A study of developing an input-oriented ratio-based comparative efficiency model

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A study of developing an input-oriented ratio-based comparative efﬁciency model

Ching-Kuo Wei

a

, Liang-Chih Chen

b

, Rong-Kwei Li

b

, Chih-Hung Tsai

c,⇑

a

Department of Health Care Administration, Oriental Institute of Technology, 58, Sec. 2, Sihchuan Road, Pan-Chiao City, Taipei County 22061, Taiwan

b

Department of Industrial Engineering and Management, National Chiao-Tung University, Hsinchu, Taiwan

c

Department of Information Management, Yuanpei University, No. 306, Yuanpei Street, Hsin-Chu, Taiwan

a r t i c l e

i n f o

Keywords:

Data envelopment analysis DEA-R-I

CCR-I

a b s t r a c t

Data envelopment analysis (DEA) is a representative method to estimate efficient frontiers and derive efficiency. However, in a situation with weight restrictions on individual input–output pairs, its suitabil-ity has been questioned. Therefore, the main purpose of this paper is to develop a mathematical method, which we call the oriented ratio-based comparative efficiency model, DEA-R-I, to derive the input-target improvement strategy in situations with weight restrictions. Also, we prove that the efficiency score of DEA-R-I is greater than that of CCR-I, which is the first and most popular model of DEA, in input-oriented situations without weight restrictions to claim the DEA-R-I can replace the CCR model in these situations. We also show an example to illustrate the necessity of developing the new model. In a nutshell, we developed DEA-R-I to replace CCR-I in all input-oriented situations because it sets a more accurate weight restriction and yields a more achievable strategy.

1. Introduction

Data envelopment analysis (DEA) is one popular method for identifying efficient frontiers and evaluating efficiency. An efficient frontier is based on the concept of a non-dominated condition, which was first expressed by the Italian economist Pareto in 1927. This concept was adapted to production byKoopmansin

1951and to evaluate efficiency byFarrellin1957(Cooper et al., 2002).Charnes, Cooper, and Rhodes (1978)applied linear program-ming (LP) to identify efficient frontiers and measure productivity. This method, which measures productivity by LP, is called ‘‘data envelopment analysis”. They derived both an output-oriented (CCR-O) model and an input-oriented (CCR-I) model, which are not only the first but also most popular models of DEA. Many scholars have used DEA as the representative method to estimate an efficient frontier and measure productivity (Amirteimoori, 2007; Jahanshahloo, Hosseinzadeh Lotfi, & Zohrehbandian, 2005a). Over the past two decades, DEA has been established as a robust and valuable methodology (Chen & Ali, 2002; Liu & Chuang, 2009).

One advantage of DEA is objective weight selection, and there are many studies that focus on weight (Bernroider & Stix, 2007; Jahanshahloo, Soleimani-damaneh, & Nasrabadi, 2004; Jahanshahloo, Memariani, Hosseinzadeh, & Shoja, 2005b; Lotﬁ, Jahanshahloo, & Esmaeili, 2007; Wang, Parkan, & Luo, 2008). However, when apply-ing the typical DEA model, which is based on ðP

v

xÞ=ðPuyÞ or

P uy

ð Þ=ðP

v

xÞ, to a situation with weight restrictions on individual input–output pairs, its suitability is questionable. We take a case in hospitals as the example of the necessity of weight restrictions. Sickbeds, physicians, outpatients, inpatients, and surgery are important variables for hospital performance evaluations, where the sickbed variable contributes only to the inpatient and surgery variables but not the outpatient variable. In this situation, it is hard to assign a suitable weight restriction to an outpatient-sickbed pair.Golany and Roll (1989)argue that input–output pairs must correspond to an isotonicity assumption to avoid this problem. However, an isotonicity assumption represents a statistical rather than a causal relationship. For example, the statistical relationship between outpatient services and sickbeds is high, but the causal relationship between them is low. Therefore, conformance to the isotonicity assumption does not always avoid this problem.Dyson et al. (2001)argue that handling weight restrictions is still a pitfall in DEA applications from a theoretical perspective.Despic, Despic and Paradi (2007)claim that this kind of problem is difﬁcult to solve with a typical DEA model and therefore developed DEA-R, a model to solve the problem of weight restriction. DEA-R is expressed as follows: ^ ^ ek¼ max cðj;iÞP0 Xs j¼1 Xm i¼1 cðj;iÞrðj;iÞk Xs j¼1 Xm i¼1 cðj;iÞrðj;iÞp61; ( for p ¼ 1; 2; . . . ; n ) : ð1Þ

However, the DEA-R model developed byDespic et al. (2007)is an output-oriented model (we call it DEA-R-O). In some situations, we 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.eswa.2010.08.036

⇑ Corresponding author. Tel.: +886 3 6102338; fax: +886 3 6102343. E-mail addresses:[email protected],[email protected](C.-H. Tsai).

