# **Advances in Understanding of Unit Operations in Non-ferrous Extractive Metallurgy 2021**

Edited by Srecko Stopic and Bernd Friedrich Printed Edition of the Special Issue Published in *Metals*

www.mdpi.com/journal/metals


**1. Introduction**

In recent years, the use of differential signals has gained increasing attention for both digital high-speed and analog microwave circuit applications [1,2]. This interest in differential devices is mainly due to their higher immunity to environmental noise, better electromagnetic compatibility, lower level of electromagnetic interference (EMI) and better signal to noise ratio performance, when compared to their single-ended counterparts. Despite all these advantages, differential devices can suffer from the presence of common-mode (CM) noise, mainly caused by amplitude unbalance and time skew of the differential signals. Therefore, to ensure the integrity of the differential signal over the frequency range of interest, strong CM rejection is highly desired. Different types of microwave devices in their differential version can be found nowadays in the literature: power dividers and combiners [3–6], diplexers [7] or passive equalizers [8]. However, differential-mode balanced bandpass filters (DM-BPFs) are, undoubtedly, the devices that have attracted the most attention in the literature. DM-BPFs with single/multiple differential passbands are required to have good differential-mode (DM) transmission within the passbands (low insertion loss, IL), good out-of-band rejection (high selectivity) and high CM rejection level (at least within the differential passbands). During the last decade, much effort has been focused on the design of balanced single-band BPFs, in such a away that a huge number of works can be found in the specialized literature. To mention just

a few, see for example, [2,9–30] and the references therein. Nevertheless, much less research has been done on balanced dual-band BPFs [27–42] when compared with single-ended versions. It is well known that balanced dual-band BPFs are key components for multi-band systems operating under several wireless standards from Institute of Electrical and Electronics Engineers (IEEE), such as IEEE 802.11 (Wi-Fi 2.45/5.3 GHz) and IEEE 802.16 (WiMAX - 2.45/3.5 GHz), as well as for Global Navigation Satellite System (GNSS) systems such as GPS (L1/L2 - 1.575/1.227 GHz) [43]. Regarding the methods developed to design balanced dual-band BPFs, the use of electrically coupled resonators is by far the most commonly used strategy, usually leading to good common-mode rejection. For example, in [27] a fourth-order dual-band balanced BPF based on half-wavelength resonators is presented. Although the filter offers good DM performance and good CM noise suppression, it is difficult to control the bandwidths of the two differential-passbands independently. In addition, it requires the introduction of lumped-elements to properly reject the common-mode signal since for the unloaded resonators CM rejection is poor. In [28,31–34], stepped impedance resonators (SIRs) are used to perform balanced dual-band operation. The impedance and length ratios of the SIRs were adjusted to realize the desired dual-band response. However, since the level of CM noise rejection for electrically coupled SIRs is relatively poor, additional elements such as lumped inductors/capacitors/resistors [28], common-mode rejection stages based on differential lines with defected ground structures (DGS) [31,32], or open-circuit stubs [33,34] must be added to the resonators in order to improve their CM performance. In all the aforementioned cases, both DM and CM exhibit good performance, but at the expense of increasing the filter size and the complexity of the design process, since two different devices must be designed: the filter itself and the additional elements to suppress common-mode transmission.

As in the cases exposed in [31,32], DGSs have also been used in [29,35] to improve CM rejection. In those papers, a multilayer structure was proposed with the ground plane located in the middle. By introducing a slotline in the ground plane, common-mode transmission is avoided since the location of the slot is not compatible with the pattern of common-mode currents. In all these cases [29,31,32,35], the use of slotted ground plane provides good CM rejection at the expense of increasing radiation losses and making difficult system integration (for many practical applications a solid ground plane is required).

Lumped or quasi-lumped microwave resonators are not the only structures used to design balanced dual-band bandpass filters. Distributed structures have also been used for this purpose. Some works based on them can be found in the literature [30,36–40]. For example, asymmetrical coupled lines [36] and substrate integrated waveguide (SIW) technology [37] are interesting recent approaches for balanced dual-band BPF design. These designs provide both good DM and CM response, but typically suffer from the problem of large electrical size. Moreover, both [36,37] require the use of a large number of via-holes, which makes their performance very dependent on the accuracy of the implementation of such vias. In [38], a second-order dual-band filter based on coupled lines loaded with a pair of SIRs and open stubs was presented showing a wide CM rejection bandwidth and high selectivity in the narrow DM passbands. However, the introduction of SIRs increased the total size. Another approach based on stub loaded resonators (SLR) was presented in [39] showing a simple and compact design, easily adaptable to two or three passbands, that presents a good level of CM noise rejection. Nevertheless, the isolation between bands is relatively poor. In [40], a via-free composite right/left-handed (CRLH) resonator was presented showing an excellent CM suppression behavior, both in terms of bandwidth and rejection level. Nonetheless, the bandwidth of the DM passbands is too small for many applications.

In [30], single-and–dual-band stub bandpass filters are presented whose input–output terminals are loaded with resistively terminated *Kth*-order bandstop filters. Thanks to resistive loading, common-mode is neither transmitted nor reflected, but absorbed in the resistors. This technique leads to good results for both DM and CM, but the prize to pay is the complexity in the design and the use of lumped resistors, which are not always desirable at high frequency applications.

The use of magnetic coupling to design balanced single-band BPFs was proposed in [26] as an alternative to the most commonly employed electric coupling. It was demonstrated that magnetic coupling provides inherent strong common-mode rejection when compared with electric coupling. For the simple case of two coupled open-loop resonators, the electric coupling occurs when the resonators are placed in such a way that the gaps of the resonators are placed face-to-face. The boundary conditions under differential- and common-mode excitations barely affect the charge distribution around the open ends of the loops, in such a way that the coupling level for both excitation modes is expected to be similar. By contrast, in the case of magnetically coupled resonators, the open sides are in the opposite orientation. In such case the coupling between resonators can be assimilated to the coupling between two microstrip transmission lines [26]. When common-mode excitation is applied, the symmetry plane of the open loop becomes a virtual open circuit (vanishing currents at that plane) and the coupling mechanism among adjacent resonators remains electrical in nature. On the contrary, a short-circuit condition appears at the same point when differential-mode excitation is applied. Currents around that point are important and the coupling mechanism between adjacent resonators becomes magnetic in nature. For a required coupling level of the differential-mode signal, which is imposed by the filter design specifications, magnetic coupling leads to larger separation between resonators when compared with the electric coupling case [41]. This automatically results into the weakening of the transmission of the common mode signal. Magnetic coupling has also been used for the design of dual-band BPF [42] by using embedded resonators printed on a monolayer structure in order to achieve dual-band operation. Once again, it was proven that magnetic coupling provided an ideal mechanism to reject the common-mode noise. However, the main drawback of the design presented in [42] was the independent control of the features of the two passbands, since both bands were dependent from each other. This is a fact inherent to the use of embedded resonators. Furthermore, in [43,44] the use of magnetic coupling has been extended to the design of balanced diplexers with excellent results that confirm the benefits and flexibility offered by magnetic coupling to design different kinds of differential components. The contribution made in [42] has motivated the authors to present in this paper a different strategy to design balanced dual-band BPFs with more flexibility and independence in what concerns the tuning the two differential passbands. As a case example, the proposed technique is used to design a filter with two differential bands corresponding to two commonly used wireless communications standards (GPS and Wi-Fi bands). It will be shown how, by using a multilayer structure, it is possible to create two differential passbands that can be independently tuned. Two magnetically coupled resonators on the top layer create the upper differential passband, while the lower differential passband is originated from another pair of resonators etched in the middle layer. These resonators are also magnetically coupled. Top layer resonators are excited by means of a edge gap capacitance between the feeding lines and the resonators. Middle layer resonators are excited thanks to the broadside capacitance created between the top and middle layers. Very good DM and CM responses can be achieved using this method, which is experimentally demonstrated for a filter based on two pairs of magnetically coupled open-loop resonators. The most relevant features of the proposed design are: (i) the two differential passbands are almost totally independent due to the asynchronous nature of the resonators and the possibility of independent control of the resonators excitation, (ii) band-to-band isolation about 36 dB can be easily achieved, and (iii) a good CM rejection level, which is inherent to magnetic coupling, can be also easily obtained (more specifically, 39/33 dB measured in the lower/upper DM passbands, respectively).

#### **2. Proposed Structure: Analysis and Design Methodology**

#### *2.1. Analysis of the Structure*

A 3-D view of the proposed multilayer structure for balanced dual-band operation is depicted in Figure 1. The two dimensional layout for each layer is depicted in Figure 2. The pattern etched on the top layer was used in [26] for the design of a single-band balanced BPF. It basically consists of a pair of open-loop resonators that are magnetically coupled (under DM operation). The middle layer also consists of a pair of open-loop resonators that are magnetically coupled when operating in DM. The difference between the two coupled resonant structures lies on the type of excitation and the values of the resonance frequencies. Thus, top layer resonators are excited via the gap capacitance associated with the gap width *st* 2, while the middle structure is excited by means of the broadside capacitance created between the top feeding lines and the middle layer conducting strips. Figure 3 illustrates the overlapping section between layers which originates the broadside capacitance (indicated by the shaded region). By setting the value of *l*over, it is possible to control the excitation level of the middle-layer resonators, which is critical to implement the response of the differential passband created by this pair of coupled resonators. Figure 4 plots the simulated electromagnetic response (simulations have been carried out using *ADS Momentum* from Keysight, stationed at Santa Rosa, California, United States of America, USA) of the entire multilayer structure for both DM and CM excitations. Two differential passbands with good CM rejection can be appreciated in such figure, demonstrating the expected behavior of the proposed structure. The design process of the multilayer balanced dual-band bandpass filter that leads to such response is described in detail in the next subsections.

