Data Envelopment Analysis and its aplications

Transcription

1 Data Envelopment Analysis and its aplications VŠB-Technical University of Ostrava Czech Republic 13. Letní škola aplikované informatiky Bedřichov

2 Content Classic Special - for the example

3 The aim to calculate and analyze the efficiency of homogenous DMUs Banks in the V4 countries Colleges in Great Britain Erasmus students in VŠB - TU Ostrava Metallurgy - emission trading Cooperation in Project Management IT Projects in Poland

4 Overview of the Literature Charnes, Cooper and Rhodes (1978) Measuring the efficiency of decision making units Banker, Charnes and Cooper (1984) Extension of basic models

5 Data Envelopment Analysis Classic Special Non-parametric technique based on linear programming for measuring the relative efficiency of a set of DMUs models differs according to orientation or assumptions on return to scale An essential topic is how to choose the appropriate inputs and outputs The analysis of non-efficient units, projection of non-efficient units on the frontier etc. There exist many improvements of the classical models - quasi fixed variables, fuzzy variables etc.

6 Classic Special the non-parametric approach for measuring the relative efficiency of the number of DMUs, let DMU k for k = 1, 2,..., T and let input and output data for DMU k be X = {x ik, i = 1, 2,...R; k = 1, 2,...T } and Y = {y jk, j = 1, 2,...S; k = 1, 2,...T }, u i for i = 1, 2,...R and v j for j = 1, 2,...S be the weights of i th input and j th output, respectively, the relative efficiency score of DMU k can be define as: e k = S j=1 v jy jk R i=1 u ix ik, for k = 1, 2,..., T. (1)

7 Classic Special The multiplier input-oriented model - CCR Charmes et al. (1978), to measure the efficiency score of the under evaluation unit, DMU Q where Q {1,..., T }: s.t. max e Q = S j=1 v jy jq, R i=1 u ix iq = 1, S j=1 v jy jk R i=1 u ix ik 0, k = 1, 2,..., T, u i 0, i = 1, 2,..., R, v j 0, j = 1, 2,..., S. (2) DMU Q is CCR-efficient if and only if e = 1 and if there exists at least one optimal solution ( u, v ) with u > 0 and v > 0, inefficient units have a degree of relative efficiency less than one.

8 Classic Special The multiplier input-oriented model - BCC Banker et al. (1984) extended the CCR model Convex envelope of data which leads to more efficient DMUs BCC model in dual multiplier form is mathematically as it follows: s.t. max e Q = S j=1 v jy jq v 0, R i=1 u ix iq = 1, S j=1 v jy jk R i=1 u ix ik v 0 0, k = 1,..., T, u i 0, i = 1,..., R, v j 0, j = 1,..., S, v 0 (, ), (3) where v 0 is the dual variable assigned to the convexity condition e T λ = 1 of envelopment form of BCC model

9 Classic Special Imprecise data envelopment analysis - I if x ik, y jk are imprecise and unknown decision variables as bounded and ordinal data, Cook et al. (1993, 1996), then model (2) becomes a non-linear and non-convex program; it is called imprecise (I), Zhu (2002, 2003);

10 Classic Special Imprecise data envelopment analysis - I weak ordinal data can be expressed as y jk y jl and x ik x il k l (4) for j DO and i DI, or to simplify the presentation y j1 y j2... y jl... y jn (j DO), (5) x i1 x i2... x il... x in (i DI ), where DO and DI represents the associated sets containing weak ordinal outputs and inputs, respectively. the strong ordinal data can be expressed as: y j1 < y j2 <... < y jl <... < y jn (j SO), (6) x i1 < x i2 <... < x il <... < x in (i SI ), where SO and SI represents the associated sets containing weak ordinal outputs and inputs, respectively.

11 Classic Special Imprecise data envelopment analysis - I model (2) involving (3)-(5) then changes into the following model: s.t. max e Q = S j=1 v jy jq, R i=1 u ix iq = 1, S j=1 v jy jk R i=1 u ix ik 0, k = 1, 2,..., T, (x ik ) θ i, i = 1, 2,..., R, (y jk ) θ + j, j = 1, 2,..., S, u i 0, v j 0, (7) where x ik θ i and y jk θ + j represents any of or all of (3)-(5); can be solved by the standard linear models vis concerning the bounded and ordinal data into exact data, Zhu (2002);

12 Classic Special Imprecise data envelopment analysis - I model (6) when θ i and θ + j are in forms of (4) and obtain a set of optimal solutions yjk and x jk with the optimal e Q is the following: s.t. where eq = max j DO v jy jq + j / DO v jy jq, i DI u ix iq + i / DI u ix iq = 1, j DO v jy jk + j / DO v jy jk i DI u ix ik i / DI u ix ik 0, k = 1, 2,..., T, u i 0, v j 0, i = 1, 2,..., R, j = 1, 2,..., S, 0 y j1 y j2... y jl... y jn 0 x i1 x i2... x il... x in where M is sufficiently large; linear CCR model; M (j DO), M (i DI ). (8)

13 Identification of relevant variables - classic Inputs - min Outputs - max Criteria Max Min Averag Input overrun on budget overrun on schedule Output percentage of scope realized customer satsifaction Table 1: Input and output variables with description

14 Identification of relevant variables - balanced scorecard (BSC) Strategic planning and management system: Financial perspective Customer perspective Internal-business-process perspective Learning and growth perspective BSC perspectives Criteria Output Financial perspective planned budget Output actual budget Input Customer perspective planned price Input actual price Input Internal-business-process perspective risk documentation Input type of document Output Learning and growth perspective experience of manager Input number of people Table 2: Input and output variables with description

15 Figure 1: Results of

16 classic I- I--BSC max min std. dev average Table 3: Descriptive statistics for the results

17 I--BSC is the best one Future work Sets of inputs and outputs should be extended DMUs should be revised, extend or divided into groups based on certain rules Different version of the model may be established

18 Thank you for your attention!