A DEA-BenchmarkingOptimization Model and Method

Based on the Theory of Maximum Entropy�

Yin-sheng Yang, Ning Li, Hai-cun Liu, and Hong-peng Guo��

School of Biological and Agricultural Engineering, Jilin University(Nanling Campus), 5988 Renmin Street, Changchun 130022, P.R. China

Abstract. Benchmarking is a technique that engages and executes a se-ries of measures to change indexes of the Decision Making Unit (DMU) toexcellent by using the gap analysis information between the DMU andbenchmark. In this paper, a DEA-Benchmarking model based on thetheory of maximum entropy is proposed and the conception of Entropy-DEA efficiency is defined. According to the optimization model basedon the theory of maximum entropy, the Entropy-DEA efficient DMUsis regarded as benchmarks, which have more advantages and directionthan DEA efficient DMUs. The measure method and existence prop-erty of Entropy-DEA efficiency are all analyzed in thispaper.

1 Introduction

DEA proposed by A. Charnes and W. W. Cooper et al. in 1978 is a non-parametrical optimization technique used to evaluate the relative efficienciesof DMUs. DEA can identify the relative efficient DMUs in multiple inputsand multiple outputs system by constructing a frontier which is composed ofexcellent DMUs and then measuring efficiency which is relative to thatfrontier by using the properties of convexity[6,7,8]. If the DMUs posited onthe frontier, the DMU is denoted as DEA efficient, otherwise the DMU isinefficient.

Benchmarking is a technique which takes the relative efficient DMUs as studystandard and obtains the improving information by using the comparison analy-sis between the improving DMUs and the benchmark[3]. The quantitative meth-ods, such as statistical algorithm and regression analysis are often used whencomparing the proceeding of benchmarking. However, because of the existenceof random errors and approximation of statistical distribution, induce the ab-sence of the methods mentioned above. The method also lacks of the ability ofdealing multiple inputs and outputs system in gap analysis. The methods can’t

� Supported by the National Science Foundation of China (No. 70571028).�� Corresponding author: Tel.:086-431-5095726, ghp@jlu.edu.cn

D.-S. Huang, K. Li, and G.W. Irwin (Eds.): ICIC 2006, LNCS 4113, pp. 875–882, 2006.c© Springer-Verlag Berlin Heidelberg 2006

876 Y.-s. Yang et al.

provide the effective information of reducing the distance between DMU andbenchmarks. The effect of benchmarking is limited by the disadvantages of themethods mentioned above.

The procession of Benchmarking is introduced in section two. In section three,the conception of information entropy and the theory of maximum entropy are de-fined, and theway of constructing aDEA-Benchmarking optimizationmodel basedon the theory ofmaximumentropy is proposed.The measuremethod and existenceproperty of Entropy-DEAefficiency are all analyzed in this section. The conclusionand prospect of the research in this field are showed in section four.

2 The Basic Procedure in Benchmarking

Benchmarking is a systemic and durative proceeding to improve the DMUs per-formance. First, we should identify the research field and corresponding in-dexes in the procedure comparing the performance of DMUs. Secondly, theexcellent performance benchmarks in the field mentioned above should be se-lected. At last, we reach the goal of improving DMUs performance and re-ducing the distance between DMUs and benchmarks by comparing their in-dexes between improving units and benchmarks. In the course of analysis howto identify the distance between DMUs and benchmarks is the core of bench-marking. The DMUs should be evaluated and compared in the view of multipleinputs and outputs system. The application of DEA to the gap analysis is effi-cient, which can calculate the relative efficiency of multiple inputs and outputsDMUs.

3 DEA-Benchmarking Model and Method Based on theTheory of Maximum Entropy

To apply DEA to benchmarking can identify the unit with excellent performanceas benchmarks in the view of technical and scale efficiency. A comprehensiveevaluation method, DEA-Benchmarking, can be constructed by integrating theadvantage of DEA in the aspect of dealing with multiple indexes and that ofbenchmarking in comparing and evaluation. The DEA-Benchmarking methodcan identify efficient benchmarks as the goal of the other inefficient DMUs in thesystem, identify corresponding benchmarks, and provide more accurate analysisand evaluation for additive DMUs. It plays an important role to perfect thefunction of the system. Two key procedures in the DEA-Benchmarking methodare (1) to identify benchmarks by integrated DEA model and benchmarking; (2)to identify the improving information of its input and output indexes by usingthe gap analysis of DMUs and benchmarks.

The key technique of DEA-Benchmarking is to obtain accurate DMUs’ im-proving information used to improve DMUs. DEA-Benchmarking model basedon the theory of maximum entropy, with integrated the maximum entropy theoryto DEA-Benchmarking, has been explored in this paper.

