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

Algorithm 4

The optimization algorithm for the solution of 1∣𝑝_[𝑗]=𝑎_[𝑗]+𝑏𝑆_[𝑗], 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗 problems for 𝑋∈{CON, DIF}.

Initialization:

                  𝑤(𝑛,1)∶=𝛾𝑎_𝑛

 For 𝑗:=1To 𝑛Do

                 𝑤(𝑗,0)∶=∑^𝑛_𝑖=𝑗𝜓_[𝑖]

Recursion:

 For 𝑗:=𝑛−1Down To 1Do

  For 𝑘:= 1To 𝑛−𝑗+1 Do

   for CON:

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

   for DIF:

        𝑤(𝑗,𝑘)=min𝑤𝑗+1,𝑘−1+𝛾(1+𝑏)^𝑘−1𝑏𝑎_𝑗,𝑤𝑗+1,𝑘+𝜓_𝑗

Optimal solution value:

               min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}.

End