Vectors of indicators and pointer function in the Multistage Bipolar Method

Abstract

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.

1 Introduction

Multistage Bipolar Method, proposed in (Trzaskalik 1987, 2021, 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, 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.

2 Overview of recent bipolar approaches in decision making

The bipolar approach to decision making can be found in various contemporary papers in the field of operational research and applied mathematics. Many of them were cited and discussed in Trzaskalik (2021, 2022). They are papers written by Michałowski and Szapiro (1992), Wojewnik and Szapiro (2010), Skulimowski 1996), Bisdorff et al. (2008), Franco et al. 2013), Aslam et al. 2014), Ezhilmaran and Sankar (2015), Bouzarour-Amokrane et al.(2015), Al-Qudah and Hassan (2017), Liu et al. 2018), Akram et al. 2018), Jana and Pal 2018), Jana et al. 2019).

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 (Akram et al. 2020).

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, 2022). The procedure will be considered in detail in the remaining part of the paper.

3 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 = (a₁,…,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 $d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right)$ and $d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right)$, as well as the values of the degree of stage failure avoidance $d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right)$ and $d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right)$ (see Trzaskalik 2021, formulas (26), (28), (33) and (35)). The following properties hold:

$$ 0 \le d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) \le 1 $$

(1)

$$ 0 \le d_{{\mathbf{G}}}{}^{ - } ({\mathbf{a}}_{t} ) \le 1 $$

(2)

$$ 0 \le d_{{\mathbf{B}}}{}^{ + } ({\mathbf{a}}_{t} ) \le 1 $$

(3)

$$ 0 \le d_{{\mathbf{B}}}{}^{ - } ({\mathbf{a}}_{t} ) \le 1 $$

(4)

Furthermore, from the assumptions and the above properties, we conclude, that:

$$ d_{{\mathbf{G}}}{}^{+} \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(5)

$$ d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(6)

$$ d_{{\mathbf{B}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(7)

$$ d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{B}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(8)

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:

$$ d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(9)

$$ d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) > 0\quad \Rightarrow \quad d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(10)

In the third phase relationships in the set of multistage alternatives are considered. For each multistage alternative values $d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}} \right)$ and $d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}} \right)$ of the multistage success achievement degree as well as the values $d_{{\mathbf{B}}}{}^{ + } \left( {\mathbf{a}} \right)$ and $d_{{\mathbf{B}}}{}^{ - } \left( {\mathbf{a}} \right)$ of the multistage failure avoidance degree are defined (see Trzaskalik 2021, formulas (37)–(40)).

Using the values $d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}} \right)$, $d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}} \right)$, $d_{{\mathbf{B}}}{}^{ + } \left( {\mathbf{a}} \right)$ and $d_{{\mathbf{B}}}{}^{ - } \left( {\mathbf{a}} \right)$, the multistage alternatives can be sorted into six classes A¹–A⁶ (see Trzaskalik 2021, formulas (42)–(47)). 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

$$ d({\mathbf{a}}{}^{(i)} ) = d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}^{(i)} } \right) - d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}^{(i)} } \right) + d_{{\mathbf{B}}}{}^{ + } \left( {{\mathbf{a}}_{}^{(i)} } \right) - d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}^{(i)} } \right) $$

(11)

$$ d({\mathbf{a}}{}^{(j)} ) = d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}^{(j)} } \right) - d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}^{(j)} } \right) + d_{{\mathbf{B}}}{}^{ + } \left( {{\mathbf{a}}^{(j)} } \right) - d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}^{(j)} } \right) $$

(12)

Within the classes the ordering of the alternatives is defined as follows:

$$ {\mathbf{a}}{}^{(i)} \;{\text{is}}\;{\text{preferred}}\;{\text{to}}\;{\mathbf{a}}{}^{{({\text{j}})}} \Leftrightarrow {d}({\mathbf{a}}{}^{{({\text{i}})}} ) > {d}({\mathbf{a}}{}^{{({\text{j}})}} ) $$

(13)

$$ {\mathbf{a}}{}^{(i)} \;{\text{is}}\;{\text{equivalent}}\;{\text{to}}\;{\mathbf{a}}{}^{{({\text{j}})}} \Leftrightarrow {d}({\mathbf{a}}{}^{{({\text{i}})}} ) = {d}({\mathbf{a}}{}^{{({\text{j}})}} ) $$

