Data Envelopment Analysis of the College of Arts & Sciences

(1)

Data Envelopment Analysis of the College of Arts & Sciences

K. Corkery, J. Miller, J. Mims, M. Silver, J. Tabat, A. Wright, and Wm. C. Bauldry ([email protected])

Appalachian State University

INFORMS Conference on O.R. Practice Orlando, FL

April 18 – 20, 2010

(2)

DEA2010: 1

Abstract

We analyzed Appalachian’s College of Arts and Sciences us- ing multiple inputs (6) and multiple outputs (10) to index the 16 disparate departments. The problem addressed is to provide a relative measure to assist resource allocation decisions. The college’s current technique uses a single stratified metric: student credit hours produced per full-time faculty position. Our methodol- ogy can be adapted to any organization that seeks to aid decision making by creating a ranking index of departments in a unit.

Our main tool is “Data Envelopment Analysis” (DEA) originated by Charnes, Cooper, & Rhodes in 1978. The linearized forms of DEA employ a large number of applications of the simplex algorithm.

We used Maple

^TM

, a computer algebra system, for computations.

(3)

Our Project Team

The graduate “Introduction to Operations Research” class is the project team.

Back row: Misty Silver, Jesse Mims, Jessica Miller, Wm Bauldry Front row: Kayla Corkery, Andrew Wright, Jenny Tabat

(4)

DEA2010: 3

What is “Data Envelopment Analysis” (DEA)?

“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 (1978)

Data Envelopment Analysis is “a performance measurement technique which can be used for evaluating the relative efficiency of decision-making units (DMUs) in organizations.”

— J.E. Beasley

DEA produces an index, or relative efficiency rating, for each DMU considered based on optimizing the weights in ratios of weighted averages of multiple output factors to weighted averages of multiple input factors.

Efficiency =weighted sum of outputs weighted sum of inputs

Essentially, we ‘envelop’ the points of the data space in a convex hull, then consider a DMU’s relative distance to the boundary or frontier of the enveloping hull. Input and output factors may have widely varying units and magnitudes.

(5)

DEA Strengths & Limitations

DEA Strengths:

• Multiple input and multiple output models

• Comparisons are against combinations of ‘peers’

• Inputs and outputs can have very different units and magnitudes

• Easy analysis of a factor’s significance to the results is possible

DEA Limitations:

• DEA is an extreme point technique—noise in the data can cause significant error

• The results estimate relative, not absolute efficiency

• No recommendations for improving a DMU’s efficiency are provided

• The procedure is computationally intensive

(6)

DEA2010: 5

Original DEA Model (Charnes, Cooper Rhodes, 1978

¹

)

All inputs and outputs are non-negative; at least one input and one output of each DMU is strictly positive. Define:

Em: efficiency of the mth DMU yjm: jth output of the mth DMU vjm: weight of yjm

xim: ith input of the mth DMU vim: weight of xim

For each DMU (m = 1, . . . , N )

max Em= PJ

j=1vjmyjm

PI

i=1uimxim

subject to

0 ≤ PJ

j=1vjmyjn

PI

i=1uimxin

≤ 1; n = 1, 2, . . . , N

vjm, uim≥ 0; i = 1, 2, . . . , I; j = 1, 2, . . . , J

1W.W. Cooper, L.M. Seiford, and J. Zhu. DATA ENVELOPMENT ANALYSIS: History, Models and Interpretations, chapter 1, pages 1–39. Springer, 2nd edition, 2004.

(7)

CCR Linear DEA Model (Charnes, Cooper Rhodes, 1978)

For each DMU (m = 1, . . . , N )

max z =

J

X

j=1

vjmyjm

subject to

I

X

i=1

uimxim= 1

J

X

j=1

vjmyjn−

I

X

i=1

uimxin≤ 0; n = 1, 2, . . . , N

vjm, uim≥ 0; i = 1, 2, . . . , I; j = 1, 2, . . . , J

(8)

DEA2010: 7

Input- and Output-Oriented Linear DEA Models

The CCR models (or BCC models when theP

jλj= 1constraint is added):

Input-Oriented CCR Model Minimize θ_ksuch that

N

X

j=1 j6=k

λjxij≤ θ_kxik, i = 1, . . . , m,

N

X

j=1 j6=k

λjyrj≥ y_rk, r = 1, . . . , t,

λj≥ 0, j = 1, . . . , N

Output-Oriented CCR Model Maximize φ_ksuch that

N

X

j=1 j6=k

λjxij≤ x_ik, i = 1, . . . , m,

N

X

j=1 j6=k

λjyrj≥ φ_kyrk, r = 1, . . . , t,

λj≥ 0, j = 1, . . . , N

Then θkand φkrepresent the DEA Input and DEA Output indices, respectively, for DMUkrelative to all data.

