Vol 3, No 6 (2012)

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Available online through

ISSN 2229 – 5046

EVALUATING THE EFFICIENCY OF DMUS WITH STOCHASTIC DATA

USING BON FERRONI INEQUALITY VIA CCR MODEL

Mehrdad Nabahat*

Sama technical and vocational training school, Islamic Azad University, Urmia branch, Urmia, Iran

Fardin Esmaeeli Sangari

Sama technical and vocational training school, Islamic Azad University, Urmia branch, Urmia, Iran

(Received on: 10-06-12; Accepted on: 30-06-12)

ABSTRACT

Data envelopment analysis (DEA) is non-parametric method for evaluating the efficiency of the decision making units

(DMUs) using mathematical programming. The models which exist are just for evaluating the performance of DMUs with deterministic data. Now the main point which can take managers attraction is that how we could evaluate the efficiency of DMUs if data were stochastically? In this article we propose a method for evaluating the efficiency of DMUs with stochastic inputs and outputs via CCR model using chance constrained programming and changing it to deterministic equivalent model. Finally, a numerical example is presented to show the application of this method.

Key Words: DEA, Stochastic programming.

1. INTRODUCTION

Stochastic programming approach is a method for modeling problems in which some of or all of the parameters of a problem expressed by random variables. These cases can appear frequently in our life such as determining the amount of profits from product sales, access to resources...

The main goal in stochastic programming is focusing on random variables that can be built with a model answer. The main idea in all the models of stochastic programming, such as Williams's method [1] is to change it into equivalent deterministic models. There are various methods for evaluating the efficiency of DMUs when the data are crisp such as Charnes, Cooper and Rodes [2] or Banker, Charnes, Cooper [3] method and also there are methods for evaluating the efficiency of DMUs with stochastic data Charnes, Cooper and Tintner [4-6] or Beale [5] which solve the quadratic form of stochastic programming or Dantzing method [7] for solving linear programming problems with uncertain data. Among the numerous models we considered the chance constrained programming. Recently formulation of the original models with the introduction of random inputs and outputs has been considered by different researchers. Cooper and Huang [8], Asgharian and Khodabakhshi [9] are working in this issue. There are also different methods for ranking the efficient DMUs with stochastic data, including Hosseinzadeh Lotfi and Nematollahi [10] ranking using coefficient variation also ranking using AP technique of Razavyan, and Tohidi [11].In this article we assume that all of the inputs and outputs of DMUs have random data. The main propose is to evaluate the efficiency of DMUs with stochastic data by using Bon Ferroni inequality via CCR model and changing the stochastic model into equivalent crisp model by using the chance constrained programming.

2. PRELIMINARS

Let us assume that there exist n, DMUs with m inputs and s outputs. Consider that

X

_j

=

(

x

₁_j

,

x

₂_j

,...,

x

_mj

)

T and T

sj j j

j

y

Y

=

(

₁

,

₂

,...,

)

are respectively input and output vectors of 𝐷𝐷𝐷𝐷𝐷𝐷_𝑗𝑗 , the production possibly set of CCR model defined as follows:

(

)

1 1

,

|

,

0

n n

CCR j j j j j

j j

T

X Y

X

λ

X

Y

λ

Y

λ

= =







=

_

≥

≤

≥

_





∑



Corresponding author:

Mehrdad Nabahat*

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Defnition1 (Bon ferroni inequality): If the set of arbitrary events

A

₁

,

A

₂

,



A

_n constitutes a partition of the

sample space

S

and

A

₁c

,

A

₂c

,



,

A

_nc are complements of events the following rule applied:

𝑃𝑃(⋂ 𝐴𝐴𝑛𝑛𝑖𝑖=1 𝑖𝑖𝑐𝑐)≥1− ∑ 𝑃𝑃𝑛𝑛𝑖𝑖=1 (𝐴𝐴𝑖𝑖) (1)

3. STOCHASTIC MODEL

Suppose that there exist n decision making unit with stochastic inputs and outputs. Consider that

n

j

y

x

_j

(

~

_j

,

~

_j

,...,

~

_mj

)

T

,

~

_j

(

~

_j

,

~

_j

,...,

~

_sj

)

T

,

1 ,

2 ,...,

~

2 1 2

1

=

are the stochastic input and output vectors of

𝐷𝐷𝐷𝐷𝐷𝐷𝑗𝑗 respectively and

x

y

j

y

j

y

j

y

sj T

j

n

T

mj j j

j

=

(

1

,

2

,...,

)

,

=

(

1

,

2

,...,

)

,

=

1 ,

2 ,...,

are deterministic equivalent input

and output vectors respectively. It is assumed that all of inputs and outputs are normally distributed.

