Vectors of indicators and pointer function in the Multistage Bipolar Method

The Multistage Bipolar Method considered in the paper deals with multistage decision processes. Multistage alternatives are not compared directly to each other, but they are confronted with the stage sets of reference objects—desirable and non-acceptable. In the paper vectors and pointer functions are defined. The aim of the paper is to apply them to classify and rank multistage alternatives and search for the final solution. Our method simplifies the procedure of finding the final solution and allows to use single criterion dynamic programming to solve the problem.


Introduction
Multistage Bipolar Method, proposed in (Trzaskalik 1987(Trzaskalik , 2021(Trzaskalik , 2022, is a modified extension of a multicriteria decision aid method named Bipolar, introduced by Konarzewska Gubała (1987,1989). The essence of Bipolar approach is that alternatives are not compared directly to each other, but reference objects from sets of good objects and bad objects are used for these comparisons. The novelty of the approach introduced in (Trzaskalik 1987(Trzaskalik , 2021 consists in the extension of analysis to multicriteria, multistage decision making processes. The Multistage Bipolar Method allows for classification of multistage alternatives, their ranking and choosing the final solution. The aim of the present paper is to introduce potential stage and multistage vector indicators and a pointer function and to show how they can be used in the Multistage Bipolar Method. These elements modify to some extend results obtained using the source description of the method presented in (Trzaskalik 2021).
The paper consists of eight sections. After a short introduction in Sect. 1, in the following Sect. 2 various bipolar approaches to decision making are presented. Section 3 provides a brief introduction to the Multistage Bipolar Method. As the full description of the method is available in the open access paper (Trzaskalik 2021), in this paper we will pay special attention to these elements of the method that are used directly in further considerations. The notation used in the present paper is consistent with that used in (Trzaskalik 2021). In Sect. 4 vectors of stage indicators are defined, while in Sect. 5 vectors of multistage indicators are proposed. In both cases actual and potential vectors of indicators are distinguished. In Sect. 6 a pointer function is introduced. Section 7 describes algorithms, using vectors of indicators and the pointer function for classification of multistage alternatives, their ranking and searching for the final solution. Numerical illustrations of the proposed procedures are given in Sect. 7. Two multistage processes are considered: the first is a three-stage process and the second is an extension of the previous one for eight stages. Discussion and conclusions can be found in Sect. 8.
The most recent papers in the field (since 2020) are recalled below. Dombi t-norm and t-conorm in the bipolar fuzzy environment to aggregate various preferences of the decision makers are used in (Jana et al. 2020).
The notions of two novel hybrid multi-attribute decision-making models: m-polar fuzzy bipolar soft set model and rough m-polar fuzzy bipolar soft set model as well as the development of efficient algorithms to solve multi-attribute decision-making problems applying these notions are presented in .
A hybrid MCDM framework applied to the ANP and TOPSIS methods under bipolar neutrosophic numbers is presented in (Abdel-Basset et al. 2020). This framework is applied for the problem of selecting the chief executive officer at the Elsewedy Electric Group in Egypt.
Bipolar fuzzy MCDM methods based on bipolar fuzzy set theory are presented in (Akram and Arshad 2020). The methods considered are TOPSIS and ELECTRE. These two powerful MCDM techniques are applied for the diagnosis of a disease in which the pairwise comparison of diseases and symptoms is considered in bipolar behavior.
A new version of the PROMETHEE method using bipolar fuzzy information: bipolar fuzzy sets or numbers, named the bipolar fuzzy PROMETHEE method is described in (Akram and Al-Kenani 2020). Imprecise information is represented in the form of trapezoidal bipolar fuzzy numbers to assign the preferences of alternatives. The entropy weighting information is employed to calculate the weights of attributes. The method is applied to solve the problem of the selection of green suppliers.
The theory of soft sets deals with uncertain, fuzzy and not clearly defined objects. As in many real-life applications the soft set needs to be specified from a parameterization viewpoint. In (Deli and Karaaslan 2020) the concept of the bipolar fuzzy parameterized soft sets and their set-theoretical operations is defined and applied to decision making.
In (Riaz and Tehrim 2020a) cubic bipolar fuzzy sets as a generalization of bipolar fuzzy sets are presented. The authors establish an innovative multi-criteria group decision making method based on cubic bipolar fuzzy set by unifying aggregation operators under geometric mean operations. A score and accuracy function to compare the cubic bipolar elements in ranking of alternatives is presented.
In (Riaz and Tehrim 2020b) the idea of bipolar fuzzy topology based on bipolar fuzzy soft set is brought out. This topology is used in decision-making by applying an algorithm to deal with unpredictability.
Bipolar fuzzy interactive multi-criteria decision-making based on the cumulative prospect theory model for dealing with the problem of multi-attribute group decision making is established in (Zhao et al. 2021). This model is applied to the selection of network security service provider selection.
The notions of two hybrid multi-attribute decision-making models: m-polar fuzzy bipolar soft set model and rough model are presented in (Akram et al. 2021). They allow to handle multipolar information under fuzziness with bipolar soft sets. Two practical application of these models: selecting a suitable employee for promotion and place for house construction, as well as developing efficient algorithms to solve multi-attribute decision-making problems are presented.
The concept of the bipolar picture fuzzy set as a hybrid structure of bipolar fuzzy set and picture fuzzy set is shown in (Riaz et al. 2021). A new MCDM approach is proposed to address uncertain real-life situations and illustrated.
In (Jana and Pal 2021) an evaluation based on distance from the average solution approach to multiple criteria group decision-making with bipolar fuzzy numbers is introduced. The proposed technique is valid and accurate for considering conflicting attributes. The proposed method is analyzed The selection of a road construction company project is analyzed by applying the proposed method.
Finally, the Multistage Bipolar Method should be mentioned. It was proposed in (Trzaskalik 1987) and developed in (Trzaskalik 2021(Trzaskalik , 2022. The procedure will be considered in detail in the remaining part of the paper.

