AHP based on scenarios and the optimism coefficient for new and risky projects: case of independent criteria

Abstract

AHP is a well-known multi-criteria procedure which has been investigated and developed by many researchers and practitioners. Some AHP modifications are designed for decision making under uncertainty. The goal of this paper is to present a new AHP approach which can be useful in the case of uncertain one-shot decisions and independent criteria. The method proposed in the article is based on scenario planning, features characteristic for the Hurwicz rule (i.e. the use of the optimism coefficient) and on a scenario set reduction. The novel procedure gives the possibility to generate a relatively small number of pairwise comparison matrices thanks to the reduction of the initial sets of scenarios. The modified version of AHP may be helpful when the decision maker’s knowledge about probabilities of the occurrence of particular scenarios is partial. Such a situation occurs in the case of innovative, innovation and risky projects for which historical data are not known. The idea of the suggested scenario-based AHP is to adjust the final choice not only to the decision makers’ preferences (concerning criteria for example), but also to their nature, attitude towards risk, predictions, expectations and fears.

1 Introduction

The Analytic Hierarchy Process (AHP) is one of the most popular multi-criteria approaches applied to diverse fields in decision analysis (e.g. business, government, education, operations research, management science, industry, healthcare, forecasting, planning and ranking) (Biloslavo et al. 2010; Longaray et al., 2015; Wu et al., 2018), and one that is arguably more accessible for new users (Durbach, 2019). The original version was designed for deterministic problems, however the necessity to operate in an uncertain environment has forced researchers and practitioners to create numerous AHP modifications enabling to take uncertainty factors into account.

This paper is also related to the Analytic Hierarchy Process and multiple criteria decision making (MCDM) under uncertainty, but our motivation to deal with that topic comes from the willingness to:

• Explore these AHP issues which are not thoroughly analysed in the literature,

• Discuss AHP problems in a different way than it is usually done.

First, we would like to investigate AHP under uncertainty in the context of independent objectives. In this case for each criterion a distinct set of scenarios may be defined. Problems with independent criteria occur quite often in real situations (see e.g. the choice of investment strategies on the basis of such features as: annual profit, bureaucracy and legal restrictions, workers satisfaction, customer loyalty, project completion time, etc.), but they have not been sufficiently analysed in the literature. That is why, a deeper examination of that aspect seems to be necessary. For clarity, let us add that the difference between dependent and independent criteria is explained for instance in (Gaspars-Wieloch, 2017b) and it will be described in the next section.

Second, we intend to concentrate on the use of AHP both for innovative and innovation projects, as well as for projects performed in very stormy periods (e.g. economic crisis, pandemic, unstable law etc.). These types of projects are very frequent nowadays. Therefore, the need to focus on that area is justified. It is worth explaning that we assume that innovation projects bring new products and new services, while innovative projects are projects managed on the basis of new methods (Spalek 2016).

Third, in connection with the fact that the paper is related to new and risky projects, we put emphasis on one-shot decisions, i.e. decisions concerning options selected and performed only once. Contrary to one-shot decisions, multi-shot decisions are connected with variants executed many times. Nevertheless, the second category of decisions does not correspond to the aforementioned projects since in this case the surrounding is too uncertain and dynamic. Thus, after the first realization of the chosen alternative (project), the decision maker may want to update some decision parameters and start the decision making process from the beginning instead of executing the selected option (project) for many years without any modifications. Note that in the case of one-shot decisions the use of objective probabilities as known primary data seems to be inappropriate because such probabilities are mathematically correct rather for repetitive events only (Mises, 1949, 1962; Gaspars-Wieloch, 2019b). Additionally, it is worth stressing that innovative and innovation projects are characterised by a very high novelty degree and usually they lack of historical data on the basis of which the likelihood could be estimated. Moreover, probabilities computed on the basis of possible historical data should be used extremely cautiously if some new future factors, being able to radically change the trend up to now, are anticipated. The investigation of AHP under uncertainty without known objective probabilities is another novelty in the article since existing papers dealing with uncertain AHP often refer to the probability calculus (Banuelas & Antony, 2006; Beynon 2002; Hauser & Tadikamalla, 1996; Levary & Wan, 1998; Paulson & Zahir, 1995; Rosenbloom, 1997).

Fourth, the subsequent reason for which it is worth analysing in detail the problem mentioned above, is the intention to find tools to reduce the AHP complexity and time-consuming nature as much as possible. Note that it is not so trivial as when handling uncertainty in the Analytic Hierarchy Process pairwise comparison matrices are required not only for criteria and decisions, but also for scenarios. Our suggestion consists in reducing the original sets of scenarios on the basis of the information on the decision maker’s attitude towards risk. This idea is not totally new since such decision rules (applied to one-criterion uncertain decision making) as the Wald’s rule (Wald, 1950a, 1950b), Hurwicz’s rule (Hurwicz, 1952), max-max rule also allow us to find the proper option by taking into consideration at most two scenarios. However, we will see that the scenario set reduction proposed here runs in a completely different manner than in the case of the aforementioned methods. Let’s recall the essence of these approaches: a) within the Wald’s rule the recommended alternative has the maximal minimum outcome; b) when using the Hurwicz’s rule the best option has the maximal value of the weighted average of the best and worst outcome; c) the max-max rule leads the decision-maker to choose the decision variant with the highest maximal outcome. Classical procedures have been often criticized by virtue of their irrational assumptions (Officer & Anderson, 1968). Researchers underline that within games against nature the last one plays a passive role and is a neutral opponent. Hence, it should not “change its mind” after the decision maker’s declaration. Meanwhile, when applying the Wald’s, Hurwicz’s or max-max rule, the index, on the basis of which final recommendations are formulated, is often computed given data coming from different scenarios, depending on the considered alternative. In order to explain this observation, let us examine data presented in Table 1. In the case of the Wald’s rule, the measure for alternative A₁ will take into account the outcome $-1.50$ (scenario S₂), but for the remaining courses of action A₂, A₃ and A₄ the index will depend on the payoff $-2.75$ (scenario S₃), 0$.50$ (scenario S₃) and $-2.00$ (scenario S₁), respectively. Within the Hurwicz’s procedure the measure for alternative A₁ will take into account the outcome $-1.50$ and 4$.50$ (scenarios S₂ and S₁), but for the remaining options A₂, A₃ and A₄ the index will depend on the payoffs $-2.75$ and $2.00$ (scenarios S₃ and S₁), 0$.50$ and $5.00$ (scenarios S₃ and S₂) and $-2.00$ and $7.00$ (scenarios S₁ and S₃), respectively. Finally, when applying the max-max approach the measure for decision variant A₁ will take into consideration the value $4.50$ (scenario S₁), but for the remaining alternatives A₂, A₃ and A₄ the index will depend on the result $2.00$ (scenario S₁),$5.00$ (scenario S₂) and $7.00$ (scenario S₃), respectively. This simulation shows that in the three classical procedures the nature behaves as a human being, i.e. it adjusts its choice to the decision maker’s (real player’s) strategy. This phenomenon is negatively perceived for example by Officer and Anderson (1968). It is regarded sometimes as the most serious drawback.

Table 1 Fictitious data

In the paper the aforementioned scenario set reduction is performed by means of the optimism coefficient (β) which can be treated as a partially subjective probability parameter. The coefficient β expresses the decision makers’ nature: their state of mind and soul as well as their predictions concerning the future. The optimism coefficient can be estimated separately for each problem or even for each alternative. It may be evaluated intuitively or by means of psychological tests (Gaspars-Wieloch, 2018). It is worth mentioning that the coefficient of optimism is applied by many researchers formulating rules supporting the decision making process (Ellsberg, 1961, 2001; Hernandez et al., 2018; Jagodziński, 2014; Perez et al., 2015). This parameter is also used in the well-known Hurwicz rule (Hurwicz, 1952). The coefficient of optimism β belongs to the interval $[\text{0,1}]$. It is equal to 0 for extreme pessimists (expecting the occurrence of scenarios with the worst outcomes) and 1 for extreme optimists (expecting the occurrence of scenarios with the best outcomes).

Note that, similarly to the Hurwicz’s rule, the approach presented in the article refers to the optimism coefficient in order to reduce the set of scenarios before making the final decision. However, in the paper we intend to treat the nature as a passive and neutral opponent. That is why, the suggested method does not allow to take into account different reduced sets of scenarios for particular options. The idea is to define a common reduced set of scenarios for each alternative.

Before the whole new methodology is described, we should start with a short explanation of the term MCDM and diverse synonyms used in the paper. A brief presentation of AHP and existing uncertain AHP modifications is also desirable.

We apply notions “alternatives”, “options”, “courses of action” and “decision variants” as equivalent words (Ravindran et al., 1987). The term “criteria” is sometimes replaced in the article by “objectives”. Both words are related to the intention to maximize or minimize a measure representing an attribute (feature) which is important for the decision maker.

Multiple Criteria Decision Making involves two groups of issues: Multiple Attribute Decision Problems (MADP) and Multiple Objective Decision Problems (MODP). In MADP the number of possible options is precisely defined at the beginning of the decision making process and the levels of considered attributes are assigned to each alternative (Singh & Gupta, 2020). Within MODP the cardinality of the set of potential decision variants is not exactly known. The decision maker (DM) only knows the mathematical optimization model, i.e. the set of objective functions and constraints that create the set of possible solutions (Ding et al., 2016; Hwang et al., 1981). The Analytical Hierarchy Process is designed for Multiple Attribute Decision Problems.

AHP consists in decomposing the problem into significant objectives and alternatives, i.e. in defining the problem hierarchy. The DM is supposed to evaluate priorities (relative importance) within appropriate criteria and alternatives pairwise comparison matrices. These steps require transforming linguistic expressions into concrete values (usually from $1/9$ to $9$ where $1/9$ signifies absolute inferiority, $1$ denotes indifference and $9$ is understood as absolute preference). After the normalization of these evaluations and the computation of their averages (being overall priorities) for each criterion and option, a weighted value can be calculated for each decision variant. Weights may be computed using various approaches. For example Blanquero et al. (2006) analyse the efficiency of diverse approaches. They notice that «the least-logarithmic-squares solution is always efficient, whereas the (widely used) eigenvector solution is not, in some cases, efficient, thus its use in practice may be questionable«. Another possible approach is the exact global optimization method suggested by Carrizosa and Messine (2006). This procedure is based on an interval analysis. Other interesting approaches can be found in Anholcer et al. (2011) or Fülöp (2008). The weighted value is used to indicate the best solution. It is worth underlining that it is usually suggested to verify the judgement consistency. If the matrices are not consistent enough (e.g. the consistency ratio exceeds $10\%$), the subjective judgements need to be revised and additional computations are necessary (Anholcer & Fülöp, 2019; Choo & Wedley, 2004; Jarek, 2016; Ozdemir, 2005; Saaty, 2000).

The main strengths and weaknesses of AHP are as follows. The first strength is the opportunity to generate a ranking for the whole set of options, which is not the case of such procedures as the interactive programming and the Pareto approach. Its second significant advantage results from the possibility to solve even problems in which criteria are difficult to quantify since in this procedure human judgements are sufficient. The third important advantage is that the criteria weights declaration is not required—they are calculated on the basis of a pairwise comparison performed by the DM. One of the most serious drawbacks connected with AHP concerns the rank reversal phenomenon (Brunelli, 2015)—it occurs when the rank-ordering of the preferability of possible decision variants (i.e. the rank order of alternatives) changes after a modification of the set of available alternatives. Another disadvantage of AHP is that it forces DMs to declare preferences for all the pairs of objectives and decision variants (Banuelas et al. 2007). As a matter of fact this weakness may be also treated as a strenght since the neccesity to define preferences so meticulously enables to check how respondents are confident of their answers.

AHP was originally developed by Saaty (1980) as a deterministic decision making tool, but it has been extensively studied and refined since then (Brunelli, 2015). For many years, researchers as well as practitioners have tried to take diverse types of uncertainty into account, which is completely justified because, usually, it is extremely difficult to precisely define ones preferences or predict future events. Here are some examples:

• 1. Beynon (2002) suggests using DS/AHP which encompasses the Dempster-Shafer theory of evidence and gives the ability to assign probability measures to groups of decisions.

• 2. Tacnet et al. (2011) combine AHP with fuzzy sets, possibility and belief functions theories in order to handle imprecise and uncertain evaluations of quantitative and qualitative criteria.

• 3. Uncertainty also can be taken into consideration on the basis of Monte Carlo AHP (Banuelas et al., 2007; Yaragi et al., 2015).

• 4. Mimović et al. (2015) use Bayesian analysis to improve the accuracy of input data for AHP.

• 5. Rezaeian et al. (2015) combine uncertain data with AHP and TOPSIS.

• 6. Lin et al. (2012) formulate an uncertain variable method and show how to check the consistency of uncertainty comparison matrices.

• 7. Ennaceur (2015) describes in his doctoral thesis new uncertain AHP methods based on the belief function theory.

• 8. Eskandari et al. (2007) suggest a stochastic approach to capture the uncertain behaviour of the global AHP weights.

• 9. Yang et al. (2013) adopt the normal Cloud model and the Delphi feedback method in order to handle the randomness and fuzziness of individual judgements.

• 10. Brozova (2004) investigates AHP under uncertainty and risk in the case of non-homogeneous problems: (1) when some alternatives are not influenced by all scenarios or (2) when some combinations decision/scenario are unreasonable.

Other interesting uncertainty issues are discussed for instance by Banuelas and Antony (2006), Brunelli (2015), Hauser and Tadikamalla (1996), Lafleur (2011), Levary and Wan (1998), Millet et al. (2002), Paulson and Zahir (1995), Rosenbloom (1997), Toth et al. (2018) and Wu et al. (2018).

As we can see, the number of papers devoted to AHP under uncertainty is impressive, but due to the assumptions adopted in this article (independent criteria, new and risky projects, one-shot decisions, objective probability avoidance, desire to reduce the original set of scenarios, willingness to take the decision makers’ nature into account on the basis of their optimism coefficient) a direct comparison of the suggested approach with procedures already created is impossible.

The paper is organized as follows. Section 2 (Methodology) develops the idea of combining AHP with scenario planning and presents a novel scenario-based AHP procedure designed for independent criteria and one-shot decisions. Section 3 (Results) contains an illustrative example and results. Discussion is led in Sect. 4 and Conclusions are gathered in the last section.

Note that the article is an extension of the investigation described by Gaspars-Wieloch (2019a). In the aforementioned paper the novel method has been already demonstrated, but due to some paper limitations the motivation and assumptions are presented there very briefly and the algorithm is outlined almost without formulas and explanations (only a shot descriptive presentation is given). The illustrative example is extremely simple and final conclusions are important, but certainly not sufficient and comprehensive. Therefore, we would like to extend and complete the first version in this article.

2 Methodology

Durbach and Stewart (2012) enumerate diverse techniques enabling handling uncertainty in MCDM such as fuzzy numbers, probabilities, probability-like quantities and explicit risk measures. However, in their opinion,»uncertainties become increasingly so complex that the elicitation of those measures becomes operationally difficult for DMs to comprehend and virtually impossible to validate«. Therefore, they encourage to construct scenarios describing possible ways in which the future might unfold and that is why, in this work we are mainly interested in methods integrating scenario planning (SP) into AHP.