(14)

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

$$ \forall {\mathbf{a}}{}^{'} \in {\mathbf{A}}{}^{m} \quad d({\mathbf{a}}{}^{ * * } ) \ge d({\mathbf{a}}{}^{'} ) $$

(15)

4 One-stage vectors of indicators

Let x be a real number. We will apply signum function

$$ {\text{sgn}} \;(x) = \left\{ {\begin{array}{*{20}l} { - 1} \hfill & {{\text{for}}\,x < 0} \hfill \\ 0 \hfill & {{\text{for}}\,x = 0} \hfill \\ 1 \hfill & {{\text{for}}\,x > 0} \hfill \\ \end{array} } \right. $$

(16)

4.1 One-stage actual vectors of indicators

For any stage alternative a_t we consider the vector

$$ {\mathbf{d}}({\mathbf{a}}_{t} ) = [d_{{\mathbf{G}}}{}^{ + } ({\mathbf{a}}_{t} ),d_{{\mathbf{G}}}{}^{ - } ({\mathbf{a}}_{t} ),d_{{\mathbf{B}}}{}^{ + } ({\mathbf{a}}_{t} ),d_{{\mathbf{B}}}{}^{ - } ({\mathbf{a}}_{t} )] $$

(17)

and apply the signum function to its components. We obtain:

$$ \sigma_{{1t}} \left( {{\mathbf{a}}_{t} } \right) = {\text{sgn}}\;d_{{\mathbf{G}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) $$

(18)

$$ \sigma_{2t} \left( {{\mathbf{a}}_{t} } \right) = sgn{\kern 1pt} \,d_{{\mathbf{G}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) $$

(19)

$$ \sigma_{3t} \left( {{\mathbf{a}}_{t} } \right) = sgn{\kern 1pt} \,d_{{\mathbf{B}}}{}^{ + } \left( {{\mathbf{a}}_{t} } \right) $$

(20)

$$ \sigma_{4t} \left( {{\mathbf{a}}_{t} } \right) = sgn{\kern 1pt} \,d_{{\mathbf{B}}}{}^{ - } \left( {{\mathbf{a}}_{t} } \right) $$

(21)

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

$$ \sigma_{t} \left( {{\mathbf{a}}_{t} } \right) = \left[ {\sigma_{{1t}} \left( {{\mathbf{a}}_{t} } \right),{\mkern 1mu} \sigma_{2t} \left( {{\mathbf{a}}_{t} } \right),{\mkern 1mu} \sigma_{3t} \left( {{\mathbf{a}}_{t} } \right),\sigma_{4t} {\mkern 1mu} \left( {{\mathbf{a}}_{t} } \right)} \right] $$

(22)

Vector σ_t(a_t) is called actual vector of stage indicators for stage realization a_t.

We have:

$$ \sigma_{{1t}} \left( {{\mathbf{a}}_{t} } \right) + \sigma_{2t} \left( {{\mathbf{a}}_{t} } \right) = 1 $$

(23)

$$ \sigma_{3t} \left( {{\mathbf{a}}_{t} } \right) + \sigma_{4t} \left( {{\mathbf{a}}_{t} } \right) = 1 $$

(24)

As stage alternatives which are simultaneously overgood and underbad are excluded from our consideration, the additional condition is fulfilled:

$$ \sigma_{{1t}} \left( {{\mathbf{a}}_{t} } \right) + \sigma_{4t} \left( {{\mathbf{a}}_{t} } \right) \le 1 $$

(25)

It results from (25), that:

$$ \sigma_{{1t}} \left( {{\mathbf{a}}_{t} } \right) = 1\quad \Rightarrow \quad \sigma_{4t} \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(26)

$$ \sigma_{4t} \left( {{\mathbf{a}}_{t} } \right) = 1\quad \Rightarrow \quad \sigma_{1t} \left( {{\mathbf{a}}_{t} } \right) = 0 $$

(27)

4.2 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.

$$ {\mathbf{p}}_{t} = [p_{1t} ,p_{2t} ,p_{3t} ,p_{4t} ] $$

(28)

$$ p_{1t} ,p_{2t} ,p_{3t} ,p_{4t} \in \{ 0,\;1\} $$

(29)

These vectors should fulfill conditions analogous to (23)–(27), i.e.:

$$ p_{1t} + p_{2t} = 1 $$

(30)

$$ p_{3t} + p_{4t} = 1 $$

(31)

$$ p_{1t} + p_{4t} \le \;1 $$

(32)

$$ p_{1t} = 1\quad \Rightarrow \quad p_{4t} = \;0 $$

(33)

$$ p_{4t} = 1\quad \Rightarrow \quad p_{1t} = \;0 $$

(34)

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).

