T.C.
GAZOSMANPAA ÜNVERSTES Bilimsel Ara³trma Projeleri Komisyonu
SONUÇ RAPORU
Proje No:2010/89
BULANIK ESNEK MATRSLER
Proje Yöneticisi Doç. Dr. Naim ÇAMAN Fen Ed. Fak. Matematik Bölümü
Serdar ENGNOLU Fen Bil. Enstitüsü
BULANIK ESNEK MATRSLER*
Bu çal³mada, bulank esnek (be) matrisler ve i³lemleri tanmland. Daha sonra,
be-matrislerin çarpmlar tanmland. Son olarak, be-karar verme metodu in³a edildi ve bir tür belirsizlik problemi üzerine uyguland.
Anahtar kelimeler: Bulank Esnek Kümeler, Bulank Esnek Matris, Bulank Esnek Matrislerin Çarpm, Bulank Esnek Max-Min Karar Verme.
(*) Bu çal³ma Gaziosmanpa³a Üniversitesi Bilimsel Ara³trma Projeleri Komisyonu tarafndan desteklenmi³tir. (Proje No: 2010/89).
ABSTRACT
FUZZY SOFT MATRICES
In this work, we de ne fuzzy soft (fs) matrices and their operations. We then de ne products of fs-matrices and their properties. We nally construct a fs- decision making method and apply to a problem that contain uncertainties.
Key words: Fuzzy Soft Sets, Fuzzy Soft Matrix, Product of Fuzzy Soft Matrices, Fuzzy Soft Max-Min Decision Making.
olarak desteklenmi³tir. Gaziosmanpa³a Üniversitesine ve tüm BAP çal³anlarna verdikleri fevkalade destekten dolay te³ekkür ederim.
Doç. Dr. Naim ÇAMAN Proje Yürütücüsü
ÇNDEKLER ÖZET . . . i ABSTRACT . . . ii TEEKKÜR . . . iii 1. GR . . . 1 2. TEMEL KAVRAMLAR . . . 2
2.1 Bulank Esnek Kümeler . . . 2
3. BULANIK ESNEK MATRSLER VE LEMLER . . . 6
3.1 Bulank Esnek Matris . . . 6
3.2 Bulank Esnek Matris ³lemleri . . . 9
3.3 be-Matris Çarpmlar . . . 12
4. BULANIK ESNEK KARAR VERME VE UYGULAMALARI . . . 14
4.1 Bulank Esnek mak-min Karar Verme . . . 14
4.2 Uygulama . . . 16
5. SONUÇ . . . 19
KAYNAKLAR . . . 20
EK: FUZZY SOFT MATRIX THEORY AND ITS APPLICATION IN DECISION MAKING . . . 21
Esnek küme teorisi; belirsiz, bulank ve açkça tanmlanmayan kavramlarla ba³a çkabilmek için, Molodtsov tarafndan ortaya atld (Molodtsov,1999). Daha sonra, bu teori üzerine baz zorluklara sahip olan bir takm çal³malar yapld. Akabinde, esnek küme i³lemleri yeniden tanmland (Ça§man ve Engino§lu, 2010a) ve bu i³lemler kullanlarak bir-kes karar verme metodu in³a edildi. Ardndan, esnek küme i³lemlerinin bilgisayarda kolayca hesaplanabilmesi için esnek matris teorisi ortaya atld ve esnek mak-min karar verme metodu in³a edildi (Ça§man ve Engino§lu, 2010b). Son zamanlarda, bulank parametreli bulank esnek (bpbe) kümeler, bulank esnek (be) matrisler, be-kümeler ve bulank parametreli esnek (bpe) kümeler üzerine çal³malar yapld (Ça§man ve ark, 2010;Ça§man ve Engino§lu, 2011; Ça§man ve ark, 2011a,2011b).
Bu çal³mada, bulank esnek (be) matrisler ve i³lemleri tanmland. Daha sonra,
be-matrislerin çarpmlar tanmland. Son olarak, be-karar verme metodu in³a edildi ve bir tür belirsizlik problemi olan karar verme problemi üzerine uyguland.
2.TEMEL KAVRAMLAR
2.1 Bulank Esnek Kümeler
Bu bölümde, be-kümeler ve i³lemleri tanld ve örneklendirildi. Esnek kümelerde, fonksiyon de§erleri klasik küme iken be-kümelerde fonksiyon de§erleri bulank kümedir. Tanm 2.1.1. U gözönüne alnan elemanlarn kümesi, E bu elemanlar niteleyen parametrelerin kümesi, A ⊆ E ve F (U), U üzerindeki tüm bulank kümelerin kümesi olsun. x /∈ A için γA(x) = ∅biçiminde tanmlanan γA : E → F (U) fonksiyonuna U
üzerinde bir be-küme denir ve γAile gösterilir.
Burada, γA(x)bulank kümesi her x ∈ E için
γA(x) = {µγA(x)(u)/u : u ∈ U, µγA(x)(u) ∈ [0, 1]}
biçiminde tanmldr. Bir be-küme
γA= {(x, γA(x)) : x ∈ E},
biçiminde sral ikililerin bir kümesi olarak gösterilebilir.
Bu çal³mada, U üzerinde E ile tanml tüm be-kümelerin kümesini BE(U) ile gösterilecektir. Tanm 2.1.2. γA ∈ BE(U)olsun. E§er her x ∈ E için γA(x) = ∅oluyorsa γA'ya bo³
be-küme denir ve γΦ ile gösterilir.
Tanm 2.1.3. γA ∈ BE(U)olsun. E§er her x ∈ A, γA(x) = Uoluyorsa γA'ya A-evrensel
be-küme denir ve γA˜ ile gösterilir. Özel olarak, bu ³art A = E için sa§lanyorsa γA'ya
evrensel be-küme denir ve γE˜ ile gösterilir.
Örnek 2.1.4. Kabul edelim ki, gözöününe alnan elemanlarn kümesi U = {u1, u2, u3, u4, u5} ve bu elemanlar niteleyen parametrelerin kümesi E = {x1, x2, x3, x4}biçiminde verilsin.
E§er A = {x2, x3, x4}, γA(x2) = {0.5/u1, 0.3/u5}), γA(x3) = ∅ ve γA(x4) = U biçiminde ise o halde γA be-kümesi γA = {(x2, {0.5/u1, 0.3/u5}), (x4, U)} ³eklinde yazlr.
