Efficiency of Green Supply Chain in the presence of non-discretionary and undesirable factors, using Data Envelopment Analysis Текст научной статьи по специальности «Строительство и архитектура»

BUSINESS INFORMATICS Vol. 15 No 3 - 2021 DOI: 10.17323/2587-814X.2021.3.78.96

Efficiency of Green Supply Chain in the presence of non-discretionary and undesirable factors, using Data Envelopment Analysis

Mehdi Shoja a ©

E-mail: [email protected]

Farhad Hosseinzadeh Lotfi b ©

E-mail: [email protected]

Amir Gholam Abri c * ©

E-mail: [email protected]

Alireza Rashidi Komijan c ©

E-mail: [email protected]

a Roodehen Branch, Islamic Azad University Address: No 4, Shahid Ahmadi Alley, Valiasr Town, Firoozkooh, Tehran 3981838183, Iran

b Science and Research Branch, Islamic Azad University

Address: Shohada Hesarak Blv, University Square, End of Shahid Sattari Highway, Tehran 1477893855, Iran

c Firoozkooh Branch, Islamic Azad University Address: First of Valiasr Town, 45 Meters Street, Firoozkooh, Tehran 3981838381, Iran

Abstract

A new approach that has dominated the production operations management field in recent years is supply chain management. A supply chain includes all the facilities, tasks and activities involved in manufacturing a product from suppliers to customers. Its various elements are planning, supply and

* Corresponding author

demand management, procurement of raw materials, production scheduling, distribution and delivery of products to the customer. Special structures in the supply chain have been less studied in previous research. In this paper, the supply chain and its performance evaluation are examined in the presence of non-discretionary, undesirable and negative data. For this purpose, another model of the network DEA is presented which evaluates performance of the chain in the presence of non-discretionary inputs and outputs, undesirable outputs and negative outputs even in its internal structure. The efficiency of the chain stages is also calculated using a dual model. Subsequently, 42 cement companies listed on the Tehran stock exchange were evaluated, each of which has a chain of four stages including suppliers, manufacturers, distributors and customers. Based on the implementation of the model, six companies were found to be efficient and the rest were introduced as inefficient. Moreover, 25 cement companies in the Supplier sector, 18 companies in the manufacturing sector, seven companies in the distribution sector and finally 17 companies in the customer service sector were found to be efficient.

Key words: supply chain efficiency; cement industry; network DEA; non-discretionary factors; undesirable outputs; negative data.

Citation: Shoja M., Lotfi F.H., Abri A.G., Komijan A.R. (2021) Efficiency of Green Supply Chain in the presence of non-discretionary and undesirable factors, using Data Envelopment Analysis. Business Informatics, vol. 15, no 3, pp. 78-96. DOI: 10.17323/2587-814X.2021.3.78.96

Introduction

In recent years, a relatively suitable approach has been presented in the intellectual, cultural and social fields on performance and productivity evaluation. In fact, today productivity is not strange for the thinking and beliefs of the community. Different applications of the supply chain in various sciences have attracted many researchers. The network supply chain consists of four parts: supplier, manufacturer, distributor and customer. The supply chain structure varies in terms of size and complexity, from a simple chain that represents independent decision-making, to the behaviors and interactions of complex companies. Therefore, proper supply chain performance evaluation is necessary to consider the network characteristics of the chain and its interactions. Generally, the larger and more complex the supply chain is, the more difficult it will be to evaluate.

Among the evaluation methods, data envelopment analysis (DEA) is widely used to evaluate the relative performance of a set of pro-

duction processes called decision-making units (DMUs). This non-parametric method evaluates DMUs by presenting different models that produce multiple outputs using multiple inputs. DEA is a very important method in supply chain management literature.

Classical DEA models do not have any theory regarding the intermediate activities of DMUs and consider them as a black box, so that just the inputs consumed and the final outputs produced are considered in calculating efficiency. This view, which does not consider intermediate activities and products of the supply chain, is suitable for a simple production process, however it doesn't have proper application in a complex network system.

To identify inefficiencies and their causes in the intermediate parts, many researchers have tried to reject the black box model and consider its internal structure in DEA models. These models are called 'network DEA' in the literature. The supply chain is one of the most important and most practical states of network DEA. Planned production of raw materials, designing

and producing appropriate products, optimal distribution and delivery, and ultimately serving customers and their satisfaction in the form of supply chain management, are of great interest. In supply chain management, all efficiency measurements seek to achieve two ultimate goals of reducing costs and increasing profits.

The important thing to consider in these situations is that, sometimes changing some inputs and outputs of the supply chain are not entirely available to the manager. These are called non-discretionary or partly-discretionary factors. In addition to non-discretionary factors, in some cases, there may be undesirable indicators within middle activities of the supply chain. Dealing with these indicators in classical DEA models has been discussed, however non-discretionary inputs and outputs, undesirable outputs and negative outputs are less discussed in previous studies performed on supply chains. Therefore, in this study, we have tried to examine these conditions in the full supply chain. In other words, the aim of this paper, is to present a model based on DEA, with a network structure to evaluate the total efficiency and performance of supply chain steps in the presence of non-discretionary inputs and outputs, undesirable and negative outputs.