Multistage Bipolar Procedure
We consider T -stage decision process, described in detail in (Trzaskalik 2021). Two kinds of alternatives: stage alternatives a t for t = 1,…,T and multistage alternatives a = (a 1 ,…,a T ) are distinguished. Sets of stage and multistage alternatives are denoted as A t and A, respectively. At each stage of the process, two reference sets are defined: the first, containing "good", desirable reference objects, named G t , and the second one, containing "bad", non-acceptable reference objects, named B t . They make up a reference system. At each stage, the reference set of good objects and the reference set of bad objects are disjoint.
The numerical procedure in the Bipolar Multistage Method consists of three phases. In the first phase a comparison of stage alternatives with stage reference objects is performed. (see (Trzaskalik 2021, formulas (4)-(22)). In the second phase the position of stage alternatives with respect to the bipolar stage reference system is defined. We calculate the values the degree of stage success achievement d G + (a t ) and d G − (a t ), as well as the values of the degree of stage failure avoidance d G + (a t ) and d G − (a t ) (see Trzaskalik 2021, formulas (26), (28), (33) and (35)). The following properties hold: Furthermore, from the assumptions and the above properties, we conclude, that: The assumptions made in (Trzaskalik 2021) exclude the possibility of the emergence of stage alternatives, which are both overgood and underbad, i.e. such which are evaluated as better than the stage reference set of good objects G t and worse than the stage reference set of bad objects B t . Therefore: In the third phase relationships in the set of multistage alternatives are considered. If k < l, then each multistage alternative from class A k is preferred over any multistage alternative from class A l .
Let a (i) and a (j) be two multistage alternatives and Within the classes the ordering of the alternatives is defined as follows: The final solution a ** is defined as the multistage alternative which belongs to the non-empty class with the lowest index m and which satisfies the relationship

One-stage vectors of indicators
Let x be a real number. We will apply signum function

