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

Algorithm 1

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

Step  1. Set ℎ=𝑛_𝐵,𝐽=𝐽^𝐴,𝑓^𝐴_max=0 and 𝑡=𝑡₀−1/𝑘∏^𝑛𝑋_𝑗=1(1−𝑘𝑎^𝑋_𝑗)+1/𝑘,; solve 𝐷^𝐵_𝑗 from

    𝑓^𝐵_𝑗(𝐷^𝐵_𝑗)=𝑈 for 𝑗=1,2,…,𝑛_𝐵 and renumber them according to the non-decreasing

   order such that 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 job 𝐽^𝐵_[ℎ] at time 𝑡, and go to Step  2;

     Else

    go to Step  3;

   Else

     go to Step  3;

Step  3. If 𝐽≠0/, then

     select the job 𝐽^𝐴_𝑙 from 𝐽 with the the smallest cost, that is,  𝑓^𝐴_𝑙(𝑡)=min_𝐽^𝐴_𝑗∈𝐽^𝐴{𝑓^𝐴_𝑗(𝑡)},

     set 𝑓^𝐴_max=max{𝑓^𝐴_max,𝑓^𝐴_𝑙(𝑡)}, 𝑡=(𝑡−𝑎^𝐴_𝑙)/(1−𝑘𝑎^𝐴_𝑙), assign job 𝐽^𝐴_𝑙 at time 𝑡, delete 𝐽^𝐴_𝑙 from 𝐽,

     and go to Step  4;

   Elseif ℎ≥1

     report that the instance is not feasible;

   Else

     go to Step  4;

Step  4. If 𝐽 is not empty or ℎ≥1, then

     go to Step  2.

   Else

     stop.