2.1 Scenario planning

SP is a tool frequently used in the decision-making process (e.g. Albornoz et al., 2020). It is very helpful when the DM deals with issues under uncertainty, i.e. situations where at least one parameter of the decision problem is not deterministic (DMU—decision making under uncertainty) (Gaspars-Wieloch, 2021). SP supports the identification of risks understood as uncertain and uncontrolled factors influencing the consequences of chosen strategies. It is useful for government planners and military analysts, companies, scientific communities, futurists and educational institutions (Mietzner & Reger, 2005; Ringland, 2006; Silber, 2017). Scenario planning supports e.g. sales forecasting, projects selection or inventory management. The scenarios may represent the alternative methods to achieve an objective or an analysis of the interaction of forces. In this paper we treat scenarios according to the second approach. Events that trigger undesired scenarios are identified as risks. Guidelines concerning the construction of scenarios are presented for example by Chermack et al. (2001), Mandel and Wilson (1993), Michnik (2013), Pomerol (2001).

Project managers eagerly use SP since it allows one to analyse problems in a more deterministic way (Schoemaker, 1995) than for example continuous probability distributions or fuzzy numbers (Durbach, 2014; Maciel et al., 2018). The strength of the scenarios is that they do not consider just one future, but multiple potential futures (Hoffmann, 2017). Thanks to SP the organisations are better prepared to handle new situations and promote proactive leadership initiatives because it recognises technological discontinuities or disruptive events and includes them into long-range planning (Mietzner & Reger, 2005).

Hoffmann (2017) stresses that the future is invariably a combination of the known and the unknown or unknowable, but the proportion of the latter tends to rise as the time-scale extends. That is why, although SP is a more modest method than exact probability-based risk measurement methods, ‘identifying a range of versions (scenarios) of what might happen’ can be a tenable and supportable basis for risk and planning. analysis, especially given the fact that SP accepts uncertainty, unpredictability and vagueness, i.e. three main features of the future. According to Probst and Bassi (2014) scenarios nurture innovative thinking about Possible future behavioural paths of the systems.

Nevertheless, in order to be objective, we must also mention some drawbacks related to SP. Scenario planning has got opponents stressing that it requires a deep understanding and knowledge of the field under investigation, which entails a very time-consuming selection of suitable experts with sufficient skills to collect, interpret and monitor data from different sources (Mietzner & Reger, 2005). Hence, if people cannot develop and apply SP properly, this method becomes useless (Gaspars-Wieloch, 2021). Probst and Bassi (2014) add that the SP team is obliged to generate plausible scenarios and continuously revise them. They also reveal that SP is inadequate for decisions made in the short-term. Other serious disadvantages of SP are highlighted by Hoffmann (2017) who draws to a conclusion that it offers some orientation only due to the aforementioned vagueness, and by Roxburgh (2009) who notices that scenarios do not cover the full range of future possibilities (Gaspars-Wieloch, 2021).

It is worth underlining that SP has also a psychological and behavioural aspect since payoffs connected with particular scenarios may be estimated by:

• 1. Experts,

• 2. Decision makers,

• 3. People being both experts and decision makers.

In the first approach the outcomes are generated in the most objective way while the second approach may lead to the most subjective predictions (Gaspars-Wieloch, 2021). Regardless of the estimation method it is suggested to convert initial values into numbers reflecting the DM’s preferences (utilities) (Ravindran et al., 1987).

2.2 Scenario planning in AHP

The combination of SP with AHP entails the necessity to apply at least four levels when describing a given decision situation as a hierarchy problem (Brozova 2004):

• 1. Goal—best alternative selection,

• 2. List of scenarios,

• 3. Set of criteria,

• 4. Decision alternatives.

Durbach discusses these issues (SP and AHP) in detail in his recent paper (Durbach, 2019). He stresses that the aggregation of objectives and scenarios may be performed in two fundamental ways: in the»meta-alternative« approach scenarios are combined with courses of action and the joint»meta-alternatives« are evaluated over attributes. On the other hand, in the»meta-attribute« approach scenarios are combined with attributes and then the decision variants are evaluated in terms of these»meta-attributes«. Both approaches use a standard implementation of the AHP, and thus are subject to the same concerns, for example regarding rank reversal and interpretability of weights. Both techniques are valuable, but the aforementioned paper does not explain the crucial application difference between them.

As a matter of fact, it is worth underlining that the first approach is designed for independent criteria—the performance of particular objectives can be analyzed totally seperately since the number ($m$) and type of scenarios can be different for each criterion: ${m}_{1},{m}_{2},\dots ,{m}_{k},\dots , {m}_{p}$, where $p$ denotes the total number of criteria and ${m}_{k}$ is the total number of scenarios for criterion ${C}_{k}$, $k=1,\dots ,p$. The second approach allows the criteria to be dependent. This time there is a strong relationship between scenarios assigned to particular criteria—the number and type of scenarios ought to be the same for each criterion considered in the decision problem and evaluation ${a}_{ij}^{k}$ can only be connected with evaluations ${a}_{ij}^{1},\dots ,{a}_{ij}^{k-1},{a}_{ij}^{k},{a}_{ij}^{k+1}\dots ,{a}_{ij}^{p}$ where ${a}_{ij}^{k}$ describes the performance of criterion ${C}_{k}$ by alternative ${D}_{j}$ provided that scenario ${S}_{i}$ happens (Gaspars-Wieloch, 2017a).

Furthermore, we would like to emphasize that the scenario-based AHP method suggested by Durbach (2019) requires the evaluation of supplementary pairwise comparison matrices related to scenarios and representing for instance how likely a scenario is to occur. For $p$ independent criteria, $n$ decision variants and ${m}_{k}$ scenarios for criterion ${C}_{k}$, $k=1,\dots ,p$ at least $\left(\left(\begin{array}{c}p\\ 2\end{array}\right)+\sum_{k=1}^{p}\left(n\times \left(\begin{array}{c}{m}_{k}\\ 2\end{array}\right)\right)+{\sum }_{k=1}^{p}\left({m}_{k}\times \left(\begin{array}{c}n\\ 2\end{array}\right)\right)\right)$ comparisons are needed. This leads to an additional effort and makes the decision making process more complex and challenging. That is why, we state that it would be desirable to integrate SP in a less time-consuming way.

Note that some researchers recommend using probabilities in scenario-based AHP (Levary et al., 1999). However, Durbach (2019) stresses that scenarios should not be treated as states of nature since the set of scenarios does not constitute a complete probability space—in a statistical sense scenario»likelihoods« are not probabilities. Moreover, the states of nature are mutually exclusive and exhaustive, they are constructed from the same underlying dimensions, which is not the case of scenarios. Durbach (2019) does not use objective probabilitites in his scenario-based procedures. Instead of them, he applies relative importance of scenarios. In our work we also do not refer to such probabilities (one reason for this is the concentration on new and risky projects which are strictly connected with one-shot decisions).

2.3 Algorithm for AHP combined with SP

In this part of the paper we only explore the problem of independent criteria (IC). Such a situation occurs when particular objectives depend on totally different factors like weather, demography, diseases, fashion, prices, political decisions.

We focus on one-shot decisions—hence, the selected variant is supposed to be performed only once. This assumption signifies that for each criterion just one scenario has the chance to occur. In such circumstances the use of the probability calculus is unjustified (Mises, 1949, 1962).

Our main goal is to create a method that recommends choices adjusted to the DMs’ state of mind and soul (i.e. attitude towards risk, attitude towards a given problem). The information on their nature may be obtained by means of the optimism coefficient characterized in the previous section. This methodology has been already applied in the literature, but mainly for one-criterion decision problems.

Additionally, we do not intend to take all the proposed scenarios into consideration till the end of the decision making process since it would make the problem too complex. Besides, in other existing procedures considering the decision maker’s nature (i.e. the Wald’s, Hurwicz’s and max-max rules) there is a tendency do make the final decision on the basis of a reduced set of scenarios. Therefore, we prefer:

• 1. Reducing the initial sets of scenarios thanks to some subjective predictions (optimism coefficients) declared by the DM and then

• 2. Choosing the final course of action on the basis of selected data.

A similar approach, but in the context of one-criterion problems or mixed strategies, is applied by Gaspars-Wieloch (2015a, 2015b), whereas Grienitz et al. (2014) suggest focusing on the most important scenario. Note that the basic difference between the classical rules (i.e. Wald’s, Hurwicz’s and max-max approaches) and the novel procedure is the reduction of the original set of scenarios to a smaller set which is common for all the decision variants within a given objective. In the classical rules this set may be different for particular alternatives.

In the paper we assume that the uncertainty results from uncertain future events, not from uncertain DM’s preferences.

The proposed method—SP/AHP(IC)—consists of the following steps:

• 1. Define the set of alternatives (D), the set of criteria (C), and $S(k)$—the set of scenarios separately for each criterion ${C}_{k}$, $k=1,\dots ,p$.

• 2. Estimate pairwise comparison matrix for criteria: ${{\varvec{A}}}^{{\varvec{C}}}=\left[{a}_{e,d}\right]$ where $e=1,..,p$ and $d=1,..,p.$

• 3. Estimate pairwise comparison matrices for particular options in terms of scenarios, separately for each criterion: ${{\varvec{A}}}^{{\varvec{k}},{\varvec{j}}}=\left[{a}_{f,g}^{k,j}\right]$ where k is related to criterion ${C}_{k}$, symbol j is connected with option ${D}_{j}$, f and g denote scenarios from $S(k)$: $f=1,\dots ,{m}_{k}$. Matrices ${{\varvec{A}}}^{{\varvec{k}},{\varvec{j}}}$ may represent, for each pair of scenarios, how much does someone prefer alternative ${D}_{j}$ within a scenario belonging to $S(k)$ to alternative ${D}_{j}$ within another scenario belonging to this set (by virtue of criterion ${C}_{k}$, of course). In Durbach (2019) such an interpretation is called “conditional assessment”. Step 3 plays a vital role—it will allow us to compute scenarios weights (i.e. their relative attractiveness) and to choose these scenarios which correspond the most to the decision maker’s nature. This selection will enable to reduce the original set of scenarios and to concentrate on scenarios which are significantly connected with the decision maker’s expectations, predictions and presumptions.

• 4. Define for each criterion ${C}_{k}$ the DM’s coefficient of optimism ${\beta }_{k}$ which belongs to the interval [0,1]. In the first section it was mentioned that this parameter may be usually estimated separately for each problem and even for each option, but in this algorithm we assume that the DM is allowed to declare a distinct optimism coefficient for each problem and each considered criterion. The opportunity to estimate different values for objectives results from the fact that the analysed criteria are independent. There is no possibility to define different optimisms parameters for each alternative within a given criterion since in this algorithm the same reduced set of scenarios for all the options is going to be found.

• 5. Normalize each value in matrices for criteria (Eq. 1) and in matrices for particular options in terms of scenarios (Eq. 2) so that the sum of transformed comparisons in each column is equal to 1:

  $$ n_{e,d}^{{}} = \frac{{a_{e,d} }}{{\mathop \sum \nolimits_{e^{\prime} = 1}^{p} a_{e^{\prime},d} }}\;e = 1, \ldots ,p;d = 1, \ldots ,p $$

  (1)
  $$ n_{f,g}^{k,j} = \frac{{a_{f,g}^{k,j} }}{{\mathop \sum \nolimits_{f^{\prime} = 1}^{{m_{k} }} a_{f^{\prime},g}^{k,j} }}\;\;\;\;k = 1,...,p;j = 1, \ldots ,n;f = 1, \ldots ,m_{k} ;g = 1, \ldots ,m_{k} $$

  (2)

  where n denotes the total number of options.

• 6. Calculate the average of normalized values for each row of all the aforementioned matrices. These averages will constitute weights (i.e. overall priorities): ${P}_{k}$ (for criteria) and ${M}_{i}^{k,j}$(for scenarios within a given criterion and option).

  $$ P_{k}^{{}} = \frac{{\mathop \sum \nolimits_{d = 1}^{p} n_{e,d} }}{p}\;k = 1,...,p_{{}} $$

  (3)
  $$ M_{i}^{k,j} = \frac{{\sum\nolimits_{g = 1}^{{m_{k} }} {n_{i,g}^{k,j} } }}{{m_{k} }}\;k = 1, \ldots ,p;\;j = 1, \ldots ,n;\;i = 1, \ldots ,m_{k} $$

  (4)

• 7. Normalize scenario weights $M_{i}^{k,j}$ (0 for the lowest weight, 1 for the highest weight), separately for each alternative and criterion, and denote them by $M\left( n \right)_{i}^{k,j}$ which represents the subjective relative attractiveness of a given scenario (0 means that the scenario, for the analysed pair criterion-alternative, is the least attractive, 1—the most attractive).

  $$ M\left( n \right)_{i}^{k,j} = \frac{{M_{i}^{k,j} - \mathop {\min }\limits_{{h = 1, \ldots ,m_{k} }} \left\{ {M_{h}^{k,j} } \right\}}}{{\mathop {\max }\limits_{{h = 1, \ldots ,m_{k} }} \left\{ {M_{h}^{k,j} } \right\} - \mathop {\min }\limits_{{h = 1, \ldots ,m_{k} }} \left\{ {M_{h}^{k,j} } \right\}}}\;k = 1,...,p;j = 1, \ldots ,n;i = 1, \ldots ,m_{k} $$

  (5)

• 8. Reduce the initial sets $S(k)$ to sets ${S(k)}^{r}$ following the rules enumerated below:

  • 1. The set ${S(k)}^{r}$ includes scenario(s) with the normalized weight ${M(n)}_{i}^{k,j}$ equal to ${\beta }_{k}$ or very close to this parameter. Let us recall that ${\beta }_{k}$ represents the DM’s attitude towards a given problem within criterion ${C}_{k}$. If its value is high, it means that the decision maker expects favourable results, i.e. favourable scenarios. On the other hand, the relative attractiveness of particular scenarios can be represented by means of ${M(n)}_{i}^{k,j}$ since this measure shows the normalized overall priority of each scenario. Furthermore, thanks to the normalization of ${M}_{i}^{k,j}$, both indices (i.e. ${M(n)}_{i}^{k,j}$ and ${\beta }_{k}$) belong to interval $\left[\text{0,1}\right]$ and may be comparable. These observations lead us to the conclusion that there is a relationship between the optimism coefficient (showing the DM’s nature, predictions and expectations) and the normalized scenario overall priority. It is up to the DM to choose the acceptable deviation between the normalized weight and parameter ${\beta }_{k}$. Note that two types of deviations ought to be considered: the upper one (i.e. the difference between ${M(n)}_{i}^{k,j}$ and ${\beta }_{k}$) and the lower one (i.e. the difference between ${\beta }_{k}$ and ${M(n)}_{i}^{k,j}$). These deviations may have different values. For moderate pessimists it is recommended to use a greater lower deviation than in the case of moderate optimists. Such suggestions result from the necessity to assure the pessimists a kind of warranty in the form of a control of less or the least attractive scenarios. The DM may also diversify the deviation levels for particular criteria. Acceptable upper deviations for extreme optimists (${\beta }_{k}$=1) should be equal to 0, since 1 is the maximal possible value both for the optimism coefficient and the scenario weight. Similarly, acceptable lower deviations for radical pessimists (${\beta }_{k}$=0) should be also equal to 0, since 0 is the minimal possible value for both parameters. Rule 8.1 covers the case of equality, but situations where there exists a scenario for which ${M(n)}_{i}^{k,j}$ is exactly equal to ${\beta }_{k}$ occur quite seldom. As we see, this step of the algorithm provides significant tools to personalize the decision making process, which is extremely important considering the goal of the paper which consists in adjusting the final choice to the decision maker’s nature.

  • 2. Scenario weights are calculated separately for each option within a given criterion. That means that a given scenario may be attractive from the point of view of one option, but it can be unattractive from the point of view of another option at the same time. Therefore, the above rule 8.1 ought to be applied separately to each alternative and then the final reduced scenario set must include scenarios indicated during the analysis of particular courses of action, because it constitutes the disjunction of all the partial sets of selected scenarios. In some cases these scenarios may be the same for each considered option, but in other cases partial checks can lead to different scenarios.