E§er Y = {x1, x4}, γA(x1) = ∅ve γA(x4) = ∅biçiminde ise o halde γY be-kümesi bo³
be-kümedir. Yani, γY = γΦ³eklindedir.
E§er Z = {x1, x2}, γZ(x1) = U ve γZ(x2) = U biçiminde ise γZ be-kümesi Z-evrensel
be-kümedir. Yani, γZ = γZ˜ ³eklindedir.
E§er A = E ve γA(x1) = U, γA(x2) = U, γA(x3) = U ve γA(x4) = U biçiminde ise o halde γAbe-kümesi evrensel be-kümedir. Yani, γA= γE˜ ³eklindedir.
Tanm 2.1.5. γA, γB ∈ BE(U)olsun. E§er her x ∈ E için γA(x) ⊆ γB(x) oluyorsa
γA'ya γB'nin be-altkümesi denir ve γA⊆γe Bile gösterilir.
Önerme 2.1.6. γA, γB ∈ BE(U)olsun. O halde,
i. γA⊆γe E˜ ii. γΦ⊆γe A iii. γA⊆γe A
iv. γA⊆γe B ve γB⊆γe C ⇒ γA⊆γe C
Tanm 2.1.7. γA, γB ∈ BE(U)olsun. E§er her x ∈ E için γA(x) = γB(x)oluyorsa γA
ve γB'ye be-e³it denir ve γA = γB ile gösterilir.
Önerme 2.1.8. γA, γB, γC ∈ BE(U)olsun. O halde,
i. γA= γBand γB = γC ⇒ γA= γC
ii. γA⊆γe B and γB⊆γe A⇔ γA= γB
Tanm 2.1.9. γA ∈ BE(U)olsun. O halde, γA'nn be-tümleyeni her x ∈ E için
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e³itli§i yoluyla elde edilir ve γ˜c
A ile gösterilir. Burada, γAc(x) kümesi γA(x)'in klasik
tümleyenidir. Yani, her x ∈ E için γc
A(x) = U \ γA(x)biçimindedir.
Önerme 2.1.10. γA∈ BE(U)olsun. O halde,
i. (γ˜c
A)˜c= γA
ii. γ˜c
Φ = γE˜
Tanm 2.1.11. γA, γB∈ BE(U)olsun. O halde, γAve γB'nin birle³imi her x ∈ E için,
γAe∪B(x) = γA(x) ∪ γB(x)
e³itli§i yoluyla elde edilir ve γA∪γe Bile gösterilir.
Önerme 2.1.12. γA, γB, γC ∈ BE(U)olsun. O halde,
i. γA∪γe A = γA
ii. γA∪γe Φ = γA
iii. γA∪γe E˜ = γE˜ iv. γA∪γe B = γB∪γe A
v. (γA∪γe B)e∪γC = γA∪(γe B∪γe C)
Tanm 2.1.13. γA, γB∈ BE(U)olsun. O halde, γAve γB'nin kesi³imi her x ∈ E için,
γAe∩B(x) = γA(x) ∩ γB(x)
e³itli§i yoluyla elde edilir ve γA∩γe Bile gösterilir.
Önerme 2.1.14. γA, γB, γC ∈ BE(U)olsun. O halde,
i. γA∩γe A = γA
iii. γA∩γe E˜ = γA
iv. γA∩γe B = γB∩γe A
v. (γA∩γe B)e∩γC = γA∩(γe B∩γe C)
Not 2.1.15. γA∈ BE(U)olsun. E§er γA6= γΦ veya γA 6= γE˜ ise o halde γA∪γe A˜c 6= γE˜
ve γA∩γe A˜c 6= γΦ.
Önerme 2.1.16. γA, γB ∈ BE(U)olsun. O halde De Morgan kurallar sa§lanr. Yani,
i. (γA∪γe B)˜c= γA˜c∩γe Bc˜
ii. (γA∩γe B)˜c= γA˜c∪γe Bc˜
Önerme 2.1.17. γA, γB, γC ∈ BE(U)olsun. O halde,
i. γA∪(γe B∩γe C) = (γAe∪γB)e∩(γA∪γe C)
3. BULANIK ESNEK MATRSLER VE LEMLER
3.1 Bulank Esnek Matris
Bu bölümde, be-kümelerin matris temsilleri olan ve be-küme i³lemlerinin bilgisayar ortamnda yaplabilmesine olanak sa§layan be-matris kavram tanmland.
Tanm 3.1.1. γA ∈ BE(U)olsun. O halde, γA'in bulank ba§nt formu
RA= {(µRA(u, x)/(u, x)) : (u, x) ∈ U × E}
biçiminde tanmlanr. Burada, µRA üyelik fonksiyonu
µRA : U × E → [0, 1] × [0, 1], µRA(u, x) = µγA(x)(u)
³eklindedir.
E§er U = {u1, u2, ..., um}, E = {x1, x2, ..., xn}ve A ⊆ E ise o halde RA
RA x1 x2 ... xn
u1 µRA(u1, x1) µRA(u1, x2) ... µRA(u1, xn) u2 µRA(u2, x1) µRA(u2, x2) ... µRA(u2, xn)
... ... ... ... ...
um µRA(um, x1) µRA(um, x2) ... µRA(um, xn)
³eklinde tablo olarak gösterilebilir. Burada, aij = µRA(ui, xj)e³lemesi yaplrsa tablo
[aij]m×n = x11 x12 · · · x1n x21 x22 · · · x2n ... ... ... ... xm1 xm2 · · · xmn
m × ntipinde matris formunda yazlabilir. Bu matrise U üzerinde bir γAbe-kümesinin
be-matrisi ad verilir.
Bu tanma göre, bir γA be-kümesi [aij]m×n matrisi yoluyla tek türlü olarak karakterize
edilebilir. Bu yüzden, bu kavramlardan biri di§eri yerine kullanlabilir.
Bu çal³mada, BEMm×n notasyonu ile U üzerinde E ile yazlan tüm m × n tipindeki
be-matrislerin kümesi kastedildi. Ayrca, i ∈ {1, 2, ...n} ve j ∈ {1, 2, ...m} için [aij]m×n ∈ BEMm×n notasyonu yerine ksalk için m × n alt indisi kullanlmadan
[aij] ∈ BEMm×ngösterimi kullanlacaktr.
Örnek 3.1.2. Örnek 2.1.4'ü gözönüne alalm. O halde, γA'nn be-ba§nt formu
RA = {0.5/(u1, x2), 1/(u1, x4), 1/(u2, x4), 1/(u3, x4),
1/(u4, x4), 0.3/(u5, x2), 1/(u5, x4)} ³eklinde yazlr. Buradan, [aij] be-matrisi
[aij] = 0 0.5 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0.3 0 1
biçiminde elde edilir.
Tanm 3.1.3. [aij] ∈ BEMm×nolsun. O halde,
1. E§er her i ve j için aij = 0oluyorsa [aij] be-matrisine bir sfr be-matrisi ad verilir
ve [0] ile gösterilir.
2. E§er her i ve j ∈ IA = {j : xj ∈ A} için aij = 1 oluyorsa [aij] be-matrisine
A-evrensel be-matris, denir ve [˜aij]ile gösterilir.
3. E§er her i ve j için aij = 1oluyorsa [aij] be-matrisine evrensel be-matris denir ve
8
Örnek 3.1.4. U = {u1, u2, u3, u4, u5}, E = {x1, x2, x3, x4}ve [aij], [bij], [cij] ∈ BEM5×4(U) verilsin.
E§er A = {x1}için γA(x1) = ∅ise o halde [aij] = [0]
[0] = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
biçiminde açkça yazlan bir sfr be-matrisidir.
E§er B = {x1, x2}, γY(x1) = U ve γY(x2) = U ise o halde [bij] = [˜bij]
[˜bij] = 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
biçiminde açkça yazlan bir B-evrensel be-matrisidir.
E§er A = E ve γA(x1) = U, γA(x2) = U, γA(x3) = U ve γA(x4) = U ise o halde [aij] = [˜eij] = [1] [1] = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
biçiminde açkça yazlan bir evrensel be-matristir. Tanm 3.1.5. [aij], [bij] ∈ BEMm×nolsun. O halde,
1. E§er her i ve j için aij 6 bij oluyorsa o halde [aij], [bij]'nin be-altmatrisidir denir
ve [aij] ˜⊆[bij]ile gösterilir.
2. E§er [aij] ˜⊆[bij] ve en az bir i ve j için aij < bij oluyorsa o halde [aij], [bij]'nin
proper be-altmatrisidir denir ve [aij] ˜⊂[bij]ile gösterilir.
3. E§er her i ve j için aij = bij oluyorsa o halde [aij]ve [bij] be-matrisleri be- e³ittir
denir ve [aij] = [bij]ile gösterilir.
3.2 Bulank Esnek Matris ³lemleri
Bu bölümde, be-kümelerin matrisler üzerinde baz i³lemler tanmland ve bir takm özelikleri incelendi.
Tanm 3.2.6. [aij], [bij] ∈ BEMm×nolsun. E§er her i ve j için
1. cij = max{aij, bij}olacak biçimde tanmlanan [cij]'ye [aij]ve [bij]'nin be-birle³imi
denir ve [aij]˜∪[bij]ile gösterilir.
2. cij = min{aij, bij}olacak biçimde tanmlanan [cij]'ye [aij]ve [bij]'nin be-kesi³imi
denir ve [aij]˜∩[bij]ile gösterilir.
3. cij = 1 − aij olacak biçimde tanmlanan [cij]'ye [aij]'nin be-tümleyeni denir ve
[aij]˜cile gösterilir.
Tanm 3.2.7. [aij], [bij] ∈ BEMm×nolsun. E§er [aij]˜∩[bij] = [0]oluyorsa o halde [aij]
ve [bij]ayrktr denir.
Örnek 3.2.8. Kabul edelim ki
[aij] = 0 0.6 0.2 0 0.1 0 1 0 0 0.3 0.8 0 0.7 0 0 0 0 1 0 0
10 [bij] = 0 0 0.7 0.4 0 0.2 0.6 1 0 0.4 0 0.9 0 0 0.5 1 0 0 0 0.3 verilsin. O halde, [aij]˜∪[bij] = 0 0.6 0.7 0.4 0.1 0.2 1 1 0 0.4 0.8 0.9 0.7 0 0.5 1 0 1 0 0.3 [aij]˜c= 1 0.4 0.8 1 0.9 1 0 1 1 0.7 0.2 1 0.3 1 1 1 1 0 1 1 biçimindedir.
Önerme 3.2.9. [aij] ∈ BEMm×nolsun. O halde,
1. [[aij]c˜]˜c= [aij]
2. [0]c˜= [1]
Önerme 3.2.10. [aij], [bij] ∈ BEMm×nolsun. O halde
1. [aij] ˜⊆[1]
2. [0] ˜⊆[aij]
3. [aij] ˜⊆[aij]
4. [aij] ˜⊆[bij]and [bij] ˜⊆[cij] ⇒ [aij] ˜⊆[cij]
Önerme 3.2.11. [aij], [bij], [cij] ∈ BEMm×nolsun. O halde,
1. [aij] = [bij]ve [bij] = [cij] ⇒ [aij] = [cij]
2. [aij] ˜⊆[bij]ve [bij] ˜⊆[aij] ⇔ [aij] = [bij]
Önerme 3.2.12. [aij], [bij], [cij] ∈ BEMm×nolsun. O halde
1. [aij]˜∩[bij] = [bij]˜∩[aij] 2. [aij]˜∪[bij] = [bij]˜∪[aij] 3. ([aij]˜∩[bij])˜∩[cij] = [aij]˜∩([bij]˜∩[cij]) 4. ([aij]˜∪[bij])˜∪[cij] = [aij]˜∪([bij]˜∪[cij]) 5. [aij]˜∩([bij]˜∪[cij]) = ([aij]˜∩[bij])˜∪([aij]˜∩[cij]) 6. [aij]˜∪([bij]˜∩[cij]) = ([aij]˜∪[bij])˜∩([aij]˜∪[cij])
Not 3.2.13. Burada, [aij] = [0]veya [aij] = [1]olmad§ durumlarda [aij]˜∩[aij]c˜ 6= [0]
ve [aij]˜∪[aij]˜c6= [1]oldu§una dikkat edilmelidir.
Önerme 3.2.14. [aij], [bij] ∈ BEMm×nolsun. O halde De Morgan kurallar sa§lanr.
1. ([aij]˜∪[bij])c˜= [aij]˜c∩[b˜ ij]c˜
2. ([aij]˜∩[bij])c˜= [aij]˜c∪[b˜ ij]c˜
spat . Her i ve j için
i. ([aij]˜∪[bij])c˜ = [max{aij, bij}]˜c
= [1 − max{aij, bij}]
= [min{1 − aij, 1 − bij}]
= [aij]˜c∩[b˜ ij]c˜
12
3.3 be-Matris Çarpmlar
Bu bölümde, be-karar verme metotlar in³a edebilmek için be-matris çarpmlar in³a edildi. Tanm 3.3.15. [aij], [bik] ∈ BEMm×nolsun. O halde, [aij]ve [bik]'nin V e-çarpm
∧ : BEMm×n× BEMm×n→ BEMm×n2, [aij] ∧ [bik] = [cip]
biçiminde tanmlanr. Burada, p = n(j − 1) + k olmak üzere cip = min{aij, bik}³eklinde
tanmldr.
Tanm 3.3.16. [aij], [bik] ∈ BEMm×nolsun. O halde [aij]ve [bik]'nin V eya-çarpm
∨ : BEMm×n× BEMm×n→ BEMm×n2, [aij] ∨ [bik] = [cip]
biçiminde tanmlanr. Burada, p = n(j − 1) + k olmak üzere cip = max{aij, bik}
³eklindedir.
Tanm 3.3.17. [aij], [bik] ∈ BEMm×nolsun. O halde, [aij]ve [bik]'nin V e-Degil-çarpm
Z : BEMm×n× BEMm×n→ BEMm×n2, [aij] Z [bik] = [cip]
biçiminde tanmlanr. Burada, p = n(j − 1) + k olmak üzere cip = min{aij, 1 − bik}
³eklindedir.
Tanm 3.3.18. [aij], [bik] ∈ BEMm×nolsun. O halde [aij]ve [bik]'nin V eya-Degil-çarpm
Y : BEMm×n× BEMm×n→ BEMm×n2, [aij] Y [bik] = [cip]
biçiminde tanmlanr. Burada, p = n(j − 1) + k olmak üzere cip = max{aij, 1 − bik}
³eklindedir.
Örnek 3.3.19. Kabul edelim ki [aij], [bik] ∈ BEM2×3
[aij] = 0 0 0.2 0 1 0.7 , [bik] = 0.5 0 0.2 0.2 0 0
olarak verilsin. O halde, [aij] ∧ [bik] 0 0 0 0 0 0 0.2 0 0.2 0 0 0 0.2 0 0 0.2 0 0
biçimindedir. [aij]∨[bik], [aij]Z[bik]ve [aij]Y[bik]çarpmlar da benzer ³ekilde bulunabilir.
Tanmlanan bu çarpmlarn de§i³meli olmad§na dikkat edilmelidir.
Önerme 3.3.20. [aij], [bik] ∈ BEMm×nolsun. O halde, De Morgan kurallar sa§lanr.
1. ([aij] ∨ [bik])◦ = [aij]◦∧ [bik]◦
2. ([aij] ∧ [bik])◦ = [aij]◦∨ [bik]◦
3. ([aij] Y [bik])◦ = [aij]◦Z [bik]◦
4. BULANIK ESNEK KARAR VERME VE UYGULAMALARI
4.1 Bulank Esnek mak-min Karar Verme
Bu bölümde, be mak-min karar verme metodu tantld. Bu metot gözönüne alnan alternati erin kümesinin bir çözüm için uygun bir alt kümesinin seçilmesini amaçlar. Bu sayede, karar vericinin çok sayda alternatif yerine daha küçük sayda alternati gözden geçirmesi sa§lanr.
Bu bölüm boyunca, ksalk için, ˜× notasyonu ∧, ∨, Z, Y notasyonlarndan bir yerine kulllanlr.
Tanm 4.1.1. γA, γB ∈ BE(U)olsun ve srasyla [aij]ve [bik]matrisleriyle temsil edilsin.
E§er IA = {r : xr ∈ A}, IB = {s : xs ∈ B} ve [aij] ˜×[bik] = [cip] olsun. O halde
makrminsve maksminroperatörleri, srasyla,
di1 = max r∈IA {min s∈IB {ci,s+(r−1)n}} olmak üzere
makrmins : SMm×n2 → SMm×1, makrmins[cip] = [di1]
ve ei1= max s∈IB {min r∈IA {ci,s+(r−1)n}} olmak üzere
maksminr : SMm×n2 → SMm×1, maksminr[cip] = [ei1]
makrminsoperatörü maksminr'nün altere operatörüdür ve alt-makrminsile gösterilir.
di§er bir ifadeyle, alt-makrmins= maksminrve alt-alt-makrmins = makrmins.
makrmins([aij] ˜×[bik]) 6= maksminr([aij] ˜×[bik])oldu§u açktr. Fakat, a³a§daki önerme
do§rudur.
Önerme 4.1.2. [aij], [bik] ∈ SMm×nolsun. O halde,
makrmins([aij] ˜×[bik]) = maksminr([bik] ˜×[aij])
spat . spat, Tanm 4.1.1'den görülebilir.
Tanm 4.1.3. [cip] ∈ SMm×n2 olsun. O halde, ˜×-çarpmlar için mak-min karar
fonksiyonu
max-min : BEMm×n2 → BEMm×1,
fonksiyonu yoluyla
max-min[cip] = makrmins[cip] ˜∪ maksminr[cip]
biçiminde tanmlanr.
Tanmlanan bu dört özel çarpmn de§i³meli olmad§ biliniyor. Fakat, a³a§daki önerme do§rudur.
Önerme 4.1.4. [aij], [bik] ∈ BEm×nolsun. O halde,
max-min([aij] ˜×[bik]) = max-min([bik] ˜×[aij])
spat . spat Önerme 4.1.2'den görülür.
Tanm 4.1.5. U = {u1, u2, ..., um}gözönüne alnan elemanlern kümesi ve max-min[cip] =
[vi1]olsun. O halde,
opt[vi1](U) = {µ(ui)opt[vi1](U )/ui : ui ∈ U, µ(ui)opt[vi1](U )= vi1}
16
Kabul edelim ki, alternati erin ve parametrelerin birer kümesi verilsin. O halde, be-mak−
minkarar verme metodu (BEMmKV ) kullanlarak U'nun uygun bir alt kümesi a³a§daki yolla bulunabilir.
Step 1 Parametre kümesinden iki alt küme seçilir.
Step 2 Seçilen bu alt kümeler kullanlarak iki esnek matris in³a edilir. Step 3 Bulank esnek matrislerin uygun bir çarpm bulunur.
Step 4 Bu çarpmdan be-mak-min karar matrisi elde edilir.
Step 5 Elde edilen bu karar matrisi yoluyla U'nun uygun bir alt kümesi tespit edilir.
