In this paper, we consider online order scheduling on two identical machines, where the maximum job size of each order is known and the job processing times are in the interval \([1,\,t]\) . The objective is to minimize the makespan. We obtain a lower bound \(\min \{\frac{5}{4},\,\min \{\frac{t+2}{3},\,\frac{3t-1}{2t}\}\}\) on the makespan. Then, we provide an optimal algorithm with a competitive ratio \(\min \{\frac{5}{4},\,\frac{3t-1}{2t}\}\) , where t is not smaller than \(\frac{3}{2}\) . We provide another optimal algorithm with a competitive ratio \(\frac{t+2}{3}\) , where t is in the interval \([1.468,\,3/2)\) .