Table 1 All the four element variations with repetitions of the two element set {0, 1}

Most of them, however, should be excluded. Vectors p_t⁽⁰⁾, p_t⁽¹⁾, p_t⁽²⁾, p_t⁽¹²⁾, p_t⁽¹³⁾, p_t⁽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:

$$ {\mathbf{P}}_{t} = \{ {\mathbf{p}}_{t}{}^{(5)} ,\;{\mathbf{p}}_{t}{}^{(6)} ,\;{\mathbf{p}}_{t}{}^{(10)} \} $$

(35)

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:

$$ \sigma ({\mathbf{A}}_{t} ) = \{ {\mathbf{p}}_{t} \in {\mathbf{P}}_{t} :\sigma ({\mathbf{a}}_{t} ) = {\mathbf{p}}_{t} \wedge {\mathbf{a}}_{t} \in {\mathbf{A}}_{t} \} $$

(36)

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.

$$ \sigma ({\mathbf{A}}_{t} ) \subset {\mathbf{P}}_{t} $$

(37)

5 Vectors of multistage indicators

5.1 Actual vectors of multistage indicators

Let

$$ {\mathbf{a}} = [{\mathbf{a}}_{1} ,...,{\mathbf{a}}_{T} ] $$

(38)

be a multistage realization of the process considered. We define a four dimensional vector

$$ \sigma ({\mathbf{a}}) = \sum\limits_{{t = 1}}^{T} {\sigma _{t} ({\mathbf{a}}_{t} )} $$

(39)

Vector σ(a) is called actual vector of multistage indicators for multistage alternative a. Its components are:

$$ \sigma _{i} ({\mathbf{a}}) = \sum\limits_{{t = 1}}^{T} {\sigma _{{it}} ({\mathbf{a}}_{t} )} $$

(40)

for i = 1, 2, 3, 4. We have:

$$ \sigma ({\mathbf{a}}) = [\sigma_{1} ({\mathbf{a}}),\sigma_{2} ({\mathbf{a}}),\sigma_{3} ({\mathbf{a}}),\sigma_{4} ({\mathbf{a}})] $$

(41)

From the definition (39), for i = 1,…,4 it holds:

$$ \sigma_{i} ({\mathbf{a}}) \in \{ 0,1,...,T\} $$

(42)

The value σ₁(a) informs by how many stages the considered multistage alternative a outranks sets G_t. Value σ₂(a) informs by how many stages sets G_t outrank the considered multistage alternative a. Value σ₃(a) informs by how many stages the considered multistage alternative a outranks sets B_t. Value σ₄(a) informs by how many stages sets B_t outrank the considered multistage alternative a.

We have:

$$ \sigma_{1} ({\mathbf{a}}) + \sigma_{2} ({\mathbf{a}}) = T $$

(43)

$$ \sigma_{3} ({\mathbf{a}}) + \sigma_{4} ({\mathbf{a}}) = T $$

(44)

$$ \sigma_{1} ({\mathbf{a}}) + \sigma_{4} ({\mathbf{a}}) \le T $$

(45)

and for k = 0,…,T:

$$ \sigma_{1} ({\mathbf{a}}) = k\quad \Rightarrow \quad \sigma_{4} ({\mathbf{a}}) \le T - k $$

(46)

$$ \sigma_{4} ({\mathbf{a}}) = k\quad \Rightarrow \quad \sigma_{1} ({\mathbf{a}}) \le T - k $$

(47)

Moreover, at each stage the conditions (26) and (27) are to be fulfilled..

5.2 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).

