An Integrated Disaster Relief Supply Chain Network Model with Time Targets and Demand Uncertainty - Regional Science Matters - page 298

h

d

S 1 , 2

1 A 1 B 1 R 1

g

f

e

Fig. 15.4 Supply chain network topology for the illustrative numerical example—Variant II

5 z p 3 þ 8 z p 4 :

ʳ R 1 ðÞ¼ 3 :

Under the assumption of positive optimal solutions, the following system of
equations had to be satisfied:

λ

∂ C p 3 x ðÞ
∂

R 1 P R 1 x p 3 þ x p 4 R 1

1 P R 1 x p 3 þ x p 4

þ λ

x p 3

g h þ g f þ g g

p 3 p 4

þ ˉ g h þ g d þ g f þ g g þ ˉ ¼ 0,

λ þ ˉ

∂ C p 4 x ðÞ
∂

R 1 P R 1 x p 3 þ x p 4 R 1

1 P R 1 x p 3 þ x p 4

p 3

þ λ g h þ g f þ g g

x p 4

g h þ g e þ g f þ g g ¼ 0,

p 4

þ ˉ

∂ʳ R 1 z ðÞ
∂ʳ p 3

p 3 ¼ 0,

ˉ

∂ʳ R 1 z ðÞ
∂ʳ p 4

ˉ p 4 ¼ 0,

T R 1 p 3 þ z p 3 x p 3 þ x p 4 g h þ g d þ g f þ g g g h þ g f þ g g ¼ 0,

T R 1 p 4 þ z p 4 x p 3 þ x p 4

g h þ g f þ g g g h þ g e þ g f þ g g ¼ 0,

where

¼ 57,

T R 1 p 3 ¼ T R 1 h p 3 ¼ T R 1 h h þ h d þ h f þ h g

¼ 61

T R 1 p 4 ¼ T R 1 h p 4 ¼ T R 1 h h þ h e þ h f þ h g :

After substitution of the partial derivatives and the other numerical values into
the above system, we have the following six equations: