On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations

: The subtraction of fuzzy numbers (FNs) is not an inverse operator to FNs addition. The family of all oriented FNs (OFNs) may be considered as symmetrical closure of all the FNs family in that the subtraction is an inverse operation to addition. An imprecise present value is modelled by a trapezoidal oriented FN (TrOFN). Then, the expected discount factor (EDF) is a TrOFFN too. This factor may be applied as a premise for invest-making. Proposed decision strategies are dependent on a comparison of an oriented fuzzy proﬁt index and the speciﬁc proﬁtability threshold. This way we get an investment recommendation described as a fuzzy subset on the ﬁxed rating scale. Risk premium measure is a special case of proﬁt index. Further in the paper, the Sharpe’s ratio, the Jensen’s ratio, the Treynor’s ratio, the Sortino’s ratio, Roy’s criterion and the Modiglianis’ coe ﬃ cient are generalised for the case when an EDF is given as a TrOFN. In this way, we get many di ﬀ erent imprecise recommendations. For this reason, an imprecise recommendation management module is described. Obtained results show that the proposed theory can be used as a theoretical background for ﬁnancial robo-advisers. All theoretical considerations are illustrated by means of a simple empirical case


Introduction
Imprecision is a natural feature of financial market information. A widely accepted way of representing an imprecise number is a fuzzy number (FN). The notion of an ordered FN was intuitively introduced by Kosiński et al. [1]. It was defined as an FN supplemented by its orientation. A significant drawback of Kosiński's theory is that there exists such ordered FNs that cannot be considered as FN [2]. This caused the original Kosiński's theory to be revised by Piasecki [3]. At present, the ordered FNs defined within Kosiński's original theory are called Kosiński's numbers [4][5][6][7]. If ordered FNs are linked to the revised theory, then they are called Oriented FNs (OFNs) [6,7].
The family of all OFNs has a symmetry axis that is equal to the family R of all real numbers. In Section 2, this axial symmetry is described in detail. The family of all OFNs may be defined equivalently with the use of the discussed axial symmetry as the symmetrical closure of all of the FNs family. Symmetry allows us to avoid problems related to the fact that FNs subtraction is not an inverse operator to FNs addition.
A robo-adviser is an internet platform providing an automated financial planning service. This service is always algorithm-driven. Therefore, no robo-adviser requires any human involvement. It implies a minimal operating cost for any robo-advisor. Due to using robo-advisers we can apply different finance models to develop algorithms editing financial advice. Implemented algorithms can inform investors of any change in the market within a short period of time. In this way, robo-advisers efficiently implement any investment strategy by using their built-in automated algorithms [8].

•
The use of FNs in financial analysis only leads to averaging the imprecision risk, • The application of OFNs in financial analysis may minimise imprecision risk.
Therefore, the main aim of this paper is an extension of the investment-making models described in [46] to the case of imprecise PV estimated by trapezoidal OFNs (TrOFNs). The first attempt of this subject was presented in [70]. Here, we use our experience gathered during our work on the other criteria. Therefore, here we present a revised approach to the considered extension.
The paper is drafted as follows. Section 2 presents OFNs with their basic properties and describes the imprecision evaluation by an energy and entropy measure. In Section 3, PV is assessed by TrOFNs. The oriented fuzzy EDF is determined in Section 4. Investment recommendations dependent on the oriented fuzzy EDF are discussed in Section 5. Profitability criteria for investments are extended in Section 6. In Section 7 we explore the management of a set of investment recommendations. Section 8 concludes the article and proposes some future research directions. In Appendix A, the optimisation algorithm used is described in detail.

Fuzzy Sets
Fuzzy sets (FSs) [73] are a suitable tool that allows to describe and process imprecise values and information. In a given space X an FS A is distinguished by its membership function µ A ∈ [0, 1] X in the following manner A = (x, µ A (x)); x ∈ X (1) By F (X) we denote the family of all FSs in a space X.
Multi-valued operations in the FSs family are defined by means of the following identities: FSs are widely used for modelling imprecise information. Following the work in [74], the imprecision is understood as a composition of ambiguity and indistinctness of. Ambiguity is defined a lack of a clear indication of one alternative among others. Indistinctness is defined as a lack of an explicit distinction between distinguished and not distinguished alternatives. More imprecise information is less useful. For this reason, it is sensible to assess the imprecision.
For the finite space X = x 1 , x 2 , . . . , x f , the suitable tool for assessing the ambiguity of an FS A ∈ F (X) is the energy measure d : F (X) → R + 0 [75] given as follows: The proper tool for measuring the indistinctness is the entropy measure e : F (X) → R + 0 [76,77] determined by the identity

