An Intelligent System for Allocating Times to the Main Activities of Managers

Abstract

Correctly allocating times to the main activities of a manager is a crucial task that directly affects the possibility of success for any company. Decision support based on state-of-the-art methods can lead to better performance in this activity. However, allocating times to managerial activities is not straightforward; the decision support should provide a flexible recommendation so the manager can make a final decision while ensuring robustness. This paper describes and assesses a novel approach where a search for the best distribution of the manager’s time is performed by an intelligent decision support system. The approach consists of eliciting manager preferences to define the value of the manager’s main activities and, by using a portfolio-like optimization based on differential evolution, obtaining the best time allocation. Aiming at applicability in practical scenarios, the approach can deal with many activities, group decisions, cope with imprecision, vagueness, ill-determination, and other types of uncertainty. We present evidence of the approach’s applicability exploiting a real case study with the participation of several managers. The approach is assessed through the satisfaction level of each manager.

1. Introduction

Time management is an issue that occupies the attention of individuals and society in general. So much importance is attached to time management that several countries have national surveys to measure it [1]. In Mexico, the National Survey on the use of time provides statistical information on the measurement of paid and unpaid work to make visible the importance of domestic work.

Managers’ expert knowledge is often the main way how time is allocated to activities. These decision makers usually dominate most of the relevant information that, implicitly and holistically, allows them to make the required set of judgments. Furthermore, expert knowledge from other managers belonging to similar contexts could imply even better decisions. However, managers often dedicate most of their time to the most urgent issues, that is, day-to-day operations. This situation has the consequence that issues with the greatest impact on strategy and the increase in business competitiveness, as time allocation, are being relegated.

A poor time allocation by the manager of an organization can affect the person, the organization, and society in general. That is why various proposals recognize the importance of this managerial ability, but it is still an unsolved problem for many managers. Stephen Covey proposed a time management matrix in 1996, in which he placed important and urgent activities in the first quadrant, important but not urgent tasks in quadrant 2, and, in the last quadrant, activities that are neither important nor urgent, but that by habit the individual performs them and unconsciously steals a large amount of time [2]. “The development of time management is often divided into four generations. In the first generation, what needs to be done (answer the question of what to do) is mainly solved. The second generation adds the question of when it is necessary to do it. The third generation extends the previous two by adding the question of how to perform the task. Finally, the fourth generation focuses mainly on man, self-knowledge and management”. The complexity of current times requires the use of information technology in this area of knowledge [3].

Computational intelligence, defined as the technology that allows to exploit computational power to solve “hard” problems, has seen a significant increment in the number of applications within many different fields during the last decades. Undoubtedly, computational intelligence has become a current of thought that has characteristics to help support in the problem of deciding how managers should allocate their time.

Improvements through computational tools help with the reduction of processing times and with optimization [4]. Tools provided by computational intelligence are useful in problems with vital impacts, such as ranging transport models of medical emergency services, prediction of meteorological data to minimize its economic impact, and problems where the result is the recommendations of tourist attractions, for example [4,5,6].

In most cases, the manager is not able/willing to determine precise proportions of his/her time to perform routine activities. However, the manager may wish to define minimum and maximum acceptable proportions of time. It is also possible that, originating in vagueness or imprecision within the manager’s mind, this minimum acceptable value is not well defined and/or the manager does not feel comfortable in expressing it as a precise value. Therefore, a decision support system that exploits computational intelligence to recommend plausible proportions of time to allocate to the diverse manager’s activities must be prepared to deal with such situations of uncertainty.