Burada, 3. admda kullanlan matris çarpmlarndan hangisinin daha uygun oldu§unu belirlemek için bir kriter bulunmamaktadr. Yani, tantm yaplan BEMmKV metodu, her zaman U'nun yetrince küçük bir alt kümesini vermeyebilir. Bu taktirde karar verici, (BEmMKV ), (BEmmKV ), (BEMMKV ) metotlarndan birini tercih edebilir.
4.2 Uygulama
Kabul edelim ki, bir emlakçnn sat³ta bulunanan evlerinin kümesi U = {u1, u2, u3, u4, u5} ve bu evleri kararkterize etmek için kulland§ parametrelerin kümesi E = {x1, x2, x3, x4} biçiminde olsun. Burada, j ∈ {1, 2, 3, 4} için xj parametreleri, srasyla, iyi konumda",
ucuz", modern", büyük" anlamna gelsin.
Örnek 4.2.6. Bay X ve Bayan X bir ev satn almak için yukarda bahsedilen emlakçya ba³vursun. Emlakç önce bu çiftin her birinden, parametreler kümesinden kendileri için uygun olanlarn seçmelerini ister. Ardndan, BEMmKV metodunu a³a§daki gibi kullanr.
Step 1: Emlakç Bay X'in ve Bayan X'in seçti§i parametrelerin kümesini A = {x2, x3, x4} ve B = {x1, x3, x4}biçiminde yazar.
Step 2: Verilen be-kümelerin matris temsillerini [aij] = 0 0 0.2 0.4 0 0.6 0.9 0.4 0 0.8 0.7 0.5 0 0.5 0 0 0 1 0 0.8 [bik] = 1 0 0.9 0.7 0.2 0 0 0.9 0.7 0 0.4 0.3 0 0 0.5 0.6 0 0 0 1
biçiminde in³a eder.
Step 3: [aij]ve [bik] be-matrislerinin V e-çarpmn a³a§daki gibi elde eder.
0 0 0 0 0 0 0 0 0.2 0 0.2 0.2 0.4 0 0.4 0.4 0 0 0 0 0.2 0 0 0.6 0.2 0 0 0.9 0.2 0 0 0.4 0 0 0 0 0.7 0 0.4 0.3 0.7 0 0.4 0.3 0.5 0 0.4 0.3 0 0 0 0 0 0 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.8
Burada, Bay X ve Bayan X'in kararlar ayn anda dikkate alnd§ için her ikisinin seçti§i özelikleri ta³yan evleri tespit etmeye olanak sa§layan V e-çarpm kullanld. Ku³ku yok ki uzman farkl bir anlam yükledi§i farkl bir çarpm da kullanabilirdi.
Step 4: Daha sonra, Mm([aij] ∧ [bik]) = [di1]'i bulur. Bunun için, IA = {2, 3, 4}ve
IB= {1, 3, 4}oldu§undan
d31 = maxr∈{2,3,4}{mins∈{1,3,4}{c3,s+(r−1)4}}
= maxr∈{2,3,4}{min{c3,1+(r−1)4, c3,3+(r−1)4, c3,4+(r−1)4}}
= max{min{c3,5, c3,7, c3,8}, min{c3,9, c3,11, c3,12}, min{c3,13, c3,15, c3,16}} = max{min{0.7, 0.4, 0.3}, min{0.7, 0.4, 0.3}, min{0.5, 0.4, 0.3}}
= max{0.3, 0.3, 0.3}} = 0.3
18
ve benzer ³ekilde d11= 0.4, d21 = 0.0, d41= 0.0ve d51 = 0.0sonucuna ula³r. Ardndan,
be-max-min karar matrisini
Mm([aij] ∧ [bik]) = [di1] = 0.4 0 0.3 0 0 biçiminde yazar.
Step 5: U üzerinde bir be-matris olan Mm([aij] ∧ [bik])'i kullanlarak
optM m([aij]∧[bik])(U) = {0.4/u1, 0.3/u3}
karar kümesini elde eder. Burada, u1 evinin di§er tercihlere göre çift için daha uygun oldu§u sonucuna vararak bu evi tavsiye eder ve ya yeniden ve daha hassas bir inceleme için tüm evler yerine U'nun {u1, u3} uygun alt kümesinin gözönüne alnmasn tavsiye edebilir.
Ancak, bu problemde en yüksek üyelik derecesinin 0.4 olmas nedeniyle çifte ba³ka bir emlakçya daha ba³vurmalar tavsiye edilebilir.
Bu çal³mada, bulank esnek (be) kümeler tantld. Ardndan, be-matrisler ve i³lemleri tanmland. Daha sonra, be-matrislerin çarpmlar da tanmland ve baz özelikleri incelendi. Son olarak, bu çarpmlar bir tür karar verme problemi üzerine uyguland. bu alanda önemli bir yer tutan bu çal³ma be-analiz ve be-cebir gibi çal³ma konular için bir ilham kayna§ olacaktr. Ayrca ³imdiye kadar yaplan çal³malarn en genel hali olarak parametreleri bulankla³trlm³ bulank esnek matris veya sezgisel parametreleri bulankla³trlm³ bulank esnek matris konular gerçekten çal³lmaya de§erdir. Bu çal³malarn di§er alanlara da rahatlkla uygulanabilece§i öngörülmektedir.
20
KAYNAKLAR
C.a§man N. and Engino§lu S., 2010. Soft set theory and uni-int decision making, European Journal of Operational Research 207, 848-855.
C.a§manN.andEngino§luS., 2010.Softmatrixtheoryanditsdecisionmaking.Computers and Mathematics with Applications, 59, 3308-3314.
C.a§man N., Ctak F. and Engino§lu S., 2010. Fuzzy parameterized fuzzy soft set theory and its applications, Turkish Journal of Fuzzy Systems, 1(1), 21-35.
C.a§man N. and Engino§lu S., 2011. Fuzzy soft matrix theory and its application in decision making. Iranian Journal of Fuzzy Systems(In Press).
C.a§man N., Engino§lu S. and Ctak F.,2011. Fuzzy Soft Set Theory and Its Applications, Iranian Journal of Fuzzy Systems, 8 (3), 137-147.
C.a§man N., Ctak F. and Engino§lu S., 2011. Fuzzy parameterized soft set theory and its applications, Annals of Fuzzy Mathematics and Informatics, 2 (2), 219-226.
Molodtsov D.A., 1999. Soft set theory- rst results, Computers and Mathematics with Applications, 37, 19-31.
