模糊環境下之定比系列現金流量

(1)

዁ጙᕗცήϞۡШفӖ౪ߜࢺ໔

д߸റ

୽ҳߏ׎׬೚Ᏸ଱ଢ଼Ψᐠడώแف

ᄢ! ौ

ၥҏ׹ၥ୰ᚠ࢐଒୛ᏰΰΙ໶ࣺ࿋१ौޟ፞ᚠȄ׹ၥПਰϞ౪ߜࢺ໔֕౪

้ШفӖ࠮ᄘ߽ငலџَڗޟȂӔёΰҥ౪ᄂᕗცܚ௥ԝڗޟၥଉ܁܁ڎര዁

ጙՓிȂӰԪȃ࡚ᄺΙৈ၌ؚ዁ጙ้ШفӖ౪ߜࢺ໔ޟПݲ࢐ϚџܖીޟȄҏ Мᔣ௰ᏲюȈӵց౥ȃ෈ኵЅ౪ߜࢺ໔ࣱ࣏዁ጙኵ঄Ȃи౪ߜࢺ໔ڎۡШفӖ

࠮ᄘਢȂڏᎌҢޟࣺᜰϴԒȂٮሄоᄂٽᇳ݂ڏٺҢПԒȄ ᜰᗤຠȈ ዁ጙ໱ӫȃ଒୛ȃۡШفӖȄ

GEOMETRIC SERIES OF CASH FLOWS IN FUZZY ENVIRONMENTS

Chun-Chieh Wang

Department of Mechanical Engineering National Huwei Institute of Technology

Yunlin, Taiwan 632, R.O.C.

Key Words: fuzzy set, finance, geometric series.

ABSTRACT

Capital investment problems are very important topics in finance.

Because the types of cash flows of investment proposals are usually geometric series and the information from the real world is fuzzy, it is essential to construct a geometric series model applicable to a fuzzy environment. In this paper, we will derive the related formulas of cash flows on the condition that interest rate, number of periods, and geometric series of cash flows are fuzzy numbers. How they can be used is illustrated.

Ιȃࠉ! ِ

׹ၥПਰޟஈ՗ޱܖؚ๊ޱӵ೎౩ၥҏ׹ၥПਰਢȂ

ོӑհΙ໶ѓࢂց౥ȃ෈ኵЅ౪ߜࢺ໔ࣱ࣏ጂۡኵ঄ޟ୅

೩Ȃӵ၎໶୅೩ԙҳޟనӇήȂپஈ՗׹ၥПਰؚ๊Ȅԃ

௴՗ΰक़ПԒ೎౩ၥҏ׹ၥؚ๊Ȃࠌོٺுؚ๊ࠢ፴Ϛᅾ ౩དܖԤѶ୑ሳȄڏкौনӰ࢐Ȉרঈܚ೎ޟᕗც࢐Ιএ ၥଉ዁ጙޟᕗცȂ፝ԃȂרঈܖ೨ོ᠙ڗܖөտΡᇳȂߜ

᚞"Ϛ࢐࡞σ"Ȃܖ᎛ೲԝΣӵΠν࿲ϯ"ΰή"้้ޟၗȂ

ΰक़Πٽࣱ࣏ڐ࠮ޟ዁ጙၥଉȄณՄȃרঈࠓငல஠၎้

዁ጙၥଉࣼ࡟ԙጂۡޟኵ঄೎౩ϞȂӰԪџ૖അԙၥଉೝ

ႆϷᙏϽܖᇲҢȂՄഅԙϚׇछޟؚ๊Ȅ

༈ಛώแငᔼᏰఀऋਪϛԤϲ໶அҏϴԒ[1-3]ȂցҢ

၎ϲ໶ϴԒџஈ՗౪঄ȃತ঄Ѕԑߜհࣺϣؑ঄ϞၼᆗȄ

Buckly[4] ౥ӑବᄇ၎ϲ໶அҏϴԒඪю዁ጙܒᘗ৤ȂณՄ

ٺҢԪᘗ৤๖ݎپஈ՗ၥҏႱᆗؚ๊ོ౰ҡΙٲϚӫ౩ޟ

౪ຫȂӰԪWang[5]ѪѴඪюΙಢᘗ৤ࡣϞϲ໶அҏϴԒȄ

Wang[6] ϷݙȈցҢ၎ಢϴԒܚ࡚ᄺޟၥҏႱᆗ዁Ԓپ೎౩

ၥҏ׹ၥ୰ᚠȂࠌϚོ౰ҡϚӫ౩ޟ౪ຫȄณՄȃ࿋רঈ

(2)

