In data envelopment analysis (DEA), input and output values are subject to change for several reasons. Such variations differ in their input/output items and their decision-making units (DMUs). Hence, DEA efficiency scores need to be examined by considering these factors. In this paper, we propose new resampling models based on these variations for gauging the confidence intervals of DEA scores. The first model utilizes past-present data for estimating data variations imposing chronological order weights which are supplied by Lucas series (a variant of Fibonacci series). The second model deals with future prospects. This model aims at forecasting the future efficiency score and its confidence interval for each DMU. We applied our models to a dataset composed of Japanese municipal hospitals.
DEA is a non-parametric methodology for performance evaluation and benchmarking. Since the publication of the seminal paper by Charnes, Cooper and Rhodes [
In this paper, we follow the principles stated in Cook, Tone and Zhu [
This paper unfolds as follows. Section 2 presents a generic methodological framework to estimate the confidence intervals of DEA scores under a past-present time frame and extends it to the past-present-future time frame. Section 3 presents a healthcare application to illustrate the proposed resampling framework. Finally, section 5 concludes the paper.
The first framework is designed for when past-present information on say m inputs and s outputs of a set of n
DMUs is available; that is,
denote the present and periods 1 though
Initialization Step
Choose an appropriate DEA model for computing the efficiency scores of DMUs;
Use the chosen DEA model to estimate the DEA scores of DMUs based on the present information; that is,
Choose an appropriate scheme, say w, to weigh the available information on the past and the present;
Choose a confidence level
Choose the number of replicas or samples to draw from the past, say B, along with any properties they should satisfy before being considered appropriate to use for generating the sampling distributions of
Set an indicator variable, say
Iterative Step
The generic nature of this framework requires a number of decisions to be made for its implementation for a particular application. Hereafter, we shall discuss how one might make such decisions.
In principle one might choose from a relatively wide range of DEA models; however, given the nature of this exercise we recommend the use of the non-oriented super slacks-based measure model (Tone [
In this paper, we regard historical data
discrete events with probability
cation process based on bootstrapping. First proposed by Efron [
For a non-correlated and homoskedastic dataset, one could for example use smooth bootstrapping or Bayesian bootstrapping, where smooth bootstrapping generates replicas by adding small amounts of zero-centered random noise (usually normally distributed) to resampled observations, whereas Bayesian bootstrapping generates replicas by reweighting the initial data set according to a randomly generated weighting scheme. In this paper, we recommend the use of a variant of Bayesian bootstrapping whereby the weighting scheme consists of the Lucas number series-based weights
On the other hand, for a correlated and/or heteroskedastic dataset, one could use one of the block bootstrapping methods, where replicas are generated by splitting the dataset into non-overlapping blocks (simple block bootstrap) or into overlapping blocks of the same or different lengths (moving block bootstrap), sampling such blocks with replacement and then aligning them in the order they were drawn. The main idea of all block bootstrap procedures consists of dividing the data into blocks of consecutive observations of length
Blocks―for an overview of bootstrapping methods, the reader is referred to [
Input: Block length
Step 1: Draw randomly and independently block labels, say
Step 2: Lay the blocks
der sampled together and discard the last
Output: Bootstrap sample
As to the choice of the number of replicas B, there is no universal rule except that the larger the value of B the more stable the results. However, one should take into consideration the computational requirements; therefore, in practice, one would keep increasing the value of B until the simulation converges; that is, the results from a run do not change when adding more iterations.
In the previous subsection, we utilized historical data
In this study we utilize a dataset concerning nineteen Japanese municipal hospitals from 2007 to 2009 to illustrate how the proposed framework works. There are approximately 1000 municipal hospitals in Japan and there is large heterogeneity amongst them. We selected nineteen municipal hospitals with more than 400 beds. Therefore, this sample may represent larger acute-care hospitals with homogeneous functions. The data were collected from the Annual Databook of Local Public Enterprises published by the Ministry of Internal Affairs and Communications. For illustration purposes, we chose for this study two inputs; namely, Doctor ((I)Doc) and Nurse ((I)Nur), and two outputs; namely, Inpatient ((O)In) and Outpatient ((O)Out).