**Figure 1.** Deployed 3-D view of the proposed planar multilayer balanced dual-band bandpass filter.

**Figure 2.** 2-D view of the two printed planes of the structure (not to scale). (**a**) Top layer; (**b**) Middle layer.

**Figure 3.** Detailed view of the broadside capacitance involved in the feeding of middle layer resonators.

**Figure 4.** Simulated response of the proposed multilayered balanced dual-band bandpass filter. The dimensions (in mm) used for the structure in Figure 2 that give rise to this response are: (i) top layer: *t* = 5.8, *w*0 = 5.1, *w*1 = 0.2, *wt* 1 = 0.2, *l t* 1 = 1.3, *l t* 2 = 5.05, *l t* 3 = 26.6, *st* 1 = 0.2; (ii) middle layer: *wb* 1 = 0.55, *wb* 2 = 0.4, *wb* 3 = 0.5, *l b* 1 = 27.2, *l b* 2 = 5.7, *l b* 3 = 4.95, *l b* 4 = 1.5, *l b* 5 = 1.575, *sb* 1 = 2.

#### *2.2. Design Methodology*

When designing a dual-band filter, it is very important that the two passbands can be independently tuned in terms of both center frequency and fractional bandwidth (FBW). In the case at hand, the first point that should be noted is that the middle layer of the structure introduces the lower passband, centered at *f d* 01 = 1.575 GHz, the L1 GPS signal frequency, while the top layer resonates at the first Wi-Fi band frequency, *f d* 02 = 2.45 GHz. This means that the upper frequency band should be allocated by setting the top layer parameters while lower frequency band must depend, exclusively, on the middle layer geometry. Let's start by studying the resonance frequency of resonators in both layers. In Figure 5, the differential return loss, |*Sdd* 11 |, have been represented when a single couple of top and middle resonators are weakly excited. The dips in the return loss correspond to the resonance frequencies of both resonators. Several parameters from each resonator were varied to show the dependence of those resonance frequencies with the resonator's length. From the simulated results (Figure 5) the following conclusions can be inferred:

From these results, it is clear that there is flexibility in locating the center frequencies of each passband as long as both frequencies are not very close to each other. In this latter case, resonators on both layers would interact significantly, thus leading to band-to-band dependence breaking the asynchronous behavior between the pair of resonators. This situation would not be desirable for designing purposes. Furthermore, the independence of the passbands is also related to the excitation mechanism of each resonator pair. As mentioned before, top layer resonators are mainly excited due to the electric field confined within the gap of width *st* 2 (gap capacitance) while middle layer resonators are excited by means of the electric field concentrated between the feeding lines and the resonator conductor strips (broadside capacitance). This fact, together with the sufficient separation of the two center frequencies, leads to the desired independence exhibited when locating the center frequencies of the differential passbands.

**Figure 5.** Differential excitation return loss (|*Sdd* 11 |) of the structure composed by a couple of top and middle resonators like those depicted in the insets when they are weakly excited: (**a**) *l b* 2 + *l b* 3 + *l b* 4 + *l b* 5 is varied; (**b**) *l t* 1 + *l t* 2 is varied. In both figures, the remaining physical parameters (see Figure 2) have been kept constant.

Once the independent control of the resonance frequencies has been demonstrated, the next electrical parameter to be considered in the design process is the correct excitation of the resonators to generate two properly matched differential passbands. In this regard, as mentioned above, the parameters that control top and middle layer resonators excitation are mainly *st* 2 and *l*over, respectively. In order to investigate on the influence of such parameters on the matching level of both differential passbands, i.e., |*Sdd* 11 |, Figure 6 shows the differential return loss of the two differential passbands as a function of *st* 2 and *l*over. From this figure it is clear that *l*over mainly influences the reflection coefficient of the lower differential passband while *st* 2 controls the reflection coefficient of the upper differential passband. Note that *st* 2 slightly modifies the value of the upper center frequency. However, this problem can be solved by barely modifying some of the other dimensions of the top layers resonators in a very simple manner, as explained in [26]. Finally, the last step in the design process is to verify that the fractional bandwidth of the two differential passbands can be set independently. If this is the case, we have two totally independent differential passbands, which represents the ideal case when designing filters with dual-band operation.

**Figure 6.** |*Sdd* 11 | for different values of (**a**) *l*over and (**b**) *st* 2.

Fractional bandwidth is related to the coupling level between resonators. The closer the resonators are the larger the fractional bandwidth is. Referring to Figure 2, the relevant geometrical parameter that controls the coupling between resonators for the ones located on the top layer is *st* 1, whereas *sb* 1 is the responsible for the coupling level between resonators in the middle layer. To verify this, fractional bandwidth as a function of *st* 1 and *sb* 1 are shown in Figure 7a,b, respectively. From Figure 7, it is clear that, as it happened with the center frequencies and the differential return loss, the fractional bandwidth of the two differential passbands are also independent from each other. This fact confers the designer full possibilities to adapt the design to other frequencies or other bandwidths that might be necessary for other requirements and specifications.

**Figure 7.** FBW with as a function of the resonators gap: (**a**) *st* 1 is adjusted; (**b**) *sb* 1 is adjusted.

In order to clarify the design process of the balanced dual-band bandpass filter proposed in this paper, the following summary is given below:

3. Finally, the matching level is adjusted by setting the values of *st* 2 and *l*over.

The proposed filter is finally designed and ready to be fabricated and measured.

#### **3. Results**

#### *3.1. Prototype Example*

In this section, an example of differential dual-band BPF filter design is reported. The first passband corresponds to the L1 GPS signal at 1.575 GHz. A fractional bandwidth of 8.86% is considered. The second passband is designed to be allocated in the first Wi-Fi frequency at 2.45 GHz, with a fractional bandwidth of 5.69%. The filter is implemented on a substrate with the following features: dielectric constant *εr*1 = *εr*2 = 3.0, and thicknesses *hsub* = 1.016 mm. As explained in detail in previous sections, the first step in the design process is to obtain the resonator dimensions to set the two desired center frequencies, which are controlled by resonator lengths (recall that resonance arises when the length of the resonators is half the guided wavelength at the desired center frequencies). Once resonators are designed, design curves in Figures 6 and 7 are used to set the desired matching level and fractional bandwidth. Following this procedure, the final dimensions of the proposed dual-band differential filter in Figure 2 are those of the caption of Figure 4, where we have represented the final simulated filter response. This figure exhibits two differential passbands well allocated. The common-mode rejection level is very satisfactory too. To verify experimentally these results, a prototype has been fabricated and measured. Results are presented in next subsection.

#### *3.2. Experimental Results*

In this section, a comparison between the results of the electromagnetic simulations and the measurements of the manufactured prototype is carried out. Figure 8 shows a photograph of the fabricated prototype, which has been fabricated using a mechanical milling machine (LPKF S103 from LPKF Laser & Electronics, stationed at Garbsen, Germany). Measurements have been carried out using Agilent PNA-E8363B with a Test-Set N4420B extension (thus, with a 4-port system), both devices from Keysight, under normal conditions of pressure, temperature and humidity. This network analyzer has a single source, so the DM and CM have been obtained using the program *Physical Layer Test System* from Keysight that allows the user load calibrations and implement a transformation between the actually measured 4 × 4 scattering matrix of the 4-port single-ended circuit and the 2 × 2 differential-differential, common-common and differential-common 2 × 2 scattering matrices of the balanced 2-port system. Simulated and measured DM and CM responses are depicted in Figure 9. This figure shows a good agreement between simulated and measured results, although the measured lower band is slightly shifted when compared with the simulated one. This fact can be explained if we take into account that multilayer structures can suffer from the presence of small air gaps that may appear if layers are not perfectly stuck. Furthermore, the mechanical milling machine might sometimes remove a thin layer of the substrate, and this affects the effective dielectric constant of the substrate. This effect can be minimized using some suitable prepreg layer, which should be included in the simulations, of course. In order to verify the effect of this hypothesized air gap, the response of the differential filter has been simulated considering a 0.025 mm air gap between the stacked dielectric substrates. This response is depicted in Figure 9b. The results show a good agreement between simulations and measurements. The use of a low permittivity substrate has the puspose of reducing the air gap effect, since the closer the dielectric constant is to that of air, the less the effective permittivity will be affected. The use of low permittivity substrates will also decrease the capacitance between layers so that the cross coupling between resonators will also be lower. The measured results show a CM rejection level greater than 39 dB and 33 dB for the first and second DM passbands, respectively. Measured insertion loss at the first and second DM center frequencies is 0.9 dB and 1.9 dB respectively. In addition, band-to-band isolation is better than 36 dB, which is satisfactory for many practical applications.