A DEA-Benchmarking Optimization Model and Method 877

3.1 The Theory of Entropy

According to information theory, entropy is a measure to scale the system’sdegree of out-of-order. To a series of random events, the bigger the uncertainty ofthe system is, the bigger output information entropy value is. On the other hand,the smaller the entropy value is, the more ordered the system is. The necessityevent’s information entropy equal to zero, then equal probable event has themaximum information entropy. In the view of information science, entropy is ameasure concerning the uncertainty of a system. If there is a random test with nkinds of states x1, x2, . . . , xn , which have corresponding occurrence probabilitydenoted by P1, P2, . . . , Pn . Although the event with bigger probability has morechance to happen, we can’t identify the certain result before the test, which isthe uncertainty and out-of-order property of random events.

Shannon defined the uncertainty as the follows mathematic form:

S = −k0

n∑

i=1

PilnPi (1)

where S is defined as information entropy, k0 is a constant connected with theunits of measurement[1].

The value of uncertainty is connected with the result of test, where an eventprobability contribution has the maximum degree of uncertainty. This meansthe system is in the most disorder state. Taynes proposed the maximum entropytheory that if and only if the entropy is maximum values, the event happeningprobability is the only unbiased estimator when inducing is based on parts ofthe important information[2].

3.2 DEA-Benchmarking Model Based on the Theory of MaximumEntropy

All aspects of effective information of DMUs can be reflected by using the the-ory of maximum entropy to construct the weights of DMUs. An objective andquantitative management method can be provided by integrating the maximumentropy theory and DEA in benchmarking, which makes the benchmarking pro-ceeding more effective[4,5].

The core technique of benchmarking is gap analysis, that is, to collect infor-mation from DMU or out of DMU, compare then, work out a plan and execute it.The procedure has direct influence on the effect of management and its power. ADEA-Benchmarking model based on the theory of maximum entropy is proposedby integrating the theory of maximum entropy and DEA model. The model canbe described as follows:

This article assumes that there are n DMU’s , and each DMU has m kindsof input indexes and s kinds of output indexes, which are denoted respectivelyas Xj = (x1j , x2j , . . . , xmj)T and Yj = (y1j , y2j , . . . , ysj)T , j = 1, 2, . . . , n . TheC2GS2 model of DEA approach to assess the technical efficiency of the DMU-j0is determined by the following linear programming:

878 Y.-s. Yang et al.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

min θ = VDMUj0

s.t.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

n∑j=1

Xjλj + S− = θX0

n∑j=1

Yjλj − S+ = Y0

n∑j=1

λj = 1, λj ≥ 0, j = 1, 2, ..., n

S+ ≥ 0, S− ≥ 0

(2)

If the optimal solution of programming (2) λ∗, S∗+, S∗−, θ∗satisfies that θ∗=1,then the DMU-j0 is DEA(C2GS2)week efficient. Under the above conditions, ifthe optimal solution also satisfies that S∗+ = 0, S∗− = 0, then the DMU-j0 isDEA(C2GS2)efficient.

The feasible solution corresponding of C2GS2 model is

T =

⎧⎨

⎩(X, Y )

∣∣∣∣∣∣

n∑

j=1

Xjλj ≤ X,

n∑

j=1

Yjλj ≥ Y,

n∑

j=1

λj = 1, j = 1, 2, ..., n

⎫⎬

⎭ (3)

The problem of multiple objective programming in set T is{

V − min(f1(X, Y ), . . . , fm+s(X, Y ))s.t.(X, Y ) ∈ T

(4)

where fk(X, Y ) =�

Xk, 1 ≤ k ≤ m−Yk−m, m + 1 ≤ k ≤ m + s

, X = (X1, X2, . . . , Xm)T , Y =

(Y1, Y2, . . . , Ys)T ,

Lemma. The sufficient and necessary condition when DMU-j0 is DEA(C2GS2)efficient is that(X*,Y*)is the Pareto effective solution of multiple objective pro-gramming (4).

In practical production and application, though every event has correspond-ing happening probability, each kind of happening has different value and availinfluence. To bring the subject value and significance into information measure,we put the weigh entropy into DEA model which can be described as follows:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

min ωT X − μT Y + 1k

(m∑

i=1ωi ln ωi +

s∑j=1

μj ln μj)

s.t.

⎧⎪⎪⎨

⎪⎪⎩

ωi > 0, μj > 0, i = 1, 2, ..., m; j = 1, 2, ..., sm∑

i=1ωi +

s∑j=1

μj = 1

(X, Y ) ∈ T

(5)

where k is a parameter which is bigger than zero.

