Scheduling

23 has displayed quick convergence and simple implementation, but it is relatively new to scheduling problems [70].

Other metaheuristics are problem-specific, and so are not defined under a distinct category [65].

Heuristics and metaheuristics have generally outperformed exact methods for classical scheduling problems (such as shifting bottleneck heuristic and GA, respectively, for the JSSP) [71]. These methods produced optimal to near-optimal solutions in reduced time, and for larger problem sizes. The early formulations of classical scheduling problems were solved with exact methods (such as B&B for the aforementioned JSSP) [72]. The classical formulation was essential to understanding the problem structure and providing a benchmark optimal solution upon which the performance of subsequent methods could be compared [73]. This was a pertinent observation for the research undertaken.

The problem investigated in the research represented a novel undertaking. The structure of the manufacturing system (Section 3.6) was formulated specifically for the on-demand FxMC in relation to a MC production system. Thus, it was decided that an exact method approach would be appropriate for the initial formulation of the problem. The formulation led to a MILP model that was solved with a B&B algorithm (see Section 6.9). The research also provided a heuristic (Section 6.10) that was developed from the problem structure identified through the exact method formulation.

24 Figure 2.11: Information flow diagram in a manufacturing system [74]

Pinedo [74] lists the prevalent objective functions in scheduling problems as: makespan; maximum lateness; total weighted completion time; and total weighted tardiness.

The exact method approach decided upon (in Section 2.7.2) warranted a deterministic scheduling formulation. The predictability of the manufacturing system behaviour for a deterministic approach provided the opportunity to focus on developing the problem structure, instead of the real-time adjustments required for a stochastic approach.

2.8.1 Job Shop Scheduling Problem

The job shop scheduling problem is an important classical scheduling problem in the manufacturing field, as demonstrated by the studies conducted on the problem itself and modifications thereof since the 1950s [75].

The classical JSSP involves the scheduling of n jobs (J = {1, 2,…, n}) on m machines (M = {1, 2,…, m}). Each job is comprised of a sequence of N operations (O = {1, 2,…, N}); these can be considered machine-job mappings. An operation must be processed on its assigned machine for the duration of its processing time p. Pre-emption is prohibited, which means that an operation must be completed without interruption. The classical JSSP does not permit parallel processing, which means that operations are uniquely assigned to machines and time periods [75].

The disjunctive Mixed Integer Programming (MIP) model created by Ku and Beck [76] for the JSSP is presented below. The model was based on the formulation developed by Manne [77], which is one of the earliest models published for the JSSP.

25

𝑀𝑖𝑛 𝐶_𝑚𝑎𝑥 (2.5)

Such that:

𝑥_𝑖𝑗≥ 0, ∀𝑗 ∈ 𝐽, 𝑖 ∈ 𝑀 (2.6)

𝑥_𝜎

ℎ 𝑗,𝑗 ≥ 𝑥_𝜎

ℎ−1 𝑗 ,𝑗+ 𝑝_𝜎

ℎ−1

𝑗 ,𝑗, ∀𝑗 ∈ 𝐽, ℎ = 2, … , 𝑚 (2.7) 𝑥𝑖𝑗≥ 𝑥𝑖𝑘+ 𝑝𝑖𝑘− 𝑉 ∙ 𝑧𝑖𝑗𝑘, ∀𝑗, 𝑘 ∈ 𝐽, 𝑗 < 𝑘, 𝑖 ∈ 𝑀 (2.8) 𝑥_𝑖𝑘 ≥ 𝑥_𝑖𝑗+ 𝑝_𝑖𝑗− 𝑉 ∙ (1 − 𝑧_𝑖𝑗𝑘), ∀𝑗, 𝑘 ∈ 𝐽, 𝑗 < 𝑘, 𝑖 ∈ 𝑀 (2.9)

𝐶_𝑚𝑎𝑥≥ 𝑥_𝜎

𝑚𝑗

,𝑗+ 𝑝_𝜎

𝑚𝑗

,𝑗, ∀𝑗 ∈ 𝐽 (2.10)

𝑧_𝑖𝑗𝑘∈ {0, 1}, ∀𝑗, 𝑘 ∈ 𝐽, 𝑖 ∈ 𝑀 (2.11) Where:

The variable xij is the integer start time of job j on machine i; the non-negative integer pij is the processing time of j on i; the list (σ^j1,…, σ^jh,…, σ^jm) denotes the processing order of operations for job j on m machines; the binary variable zijk is equal to 1 if job j precedes job k on machine i. The Objective Function (2.5) minimises maximum completion time Cmax, i.e. makespan. Constraint (2.6) ensures the start time is non-negative. Constraint (2.7) ensures the operation sequence is upheld. Constraints (2.8) and (2.9) ensure that there is only one job per machine for any given time period. V is a number large enough to ensure validity of (2.8) and (2.9). Constraint (2.10) ensures that the makespan is at least the largest completion time of the final operation of every job. Constraint (2.11) enforces the binary condition of zijk.