Let p ∈ P(T)

$$ {\mathbf{p}} = [p_{1} ,p_{2} ,p_{3} ,p_{4} ] $$

(48)

$$ p_{1} ,p_{2} ,p_{3} ,p_{4} \in \{ 0,...,T\} $$

(49)

These vectors should fulfill conditions analogous to (43)–(47), i.e.:

$$ p_{1} + p_{2} = T $$

(50)

$$ p_{3} + p_{4} = T $$

(51)

$$ p_{1} + p_{4} \le T $$

(52)

and for k = 0,…,T:

$$ p_{1} = k\quad \Rightarrow \quad p_{4} \le T - k $$

(53)

$$ p_{4} = k\quad \Rightarrow \quad p_{1} \le T - k $$

(54)

The set σ(A) is defined as follows:

$$ \sigma ({\mathbf{A}}) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma ({\mathbf{a}}) = {\mathbf{p}} \wedge {\mathbf{a}} \in {\mathbf{A}}\} $$

(55)

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.

$$ \sigma ({\mathbf{A}}) \subset {\mathbf{P}}(T) $$

(56)

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.

Table 2 All the 4-elements variations with repetitions of three element set {0, 1,2}

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:

$$ {\mathbf{P}}(2) = \{ {\mathbf{p}}{}^{(21)} ,{\mathbf{p}}{}^{(23)} ,{\mathbf{p}}{}^{(25)} ,{\mathbf{p}}{}^{(41)} ,{\mathbf{p}}{}^{(43)} ,{\mathbf{p}}{}^{(61)} \} $$

(57)

Vector p⁽⁶¹⁾ = [2, 0, 2, 0] characterizes all the multistage alternatives whose stage alternatives outrank both sets G₁ and G₂ and at the same time outrank both sets B₁ and B₂. 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 G₁ or G₂ and at the same time outrank both sets B₁ and B₂. 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 G₁ or G₂ and at the same time outrank one of sets B₁ or B₂. 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 G₁ or G₂ and at the same time outrank both sets B₁ and B₂. 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 G₁ or G₂ and at the same time outrank one of sets B₁ or B₂. 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 G₁ or G₂ and at the same time they are outranked by sets B₁ or B₂. It is the worse situation.

Table 3 Components of multistage, potential vectors of indicators and values of pointer functions

For T ≥ 2 the potential vectors of multistage indicators can be sorted into classes P¹(T) – P⁶(T) defined as follows:

$${\mathbf{P}}^{1} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma _{1} ({\mathbf{p}}) = T , \quad \sigma _{2} ({\mathbf{p}}) = 0 , \quad \sigma _{3} ({\mathbf{p}}) = T , \quad \sigma _{4} ({\mathbf{p}}) = 0\} $$

(58)

$$ \begin{aligned} &{\mathbf{P}}^{2} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma _{1} ({\mathbf{p}}) \in \{ 1,...,T - 1\} , \; \sigma _{2} ({\mathbf{p}}) \in \{ 1,...,T - 1\},\\ &\quad \sigma _{3} ({\mathbf{p}}) = T , \; \sigma _{4} ({\mathbf{p}}) = 0\;\text{and condition}\;(50)\;\text{is fulfilled}\} \end{aligned} $$

(59)

$$ \begin{aligned} &{\mathbf{P}}^{3} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma _{1} ({\mathbf{p}}) \in \{ 1,...,T - 1\} , \; \sigma _{2} ({\mathbf{p}}) \in \{ 1,...,T - 1\},\\ &\quad \sigma _{3} ({\mathbf{p}}) \in \{ 1,...,T - 1\} , \; \sigma _{4} ({\mathbf{p}}) \in \{ 1,...,T - 1\}\\ &\quad \text{and conditions}\;(50) - (54)\;\text{are fulfilled}\} \end{aligned} $$

(60)

$$ {\mathbf{P}}^{4} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma_{1} ({\mathbf{p}}) = 0 , \; \sigma_{2} ({\mathbf{p}}) = T , \; \sigma_{3} ({\mathbf{p}}) = T , \; \sigma_{4} ({\mathbf{p}}) = 0\} $$

(61)

$$ \begin{aligned} &{\mathbf{P}}^{5} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma_{1} ({\mathbf{p}}) = 0 , \; \sigma_{2} ({\mathbf{p}}) = T , \; \sigma_{3} ({\mathbf{p}}) \in \{ 1,...,T - 1\},\\ &\quad \sigma_{4} ({\mathbf{p}}) \in \{ 1,...,T - 1\}\;\text{and condition}\;(51)\;\text{is fulfilled}\} \end{aligned} $$

(62)

$$ {\mathbf{P}}^{6} (T) = \{ {\mathbf{p}} \in {\mathbf{P}}(T):\sigma_{1} ({\mathbf{p}}) = 0 , \; \sigma_{2} ({\mathbf{p}}) = T , \; \sigma_{3} ({\mathbf{p}}) = 0 , \; \sigma_{4} ({\mathbf{p}}) = T\} $$

(63)

Classes P¹(T)—P⁶(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).