೎౩ၥҏ׹ၥ୰ᚠਢȂ༉Ԥ௰৤ࡣϞϲ໶அҏϴԒٮϚٗ

ஊȂӰ࣏ۦԤڍ໶੫੆࠮ᄘϞ౪ߜࢺ໔໸ौհ዁ጙܒϞᘗ

৤ȂڏϛΙᆍᆎ้࣏৯فӖ౪ߜࢺ໔ȂѪΙᆍࠌ้࣏Шف Ӗ౪ߜࢺ໔Ȅд߸റ[7]ϐׇԙ้৯فӖ౪ߜࢺ໔Ϟᘗ৤Ȃ Մ้ШفӖ౪ߜࢺ໔ۦҐԤᏰޱ௤ଆȂӰԪҏМᔣ௤ଆ዁

ጙၥଉᕗცήϞ้ШفӖ౪ߜࢺ໔Ȅζ൷࢐ᇳȂҏМ୅೩

׹ၥПਰϞέ໶ᡐኵȞ౪ߜࢺ໔ȃց౥Ѕፒց෈ኵȟࣱ࣏

዁ጙኵᐃȂиӨ෈౪ߜࢺ໔֕౪้ШفӖ࠮ᄘȂᔖҢ Wang[5] ܚ࡚ҳޟϲ໶዁ጙஅҏϴԒپؑڥڏ዁ጙ౪঄ϴ ԒȂٮᖞٽᇳ݂ԃդၼҢȄ

ΠȃѠڐۡШفӖ౪ߜࢺ໔

ӵհၥҏ׹ၥؚ๊ਢȂԤਢঐרঈོ࿦ڗ้ШفӖ࠮

ᄘϞ౪ߜࢺ໔ȄԃݎӨ෈౪ߜࢺ໔ӵ׹ၥॎგ෈໢֕౪࢚

ھۡԻϷШޟ׽ᡐȂࠌרঈᆎ၎࠮ᄘ౪ߜࢺ໔فӖ้࣏Ш فӖ[1-3]Ȅх E ߒۡШ঄ȂD ߒ಑Ι෈Ϟ౪ߜࢺ໔ȂA

n

ߒ

಑ n ෈Ϟ౪ߜࢺ໔Ȃࠌ้Ш౪ߜࢺ໔فӖϞӨ෈౪ߜࢺ໔ џоήԒߒϞȄ

( 1 + )

⁻¹

=

ⁿ

n

D E

A (1)

ԃݎ E>0 ࠌ၎࠮౪ߜࢺ໔֕౪ሎቨ౪ຫȇІϞȂE<0Ȃ ࠌ၎࠮౪ߜࢺ໔֕౪ሎ෵౪ຫȄۡШفӖ౪ߜࢺ໔فӖϞ

ࣺᜰϴԒӓԤڍ໶ȂΙ໶࣏ۡШ঄ E ᇄց౥ i ࣺӣϞϴԒȂ ѪΙ໶࣏ۡШ঄ E ᇄց౥ i ϚӣϞϴԒȂ౪ϷտߒҰԃή [1-3] Ȉ

( ) ( )

i E

i E D P

n n

−

 





 



 −

+ +

⋅

= 1 1 1

ʳ E ≠ (2) i

E D n P = ⋅ +

1 ! ! ! ! E = (3) i

έȃ዁ጙኵᏰஅᙃ

ԃ಑Ι࿽ࠉِܚक़Ȃ࿋רঈஈ՗׹ၥॎგຟեਢȂ҆

໸ᇔ໱ЅຟեࣺᜰᡐኵȂҥܻ౪ᄂΰܚ೎౩ޟၥଉԤࣺ࿋

σޟШٽࣥܖӒഋࣱ࣏዁ጙኵ঄ȂӰԪ҆໸ӑጂҳڎരդ ᆍלᄘኵ঄࣏዁ጙኵȄॶӑᄇ዁ጙኵȞfuzzy numberȟհۡ

ဎȂՄࡣӔۡဎᝒਿғȞ॒ȟ዁ጙኵȞstrictly positive Ȟnegativeȟ fuzzy number, SPȞNȟFNȟȄ

ۡဎ 1 [5]Ȉ A ~ ࣏ᄂኵጣΰϞ዁ጙ໱ӫȂ A ~

α

ߒ A ~ Ϟ α ᄠ໱

ӫȂԃݎ A ~ ᅖٗήӖనӇȂࠌרঈᆎ A ~ ࣏዁

ጙኵȄ

(1) A ~

_α

࣏ഖ໱ӫȂ ∀ α ∈ ( 0 , 1 ] Ȅ (2) A ~

_α

࣏Ԥࣨ໱ӫȂ ∀ α ∈ ( 0 , 1 ] Ȅ (3) A ~

_α

࣏я໱ӫȂ ∀ α ∈ ( 0 , 1 ] Ȅ

(4) Ԇӵ a ∈ Ȃٺு R A ~ ( ) a = 1 Ȅ

ۡဎ 2 [5]Ȉ A ~ ࣏዁ጙኵȄԃݎᄇӈդ a ≤ a 0 ≥ ( 0 ) Ȃོٺு

0 )