Fuzzy Numbers
The fuzzy number (FN) may be intuitively defined as FS in the real line R. The most general FN definition is proposed by Dubois and Prade [78]. Any FN may be defined in an equivalent way as follows [79]: Theorem 1. For any FN L there exists such a non-decreasing sequence (a, b, c, d) ⊂ R that L(a, b, c, d, L L , R L ) = L ∈ F (R) is determined by its membership function µ L (·|a, b, c, d, L L , R L ) ∈ [0, 1] R described by the identity where the left reference function L L ∈ [0, 1] [a,b] and the right reference function R L ∈ [0, 1] [c,d] are upper semi-continuous monotonic ones meeting the conditions: The family of all FNs is denoted by the symbol F. The symbol * denotes any arithmetic operation defined on R. By the symbol we denote such extension of operation * to F that it is coherent with Zadeh's Extension Principle [80]. It means that, for each pair (K, L) ∈ F 2 described by their membership functions µ K , µ L ∈ [0, 1] R , the FN is described by membership function µ M ∈ [0, 1] R determined by the identity: A special case of FNs is trapezoidal FNs (TrFNs). Due to their simplicity and ease of performing operations on them, they are often used in applications. A suitable definition of trapezoidal fuzzy numbers is given in [81]:

Oriented Fuzzy Number
Ordered FN was defined by Kosiński et al. [1] as an extension of the FN concept. An important disadvantage of Kosiński's theory is that there exists such ordered FNs that cannot be represented by a membership function [2]. On the other hand, ordered FNs' usefulness is a result of their interpretation as FN supplemented by its orientation. The ordered FN orientation describes a forecast of the nearest future changes of FN. This caused Kosiński's theory to be revised by Piasecki [3]. An ordered FN linked to the revised theory is called Oriented FN (OFN) [6,7]. In a general case, OFNs are defined as follows: Definition 2 [3]. For any monotonic sequence (a, b, c, d) ⊂ R, the oriented fuzzy number OFN ↔ L(a, b, c, d, S L , E L ) = ↔ L is a pair of an orientation → a, d = (a, d) and a fuzzy set L ∈ F (R) described by a membership function µ L (·|a, b, c, d, S L , E L ) ∈ [0, 1] R given by the identity The symbol K denotes the space of all OFNs. If a < d, then any ↔ L(a, b, c, d, S L , E L ) is a positively oriented OFN. Any positively oriented OFN may be interpreted as such FN, which can increase in the near future. The symbol K + denotes the space of all positively oriented OFN. If a > d, then any ↔ L(a, b, c, d, S L , E L ) is a negatively oriented OFN, which may be interpreted as decreasing FN. The symbol K − denotes the space of all negatively oriented OFN. For a = d, ↔ L(a, b, c, d, S L , E L ) = a represents the unoriented crisp number a ∈ R. Summing up, we see that Let us consider the mapping U : K → K given by identity This mapping meets following conditions: It shows that the mapping (15) is axial symmetry on the space K of all OFNs. Then the symmetry axis is identical with family R of all real numbers. Moreover, Theorem 1 together with Definition 2 implies that the space F of FNs and the space K + of all positively oriented OFNs are isomorphic. Therefore, we can say that the space K may be determined as symmetry closure of the space F.
In the studies planned here, we limit discussion to a special kind of OFNs defined as follows.
The symbol K Tr denotes the space of all TrOFNs.On the space K Tr , a relation ↔ K . GE.L was defined as follows This is a fuzzy preorder GE ∈ F (K Tr × K Tr ) described by membership function ν GE ∈ [0, 1] K Tr ×K Tr described in detail in [6]. Due to these results, for any pair ( ↔ Tr(a, b, c, d), h) ∈ K Tr × R we have:

Oriented Present Value
The present value (PV) is defined as a current equivalent of a payment due at fixed point in time [41]. Therefore, we commonly accept that PV of future payments may be imprecise. This means that PV should be assessed with FNs. Such PV is called a fuzzy one. Buckley [52], Gutierrez [55], Kuchta [56] and Lesage [57] show the soundness of using TrFNs as an imprecise financial arithmetic tool. Moreover, PV estimation should be supplemented by a forecast of PV closest price changes. These price changes may be subjectively predicted. Moreover, closest price changes may be predicted with the help of the prediction tables presented in [82]. For these reasons, an imprecise PV should be evaluated by OFN [7,70]. Such PV is called an oriented PV (O-PV). Any O-PV is estimated by TrOFN where the monotonic sequence V s , V f ,P, V l , V e is defined as follows is the set of all values that do not noticeably differ from the quoted priceP.
If we predict a rise in price then O-PV is described by a positively oriented TrOFN. If we predict a fall in price, then O-PV is described by a negatively oriented OFN. Example 1. We observe the portfolio π composed of company shares included in WIG20 quoted on the Warsaw Stock Exchange (WSE). Based on a session closing on the WSE on 28 January 2020, for each observed share we assess its O-PV equal to TrOFN describing its Japanese candle [83]. Shares' O-PVs, obtained in such a manner, are presented in Table 1. For each portfolio componentŜ, we determine its quoted priceP s as an initial price on 29.01.2020.

Oriented Expected Discount Factor
We assume that duration t > 0 of an investment is fixed. Then, the considered security is determined by two values: a foreseen FV = V t and an estimated PV = V 0 . The benefits from owning this security are characterised by the simple return rate (RR) defined by the identity where the simple RR r t : Ω → R is determined for PV assessed as the quoted priceP. After Markowitz [9] we assume that the RR r rate is the gaussian probability distribution N(r, σ).

In our case PV is determined as O-PV
According to (10), the simple RR calculated for the O-PV is a fuzzy probabilistic set represented by membership function ρ ∈ [0; 1] R×Ω given as follows Then, the membership function ρ ∈ [0; 1] R of the expected RR is computed in the following manner In [48] it is shown that the fuzzy expected discount factor (EDF) is a better tool for appraising any securities than the expected fuzzy RR. Therefore, we determine EDF for the case of O-PV. In general, for a given RR r t , the discount factor v t is explicitly determined by the identity v t = 1 1 + r t (29) We consider the EDF v ∈ R defined by the identity: In line with (28), the membership function δ ∈ [0, 1] R of an oriented fuzzy EDF (O-EDF) ↔ V ∈ K is given by the identity: Then, O-EDF is given as follows: Example 2. All considerations in the paper are run for the quarterly period of the investment time t = 1 quarter. We research the components of the portfolio π presented in Table 1. Using the one-year time series of quotations, for each portfolio componentŜ we calculate the following parameters: With the application of (30) and (32), we calculated quarterly O-EDF for each component of the portfolio π. All evaluations obtained in this way are presented in Table 2.
The O-EDF of a security described in this way is a TrOFN with the identical orientation as the O-PV used for estimation. It is worth stressing that the maximum criterion of the expected RR can be equivalently replaced by the minimum criterion of the EDF.

Investment Recommendations
We understand an investment recommendation as a counsel given by the advisors to the investor. After evaluating the stocks, the advisor compares the obtained assessment with the current market value of the stocks. The difference between those values determines the potential of the investment return rate. Advisors give various recommendations depending on the volume of the return rate potential and its direction. Experts also define the potential of the return rate in different ways. We will here consider the collection of standardised recommendations, which are applied in [46]. The rating scale is given as the A ++ denotes the advice Buy suggesting that the expected price is well above the current quoted price, • A + denotes the advice Accumulate suggesting that the expected price is above the current quoted price, • A 0 denotes the advice Hold suggesting that the expected price is similar to the current quoted price, • A − denotes the advice Reduce suggesting that the expected price is below the current quoted price, • A −− denotes the advice Sell suggesting that the expected price is well below the current quoted price.
The investor attributes each recommendation with the appropriate way of entering the transaction and the value of its volume. The way of entering the transaction describes the investment strategy. Investors can differ among one another by the implemented strategies.
Let fixed securityŠ be represented by the pair (r s , s ), where r s is an expected RR onŠ and s is other parameter characterisingŠ. The symbol S denotes the set of all considered securities. Adviser's counsel depends on the expected RR. The criterion for a competent choice of advice can be presented as a comparison of the profit index g(r s s ) and the profitability threshold (PT)Ǧ, where g(·| s ) : R → R is an increasing function of the expected RR. The advice choice function Λ : S × R → 2 A was given in the following way [46] This way, the advice set Λ Š ,Ǧ ⊂ A was assigned. We interpret the advice set Λ Š ,Ǧ as the investment recommendation given for the securityŠ.
The securityŠ may be equivalently represented by the ordered pair (v s , s ), where v s is the EDF determined by (30). Then the identity (30) implies The value H s is used as a specific profitability threshold (SPT) appointed for the securityŠ. Then, the advice choice function Λ : S × R → 2 A is equivalently described in the following way We consider the case when the securityŠ is characterised by the ordered pair ( (32). Then the advice choice function Λ Š ,Ǧ is FS described by (44) in the following way: where ν GE : K Tr × K Tr → [0, 1] is membership function of relation "less than or equal" (20). The required values of this function are computed with the use of (21) and (22). From the point of view of invest-making, the value λ A Š ,Ǧ is understood as a recommendation degree of the advice A ∈ A, i.e., a declared participation of the advisor's responsibility in the case of a final invest-made according to the advice A ∈ A. It implies that the investment recommendation Λ Š ,Ǧ is emphasised as a FS in the rating scale A.
In turn, the final decision is taken by the investors. Their personal responsibility for taking this investment decision decreases along with the increase in the recommendation degree related to the decision taken.
The increase in the ambiguity of the recommendation Λ Š ,Ǧ ∈ F (A) suggests a higher number of alternative recommendations to choose from. This is an increase in the risk of choosing an incorrect decision from recommended ones. This may result in obtaining a profit lower than maximal, that is with a loss of chance. Such risk is called an ambiguity risk. The ambiguity risk burdening the recommendation Λ Š ,Ǧ ∈ F (A) is assessed with an energy measure d Λ Š ,Ǧ computed with the use of (5).
An increase in the indistinctness of the recommendation Λ Š ,Ǧ ∈ F (A) suggests that the explicit distinction between recommended and not recommended decisions is more difficult. This causes an increase in the indistinctness risk understood as risk of choosing a not recommended decision. The indistinctness risk burdening the recommendation Λ Š ,Ǧ ∈ F (A) is measured by the entropy measure e Λ Š ,Ǧ computed with the use of (6).
An imprecision risk is always determined as a combination of indistinctness and ambiguity risks combined.