**Figure 8.** Photograph of the manufactured prototype: (**a**) Top layer; (**b**) middle layer.

**Figure 9.** Simulated and measured response of the proposed dual-band bandpass filter: (**a**) without gap; (**b**) with an air gap of 0.025 mm.

#### **4. Discussion**

In order to illustrate the benefits of the proposed structure, a comparison between the proposal in this paper and a number of contributions available in the specialized literature is provided in Table 1.

**Table 1.** Comparison with reported balanced dual-band bandpass filters.

\* Guided wavelength (*λg*) at lowest frequency of operation (*f d* 0 ).

In terms of the different relevant parameters, several conclusions can be inferred from the comparison between our design and those reported in the literature (Table 1):

In brief, the design presented in this work is one of the smallest reported in the literature, it has low complexity and allows for easy configuration of the two passbands (both, center frequency and FBW). At the same time, leads to high CMRR values without relying on additional elements that might disrupt the differential-mode response. The main deawback of our proposal could be the lack of transmission zeroes in the frequency region between passbands. Nevertheless, the band-to-band isolation level presented in this work is satisfactory for most applications.


**2. Background**

In this section, we give some background about various pertinent topics. Firstly, we provide some basic knowledge about the theory and practice of intertemporal choices, inclusive of a fully developed real example. Secondly, we give a general overview of fuzzy sets and other related models of uncertain information. Finally, we focus on the specific characteristics of the framework where we develop our contribution, namely, fuzzy soft sets and their decision making.

#### *2.1. Intertemporal Choice: Theory and Practice*

Intertemporal choices are decisions whose consequences (costs and benefits) are distributed over time [2]. Decisions about investments, spending and savings are standard examples of monetary intertemporal choices. However, there are also non-monetary intertemporal choices such as decisions related with sustainability (environmental issues such as forestry [3], climate policy [4] or the use of energy-using durables [5]), health (diet, exercise, and addictions [6,7]), job search [8], or work effort [9].

Discounted utility theory is the normative theory for intertemporal choice, or choices between outcomes accruing at different points in time; usually between immediate and delayed outcomes [10]. Since its introduction by Samuelson [11] in 1937, the discounted utility (DU) model has dominated the economic analysis of intertemporal choice (e.g., the aforementioned [4,5,9]). DU model was completed by Koopmans [12] who clarified its logic and main assumptions. This model presumes that people evaluate the pleasures and pains resulting from a decision in a similar way that financial markets evaluate gains and losses spread out over time. Anyone prefers to get 1000 dollars now rather than 1000 dollars in a year. However, people behave differently if they have to choose between receiving 1000 dollars now or 1100 dollars in a year. To compare choices made in different moments of time, under DU, it is assumed that agents exponentially discount these costs and benefits according to how delayed they are in time [13]. Although there is experimental evidence showing that this is not always the case [2,14,15], a fact that prompted the appearance of other explanatory models such as hyperbolic discounting [16–18] or q-exponential discounting [19–21], the DU model is nevertheless used as the common tool for public policy in the evaluation of public projects. The model can be calibrated with suitable discount factors, for example, using a decreasing sequence of discount rates for projects with very long-term impacts to account for intergenerational equity [22]. The governments of the United Kingdom and France, in line with the proposals of several authors for long-term valuations [23], recommend the use of decreasing discount rates in public projects with long time horizons [24,25].

In real practice, and provided that the DU model is adopted for the evaluation of intertemporal projects, the expected cash-flows of a project are always discounted to obtain its Net Present Value (NPV) at instant 0 (time of evaluation), see for example [26]. To that purpose, a well-known formula is applied which requires using an appropriate discount rate. For private projects, the weighted average cost of capital or a required profitability are usually used as a discount rate. For public projects, the social time preference rate is employed to calculate the discount rate, which is then called *social discount rate* (SDR). The social time preference rate is a rate used for discounting future benefits and costs, and it is based on comparisons of utility across different points in time or different generations [24]. The SDR "is used by society to give relative weight to social consumption or income accruing at different points in time" [3].

Since we are applying our intertemporal model of choice to concrete examples, we need to fix an appropriate discount rate. Our choice is not arbitrary. On the contrary, we take advantage of the fact that, for EU funded projects, the European Commission [26] recommends the use of the exponential discounting model and a constant 5% European social discount rate for the Cohesion Fund eligible countries and 3% for the others (countries non-eligible for the Cohesion Fund). Therefore, 5% is our reference rate unless otherwise stated.

According to the previous discussion, discounted utility computations made for choices in the present and at various moments in the future (*x*0, *x*1,..., *xT*) adopt the form

$$\sum\_{t=0}^{T} \beta^t u(\mathbf{x}\_t) \tag{1}$$

Here, *xt* is the choice made at moment *t*, whose utility at that moment is *u*(*xt*); *βt* is the discount factor for a time period of *t* periods, usually years (for example, *β* = 1 1+0.05 under our assumption for the reference rate); choices are made along periods *t* = 0 (the present), *t* = 1, 2, ... , *T*; and *T* may be +∞. Put otherwise, if we want to assess the temporal sequence (*x*0, *x*1, ... , *xT*) and the utility *u* gives us the degree of satisfaction of each choice *xt*, which is of course *u*(*xt*), then ∑*T t*=0 *βt u*(*xt*) gives the discounted utility of this temporal sequence. The *β* parameter accounts for the fact that people prefer to enjoy utility as soon as possible.

Let us now give a real example that illustrates the application of our reference model for intertemporal choice.

**Example 1.** *On 7 November 2017, the Spanish infrastructure operator Ferrovial published a press release (see https://www.ferrovial.com/en/press-room/press\_releases/500-million-euro-2-124-perpetual-hybrid-bond/, retrieved 18 August 2018.) It stated: "Taking advantage of a favorable market environment with low interest rates, Ferrovial today successfully priced a 500 million euro perpetual hybrid bond. The issue pays a* 2.124% *annual coupon until 14 May 2023. Subsequently, it will pay a fixed coupon equal to the applicable swap rate plus a spread of* 2.127% *until 14 May 2043 and of* 2.877% *thereafter. The swap rate will be updated every five years."*

*Therefore, Ferrovial perpetual bonds, with a face value of* 100 *euros, will pay a* 2.124% *annual coupon during the first six years,* 2.127% *(supposing a swap rate of* 0%*) the following 20 years, and a* 2.877% *thereafter. Table 1 expresses this intertemporal situation and gives the computations that produce the NPV at time 0 of such a bond when we assume a discount rate of* 3%*, hence β* = (1 + 0.03)−1 = 0.97087*.*

**Table 1.** Detailed computations of the NPV, assuming a discount rate of 3%, in the real Example 1. It is the sum of the present values at the right column of the table.

#### *2.2. A Concise Presentation of Fuzzy Sets and Related Notions*

Since Zadeh [27] laid the foundations of fuzzy set theory, whose main feature is the introduction of partial membership degrees, many authors produced a large amount of literature on their advantages and potential applications in decision making. Mardani et al. [28] gave a summary of articles about fuzzy multi-criteria decision making from the period 1994–2014. Other classical references for the fundamentals of decision making in fuzzy set theory include Tanino [29] and Fodor and Roubens [30].

When imprecise individual or collective knowledge cannot be faithfully represented by fuzzy sets, extensions of this concept and multiple variations offer more suitable models. Atanassov [31,32] presented the idea of intuitionistic fuzzy sets. Afterwards, Chen et al. [33] or Wei [34] produced intuitionistic fuzzy multi-attribute group decision making methods, and De Miguel et al. [35] applied interval-valued Atanassov intuitionistic fuzzy sets in multi-expert decision making. Pythagorean fuzzy sets are surveyed in Peng and Selvachandran [36], and interval-valued Pythagorean fuzzy sets were studied by Peng and Yang [37] and Peng [38]. Hesitancy was first merged with fuzzy sets by Torra [39] (for more information, a good source is Rodríguez et al. [40]; see also Alcantud and Torra [41] for the first decomposition theorems and extension principles in the framework of hesitant fuzzy sets).

From a different position, rough set theory was established by Pawlak [42] and, in his first formulation, an equivalence binary relation is the source of granulation of the set of alternatives.

It is at this junction that a different, parameterized description of the alternatives made its appearance. The idea produced soft sets, extensions and hybrid models. Since they are a benchmark in our paper, we proceed to describe them succinctly in the next subsection.

#### *2.3. Soft Sets, Extensions and Hybrid Models*

The theory of soft sets originates with the seminal paper [43]. Feng and Zhou ([44], Section 1) cleverly described soft set theory in the following terms: it "is considered as a new mathematical tool for dealing with uncertainties which is free from the inadequacy of parameter tools. In soft set theory, the problem of setting the membership function simply does not arise as in fuzzy set theory, which makes the theory convenient and easy to use in practice." Its relevancy to decision making in various fields was already pointed out in [43], which also explained that the models by fuzzy sets and soft sets are linked to each other. Further relationships are proven in [45–48]. The early works of Maji et al. [49] and Akta¸s and Ça ˘gman [50] among others expanded the basic theory of soft sets. Khameneh and Kılıçman [51] systematically reviewed multi-attribute decision-making based on soft set theory. Zhan and Alcantud [52] recently summarized parameter reduction of soft sets, a thriving area that allows for approaches in extended models.