As we know, recently the application of data envelopment analysis has attracted a lot of attention in various sciences, and many studies have been carried out in the field of data envelopment analysis with a network structure, all of which attempts to analyze different states of a network. In this regard, one of the first studies was presented about twenty years ago by Fare and Grosskopf [1, 2]. They considered the production possibility set with original standard principles in the variable returns-to-scale scheme for the general structure of a network, and then created the production possibility set of the supply chain by combining the sets of production possibilities for its internal parts. Tone and Tsutsui [3] showed that these assumptions need more discussion. For example, if we consider labor, raw materials and capital as inputs, the same decrease

in all of them may not be possible. Therefore, they suggested a network DEA model that used a slack-based measurement method to evaluate the efficiency of different parts.

Xu et al. [4] evaluated the supply chain performance of furniture manufacturing industries in southwestern China by presenting a rough DEA model. They identified the main uncertainty factors that affect the evaluation process, and then, these cases formed a model and were analyzed using rough DEA (RDEA) models.

As we know, a proper measurement of the supply chain efficiency must take into account both the network properties of the chain and the relationship among supplier, manufacturer, distributor, and customer. Taking into account these considerations, Saranja and Moser [5], and Chen and Yan [6] presented various models with different supply chain structures.

Azadeh and Alem [7] presented three models of supplier selection in the supply chain in certainty, uncertainty and probabilistic circumstances. These models include data envelopment analysis (DEA), fuzzy data envelopment analysis (FDEA), and chance constrained data envelopment analysis (CCDEA). Babazadeh et al. [8] designed a network supply chain for organizing a biodiesel fuel type in Iran using a combined model of data envelopment analysis and mathematical programming model. Badiezadeh et al. [9] discussed the supply chain in the presence of big data and undesirable outputs. Boudaghi and Saen [10] provided a combined model of data envelopment analysis and discriminant analysis (DEA-DA) to predict group membership of suppliers in the sustainable supply chain. The proposed model could predict this, considering the nature of factors, such as inputs, outputs and performance of each supplier. Fathi and Saen [11] presented a model of bidirectional network DEA in assessing the sustainability of distribution supply chains, to rank Iran's transportation companies and propose improvement solutions. Goodarzi and Saen [12] also considered unde-

sirable outputs by developing the SBM model in network DEA. Grigoroudis et al. [13] presented a recursive DEA algorithm (RDEA) that introduces a different method to design a supply chain network. Huang [14] examined the tourism industry in the framework of supply chain and calculated the efficiency of each sector as well as overall efficiency. One of the characteristics that distinguishes this research from other research is the division of model inputs into variable and semi-variables and using it in mathematical programming. Izadikhah and Saen [15] evaluated the stability of supply chains with a two-stage model in the presence of negative data. Izadikhah et al. [16] developed conventional DEA models to a method to assess the sustainability of suppliers in the presence of internal and fuzzy data. Kalantari and Saen [17] examined the sustainability of supply chains with a model of reverse dynamic network DEA. Khodakarami et al. [18] analyzed the evaluation of 27 Iranian companies in the context of supply chain management sustainability based on the development of a two-stage model. Mirhedayat-ian et al. [19] assessed green supply chain management in the presence of undesirable outputs and fuzzy data. Sarah and Khalili-Damghani [20] evaluated the natural gas supply chain with the fuzzy type-II De-Novo program for allocating resources and setting the target in network DEA. Shafiee et al. [21] used a combination of network DEA and balanced scorecard approach to evaluate supply chain performance. In this method, the combination of Balanced Score-card (BSC) and DEMATEL method was used for network structure. Then the structure of this network was expressed in the form of DEA, and was evaluated in this structure. Tajbakhsh and Hassini [22] provided a method for evaluating the sustainability of supply chain networks. Their focus was on three issues: to maximize economic return, to minimize environmental impact and to achieve social expectations. They applied the proposed model to evaluate two case studies, one in the manufacturing sector and the other in the banking sector.

Tavana et al. [23] presented a two-stage methodology for assessing the performance of the three-part supply chain, including supplier, manufacturer and distributor.

Tavana et al. [24] evaluated the supply chain performance by providing a network epsilon — based measure (NEBM) and emphasized the radial and non-radial (hybrid) simultaneous changes in inputs and outputs in the network. Tavassoli and Saen [25] evaluated the prediction of membership in a sustainable supplier group using the DEA and Discriminant Analysis. Yousefi et al. [26] presented improvement ways and optimal solutions, using a combination model of goal programming and data envelopment analysis with network structure. Zhai et al. [27] measured the efficiency of energy supply chains with business plan, by presenting a two-stage frontier-shift DEA model.

In this research, the application of data envelopment analysis is examined for evaluating supply chain management in the presence of non-discretionary, undesirable and negative data. A model is presented with the inclusion of the above particular circumstances, according to which the performance of listed cement companies in the stock exchange is calculated. Therefore, the innovations of this research are as follows:

♦ development of network models in the data envelopment analysis with undesirable intermediate and final outputs, and negative outputs;

♦ development of network models in data envelopment analysis with non-discretionary and partly-discretionary factors;

♦ application of the models so developed in the cement industry, taking into account the specific conditions of the indicators.

The rest of this paper includes the following sections: reviewing basic concepts, presenting a suitable model in the presence of non-discretionary, undesirable and negative indicators, presenting a case study in the field of the cement industry with the structure of the proposed model. Finally, the results are investigated.

1. Basic concepts

Suppose that there are n decision-making units so that Yj = (ylj}..., ysj)', X} = (Xy,..., xmj)' are DMU. input and output vectors respectively, and X. > 0, X. * 0, Y. > 0, Y. * 0.

