Copyright © 2014 The Authors. Published by VGTU Press. This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC 4.0) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The material cannot be used for commercial purposes. B u s i n e s s, Ma n ag e M e n t a n d e d u c at i o n ISSN 2029-7491 / eISSN 2029-6169 2014, 12(2): 283–302 doi:10.3846/bme.2014.242 THE USE OF A MODIFICATION OF THE HURWICZ’S DECISION RULE IN MULTICRITERIA DECISION MAKING UNDER COMPLETE UNCERTAINTY Helena GASPARS-WIELOCH Department of Operations Research, Faculty of Informatics and Electronic Economy, Poznań University of Economics, al. Niepodległości 10, 61-875 Poznań, Poland E-mail:helena.gaspars@ue.poznan.pl Received 18 October 2014; accepted 08 November 2014 Abstract. The paper concerns multicriteria decision making under uncertainty with scenario planning. This topic is explored by many researchers because almost all real-world decision problems have multiple conflicting criteria and a determin- istic criteria evaluation is often impossible (e.g. mergers and acquisitions, new product development). We propose two procedures for uncertain multi-objective optimization (for dependent and independent criteria matrices) which are based on the SAPO method – a modification of the Hurwicz’s rule for one-criterion prob- lems, recently presented in another paper. The new approaches take into account the decision maker’s preference structure and attitude towards risk. It considers the frequency and the level of extreme evaluations and generates logic rankings for symmetric and asymmetric distributions. The application of the suggested tool is illustrated with an example of marketing strategies selection. Keywords: multicriteria decision making, optimization, uncertainty, scenario plan- ning, one-shot decision, attitude towards risk, criteria weights, normalization tech- nique, SAPO method, marketing strategies selection, business objectives. JEL Classification: C44, C61, D81, L21, L25, M31. 1. Introduction The paper deals with multiple criteria decision making for cases where attribute (criteri- on) evaluations are uncertain. This topic is investigated by many researchers. (Durbach, Stewart 2012) provide an impressive review of possible models, methods and tools used to support uncertain multicriteria decision making (e.g. models using probabil- ities or probability-like quantities, models with explicit risk measures, models with fuzzy numbers, models with scenarios). The number of various contributions devoted to uncertain multiobjective optimization is evidence of the theoretical (Dominiak 2009; Eiselt, Marianov 2014; Janjic et al. 2013; Michnik 2013b) and practical (Aghdaie et al. 2013; Ginevičius, Zubrecovas 2009; Hopfe et al. 2013; Korhonen 2001; Lee 2012; mailto:helena.gaspars@ue.poznan.pl 284 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... Michnik 2013b; Mikhaidov, Tsvetinov 2004; Montibeller et al. 2006; Ram et al. 2010; Suo et al. 2012; Tsaur et al. 2002; Wang, Elhag 2006; Watkins et al. 2000) importance of this topic. In this contribution two new procedures (for dependent and independent criteria matrices) designed for multicriteria decision making with scenario planning are presented. The goal of these approaches is to select optimal decisions and generate rankings of alternatives for one-shot decision problems. The proposed methods take into consideration decision makers’ preferences and attitude towards risk. The paper is organized as follows. Section 2 deals with the main features of DMU (decision making under uncertainty) with scenario planning. Section 3 briefly de- scribes the essence of multiobjective decision making (MDM) and its discrete version (DMDM). Section 4 is devoted to methods applied to multicriteria decision making under uncertainty with scenario planning (MDMU+SP). Section 5 demonstrates how the SAPO method, a modification of the Hurwicz’s decision rule which is presented in (Gaspars-Wieloch 2014d), can be used as a tool in multicriteria optimization under uncertainty. Section 6 provides a case study related to the choice of an optimal market- ing strategy. Conclusions are gathered in the last part. 2. Decision making under uncertainty with scenarios According to the Knightian definition (Knight 1921), decision making under uncertainty (DMU), in contrast to decision making under certainty (DMC) or risk (DMR), is charac- terized by a situation where future factors (quantitative and qualitative) are neither deter- ministic nor probabilistic at the time of the decision (complete uncertainty, uncertainty without probabilities). Actually the decision maker (DM) has to choose the appropriate alternative (decision, act, project, strategy) on the basis of some scenarios (events, states of nature) whose probabilities are not known (Chronopoulos et al. 2011; Dominiak 2006; Groenewald, Pretorius 2011; Render et al. 2006; Sikora 2008; Trzaskalik 2008; von