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Composite Strategy for Multicriteria Ranking/Sorting (methodological issues, examples)

The paper addresses the modular design of composite solving strategies for multicriteria ranking (sorting). Here a 'scale of creativity' that is close to creative levels proposed by Altshuller is used as the reference viewpoint: (i) a basic object, (ii) a selected object, (iii) a modified object, and (iv) a designed object (e.g., composition of object components). These levels maybe used in various parts of decision support systems (DSS) (e.g., information, operations, user). The paper focuses on the more creative above-mentioned level (i.e., composition or combinatorial synthesis) for the operational part (i.e., composite solving strategy). This is important for a search/exploration mode of decision making process with usage of various procedures and techniques and analysis/integration of obtained results. The paper describes methodological issues of decision technology and synthesis of composite strategy for multicriteria ranking. The synthesis of composite strategies is based on 'hierarchical morphological multicriteria design' (HMMD) which is based on selection and combination of design alternatives (DAs) (here: local procedures or techniques) while taking into account their quality and quality of their interconnections (IC). A new version of HMMD with interval multiset estimates for DAs is used. The operational environment of DSS COMBI for multicriteria ranking, consisting of a morphology of local procedures or techniques (as design alternatives DAs), is examined as a basic one.

In the field of algorithms/algorithm systems the following interesting trends can be pointed out: 1. Automatic algorithm design as a combinatorial meta-problem [169]. Here typical entries (for algorithm design) are: (i) choice of problem variables, (ii) choice of constraints, and (iii) choice of search method and constraint behavior. The approach realizes a joint design of problem formulation and algorithm.
In this paper, an operational part of DSS for multicriteria ranking (sorting) and composition of corresponding composite solving strategies is examined. Usually, planning the decision support process is based on the selection and integration of models from a model base. These procedures use special model knowledge including descriptions of basic submodels and their connections, etc. (e.g., [35], [77], [115], [168]). In general, four "creative levels"can be considered: (i) a basic object, (ii) a selected object, (iii) a modified object, and (iv) a designed composite object (as composition of local elements/components). The levels are close to a "scale of creativity" that has been suggested by G.S. Altshuller (e.g., [4]) The levels may be applied to various parts of DSS (information, operations, user). Here, composition of solving strategies for multicriteria ranking (sorting) from some local techniques or procedures is studied. Our morphological composition approach is close to the strategy-generation table suggested by Howard [68]. Note here issues of creativity in decision making are not considered [78].
The morphological approach for the design/planning of composite solving strategies is based on "hierarchical morphological multicriteria design" (HMMD) which involves a series and/or parallel composition of the design alternatives (DAs) for data processing while taking into account interconnections (IC) among DAs ( [98], [99], [102], [105]). In the case of model management/engineering, DAs correspond to alternative models or algorithms/interactive procedures. HMMD is based on the following assumptions: (1) tree-like structure of a designed system; (2) system effectiveness (excellence) is represented as an aggregation of two parts: effectiveness of subsystems (components), and effectiveness of compatibility among subsystems; (3) monotone criteria (Cr) for DAs; (4) ordinal coordinated priorities of DAs (ι = 1, l, 1 corresponds to the best one); and (5) ordinal coordinated estimates of compatibility for each "neighbor" pair of DAs (w = 1, ν, ν corresponds to the best one). In the case of interval multiset estimates (a generalization of ordinal estimates [106]), qualities of DAs and/or their compatibility are evaluated with usage of this kind of estimates (i.e., a special new type of poset-like scales).
Our study is based on the experience in the design and implementation of DSS COMBI for multicriteria ranking (joint project of the author and Andrew A. Michailov; 1984...1991) ( [93], [99], [110]). This DSS was based on a series-parallel transformation of preference relations ( [92], [93], [110]). Operational environment of DSS COMBI includes a morphology of the composite solving strategy for forming/transforming the preference relations, linear orderings, and rankings. DSS COMBI corresponds to DSS generation 4.5 of 5-level classification of Jelassi ([72], [159]). The following is described: (i) methodological issues in decision making technology for multicriteria ranking, (ii) a scheme for designing a series solving strategy.