~ ( a =

A Ȃࠌרঈᆎ A ~ ࣏ᝒਿғȞ॒ȟ዁ጙኵȄ ҥܻ዁ጙኵϞၼᆗПԒԤӻᆍ[8]ȂҏМ௴Ңശೝኄ࣏

ٺҢޟПԒȂհ࣏዁ጙኵѲࠌၼᆗϞПݲȂ૭ۡဎԃήȄ

ۡဎ 3 [9,10]Ȉ A ~ ȃ B ~ ࣱ࣏዁ጙኵȂ ) A ~ ( x Ѕ B ~ y ( ) Ϸտ࣏

ڏ೿៉ᗵ឴࡙ڒኵȂցҢ Zadeh Ϟᘗ৤ন ౩[9]Ȃ A ~ ~ ~ ∗ B Ϟᗵ឴࡙ڒኵџҥήԒॎᆗ Մு

{ ^~ ⁽ ^), ^~ ⁽ ⁾ }

min sup )

~ (

~ ~ B z A x B y A ∗ =

z ∗=x y

዁ጙኵၼᆗӵஈ՗ΰٮϚৠܾȂ࣏ஈ՗ၼᆗώհоЅ ၼᆗࡣ዁ጙኵყלޟᛲᇧȂџо஠ҥӻᡐኵ዁ጙኵಢԙޟ ڒኵȂоୢ໢ϷݙȞinterval analysisȟޟПԒپ໌՗Ȃ໌՗

ПԒԃήȄ

ԃݎ f ࣏ۡဎӵ n ࡙ު໢ X

₁

× X

₂

× L × X

_n

ΰࢎৢڗᄂ ኵΰ R ϞڒኵȂи X ~ , X ~ , X ~

_n

2

1

L Ϸտߒ X

₁

, X

₂

, L X

_n

Ϟ዁ጙ

໱ӫȂԃݎ X ~ , X ~ , X ~

_n

2

1

L ࣏ۡဎ 1 ܚࣨۡޟ዁ጙኵȂࠌή ӖȞ4ȟԒ҆ۡԙҳ[11]Ȅ

α α

α

, ( ~ ) , , ( ~ ) ] [ ( ~ , ~ , , ~ )]

~ )

[( X

₁

X

₂

X

_n

f X

₁

X

₂

X

_n

f L = L (4)

ҥܻҏМٺҢڗ዁ጙԝᕻޟ྅܈Ѕ׬ѽȂӰԪӵή७ רঈϭಝ዁ጙԝᕻᆗυȞcontraction operatorȟޟۡဎȄ዁

ጙԝᕻϷ࣏ڍ࠮Ȃё࠮዁ጙԝᕻЅॸ࠮዁ጙԝᕻȂҥܻר ঈ༉ցҢڗॸ࠮዁ጙԝᕻυȂӰԪ༉ۡဎॸ࠮዁ጙԝᕻᆗ υԃήȈ

ۡဎ 4 [5]Ȉ A ~ ࣏ӈཎ዁ጙኵȂ r~ ࣏ SPFNȂх a

₁⁽^α⁾

, a

₂⁽^α⁾

Ϸ տߒ A ~ Ϟ α ᄠ໱ӫ A ~

α

ϞѾᆒᘈЅѡᆒᘈȂи

] ,

~ [

~

( )

) 2 1( ] 1 , 0 ( ]

1 , 0 (

α α α α

α

α A α a a

A = ⋅ = ⋅

∈

U U ȇ᜸խӴȂ

r

1(α)

, r

2(α)

Ϸտߒ r~ Ϟ α ᄠ໱ӫ r~ ϞѾᆒᘈЅѡ

_α

ᆒᘈȂи = ⋅ =

∈ α

α

α r

r ~

~

] 1 , 0

U

(

⋅

∈

α

U

(0,1]

[ r

₁⁽^α⁾

, r

₂⁽^α⁾

] Ȅ ԃݎήӖనӇԙҳ

α

α α α α α α α

α α α α α α

2

) (2 ) 1(

) 2( ) 2( ) 2(

) 1( )

1( ) 2(

) 1( ) 1( ) 1(

) 2(

1

max , min ,

a

r a a r r a r

r a r r a r

=

 



 





≤ 

 



 





= 

и a

1α

࣏ሎቨȂa

2α

ሎ෵Ȃࠌ

 





 





 





 





 





 



