Scheduling is a decision-making process about how we allocate limited resources, including time, facilities, and work force, to tasks or activities. This type of the decision-making processes is important and common in procurement and production.
Allocating limited resources may face different conditions and different objectives in each company. Therefore, the scheduling literature covers a variety of objectives, including for example makespan (Cmax) and total completion time (
Ci
∑ ). The theoretical results of machine scheduling have abounded and applied since the 1950s. In this thesis, we consider a scheduling issue that arises in the production line of industry.
Production models can be categorized into several types, including single-machine, parallel-machine, flowshop, job shop and open shop. Among the shop models, flowshop scheduling is the most widely discussed due to its real-world applications and several interesting structure properties. Before we formally define the flowshop problem, we provide an example to illustrate its configuration and properties. There are two tasks (operations), washing and waxing, for each car (job) to be processed in a car washing site. Each car needs to wash before to wax. We assume machine one is for washing and machine two is for waxing.
An ordering of all cars on machine one must be the same as that on machine two. This is called a two-machine flowshop in which all jobs need to visit machine one and then machine two, and a machine-two operation cannot start until the corresponding machine-one operation is finished and machine two is not occupied. With the two-machine model, we can generalize it to a multiple-machine model where a number of operations have to be done for every job and the operations have a fixed sequence of machines to visit, and the sequence is same to all jobs.
This thesis focuses on the flowshop model with two machines. The objective we investigate is the maximum completion time among the jobs, i.e. the makespan. Scheduling to minimize makespan in a two-machine flowshop is commonly denoted by the three-field notation F2||Cmax, in which F2 indicates the two-machine flowshop environment and Cmax specifies the objective to optimize. The problem was proposed and investigated by Johnson (1954). In his seminal work, he proposed an O(n log n) algorithm to solve the problem to optimality. This algorithm is referred to as Johnson’s algorithm. While the F2||Cmax problem
can be solved in polynomial time, this thesis investigates a more complicated variant where multiple independent flowshops are available for processing the jobs. We denote the problem by lF2||Cmax, where lF2 specifies the machine configuration with l independent two-machine flowshops.
It is common for a plant to have more than one production line. Moreover, a plant can rent or borrow production lines from its peers in the industry to fulfill the production demand.
Therefore, we the multiple independent flowshop models can better reflect the real-world applications. From the theoretical point of view, the proposed model can be regarded as a hybridization of the standard F2 model and the standard parallel-machine model. In a parallel-machine model (P2), each job has exactly one operation that can be processed on any of the machines. It is easy to see that P2||Cmax is NP-hard in the ordinary sense because it is can be reduced from the PAPTITION problem. Therefore, the problem we want to study is also NP-hard.
To the best of our knowledge, there is no previous work on the lF2||Cmax problem. In the following, we introduce previous results on flowshop scheduling and parallel-machine scheduling. The solution algorithm for the F2||Cmax problem was proposed by Johnson (1954).
He resolved the two machines of flowshop problem and then identified two special cases of three-machine flowshops which can be solved in polynomial time. An integer programming formulation of Fm||Cmax is presented by Wagner (1959). The NP-hardness proof of F3|| Cmax
was presented by Garey, Johnson, and Sethi (1976). Monma and Rinnooy Kan (1983) gave an overview of Fm||Cmax models to elaborate the special structures that make the problem polynomially solvable.
To minimize the makespan in a parallel-machine environment, the shortest processing time first (SPT) rule is the most well studied heuristic. The worst case analysis of the LPT rule for Pm||Cmax was presented by Graham (1969). Graham gives a general bound, a modified system and some special case. Finally, he provides an algorithm and some theorems for list scheduling. The first examples of worst case analyses of heuristics are presented by Graham (1966). It also provides a worst case analysis of an arbitrary list schedule for Pm||Cmax. A more sophisticated heuristic for Pm || Cmax, with a tighter worst case bound, called MULTIFIT, was proposed by Coffman, Garey, and Johnson (1978) and Friesen (1984a). MULTIFIT is well known to have an error bound better than the LPT rule at the cost of a longer running time. Lee and Massey (1988) analyze a heuristic that is based on the LPT
rule as well as on the MULTIFIT. They design the combined algorithm in order to integrate each advantage.
In this thesis, we design a dynamic programming algorithm to optimally solve the multiple flowshops problem. Furthermore, we also design a lower bound and two heuristics, which are used to produce an initial solution of the tabu search approach. The rest of this thesis is organized as follows. We define the multiple flowshops problem and design the dynamic programming algorithm in Chapter 2. In Chapter 3, we introduce the existing lower bounds, develop two heuristics and design our tabu search approach. Computational experiments and analysis of the results are included in Chapter 4. Finally, we give our conclusion in Chapter 5.