Data Envelopment Analysis with Maple 13 Bill Bauldry Boone, NC [email protected]

ACA ’09 ´ ETS, Montreal

Data Envelopment Analysis with Maple 13

Abstract We will present the “Data Envelopment Analysis” (DEA) technique using Maple 13. DEA is an operations research tool that uses the simplex algorithm to evaluate the relative efficiency of ‘decisionmaking units’ (DMU’s) in an organization. We will illustrate DEA by applying the method to analyze the 16 departments of the College of Arts & Sciences at Appalachian State University. Computer algebra is best at easily handling the large scale computations intrinsic to DEA studies. The College of Arts & Sciences DEA analysis, a simplified example, requires 16 simplex optimizations each over 16 variables with 3 constraints. A significant DEA study would likely involve 30 or more simplex optimizations with a much larger number of variables and of constraints for each run. The original Input-Output ratio form of DEA is nonlinear and is quite difficult to compute; Maple makes it feasible to attempt. ACA ’09: 2 – 35

Data Envelopment Analysis with Maple 13 Introduction

What is Data Envelopment Analysis? “Data Envelopment Analysis (DEA), occasionally called frontier analysis, was first put forward by Charnes, Cooper and Rhodes in 1978. It is a performance measurement technique which can be used for evaluating the relative efficiency of decision-making units (DMU’s) in organisations.” — John Beasley

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Data Envelopment Analysis with Maple 13 Introduction

What is Data Envelopment Analysis? “Data Envelopment Analysis (DEA), occasionally called frontier analysis, was first put forward by Charnes, Cooper and Rhodes in 1978. It is a performance measurement technique which can be used for evaluating the relative efficiency of decision-making units (DMU’s) in organisations.” — John Beasley “From the field of combustion engineering, ‘efficiency is the ratio of the actual amount of heat liberated . . . to the maximum amount which could be liberated’.” — A. Charnes, W. Cooper, and E. Rhodes

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Data Envelopment Analysis with Maple 13 Example

Batter Data Whom to trade?

Choose from among three players:

Data: Player A: 100 at-bats for 40 singles & 0 home runs Player B: 100 at-bats for 20 singles & 5 home runs Player C: 100 at-bats for 10 singles & 20 home runs

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Data Envelopment Analysis with Maple 13 Example

Batter Data Whom to trade?

Choose from among three players:

Data: Player A: 100 at-bats for 40 singles & 0 home runs Player B: 100 at-bats for 20 singles & 5 home runs Player C: 100 at-bats for 10 singles & 20 home runs Analysis: Player A: no combination of B and C can equal A

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Data Envelopment Analysis with Maple 13 Example

Batter Data Whom to trade?

Choose from among three players:

Data: Player A: 100 at-bats for 40 singles & 0 home runs Player B: 100 at-bats for 20 singles & 5 home runs Player C: 100 at-bats for 10 singles & 20 home runs Analysis: Player A: no combination of B and C can equal A Player C: no combination of A and B can equal C ACA ’09: 7 – 35

Data Envelopment Analysis with Maple 13 Example

Batter Data Whom to trade?

Choose from among three players:

Data: Player A: 100 at-bats for 40 singles & 0 home runs Player B: 100 at-bats for 20 singles & 5 home runs Player C: 100 at-bats for 10 singles & 20 home runs Analysis: Player A: no combination of B and C can equal A Player B: 44%A + 25%C = B for a 69% “efficiency index” Player C: no combination of A and B can equal C ACA ’09: 8 – 35

Data Envelopment Analysis with Maple 13 Example

Graphical Analysis (2-D)

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Data Envelopment Analysis with Maple 13 Strengths & Limitations

Data Envelopment Analysis: Strengths & Limitations DEA Strengths: Multiple input and multiple output models Comparison against combinations of peers Inputs and outputs can have very different units

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Data Envelopment Analysis with Maple 13 Strengths & Limitations

Data Envelopment Analysis: Strengths & Limitations DEA Strengths: Multiple input and multiple output models Comparison against combinations of peers Inputs and outputs can have very different units DEA Limitations: Extreme point technique—noise in the data can cause significant error Estimates relative, not absolute efficiency Computationally intensive ACA ’09: 11 – 35

Data Envelopment Analysis with Maple 13 Data Envelopment Analysis Defined

DEA: Nonlinear Representation (‘I-O Ratio’) Let: n number of DMUs nin number of input measures ui weight factor for input i

