Fuzzy Data Envelopment Analysis (DEA)

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GHD lv dq dssurdfk irfxvhg rq iurqwlhuv lqvwhdg ri fhqwudo whqghqflhv1 Lw hydoxdwhv wkh h!flhqf| ri hdfk GPX uhodwlyh wr vlplodu GPXv1 Wkxv/ lw surylghv dq h!flhqw iurqwlhu ru hqyhorsh iru doo frqvlghuhg GPXv udwkhu wkdq wwlqj d uhjuhvvlrq sodqh wkurxjk wkh fhqwhu ri wkh gdwd1

(19)

wkdw/ wdujhw/ GPX1

Dowhuqdwlyhv iru pdnlqj hdfk lqh!flhqw GPX h!flhqw fdq eh vhhq e| surmhfwlqj wkhp rqwr wkh h!flhqw iurqwlhu1

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+4,Lqsxw0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr plqlpl}h wkh lqsxw xvdjh wr surgxfh jlyhq rxwsxw ohyhov iru hdfk GPX1

+5,Rxwsxw0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr surgxfh wkh kljkhvw srvvleoh rxwsxw ohyhov iru d jlyhq lqsxw xvdjh iru hdfk GPX1

+6,Edvh0Rulhqwhg1 Zlwk wklv rulhqwdwlrq/ zh wu| wr surgxfh wkh rswlpdo xvdjh ri lqsxwv dqg wkh rswlpdo surgxfwlrq ri rxwsxwv vlpxowdqhrxvo|1 Erwk lqsxwv dqg rxwsxwv fdq eh frqwuroohg lq d Edvh0Rulhqwhg prgho1 Iljxuh 514 looxvwudwhv wkh surmhfwlrq iru hdfk rulhqwdwlrq iru wkh fdvh ri rqh lqsxw dqg rqh rxwsxw1

I: Input-Oriented Projection O: Output-Oriented Projection B: Base-Oriented Projection

O u t p u t

I n p u t

B I

O Efficient Frontier

(20)

51414 Gdwd hqyhorsphqw dqdo|vlv prghov

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v ' d+&oof? lv wkh pdwul{ ri rxwsxwv/

(21)

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Lqsxw0rulhqwhg FFU prgho

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(22)

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(23)

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Lqwhusuhwdwlrq ri GHD prghov

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(24)

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+4, wW ' 

(25)

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(26)

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(27)

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lqfuhdvlqj wkh rxwsxwv )J e| wkh rxwsxw vkruwidoov/ tnW

OhwE R

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Wkh surmhfwhg srlqw lv rewdlqhg iurp=

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lpsuryhphqw phwkrg lv fdoohg wkh FFU surmhfwlrq1

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(28)

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(29)

zklfk lv wkh EFF prgho e| Edqnhu/ Fkduqhv/ dqg Frrshu ^7`1 Wkh surgxfwlrq iurqwlhu +hqyhorsh,iru wkh EFF prgho lv vsdqqhg e| wkh frqyh{ kxoo ri doo GPXv1 Wkh iurqwlhu lv slhfhzlvh olqhdu dqg frqfdyh/ zklfk ohdgv wr yduldeoh uhwxuqv wr vfdoh1 Iljxuh 515 dqg Iljxuh 516 vkrz wkh h!flhqw iurqwlhuv ri wkh FFU dqg wkh EFF prghov/ uhvshfwlyho|/ zkhq wkhuh lv rqo| rqh lqsxw dqg rqh rxwsxw1

Output

Input Production Frontier

Production Possibility Set

Iljxuh 515= Surgxfwlrq Iurqwlhu +Hqyhorsh,ri wkh FFU prgho

Output

Input Production Possibility Set Production Frontier

IRS

DRS

CRS

(30)

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Lqsxw0rulhqwhg EFF prgho

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frxqwhusduwv +FFU dqg GFFU prghov,1 Wkhuhiruh/ wkh h!flhqf| ydoxhv zloo eh fulvs qxpehuv1 Wkh fulvs frxqwhusduwv ri wkh IFFU dqg GIFFU prghov duh