  • 3. The reduced scenario set should contain at least one scenario by virtue of particular options. If the current deviation levels are too small to catch at least one scenario for each alternative, the DM is supposed to slightly increase them, just to generate nonempty sets ${S(k)}^{r}$ for each criterion.

• 9. Estimate pairwise comparison matrices in terms of alternatives only for scenarios from sets ${S(k)}^{r}$, i.e. ${{\varvec{A}}}^{{\varvec{k}},{\varvec{i}}({\varvec{r}})}=\left[{a}_{h,l}^{k,i(r)}\right]$ where $h=1,..,n$ and $l=1,..,n$. Symbol $i(r)$ is related to scenarios belonging to ${S(k)}^{r}$. This step is only connected with selected scenarios, which is a relevant facilitation since the options are not supposed to be compared by virtue of all the scenarios originally defined. Such an improvement enables the DM to save a lot of time.

• 10. Normalize values ${a}_{h,l}^{k,i(r)}$ applying the same method as in step 5.

  $$ n_{h,l}^{k,i\left( r \right)} = \frac{{a_{h,l}^{k,i\left( r \right)} }}{{\mathop \sum \nolimits_{h = 1}^{n} a_{h,l}^{k,i\left( r \right)} }}\;k = 1,...,p;S_{i\left( r \right)}^{k} \in S\left( k \right)^{r} ; h = 1, \ldots ,n;l = 1, \ldots ,n;_{{}} $$

  (6)

• 11. Compute alternative weights, i.e. the averages of the normalized values for each row ($N_{j}^{k,i\left( r \right)}$).

  $$ N_{j}^{k,i\left( r \right)} = \frac{{\mathop \sum \nolimits_{l = 1}^{n} n_{h,l}^{k,i\left( r \right)} }}{n}\;k = 1,...,p;S_{i\left( r \right)}^{k} \in S\left( k \right)^{r} ;j = 1, \ldots ,n_{{}} $$

  (7)

  • 1. If set $S\left( k \right)^{r}$ is a singleton, these averages do not need to be modified: $N_{j}^{k,i\left( r \right)}$ = $N_{j}^{k*}$ where $N_{j}^{k*}$ denotes final alternative weights within a given criterion.

  • 2. If a given reduced set includes more than one scenario, compute the mean alternative weights on the basis of the selected scenarios in the following way:

    $$ N_{j}^{k*} = \frac{1}{{\left| {S\left( k \right)_{{}}^{r} } \right|}}\mathop \sum \limits_{{i\left( r \right)^{k} \in S\left( k \right)_{{}}^{r} }} N_{j}^{k,i\left( r \right)} \;k = 1,...,p;j = 1, \ldots ,n_{{}} $$

    (8)

    $\left| {S\left( k \right)^{r} } \right|$ is the cardinality of $S\left( k \right)^{r}$.

• 12. Choose the decision variant for which the following measure has the highest value.

  $$ N_{j}^{*} = \mathop \sum \limits_{k = 1}^{p} \left( {N_{j}^{k*} \times P_{k} } \right) $$

  (9)

Note that we omit the consistency analysis in the procedure description since we assume that all the matrices are consistent enough. Otherwise, an appropriate matrices transformation may be required in steps 2, 3 and 9 of the aforementioned algorithm (Alonso et al., 2006; Anholcer et al., 2011, Blanquero et al., 2006; Carrizosa & Messine, 2006 Deturck, 1987; Franek et al., 2014; Fülöp, 2008, Ishizaka et al., 2004; Shyam Prasad et al., 2017).

When analysing carefully the subsequent steps of the procedure, we can observe that steps 1 (partially), 2 (entirely), 3 (partially), 5 (partially), 6 (partially), 9 (partially), 10 (partially), 11 (partially), 12 (entirely) come from the original AHP for deterministic problems. The rest, i.e. stages connected with sets of scenarios, the declaration of the optimism coefficient, the scenario set reduction and the scenario and alternative weights computations are novel.

Step 1 may be done by experts, other can be implemented and run automatically (steps 5, 6, 7, 10, 11, 12), but the evident involvement of the decision maker is characteristic of steps 2, 3, 4, 8, 9.

3 Example and results

Let us assume that the decision problem includes 2 alternatives (D₁, D₂), for instance 2 innovation projects, and 2 independent criteria (C₁, C₂)—a similar example is presented in (Gaspars-Wieloch, 2019a), but the initial data and final results are different. The DM considers 3 scenarios for the first criterion (S¹₁, S¹₂, S¹₃) and 2 scenarios for the second one (S²₁, S²₂)—see step 1. Pairwise comparison matrices for (1) criteria and (2) particular options in terms of scenarios are presented in Tables 2, 3 and 4 (steps 2–3). Let us analyse the case of a moderate pessimist who declares the following coefficient values: ${\beta }_{1}=0.45$ and ${\beta }_{2}=0.25$ (step 4). Steps 5–7 have been already done (see the second part of Table 2 and values introduced in brackets next to the original preferences in Tables 3, 4). Averages ${M}_{i}^{k,j}$ and normalized averages ${M(n)}_{i}^{k,j}$ are gathered in additional columns. Let us recall that they show the subjective attractiveness of particular scenarios.

Table 2 Criteria comparison matrices

Table 3 Scenario comparison matrices for options D₁ and D₂ within criterion C₁

Table 4 Scenario comparison matrices for options D₁ and D₂ within criterion C₂

Now (step 8) initial sets $S\left(1\right)=\left\{{S}_{1}^{1},{S}_{2}^{1},{S}_{3}^{1}\right\}$ and $S\left(2\right)=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$ are reduced to ${S(1)}^{r}=\left\{{S}_{2}^{1},{S}_{3}^{1}\right\}$ and ${S(2)}^{r}=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$. Let us explain in detail how this reduction has been performed. First, we see that there is no scenario with a normalized weight equal to 0.45 (in the case of criterion C₁) or 0.25 (in the case of criterion C₂). Thus, the case of equality is not applicable here. Due to the pessimistic nature of the decision maker we assume that they have chosen the following levels of upper and lower deviations (the same for both criteria): 0.15 and 0.25, respectively (rule 8.1).

According to rule 8.2 we must select scenarios within a given criterion, but separately for each alternative. We start with the analysis of objective C₁.

For option D₁ scenario ${S}_{2}^{1}$ belongs to ${S(1)}^{r}$ since its normalized weight 0.308 is the closest to 0.45 and fulfils the DM’s conditions (lower deviation = $0.45-0.308\le 0.25$). The remaining scenarios ${S}_{1}^{1}$ and ${S}_{3}^{1}$ are too “far” (lower deviation = $0.45-0.00>0.25$; upper deviation = $1.00-0.45>0.15$), thus they do not reflect the DM’s predictions and presumptions.

For option D₂ scenario ${S}_{3}^{1}$ belongs to ${S(1)}^{r}$ since its normalized weight 0.209 is the closest to 0.45 (lower deviation = $0.45-0.209\le 0.25$). The remaining scenarios ${S}_{1}^{1}$ and ${S}_{2}^{1}$ are too “far” (upper deviation = $1.00-0.45>0.15$; lower deviation = $0.45-0.00>0.25$), thus, again, they do not reflect the DM’s predictions and presumptions.

Finally, the whole reduced scenario set ${S(1)}^{r}$ includes ${S}_{2}^{1}$ and ${S}_{3}^{1}$ (rule 8.2). One may formulate the following question: why is scenario ${S}_{3}^{1}$ not selected when considering option D₁ and why is scenario ${S}_{1}^{1}$ not selected when considering option D₂. In these cases scenario weights (i.e. their attractiveness) are equal to 1. However, note that the aforementioned measures represent only the status of particular scenarios, but the optimism coefficient shows how does the decision maker assess the prospects for the occurrence of the best (the most attractive) scenario in the future.