The CCR model for evaluating the efficiency of DMUs would be as follows:

𝐷𝐷𝑖𝑖𝑛𝑛 𝜃𝜃

𝑠𝑠.𝑡𝑡 � 𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑥𝑥𝑖𝑖𝑗𝑗 ≤ 𝜃𝜃𝑥𝑥𝑖𝑖𝑖𝑖 ,𝑖𝑖= 1, … ,𝑚𝑚

� 𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑦𝑦𝑟𝑟𝑗𝑗 ≥ 𝑦𝑦𝑟𝑟𝑖𝑖 ,𝑟𝑟= 1, … ,𝑠𝑠

𝜆𝜆𝑗𝑗 ≥0 ,𝑗𝑗= 1, … ,𝑛𝑛

Here we want to evaluate the efficiency of DMUs with stochastic data by using CCR model and the method of transforming the stochastic model into deterministic model will be explained.

𝑠𝑠.𝑡𝑡 𝑃𝑃�∑𝑛𝑛𝑗𝑗=1𝜆𝜆𝑗𝑗𝑥𝑥�𝑖𝑖𝑗𝑗 ≤ 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖 ,∑𝑛𝑛𝑗𝑗=1𝜆𝜆𝑗𝑗𝑦𝑦�𝑟𝑟𝑗𝑗 ≥ 𝑦𝑦�𝑟𝑟𝑖𝑖� ≤1− 𝛼𝛼 (2)

𝜆𝜆𝑗𝑗 ≥0 ,𝑗𝑗= 1, … ,𝑛𝑛

Where in the above models, P means "probability" and 𝛼𝛼 ∈[0,1] is a level of error which is a predefined number,

1− 𝛼𝛼Represents acceptance of constraints. We are explaining the steps of transforming to deterministic model now.

Suppose that:

𝐴𝐴𝑖𝑖 =� 𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑥𝑥�𝑖𝑖𝑗𝑗 ≥ 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖 𝑖𝑖= 1,2, … ,𝑚𝑚

𝐴𝐴𝑚𝑚+𝑟𝑟 =∑𝑛𝑛𝑗𝑗=1𝜆𝜆𝑗𝑗𝑦𝑦�𝑟𝑟𝑗𝑗 ≤ 𝑦𝑦�𝑟𝑟𝑖𝑖 𝑟𝑟= 1, … ,𝑠𝑠 (3)

Through equations (1) and (3), we have:

𝑃𝑃(⋂ 𝐴𝐴𝑛𝑛𝑖𝑖=1 𝑖𝑖𝑐𝑐) =𝑃𝑃�∑𝑛𝑛𝑗𝑗=1𝜆𝜆𝑗𝑗𝑥𝑥�𝑖𝑖𝑗𝑗 ≤ 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖 ,∑𝑛𝑛𝑗𝑗=1𝜆𝜆𝑗𝑗𝑦𝑦�𝑟𝑟𝑗𝑗 ≤ 𝑦𝑦�𝑟𝑟𝑖𝑖� ≥1− 𝛼𝛼 (4)

And through equations (1) and (4), we can write:

𝑃𝑃(𝐴𝐴𝑖𝑖)≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 (5)

𝑆𝑆𝑖𝑖𝑛𝑛𝑐𝑐𝑆𝑆 � 𝑃𝑃(𝐴𝐴𝑖𝑖) 𝑚𝑚+𝑠𝑠

𝑖𝑖=1

≤ �_𝑚𝑚𝛼𝛼₊_𝑠𝑠

𝑚𝑚+𝑠𝑠

𝑖𝑖=1

=𝛼𝛼

So we have:

𝑃𝑃 ��𝐴𝐴𝑐𝑐𝑖𝑖 𝑛𝑛

𝑖𝑖=1

� ≥1− � 𝑝𝑝(𝐴𝐴𝑖𝑖 𝑚𝑚+𝑠𝑠

𝑖𝑖=1

(3)

From (5)

𝑃𝑃(𝐴𝐴𝑖𝑖)≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 ⟹

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧

𝑃𝑃 ��𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑥𝑥�𝑖𝑖𝑗𝑗 ≥ 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖� ≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 (6)

𝑃𝑃 �� 𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑦𝑦�𝑟𝑟𝑗𝑗 ≤ 𝑦𝑦�𝑟𝑟𝑖𝑖� ≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 (7)

From (6):

𝑗𝑗=1

𝑥𝑥�𝑖𝑖𝑗𝑗 ≥ 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖� ≤_𝑚𝑚𝛼𝛼₊_{𝑠𝑠 ⟹ 𝑃𝑃}�� 𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

𝑥𝑥�𝑖𝑖𝑗𝑗 − 𝜃𝜃𝑥𝑥�𝑖𝑖𝑖𝑖 ≥0�=𝑃𝑃

⎝ ⎜ ⎛

� 𝜆𝜆𝑗𝑗𝑥𝑥�𝑖𝑖𝑗𝑗 𝑗𝑗≠𝑖𝑖

+ (𝜆𝜆𝑖𝑖− 𝜃𝜃)𝑥𝑥�𝑖𝑖𝑖𝑖

��

ℎ𝑖𝑖

≥0

⎠ ⎟ ⎞

≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 ,

𝑖𝑖 = 1, … ,𝑚𝑚

Assuming:

ℎ𝑖𝑖 =� 𝜆𝜆𝑗𝑗𝑥𝑥�𝑖𝑖𝑗𝑗 𝑗𝑗 ≠𝑖𝑖

+ (𝜆𝜆𝑖𝑖− 𝜃𝜃)𝑥𝑥�𝑖𝑖𝑖𝑖 ,𝑖𝑖= 1, … ,𝑚𝑚

We can obtain mean and variance of stochastic variableℎ_𝑖𝑖 through:

𝐸𝐸(ℎ𝑖𝑖) =𝐸𝐸 �� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑥𝑥�𝑖𝑖𝑗𝑗+ (𝜆𝜆𝑖𝑖− 𝜃𝜃)𝑥𝑥�𝑖𝑖𝑖𝑖�=� 𝜆𝜆𝑗𝑗 𝑗𝑗 ≠𝑖𝑖

𝑥𝑥̅𝑖𝑖𝑗𝑗 + (𝜆𝜆𝑖𝑖− 𝜃𝜃)𝑥𝑥̅𝑖𝑖𝑖𝑖 ,𝑖𝑖= 1, … ,𝑚𝑚

𝑣𝑣𝑣𝑣𝑟𝑟(ℎ𝑖𝑖) =� � 𝜆𝜆𝑗𝑗 𝑘𝑘≠𝑖𝑖 𝑗𝑗≠𝑖𝑖

𝜆𝜆𝑘𝑘𝑐𝑐𝑖𝑖𝑣𝑣�𝑥𝑥�𝑖𝑖𝑗𝑗,𝑥𝑥�𝑖𝑖𝑘𝑘�+ 2(𝜆𝜆𝑖𝑖− 𝜃𝜃)� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑐𝑐𝑖𝑖𝑣𝑣�𝑥𝑥�𝑖𝑖𝑗𝑗,𝑦𝑦�𝑖𝑖𝑖𝑖�+ (𝜆𝜆𝑖𝑖− 𝜃𝜃)2𝑣𝑣𝑣𝑣𝑟𝑟(𝑥𝑥�𝑖𝑖𝑖𝑖)

Consider 𝜎𝜎_𝑖𝑖𝐼𝐼=�𝑣𝑣𝑣𝑣𝑟𝑟(ℎ_𝑖𝑖) ,𝑖𝑖= 1, … ,𝑚𝑚be Standard deviation of ℎ_𝑖𝑖 so we have:

𝑃𝑃(ℎ𝒊𝒊− 𝐸𝐸_𝜎𝜎 (ℎ𝑖𝑖) 𝑖𝑖𝐼𝐼 ≥

0− 𝐸𝐸(ℎ𝑖𝑖)

𝜎𝜎𝑖𝑖𝐼𝐼 )≤

𝛼𝛼 𝑚𝑚+𝑠𝑠

ℎ𝒊𝒊−𝐸𝐸(ℎ𝑖𝑖)

𝜎𝜎_𝑖𝑖𝐼𝐼 Have standard normal distribution like this: ℎ𝒊𝒊−𝐸𝐸(ℎ𝑖𝑖)

𝜎𝜎_𝑖𝑖𝐼𝐼 ~𝑁𝑁(0,1)

However, when Z has standard normal distribution we can write 𝑃𝑃(𝑍𝑍 ≥ −𝑧𝑧) =𝑃𝑃(𝑍𝑍 ≤ 𝑧𝑧) =𝜙𝜙(𝑧𝑧) and:

𝑃𝑃 �ℎ𝒊𝒊− 𝐸𝐸_𝜎𝜎 (ℎ𝑖𝑖)

𝑖𝑖𝐼𝐼 ≥

−𝐸𝐸(ℎ𝑖𝑖)

𝜎𝜎_𝑖𝑖𝐼𝐼 � ≤_𝑚𝑚𝛼𝛼₊_{𝑠𝑠 ⟹ 𝑃𝑃}�ℎ𝒊𝒊− 𝐸𝐸 (ℎ𝑖𝑖)

𝜎𝜎_𝑖𝑖𝐼𝐼 ≤𝐸𝐸 (ℎ𝑖𝑖)

𝜎𝜎_𝑖𝑖𝐼𝐼 � ≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠

⟹ 𝜙𝜙 �𝐸𝐸(_𝜎𝜎ℎ𝑖𝑖)

𝑖𝑖𝐼𝐼 � ≤

𝛼𝛼 𝑚𝑚+𝑠𝑠

Suppose𝜙𝜙 �−𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠�=

𝛼𝛼

𝑚𝑚+𝑠𝑠, since 𝜙𝜙 is an inverse able function we have:

𝜙𝜙 �𝐸𝐸(_𝜎𝜎ℎ𝑖𝑖)

𝑖𝑖𝐼𝐼 � ≤

𝛼𝛼

𝑚𝑚+𝑠𝑠=𝜙𝜙(−𝐾𝐾𝑚𝑚𝛼𝛼+𝑠𝑠)⟹

𝐸𝐸(ℎ𝑖𝑖)

𝜎𝜎𝑖𝑖𝐼𝐼 ≤ −𝐾𝐾 𝛼𝛼

𝑚𝑚+𝑠𝑠⟹ 𝐸𝐸(ℎ𝑖𝑖)≤ −𝐾𝐾𝑚𝑚𝛼𝛼+𝑠𝑠𝜎𝜎𝑖𝑖 𝐼𝐼

� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑥𝑥̅𝑖𝑖𝑗𝑗+ (𝜆𝜆𝑖𝑖− 𝜃𝜃)𝑥𝑥̅𝑖𝑖𝑖𝑖 ≤ −𝐾𝐾_𝑚𝑚𝛼𝛼₊_𝑠𝑠𝜎𝜎𝑖𝑖𝐼𝐼 ⟹ � 𝜆𝜆𝑗𝑗𝑥𝑥̅𝑖𝑖𝑗𝑗 𝑛𝑛

𝑗𝑗=1

+𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠𝜎𝜎𝑖𝑖

𝐼𝐼_{≤ 𝜃𝜃𝑥𝑥̅}

𝑖𝑖𝑖𝑖 (8)

If the same explained procedure applied for outputs too, we have:

𝑗𝑗=1

𝑦𝑦�𝑟𝑟𝑗𝑗 ≤ 𝑦𝑦�𝑟𝑟𝑖𝑖� ≤_𝑚𝑚𝛼𝛼₊_{𝑠𝑠 ⟹ 𝑃𝑃}��𝜆𝜆𝑗𝑗 𝑛𝑛

𝑗𝑗=1

(4)

⟹ 𝑃𝑃

⎝ ⎜ ⎛

�𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑦𝑦�𝑟𝑟𝑗𝑗 + (𝜆𝜆𝑖𝑖−1)𝑦𝑦�𝑟𝑟𝑖𝑖

��

ℎ𝑟𝑟

≤0

⎠ ⎟ ⎞

≤_𝑚𝑚𝛼𝛼₊_𝑠𝑠 ,𝑟𝑟= 1, … ,𝑠𝑠 (9)

Assuming:ℎ_𝑟𝑟=∑_{𝑗𝑗≠𝑖𝑖}𝜆𝜆_𝑗𝑗𝑦𝑦�_{𝑟𝑟𝑗𝑗}+ (𝜆𝜆_𝑖𝑖−1)𝑦𝑦�_{𝑟𝑟𝑖𝑖},𝑟𝑟= 1, … ,𝑠𝑠We can obtain mean and variance of stochastic variableℎ_𝑟𝑟 through:

𝐸𝐸(ℎ𝑟𝑟) =𝐸𝐸 �� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑦𝑦�𝑟𝑟𝑗𝑗 + (𝜆𝜆𝑖𝑖−1)𝑦𝑦�𝑟𝑟𝑖𝑖�=� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑦𝑦�𝑟𝑟𝑗𝑗+ (𝜆𝜆𝑖𝑖−1)𝑦𝑦�𝑟𝑟𝑖𝑖 ,𝑟𝑟= 1,2, … ,𝑠𝑠 (10)

𝑣𝑣𝑣𝑣𝑟𝑟(ℎ𝑟𝑟) =𝑣𝑣𝑣𝑣𝑟𝑟 �� 𝜆𝜆𝑗𝑗 𝑗𝑗≠𝑖𝑖

𝑦𝑦�𝑟𝑟𝑗𝑗 + (𝜆𝜆𝑖𝑖−1)𝑦𝑦�𝑟𝑟𝑖𝑖�=� � 𝜆𝜆𝑗𝑗 𝑘𝑘≠𝑖𝑖 𝑗𝑗 ≠𝑖𝑖

𝜆𝜆𝑘𝑘𝑐𝑐𝑖𝑖𝑣𝑣�𝑦𝑦�𝑟𝑟𝑗𝑗,𝑦𝑦�𝑟𝑟𝑘𝑘�+ 2(𝜆𝜆𝑖𝑖−1)

𝑐𝑐𝑖𝑖𝑣𝑣�𝑦𝑦�𝑟𝑟𝑗𝑗,𝑦𝑦�𝑟𝑟𝑖𝑖�+ (𝜆𝜆𝑖𝑖−1)2𝑣𝑣𝑣𝑣𝑟𝑟(𝑦𝑦�𝑟𝑟𝑖𝑖) (11)

Consider 𝜎𝜎_𝑟𝑟𝑖𝑖=�𝑣𝑣𝑣𝑣𝑟𝑟(ℎ_𝑟𝑟) ,𝑟𝑟= 1, … ,𝑠𝑠be Standard deviation of ℎ_𝑟𝑟 so from (9) we have:

𝑃𝑃 �ℎ𝑟𝑟− 𝐸𝐸_𝜎𝜎 (ℎ𝑟𝑟)

𝑟𝑟𝑖𝑖 ≤

0− 𝐸𝐸(ℎ𝑟𝑟)

𝜎𝜎𝑟𝑟𝑖𝑖 � ≤

𝛼𝛼

𝑚𝑚+𝑠𝑠 ⟹ 𝜙𝜙�− 𝐸𝐸(ℎ𝑟𝑟)

𝜎𝜎𝑟𝑟𝑖𝑖 � ≤

𝛼𝛼

𝑚𝑚+𝑠𝑠 (12)

Suppose𝜙𝜙 �−𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠�=

𝛼𝛼

𝑚𝑚+𝑠𝑠, and (12) then𝜙𝜙 �− 𝐸𝐸(ℎ𝑟𝑟)

𝜎𝜎𝑟𝑟𝑖𝑖 � ≤ 𝜙𝜙(−𝐾𝐾𝑚𝑚𝛼𝛼+𝑠𝑠), since 𝜙𝜙 is an inverse able function we have:

−𝐸𝐸(_𝜎𝜎ℎ𝑟𝑟)

𝑟𝑟𝑖𝑖 ≤ −𝐾𝐾 𝛼𝛼

𝑚𝑚+𝑠𝑠 ⟹ 𝐸𝐸(ℎ𝑟𝑟)≥ 𝐾𝐾𝑚𝑚𝛼𝛼+𝑠𝑠𝜎𝜎𝑟𝑟 𝑖𝑖

From (10):

𝑦𝑦�𝑟𝑟𝑗𝑗 + (𝜆𝜆𝑖𝑖−1)𝑦𝑦�𝑟𝑟𝑖𝑖≥ 𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠𝜎𝜎𝑟𝑟

𝑖𝑖_{⟹ � 𝜆𝜆} 𝑗𝑗𝑦𝑦�𝑟𝑟𝑗𝑗 𝑛𝑛

𝑗𝑗=1

− 𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠𝜎𝜎𝑟𝑟

𝑖𝑖_{≥ 𝑦𝑦�}

𝑟𝑟𝑖𝑖,𝑟𝑟= 1, … ,𝑠𝑠 (13)

From (2), (8), and (13) deterministic CCR model will be as follows:

𝑠𝑠.𝑡𝑡 � 𝜆𝜆𝑗𝑗𝑥𝑥̅𝑖𝑖𝑗𝑗 𝑛𝑛

𝑗𝑗=1

+𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠𝜎𝜎𝑖𝑖

𝐼𝐼_{≤ 𝜃𝜃𝑥𝑥̅}

𝑖𝑖𝑖𝑖 ,𝑖𝑖= 1, … ,𝑚𝑚

� 𝜆𝜆𝑗𝑗𝑦𝑦�𝑟𝑟𝑗𝑗 𝑛𝑛

𝑗𝑗=1

− 𝐾𝐾 𝛼𝛼 𝑚𝑚+𝑠𝑠𝜎𝜎𝑟𝑟

𝑖𝑖 _{≥ 𝑦𝑦�}

𝑟𝑟𝑖𝑖 ,𝑟𝑟= 1, … ,𝑠𝑠 (14)

𝜆𝜆𝑗𝑗 ≥0 ,𝑗𝑗= 1, … ,𝑛𝑛

4. NUMERICAL EXAMPLE

(5)

Table1. Mean of inputs (payable profit and personal)

Table 2. Mean of outputs (Facilities and Received profit)

Table 3. Variance of inputs (payable profit and personal)

Table 4. Variance of outputs (Facilities and Received profit)

E D

C B

A

(

_rj

)

var y



46223196999 15248173088 901499969.9 1157943092 10438342998 1

(

j

)

var y



31520733945 9055718955 336854494.1 102193533.9 319349741.2 2

(

j

)

var y



The covariance of input and output values in tables (5) to (8) is listed.

Table 5. Covariance of the first input cov(X₁, X₁)

15

x



14

x



13

x



12

x



11

x



1 1

(

_j

,

_j

)

cov x



x



104738576.3 32328417.12 39947161.84 46273169.87 59201384.69 11

x



82060967.48 25257222.19 31254093.77 36234591.73 46273169.87 12

x



70772451.89 21840390.15 26991823.4 31254093.77 39947161.84 13

x



57179560.38 17704897.74 21840390.15 25257222.19 32328417.12 14

x



185942039.4 57179560.38 70772451.89 82060967.48 104738576.3 15

x



Table 6. Covariance of the second input cov(X₂, X₂)

25

x



24

x



23

x



22

x



21

x



2 2

(

_j

,

_j

)

cov x



x



1.57 1.27 1.52 1.20 11.78 21

x



2.21 2.58 2.56 1.77 1.20 22

x



3.28 4.37 3.91 2.56 1.52 23

x



3.36 6.22 4.37 2.58 1.27 24

x



2.95 3.36 3.28 2.21 1.57 25

x



E D C B A

( )

ij ij

E x



=

x

10992.61 3187.22

16264.77 4937.71

6214.71

( )

1j 1j

E x



=

x

11.88 16.50

13.87 12.43

13.15

( )