(9)

The College of Arts & Sciences at Appalachian

About Us

The College of Arts & Sciences at Appalachian State University is home to sixteen academic departments spanning the Humanities, Social Sciences, and the Mathematical and Natural Sciences. The College is dedicated to providing instruction and research essential to the University’s mission, and seeks to cultivate the habits of inquiry, learning, and service among all its constituents.

• Anthropology

• Biology

• Chemistry

• Computer Science

• English

• Foreign Lang & Lit

• Geography & Planning

• Geology

• Gov’t & Justice Studies

• History

• Mathematical Sciences

• Philosophy & Religion

• Physics & Astronomy

• Psychology

• Social Work

• Sociology

http://www.cas.appstate.edu

(10)

DEA2010: 9

College of Arts & Sciences DEA Input- and Output Factors

A total of6input measures.

• Financial Resources

1. Discretionary Operating Budget 2. Supplemental Instructional Funds

(PT faculty & graduate students) 3. Fulltime Faculty Payroll

• Number of Faculty 4. Fulltime Tenure Track 5. Fulltime Non-tenure Track 6. Parttime Non-tenure Track A total of10output measures.

• Number of Majors 1. Lower Division 2. Upper Division

• Number of Degrees Awarded 3. Bachelors

• Number of Class Sections Offered 4. Lower Division

5. Upper Division

• Number of Student Credit Hours (SCH) Produced

6. Lower Division 7. Upper Division

• Research / Scholarship 8. Number of Publications 9. Number of Presentations

• Externally Funded Proposals 10.Total Awards ($)

Normalization via: xi7→ xi/P

jxj.

(11)

College of Arts & Sciences DEA Results

Ouput- and Input-Oriented CCR Model Efficiency Index for Normalized Data

DMU Output Input DMU Output Input

ANT 53% 100% GJS 36% 100%

BIO 58% 90% HIS 55% 67%

CHE 82% 71% MAT 77% 51%

C S 52%¹ 100% P&R 12% 99%

ENG 36% 61% PHY/AST 64% 86%

FLL 78% 50% PSY 57% 100%

GHY & PLN 41% 100% S W 29% 100%

GLY 46% 98% SOC 38%¹ 38%

Data sets (not normalized) are available at:

http://www.mathsci.appstate.edu/∼wmcb/INFORMS/CAS DEA Data.pdf

1BCC model results.

(12)

DEA2010: 11

Future Work

• Perform Sensitivity Analysis on Data and Input / Output Factors

• Use “Value Judgements” (via weight-parameter restrictions) on I / O Factors

• Incorporate Multiple Year Data

• Investigate DMU “Best Direction / Strategy for Improvement” Metrics

This poster is available as a PDF file at:

http://www.mathsci.appstate.edu/∼wmcb/INFORMS/DEA2010.pdf

(13)

Bibliography

• J. Beasley. “Determining teaching and research efficiencies.” J. of the Operational Research Soc,46(4):441–452, 1995.

• J. Beasley. “Comparing university departments.” Omega,18:171–183, 1990.

• A. Charnes, W. Cooper, and E. Rhodes. “Measuring the efficiency of decision making units.” European J. of Operational Research,2(6):429–444, 1995.

• W. Cooper, L. Seiford, and J. Zhu. “Data Envelopment Analysis: History, Models and Interpretations,” Chapter 1, pages 1–39. Springer, New York, 2nd ed., 2004.

• J. Gauder. “Academic research and teaching productivities: A case study.”

Technological Forecast and Social Change,49(3):319–328, 1995.

• J. Johnes. “Data envelopment analysis and its application to the measurement of efficiency in higher education.” Econ. of Education Review,25:273–288, 2006.

• A. Lopes and E. Lanzer. “Data envelopment analysis: DEA and fuzzy sets to assess the performance of academic departments: A case study at Federal Univ. of Santa Catarina—UFSC.” Pesquisa Operacional,22(2):217–230, 2002.

• S. Talluri. “Data envelopment analysis: Models and extensions.” Decision Line, 31(3):8–11, May 2000.