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need an input-oriented model to provide an input-target improve-ment strategy with weight restrictions. Using Taiwan’s private hos-pitals as an example again, the output was bounded by National Health Insurance; they have to adopt an input-targeted improve-ment strategy (reduce inputs), rather than an output-targeted strat-egy, to improve their efﬁciency. Therefore, a new mathematical method of deriving the input strategy (we call it input-oriented DEA-R, or DEA-R-I) has been developed.

In addition, the DEA-R-I seems to substitute for CCR-I in input-oriented situations without weight restrictions because the effi-ciency score of DEA-R-I is greater than or equal to than CCR-I when the relationship between DEA-R-O and CCR-O is unclear. According toDespic et al. (2007), the efficiency score of DEA-R-O with no weight restrictions is sometimes higher and sometimes lower than the efficiency score of CCR-O. This drawback prevents DEA-R-O from replacing CCR-O in a situation without weight restrictions. But, based on our study, we found two factors that cause this effi-ciency score discrepancy. The first is a more flexible selection of optimum weight, which affects the efficiency score of DEA-R-O higher than the efficiency score of CCR-O, while the second is the sum of the output-oriented ratioP w y

x

, which affects the effi-ciency score of DEA-R-O less than the effieffi-ciency score of CCR-O. Since we will use the sum of the input-oriented ratioP w x

y

to replace the sum of the output-oriented ratioP w y x

in com-puting the efficiency score in the DEA-R-I mathematical method, we suggest that the efficiency score of DEA-R-I will always be greater than or equal to the efficiency score of CCR-I (CCR input-oriented). This also means that the strategies of DEA-R-I are easier to achieve than the strategies of CCR-I because the strategy derived from the higher efficiency score needs fewer changes. If we can prove this hypothesis, the CCR-I model can be replaced by DEA-R-I because DEA-DEA-R-I provides a more accurate efficiency score in situations with weight restrictions and a better strategy in situa-tions without weight restricsitua-tions.

Because of the above reasons, the ﬁrst goal of this paper is to de-velop a mathematical method (we call it DEA-R-I) to derive the in-put-target improvement strategy in a situation with weight restrictions. The second goal is to prove that the input-target improvement strategy developed by DEA-R-I is always better than the CCR-I model in a situation without weight restrictions. There-fore, we can claim that the DEA-R-I model can replace the CCR-I model in all input-oriented situations.

2. Mathematical method to evaluate efﬁciency scores and derive input-target improvement strategies

Because there are no suitable input-oriented models for situa-tions with weight restricsitua-tions on single I/O pairs, we developed a new model to evaluate the efficiency score and derive the input-oriented strategy. We applied a new model to calculate the effi-ciency score and then derive the input-target strategy from the efficiency score:

Step 1: Compute the efﬁciency score

DEA-R-I, the mathematical model for computing the efﬁciency score of a DMU object ^ho, is expressed as follows:

Max ^ho ð2Þ st: X m i¼1 Xs r¼1 Wir ðXij=YrjÞ ðXio=YroÞ P ^ho j ¼ 1; . . . ; n ð3Þ Xm i¼1 Xs r¼1 Wir¼ 1 ð4Þ WirP0; ^hoP0 ð5Þ

Xi j: ith input variable of the jth DMU. Yr j: rth output variable of the jth DMU. Xi o: ith input variable of the object. Yr o: rth output variable of the object.

Wir: The weight of the ratio ofthe rth output variable Ythe ith input variable Xir.

Pm i¼1

Ps

r¼1WirðXij=YrjÞ=ðXio=YroÞ : the relative efﬁcient score with jth DMU’s.