FUZZY SOFT MATRIX THEORY AND ITS APPLICATION IN DECISION MAKING
N. C. A ˘GMAN AND S. ENGINO ˘GLU
Abstract. In this work, we define fuzzy soft (f s) matrices and their opera-tions which are more functional to make theoretical studies in the f s-set theory. We then define products of f s-matrices and study their properties. We finally construct a f s-max-min decision making method which can be successfully applied to the problems that contain uncertainties.
1. Introduction
Soft set theory [33] was firstly introduced by Molodtsov in 1999 as a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. After presentation of the operations of soft sets [29], the properties and applications on the soft set theory have been studied increasingly [4, 12, 13, 25, 26, 30, 31, 35, 37, 38, 40, 46, 47, 48, 51]. The algebraic structure of soft set theory has also been studied in more detail [1, 3, 5, 11, 14, 15, 16, 17, 18, 19, 20, 22, 36, 41, 44]. In recent years, by embedding the ideas of fuzzy sets [52] many interesting applications of soft set theory have been done [2, 8, 9, 10, 21, 23, 24, 27, 28, 32, 34, 39, 42, 43, 45, 49, 50]. To develop the soft set theory, operations of the soft sets are redefined to improve several new results and uni-int decision making method is constructed by using these new operations [6]. To make easy computation with the operations of soft sets, the soft matrix theory is presented and soft max-min decision making method is set up [7]. These decision making methods are more practical and can be successfully applied to many problems that contain uncertainties.
In [9], a fuzzy soft (f s) set theory is defined. It allows constructing more ef-ficient decision making method. In this paper, we first define f s-matrices which are representation of the f s-sets. This representation has several advantages. It is easy to store and manipulate matrices and hence the f s-sets represented by them in a computer. Here, we also construct a f s-decision making method which is more practical and can be successfully applied to many problems. We finally give an example which shows that the method successfully works.
Received: July 2010; Revised: ; Accepted:
Key words and phrases: Fuzzy soft sets, fuzzy soft matrix, products of fuzzy soft matrices,
52 N. C. a˘gman and S. Engino˘glu
2. Fuzzy Soft Matrices
In this section, we define f s-matrices which are representations of the f s-sets. This style of representation is useful for storing a soft set in a computer mem-ory. The operations can be presented by the matrices which are very useful and convenient for application.
From now on, a set of all fuzzy sets over U will be denoted by F (U ). ΓA,
ΓB, ΓC,..., etc. and γA, γB, γC,..., etc. will be used for f s-sets and their fuzzy
approximate functions, respectively.
Definition 2.1. [9] Let U be an initial universe, E be the set of all parameters,
A ⊆ E and γA(x) be a fuzzy set over U for all x ∈ E. Then, an f s-set ΓA over U
is a set defined by a function γArepresenting a mapping
γA: E → F (U ) such that γA(x) = ∅ if x /∈ A
Here, γA is called fuzzy approximate function of the f s-set ΓA, the value γA(x)
is a fuzzy set called x-element of the f s-set for all x ∈ E, and ∅ is the null fuzzy set. Thus, an f s-set ΓA over U can be represented by the set of ordered pairs
ΓA= {(x, γA(x)) : x ∈ E, γA(x) ∈ F (U )}
Note that from now on, the sets of all f s-sets over U will be denoted by F S(U ). Example 2.2. Assume that U = {u1, u2, u3, u4, u5} is a universal set and E = {x1, x2, x3, x4} is a set of all parameters.
If A = {x2, x3, x4}, γA(x2) = {0.5/u2, 0.8/u4}, γA(x3) = ∅ and γA(x4) = U ,
then the f s-set ΓA is written by ΓA= {(x2, {0.5/u2, 0.8/u4}), (x4, U )}.
Definition 2.3. Let ΓA∈ F S(U ). Then a fuzzy relation form of ΓA is defined by
RA= {(µRA(u, x)/(u, x)) : (u, x) ∈ U × E}
Where the membership function of µRA is written by
µRA : U × E → [0, 1], µRA(u, x) = µγA(x)(u)
If U = {u1, u2, ..., um}, E = {x1, x2, ..., xn} and A ⊆ E, then the RA can be
presented by a table as in the following form
RA x1 x2 ... xn u1 µRA(u1, x1) µRA(u1, x2) ... µRA(u1, xn) u2 µRA(u2, x1) µRA(u2, x2) ... µRA(u2, xn) .. . ... ... . .. ... um µRA(um, x1) µRA(um, x2) ... µRA(um, xn)
If aij = µRA(ui, xj), we can define a matrix
[aij]m×n= x11 x12 · · · x1n x21 x22 · · · x2n .. . ... . .. ... xm1 xm2 · · · xmn which is called an m × n f s-matrix of the f s-set ΓA over U .
Therefore, we shall identify any f s-set with its f s-matrix and use these two concepts as interchangeable.
The set of all m × n f s-matrices over U will be denoted by F SMm×n. From now
on we shall delete the subscripts m × n of [aij]m×n, we use [aij] instead of [aij]m×n,
since [aij] ∈ F SMm×nmeans that [aij] is an m × n f s-matrix for i = 1, 2, ...m and
j = 1, 2, ...n.
Example 2.4. Let us consider Example 2.2. Then the relation form of ΓA is
written by
RA = {0.5/(u2, x2), 0.8/(u4, x2), 1/(u1, x4)
1/(u2, x4), 1/(u3, x4), 1/(u4, x4), 1/(u5, x4)}
Hence, the f s-matrix [aij] is written by
[aij] = 0 0 0 1 0 0.5 0 1 0 0 0 1 0 0.8 0 1 0 0 0 1 Definition 2.5. Let [aij] ∈ F SMm×n. Then [aij] is called
(1) a zero f s-matrix, denoted by [0], if aij = 0 for all i and j.
(2) an A-universal f s-matrix, denoted by [˜aij], if aij = 1 for all j ∈ IA= {j :
xj∈ A} and i.
(3) a universal f s-matrix, denoted by [1], if aij = 1 for all i and j.
Example 2.6. Assume that U = {u1, u2, u3, u4, u5} is a universal set, E = {x1, x2, x3, x4}
be a set of parameters, A ⊆ E, γA(x) be a fuzzy set over U for all x ∈ E and
[aij], [bij], [cij] ∈ F SM5×4.
