A branch and bound algorithm for minimizing total completion time on a single batch machine with incompatible job families and dynamic arrivals Shiqing Yao, Zhibin Jiang n , Na Li Department of Industrial Engineering & Logistics Management, Shanghai Jiao Tong University, Shanghai, China article info Available online 1 July 2011 Keywords: Branch and bound algorithm Dynamic arrivals Batch scheduling Incompatible job families abstract In this paper, we consider a single batch machine scheduling problem with incompatible job families and dynamic job arrivals. The objective is to minimize the total completion time. This problem is known to be strongly NP-hard. We present several dominance properties and two types of lower bounds, which are incorporated to construct a basic branch and bound algorithm. Furthermore, according to the characteristics of dynamic job arrivals, a decomposed branch and bound algorithm is proposed to improve the efficiency. The proposed algorithms are tested on a large set of randomly generated problem instances. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction This paper addresses a batch processor scheduling problem. Jobs arrive dynamically and thus have unequal ready times. Jobs in the same family have the same process time and could be processed simultaneously in batches. The batch size may vary by jobs while the maximum size is given. The objective is to minimize the total completion time. This problem arises from the batch scheduling problem in wafer fabrications, where typical furnace machines serve as batch tools. In general, the capacity of furnace machines is six lots with each containing 25 wafers. Lots belonging to different process recipes cannot be batched together, since their process parameters (i.e., pressure and temperature) are quite different. This leads to incompatible job families. Referring to [1], we denote the problem of interest as 19p-batch, r j , b on, incompat9 X C j This problem belongs to NP-hard in strong sense based on reductions to the known NP-hard problem in strong sense: minimizing the total completion time on a single machine with unequal ready times [2].The problem we are interested in involves two interrelated decision making problems: (1) when to group jobs into batches; and (2) how to sequence the batches after they form. Due to dynamic arrivals, when to batch jobs indirectly affects sequencing decisions. On the other hand, the sequencing results also influence batching decisions. In this paper, we study the structural properties of the problem, develop two types of lower bounds, and propose a branch and bound (B&B) algorithm and its improved version. 2. Literature review Batch scheduling problems can be categorized from different aspects. Readers could refer to [3,4] for the details. In this section, we review the highly related works. As for identical job family, many relevant problems are solvable. Glassey and Weng [5] proposed a dynamic scheduling algorithm for minimizing the average waiting time. Webster and Baker [6] developed a dynamic programming algorithm for 19p-batch, b on, r j , p j ¼ p9 P C j with O(n 3 ) overall worst-case time complexity. It is worthwhile to note that this problem is a special case of the problem of interest. Furthermore, Baptiste [7] showed that 19p-batch, b on, r j , p j ¼ p9 P w j C j also belongs to P. As for multiple job families, there exist two batch types, i.e., compatible job families and incompatible job families. For compatible job families, jobs from different families can be processed together. Jobs in a batch start and complete at the same time, and the process time of a batch is equal to the largest process time among the jobs in its batch. Chandru et al. [8] studied the optimality structure of 19p  batch, b on9 P C j and proposed a B&B algorithm. For the case of m jobs families, Chandru et al. [8] presented an O(m 3 b mþ 1 ) time dynamic programming algorithm, and Brucker et al. [9] designed a dynamic programming algorithm with a further gain in efficiency, which only requires O(b 2 m 2 2 m ) time. In consideration of weighted jobs, Uzsoy and Yang [10] developed a B&B algorithm and several heuristics for 19p  batch, b on9 P w j C j . In recent years, Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2011.06.003 n Corresponding author. Tel.: þ86 13918891152; fax: þ86 2134206065. E-mail address: zbjiang@sjtu.edu.cn (Z. Jiang). Computers & Operations Research 39 (2012) 939–951