One-stage actual vectors of indicators
For any stage alternative a t we consider the vector and apply the signum function to its components. We obtain: The values σ 1t (a t ), σ 2t (a t ), σ 3t (a t ), σ 4t (a t ) are called actual stage indicators for the stage realization a t . According to (1)-(4) the values of stage indicators are 0 or 1.
Consider the vector Vector σ t (a t ) is called actual vector of stage indicators for stage realization a t . We have: As stage alternatives which are simultaneously overgood and underbad are excluded from our consideration, the additional condition is fulfilled: It results from (25), that:

One-stage potential vectors of stage indicators
We will answer the question which zero-one vectors from the four-dimensional space can be vectors of stage indicators. These vectors will be called potential vectors of stage indicators. Let P t .be the set of all potential vectors of stage indicators and p t ∈ P t .
These vectors should fulfill conditions analogous to (23)-(27), i.e.: Vector To create this set, let us consider all four element variations with repetitions of the two element set {0, 1}. We will consider all the possibilities (see Table 1).
Most of them, however, should be excluded. 14) and p t (15) are excluded by condition (30). Vectors p t (3) , p t (4) , p t (7) , p t (8) and p t (11) are excluded by condition (31). Vector p t (9) is excluded (conditions (33) and (34) are not fulfilled) because it describes a stage alternative which is simultaneously overgood and underbad. Thus for t = 1,…,T the set P t contains three elements: Vector p t (10) = [1, 0, 1, 0] characterizes all the stage alternatives that outrank the set G t and at the same time outrank the set B t . This is the most favorable situation.
Vector p t (6) = [0, 1, 1, 0] characterizes all the stage alternatives that are outranked by the set G t and at the same time outrank the set B t . This situation is less favorable than the previous one, but better than the situation described below.
Vector p t (5) = [0, 1, 0, 1]] characterizes all the stage alternatives that are outranked by the set G t and at the same time are outranked by the set B t . This is the least favorable situation.
The set σ(A t ) is defined as follows: The set σ(A t ) contains all the potential stage vector indicators, which can occur for the considered set of the stage alternatives A t . The inclusion holds.

Actual vectors of multistage indicators
be a multistage realization of the process considered. We define a four dimensional vector Vector σ(a) is called actual vector of multistage indicators for multistage alternative a. Its components are: for i = 1, 2, 3, 4. We have: From the definition (39), for i = 1,…,4 it holds: The value σ 1 (a) informs by how many stages the considered multistage alternative a outranks sets G t . Value σ 2 (a) informs by how many stages sets G t outrank the considered multistage alternative a. Value σ 3 (a) informs by how many stages the considered multistage alternative a outranks sets B t . Value σ 4 (a) informs by how many stages sets B t outrank the considered multistage alternative a.
We have: and for k = 0,…,T : Moreover, at each stage the conditions (26) and (27) are to be fulfilled..