Neumann, Morgenstern 1944). Apart from DMU, DMR and DMC, there is a forth category – decision making with partial information (DMPI) – characterized by prob- ability distributions not known completely (Cannon, Kmietowicz 1974; Dubois, Prade 1988; Guo 2011; Kapsos et al. 2014; Klir, Folger 1988; Kmietowicz, Pearman 1984; Kofler, Zwiefel 1993; Michalska 2014; Weber 1987). There are many classical decision rules designed for DMU, such as the Wald’s cri- terion (Wald 1950), the maximax criterion presented e.g. in (Pazek, Rozman 2009), the Hurwicz’s criterion (Hurwicz 1952), the Savage’s criterion (Savage 1961), the maximin joy criterion (Hayashi 2008), the Bayes (Laplace’s) criterion presented e.g. in (Render et al. 2006). The literature also offers many diverse extensions or hybrids of those methods, e.g. (Basili 2006; Basili et al. 2008; Basili, Chateauneuf 2011; Ellsberg 2001; Etner et al. 2012; Gaspars 2007; Gaspars-Wieloch 2012, 2013, 2014a, 2014c, 2014d, 2014e, 2015; Ghirardato et al. 2004; Gilboa 2009; Gilboa, Schmeidler 1989; Marinacci 2002; Piasecki 1990; Schmeidler 1986; Tversky, Kahneman 1992). Nevertheless, the 285 Business, Management and Education, 2014, 12(2): 283–302 majority of the extended rules refer to the probability calculus (for instance expected utility maximization, maximin expected utility, α-maximin expected utility, restricted Bayes/Hurwicz, prospect theory, cumulative prospect theory, Choquet expected utility), which is rather characteristic of DMR (or DMU with probabilities). Let us remind that according to the Knight’s definition the uncertainty occurs when we do not know (i.e. we can not measure) the probabilities of particular scenarios: “when the uncertainty may be measured, it is called “risk” (Knight 1921). As it was mentioned before, scenario planning (SP) can be used within the frame- work of DMU (Dominiak 2006; Pomerol 2001). SP is a popular and comprehensive de- cision support tool for considering future uncertainties. It is a technique for facilitating the process of identifying uncertain and uncontrollable factors which may influence the effects of decisions in the strategic management context. The way how scenarios should be constructed is described e.g. in (Dominiak 2006; Van der Heijden 1996). The result of the choice made under uncertainty with scenario planning depends on two factors: which decision will be selected and which scenario will occur. DMU can be presented by means of a payoff matrix where m (the number of rows) denotes the number of mutually exclusive scenarios (S1, …, Si, … Sm), n (the number of columns) stands for the number of decisions (D1, …, Dj, …, Dn) and aij is the profit connected with scenario Si and alternative Dj. We assume in this paper that the distribution of payoffs related to a given decision is discrete and that the set of those profits can be a multiset (a bag). The contribution concerns searching an optimal pure strategy. A pure strategy is a solution assuming that the DM chooses and completely executes only one alternative. Meanwhile the mixed strategy allows him or her to select and perform a weighted combination of several accessible alternatives (Gaspars-Wieloch 2014b; Officer, Anderson 1968; Puppe, Schlag 2009; Sikora 2008). We recognize both uncertainties: internal (related to DM’s values and judgments) and external (related to imperfect knowledge concerning consequences of action), but in this paper we focus on the latter (Durbach, Stewart 2012; Stewart 2005). Existing decision rules differ one from another with respect to the DM’s attitude towards risk which can be measured, for instance, by means of the coefficient of pes- simism (α) or the coefficient of optimism (β). Note that in this context we do not treat risk as a situation where the probability distribution of each parameter of the decision problem is known, but we mean the possibility that some bad circumstances will happen (Dominiak 2006, 2009; Fishburn 1984). It is worth emphasizing that some rules find application when the DM intends to perform the selected strategy only once, i.e. the decision is experienced only once (e.g. Wald’s criterion, Hurwicz’s criterion, Savage’s criterion, maximax criterion, maximin joy criterion). Others are recommended for people contemplating realization of the chosen variant many times (Laplace’s criterion). In the first case, alternatives are called one-shot (one-time) decisions. In the second case, one deals with multi-shot decisions. 286 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... This paper focuses on one-shot decision problems which are commonly encountered in business, economics and social systems and have been receiving increasing research interest because of the growing dominance of service industries for which such prob- lems are particularly applicable. Typical examples of one-shot decisions include merg- ers and acquisitions (M&A), emergency management for irregular events, supply chain management of products with short life cycles, new product development and private real-estate investment (Guo 2010, 2011, 2013, 2014; Liu, Zhao 2009). 