Super-efficiency Models, Part II

Transcription

1 Super-efficiec Mdels, Part II Emilia Niskae The 4th f Nvember S steemiaalsi

2 Ctets. Etesis t Variable Returs-t-Scale (0.4) S steemiaalsi Radial Super-efficiec Case Prblems with Radial Super-efficiec Case N-radial Super-efficiec 2. Eample (Prblem 0.) 3. Summar 4. Hme assigmet

3 . Etesis t Variable Returs-t- Scale S steemiaalsi

4 I this part: The super-efficiec mdels are eteded t variable returs-t-scale mdels, The we will discuss abut the ifeasibilit prblems f these mdels, Ad we will itrduce a -radial mdel which des t have these ifeasibilit prblems S steemiaalsi

5 Etedig t Variable Retur-t- Scale mdels We eted ur aalsis t the variable returs-t-scale case b adiig the fllwig cveit cstraits t the mdels: 0, (0.5) S steemiaalsi

6 Radial Super-efficiec Case /3 SuperRadial-I-V (Iput-rieted Varibale RTS) * θ miθ θ ( i,...,m ) i subect t i,, r ( r,...,m ) r (0.6) (0.7) (0.8), 0, (0.9) S steemiaalsi

7 Radial Super-efficiec Case 2/3 Fr cmparis, the rigial Super Radial-I- C (0.) * θ mi θ ε es + θ,, s, Subect t s θ + + s, s, + All cmpets f, s- ad s+ are cstraied t be -egative, ε >0 is the usual - Archimedea elemet ad e is rw vectr with uit fr all elemets S steemiaalsi

8 Radial Super-efficiec Case 3/3 SuperRadial-O-V (Output-rieted Variable RTS) * η mi η (0.20), Subect t i i,..., m (0.2) i ( ), η r, r ( r,..., m ) 0, (0.22) (0.23) S steemiaalsi

9 Prblems with Radial mdels /3 The previusl preseted tw ew mdels suffer frm ifeasibilit uder the fllwig cditis: If i ma >, { } i The the cstrait (0.8) i SuperRadial-I-V is ifeasible fr r i, b dit f the cstrait (0.9). S steemiaalsi

10 Prblems with Radial mdels 2/3 Likewise, If i mi <, { } i The the cstrait (0.2) i SuperRadial-O- V is ifeasible fr ri b dit f the cstrait (0.23) S steemiaalsi

11 Prblems with Radial mdels 3/3 Prpsiti 0.4 SuperRadial has feasible sluti if there eists r such satisfies the cditis preseted i the last tw slides The ifeasibilit ma ccur i ther cases t. S steemiaalsi

12 N-radial Super-efficiec Case /3 SuperSBM-V (The -radial super-efficiec uder variable returs-t-scale) is evaluated b slvig the fllwig prgram: m m i i δ * mi δ s s r i r r (0.24) Subect t, S steemiaalsi, 0, ad 0 ad 0,

13 N-radial Super-efficiec Case 2/3 Uder the assumptis X > 0 ad Y > 0, the SuperSBM-V is alwas feasible ad has a fiite ptimum i ctrast t the radial superefficiec. Prf: We chse a DMU ( ) with (, ), ad set ad 0 (k ). Usig this (, ) we defie: ~ ~ i r ma mi { i, i}( i,..., m ) {, }( r,..., s) r ( ) r k Thus, the set ~, ~, is feasible fr the SuperSBM-V. Hece, SuperSBM-V is alwas feasible with a fiite ptimum. S steemiaalsi

14 N-radial Super-efficiec Case 3/3 Therem 0. The -radial super-efficiec mdel uder the variable retur-t-scale evirmet, is alwas feasible ad has a fiite ptimum We ca defie Iput (Output)-Orieted Variable RTS mdel similar t SuperSBM-I-C (SuperSBM- O-C). We tice that SuperSBM-I-V ad SuperSBM-O-V mdels cfrt the same ifeasible LP issues as the SuperRadial-I-V ad SuperRadial-O-V. S steemiaalsi

15 S steemiaalsi Eample

16 Prblem 0. /2 Table displas data fr 6 DMUs (A, B, C, D, E, F) with tw iputs (, 2 ) ad tw utputs (, 2 ). Obtai ad cmpare the super-efficiec scres usig the attached DEAslver A B C F D Iput 2 / O utput E Iput / Output Table D M U A B C D E F I p u t ( ) ,5 I p u t 2 ( 2 ) ,5 O u tp u t ( ) O u tp u t 2 ( 2 ) S steemiaalsi

17 Prblem 0. 2a/2 Super-CCR-I θ, mi, s, s + θ ε es θ, + s, s +, s,, ε > 0 s + N. DMU Scre Rak Referece set (lambda) A, B, B, A 0,66765 C,47E-02 F 0, C, B 0,28574 D 0,42857 F 0, D,25 3 C 5 E 0,75 6 C 0,5 D 0,5 6 F 5 B 0,5 C 0,5 S steemiaalsi

18 Prblem 0. 2b/2 The prblem is the same as i eample i the chapter + utput I the eample: B>D>C>F>A>E Ad the crrespdig ratis,32 >,25-7,5ε >,2 > > - 4ε > 0,75 DMU A B C D E F Iput ,5 Iput ,5 Output A B F C E Iput Iput D S steemiaalsi

