An Integrated Disaster Relief Supply Chain Network Model with Time Targets and Demand Uncertainty - Regional Science Matters - page 294

5 z p 1 þ 8 z p 2 : ʳ R 1 ðÞ¼ 3 :

Solution: For the variational inequality ( 15.24 ) to hold for all ( x , z , ˉ ) ∈ K , at the

optimal solution ( x ∗ , z ∗ , ˉ ∗ ), we must have that the term in each of the three left-

hand-side brackets is equal to zero, assuming that the optimal value of each of the
variables is positive. Thus, we have the following six equations with
x p 1 , x p 2 , z p 1 , z p 2 , ˉ p 1 , and ˉ p 2 as unknowns:

λ

∂ C p 1 xðÞ

R 1 P R 1 x p 1 þ x p 2 R 1

1 P R 1 x p 1 þ x p 2 þ λ

∂ x p 1

þ ˉ p 1 þ ˉ p 2

g d þ g f þ g g g f þ g g ¼ 0,

λ þ ˉ p 1

∂ C p 2 x ðÞ
∂

R 1 P R 1 x p 1 þ x p 2 R 1

1 P R 1 x p 1 þ x p 2 þ λ g f þ g g

x p 2

¼ 0,

þ ˉ p 2 g e þ g f þ g g

zðÞ

∂ʳ R 1

ˉ p 1 ¼ 0,

∂ z p 1

zðÞ

∂ʳ R 1

z p 2 ˉ p 2 ¼ 0, ∂

T R 1 p 1 þ z p 1 x p 1 þ x p 2

g d þ g f þ g g g f þ g g ¼ 0, and

þ x p 2 ¼ 0

T R 1 p 2 þ z p 2 x p 1 g f þ g g g e þ g f þ g g :

Note that, in the first two and the last two equations above, g a ¼g b ¼g c ¼ 0,
and several ʴ ap 's and ʴ aq 's are zero. Also, note that:

¼ 60

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

and

¼ 64

T R 1 p 2 ¼ T R 1 h p 2 ¼ T R 1 h e þ h f þ h g :

Next, using ( 15.25 ) to calculate the partial derivatives of the total path costs,