The Profitability Criteria for Investments
We evaluate chosen securities traded on a fixed capital market. We always assume that there exists a risk-free bond instrument represented by the pair (r 0 , 0). Moreover, we distinguish the market portfolio represented by the pair (r M , σ M ).
Example 3. We focus on the WSE. We take into account a risk-free bound instrument determined as quarterly treasure bonds with a risk-free RR r 0 = 0.0075. The market portfolio is determined as the portfolio determining a stock exchange index WIG. The RR from WIG has the normal distribution N r M , σ M 2 = N(0.0200, 0.000025).

Sharpe Ratio
The profit index is defined as Sharpe's ratio estimating the amount of the premium per overall risk unit. Then Sharpe's PT is equal to the unit premium of the market portfolio risk [84].
If the securityŠ is represented by the pair (r s , σ s ), then, in line to Sharpe, the profit index g(·|σ s ) : R → R and the PTǦ are defined as follows: We compute SPT H s with the use of (38) in the following manner: Example 4. Using Equation (52), we compute an SPT H s for all components of the portfolio π described in Examples 1 and 2. Obtained SPT values are compared with O-EDFs in Table 3. If we estimate PV by TrOFN presented in Table 1, then using the Sharpe criterion is simply comparing an imprecise O-EDF with the precise SPT. By means of Equations (45)-(49), we compute the values of the recommendation choice function presented in Table 4. Table 4 also presents information on the imprecision risk burdening individual recommendations. That information will be used to choose the recommendation.
Investment recommendations for ALR, CCC, OPL and PKN are burdened with the increased ambiguity risk. Moreover, the recommendations for CCC, OPL and PKN carry the indistinctness risk. For that reason, those recommendations are rejected. Eventually, only the following stocks are attributed with "Buy" or "Accumulate" recommendation: CDR, CPS, DNP, JSW, KGH, LTS, LPP, MBK, PEO, PGE, PGN, PKO, PLY, PZU, SPL and TPE. Thus, the disclosure of imprecision of PV estimations allows rejecting riskier recommendations.  Table 4. Imprecise recommendations determined with the use of the Sharpe ratio.

Jensen's Alpha
The profit index is defined as Jensen's alpha [85], estimating the amount of the premium for market risk. The securityŠ is represented by the pair (r s , β s ), where β s is the directional factor of the CAPM model assigned to this instrument. Then, the profit index g(·|σ s ) : R → R and the PTǦ are defined as follows: We calculate SPT H s with the use of (38) in the following manner Example 5. Using (55), we calculate a specific profitability threshold SPT H s for all components of the portfolio π described in Examples 1 and 2. The CAPM directional factors for each portfolio component are presented in Table 2. Evaluations obtained in this way are presented in Table 5. If now we estimate PV with the use of TrOFN presented in Table 1 then using the Jensen's alpha goes down to the comparison of an imprecise O-EDF with the precise SPT. By means of (45)-(49) we estimate the values of a recommendation choice function presented in Table 6.
Investment recommendations for ALR, CCC, CPS, OPL, PGE, PKN and TPE are burdened with the increased ambiguity risk. Moreover, the recommendations for ALR, CCC, CPS and PGE carry an indistinctness risk. For that reason, those recommendations are rejected. Eventually, only the following stocks are attributed with "Buy" or "Accumulate" advice: CDR, DNP, JSW, KGH, LTS, LPP, PEO, PGN, PKO, PLY, PZU and SPL. Advice "Sell" or "Reduce" were associated with MBK. Thus, the disclosure of imprecision of PV estimations allows rejecting riskier recommendations.

