Design of an Analysis Model for Strategic Behavior in the Digital Economy

Nowadays, multi-criteria decision-making techniques are highly developed, and are widely applied in multiple fields. They model and solve decisional problems by optimising multiple conflicting objectives. These techniques are very useful because they simultaneously analyse all the different criteria, and select the best alternatives according to the decision-maker’s objectives and preferences. An important issue in this context is the adequacy of the structure of corporate long-term financing and its potential impact on the sustainable development of the long-term business plan. The purpose of this study is to advance the analysis of these strategic decisions, measuring the a posteriori results and analysing their coherence with the strategies followed a priori. To do this, sustainable strategic decisions will be mathematically modelled and parametrised, creating a system to study the preferences followed and to describe the corporate behaviour. This system is applied as a case example for two leading companies in the digital sector, and the corresponding results over the last few years are evaluated.

The Linke and Withford model pioneered the use of multi-criteria techniques to develop a financial planning model to suggest efficient rates for the electricity market. In 1989, Batson's compilation work had already described up to 10 successful applications in economic-financial spheres that applied the goal programming methodology. However, it was limited by the amount of time taken to collect the data, and the indivisibility of the projects, which limited the different models to a simple "conflict resolution" tool.
These developments allowed for the expansion of more complex problems to more specialised environments, solving difficult problems. Thus, in 1995, the Goedhart and Spronk model of financial planning with fractional objectives emerged, which enabled the linear goal programming algorithms to be extended to financial objectives. In 1997, the Maranas et al. model   Given the potential of multi-criteria decision-making techniques in these areas, the inverse question could be presented as a hypothesis of this study. Can it be verified that a company's strategy can be quantified based on the results it presents? Would it be reasonable to assume that, as maintained by economic behaviour theories, companies follow criteria other than those of profit maximisation as the sole strategic objective to guide their strategy?
Taking different models of strategic planning developed by and as a starting point, this work aims to give continuity to the hypotheses of these models and apply to the business environment the methodology developed by Sumpsi et al. in the field of agricultural research, which, based on the results obtained, evaluates the fulfilment of a company's objectives and the strategy followed. In general, given that any financial decision involves a balance or equilibrium between conflicting elements with different criteria (for example, increasing dividends to improve internal financing; increasing financial leverage for solvency; choosing whether to finance using internal or external resources, etc.), an issue of notable practical interest for decision-making techniques is raised when observing the results obtained by companies when different conflicting criteria are found.

Methodology: Mathematical Formulation of an Analysis Model for Strategic Behavior
The general formulation of a multi-criteria problem in which q represents relevant conflicting objectives is usually presented as follows: (1) Subject to x  F where "Eff" is an operator that indicates the search for efficient or Pareto-optimal solutions; x is a decision variables vector for the problem; f j (x) the mathematical expression of the j-th objective; and F is the feasible set, or those solutions that provide an answer to the problem's restrictions.
For the particular case of weighted goals, the following inputs can also be defined: -: Weight associated with the priorty of the j-th objective.
-: Ideal or optimal value of the j-th objective.
-: Observed value for the j-th objective.
-: Value obtained for the j-th objective, when the i-th objective is optimised (i.e., ).
-: Positive and negative deviations of the j-th objective with respect to the observed value.

Mathematical Formulation of a Strategic Planning Model
The multi-objective strategic planning problem proposed by Martin et al. focuses on five main strategic objectives of long-term sustainability (company expansion or growth, solvency, theoretical value of the share, shareholder dividends and operational efficiency), which are calculated from six strategic decision variables linked to the company's equity increases in its long-term balance sheet:  Establishing the following restrictions to maintain the company's long-term sustainability: -The solvency of the company must be higher than the prior value of solvency: -The theoretic value of the share must grow: -The company must distribute a minimum share, at least  per cent of the capital, depending on each company: -The annual cost of amortisation must not exceed established legal limits: -The annual increase in assets must be covered by permanent resources, i.e., capital increases, reserves and debts: -The company's gross profit must be distributed in reserves and dividends: -The internal rate of return of the benefit after tax must be higher than a certain level: -The decision variables cannot be negative,