We solved the non-oriented super slacks-based measure model year by year and obtained the super-efficiency scores in
2007 | 2008 | 2009 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
DMU | (I)Doc | (I)Nur | (O)In | (O)Out | (I)Doc | (I)Nur | (O)In | (O)Out | (I)Doc | (I)Nur | (O)In | (O)Out |
H1 | 108 | 433 | 606 | 1239 | 114 | 453 | 617 | 1244 | 116 | 545 | 603 | 1295 |
H2 | 125 | 448 | 642 | 1363 | 133 | 499 | 638 | 1310 | 136 | 482 | 618 | 1300 |
H3 | 118 | 567 | 585 | 1072 | 121 | 600 | 569 | 1051 | 125 | 616 | 561 | 1071 |
H4 | 138 | 541 | 699 | 1210 | 138 | 531 | 704 | 1194 | 140 | 554 | 679 | 1182 |
H5 | 138 | 613 | 653 | 1195 | 142 | 616 | 644 | 1147 | 137 | 633 | 622 | 1147 |
H6 | 99 | 569 | 716 | 1533 | 106 | 592 | 701 | 1478 | 109 | 613 | 651 | 1457 |
H7 | 94 | 498 | 540 | 1065 | 103 | 494 | 551 | 1067 | 101 | 491 | 540 | 1067 |
H8 | 106 | 461 | 496 | 1051 | 118 | 490 | 504 | 1033 | 133 | 479 | 505 | 1081 |
H9 | 109 | 450 | 483 | 851 | 119 | 483 | 487 | 877 | 121 | 501 | 486 | 904 |
H10 | 102 | 540 | 581 | 1268 | 106 | 558 | 565 | 1278 | 148 | 611 | 586 | 1321 |
H11 | 92 | 495 | 490 | 1217 | 101 | 497 | 501 | 1146 | 102 | 501 | 479 | 1113 |
H12 | 148 | 721 | 771 | 1637 | 147 | 710 | 723 | 1657 | 158 | 737 | 743 | 1714 |
H13 | 103 | 593 | 679 | 2011 | 106 | 673 | 642 | 1883 | 120 | 697 | 634 | 1872 |
H14 | 101 | 500 | 613 | 1868 | 110 | 519 | 617 | 1894 | 116 | 517 | 623 | 2009 |
H15 | 159 | 793 | 964 | 2224 | 160 | 801 | 906 | 2148 | 166 | 817 | 877 | 2155 |
H16 | 77 | 354 | 410 | 1047 | 68 | 359 | 391 | 916 | 81 | 378 | 406 | 897 |
H17 | 111 | 663 | 717 | 1674 | 112 | 645 | 702 | 1774 | 112 | 663 | 709 | 1733 |
H18 | 62 | 388 | 480 | 913 | 64 | 385 | 467 | 907 | 63 | 381 | 463 | 872 |
H19 | 98 | 323 | 508 | 1192 | 95 | 314 | 483 | 1018 | 95 | 320 | 490 | 1034 |
2007 | 2008 | 2009 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
(I)Doc | (I)Nur | (O)In | (O)Out | (I)Doc | (I)Nur | (O)In | (O)Out | (I)Doc | (I)Nur | (O)In | (O)Out | |
Min | 62 | 323 | 410 | 851 | 64 | 314 | 391 | 877 | 63 | 320 | 406 | 872 |
Max | 159 | 793 | 964 | 2224 | 160 | 801 | 906 | 2148 | 166 | 817 | 877 | 2155 |
Avg | 110 | 524 | 612 | 1349 | 114 | 538 | 601 | 1317 | 120 | 555 | 593 | 1328 |
