# Multivariate Mathematical Methods in Decision Making

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Decision making (DM) process supports experts, managers, practitioners, and many other professionals (among others) to address concerns and solve problems by investing available alternative choices (Reimann & Bechara, 2010). This shows that making rational decisions is important, and also that using scientific methods in DM is prudent. Simple decisions are easier to make, which is often not the case with complex ones. In order to demonstrate the details required in solving a real life complex problem, Guo (2008) defines DM as the thought process of picking a rational or logical choice from the available options. When attempting to make a skillful decision, a decision maker should evaluate and compare the advantages and disadvantages of each option, and consider all the alternatives. Effective DM requires the decision maker to forecast the outcome of each option involved. Moreover, based on all these items, the decision maker should determine the option that outsmart all others for that particular situation.

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According to Brunton, Botvinich and Brody (2013), when decision makers are rational and unrestricted to make decisions, then they would probably apply rational choice theory (RCT) which uses scientific methods in DM. Some decisions made in high profile instances seem to lack rationality. Losses and wastages are experienced in many instances due to lack of rationality. Lack of rationality is a drawback in economic development. For brevity of a research proposal, rationality shall imply logical or scientific approach.

Decision making (DM) algorithms and procedures are designed for univariate statistical methods while most applications occur in multivariate forms. As a result, standard multivariate methods are lacking and then ad hoc methods are used often as a compromise. When these methods are used, monitoring of the correctness of results or outcomes become cumbersome, because manipulations used in deriving solutions may suit the user instead of the scientific technique. Solutions that lack a science can also evade ethics and legislations, since science is the method that enable systematic observation, measurement, experiment, formulation, testing, and modification of hypotheses (Kuhn, 2012). Making ad hoc decisions limit usage of science while existence of standard methods enable increased usage and more learning. Cases where standard methods lack have limitations in solution provision. For example, it may be difficult to judge if the solution is adequate or substandard.

DM is an important activity in almost all aspects of real life. Existing approaches where multivariate decisions are made are classified under multiattribute (MA), multicriteria (MC) and multiobjective (MO) (Reimann & Bechara, 2010). These are then termed multiattribute decision making (MADM), multicriteria decision making (MCDM) and multiobjective decision making (MODM). Users of these three methods tend to believe that the methods use exactly the same mathematical procedures to address decision problems. This is because of lack of standard methods, which need to be developed. In the proposed study, multivariate analysis are used to determine the standard methods for adressing each of the items. The methods developed intend to show that attributes, criteria and objectives are generally not necessrily the same, and have different roles in DM problems.

### The aim of the study is to develop multivariate DM models using matrices and vectors. The objectives of the study are to determine:

• attributes in multiple settings;
• mathematical formats for fulflment of multiple criteria; and
• mathematical methods for multivariate fulfilment of multiple objectives.

### Rationale or motivation

A complex environment is an environment with a large number of different possible states which come and go over time. Multivariate methods are important because studies (Barbey, Colom & Grafman, 2014; Moutsiana, Garrett, Clarke, Lotto, Blakemore & Sharot, 2013; Reyna, 2013) show that more complex environments correlate with higher cognitive function, which means that a decision can be influenced by external factors.

Naturalistic decision-making (DM) in situations with higher time pressure, higher stakes, or increased ambiguities show that experts may use intuitive DM which may lead to flaws when compared to structured approaches. Experience shows that following recognised primed decision that suits the experiences without weighing alternatives may lead to a course of action that does not provide desirable results.

This study represents innovation in mathematical sciences through the use of operations research approaches with a foundation in Statistics. Due to the ambition to improve practice, operations research is the field in which the study rotates. Matrix theory will serve the notation because many variables are involved. The mathematical properties in matrices will be reviewed as to what they mean or imply in practice.

Mathematics provides exact solutions to abstract problems in the sense that the definitions of maths are unique and the solutions are either precise or incorrect (Tokuhama-Espinosa, 2010). The main proven principles of maths occur as theorems. These are the principles necessary in the development of the solutions envisaged in this study. In this study, matrix methods in which matrices and vectors are integrated in order to derive formulae for mathematical methods for dealing with them in MADM, MCDM and MODM are considered similar. However, there are unique differences reflecting each concept. The study unpacks various methods and exposes some of the differences. Attributes in MADM are features. They are defined as characteristics. Criteria in MCDM are measures that give the required levels of the characteristics. Then the objectives in MODM are the required outcomes.

