Data envelopment analysis

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Data envelopment analysis

  1. 1. DEPARTMENT OF COMMERCE DEA NORTH BENGAL UNICVERSITY 26-27 FEBRUARY, 2010DATA ENVELOPMENT ANALYSIS A QUANTITATIVE TECHNIQUE TO MEASURE EFFICIENCY Click Mouse for Next
  2. 2. Single Input and Single OutputUnits Inputs Outputs Output/ Input A 2 1 0.5 B 3 3 1 C 3 2 0.666667 D 4 3 0.75 E 5 4 0.8 F 5 2 0.4 G 6 3 0.5 H 8 5 0.625 66 6 5 55 Efficient 4 Frontie r Output 44 Regres sion Line Output 3 33 2 1 22 0 11 Input Input 0 2 4 6 8 10 00 0 0 11 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9
  3. 3. Two Input and Single Output Units A B C D E F G H IInput-1(x1) 4 7 8 4 2 5 6 5.5 6Input-2(x2) 3 3 1 2 4 2 4 2.5 2.5Output (y) 1 1 1 1 1 1 1 1 1
  4. 4. 4.5 4.5 4 G 4 E G 3.5 E 3.5 Ax2/y Ax2 /y 3 3 B B H 22.5 .5 H I I 2 2 F F D D Efficient Frontier Line 1. 5 1.5 1 1 C 0.5 C 0.5 0 00 1 2 3 4 5 6 7 8 9 0 2 4 x1/y x1/y 6 8 10
  5. 5. 4.5 4 G E 3.5 Ax2/y 3 B 2.5 P H I 2 F 1 .5 D 1 C 0.5 0 0 1 2 3 4 5 6 7 8 9 x 1/ y Efficiency of A = OP OA = 0.8571
  6. 6. Production Possibility Set
  7. 7. 4.5 4 G E 3.5 Ax2/y 3 A1 B 2.5 P H I 2 F 1 .5 D 1 C 0.5 0 0 1 2 3 4 5 6 7 8 9 x 1/ y Improvement of efficiency
  8. 8. ONE INPUT AND TWO OUTPUTSUnits A B C D E F GInput-1(x1) 1 1 1 1 1 1 1Output y1 1 2 3 4 4 5 6 Output y2 5 7 4 3 6 5 2
  9. 9. 8 y2/x7654321 y1/x0 0 2 4 6 8
  10. 10. 8 y2/x7 Efficient Frontier654321 y1/x0 0 2 4 6 8
  11. 11. OD B Efficiency − of − D = OP = o.750Q OA Efficiency − of − A = = 0.714 E OQ FA P C D G Production Possibility Set
  12. 12. MULTIPLE INPUTS AND MULTIPLE OUTPUTS Inputs Outputs Units Vector Vectors 1 x1 y1 2 x2 y2 − − − − − − n xm ys•Units in DEA are known as DMUs- Decision Making Units•Generally a DMU is regarded as the entity responsible for converting inputs into outputs.•Data are assumed to be positive•Measurement is unit invariant
  13. 13. ( DMUs j = 1 2 ...... n )  x11 x12 ... x1n     x21 x22 ... x21  X = . . ... .   x   m1 xm1 ... xmn    y11 y12 ... y1n     y21 y22 ... y21  Y = . . ... .   y   s1 ys2 ... ysn  
  14. 14. For each DMU we have to compute a ratioOutput Virtual Output = Input Virtual Input s ∑r yrk u θ= 1 m ; k = ,2,...., n 1 ∑i xik v 1 ur r = 1,2,..., s Output weights vi i = 1,2,..., m Input weights
  15. 15. For k-th DMU DMU k u1 y1k + u2 y2 k + .... + us yskθ= v1 x1k + v2 x2 k + ... + vm xmk
  16. 16. Fractional Programming ProblemMaximize u1 y1k +u2 y2 k +.... +u s ysk θ= v1 x1k + v2 x2 k +... + vm xmkSubject to u1 y11 + 2 y21 + + s y s1 u .... u ≤1 v1 x11 + 2 x21 + + m xm1 v ... v u1 y12 + 2 y22 + + s y s 2 u .... u ≤1 v1 x12 + 2 x22 + + m xm 2 v ... v ... ... ... ... ... ... u1 y1n + 2 y2 n + + s y sn u .... u ≤1 v1 x1n + 2 x2 n + + m xmn v ... vu1 , u2 , ..., us ≥ 0 v1 , v2 , ..., vm ≥ 0
  17. 17. Linear Programming Problem Of kth DMUMaximize θ = µ1 y1k + µ 2 y2 k + .... + µ s yskSubject to υ1 x1k + υ 2 x2 k + ... + υ m xmk = 1 µ1 y11 + µ 2 y21 + .... + µ s ys1 ≤ υ1 x11 + υ2 x21 + ... + υm xm1 µ1 y12 + µ2 y22 + .... + µ s ys 2 ≤ υ1 x12 + υ2 x22 + ... + υm xm 2 ... ... ... ... ... ... µ1 y1n + µ 2 y2 n + .... + µ s ysn ≤ υ1 x1n + υ2 x2 n + ... + υm xmn µ, 1 µ, 2 ..., µs ≥ 0 υ, 1 υ, 2 ..., υm ≥ 0
  18. 18. • This is what is known as CCR model of DEA • Proposed by Charnes, Cooper and Rhodes in 1978CCR Efficiency : DMUk is CCR-efficient if θ = 1 *1. and there exists at least one optimal (v*,u*) with v* >0 and u* >02. Otherwise DMU is CCR-inefficient
  19. 19. Numerical Example DMU A B C D E F Input x1 4 7 8 4 2 10 x2 3 3 1 2 4 1 Output y 1 1 1 1 1 1 LPP of DMU A C BMax θθθ=u uMax = =u Maxsubject to subject to84v1++v3v=2==117v1 32 2 1 v u u≤≤441v+++33222 u≤ 4vv11 3vvv u ≤ 7v1 + 3v2 u ≤ 7v1 + 3v2 u ≤ 7v + 3v u ≤ 8v1 + v 2 u ≤ 8v1 + v 2 1 2 u ≤ 10v1++2v22 u ≤ 4v1 + 2v2 4v1 v u ≤≤22v1++44v2 u v1 v2 u ≤≤4v1v+ + v2 u 10 2 v 1 2