⋅ 

=

∈

) 2( ) 1(

) 2( ) 2( ) 2(

) 1( ] 1 , 0 (

) 1( ) 2(

) 1( ) 1( ) 1(

) 2(

~

, min

, ,

max

~ ) (

α α α α α α α

α α α α α

α

r a a r r r

r a a r r A r

C

r

U

ᆎ࣏ԝᕻ้઻࣏ r~ Ϟॸ࠮዁ጙԝᕻᆗυȄ

࿋ۡဎ 4 ϛϞ A ~ ࣏ SPFN ਢȂڏॸ࠮዁ጙԝᕻᆗυџ

(3)

0 1 2 3 4 n~

D~

^D^~^~^×

( )

¹^~⁺^E^~¹^~

^D^~^~^×

( )

¹^~⁺^E^~^~²

^D^~^~^×

( )

¹^~⁺^E^~³^~

^…

…

^D^~^~^×

( )

¹^~⁺^E^~^}ⁿ^~⁻¹

ᙏϽԃή[5]Ȉ

( )

^{( )}( ) ( ) ( ) ( ) ( ) ( ]

U

0,1 2 2 1 1 1

~

2

,

∈

 





 



⋅ 

=

α

α α α α α

α

a

r a r r A r

C

r

(5)

Ѳȃ዁ጙۡШ౪ߜࢺ໔فӖ

1.౪঄ᗵ឴࡙ڒኵϞ௰Ᏺ

዁ጙۡШ౪ߜࢺ໔فӖ዁Ԓ߽Ѡڐ౪ߜࢺ໔فӖ዁Ԓ ϞΙૡϽȞgeneralizationȟȂζ൷࢐ᇳȂ዁ጙۡШ౪ߜࢺ໔ فӖ዁ԒӵܚԤᡐኵࣱ࣏ጂۡ঄Ϟ੫ۡޑݷήȂ։࣏Ѡڐ ౪ߜࢺ໔فӖ዁ԒȄ዁ጙۡШ౪ߜࢺ໔فӖ዁ԒϞ୅೩ԃ ήȈ

( Ι) ዁ጙۡШفӖ౪ߜࢺ໔዁ԒήϞܚԤᡐኵࣱ࣏዁ጙ ኵȇζ൷࢐ᇳȂց౥ȃፒց෈ኵȃ಑ 1 ~ ෈౪ߜࢺ໔о ЅۡШ঄Ȃࣱ࣏዁ጙኵȄ

( Π) ዁ጙց౥ i~ ȃ዁ጙፒց෈ኵ n~ оЅ಑ 1 ~ ෈౪ߜࢺ໔ D~ Ȃࣱ࣏ SPFNȇՄ዁ጙۡШ঄ E ~ Ϛ࢐ SPFNȂ൷࢐

SNFN Ȅ

( έ) ௃಑ 1 ~ ෈ଔՍ಑ n ~ ( = n ~ × 1 ~ ) ෈ХȂؐΙ෈Ϟ዁ጙ౪ߜ ࢺ໔ၶࠉΙ෈Ϟ዁ጙ౪ߜࢺ໔ቨё E ~ ॻȄ

( Ѳ) ዁ጙۡШفӖ౪ߜࢺ໔዁Ԓ௴෈ґᄛٽȞend-of-period convention ȟȄ

፜ݧཎȂҥ୅೩ 3 ޣȂؐΙ෈Ϟ዁ጙ౪ߜࢺ໔ၶࠉΙ

෈Ϟ዁ጙ౪ߜࢺ໔ቨё E ~ ॻȂ࢈዁ጙ้Ш౪ߜࢺ໔فӖϞ Ө෈౪ߜࢺ໔џߒ࣏

}^~

~

~ ~ ( 1 ~ ~ )

1

~

_n

= D × + E

ⁿ−

A (6)

ڧ३ܻጡᒮ೺ᡝђ૖ϚٗȂ࣏ᗗջᇲ၌Ȟ6ȟԒޟཎဎȂ घۡߒ }

^~

B ߒ B~ Ȃڏϛ B ३࣏ۡΙߝၼᆗԒȂٽԃ Bɶ3ɮ6 ܖ BɶKɮ1Ȅਲ਼ᐃΰक़୅೩Ȟ1ȟՍ୅೩Ȟ4ȟȂרঈ஠዁ጙ

ۡШفӖ౪ߜࢺ໔዁ԒყҰ࣏ყ 1Ȅ

ӈΙ๋዁ጙ౪ߜࢺ໔ F~ Ȃڏ n~ ෈ࠉϞΙԩЛп౪঄

P~ ȂџҥήӖϴԒॎᆗ[5-7]Ȉ

( )

( ) ^ ^ 



 





= +

+i

i

n

C F

P

1~~_n

1 ~ ~

~

~ ~

~

(7)

ӰԪȂ዁ጙۡШفӖ౪ߜࢺ໔Ϟ౪঄џоήӖПแԒ ёоॎᆗȈ

( )