By accepting the principles of Inclusion of observations, fixed-scale returns, convexity, feasibility, and the minimum of interpolation, the Production Possibility Set is as follows:

71 =

r v\

J J

7=1

(1)

Y^AjYy, Aj>0;j = 1,2, .

By eliminating the assumption of constant returns-to-scale from the above assumptions, the Production Possibility Set turns into below for which its returns-to-scale is variable:

Jj

7=1 7=1

(2)

0; y = 1, 2, ..., n .

The following input-oriented model, should be solved in order to evaluate DMU:

o

n

XW / = 1> 2> •••»

7=1

7=1

(3)

>0,y=l, 2, ..., n.

Model (3) that is called an input-oriented BCC model, is introduced by Banker, Charnes and Cooper [28].

It's clear that DMU is Pareto efficient if and

o

only if Q = 1 and the value of all auxiliary variables is zero, in each optimal solution of the model (3).

Definition 1: Let (X, Y) be input and output vector corresponding any DMU. (X', Y) is the transformed form of this unit with respect to constant values of ai and i = 1, 2, ..., m and /?r, r = 1, 2, ..., s, if:

4 = Xf+a,, i = 1, 2, ..., m, j = 1, 2, ..., n; y'n=yri+Pr> r = l> 2> •••> s> j = h •••>"■

(4)

Definition 2: A DEA model is called stable to transmission if the efficiency of the DMUs (the value of model objective function) when transforming inputs and outputs of them is not changed.

Theorem 1: The input-oriented BCC model is stable with respect to the output transmission [29].

The above theorem indicates that if the outputs are transmitted in the input-oriented BCC model, its objective function or efficiency score will not change.

A centralized control model was developed by Chen and Yan [6] to evaluate the performance of the two-stage supply chain. For simplicity, they assumed a supplier- manufacturer chain, as in Figure 1.

Where, S is supplier, M1 and M2 represent first and second manufacturers, respectively. X = (xp ..., xm) is input vector of supplier S and Yl=(yl,..., y\), Y2={yl...,y2k.) are output vectors of it which are also input vectors of manufacturers M1 and M2. Z1 =(z}, ■■■, z]) and Z2 = (zf, ..., ZsO are output vectors corresponding to M1 and M2 respectively.

SSuppose that DMU1, DMU2, ..., DMU, are decision-making units corresponding to the 1-st chain, 2-nd chain, ..., n-th chain, respectively. The model presented by Chen and Yan for measuring overall efficiency of corresponding chain of DMUo with constant returns to scale assumption is as follows:

X

MANUFACTURER Z1

Mi

Fig. 1. Two-stage supply chain

miné?;

n

XaJ ^ - 0x*» i=l> 2> •••' m>

j=1

t' = l,2,...,k'; (5)

M J=1

> Z^, r = l, 2,

j=1

¿^Z^. >Z^0, r' = l, 2, 5'; j -1, 2,

The main discussion of model (5) is how to evaluate the supply chain with regard to intermediate products in DEA models. Using the above model, the supply chain being evaluated is efficient if and only if 0*= 1.

2. Methodology

Examining supply chain performance in industries is considered a sample of multi-stage and network decision-making units that should include intermediate products and the relationship between activities within the various parts of its system. In these systems, the outputs of a stage (process) are considered as inputs of the next stage, which is called intermediate data [30, 31]. Moreover, in each stage there may be independent inputs for that stage which should be considered. Managers must try to identify the factors affecting supply chain performance by developing appropriate methods, use them to measure the overall efficiency and performance of various sectors. Some of these effective factors are non-discretionary inputs and outputs, undesirable outputs and negative outputs that need to be identified and considered in developing the model for

evaluation. In this research, the supply chain of the cement industry is evalu-ated in the presence of non-discretionary, undesirable and negative factors. It will have the following structure:

In the above chain:

♦ S1, S2, S3 and S4 represent supplier, manufacturer, distributer and costumer, respectively;

♦ x/-

C f ■ *9> 1

Df * NDf

for f = 1, 2, 3, 4, is

input vector of DMU including discretionary and non-discretionary inputs to stage Sf;

¿1,^+1 _

1

\Zy

l G jVZ)*"*>+1

for k1 = 1, 2, 3

are intermediate data from stage S^ to stage S^j of j-th unit which includes discretionary and non-discretionary data, or in other words represents the output vector of stage S^ of j-th unit, including discretionary and non-discretionary outputs which also are the input of stage for that unit;

♦ Y*1 = (yJ1-, rsR^) for k2 = 1, 2, 3 are the undesirable output of stage S^;

♦ Y= (y 1, y2, ..., y) is the output vector of stage S4;

♦ D f and ND f for f = 1, 2, 3, 4 are set of discretionary and non-discretionary input indices, respectively;

♦ D^^1 and NDh'h+l for k1 = 1, 2, 3 are a set of discretionary and non-discretionary indicators indices, respectively;

♦ i?*2 for k2 = 1, 2, 3 show a set of undesirable output indices for level S^.

Consider n identical supply chains similar to chains in Figure 2 that are called n decision-

X1

I

Z

12

t

X2

y1

Z

.23

t

X3

Z

34

t

X4

S

S4

Y

ly2 yy3

Fig. 2. Four-stage supply chain

2

3

making units in the data envelopment analysis literature which are denoted as DMU1, DMU, ..., DMU. A network DEA model with

V ' n

series structure is required to calculate their performance. The BCC model can't specifically identify the function of the corresponding black box of the supply chain in Figure 2, because it only takes into account the inputs and outputs of the supply chain and ignores the intermediate products resulting from supply chain processes and factors affecting its performance. Therefore, a suitable model is required to be developed for evaluating the above supply chain.

