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This work proposes a complexity metric which maps internal connections of the system and its relationship with the environment through the application of sensitivity analysis. The proposed methodology presents (i) system complexity metric, (ii) system sensitivity metric, and (iii) two models as case studies. Based on the system dynamics, the complexity metric maps the internal connections through the states of the system and the metric of sensitivity evaluates the contribution of each parameter to the output variability. The models are simulated in order to quantify the complexity and the sensitivity and to analyze the behavior of the systems leading to the assumption that the system complexity is closely linked to the most sensitive parameters. As findings from results, it may be observed that systems may exhibit high performance as a result of optimized configurations given by their natural complexity.

The scientific and technological advances of the second half of the twentieth century have generated significant changes in the dynamics of human civilization. The creation of electronic systems and their network structure revolutionized communication systems, modifying the social and economic relations in the world. The systems became more integrated and interdependent, consequently, more complex; since the network structure was not restricted to the computational systems, it is embedded in the human relationships.

Bar-Yam [

The network structure assigns a prominent role to the interactions that, in turn, are responsible for the holistic approach in the study of systems [

Likewise, in engineering studies, researchers have realized that subdividing systems to analyze them may cause significant losses in the internal structure of the original system [

This holistic approach to systems is based on a philosophical assertion that the whole is more than the sum of the parts. According to Simon [

According to Holland [

Lloyd [

Based on the fractal dimension as measure for self-similar objects, Balaban et al. [

Given the relevance of geometrical and computational frameworks, Joosten et al. [

Among the complexity metrics, fractal dimension is frequently applied to the analysis of textures, shapes, and network structures [

Regarding the connections, the complexity may be measured from the evolution of the system over time, considering the active connections in each state. The connections depend mainly on (i) the transition from one state to another due to the occurrence of events and (ii) the change of the input parameters that lead to variation in the system output [

Sensitivity analysis is relevant to the study of complexity because certain variables may eventually emerge and have a significant impact on the system. Even if the variables are hidden, the relevance of each one may be defined previously by means of its sensitivity and so anticipate strategies if such variables emerge. According to Holland [

Here, we focus on the degree of system complexity as a measure that comprises mechanisms related to internal and external interactions of the system. Thus considering (i) the increase in complexity, (ii) the holistic approach to systems, (iii) complexity as a unifying variable, and (iv) the absence of a practical and representative quantitative definition of complexity, we propose a complexity metric based on connections, which may be weighted according to the relevance of each one. This metric is applicable to any system that may be modeled and simulated from its input parameters and output variables.

The proposed metric covers a wide range of systems in the physical world. Using this metric, it is possible (i) to say how complex a particular system is or how much more complex one system is than another, (ii) to use the complexity in the objective function of optimization process, in order to minimize it, or as a constraint, in order not to exceed the value defined as a reference, and (iii) to support decision making.

In order to apply the proposed metrics, Section

Several metrics to calculate the complexity have been developed based on the size of the system, entropy, information, cost, hierarchy, organization, and other criteria [

Some complexity metrics are proposed based on the information entropy [

Based on the Paiva’s [

In (

Expressions (

The proposed metric maps the active connections related to entities, resources, and queues at any given time

Representation of the connections established by the entities

In Figure

In this work, a quantitative analysis is proposed of the curves that express the impact generated by the changes of the input parameters. These changes are carried out from the base values of the parameters, which may be defined as optimized solution or as the best bet for parameters. While one parameter is changed, the others are held at their base values and the output is measured. This approach is known as one-at-a-time measures.

The system output is given by the function

Example of area delimitation for the parameters

The proposed method based on Saraiva [

This sensitivity metric has mathematical properties of entropy related to the lack of information about system behavior. Likewise the fact that the entropy is maximum when the most uncertain situation is expressed in terms of probabilities [

In order to determine the parameters sensitivity, the proposed metric requires few one-at-a-time measures (set of parameters values and the respective output value), which may be obtained either by experiments performed in the real system or by simulations performed in the model. In both cases, the system is observed as a whole, leading us to believe this may be an appropriate mechanism to measure complexity. Some researches have showed the relationship between complexity and sensitivity [

The parameters influence on the model output may be quantified by the sensitivity indices. Here, we define sensitivity index

Figure

The distribution center of a company consists of the logistics of products delivery. After separating the products, an order of delivery is generated. During the center operation, the following steps are performed: (1) the distribution center generates the orders of delivery; (2) the orders are kept in queue until the resources are available; (3) the truck stays on the dock while the loading process is performed; (4) the truck leaves for delivery, releasing the dock and the group of workers for new loading; (5) the products are transported to their destinations; (6) the truck comes back to the distribution center for new deliveries. In the model, the duration at each stage corresponds to values given by probability distributions.

