Two-agent scheduling on a single parallel-batching machine to minimize the weighted sum of the agents’ makespans

We schedule the jobs from two agents with equal processing times and non-identical job sizes on a single parallel-batching machine. The objective is to minimize the weighted sum of the two makespans of the jobs from two agents. We give a lower bound on the optimal solution, and present a polynomial-time algorithm H for solving the problem. Furthermore, we prove that both the absolute worst-case ratio and the asymptotic worst-case ratio of algorithm H are the functions of the weight of one agent, $$\alpha$$α. Specifically, (1) for the absolute worst-case ratio, $$H(I)\le f_1(\alpha )\times OPT(I)$$H(I)≤f1(α)×OPT(I), where $$f_1(\alpha )=(3-3\alpha +3\alpha ^2)/(2-4\alpha +4\alpha ^2)\le 9/4$$f1(α)=(3-3α+3α2)/(2-4α+4α2)≤9/4; (2) for the asymptotic worst-case ratio, $$H(I)\le f_2(\alpha ) \times OPT(I)+4/3$$H(I)≤f2(α)×OPT(I)+4/3, where $$f_2(\alpha )=(11-11\alpha +11\alpha ^2)/(9-18\alpha +18\alpha ^2)$$f2(α)=(11-11α+11α2)/(9-18α+18α2) and $$11/9 \le f_2(\alpha )\le 11/6$$11/9≤f2(α)≤11/6. The effectiveness of algorithm H is demonstrated by using a set of computational experiments. The results show that algorithm H performs well in practice and tends to perform better when facing the large-scale instances.