For each DMUj , j = 1..n, define the nonlinear program: Choose ~u, ~v to maximize ej subject to nout X

xik input i for DMUk ek = nout number of output measures vi weight factor for output i yik output i for DMUk ek efficiency of DMUk

i=1 nin X

vi yik k = 1..n ui xik

i=1

0 ≤ ek ≤ 100%

k = 1..n

uk ≥ 0

k = 1..nin

vk ≥ 0

k = 1..nout ACA ’09: 12 – 35

Data Envelopment Analysis with Maple 13 Data Envelopment Analysis Defined

DEA: As a Linear Program (‘Input Oriented’) For each DMU, i = 1..n, define the linear program minimize θ subject to ~ Inputs · ~λ − X ~ iθ ≤ 0 X ~ Outputs · ~λ − M ~ Outputs ≥ 0 M j j,i

j = 1..n

and λi ≥ 0

i = 1..n

(λi can be interpreted as a dual variable.) ACA ’09: 13 – 35

Data Envelopment Analysis with Maple 13 Arts & Sciences at ASU The Problem

Arts & Sciences at Appalachian

Task: Analyze the 16 disparate departments. ACA ’09: 14 – 35

Data Envelopment Analysis with Maple 13 Arts & Sciences at ASU Initial Data

Initial Arts & Sciences Data

Fall, 2006. Source: Institutional Research, Assessment, & Planning, ASU ACA ’09: 15 – 35

Data Envelopment Analysis with Maple 13 Arts & Sciences at ASU Scaled Data

Linearly Scaled Data

Return to Results ACA ’09: 16 – 35

Data Envelopment Analysis with Maple 13 Arts & Sciences at ASU Data Graph

Graphical Analysis (2-D)

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Data Envelopment Analysis with Maple 13 Linear Program

Linear Programming Analysis The Math Sciences (DMU #11) linear program is: minimize θ subject to

~ Outputs M j

~ Inputs · ~λ − θ ≤ 0 X ~ Outputs · ~λ − M ≥ 0, j,11 λi ≥ 0,

j = 1..16

i = 1..16

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple Arts & Sciences DEA Maple Setup: Use simplex[minimize] or Optimization[LPSolve] DMU := ["Anthropology", "Biology", ... N := nops(DMU):

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple Arts & Sciences DEA Maple Setup: Use simplex[minimize] or Optimization[LPSolve] DMU := ["Anthropology", "Biology", ... N := nops(DMU): MO := Matrix([[610, 204, 3.56], [734, ..., [702, 231, 4.35]]):

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple Arts & Sciences DEA Maple Setup: Use simplex[minimize] or Optimization[LPSolve] DMU := ["Anthropology", "Biology", ... N := nops(DMU): MO := Matrix([[610, 204, 3.56], [734, ..., [702, 231, 4.35]]): eq1 := sum(λ[i],i=1..N) - θ ≤ 0:

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple Arts & Sciences DEA Maple Setup: Use simplex[minimize] or Optimization[LPSolve] DMU := ["Anthropology", "Biology", ... N := nops(DMU): MO := Matrix([[610, 204, 3.56], [734, ..., [702, 231, 4.35]]): eq1 := sum(λ[i],i=1..N) - θ ≤ 0: OV := Vector[row](N,symbol=λ). MO:

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple, II Arts & Sciences DEA Maple Computation: Results := NULL: for n from 1 to N do eq2 := OV[1] - M[n,1] ≥ 0: # credit hours eq3 := OV[2] - M[n,2] ≥ 0: # num students eq4 := OV[3] - M[n,3] ≥ 0: # degrees sys := {eq1,eq2,eq3,eq4}: s := simplex[minimize](θ, sys, NONNEGATIVE); Maple Output Form

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Data Envelopment Analysis with Maple 13 Maple

Simplex in Maple, II Arts & Sciences DEA Maple Computation: Results := NULL: for n from 1 to N do eq2 := OV[1] - M[n,1] ≥ 0: # credit hours eq3 := OV[2] - M[n,2] ≥ 0: # num students eq4 := OV[3] - M[n,3] ≥ 0: # degrees sys := {eq1,eq2,eq3,eq4}: s := simplex[minimize](θ, sys, NONNEGATIVE); Results := Results, [DMU[n],select(x→(rhs(x)0),s))]; end do: Matrix([Results]);