Indeed, suitable extensions of soft sets come up from the incorporation of ideas such as the aforementioned fuzziness and hesitancy. Fuzzy soft sets were designed by Maji, Biswas and Roy [53]. Parameter reduction in the context of fuzzy soft sets was developed, e.g., by Khameneh and Kılıçman [54]. Wang, Li and Chen [55] produced hesitant fuzzy soft sets by adding up hesitancy to the latter concept. Because data collection and measurement often produce errors or are restricted, some studies (e.g., [56–58]) are concerned with another extended form of soft sets called incomplete soft sets, while other [59,60] are concerned with the natural extension called incomplete fuzzy soft sets.

Let us now consider fuzzy soft set based decision making. The performance of the pioneering analysis by Roy and Maji [61] has been improved by [62,63]. Feng et al. [64] put forward a flexible method based on level soft sets. They explained that [65] challenges the position in [61] by claiming that the criterion for making a decision should use scores instead of fuzzy choice values, a point of view that found little support among other scholars. Liu et al. [66] gave another methodology for fuzzy soft set based decision-making based on an ideal solution. Feng and Guo [67] intended to effectively resolve the natural group decision-making problem in the context of fuzzy soft sets. To achieve their goal, they first design another adjustable method for solving fuzzy soft set based decision-making problems.

Hybrid models combine the spirit of soft sets with other methodologies. Peng, Dai and Yuan ([68] and the references therein) contributed to interval-valued fuzzy soft decision making. Park, Kwun and Son [69] gave an approach to decision making problems based on generalized intuitionistic fuzzy soft sets (see also [70] for the operations in that framework). Feng et al. [48] used soft approximation spaces instead of binary relations in rough set theory. Zhan and Wang [71] built five new different types of soft coverings based rough sets and investigated relationships between soft rough sets and soft covering based rough sets. Zhan and Alcantud [72] designed a soft rough covering by means of soft neighborhoods, which they utilized to improve decision making in a multi-criteria group environment. Ma et al. [73] presented an updated summary of decision making methodologies based on two classes of hybrid soft set models. Fatimah et al. [74] studied the decision-making implications of probabilistic and dual probabilistic soft sets.

Now, we proceed to state the formal definitions that we shall use in the remaining of this paper.

#### **3. Definitions: Soft Sets and Fuzzy Soft Sets**

In soft set theory and their extensions, we start with *U*, which is a set of alternatives (or a universe of objects), and *E*, which is a universal set of attributes, parameters or characteristics. We let P(*U*) denote the set of parts of *U*, i.e., the set formed by all the subsets of *U*.

#### **Definition 1** (Molodtsov [43])**.** *A soft set over U is a pair* (*F*, *A*) *with A* ⊆ *E and F* : *A* −→ P(*U*).

A soft set over *U* is a parameterized family of subsets of *U*, and the set *A* contains the relevant parameters. For every *e* ∈ *A*, the subset *F*(*e*) is the set of *e*-approximate elements, or also, the subset of *U* approximated by *e*. For example, if *U* = {*s*1,*s*2,*s*3} is a universe of shirts and *A* contains the parameter *e* that describes "blue color" and the parameter *e* that describes "silk fabric" then *F*(*e*) = {*s*1} means that the only shirt with blue color is *s*1 and *F*(*e* ) = {*s*1,*s*3} means that exactly *s*1 and *s*3 have silk fabric.

To model more general situations, the following notion is subsequently proposed and investigated in [53]:

**Definition 2** (Maji, Biswas and Roy [53])**.** *Let* **FS**(*U*) *denote the set of all fuzzy sets on U*. *The pair* (*F*, *A*) *is a fuzzy soft set (FSS) over U when A* ⊆ *E and F* : *A* −→ **FS**(*U*).

Needless to say, soft sets are an example of fuzzy soft sets. However, fuzzy soft sets are better suited to model subjectively perceived properties, since partial memberships are designed to express such subjectivity.

In the standard instance with finite *U* and *A*, both soft and fuzzy soft sets can be displayed in the form of a table, where rows correspond to the alternatives in *U*, columns correspond to the attributes in *A*, and there is a number from [0, 1] in each cell. Of course, the cells of these matrices contain either 0 or 1 when the fuzzy soft set is a soft set.

To illustrate these ideas and motivate the subsequent decisional analysis, let us put forward an example in terms of an object recognition problem:

**Example 2.** *A collection of objects U* = {*o*1, ..., *o*6} *is characterized in terms of a space of attributes which is denoted as A* = {*p*1, ... , *p*7}*. Here, the attributes represent the relevant combinations of characteristics. The fuzzy soft set that describes the objects is* (*F*, *A*)*, which is given by the tabular representation in Table 2. For illustration, number* 0.650 *at the junction of Row o*1 *and Column p*1 *means that the degree of membership of o*1 *to the objects that verify characteristic p*1 *is* 0.650*.*

**Table 2.** Tabular representation of the fuzzy soft set (*F*, *A*) in Example 2.

#### *Fuzzy Soft Set Based Decision Making*

Soft set based decision making relies on [44,75,76]. However, since the appearance of the seminal [61], there have been many remarkable approaches to decision making in the framework of fuzzy soft sets. The most successful contributions include [61–67]. We do not describe them all in detail here. For our purposes, it should suffice to know their general features and relative advantages, which are summarized in Table 3. It compares various noteworthy criteria with respect to their main characteristics.

**Table 3.** A critical summary of the main fuzzy soft set based decision making procedures.

As explained in Section 1, ultimately we need to make choices from a fuzzy soft set to solve the intertemporal choice problem for fuzzy soft sets that we present in Section 4. Therefore, it is convenient to be familiar with the machinery of at least one such procedure. For the sake of clarity, to apply fuzzy soft sets in decision making practice, we focus on the proposal by Alcantud [62], who stated a feasible algorithmic solution to solve problems in the format of Example 2.

To make this paper self-contained, we recall that the application of Alcantud's algorithm proceeds as follows (afterwards Example 3 illustrates the application of Algorithm 1 below to a concrete situation):

#### **Algorithm 1** (Alcantud [62])

*Input*: a fuzzy soft set (*F*, *A*), which we place in the form of a table. Its cell (*i*, *j*) is represented by *tij*

1: For every attribute *j*, let *Mj* denote the maximum membership value of the alternatives, i.e., *Mj* = max*i*=1,...,*k tij* for each *j* = 1, ..., *q*.

Produce a *k* × *k* comparison matrix *A* = (*aij*)*k*×*k* as follows: for every *i*, *j*, *aij* is the sum of the non-negative values in the following finite sequence:

$$\frac{t\_{i1} - t\_{j1}}{M\_1}, \frac{t\_{i2} - t\_{j2}}{M\_2}, ..., \dots, \frac{t\_{iq} - t\_{jq}}{M\_q}.$$

We can display this matrix as a comparison table.

2: For each *i* = 1, ..., *k*, calculate *Ri* as the sum of the elements in row *i* of *A*, and *Ti* as the sum of the *i*.

**Example 3.** *Let us assume that we have the input data of Example 2. Its comparison table is given in Table 4, as computed by Algorithm 1 above. Then, Table 5 shows its associated scores.*

*As a result, one concludes that o*4 *should be selected when we consider the input data of Example 2 and we adhere to Algorithm 1.*

**Table 4.** Comparison table of the fuzzy soft set (*F*, *A*) in Example 2 using Algorithm 1.

**Table 5.** Score table of the fuzzy soft set (*F*, *A*), derived from its Comparison table by Algorithm 1.

#### **4. A New Model: Intertemporal Fuzzy Soft Sets**

Thus far, the literature has dealt with alternatives with a very simple structure: for each characteristic, we know the degree of membership of the alternative to the set of elements that verify the characteristic. However, it is not difficult to find examples where the performance of the options is far more complex. Particularly, in this paper, we are concerned with an intertemporal setting. Indeed, in the general framework of project selection (e.g., solar energy projects [77], environmental impact assessment [78], financial portfolio [79], etc.), each alternative has a performance along an indefinite number of periods, typically years. Therefore, if we want to decide which of a list of projects

should be selected, we face an infinite number of fuzzy soft sets (one that represents each possible period). Neither of the existing approaches to fuzzy soft set based decision making can deal with this potentially infinite structure.

To tackle this new problem, now we proceed to formalize our model. It accounts for the intertemporal setting that we have motivated. Afterwards, we interpret the formal statement of the model in terms of tables when the number of alternatives and attributes is finite, which facilitates their computational manipulation. In the next section, we give a procedure for making decisions in this novel framework. In addition, in that section, an example illustrates the decision making algorithm in a situation motivated by public projects evaluation. For comparison, recall that Example 1 shows a recent, real situation in a financial environment with crisp data.

#### *4.1. The Structure of Intertemporal Fuzzy Soft Sets*

Molodtsov's notion of parameterized descriptions of the universe has already been combined with features that do not pertain to the original formulation of soft sets. Here, we propose a model where for each of a possibly infinite number of periods, each attribute produces a possibly different fuzzy parameterization of the universe.

