A fuzzy MCDM approach for stock selection

Abstract

A fuzzy MCDM approach is applied to the stock selection problem, where the proposed approach can deal with qualitative information in addition to quantitative information. A hierarchy of major–sub criteria is then established to reduce the dependence between criteria. The ratings of alternatives versus qualitative sub-criteria and the weights of major- and sub-criteria are assessed in linguistic terms represented by fuzzy numbers. Each sub-criterion is in a benefit, cost, or balanced nature. New standardization methods for fuzzy numbers in the cost and balanced nature are presented. The algorithms of membership functions of the final aggregation are completely developed instead of approximation. The final aggregations in fuzzy numbers are then defuzzified to crisp values in order to rank the performance of alternatives. Moreover, the ratio of market price to performance (PP) is suggested to filter the over/under-pricing of alternatives. A set of buying/selling strategies are recommended according to the performance and PP. An empirical example then demonstrates the processing of the proposed approach.

Appendix

Appendix presents the proofs of Property 1–Property 4.

Proof of Property 1

• For x _i1k ′∈R, let x _i1k ′=μ _1k ′+ɛσ _1k ′, ɛ∈R. If R is cut into three segments, ɛ is in one of the situations:

  • 1) ɛ⩽−2

  • 2) −2<ɛ<2 (or −2<ɛ⩽0, 0⩽ɛ<2)

  • 3) ɛ⩾2

  It is to prove that 0⩽x _i1k ⩽1 for x _i1k ′∈{B, C, M}.Using Equation (10) for x _i1k ′∈B:

  When ɛ⩽−2, μ _1k ′+ɛσ _1k ′⩽μ _1k ′−2σ _1k ′ → x _i1k =0.

  When −2<ɛ<2,

  

  When ɛ⩾2, μ _1k ′+ɛσ _1k ′⩾μ _1k ′+2σ _1k ′ → x _i1k =1.Using Equation (11) for x _i1k ′∈C:

  When ɛ⩽−2, μ _1k ′+ɛσ _1k ′⩽μ _1k ′−2σ _1k ′ → x _i1k =1.

  When −2<ɛ<2,

  

  When ɛ⩾2, μ _1k ′+ɛσ ₁₁′⩾μ _1k ′+2σ _1k ′ → x _i1k =0.Using Equation (12) for x _i1k ′∈M:

  When ɛ⩽−2, μ _1k ′+ɛσ _1k ′⩽μ _1k ′−2σ _1k ′ → x _i1k =0.

  When −2<ɛ⩽0,

  

  When 0⩽ɛ<2,

  

  When ɛ⩾2, μ _1k ′+ɛσ _1k ′⩾μ _1k ′+2σ _1k ′ → x _i1k =0.

Proof of Property 2

• Suppose A _i ′=(o _i ′, p _i ′, q _i ′, r _i ′)∈C, where i=1∼n: n is the number of fuzzy numbers. Using Equation (15), the transformed numbers A _i =(o _i , p _i , q _i , r _i ).

  (1) It is to prove

  (r ₁−o ₁):(r ₂−o ₂): … :(r _n −o _n )=(r ₁′−o ₁′):(r ₂′−o ₂′): … :(r _n ′−o _n ′) if the relative dispersions of the transformed numbers are equal to those of the original numbers:

  

  (2) It is to prove

  (p _i −o _i ):(q _i −p _i ):(r _i −q _i )=(r _i ′−q _i ′):(q _i ′−p _i ′):(p _i ′−o _i ′) if the transformed shapes are a reverse to the original shapes:

  

Proof of Property 3

• Let A ₁′=(o ₁′, p ₁′, q ₁′, r ₁′), A ₂′=(o ₂′, p ₂′, q ₂′, r ₂′), A ₁′, A ₂′⊂A _i ′, and A _i ′=(o _i ′, p _i ′, q _i ′, r _i ′)∈M. Using Equation (16) A ₁′ and A ₂′ are standardized to be A ₁=(o ₁, p ₁, q ₁, r ₁) and A ₂=(o ₂, p ₂, q ₂, r ₂). It is to prove (1) if A ₁′ is closer to max _i r′/2 than A ₂′, then A ₁>A ₂. 2. A _i ∈[0, 1].

  (1) If A ₁′ is closer to max _i r′ _i /2 than A ₂′ then the distance from A ₁′, to max _i r′ _i /2 is shorter than that from A ₂′ to max _i r′ _i /2:

  

  Multiply the above equation by 2/max _i r′ _i , where max _i r′ _i >0,

  

  (2)

  

Proof of Property 4

• • 1) For G _i =[Q _i , R _i , Y _i , Z _i ],   (x=Q _i )=0, (x=R _i )=1, (x=Y _i )=1, and (x=Z _i )=0.

  • 2) (x) is strictly increasing on x∈[Q _i , R _i ] and (x) is strictly decreasing on x∈[Z _i , Y _i ].

  (1.1) To prove (x=Q _i )=A _1i ^1/3+B _1i ^1/3−F _1i /3E _1i =0 is to prove A _1i ^1/3+B _1i ^1/3=F _1i /3E _1i , when x=Q _i .

  

  Solving the above equation, the real-number root of (A _1i ^1/3+B _1i ^1/3) is F _1i /3E _1i .

  

  (1.2) To prove (x=R ⁱ)=A _1i ^1/3+B _1i ^1/3−F _1i /3E _1i =1 is to prove A _1i ^1/3+B _1i ^1/3=1+F _1i /3E _1i , when x=R _i .

  

  Solving the above equation, the real-number root of (A _1i ^1/3+B _1i ^1/3) is (1+F _1i /3E _1i ).

  

  (1.3) It can be proven that (x=Y _I)=1 and (x=Z _i )=0 by the same logic. The proofs are omitted.

  (2) Suppose Q _i <x _1i <x _2i <R _i and Y _i <x _3i <x _4i <Z _i . To prove strictly increasing is to prove 0=(Q _i )<(x _1i )<(x _2i )<(R _i )=1. To prove strictly decreasing is to prove 1=(Y _i )>(x _3i )>(x _4i )>(Z _i )=0.

  

  It can be proven that 1=(Y _i )>(x _3i )>(x _4i )>(Z _i )=0 by the same logic. The proofs are omitted.