The study also intends to elaborate on the assumed similarities for mathematical methods of MADM, MCDM and MODM in order to enhance an improvement in DM. Attributes are features defining the premise or proposition for the DM setting. While there are criteria for final decisions, the issue of ‘Criteria’ is an important component in DM. A DM processes usually involve Attributes, Criteria and Objectives (ACO). In the discussion, ACO shall refer to cases where Attributes, Criteria and Objectives, respectively, are emphasised or acknowledged.

### Matrix Theory

Notation is important in mathematical formations, especially when there are many variables involved in instances known as multivariate or multidimensional settings. Without sensible notation, mathematical principles are not easy to develop because appropriate definitions become impossible to generate. Further, the convenience of notation is also beneficial for thinking fast (Kahneman, 2011).

Notation of univariate instances is easier to design, but when there are many variables, it can be difficult to define properly. Real life existences show that multivariate settings are more realistic than univariate because every existing entity exists around others, even of different formation. Huang and Ma (2004) point out that the multidimensional environment should be appraised and recognised by including methods that are useful in problem solving, including DM. Multivariate methods are based on many variables, which, when stacked together, form a vector (Hur, Lee & Kwon, 2005). Multivariate statistical methods show that derivations emerging in matrix methods include matrices, such as the covariance and correlation matrices.

### Calculus

Calculus, on the other hand, is viewed as the mathematical study of continuous change (Katz, 2008). According to Zill, Wright and Wright (2009), calculus methods are used in optimisation theory. For this study, incorporation of calculus with matrices is aimed at enabling adaptation of the derived methods due to the expectation that some settings are unique and when conditions change in the future, the solutions derived should also be adaptable.

### Rational Choice Theory

McKinnon (2013) defines rational choice theory (RCT) as a charter for understanding and often formally modelling social and economic patterns. Due to the purpose of RCT to inspire actual application, RCT is also known as rational action theory. The basic premise of RCT is that amassed shared conduct or performance results from the activities of individual variables involved, or actors, each of whom is making their individual decisions. The theory also focuses on the determinants of the individual choices. RCT therefore, assumes that an individual has preferences among the available choice alternatives that permit them to state the option that they prefer. According to Nell and Errouaki (2011), these preferences are also assumed to be complete. This means that the person can indicate the alternatives that they consider preferable, or even when neither is preferred to the other. It is also transitive, which means that if option A is preferred over option B and option B is preferred over option C, then A is preferred over C. RTC is based in probability theory as well. The rational agent of RCT takes account of available information, probabilities of events, and potential costs and benefits in determining preferences. It also ensures consistent performance when choosing the self-determined best choice of action.

### Decision Making

Decision making (DM) is the cognitive process resulting in the selection of a belief or a course of action among several alternative possibilities (Sharot, 2011). Sharot, Korn and Dolan (2011) regard DM as a problem-solving activity that concludes with a solution that is considered to be optimal, or at least satisfactory. Every DM process produces a final choice, which may or may not prompt action. Thus, DM entails the process of identifying and choosing alternatives based on the values, preferences and beliefs of the decision-maker. Forsyth (2014) enlightens that one main DM component entails the analysis of a finite set of alternatives described in terms of evaluative criteria. The task may require ranking of these alternatives in terms of their attractiveness to the decision-maker(s) when considering all the criteria simultaneously. Another possibility is to find the best alternative, or to determine the relative total priority of each alternative when considering all the criteria simultaneously. Solving such problems is the focus of MADM, MCDM and MODM. Rational DM is vital in all science-based professions, where specialists apply their knowledge in an area of concern to make informed decisions.

### Methodology and analytical procedures

The study will interrogate univariate MADM, MCDM and MODM methods and then create their extensions in multivariate setting. Properties of emerging matrices will be proved and emerging relations of variables in the newly formed matrices such as covariance matrices will be derived. Premises in mathematical methods start with abstract assumptions.

### Analytical Processes

Matrix methods such as their additions, properties of matrices, and special matrix will be brought in and the meaning or implications derived for MADM, MCDM and MODM practices. Covariance and correlation matrices play an important role in analysing relationships of variables. They will be given special roles in the study. Also, this study does not involve a data collection tool.

### Scientific contribution

According to Thurs (2011; 2015), new mathematical models and principles make a scientific contribution. This study plans to develop new models in mathematical science. In addition, the benefits/contributions will enable methodical practices. Integrating common DM methods with matrix methods will lead to more theorems in matrix theory and applications. Due to the practical focus and capability, the study therefore will provide a comesnurate use of previously dissimilar aspects in DM. In addition, the roles of each of the components of ACO will be clarified and amplified in clearer details. Moreover, MADM, MCDM and MODM mathematical methods will be derived and compared in multivariate statistical theory.

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