For the two stage process considered we have:

$$ {\mathbf{p}}{}^{{\left( {{61}} \right)}} = \, \left[ {{2}, \, 0,{ 2}, \, 0} \right] \in {\mathbf{P}}{}^{{1}} \left( {2} \right) \,\,\,\,\,\,\,\,{\mathbf{p}}{}^{{\left( {{43}} \right)}} = \left[ {{1},{ 1},{ 2}, \, 0} \right] \in {\mathbf{P}}{}^{{2}} \left( {2} \right)$$

$$ {\mathbf{p}}{}^{{\left( {{41}} \right)}} = \left[ {{1},{ 1},{ 1},{ 1}} \right] \in {\mathbf{P}}{}^{{3}} \left( {2} \right) \,\,\,\,\,\,\,\,{\mathbf{p}}{}^{{\left( {{25}} \right)}} = \left[ {0,{ 2},{ 2}, \, 0} \right] \in {\mathbf{P}}{}^{{4}} \left( {2} \right)$$

$$ {\mathbf{p}}{}^{{\left( {{23}} \right)}} = \left[ {0,{ 2},{ 1},{ 1}} \right] \in {\mathbf{P}}{}^{{5}} \left( {2} \right) \,\,\,\,\,\,\,\,{\mathbf{p}}{}^{{\left( {{21}} \right)}} = \left[ {0,{ 2}, 0,{ 2}} \right] \in {\mathbf{P}}{}^{{6}} \left( {2} \right)$$

6 Pointer functions

For the given potential multistage alternative p a pointer function is defined in the following way:

$$ \delta ({\mathbf{p}}) = \sigma_{1} ({\mathbf{p}}) - \sigma_{2} ({\mathbf{p}}) + \sigma_{3} ({\mathbf{p}}) - \sigma_{4} ({\mathbf{p}}) $$

(64)

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 P¹(T)–P⁶(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).

Table 4 A possible ordering of subclasses within classes of potential vectors of multistage indicators

Let a⁽ⁱ⁾, a^(j) be two multistage alternatives and σ(a⁽ⁱ⁾) ∈ P^k,m, σ(a^(j)) ∈ P^l,n. We propose a modified ordering of alternatives in the Multistage Bipolar Method:

$$ k < l\quad \Rightarrow \quad {\mathbf{a}}{}^{(i)} \text{ is\;preferred\;to}\;{\mathbf{a}}{}^{(j)} $$

(65)

$$ k < l \wedge m < n\quad \Rightarrow \quad {\mathbf{a}}{}^{(i)} \text{ is\;preferred\;to}\;{\mathbf{a}}{}^{(j)} $$

(66)

$$ k = l ,\; m = n ,\; d({\mathbf{a}}{}^{(i)} ) > d({\mathbf{a}}{}^{(j)} ) \Rightarrow {\mathbf{a}}{}^{(i)} \;\text{ is\;preferred\;to}\;{\mathbf{a}}{}^{(j)} \} $$

(67)

$$ k = l ,\; m = n ,\; d({\mathbf{a}}{}^{(i)} ) = d({\mathbf{a}}{}^{(j)} ) \Rightarrow {\mathbf{a}}{}^{(i)} \;\text{ is\;equivalent\;to}\;{\mathbf{a}}{}^{(j)} \} $$

(68)

7 Application of potential vectors of multistage indicators and pointer function in solution procedures

7.1 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 2.

For each t = 1,…,T we find sets σ(A_t).

Step 3.

We find the value

$$ \delta{}^{0} = \max \delta (\sigma ({\mathbf{a}})):{\mathbf{a}} \in {\mathbf{A}}\} $$

(69)

The calculation can be performed using single criterion dynamic programming.

Step 4.

We identify the potential vector of indicators p⁰ ∈ P(T) such that δ(p⁰) = δ^o, which belongs to the class with the lowest number k and the number of subclass P^k,m(T) such that p⁰ ∈ P^k,m(T).

Step 5.

