Evolutionary Data Envelopment Analysis
8.4 Testing EDEA
suboptimal solutions presented in Section 8.1.1 is for several disadvantageous operations to be carried out consecutively on the same individual. However, the probability that this individual with a (temporarily) very bad fitness will survive in the next generation is very small,especially during the end of the run (when the variance in the fitness values is small).
To overcome these weaknesses the author proposes an additional mutation operator which allows the simultaneous variation of two genes. The procedure starts by selecting two genes randomly for information exchange. If svt= {v1, . . . ,vm} is a chromosome and thekth andlth components are the selected genes, the offspring issvt+ 1= {v1, . . . ,v′k, . . ., v′l, . . . ,vm}, where
′ = − ⋅ ⋅
′ =
+ ⋅ ⋅
v v =
v r a v v v
v r a v
v v
l l
l l
k k
l l
k l
,
,
,
and , if
else S
∈ (8.13)
andris a random number from [0..1]. Ifv′kandv′lare simply modified,then it would be very likely that this would produce an offspring outside the convex solution spaceS. To avoid this problem,the property of the convex solution space can be used,in analogy to the simple crossover operator (see p. 174), where there existsa∈[0..1] so that Equation 8.13 produces a feasible solution.
Whena= 1,the two genes will exchange information in the way that a proportion of thevlweight is transferred tovk; whena= 0,no changes are made. In order to achieve the highest possible information exchangeEDEA
performs a stepwise search starting witha= 1 and reducing stepwise by a user-defined constant until a feasible solution is found ora= 0. An illustration of how the conditional mutation is coded inEDEAis presented by Fig. 8.18. In order to raise the effectiveness of this operator,the program selects genel amongvm> 0.
1. EDEA finds similar solutions compared to the global optimum found by the simplex-DEA.
2. EDEA produces high-quality solutions faster than the simplex-DEA approach.
3. EDEA is less sensitive to differences in problem characteristics, data quality, or tuning parameters than the simplex-DEA.
4. EDEA is easier to implement than the simplex-DEA.
5. EDEA has a wider range of applicability compared to the simplex-DEA.
For the experimental tests an input-oriented constant returns to scale model and data for the fiscal years 1991–1997 from the AHRP database were
Fig. 8.18. Conditional mutation.
selected. The data for 61 Austrian hotels includes three output variables and four input variables; three of the input variables were defined as non- discretionary variables.
In order to obtain an optimal mixture of parameter settings for EDEA, several tests have been performed. The final settings,which were held constant during theEDEAruns, are illustrated in Table 8.5.
8.4.1 Does EDEA find similar solutions compared to LP-DEA?
For the EDEA/LP-DEA comparison,multiple runs were made under different values of parameters and the best result was chosen (the parameter settings in Table 8.5 refer to this best solution). InEDEA,the population size was 40 and the maximum number of generations was set to 100,000 throughout all trials.
There are basically two possibilities for comparing EDEA and LP-DEA solu- tions: (i) using the efficiency scores or (ii) using the recommended comparison partners.
Comparing efficiency scores
The comparative findings concerning the efficiency scores are summarized in Appendix Table A.11 and displayed in Fig. 8.19. Obviously there is a high correlation between the EDEA and LP-DEA findings. Some small deviations demand explanations.
Generally speaking,the EDEA efficiency scores for almost every company are slightly higher than the LP-DEA efficiency scores. Moreover,29 companies (8.9%),which have been identified as inefficient by LP-DEA,are classified as best-practice companies in EDEA. This is either caused by EDEA not finding the global optimum (minimum) or by finding the global optimum in one
Parameter Value Description pop_size
pm_um pm_bm pm_nm pm_cm pm_sc pm_sa pm_wa ab
fmultiple
40.02 0.020.01 0.020.08 0.100.10 0.100.25 2.02 4.02
Population size
Probabilityof uniform mutation Probabilityof boundarymutation Probabilityof non-uniform mutation Probabilityof conditional mutation Probabilityof simple crossover
Probabilityof single arithmetical crossover Probabilityof whole arithmetical crossover Coefficient used bythe whole arithmetical crossover
Coefficient used for the simulated annealing function in the non-uniform mutation
Coefficient used bythe scaling procedure Table 8.5. Parameter specifications inEDEAtest runs.
generation but then dropping it again in order to search for an even better solution (which does not exist).
The Pearson correlation coefficient between EDEA and LP-DEA results are 0.970 (sign. < 0.01),and 0.960 (sign. < 0.01) when excluding the efficient companies from the evaluation. Note that the LP-DEA results for efficient companies are achieved by running an additional (slightly modified) DEA model,whereas in EDEA the scores for efficient companies are calculated in one optimization run.
Comparing benchmarking partners
The comparison of the efficiency scores calculated by LP-DEA and EDEA have resulted in small differences between the two methodologies. But what about the recommended benchmarking partners? For the selection of comparison partners it is less important that the overall efficiency scores are exactly the same. In this situation,it is more important that the comparison of the recom- mended benchmarking partners leads to similar managerial recommenda- tions. Furthermore,a manager who wants to benchmark his/her company may be more interested to learn about the superimposed (virtual) company,which offers insights into potential improvements for the company under evaluation.
By comparing the lambda values calculated by both methodologies analysis of EDEA and LP-DEA benchmarking partners can best be performed.
The Pearson correlation coefficient between EDEA and LP-DEA lambda values
Fig. 8.19. Comparing EDEA and LP-DEA efficiency scores.
is 0.699 (sign. < 0.01),indicating a relatively poor relationship when consid- ering the high correspondence of the efficiency scores (see p. 184). In fact, the recommendations of ‘best-practising’ partners evaluated by the two meth- odologies can lead to very different sets of peers. An example for this quite different selection of peers is illustrated in Fig. 8.20 and Table 8.6.
Figure 8.20 compares the EDEA and LP-DEA peers for hotel no. 396 in 1995. The overall inefficiency scores for hotel no. 396 calculated by EDEA and LP-DEA are 0.40 and 0.39,respectively. The LP-DEA analysis indicates that hotel no. 396 has the potential to improve its gross profit by?526,000 when it reduces it expenditures to?331,000 (a decrease of 61%), whereas the EDEA analysis indicates that it can improve its gross profit by?514,000 when reducing its expenditures to?343,000 (a decrease of 60%). Although both programs show almost identical scores and target values,the recom- mended comparison partners and their lambda values are very different.
Fig. 8.20. Comparing EDEA and LP-DEA peers for hotel no. 396 (1995).
Hotel
λ no. 1353 no. 2044 no. 2099 no. 2771 no. 2788 no. 2902
LP-DEAa
EDEAb 0.358
0.142 0.001 0.001 0.056
0.310 0.371 0.731
0.360
ae396= 0.39;be396= 0.40 after 100,000 iterations.
Table 8.6. Comparing EDEA and LP-DEA peers for hotel no. 396 (1995).