The statement of the model is simple but powerful:

**Definition 3.** *An intertemporal fuzzy soft set (ItFSS) over U is a sequence* F = {(*Fi*, *A*)}*i*∈N *of fuzzy soft sets over the common universe U.*

In Section 4.2, we specify two alternative formulations of this definition, that are amenable for calculations with computers.

The basic idea is that the fuzzy parameterization of the universe is allowed to vary with time. According to this model, a fixed set of attributes is given, and then for each period *i* a parameterized description of the common universe produces a fuzzy soft set (*Fi*, *A*) which accounts for the situation at that period. To emphasize differences, on occasions, we refer to these standard FSSs as *static* FSSs.

**Remark 1.** *When the sequence in Definition 3 is constant (in other words, we use* {(*Fi*, *A*)}*i*∈N *with Fi* = *F for each i), then we identify the ItFSS with a standard FSS. Therefore, our model of ItFSSs is a natural extension of FSSs in Definition 2.*

Henceforth, we assume that the common attributes in *A* are all positive (otherwise, we can proceed as in [80] to convert the input into this format). Therefore, the higher is the membership degree, the better. The decision problem that arises consists of determining an optimal alternative from the list *U*. We propose a flexible solution in Section 5.

The next motivating example illustrates the structure of our model above. Observe that in this simplified statement it is possible to display the information pertaining to an intertemporal fuzzy soft set in one table, even though it concerns an infinite number of periods:

**Example 4.** *A civil project (e.g., building a bridge or a dam) will have a long-term impact on the population of a certain geographical area. There are two targets that should be achieved: environmental effects (e*1*) and economic development (e*2*). Let A* = {*e*1,*e*2}*. There are two possible projects, namely, p*1 *and p*2*. Their respective yearly effects are captured by the ItFSS* G = {(*Gi*, *A*)}*i*∈N *over U shown in Table 6, where U* = {*p*1, *p*2}*.*

*It is not difficult to check that, in every period i, the positive effects of p*1 *exceed the effects of p*2 *at any of the two target attributes:*

$$0.3 + \frac{1}{i+1} > 0.2 + \frac{1}{i+2} \text{ for each } i$$

$$0.4 + \frac{1}{i+1} > 0.3 + \frac{1}{i+2} \text{ for each } i$$

*Therefore, project p*1 *should be selected.*

**Table 6.** A tabular representation of the intertemporal fuzzy soft set G = {(*Gi*, *A*)}*i*∈N in Example 4.

The streamlined Example above is very simple for two reasons. Firstly, even though there are an infinite number of different FSSs for the infinite periods, the ItFSS can be represented by one parametric table, which is often not the case. Secondly, we do not need to use the theoretical contribution of Section 5 because the decision about which project should be selected is trivial: there is a sort of domination of the first project over the second in all attributes and for all periods that makes the decision obvious. Now, we proceed to give a different example where the modelling power of ItFSSs is more apparent. We can also infer the need for a formal analysis of decision making in that context, hence the next example motivates Section 5.

**Example 5.** *A new regulation will have long-term effects on the population of a country. There are three groups that are potentially affected in terms of satisfaction: students (e*1*), working class (e*2*) and retired people (e*3*). There are two possibilities, namely, passing the law (p*1*) and retaining the current regulation (p*2*). Their respective yearly effects on the satisfaction across groups are captured by Table 7, which embodies an ItFSS* F = {(*Fi*, *A*)}*i*∈N *over U* = {*p*1, *p*2} *where A* = {*e*1,*e*2,*e*3}*.*

*We read, for example, that in year* 1 *the students' degree of membership to being satisfied with the new regulation is* 0.3*, and it is* 0.2 *under the current law. For the working class, the respective degrees of satisfaction are* 0.4 *and* 0.6*. For retired people, the respective degrees of satisfaction are* 0.7 *and* 0.8*.*

*From the third year onwards, the students' degree of membership to being satisfied with the new regulation is* 0.6*, and it is* 0.3 *under the current law. For the working class, the respective degrees of satisfaction are* 0.6 *and* 0.4*. For retired people, the respective degrees of satisfaction are* 0.8 *and* 0.6*.*

**Table 7.** Tabular representation of the intertemporal fuzzy soft set in Example 5.

Example 5 shows the intrinsic difficulty of dealing with these problems. The snapshots at different moments can vary substantially from each other, and of course they are not always as obvious as the situation of Example 4. Additionally, the "attributes" can have different weights, for example because they represent characteristics of groups with different proportions in the society.

For the purpose of favoring implementability, now we proceed to state an equivalent formulation of our intertemporal model. It allows us to work with a tabular format that is amenable for computations.

#### *4.2. An Alternative Representation of Intertemporal Fuzzy Soft Sets*

In practical terms, when both *U* = {*o*1, ... , *om*} and *A* = {*e*1, ... ,*en*} are finite, we can represent the information that describes an ItFSS in a table where the cells are either finite or infinite sequences of membership degrees. Table 8 gives the general form of such a representation.

**Table 8.** The tabular representation of our novel intertemporal model for fuzzy soft sets.

Let us analyze this alternative description. To that purpose, the set of infinite sequences of numbers from [0, 1] (or infinite utility streams [81–83]) is denoted by S. Our intertemporal model of fuzzy soft sets over *U* can also be defined by *F*¯ : *A* −→ **S**(*U*) where **S**(*U*) represents the mappings *U* −→ S. Consequently, for each attribute, we capture the degree of membership of any alternative in each moment of time.

Indeed, in Table 8, we can define *F*¯(*ej*)(*oi*)=(*u*1*ij*, *u*2*ij*, ... , *utij*, ...) ∈ S, hence *utij* means the degree of membership of alternative *oi* to the fuzzy set of elements that verify attribute *ej* in period *t*. Conversely, every *F*¯ : *A* −→ **S**(*U*) produces a table with the structure of Table 8 under the finiteness restriction for both *A* and *U*.

We can also swap between the tabular form and the notation of Definition 3.

From the aforementioned tabular representation of *F*¯ : *A* −→ **S**(*U*), we can define the corresponding ItFSS over *U* as the sequence F = {(*Fi*, *A*)}*i*∈N where the tabular form of each FSS (*Fi*, *A*) is described in Table 9.

**Table 9.** The tabular representation of the fuzzy soft set (*Fi*, *A*) corresponding to moment *i* in the ItFSS F = {(*Fi*, *A*)}*i*∈N represented by Table 8.

Conversely, an ItFSS given by the construction in Section 4.1 can be trivially transformed into the tabular form presented in this subsection. We do this in Table 10 for the case of G in Example 4.

**Table 10.** The tabular representation of G = {(*Gi*, *A*)}*i*∈N in Example 4.

#### **5. Choices from Intertemporal Fuzzy Soft Sets**

When it comes to prioritizing alternatives in the framework of intertemporal fuzzy soft sets, the most natural course of action is to associate a FSS with our original ItFSS and then apply Algorithm 1 (or any other existing proposal, see Table 3) to the latter fuzzy soft set.

To implement this solution, now we proceed to describe some procedures that from each ItFSS produce a static or standard fuzzy soft set (cf., Section 5.1). Afterwards, we show how we can integrate these procedures with decision making based on FSSs to design an intertemporal fuzzy soft set based decision making procedure (cf., Section 5.2).

#### *5.1. Static FSSs Associated with an Intertemporal FSS*

In this subsection, we formalize some methods that associate a static FSS (Definition 2) with any intertemporal FSS (Definition 3). We refer to these methods as *reduction mechanisms*.

The simplest reduction mechanisms act cell-by-cell on the tabular representation. For instance, one can pinpoint the evaluation at a distinguished moment (e.g., the first period); or, in the case of finite time horizon, their lowest or highest evaluation, their (either arithmetic or geometric) average, etc. Under a genuine infinity of periods, we can use the natural modifications by infimum, supremum or discounted sums for the same purpose.

Let us now formalize a few explicit reduction mechanisms. We fix F = {(*Fi*, *A*)}*i*∈N, an intertemporal fuzzy soft set over *U*. We can obtain a *reduced* FSS associated with F by the application of one of the following expressions:

$$F^\delta(A) = \frac{1-\delta}{\delta} \sum\_{i=1}^\infty \delta^i F\_i(a). \tag{2}$$

Observe that these definitions are correct: the only non-trivial case is justified in the following auxiliary result.

**Lemma 1.** *The δ-discounted FSS is well-defined, i.e., it is a FSS.*

**Proof.** We just need to observe that, because each (*Fi*, *A*) is a FSS, ∑∞*i*=1 *δiFi*(*a*) is bounded above by ∑∞*i*=1 *δi* = *δ* 1−*δ* . Hence, ∑∞*i*=1 *δiFi*(*a*) converges for each *a* ∈ *A*, and 1−*δ δ* ∑∞*i*=1 *δiFi*(*a*) 1. Obviously, 1−*δ δ* ∑∞*i*=1 *δiFi*(*a*) 0.