StdDev | 23.75 | 120.41 | 130.51 | 378.24 | 24.15 | 121.43 | 119.57 | 380.07 | 25.58 | 126.78 | 113.05 | 389.49 |
We applied the proposed procedure to the historical data of nineteen hospitals for the two years 2008-2009 in
2007 | 2008 | 2009 | |
---|---|---|---|
H1 | 0.883 | 0.905 | 0.754 |
H2 | 0.875 | 0.801 | 0.779 |
H3 | 0.623 | 0.615 | 0.592 |
H4 | 0.700 | 0.765 | 0.680 |
H5 | 0.619 | 0.620 | 0.604 |
H6 | 1.004 | 0.942 | 0.848 |
H7 | 0.719 | 0.732 | 0.725 |
H8 | 0.676 | 0.651 | 0.631 |
H9 | 0.588 | 0.583 | 0.568 |
H10 | 0.758 | 0.764 | 0.631 |
H11 | 0.757 | 0.740 | 0.698 |
H12 | 0.711 | 0.741 | 0.714 |
H13 | 1.034 | 1.025 | 0.831 |
H14 | 1.039 | 1.107 | 1.145 |
H15 | 0.858 | 0.857 | 0.811 |
H16 | 0.831 | 0.847 | 0.742 |
H17 | 0.847 | 0.948 | 0.937 |
H18 | 1.034 | 1.050 | 1.074 |
H19 | 1.071 | 1.072 | 1.100 |
Avg | 0.822 | 0.830 | 0.782 |
2007 | 2008 | 2009 | |
---|---|---|---|
H1 | 0.883 | 0.833 | 0.727 |
H2 | 0.875 | 0.750 | 0.745 |
H3 | 0.623 | 0.584 | 0.571 |
H4 | 0.700 | 0.712 | 0.654 |
H5 | 0.619 | 0.590 | 0.584 |
H6 | 1.004 | 0.860 | 0.783 |
H7 | 0.719 | 0.696 | 0.699 |
H8 | 0.676 | 0.620 | 0.613 |
H9 | 0.588 | 0.556 | 0.551 |
H10 | 0.758 | 0.726 | 0.610 |
H11 | 0.757 | 0.703 | 0.672 |
H12 | 0.711 | 0.704 | 0.688 |
H13 | 1.034 | 0.871 | 0.794 |
H14 | 1.024 | 0.950 | 1.020 |
H15 | 0.858 | 0.812 | 0.779 |
H16 | 0.831 | 0.798 | 0.715 |
H17 | 0.847 | 0.872 | 0.855 |
H18 | 1.028 | 0.929 | 0.922 |
H19 | 1.042 | 0.920 | 0.924 |
Avg. | 0.820 | 0.763 | 0.732 |
Doc | Nurse | Inpatient | Outpatient | |
---|---|---|---|---|
Doc | 1 | 0.7453 | 0.7372 | 0.5178 |
Nurse | 0.7453 | 1 | 0.8610 | 0.7387 |
Inpatient | 0.7372 | 0.8610 | 1 | 0.8264 |
Outpatient | 0.5178 | 0.7387 | 0.8264 | 1 |
Lower bounds | |||||
---|---|---|---|---|---|
Doc | Nurse | Inpatient | Outpatient | ||
Doc | 0.4400 | 0.4255 | 0.0832 | ||
Upper | Nurse | 0.8961 | 0.6681 | 0.4281 | |
bounds | Inpatient | 0.8926 | 0.9455 | 0.5959 | |
Outpatient | 0.7869 | 0.8932 | 0.9311 |
0.5178 and its 95% lower/upper bounds are respectively 0.0832 and 0.7869. In addition, we report Fisher 20% confidence lower/upper bounds in
In the Fisher 95% (ζ95) case, we found no discarded samples, whereas in the Fisher 20% (ζ20) case, 1945 samples were discarded before getting 500 replicas.
Note that one resample produces one efficiency score for each DMU. We compared 500 and 5000 replicas and obtained the 95% confidence interval as exhibited in
Lower bounds | |||||
---|---|---|---|---|---|
Doc | Nurse | Inpatient | Outpatient | ||
Doc | - | 0.71578 | 0.70695 | 0.46998 | |
Upper | Nurse | 0.77214 | - | 0.8437 | 0.70854 |
bounds | Inpatient | 0.76482 | 0.87652 | - | 0.80525 |
Outpatient | 0.56266 | 0.76614 | 0.84547 | - |
97.50% | DEA (2009) | Average | 2.50% | Rank (Avg) | |
---|---|---|---|---|---|
H1 | 0.9228 | 0.754 | 0.8047 | 0.724 | 8 |
H2 | 0.8279 | 0.7787 | 0.7865 | 0.7415 | 9 |
H3 | 0.6285 | 0.5918 | 0.5999 | 0.573 | 18 |
H4 | 0.7574 | 0.6802 | 0.709 | 0.6694 | 14 |
H5 | 0.6375 | 0.6042 | 0.6088 | 0.5792 | 17 |
H6 | 0.9384 | 0.8475 | 0.8758 | 0.8159 | 6 |
H7 | 0.762 | 0.725 | 0.7284 | 0.6998 | 11 |
H8 | 0.6902 | 0.6311 | 0.6365 | 0.6002 | 16 |
H9 | 0.603 | 0.5681 | 0.5732 | 0.5452 | 19 |
H10 | 0.7963 | 0.6308 | 0.6818 | 0.6032 | 15 |
H11 | 0.7433 | 0.6985 | 0.7116 | 0.6808 | 13 |
H12 | 0.7684 | 0.714 | 0.7237 | 0.6849 | 12 |
H13 | 1.0465 | 0.831 | 0.8978 | 0.8081 | 5 |
H14 | 1.1564 | 1.1448 | 1.1329 | 1.1037 | 1 |
H15 | 0.8692 | 0.8107 | 0.8277 | 0.7886 | 7 |
H16 | 0.8792 | 0.7418 | 0.7782 | 0.714 | 10 |
H17 | 1.0142 | 0.9368 | 0.9542 | 0.9076 | 4 |
H18 | 1.0837 | 1.0745 | 1.0708 | 1.0497 | 3 |
H19 | 1.1194 | 1.0996 | 1.0897 | 1.0618 | 2 |
ζ20 - ζ95 | ζ20 - ζ95 | ||||||||
---|---|---|---|---|---|---|---|---|---|
ζ20 | 97.50% | DEA | 2.50% | ζ95 | 97.50% | DEA | 2.50% | 97.50% | 2.50% |
H1 | 0.9061 | 0.754 | 0.724 | H1 | 0.9228 | 0.754 | 0.724 | −0.017 | 0.000 |
H2 | 0.8247 | 0.7787 | 0.7419 | H2 | 0.8279 | 0.7787 | 0.7415 | −0.003 | 0.000 |
H3 | 0.6279 | 0.5918 | 0.5757 | H3 | 0.6285 | 0.5918 | 0.573 | −0.001 | 0.003 |
H4 | 0.7476 | 0.6802 | 0.6684 | H4 | 0.7574 | 0.6802 | 0.6694 | −0.010 | −0.001 |
H5 | 0.6375 | 0.6042 | 0.5832 | H5 | 0.6375 | 0.6042 | 0.5792 | 0.000 | 0.004 |
H6 | 0.9382 | 0.8475 | 0.8168 | H6 | 0.9384 | 0.8475 | 0.8159 | 0.000 | 0.001 |
H7 | 0.7611 | 0.725 | 0.6989 | H7 | 0.762 | 0.725 | 0.6998 | −0.001 | −0.001 |
H8 | 0.6905 | 0.6311 | 0.6011 | H8 | 0.6902 | 0.6311 | 0.6002 | 0.000 | 0.001 |
H9 | 0.6023 | 0.5681 | 0.5467 | H9 | 0.603 | 0.5681 | 0.5452 | −0.001 | 0.001 |
H10 | 0.7903 | 0.6308 | 0.6044 | H10 | 0.7963 | 0.6308 | 0.6032 | −0.006 | 0.001 |
H11 | 0.7469 | 0.6985 | 0.6808 | H11 | 0.7433 | 0.6985 | 0.6808 | 0.004 | 0.000 |
H12 | 0.767 | 0.714 | 0.6828 | H12 | 0.7684 | 0.714 | 0.6849 | −0.001 | −0.002 |
H13 | 1.0445 | 0.831 | 0.8081 | H13 | 1.0465 | 0.831 | 0.8081 | −0.002 | 0.000 |
H14 | 1.1568 | 1.1448 | 1.1041 | H14 | 1.1564 | 1.1448 | 1.1037 | 0.000 | 0.000 |
H15 | 0.867 | 0.8107 | 0.7886 | H15 | 0.8692 | 0.8107 | 0.7886 | −0.002 | 0.000 |
H16 | 0.8747 | 0.7418 | 0.7222 | H16 | 0.8792 | 0.7418 | 0.714 | −0.004 | 0.008 |
H17 | 1.0121 | 0.9368 | 0.9058 | H17 | 1.0142 | 0.9368 | 0.9076 | −0.002 | −0.002 |
H18 | 1.0837 | 1.0745 | 1.0491 | H18 | 1.0837 | 1.0745 | 1.0497 | 0.000 | −0.001 |
H19 | 1.1195 | 1.0996 | 1.063 | H19 | 1.1194 | 1.0996 | 1.0618 | 0.000 | 0.001 |
500 Replica | 5000 Replica | Difference | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
500 | 97.50% | DEA | 2.50% | 5000 | 97.50% | DEA | 2.50% | 97.50% | 2.50% | |
H1 | 0.9228 | 0.754 | 0.724 | H1 | 0.9184 | 0.754 | 0.7227 | 0.0044 | 0.0013 | |
H2 | 0.8279 | 0.7787 | 0.7415 | H2 | 0.8266 | 0.7787 | 0.7412 | 0.0013 | 0.0003 | |
H3 | 0.6285 | 0.5918 | 0.573 | H3 | 0.6291 | 0.5918 | 0.5719 | −0.0006 | 0.0011 | |
H4 | 0.7574 | 0.6802 | 0.6694 | H4 | 0.7581 | 0.6802 | 0.6679 | −0.0007 | 0.0015 | |
H5 | 0.6375 | 0.6042 | 0.5792 | H5 | 0.6379 | 0.6042 | 0.5801 | −0.0004 | −0.0009 | |
H6 | 0.9384 | 0.8475 | 0.8159 | H6 | 0.9423 | 0.8475 | 0.8164 | −0.0039 | −0.0005 | |
H7 | 0.762 | 0.725 | 0.6998 | H7 | 0.7615 | 0.725 | 0.6985 | 0.0005 | 0.0013 | |
H8 | 0.6902 | 0.6311 | 0.6002 | H8 | 0.6907 | 0.6311 | 0.5998 | −0.0005 | 0.0004 | |
H9 | 0.603 | 0.5681 | 0.5452 | H9 | 0.603 | 0.5681 | 0.5456 | 0 | −0.0004 | |
H10 | 0.7963 | 0.6308 | 0.6032 | H10 | 0.7942 | 0.6308 | 0.6055 | 0.0021 | −0.0023 | |
H11 | 0.7433 | 0.6985 | 0.6808 | H11 | 0.7447 | 0.6985 | 0.6808 | −0.0014 | 0 | |
H12 | 0.7684 | 0.714 | 0.6849 | H12 | 0.7684 | 0.714 | 0.6828 | 0 | 0.0021 | |
H13 | 1.0465 | 0.831 | 0.8081 | H13 | 1.046 | 0.831 | 0.8081 | 0.0005 | 0 | |
H14 | 1.1564 | 1.1448 | 1.1037 | H14 | 1.1565 | 1.1448 | 1.1026 | −1E−04 | 0.0011 | |
H15 | 0.8692 | 0.8107 | 0.7886 | H15 | 0.8726 | 0.8107 | 0.7886 | −0.0034 | 0 | |
H16 | 0.8792 | 0.7418 | 0.714 | H16 | 0.8785 | 0.7418 | 0.7198 | 0.0007 | −0.0058 | |
H17 | 1.0142 | 0.9368 | 0.9076 | H17 | 1.0141 | 0.9368 | 0.9051 | 1E−04 | 0.0025 | |
H18 | 1.0837 | 1.0745 | 1.0497 | H18 | 1.0837 | 1.0745 | 1.0459 | 0 | 0.0038 | |
H19 | 1.1194 | 1.0996 | 1.0618 | H19 | 1.1193 | 1.0996 | 1.0618 | 1E−04 | 0 | |
Max | 0.0044 | 0.0038 | ||||||||
Min | −0.0039 | −0.0058 |
inputs, outputs and DMUs. Hence, we need to check the variations of scores by increasing the number of replicas.