For each DMU object, the model first computes its relative effi-ciency score for each specified weight, and the smallest is selected as the efficiency score of this set of weights. Second, by adjusting the weighting set, a maximum efficiency score will be selected as the efficiency score ^ho of the object. Since each DMU can get its optimal weight, we can say objectively that the DMU is inefficient if the efficiency score of this DMU is less than one. It is necessary to provide the improved strategy for this DMU, which we will discuss in next part. Using the data inTable 1as an example, if we want to calculate the efficiency score of the DMU1 ofTable 1, we must first find four relative efficiency scores for the DMU 1 in each weight.

When the weight set is W11= 1 and W12= 0, the relative efﬁ-ciency scores of DMU 1 with DMUs 1–4 are 1 2

4 = 2 4 þ 0 2 3 = 2 3 ¼ 1:00; 1 2 3 =2 4 þ 0 2 5 = 2 3 ¼ 1:33; 1 2 4:2 =2 4 þ 0 2 4:2 = 2 3 ¼ 0:95, and 1 2 5 = 2 4 þ 0 2 3 = 2 3 ¼ 0:80, respectively (the right-most points of lines 1, 2, 3, and 4 shown inFig. 1a). The relative efficiency score of DMU 1 with DMU 4 in weight W11= 1 and W12= 0 is 0.8, which means that if we need one unit of X1from DMU 1 to produce one unit of Y1, only 0.8 units of X1 from DMU 4 are needed to produce one unit of Y1. Repeating the computation, the relative efficiency scores of DMU1 with each DMU in different weight sets are shown inFig. 1a. InFig. 1a, when W11is between 0.000 and 0.231, the lowest value of the four rela-tive efficiency scores is the relarela-tive efficiency score with DMU 2.

Table 1

One input and two-outputs.

DMU Input Output

X1 Y1 Y2

1(A) 2.0 4.0 3.0

2(B) 2.0 3.0 5.0

3(C) 2.0 4.2 4.2

4(D) 2.0 5.0 3.0

Fig. 1a. Relative Efﬁciency Score of DMU 1 with DMUs in Different Weights and the Efﬁciency curve of DMU 1.

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When W11is between 0.231 and 0.652, the lowest value of the four relative efficiency scores is the relative efficiency score with DMU 3. Finally, when W11is between 0.652 and 1.000, the lowest value of the four relative efficiency scores is the relative efficiency score with DMU 4. The graph of the lowest values of the four relative effi-ciency scores in different weight sets is the effieffi-ciency curve of DMU 1 (it is not the efficiency frontier of all DMUs but the efficiency of DMU 1 in different weight sets). Since the efficiency score of point r (the least relative efficiency score is 20/23 (about 0.870) when W11= 0.652 and W12= 0.348) on the efficiency curve is the highest relative efficiency score on the curve, we select it as the efficiency score of DMU 1. Repeating the model, the efficiency scores of DMUs 2, 3, and 4 can be found and are shown inFigs. 1b, 1c and 1d, respectively. The efficiency scores of all DMUs are shown in

Table 2.

Step 2: Derive the input-target improvement strategy

As we stated above, if the efficiency score of a DMU is less than one, it is inefficient. The improvement now becomes indispensable. We can simply replace each input variable with a new DMU that equals its original data times its efficiency score, without changing the output variables. We call the new DMU our improved strategy. If we replace the inefficient DMU with this new DMU, then the effi-ciency score of this new DMU is one, and the effieffi-ciency score of the others is the same as before.Table 3shows the improved strategy of the DMU1. The table indicates that in order to make the DMU efficient, the input variable X1should be lowered from 2.0 to 40/ 23 to get the same outputs Y1and Y2.

3. Mathematical proof that the efﬁciency score of DEA-R-I is greater than or equal to the efﬁciency of CCR-I

After studying the relationship between DEA-R-O and CCR-O, we suggest that the efficiency score of DEA-R-I is greater than or equal to the efficiency of CCR-I. If this hypothesis holds, then we can claim that DEA-R-I can replace CCR-I because DEA-R-I is more efficient in situations with weight restrictions and more achievable in situations without weight restrictions.

Inspired by Despic et al. (2007), we deﬁne the efﬁciency of CCR-I

Fig. 1b. Relative Efﬁciency Score of DMU 2 with DMUs in Different Weights and the Efﬁciency curve of DMU 2.

Fig. 1c. Relative Efﬁciency Score of DMU 3 with DMUs in Different Weights and the Efﬁciency curve of DMU 3.

Fig. 1d. Relative Efﬁciency Score of DMU 4 with DMUs in Different Weights and the Efﬁciency curve of DMU 4.