As typical discrete event system, the distribution center is modelled in terms of entities, queues, and resources. The entities are the orders of delivery, which are waiting in line for the availability of the resources: docks, trucks, and group of workers. The set of discrete states concerning the orders are (i) waiting in the queue, (ii) being loaded, and (iii) being transported. Based on the states, how many resources are being used at every instant of time

The distribution center works like an open system, in which new entities may be integrated at any moment; hence the number of entities (orders of delivery) and the demand for resources (docks, trucks, and groups of workers) vary over time. Different performance measures may be chosen for the problem of the distribution center, e.g., the average waiting time in queue, the average time for transportation, or the percentage of use of the system resources.

At this work, the delivery time

The measure for complexity is calculated by (

The probability of connection occurrence in the distribution center problem is given by (

Considering that there are 2 docks, 3 trucks, and 2 groups of workers into the distribution center, the system configuration may occur at any instant

In (

The medical center consists of the processes of medical care and basic procedures. The flow at medical center is as follows: (1) the patients arrive at the medical center; (2) the patients wait in queue for medical assistance; (3) the patients get the medical appointment; (4) after the appointment, some of the patients are released and others are forwarded to (5) perform basic procedures (receiving medication or doing medical exams); (6) after medication or exams, some of the patients are released and others perform new exams or take more medication, returning to step 5, and some of them wait in queue to come back to the physician; (7) after a new medical appointment, some of the patients are directed to further exams or medication, returning to step 5, and the others leave the center.

During the care process, queues of people may be generated in order to wait for (i) medical appointment, (ii) medication, and (iii) medical exams. The set of discrete states concerning patients are (i) waiting in queue, (ii) having medical appointment, (iii) receiving medication, and (iv) doing exams. The resources are used according to the demand.

The medical center is also an open system. It means that the number of entities (patients) and the demand for resources (physicians, nurses, and technicians) vary over time. In this model, the time between arrival and leaving of patients

The complexity measure is calculated by expression (

The connection probability in the medical center is given by expression (

Considering the medical center have 2 physicians, 2 nurses, and 1 technician in its staff, the configuration presented in matrix

In (

The simulation of the distribution center model presented in Section

The number of resources used in the simulation ranged from 1 to 10 for docks, 1 to 15 for trucks, and 1 to 10 for groups of workers, compounding

The normalized values of

Delivery time

Figure

Scenarios related to the lowest and greatest values of

| | | dock | truck | group of workers |
---|---|---|---|---|---|

| | 0.007 | 10 | 15 | 10 |

| 0.686 | | 2 | 6 | 2 |

| | 0.222 | 1 | 1 | 1 |

| 0.617 | | 10 | 1 | 10 |

The complexity measure is a way of seeing the system as a whole. The connections mapping concerning the states expresses the system configuration. Thus the relation

The lowest (in bold) and greatest (in italic) values of

Scenarios related to the lowest and greatest values of

| | | physician | technician | nurse |
---|---|---|---|---|---|

| 0.252 | 0.082 | 8 | 10 | 13 |

| | 0.089 | 8 | 10 | 15 |

| 0.359 | | 7 | 7 | 5 |

| 0.493 | 0.908 | 3 | 3 | 15 |

| | 0.490 | 3 | 2 | 5 |

| 0.410 | | 3 | 8 | 15 |

The model of the medical center presented in Section

The number of physicians, technicians, and nurses used in the simulation ranged from 3 to 8, 2 to 10, and 5 to 15, respectively, compounding 594 different scenarios. The simulation was performed for 180 days by scenario, considering 24 daily hours of operation. The medical care time

The lowest complexity obtained from all scenarios

Figure

Resource utilization and complexity

In the scenario with the lowest complexity, the time

Considering the relation between performance and complexity measures for medical center, expressed by (

In Table

The complexity of the distribution center and medical center was analyzed in Section