4@ AU+ J

AU% J ' 

A%nA+ f

f f

(57)

4? w r| wU%

J %f

+U+ J

f

+617,

Wkh ghix}}lfdwlrq dssurdfk zloo eh looxvwudwhg e| xvlqj dq h{dpsoh wdnhq iurp Jxr dqg Wdqdnd ^5<`1 Wdeoh 614 surylghv wkh gdwd iru wkh h{dpsoh1 Lw frqvlvwv ri wzr ix}}| lqsxwv dqg wzr ix}}| rxwsxwv/ doo ri zklfk kdyh v|pphwulfdo wuldqjxodu phpehuvkls ixqfwlrqv1 Wkh phpehuvkls ixqfwlrqv duh ghqrwhg e| ESc _ zkhuh S lv wkh fhqwhu dqg _ lv wkh vsuhdg ri wkh phpehuvkls ixqfwlrqv1

Wdeoh 614= GPXv zlwk wzr ix}}| lqsxwv dqg wzr ix}}| rxwsxwv1

GPXE 4 5 6 7 8

lqsxw 4 +713/318,+51</313,+71</318,+714/31:,+918/319, lqsxw 5 +514/315,+418/314,+519/317,+516/314,+714/318, rxwsxw 4 +519/315,+515/313,+615/318,+51</317,+814/31:, rxwsxw 5 +714/316,+618/315,+814/31;,+81:/315,+:17/31<,

Iurp Wdeoh 614/ wkh ghix}}lfdwlrq ri wkh ix}}| lqsxwv dqg rxwsxwv surgxfhv wkh fulvs lqsxwv dqg rxwsxwv dv vhhq lq Wdeoh 6151

Wdeoh 615= Fulvs ydoxhv iurp ghix}}lfdwlrq ri ix}}| lqsxwv dqg rxwsxwv ri doo GPXv1

GPX E 4 5 6 7 8

lqsxw 4 713 51< 71< 714 918

lqsxw 5 514 418 519 516 714

rxwsxw 4 519 515 615 51< 814

rxwsxw 5 714 618 814 81: :17

(58)

iurp Wdeoh 616 vkrz wkdw wkh h!flhqw GPXv +l1h1/ wW '  zlwk }hur vodfn,duh GPXv

5/ 7 dqg 8/ zkloh wkh lqh!flhqw GPXv duh GPXv 4 dqg 61 Wkh uhihuhqfh vhw ri GPX 4 lv frpsrvhg ri GPXv 5 dqg 8/ zkloh wkh uhihuhqfh vhw ri GPX 6 lv frpsrvhg ri GPXv 5/ 7 dqg 81

Wdeoh 616= Rswlpdo vroxwlrqv iru doo GPXv xvlqj fulvs lqsxwv dqg fulvs rxwsxwv1

GPXE wW b b2 b be bD

4 31;88 3 413<; 3 3 31369

5 4W 3 4 3 3 3

6 31;94 3 4156< 3 31384 31397

7 4W 3 3 3 4 3

8 4W 3 3 3 3 4

- Vodfn yduldeohv/ Er3Wc rnWc ri GPXv 5/ 7 dqg 8 duh }hur1

Frpsduhg zlwk wkh ix}}| udqnlqj phwkrg ri Jxr dqg Wdqdnd ^5<`/ wkh dgydqwdjhv ri wkh ghix}}lfdwlrq dssurdfk duh vlpsolflw| lq lpsohphqwdwlrq dqg lqwhusuhwdwlrq ri wkh uhvxowv1 Krzhyhu/ wkh xqfhuwdlqw| lq lqsxwv dqg rxwsxwv +l1h1/ srvvleoh udqjh ri ix}}| lqsxwv dqg rxwsxwv dw glhuhqw k ohyhov,lv hhfwlyho| ljqruhg1

61515

k

,ee,

edvhg dssurdfk

(59)