There are several ways to introduce uncertainty into modeling. Fuzzy numbers are a plausible way to model the situation where a number x is not precisely known, but the range where it lies and its distribution are known. This type of numbers is used in many areas, from robotics to finance. In [7] they evaluated the risk associated with investments and in [8] to rank alternatives. There is another way to introduce uncertainty that is potentially simpler than fuzzy numbers, interval numbers. Interval numbers characterize the range where an unknown quantity may be by using uniform distributions. Interval theory allows to determine if a given interval number is greater than or equal to another interval number in a straightforward and intuitive manner [9]. In [10], interval numbers were used to assess risks based on the probability of fails. In [11], a drought risk model is built and evaluated through interval numbers. Solares et al. [12] exploited the ordering properties of interval numbers to characterize and compare investment portfolios. Similarly, Fernandez et al. [13,14,15] not only characterized investment portfolios through interval numbers, but also the preferences of the investors. Interval numbers are remarkably related to fuzzy numbers [16,17,18].

Another important issue addressed in this paper is the modelling of preferences. Modelling of preferences has been used in different ways, such as in [19], where a fuzzy technique is applied for order preference and ranking of the alternatives concerning the reverse logistic problem; or by [20] where preference modelling is used to generate individualized interactive narratives based on the preferences of users to improve user satisfaction and experience. One way to model preferences is by using value functions. These types of functions are the result of a very well-studied and well-known theory (see e.g., [21,22,23]). Recently, value functions have been used to assess project portfolios [24], providers [25], and market segmentations [26], for example. There have been important efforts in creating effective methods to elicit the parameter values of value functions [27,28,29,30].

On the other hand, a very effective tool for optimization problems whose decision variables are continuous is differential evolution (DE). This is a tool commonly used for solving engineering problems and it has become very popular among computer scientists and practitioners [31]. Recently, this technique has been introduced in other areas such as finance, marketing, and customer segmentation, as in [15,32].

Regarding the case study of this work, namely the time allocation for managers’ activities, some researchers have intended to analyze the impact and importance of this issue, such as in [33], where it is shown how time allocation for sales managers affects the sales team performance. Their findings underscore the importance of effective time management for sales managers across a core set of leader behaviors. Moreover, they argue that when managing more (less) experienced teams, managers should focus on spending more time on managing people (customer interaction). In [34], the influence of entrepreneurs’ individual entrepreneurial orientation (IEO) on their time allocation behavior is studied. The findings indicate that proactiveness and risk-taking are associated with specific time allocation behaviors. In [35], it is identified that a core micro-foundation of dynamic managerial capabilities is the ability of the manager to allocate their own time; and it is illustrated that failure to allocate time to capability enactment can lead to capability vulnerability. In [36], it is said that time management practice can facilitate productivity and success, contributing to work effectiveness, maintaining balance, and job satisfaction. Furthermore, the decision support provided to managers must offer flexible recommendations such that the manager can make a final decision according to his/her specific context, but without affecting the approach’s effectiveness. In this work, we intend to achieve all these goals.

To the best of our knowledge, there are not published methods that explicitly allocate times to managerial activities. This is likely due to the difficulty for managers to follow strict schedules in ever-changing environments. The problem is, of course, that neglecting the support of recent advances in technology and modeling is discouraged. Traditional approaches to this type of problem consist of mathematical models exploited by exhaustive optimization methods that lead to precise output values where strict schedules are recommended. The manager is not given the option to make a final decision where his/her expert knowledge and that of other managers is used. The recommendation also does not have the flexibility to adapt to unexpected situations. This implies a lack of robustness of traditional approaches. Therefore, in this work, we aim not only to propose a time allocation that best suits the manager (from the perspective of a given objective function), but also to provide sufficient flexibility for the manager to adapt to real-world situations such as considering group decisions and uncertainties. We present a novel way to hybridize some state-of-the-art methods from some theories of the literature (see Section 2).

The paper is structured as follows. In Section 2, we describe the materials and methods used in this work. Section 3 presents the methodology of the proposed approach and details the design of experiments. Section 4 presents and discusses the obtained results. Finally, Section 5 concludes this paper.

2. Materials and Methods

We propose using value functions to model the preferences of the managers (i.e., what the value of each activity is), interval numbers to encompass the uncertainty related to such preferences and to the ideal solution (i.e., on what range the ideal time allocation is), and differential evolution to determine such an ideal solution.