In the case of criterion C₂ it is suggested to keep both scenarios for further analysis since for option D₁ scenario ${S}_{2}^{2}$ has got a zero normalized weight which is the closest to 0.25 (lower deviation = $0.25-0.00\le 0.25$) and for alternative D₂ scenario ${S}_{1}^{2}$ also has got a zero normalized weight which is the closest to parameter ${\beta }_{2}$ (lower deviation = $0.25-0.00\le 0.25$).

We can observe that for ${\beta }_{2}>0.25$ the DM would be forced to increase the lower deviation level in order to obtain an nonempty reduced scenario set.

In steps 9–11 new DM’s estimations (i.e. pairwise comparison matrices in terms of alternatives only for scenarios from sets ${S(k)}^{r}$) are used to calculate option weights (Table 5), separately for each criterion and scenario.

Table 5 Alternative comparison matrices for scenarios ${\text{S}}_{2}^{1}$,${\text{S}}_{3}^{1}$ (within C₁) and ${\text{S}}_{1}^{2}$, ${\text{S}}_{2}^{2}$ (within C₂)

Our reduced sets are not single-element sets. That is why, the use of Eq. (8) is recommended to obtain a single alternative weight for each option within particular criteria: ${N}_{1}^{1*}=\frac{1}{2}\times \left(0.80+0.75\right)=0.78$; ${N}_{2}^{1*}=0.23$; ${N}_{1}^{2*}=0.39$; ${N}_{2}^{2*}=0.61$ (values are rounded to two decimal places).

The use of Eq. (9) allows us to indicate the final option (step 12): ${N}_{1}^{*}=0.78\times 0.67+0.39\times 0.33=0.65$; ${N}_{2}^{*}=0.35$. The DM should select project D₁.

Before considering another example, it would be desirable to check how the recommendations depend on the optimism coefficient.

For ${\beta }_{1}=0.55$, ${\beta }_{2}=0.50$ (moderate decision maker), upper and lower deviations equal to 0.25 and 0.25, respectively, initial sets $S\left(1\right)=\left\{{S}_{1}^{1},{S}_{2}^{1},{S}_{3}^{1}\right\}$ and $S\left(2\right)=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$ are reduced to ${S(1)}^{r}=\left\{{S}_{2}^{1},{S}_{3}^{1}\right\}$ and ${S(2)}^{r}=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$, but under the condition that the DM significantly increases the lower deviation in the case of criterion C₂, that is from 0.25 to 0.50! Such a modification allows the decision maker to find at least one scenario for each option, within each analysed objective. In connection with the fact that the reduced sets are the same as for the more pessimistic decision maker, the final option recommended by SP/AHP(IC) is also the same—it is D₁.

For ${\beta }_{1}=0.95$, ${\beta }_{2}=0.85$ (optimist), upper and lower deviations equal to 0.05 and 0.10, respectively, initial sets $S\left(1\right)=\left\{{S}_{1}^{1},{S}_{2}^{1},{S}_{3}^{1}\right\}$ and $S\left(2\right)=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$ are reduced to ${S(1)}^{r}=\left\{{S}_{1}^{1},{S}_{3}^{1}\right\}$ and ${S(2)}^{r}=\left\{{S}_{1}^{2},{S}_{2}^{2}\right\}$. However, due to the very small initial set of scenarios for criterion C₂, the upper deviation had to be increased to 0.15 (without this modification, the reduced set for C₂ would be empty). This time, the reduced set for objective C₁ has changed in comparison with set ${S(1)}^{r}$ generated for two previous cases, so further computations will also change. Table 6 contains alternative comparison matrices for current selected scenarios. This table is very similar to Table 5, but its first part concerns scenario ${S}_{1}^{1}$ instead of ${S}_{2}^{1}$.

Table 6 Alternative comparison matrices for scenarios ${\text{S}}_{1}^{1}$, ${\text{S}}_{3}^{1}$ (within C₁) and ${\text{S}}_{1}^{2}$, ${\text{S}}_{2}^{2}$ (within C₂)

The reduced sets are not single-element sets. Therefore, the use of Eq. (8) is recommended again to obtain a single alternative weight for each option within particular objectives: ${N}_{1}^{1*}=\frac{1}{2}\times \left(0.33+0.75\right)=0.54$; ${N}_{2}^{1*}=0.46$; ${N}_{1}^{2*}=0.39$; ${N}_{2}^{2*}=0.61$ (values are rounded to two decimal places).

The use of Eq. (9) enables us to find the final course of action (step 12): ${N}_{1}^{*}=0.54\times 0.67+0.39\times 0.33=0.49$; ${N}_{2}^{*}=0.51$. The DM should select project D₂. Nevertheless, measure values are extremely similar. That’s why, project D₁ may be also treated as a good choice.

The above sensitivity analysis shows that depending on the optimism coefficient and acceptable deviations recommendations may change. This phenomenon results from the fact that both factors have an impact on the structure of the reduced sets which influences the data used to compute measure values for each alternative.

4 Discussion

AHP under uncertainty is obviously more complex than the original deterministic version of the method. Nevertheless, there are ways to reduce the number of estimated pairwise comparison matrices. One of them consists in focusing on scenarios reflecting the most the state of mind and soul of the decision maker. It can be done for instance by means of the optimism coefficient declared by them. Let us examine the following case in order to explain this observation. We assume that the case contains 3 criteria (C₁, C₂, C₃), 4 alternatives (D₁, D₂, D₃, D₄), e.g. 4 new, very risky projects, and 5 scenarios for each objective $S\left(1\right)=\left\{{S}_{1}^{1},{S}_{2}^{1},{S}_{3}^{1},{S}_{4}^{1},{S}_{5}^{1}\right\}$, $S\left(2\right)=\left\{{S}_{1}^{2},{S}_{2}^{2},{S}_{3}^{2},{S}_{4}^{2},{S}_{5}^{2}\right\}$, $S\left(3\right)=\left\{{S}_{1}^{3},{S}_{2}^{3},{S}_{3}^{3},{S}_{4}^{3},{S}_{5}^{3}\right\}$. We present very few numerical data, because this time we focus on the size of the problem and subsequent steps of the algorithm:

• 1. The cardinality of sets D, C and S(k) is already known: $\left|D\right|=4, \left|C\right|=3, \left|S(1)\right|=\left|S(2)\right|$=$\left|S(3)\right|=5$

• 2. The criteria pairwise comparison matrix for criteria contains 9 comparisons, i.e. 3 main rows and 3 main columns since there are 3 objectives.

• 3. There are 12 pairwise comparison matrices for particular options in terms of scenarios, separately for each criterion: 4 for criterion C₁, 4 for criterion C₂ and 4 for criterion C₃. Each matrix contains 25 comparisons, i.e. 5 main rows and 5 main columns because there are 5 scenarios for each objective.

• 4. The DM defines 3 coefficients of optimism β₁, β₂ and β₃ (for each criterion). Let us assume that the decision maker is a moderate optimist.

• 5. The normalization for each matrix (i.e. 12 + 1 = 13 matrices) is performed.

• 6. Criteria weights are calculated: P₁, P₂, P₃. Scenario weights are also computed (within a given criterion and option), that is: ${M}_{1}^{\text{1,1}}$, ${M}_{2}^{\text{1,1}}$, ${M}_{3}^{\text{1,1}}$, ${M}_{4}^{\text{1,1}}$, ${M}_{5}^{\text{1,1}}$, ${M}_{1}^{\text{1,2}}$, ${M}_{2}^{\text{1,2}}$, ${M}_{3}^{\text{1,2}}$, ${M}_{4}^{\text{1,2}}$, ${M}_{5}^{\text{1,2}}$, …, ${M}_{1}^{\text{3,4}}$, ${M}_{2}^{\text{3,4}}$, ${M}_{3}^{\text{3,4}}$, ${M}_{4}^{\text{3,4}}$, ${M}_{5}^{\text{3,4}}$ (5 $\times$ 12 = 60 all together)

• 7. The normalization of scenario weights ${M}_{i}^{k,j}$ is executed (for 60 results).