Decision Making Framework, Solving Scheme, Problems
Generally, decision making process is based on the following basic parts (e.g., [99], [102], [157]): (i) alternatives, (ii) criteria and estimates of the alternatives upon the criteria, (iii) preferences over the alternatives, (iv) solving method(s), (v) decision(s), and (vi) specialists (decision maker, support experts). Fig. 1 depicts a basic framework of the decision making process.   Stage 3. Obtaining the initial information (estimates of the alternatives, preferences over the alternatives).
The following numerical example illustrates the examined multicriteria problem.
} be a set of initial alternatives. The alternatives are evaluated upon 2 criteria K = {K 1 , K 2 }, an ordinal scale is used for each criterion ([0, 1, 2, 3, 4] and 4 corresponds to the best level). The ordinal estimates of alternatives are presented in Table 1, a space of estimates is depicted in Fig. 7. Three types of solutions are presented: (a) linear ordering B (Fig. 8; e.g., additive utility function is used); (b) ranking (sorting, four linear ordered subsets) B ( Fig. 9; e.g., expert judgement is used); and (c) "fuzzy" ranking B ( Fig. 10; e.g., expert judgement is used). Table 2 integrates the obtained priorities of alternatives. Table 1. Estimates Alternative An approach for decision making as series or series-parallel processing (transformation, articulation) of preferences (combinatorial models for decision making) for suggested in [91]. In DSS COMBI, a functional graph has been suggested as impementation of the approach for multicriteria ranking ( [93], [99], [110]). The graph was realized as a graphical menu. In this case, the solving strategy is combined from a set of basic operations (local techniques or procedures) (e.g., forming preference relations over the alternatives, forming the intermediate linear ordering of the alternatives, forming the resultant decision structure over the alternatives) (Fig. 11).
Alternative Linear ordering

Towards Reconfigurable Problem Solving Framework
Here a generalized problem solving framework is examined. An algorithm system (e.g., for data processing) is considered as a basic example.
The basic parts of the framework are the following: (a) problem (problem structuring/formulation) (b) algorithm (operational part) (c) data part (obtaining data, elicitation of preferences, ...) (d) human part (preliminary learning, learning based on solving process) First, a statical structure of the algorithm system is analyzed. The following versions the algorithm/algorithm systems can be considered: 1. usage of the basic algorithm(s) (Fig. 12); 2. selection of the best algorithm from an algorithm base (while taking int account an input information) and its usage (Fig. 13); 3. modification of the basic algorithm(s) (while taking int account an input information) and its usage (Fig. 14); and 4. design of the new algorithm(s) (while taking into account an input information) and its usage (Fig.  15).
A simplified hierarchy of solving process components is depicted in Fig. 16.
In general, the following problem solving frameworks can be considered: 1. One-phase framework: problem solving.

Problem reformulation/ restructuring
The following three illustrative examples of multistage solving strategies are presented for set consisting of nine initial alternatives (Table 1): (i) two-stage series ranking strategy with preliminary linear ordering (Fig. 17); (ii) two-stage series ranking strategy with preliminary preferences (Fig. 18); (iii) three-stage series ranking strategy (Fig. 19); and (iv) three-stage series-parallel strategy (with aggregation) (Fig.  20). Fig. 21 depicts adaptive three-stage framework for problem solving. In addition, it is possible to consider a solving framework in which problem formulation/structuring is executed during the solving process for a preliminary problem.
The stage "Reconfiguration of problem solving process" can be considered as the following:: (i) selection of another algorithm (solving scheme) (Fig. 13), (ii) modification of the algorithm (solving scheme) (Fig.  14), (iii) design of a new algorithm (solving scheme) (Fig. 15). Concurrently, information operations (searching/acquisition/usage) can be used (e.g., new/additional preferences, new/additional reference points of intermediate decisions).
In the case of a man-machine procedure, it is possible to examine change or re-organization of an expert team and types (e.g., mode, support procedure(s)) of man-machine interaction.
The stage "Problem reformulation/restructuring" can be considered as the following:: (i) change of the initial problem, (ii) modification of the initial problem (e.g., change of the problem parameters, initial data), (iii) building of a new problem framework (i.e., a composite problem). Fig. 22 depicts an example to illustrate a series process with problem reformulation. Fig. 17. Two-stage series ranking strategy (with linear ordering) Fig. 18. Two-stage series ranking strategy (with preferences) Fig. 19. Three-stage series ranking strategy Initial set of alternatives Fig. 20. Three-stage series-parallel ranking strategy (with aggregation) Initial set of alternatives