Solution methods for the JSSP include the B&B algorithm, disjunctive graph, priority rules and local search methods [75]. The shifting bottleneck heuristic is a prevalent heuristic solution method for the JSSP. This method creates a graph with conjunctive arcs only, analyses the machines to be scheduled, and determines the most disruptive disjunctive arc that could be implemented to continue constructing the graph, i.e. the bottleneck. The heuristic then optimises the bottleneck condition in isolation, such that it is no longer the bottleneck for the main problem [78].

The Flow Shop Scheduling Problem (FSSP) is a derivation of the JSSP, where jobs exhibit the same sequence as each other, i.e. every job utilises the same machines in the same sequence, thus relaxing the classical JSSP [79]. Johnson’s rule is an algorithm developed to minimise makespan for the FSSP [75]. Johnson’s rule schedules the jobs based on the duration of the operation time and the machine it relates to. For two machines, the algorithm continues selecting the shortest operation times from the list (updated after every iteration to eliminate the shortest time) and places them either at the start (if related to first machine) or the end (if related to second machine) of the schedule, until the schedule is complete.

The Flexible Job Shop Scheduling Problem (FJSSP) is another alternative to the JSSP, where multiple identical machines are available for processing of jobs, such that machine-job mappings are not necessarily exclusive [79]. Jobs can be processed in parallel, and decisions are made for which machine

26 is utilised for a given operation. The complexity of the FJSSP is increased in comparison to the JSSP, which has led to numerous recent studies that focus on this variation of the problem [73].

The problem formulated for the research (Section 6.3) was advised by the JSSP. The production system workflow (Section 3.6) exhibited characteristics of a flow shop, whereby a consistent job flow from Cell 1 to Cell 2 was observed. However, the final problem structure was unique to the research problem, and the formulation specific to it. Nevertheless, the classical scheduling problems did provide an initial background, which advised the final scheduling method that was developed.

2.8.2 Research Studies

Research studies pertaining to scheduling and optimisation of relevant production systems were investigated. A selection of the studies reviewed are presented below.

Evolutionary algorithms have emerged as the state-of-the-art in the scheduling field. Scheduling studies within the manufacturing environment commonly comprise of the JSSP and modifications thereof [73].

Algorithms explored include: ACO, co-evolutionary algorithm, classifier system, differential evolution, estimation of distribution algorithms, evolutionary programming, evolution strategies, evolvable hardware, GA, genetic programming, interactive evolutionary computation, linkage learning GA, memetic algorithm, parallel GA, probabilistic model building GA, and PSO [63].

Birgin et al. [80] created an extension to the FJSSP by generating the order of operations for the jobs with an arbitrary acyclic graph instead of using a linear order. The objective function was to minimise the makespan. A list scheduling algorithm was used together with a beam search method to find the optimal solution.

Mencia et al. [81] developed a genetic algorithm for JSSP, with weak Lamarckian evolution used to enhance chromosomes, together with search space narrowing to improve efficiency. The objective was to minimise makespan.

Ku and Beck [76] investigated the performance of MIP models for the classical JSSP, as it was found that evaluation of these models with modern software packages had not been adequately investigated.

The authors revealed that MIP models for scheduling problems are prevalent in current literature, despite the onset of advanced metaheuristics. Two disjunctive models, a time-indexed model and a rank-based model were tested and compared; along with the use of multi-threading and parameter tuning to improve performance. The software packages utilised for the tests were CPLEX®, GUROBI®

and SCIP®. The results varied, depending on the problem size and the software used. It was concluded that MIP models can be solved for moderately-sized problems (up to the 15-job×15-machine problem) in reasonable time (within 3600 seconds) with modern scheduling software. The 20×20 problem was tested but not solved within the time limit for the instances tested.

Jalilvand-Nejad and Fattahi [82] used a MILP model to solve a FJSSP with cyclic jobs. The manufacturing system implemented a kanban policy. A total cost function was created, which included setup cost, delay cost, finished product holding cost and WIP holding cost. The objective was to minimise total cost. A genetic algorithm and simulated annealing algorithm were developed for larger- sized problems, due to the NP-hardness of the JSSP. The GA outperformed the SA algorithm.

