Categorical Decision Making Units and Comparison of Efficiency between Different Systems Mat-2.4142 Optimointiopin Seminaari Source William W. Cooper, Lawrence M. Seiford, Kaoru Tone: Data Envelopment nalysis: Comprehensive Text with Models, pplications, References and DE-Solver Software.
Topics discussed Data Envelopment nalysis with categorical DMUs Controllable and non-controllable categories Comparison of efficiency between different systems Loosening the convexity assumption Graphical presentation of the efficiency frontier formed by different systems Rank-sum statistics comparing two DE data groups
DE with Categorical DMUs There are managerial situations over which the management doesn t have control e.g. laws of particular states There can be managerial situations where relative dedication of different units to the business activity needs to be quantified e.g. bonus system for the relative service level Service level has an effect on the efficiency of sales, how large is the bonus for a good service per business unit (ceteris paribus)?
DE with Categorical DMUs Categorical DMUs can be defined non-controllable or controlled non-controllable by DMs Laws, population density, sales environment, etc. controlled by DMs Service level, prices, floor area, etc.
Categorical DMUs non-controllable by DMs: example Stores in three areas Each area has a different competition level: severe, normal or advantageous Stores are categorized according to the competition level Straight evaluation would be unfair to stores in areas of higher competition handicap is introduced to balance this
Categorical DMUs non-controllable by DMs: method 1. The stores in category severe are evaluated within the group 2. The stores in category normal are evaluated with reference to the groups severe and normal 3. The stores in category advantageous are evaluated within all groups in the model - Every evaluation should be done with the same, freely chosen DE model
Categorical DMUs controllable by DMs: introduction The method for controllable categories differs only a little from that of non-controllables DE model can still be chosen freely n algorithm can be given to solve the category levels and reference set of each unit In the next slide the algorithm is considered for DMU0, which is currently at the level l ( 1 l L )
Categorical DMUs controllable by DMs: algorithm For h=l, l+1,,l, repeat the following steps: Step 1. Organize a set of DMUs composed of level h or higher and the DMU0 to be evaluated. Evaluate the efficiency of DMU0 with respect to this group Step 2. (i) If DMU0 is efficient, goto step 3. (ii) If DMU0 is inefficient, record its reference set and reference point on the frontier. Step 3. Examine the reference set, reference point and category level obtained. Choose the most appropriate category level for DMU0.
DMU C D E F G H I Categorical DMUs controllable by DMs: example Input 3 7 12 4 6 11 8 9 13 Category Output 1 7 6 5 10 11 11 13 15 poor poor poor aver. aver. aver. good good good Reference set (1) D(.6),E(.4) D(.8),E(.2) G(1) D(1) E(1) E(.67),H(.33) G(1) G(1) H(1) I(1) output 16 14 12 10 8 6 4 2 0 Frontiers of different levels 0 2 4 6 8 10 12 14 input poor average good
Comparison of efficiency between different systems Different activities usually lead to different outputs with different inputs Methods of producing services, consumer behaviour in a particular user group etc. It is not always reasonable to compare different DMUs as if any percentage combination of activities is a valid optimum There might be different systems of production in place, creating implicit categories
Comparing the efficiency: loosening the convexity assumption If two activities, (x1,y1) and (x2,y2) belong to a convex set P, then every point on the line segment m connecting these activities belong to P.* Loosening the convexity assumption means that the points on the connecting line segment m are not necessarily valid activities * with the exception of FDH and multiplicative models in chapters 4 & 5.
Formulation of the optimization problem with multiple systems (1/2) Two systems, and are proposed Convexity assumption holds within the same system but does not hold between the two systems Inputs and outputs X and Y are divided: X={X,X}, Y={Y,Y} Search for the optimal efficiency can be formulated as an mixed integer LP problem:
Formulation of the optimization problem with multiple systems (2/2) min s. t. θx 0 y 0 Lz Lz z θ Y z X, z + λ z eλ eλ + Uz Uz = 1 X λ + Y λ 0, λ 0 = {0,1}. where z, z are binary variables that assume the values of 0 or 1 λ λ (1)
Computation of efficiency Formulation (1) can be solved by enumeration rather than using a mixed integer 0-1 program. 1. Set z=1 and z=0 and solve the resulting ordinary LP problem. Let the optimal objective value be θ. 2. Set z=0 and z=1 and solve the resulting ordinary LP problem. Let the optimal objective value be θ. 3. The efficiency of DMU (x0,y0) is obtained by θ = min{ θ, θ } * 0
Comparison between two systems Two Systems/Sales areas Profit(1000 )/Salesman 2 1,8 1,6 1,4 1,2 1 Efficient frontier Convexity between the systems (not assumed) 9 11 13 15 17 Customer/Salesman Convexity within a syste
Rank-sum statistics and DE Testing the difference between two groups using the DE scores Is there a difference between two groups and in terms of efficiency? Distribution of DE scores is usually unknown Nonparametric statistics Wilcoxon-Mann-Whitney Rank-sum test can be used oth inter and within uses in DE group comparison
Summary Categories for DMUs can be both controllable and non-controllable for the DMs Solutions for relative efficiency can be reached in a similar manner Efficiency frontier becomes shorter in smaller groups(!) Comparison of efficiency between different systems Dividing the inputs and outputs between the systems Efficiency frontier becomes more complex Two data groups can be tested for belonging to the same system using Rank-sum statistics
Home assignment: Categorization C1 C2 C3 C4 Enumerate the reference group for each category group Ck, k=1,2,3,4 in above figure (10p)