Results ACA ’09: 24 – 35

Data Envelopment Analysis with Maple 13 Maple

DEA: Maple Simplex Output n:= 5: eq2 := OV[1] ≥ MO[n, 1]: eq3 := OV[2] ≥ MO[n, 2]: eq4 := OV[3] ≥ MO[n, 3]: sys := {seq(eq||i, i=1..4)}: DMU[n], simplex[minimize](θ,sys, NONNEGATIVE); ”English”, {θ = .7385773744, λ1 = 0., λ2 = 0.01004339126, λ3 = 0., λ4 = 0., λ5 = 0., λ6 = 0., λ7 = 0., λ8 = 0., λ9 = 0., λ10 = 0., λ11 = 0., λ12 = .5213512669, λ13 = 0., λ14 = .2071827164, λ15 = 0., λ16 = 0.} Back to Code ACA ’09: 25 – 35

Data Envelopment Analysis with Maple 13 Maple Results

DEA: Maple Results for Arts & Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

DMU Anthropology Biology Chemistry Computer Science English Foreign Lang & Lit Geography & Planning Geology History Interdisc Studies Mathematical Sci Philosophy & Religion Physics & Astronomy Poli Sci/Crim Justice Psychology Soc & Social Work

θ% 79 100 74 49 74 87 74 63 91 51 79 100 73 100 86 92

~λ = 0.43

λ12 = 0.35, λ14 λ2 = 1.00 λ2 = 0.74 λ2 = 0.08, λ14 = 0.41 λ2 = 0.01, λ12 = 0.52, λ14 λ12 = 0.53, λ14 = 0.34 λ2 = 0.20, λ12 = 0.26, λ14 λ2 = 0.63 λ12 = 0.61, λ14 = 0.30 λ12 = 0.04, λ14 = 0.47 λ12 = 0.79 λ12 = 1.00 λ2 = 0.44, λ12 = 0.27, λ14 λ14 = 1.00 λ2 = 0.06, λ12 = 0.11, λ14 λ12 = 0.38, λ14 = 0.54

= 0.21 = 0.28

= 0.02 = 0.69 ACA ’09: 26 – 35

Data Envelopment Analysis with Maple 13 Maple Results

Sorted Results

Back to Data ACA ’09: 27 – 35

Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups

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Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; number of offices; total assignable sq ft; etc. (14)

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Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; 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 of publications/books; number of conference presentations; number of grant proposals: submitted, awarded; number of external committees; number of professional org offices held; etc. (10)

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Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; 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 of publications/books; number of conference presentations; number of grant proposals: submitted, awarded; number of external committees; number of professional org offices held; etc. (10) Perform a data envelopment analysis ACA ’09: 31 – 35

Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; 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 of publications/books; number of conference presentations; number of grant proposals: submitted, awarded; number of external committees; number of professional org offices held; etc. (10) Perform a data envelopment analysis Give a copy to the Dean.

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Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; 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 of publications/books; number of conference presentations; number of grant proposals: submitted, awarded; number of external committees; number of professional org offices held; etc. (10) Perform a data envelopment analysis Give a copy to the Dean. Run.

ACA ’09: 33 – 35

Data Envelopment Analysis with Maple 13 A Project for Brave Students

A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups Define inputs: number of faculty: professor, associate, assistant; number of non-tenure-track faculty: full-time, part-time; number of graduate students: GTAs, RAs, fellows; number of staff; operating budget; number of classrooms; number of laboratories; 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 of publications/books; number of conference presentations; number of grant proposals: submitted, awarded; number of external committees; number of professional org offices held; etc. (10) Perform a data envelopment analysis Give a copy to the Dean. Run. Hide.

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Data Envelopment Analysis with Maple 13 Bibliography

Selected References Charnes A., Cooper W.W. and Rhodes E. (1978) “Measuring the efficiency of decision making units,” Eur. J. Opl. Res. 2, 429–444. Charnes, A., Cooper, W.W., Lewin, A.Y., Seiford, L.M. (Eds.) (1995) Data Envelopment Analysis: Theory, Methodology and Applications, Springer. Tavares G. (2002), A Bibliography Of Data Envelopment Analysis (1978–2001), RUTCOR Research Report, RRR 01-02. Rutgers University. http://rutcor.rutgers.edu/pub/rrr/reports2002/1 2002.pdf (3203 entries) ACA ’09: 35 – 35