27 Scheduling within the cellular manufacturing field was investigated.

Sakhaii et al. [83] created an integrated MILP model for a dynamic CM system with unreliable machines, together with a production planning problem. The objective was to minimise the costs of machine breakdown and relocation, operator training and hiring, inter-intra cell part trip, and shortage of inventory.

Liu et al. [84] used a Discrete Bacteria Foraging Algorithm (DBFA) to simultaneously solve cell formation and task scheduling in a CM system. The objective function was to minimise material handling costs, and both fixed and operating costs of machines and workers. The DBFA was compared to a GA, and the results of the former were found to be superior in this study.

Raminfar et al. [85] developed an integrated model for production planning and cell formation. A MIP model was developed to solve the production planning and cell formation problems for a CM system, simultaneously. The objective function was to minimise inter-cell material handling cost, machine operating cost, production set-up costs and part inventory cost.

Studies regarding the utilisation of fixtures in an optimisation model were of particular interest to the research.

Thörnblad et al. [7] conducted a study on a multi-task cell, defined as a FJSSP. A time-indexed formulation of the problem was used. Side constraints factored in preventative maintenance, fixture availability and unmanned night shifts. A generic iterative procedure was used to solve the problem, together with a non-generic squeezing procedure. The objective function was to minimise the total weighted tardiness, where the weighting increased as the delay for a job increased. The fixture constraints in the study were to assign a particular fixture to a particular job, and to limit the number of fixtures of each type.

Yu et al. [9] conducted a study on a RMS with multiple process plans and limited pallets/fixtures. The goal was to determine the input sequencing and scheduling of the RMS. A deterministic schedule was created to be tested on, which included multiple process plans for the various jobs. The problem was solved using a priority rule based scheduling approach. Multiple objectives were investigated:

minimising makespan, minimising mean flow time, and minimising mean tardiness. The practical constraint of releasing a job only when the relevant pallet/fixture was available, was included in the study.

Doh et al. [10] expanded on the work of Yu et al. [9] by conducting a study on a FJSSP with reconfigurable manufacturing cell. The decisions were based on: finding operation/machine pairs for processing parts; sequencing of parts to be sent through the reconfigurable manufacturing cell; and sequencing the parts assigned to each machine. The objectives and solution technique were the same as for the precursor work, which yielded sub-optimal solutions. The study also similarly regarded fixtures by constraining the part flow based on availability of fixtures for parts.

Da Silva et al. [86] conducted a case study on the scheduling of assembly fixtures in the aeronautical industry. The fixtures comprised of multiple workstations to hold large aircraft parts during the assembly process. The arrangement of workstations resulted in adjacency constraints, which prevented

28 access to available workstations on the factory floor. Mathematical models were developed to optimise the production scheduling for the assembly problem, yielding improved results over the traditional methods in practice.

Wong et al. [8] solved a resource-constrained assembly JSSP with lot streaming technique by utilising a GA with job-based order crossover. The model objective was the minimisation of total lateness cost.

Resource constraints were used to place limits on the tools and fixtures used in the process. The resources were recyclable in the system.

Metaheuristics (evolutionary algorithms in particular) are prevalent in recent scheduling research studies. However, the relevance of exact methods and integer programming formulations remain current. Integer programming methods and exact solution techniques are required to find the optimal solutions and benchmarks for novel problem formulations. The JSSP and its alternatives are the scheduling problems that are predominantly investigated. The JSSP provided a background from which the MILP model formulation (Section 6.9) was advised; however, the intricacies introduced through reconfigurable fixtures and the manufacturing system workflow meant that the scheduling method was designed as a problem-specific solution. The details of the scheduling method formulation is elaborated in Chapter 6.

Optimisation research in the reconfigurable fixture area predominantly comprised of improvements in the design of the fixtures themselves [22], [23]. Scheduling studies that did consider fixtures were scarce; the studies found were included in this section. It was established that fixture utilisation in a manufacturing environment was limited to regarding fixtures (standard, not reconfigurable) as a constant resource through a single constraint. The research undertaken aimed to investigate this problem more comprehensively than the studies conducted to-date. The fixture manufacturing cell required an optimisation method that considered reconfigurable fixtures, and the recirculation of those fixtures for customised parts; a problem of this type was not explored by the studies reviewed. Bi et al. [12] declared the need for efficient scheduling of modular fixture components with the manufacturing system; this finding was confirmed by the absence of equivalent studies corresponding to this problem in the literature.