By accepting the principles of Inclusion of observations, convexity, feasibility (except for non-discretionary intermediate products) and the minimum of interpolation, PPS for stages and total PPS for the supply chain of Figure 2 are as follows:

r=

rXw ^

glND

znD

g\2ND

Y1

7=i

' j=i

n

1 _12A® ryXWD,

¿jZj =z ,

7=1

7=1

T2=)

iX2D \

Z12D gllND

z23fl

gThND

J" ,

7=1

7=1

7=1

n

2 —23D ^ -7-232). 7 j '

7=1

12ND ryVWD.

7=1

Z 1 2 23 ND rj21ND.

AjZ] =z ,

7=1

7=1

=i;

7=1

T?=

rxw\

jfSND Z2W gllND

z140

gMND

P

\ J

7=1 7=1

n n

7 7 -z 'zA*/ -z ;

7=1 7=1

23/Vfl ryTiND

Z23ND V^ 1 3 34/Vfl ryiAND.

;2JjZj =z

7=1 7=1

7=1

7=1

rj* 4_

j^WD

Z34D

7=1 7=1

El 4 342) ^ <7-34A V 1 4 34® -7:

' 7=1 7=1

34JVD.

v v 7=1

7=1

(6)

T =

-* V

ijfkD \ gkND gk(k+\)D

Yk X4D

j^AND

7=1 7=1

n

y j^k ^k(k+l)D > ^k(k+\)D . 7=1

. ~y^ ^kzk(k+l)ND _ gk(k+l)ND . 7=1 7

7=1

7=1 n

k+\ k(k+\)ND _ ryk(k+\)ND.

Aj ZJ -Z '

/ /=1

7=1

7=1

7 1=

7=1 7=1 7=1

A*>0; A4> 0; it = 1, 2, 3

To evaluate the performance of DMU, the following input-oriented radial network model is suggested that is written according to total PPS:

7=1

7=1

ZW * Z4 42>

7=1

7=1

7=1

7=1

< yl, re R1;

7=1

7=1

< x2,, /eJV»2;

7=1

7=1

7=1

Z«3 /-^o23;

7=1

7=1

7=1

< Oxl, izD3;

7=1

^ xl ieNtf ;

7=1

7=1

7 1=

7=1

5>,3Z,M = Z/o34, /eiW34;

7=1

Z* = Zl\ leND34 ;

7=1

2>3J,3 ^ y3m,rzR3;

7=1

7=1

< X4, /eiVD4;

7=1

Z^tf ^ ym> r = \,2, ..., s;

7=1

(7)

14= i;

7=1 7=1 7=1 7=1

X), X) >0, j = l, ..., n, 6: free.

where:

1. For f = 1, 2, 3, 4, constraints

fx'xf < 0x{o, i<=Df

7=1

are written corresponding to discretionary inputs and constraints

±AfXs < x{, i g NDf

7=1

are written corresponding to non-discretionary inputs of stage Sf.

2. For k1 = 1, 2, 3, constraints

7=1 7=1

are written corresponding to Discretionary outputs stage S^ which shows that convex composition of these outputs as inputs of stage S^ must be equal to or less than pro-duct of stage S^.

3. For k1 = 1, 2, 3, constraints

Y^XfZ^^Yxf'Z^1 = ZtK+\ lgNDk'

7=1 7=1

ki+1

are written corresponding to non-discretionary outputs of stage S^ and shows that the product of stage S^ as non-discretionary output, must be equal to input of stage and non-discretionary output of DMUo in stage S^.

4. For a = 1, 2, 3 and r e Ra, constraints

ZI'y'n * yaro

7=1

are written corresponding to undesirable outputs of stage Sa and shows that for r e Ra, convex composition of undesirable output of DMUs in stage Sa must be equal to or less than r-th output of DMUo in this stage S^.

5. For r = 1, 2, ..., s, constraints

tÀ>a * ym

7=1

are written corresponding to final outputs.

Theorem 2: Model (7) is always feasible.

Argument: Since the solution below is a feasible solution to the model (7), so this model will be feasible:

0 = 1;

Aj = (41,..., ..., ^ ) = (0, ...,0,1,0, ...,0); k2 = (A2,.... 4?,..., A„2) = (0, ...,0,1,0, ...,0); (8) k3 = (A?,..., k03,..., k3) = (0, ...,0,1,0, ...,0); = (A4, A04, 4,4) = (0, ...,0,1,0, ...,0);

Definition 3: The corresponding supply chain DMUo is efficient in evaluating with model (7), if d* = 1.

In the following, measuring efficiency of the stages and the relationship between the efficiency of each stage with the overall efficiency are discussed. In order to evaluate the efficiency of each stage in the supply chain, we consider the dual form of model (7) as follows:

maxZo = - ZvX+2>2z/l2 + 2>/3zI2 -

leiW)1 Id®12 /eM)12

-Ewi. - Z^ + ÏMnZl3 + -

re J?1 isND2 IsND23 /e №)23

leJVD3 /eJVD34 /e iVD34 _ s

re«'

teS'

I'eM)

r=1

+ M10+M20+M30+M40;

l'e Ö1 ¡e /Vfl1 /efl12 iejra12

o. y = i. 2. »;

reü

- ZM3Z23- Zv,x3 - +

/E fl23 /E M)23 IE B3 iE M)3

+ Za2z34- Z"^3 +«3o ^ 0,

Es Du le NDM re Ä3

y = l, 2, ..., n;

- Z <PnZy + z <p*zv - Z^ - z w2 +

/el)12 feM)12 le/)2 i'eJVZ)2

+ Z Ah^T + - Z"r2^- +"2o < 0,

/effl)3 fsR2

j = 1, 2, ..., « ;

/eJ1

+ -

/eß34 /eM)34 jeD4 ieATD4

5

+M4„ ^ 0, y = 1, 2, ..., n ;

r=l

Zv/ixi+Zv/2^ + Zv-3^+Zv<4< =1 ;

leD1 isD2 "4

leD3 ieB

vif>0, ieDf, f = 1, 2, 3, 4;

le^D7, / = 1, 2, 3, 4; çn>0,UD12; (9)

//„>0, /eZ)23 ; pn>0, /eD34; wM>0, A = l, 2, 3;

wr >0, r = l, 2, ..., 5; <pl2,(pn \free, I <=NDn ; Hn,Hn : ./ree, / e ND23 ; pl2,pn : free, I e 7VD34 ; pl2,pn:free, UND34 ; w,0 : ./ree, i = 1, 2, 3, 4 .

Suppose that Z0* is the value of objective function in model (9) for each optimal solution. In this case, overall efficiency of supply chain corresponding to DMU is equal to:

(10)

iE A®1 /eJVD12 /eM)12

-Z« + Z^z/23 +

reÄ1 isND1 UND13

+ Z /^z,23 -Z«^™ -z«+

/eM)23 reÄ2 ieJVD3

/eM)34 /eM)34 /eM)34

+ Z^zr-Z«^™ - Z^*4.<,+

¡Effl* refl3 ieM>4

j

/ .Wrym

Definition 4: If Z* = 1 then DMU or the chain

0 o

corresponding to it, is efficient, otherwise it will be inefficient.

It is concluded from the series of first, second, third and fourth constraints in model (9) and the last constraint that for the optimal solution and for j = 0:

ZVX - Z VX + Z <PnZïo +

ieDl ie ND1 leD12

+

iE®"

-z«+Z^-Z«

+

leDu IeND12 ieD2 inND£

+Z^3 +<4^0;

(11)

(12)

leD

IeND

reR'

- ZmX+Z^z"- Z^ - z« +

isD23 irf iefl3 ieM}1

+ Zhfî + Z^^4- + ^ 0; ()

/eZ)34 îeND34 rei3

ZKzr + Z^r-Z«,-

D34 ZeJWD34 ZeD4 (14)

;_^ s

- Z^*4Xio + ZW>™ + U4o ^ 0;

r=l

ZvX+Zv^ +Zvm4=(15)

ieZJ1 leD

|2 n3 ni

From inequalities (11), (12), (13) and (14), it resulted that:

/efl1'

leND"

Hv*A+Hvnxl + Z w,

ie £>' ieND1

T * (16)

refl1

^ — < 1; (17)

leND le Di:

T<Pnzb+Zv*2xt+Zv^+ZM,>i

leDt2 ie D2 ieND2 reR2

IeND

leD

leND"

leD

ieD3 ieMD3 re*2

■s

< 1; (18)

4o

(si®*

Z#1Z/»4 + ZV*4^ + ZV.4X-

< l; (19)

leD"

ieD ieND*

The deduction of the first side of the inequality (16), (17), (18) and (19), will respectively determine the efficiency of stages S1, S2, S3 and S4 of chain corresponding to DMUo which are represented by symbols Zl0, Z2o, Z3o and Z4o.

Definition 5: If to, then the chain corresponding to DMUo , is efficient in stage St .

Theorem 3: DMUo or the chain corresponding to it, is efficient if and only if Z'„ = 1, ZL = 1, ZL = 1 and Zl = 1.

Argument:

Suppose that Z*0-1 .

Assume for proof by contradiction that Z*0 < 1, or 2 < 1,

or Z30 < 1, or Zto < 1.

Without loss of the generality of the argument, assume that Z*0 < 1, or 2 < 1, or 3 < 1, or z; < 1.

Hence, with respect to equations (16), (17), (18) and (19) we have:

- Zv/X - Z + Z^n Z\o +

ieD1 ieND1 ¡Efl1' (20)

+ Z <Pnzl2 - +«i*0 <

leND"

reR'

- z<Pnz" + - z^ -

leDn leND'2 ieD2

- Z « + Z Mn Z" + Z - (21)

ieND2 leDn leND3

- + "L ^ 0;

reR2

leD23 leND23 lei)3

-Z«+Z^X4 + Z^4 - (22)

ieND3 leD34 leNDM

- Z"*3 3d + «L ^ o

TEfl3

-z^^r+z^^-z«-

/eZT

ieD"

- Z + + M4*0 ^ 0 •

(23)

ieND*

r= 1

Now by summing the sides of the four inequalities and the equation

Z* 4 + IX*"+Zv*3xl + Zv*4<=1

is D' is D7 is D3 is D4

we have:

ieZ)1 /e A®12 /e JVD12 re Ü1

- Z« + + S^Z» - (24)

is ND2 Is ND23 Is ND23

-Zu*riyl - Z^X + Z#2z/»4-

is ND3 te MT

So we have Z0* < 1 which is against the original assumption, therefore Z*a= Z2o = Z3o = Z4o = 1. Now inversely suppose that Z*u = Z2/ = Z3* =

= Z -1

4/ _ 1 •

From equations (16), (17), (18) and (19) we have:

- ZVX - Z vX + z^ +

isND IsD

+ Z - Ztó +ul = o;

(25)

IsNDu

re.fi1

-z^+z^-z«-

IsD11 IsND12 isD2

-Z« + z^3+ (26)

isND2 Is D23 IsND3

-Turym+u2o= 0;

rsR2

IsD23 IsND23 isD3

-Z^+ZK^+Z^- (27)

isND3 Is D14 IsND34

"Z«+«3> 0;

re«3

isD

(28)

IsND3

o;

By summing sides of the four above equations and

Z^i*!» + Zv*2^+ Zv*3xl + Zv*4< =1,

ieO'

ie Dl is D'

is D

we conclude that Z*=l, so DMU or the cor-

0 ' 0

responding chain, is efficient.

(29)

In order to provide an expert opinion for some of the indicators preference over each other and the influence of this approach on the efficiency of the supply chain, we attempted to determine the weights for the indicators using the fuzzy decision-making technique and expert opinion survey. We add the considered weight constraints (equations 29) to the model (9) to help more accurately estimate the supply chain efficiency:

«u > (0.165 / 0.132) • <piX ;

p31 > (0.132 / 0.116) • <p5l ;

q>5l > (0.116 / 0.103) • v12 ;

p2l > (0.147 / 0.146) ■ pn ;

pn >(0.146 / 0.140)./>41;

p4l > (0.140 / 0.122) • v24 ;

vn > (0.185 / 0.132) ■ v51 .

3. Case study

Green supply chain management is one of the critical issues in organizations that managers need to design appropriate models for performance evaluation. In this regard, in some cases, we may encounter undesirable outputs such as environmental pollutants. In addition to special indicators, undesirable and environmental factors must be considered for correct evaluation of Cement companies in the framework of the supply chain.

The evaluation of cement factories in the country in the form of complete supply chains and attention to these undesirable and negative factors, as well as Non-Discretionary factors that are partly or fully out of management control, was one incentive to perform this research. For this purpose, according to the opinion of experts and based on theoretical studies conducted in previous research and methods for selecting the indicators, suitable indicators for the Iranian cement industry were determined in accordance with the following tables (Tables 1—4). Also, among these performance indicators, special indicators such as non-discretionary inputs and outputs, undesirable outputs and negative outputs have been specified.

To illustrate the application of the proposed method, we consider the actual data of the 42 cement companies listed on the Tehran stock exchange during one year with the inputs, outputs, and intermediate data in accordance with the above tables. We ran the model (9) by defining weight constraints for some of the indicators based on the expert opinion and by considering theorem 1 for the negative outputs of the fourth stage. By applying this model, the efficiency of each company was determined as well as the efficiency of each stage of their corresponding chains based on the explanations given in the previous sections. The results are shown in Table 5.

We calculate the efficiency correlation coefficient of each company with its distance to the center of the soil mine (silica) and its distance to the center of the province. Tables 7 and 8 show this correlation coefficient.

Also, the variance of the total efficiency and the efficiency of each stage are calculated for companies and are presented in Table 9.

The efficiency correlation coefficient of the supplier stage with the distance from the company to the soil mine is equal to 0.261, which is greater than the correlation coefficient of the efficiency of other stages and the total efficiency with the distance to the mine. This shows that this stage has the highest relationship with the distance between companies and the mine compared to other stages. The low correlation coefficient of customer stage efficiency (—0.011) with the distance to the mine shows that there is no significant relationship between this stage and the distance of companies to the mine.

The high correlation coefficient of the efficiency of the distributor stage with the distance of companies to the center of the province (0.385) indicates that for the distribution of the product, the distance from the factory to the center of the province is very important and there is a significant relationship between the distributor stage and this distance. There is no significant relationship between the supplier stage and the dis-

tance between the factories and the center of the province because the correlation coefficient of this stage is the lowest value, 0.063.

The variance of the manufacturer stage is 0.0005. The difference between the maximum and minimum values of the efficiencies at this stage is a small number of 0.07 (1 — 0.93 = 0.07). Therefore, all companies in cement production have similar performance and there is no significant difference between their efficiency at this stage.

The efficiency variance of cement companies in the customer stage (0.0733) is greater than the efficiency variance of other stages and the total efficiency variance. This indicates that the efficiency dispersion of the units is high at this stage. The value of variance for the supplier and distributor stages shows that the performance of the chains in these two stages is close to each other and the value of variance of total efficiency (0.0153) indicates that the dispersion of total efficiency of 42 supply chains is relatively high.

Conclusion

In this study, the status of the green supply chain including four stages of supplier, manufacturer, distributor and customer is investigated. In addition, the authors have attempted to evaluate the efficiency of this chain with inputs and outputs that have a specific structure, including non-discretionary inputs and outputs, undesirable outputs, and negative outputs which are investigated. The results show that among 42 green supply chains with this structure, there are six efficient and 36 inefficient chains. In addition, chains that have been fully efficient and have achieved an efficiency score of 1 are efficient in all four stages. Inefficient chains are inefficient at least in one stage, which also confirms the stated proposition. The most inefficient supply chain relates to the Sepahan factory (Unit 11) whose efficiency score is d* = 0.5352. As can be seen, this unit is only efficient in the supplier stage and it is inefficient in the other three stages, as it has very poor performance in the fourth stage. The efficiency score of all companies in the manufacturer stage

Table 1.