Definition 1. For a given parameter k > 0, if the optimal solution of program-ming (4) is denoted as X, Y , ωi, μj(i = 1, 2, . . . , m; j = 1, 2, . . . , s) , then theDMU identified by vector (X, Y )is Entropy DEA efficient.

A DEA-Benchmarking Optimization Model and Method 879

Theorem 1. If (X, Y ) is Entropy DEA efficient, the DMU must be DEAefficient.

Proof: Based on the Lemma, we just need to prove that (X, Y ) is the Paretoeffect solution of multiple objective programming (4). By using reduction toabsurdity, we assume that (X, Y ) is not the Pareto effect solution of multipleobjective programming (4), then there must be a vector (X ′, Y ′) ∈ T whichsatisfies that (

X ′

− Y ′

)<

(X

− Y

)(6)

Because (X, Y ) is Entropy DEA efficient, there must be a group of weight vectors

ω = (ω1, ω2, ..., ωm) > 0, μ = (μ1, μ2, ..., μs) > 0 , andm∑

i=1ωi+

s∑j=1

μj = 1 satisfied

the nonlinear programming (5), where (X, Y , ω, μ) is the optimal solution ofnonlinear programming (5), then

ωT X−μT Y +1k

(m�

i=1

ωi ln ωi+s�

j=1

μj ln μj) ≤ ωT X ′−μT Y ′+1k

(m�

i=1

ωi ln ωi+s�

j=1

μj ln μj)

ωT X − μT Y ≤ ωT X ′ − μT Y ′ (7)

Left-handed multiply weight vector (ω, μ) into (6), where ω > 0, μ > 0 , then

ωT X − μT Y > ωT X ′ − μT Y ′ (8)

There is an antinomy between the inequality (7) and (8), so (X, Y ) must be thePareto effect solution of multiple objective programming (4), which also meansthe corresponding DMU is efficient. �

The Theorem 1 indicates the Entropy DEA efficient DMU must be DEA efficientDMU, but the contrary proposition is not true. So the Entropy DEA efficient isa more significant conception than DEA efficient.

3.3 The Measure of Entropy-DEA Efficiency

Although the Entropy DEA efficient has more application significance in ap-plication than DEA efficient, the DEA model based on maximum entropy (5)is a nonlinear programming, which brings a lot of inconvenience in calculation.We introduce Lagrange multiplication operator to discuss the solution of pro-gramming (5), then the programming (5) can be changed into an equivalentexponential programming problem which is a measure of DEA efficiency.

We introduce Lagrange multiplication operator λ into programming (5), thena Lagrange function can be constructed as follows:

Lk(ω, μ, X, Y, λ) = ωT X −μT Y +1k

(m�

i=1

ωi ln ωi +s�

j=1

μj ln μj)+λ(m�

i=1

ωi +s�

j=1

μj − 1)

880 Y.-s. Yang et al.

Because the stationary point condition is∂Lk

∂ωi= 0,

∂Lk

∂μj= 0, we can obtain the

formula as follows:

Xi +1k

(ln ωi + 1) + λ = 0 i = 1, 2, ..., m (9)

− Yj +1k

(ln μj + 1) + λ = 0 j = 1, 2, ..., s (10)

The other stationary point condition is∂Lk

∂λ= 0 , we can draw the criterion

constraint as follows:m∑

i=1

ωi +s∑

j=1

μj = 1 (11)

By associating calculate the formula (9), (10) and (11), we can obtain theformulas as follows:

ωi = Ce−kxi i = 1, 2, ..., mμj = Cekyj j = 1, 2, ..., s

where C =1

m∑i=1

e−kxi +s∑

j=1ekyj

Put ωi and μj into the formula of Lagrange function, we get the result :

Lk(ω, μ, X, Y, λ) = ωT X − μT Y +1k

[m∑

i=1ωi ln(Ce−kxi) +

s∑j=1

μj ln(Ce−kyj )

]

= ωT X − μT Y +1k

[m∑

i=1ωi ln C +

m∑i=1

ωi ln e−kxi +s∑

j=1μj ln C +

s∑j=1

μj ln e−kyj

]

= ωT X − μT Y +1k

[ln C(

m∑i=1

ωi +s∑

j=1μj) − ωT X + μT Y

]=

1k

ln C

So the problem of calculate the function Lk is equal to the formula as follows:

max(m∑

i=1

e−kxi +s∑

j=1

ekyj ) (X, Y ) ∈ T (12)

Then we can draw the conclusions:

Theorem 2. The solution of (5) is equal to the solution of (12).

Using Theorem 1 and Theorem 2 , we can draw the conclusion below:

Theorem 3. For all k > 0, the ∀(X, Y ) of the solution of programming (12) isEntropy-DEA efficient, that is DEA efficient.

