Abstract
We consider the online scheduling problem on a single machine with the assumption that all jobs have their processing times in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mo stretchy="false">[</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mn>5</mml:mn></mml:msqrt><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math>. All jobs arrive over time, and each job and its processing time become known at its arrival time. The jobs should be first processed on a single machine and then delivered by a vehicle to some customer. When the capacity of the vehicle is infinite, we provide an online algorithm with the best competitive ratio of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mn>5</mml:mn></mml:msqrt><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math>. When the capacity of the vehicle is finite, that is, the vehicle can deliver at most<jats:italic>c</jats:italic>jobs at a time, we provide another best possible online algorithm with the competitive ratio of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mo stretchy="false">(</mml:mo><mml:msqrt><mml:mn>5</mml:mn></mml:msqrt><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math>.
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