Treynor Ratio
The profit index is defined as the Treynor ratio [86], which estimates the amount of premium for the market risk. The securityŠ is represented by the pair (r s , β s ), where β s is the directional factor of the CAPM model assigned to this instrument. Then the profit index g(·|σ s ) : R → R and the PTǦ are defined as follows: g(r s β s ) = r s − r 0 β s (56) We compute SPT H s with the use of (38) in the following manner Example 6. Using (58), we calculate SPT for all components of the portfolio π described in Examples 1 and 2.
Evaluations obtained in this way are presented in Table 7.
If we estimate PV with the use of TrOFN presented in Table 1 then using the Treynor ratio criterion goes down to the comparison of an imprecise O-EDF with the precise SPT. By means of (45)-(49) we estimate the values of the recommendation choice function presented in Table 8.
Investment recommendation for CCC is burdened with an increased ambiguity risk and carries an indistinctness risk. For that reason, this recommendation is rejected. Eventually, only the following stocks are attributed with "Sell" or "Reduce" advice: ALR, CDR, CPS, DNP, KGH, LTS, LPP, MBK, OPL, PEO, PGE, PGN, PKN, PKO, PLY, PZU, SPL and TPE. Advice "Buy" or "Accumulate" were associated just with the stock of JSW. Thus, the disclosure of imprecision of PV estimations allows rejecting riskier recommendations.

Sortino Ratio
The Sortino ratio [87] is a tool for risk management under a financial equilibrium. In this model we compare the expected RR r s from considered security and the expected return rate r M from the distinguished market portfolio. We consider the advice choice function where the profit index and the limit value are determined by the Sortino ratio. Then, the profit index evaluates the amount of a specific unit premium for the loss risk. Moreover, the limit value evaluates an amount of the market unit premium for the loss risk. The benchmark of our assessment is a market portfolio represented by such an ordered pair r M , ς M 2 , where the downside semi variance ς M 2 evaluates the market loss risk.
The reference point is a risk-free bond instrument represented by the ordered pair (r 0 , 0), where r 0 is a risk-free return rate. The considered securityŠ is represented by the ordered pair r s , ς 2 S , where downside semi variance ς S 2 evaluates the loss risk. Then, Sortino and Price (1997) define the profit index g(·|ς s ) : R → R and the limit value PTǦ as follows: We compute SPT H s Ǧ with the use of (38) in the following manner Example 7. The market portfolio is represented by the ordered pair r M , ς M 2 = (0.0200, 0.000015). Using (61), we calculate SPT for all securities belonging to the portfolio π described in Examples 1 and 2. Evaluations obtained in this way are presented in Table 9. For each considered security, by means of (45)-(49) we calculate membership functions of investment recommendations presented in Table 10. Table 10. Imprecise recommendations determined with the use of the Sortino ratio.

Modiglianis' Coefficient
In the crisp case, the Modiglianis' Coefficient Criterion is equivalent to Sharpe Ratio Criterion. In this model, the compared values are the expected RR on a security and the expected RR on the market portfolio. Modiglianis' profit coefficient estimates the bonus over market profits. Modiglianis' limit value equals zero.
If the securityŠ is represented by the pair r s , σ 2 S , then Modigliani [88] defines the profit index g(·|σ s ) : R → R and the PTǦ as follows: We compute SPT H s with the use of (38) in the following manner We see that in a fuzzy case, the Modiglianis' Coefficient Criterion is also equivalent to the Sharpe Ratio Criterion. In this case, the recommendations obtained with the use of Modiglianis' Coefficient can be found in Table 4.