α µ

input 0

X~

[ L

] U

Iljxuh 616= Lqwhuydo ydoxh iru d ix}}| lqsxw dw k ohyho1

Phwkrg 4= Ehvw0Ehvw

Zlwk wklv phwkrg/ hyhu| GPX lv frqvlghuhg lq dq rswlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh vpdoohvw lqsxwv dqg wkh odujhvw rxwsxwv duh xvhg iru hyhu| GPX1 Wklv uhsuhvhqwv wkh ehvw srvvleoh frpelqdwlrqv ri lqsxwv dqg rxwsxwv iru hyhu| GPX1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw dq| jlyhqk ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w 'AE)h JLk

AEh

Juk ' 

AEh

JuknAE)hJLk f iru GPXJ

AEh

uk nAEh)Lk fc ' c c ? dqg9'J

f fc

+618,

zkhuhEhu

k lv wkh froxpq yhfwru ri wkh plqlpxp ydoxhv ri wkh fruuhvsrqglqj ix}}| vhwv

dfklhyhg dw wkh jlyhqkohyho/ dqg vlploduo|/EhL

(60)

wkh GIFFU prgho iru dq| k ohyho fdq eh rewdlqhg lq d vlplodu zd|/ dqg wkhuhiruh lv qrw vkrzq khuh1

Phwkrg 5= Zruvw0Zruvw

Xvlqj wklv phwkrg/ hyhu| GPX lv frqvlghuhg lq d shvvlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh odujhvw lqsxwv dqg wkh vpdoohvw rxwsxwv duh xvhg iru hyhu| GPX1 Wklv uhsuhvhqwv wkh zruvw srvvleoh frpelqdwlrqv ri lqsxwv dqg rxwsxwv iru hyhu| GPX1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw d jlyhq k ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w'AEh) Juk

AEh

JLk ' 

AEh

JLk nAE)hJuk f iru GPXJ

AEh

Lk nAE)huk fc ' c c ? dqg9'J

f f

Phwkrg 6= Ehvw0Zruvw

Xqghu wklv phwkrg/ GPXJ lv frqvlghuhg lq dq rswlplvwlf zd| exw doo rwkhu GPXv

duh frqvlghuhg lq d shvvlplvwlf zd|1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfk k ohyho/ wkh vpdoohvw lqsxwv dqg wkh odujhvw rxwsxwv duh xvhg iru GPXJczkloh wkh odujhvw

lqsxwv dqg vpdoohvw rxwsxwv duh xvhg iru doo rwkhu GPXv1 Wkxv/ zlwk wkhvh lqsxwv dqg rxwsxwv/ GPXJ zloo kdyh wkh odujhvw srvvleoh h!flhqf| ydoxh zkhq frpsduhg

(61)

4@ w 'AE)h JLk

AEh

Juk ' 

AEh

JuknAE)hJLk f iru GPXJ

AEh

Lk nAE)huk fc ' c c ? dqg9'J

f f

+619,

Phwkrg 7= Zruvw0Ehvw

Lq frqwudvw wr Phwkrg 6/ wkh qdo phwkrg lv edvhg rq d shvvlplvwlf ylhzsrlqw iru GPXJ1 Iurp wkh lqsxw dqg rxwsxw lqwhuydov dw hdfkk ohyho/ wkh odujhvw lqsxwv dqg

vpdoohvw rxwsxwv duh xvhg iru GPXJc zkloh wkh vpdoohvw lqsxwv dqg odujhvw rxwsxwv

duh xvhg iru doo rwkhu GPXv1 Wkxv/ GPXJ zloo kdyh wkh vpdoohvw srvvleoh h!flhqf|

ydoxh dprqj wkh irxu phwkrgv1 E| xvlqj wklv phwkrg/ wkh IFFU prgho dw d jlyhqk ohyho ehfrphv wkh iroorzlqj olqhdu surjudpplqj prgho1

4@ w'AEh) Juk

AEh

JLk ' 

AEh

JLk nAE)hJuk f iru GPXJ

AEh

uk nAEh)Lk fc ' c c ? dqg9'J

f f

+61:,

Wr looxvwudwh wkh k,ee, edvhg dssurdfk/ wkh vdph h{dpsoh dv jlyhq lq Wdeoh 614 lv xvhg1 Wkh uhvxowv iurp wkh diruhphqwlrqhg irxu phwkrgv iru yh glhuhqw k ohyhov +3/ 3158/ 318/ 31:8/ 4,duh surylghg lq Wdeohv 617/ 618/ 619/ dqg 61:/ uhvshfwlyho|1

(62)

Wdeoh 617= Uhvxowv ri Ehvw0Ehvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31;<< 31<;8 41333 41333 41333 3158 31<4< 41333 41333 41333 41333 3183 31<3< 41333 31<8; 41333 41333 31:8 31;;: 41333 31<3: 41333 41333 4133 31;88 41333 31;94 41333 41333

Wdeoh 618= Uhvxowv ri Zruvw0Zruvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31:;; 41333 319<< 41333 31;4: 3158 31;33 41333 31:67 41333 31;9; 3183 31;46 41333 31::5 41333 31<54 31:8 31;59 41333 31;47 41333 31<:9 4133 31;88 41333 31;94 41333 41333

Dw hdfkk ohyho/ wkh lqwhuydo h!flhqf| fdq eh frqvwuxfwhg iurp wkhwW ydoxhv iurp

Phwkrg 6 dqg Phwkrg 71 Wkhq/ wkhvh lqwhuydov iru glhuhqw k ohyhov fdq eh xvhg wr exlog wkh fruuhvsrqglqj ix}}| h!flhqf| phpehuvkls ixqfwlrqv iru hdfk GPX1 Wkh frqvwuxfwlrq ri wkh phpehuvkls ixqfwlrq xvlqj wkh wW ydoxhv iurp Phwkrg 6 dqg

Phwkrg 7 lv vkrzq lq Iljxuh 6171 Wkh phpehuvkls ixqfwlrqv lq Iljxuh 617 vkrz wkh odujhvw srvvleoh vsuhdgv iru doo GPXv ehfdxvh zh frpelqh wkh prvw shvvlplvwlf dqg rswlplvwlf ylhzsrlqwv1

(63)

Wdeoh 619= Uhvxowv ri Ehvw0Zruvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 41333 41333 41333 41333 41333 3158 41333 41333 41333 41333 41333 3183 31<96 41333 41333 41333 41333 31:8 31<37 41333 31<65 41333 41333 4133 31;88 41333 31;94 41333 41333

Wdeoh 61:= Uhvxowv ri Zruvw0Ehvw phwkrg ri k,ee, edvhg dssurdfk1

k wW

GPX4 GPX5 GPX6 GPX7 GPX8 3133 31957 31;69 318:4 31;88 3196; 3158 31:35 31<3; 31975 31<76 31:68 3183 31:8; 31<<3 31:49 41333 31;78 31:8 31;3: 41333 31:<4 41333 31<9< 4133 31;88 41333 31;94 41333 41333

E| xvlqj wkh k,ee, edvhg dssurdfk/ irxu dowhuqdwlyhv +phwkrgv,iru d ghflvlrq pdnhu wr hydoxdwh wkh h!flhqf| ri GPXv duh=

Li wkh ghflvlrq pdnhu lv rswlplvwlf derxw hdfk GPX/ wkhq Phwkrg 4 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/

Li wkh ghflvlrq pdnhu lv rswlplvwlf derxw wkh wdujhw GPX +GPXJ, / wkhq

Phwkrg 6 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/ +l1h1/ wkh prvw rswlplvwlf phwkrg,

Li wkh ghflvlrq pdnhu lv shvvlplvwlf derxw hdfk GPX/ wkhq Phwkrg 5 fdq eh xvhg wr hydoxdwh wkh h!flhqf|/ dqg

Li wkh ghflvlrq pdnhu lv shvvlplvwlf derxw wkh wdujhw GPX +GPXJ, / wkhq

(64)

ylhz,1

Wkhk,ee, edvhg dssurdfk surylghv ghflvlrq pdnhuv zlwk dowhuqdwlyhv wr hydo0 xdwh wkh h!flhqf| ri GPXv1 Lw fdq surylgh ix}}| h!flhqf| exw uhtxluhv wkh udqnlqj ri ix}}| h!flhqf| vhwv1

(65)

Figure

Updating...

References