Behaviour Analysis Model Based on the Observed Results
The methodology proposed by Sumpsi et al. allows us to estimate the decision-maker's observed behaviour "as if" it were subject to the defined mathematical model. To do this, we start with the individual estimation of the preferences based on the payment matrix and follow with a posterior approximation of the behavioural vectors with a weighted goals programming scheme.
Given that the payment matrix is by definition one in which the columns (j=1..q) correspond to the vectors associated with the individual optimisation of each of the objectives that meet the problem's restrictions, its calculation can identify the degree of conflict between each of the objectives: The transposition of this matrix allows a system of equations to be presented such that if the vector of preferences w was canonical of its j-th component (for example, if w=[1,0,…,0] T ), the solution obtained by this system would correspond exactly with the optimisation of the j-th objective.
Therefore, a linear combination of these possible solutions associated with a different weighting of the w preferences would generate the following system: that is, that the combination of weights w=(w 1 ,...,w j ) which, on the basis of the preferences shown in the payment matrix, generates the observations vector f=(f 1 ,...,f j ).
This system, in general, has no solution because of the problem's restrictions and the existing conflict between the objectives. However, it is possible to seek an approximate solution under Goal Programming conditions by considering the observed values as the goals to be achieved. In this case, the achievement function would correspond to the minimisation of the sum of the deviation variables normalised to the value observed in each of the objectives.
The formulation of the problem of behaviour analysis would correspond to solving the following system of equations: Subject to the equations system: In general, the last restriction is proposed so that the set of preferences is normalised. The resolution of this system would allow the decision-maker's behaviour and preferences to be identified, based on the observed results and the approximate solution.

Application Scenario
To study the applicability of this predictive analytical model (6)    It can be observed that both companies are well financed in the long term, because the value of their resources is higher than that of their assets, and financial leverage has increased over the period.

Result
The first step is to obtain the payment matrices (3) associated with the two companies. The results for Apple Inc. are presented in Table 3, and after the application of the approximation model, the preferences vectors are displayed in Table 4.   Tables 5 and 6 display respectively the payment matrices and preferences for Microsoft Inc. The results from Tables 4 and 6 show a reasonable margin of error in both cases, so long as the real values remain within the ideal-anti-ideal range. When this is not the case, for example in the year 2015, the prediction error rises noticeably.
The main reason for this is that the model presents long-term sustainability as a starting hypothesis, based on the annual improvement of solvency and value creation ratios. If this does not occur, as in the case of solvency owing to the increase in financial leverage, the value obtained is outside the approximation range. The rest of the objectives (growth, dividend and efficiency) must maintain positive values, as is the case in reality, and so do not present a problem.
The data obtained show that the main difference between the two companies is that the creation of measured value as a ratio of Net Equity to Capital increases in the period for Microsoft and decreases for Apple. As expected, the prediction error is lower for Microsoft because its data fits the analysis model better. As a consequence, the forecast is of better quality during the years in which the actual solvency level is closer to the estimated range. Therefore, the hypothesis presented regarding the validity of the behaviour analysis study based on the results will depend primarily on whether the company fits the hypothesis of the proposed mathematical model.

Conclusions
In this article we have investigated the possibility of quantifying a company's strategy based on the presented results. Starting from this hypothesis, we have applied MCDM methodology to analyse the a posteriori decision-maker's behaviour "as if" the company were following a multi-criteria decision strategy model based on 5 key drivers in conflict to achieve long-term sustainability success: company expansion, solvency, theoretical value of the share, shareholder dividends and operational efficiency.
By approaching the percentage of success achieved for each objective from the observed results in the payment matrix, we have been able to evaluate the corporate's behaviour according to the model. Our first conclusion is to note the high correlation shown by the payment matrix in all the objectives except the dividend. The maximisation of the dividend objective creates the anti-ideals values of the matrix, while the ideal values are obtained simultaneously, optimising any of the remaining objectives.
This result could be expected a priori because, if the resources generated are paid directly to the shareholder, the company's self-financing and potential for growth is reduced in the rest of the objectives.
Secondly, setting to one side the year 2015 in both cases because of the high margin of error, similar behaviour can be observed in the behaviour models for both companies because they prioritise efficiency and profitability over the dividend objective. For Apple, the weight that maintains the objective of efficiency reaches 97%, while for Microsoft it is 70% compared to 30% of the target dividend on average over the last three years.
Finally, it is important to highlight the improvement in these results with regard to the supposed equipreferential (i=1..5), common in cases where there is little information about the decision-maker's habits. The prediction error rate has, in this instance, increased in the results that were obtained with the goal approach methodology.
These results lead us to believe that, in order to properly analyse a company and be able to make correct future predictions about it, it is essential to understand management's prior behaviour in order to subsequently create the strategic proposals that weigh up the company's preferences or new future strategy in the most effective way possible. This behavioural analysis model allows us to identify this information and present future objectives in a coherent way that can be complementary to multi-criteria interactive methods by analysing the previous behaviour of the decision-making centre before the objectives are set.
In terms of future lines of research in this area, the extension of this behaviour analysis model to other more precise financial planning models could be considered, whether in the long term or in the accounting balance that best adapts to the behaviour of the companies analysed during the study period.