Roy's Criterion
Roy [89] has consider a fixed securityŠ, represented by the pair (r s , σ s ), where r s is an expected return onŠ and σ 2 S is the variance of a return rate of the considered financial instrument. After Markowitz [9] we assume that the considered securityŠ has a simple return rate with Gaussian distribution N(r s , σ s ). This distribution is described by its increasing and continuous cumulative distribution function F · r s , σ s : R → [0; 1] given by the identity where the function Φ : R → [0; 1] is the cumulative distribution function of the Gaussian distribution N(0, 1). The Safety Condition [89] is given as follows: where • L-a minimum acceptable RR, • ε-the probability of RR realisation below the minimum acceptable rate.
The RR realisation below the minimum acceptable rate is identified with a loss. The Roy's criterion minimises the probability of a loss for a set minimum acceptable rate of return [46]. Additionally, the investor assumes the maximum level ε * of the loss probability. Then the Roy's criterion is described by the inequality In line with (38), SPT is given as follows Example 8. We study recommendations implied by Roy's criterion all components of portfolio π described in Example 1. The investor assumes the minimal acceptable RR L = 0.0075. Additionally, the investor assumes the maximum level of a loss probability ε * = 0.05. Then, we have Φ −1 (0.05) = −1.64. Table 2 lists the values of O-EDF. Using (69), we compute SPT for all components of the portfolio π described in Examples 1 and 2. Evaluations obtained in this way are presented in Table 11.
If we estimate PV with the use of TrOFN presented in Table 1 then using the Roy's criterion goes down to the comparison of an imprecise OEF with the precise SPT [70]. By means of (45)-(49) we then estimate the values of a recommendation choice function presented in Table 12.
Investment recommendations for ALR and CCC are burdened with an increased ambiguity risk. Moreover, the recommendations for CCC carry the indistinctness risk. For that reason, those recommendations are rejected. Eventually, only the following stocks are attributed with "Buy" or "Accumulate" advice: CDR, CPS, DNP, JSW, KGH, LTS, LPP, MBK, OPL, PEO, PGE, PGN, PKN, PKO, PLY, PZU, SPL and TPE. Thus, the disclosure of imprecision of PV estimations allows rejecting riskier recommendations.

Discussions
This chapter presented the recommendations obtained by means of ratios representing various criteria of assessment of the current financial efficiency of a considered asset. Here we have

•
Sharpe ratio and Sortino ratio used to maximise the premium per overall risk unit, • Jensen's alpha and Treynor ratio used to maximise the premium for market risk.

•
Roy's criterion used to minimise the probability of bearing the unacceptable loss.
This opulence of the used criteria explains to some extent the variety of recommendations attributed by the mentioned criteria to the same financial instrument. However, this is not the only reason of the differentiation between those recommendations. We should pay attention to a big differentiation of the recommendations established by Jensen's alpha and Treynor ratio used to maximise the premium for risk. This phenomenon is difficult to explain substantively. Hence, we deduce that while managing the chosen financial instruments, we should take into account a fixed set of recommendations that attributed to them. The next chapter will be dedicated to the issue of managing the fixed set of investment recommendations.

Management of Investment Recommendation Set
In Sections 5 and 6, the proposed procedure for recommendations was always considered in the case of one established criterion. Due to that we could mark all recommendations with a single symbol. In this chapter we will consider the relations between recommendations with various criteria attributed to them. For a bigger transparency of those considerations we will introduce a modified system of recommendation labels.
Any FS Λ γ ∈ F (A) is called a recommendation. The subscript γ means any set of symbols identifying the kind of distinguished recommendation. Any recommendation Λ γ is represented by its membership function λ γ : A → [0, 1] . Also, each recommendation can be noted as In special cases we have Moreover, in notation (70) of recommendation Λ γ we can omit every advice A ∈ A satisfying the condition µ γ (A) = 0. Each securityŠ is assigned a recommendation Λ S,1 , Λ S,2 . . . , Λ S,5 ∈ F (A), where • Λ S,1 -recommendations obtained with the use of the Sharpe ratio, • Λ S,2 -recommendations obtained with the use of Jensen's alpha, • Λ S,3 -recommendations obtained with the use of the Treynor ratio, • Λ S,4 -recommendations obtained with the use of the Sortino ratio, • Λ S,5 -recommendations obtained with the use of Roy's criterion.
Example 9. Table 12 presents the recommendation Let us note that various criteria assign different recommendations to the same security. Each recommendation can bear a different imprecision risk. We propose to limit the acceptable recommendations to those that are characterised by the minimal risk of imprecision. However, imprecision is evaluated by the means of two indices, which should be minimised. In this case, to minimise the risk, a multicriterial approach was implemented.
Each recommendation Λ S,i is given a pair d Λ S,i , e Λ S,i where d Λ S,i and e Λ S,i respectively mean energy and entropy measures. On the recommendation set we define two preorders " Λ S,i is more acceptable than Λ S,j " : Those preorders are formal models of ambiguity and indistinctness of information minimisation criterion. A multicriterial comparison defined by the preorders Q 1 and Q 2 is a model of satisfying the postulate of minimisation of both factors.
Using the multicriterial comparison (73) and (74) for each securityŠ we determine the Pareto optimum O s which includes all acceptable recommendations. To solve this optimisation task, we use an algorithm described in Appendix A.

