Discrete OptimizationSingle machine batch scheduling with jointly compressible setup and processing times
Introduction
The problem of batching and scheduling n jobs on a single machine is studied. Jobs assigned to the same batch are processed contiguously and are completed together upon the completion of the last job in the batch. The batch processing time is equal to the sum of the processing times of its jobs. A common machine setup time precedes the processing of each batch.
The processing times of the jobs can be jointly compressed through the allocation of a same amount of a resource to them. The setup times can also be jointly compressed by using the same or another resource. The resources are assumed to be continuously divisible and the corresponding dependency functions are assumed to be linearly decreasing.
The objective is to find values of the resources and a batch schedule so as to minimize total job completion time, subject to an upper bound on total weighted resource consumption (problem TC) or to minimize total weighted resource consumption, subject to an upper bound on total job completion time (problem ITC).
Biskup and Jahnke [3] give the following examples where job processing times are jointly compressible. In steel production, a furnace is heated to a specific temperature every day before the processing of an order of ingots (jobs) starts. It is not beneficial to change the temperature for every single job. A higher temperature reduces the processing time of the jobs but incurs a higher fuel consumption cost. A similar situation is observed for a machine on which some tools have to be periodically changed. When a new tool is installed, a decision about the production power and energy consumption of the tool to be used in the coming time period has to be made. For example, a drilling machine can run with a diamond drill, a high- or low-quality steel drill. If a diamond drill is set up, then the jobs can be carried out faster than with a steel drill while incurring higher costs. Another example is an assembly line the speed of which depends on the number of workers and tools available. It is generally not possible or advantageous to change the speed of an assembly line during a shift.
In all of the above situations, the jobs are processed on the machine and delivered on completion in batches. The processing of a batch requires loading/unloading and the loading/unloading time, i.e., setup time, depends on the tools used for performing these operations.
Problems TC and ITC combine scheduling, batching and resource allocation decisions. Reviews of batch scheduling are provided by Potts and Van Wassenhove [18], Webster and Baker [23], Allahverdi, Gupta and Aldowaisan [2], and Potts and Kovalyov [17]. The batching model studied in this paper is called the batch availability model in [17]. The batch availability model applies to situations where jobs move between the processing facilities in containers such as pallets, boxes or carts.
Scheduling problems with compressible job processing times are observed in steel production, part manufacturing and project management. The results on such problems can be found, for example, in Williams [22], Vickson [20], [21], Van Wassenhove and Baker [19], Janiak [13], Li, Sewell and Cheng [16], Błazewicz, Ecker, Pesch, Schmidt and Weglarz [4], Janiak and Kovalyov [14], Cheng, Oguz and Qi [10], Chen, Lu and Tang [9], Cheng, Chen, Li and Lin [5], and Biskup and Jahnke [3].
Clearly, there are real-life scheduling problems where batch scheduling and resource allocation decisions have to be taken simultaneously. We are only aware of the papers by Cheng and Kovalyov [6], Cheng, Janiak and Kovalyov [8] and Janiak and Lichtenstein [15] in which such a problem is studied. The former two papers consider batch availability model while Janiak and Lichtenstein consider the so-called job availability model, in which each job is completed immediately when its processing is finished. The quality of a schedule is evaluated by maximum job lateness in [6], [8]. Minimizing total job completion time is another important scheduling criterion aimed to achieve two purposes: it seeks to minimize work-in-process inventories and to yield an average earliest job delivery to downstream operations. Problems TC and ITC studied in this paper are concerned with this criterion.
In the following section, we give formal definitions of problems TC and ITC. Some properties of the optimal batch sequences for these problems are established in Section 3. 4 Problem TC(, 5 Problem ITC( present an O(n3) and O(n3logPmax) time algorithm for problem TC and ITC, respectively, which are based on these properties, where Pmax is the maximum numerical parameter of problem ITC. A numerical example is given in Section 6. The paper concludes with some remarks, extensions and suggestions for future research. The Appendix A contains a description of the set of all shortest processing time (SPT) sequences of jobs to be defined in Section 3.
Section snippets
Problem formulation
The problem studied in this paper can be formulated as follows: There are one machine, and n independent and non-preemptive jobs available for processing at time zero. The jobs have to be batched and sequenced before processing on the machine. A common machine setup time s precedes the processing of each batch. The batch availability model applies to determine job completion times.
The processing time of job j is a linearly decreasing function of an amount x of a continuously divisible resource
Properties of optimal solutions
It is clear that problem TC or problem ITC can be solved by enumerating all possible numbers of batches k=1,…,n. Denote problem TC and problem ITC with a fixed number of batches k by TC(k) and ITC(k), respectively.
We call the batch sequence (B1,…,Bk) an SPT-batch sequence if for any batches Bl and Bm, where l<m, the inequality pi⩽pj holds for any jobs i∈Bl and j∈Bm. Lemma 1 Any optimal batch sequence for problem TC(k) or for problem ITC(k) with optimal resource values (x*,y*)≠(0,0) is an SPT-batch
Problem TC(k)
Problem TC(k) is to minimize ∑j=1nCj, subject toLet x(k) and y(k) be the optimal resource values for this problem.
Assume that Q=(j1,…,jn) is an optimal job sequence for problem TC(k) and its optimal partition into k batches is expressed asFor such a batch sequence and resource values x and y, calculatewhere l0=0, lk=n,
Problem ITC(k)
Given an optimal batch sequence described by (2), rewrite ∑j=1nCj⩽V as K−Lx−My⩽V, where K, L and M are positive integer numbers dependent on batch sequence and independent of x and y, see (4). Then problem ITC(k) is to minimize wnx+vky, subject to Lx+My⩾K−V, x+y⩽E, 0⩽x⩽xmax and 0⩽y⩽ymax.
A bisection search in the range of the objective function values can be used to solve this problem. Again, let x(k) and y(k) denote the optimal resource values and ITC(k)=wnx(k)+vky(k). It is clear that
A numerical example
Consider the problem with four jobs. The setup time is given byand the job processing times are given byResource constraints are
In problem TC, we would like to minimize total job completion time, subject to the upper bound on total weighted resource consumption: ∑Cj→min, subject to (7) and 12x+3ky⩽72, where k is the unknown number of batches.
In problem ITC, we would like to minimize total weighted resource consumption, subject to the
Conclusions and extensions
Single machine batch scheduling problems TC and ITC with jointly compressible setup and processing times have been studied. Batches are processed under the batch availability assumption. In problem TC, the objective is to minimize total job completion time, subject to an upper bound on total weighted resource consumption. The inverse problem ITC is to minimize total weighted resource consumption, subject to an upper bound on total job completion time. The problems are solved in O(n3) and O(n3log
Acknowledgments
This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grant Council of Hong Kong under grant number PolyU 5191/01E. M.Y. Kovalyov is also supported by INTAS under grant number 00-217.
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