Introducing indicators and their definition for j-th decision-making unit in inputs and intermediate data

Symbols Discretionary or non-discretionary indicators Indicator classification Indicator title

K Total current or operational costs

K Total initial investment in mine exploitation and plant process

K D Original input Total debt of the factory

K of chain Total financial expenses

Total amount of salary

n ND Quality of suppliers in terms of sustainability in the supply of minerals and consumables

K2 Total mineral resources available

712 A/ Intermediate data (output of stage 1 and input of stage 2) Total tonnage of raw materials harvested from mines, which should be consumed in the production process

zf z12 A; D The tonnage of other chemical and mineral substances consumed in the process of production

z12 Ay Total mineral raw materials stored for use in cold season

Z12 A/ Total R&D expenses

z12 A; ND Real industry capacity

z23 723 A j D Intermediate data Total clinker production capacity

Ay (output of stage 2 and input of stage 3) Total cement production capacity

734 A j Total value of assets and inventories ready for sale

y 14 ^lj 734 Aj ND Intermediate data (output of stage 3 and input of stage 4) Total tonnage of packaged and bulk cement sales in the domestic market and exports

734 A; Total clinker sales tonnage

734 Aj The cost of the product

is between 1 and 0.93. Therefore, at this stage, all companies have similar performance and there is no significant difference between their efficiencies.

The efficiency correlation coefficient of the supplier stage with the distance to the center of the mine shows that the highest relationship

is between the supplier and the distance to the mine, which means that the performance of suppliers will be higher due to closer proximity to the mine. Since most suppliers were at a short distance from the mine, the efficiency of all cement companies was high at the supplier stage. The efficiency correlation coefficient of

Table 2.

Introducing indicators and their definition for j-th decision-making unit in final output

Symbols Status of indicator Indicator classification Indicator title

yj Total current assets

y2j Non-negative Brand competitiveness and globalization

Yj y3j Final output Customer satisfaction

Total profit

y5j Negative or non-negative Annual growth rate based on performance

Equity return

Table 3.

Introducing indicators and their definition for j-th decision-making unit in independent inputs of each stage

Symbols Discretionary or non-discretionary indicators Indicator classification Indicator title

K ND Independent inputs of Stage 2 Total cost of energy payment

K D Independent inputs of Stage 3 Total cost for increasing reliability in the supply chain

K D Independent inputs of Stage 4 Cost of environmentally friendly design

K Total marketing fee

Table 4.

Introducing indicators and their definition for j-th decision-making unit in undesirable outputs

Symbols Status of indicators Indicator classification Indicator title

V, y'j U.D Undesirable outputs of Stage1 Causing destructive environmental effects in harvesting of mines

yl Total produced dust particles (mg / m3)

yl Average annual emissions of NOx

yl Undesirable outputs of Stage2 Average annual emissions of CO2

n yl U.D Average annual emissions of SO2

yl Effect of total water and sewage intake in groundwater

yl Impact of factory performance on the creation of negative conditions in the ecosystem

Table 5.

Efficiency of all companies and efficiency of each chain stages corresponding to them

Number Company Total efficiency Supplier stage efficiency Manufacturer stage efficiency Distributor stage efficiency Customer stage efficiency

1 Abadeh 0.8679 1.00 0.98 0.95 0.55

2 Abbey 0.8955 1.00 1.00 0.85 1.00

3 Darab 0.7090 0.80 0.96 0.89 0.57

4 Ardebil 0.8308 1.00 0.93 0.87 0.68

5 Oroumiyeh 0.6781 1.00 0.95 0.83 0.46

6 Saveh 0.7078 1.00 1.00 0.95 0.29

7 Bagheran 0.6810 0.77 0.97 0.80 0.50

8 Bojnourd 0.8716 0.91 0.97 0.87 1.00

9 Sabzevar Lar 0.9452 1.00 1.00 0.97 1.00

10 Behbahan 0.9494 1.00 0.93 1.00 1.00

11 Sepahan 0.5352 1.00 0.93 0.87 0.38

12 Tehran 1 1.00 1.00 1.00 1.00

13 West Asia Complex 0.6978 1.00 1.00 0.76 0.33

14 Khash 0.8124 0.95 0.95 0.91 0.75

15 Khorramabad 0.6981 0.89 0.97 0.88 0.27

16 Khazar 0.8304 0.94 1.00 0.84 0.60

17 Majd Khaf 1 1.00 1.00 1.00 1.00

18 Khoozestan 1 1.00 1.00 1.00 1.00

19 Dashtestan Ce 0.9697 1.00 1.00 0.99 1.00

20 Doroud 0.7840 1.00 1.00 0.80 0.36

21 Shahroud 0.7381 0.85 1.00 0.84 0.54

22 Shargh 0.8456 1.00 1.00 0.89 0.67

23 Shomal 0.8522 0.92 0.96 0.88 1.00

24 Isfahan 0.6949 0.84 0.95 0.89 0.53

25 Soufian 0.7976 0.92 0.99 0.82 0.76

26 Gharb 0.7947 0.91 0.98 0.85 0.69

27 Fars Cement 0.8577 1.00 0.98 0.88 0.64

28 Faraz Firozkouh 0.8291 0.93 0.97 0.88 0.70

29 Urmia White 0.9405 1.00 0.96 0.97 0.86

30 Siman Fars Noe 0.8854 0.92 0.98 0.94 1.00

31 Firozkouh 0.8395 1.00 0.97 0.87 0.79

32 Qayen 1 1.00 1.00 1.00 1.00

33 Karoon 0.9064 1.00 1.00 0.88 0.72

Number Company Total efficiency Supplier stage efficiency Manufacturer stage efficiency Distributor stage efficiency Customer stage efficiency

