Approximation Algorithms and an FPTAS for the Single Machine Problem with Biased Tardiness Penalty

Algorithm 1

DP algorithm.

(1) For each 𝛼={1,2,…,𝑛}

 (1.1) Set 𝜈₀^(𝛼)={[0,0]}

 (1.2) For each 𝑘={1,2,…,𝛼−1,𝛼+1,…,𝑛}

  (i) Consider every state 𝑡,𝑓 in 𝜈^(𝛼)_𝑘−1

   (a) If 𝑡+𝑝_𝑘≤𝑑 then add 𝑡+𝑝_𝑘,𝑓 to 𝜈_𝑘^(𝛼)

   (b) If 𝑃_sum−∑^𝑘_𝑖=1𝑝_𝑖+𝑡>𝑑 then add 𝑡,𝑓+𝑢_𝑘+𝑤_𝑘𝑃_sum−∑^𝑘−1_𝑖=1𝑝_𝑖+𝑡−𝑑 to 𝜈_𝑘^(𝛼)

  (ii) Delete the state space 𝜈^(𝛼)_𝑘−1

 (1.3) Set 𝑍^∗_𝛼=min_{[𝑡,𝑓]∈𝜈𝑛}^(𝛼)𝑓+𝑢_𝛼+𝑤_𝛼𝑡+𝑝_𝛼−𝑑

(2) Calculate the optimal value from 𝑍^∗=min_𝛼∣𝑍^∗_𝛼≠∅𝑍^∗_𝛼