2.1. Value Functions

Value functions [37] can be used to represent the preferences of the managers over a set of decision alternatives regarding time allocation. Value functions are a simple yet effective and very useful way of modeling preferences. This paradigm is particularly relevant here because of the difficulty of defining a mathematical function whose inputs are the times allocated to the activities. It is not straightforward to define tangible impacts on the organizations’ objectives for such function. Therefore, the expert opinion of the managers regarding what they consider most convenient is valuable. Value functions are therefore used to represent such opinions by defining “how desirable it is to spend time on a given activity” from the perspective of experts.

The main goal of this type of function is to build a way of relating a real number with each alternative, such that an order on the alternatives can be produced that is consistent with the decision maker’s preferences. To achieve it, the theory of value functions assumes the existence of a real-valued function representing the preferences of the decision maker. This function is used to transform the attributes of each alternative into a real number.

2.2. Interval Numbers

An interval number (see [38,39]) represents a numerical quantity whose exact value is not well defined, that is, it is not exactly known. However, it is known that the quantity is within a range of numbers. Let r be a real value lying between r^– and r⁺. The interval number representing r is therefore R = [r⁻, r⁺]. A real number s can be represented by an interval number as [s, s]. For clarity purposes, we use bold italic font to denote an interval number.

Notice how the nature of the uncertainty modeled by interval numbers is different from that modeled by fuzzy logic. In the former, knowing that the quantity is within a range of numbers is the only known information; in the latter, usually more information about the quantity and range are stated. Therefore, fuzzy logic could handle the proposed approach information in a more sophisticated way than interval theory. However, such sophistication comes at a price, complexity. For example, it is straightforward to determine if an interval number is greater than or equal to another (see function Poss below), which usually is not the case for fuzzy numbers. Here we focus on the scenario where the interval numbers are sufficient to model the problem and defer the modeling of such information through fuzzy logic to future research work where the authors are already working.

Basic arithmetic operations that can be performed using interval numbers are addition, subtraction, and multiplication. Let A = [a⁻, a⁺], B = [b⁻, b⁺] be two interval numbers. The arithmetic operations between these numbers are defined by:

A + B = [a⁻ + b^–, a⁺ + b⁺],

A− B = [a⁻ − b⁺, a⁺ − b⁻],

A × B = [min{a⁻b⁻, a⁻b⁺, a⁺b⁻, a⁺b⁺}, max{a⁻b⁻, a⁻b⁺, a⁺b⁻, a⁺b⁺}].

Evidently, it is not possible to precisely define the order of two interval numbers when there is overlap between them. Thus, a possibility function was defined in [9] to determine the “credibility that one of the interval numbers is at least as great as the another”. The possibility function is defined as:

$$Poss\left(\mathit{A}\ge \mathit{B}\right)=\{\begin{array}{c}1 \mathrm{if} p_{AB}>1,\\ p_{AB} \mathrm{if} 0\le p_{AB}\le 1,\\ 0 \mathrm{if} p_{AB}<0\end{array}$$

where $p_{AB}=\frac{a^{+}-b^{-}}{\left(a^{+}-a^{-}\right)+\left(b^{+}-b^{-}\right)}$.

When $a^{+}=a^{-}=a$ and $b^{+}=b^{-}=b$, $P\left(\mathit{A}\ge \mathit{B}\right)=\{\begin{array}{c}1 \mathrm{if} a\ge b, \\ 0 \mathrm{otherwise}.\end{array}$.

In [13,14], $\mathrm{the} \mathrm{value} Poss\left(\mathit{A}\ge \mathit{B}\right)$ is interpreted as “the credibility of a being greater than or equal to b, where a and b are two realizations from A and B”, where a realization is a given real number within the interval number.

2.3. Differential Evolution

Differential evolution [40] is an optimization tool highly effective and efficient when addressing optimization problems whose decision variables are continuous. It is characterized by approximating the optimal solution by improving multiple possible solutions at the same time. Differential evolution (DE) is especially convenient when the optimization problem does not satisfy the requirements of exhaustive mathematical optimization methods. However, it does not ensure to find the overall best solution, but only suboptimal solutions. The found solutions are generally good enough to be accepted by the user.