Potential vectors of multistage indicators
We will answer the question which four-dimensional vectors from the four dimensional space can be vectors of multistage indicators. These vectors will be called potential vectors of multistage indicators. The set of all potential vectors of multistage indicators will be denoted as P(T ).
These vectors should fulfill conditions analogous to (43)-(47), i.e.: and for k = 0,…,T : The set σ(A) is defined as follows: The set σ(A t ) contains all the potential multistage vector indicators, which can occur for the considered set A t of the multistage alternatives. The inclusion holds.
To illustrate how to create sets P(T ) we will determine the set P(2). Consider all four-element variations with repetitions of the three-element set {0, 1, 2}. All possibilities are listed in Table 2.
Most of them, however, should be excluded, applying formulas (50)-(54). Vectors p (39) , p (57) and p (59) are excluded as multistage realizations which are simultaneously overgood and underbad. Thus, set P(2) contains six elements:  Vector p (61) = [2, 0, 2, 0] characterizes all the multistage alternatives whose stage alternatives outrank both sets G 1 and G 2 and at the same time outrank both sets B 1 and B 2 . This is the most favorable situation.
Vector p (43) = [1, 1, 2, 0] characterizes all the multistage alternatives whose stage alternatives outrank one of the sets G 1 or G 2 and at the same time outrank both sets B 1 and B 2 . This situation is less favorable than the previous one, but better than the situations described below.
Vector p (41) = [1, 1, 1, 1] characterizes all the multistage alternatives whose stage alternatives outrank one of the sets G 1 or G 2 and at the same time outrank one of sets B 1 or B 2 . At the same time conditions (37) and (38) have to be fulfilled. This situation is less favorable than the previous ones.
Vector p (25) = [0, 2, 2, 0] characterizes all the multistage alternatives whose stage alternatives are outranked by sets G 1 or G 2 and at the same time outrank both sets B 1 and B 2 . This situation is less favorable than the previous ones.
Vector p (23) = [0, 2, 1, 1] characterizes all the multistage alternatives whose stage alternatives are outranked by sets G 1 or G 2 and at the same time outrank one of sets B 1 or B 2 . This situation is less favorable than the previous ones.
Vector p (21) = [0, 2, 0, 2] characterizes all the multistage alternatives whose stage alternatives are outranked by sets G 1 or G 2 and at the same time they are outranked by sets B 1 or B 2 . It is the worse situation.
For T ≥ 2 the potential vectors of multistage indicators can be sorted into classes P 1 (T ) -P 6 (T ) defined as follows: Classes P 1 (T )-P 6 (T ) are defined so that if p (k) ∈ P (k) , p (l) ∈ P (l) , and k < l then p (k) is preferred over p (l) .

Pointer functions
For the given potential multistage alternative p a pointer function is defined in the following way: A pointer function is a synthetic measure that indicates for the entire process: • in how many stages the components of a potential multistage alternative outrank or are outranked by the sets G t • in how many stages the components of a potential multistage alternative outrank or are outranked by the sets B t .
It results from the definition of the function δ that a pointer function can take the largest value equal to 2 T and the smallest value equal to − 2 T (Table 3).
We want to associate the potential vectors of indicators P(T ) and the values of pointer functions with classes P 1 (T )-P 6 (T ). Let i, j = 1,…, T -1.
Within classes we create subclasses that we order according to the decreasing values of the pointer function (see Table 4).

Searching for the final solution
Applying properties of vectors of indicators and pointer function, we propose an algorithm for searching for the final solution in the Multistage Bipolar Method. Algorithm 1 Step 1. We consider a T -stage process. The set of potential vectors of multistage indicators P(T ) is generated.
Step 3. We find the value The calculation can be performed using single criterion dynamic programming.
Step 4. We identify the potential vector of indicators p 0 ∈ P(T ) such that δ(p 0 ) = δ o , which belongs to the class with the lowest number k and the number of subclass P k,m (T ) such that p 0 ∈ P k,m (T ).
Step 5. We find the set of multistage realizations A k,m such that Step 6. From A k,m we choose a multistage realization a ** such that

Classification and ranking
The values of the pointer function allow to split the set P(T ) into subclasses and to sort the actal multistage alternatives into these subclasses. We sort the multistage alternatives according to the sequence shown in Table 4 and formulas (65)-(68). The procedure is described in Algorithm 2. Algorithm 2 Step 1. We consider T -stage process. Set P(T ) is found and divided into subclasses P k,m (T ).
Step 2. The set of actual multistage alternatives A and the set σ(A) are found.
Step 3. Multistage alternatives are sorted into classes according to vectors σ(a) ∈ P k (T ). Within the classes the multistage alternatives are sorted into subclasses according to vectors σ(a) ∈ P k,m (T ).