If A = {x1, x3} and γA(x1) = ∅, γA(x3) = ∅, then [aij] = [0] is a zero f s-matrix
written by [0] = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
If B = {x1, x2} and γB(x1) = U , γB(x2) = U , then [bij] = [˜bij] is a B-universal
f s-matrix written by [˜bij] = 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
54 N. C. a˘gman and S. Engino˘glu
If C = E, and γC(xi) = U for all xi ∈ C, i = 1, 2, 3, 4, then [cij] = [1] is a
universal f s-matrix written by
[1] = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Definition 2.7. Let [aij], [bij] ∈ F SMm×n. Then
(1) [aij] is a f s-submatrix of [bij], denoted by [aij] ˜⊆[bij], if aij 6 bij for all i
and j.
(2) [aij] is a proper f s-submatrix of [bij], denoted by [aij] ˜⊂[bij], if aij 6 bij for
all i and j and for at least one term aij < bij.
(3) [aij] and [bij] are f s-equal matrices, denoted by [aij] = [bij], if aij = bij for
all i and j.
Definition 2.8. Let [aij], [bij] ∈ F SMm×n. Then the f s-matrix [cij] is called
(1) union of [aij] and [bij], denoted [aij]˜∪[bij], if cij = max{aij, bij} for all i
and j.
(2) intersection of [aij] and [bij], denoted [aij]˜∩[bij], if cij = min{aij, bij} for
all i and j.
(3) complement of [aij], denoted by [aij]◦, if cij = 1 − aij for all i and j.
Definition 2.9. Let [aij], [bij] ∈ F SMm×n. Then [aij] and [bij] are disjoint, if
[aij]˜∩[bij] = [0] for all i and j.
Example 2.10. Assume that
[aij] = 0 0.6 0 0 0.1 0 1 0 0 0.3 0.8 0 0.7 0 0 0 0 1 0 0 , [bij] = 0 0 0.7 0.4 0 0.2 0 1 0 0 0 0.9 0 0 0.5 1 0 0 0 0.3 Then, [aij]˜∩[bij] = [0] and [aij]˜∪[bij] = 0 0.6 0.7 0.4 0.1 0.2 1 1 0 0.3 0.8 0.9 0.7 0 0.5 1 0 1 0 0.3 , [aij] ◦= 1 0.4 1 1 0.9 1 0 1 1 0.7 0.2 1 0.3 1 1 1 1 0 1 1 Proposition 2.11. Let [aij] ∈ F SMm×n. Then
(1) [[aij]◦]◦= [aij]
(2) [0]◦= [1]
Proposition 2.12. Let [aij], [bij] ∈ F SMm×n. Then
(4) [aij] ˜⊆[bij] and [bij] ˜⊆[cij] ⇒ [aij] ˜⊆[cij]
(5) [aij] ˜⊆[bij] ⇔ [aij]˜∩[bij] = [aij] ⇔ [aij]˜∪[bij] = [bij]
Proposition 2.13. Let [aij], [bij], [cij] ∈ F SMm×n. Then
(1) [aij] = [bij] and [bij] = [cij] ⇔ [aij] = [cij]
(2) [aij] ˜⊆[bij] and [bij] ˜⊆[aij] ⇔ [aij] = [bij]
Proposition 2.14. Let [aij], [bij], [cij] ∈ F SMm×n. Then
(1) [aij]˜∩[bij] = [bij]˜∩[aij] (2) [aij]˜∪[bij] = [bij]˜∪[aij] (3) ([aij]˜∩[bij])˜∩[cij] = [aij]˜∩([bij]˜∩[cij]) (4) ([aij]˜∪[bij])˜∪[cij] = [aij]˜∪([bij]˜∪[cij]) (5) [aij]˜∩([bij]˜∪[cij]) = ([aij]˜∩[bij])˜∪([aij]˜∩[cij]) (6) [aij]˜∪([bij]˜∩[cij]) = ([aij]˜∪[bij])˜∩([aij]˜∪[cij])
Note that, [aij]˜∩[aij]◦6= [0] and [aij]˜∪[aij]◦6= [1]
Proposition 2.15. Let [aij], [bij] ∈ F SMm×n. Then De Morgan’s laws are valid
(1) ([aij]˜∪[bij])◦= [aij]◦∩[b˜ ij]◦
(2) ([aij]˜∩[bij])◦= [aij]◦∪[b˜ ij]◦
Proof. For all i and j,
i. ([aij]˜∪[bij])◦ = [max{aij, bij}]◦
= [1 − max{aij, bij}]
= [min{1 − aij, 1 − bij}]
= [aij]◦∩[b˜ ij]◦
ii. It can be proved similarly. ¤
3. Products of f s-matrices
In this section, we define four special products of f matrices to construct f s-decision making methods.
Definition 3.1. Let [aij], [bik] ∈ F SMm×n. Then And-product of [aij] and [bik] is
defined by
∧ : F SMm×n× F SMm×n → F SMm×n2, [aij] ∧ [bik] = [cip]
where cip= min{aij, bik} such that p = n(j − 1) + k.
Definition 3.2. Let [aij], [bik] ∈ F SMm×n. Then Or-product of [aij] and [bik] is
defined by
∨ : F SMm×n× F SMm×n → F SMm×n2, [aij] ∨ [bik] = [cip]
56 N. C. a˘gman and S. Engino˘glu
Definition 3.3. Let [aij], [bik] ∈ F SMm×n. Then And-N ot-product of [aij] and
[bik] is defined by
Z : F SMm×n× F SMm×n → F SMm×n2, [aij] Z [bik] = [cip]
where cip= min{aij, 1 − bik} such that p = n(j − 1) + k.
Definition 3.4. Let [aij], [bik] ∈ F SMm×n. Then Or-N ot-product of [aij] and
[bik] is defined by
Y : F SMm×n× F SMm×n → F SMm×n2, [aij] Y [bik] = [cip]
where cip= max{aij, 1 − bik} such that p = n(j − 1) + k.
Example 3.5. Assume that [aij], [bik] ∈ F SM2×3 are given as follows
[aij] = · 0 0 0.3 0 1 0.7 ¸ , [bik] = · 0.5 0 0.2 0.2 0 0 ¸
To calculate [aij] ∧ [bik] = [cip], we have to find cip for all i = 1, 2 and p = 1, 2, ..., 9.