Table 3

Original data, efﬁciency score and strategy of DMU 1 inTable 1.

DMU Original data Efﬁciency score Input-target strategy

Input Output Input Output

X1 Y1 Y2 X1 Y1 Y2

1(A) 2.0 4.0 3.0 20/23 40/23 4.0 3.0

Table 2

The efﬁciency score of DMUs.

DMU Efﬁciency score

1(A) 0.870

2(B) 1.000

3(C) 1.000

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ho¼Pmax ivi¼1;viP0 P rur¼1;urP0 min j P i

v

i Xij Xio P rur Yrj Yro

and the efﬁciency of CCR-I-Harmonic

h o¼Pmax ivi¼1;viP0 P rur¼1;urP0 min j X i

v

i Xij Xio X r ur Yro Yrj

to help us prove our claim. The proof requires two steps: Step 1: Proof that the efﬁciency of CCR-Harmonic h

o is always greater than or equal to CCR ho*.

By replacing X0 ij, Y0ijwith Xij Xio; Yrj Yroand multiplying P rurYYrorj P rurY_Yro rj ¼ P rurY1_rj0 P rurY1_rj0

in formulation, we can transpose h oto max P ivi¼1;viP0 P rur¼1;urP0 min j P i

v

iX0ij P rurY10 rj P rurY0rj P rur_Y10 rj and h o¼Pmax ivi¼1;viP0 P rur¼1;urP0 min j X i

v

iX0ij X r ur 1 Y0 rj :

The difference between h

oand hois 1 P rurY0rj P rur_Y10 rj : Because X r urY0rj X r ur 1 Y0 rj ¼ X r¼1::s t¼rþ1...s u2 rþ urut Y0 rj Y0 tj þY 0 tj Y0 rj ! ¼ X r¼1::s t¼rþ1...s u2 rþ 2urut 2urutþ urut Y0 rj Y0tj þY 0 tj Y0rj ! ¼ X r ur !2 þ X r¼1::s t¼rþ1...s urut Y0 rjY 0 tj Yrj0 Ytj0 2 ¼ 1 þ X r¼1::s t¼rþ1...s urut Y0 rjY 0 tj Y0 rj Y 0 tj 2 P1; we get P 1 rurY0rj P rur_Y10 rj 6_{1 if}

₈

_u_r;Y0 rjP0

and infer that h

o 6 ho.

Step 2: Proof that the efﬁciency of DEA-R-I ^h

ois always greater than or equal to CCR-Harmonic h

o.

We can rewrite the efﬁciency of DEA-R-I as ^h

o¼Pmax i P rw 0 ir¼1 ;w0 irP0 min j P i P rwir X0 ij Y0 rj and CCR harmonic h o¼Pmax ivi¼1;viP0 P rur¼1;urP0 min j P i P r

v

iur X0 ij Y0 rj:. Because P i

v

i¼ 1 andPrur¼ 1, we getPi P r

v

iur¼ 1, and we argue that h

o is a special case of ^ho. The best we obtain is: h

o6 ^ho. With the two steps, we obtain: ho6 ho6 ^ho.

4. An example

Like other studies (Ballestero & Maldonado, 2004; Katharaki, 2008), this study takes a hospital as an example to show the neces-sity of developing DEA-R-I and the advantages of this model. Two-inputs- two-outputs simpliﬁed data for four hospitals are shown in

Table 4. Because only DMU 5 is inefﬁcient, we show the efﬁciency scores, improvement strategies, and optimal weight sets that are derived by different models with two oriented situations inTable 5. The upper part of Table 5 shows the results without weight restrictions. Because the sickbed variable has no direct contribu-tion to the outpatient variable, the weight restriccontribu-tion is needed. Although there are not suitable restrictions for CCR, we take the most approximated restriction, v1x1

v1x1þv2x2

6 u2y2

u1y1þu2y2, as the weight

restriction of CCR in this case to compare with other models. For the DEA-R models, we set w11= 0 as a weight restriction. The results with weight restrictions are shown in the lower part of

Table 5.