^{( )}

( ) ( )

( )

^}

( ) _ ^ ^







 



 +

+ + ×

 +













 +

+ + ×

















= +

− +

+ +

n n i

i i

i E C D

i C D P

n ~

1

~~ 1

2~

~~ 1 1

~ ~ 1~

~ ~ 1

~ ~

~ 1

~ ~

~ ~ 1 ~

~ ~

~ 1

~ ~

~ ~ 1

~ ~

~

2~ 1~

L

(8)

ყ 1! ዁ጙۡШفӖ౪ߜࢺ໔ყ

ցҢȞ5ȟԒЅୢ໢ϷݙၼᆗПԒȂџо஠Ȟ8ȟԒӵ ӈཎᗵ឴࡙ ^α ^∈ [ ] ⁰ ^, ¹ ਢϞ߬ᒦୢ໢Ȟinterval of confidenceȟ Ͻᙏ࣏Ȉ

( )

(

( )

)

^{( )}

( ) ( )

(

( )

)

^{( )}

 







+

−

⋅ +

 







 







 −













 +

+ ,

1 1 1

1 1

1

1 1

1

1 α α

α α

α

i e

d i

e

n

( )

(

( )

)

^{( )}

( ) ( )

(

( )

)

^{( )}

^ ^ ^ _

 +

−

⋅ +

 







 







 −













 +

+

α

α α α

α α

α

2

2 1

2 2

2 1

2 2

1 1 1

1 1

i e

d i

e

n

! ! (9)

ҥܻȞ9ȟԒޟѾᆒᘈӵ 1 + e

₁⁽^α⁾

= ( 1 + i

₁⁽^α⁾

)

¹⁽¹^α⁾

ਢฒཎ ဎȇ᜸խӴȞ9ȟԒޟѡᆒᘈӵ 1 + e

₂⁽^α⁾

= ( 1 + i

₂⁽^α⁾

)

¹⁽²^α⁾

ਢһฒ ཎဎȂ࢈ԪΠᆍޑݷȂџցҢᛳ҆ႀݲࠌȞL’Hospital Ruleȟ [12] پ೎౩ȂџϷտᡐ࣏ nd

₁⁽^α⁾

1 + e

₁⁽^α⁾

Ѕ nd

₂⁽^α⁾

1 + e

⁽₂^α⁾

Ȃᆣ ӫΰक़Ϸݙ๖ݎுޣȂۡШفӖ౪ߜࢺ໔౪঄ȂོӰ዁ጙ

ۡШ঄Ϟᗵ឴࡙ڒኵᇄ዁ጙց౥Ϟᗵ឴࡙ڒኵࣺҺᇄ֏Մ ԤϚӣޟॎᆗПԒȂרঈџ஠ڏϷ࣏έᆍޑݷȄή७Ӗю

၎έᆍޑݷӵӈΙᗵ឴࡙ α ήϞ߬ᒦୢ໢Ȉ

(1) ࿋ 1+ ~ E~ ޟѾᗵ឴࡙ڒኵᇄ ( 1 ~ + ~ i )

¹^~

ޟѾᗵ឴࡙ڒኵϚࣺ

ҺȂи 1+ ~ E~ ޟѡᗵ឴࡙ڒኵᇄ ( 1 ~ + ~ i )

¹^~

ޟѡᗵ឴࡙ڒኵ һϚࣺҺਢ

( )

(

( )

)

^{( )}

( ) ( )

(

( )

)

^{( )}

 







+

−

⋅ +

 







 







 −













 +

+ ,

1 1 1

1 1

1

1 1

1

1 α α

α α

α

i e

d i

e

n

( )

(

( )

)

^{( )}

( ) ( )

(

( )

)

^{( )}

^ ^ ^ _

 +

−

⋅ +

 







 







 −













 +

+

α

α α α

α α

α

2

2 1

2 2

2 1

2 2

1 1 1

1 1

i e

d i

e

n

! ! (10)

(2) ࿋ 1+ ~ E~ ᇄ ( 1 ~ + ~ i )

¹^~

१᠒ਢ

 





 





+ +

₂⁽ ⁾

) 2( ) 1(

) 1(

, 1

1

^α

α α α

e nd e

nd (11)

(4)

(3) ࿋ 1+ ~ E~ ޟѾᗵ឴࡙ڒኵᇄ ( 1 ~ + ~ i )

^~¹

ޟѾᗵ឴࡙ڒኵࣺ

ҺȂܖ 1+ ~ E~ ޟѡᗵ឴࡙ڒኵᇄ ( 1 ~ + ~ i )

¹^~

ޟѡᗵ឴࡙ڒኵ

ࣺҺਢ

( )

(

( )

)

^{( )}

( ) ( )

(

( )

)

