A Simulated Annealing Approach to the Solution of Flexible Labour Scheduling Problems

This paper presents a new solution approach to the discontinuous labour tour scheduling problem where the objective is to minimize the number of full-time employees required to satisfy forecast demand. Previous heuristic approaches have often limited the number of allowable tours by restricting labour scheduling flexibility in terms of shift length, shift start-times, days-off, meal-break placement, and other factors. These restrictions were essential to the tractability of the heuristic approaches but often resulted in solutions that contained a substantial amount of excess labour. In this study, we relaxed many of the restrictions on scheduling flexibility assumed in previous studies. The resulting problem environment contained more than two billion allowable tours, precluding the use of previous heuristic methods. Consequently, we developed a simulated annealing heuristic for solving the problem. An important facet of this new approach is an ‘intelligent’ improvement routine which eliminates the need for long run-times typically associated with simulated annealing algorithms. The simulated annealing framework does not rely on a special problem structure and our implementation rapidly converged to near-optimal solutions for all problems in the test environment.     

The timetable constrained distance minimization problem

We define the timetable constrained distance minimization problem (TCDMP) which is a sports scheduling problem applicable for tournaments where the total travel distance must be minimized. The problem consists of finding an optimal home-away assignment when the opponents of each team in each time slot are given. We present an integer programming, a constraint programming formulation and describe two alternative solution methods: a hybrid integer programming/constraint programming approach and a branch and price algorithm. We test all four solution methods on benchmark problems and compare the performance. Furthermore, we present a new heuristic solution method called the circular traveling salesman approach (CTSA) for solving the traveling tournament problem. The solution method is able to obtain high quality solutions almost instantaneously, and by applying the TCDMP, we show how the solutions can be further improved.     

The Scheduling of Continuous Flow Prodcution: A Separable Programming Approach

Zero-one integer programming formulations have been described in the literature to solve a wide range of problems in areas as diverse as capital budgeting, allocation, machine sequencing, etc., but as yet large-scale realistic problems can be very expensive to compute using the standard branch-and-bound extensions to linear programming packages. Many authors, noting that there are many situations in which low-cost approximate solutions may be very acceptable, suggest the use of approximating methods, such as the corresponding linear programme or "effective gradients". This paper describes a complex production scheduling problem for which a near-optimal solution is obtained using a separable programming algorithm.By utilizing the special features of separable programming, the model is able to include considerations of setup times in the optimization. Computational experience suggests that the method is a considerable improvement on heuristic attempts, giving improvements in throughput of approximately 30 per cent.     

A GRASP for simultaneously assigning and sequencing product families on flexible assembly lines

This paper introduces a new model and solution methodology for a real-world production scheduling problem arising in the electronics industry. The production environment is a high volume, just-in-time, make-to-order facility with volatile demand over many product families that are assembled on flexible lines. A distinguishing characteristic of the problem is the presence of non-traditional sequence-dependant setup costs, which complicate our ability to find high-quality solutions. The scheduling problem arose when product variety exceeded the mix that the existing lines could accommodate. A nonlinear integer programming formulation is presented for the problem of minimizing setup costs, and a greedy randomized adaptive search procedure (GRASP) is developed to find solutions. To select the GRASP parameter values, an efficient, space-filling experimental design method is used based on nearly orthogonal Latin hypercubes. The proposed methodology is tested on actual factory data and compared to a prior heuristic presented in the literature; our heuristic provides a cost savings in 7 out of the 10 cases examined, and an average improvement of 17.39 % which is shown to be highly statistically significant. This improvement is due in part to the introduction of a pre-processing step to determine preferential and non-preferential line assignment information.     

Order selection on a single machine with high set-up costs

We study the problem of allocating a limited quantity of a single manufacturing resource to produce a subset of possible part-types. Customer orders require one or more part-types. We assume that revenue is received for an order only if it is completely filled, and that set-up costs and order revenues dominate the variable costs of production. We present a heuristic for the solution of our problem, as well as families of cutting-planes for an integer programming formulation. Computational results on a set of random test problems indicate that the heuristic is quite effective in producing near optimal solutions. The cutting-planes appear to be quite useful in reducing the number of linear programming solutions required by branch-and-bound.     

