Stochastic multi-criteria decision-making: an overview to methods and applications
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The alternatives selection problem with multi-criteria in stochastic form variables is called as stochastic multi-criteria decision-making. The stochasticity of the criteria is considered using stochastic dominance, prospect theory, and regret theory.
In this paper, a total 61 papers are reviewed and analyzed based on method(s) used in stochastic multi-criteria decision-making problem, method used in stochasticity, specific objective, application area, and so on classification. All papers with respect to classification aspects are examined their real or empirical applications. Moreover, the studies are statistically investigated to present the latest trends of stochastic multi-criteria decision-making.
This detailed review study ensures a comprehension for researchers on stochastic multi-criteria decision-making in respect of showing up-to-date literature and potential research areas to be concentrated in the future. It is observed that the stochastic multi-criteria decision-making problem has an attractive approach by researchers.
KeywordsStochastic decision-making Probability Stochastic dominance Regret theory Prospect theory
Stochastic analytic hierarchy process
Stochastic analytic network process
Stochastic the evaluation based on distance from average solution
Stochastic elimination et choix traduisant la realité
Stochastic multi-criteria decision-making
Stochastic preference ranking organization method for enrichment of evaluations
Stochastic technique for order of preference by similarity to ideal solution
Stochastic Visekriterijumska Optimizacija I Kompromisno Resenje
Summary of MCDM approaches
Analytic hierarchy process
A hierarchical pairwise comparison considering stochastic variables
Analytic network process
Evaluation of the dynamic multi-directional relationship between the decision criteria using stochastic variables
Technique for order of preference by similarity to ideal solution
A MCDM technique based on the concept of choosing the solution with distance from ideal solution considering stochastic variables
Preference ranking organization method for enrichment of evaluations
An outranking method based on a pairwise comparison of alternatives to defined criterion using stochastic variables
Elimination et choix traduisant la realité
An outranking method based on pairwise comparisons to determine the concordance and discordance sets using stochastic variables
Visekriterijumska Optimizacija IKompromisno Resenje
Method for determining the compromise ranking-list of a set of alternatives using stochastic variables
The evaluation based on distance from average solution
It is based on distances of each alternative from the average solution with respect to each criterion
Literature-related SMCDM, which a total of 61 papers, were analyzed ranged from 1996 to December 2018. The main contributions of our paper are summarized as follows: (1) it determines the SMCDM approaches that have been combined with stochastic parameters, (2) it represents method(s) used in SMCDM problem: AHP, TOPSIS, PROMETHEE, ELECTRE, VIKOR, AHP-TOPSIS hybrid methods, ANP, (3) which stochasticity used in SMCDM problems as stochastic dominance (SD) degree, prospect theory (PT), regret theory (RT), and others that have been further used by SMCDM approaches, (4) it shows the countries of the published papers, and (5) the trend of SMCDM is also determined for future studies.
The rest of the paper is given as follows: a summary overview of the fundamentals of SMCDM is given in Sub-section 1. While Section 2 presents the review methodology, the stochastic MCDM methods and applications are analyzed in Section 3. Results and discussions are detailed in Section 4. Lastly, limitations, recommendations, and conclusions are presented for future directions in Section 5.
1.1 The fundamentals of SMCDM
1.2 Regret theory
RT is firstly developed by Bell  and Loomes and Sugden . The RT is a novel significant reasoning method and the preferences are not required to be transitive. Regret theory is a nontransitive model describing preferences by a bivariate utility function. The details of the basic concept of the utility function can be analyzed from the article of [6, 8, 30, 39, 68].
1.3 Prospect theory
The PT is firstly proposed by Kahneman and Tversky . The optimal alternative is selected with respect to the prospect values of all alternatives. It is defined by the value and the probability weight function. The outcome is defined as the gain when the existing wealth surpasses the reference point. On the other hand, the outcome is defined as the loss. The PT underlines the difference between expectation and result, rather than the result itself; hence, the selection of reference point is very important [23, 53].
1.4 Stochastic dominance
Two groups for two classes of utility functions classify the rules of SD . While the first group comprises of first, second, and third-degree stochastic dominance, the second group comprises first-degree stochastic dominance, second inverse stochastic dominance, third inverse SD of the first type and third inverse SD of second type. The first group is utilized in the gains domains, but the second group is used in the losses domain . The description of SD rules can be analyzed in Zhang et al. .
2 Review methodology
Year: publication year;
Journal: journal title;
Country: country where the study was being conducted (In general, country of the first author is considered);
Method(s) used in SMCDM problem: AHP, TOPSIS, PROMETHEE, ELECTRE, VIKOR, AHP-TOPSIS hybrid methods, ANP;
Method used in stochasticity: SD degree, PT, RT, and etc.;
Specific objective: short aim of the study
Application area: applied areas are construction (C), education (ED), energy (EN), environment (ENV), finance (F), healthcare (H), information technology (IT), logistics (L) and manufacturing (M);
Statistical distribution type used in SMCDM problem.
Second, a classification is performed according to applied methods used for SMCDM problem. Ultimately, we analyze the studies by considering statistical results the studies distributions and concluding remarks of future directions.
3 Stochastic MCDM methods and applications
3.1 Stochastic AHP and ANP methods and applications
AHP is based on the hierarchical MCDM problem that comprises attributes, alternatives, and goal. Pairwise comparisons are applied in each hierarchical level with judgments using real values received from the scale of Saaty . In SMCDM knowledge, imprecise preferences of decision-makers must be converted into the stochastic pairwise comparisons . To get crisp values of a stochastic pairwise comparison, the conversion is applied with respect to the probability density functions with related parameters. On the other hand, ANP can be used to model SMCDM problems. It is an appropriate approach for solving decision-making problems with the inclusion of interaction and dependence among criteria and sub-criteria . In SMCDM literature, several papers contributed to both methodologically by proposing stochastic based AHP and its variations and applicably by finding solutions in different areas. The following studies were retrieved in terms of application novelty in SMCDM knowledge using AHP, FAHP, or ANP.
Ramanathan  adapted stochastic programming to multiplicative AHP context. The process of weight derivation using multiplicative AHP was considered. Stochastic goal programming is used for developing to derive the maximum likelihood values of weights. Stam and Silva  proposed two measures of rank reversal probabilities in the AHP resulting from pairwise judgments. Van den Honert  examined the effect of uncertainty in the pairwise judgements or ratings of alternatives as a probability distribution. Cobuloglu and Büyüktahtakın  presented SAHP for biomass selection problem. They used the beta distribution and approximating its median. The logarithmic least squares method is applied to measure the consistency. Ubando et al.  applied SAHP in algal cultivation systems assessment for sustainable production of biofuel. Zhao and Li  proposed a model to assess the performance of strong smart grid based on the SAHP and fuzzy TOPSIS. A sensitivity analysis was also implemented to prove the robustness of the proposed approach as in Ubando et al. . Zhang et al.  presented a stochastic multi-criteria assessment developed by applying the SANP-GCE weight calculation approach. The proposed SANP—game cross-evaluation (GCE) handled the uncertainties and inconsistencies of expert opinions. Finally, the use of ArcGIS helped to visualize vulnerabilities and sensitivities spatially, thus making the decision process more intuitive. Moreover, the criteria weights constituting Nash equilibrium points that determined by GCE improved the objectivity of SANP. Rabelo et al.  used hybridized SD–DES simulation models and AHP for value chain analysis. Banuelas and Antony  applied SAHP for selecting the best suitable technology for the domestic appliance platform. Four design concepts and eight criteria were considered.
Kim et al.  applied SAHP and knowledge-based experience curve (EC) to rank restoration needs. AHP and SAHP are compared for ordering restoration needs of cultural heritage. Minmin and Li  proposed SAHP and fuzzy AHP for credit evaluation. Jing et al. [20, 21] contributed to the SAHP application domains. In the first paper, they incorporated stochastic and fuzzy uncertainty into the traditional AHP as fuzzy SAHP. In the second one, they proposed a hybrid stochastic-interval AHP method to reflect uncertainty by combining lexicographic goal programming, probabilistic distribution, interval judgment, and Monte Carlo simulation.