  20. 20. LPP SOLUTION OF ALL DMUSDMU CCR(θ*) Reference v1 v2 u SetA 0.8571 D,E 0.1429 0.1429 0.8571B 0.6316 C,D 0.0526 0.2105 0.6316C 1 C 0.0833 0.3333 1D 1 D 0.1667 0.1667 1E 1 E 0.2143 0.1429 1F 1 C 0 1 1
  21. 21. • Optimal weights for an efficient DMU need not be unique• Optimal weights for inefficient DMUs are unique except when the line of the DMU is parallel to one of the boundaries.• If an activity (x,y)εP (Production Possibility set), then the activity (tx.ty) belongs to P for any positive scalar t.
  22. 22. CCR model and importance of Dual PRIMAL DUALmax uyk min θsubject to subject tovxk = 1 θxk − Xλ ≥ 0− vX + uY ≤ 0 Yλ ≥ ykv≥0 u≥0 λ ≥ 0 θ unrestricted
  23. 23. Input excesses and output shortfallsmin θsubject toθ k −Xλ≥0 xYλ≥ ykλ≥0 θ unrestrictmin θsubject toθ k −Xλ−s =0 x −Yλ−s + =ykλ≥0 θ unrestrict
  24. 24. LPP of shortfalls and excesses − +max w = es +essubject to −s =θ k − Xλ x k +s =Yλ − yk − +λ ≥0 s ≥0 s ≥0
  25. 25. COMPUTATIONAL PROCEDURE OF CCR MODELPhase-I: We solve the dual first to obtainӨ*. This Ө* is called “Farrell Efficiency” andoptimal objective value of LPPhase-II: Considering this Ө* as given, wesolve the LPP of output shortfalls andInput excesses. (max slack solution)
  26. 26. Refine Definition of CCR-EfficiencyIf an optimal solution (θ*,λ*,s-*,s+*) of twoLPs satisfies θ*=1 and s-*=0 and s+*=0 thenDMUk is CCR-Efficient.DMUDMU CCR(θ*) CCR(θ*) Reference Reference v1 v Excess2 u Shortfall Set Set S1 - S2 - S+AA 0.8571 0.8571 D,E D,E 0.1429 0 0.1429 0 0.8571 0BB 0.6316 0.6316 C,D C,D 0.0526 0 0.2105 0 0.6316 0CC 1 1 C C 0.0833 0 0.3333 0 1 0DD 1 1 D D 0.1667 0 0.1667 0 1 0EE 1 1 E E 0.2143 2 0.1429 0 1 0FF 1 1 C C 0 1 .6667 1 0
  27. 27. EXTENSION OF TWO-PHASETwo-Phase process aims to obtain the maximum sum of slacks(input excesses+output shortfalls). For the projection of aninefficient DMU on efficient frontier,Itxmay result a mix which is far DMU x1 x1 1 yfrom the observed mix A 1 2 1 1 B 1 1 2 1 C 2 10 5 1The Phase-III process is recently proposed (Tone K., “AnExtension of Two Phase Process”,ORS 1 D 2 5 10 Con, 1999 ) andimplemented in few software 2 10 groups similar DMUs in a E which 10 1subset within peer set.
  28. 28. INPUT ORIENTED AND OUTPUT ORIENTED MODELSWe discuss so far input-oriented models whose objectiveis to minimize inputs while producing atleast given level ofOutputs.There is another type of model that attempts to maximize outputs while using no more than the given (observed)inputs. This is known as output-oriented model.
  29. 29. Comparing Input-oriented CCR Model with Output-oriented CCR modelIf µ, η are variable vector of output oriented CCR problemthen at the optimum the following are the equivalences withOptimal solution of Input-oriented CCR problem λ* and θ*: * 1 η= * = λ µ θ * * θ *Slacks of the output-oriented model are related to the slacksof input-oriented models − t −* =s * t +* =s+* θ * θ*
  30. 30. It suggests that an input-oriented CCR model will be efficientfor any DMU iff it is also efficient when the output –orientedCCR model is used to evaluate its performance.
  31. 31. BCC MODELCCR model has been developed on the assumption of constantreturn to scale. A variation of DEA model is BCC (Banker,Charnes and Cooper) which is based on variable return to scale.• Increasing return to scale• Decreasing return to scale• Constant return to scaleThe BCC model has its production frontiers spanned by theConvex Hull of the existing DMUs
  32. 32. Production Frontier of BCC Model 6 6 Production Frontier of BCC Model 4 4Output 2 2 0 Input 0 0 5 10 0 2 4 I n put 6 8 10
  33. 33. Acknowledgement:Cooper, Seiford & Tone: Data Envelopment Analysis – A ComprehensiveText with Models, Applications, ReferencesKluwer Academic Publishers, London (Fifth Ed, 2004) Thanks

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