Mat-2.4142 Alternative DEA Models 19.9.2007
Table of Contents Banker-Charnes-Cooper Model Additive Model Example Data Home assignment
BCC Model (Banker-Charnes-Cooper) production frontiers spanned by convex hull of existing DMUs Includes convexity condition n j=1 j =1, j 0, j Piecewise linear and concave characteristics
Production frontier of CRR / BCC Models CCR Model BCC Model => subset of CCR Model
Evaluating efficiency CCR Model PQ PD = 2,25 4 =0,5625 BCC Model PR PD = 2,6667 =0,6667 4 generally: CCR-efficiency does not exceed BCC-efficiency
Output orientated BCC Model ST DT = 5 3 =1,6667 achievement of efficiency: D needs increase output by 1.6667 3=5 comparable CCR model: reciprocal of its input inefficiency Note: 1/0,5625=1,7778 simple reciprocal relation between input an output not available at BCC model!
The BCC Model Production possibility set defined by: P B ={ x, y x Xλ, yyλ, eλ=1, λ0} where X = x j R m n and Y =Y j R s n are given data set R n e and is a row vector with all elements equal to 1. n Adjunction of condition: j =1 j=1 can be written as e where e is a row vector with all elements unity and is a column vector with all elements non negative
The BCC Model Together with the condition j 0 this imposes a convexity condition, for all j Linear program to solve input-orientated BCC model
The BCC Model The dual multiplier form of this linear program is expressed as: where v and u are vectors and z and u 0 are scalars u 0 free in sign. Main difference between CCR/BBC models: free u 0
Solving BCC Model Similar CCR model: solved in 2 Steps 1) Minimize B 2) Maximize the sum of the input excesses and * output shortfalls, keeping B = B B * (optimal value found in phase one) is not less then then optimal value * (CCR model)
BCC-Efficiency A optimal Solution for B *, *, s *,s * satisfies B * =1 and has no slack s * =0,s * =0 => then the DMU is BCC-efficient otherwise the DMU is called BCC-inefficient
The output-orientated BCC model basic model: dual (multiplier) form:
The additive Model (ADD) Combines both (input-orientated and outputorientated models) Same production possibility set as CCR + BCC models and their variants Treats the slacks (input excesses and output shortfalls) directly in the objective function
The additive Model (ADD) The basic additive model selected: input excesses and output shortfalls included in the target function
The additive Model (ADD) The dual problem can be expressed as follows:
The additive Model (ADD) - maximal value of s - and s + is attained at B - considers input excess and output shortfall simultaneously
The additive Model (ADD) Definition ADD-efficient DMU DMU is ADD-efficient if and only if: s * =0 and s * =0
Data and Results Example DMU Input Output BCC Additive Model x y * s * Ref. A 2 1 1 0 1 C B 3 3 1 0 0 B C 2 2 1 0 0 C D 4 3 0.75 1 0 B E 6 5 1 0 0 E F 5 2 0.4 3 0 C G 6 3 0.5 3 0 B H 8 5 0.75 2 0 E s *
Data and Results Example DMU Input Output BCC Additive Model x y * s * Ref. A 2 1 1 0 1 C B 3 3 1 0 0 B C 2 2 1 0 0 C D 4 3 0.75 1 0 B E 6 5 1 0 0 E F 5 2 0.4 3 0 C G 6 3 0.5 3 0 B H 8 5 0.75 2 0 E s * BCC + ADD efficient DMUs
Home assignment Sometimes it is suggested that the weak efficiency value,, * be used to rank DMU performances as determined from CCR or BCC models. Assignment: Discuss possible shortcomings in this approach. (Problem 4.1)