Stock Company
Pareto Optimum The results obtained in Example 10 show that in the case of many securities there is a big variety in the sets of optimum recommendations.
To unify the final recommendations for each securityŜ we determine: • A weakly justified recommendation (WJR) Λ S,WJR defined as the union of such Pareto optimal recommendations, which are linked to the securityŜ; • A strongly justified recommendation (SJR) Λ S,SJR defined as the intersection of such Pareto optimal recommendations, which are linked to the securityŜ.
The WJR Λ S,WJR and the SJR Λ S,SJR are determined respectively by their membership functions given as follows Example 11. Separatelyfor each securityŜ described in Example 1, imprecise recommendations Λ S,1 , Λ S,2 , Λ S,3 , Λ S,4 , Λ S,5 are compared in more detail in Tables 14-33. In the two bottom rows of Tables 14-33 WJRs and SJRs are given along with their imprecision estimates. All Tables 14-33 are linked to the comments by using the names of discussed stock companies. Table 14. Imprecise and Pareto optimal recommendations for ALR.
For the ALR shares only the following recommendation is Pareto optimal which was obtained by means of the Treynor ratio. In such situation, for this recommendation, WJR and SJR are identical. The advice Sell and Reduce is recommended by the advisor with the degree that equals 1. It means that the advisor is prepared to take full responsibility for making the investment decisions resulting from the suggested advice. The other recommendations are rejected by the advisor. In such case it is the investor who takes full responsibility for making any other decision resulting from the rejected recommendations of Buy, Accumulate and Hold.
From the distribution of a recommendation degree represented by WJR it shows that the advisor definitely rejects the Sell recommendation. WJR also tells us that the Accumulate and Buy recommendations can be taken into consideration. Additional information on the distribution of the responsibility for the decisions taken is reflected in SJR. It supplements the picture with the following information: • The investor bears almost all the responsibility for making an investment decision resulting from the Reduce and Hold recommendations, • The advisor bears full responsibility for making an investment decision resulting from the Accumulate recommendations, • The investor and the advisor share the responsibility among themselves for making the investment decision based on the Buy recommendation, however, the advisor bear approximately two-thirds of that responsibility.
After analysing the information and interpretations, the investor takes the decision. We can suspect that the investor characterised by risk-aversion will choose the Accumulate recommendation while the investor who is a risk-taker will choose the Buy advice. Table 16. Imprecise and Pareto optimal recommendations for CDR.

Criterion
A For the CDR shares the following WJR and SJR were determined From the distribution of a recommendation degree represented by WJR it shows that the advisor definitely rejects the Hold recommendation. It means that the advisor recommends an investment activity without defining its kind. SJR shows that it is the investor who bears full responsibility for any decisions made. It is obvious that such a recommendation is not useful so in such a situation we state that there is no useful recommendation.
For the CPS shares, WJR and SJR are determined by (80) and (81). In this situation we state that there is no useful recommendation. Table 18. Imprecise and Pareto optimal recommendations for DNP.
For the DNP shares, WJR and SJR are determined by (80) and (81). In this situation we state that there is no useful recommendation. Table 19. Imprecise and Pareto optimal recommendations for JSW.

Criterion
A For the JSW shares, the following WJR and SJR were determined From the distribution of a recommendation degree represented by WJR it shows that the advisor definitely rejects the Sell, Reduce and Hold recommendations. The advisor strongly recommends Accumulate or Buy. SJR shows that it is the advisor who is willing to take full responsibility for taking the investment decisions resulting from the advised recommendations. Table 20. Imprecise and Pareto optimal recommendations for KGH.
For the KGH shares, WJR and SJR are determined by (80) and (81). In this situation we state that there is no useful recommendation. Table 21. Imprecise and Pareto optimal recommendations for LTS.
For the LTS shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 22. Imprecise and Pareto optimal recommendations for LPP.

Criterion
A For the LPP shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 23. Imprecise and Pareto optimal recommendations for MBK.
For the MBK shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation.  (77) is only the one determined by the Treynor ratio. From the distribution of a recommendation degree represented by WJR it shows that the advisor definitely rejects the Hold, Accumulate and Buy recommendations. The advisor strongly recommends Sell or Reduce. SJR shows that the advisor is willing to take full responsibility for taking the investment decisions resulting from the advised recommendations. Table 25. Imprecise and Pareto optimal recommendations for PEO.
For the PEO shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation.
For the PGE shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 27. Imprecise and Pareto optimal recommendations for PGN.
For the PGN shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation.  (80) and (81). In this situation there is no useful recommendation. Table 29. Imprecise and Pareto optimal recommendations for PKO.