3. Multiobjective (multicriteria) decision making The multiobjective optimization or multiple objective decision making (MDM) is a topic discussed in many contributions because decision making is a part of our life and almost all real-world decision problems have multiple conflicting criteria. Usually those problems do not have a single global solution. Theoretical issues concerning multigoal optimization are comprehensively treated for instance in (Ehrgott 2005; Jahn 2004; Marler, Arora 2004; Michnik 2013b; Sawaragi et al.1985; Trzaskalik 2014). MDM has diverse goals, such as selecting a preferred alternative, ranking alterna- tives from the best to the worst, sorting the alternatives into ordered classes such as “good” and “bad” (Durbach, Stewart 2012). There are various methods enabling one to find the set of satisfactory solutions and the compromise solution (i.e. the final choice among efficient solutions) for multiobjective problems, such as AHP – analytic hier- archy process (Saaty 1980; Podvezko 2009), ANP – analytic network process (Saaty 1996; Azimi et al. 2011), COPRAS-G method (Zavadskas, Kaklauskas 1996), TOPSIS (Hwang, Yoon 1981; Azimi et al. 2011), SAW – simple additive weighting method (Churchman, Ackoff 1954), goal programming (Charnes et al. 1955), e-constraint meth- od (Chankong, Haimes 1983), lexicographic method (Khorram et al. 2010), VIKOR-S (Michnik 2013b), WINGS (Michnik 2013a) and diverse hybrids (e.g. Haeri, Tavakkoli- Moghaddam 2012; Hsu 2014; Wu et al. 2013). Multicriteria methods differ according to how they (a) evaluate performances on each attribute and (b) aggregate evaluations across attributes to arrive at an overall global evaluation (Durbach, Stewart 2012; Greco et al. 2010). (Marler, Arora 2004) divide multi-objective optimization concepts and methods into three categories: (a) methods with a priori articulation of preferences (MPAP), (b) meth- ods with a posteriori articulation of preferences (MPSAP) and (c) methods with no ar- ticulation of preferences (MNAP). In MPAP the user indicates the relative importance of the objective functions or desired goals (by means of parameters which are coefficients, exponents, constraint limits) before running the optimization algorithm. The preference structure can be defined, among other things, on the basis of aspirations levels (Lotfi et al. 1997), utility functions (Chang 2011) and weights representing the importance of each criterion (e.g. Gaspars-Wieloch 2011). MPSAP entail selecting a single solution 287 Business, Management and Education, 2014, 12(2): 283–302 from a set of mathematically equivalent solutions. It means that the DM imposes prefer- ences directly on a set of potential final solutions. Within MDM, the set of alternatives may be either explicitly defined and discrete in number or implicitly defined via constraints in a mathematical programming formula- tion (Ehrgott 2005). In the first case the problem is discrete (DMDM), in the second one – we deal with the continuous version of multicriteria decision making (CMDM). This contribution is devoted to DMDM. The discrete problem consists of n decisions (D1, …, Dj, …, Dn), each evaluated on p criteria denoted by: C1, …, Ck, …, Cp. There are also n×p evaluations of alternatives (Dj) in terms of particular criteria (Ck), accord- ing to some suitable performance measures. Let us denote them by aj k. The multi-objective optimization may be analyzed for the deterministic (MDMC – multi-criteria decision making under certainty) or non-deterministic (MDMU, MDMR, MDMPI) case. In the second situation the evaluation of alternatives is complicated by their performance on at least some attributes not being known with certainty (Durbach, Stewart 2012). In this paper we consider the case of MDMU. 4. Multicriteria decision making with scenario planning Many MDM models and methods are based on essentially deterministic evaluations of the consequences of each action in terms of each criterion, possibly subjecting final results and recommendations to a degree of sensitivity analysis. In many situations, such an approach can be justified when the primary source of complexity in decision making relates to the multicriteria nature of the problem rather than to the stochastic nature of individual consequences. Nevertheless, situations do arise, especially in strategic plan- ning problems, when uncertainty is as critical as the issue of conflicting management goals. In such cases, approaches designed for MDMU become necessary, for instance (a) the multi-attribute utility theory, (b) pairwise comparisons of probability distribu- tions, (c) the use of surrogate risk measures (quantiles, variances) as additional deci- sion criteria, (d) models combining fuzzy numbers