^{1}

The Pareto optimality is a widely used concept for the multicriteria decision-making problems. However, this concept has a significant drawback—the set of Pareto optimal alternatives usually is large. Correspondingly, the problem of choosing a specific Pareto optimal alternative for the decision implementation is arising. This study proposes a new approach to select an “appropriate” alternative from the set of Pareto optimal alternatives. The proposed approach is based on ranking-theory methods used for ranking participants in sports tournaments. In the framework of the proposed approach, we build a special score matrix for a given multicriteria problem, which allows the use of the mentioned ranking methods and to choose the corresponding best-ranked alternative from the Pareto set as a solution of the problem. The proposed approach is particularly useful when no decision-making authority is available, or when the relative importance of various criteria has not been evaluated previously. The proposed approach is tested on an example of a materials-selection problem for a sailboat mast.

This paper considers a novel approach for solving a multicriteria decision-making (MCDM) problem, with a finite number of decision alternatives and criteria. The multicriteria formulation is the typical starting point for theoretical and practical analyses of decision-making problems. Thus, the definition of Pareto optimality and a vast arsenal of different Pareto optimization methods can be used for decision-making purpose.

However, unlike single-objective optimizations, a characteristic feature of Pareto optimality is that the set of Pareto optimal alternatives (i.e., set of efficient alternatives) is usually large. In addition, all these Pareto optimal alternatives must be considered as mathematically equal. Correspondingly, the problem of choosing a specific Pareto optimal alternative for implementation arises, because the final decision usually must be unique. Thus, additional factors must be considered to aid a decision-maker the selection of specific or more-favorable alternatives from the set of Pareto optimal solutions.

The proposed approach is based on ranking-theory methods that used to rank participants in sports tournaments. In the framework of the proposed approach, we build a special score matrix for a given multicriteria problem, which allows us to use the mentioned ranking methods and choose the corresponding best-ranked alternative from the Pareto set as a solution of the problem. Note that the score matrix is built by the quite natural way—it is composed on the simple calculations of how many times one alternative is better than the other for each of the criteria. Hence, there is hope that the proposed approach yields a “notionally objective” ranking method and provides an “accurate ranking” of the alternatives for MCDM. The proposed approach is particularly useful when no decision-making authority is available, or when the relative importance of various criteria has not been evaluated previously.

To demonstrate viability and suitability for applications, the proposed approach illustrated using an example of a materials-selection problem for a sailboat mast. This problem has been addressed by several researchers using various methods and, thus, can be considered as a kind of a benchmark problem. This illustration sheds light on the ranking approach’s applicability to the MCDM problems. Particularly, it is shown that the solutions of the illustrative example obtained by the proposed approach are quite competitive.

The rest of this paper is structured as follows. In Section

In what follows, for a natural number

The following notation is drawn from a general treatment of multicriteria optimization theory [

We say furthermore that

Pareto optimality is an appropriate concept for the solutions of MCDM problems. In general, however, the set

This section gives a brief overview of the basic concepts of ranking theory. References [

For natural

Assume now that

The following procedure is used for solving MCDM problem

For the MCDM problem

Using the score matrix

The alternative from the Pareto set,

Obviously, it would suffice to rank the Pareto set if Pareto set is known at the beginning of the proposed procedure. Nevertheless, we prefer given above description because it is more convenient in the cases when Pareto set is not known (or partially/approximately known), as it took place usually for the complex MCDM problems.

It is clear that, instead of the MCDM problem

This section discusses the example problem that was solved to demonstrate the practicality of the proposed in Section

The component to be optimized, the mast, is modeled as a hollow cylinder that is subjected to axial compression. It has a length of 1,000 mm, an outer diameter ≤ 100 mm, an inner diameter ≥ 84 mm, a mass ≤ 3 kg, and a total axial compressive force of 153 kN [

The following methods were used to solve the problem by previous investigators: WPM (weighted-properties method), VIKOR (multicriteria optimization through the concept of a compromise solution), CVIKOR (comprehensive VIKOR), FLA (fuzzy-logic approach), MOORA (multiobjective optimization based on ratio analysis), MULTIMOORA (a multiplicative form of MOORA), RPA (the reference-point approach), and a recently proposed game-theoretic method GTM [

Direct calculations show that the score matrix

Materials ranked by proposed methods.