DE uses a generational improvement that simulates biological evolution. At any given moment it deals with a set of potential solutions called individuals or agents; the set of individuals is called population. The parameters used by DE consist of a crossover probability, CR ∈ [0, 1], a differential weight, F ∈ [0, 2], and a number of individuals in the population, P_size ≥ 4. Each individual in the population is represented by a real-valued vector z = [z₁, z₂, …, z_m]^T, where z_i is the value assigned to the ith decision variable and m is the number of decision variables.

Let y ∈ ℝ^m be a temporary solution. DE must perform the following steps:

• Initialize individuals placing them in a random position within the search space. This means that a random value must be assigned to each decision variable respecting the constraints associated with the variable.
• Perform the following steps until a termination criterion is reached. This criterion is defined here as a number of iterations, N_iterations.
  • Perform the following steps for each individual z in the population
    • Let a, b, c be individuals from the population chosen randomly such that z, a, b, c are all different.
    • Randomly define r ∈ {1, …, m}
    • Perform the following steps for each i ∈ {1, …, m}
      • Randomly define u ∈ [0, 1].
      • If u < CR or i = r, set y_i = a_i + F(b_i − c_i), otherwise set y_i = z_i.
    • If f(z) ≤ f(y), then replace z for y in the population. f(·) is a given fitness function (see Equation (1) below).
• Select the individual z of the population with the best performance f(z).

3. Methodology

Assume the existence of a set of activities, A = {a₁, a₂, …, a_n}, relevant for the manager of a given company. Let $x_i$ denote the proportion of time allocated to a_i ∈ A; ${\sum }_{i=1}^n{x}_i=1$ and x_i = t_i/T, where t_i is the time assigned to a_i and T is the total time that the manager assigns to the activities in A. An approach based on computational intelligence that is focused on determining the precise value of x_i is unrealistic since day-to-day operations and urgent issues may alter scheduled activities; moreover, the manager may not feel comfortable with the advised values of x_i. However, such unprevented operations and issues usually imply significant deviations from the ideal allocation of the manager’s time. Therefore, we describe here a hybrid approach based on computational intelligence and multicriteria decision aid that exploits interval theory to propose a flexible allocation of the manager’s time.

The idea is to provide the manager with the flexibility to choose, among a set of plausible values of x_i, the most convenient one according to his/her specific context and preferences. Let x_i = [$x_i^{-}, x_i^{+}$] denote an interval number representing the set of plausible values from which the manager can choose what he/she considers the most convenient one. In order to define a plausible set of values of x_i, we perform an elicitation of preferences based on multicriteria decision aid that ensures the maximization of the manager’s preferences.

The proposed approach consists of four main stages:

• Determine the set of activities in A.
• Assign a relative value $v_i$ to each $a_i$ ∈ A; this value represents “how desirable it is to spend time on activity $a_i$”.
• Determine the set of constraints imposed to the time allocation.
• Optimize the total value of the performed activities by defining $x_i$, $i=1,\dots ,n$.

3.1. Determining the Set of Activities in A

On a preliminary survey of both expert opinions and literature, we found out that activities commonly performed by managers can be assigned to, at least, the following classes:

• Supplier Management
• Marketing and Sales Management
• Strategic Planning
• General administration (Internal processes)
• Inventory Management
• Financial Capital Management (Cash Flow)
• Strategy Management
• Quality and Service Management
• Human Capital Management

Evidently, the specific context of the manager would determine if a different set of activities is considered during the decision-making process. Therefore, a particular analysis of the context handled should be carried out.

