Flow shop scheduling with earliness, tardiness and intermediate inventory holding costs

We consider the problem of scheduling customer orders in a flow shop with the objective of minimizing the sum of tardiness, earliness (finished goods inventory holding) and intermediate (work-in-process) inventory holding costs. We formulate this problem as an integer program, and based on approximate solutions to two di erent, but closely related, Dantzig-Wolfe reformulations, we develop heuristics to minimize the total cost. We exploit the duality between Dantzig-Wolfe reformulation and Lagrangian relaxation to enhance our heuristics. This combined approach enables us to develop two di erent lower bounds on the optimal integer solution, together with intuitive approaches for obtaining near-optimal feasible integer solutions. To the best of our knowledge, this is the first paper that applies column generation to a scheduling problem with di erent types of strongly NP-hard pricing problems which are solved heuristically. The computational study demonstrates that our algorithms have a significant speed advantage over alternate methods, yield good lower bounds, and generate near-optimal feasible integer solutions for problem instances with many machines and a realistically large number of jobs

Heuristic Solutions for Loading in Flexible Manufacturing Systems

Production planning in flexible manufacturing system deals with the efficient organization of the production resources in order to meet a given production schedule. It is a complex problem and typically leads to several hierarchical subproblems that need to be solved sequentially or simultaneously. Loading is one of the planning subproblems that has to addressed. It involves assigning the necessary operations and tools among the various machines in some optimal fashion to achieve the production of all selected part types. In this paper, we first formulate the loading problem as a 0-1 mixed integer program and then propose heuristic procedures based on Lagrangian relaxation and tabu search to solve the problem. Computational results are presented for all the algorithms and finally, conclusions drawn based on the results are discussed

One Benders cut to rule all schedules in the neighbourhood

Logic-Based Benders Decomposition (LBBD) and its Branch-and-Cut variant, namely Branch-and-Check, enjoy an extensive applicability on a broad variety of problems, including scheduling. Although LBBD offers problem-specific cuts to impose tighter dual bounds, its application to resource-constrained scheduling remains less explored. Given a position-based Mixed-Integer Linear Programming (MILP) formulation for scheduling on unrelated parallel machines, we notice that certain $k-$OPT neighbourhoods could implicitly be explored by regular local search operators, thus allowing us to integrate Local Branching into Branch-and-Check schemes. After enumerating such neighbourhoods and obtaining their local optima - hence, proving that they are suboptimal - a local branching cut (applied as a Benders cut) eliminates all their solutions at once, thus avoiding an overload of the master problem with thousands of Benders cuts. However, to guarantee convergence to optimality, the constructed neighbourhood should be exhaustively explored, hence this time-consuming procedure must be accelerated by domination rules or selectively implemented on nodes which are more likely to reduce the optimality gap. In this study, the realisation of this idea is limited on the common 'internal (job) swaps' to construct formulation-specific $4$-OPT neighbourhoods. Nonetheless, the experimentation on two challenging scheduling problems (i.e., the minimisation of total completion times and the minimisation of total tardiness on unrelated machines with sequence-dependent and resource-constrained setups) shows that the proposed methodology offers considerable reductions of optimality gaps or faster convergence to optimality. The simplicity of our approach allows its transferability to other neighbourhoods and different sequencing optimisation problems, hence providing a promising prospect to improve Branch-and-Check methods

Survey on Combinatorial Register Allocation and Instruction Scheduling

Register allocation (mapping variables to processor registers or memory) and instruction scheduling (reordering instructions to increase instruction-level parallelism) are essential tasks for generating efficient assembly code in a compiler. In the last three decades, combinatorial optimization has emerged as an alternative to traditional, heuristic algorithms for these two tasks. Combinatorial optimization approaches can deliver optimal solutions according to a model, can precisely capture trade-offs between conflicting decisions, and are more flexible at the expense of increased compilation time. This paper provides an exhaustive literature review and a classification of combinatorial optimization approaches to register allocation and instruction scheduling, with a focus on the techniques that are most applied in this context: integer programming, constraint programming, partitioned Boolean quadratic programming, and enumeration. Researchers in compilers and combinatorial optimization can benefit from identifying developments, trends, and challenges in the area; compiler practitioners may discern opportunities and grasp the potential benefit of applying combinatorial optimization

A hybrid shifting bottleneck-tabu search heuristic for the job shop total weighted tardiness problem

In this paper, we study the job shop scheduling problem with the objective of minimizing the total weighted tardiness. We propose a hybrid shifting bottleneck - tabu search (SB-TS) algorithm by replacing the reoptimization step in the shifting bottleneck (SB) algorithm by a tabu search (TS). In terms of the shifting bottleneck heuristic, the proposed tabu search optimizes the total weighted tardiness for partial schedules in which some machines are currently assumed to have infinite capacity. In the context of tabu search, the shifting bottleneck heuristic features a long-term memory which helps to diversify the local search. We exploit this synergy to develop a state-of-the-art algorithm for the job shop total weighted tardiness problem (JS-TWT). The computational effectiveness of the algorithm is demonstrated on standard benchmark instances from the literature

A Primal-Dual Approach for Large Scale Integer Problems

This paper presents a refined approach to using column generation to solve specific type of large integer problems. A primal-dual approach is presented to solve the Restricted Master problem belonging to the original optimization task. Firstly, this approach allows a faster convergence to the optimum of the LP relaxation of the problem. Secondly, the existence of both an upper and lower bound of the LP optimum at each iteration allows a faster searching of the Branch-and-Bound tree. To achieve this an early termination approach is presented. The technique is demonstrated on the Generalized Assignment problem and Parallel Machine Scheduling problem as two reference applications