First, we show the necessity of weight restrictions. Compare the upper part ofTable 5with the lower part. The results show that the efficiency scores of the models without weight restriction are all greater than the same efficiency scores of the models with weight restrictions. Thus, weight restrictions in different models are nec-essary. Then compare the optimal weight sets of different models with weight restrictions. Although the efficiency score of DEA-R-O with restriction is the same as CCDEA-R-O, we show the difference between the optimal weight sets in two output-oriented models. In the input-oriented situation, DEA-R-I derives not only a different efficiency from CCR-I but also a different optimal weight set. In addition, the weight sets derived by DEA-R models are more easily understood than those of CCR because the weight restriction of DEA-R always holds, but the same is not true for CCR. Therefore, we claim that the DEA-R model is more suitable under weight restrictions.

We compare DEA-R-I with DEA-R-O after comparing DEA-R with CCR. Compare the improvement strategies. The left part of Ta-ble 5shows that the strategies of the input-oriented models are different from output-oriented models, both CCR and DEA-R. Then compare the efficiency scores of DEA-R in different orientations. The efficiency scores of DEA-R-I are different from those of DEA-R-O whether or not there are weight restrictions. Unlike the difference between two CCR models, which is only different improvement strategies, the differences between DEA-R-I and DEA-R-O are in both strategies and efficiency. This means that DEA-R-I cannot be replaced with DEA-R-O and that the develop-ment of DEA-R-I is necessary.

Finally, compare the DEA-R-I with CCR-I in the situation with-out weight restrictions. The left part ofTable 5shows that DEA-R-I without weight restrictions has a greater value than CCDEA-R-I. This result does not contradict our proof. The middle part ofTable 5

shows the improvement strategies, which are translated from the Table 4

Two-inputs-two-outputs.

DMU Input Output

Sickbed Doctor Outpatient Inpatient

5(E) 2.0 3.0 4.0 3.0

6(F) 2.0 2.7 3.0 5.0

7(G) 2.0 2.7 4.2 4.2

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efﬁciency score. DEA-R-I without weight restrictions shows that we need a change of X0

1from 2 to 1.740 (i.e. 2,000 beds to 1,740 beds) and X0

2from 3 to 2.610 (that is, 300 doctors to 261 doctors) and keep the same output. CCR-I shows that values of X0

1= 1.714 (1,714 beds) and X0

2= 2.571 (2,571 doctors) are needed. This result shows that the changes of DEA-R-I are less than those of CCR-I in the situation without weight restrictions. Based on these results, we claim that the DEA-R-I strategy is always easier to achieve than the CCR-I strategy. We can conclude that DEA-R-I can replace CCR-I in all input-target situation.

5. Conclusion

In this paper, we developed an input-oriented ratio-based mod-el (DEA-R-I) for calculating efﬁciency scores and identifying input-target improvement strategies in situations with weight restric-tions. We also show further proof of our model in order to claim that this model can replace the CCR-I model in situations without weight restrictions. A numerical example shows the difference be-tween DEA-R-O and DEA-R-I to support our claim that the develop-ment of the DEA-R-I model is necessary for input-oriented situations with weight restrictions. This example further supports the claim that DEA-R-I can also provide easier improvement strat-egies than CCR-I in situations without weight restrictions. Because of its accuracy in situations with weight restrictions and its better strategy, we claim that DEA-R-I can replace CCR-I in all input-ori-ented situations.

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Table 5

Efﬁciency scores, strategies and optimal weight sets for DMU 5(E).

Without restriction ho X01 X02 Y01 Y02 v1x1 v2x2 u1y1 u2y2 CCR-I 0.857 1.71 2.57 4.00 3.00 1.000 0.000 0.571 0.286 CCR-O 0.857 2.00 3.00 4.67 3.50 1.167 0.000 0.667 0.333 ho X01 X02 Y01 Y02 w11 w12 w21 w22 DEA-R-I 0.870 1.74 2.61 4.00 3.00 0.652 0.348 0.000 0.000 DEA-R-O 0.857 2.00 3.00 4.67 3.50 0.667 0.333 0.000 0.000 With Restriction ho X01 X02 Y01 Y02 v1x1 v2x2 u1y1 u2y2 CCR-I 0.800 1.60 2.40 4.00 3.00 0.333 0.667 0.533 0.267 CCR-O 0.800 2.00 3.00 5.00 3.75 0.417 0.833 0.667 0.333 ho X0₁ X0₂ Y0₁ Y0₂ w11 w12 w21 w22 DEA-R-I 0.811 1.62 2.43 4.00 3.00 0.000 0.324 0.676 0.000 DEA-R-O 0.800 2.00 3.00 5.00 3.75 0.000 0.357 0.643 0.000