History of DSS COMBI
A preliminary version of DSS COMBI was implemented as a set of multicriteria techniques (Fortran, mainframe, methods: several types of utility functions, Electre-like technique). Analysis, comparison and aggregation of results, obtained via different techniques, was widely used. Further, DSS with method composition was designed. DSS COMBI was targeted to three type of resultant decisions (Fig. 23) Table 3 presents a brief description of DSS COMBI generations with method composition ( [93], [99], [110]). A functional graph menu was realized in DSS COMBI (since generation 1) (Fig. 24).

Main Components of DSS
Here, the following DSS parts are considered: 1. information part as data, knowledge; 2. operational part as tools for information processing); and 3. human part as user or group of users (including experts, decision maker, etc.).
Usually information part includes the following: (1) data (alternatives or basic items, criteria, multicriteria estimates of alternatives upon criteria, preference relations); (2) tools for maintaining data (DBMS and interfaces with other commercial DBMSs); (3) support information for learning (e.g., a helper, etc.).
In this paper (as in DSS COMBI), the following series (or series-parallel) framework of information (a preference relation or a matrix) processing is examined ( [93], [99], [110]): (1) basic data as alternatives (items) (A), criteria (K), multicriteria estimates of alternatives upon criteria (Z); (2) preference relation of alternatives (G); (3) an intermediate linear ordering of alternatives (B) (4) an preliminary ranking of alternatives as a layered structure(s) (B ′ ); and (5) resultant ranking of alternatives as layered structure (B) (an ordinal priority for each alternative) or fuzzy ranking ( B) (with intersection of layers).
The following kinds of basic operations are considered: (i) data processing (series and/or parallel); (ii) data aggregation. In addition, parallelization of the solving process on the basis of various components (alternatives, criteria, techniques, experts) can be used. The solving process may be presented as a hierarchy with the following functional/operatiopnal layers ( [93], [99], [110]): (1) algorithms and man-machine interactive procedures for data transformation (bottom layer); (2) strategies (step-by-step schemes of data transformation, particularly series-parallel ones); (3) scenarios (complexes of strategies with their analysis and feedback).
Thus, the following alternative series frameworks (composite solving strategies) of information processing are considered (Fig. 11, Fig. 24): 1. Basic series framework: 2. Compressed series frameworks: 3. Direct solving process: D : A ⇒ B or B. Thus, the solving process can be considered as the following four-part system (Fig. 25): S = H ⋆ T ⋆ U ⋆ X, where the following four stages are basic ones: (i) stage H corresponds to forming a preliminary preference relation G (over alternatives A) (an algorithm or a procedure); (ii) stage T corresponds to forming a preliminary linear ranking B (an algorithm or a procedure); (iii) stage U corresponds to forming some preliminary rankings {B} (an algorithm or a procedure); and (iv) stage X corresponds to aggregation of the preliminary rankings {B} into the resultant decisions B (or B) (an algorithm or a procedure). The following basic local techniques (as processing units) have been used in DSS COMBI ( [93], [99], [110]): I. Stage H: absent (H 0 ), pairwise comparison (a simple expert-based procedure) (H 1 ), dominance by of Pareto-rule (H 2 ) (e.g., [124], [132]), outranking Electre-like technique (a special Electre-based interactive procedure with feedback, designed by M.Sh. Levin [110]) (H 3 ).
Here the index 0 corresponds to absence of the precessing at the stage. Thus, the following examples can be considered (symbol & corresponds to parallel integration of processing operations): (1) a series solving strategy S 1 = H 1 ⋆ T 2 ⋆ U 2 ⋆ X 0 ; (2) a series-parallel solving strategy: Parallelization of the solving process is based on three approaches: (1) concurrent usage of different experts in the same interactive procedure (e.g., in U 4 ); (2) concurrent usage of different methods at the same solving stage (e.g., H 3 ); (3) concurrent usage of the same method with different parameters at the same solving stage (e.g., in U 4 ).
The design of solving strategies consists of two problems [99]: (a) selection of local techniques and (b) combinatorial synthesis of the selected techniques. The requirements to the composite solving strategies are based on the following: (i) kinds of task and of users; (ii) available resources (e.g., human, computer, time); (iii) features of the decision situation (e.g., kind of uncertainty, required precision and robustness of result). Thus, the following six criteria are examined [99]: (1) required computer resources; (2) required human resources; (3) quality of ranking (robustness, etc.); (4) possibility for data representation; (5) possibility for an analysis of intermediate data; and (6) usability (easy to learn, understanding, etc.).
Note, some traditional approaches to build a solving process for decision making are oriented to the selection of the best method (or algorithm, model) (e.g., [8], [61], [131], [156]). A model composition on the basis of a filter space is described in [25]. A non-linear recursive process consisting of four steps is analyzed for multicriteria decision aid in [61].