The first two reduction mechanisms are symmetric, in the standard sense: when F*σ* = {(*Fσ*(*i*), *A*)}*i*∈N is the ItFSS derived from a permutation of the periods *σ* : N −→ N, one has *Fpσ* = *Fp* and *Foσ* = *Fo*. However, the *δ*-discounted reduction mechanism violates the symmetric treatment of the periods, i.e., the statement *Fδσ* = *Fδ* is in general false in the aforementioned conditions.

However, the *δ*-discounted reduction mechanism has an important advantage over the pessimistic and optimistic reduction mechanisms. The pessimistic and optimistic reduction mechanisms produce a considerable loss of information because they discard the data about the degrees of membership that are not minimal and maximal respectively, whereas the *δ*-discounted reduction mechanism uses all the information available to produce the reduced FSS.

Our next remark insists on the importance of this mechanism in decision making.

**Remark 2.** *The deduction of the δ-discounted FSS is motivated by a successful solution to the problem of aggregating intergenerational utilities [81–83]. In Section 2.1, we explained that according to the popular DU model [11,12,84], decision-makers evaluate the alternatives on the basis of the weighted addition of utilities, these weights being discount factors based on temporal delays. In our benchmark case, these delays can extend to infinity.*

Due to the aforementioned advantages, henceforth we adopt the *δ*-discounted reduction mechanism as the standard mechanism for transforming an ItFSS into a FSS in practical problems.

#### *5.2. Decision Making in Intertemporal FSSs*

We are ready to put forward a procedure for ranking a finite list of alternatives when the decision-making problem is characterized by an intertemporal FSS. It consists of three basic steps.

Put shortly, the algorithm suggests to reduce the ItFSS to a FSS (Step 1) and then order the alternatives according to standard decision-making in this framework (Step 2). The ordering in the reduced FSS carries forward in the ItFSS for which it is a natural representation (Step 3), which solves our problem. The next subsection illustrates how this can be put into practice in a concrete example of project appraisal.

#### *5.3. An Example of Decision Making in the Framework of ItFSSs*

In this section, we develop an example that serves two purposes. Firstly, we describe the structure of the problem in a practical fashion that is different from the description in Section 4.2. Secondly, it illustrates the application of Algorithm 2.

#### **Algorithm 2** Algorithm for decision making

*Inputs*: An intertemporal table of fuzzy soft sets (in the notation of Table 8 or, otherwise, see Section 5.3). A reduction mechanism (e.g., from Section 5.1). A fuzzy soft set decision making procedure (e.g., from Table 3).

**Example 6.** *For the convenience of presentation, we are going to evaluate two alternative portfolios of projects that a public administration may undertake. Portfolios 1 and 2 are parameterized in terms of four attributes along an infinite number of periods, and each of these characteristics can also be regarded as a project on its own (e.g. bike lanes, urban parks, sports facilities, and sewage treatment plants). The objective of this evaluation is to choose the best alternative.*

*For each project, a value for its social suitability in each period is assigned. To simplify, we consider projects whose utilities follow the following patterns: increasing and then constant; decreasing and then constant; decreasing, then increasing and finally constant; and constant.*

*Tables 11 and 12, respectively, describe these two portfolios P*1 *and P*2*. At their bottoms they contain additional values whose meaning we explain below.*

*These tables jointly define an ItFSS that we have displayed in Table 13, albeit in incomplete form due to obvious restrictions: we cannot display the infinite digits that appear at each cell. According to Step 1 of Algorithm 2, with this element we associate a standard FSS that we denote as* (*S*, *P*)*, by the application of the DU reduction mechanism with a* 0.05 *discount rate, hence δ* = 1 1.05 ≈ 0.952381*. The suitability of this rate has been argued in Section 2.1. Let us denote* **T** = 1−*δ δ* = 0.05*. The reduced fuzzy soft set* (*S*, *P*) *is displayed in Table 14. The results of the computations that produce it already appear at the bottom of Tables 11 and 12, and now we explain how we have obtained them (see Equation (2), and Remark 2 for inspiration). In the case of Portfolio1,*

$$\mathbf{A}\_{1} = 13.085321 = 0.05\delta + 0.10\delta^2 + 0.15\delta^3 + 0.20\delta^4 + \dots$$

$$\mathbf{A}\_{2} = 10.60642701 = 1\delta + 0.95\delta^2 + 0.90\delta^3 + 0.85\delta^4 + \dots$$

$$\mathbf{A}\_{3} = 14 = 0.70\delta + 0.70\delta^2 + 0.70\delta^3 + 0.70\delta^4 + \dots$$

$$\mathbf{A}\_{4} = 18.3258941 = 0.05\delta + 0.10\delta^2 + 0.15\delta^3 + 0.20\delta^4 + \dots + 0.70\delta^9 + 0.75\delta^{10} + \dots$$

*Therefore, in the reduced FSS* (*S*, *P*)*, Equation (2) states that the degrees of membership for Portfolio 1 are* **T** ∗ **A***i. We proceed similarly to obtain the degrees of membership for Portfolio 2.*

*We now apply Algorithm 1 to the FSS* (*S*, *P*)*. Table 15 gives the necessary computations. From the last computation in Table 15, we conclude that the first portfolio should be selected.*

**Table 11.** Portfolio 1.

**Table 12.** Portfolio 2.

**Table 13.** A (necessarily incomplete) tabular representation of the intertemporal fuzzy soft set in Example 6. Each cell actually contains an infinite sequence.

**Table 14.** Tabular representation of the reduced fuzzy soft set (*S*, *P*) in Example 6.

**Table 15.** Computing the Comparison table and scores of the reduced fuzzy soft set (*S*, *P*) through Algorithm 1.

#### **6. Discussion and Concluding Remarks**

We have designed a pioneering framework for making choices in soft computing models. For the first time in this broad area, we have considered the situation where the consequences of a decision extend along an unlimited number of periods, such as a financial investment or a social project. Existing models universally refer to a finite framework, hence they are incapable of dealing with these practical issues. We have set the grounds for a correct extension to this critical aspect of decision making.

In this paper, our reference model for uncertainty has been fuzzy soft sets, which allows for fuzzy parameterized description of the alternatives in terms of a list of attributes. We have opted for working with this environment because fuzzy soft sets are especially amenable for decision making, with plenty of interesting approaches in the literature. Future research should expand the scope of the intertemporal analysis that we have founded to other frameworks such as incomplete fuzzy soft sets, rough sets, hesitant fuzzy sets, or hesitant fuzzy soft sets among many others. Whatever the selected format for the input data, when choices extend along an infinite number of periods, the fundamental roadmap for making decisions has been established in this paper.

Obviously, it may also be possible to approach the exact problem that we have described in this paper by alternative methodologies to improve the performance of our proposal, or make it more faithfully adapted to the circumstances of the problem under inspection.

Overall, we believe that the intertemporal modelization may become a thriving area of research in the extended theories of fuzziness, vagueness and uncertainty.

**Author Contributions:** Conceptualization of the model, J.C.R.A.; Methodology, Validation and Formal Analysis, J.C.R.A. and M.J.M.T.; Writing—Original Draft Preparation, Review and Editing, J.C.R.A. and M.J.M.T.; and Funding Acquisition, M.J.M.T.

**Funding:** The research of Muñoz Torrecillas was funded by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund ERDF/FEDER-UE (National R&D Project ECO2015-66504 and National R&D Project DER2016-76053-R).

**Acknowledgments:** We are grateful to three anonymous referees and the Academic Editor for their detailed comments on the original draft of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.


**1. Introduction**

This chapter answers the following question: what relationship is there between the datagathering and statistically oriented empirical research and the theoretical formulation and explanations of the social sciences? Even though it might seem obsolete after the discussions promoted by the rise of positivism, this question has obtained renewed relevance, due,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

among other reasons, to the predominance of the comparative-descriptive method in sociological and psychological research and that it has become one of the main analysis methodologies for society. Away from other debates, the comparative-descriptive analysis's only justification is its empirical reference; its goal doesn't offer anything beyond a picture of the similarities and differences observed in national societies. For example, there are investigations about the differences in moral values or unemployment, birth, marriage, divorce, mortality rates, and so on. This stems from the conviction that the comparison between social groups or between regions within a nation facilitates getting to know a society. Indeed, its results make it clear that there are differences within societies, for example, between north and south, east and west, and so on. However, these differences between one region and another, between the north and south of Italy, for example, tell very little about what makes the Italian society a society. This type of research appeals to empirical researchers or sociologists, although it barely helps to explain the emergence of society as a rising phenomenon. Society does not result from it. Many of its parts are classified and assessed, but their sum does not make a society [1].

agencies or institutions, gather valuable information about nation states´ populations and analyze their similarities and differences. However, as indicated before, these comparisons help to understand that there are significant differences between countries, for example, they help us to understand that the economies of Mexico and Germany are very different, but their contribution to understand the world order's reorganization or the globalization processes is not very significant. The challenge for quantification does not derive from the comparison between countries with similar social structures but between large regions. Undoubtedly, the European national societies can be compared to each other since they are units with multiple similar internal factors in a homogeneous social space and can be seen as elements of an institutionally defined space that encompasses them. However, what can Europe, for example, be compared with? Obviously, only with something comparable; with the United States, or China, or Latin America, probably [1]. However, due to the large dimensions of these units, their complex internal structures of subunits, which interact in a very particular manner, and their lack of a common space, it is difficult to establish comparison points. In addition, the way to compare regions in a way that contributes to understand the formation process of a

The Relationship between Theory, Scientific Explanation and Statistics in the Social Sciences

http://dx.doi.org/10.5772/intechopen.75117

9

To identify the contribution of quantitative strategies to the formulation of theory, we must pay attention to the conditions social sciences thought theories must fulfill. For this, we must first explain what is meant by "modern thinking." From this follows a guide for the construction of any social theory. Once the logical cognitive scheme every scientific explanation should be based upon is presented, the chapter describes the requirements that should be covered by any explanatory or theoretical reflection. First, it points out the need to resort to the conditions in which a phenomenon emerges. Any object or event results from a formation process that must be causally observed and reconstructed to achieve an explanation. This is also true in natural sciences, indispensable to understand diseases or the characteristics of a territory, for example.