3.2. Assigning a Relative Value v_i to Each a_i ∈ A

During this stage, a value representing “how desirable it is to spend time on activity $a_i$” is defined relatively to the values assigned to the rest of activities in $A$. Different techniques can be followed to determine such values (cf. [41,42]). The chosen technique must define cardinal values as opposed to ordinal ones that some elicitation procedures define (e.g., [43]). A plausible way to assign such values is then through an elicitation procedure, such as the Swing method [27,28], where the preferences of the manager (or a group of managers from the same sector, region or expertise) are properly defined. If a group of managers is consulted to define the value of the ith activity, then multiple values will be surely determined. To deal with this imprecision, the value of the activity can be defined as an interval number v_i = [$v_i^{-}$,$v_i^{+}$] where $v_i^{-}$ and $v_i^{+}$ are the minimum and maximum values assigned to the ith activity by the group of managers.

3.3. Determining the Set of Constraints Imposed to the Time Allocation

Common constraints considered when scheduling the time of managers are:

• avoid spending too little time on some activities
• avoid spending too much time on some activities
• avoid spending too much time on all the considered activities
• avoid spending too little time on all the considered activities
• address a maximum number of activities
• address at least a certain number of activities

The previous constraints can be formalized, respectively, as follows:

x_i ≥ l_iy_i

x_i ≤ u_iy_i

$${{\displaystyle \sum }}_{i=1}^n{x}_i \le \mathit{U}$$

$${\sum }_{i=1}^n{x}_i\ge \mathit{L}$$

$${\sum }_{i=1}^n{y}_i \le N^{+}$$

$${\sum }_{i=1}^n{y}_i \ge N^{-}$$

y_i = {0, 1}

where l_i and u_i are the lowest and highest proportions of time that the manager is willing to allocate to ith activity. L and U are the lowest and highest proportions of time that the manager is willing to allocate to all the considered activities. N⁻ and N⁺ are the minimum and maximum number of activities to which the manager allocates some of his/her time. Additionally, y_i = 1 if the manager allocates some time to the ith activity and y_i = 0 otherwise. Note how l_i, u_i, L, and U are all defined as interval numbers, providing the manager with additional flexibility when defining the optimization model.

3.4. Defining an Objective Function

Let X = (x₁, x₂, …, x_n) and V = (v₁, v₂, …, v_n) be the sequences of time proportions and activity values, respectively; and let f(X, V) be a function that determines the total value of the time allocation made by the manager. Then, solving the following general statement implies finding the best time allocation:

$$\underset{{\mathit{x}}_{\mathbf{1}}, {\mathit{x}}_{\mathbf{2}}, \dots , {\mathit{x}}_{\mathit{n}}}{\mathrm{maximize}}f\left(X,V\right)$$

(1)

Subject to

x_i ≥ l_iy_i

$${{\displaystyle \sum }}_{i=1}^n{x}_i \le \mathit{U}$$

$${{\displaystyle \sum }}_{i=1}^n{x}_i\ge \mathit{L}$$

$${{\displaystyle \sum }}_{i=1}^n{y}_i \le N^{+}$$

$${{\displaystyle \sum }}_{i=1}^n{y}_i \ge N^{-}$$

y_i = {0, 1}

We propose to define f(X, V) as a weighted sum, that is:

$$f\left(X, V\right)={\displaystyle \sum }_{i=1}^n{\mathit{v}}_i{\mathit{x}}_i$$

We also propose to use differential evolution to address Equation (1).

3.5. Optimizing the Total Value of the Performed Activities

Multiple alternative techniques can be used to optimize Equation (1). However, given the interval-based feature of such equation (necessary in the approach to provide the required flexibility) and its definition as an NP-problem (cf. [44]), a metaheuristic optimization technique should be used. Metaheuristics are optimization techniques of the area of computational intelligence that have shown to be reliable when addressing hard problems and that can be flexible to adapt to situations as the stated above. It has been proved that differential evolution (DE) (e.g., [45]), a highly efficient metaheuristic, often outperforms other optimization techniques when addressing problems similar to that stated in previous sections; particularly, when the optimization problem is mono-objective and with real-valued decision variables. Therefore, we use here differential evolution to define the most convenient values associated to the manager’s activities.

3.6. Experimental Design

To assess the proposed approach, we have applied the methodology to a set of nine micro, small and medium-sized organizations in the commerce sector. Each of the five steps in the methodology described above was applied to the organizations, creating a single case study.