Genetic algorithm-based subproblem solution procedures for a modified shifting bottleneck heuristic for complex job shops

In this paper, we consider a modified shifting bottleneck heuristic for complex job shops. The considered job shop environment contains parallel batching machines, machines with sequence-dependent setup times and reentrant process flows. Semiconductor wafer fabrication facilities (Wafer Fabs) are typical examples for manufacturing systems with these characteristics. Our primary performance measure is total weighted tardiness (TWT). The shifting bottleneck heuristic uses a disjunctive graph to decompose the overall scheduling into scheduling problems for single tool groups. The scheduling algorithms for these scheduling problems are called subproblem solution procedures (SSPs). In previous research, only subproblem solution procedures based on dispatching rules have been considered. In this paper, we are interested in how much we can gain in terms of TWT if we apply more sophisticated subproblem solution procedures like genetic algorithms for parallel machine scheduling. We conduct simulation experiments in a dynamic job shop environment in order to assess the performance of the suggested subproblem solution procedures. It turns out that using near to optimal subproblem solution procedures leads in many situations to improved results compared to dispatching-based subproblem solution procedures.     

Stochastic scheduling to minimize expected maximum lateness

This paper is concerned with the problems in scheduling a set of jobs associated with random due dates on a single machine so as to minimize the expected maximum lateness in stochastic environment. This is a difficult problem and few efforts have been reported on its solution in the literature. In this paper, we first derive a deterministic equivalent to the expected maximum lateness and then propose a dynamic programming algorithm to obtain the optimal solutions. The procedures to compute optimal solutions are initially developed in the case of deterministic processing times, and then extended to stochastic processing times following arbitrary probability distributions. Moreover, several heuristic rules are suggested to compute near-optimal solutions, which are shown to be highly efficient and accurate by computer-based experiments.     

A global MILP model for FMS scheduling

Flexible manufacturing systems (FMS) require intelligent scheduling strategies to achieve their principal benefit — combining high flexibility with high productivity. A mixed-integer linear programming model (MILP) is presented here for FMS scheduling. The model takes a global view of the problem and specifically takes into account constraints on storage and transportation. Both of these constrained resources are critical for practical FMS scheduling problems and are difficult to model. The MILP model is explained and justified and its complexity is discussed. Two heuristic procedures are developed, based on an analysis of the global MILP model. Computational results are presented comparing the performance of the different solution strategies. The development of iterative global heuristics based on mathematical programming formulations is advocated for a wide class of FMS scheduling problems.     

An Experimental Investigation of Some Heuristic Strategies for Component Placement

The purpose of this study is to examine the relative efficacy of several promising heuristic approaches to a classic problem of component placement. Four "construction" and nine "improvement" algorithms were chosen for investigation and compared experimentally on a CDC 6400 computer. The improvement methods were selected to test some basic strategies of pairwise-interchanging of components and the construction procedures were chosen primarily to evaluate the effects of the quality of starting solution on the improvement methods. The algorithms were tested on 75 problems generated from the literature and compared with respect to the produced solution quality and CPU run-time requirements. A construction approach due to Graves and Whinston produced the best results, both when used to generate starting solutions for the improvement methods and when evaluated on its own merit against the improvement methods using other starts. Construction approaches have previously been regarded in the the past as relatively inferior techniques.     

A Hybrid Approach to Scheduling with Earliness and Tardiness Costs

A hybrid technique using constraint programming and linear programming is applied to the problem of scheduling with earliness and tardiness costs. The linear model maintains a set of relaxed optimal start times which are used to guide the constraint programming search heuristic. In addition, the constraint programming problem model employs the strong constraint propagation techniques responsible for many of the advances in constraint programming for scheduling in the past few years. Empirical results validate our approach and show, in particular, that creating and solving a subproblem containing only the activities with direct impact on the cost function and then using this solution in the main search, significantly increases the number of problems that can be solved to optimality while significantly decreasing the search time.     

