Some Single-Machine Scheduling Problems with Learning Effects and Two Competing Agents

Algorithm 3

Input: 𝑛_𝐴,𝑛_𝐵,𝑈,𝑟, 𝑤^𝐴=(𝑤^𝐴₁,𝑤^𝐴₂,…,𝑤^𝐴_𝑛𝐴), 𝑝^𝐴=(𝑝^𝐴₁,𝑝^𝐴₂,…,𝑝^𝐴_𝑛𝐴) and 𝑝^𝐵=(𝑝^𝐵₁,𝑝^𝐵₂,…,𝑝^𝐵_𝑛𝐵).

Step  1. Set 𝑡=𝑡₀−1/𝑘∏^𝑛𝑋_𝑗=1(1−𝑘𝑎^𝑋_𝑗)+1/𝑘,𝑙=𝑛_𝐴,ℎ=𝑛_𝐵, and ∑𝑤^𝐴_𝑗(1−𝑒^{−𝑟𝐶𝐴𝑗})=0;

   sort the 𝐴-jobs in non-decreasing order of (1−𝑒^{−𝑟𝑎𝐴𝑗})/(𝑤^𝐴_𝑗𝑒^{−𝑟𝑎𝐴𝑗}),

   that is, (1−𝑒^{−𝑟𝑎𝐴[1]})/(𝑤^𝐴_[1]𝑒^{−𝑟𝑎𝐴[1]})≤(1−𝑒^{−𝑟𝑎𝐴[2]})/(𝑤^𝐴_[2]𝑒^{−𝑟𝑎𝐴[2]})≤⋯≤(1−𝑒^{−𝑟𝑎𝐴𝐴][𝑛})/(𝑤^𝐴_𝑛𝐴𝑒^{−𝑟𝑎𝐴𝐴][𝑛});

   calculate the deadlines of the 𝐵-jobs from 𝑓^𝐵_𝑗(𝐷^𝐵_𝑗)=𝑈 and

   renumber them according to the non-decreasing order, that is, 𝐷^𝐵_[1]≤𝐷^𝐵_[2]≤⋯≤𝐷^𝐵_[𝑛𝐵];

Step  2. If ℎ≥1, then

     If 𝑡≤𝐷^𝐵_[ℎ], then

    set ℎ=ℎ−1, 𝑡=(𝑡−𝑎^𝐵_[ℎ])/(1−𝑘𝑎^𝐵_[ℎ]), assign 𝐽^𝐵_[ℎ] at time 𝑡, and go to Step  2;

     Else

    go to Step  3;

   Else

     go to Step  3;

Step  3. If 𝑙≥1, then

     set 𝑙=𝑙−1, ∑𝑤^𝐴_𝑗(1−𝑒^{−𝑟𝐶𝐴𝑗})=∑𝑤^𝐴_𝑗(1−𝑒^{−𝑟𝐶𝐴𝑗})+𝑤^𝐴_[𝑙]1−𝑒^−𝑟𝑡, 𝑡=(𝑡−𝑎^𝐴_[𝑙])/(1−𝑘𝑎^𝐴_[𝑙]),

     assign 𝐽^𝐴_[𝑙] at time 𝑡, and go to Step  4;

   Elseif ℎ≥1

     output that the instance is not feasible;

   Else

     go to Step  4;

Step  4. If ℎ≥1 or 𝑙≥1, then

     go to Step  2.

   Else

     stop.