The application of the proposal to this case study is twofold. On the one hand, it will allow us to show how the proposed approach can work with a group of decision makers, and how each of them can take advantage of the other decision makers’ expert knowledge. On the other hand, the assessment will shed light on the performance of the proposed approach, and we will be able to determine how the proposed approach’s recommendations satisfied each manager’s preferences. The performance of the proposed approach, assessed through the satisfaction level of each manager, is compared to the performance of a benchmark.

4. Results and Discussions

This section provides (i) the general recommendations provided to the managers in the case study, and (ii) a comparison of each manager’s satisfaction between the recommendations provided by the proposed approach and an approach from the literature.

4.1. Results Obtained by the Proposed Approach

The results of the first stage of the methodology described in the previous section showed that the managers in the case study determined the following activities as the ones that should be prioritized when allocating resources:

• Supplier Management
• Marketing and Sales Management
• Strategic Planning
• General Administration (internal processes)
• Inventory Management
• Financial Capital Management (cash flow)
• Strategy Management
• Quality and Service Management
• Human Resource Management

A wide description of these activities is presented in Appendix A.

Regarding stage two of the methodology, we used the so-called Swing method to elicit from the managers the values in the additive value function denoted by Equation (1) (See [27,46], to see a more in-depth description of the method). The results obtained are shown in

Table 1. Values assigned by the managers to the activities regarding “how desirable it is to spend time on activity $a_i$”.

Activity	Manager 1	Manager 2	Manager 3	Manager 4	Manager 5	Manager 6	Manager 7	Manager 8	Manager 9
Supplier Management	7%	22%	0%	9%	0%	5%	13%	11%	13%
Marketing and Sales Management	23%	2%	10%	14%	21%	5%	12%	19%	19%
Strategic Planning	10%	18%	32%	1%	19%	53%	14%	11%	9%
General administration (internal processes)	3%	16%	6%	29%	0%	5%	13%	21%	11%
Inventory Management	7%	9%	3%	3%	0%	5%	13%	17%	10%
Financial Capital Management (cash flow)	14%	7%	26%	3%	0%	11%	14%	14%	11%
Strategy Management	16%	2%	10%	3%	20%	11%	13%	0%	9%
Quality and Service Management	10%	11%	10%	19%	20%	3%	7%	0%	9%
Human Resource Management	10%	13%	3%	19%	20%	3%	0%	7%	11%

.

After defining the weights that each manager assigned to the activities regarding “how desirable it is to spend time on activity a_i”, we intend to embrace the whole set of preferences by defining a unique interval number for each activity whose boundaries are set by the minimum and maximum weights provided by the managers, as shown in

Table 2. Values assigned by the managers to the activities.

Activity	Minimum Value (%)	Maximum Value (%)
Supplier Management	5	22
Marketing and Sales Management	2	23
Strategic Planning	1	53
General administration (internal processes)	3	29
Inventory Management	3	17
Financial Capital Management (cash flow)	3	26
Strategy Management	2	20
Quality and Service Management	3	20
Human Resource Management	3	20

. The goal of this aggregation of the managers’ preferences is to deal with the whole set of organizations (with similar characteristics in very similar contexts) as a single case study; thus, implying that the combined experience of the managers is convenient to dictate how the organizations’ managers should allocate their times.

Stage three of the methodology in Section 3 requires determining the constraints that the managers want to impose to the problem in Equation (1). For the case study carried out here, we found that the managers are only interested in defining boundaries to the proportion of times allocated to the activities. This is originated in the idea none of the nine fundamental activities mentioned above should be performed during very short or very long periods of time. The constraints established by the managers are shown in

Table 3. Lowest and highest proportions of time allowed per activity.

Activity	Lowest Proportion (%)	Highest Proportion (%)
Supplier Management	2	25
Marketing and Sales Management	4	8
Strategic Planning	8	17
General administration (internal processes)	4	21
Inventory Management	2	8
Financial Capital Management (cash flow)	4	12
Strategy Management	4	8
Quality and Service Management	8	25
Human Resource Management	12	25

.