^{( )}

 







+

−

⋅ +

 







 







 −













 +

+ ( լઌٌរ ) ࢨ

1 1 1

1 1

1

1 1

1

1 α α

α α

α

i e

d i

e

n

( ) ( )

( )

(

( )

)

^{( )}

^ ^ ^ _



 







 −













 +

+

+ 1

1 ), 1

1 (

¹²

2 2 1

1

n

i e e

nd

α α

α

ࣺҺᘈ

( ) ( )

(

( )

)

^{( )}

( )

 



 + +

−

⋅ + ( )

) 1 1 (

1

²

2 1

2 2

2

ϚࣺҺᘈ ܖ

^α_α

ࣺҺᘈ

α α

α

e

nd i

e d

(12) ӰԪభؑ዁ጙۡШفӖ౪ߜࢺ໔౪঄ყȂџࡸ 1+ ~ E~ ᇄ

1~

~ ) 1 ~

( + i ࣺҺޑݷȂپᘈᒵΰक़έᆍޑݷޟ࢚ΙಢϴԒپॎ

ᆗӈΙᗵ឴࡙ α ޟѾȃѡᆒᘈȄ೿௥ܚԤ α ঄ޟѾᆒᘈȂ

։џᛲюѾᗵ឴࡙ڒኵȇ೿௥ܚԤ α ঄ޟѡᆒᘈȂ։џᛲ юѡᗵ឴࡙ڒኵȄα ঄ޟᒵڥູ஝໱Ȃࠌܚᛲᇧᗵ឴࡙ڒ ኵζཕҁྤȄ

2.዁ጙ౪঄዁ԒϞᓺؾܒ

ҥጂۡኵ঄ܚॎᆗՄுϞ዁ጙۡШفӖ౪ߜࢺ໔౪঄

࣏Ιጂۡ঄ȂณՄҥ዁ጙኵ঄ܚॎᆗՄுϞ዁ጙۡШفӖ ౪ߜࢺ໔౪঄ࠓ࣏Ι዁ጙኵ঄Ȅҥܻרঈџ૖ᕕுϞၥଉ ငல࣏዁ጙኵ঄ȂӰԪࠎ஼о௰กܖեॎՄுϞጂۡኵ঄

پեᆗ౪঄Ȃࠌџ૖ོഅԙᒿᇲޟؚ๊ȂӰ࣏೨ӻџ૖ี

ҡиԤҢϞၥଉȂӵௌоጂۡܒ዁Ԓپեᆗ౪঄ਢȂϐೝ

ᙏϽΟȂӰԪܚॎᆗՄுϞ౪঄Ȟጂۡ঄ȟོԤᇲᏲؚ๊

ޱϞџ૖ȄࣺІޟȂԃݎௌоᎌҢܻ዁ጙᕗცϞ዁Ԓپե ᆗ౪঄ਢȂܚӖΣॎᆗϞኵ঄ϑѓ֤࿋ਢܚԤޟၥଉȂӰ ԪϚོԤᇲᏲؚ๊ޱϞџ૖ܒȄ

ࢋณоᎌҢܻ዁ጙᕗცϞ዁Ԓپեᆗ౪঄ၶ࣏࡬࿋Ȃ רঈᔖԃդپஈ՗ڹȉרঈޣၾȂӈཎ዁ጙኵϞ α ᄠ໱

ӫȂߒ၎዁ጙኵϞ߬ᒦ࡙࣏ α Ϟୢ໢ጒ൜Ȃһߒ၎௰ۡϞ

઻ኵ࣏ α [13]ȂӰԪרঈџоࡸএਰޑݷȂᎌ࡙Ӵᒵᐅᎌ ӫޟ௰ۡ઻ኵ α پॎᆗ዁ጙۡШفӖ౪ߜࢺ໔౪঄Ϟ߬ᒦ

࡙ୢ໢Ȃоց׹ၥؚ๊Ȅ

Ϥȃភ! ٽ

ή७רঈᖞΙএٽυᇳ݂ԃդॎᆗٮᛲᇧ዁ጙۡШ౪ ߜࢺ໔فӖ౪঄Ϟᗵ឴࡙ڒኵȄ

೩࢚׹ၥПਰϞࣺᜰኵᐃࣱ࣏዁ጙኵ঄ȂϷտԃήȈ

170000 17500 18000 18500 19000 19500 20000 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ყ 2! ዁ጙۡШفӖ౪ߜࢺ໔౪঄ყ

಑ 1 ~ ԑۻϞ዁ጙ౪ߜࢺ໔ D~ Ȉ

 

 



≤

−

≤

= −

200 , 2

$ 000 , 2 200 $ 11 1

000 , 2

$ 800 , 1

$ 200 9

~ 1

x x

x

D x !

