Appendixes
Appendix 1
Constraint (5) is derived in detail, which represents the capacity restriction of DCs.
Pr( Ri  D( LTi )  Qi  uia Cia )  
i  1,..., m .
(A-1)
The expression on the left side of Constraint (A-1) can be transformed into:
Pr  Ri  D  LTi   Qi  uia Cia   Pr  D( LTi )  Ri  Qi  uia Cia 
 D( LTi )  i' LTi Ri  Qi  uia Cia  i' LTi


LTi  i'
LTi  i'




 Pr 
 D( LTi )  i' LTi
 Pr 


LTi  i
'
 D( LTi )  i' LTi
 Pr 


LTi  i
'
LTi ui  Z 
'
'
LTi  i  Qi  uia Cia  i LTi 
'


i



LTi  i
'
Z
i
(A-2)
LTi  i'  Qi  uia Cia 
LTi  i
'



Hence, Constraint (A-1) can be re-arranged as follows:
 D( LT )   ' LT Z LTi  i'  Qi  uia Cia
i
i
i
Pr 
 i
'

LT

LTi  i'
i i


  1 


(A-3)
This constraint can be reformulated as a deterministic nonlinear constraint (which assures that the
probabilistic constraint is respected) as follows:
Zi LTi  i'  Qi  uia Cia
LTi  i'
 Z1
(A-4)
Therefore, Constraint ( A-4) can be simplified as follows:
Qi  Zi LTi  i'  Z LTi  i'  uia Cia
(A-5)
As mentioned above, Constraint (4) can be equivalently transformed into Constraint (5).
Appendix 2
Expression (8) is derived, which represents the expected unfulfilled demand during order cycle.
By definition, the expected unfulfilled demand of DC i can be obtained:



Ri
Ri
Ri
  LQi    ( x  Ri ) fi  x  dx   xf i  x  dx   Ri f i  x  dx ,
(A-6)

1
where fi  x  
2 LTi 
'
i
e
 x  LT u 
2 LT  
' 2
i i
' 2
i
i
(A-7)
To compute this expression, a variable change as in Eq. (A-6) is first considered, logically satisfying
Condition (A-8).
y
 x  LT u 
'
i i
(A-8)
LTi  i'
x  y LTi  i'  LTi ui' ; dx  LTi  i dy
(A-9)
Replacing (A-8) with (A-6), the equation can be written as:
  LQi  

  x  R  f  x  dx
i
Ri


( y LTi  i'  i LTi )
2 LTi  i'
i

Ri

2 LTi  i'
i
where  i
e

y2
2
e

y2
2
LTi  i dy
LTi  i' dy
(A-10)
 R  LT u 

'
i i
i
(A-11)
LTi  i'
Rearranging Expression (A-10):

  LQi    ( x  Ri ) f  x  dx 
Ri



( y LTi  i'  LTi ui' )
2
i
e

y2
2

Ri
dy  
2 LTi  i'
i
e

y2
2
LTi  i' dy
 i2
1
where ( i )     x  dx ,   i  
e 2 .
2


(A-12)
i
(A-13)
Finally, the mean of the unfulfilled demand can be computed as:
  LQi   LT  i'   i    i   i    i 



i

( y LTi  i' )
2
 LTi  i'
2
e

y2
2

dy  
i

  ye
i

y2
2
LTi ui'
2
e

y2
2

dy  
Ri
i

dy   LTi ui'  Ri  
i
1
2
2
e

e
y2
2

y2
2
dy
dy

 LTi  i'
2
e

y2
2
 i 1  y

  LTi ui'  Ri  1  
e 2 dy 


 0 2

2

i

 i
 LTi  i' 
0  e 2
2 

   LTi ui'  Ri  1    i  


2

LTi  i'

2
e

i 2
2
  LTi ui'  Ri  1   i  
 LTi  i'   i    i   i    i 
(A-14)
Appendix 3
Expression (9) is derived, which represents the variance of unfulfilled demand.
By definition, the variance of unfulfilled demand at DC i can be written as follows:
Var  LQi  


  x  R     LQ  f  x  dx    x  R 
2
i
i
i
i
Ri
Ri
2
 

f i  x  dx     x  Ri  f i  x  dx 
R

 i

2
（A-15）
To compute this expression, a variable change as in Eq. (A-15) is considered first, thereby logically
satisfying Condition (A-16).
v
 x   LT 
'
i
i
(A-16)
LTi  i'
x  v LTi  i'  i' LTi ; dx  LTi  i' dv
(A-17)
Replacing (A-17) with (A-15), and considering Equation (A-11), the following can be obtained:
Var  LQi  

 x  R 
2
i
Ri




v
 

fi  x  dx     x  Ri  f i  x  dx 
R

 i

LTi    i LTi 
'
i
2 LTi 
i
 LTi 

 
' 2
i
i

'
i
 v  i 
2
'
i
2
e

v2
2
2

e
v

2 LT  
LTi  i'
i
 v LTii' 



2
  v LTi  i'   i LTi  i'

2 LTi  i' 

LTi  i' dv   
e
dv
 i

2 LTi  i'




2

' 2
i
dv  LTi 
2

' 2
i

2

v2
v2
 



i
v
2
2
e
dv

e
dv 


'
'
  2 LT 

2 LTi  i
i
i k
 i

2
2
2
2
 LTi  i'  1   i   i i   i   i   i   LTi  i'    i    i 1    i  


2
2
2
2
2
2
2
2
 LTi  i'  1  i i    i   i   i   i    i     i    2 i   i   i 


Finally, the variance of unfulfilled demand of DC i can be simplified as follows:
2
(A-18)
2
2
2
2
2
2
Var  LQi   LTi  i'  1  i i    i   i   i   i    i     i    2 i   i   i  (A-19)