Apart from application novelties of reviewed SAHP-related papers, some are available in the current knowledge which includes methodological novelties. They are summarized as follows: Phillips-Wren et al.  presented SAHP in the context of a real-time threat criticality detection decision support systems. Hahn  proposed two stochastic formulations of the AHP using Bayesian categorical data. While the first model used a multinomial logit model, the second one used independent multinomial probit model. Eskandari and Rabelo  presented a stochastic approach for calculating the variances of the AHP weights using Monte Carlo simulation. Wanitwattanakosol et al.  used AHP for input feature selection in logistics management. Ramanujan et al.  developed a SAHP approach and implemented it for prioritizing design for environment strategies. Jalao et al.  proposed an AHP model changing stochastic preferences of the decision-maker. AHP with stochastic multi-criteria acceptability analysis (SMAA) is combined by Durbach et al. . The consistency of judgements is analyzed using a simulation experiment.
3.2 Stochastic outranking methods and applications
PROMETHEE method was proposed by Brans et al. . Stochastic PROMETHEE (SPROMETHEE) is a solid member of SMCDM methods. The probability distributions are used for the input parameters instead of real values . In this category, we can also mention ELECTRE and its family with various versions. SMCDM method, which is based on the SD degree using the simple additive weighting method, was proposed by Zhang et al. . PROMETHEE-II was proposed to acquire the alternatives ranking result based on SD degree. Hyde and Maier  presented a stochastic uncertainty and distance-based analysis in Excel using Visual Basic. While Marinoni  proposed SPROMETHEE in GIS, Marinoni  compared the results of a stochastic multivariate PCA and the results of stochastic outranking evaluations. Maciej Nowak  showed how to employ the concept of the threshold in the stochastic case using stochastic dominance. The concept of pseudo-criteria was used. Zaras  suggested an approach using SD for a reduced number of attributes. Rogers and Seager  presented a method based on stochastic multi-attribute life cycle impact assessment. Random variables with probability distributions used the consequence of the alternative according to criteria by Liu et al. [28, 29]. At first, the alternative pairwise comparisons dominance degree matrix according to each criterion was implemented with probability distributions comparison. Then, an overall dominance degree matrix was constructed using PROMETHEE II. Zhou et al.  proposed a gray SMCDM approach based on a combination of SMAA-ELECTRE, with criteria values that extended gray random variables. With this approach, it contributes a new way to solve SMCDM problems with imprecise, uncertain, and/or missing preference information, and also they determine that gray number is a powerful tool to express uncertainty in MCDM problems. Keshavarz Ghorabaee et al.  proposed a stochastic EDAS method using the normal distribution.
3.3 Stochastic dominance-based methods and applications
SD aims to choose the best alternative that dominates another. Some papers on SD-based methods have been proposed. Nowak  combined SD and interactive approach to suggest a new procedure for a discrete SMCDM problem. Nowak  aimed to present how to use the concept of the threshold in the stochastic case. Unlike mean-risk analysis, SD can be implemented into models of preferences versus risks. Zaras  recommended the multi-criteria SD to reduce attributes number. Zaras  made the standardization by the dominance notion extension to evaluate all types (fuzzy or probabilistic, deterministic). Deterministic, stochastic, or fuzzy are examined as three kinds of evaluations that are defined as mixed-data dominances. Zaras  proposed a rough sets methodology for the preferential information analysis. Xiong and Qi  applied interval estimation for converting SMCDM to IMCDM using TOPSIS. Zhang et al.  used a simple additive weighting method in SD degree matrix for PROMETHEE-II. Mousavi et al.  presented a fuzzy-stochastic VIKOR approach. Triangular fuzzy numbers and associated linguistic variables were used in MCDM problem. The performance distribution is generated by applying Monte Carlo simulation. Lastly, VIKOR was implemented to assess probability distributions for each alternative on each criterion. Jiang et al.  used SD rules in the classical TOPSIS method. The probability distributions for both stochastic and discrete variables are defined and determined. Tavana et al.  extended the VIKOR method and improve a methodology to solve problems of MCDM with stochastic data. They presented a case study to evaluate 22 bank branches performance efficiency using SVIKOR. Zhao and Li  proposed fuzzy TOPSIS and stochastic AHP to evaluate the strong smart grid performance. While fuzzy TOPSIS method is applied to evaluate the performance of the smart grid, stochastic AHP method is used to get the sub-criteria weights. Yang and Huang  presented a dynamic stochastic decision-making method. Firstly, the proposed approach obtained time-sequence weights by combining time-degree theory and TOPSIS. Attribute weights were determined based on the characteristics of