34 Kordestan Ce. Co 0.5810 0.81 0.98 0.84 0.15

35 Kerman 0.9603 1.00 0.98 0.95 1.00

36 Larestan 1 1.00 1.00 1.00 1.00

37 Mazandaran 0.9383 1.00 1.00 0.85 1.00

38 Momtazan kerman 0.9011 0.89 0.94 0.94 1.00

39 Neiriz 1 1.00 1.00 1.00 1.00

40 Hormozgan 0.9314 1.00 1.00 0.89 1.00

41 Hegmatan 0.7213 0.93 0.98 0.83 0.45

42 Ilam 0.6458 0.85 0.97 0.80 0.27

Table 6.

Distance of factory to center of province and soil mine (silica)

Number Company Distance to soil mine (silica) (Km) Distance to center of province (Km) Number Company Distance to soil mine (silica) (Km) Distance to center of province (Km)

1 Abadeh Cement 10 228 22 Shargh Cement 6 259

2 Abik Cement 4 68 23 Shomal Cement 5 119

3 Darab Cement 5 350 24 Esfahan Cement 10 135

4 Ardebil Cement 14 169 25 Soufian Cement 9 150

5 Urumia Cement 10 201 26 Gharb Cement 8 126

6 Saveh Cement 9 145 27 Fars Cement 10 155

7 Bagheran Cement 8 494 28 Faraz Firouzkouh Cement 5 120

8 Bojnourd Cement 6 208 29 Urumia Cement 6 178

9 Lar Sabzavar Cement 8 355 30 Farsno Cement 4 202

10 Behbahan Cement 9 239 31 Firouzkouh Cement 5 120

11 Sepahan Cement 10 132 32 Qaen Cement 9 356

12 Tehran Cement 15 81 33 Karoun Cement 7 296

13 West Asia Cement Complex 10 316 34 Kordestan Cement 6 152

14 Khash Cement 8 515 35 Kerman Cement 8 281

15 Khoramabad Cement 6 203 36 Larestan Cement 9 406

16 Khazar Cement 10 121 37 Mazandaran Cement 5 153

17 Majd Khaf Cement 7 264 38 Momtazan Kerman Cement 8 283

18 Khouzestan Cement 15 272 39 Sefid Neyriz Cement 6 341

19 Dashdestan Cement 16 178 40 Hormozgan Cement 9 416

20 Doroud Cement 8 171 41 Hegmatan Cement 10 178

21 Shahroud Cement 7 254 42 Ilam Cement 9 132

Table 7.

Efficiency correlation coefficient with its distance to the center of the soil mine (silica)

Efficiency Total Supplier stage Manufacturer stage Distributor stage Customer stage

Correlation coefficient 0.099 0.261 0.023 0.212 -0.011

Table 8. Efficiency correlation coefficient with its distance to the center of the province

Efficiency Total Supplier stage Manufacturer stage Distributor stage Customer stage

Correlation coefficient 0.197 0.063 0.236 0.385 0.096

Efficiency variance Table 9.

Efficiency Total Supplier stage Manufacturer stage Distributor stage Customer stage

Variance 0.0153 0.0045 0.0005 0.0046 0.0733

the distributor stage with the distance of the company to the center of the province where it is located indicates that the highest relationship is between the distributor and the distance to the center of the province. Therefore, the distance of companies to the geographical location of cement sales will have a significant impact on the performance of the distributor stage.

The authors focus in this study was to provide a radial model to evaluate the performance of green supply chain with inputs and outputs having a specific structure. Therefore, the authors suggest research to investigate the following: (1) provide non-radial models with this structure and (2) provide a model to determine the cause of supply chain inefficiency and provide solutions to improve their efficiency. ■

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About the authors

Mehdi Shoja

Ph.D. Student, Industrial Engineering Department, Roodehen Branch, Islamic Azad University, No 4, Shahid Ahmadi Alley, Valiasr Town, Firoozkooh, Tehran 3981838183, Iran; E-mail: [email protected] ORCID: 0000-0001-9221-8071

Farhad Hosseinzadeh Lotfi

Professor, Mathematics Department, Science and Research Branch, Islamic Azad University, Shohada Hesarak Blv, University Square, End of Shahid Sattari Highway, Tehran 1477893855, Iran;

E-mail: [email protected]

ORCID: 0000-0001-5022-553X

Amir Gholam Abri

Associate Professor, Mathematics Department, Firoozkooh Branch, Islamic Azad University, First of Valiasr Town, 45 Meters Street, Firoozkooh, Tehran 3981838381, Iran;

E-mail: [email protected]

ORCID: 0000-0003-1981-9756

Alireza Rashidi Komijan

Associate Professor, Industrial Engineering Department, Firoozkooh Branch, Islamic Azad University, First of Valiasr Town, 45 Meters Street, Firoozkooh, Tehran 3981838381, Iran; E-mail: [email protected] ORCID: 0000-0001-7705-980X