with the analytic hierarchy process, (e) fuzzy TOPSIS, and (f) the integration of MDMU and scenario planning (SP) (Ben Amor et al. 2007; Dominiak 2009; Durbach 2014; Durbach, Stewart 2012; Keeney, Raiffa 1993; Liu et al. 2011; Michnik 2013b; Stewart 2005; Triantaphyllou, Lin 1996; Urli, Nadeau 2004; Watkins et al. 2000; Xu 2000; Yu 2002; Zaras 2004). (Durbach, Stewart 2012) state that uncertainties become increasingly so complex that the elicita- tion of measures such as probabilities, belief functions or fuzzy membership functions becomes operationally difficult for DMs to comprehend and virtually impossible to validate. Therefore, in such contexts it is useful to construct scenarios which describe possible ways in which the future might unfold. Hence, the last example mentioned in the previous paragraph (MDMU+SP: multicriteria decision making under uncertainty with scenario planning) will be considered in this paper. When MDMU+SP is taken into account, the problem can be discrete (the number of possible decisions is finite and 288 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... countable) or continuous (the set of decisions is given through constraints), but here, as already mentioned, we only discuss the discrete type. The discrete version of MDMU+SP (DMDMU+SP) consists of n decisions (D1, …, Dj, …, Dn), each evaluated on p criteria denoted by: C1, …, Ck, …, Cp and on m, scenarios (S1, …Si, … Sm). Hence, the problem can be presented by means of p payoff matrices (one for each criterion) and p×n×m evaluations. Each payoff matrix contains n×m evaluations, say aij k, which signify the performance of criterion Ck result- ing from the choice of decision Dj and the occurrence of scenario Si . According to (Durbach, Stewart 2012; Michnik 2013b) MDMU+SP models can be divided into two classes. The first one (A) includes two-stage models in which evalua- tions of particular alternatives are estimated in respect of scenarios and criteria in two separate stages. Class A contains two subclasses: A-CS and A-SC. Subclass A-CS de- notes the set of approaches considering decisions separately in each scenario before and setting a n×m table giving the aggregated (over attributes/criteria) performance of alternative Dj under scenario Si. These evaluations are then aggregated over scenarios. In subclass A-SC the order of aggregation is reversed – performances are generated across scenarios and then measures are calculated over criteria. The second class (B) consists of one-stage procedures considering combinations of scenarios and attributes (scenario-criterion pairs) as distinct meta-criteria. There is currently no consensus on the best way to solve uncertain multigoal problems. Within the framework of the discrete multicriteria optimization with scenarios, re- searchers have already proposed, among others, the following techniques: 1. additive aggregation giving a scenario-based utility (Stewart 2005); 2. multiattribute value modelling (Goodwin, Wright 2001); 3. results aggregation over scenarios using a relative likelihood (Korhonen 2001); 4. maximization of the worst performance across scenarios (Montibeller et al. 2006; Ram et al. 2010); 5. dominance relation based on Wald’s rule (Dominiak 2006); 6. hierarchy and quasi-hierarchy approach – when the DM is able to formulate his preferences in the form of order of criteria (Dominiak 2006); 7. distance function between alternatives and different reference points, such as the ideal pessimistic or ideal optimistic point – when the DM can describe weights of criteria (Dominiak 2006; Michnik 2013b); 8. interactive approach based on the Interactive Multiple Goal Programming and potency matrices with criteria evaluation of the ideal optimistic (max-max), ide- al pessimistic (max-min) and current pessimistic (min-min) solution (Dominiak 2006); 9. interactive approach applying Monte Carlo simulation (Dominiak 2009); 10. combination of Hurwicz’s rule with MDM (Ravindran 2008; Michnik 2013b). 289 Business, Management and Education, 2014, 12(2): 283–302 Some of these methods are a little criticized – for instance the approach used by (Korhonen 2001) due to the fact that the set of scenarios does not constitute a complete probability space. According to (Durbach, Stewart 2012), one should not aggregate scenario “probabilities”. In the case of dominance relation, hierarchy, quasi-hierarchy and distance function described in (Dominiak 2006) only the worst evaluations of par- ticular alternatives are taken into account, which means that those procedures are rather devoted to a radical pessimist and that they do not fit the solution to the DM’s attitude towards risk (understood as the possibility that some bad circumstances will happen). Furthermore, some rules assume that the occurrence of a given scenario with respect to criterion Ck does not mean that this state of nature will be the true one in terms of another criterion (Ravindran 2008). In such approaches, evaluations from one payoff matrix do not depend on evaluations from other matrices (payoff matrices are totally independent), which is rather rarely found in real decision problems. On the other hand, interactive approaches proposed by (Dominiak 2006, 2009) are much desired since they are very flexible – they can be used without any a priori knowledge about DM’s prefer- ences and can also be applied when criteria are on the ordinal scale. In Section 5, a new approach for generating rankings of decisions under uncertainty will be presented. We will notice that this procedure has many advantages. Firstly, it adjust the recommended solution not only to the DM’s preference structure concern- ing particular criteria, but also to the DM’s attitude towards risk (measured by the coefficient of pessimism). Secondly, if necessary, it allows to treat matrices related to particular attributes as dependent. Thirdly, it enables to make a multi-criteria analysis even in problems with objectives defined in different dimensions and scales. Fourthly, the new method copes with asymmetric distributions of (aggregated) evaluations. 5. The SAPO method for multicriteria decision making under uncertainty The procedure proposed in this Section refers to a two-stage model (see Section 4). We will consider two cases (I and II). In the first case (Section 5.1.) we assume that payoff matrices related to particular attributes are dependent. Thus, for instance, evaluation aij k can be only connected with evaluations aij 1, …, aij k-1, aij k+1, …, aij p-1 and aij p. Those values describe the performance of each criterion by decision Dj provided that scenario Si happens. There is no possibility that, for a given alternative, evaluations concerning particular criteria, come from different scenarios. This assumption implies the neces- sity of using A-CS model. In the second case (Section 5.2) we treat values related to one criterion as independent of evaluations of other criteria. That means that evaluation aij k may be connected with any evaluation aij 1 (i = 1, …, m), any evaluation aij 2 (i = 1, …, m), … and any evaluation aij p (i = 1, …, m). Those values describe the performance of each criterion by decision Dj assuming that any scenario occurs for criteria C1, …, Ck-1, Ck+1, …, Cp. The second case (II) allows us to apply A-SC model. 290 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... In both approaches, we will take advantage of an aggregate objective function (see SAW, Section 3) and a modification of the Hurwicz’s decision rule, which is described and justified in (Gaspars-Wieloch 2014d). This modification is called “SAPO method”. In contrast to the original version of the Hurwicz’s criterion, the SAPO method copes with asymmetric distribution of payoffs. Even in that case, it recommends logic rank- ings and provides answers reflecting the DM’s preferences. The procedure aforemen- tioned is designed for one-criterion decision problems and its goal is to find an optimal pure strategy. The essence of the SAPO method is to multiply the coefficient of opti- mism by all payoffs belonging to a suitably defined range of good results (not only the highest payoff) and to multiply the coefficient of pessimism by all values belonging to a properly set range of bad results (not only the worst payoff), according to the level of risk aversion and on the basis of some additional bounds or deviation degrees. The SAPO rule takes into account both the level and the frequency of extreme values. The first steps of the procedure combining SAW with SAPO are common in both cases (A-CS and A-SC): 1. Present the multicriteria problem by means of p tables containing n×m evaluations aij k (where i = 1, …, m and j = 1, …, n). 2. If criteria are defined in different scale or/and different units, use a normalization technique for each attribute separately (Equation 1 for maximized targets, Equation 2. for minimized targets), (Gaspars-Wieloch 2012, 2015). Otherwise, go to step 3. { } { } { } 1,..., 1,..., 1,...,1,..., 1,...,1,..., min ( ) max min k k ij ij i m j nk ij k k ij ij i mi m j nj n a a a n a a = = == == − = − 1,..., ; 1,..., ; 1,...,k p i m j n= = = ; (1) { } { } { } 1,..., 1,..., 1,...,1,..., 1,...,1,..., max ( ) max min k k ij ij i m j nk ij k k ij ij i mi m j nj n a a a n a a = = == == − = − 1,..., ; 1,..., ; 1,...,k p i m j n= = = . (2) 3. Define weights for each target (criterion): wk (k = 1, …, p) and declare the coef- ficient of pessimism (α). 5.1. The SAPO procedure based on A-CS model (case I) Let us analyze further steps of the SAPO(CS) approach – in the first place, criteria are aggregated within scenarios and then obtained values are calculated over scenarios: 4. Compute an aggregated measure for each pair of decision Dj and scenario Si (for each combination (Dj,Si)) and generate a table with n×m aggregated measures Aij or A(n)ij (Equation 3 for problems that do not require normalization, Equation 4 for problems with normalized evaluations): 291 Business, Management and Education, 2014, 12(2): 283–302 1 p k k ij ij k A w a = = ⋅∑ 1,..., ; 1,...,i m j n= = ; (3) 1 ( ) ( ) p k k ij ij k A n w a n = = ⋅∑ 1,..., ; 1,...,i m j n= = . (4) 5. Present the aggregated measures as a non-increasing sequence 1( ,..., ,..., )j j sj mjSq A A A= containing m terms (where m still denotes the number of scenarios, s is the number of the term in the sequence and A1j > Amj) for each alternative Dj. If the normali- zation technique was applied in step 2, then, instead of values Asj, use measures A(n)sj in step 5 and all further steps. 6. Generate the subsequence of good results (SSqj max) and the subsequence of bad results (SSqj min) for each alternative: max max 1 1 1{ : ( ( ) )sj j j j mj sj jjSSq A Sq A d A A A A= ∈ − − ≤ ≤ ( )max ( max)}sjjSSq C A∧ ≤ ∧ → 1,...,j n= , (5) min min 1{ : ( ( ))j sj j mj sj mj j mjSSq A Sq A A A d A A= ∈ ≤ ≤ + − ( )min ( min)}j sjSSq C A∧ ≤ ∧ → 1,...,j n= , (6) where d max and d min signify the allowable degrees of deviation from the highest (A1j) and the lowest (Amj) aggregated measure, respectively. The deviation degrees can be set separately for each decision and then, instead of d max and d min, parameters dj max and dj min are applied. The deviation degrees are set arbitrarily by the DM. These parameters ought to satisfy the following conditions: max min 1d d+ < , (7) max min, 0d d ≥ . (8) |SSqj max| and |SSqj min| denote final cardinalities of both subsequences. C is the maximal number of good and bad results computed according to constraint (9): max{1, min{ ,1 } }C m= ⋅ α − α   . (9) Equation (5) allows the DM to include in subsequence SSqj max only the elements of the whole sequence which belong to the range determined by d max. For example, if d max = 0.2, A1j = 20, Amj = 5, then the elements of SSqj max should satisfy the following constraint [20 0.2 (20 5);20] [17;20]sjA ∈ − ⋅ − = . Note that the final cardinality of SSqj max is additionally limited by C which depends on the pessimism and optimism indices. Closer to 0 and 1 they are, fewer elements subsequence SSqj max can contain. Such a rela- tion may be explained by the fact that more radical the decision maker is, more likely, in his or her opinion, one of the extreme values is. If m = 10 and α = 0.2, SSqj max may constist of at the most two elements which, due to the last part of Equation (5), must 292 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... be the highest. Thanks to parameters dmax and α the DM is able to set a subsequence SSqj max which, from his or her point of view, is composed of appropriate values, because formula (5) considers both the subjective evaluation of good values and the DM’s risk aversion. Equation (6) has an analogical interpretation. 7. Compute averages of good and bad results for each decision: max max max 1 sj j sjj A SSqj AV A SSq ∈ = ∑ 1,...,j n= , (10) min min min 1 sj j j sj A SSqj AV A SSq ∈ = ∑ 1,...,j n= . (11) 8. Calculate the SAPO(CS) measure (SCSj) for each alternative: maxmin min max 11 (1 ) j jCS jj j m SSqm SSq S AV AV m m − ++ − = α ⋅ + − α ⋅ 1,...,j n= . (12) Parameters min max, ,j jm SSq SSq inserted in condition (12) enable one to take into consideration the size of both subsequences, i.e. the frequency of extreme aggregated measures. SCSj is proportional to the number of good values, i.e. the final cardinality of SSqj max, and inversely proportional to the number of bad values, i.e. the final cardinality of SSqj min, because a given alternative is more attractive when it contains many high aggregated measures and few low results. Fractions min1 jm SSq m + − and max1 jm SSq m − + are equal to 1 when particular subsequences consist of one term. If |SSqj min| increases, then the first fraction is less than 1, but greater than 0.5. Weight min1 jm SSq m + − is a kind of punishment for the alternative whose number of bad results is high because such a distribution of measures is not desirable for the decision maker. On the other hand, if |SSqj max| increases, then the second fraction is greater than 1, but less than 1.5. Weight max1 jm SSq m − + is a bonus for the alternative whose number of good results is high since such a value distribution is much desired. 9. Select the decision according to Equation (13): * arg maxCS CSj j j D S= . (13) 5.2. The SAPO procedure based on A-SC model (case II) Now, let us check how the remaining steps of the SAPO(SC) procedure should be for- mulated. We assume that evaluations from particular matrices are independent. 4. Present evaluations as a non-increasing sequence 1( ,..., ,..., ) k k k k j j sj mjSq a a a= contai- ning m terms (where m denotes the number of scenarios, s is the number of the term in the sequence and ak1j > a k mj) for each alternative Dj and within each crite- 293 Business, Management and Education, 2014, 12(2): 283–302 rion. If the normalization technique was applied in step 2, then, instead of values aksj, use measures a(n) k sj in step 4 and all further steps. 5. Generate the subsequence of good results (SSqj k,max) and the subsequence of bad results (SSqj k,min) for each alternative and within each criterion: ( ),max ,maxmax1 1 1{ : ( ( ) )k kk k k k k k kj sj j j j mj sj j jSSq a Sq a d a a a a SSq C= ∈ − − ≤ ≤ ∧ ≤ ( max)}ksja∧ → 1,..., ; 1,...,k p j n= = , (14) ( ),min ,minmin 1{ : ( ( ))k kk k k k k k kj sj j mj sj mj j mj jSSq a Sq a a a d a a SSq C= ∈ ≤ ≤ + − ∧ ≤ ( min)}ksja∧ → 1,..., ; 1,...,k p j n= = , (15) where dmax and dmin signify the allowable degrees of deviation from the highest (ak1j) and the lowest (akmj) evaluation, respectively. The deviation degrees can be set sepa- rately for each decision (or for each criterion / and for each criterion) and then, instead of dmax and dmin, parameters dj max and dj min (d k,max and dk,min / dj k,max and dj k,min) are applied. As in case I, the deviation degrees are set arbitrarily by the DM and satisfy Equations (7)–(8). |SSqj k,max| and |SSqj k,min| denote final cardinalities of both subse- quences and C is calculated following Equation (9). 6. Compute averages of good and bad results for each decision within each attribute: ,max ,max ,max 1 kk sj j k k j sjk a SSqj av a SSq ∈ = ∑ 1,..., ; 1,...,k p j n= = , (16) ,min ,min ,min 1 kk sj j k k j sjk a SSqj av a SSq ∈ = ∑ 1,..., ; 1,...,k p j n= = . (17) 7. Calculate the criterion-dependent SAPO measure for each alternative: ,min ,max ,min ,max 1 1 (1 ) k k j jk kk j j j m SSq m SSq S av av m m + − − + = α ⋅ + − α ⋅ 1,..., ; 1,...,k p j n= = . (18) Skj is proportional to the number of good evaluations connected with Dj, and inversely proportional to the number of bad evaluations related to Dj. Fractions ,min1 kjm SSq m + − and ,max1 kjm SSq m − + have a similar interpretation as in case I. 8. Compute an aggregated measure, the SAPO(SC) measure, for each decision Dj: 1 p SC k k j j k S w S = = ⋅∑ 1,...,j n= . (19) 9. Select the decision according to Equation (20): * arg maxSC SCj j j D S= . (20) Note that the number of scenarios considered for particular attributes can be different in case II, since events from each matrix are totally independent. If such a situation 294 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... takes place, it is recommended to apply separate notation for each set of scenarios. For instance, m(k) may denote the number of states of nature assigned to attribute Ck and {Sk1, …, S k i, …, S k m(k))} may be the set of scenarios connected with this criterion. SAPO(CS) and SAPO(SC) are methods with a priori articulation of preferences. 6. Case study We will illustrate the use of the approach described in the previous section on the basis of a case of marketing strategies (activities) selection and by referring to a very interest- ing paper concerning the so-called 4P marketing model including product, price, pro- motion and place (Ginevičius et al. 2013). Results gathered by (Ginevičius et al. 2013) enable one to assess the effectiveness of marketing strategies thanks to a multicriteria evalutation procedure based on the sum of products of criteria values and their weights properly estimated. Authors of that contribution develop a hierarchical structure of cri- teria describing enterprise marketing system. They distinguish 8 criteria of product (e.g. innovations, product design, quality, brand), 7 criteria of price (e.g. initial price, terms of payment, price differentiation), 7 criteria of promotion (e.g. advertising, increase of sales, corporate identity) and 5 criteria of place (e.g. place of sale, sales online). All those criteria are integrated into one generalized quantity and are applied in order to quantitatively evaluate marketing activities. Note that if a company is trying to choose the best strategy among a set o potential activities, all criteria values are known before the choice made by the enterprise, since this data constitutes initial parameters of par- ticular strategies. This set of criteria describes factors which are planned and controlled by the company. Now, let us analyze possible further steps performed by a fictitious enterprise, say enterprise “E”, which has already evaluated potential marketing activities (connected with a new product development) and selected 4 strategies with the highest measures proposed in (Ginevičius et al. 2013). Assume that enterprise “E” contemplates realiza- tion of one of these activities, but it aims to choose the strategy which will maximize the future annual profit and the market share. Increasing these criteria is one of the most important objectives of business. Hence, we have two new criteria (apart from 27 initial attributes enumerated by the authors mentioned), but, this time, they are a little beyond the control of the company. They depend not only on the decisions made by the company, but also on some macroeconomic, microeconomic and environmental factors (demand, fashion, population structure, tax mechanism, competitors’ strategies, misfortune, weather and so on). For such criteria (profit and market share) the exact estimation is rather complicated. Therefore, instead of using deterministic parameters, possible states of nature may be predicted, for instance by experts. In our example we assume that there are 5 possible scenarios. Forecasted (step 1) and normalized (step 2, Equation 1) criteria values for each pair (Dj,Si) are given in Table 1. Enterprise “E” declares attribute weights, e.g. w1 = 0.4, w1 = 0.6, and the level of pessimism, e.g. 295 Business, Management and Education, 2014, 12(2): 283–302 α = 0.43 (step 3). Now, one should choose the appropriate procedure: SAPO(CS) or SAPO(SC). Profits and market shares certainly depend on each other, thus one cannot consider scenario evaluations independently, which means that the first approach is cor- rect. We calculate aggregated measures (step 4, Equation 4, Table 2, rows 1–5), which are very similar to formulas applied by (Ginevičius et al. 2013). Then, non-increasing sequences of normalized synthetic values can be generated (step 5, Table 2, row 6). In order to define subsequences of good and bad values (step 6), we have to calculate { }{ }max 1, 5 min 0.43,0.57 3C  = ⋅ =  and set deviation degrees, e.g. d max = d min = 0.3. The use of Equations 5–6 enables us to find the elements of both subsequences (Table 2, rows 7–8). For instance, { }max1 0.600,0.554,0.532SSq = since max1 3SSq C≤ = and ( )0.600 0.3 0.600 0.326 0.600,0.554,0.532 0.600.− ⋅ − ≤ ≤ Table 1. Annual profit (mln euro) and market share (%): initial (akij) and normalized (a(n) k ij) values (Source: created by the author) No Annual profit (C1) Market share (C2) D1 D2 D3 D4 D1 D2 D3 D4 S1 2.5/0.38 4.0/0.84 4.5/1.00 3.0/0.53 20/0.29 22/0.41 15/0.00 21/0.35 S2 1.3/0.00 2.5/0.38 3.5/0.69 3.0/0.53 32/1.00 18/0.18 19/0.24 17/0.12 S3 1.6/0.09 3.0/0.53 4.3/0.94 2.0/0.22 29/0.82 19/0.24 16/0.06 18/0.18 S4 1.7/0.13 3.0/0.53 2.0/0.22 2.5/0.38 28/0.76 15/0.00 23/0.47 24/0.53 S5 1.5/0.06 3.5/0.69 4.2/0.91 4.0/0.84 30/0.88 17/0.12 16/0.06 24/0.53 Table 2. Aggregated measures A(n)ij (Source: created by the author) No D1 D2 D3 D4 S1 0.326 0.585 0.400 0.424 S2 0.600 0.256 0.416 0.283 S3 0.532 0.354 0.410 0.193 S4 0.509 0.213 0.370 0.468 S5 0.554 0.346 0.398 0.655 Sqj 0.60;0.55;0.53;0.51;0.33 0.58;0.35;0.35;0.26;0.21 0.42;0.41;0.40;0.40;0.37 0.66;0.47;0.42;0.28;0.19 SSqmaxj 0.600; 0.554; 0.532 0.585 0.416; 0.410 0.655 SSqminj 0.326 0.256; 0.213 0.370 0.283; 0.193 AVmaxj 0.562 0.585 0.413 0.655 AVminj 0.326 0.234 0.370 0.238 SCSj 0.589 0.414 0.442 0.445 296 H. Gaspars-Wieloch. The use of a modification of the Hurwicz’s decision rule in multicriteria decision... Now, let us compute the averages of good and bad results (step 7, Equations 10–11, Table 2, rows 9–10) and the SAPO(CS) measure (step 8, Equations 12, Table 2, row 11). For instance: 1 5 1 1 5 1 3 0.43 0.326 0.57 0.562 0.589. 5 5 CSS + − − +    = ⋅ ⋅ + ⋅ ⋅ =       According to Equation 13 (step 9) the first marketing strategy (D1) should be selected. The ranking is: D1, D4, D3, D2. Note that the coefficient of pessimism has a significant impact on the final decision. If α = 0.90, the ranking is as follows: D3, D1, D4, D2. Of course, the change of criteria weights also influences the order of activities (when w1 = 0.6, w1 = 0.4, the order is D3, D4, D2, D1 for α = 0.43 and D3, D1, D2, D4 for α = 0.90). 7. Conclusions Many rules for the discrete version of uncertain multicriteria decision making with SP have been already developed. Methods proposed in this paper: SAPO(CS) and SAPO(SC) for dependent and independent criteria matrices respectively – are based on the concept presented in (Ravindran 2008), i.e. they consider the DM’s risk aversion and they refer to SAW and the Hurwicz’s rule. Nevertheless, in the new approaches the additive weighting method is combined with a modification of the Hurwicz’s criterion – the SAPO method, which leads to logic rankings even in problems with asymmetric payoff distributions, because it analyzes the frequency of good and bad results related to particular decisions. SAPO(CS) and SAPO(SC) allow the DM to define precisely his or her preferences concerning the importance of considered goals, the attitude towards risk and the evaluation of extreme results related to particular alternatives. They do not require any information about the probabilities of states of nature, which is certainly a significant advantage. Both procedures can be successfully applied in any business or management multi-objective problem provided that future economic consequences are presented by means of scenario planning which is a comprehensive decision support tool for considering uncertainties. 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