We find the set of multistage realizations A^k,m such that

$$ {\mathbf{A}}{}^{k,m} = \{ {\mathbf{a}}{}^{0} \in {\mathbf{A}}:\delta ({\mathbf{a}}{}^{0} ) = \delta{}^{0} \wedge \sigma ({\mathbf{a}}{}^{0} ) \in {\mathbf{P}}^{0} \} $$

(70)

Step 6.

From A^k,m we choose a multistage realization a^** such that

$$ d({\mathbf{a}}{}^{ * * } ) = \max\{d({\mathbf{a}}{}^{0} ):{\mathbf{a}}{}^{0} \in {\mathbf{A}}{}^{k,m} \} $$

(71)

7.2 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).

8 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).

Algorithm 1

Step 1.

The set of potential vectors of multistage indicators P(3) is generated and divided into classes P¹(3)—P⁶(3). It is shown in Table 6 together with values of pointer function.

Fig. 1

The graph of the exemplary process

Table 5 Numerical values for Example 1

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 3.

Applying single criterion dynamic programming algorithm we find the value.

δ^o = max {δ(σ(a)): a ∈ A)} = 6.

Step 4.

We identify the only one existing potential vector of indicators p⁰ ∈ P(3) such, that δ(p⁰) = 6. It belongs to the class P¹(3) (subclass P^1,1(3). Its components are as follows: p⁰ = [3, 0, 3, 0].

Step 5.

The set A^1,1 consists of the single element: a^** = [(0, 1), (1, 0), (0,0)].

Step 6.

Realization a^** is the final solution.

Table 6 Set of potential vector indicators and values of pointer functions for T = 3

Table 7 Sets σ(A_t) for t = 1,2,3

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

Table 9 Numerical values for stages 4 and 5 in the Example 2

Fig. 2

The graph of the process considered in Example 2 together with values δ_t(σ(a_t))

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.

Step 2.

For each t = 1,…,T we find sets σ(A_t) shown in Table 11.

Table 10 Set of potential vector indicators and values of pointer function for T = 8

Table 11 Sets σ(A_t) for T = 8

Step 3.

Applying the single criterion dynamic programming algorithm we find the value.

δ^o = max {δ(σ(a)): a ∈ A)} = 14.

Step 4.

We identify potential vector of indicators p⁰ ∈ P(3) such, that δ(p⁰) = 14. It belongs to the class P^2,1. Its components are as follow: p⁰ = [7, 1, 8, 0].

Step 5.

We find the set of multistage realizations A^2,1 such that

$$ {\mathbf{A}}{}^{{{2},{1}}} = \, \{ {\mathbf{a}}{}^{0} \in {\mathbf{A}}{}^{{2}} :\delta \left( {{\mathbf{a}}{}^{0} } \right) \, = {14}\} ,\; \sigma \left( {{\mathbf{a}}{}^{0} } \right) = \, \left[ {{7},{ 1},{ 8}, \, 0} \right]\} $$

The set A^2,1 consists of two elements:

$$ \begin{gathered} {\mathbf{a}}{}^{{(}{{69})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} , {\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{{({1})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{13}{-}{14} - {16}} \right) \hfill \\ {\mathbf{a}}{}^{{(}{{77})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} , {\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({3})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{11}{-}{13}{-}{14} - {16}} \right) \hfill \\ \end{gathered} $$

Step 6.

We have d(a ⁽⁶⁹⁾) = 1.19 and d(a⁽⁷⁷⁾) = 1.15. Since

argmax {d(a⁽⁶⁹⁾), d(a ⁽⁷⁷⁾)} = a ⁽⁶⁹⁾ and multistage realization a⁽⁶⁹⁾ is pointed as a final solution.

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:

$$ \begin{aligned}{\mathbf{a}}{}^{{(}{{69})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{{({1})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{13}{-}{14}{-}{16}} \right)\end{aligned} $$

and

$$\begin{aligned} {\mathbf{a}}{}^{{(}{{77})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({3})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{11}{-}{13}{-}{14}{-}{16}} \right) \end{aligned}$$

such that δ(a⁶⁹) = δ(a⁷⁷) = 14. As δ(σ(a⁶⁹)) = δ(σ(a⁷⁷)) = [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:

$$\begin{aligned} {\mathbf{a}}{}^{{(}{{197})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({3})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{{({1})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{2}{-}{4}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{13} - {14}{-}{16}} \right)\end{aligned} $$

and

$$\begin{aligned} {\mathbf{a}}{}^{{(}{{2}0{5})}} = \left( {{\mathbf{a}}_{{1}}{}^{{({3})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({3})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = ({2}{-}{4} - {5}{-}{7}{-}{9}{-}{11}{-}{13}{-}{14}{-}{16})\end{aligned} $$

As δ(σ(a¹⁹⁷)) = δ(σ(a²⁰⁵)) = [6,2,8,0], they belong to the subclass P^2,2. Calculating values d(a¹⁹⁷)) = 1,039 and d(a²⁰⁵)) = 1,008 we see, that a⁽¹⁹⁷⁾ is a better multistage realization than a^(205).

The third highest value of the function δ is 10. We have 6 multistage alternatives for which function δ takes this value:

$$ \begin{gathered} {\mathbf{a}}{}^{{\left( {5} \right)}} = \, \left( {{\mathbf{a}}_{{1}}{}^{(0)} ,{\mathbf{a}}_{{2}}{}^{(0)} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{{({1})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{3}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{13}{-}{14}{-}{16}} \right) \hfill \\ {\mathbf{a}}{}^{{\left( {{13}} \right)}} = \, \left( {{\mathbf{a}}_{{1}}{}^{(0)} ,{\mathbf{a}}_{{2}}{}^{(0)} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({3})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{3}{-}{5}{-}{7}{-}{9}{-}{11}{-}{13}{-}{14}{-}{16}} \right) \hfill \\ {\mathbf{a}}{}^{{\left( {{65}} \right)}} = \, \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{(0)} ,{\mathbf{a}}_{{7}}{}^{(0)} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{12}{-}{14}{-}{16}} \right) \hfill \\ {\mathbf{a}}{}^{{\left( {{7}0} \right)}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{(0)} ,{\mathbf{a}}_{{6}}{}^{{({1})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{{({1})}} } \right) \, = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{1}0{-}{13}{-}{14}{-}{17}} \right) \hfill \\ {\mathbf{a}}{}^{{\left( {{73}} \right)}} = \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({2})}} ,{\mathbf{a}}_{{7}}{}^{(0)} ,{\mathbf{a}}_{{8}}{}^{(0)} } \right) = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{11}{-}{12}{-}{14}{-}{16}} \right) \hfill \\ {\mathbf{a}}{}^{{\left( {{78}} \right)}} = \, \left( {{\mathbf{a}}_{{1}}{}^{{({1})}} ,{\mathbf{a}}_{{2}}{}^{{({2})}} ,{\mathbf{a}}_{{3}}{}^{(0)} ,{\mathbf{a}}_{{4}}{}^{(0)} ,{\mathbf{a}}_{{5}}{}^{{({1})}} ,{\mathbf{a}}_{{6}}{}^{{({3})}} ,{\mathbf{a}}_{{7}}{}^{{({2})}} ,{\mathbf{a}}_{{8}}{}^{{({1})}} } \right) \, = \left( {{1}{-}{4}{-}{5}{-}{7}{-}{9}{-}{11}{-}{13}{-}{14}{-}{17}} \right) \hfill \\ \end{gathered} $$

Based on δ(σ(a)) and d(a), the multistage alternatives are assigned to the appropriate subclasses (Table 12).

Table 12 Sorting and ranking alternatives in Example 2

9 Discussion and conclusions

In the paper actual and potential stage and multistage vectors of indicators and pointer function have been introduced and used for searching for the final multistage realization, as well as sorting and ranking of the multistage alternatives. The solution of the problem of searching for the final multistage realization proposed in the paper and based on single criterion dynamic programming seems much simpler than the solution, proposed in (Trzaskalik 2022), based on multiobjective dynamic programming. Moreover, applying single criterion dynamic programming algorithm of finding near-optimal solutions described in (Trzaskalik 2015) gives the possibility to find subsequent values of pointer function and to sort multistage alternatives to subsequent classes.

Two numerical examples presented in the paper show properties of the proposed approach. In Example 1 a multistage alternative belonging to the “best” class of alternatives A¹ was found and proposed as a final solution. There was only one such a multistage alternative. In Example 2 it such an alternative does not exist in class A¹. As the final solution two multistage alternatives were proposed, belonging to the less desirable class A². 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 A¹, A⁴ and A⁶ 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.