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

Algorithm 3

A unified optimization algorithm for solving the 1∣𝑝_[𝑗]=(𝑎_[𝑗]+𝑏𝑆_[𝑗])𝑗^𝑐, 𝑋∣∑^𝑛_𝑗=1𝜓_𝑗𝑈_𝑗+𝛾∑^𝑛_𝑗=1𝑑_𝑗+𝛿𝐶_max problems for 𝑋∈{CON, SLK, DIF}.

Step  1. Apply Algorithm 2 where 𝜉_𝑗 is calculated by (15) for the CON method, by (19) for the SLK method, and by (11) for the

   DIF method.

Step  2. For the CON method, assign 𝐶_[𝑙^∗_] to all 𝑑^∗_[𝑗]. For the SLK method, assign slack value 𝑠^∗=𝐶_[𝑙^∗_−1] to all jobs. For the DIF

   method, let 𝑑^∗_[𝑗]=𝐶_[𝑗] for 𝑗=1,2,…,𝑙^∗, and 𝑑^∗_[𝑗]=0 for 𝑗=𝑙^∗+1,…,𝑛.