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

Algorithm 6

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

Initialization:

                 𝑤(1,1)∶=𝛾𝑎₁

 For 𝑗∶=1To 𝑛Do

                𝑤(𝑗,0)∶=∑^𝑖_𝑖=𝑖𝜓_[𝑖]

Recursion:

 For 𝑗∶=1To 𝑛Do

  For 𝑘∶=0To 𝑗Do

         𝑤𝑗,𝑘=min𝑤𝑗−1,𝑘−1+𝑛𝑗^𝑐𝑎_𝑗,𝑤𝑗−1,𝑘+𝜓_𝑗

            𝑊(𝑗)=min_{0≤𝑘≤𝑗−1}𝑤(𝑗−1,𝑘)+∑^𝑛_𝑖=𝑗+1𝜓_𝑖

Optimal solution value:

 For CON:

                min{𝑤(1,𝑘)∣0≤𝑘≤𝑛}.

 For SLK:

                 𝑊=min_{1≤𝑗≤𝑛}𝑊(𝑗).

 End