Material | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

| | | | | ||||||

Rating | Rank | Rating | Rank | Rating | Rank | Rating | Rank | Rating | Rank | |

1 | 0,3529 | 14 | 0,1666 | 14 | 0,8752 | 14 | 0,0335 | 14 | -3,408 | 14 |

| | | | | | | | | | |

| | | | | | | | | | |

| | | | | | | | | | |

5 | 0,2340 | 15 | 0,1164 | 15 | 0,7342 | 15 | 0,0201 | 15 | -4,072 | 15 |

6 | 0,5870 | 7 | 0,2716 | 6 | 1,0694 | 7 | 0,0822 | 6 | -2,532 | 7 |

| | | | | | | | | | |

8 | 0,3673 | 13 | 0,1767 | 13 | 0,8888 | 13 | 0,0371 | 13 | -3,349 | 13 |

| | | | | | | | | | |

10 | 0,4082 | 11 | 0,1931 | 10 | 0,9288 | 10 | 0,0425 | 10 | -3,167 | 11 |

| | | | | | | | | | |

| | | | | | | | | | |

| | | | | | | | | | |

| | | | | | | | | | |

| | | | | | | | | | |

Note: italic corresponds to the Pareto optimal (efficient) alternatives.

Table

For comparison, Table

Correlation between methods.

^{ S } | ^{ N } | ^{ B } | ^{ fb } | ^{ ml } | |
---|---|---|---|---|---|

MOORA | 0,564286 | 0,603571 | 0,578571 | 0,603571 | 0,564286 |

MULTIMOORA | 0,496429 | 0,503571 | 0,521429 | 0,503571 | 0,496429 |

RPA | 0,467857 | 0,492857 | 0,485714 | 0,492857 | 0,467857 |

FLA | 0,764286 | 0,717857 | 0,792857 | 0,717857 | 0,764286 |

Wpm | 0,403571 | 0,410714 | 0,442857 | 0,410714 | 0,403571 |

CVIKOR | 0,742857 | 0,646429 | 0,739286 | 0,646429 | 0,742857 |

VIKOR | 0,892857 | 0,871429 | 0,907143 | 0,871429 | 0,892857 |

GTM | -0,12143 | -0,07857 | -0,09286 | -0,07857 | -0,12143 |

Sources

Decision matrix for selecting material for a sailing boat mast.

# | Material | Criteria | |||
---|---|---|---|---|---|

Specific strength (MPa) | Specific modulus (GPa) | Corrosion resistance | Cost Category | ||

SS | SM | CR | CC | ||

1 | 2 | 3 | 4 | ||

1 | AISI 1020 | 35.9 | 26.9 | 1 | 5 |

2 | AISI 1040 | 51.3 | 26.9 | 1 | 5 |

3 | ASTM A242 type 1 | 42.3 | 27.2 | 1 | 5 |

4 | AISI 4130 | 194.9 | 27.2 | 4 | 3 |

5 | AISI 316 | 25.6 | 25.1 | 4 | 3 |

6 | AISI 416 heat treated | 57.1 | 28.1 | 4 | 3 |

7 | AISI 431 heat treated | 71.4 | 28.1 | 4 | 3 |

8 | AA 6061 T6 | 101.9 | 25.8 | 3 | 4 |

9 | AA 2024 T6 | 141.9 | 26.1 | 3 | 4 |

10 | AA 2014 T6 | 148.2 | 25.8 | 3 | 4 |

11 | AA 7075 T6 | 180.4 | 25.9 | 3 | 4 |

12 | Ti–6Al–4V | 208.7 | 27.6 | 5 | 1 |

13 | Epoxy–70% glass fabric | 604.8 | 28.0 | 4 | 2 |

14 | Epoxy–63% carbon fabric | 416.2 | 66.5 | 4 | 1 |

15 | Epoxy–62% aramid fabric | 637.7 | 27.5 | 4 | 1 |

Source: [

CC scale: 1 = very high; 2 = high; 3 = moderate; 4 = low; 5 = very low.

Normalized decision matrix for the material selection problem.