዁ጙց౥ i~ Ȉ

 



−

= −

x i x

100 8

6 ~ 100 !

08 . 0 07 . 0

07 . 0 06 . 0

≤

≤ x x

዁ጙፒց෈ኵ n ~ = n × ~ 1 ~ Ȉ

 



−

= − x n x

11 ~ 9 !

11 10

10 9

≤

≤ x x

዁ጙۡШ঄ E~ Ȉ

 



−

= −

x E x

100 8

6 ~ 100 !

08 . 0 07 . 0

07 . 0 06 . 0

≤

≤ x x

؏᡽ΙȈॶӑॎᆗ 1+ ~ E~ ޟѾȞѡȟᗵ឴࡙ڒኵ࢐֏ᇄ

1~

~ ) 1 ~

( + i ޟѾȞѡȟᗵ឴࡙ڒኵࣺҺȄ

ҥΰ७ܚ๝ޟኵᐃޣ 1+ ~ E~ ޟѾȃѡᗵ឴࡙ڒኵϷտ࣏

106 100 −

= x

α ʳ 1 . 06 ≤ x ≤ 1 . 07 (13) x

100 108 −

α = ʳ 1 . 07 ≤ x ≤ 1 . 08 (14) Մ ( 1 ~ + ~ i )

¹^~

ޟѾȃѡᗵ឴࡙ڒኵฒݲоᡗڒኵߒҰȂՄ

ѫ૖оᗴڒኵϷտߒҰ࣏

0 )

06 . 1 01 . 0

( α +

⁰^.¹^α⁺⁰^.⁹

− x = ! 1 . 06

⁰^.⁹

≤ x ≤ 1 . 07 (15) 0

) 01 . 0 08 . 1

( − α

¹^.¹⁻⁰^.¹^α

− x = ! 1 . 07 ≤ x ≤ 1 . 08

¹^.¹

(16) ҥ(13)ԒЅ(15)Ԓؑ၌џு x = 1 . 07 , α = 1 ȇӔҥȞ14ȟ ԒЅȞ16ȟԒؑ၌һџு x = 1 . 07 , α = 1 Ȅ࢈ x = 1 . 07 , α = 1

࣏ 1 + ~ E~ ᇄ ( 1 ~ + ~ i )

¹^~

Ϟ୲ΙࣺҺᘈȄ

؏᡽ΠȈᛲᇧ዁ጙۡШ౪ߜࢺ໔فӖ౪঄Ϟᗵ឴࡙ڒኵ

(5)

ҥ؏᡽ΙϷݙޣȂ዁ጙۡШ౪ߜࢺ໔فӖ౪঄ϞѾȃ ѡᗵ឴࡙ڒኵӵᗵ឴࡙ α = 1 ਢԤ 1 + ~ E~ ᇄ ( 1 ~ + ~ i )

¹^~

ԤࣺҺ ϞޑݷȂ࢈רঈџޢ௥ٺҢ಑Ѳ࿽ϛ಑έᆍޑݷϞᅋᆗ ݲȄӵᗵ឴࡙ α = 1 ਢޟѾȃѡᆒᘈϷտ࣏ 18691.6 Ѕ 18691.6 ȇ ӵ ᗵ ឴ ࡙ α = 0 . 9 ਢ ޟ Ѿ ȃ ѡ ᆒ ᘈ ঄ Ϸ տ ࣏ 18590.11 Ѕ 18789.89Ȅҽΰक़հݲȂӔϷտؑюᗵ឴࡙࣏

0.8 Ȃ0.7Ȃ...Ȃ0.1 ਢϞѾȃѡᆒᘈȄՄࡣ೿௥ܚԤ α ঄ޟ ѾᆒᘈȂ։џᛲюѾᗵ឴࡙ڒኵȂ೿௥ܚԤ α ঄ޟѡᆒᘈȂ

։џᛲюѡᗵ឴࡙ڒኵȂ࿋ณȂα ঄ޟᒵڥູ஝໱Ȃࠌܚ ᛲᇧᗵ឴࡙ڒኵζཕҁྤȄ஠၎዁ጙۡШفӖ౪ߜࢺ໔౪

঄Ϟᗵ឴࡙ڒኵყҰԃყ 2Ȅ

ϲȃ๖! ፣

רঈܚ೎ޟᕗც࢐Ιএၥଉ዁ጙޟᕗცȂӰԪ࣏հؚ

๊Մᇔ໱ڗޟၥଉȂ܁܁ڎ዁ጙ੫፴Ȅӵ዁ጙၥଉޟᕗც ήȂԃݎרঈభ೎౩ޟၥҏႱᆗ୰ᚠȂڏ౪ߜࢺ໔֕౪้

ШفӖ࠮ᄘȂо༈ಛၥҏ׹ၥϷݙПԒ܁܁؂Йฒ๊ȂӰ ԪҏМ࡚ᄺΙৈ၌ؚ዁ጙ้ШفӖ౪ߜࢺ໔ޟПݲȄৈҢ ӵ಑Ѳ࿽ϛܚ௰ᏲՄுޟέ໶ϴԒȂ։Ϛᜲؑю዁ጙ้Ш فӖ౪ߜࢺ໔౪঄ӵӨᗵ឴࡙ϞѾѡᆒᘈȂ໌Մؑю዁ጙ

