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. 2017 May 17;17(5):1144. doi: 10.3390/s17051144
Model 1. Integer Programming
Minimize: ∑i∈V∑j∈V(yij×tij) (0)
Subject to:
∑i∈{0}∪Vsxij=∑i∈{0}∪Vs xji≤1, ∀j∈{0}∪Vs, (1)
∑i∈Vsxi0=∑i∈Vsx0i=1, (2)
∑i∈S∑j∈Sxij<|S|, ∀S⊂{0}∪Vs (3)
zijk≤xjk, ∀i∈Vt,∀j,k∈{0}∪Vs, (4)
∑j∈{0}∪Vs∑k∈{0}∪Vszijk=1, ∀i∈Vt, (5)
∑i∈Vyij=∑i∈Vyji=1, ∀j∈Vt, (6)
∑i∈Vyij=∑i∈Vyji≤1, ∀j∈{0}∪​Vs, (7)
∑i∈{0}∪Vs∑j∈Vtyij=∑i∈{0}∪Vs∑j∈Vtyji=∑i∈{0}∪Vs∑j∈{0}∪Vsxij, (8)
∑j∈Vtyij=∑j∈{0}∪Vsxij, ∀i∈{0}∪Vs, (9)
∑k∈{0}∪Vszijk≥yji,∀i∈Vt, ∀j∈{0}∪Vs, (10)
∑j∈{0}∪Vszijk≥yik,∀i∈Vt, ∀k∈{0}∪Vs, (11)
2×yij≤max{ziab+zjab}, ∀i,j∈Vt, ∀a,b∈{0}∪Vs, (12)
∑j∈S1∑j∈S1yij<|S1|, ∀S1⊂V, (13)
tab×xab≤∑i(ziab×si)+∑i∈R∪{a,b}∑j∈R∪{a,b}(yij×tij)≤θ, ∀a,b∈{0}∪Vs, R=f(a,b), (14)
xii=0, ∀i∈{0}∪Vs, (15)
yii=0, ∀i∈V, (16)
yij=0, ∀i,j∈{0}∪Vs, (17)
xij, yij, zijk∈{0,1}, ∀i ∈Vt, j,k∈{0}∪Vs.