Finally, an effective optimization tool based on metaheuristics, differential evolution, was used to determine the best allocation of times. Common parameter values were assigned to this optimizer. The crossover probability, CR, was set to 0.9; the differential weight, F, was set to 0.8; the population size, P_size, was set to 200; and the number of iterations, N_iterations, was set to 100.

Table 4. Recommendations of the system. If the manager spends a proportion of time indicated by the corresponding interval on a given activity, the manager would be maximizing the total value of the time allocation.

Activity	Best Time Recommended (%)
Supplier Management	[12, 21]
Marketing and Sales Management	[4, 6]
Strategic Planning	[10, 13]
General administration (internal processes)	[15, 18]
Inventory Management	[6, 7]
Financial Capital Management (cash flow)	[4, 9]
Strategy Management	[6, 7]
Quality and Service Management	[19, 25]
Human Resource Management	[15, 17]

shows the results obtained when exploiting the proposed approach regarding the case study.

As can be seen from Table 4, the summatory of lowest values is lower than 100%, that is, ∑v_i⁻ < 100. Similarly, the summatory of highest values is greater than 100%, that is, ∑v_i⁺ > 100. This is a desirable feature of the proposed approach regarding pragmatism. When exploiting a decision support system, it is convenient for the manager to have a sufficiently flexible recommendation to generate the final decision. Furthermore, such final decision still must ensure robustness. Through recommendations provided in the form of ranges, where 100% of the time dedicated by the manager to the activities is within the recommended boundaries, the proposed approach satisfies both requirements allowing the manager to decide the precise times dedicated to the activities.

4.2. Comparison with a Benchmark Approach

Once the proposed approach outputted general recommendations for the group of managers (Table 4), the recommendations and the individual preferences stated in Table 1 are used to create a “satisfaction level” per manager by exploiting Equation (1). Such satisfaction level is compared to that obtained by an optimizer from the literature addressing a simplified version of the problem. (Note that there are not published methods that can address the whole complexity of the problem stated in Section 4 in a straightforward manner, so we built a simplified version of the problem so it could be addressed by methods from the literature whose results could be used as a benchmark.) The simplified version of the problem is defined as follows.

Since, it can be shown that, for any interval numbers E and D, if e and d are, respectively, the middle points of E and D, then $\mathit{E}>\mathit{D}\iff e>d$ and $\mathit{D}=\mathit{E}\iff d=e$ (dictatorship of the middle point), we redefine the problem in the case study so now the values “assigned” by the managers to the activities correspond to the middle points of those actually provided by the managers and shown in Table 2. Such values must be normalized. The new values are shown in

Table 5. Simplification of the values assigned by the managers to the activities. The middle points of the original intervals have been normalized.

Activity	Value (%)
Supplier Management	11
Marketing and Sales Management	10
Strategic Planning	21
General administration (internal processes)	13
Inventory Management	8
Financial Capital Management (cash flow)	11
Strategy Management	8
Quality and Service Management	9
Human Resource Management	9

.

The original set of activities and constraints are maintained to ensure fairness. The objective function and the decision variables are now precise; that is, they are real-valued.

Applying the Simplex method to this simplified version of the problem provides the recommendations shown in

Table 6. Best solution found by a benchmark approach to a simplified version of the problem.

Activity	Best Time Recommended (%)
Supplier Management	25
Marketing and Sales Management	4
Strategic Planning	17
General administration (internal processes)	21
Inventory Management	2
Financial Capital Management (cash flow)	7
Strategy Management	4
Quality and Service Management	8
Human Resource Management	12

.

