What is Data Envelopment Analysis?

1 data Envelopment analysis with Maple in OR and Modeling CoursesIntroductionWhat is data Envelopment analysis ? data Envelopment analysis (DEA), occasionally calledfrontieranalysis, was first put forward by Charnes, Cooper and Rhodesin 1978. It is a performance measurement technique which canbe used for evaluating the relative efficiency of decision-makingunits (DMU s) in organisations. John Beasley From the field of combustion engineering, efficiency is theratio of the actual amount of heat liberated .. to the maximumamount which could be liberated . A. Charnes, W. Cooper,and E. RhodesData Envelopment analysis with Maple in OR and Modeling CoursesExampleBatter DataWhom to trade?

2 Choose from among three players: data :Player A: 100 at-bats for 40 singles & 0 home runsPlayer B: 100 at-bats for 20 singles & 5 home runsPlayer C: 100 at-bats for 10 singles & 20 home runsAnalysis:Player A: no combination ofBandCcan equalAPlayer B:44%A+ 25%C=Bfor a69% efficiency index Player C: no combination ofAandBcan equalCData Envelopment analysis with Maple in OR and Modeling CoursesExampleGraphical analysis (2-D) data Envelopment analysis with Maple in OR and Modeling CoursesStrengths & LimitationsData Envelopment analysis :Strengths & LimitationsDEA Strengths:Multiple input and multiple output modelsComparison against combinations of peersInputs and outputs can have very different unitsDEA Limitations:Extreme point technique noise in the data can causesignificant errorEstimates relative, not absolute efficiencyComputationally intensiveData Envelopment analysis with Maple in OR and Modeling CoursesData Envelopment analysis DefinedDEA: Nonlinear Representation ( I-O Ratio )Let:nnumber ofDMUsninnumber of input measuresuiweight factor for inputixikinputiforDMUknoutnumber of output measuresviweight factor for outputiyikoutputiforDMUkekefficiency ofDMUkFor eachDMUj,j= ,definethe nonlinear program.

3 Choose~u,~vto maximizeejsubject toek=nout i=1viyiknin i=1uixikk= ek 100%k= 0k= 0k= Envelopment analysis with Maple in OR and Modeling CoursesData Envelopment analysis DefinedDEA: As a Linear Program ( Input Oriented )For each DMU,i= ,define the linear programMinimize subject to~XInputs ~ Xi 0~MOutputsj ~ MOutputsj,i 0j= i 0i= ( ican be interpreted as a dual variable.) data Envelopment analysis with Maple in OR and Modeling CoursesArts & Sciences at ASUThe ProblemArts & Sciences at AppalachianTask: Analyze the 16 disparate Envelopment analysis with Maple in OR and Modeling CoursesArts & Sciences at ASUI nitial DataInitial Arts & Sciences DataFall.

4 Institutional Research, Assessment, & Planning, ASUData Envelopment analysis with Maple in OR and Modeling CoursesArts & Sciences at ASUS caled DataLinearly Scaled DataReturn to ResultsData Envelopment analysis with Maple in OR and Modeling CoursesArts & Sciences at ASUData GraphGraphical analysis (2-D) data Envelopment analysis with Maple in OR and Modeling CoursesLinear ProgramLinear Programming AnalysisTheMath Sciences(DMU #11) linear program is:Minimize subject to~XInputs ~ 0~MOutputsj ~ MOutputsj,11 0, j= i 0, i= Envelopment analysis with Maple in OR and Modeling CoursesMapleSimplex in MapleArts & Sciences DEA Maple Setup:Usesimplex[minimize]orOptimization [LPSolve]DMU := ["Anthropology", "Biology".]