normally distributed vertical projection distance and stochastic variable variances. Decision-making information is then integrated from time-sequence weights and the attribute via related operators, to obtain the stochastic normally distributed comprehensive decision-making matrix constituted by target single dimensions. Finally, the priority sequence of alternative solutions was provided using order relation criteria. Kolios et al.  proposed stochastic TOPSIS in selecting offshore wind turbines support structures. A TOPSIS-based method considering stochastic inputs (statistical distributions) was proposed for an offshore wind turbine supports the structure selection process. Based on the collected data, a sensitivity analysis was illustrated the required number of simulations for the required accuracy and performed an assessment of the results based on weighting of the respondents’ perceived expertise. Liang et al.  presented a new method based on disappointment SD with respect to the SMCDM problem with criterion 2-tuple aspirations. The overall disappointment SD each alternative degree over the aspiration alternative is calculated to determine the ranking result. Wu et al.  proposed an interval number explanation with the distribution of probability.
3.4 Stochastic regret theory-based methods and applications
RT is a novel significant reasoning method that does not involve preferences to be transitive. It is a nontransitive model to show preferences by a bivariate utility function, which takes the feelings of regret and rejoice into consideration . The number of RT-based methods is scarce and the number of paper should be increased. Zhou et al.  proposed a gray stochastic MCDM approach based on TOPSIS and RT. Discrete and continuous gray numbers were proposed to represent the values of criteria. At first, RT was applied to get the utility and regret value concerning the criteria. Then, the TOPSIS method was applied to rank the alternatives with respect to the overall perceived utility intervals. Two algorithms are proposed which take decision-makers prospect preference and regret aversion by Peng and Yang . The score function based on regret and PT is proposed for two new interval-valued fuzzy soft approaches. A novel interval-valued fuzzy distance measure axiomatic definition is constructed.
3.5 Stochastic prospect theory-based methods and applications
PT assumes that the decision-maker(s) will opt for the optimum alternative with respect to all alternative prospect value. It is decided with probability weight function and the value. Peng and Yang  used PT to calculate score function. Liu et al. [28, 29] developed a MCDM based on PT. It is compared with classical MCDM methods. The result of the proposed method based on PT is compared with expected utility theory. Tan et al.  aimed to develop a new method based on combining PT with stochastic dominance. The proposed approach is compared with other SMCDM methods based on stochastic dominance. Hu and Yang  proposed a dynamic SMCDM based on cumulative PT and set pair analysis. Zhou et al. (2017) proposed a gray SMCDM approach based on distance measures and PT that is integrated with discrete gray numbers. The proposed approach is TODIM that aims to select the best alternative. Gao and Liu  proposed an approach to solving the interval-valued intuitionistic fuzzy SMCDM problem. A new precision score function was suggested based on the hesitation, non-membership, and membership degrees to transform the interval-valued intuitionistic fuzzy number into a computational numerical value. A new criteria weighting model was put forward based on the least square method, the maximizing deviation method, and PT.
Some papers are not compatible with subtitle as RT, SD degree, and etc. Zarghami et al.  presented fuzzy-stochastic MCDM approach by combining the stochastic and fuzzy sets for OWA operator. Random variables with probability mass functions or known probability density functions in SMCDM approach were used by Fan et al. . They applied pairwise comparison for evaluating alternatives with a random variable. They used identification rule, superior, indifferent, and inferior probabilities on pairwise comparison. Ren et al.  proposed a SMCDM approach using differences between the superiorities and the inferiorities. Zarghami and Szidarovszky  presented a new approach fuzzy-stochastic-revised ordered weighted averaging. The stochastic and fuzzy sets are combined in a revised OWA operator. Zarghami and Szidarovszky  proposed stochastic fuzzy ordered weighted averaging approach. Simulation model and fuzzy linguistic quantifiers are applied to the inputs of the approach and obtaining the optimism degree of the decision-maker(s), respectively. Prato  considered probability distributions and the other information required to implement the method for SMCDM method. The method can be applied to order any set of management actions for which the stochastic attributes of outcomes can be is willingly suitable. Wang et al.  proposed gray SMCDM problems with incompletely uncertain criteria weights. An optimal programming model based on the sorting vector closeness degree is constructed. It is solved using a genetic algorithm to get optimum criteria weights when the criteria weights were uncertain.