Subsequently, the chapter addresses the distinction between first- and second-order observations, which allows us to distinguish between the empirical research activity and a reflection that relates the results of this activity to the wealth of conceptualizations and theories, which constitute a solid knowledge and have achieved the understanding of other phenomena, or,

Separating the intentions of the members from their effects has resulted essential for the study of societies. Society is no longer seen as the result of a collective idea inspired on the achievement of the common good. Therefore, in the next point, this distinction is addressed, which allows for the entry of categories such as power, competence, or conflict, to explain the dynamics of social order and places morality as a relevant means only in a community where

Then, we identify three requirements that may be considered the starting point for any explanation and theorization. In each of them, the role played by quantification is indicated. Given the results of statistical instruments that are often linked to assumptions that escape empirical verification, I present a sociobiology study at the end as example, which is supported by a

the neighborly, kinship, or face-to-face relationship ties prevail.

global society is an open question.

in any case, constructs it.

Another reason why it is important to ask the question about the relationship between the statistical data elaboration method and theory is the current conditions in social science research. Subjected to the logic of project financing, many observers of society are forced to present "useful" results, inspired by logical positivism. At the end of their work, they invariably maintain that their conclusions have been rigorously tested, that they are based on first-hand experiences, and that they have resulted from logical analysis. The result has been a colossal amount of empirical knowledge, which is hardly the object of a second-order observation, with explanatory aims, or rebuttals that lead to social theories.

In terms of Piaget's theory, it could be said that a formal-operational thinking exercise, which considers the thought and not only reality itself, is needed [2]. Data alone say nothing. To understand it, a reasoning is necessary that thoughtfully relates it to ideas that give it a meaning. Concisely, it can be said that there is too much sociological research and a growing theoretical deficit. One of the reasons that have led to this is easily identified: in terms of projects financing, the institutional preference to support quantitative research that provides immediate political usefulness instead of the thoughtful sociological observation.

Another reason why we are inclined to consider the relationship between empirical research and the scientific and theoretical explanation is the recent emergence of a global society and regional societies of considerable dimensions, for example, the European society, which represents a challenge for quantification and theorization [3]. So far, the comparative-descriptive method and its statistical resources in sociological research had application within the limits of national borders [4]. When we look at the reports provided by the institutes devoted to statistical information, as well as the publications resulting from empirical research in social sciences, it is possible to argue we are before a "nationalist" methodology [4]. The nation state constitutes the basic unit for the calculation of every social research [5]. However, comparative-descriptive research has broadened its application area. Studies, like those performed by the OECD in its member states or those done by the European Union through academic agencies or institutions, gather valuable information about nation states´ populations and analyze their similarities and differences. However, as indicated before, these comparisons help to understand that there are significant differences between countries, for example, they help us to understand that the economies of Mexico and Germany are very different, but their contribution to understand the world order's reorganization or the globalization processes is not very significant. The challenge for quantification does not derive from the comparison between countries with similar social structures but between large regions. Undoubtedly, the European national societies can be compared to each other since they are units with multiple similar internal factors in a homogeneous social space and can be seen as elements of an institutionally defined space that encompasses them. However, what can Europe, for example, be compared with? Obviously, only with something comparable; with the United States, or China, or Latin America, probably [1]. However, due to the large dimensions of these units, their complex internal structures of subunits, which interact in a very particular manner, and their lack of a common space, it is difficult to establish comparison points. In addition, the way to compare regions in a way that contributes to understand the formation process of a global society is an open question.

among other reasons, to the predominance of the comparative-descriptive method in sociological and psychological research and that it has become one of the main analysis methodologies for society. Away from other debates, the comparative-descriptive analysis's only justification is its empirical reference; its goal doesn't offer anything beyond a picture of the similarities and differences observed in national societies. For example, there are investigations about the differences in moral values or unemployment, birth, marriage, divorce, mortality rates, and so on. This stems from the conviction that the comparison between social groups or between regions within a nation facilitates getting to know a society. Indeed, its results make it clear that there are differences within societies, for example, between north and south, east and west, and so on. However, these differences between one region and another, between the north and south of Italy, for example, tell very little about what makes the Italian society a society. This type of research appeals to empirical researchers or sociologists, although it barely helps to explain the emergence of society as a rising phenomenon. Society does not result from it. Many of its parts are classified and assessed, but their sum

Another reason why it is important to ask the question about the relationship between the statistical data elaboration method and theory is the current conditions in social science research. Subjected to the logic of project financing, many observers of society are forced to present "useful" results, inspired by logical positivism. At the end of their work, they invariably maintain that their conclusions have been rigorously tested, that they are based on first-hand experiences, and that they have resulted from logical analysis. The result has been a colossal amount of empirical knowledge, which is hardly the object of a second-order observation,

In terms of Piaget's theory, it could be said that a formal-operational thinking exercise, which considers the thought and not only reality itself, is needed [2]. Data alone say nothing. To understand it, a reasoning is necessary that thoughtfully relates it to ideas that give it a meaning. Concisely, it can be said that there is too much sociological research and a growing theoretical deficit. One of the reasons that have led to this is easily identified: in terms of projects financing, the institutional preference to support quantitative research that provides immedi-

Another reason why we are inclined to consider the relationship between empirical research and the scientific and theoretical explanation is the recent emergence of a global society and regional societies of considerable dimensions, for example, the European society, which represents a challenge for quantification and theorization [3]. So far, the comparative-descriptive method and its statistical resources in sociological research had application within the limits of national borders [4]. When we look at the reports provided by the institutes devoted to statistical information, as well as the publications resulting from empirical research in social sciences, it is possible to argue we are before a "nationalist" methodology [4]. The nation state constitutes the basic unit for the calculation of every social research [5]. However, comparative-descriptive research has broadened its application area. Studies, like those performed by the OECD in its member states or those done by the European Union through academic

with explanatory aims, or rebuttals that lead to social theories.

ate political usefulness instead of the thoughtful sociological observation.

does not make a society [1].

8 Statistics - Growing Data Sets and Growing Demand for Statistics

To identify the contribution of quantitative strategies to the formulation of theory, we must pay attention to the conditions social sciences thought theories must fulfill. For this, we must first explain what is meant by "modern thinking." From this follows a guide for the construction of any social theory. Once the logical cognitive scheme every scientific explanation should be based upon is presented, the chapter describes the requirements that should be covered by any explanatory or theoretical reflection. First, it points out the need to resort to the conditions in which a phenomenon emerges. Any object or event results from a formation process that must be causally observed and reconstructed to achieve an explanation. This is also true in natural sciences, indispensable to understand diseases or the characteristics of a territory, for example.

Subsequently, the chapter addresses the distinction between first- and second-order observations, which allows us to distinguish between the empirical research activity and a reflection that relates the results of this activity to the wealth of conceptualizations and theories, which constitute a solid knowledge and have achieved the understanding of other phenomena, or, in any case, constructs it.

Separating the intentions of the members from their effects has resulted essential for the study of societies. Society is no longer seen as the result of a collective idea inspired on the achievement of the common good. Therefore, in the next point, this distinction is addressed, which allows for the entry of categories such as power, competence, or conflict, to explain the dynamics of social order and places morality as a relevant means only in a community where the neighborly, kinship, or face-to-face relationship ties prevail.