2j 2j

E x



=

x

E D

C B

A

( )

rj rj

E y



=

y

862602.45 8554755.15

271150.02 180786.77

282125.65

( )

1j 1j

E y



=

y

157820.95 84894.71

16774.20 9441.14

40742.8

( )

2j 2j

E y



=

y

E D

C B

A

(

_ij

)

var x



180617623.7 18474675.91 28165380.94 37810008.77 61775357.94 1

(

_j

)

var x



3.07 6.49 4.08 1.93 112.29 2

(

_j

)

(6)

Table 7. Covariance of the first output cov(Y₁, Y₁)

Table 8. Covariance of the second output cov(Y₂, Y₂)

Considering α=0.4, 0.9, using model (14) and Lingo the following results can occur:

5. CONCLUSION

Previous models of data envelopment analysis which evaluate the relative efficiency of DMUs are restricted to crisp inputs and outputs. Finally the created model is linear programming. Here a method is proposed for evaluating the efficiency of DMUs by using Bon Ferroni inequality when all the data are stochastic. The created model is a nonlinear programming and for different values of 𝛼𝛼 the relative efficiency of DMUs can be evaluated.

REFERENCES

[1] Avriel, M. and Williams, A., The value of information and stochastic programming. Operational Research, 18(1970), 947–954.

[2] Charnes, A., Cooper, W.W., Rhodes, E., Measuring the efficiency of decision-making units, European Journal of Operations Research, 2 (1978), 429–444.

[3] Banker, R.D., Charnes, A., Cooper W.W., Some models of estimating technicaland scale inefficiency in data envelopment analysis, Management sciences, 30(1984).

[4] Tintner, G., Stochastic linear programming with applications to agricultural economics. In Antosiewicz H. (Ed)

Proc. 2nd Symp. Linear Programming, 2(1955), 197–228. National Bureau of Standards, Washington D.C.

[5] Beale, E. M. L, The use of quadratic programming in stochastic linear programming, Rand Report (1961) 2404, the RAND Corporation.

[6] Charnes A, Cooper WW. , Deterministic equivalents for optimizing and satisfying under chance constraints,

Management, 11 (1963), 18-39.

[7] Danzig, G. B., Linear programming under uncertainty, Managementsciences,1(1955), 197–206.

15

y



14

y



13

y



12

y



11

y



1 1

(

j

,

j

)

cov y



y



-4415922686 -3200565967

-1065092995 -2480925023

9984501998

11

y



2454473916 2072749896

552393950.1 1107597740

-2480925023

12

y



4426989120 3142740309

862304319 552393950.1

-1065092995

13

y



21495611751 14585209041

2072749896 2072749896

-3200565967

14

y



44213492781 21495611751

4426989120 2454473916

-4415922686

15

y



25

y



24

y



23

y



22

y



21

y



2 2

(

_j

,

_j

)

cov y



y



2921229784 1571499966

1628811015 165211140.2

305464969.9

21

y



3063433415 878611276.7

175234655.6 97750336.75

165211140.2

22

y



3063433415 1646401317

322208646.5 175234655.6

1628811015

23

y



16151354522 8661992044

1646401317 878611276.7

1571499966

24

y



30150267252 16151354522

3063433415 3063433415

2921229784

25

y



E D C

B A

j

DMU

1.000 1.000 0.9998 0.9999

1.000

0.4 α

=

1.000 1.000 1.000

0.9999 1,000

0.9

(7)

[8] Cooper, W. W., Deng, H., Huang, Z., & Li, S. X., Chance constrained programming approaches to congestion in stochastic data envelopment analysis, European Journal of Operational Research, 155(2004), 487-501.

[9] Asgharian, M., Khodabakhshi, M., & Neralic, L., Congestion in stochastic data envelopment analysis: An input relaxation approach, International Journal of Statistics and Management System, 5(2010), 84-106.

[10] Hosseinzadeh Lotfi, F., Nematollahi, N., & Behzadi, M.H., Ranking Decision making units with stochastic data by using coefficient of variation, Mathematical and Computational Application, 15(2010), 148-155.

[11] Razavyan, Sh., Tohidi, Gh., Ranking of efficient DMUs with stochastic data, International Mathematical Forum, 3(2008), 79-83.