• 8. We do not know exactly the level of the DM’s optimism, but we may assume that the sum of the upper and lower acceptable deviations is not too large (let us recall that the decision maker is a moderate optimist) and that the reduced scenario sets are as follows: ${S(1)}^{r}=\left\{{S}_{4}^{1}\right\}$, ${S(2)}^{r}=\left\{{S}_{1}^{2},{S}_{3}^{2}\right\}$, ${S(3)}^{r}=\left\{{S}_{4}^{3},{S}_{5}^{3}\right\}$. One can ask the following question—is it possible to reduce the original scenario sets so much? There were 15 scenarios and we only keep 5 for further analysis? Indeed, there is such a possibility. We can observe that it occurs when:

• • The aforementioned sum of both types of deviations is relatively small (this allows one not to catch too many scenarios),

• • The total number of scenarios defined for each objective is larger than 2 or 3 (this enables one to obtain varied scenario weighs, not only 1.00 and 0.00, hence the chance to find a suitable scenario increases),

• • Scenario weights are evenly distributed within particular options and criteria (this allows one to find a scenario with a weight corresponding to the DM’s optimism coefficient),

• • Within a given objective there exists a common scenario (for each alternative) with a weight close to the optimism coefficient.

• 9. Pairwise comparison matrices in terms of alternatives are estimated only for scenarios from sets ${S(k)}^{r}$. There are 5 alternative comparison matrices. Each of them contains 16 comparisons, i.e. 4 main rows and 4 main columns since the DM considers 4 options. If the reduction has not been made, the DM would have to present 15 alternative comparison matrices, which would be extremely time-consuming.

• 10. The normalization of values ${a}_{h,l}^{k,i(r)}$ generates 80 new results $\left(5\times 4\times 4\right)$.

• 11. Alternative weights are calculated. For criterion ${C}_{1}$ the obtained averages are the final ones since only one scenario has been selected: ${N}_{1}^{\text{1,4}}={N}_{1}^{1*}$, ${N}_{2}^{\text{1,4}}={N}_{2}^{1*}$, ${N}_{3}^{\text{1,4}}={N}_{3}^{1*}$, ${N}_{4}^{\text{1,4}}={N}_{4}^{1*}$. In the case of the remaining criteria computations are more complex: ${N}_{1}^{2*}=\left({N}_{1}^{\text{2,1}}+{N}_{1}^{\text{2,3}}\right)/2$, ${N}_{2}^{2*}=\left({N}_{2}^{\text{2,1}}+{N}_{2}^{\text{2,3}}\right)/2$, ${N}_{3}^{2*}=\left({N}_{3}^{\text{2,1}}+{N}_{3}^{\text{2,3}}\right)/2, {N}_{4}^{2*}=\left({N}_{4}^{\text{2,1}}+{N}_{4}^{\text{2,3}}\right)/2,$ ${N}_{1}^{3*}=\left({N}_{1}^{\text{3,4}}+{N}_{1}^{\text{3,5}}\right)/2,$ ${N}_{2}^{3*}=\left({N}_{2}^{\text{3,4}}+{N}_{2}^{\text{3,5}}\right)/2,$ ${N}_{3}^{3*}=\left({N}_{3}^{\text{3,4}}+{N}_{3}^{\text{3,5}}\right)/2,$ ${N}_{4}^{3*}=\left({N}_{4}^{\text{3,4}}+{N}_{4}^{\text{3,5}}\right)/2,$

• 12. In the last step, before making the final choice, 4 measures have to be calculated, one for each project: ${N}_{1}^{*}={N}_{1}^{1}{\times P}_{1}+{N}_{1}^{2}\times {P}_{2}+{N}_{1}^{3}\times {P}_{3}+{N}_{1}^{4}\times {P}_{4}$ etc.

The case described above allows us to note that when a complete AHP under uncertainty is performed (Durbach, 2019), at least $\left(\left(\begin{array}{c}p\\ 2\end{array}\right)+\sum_{k=1}^{p}\left(n\times \left(\begin{array}{c}{m}_{k}\\ 2\end{array}\right)\right)+{\sum }_{k=1}^{p}\left({m}_{k}\times \left(\begin{array}{c}n\\ 2\end{array}\right)\right)\right)$ comparisons are needed. On the other hand, thanks to the scenario set reduction, it is possible to estimate only $\left(\left(\begin{array}{c}p\\ 2\end{array}\right)+\sum_{k=1}^{p}\left(n\times \left(\begin{array}{c}{m}_{k}\\ 2\end{array}\right)\right)+\sum_{k=1}^{p}\left(\left|S{(k)}^{r}\right|\times \left(\begin{array}{c}n\\ 2\end{array}\right)\right)\right)$ comparisons (see point 9). If, e.g., $p=5$, $n=6$, ${m}_{1}=2$, ${m}_{2}=5$, ${m}_{3}=4$, ${m}_{4}=6$, ${m}_{5}=3$, $\left|S{\left(1\right)}^{r}\right|=1$, $\left|S{\left(2\right)}^{r}\right|=2$, $\left|S{\left(3\right)}^{r}\right|=2$, $\left|S{\left(4\right)}^{r}\right|=2$, $\left|S{\left(5\right)}^{r}\right|=1$, the values obtained thanks to the first and second formula equal 520 and 340, respectively, which reflects the superiority of the novel approach.

Note that the reduced set of scenarios should not be too narrow especially in the case of pessimist decision makers who prefer to control more non-attractive scenarios. That is why, rule 1 in step 8, allowing one to set the size of particular types of acceptable deviations, constitutes a significant warranty for the DM. This warranty may be particularly useful in the case where the real scenario (i.e. this one that will really occur in the future) is not the one with the weight being the closest to the optimism coefficient. If the decision makers make the final decision on the basis of more scenarios, they may be better prepared for the real scenario. On the other hand, let us recall that the number of operations performed increases with the increase of the size of the reduced sets. That is why, it is irrational to include too many scenarios in the reduced sets.

It is worth emphasizing that in the illustrative example presented in the previous section the scenario set reduction was not significant (from 5 to 4—it is a decrease of 20%). It mainly results from the fact that the original number of scenarios defined for each criterion was small. However, when the problem increases and the sets of scenarios are more numerous, the final number of scenarios for each objective can be considerably lower than the first one (in the case investigated in this section the total number of scenarios has decreased from 15 to 5—it is a decrease of 67%), but let us recall that the cardinality of the reduced set is strictly related to the DM’s state of mind and soul. People intending to control more scenarios will be keen to set larger acceptable deviations. People being sure of their predictions may want to declare smaller deviations.

For clarity, we would like to remind that the optimism coefficient does not affect the attractiveness of particular scenarios (i.e. scenario weights). It only allows finding scenarios reflecting the most the decision maker’s attitude risk.

The problem of acceptable deviations requires additional discussion. In this paper we recommend a conscious deviation estimation to the DMs, which gives the possibility to take thoroughly their nature into consideration. On the other hand, there are no serious obstacles to replace the aforementioned custom step with a more automatic step within which strict, homogeneous regulations concerning the size of particular deviations are given and applied without the excessive involvement of decision makers. Such an improvement is certainly desirable for people (project managers) who are not familiar enough with the idea of the whole algorithm or are too busy to actively participate in the decision making process.

The literature puts emphasis on cases where the set of scenarios is the same for each criterion, which means that the authors (consciously or not) assume that the considered objectives are dependent. Here, we investigate examples which from the literature point of view may be treated as quite unusual, but, as a matter of fact, they are very realistic: is the set of scenarios defined, for instance, for the profit maximization always the same as the set of scenarios essential for the minimization of the number of formalities supposed to fulfil? Of course, not. Both criteria depend on, at least partially, different factors.

