Scheduling a Proportionate Flow Shop of Batching Machines

Christoph Hertrich

doi:10.1007/978-3-030-48439-2_4

Abstract

In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number $$m \ge 2$$ of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines, the makespan and the sum of completion times can be minimized in polynomial time. Furthermore, we show that the obtained algorithm can also be used to minimize the weighted total completion time, maximum lateness, total tardiness and (weighted) number of late jobs in polynomial time if all release dates are 0. Previously, polynomial time algorithms have only been known for two machines.

Highlights

Modern medicine can treat some serious illnesses using individualized drugs, which are produced to order for a specific patient and adapted to work only for that unique patient and nobody else
We show that for proportionate flow shop of batching machines (PFB) with release dates and a fixed number m of machines, the makespan and the total completion time can be minimized in polynomial time
Theorem 19 For a PFB without release dates and a fixed number m of machines, the weighted number of late jobs can be minimized in O(nm2+2m−1) time

Summary

Introduction

Modern medicine can treat some serious illnesses using individualized drugs, which are produced to order for a specific patient and adapted to work only for that unique patient and nobody else. Each job J j has a release date r j ≥ 0, denoting the time at which the job J j is available for processing at the first machine M1. All jobs in one batch Bk(i) on some machine Mi have to start processing on Mi at the same time. Example 1 Consider a PFB instance with m = 2 machines, n = 5 jobs, maximum batch sizes b1 = 3 and b2 = 4, processing times p1 = 2 and p2 = 3, and release dates r1 = r2 = 0, r3 = r4 = 1, and r5 = 2. 2 we prove that permutation schedules with jobs in earliest release date order are optimal for PFBs with the makespan and total completion time objectives.

Literature

Our results

Optimality of permutation schedules

Dynamic programming to find an optimal schedule for a given job permutation

Improved running time for makespan minimization

Equal release dates

Conclusions and future work