Column Generation for a Multitrip Vehicle Routing Problem with Time Windows, Driver Work Hours, and Heterogeneous Fleet

Algorithm 2

Feasibility criteria for combining forward and backward labels.

(1)  𝑟₁≤𝑟₂;

(2)  There is arc 𝑖,𝑗;

(3)  𝑆₁[𝑣]+𝑆₂[𝑣]≤1  ∀𝑣∈𝑁

(4)  𝑞₁+𝑞₂≤1, that is, total capacity is less than or equal to 100% of 𝑄_𝑘;

(5)  𝑡_𝑖+𝑠_𝑖+𝑡̃^𝑘_𝑖𝑗+𝑡_𝑗≤𝑏_DC;

(6)  If 𝑅>1,  𝑖=𝑂₁ and 𝑗=𝑂₂ then

     𝑟₂=𝑅 must be true.

   End

(7) After calculating 𝑡_idle=𝑏_DC−𝑡_𝑗+𝑠_𝑗−𝑡_𝑖+𝑠_𝑖+𝑡̃^𝑘_𝑖𝑗

   𝑡_idle≥0 must be true.

   If (𝑡_idle=0), then 𝑤_𝑖+𝑤_𝑗+𝑏_DC−𝑡_𝑗−𝑡_𝑖≤𝑊_max,𝑘 must be true.

   Else

     Let Δ=Δ₁+Δ₂

     If (Δ≥𝑡_idle), then 𝑤_𝑖+𝑤_𝑗+𝑏_DC−𝑡_𝑗−𝑡_𝑖−𝑡_idle≤𝑊_max,𝑘 must be true.

     Else, 𝑤_𝑖+𝑤_𝑗+𝑏_DC−𝑡_𝑗−𝑡_𝑖−Δ≤𝑊_max,𝑘 must be true.

     End

   End