Finally, we would like to draw the reader’s attention to the fact that, in some applications, one might set weights to inputs and outputs. Actually, if costs for inputs and incomes from outputs are available, we can evaluate the comparative cost performance of DMUs. In the absence of such information, instead, we can set weights to inputs and outputs. For example, the weights to Doc and Nurse are assumed to be 5 to 1 (on average), and those of Outpatient to Inpatient are 1 to 10 (on average). We can solve this problem via the Weighted-SBM model, which will enhance the reliability and applicability of our approach.
Hereafter, we shall present numerical results for the past-resent-future framework. In this case we regard 2007- 2008 as the past-present and 2009 as the future. In our application, we used three simple prediction models to forecast the future; namely, a linear trend analysis model, a weighted average model with Lucas weights, and a hybrid model that consists of averaging their predictions.
DEA, originated by Charnes and Cooper (Charnes et al. [
DMU | (I)Doc | (I)Nurse | (O)Inpatient | (O)Outpatient |
---|---|---|---|---|
H1 | 120 | 473 | 628 | 1249 |
H2 | 141 | 550 | 634 | 1257 |
H3 | 124 | 633 | 553 | 1030 |
H4 | 138 | 521 | 709 | 1178 |
H5 | 146 | 619 | 635 | 1099 |
H6 | 113 | 615 | 686 | 1423 |
H7 | 112 | 490 | 562 | 1069 |
H8 | 130 | 519 | 512 | 1015 |
H9 | 129 | 516 | 491 | 903 |
H10 | 110 | 576 | 549 | 1288 |
H11 | 110 | 499 | 512 | 1075 |
H12 | 146 | 699 | 675 | 1677 |
H13 | 109 | 753 | 605 | 1755 |
H14 | 119 | 538 | 621 | 1920 |
H15 | 161 | 809 | 848 | 2072 |
H16 | 59 | 364 | 372 | 785 |
H17 | 113 | 627 | 687 | 1874 |
H18 | 66 | 382 | 454 | 901 |
H19 | 92 | 305 | 458 | 844 |
DMU | 97.50% | Forecast (2009) | Actual (2009) | 2.50% |
---|---|---|---|---|
H1 | 1.0237 | 0.9338 | 0.754 | 0.8245 |
H2 | 1.0027 | 0.787 | 0.7787 | 0.722 |
H3 | 0.6649 | 0.6148 | 0.5918 | 0.5641 |
H4 | 0.8816 | 0.8581 | 0.6802 | 0.7319 |
H5 | 0.6814 | 0.6421 | 0.6042 | 0.5771 |
H6 | 1.0213 | 0.8768 | 0.8475 | 0.8062 |
H7 | 0.8292 | 0.7586 | 0.725 | 0.6945 |
H8 | 0.7641 | 0.6725 | 0.6311 | 0.6066 |
H9 | 0.6983 | 0.6213 | 0.5681 | 0.539 |
H10 | 0.8422 | 0.7781 | 0.6308 | 0.7111 |
H11 | 0.8425 | 0.7206 | 0.6985 | 0.6679 |
H12 | 0.8136 | 0.7716 | 0.714 | 0.7068 |
H13 | 1.0814 | 1 | 0.831 | 0.8276 |
H14 | 1.1575 | 1.0909 | 1.1448 | 1.0281 |
H15 | 0.9467 | 0.8541 | 0.8107 | 0.7902 |
H16 | 1.0376 | 0.9444 | 0.7418 | 0.7258 |
H17 | 1.0387 | 1.0348 | 0.9368 | 0.8982 |
H18 | 1.0899 | 1.0537 | 1.0745 | 0.9692 |
H19 | 1.1354 | 1.0594 | 1.0996 | 1.0113 |
DMU | (I)Doc | (I)Nurse | (O)Inpatient | (O)Outpatient |
---|---|---|---|---|
H1 | 112 | 446 | 613 | 1242 |
H2 | 130 | 482 | 639 | 1328 |
H3 | 120 | 589 | 574 | 1058 |
H4 | 138 | 534 | 702 | 1199 |
H5 | 141 | 615 | 647 | 1163 |
H6 | 104 | 584 | 706 | 1496 |
H7 | 100 | 495 | 547 | 1066 |
H8 | 114 | 480 | 501 | 1039 |
H9 | 116 | 472 | 486 | 868 |