From now on, we can construct a mapping in JD for DEA efficient DMUs:

H : JD → R+1

∀(X, Y ) ∈ JD , defineH(X, Y )Δ=m∑

i=1e−kxi +

n∑j=1

ekyj as the exponential measure

of DEA efficient DMU (X,Y).

A DEA-Benchmarking Optimization Model and Method 881

3.4 The Existence of Entropy-DEA Efficient DMUs

Theorem 4. Assume that(Xj0,Yj0)is the maximum solution tom∑

i=1e−kxi+

s∑j=1

ekyj ,

where (X, Y ) ∈ T = {(X1, Y1), ..., (Xn, Yn)}andX = (X1, X2, ..., Xm)T , Y =(Y1, Y2, ..., Ys)T , then (Xj0 , Yj0)must be DEA efficient.

Proof: Based on the Theorem 3, we just need to prove (Xj0 , Yj0) is also thesolution of

max(m∑

i=1e−kxi +

s∑j=1

ekyj ) s.t.(X, Y ) ∈ T

In fact, because (Xj0 , Yj0) is the solution of

max(m∑

i=1e−kxi +

s∑j=1

ekyj ) s.t.(X, Y ) ∈ T (∀j = 1, 2, . . . , n)

Thenm∑

i=1

e−kx(j)i +

s∑

j=1

eky(j)r ≤

m∑

i=1

e−kx(j0)i +

s∑

j=1

eky(j0)r (13)

where Xj = (x(j)1 , x

(j)2 , ..., x

(j)m ), Yj = (y(j)

1 , y(j)2 , ..., y

(j)s )

For ∀(X, Y ) ∈ T ,n∑

j=1Xjλj ≤ X,

n∑j=1

Yjλj ≥ Y , because of T ⊂ T , it also means

Xi ≥n∑

j=1x

(j)i λj , Yr ≤

n∑j=1

y(j)r λj , i = 1, 2, ..., m; j = 1, 2, ...s

Based on the convexproperty of function ex, we can draw the conclusion as follows:

e−kXi ≤ e

n�

j=1λj(−kx

(j)i )

≤ λ1e−kx

(1)i + λ2e

−kx(2)i + ... + λne−kx

(n)i

wherem∑

i=1

e−kXi ≤m∑

i=1

λ1e−kx

(1)i +

m∑

i=1

λ2e−kx

(2)i + ... +

m∑

i=1

λne−kx(n)i (14)

Because ofs∑

r=1

ekYr ≤ λ1

s∑

r=1

eky(1)r + λ2

s∑

r=1

eky(2)r + ... + λn

s∑

r=1

eky(n)r (15)

Calculate the formula (14)+(15) and use the formula (13), we can draw thefollowing conclusion:

m�i=1

e−kx(j)i +

s�j=1

eky(j)r ≤ λ1

�m�

i=1e−kx

(1)i +

s�r=1

eky(1)r

�+ λ2

�m�

i=1e−kx

(2)i +

s�r=1

eky(2)r

�

+... + λn

�m�

i=1e−kx

(n)i +

s�r=1

eky(n)r

�≤ λ1

�m�

i=1e−kx

(j0)i +

s�r=1

eky(j0)r

�+ ...+

λ2

�m�

i=1e−kx

(j0)i +

s�r=1

eky(j0)r

�= (λ1 + ... + λn)

�m�

i=1e−kx

(j0)i +

s�r=1

eky(j0)r

�

=m�

i=1e−kx

(j0)i +

s�r=1

eky(j0)r

�

882 Y.-s. Yang et al.

From the Theorem 4, we know that the solution of programming (12) must bein the set T = {(X1, Y1), ..., (Xn, Yn)}, which indicates that there at least exist aDMU in the evaluation DMUs is Entropy-DEA efficient, and the DMU is DEAefficient. It is an embodiment of relative efficiency of Entropy-DEA efficient.

4 Conclusions

The core of benchmarking needs some accurate quantitative methods to ob-tain efficient information, so we proposed a DEA-Benchmarking model with themaximum entropy and defined Entropy-DEA efficiency. The measure and the ex-istence of Entropy-DEA efficient DMU were also strictly induced in this paper.The Entropy-DEA efficient DMU is a more complete and objective in the viewof uncertainty of DMUs information, which has more advantage property overDEA efficient DMUs. In the proceeding of benchmarking, we accept the eval-uation method based on Entropy-DEA efficiency and select the Entropy-DEAefficient DMUs as benchmarks for gap analysis between the indexes of DMUsand benchmarks, which have more accurate and operative property in manage-ment. The optimization method provides a more efficient means to search inselect DEA efficient DMUs within feasible solution space.

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