Criterion
A For the PKO shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 30. Imprecise and Pareto optimal recommendations for PLY.
For the PLY shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 31. Imprecise and Pareto optimal recommendations for PZU.
For the PZU shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation. Table 32. Imprecise and Pareto optimal recommendations for SPL.

Criterion
A For the SPL shares, WJR and SJR are determined by (80) and (81). In this situation there is no useful recommendation.
and (81). WJR informs us that the advisor does not exclude any recommendation. SJR shows that full responsibility for taking any investment decision goes to the investor. Therefore, there is no useful recommendation. Summing up, for the public companies considered in the examples, in most cases there was no useful recommendation. Such a situation occurred in the case of CDR, CPS, DNP, KGH, LTS, LPP, MBK, PEO, PGE, PGN, PKN, PKO, PLY, PZU, SPL and TPE. Only for three following companies: ALR, CCC and OPL the recommendations could be considered useful. This situation does not differ from the real phenomena in financial markets. The number of useless recommendations can be decreased by limiting the number of assessment criteria. Also, another set of criteria can be implemented. The solution to those problems should be searched based on finance.
An observation can be useful that each pair of WJR and SJR might be presented as an intuitionistic fuzzy set [90] representing a justified recommendation (JR). Then any JR is defined by its membership function equal to the SJR membership function and by its non-membership function equal to the membership function of WJR complement.

Conclusions
In the subject literature it is shown that OFNs are a more convenient tool for financial analysis than FNs. Therefore, the most important achievement of this work is the implementation of OFNs into the algorithmic system supporting investment decisions. In my best knowledge, the obtained algorithmic system is the only one that applies any set of profitability criteria evaluated with the use of OFNs. Until now, only an analogous system was known to be linked to the Sharpe's criterion. For any security, this simple system assigns exactly one imprecise recommendation. The algorithmic system described in Sections 5 and 6 assigns each security many different imprecise recommendations. For this reason, in Section 7, the proposed system is equipped with an imprecise recommendation management module.
Obtained results may provide theoretical foundations for constructing a robo-advice system supporting investment decisions. Then, we can use determined recommendations as behavioural premises for investment decisions. The attempt to use chosen recommendations multiple times leads to establishing an investing strategy. In Example 11, the interpretation of determined recommendations was presented. The shown case study is the reflection of only a little share of the set of all possible recommendations. Therefore, taking up research on a wider spectrum of recommendations established by the described algorithms seems justified. It should be an empirical research leading to establishing the heuristic investment strategy.
In financial practice, we can meet with the situation when part of PV securities is imprecisely evaluated without a subjective forecast of future quotation changes. Such PV should be evaluated by unoriented FNs. Against imprecisely evaluated PV, other securities may be equipped with a subjective forecast of rise in quotation. Such PV should be evaluated by positively oriented OFNs. In both of these cases the membership functions are identical. This results in the impossibility of a simultaneous comparison of oriented PV and unoriented PV. This is a significant disadvantage of the proposed algorithmic system supporting invest-making. The intention to deal with this inconvenience points to another direction of research into the OFNs theory.
The obtained results may as well be a starting point for future research on the impact of the PV imprecision and orientation on the investment recommendation determined with the use of algorithms presented in this paper. The implementation of intuitionistic fuzzy sets should be preceded by a theoretical and empirical research of the expediency of such approach for a representation of the justified recommendations mentioned in Section 7. the subset of the most acceptable recommendation systems is distinguished as the Pareto's optimum, determined as a two-criteria comparison of minimisation recommendation ambiguity and minimisation recommendation indistinctness.
The ambiguity of recommendation Λ S,γ is valued by energy measure d Λ S,γ calculated with the use of (5). The indistinctness of recommendation Λ S,γ is valued by energy measure d Λ S,γ determined by (6). Therefore, we represent each recommendation by the pair d Λ S,γ , e Λ S,γ = d S,γ , e S,γ On the recommendation set we define two preorders " Λ S,i is more than Λ S, j " : The set of all acceptable recommendations we appoint as Pareto's optimum O S determined by multi-criterial comparison Q 1 ∩ Q 2 . To solve this optimisation task, we adapt an analogous algorithm presented in [91]. In order to determine the Pareto optimum O S , we execute the following algorithm: In this way, we obtain the sequence O S of partial optima of Pareto.