Let us find c17. Since n = 3, i = 1 and p = 7, we get j = 3 and k = 1 from
7 = 3(j − 1) + k. Hence c17 = min{a13, b11} = min{0.3, 0.5} = 0.3. If the other
entries of [cip] can be found similarly, then we can obtain the matrix as follows;
[aij] ∧ [bik] =
·
0 0 0 0 0 0 0.3 0 0.2 0 0 0 0.2 0 0 0.2 0 0
¸
Similarly, we can also find the others products [aij] ∨ [bik], [aij] Z [bik] and [aij] Y
[bik].
Note that the commutativity is not valid for the products of f s-matrices. Proposition 3.6. Let [aij], [bik] ∈ F SMm×n. Then the following De Morgan’s
types of results are true.
(1) ([aij] ∨ [bik])◦= [aij]◦∧ [bik]◦
(2) ([aij] ∧ [bik])◦= [aij]◦∨ [bik]◦
(3) ([aij] Y [bik])◦= [aij]◦Z [bik]◦
(4) ([aij] Z [bik])◦= [aij]◦Y [bik]◦
4. f s-max-min decision making
In this section, we construct an f s-max-min decision making (F SM mDM ) method by using f s-max-min decision function which is also be defined here. The method selects optimum alternatives from the set of the alternatives.
Definition 4.1. Let [cip] ∈ F SMm×n2, Ik = {p : ∃i, cip 6= 0, (k − 1)n < p 6 kn}
for all k ∈ I = {1, 2, ..., n}. Then f s-max-min decision function, denoted M m, is defined as follows M m : F SMm×n2 → F SMm×1, M m[cip] = [di1] = [max k {tik}] where tik= ½ minp∈Ik{cip}, if Ik6= ∅, 0, if Ik= ∅.
opt[di1](U ) = {di1/ui: ui∈ U, di16= 0}
which is called an optimum fuzzy set on U .
Now, by using the definitions we can construct a F SM mDM method by the following algorithm.
Step 1: choose feasible subsets of the set of parameters, Step 2: construct the f s-matrix for each set of parameters, Step 3: find a convenient product of the f s-matrices, Step 4: find a max-min decision f s-matrix,
Step 5: find an optimum fuzzy set on U .
Note that, by the similar way, we can define f s-min-max, f s-min-min and
f s-max-max decision making methods which may be denoted by (F SmM DM ),
(F SmmDM ), (F SM M DM ), respectively. One of them may be useful than others according to the type of the problems.
5. Applications
Assume that a real estate agent has a set of different types of houses U =
{u1, u2, u3, u4, u5} which may be characterized by a set of parameters E = {x1, x2, x3, x4}. For j = 1, 2, 3, 4 the parameters xj stand for “in good location”, “cheap”,
“modern”, “large”, respectively. Then we can give the following examples.
Example 5.1. Suppose that a married couple, Mr. X and Mrs. X, come to the real estate agent to buy a house. If each partner has to consider their own set of parameters, then we select a house on the basis of the sets of partners’ parameters by using the F SM mDM as follows.
Assume that U = {u1, u2, u3, u4, u5} is a universal set and E = {x1, x2, x3, x4}
is a set of all parameters.
Step 1: First, Mr. X and Mrs. X have to choose the sets of their parameters, A = {x2, x3, x4} and B = {x1, x3, x4}, respectively.
Step 2: Then we can write the following f s-matrices which are constructed
according to their parameters.
[aij] = 0 0 0.2 0.4 0 0.6 0.9 0.4 0 0.8 0.7 0.5 0 0.5 0 0 0 1 0 0.8 [bik] = 1 0 0.9 0.7 0.2 0 0 0.9 0.7 0 0.4 0.3 0 0 0.5 0.6 0 0 0 1
58 N. C. a˘gman and S. Engino˘glu
Step 3: Now, we can find a product of the f s-matrices [aij] and [bik] by using
And-product as follows 0 0 0 0 0 0 0 0 0.2 0 0.2 0.2 0.4 0 0.4 0.4 0 0 0 0 0.2 0 0 0.6 0.2 0 0 0.9 0.2 0 0 0.4 0 0 0 0 0.7 0 0.4 0.3 0.7 0 0.4 0.3 0.5 0 0.4 0.3 0 0 0 0 0 0 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.8 Here, we use And-product since both Mr. X and Mrs. X’s choices have to be considered.
Step 4: To calculate M m([aij] ∧ [bik]) = [di1], we have to find di1 for all i ∈
{1, 2, 3, 4, 5}. For instance, let us find d31. Since i = 1 and k ∈ {1, 2, 3, 4}, d31= maxk{t3k} = max{t31, t32, t33, t34}
Let us find t31and t32. For k = 1 and n = 4, I1= {p : cip6= 0, 0 < p ≤ 4} = ∅ and
for k = 2 and n = 4, I2= {p : cip6= 0, 4 < p ≤ 8} = {5, 7, 8}. Hence t31= 0 and t32= min{c35, c37, c38} = min{0.7, 0.4, 0.3} = 0.3
Similarly, we can find as t33= 0 and t34= 0. Thus, d31= max{0.0, 0.3, 0.3, 0.3} = 0.3
Similarly we can find d11= 0.4, d21= 0.0, d41= 0.0 and d51= 0.0. Finally, we can
obtain the f s-max-min decision f s-matrix as
M m([aij] ∧ [bik]) = [di1] = 0.4 0 0.3 0 0
Step 5: Finally, we can find an optimum fuzzy set on U according to M m([aij] ∧
[bik])
optM m([aij]∧[bik])(U ) = {0.4/u1, 0.3/u3}
where u1is an optimum house to buy for Mr. X and Mrs. X.
Similarly, we can also use the others products [aij]∨[bik], [aij]Z[bik] and [aij]Y[bik]
for the other convenient problems.
6. Conclusion
The f s-set theory is being applied to many fields varying from theoretical to practical. In this paper, we define f s-matrices which are matrix representation of the f s-sets. We then define the set-theoretic operations of f s-matrices which are more functional to improve several new results. Afterwards, we construct a
f s-decision making model on the f s-set theory. This new decision making method
depends on the ideas of fuzzy and soft sets. The main idea of it is similar to the de-cision making method, given in [7], which depends on only soft sets. Therefore, this
Acknowledgements. The authors are grateful for financial support from the Research Fund of Gaziosmanpasa University under grand no: 2010/89.
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Naim Cagman, Department of Mathematics, Faculty of Arts and Sciences, Gazios-manpasa University, 60250 Tokat, Turkey
E-mail address: ncagman@gop.edu.tr
Serdar Enginoglu, Department of Mathematics, Faculty of Arts and Sciences, Gazios-manpasa University, 60250 Tokat, Turkey