H10 | 105 | 552 | 570 | 1275 |
H11 | 98 | 496 | 497 | 1170 |
H12 | 147 | 714 | 739 | 1650 |
H13 | 105 | 646 | 654 | 1926 |
H14 | 107 | 513 | 616 | 1885 |
H15 | 160 | 798 | 925 | 2173 |
H16 | 71 | 357 | 397 | 960 |
H17 | 112 | 651 | 707 | 1741 |
H18 | 63 | 386 | 471 | 909 |
H19 | 96 | 317 | 491 | 1076 |
97.50% | Forecast (2009) | Actual (2009) | 2.50% | |
---|---|---|---|---|
H1 | 1.0001 | 0.8974 | 0.754 | 0.8469 |
H2 | 0.9329 | 0.8527 | 0.7787 | 0.797 |
H3 | 0.6448 | 0.6218 | 0.5918 | 0.5987 |
H4 | 0.7855 | 0.7618 | 0.6802 | 0.7303 |
H5 | 0.6584 | 0.64 | 0.6042 | 0.62 |
H6 | 1.0101 | 0.9604 | 0.8475 | 0.9123 |
H7 | 0.7813 | 0.7347 | 0.725 | 0.7006 |
H8 | 0.7201 | 0.6867 | 0.6311 | 0.6596 |
H9 | 0.6578 | 0.6177 | 0.5681 | 0.5894 |
H10 | 0.8109 | 0.7829 | 0.6308 | 0.7441 |
H11 | 0.8101 | 0.7573 | 0.6985 | 0.7171 |
H12 | 0.7623 | 0.7336 | 0.714 | 0.712 |
H13 | 1.059 | 1.0286 | 0.831 | 1 |
H14 | 1.1306 | 1.0868 | 1.1448 | 1.0409 |
H15 | 0.912 | 0.8665 | 0.8107 | 0.8263 |
H16 | 0.9296 | 0.8488 | 0.7418 | 0.7869 |
H17 | 0.9731 | 0.9427 | 0.9368 | 0.8984 |
H18 | 1.0686 | 1.0443 | 1.0745 | 1.0115 |
H19 | 1.1075 | 1.0769 | 1.0996 | 1.0417 |
Trend | Lucas | Average of Trend and Lucas | |
---|---|---|---|
No. of fails | 3 | 15 | 15 |
of DMUs and propose a plan to improve the inputs/outputs of inefficient DMUs. This function is difficult to achieve with similar models in statistics, e.g., stochastic frontier analysis. DEA scores are not absolute but relative. They depend on the choice of inputs, outputs and DMUs as well as on the choice of model for assessing DMUs. DEA scores are subject to change and thus data variations in DEA should be taken into account. This subject should be discussed from the perspective of the itemized input/output variations. From this point of view, we have proposed two models. The first model utilizes historical data for the data generation process, and hence this model resamples data from a discrete distribution. It is expected that, if the historical data are volatile widely, confidence intervals will prove to be very wide, even when the Lucas weights are decreasing depending on the past-present periods. In such cases, application of the moving-average method is recommended. Rolling simulations will be useful for deciding on the choice of the length of the historical span. However, too many past year data are not recommended, because environments, such as healthcare service systems, are changing rapidly. The second model aims to forecast the future efficiency and its confidence interval. For forecasting, we used three models; namely, the linear trend model, the weighted average, and their average. On this subject, Xu and Ouenniche [
KaoruTone,JamalOuenniche, (2016) DEA Scores’ Confidence Intervals with Past-Present and Past-Present-Future Based Resampling. American Journal of Operations Research,06,121-135. doi: 10.4236/ajor.2016.62015