้ШفӖ౪ߜࢺ໔Ϟ౪঄Ѕ౪঄ყȂڏܚு๖ݎԤցܻӵ

዁ጙၥଉᕗცήȂհюၶ༈ಛПԒ؁ӫ౩ᇄ؁ᆠጂޟၥҏ Ⴑᆗؚ๊Ȅ

ಒဴષЕ

P ౪ߜࢺ໔Ϟ౪঄

A

n

಑ n ෈౪ߜࢺ໔ D ಑Ι෈Ϟ౪ߜࢺ໔

E ౪ߜࢺ໔ം෈ԙߝޟԻϷШȂܖᆎۡШ঄

i ց౥Ȃܖശճџ௥ڧൢႍ౥

n

ፒց෈ኵ 1~ ዁ጙ 1

P~ ዁ጙ౪ߜࢺ໔Ϟ౪঄

F~ ዁ጙ౪ߜࢺ໔Ϟತ঄

( ) a

A~ ዁ጙኵ A~ ӵ a ঄Ϟᗵ឴࡙

A~

α

ӈཎ዁ጙኵ A~ Ϟ α ᄠ໱ӫ A ~

n^~

಑ n~ ෈౪ߜࢺ໔

D~ ಑ 1~ ෈Ϟ዁ጙ౪ߜࢺ໔ E~ ዁ጙۡШ঄

i~ ዁ጙց౥Ȃܖ዁ጙശճџ௥ڧൢႍ౥

n~ ዁ጙፒց෈ኵ

( )α

d

1

D~ Ϟ α ᄠ໱ӫޟѾᆒᘈ

( )α

d

2

D~ Ϟ α ᄠ໱ӫޟѡᆒᘈ

( )α

e

1

E~ Ϟ α ᄠ໱ӫޟѾᆒᘈ

( )α

e

2

E~ Ϟ α ᄠ໱ӫޟѡᆒᘈ

( )α

i

1

i~ Ϟ α ᄠ໱ӫޟѾᆒᘈ

( )α

i

2

i~ Ϟ α ᄠ໱ӫޟѡᆒᘈ

( )α

1

1~ Ϟ α ᄠ໱ӫޟѾᆒᘈ

( )α

`2

1 1~ Ϟ α ᄠ໱ӫޟѡᆒᘈ

୤ՃМᝦ

1. Blank, L. T. and Tarquin, A. J., Engineering Economy, McGraw-Hill, NY (1998).

2. Park, C. S., Contemporary Engineering Economics, Addison Wesley, Menlo Park (1997).

3. Steiner, H. M., Engineering Economic Principles, McGraw-Hill, NY (1996).

4. Buckley, J. J., “The Fuzzy Mathematics of Finance,”

Fuzzy Sets and Systems 21, Vol. 21, pp.257-273 (1987).

5. Wang, C. C., “The Fuzzy Mathematics of Engineering Economy,” Fuzzy Sets and Systems (revised), (2000).

6. Wang, C. C., “Some Appropriate Capital Budgeting Methods Applying to the Fuzzy Environment,” Submitted to the International Journal of Production Economics, (2000).

7. д߸റȂȶ዁ጙኵᐃήϞᝒਿۡ৯فӖ౪ߜࢺ໔ȷȂᆓ ౩ᇄفಛȂ಑ΜڢȂ಑Ѳ෈Ȃ಑505-532ॲȂ(2000)Ȅ 8. Klir, G. J. and Yuan, B., Fuzzy Sets and Fuzzy Logic:

Theory and Applications, Prentice Hall, NJ (1995).

9. Dubois, D. and Prade, H., “Fuzzy Real Algebra: Some Results,” Fuzzy Sets and Systems, Vol. 2, pp.327-348 (1979).

10. Zimmerman, H. J., Fuzzy Set Theory and Its Applications, Kluwer, Boston (1991).

11. Calzi, M. L., “Towards a General Setting for the Fuzzy Mathematics of Finance,” Fuzzy Sets and Systems, Vol. 35, pp.265-280 (1990).

12. ྆ᆰ঱ȂཌᑖϷȂ಑167ॲȂέҕਪ׋ȂѮѕ(1991)Ȅ 13. Kaufmann, A. and Gupta, M. M., Introduction to Fuzzy