Scheduling with Learning Effects and/or Time-Dependent Processing Times to Minimize the Weighted Number of Tardy Jobs on a Single Machine

Algorithm 5

The optimization algorithm for the solution of 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗],SLK∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problem.

Initialization:

                𝑤(𝑛,1)∶=𝛾𝑎_𝑛

 For 𝑗∶=1To 𝑛Do

               𝑤(𝑗,0)∶=∑^𝑛_𝑖=𝑗𝜓_[𝑖]

Recursion:

 For 𝑗∶=𝑛−1Down To 1Do

     𝑤(𝑗,1)=min𝑤𝑗+1,0,𝑤𝑗+1,1+𝜓_𝑗=min∑^𝑛_𝑖=𝑗+1𝜓_[𝑖],𝑤𝑗+1,1+𝜓_𝑗

For 𝑗:=𝑛−1Down To 1Do

   For 𝑘:= 2 To 𝑛−𝑗+1Do

       𝑤(𝑗,𝑘)=min𝑤(𝑗+1,𝑘−1)+𝑛𝛾(1+𝑏)^𝑘−1𝑎_𝑗,𝑤(𝑗+1,𝑘)+𝜓_𝑗

Optimal solution value:

          min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}.

 End