The scenario-based AHP is basically designed for one-shot decisions (innovative or innovation projects) which often occur in our very dynamic world. Circumstances are changing so unexpectedly that multi-shot decisions (i.e. decisions concerning projects performed many times) are made more rarely. Nevertheless, we are obviously aware of the fact that in numerous domains (production, finance, employment, medicine etc.) the selected solution could be implemented many times, but due to constant changes it is desirable to revise the chosen option systematically. The proposed method—SP/AHP(IC)—gives such opportunities. After the realization of the selected variant the DM is able to check what the real scenario is and what the real results are. If they need to choose a suitable strategy for the next periods, it is possible to modify some AHP parameters (e.g. the optimism coefficient or some pairwise comparisons) and establish an updated solution.

As it was mentioned in Sect. 2, we omit the consistency analysis in the procedure description. However, we admit that if at least one matrix is not consistent enough, the SP/AHP(IC) must be properly developed. In such cases matrix modifications may be necessary in steps 2, 3 and 9. Then, further steps of the suggested algorithm can be performed as usual. Note that the optimism coefficient and acceptable deviations do not affect the matrix consistency. Matrices may be inconsistent only if comparisons are made incorrectly.

Within this section, it would be also desirable to compare the effectiveness of the new procedure to the effectiveness of other methods designed for the same purpose. Although numerous approaches for handling uncertainty in AHP have been mentioned in the Introduction, as a matter of fact they allow to solve different problems (and reach different goals). They also use different initial data because they refer to different measures (probabilities, fuzzy sets, belief functions, instead of scenario planning). That is why, a direct comparison is rather impossible. The largest number of similarities can be seen when comparing the novel method to the approach presented in Durbach (2019) thanks to the use of scenario planning. Nevertheless, further steps of both algorithms are not common any more. The first reason is that in the procedure described in Durbach (2019) all the scenarios have to be considered till the end of the method while in the new approach such requirement is not in force. Second, in Durbach (2019) the decision maker’s nature and preferences can be declared in pairwise comparison matrices only while here it is possible to refer to the optimism coefficient (a quite popular parameter applied to other decision rules) and acceptable deviations. Third, comparisons for subsequent pairs of scenarios have got a totally different meaning in particular procedures. In Durbach (2019) they may signify how likely a scenario is to occur while in the suggested method they show how much does someone prefer alternative ${D}_{j}$ within a scenario to alternative ${D}_{j}$ within another scenario (by virtue of a given criterion). Fourth, when analysing the approach described in Dubarch (2019), we cannot clearly state for what type of decisions (one-shot or multi-shot) and criteria (dependent or independent) it has been developed. As we see, even for this pair of procedures, there are too many differences. However, the most visible difference is related to the opportunity to reduce the scenario sets and to examine fewer comparisons.

The above conclusion may raise doubts since, as a matter of fact, one of the steps (step 3) of SP/AHP(IC) involves the estimation of pairwise comparison matrices for particular options in terms of scenarios, separately for each criterion. Indeed, this step is necessary to assess the relative scenario attractiveness, but just after the scenario set reduction the estimation of pairwise comparison matrices in terms of alternatives are made only for selected scenarios, not for all the initial scenarios (step 9).

It is worth stressing that the whole proposed procedure seems to be complex due to the number of steps, but note that many of them can be performed automatically, without the DM’s involvement. However, in connection with the fact that still many pairwise comparisons are needed, we suggest applying the method mainly to important decision problems which contain a reasonable number of alternatives, criteria and scenarios. Furthermore, SP/AHP(IC) is designed for decisions which are not too urgent since the pairwise comparisons are quite time-consuming.

The example analysed in the previous section has been supplemented by a sensitivity analysis connected with the optimism level. We encourage the user to always precede the final choice by this stage. The DM may check how the SP/AHP(IC) recommendations change depending on the values of diverse parameters (not only the optimism coefficient). Such an analysis enables one to make a more conscious choice.

Here, we have used the presented decision rule in order to find the best solution, but it is worth stressing that the new approach, similarly to the original AHP, may be also applied to create rankings. Note that we consciously do not use the term “optimal solution” since the goal of the scenario-based AHP is not to find optimal options, but to indicate decision variants reflecting the decision maker’s preferences and predictions.

5 Conclusion

The paper describes SP/AHP(IC)—a novel scenario-based AHP approach designed for uncertain one-shot decisions and independent criteria, i.e. for situations occurring quite often in our dynamic environment. The procedure is based on scenario planning and the optimism coefficient.

A very short presentation of this method has been already given by Gaspars-Wieloch (2019a), but the previous description does not explain enough the motivation to create a new procedure, the assumptions of the suggested approach, the way to execute in practice subsequent steps of the algorithm and all the essential advantages of the decision rule. That is why, a significant extension of the aforementioned paper was absolutely mandatory to understand the idea of SP/AHP(IC).

The proposed decision rule constitutes a combination of many existing approaches. The method refers to the methodology suggested mainly by:

• Pomerol (2001)—in connection with a similar interpretation of scenarios,

• Durbach (2019), Brozova (2004)—due to the use of scenario planning in AHP,

• Grienitz et al. (2014), Hurwicz (1952), Wald (1950a, 1950b)—due to the idea to focus on selected scenarios

• Hurwicz (1952)—by virtue of the application of the optimism coefficient,

• Saaty (1980)—certain basic steps of the original AHP are the same as the steps belonging to SP/AHP(IC).

The advantages of the new decision rule are numerous. The SP/AHP(IC):

• 1. Gives the possibility to generate a relatively small number of pairwise comparison matrices thanks to the reduction of the initial sets of scenarios (this reduction increases with the increase of the size of the problem),

• 2. Can be applied to different decision makers (optimist, pessimist, moderate) since one of its steps consists in estimating the optimism coefficient,

• 3. Offers a kind of warranty (before the final decision is selected), especially to pessimists, because the decision makers, thanks to acceptable deviations, may control both scenarios with a weight (attractiveness measure) close to the optimism coefficient and scenarios reflecting their attitude towards risk to a lesser degree—the possibility to reduce scenario sets carefully and in a controlled way can be a reassuring factor, for pessimists in particular,

• 4. Gives the opportunity to choose an option even if the decision maker’s knowledge about scenario probabilities is merily partial—hence, it enables analyzing innovative, innovation and risky projects,

• 5. Can be successfully applied both to the best solution searching as well as to generate rankings of banks, airlines, investment strategies, mortgage credits etc.,

• 6. Allows handling imprecise and uncertain evaluations of both quantitative and qualitative criteria.

What is new in the algorithm in relation to existing AHP procedures handling uncertainty? The major novelties are given below:

• The use of the optimism coefficient in AHP (as a way to include the decision maker’s nature).

• Its application to one-shot decisions and independent criteria.

• The chance to reduce scenario sets and the tools applied to reach it (scenario attractivenes measures, optimism coefficient, acceptable deviations) in such a way that the nature, i.e. the decision maker’s opponent, still keeps its neutral and passive character.

The scenario-based AHP may seem to be a little controversial due to the reduction of the number of scenarios, but such an approach partially considers the one-shot character of decisions within which only one scenario have the chance to occur in the future.

In connection with the fact that SP/AHP(IC) data are based on subjective human judgements, preferences and predictions, the algorithm cannot be evaluated considering the future real scenario since the direct goal of the procedure is not to minimize the difference between the result connected with the solution recommended by the method and the result related to the solution that could be selected if the decision maker knew the real scenario from the beginning. Let us recall that the direct goal of SP/AHP(IC) is to adjust the choice of the decision makers to their state of mind and soul, to their nature, i.e. to choose such a solution which, on the basis of current information, reflects the best their attitude towards risk.

As a part of further research it would be desirable to create an analogous procedure for independent decisions, i.e. for options which are so different that they depend on different scenarios even within a given criterion.