Combinatorial Synthesis with Interval Multiset estimates
Interval multiset estimates have been suggested by M.Sh. Levin in [106]. The approach consists in assignment of elements ( . This number corresponds to possible estimates (without taking into account interval condition 2). In the case of condition 2, the number of estimates is decreased. Generally, assessment problems based on interval multiset estimates can be denoted as follows: P l,η . A poset-like scale of interval multiset estimates for assessment problem P 3,4 is presented in Fig. 26. The assessment problem will be used in our applied numerical examples.
In addition, operations over multiset estimates are used [106]: integration, vector-like proximity, aggregation, and alignment.
The next version of combinatorial synthesis problem will be the following. Let w(i, j) will be an interval multiset estimate for compatibility between two selected design alternatives S i and S j in solution S (the same type of interval multiset estimates is used for the elements and for their compatibility). Then, the problem is: Here w 0 is a constraint (a bottom bound) for compatibility. Evidently, enumeration methods or heuristics can be used here as well.

Example of Composite Strategy
The four-stage (four-part)) morphological scheme of solving strategy for multicriteria ranking (sorting) to obtain the layered structure B is presented in Fig. 24: S = H ⋆T ⋆U ⋆X. Generally, design alternatives (local techniques, DAs) are evaluated upon six criteria [99] and the vector estimates are transformed into interval multiset estimates. In the example, expert judgment was used and illustrative estimates are presented in Table 4 (Fig. 27). Note, the estimates of DAs correspond to a certain applied situation, e.g., professional level of expert(s).
Here the design of series solving strategy is based on the compressed morphology (without aggregation, i.e., without local techniques X 1 and X 2 ) (Fig. 27). Table 5 contains estimates of compatibility.

CONCLUSION
In the paper, a modular approach to solving strategies in DSS for multicriteria ranking (sorting) has been described. First, methodological issues in architectural design of DSS are examined (decision making framework, typical problems, approaches to configuration of solving strategies). Second, combinatorial synthesis for design of series composite strategy for multicriteria ranking (sorting)) is presented. The study is based on DSS COMBI for multicriteria ranking (1984...1991). Evidently, an important and prospective direction consists in aggregation (fusion) of preference relations which are obtained from different sources (experts, algorithm / procedures). This leads to series-parallel solving strategies (e.g., [99]). Here it may be reasonable to take into account the following two research directions: (1) routing in And-Or graphs (e.g., [1], [34]); (2) activity nets [40]). In addition, it is reasonable to point out the significance of an extended composite solving framework that involves a preliminary stage for problem formulation, a final stage for analysis of results, and on-line monitoring of the solving process (Fig. 28). Here, the same design approaches may be applied.  Finally, it is reasonable to emphasize the significance of our approach for teaching of multicriteria analysis and system design (techniques, case studies, applied examples, projects) (e.g., [95], [100], [103], [104]).