5 N := nops(DMU):MO := Matrix([[610, 204, ], [734, ..,[702, 231, ]]):eq1 := sum( [i],i= ) - 0:OV := Vector[row](N,symbol= ).MO: data Envelopment analysis with Maple in OR and Modeling CoursesMapleSimplex in Maple, IIArts & Sciences DEA Maple Computation:Results := NULL:for n from 1 to N doeq2 := OV[1] - M[n,1] 0: #credit hourseq3 := OV[2] - M[n,2] 0: #num studentseq4 := OV[3] - M[n,3] 0: #degreessys :={eq1,eq2,eq3,eq4}:s := simplex[minimize]( , sys, NONNEGATIVE);Maple Output FormResults := Results,[DMU[n],select(x (rhs(x)<>0),s))];end do:Matrix([Results]);ResultsData Envelopment analysis with Maple in OR and Modeling CoursesMapleDEA: Maple Simplex Outputn:= 5:eq2 := OV[1] MO[n, 1]:eq3 := OV[2] MO[n, 2]:eq4 := OV[3] MO[n, 3]:sys :={seq(eq||i, i= )}:DMU[n], simplex[minimize]( ,sys, NONNEGATIVE); English ,{ =.]}

6 7385773744, 1= 0., 2= , 3=0., 4= 0., 5= 0., 6= 0., 7= 0., 8= 0., 9= 0., 10= 0., 11= 0., 12= .5213512669, 13= 0., 14= .2071827164, 15= 0., 16= 0.}Back to CodeData Envelopment analysis with Maple in OR and Modeling CoursesMaple ResultsDEA: Maple Results for Arts & SciencesMatrix([Results]); Anthropology = 12= 14= Biology = 2= Chemistry = 2= Computer Science = 14= English = 12= 14= Foreign Lang & Lit = 12= 14= Geography & Planning = 2= 12= 14= Geology = 2= History = 12= 14= Interdisc Studies = 14= Mathematical Sci = 12= Philosophy & Religion = 12= Physics & Astronomy = 2= 12= Poli Sci/Crim Justice = 14= Psychology = 12= 14= Soc & Social Work = 12= 14= data Envelopment analysis with Maple in OR and Modeling CoursesMaple ResultsDEA.

7 Maple Results for Arts & SciencesDMU %~ 1 Anthropology79 12= , 14= Biology100 2= Chemistry74 2= Computer Science49 2= , 14= English74 2= , 12= , 14= Foreign Lang & Lit87 12= , 14= Geography & Planning74 2= , 12= , 14= Geology63 2= History91 12= , 14= Interdisc Studies51 12= , 14= Sci79 12= Philosophy & Religion100 12= Physics & Astronomy73 2= , 12= , 14= Poli Sci/Crim Justice100 14= Psychology86 2= , 12= , 14= Soc & Social Work92 12= , 14= Envelopment analysis with Maple in OR and Modeling CoursesExcelResults in ExcelFor implementing simplex using Solver in Excel, see MacDonald, or Beasley, Envelopment analysis with Maple in OR and Modeling CoursesExcel: Sorted ResultsExcel Sorted ResultsBack to DataData Envelopment analysis with Maple in OR and Modeling CoursesStudent ProjectsA Simple Student ProjectChoose a college, division, or school:Identify DMUs:colleges, divisions, or departmentsDefine inputs:Frequency Distribution of SAT Verbal, Math,and Writing Scores for Enrolling FreshmenDefine outputs:number of majors and degrees awardedPerform a data Envelopment analysis with 3 inputs and2 outputsData Envelopment analysis with Maple in OR and Modeling CoursesStudent ProjectsA Fun Student ProjectChoose a college, division, or schoolIdentify DMUs.

8 Departments, academic areas, or groupsDefine inputs:number of faculty: professor, associate, assistant;number of non-tenure-track faculty: full-time, part-time; numberof graduate students: GTAs, RAs, fellows; number of staff;operating budget; number of classrooms; number oflaboratories; number of offices; total assignable sq ft; etc. (14)Define outputs:number of majors; number of degrees awarded;number of sections; student credit hours produced; number ofpublications/books; number of conference presentations;number of grant proposals: submitted, awarded; number ofexternal committees; number of professional org offices held;etc. (10)Perform a data Envelopment analysisGive a copy to the Envelopment analysis with Maple in OR and Modeling CoursesBibliographySelected ReferencesCharnes A.

9 , Cooper and Rhodes E. (1978) Measuring the efficiency of decision making units, Eur. , 429 , A., Cooper, , Lewin, , Seiford, (Eds.) (1995) data Envelopment analysis : Theory,Methodology and Applications, G. (2002),A Bibliography Of data EnvelopmentAnalysis (1978 2001),RUTCOR Research Report, RRR01-02. Rutgers (3203 entries)