We now compare the satisfaction level of each manager produced by the recommendations in Table 4 and Table 6. This comparison will shed light on the performance of the proposed approach to encompass and exploit the expert knowledge of a group of decision makers from the perspective of each manager. With this purpose, the value function in Equation (1) is used to aggregate the value assigned by each manager in Table 1 with the recommendations of Table 4 first (to create the satisfaction level produced by the proposed approach), and with the recommendations of Table 6 later (to create the satisfaction level produced by the benchmark approach). Such satisfaction levels are shown in

Table 7. Satisfaction level of each manager regarding the recommendations provided by the benchmark and proposed approaches.

Manager	Benchmark	Proposal
1	8.76	[8.55, 11.76]
2	15.19	[11.9, 16.32]
3	10.54	[8.67, 12.1]
4	12.26	[13.03, 16.75]
5	8.87	[10.74, 13.53]
6	13.42	[9.27, 12.51]
7	11.16	[8.84, 12.44]
8	11.95	[8.96, 12.3]
9	11.22	[9.81, 13.46]

(note that, since the recommendations provided by the proposed approach consist of interval numbers, the satisfaction levels are given also as interval numbers).

From Table 7, we can see that each satisfaction level provided by the benchmark approach is contained within the corresponding satisfaction level provided by the proposed approach (except for Managers 4 and 6). This means that the recommendations of the proposed approach combined with good final decisions by the manager could imply better satisfaction levels for him/her. Of course, this combination could also imply worse satisfaction levels. So, here is where the synergy between the expert knowledge of each manager and the advances in computational intelligence and decision support systems can take place. The proposed approach exploits the experience of the group through computational intelligence and gives the opportunity for each manager to provide a final decision where his/her particular knowledge and preferences could lead to better results.

Finally, note how the simplified version of the problem is unrealistic. The managers would never be satisfied following recommendations of strict schedules; so, an approach that depends on the managers following such schedules would lack pragmatism.

Since the paper’s goal is to present a novel approach to deal with the realistic problem through the exploitation of a computational intelligence-based system for the first time, it is out of the scope of this work to perform in situ experiments to capture indicators of the organizations.

5. Conclusions

This work presents a novel idea to allocate the times of managers to their main activities. The idea exploits the so-called computational intelligence within a decision support system that provides recommendations about how managers should allocate their times.

Even when computational intelligence has been widely exploited in a plethora of fields, to the best of our knowledge, it has never been applied to the problem of supporting organization managers to allocate their times. We believe this is due to the “hardness” often imposed to models that would represent this problem. That is, models based on computational intelligence usually provide strict recommendations that would not allow the manager to deviate from such recommendations with robustness of effectivity. This is not ideal since managers usually make decisions on the progress of the activities. Therefore, here, a novel hybrid approach that integrates value function theory, interval theory and evolutionary algorithms, intends to give the manager flexibility regarding the times that he/she should dedicate to his/her activities. Furthermore, we ensure that the effectiveness of the approach is maintained as long as the actual allocated times remain within the recommended slack.

The proposed approach was applied in a case study to the managers of nine micro, small and medium-size organizations that participated during the whole process of the methodology described in Section 3. Table 1, Table 2 and Table 3 show the information provided by the managers that worked as inputs to our approach. Table 4 provides the results obtained. Such results proved to fulfill the constraints imposed by the managers and to maximize the total value of time allocated to the activities from the perspective of group decision. Table 5, Table 6 and Table 7 show a comparison between the results of our approach and those of a method from the literature. Since the benchmark method is not able to deal with the whole complexity of the case study’s problem, a simplified version of the problem was created. The results of both approaches are assessed from the perspective of each manager within the group. The comparison shed light on how the proposed approach could provide a higher satisfaction level to the managers if they make a convenient final decision. Therefore, we conclude that the proposed approach could be of pragmatical relevance to support the decisions of the managers; at least of those in charge of micro-, small- and medium-sized organizations.

Future research lines include (i) provide more in-depth analysis of the approach’s impact by assessing the organizations’ performances before and after implanting the approach; (ii) use more sophisticated techniques to model the manager’s preferences, for example, exploiting the so-called outranking approach (see [47]) and Fuzzy Logic [48]; (iii) follow a statistical procedure to define the parameter values of the approach.