4 Results and discussions
4.1 Classification of papers
A total of 61 papers on SMCDM approaches were analyzed in this literature review. The majority of the 57(94%) belong to journal articles, a number of 3(5%) are presented at selected congress proceedings, and very few 1(2%) are published as a book chapter.
Statistical probability distributions used in SMCDM studies
Distributions used in SMCDM problem
Xiong and Qi ; Ramanathan ; Peng and Yang ; Tavana et al. ; Eskandari and Rabelo ; Kim et al. ; Szidarovszky and Szidarovszky (2009); Marinoni ; Zhang et al. ; Yang and Huang ; Zhou et al. ; Kolios et al. ; Shengbao and Chaoyuan ; Keshavarz Ghorabaee et al. 
Hyde and Maier ;
Van den Honert ;
Hyde and Maier ;
Others (3-parameter Weibull, Smallest extreme value, Chi-Square, Logbeta, Posterier, Multinomial, PERT, InvGauss, Pearson 5, Gaussian, Dirac’s delta function)
4.2 Discussion and future remarks
In literature, the SD degree proposed in the literature is mostly based on the first-degree SD rule. Hence, the higher-order SD degrees for different risk preference styles are also interesting for further studies. In literature, the researcher mostly presented empirical studies rather than a real case study. Hence, a more real case study should be presented for analyzing the proposed SMCDM approaches. Developing a decision support system and open-access source for the proposed approaches are suggested to analyze and improve the SMCDM. Interval-valued intuitionistic fuzzy set, interval-valued fuzzy soft sets, the trapezoidal fuzzy number, Triangular fuzzy numbers are combined with stochastic MCDM approaches. Interval type-2 fuzzy sets, Pythagorean fuzzy sets, hesitant fuzzy sets, neutrosophic fuzzy sets should be combined with stochastic MCDM approaches. The number of RT-based methods is scarce and the number of paper should be increased.
The importance of the weight for the criteria can be calculated using AHP, ANP, best-worst method, SWARA, SAW, and DEMATEL approaches. While some extension of stochastic AHP and ANP is applied in literature, the extension of the best-worst method, SWARA, SAW, and DEMATEL based on stochasticity should be developed for future studies. On the other hand, the rankings of the alternatives are calculated proposing TOPSIS, VIKOR, PROMETHEE, and ELECTRE using stochasticity as RT, SD, and PT. For further studies, TODIM, COPRAS, GRA, Qualiflex, information axiom, and Choquet integral should be developed. As a conclusion, SMCDM approaches should receive greater attention in the future since they offer better insight into multi-criteria evaluation results .
In this paper, we presented a comprehensive review on SMCDM applications and approaches. SMCDM have increased popularity in MCDM problems in an extensive range of applications and approaches because of its ability to implement higher degrees of ambiguity and uncertainty in recent years. We contribute several standpoints to the literature as follows: (1) SMCDM approaches are determined that have been integrated with stochastic parameters, (2) it represents method(s) used in SMCDM problem: AHP, TOPSIS, PROMETHEE, ELECTRE, VIKOR, AHP-TOPSIS hybrid methods, ANP, (3) which stochasticity used in SMCDM problems as SD degree, PT, RT, and others that have been further used by SMCDM approaches, (4) the countries of the author(s) related published papers are presented, and (5) the trend of SMCDM is determined how it will continue in the future. We observe and expect that the number of SMCDM approaches and applications will increase because of the complexity and advanced degrees of vagueness, ambiguity, and uncertainty in MCDM problems.
EC, MG, MY, and SM analyzed the review, performed the statistical analysis, and wrote the draft paper. All authors contributed equally to all sections of the paper. All authors read and approved the final manuscript.
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The authors declare that they have no competing interests.
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