DATA ENVELOPMENT ANALYSIS

Mat-2.4142 Seminar n Optimizatin DATA ENVELOPMENT ANALYSIS Scale Elasticity and Cngestin 14.11.2007

Cntents Intrductin Scale Elasticity in Prductin Cngestin Strng Cngestin Hme Assignment

A quantitative analysis n the subject f scale elasticity (=returns t scale) in prductin is intrduced Situatin: DMUs face increasingly limited supply f resurces Cmpetitin is fierce and persistent Cnsequences: Significant emphasis n the efficient utilizatin and allcatin f resurces Larger perating units t achieve the advantages f scale ecnmies Estimatin f scale elasticity (returns t scale) parameter is f particular imprtance cncerning whether there is any scpe fr increased prductivity by e.g. expanding/cntracting

Scale elasticity in prductin is defined as the rati f marginal prduct (MP) and average prduct (AP) single input/utput case ρ > 1 increasing returns t scale ρ = 1 cnstant returns t scale ρ< 1 decreasing returns t scale Scale elasticity is well defined at a pint n the efficient prtin f input-utput crrespndence (e.g. the pint A) The pints n which inefficient DMUs are perating are prjected t the efficient frntier

Measuring the scale elasticity f DMUs in multiple input-utput envirnment Prductin pssibility set P BCC related t the multiple input-utput crrespndence is defined as fllws: Analgus t the single input-utput case, we measure the scale elasticity f DMUs psitined n the efficient prtin f P BCC Such prjectin can be realized by slving the fllwing utput-riented BCC mdel fr each DMU T slve the BCC prblem tw-stage prcess is emplyed: 1.Obtain θ * BCC 2. Maximize the sum f by slving the prgram[bcc - O] input slacks s and utput slacks s while keeping the bjective functin value at θ * BCC - i + r (see ch.3)

Prjecting inefficient DMUs nt the efficient frntier Let an ptimal slutin f * [BCC - O] be ( θ BCC *, λ, s *, s + * ) DMU (x θ * BCC, = 1, s y * ) is called"strngly efficient"if = 0 and s + * and nly if = 0 hld fr every ptimal slutin fr [BCC - O] Otherwise, DMU (x, y) can be brught t nt the strngly efficient status by the BCC - O prjectin defined as fllws: Hereafter, it is assumed that this peratin has been perfrmed s that DMU (x, y ) is strngly efficient

The scale elasticity measure is derived frm the dual prgram f BCC mdel * * * Als, fr ptimal slutin (v,u, ω ) : We define the virtual input ( ξ ) and virtual utput ( η) assciated with (x, y) P BCC by :

Measuring scale elasticity in the presence f multiple ptima The upper (lwer) scale elasticity in prductinρ ( ρ) is calculated by :

Cngestin means decreased utput(s) with increased input(s) S far input slacks (excesses) have been cnsidered free The set PBCC allws input x Xλ fr prducing an utput y= Yλ Under this assumptin the scale elasticity is nnnegative Hwever, there are cases in which an increase in ne r mre inputs causes the wrsening f ne r mre utputs In rder t deal with such situatins, prductin pssibility set is mdified as fllws:

The degree f cngestin can be measured by ρ Inefficient DMUs are prjected nt the efficient frntier: The dual prgram f [Cngestin-strng] is described as fllws:

The multiple ptima case The dual prgram t [Upper-w] can be described as fllws:

(Strng) cngestin is recgnized by having negative prductin elasticity If ρ< 0 : Next, we make the fllwing definitins: Then, we get: DMU (x, y different activity (x ) is in the regin f "cngestin" where there exists ^ ^, y ) that uses less inputs t prduce mre utputs

Cnclusins Definitin 12.1 (Strng Cngestin) A DMU (x ~ such that x Therem 12.2 A DMU (x, = αx, y y ) is strngly cngested if (with 0< α < 1) and y ~ there exists an activity (x βy (with β > 1). ) is in the status f strng cngestin, if and nly if the upper scale elasticity ( ρ) as measured by scheme[upper - w]is ~, ~ y ) P cnvex negative.

Hme Assignment Slve the [BCC-O] mdel * t btain θ BCC values fr DMUs B and F (5p) Are DMUs B and F BCC-efficient Cngested? (5p) DMU A B C D E F G Input (x) 2 3 4 6 7 8 9 Output (y) 1 3 4 5 5 4 2