EXP     CRE    Mean Var Mean Var   Mean Var Mean Var Stochastic efficiency (a = 0.05)

1 248 8 9 0.3   44 10 100 2.5 0.4

2 272 7 12 0.6   95 21 70 3 0.36

3 330 10 16 0.2   82 20 50 5 0.29

4 327 11 10 0.4   74 15 70 4 0.3

5 330 8 15 0.7   49 10 140 2.4 0.3

6 253 5 10 0.4   45 6 55 2 1

7 268 7 14 0.8   100 15 65 4 0.74

8 259 4 8 0.1   35 9 45 1.5 0.48

9 180 4 6 0.4   92 12 130 2.25 0.36

10 257 6 7 0.4   38 8 50 3.75 0.57

11 95 4 70 3   100 22 47 1.5 0.53

12 130 2.25 50 5   95 20 26 3.75 0.37

I standard deviation of ith input,i

a risk that is between zero and 1Var(yro) rth output variance of the DMUo

Var(xio) ith input variance of the DMUo

To ensure sustainability of the constraints, we must considerthe probability of these constraints (1 a) to be sustainable, for which the higher this value, the more likely the satisfaction of

~xik

ith input estimation of the DMUk

these constraints

~xio ith input estimation of the DMUo

y~rk rth output estimation of the DMUk

min ho

( K )

y~ro rth output estimation of the DMUo

w~ k dual-role factor estimation of the DMUk

s:t: P ~xio ho X ~xik kkk¼1

P 1 a i ¼ 1; .. . ; I;

w~ o dual-role factor estimation of the DMUo

k [kk] vector of DMU loadings, determining best practice for theDMUo

ho radial input shrinkage factor

( K )P X y~rk kk y~rok¼1

P 1 a r ¼ 1; ... ; R;

ð4Þ

lr weight given to output r

( K )

ti weight given to input iZ~ standard normal random variable

P X w~ dk kk w~ ok¼1

¼ 1 a d ¼ 1 .. . ; D;

P X y~rk kk y~ro P sþk¼1

P 1 a r ¼ 1; .. . ; R;

ð5Þ

input i. DMUo is the DMU under consideration. DMUo consumes xio (i = 1, ... , I), the amount of input i, to produce y ro (r = 1, ... , R), the amount of output r. Assume that some factors are held by each

( KP Xk¼1

)w~ dk kk w~ o

¼ 1 a d ¼ 1 .. . ; D;

DMU in the amount wk and serve as both an input and output factor.Dual problem of Model (1) is as below:min hoK

s:t: xio ho X xik kk P 0 i ¼ 1; .. . ; I;k¼1K

X yrk kk P yro r ¼ 1; ... ; R;

kk P 0 k ¼ 1; .. . ; K ;ho unrestricted in sign:Assume that distribution function of the inputs, outputs, and dual-role factors are normal. Thus mean parameter and the devia- tion (variance) are known. By using the normal distribution stan- dardization, Model (5) is converted to a deterministic model, which is discussed as

below.For the second set of constraints of Model (5), there is2

k¼1K

X wk kk ¼ wo ;

ð2Þ

( K

a ¼ P X y~k¼1

rk kk

/y~ro

)

r ð/; kÞ ¼i.e.,3k

U 1

ðaÞ;

who intends to select the best 3PL provider, such measures may play the role of proxy for ‘‘high quality of services’’, thus can sensi- bly be classified as outputs. On the other hand, from the viewpoint of 3PL provider that intends to supply reverse logistics services,

they can be considered as inputs that assist the 3PL provider in

/yro X X yrk kk þ sþ U ðaÞrr ð/; kÞ ¼ 0; ð6Þ1 ork¼1

The deterministic equivalent of Model (5) can be represented bymin hoK

s:t: /yro X yrk kk þ sþ U ðaÞr ð/; kÞ ¼ 0; r ¼ 1; ... ; R;1 or rk¼1I

X xik kk þ s U 1 ðaÞrI ðkÞ ¼ xio ; i ¼ 1; .. . ; I;i ii¼1

obtaining more patrons. The reason why EXP and CRE are consid- ered as inputs is that they are as a means to attract more customers for 3PL providers. Notice that, in this paper, 3PL provider is consid- ered as a DMU.Table 3 depicts the 3PL provider’s attributes. In Table 3, the data set for input, output, and dual-role factors are presented for twelve3PL providers. Random numbers were generated by Microsoft Ex- cel. Also, the last column of Table 3 shows stochastic efficiency scores of the twelve 3PL providers which are measured witha = 0.05. These results have been obtained by using Model (7).The 3PL provider 6 is efficient stochastically. This 3PL provider is

D efficient, because / ⁄

X 1

= 1.

wdoþK

d¼1

wdk kk U ðaÞrd ð/; kÞ ¼ 0; d ¼ 1; ... ; D;

Fig. 1 describes a visual result of Model (7). Fig. 1 has two coor- dinates (DMUs and stochastic efficiency) in which all the observa-tions are depicted.

X kk ¼ 1; kk P 0; sþ P 0; s P 0; k ¼ 1; ... ; K ;r ik¼1

and

r ¼ 1; ... ; R; i ¼ 1; .. . ; I; d ¼ 1; .. . ; D2

ð7Þ

4. Managerial implicationsOver the past few decades, increased rivalry caused by global- ization and rapid technological advances has motivated firms toimprove their efficiency in supply chain management. Increasing