular application of the model is frequently called or sometimes the model. The common methods of obtaining the parameter values are graphical, statistical (usually involving least-squares estimation), and mathematical (usually requiring the solution of linear or nonlinear equations) methods. Parameters are used to identify characteristics, features, and measurable factors that can help in defining a specific system. They are essential elements to take into consideration for the evaluation of an event, a project, or any situation.

The optimization of asset management is influenced by a large number of processes, which has been modelled by many researchers with various mathematical methods based on different theories. Tunjo and Zoran [19] worked on financial structure optimization using a goal programming approach which proposes a new methodology for solving multiobjective fractional linear programming problems using Taylor’s series formula, Xiaoxia [20] worked on portfolio selection with a new definition of risk in which a new type of model was proposed based on his new definition, and Schyns [21] worked on financial data and portfolio optimization problems where he deals with an extension of Merkowitz model and takes into account some of the side-constraints faced by a decision-maker when compromising an investment portfolio. There are many approaches which researchers have used in optimizing the asset management and measure risk. These are not limited to fuzzy approach, log robust approach, heuristic approach, stochastic approach, and uncertainty approach.

In most practical problems, the interest is not just in the solution of the problem, but also in how the solution changes when the parameters of the problem change. Parameter sensitivity analysis is thus used to decide how sensitive the results of propagation of a parameter by varying the estimation of the parameter in a model. A mathematical model comprises parameters, in which the more the parameters existing in the model, the higher the dimension of the model and the more complex to solve. Thus, the sensitivity of parameters makes it easier in determining the relative significance of each parameter, thus modifying the model by reducing the model’s dimension which reduces the complexity.

The optimal control model of risky capital assets based on uncertainty theory, however, requires over 15 input parameters. These parameters help in quantifying some factors utilized in the model. It is therefore necessary to survey available values for these input parameters and ascertain the sensitivity of the model to changes in every one of the parameters.

The sensitivity of parameters helps in classifying modelling parameters into different types on the relative significance of each parameter related to a model. The classification can be done using the information in Table 1.