In order to obtain the one-at-a-time measures, the configuration given by the lowest value of

Data for sensitivity analysis of distribution center and medical center.

resource | range | base values |
---|---|---|

dock | | |

truck | | |

group of workers | | |

physician | | |

technician | | |

nurse | | |

The results obtained by the sensitivity analysis are presented in the following sections. The metric proposed in Section

The one-at-a-time measures for the distribution center are presented in Figure

One-at-a-time measures related to the output

According to the values presented in Table

Sensitivity index for distribution center related to output

interval | | | |
---|---|---|---|

| | | |

| | | |

The resources dock and group of workers presented the same values for sensitivity indices, equal to

The optimized solution for distribution center, i.e., 2 docks, 6 trucks, and 2 groups of workers, may be regarded robust for parameter change between

Based on the medical center simulation, Figure

One-at-a-time measures related to the output

The resource physician obtained the higher sensitivity indices as presented in Table

Sensitivity index for medical center related to the output

Interval | | | |
---|---|---|---|

| | | |

| | | |

Between

The system complexity measure contributes to the knowledge of the system as a whole. The simulated scenarios for the distribution center and the medical center have shown that the system becomes less complex as the number of resources increases. However, complexity saturates from a certain number of resources, indicating idleness in the system.

As the complexity

The lowest value of

In addition to the system overload, the complexity peaks have indicated the most sensitive parameters. The sensitivity analysis has been performed and has confirmed this point. In this paper, the sensitivity analysis contributed to (i) quantifying the influence of the parameters, (ii) understanding relationships between input and output variables, and (iii) checking the robustness of optimized solution. Besides that we here propose that the sensitivity indices are used to quantify the coupling between components of a system, since its inner structure reveals the system relationship with its environment.

So far we have measured the complexity based only on system connections regardless of their relevance. In order to make a complexity metric more comprehensive, this work proposes the use of sensitivity analysis in the metric of system complexity by the inclusion of relevance factor of connection

Thus, we propose updating expression (

In this paper, we presented a local sensitivity metric, called method of the area. However the local approach may make the complexity analysis unstable if the parameters base values are in a region of system instability. In order to overcome this problem, the global sensitivity analysis should be performed to comprise regions of instability and stability of the analyzed system. In this way, the impact of the nonlinearity in every region of operation of the system will be considered.

The proposed complexity metric abstracts aspects related to the spatial arrangement of the system, taking into account spatiotemporal interactions, as an alternative to Koorehdavoudi and Bogdan [

The sensitivity analysis was chosen because even parameters considered as less important by the uncertainty analysis (i.e., with low variability) may lead to significant changes on the output model due their sensitivity [

For instance, if we apply the proposed metric to a control system of DC motor, whose model presents continuous variables and continuous time, we could observe that when the motor is running, all connections are active and the probabilities of occurrence of all connections are equal to 1; thus the second part of the expression (

This example of control system of DC motor shows that some metrics such as fractal dimension could not be used due the absence of geometrical patterns described in phase space. Even for other types of systems, such as those analysed in the case studies (distribution center and medical center), the fractal dimension would be not effective, since the arrangements observable during simulation are abstractions made from system operation, besides the fact that they would not probably have self-similarity. Therefore the proposed metric is able to quantify complexity related to system dynamics in several contexts.

The complexity metric

In future works, we intend to apply the proposed complexity metric

In this paper, the main contribution is the proposal of integrating several system characteristics (configuration, arrangement, performance, and workload) into the complexity metric

This paper has proposed sensitivity and complexity metrics based on one-at-a-time values and system connections, respectively. It has been observed that complexity may indicate (i) most sensitive parameter, (ii) idleness or overload in the system, and (iii) lowest or greatest number of resources. The relation between performance and complexity has led to scenarios with optimized configuration for meeting the demand. Considering these cases, the paper has established that systems have their proper level of complexity, denoted natural complexity. Regarding the different types of couplings in the system, the use of sensitivity analysis has been proposed in order to determine the relevance factor of connection, contributing to more accurate measurement of the system complexity.

The CSV file with data used to support the findings of this study are available from the corresponding author upon request. The data are results from the simulation of the models.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank the National Council for Scientific and Technological Development (CNPq), the Foundation for Research Support of the State of Goias (FAPEG), and the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.