*Article* **Online Batch Scheduling of Simple Linear Deteriorating Jobs with Incompatible Families**

**Wenhua Li 1,\*, Libo Wang 1, Xing Chai <sup>2</sup> and Hang Yuan <sup>3</sup>**


Received: 31 December 2019; Accepted: 24 January 2020; Published: 1 February 2020

**Abstract:** We considered the online scheduling problem of simple linear deteriorating job families on *m* parallel batch machines to minimize the makespan, where the batch capacity is unbounded. In this paper, simple linear deteriorating jobs mean that the actual processing time *pj* of job *Jj* is assumed to be a linear function of its starting time *sj*, i.e., *pj* = *αjsj*, where *α<sup>j</sup>* > 0 is the deterioration rate. Job families mean that one job must belong to some job family, and jobs of different families cannot be processed in the same batch. When *m* = 1, we provide the best possible online algorithm with the competitive ratio of (1 + *α*max)*<sup>f</sup>* , where *f* is the number of job families and *α*max is the maximum deterioration rate of all jobs. When *m* ≥ 1 and *m* = *f* , we provide the best possible online algorithm with the competitive ratio of 1 + *α*max.

**Keywords:** online algorithm; batch scheduling; linear deterioration; job families; competitive ratio

#### **1. Introduction**
