Data Envelopment Analysis (DEA) with Maple in Operations Research and Modeling Courses

Transcription

1 Data Envelopment Analysis (DEA) with Maple in Operations Research and Modeling Courses Wm C Bauldry BauldryWC@appstate.edu wmcb March, 2009

2 Abstract We will present the Data Envelopment Analysis technique using Maple. This topic is natural for introductory Operations Research or Math Modeling courses. A sample student project (that can raise a dean s eyebrows) will be given. ICTCM-21: 2 41

3 Introduction What is Data Envelopment Analysis? Data Envelopment Analysis (DEA), occasionally called frontier analysis, was first put forward by Charnes, Cooper and Rhodes in 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 ICTCM-21: 3 41

4 Introduction What is Data Envelopment Analysis? Data Envelopment Analysis (DEA), occasionally called frontier analysis, was first put forward by Charnes, Cooper and Rhodes in 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 ICTCM-21: 4 41

5 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 ICTCM-21: 5 41

6 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 ICTCM-21: 6 41

7 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 ICTCM-21: 7 41

8 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 ICTCM-21: 8 41

9 Example Graphical Analysis (2-D) ICTCM-21: 9 41

10 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 ICTCM-21: 10 41

11 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 ICTCM-21: 11 41

12 Data Envelopment Analysis Defined DEA: Nonlinear Representation ( I-O Ratio ) Let: n number of DMUs For each DMU j, j = 1..n, define the nonlinear program: n in u i x ik n out v i y ik e k number of input measures weight factor for input i input i for DMU k number of output measures weight factor for output i output i for DMU k efficiency of DMU k Choose u, v to maximize e j subject to n out v i y ik i=1 e k = n in u i x ik i=1 k = 1..n 0 e k 100% k = 1..n u k 0 k = 1..n in v k 0 k = 1..n out ICTCM-21: 12 41

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 X Inputs λ X i θ 0 M Outputsj λ M Outputsj,i 0 j = 1..n and λ i 0 i = 1..n (λ i can be interpreted as a dual variable.) ICTCM-21: 13 41

14 Arts & Sciences at ASU The Problem Arts & Sciences at Appalachian Task: Analyze the 16 disparate departments. ICTCM-21: 14 41

15 Arts & Sciences at ASU Initial Data Initial Arts & Sciences Data Fall, Source: Institutional Research, Assessment, & Planning, ASU ICTCM-21: 15 41

16 Arts & Sciences at ASU Scaled Data Linearly Scaled Data Return to Results ICTCM-21: 16 41

17 Arts & Sciences at ASU Data Graph Graphical Analysis (2-D) ICTCM-21: 17 41

18 Linear Program Linear Programming Analysis The Math Sciences (DMU #11) linear program is: minimize θ subject to X Inputs λ θ 0 M Outputsj λ M Outputsj,11 0, j = λ i 0, i = ICTCM-21: 18 41

19 Maple Simplex in Maple Arts & Sciences DEA Maple Setup: Use simplex[minimize] or Optimization[LPSolve] DMU := ["Anthropology", "Biology",... N := nops(dmu): ICTCM-21: 19 41

20 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]]): ICTCM-21: 20 41

21 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: ICTCM-21: 21 41

22 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: ICTCM-21: 22 41

23 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 ICTCM-21: 23 41

24 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: eq3 := OV[2] - M[n,2] 0: eq4 := OV[3] - M[n,3] 0: sys := {eq1,eq2,eq3,eq4}: # credit hours # num students # degrees s := simplex[minimize](θ, sys, NONNEGATIVE); Results := Results, [DMU[n],select(x (rhs(x)<>0),s))]; end do: Matrix([Results]); Results ICTCM-21: 24 41

25 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 = , λ 1 = 0., λ 2 = , λ 3 = 0., λ 4 = 0., λ 5 = 0., λ 6 = 0., λ 7 = 0., λ 8 = 0., λ 9 = 0., λ 10 = 0., λ 11 = 0., λ 12 = , λ 13 = 0., λ 14 = , λ 15 = 0., λ 16 = 0.} Back to Code ICTCM-21: 25 41

26 Maple Results DEA: Maple Results for Arts & Sciences Matrix([Results]); Anthropology θ = λ 12 = λ 14 = Biology θ = 1.0 λ 2 = Chemistry θ = λ 2 = Computer Science θ = λ 14 = English θ = λ 12 = λ 14 = Foreign Lang & Lit θ = λ 12 = λ 14 = Geography & Planning θ = λ 2 = λ 12 = λ 14 = Geology θ = λ 2 = History θ = λ 12 = λ 14 = Interdisc Studies θ = λ 14 = Mathematical Sci θ = λ 12 = Philosophy & Religion θ = 1.0 λ 12 = Physics & Astronomy θ = λ 2 = λ 12 = Poli Sci/Crim Justice θ = 1.0 λ 14 = Psychology θ = λ 12 = λ 14 = Soc & Social Work θ = λ 12 = λ 14 = ICTCM-21: 26 41

27 Maple Results DEA: Maple Results for Arts & Sciences DMU θ% λ 1 Anthropology 79 λ 12 = 0.35, λ 14 = Biology 100 λ 2 = Chemistry 74 λ 2 = Computer Science 49 λ 2 = 0.08, λ 14 = English 74 λ 2 = 0.01, λ 12 = 0.52, λ 14 = Foreign Lang & Lit 87 λ 12 = 0.53, λ 14 = Geography & Planning 74 λ 2 = 0.20, λ 12 = 0.26, λ 14 = Geology 63 λ 2 = History 91 λ 12 = 0.61, λ 14 = Interdisc Studies 51 λ 12 = 0.04, λ 14 = Mathematical Sci 79 λ 12 = Philosophy & Religion 100 λ 12 = Physics & Astronomy 73 λ 2 = 0.44, λ 12 = 0.27, λ 14 = Poli Sci/Crim Justice 100 λ 14 = Psychology 86 λ 2 = 0.06, λ 12 = 0.11, λ 14 = Soc & Social Work 92 λ 12 = 0.38, λ 14 = 0.54 ICTCM-21: 27 41

28 Excel Results in Excel For implementing simplex using Solver in Excel, see MacDonald, or Beasley, etc. ICTCM-21: 28 41

29 Excel: Sorted Results Excel Sorted Results Back to Data ICTCM-21: 29 41

30 Student Projects A Simple Student Project Choose a college, division, or school: Identify DMUs: colleges, divisions, or departments ICTCM-21: 30 41

31 Student Projects A Simple Student Project Choose a college, division, or school: Identify DMUs: colleges, divisions, or departments Define inputs: Frequency Distribution of SAT Verbal, Math, and Writing Scores for Enrolling Freshmen ICTCM-21: 31 41

32 Student Projects A Simple Student Project Choose a college, division, or school: Identify DMUs: colleges, divisions, or departments Define inputs: Frequency Distribution of SAT Verbal, Math, and Writing Scores for Enrolling Freshmen Define outputs: number of majors and degrees awarded ICTCM-21: 32 41

33 Student Projects A Simple Student Project Choose a college, division, or school: Identify DMUs: colleges, divisions, or departments Define inputs: Frequency Distribution of SAT Verbal, Math, and Writing Scores for Enrolling Freshmen Define outputs: number of majors and degrees awarded Perform a data envelopment analysis with 3 inputs and 2 outputs ICTCM-21: 33 41

34 Student Projects A Fun Student Project Choose a college, division, or school Identify DMUs: departments, academic areas, or groups ICTCM-21: 34 41

35 Student Projects 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) ICTCM-21: 35 41

36 Student Projects 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) ICTCM-21: 36 41

37 Student Projects 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 ICTCM-21: 37 41

38 Student Projects 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. ICTCM-21: 38 41

39 Student Projects 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. ICTCM-21: 39 41

40 Student Projects 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. ICTCM-21: 40 41

41 Bibliography Selected References Charnes A., Cooper W.W. and Rhodes E. (1978) Measuring the efficiency of decision making units, Eur. J. Opl. Res. 2, 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 ( ), RUTCOR Research Report, RRR Rutgers University pdf (3203 entries) ICTCM-21: 41 41