Numerical illustrations
Example 1 We consider a three-stage decision process. There are two states at each stage of the process and two decisions in each state. At each stage we have four stage alternatives and we consider two reference sets at each stage: G t and B t . Numerical values for Example 1 are given in Table 5. Sets of potential vectors of multistage indicators and values of pointer functions for T = 3 are given in Table 6. The structure of the process together with values δ(a t ) is shown in Fig. 1. The full description of the process in question can be found in (Trzaskalik 2021).    Fig. 1 The graph of the exemplary process Algorithm 1 Step 1. The set of potential vectors of multistage indicators P(3) is generated and divided into classes P 1 (3)-P 6 (3). It is shown in Table 6 together with values of pointer function.
Step 2. For each t = 1,…,T we find sets σ(A t ) for t = 1,…,T (formula 36). They are shown in Table 7 (together with the values of the pointer function).
Step 6. Realization a ** is the final solution.

Algorithm 2
Step 1. The set P(T ) divided into classes P m (T ) for m = 1,…,6 and subclasses had been previously found and is shown in Table 6.
Step 2. Sets A and σ(A) are found.
Step 3. Multistage alternatives are sorted into classes P m according to vectors σ(a) ∈ P m (T ). Within the classes the multistage alternatives are sorted into subclasses according to vectors σ(a) ∈ P m,k (T ). The results are shown in Table 8.

Example 2
We consider an eight stage decision process consisting of three subprocesses. The first subprocess includes stages 1 -3 and is a replication of the process from Example 1. The second subprocess includes stages 4 and 5. The sets of good reference objects and the sets of bad reference objects in both stages consists of two objects each. Numerical data is given in Table 9. The third subprocess includes stages 6 -8 and is a replication of the process from Example 1.
The graph of the considered process together with the values of δ t (a t ) is given in Fig. 2. Let us notice that we use the continuous numbering of stage in consecutive stages that starts at 1 in the stage 1 and ends at 17 in the stage 8.

Table 8
Sorting and ranking alternatives in Example 1 (3)

Table 9
Numerical values for stages 4 and 5 in the Example 2  Algorithm 1 Step 1. The set of potential vector indicators P(6) is generated. It is shown in Table 10 together with values of the pointer function.
For each t = 1,…,T we find sets σ(A t ) shown in Table 11.
Algorithm 2 In our eight stage process the set A consists of 256 actual multistage alternatives. We want to classify and sort them into classes and subclasses. We will present several best multistage alternatives, using values of the function δ and multistage potential vectors of indicators.
The maximum value of δ is14. Applying algorithm 1 we found that there exist two multistage alternatives:  such that δ(a 69 ) = δ(a 77 ) = 14. As δ(σ(a 69 )) = δ(σ(a 77 )) = [7,1,8,0], both of them belong to the subclass P 2,1 . As d(a (69) ) = 1,216 and d(a (77 ) = 1,185, the better of them is a (69) . The second highest value of the function δ is 12. Two multistage alternatives correspond to this value: Table 11 Sets As the final solution two multistage alternatives were proposed, belonging to the less desirable class A 2 . Their order is determined by the values of the function d(a).
Comparing the ordering proposed in the present paper with the ordering shown in (Trzaskalik 2021) it is worth noting that the order of multistage alternatives in classes A 1 , A 4 and A 6 does not change. This is because the potential sets of vectors of multistage indicators in these classes are not divided into subclasses. A possible change of order results from the fact that within the classes the multi-stage alternatives are ordered on the basis of the values of function d(a).
It may happen that some multistage alternatives for which the values of pointer functions are the same will be assigned to different classes. Such a situation was illustrated in the Example 2. There are multistage alternatives for which the pointer function has a value of 10. Depending on the corresponding potential vectors of multistage indicators there were assigned to classes P 2,3 or P 3,2 , respectively.
The Multistage Bipolar Method can be applied to solve the sustainable regional development problem, as described in (Trzaskalik 2022). The method of sorting multistage alternatives proposed in the current paper may be useful there.

Conflict of interest
The author has no financial or proprietary interests in any material discussed in this article.
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