Then, we identify three requirements that may be considered the starting point for any explanation and theorization. In each of them, the role played by quantification is indicated. Given the results of statistical instruments that are often linked to assumptions that escape empirical verification, I present a sociobiology study at the end as example, which is supported by a rigorous statistical exercise but whose premises and conclusions lack an analytical foundation. Absent this, most thesis, no matter how crazy, can find a statistical basis to give it some credibility.

a paradigm for another. According to the old logic, explaining meant to part from the concrete object or phenomenon—natural or social—and to assign it an ultimate origin to claim it as its cause. This way of proceeding forced us to think of the cause as an origin analogous to what

The Relationship between Theory, Scientific Explanation and Statistics in the Social Sciences

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11

The attempt to find explanations through this process lost strength for the same question about the origin that could not be answered by resorting to an absolute origin anymore, not

Accordingly, every form of explanation where the explainer is present in the initial explanation from which it emerges is unsatisfactory, for example, in the explanation of a social phenomenon where the said phenomenon is already embedded in the actors' wills, or in the explanation

Modern thinking imposes then a first requirement for sociological research: it is not feasible to opt for a research strategy where the concept is defined beforehand to apply it then to the phenomenon being researched; we would rather have to work the other way, we must per-

Nevertheless, in many modern sociological and psychological theoretical approaches survive remnants of the absolutist and substantialist logic, which are especially visible in sociology's classical theories and authors because during the historical transition from one logic to another—and despite their attempt to abandon the old explanatory scheme—they retain the idea of an absolute as explanation, but under a different tag. In Durkheim's sociology of religion, for example, "society" is attributed to all forms of religious thought, but this concept in his theory does not provide any information about the creation process of mental constructions. Society is the only origin of the social. With regard to religious phenomena, Durkheim reaffirms the formula that inspired his sociological theory and is based on the identity logic that resort to the Absolute. In it, "society" appears as the origin and cause of all phenomena

The general result of the book (The elementary forms of religious life) is that religion is a prominently social issue, religious ideas are collective ideas that express collective realities: rites are behaviors that can only arise within groups and must be useful to maintain or restore certain psychic states of these groups [8]. In a different place: "just like the concept of religious power and divinity, the concept of soul is not devoid of reality…since society – the only source of everything sacred – is not content with moving us from outside temporarily; it is

As shown, Durkheim's concept of society is an ontological construction placed in an allexplaining position by theory, and society precedes everything which sabotages any other possible explanation. The best proof that we are dealing with a secularized form of the metaphysical structure of thought is provided by Durkheim himself, when he declares that it will

be necessary to choose between god and society as explanation [9].

of specific social phenomena that are already in the own qualities of that society [4].

would emerge from it. Cause and effect were (partially) identical.

form an observation and then specify how it developed.

*2.1.1. The survival of the old explanatory scheme*

that require explanation.

permanently installed in us." [8].

without falling into an endless return.

Surely, at this point, specialists in empirical research await an explanation of the observations´ methodology followed in this chapter, but methodological strategy is a procedure proper of the first-order observations, as we will see hereunder.


**11. Recommendations for vancomycin dosing**

In case of bacteremia, the dose of IV vancomycin is 15–20 mg/kg/day divided in two or three doses in order to conserve normal renal function. For seriously ill patients such as those with sepsis, meningitis, pneumonia, or infective endocarditis with suspected MRSA infec‐ tion, a loading dose of 25–30 mg/kg (actual body weight) may be considered. Monitoring of vancomycin trough levels is necessary to guide the dosing of this antibiotic. Serum trough levels should be measured prior to the fourth or fifth dose. For serious infections such as bacteremia, infective endocarditis, meningitis, pneumonia, and necrotizing fasci‐ itis due to MRSA*,* vancomycin trough concentrations of 15–20 μg/mL are recommended. Vancomycin trough monitoring is recommended for serious infections, patients who are morbidly obese have renal dysfunction or have fluctuating volumes of distribution. For isolates with a vancomycin MIC ≤ 2, the patient's clinical response should determine the continued use of vancomycin; however, if the patient has not had a clinical or microbio‐ logic response to vancomycin despite adequate debridement and removal of other foci of infection, an alternative to vancomycin is recommended regardless of MIC. For the isolates with a vancomycin MIC >2 μg/mL (e.g., VISA or VRSA), an alternative to vancomycin should be used.

## **12. Prevention**

Decolonization is important to achieve prevention of *S. aureus* bacteremia and other infec‐ tions. The role of decolonization in controlling the spread of *S. aureus* is still unclear. It is also unclear what the optimal regimen is. Options include agents for nasal decolonization such as mupirocin and topical body decolonization with an agent such as chlorhexidine gluconate to target the extra nasal sites. Systemic oral antibiotics can be used for decolonization; however, there are issues that are very important to consider for decolonization, recolonization, and development of resistance. The current guidelines suggest that decolonization be considered in patients with recurrent skin infections or ongoing transmission occurring among house‐ hold contacts despite optimizing wound care and hygiene measures. Hand hygiene consists of soap and water or an alcohol‐based hand rub before and after contact with infected areas. Sharing personal items is discouraged.

As for hospitals, infection control and prevention strategies should include hand hygiene, active surveillance to identify *S. aureus* colonization, and environmental cleaning. Patient bathing with chlorhexidine gluconate in intensive care units leads to a reduction in *S. aureus* colonization and infection. It is felt that bathing with chlorhexidine gluconate is a measure for source control that may lead to less contamination of health care worker hands, thus less con‐ tamination of the environment and the spread of infection to other patients. One additional infection control strategy for years has been to create a vaccination against *S. aureus*. So far, attempts have been unsuccessful, but there is much research in this area.


6. Prognosis

#### 6.1. International Prognostic Index (IPI)

Staging in DLBCL has a limited value. The original IPI was developed to identify variables that could predict OS and PFS (Table 2). The factors included were age >60 years, elevated serum lactate dehydrogenase (LDH), Eastern Cooperative group (ECOG) performance status ≥2, clinical stage III or IV, more than 1 involved extranodal disease sites. One point was given for each of the characteristics ranging from zero to five. Five-year survival rates worsened as the scores increased. Five-year overall survival rates for patients with scores of zero to one, two, three, four to five were 73, 51, 43, 26%, respectively [58] (project TIN-HsLPF). This was before the rituximab era. The utility of IPI was reassessed in a retrospective analysis of patients with DLBCL treated with R-CHOP in the province of British Columbia to assess the value of IPI [5]. They redistributed the IPI into revised IPI which identified 3 distinct prognostic groups of very good (score-0), good (scores 1, 2) and poor (scores 3, 4, 5) with a 4 year OS of 94, 79 and 55%, respectively. The NCCN incorporated detailed information about the variables in the original IPI and the location of extranodal disease is used rather than the number of extranodal disease as the lymphomatous involvement in major organs (bone marrow, CNS, liver/GI tract or lung) appeared to be a stronger predictor for a worse prognosis. It stratifies patients into four risk groups low—0–1 points, low-intermediate risk—2–3 points, high-intermediate risk—4–5 points and high risk—≥6 points with OS of 94, 72, 54, 35%, respectively. Both the R-IPI and the NCCN-IPI predict clinical outcome with accuracy and the use of R-IPI or the NCCN-IPI

Recent Advances in Diffuse Large B Cell Lymphoma http://dx.doi.org/10.5772/intechopen.74263 53

routinely to better understand the prognosis of these patients is recommended [59].

Several strategies with intense regimens have been tried to mitigate this risk. Table 3 illustrates the studies of these regimens. Additional randomized control trials are needed to evaluate the efficacy of intense regimens. Based on these small data sets, R-CHOP is associated with inferior outcome. Current literature indicates better outcomes when treated with DA-EPOCH-R in this group of patients [60]. The regimen is a dose adjusted regimen combining etoposide, prednisone, vincristine, cyclophosphamide and doxorubicin in an infusional manner targeting prolonged drug exposure to reduce resistance and dynamic dose adjustments allowing for

Treatment PFS OS

10.9 months

p = 0.001)

12.1 months 22.2 months\* 18.9 months 18.9 month

25% 32% 67%

PFS (R-CHOP): 7.8 months vs. 26.6 months (intensive regimen 21.9 months (p = 0.14)

OS with R-EPOCH vs. R-CHOP (p = 0.057)

6.2. Double-/triple-hit lymphoma

highest acceptable doses.

Study Number of

Petrich et al. [44] Induction therapy with HDSCT

patients

Oki et al. [43] 129 R-CHOP

Howlett et al. [46] 394 R-CHOP

\*p = 0.032 (significant at p <0.05).

311 R-CHOP (32%)

Hyper CVAD/MA (2%) R-EPOCH (21%) R-CODOX-M/IVAC (14%)

R-Hyper CVAD/MA R-EPOCH

Table 3. Intense regimen studies for MYC rearranged with/without BCL2 rearrangement.

R-EPOCH R-Hyper-CVAD/MC R-CODOX-M/R-IVAC

Dunleavy et al. [45] 52 DA-REPOCH 79% (14 months follow up) 77%

Table 2. Original and modified International Prognostic Index.

three, four to five were 73, 51, 43, 26%, respectively [58] (project TIN-HsLPF). This was before the rituximab era. The utility of IPI was reassessed in a retrospective analysis of patients with DLBCL treated with R-CHOP in the province of British Columbia to assess the value of IPI [5]. They redistributed the IPI into revised IPI which identified 3 distinct prognostic groups of very good (score-0), good (scores 1, 2) and poor (scores 3, 4, 5) with a 4 year OS of 94, 79 and 55%, respectively. The NCCN incorporated detailed information about the variables in the original IPI and the location of extranodal disease is used rather than the number of extranodal disease as the lymphomatous involvement in major organs (bone marrow, CNS, liver/GI tract or lung) appeared to be a stronger predictor for a worse prognosis. It stratifies patients into four risk groups low—0–1 points, low-intermediate risk—2–3 points, high-intermediate risk—4–5 points and high risk—≥6 points with OS of 94, 72, 54, 35%, respectively. Both the R-IPI and the NCCN-IPI predict clinical outcome with accuracy and the use of R-IPI or the NCCN-IPI routinely to better understand the prognosis of these patients is recommended [59].