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

Algorithm 2

FPTAS 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

  (iii) Let 𝑡,𝑓_𝑚 be a state in 𝜈_𝑘^(𝛼)# such that 𝑓∈𝐼_𝑚 with the smallest 𝑡 (break ties

    by choosing the state of the smallest 𝑓). Set 𝜈_𝑘^(𝛼)#=𝑡,𝑓_𝑚∣1≤𝑚≤𝐿+1

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

(2) Calculate the final solution of FPTAS from 𝑍^#=min_𝛼∣𝑍^#_𝛼≠∅𝑍^#_𝛼