19 Prblem 0. 3/2 Super-CCR-O * θ φ * Ad *, s - *, s + * are adusted with θ* N. DMU Scre Rak Referece set (lambda) A, B 2 B, A 0, C,6E-02 F 0, C, B 0, D 0, F 0, D,25 3 C 0,8 5 E 0,75 6 C 0, D 0, F 5 B 0,5 C 0,5 S steemiaalsi

20 Prblem 0. 4/2 Super-SBM-I-C δ * I, mi i φ, + m i φ i m i φ i i ( i,..., m ) φ i,, r 0( i ) r ( r,... s ) N. DMU Scre Rak Referece set (lambda) A, B, B,25 A 3 C, B 0, D 0,30345 F 0, D,25 3 C 5 E 0, C 6 F 5 B 0,5 C 0,5 S steemiaalsi

21 Prblem 0. 5/2 Super-SBM-O-C δ * I, mi i ψ, s i s i ψ i ( i,..., m ) φ i,, r + 0( i ) r ψ i r ( r,... s ) N. DMU Scre Rak Referece set (lambda) A, B 2 B,2 2 A 0,6 C 0,6 3 C, B 0, D 0, F 5,4E-02 4 D,25 C 0,8 5 E 0,5 6 C 0, D 0, F 0, B 0,5 C 0,5 S steemiaalsi

22 Prblem 0. 6/2 SuperSBM-C δ * mi r i s,,,, m s m r i i r, 0 N. DMU Scre Rak Referece set (lambda) A, B 2 B,63793 A 0,6875 C 0,25 3 C, B 0, D 0, F 5,4E-02 4 D,25 3 C 5 E 0,42 6 B 0, C 0, F 0, B 0,5 C 0,5 S steemiaalsi

23 Prblem 0. 7/2 Super-BCC-I Ifeasibilit prblem: DMU A des t have a scre because it has the biggest value * θ miθ θ ( i,...,m ) i subect t i,, r, ( r,...,m ) r 0, N. DMU Scre Rak Referece set (lambda) A 4 Ifeasible LP 2 B, A 0,66765 C 0,04706 F 0, C, B 0,28574 D 0,42857 F 0, D,25 2 C 5 E 0,75 6 C 0,5 D 0,5 6 F 4 B 0,5 C 0,5 S steemiaalsi

24 Prblem 0. 8/2 Super-BCC-O Ifeasibilit prblems: DMU D has the smallest 2 value DMU B ad C (ust) d t have feasible sluti * η mi η η, i, r, i r ( i,..., m ) ( r,..., m ) 0, N. D M U S cre R a k R e fere ce set (la m b da) A, B 2 B 2 I feasib le L P 3 C 2 I feasib le L P 4 D 2 I feasib le L P 5 E 2 D 0,2 F 0,8 6 F 2 B 0,5 C 0,5 S steemiaalsi

25 Prblem 0. 9/2 Super-SBM-I-V Same feasibilit prblems as i Super-BCC-I N. DMU Scre Rak Referece set (lambda) A 4 Ifeasible LP 2 B,25 A 3 C, B 0, D 0,30345 F 0, D,25 2 C 5 E 0, C 6 F 4 B 0,5 C 0,5 S steemiaalsi

26 Prblem 0. 0/2 Super-SBM-O-V Same feasibilit prblems tha Super-BCC-O N. D M U S c re R a k R e fe re c e s e t (la m b d a ) A, B 2 B 2 I fe a s ib le L P 3 C 2 I fe a s ib le L P 4 D 2 I fe a s ib le L P 5 E 0, B 0,5 D 0,5 6 F 0, B 0,5 C 0,5 S steemiaalsi

27 Prblem 0. /2 Super-SBM-V N feasibilit prblems! δ m s r i m i * mi δ s r,, i r N. DMU Scre Rak Referece set (lambda) A, B 2 B, A 0, 3 C, B 0,34483 D 0,3035 F 0, D,25 3 C 5 E 0,42 6 B 0, C 0, F 0, B 0,5 C 0,5 0 0, S steemiaalsi

28 Prblem 0. 2/2 All the ratis cllected tgether: A B C D F F Super-CCR-I,333333,264706,7429,25 0,75 Super-CCR-O,333333,264706,7429,25 0,75 Super-SBM-I-C,66667,25,2438,25 0, Super-SBM-O-C,42857,2,088235,25 0,5 0,57429 Super-SBM-C,42857,63793,088235,25 0,42 0,57429 Super-BCC-I NA,264706,7429,25 0,75 Super-BCC-O, NA NA NA Super-SBM-I-V NA,25,2438,25 0, Super-SBM-O-V,42857 NA NA NA 0, ,57429 Super-SBM-V,42857,249994,2436,25 0,42 0,57429 S steemiaalsi

29 S steemiaalsi Summar

30 Summar We eteded ur super-efficiec mdels t the variable returs-t-scale mdels, Ad discussed abut the feasibilit prblems, Ad fiall we itrduced -radial superefficiec uder variable returs-t-scale Which des t suffer ifeasibilit-prblems! Ad we leart that DEA-slver is mre useful tha Ecel S steemiaalsi

31 S steemiaalsi Hme assigmet

32 Hme assigmet a) Eplai which e d u thik is mre efficiet i the eample, DMU A r B? (5 pits) b) Which e f the methds d u thik is best e t u use ad wh? (5 pits) Or Hw abut ew DMU G? Calculate the Super- V ratis whe (, ) are (7 3;2 ) (0 pits) S steemiaalsi