Criteria | |||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||

Materials | 1 | 0.9832 | 0.9565 | 1.0000 | 0.0000 |

| 0.9580 | 0.9565 | 1.0000 | 0.0000 | |

| 0.9727 | 0.9493 | 1.0000 | 0.0000 | |

| 0.7234 | 0.9493 | 0.2500 | 0.5000 | |

5 | 1.0000 | 1.0000 | 0.2500 | 0.5000 | |

6 | 0.9485 | 0.9275 | 0.2500 | 0.5000 | |

| 0.9252 | 0.9275 | 0.2500 | 0.5000 | |

8 | 0.8753 | 0.9831 | 0.5000 | 0.2500 | |

| 0.8100 | 0.9758 | 0.5000 | 0.2500 | |

10 | 0.7997 | 0.9831 | 0.5000 | 0.2500 | |

| 0.7471 | 0.9807 | 0.5000 | 0.2500 | |

| 0.7009 | 0.9396 | 0.0000 | 1.0000 | |

| 0.0537 | 0.9300 | 0.2500 | 0.7500 | |

| 0.3619 | 0.0000 | 0.2500 | 1.0000 | |

| 0.0000 | 0.9420 | 0.2500 | 1.0000 |

Note: italic denotes Pareto optimal (efficient) alternatives.

Materials ranked by comparable methods.

Material | MOORA | MULTIMOORA | RPA | FLA | Wpm | CVIKOR | VIKOR | GTM |
---|---|---|---|---|---|---|---|---|

1 | 14 | 14 | 14 | 14 | 14 | 12 | 14 | 14 |

2 | 15 | 15 | 13 | 13 | 13 | 6 | 11 | 10 |

3 | 13 | 13 | 12 | 15 | 15 | 9 | 13 | 11 |

4 | 12 | 12 | 15 | 4 | 11 | 4 | 4 | 2 |

5 | 4 | 4 | 4 | 11 | 10 | 15 | 15 | 9 |

6 | 7 | 11 | 11 | 9 | 9 | 14 | 10 | 8 |

7 | 6 | 10 | 10 | 10 | 8 | 11 | 5 | 7 |

8 | 11 | 9 | 9 | 8 | 7 | 13 | 12 | 5 |

9 | 10 | 7 | 8 | 12 | 2 | 8 | 7 | 4 |

10 | 9 | 6 | 7 | 7 | 4 | 10 | 9 | 3 |

11 | 5 | 8 | 6 | 6 | 6 | 5 | 6 | 1 |

12 | 8 | 5 | 2 | 5 | 3 | 7 | 8 | 12 |

13 | 2 | 2 | 3 | 3 | 12 | 2 | 2 | 6 |

14 | 3 | 3 | 1 | 2 | 1 | 1 | 1 | 15 |

15 | 1 | 1 | 5 | 1 | 5 | 3 | 3 | 13 |

Sources:

In this study, we have proposed a new approach for solving MCDM problems. The proposed approach is based on ranking-theory methods which are used in the competitive sports tournaments. In the framework of the proposed approach, we build a special score matrix for a given multicriteria problem, which allows us to use an appropriate ranking method and choose the corresponding best-ranked alternative from the Pareto set as a solution of the MCDM problem. The proposed approach is particularly useful when no decision-making authority is available, or when the relative importance of various criteria has not been evaluated previously.

To demonstrate the viability and suitability for applications, the proposed approach illustrated using an example of a materials-selection problem. It is shown that the solutions of the illustrative example obtained by the proposed approach are quite competitive. Note also that the proposed approach seems numerically efficient. Namely, our preliminary numerical experiments (unpublished) show that that MCDM problems with the number of alternatives of the order of 1.5 hundred and with the number of criteria of the order of ten can be solved by the proposed method in a few minutes (~5 min, the calculations were conducted on a laptop with 2.59GHz, 8GB RAM, 64-bit operation system, MATLAB environment, and not making any effort to optimize the code).

Due to the simplicity and flexibility of the implementation, the proposed approach can be also used in a few interesting directions. For example, if we consider the “transposed” MCDM problem (i.e., the problem, for which the criteria of the original problem are alternatives and the alternatives of the original problem are criteria), the proposed approach also allows ranking the criteria and identified a “leading criterion”. On the other hand, an “objective” ranking of the criteria may stimulate the development of other instruments for the Pareto optimization. It also seems possible that the proposed approach will find applications in the (e.g., evolutionary) Pareto optimization algorithms. However, we will limit ourselves here only to mention these directions for further investigations.

See Tables

Previously reported data were used to support this study. These prior studies are cited at relevant places within the text as references.

The author declares that he has no conflicts of interest.