A heuristic for scheduling problems,especially for scheduling farm operations

Scheduling problems in agriculture are often solved using techniques such as linear programming (the multi-period formulation) and dynamic programming. But it is difficult to obtain an optimal schedule with these techniques for any but the smallest problems, because the model is unwieldly and much time is needed to solve the problem. Therefore, a new algorithm, a heuristic, has been developed to handle scheduling problems in agriculture. It is based on a search technique (i.e. hill-climbing) supported by a strong heuristic evaluation function. In this paper the heuristic performance is compared with dynamic programming. The heuristic offers near-optimal solutions and is much faster than the dynamic programming model. When tested against dynamic programming the difference in results was about 3%. This heuristic could probably also be applied in an industrial environment (e.g. agribusiness or road construction).     

A decision theory based scheduling procedure for single-machine weighted earliness and tardiness problems

We consider the dynamic single-machine scheduling problem where the objective is to minimize the sum of weighted earliness and weighted tardiness costs. A single pass heuristic, based on decision theory, is developed for constructing schedules. The heuristic permits schedules with idle time between jobs and behaves like a dispatching procedure. The performance of the new heuristic is examined using 116 published problems for which the optimum solution is known. Its performance is also investigated using 540 randomly generated problems covering a variety of conditions by comparing it to two well known dispatching procedures, adapted for dynamic early/tardy problems. The results indicate that the heuristic performs very well.     

Joint rolling-horizon scheduling of materials processing and lot-sizing with sequence-dependent setups

A lot sizing and scheduling problem from a foundry is considered in which key materials are produced and then transformed into many products on a single machine. A mixed integer programming (MIP) model is developed, taking into account sequence-dependent setup costs and times, and then adapted for rolling horizon use. A relax-and-fix (RF) solution heuristic is proposed and computationally tested against a high-performance MIP solver. Three variants of local search are also developed to improve the RF method and tested. Finally the solutions are compared with those currently practiced at the foundry.     

Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs

Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data uncertainty appears in the problem is made, and it is shown how a valid inequality derived for one scenario can be made valid for other scenarios, potentially reducing solution time. Computational results amply demonstrate the effectiveness of disjunctive cuts in solving several large-scale problem instances from the literature. The results are compared to the computational results of disjunctive cuts based on the subproblem space of the formulation and it is shown that the two methods are equivalently effective on the test instances.     

A distributed genetic algorithm for deterministic and stochastic labor scheduling problems

A recurring operational decision in many service organizations is determining the number of employees, and their work schedules, that minimize labor expenses and expected opportunity costs. These decisions have been modeled as generalized set covering (GSC) problems, deterministic goal programs (DGP), and stochastic goal programs (SGP); each a challenging optimization problem. The pervasiveness and economic significance of these three problems has motivated ongoing development and refinement of heuristic solution procedures. In this paper we present a unified formulation for these three labor scheduling problems and introduce a distributed genetic algorithm (DGA) that solves each of them.Our distributed genetic algorithm operates in parallel on a network of message-passing workstations. Separate subpopulations of solutions evolve independently on each processor but occasionally, the fittest solutions migrate over the network to join neighboring subpopulations. With its standard genetic operators, DGA frequently produces infeasible offspring. A few of these are repaired before they enter the population. However, most enter the population as-is, carrying an appropriate fitness penalty. This allows DGA to exploit potentially favorable adaptations that might be present in infeasible solutions while orienting the locus of the search near the feasible region.We applied the DGA to suites of published test problems for GSC, DGP, and SGP formulations and compared its performance with alternative solution procedures, including other metaheuristics such as simulated annealing and tabu search. We found that DGA outperformed the competing alternatives in terms of mean error, maximum error, and percentage of least cost solutions. While DGA is computationally intensive, the quality of its solutions is commensurate with the effort expended. In plots of solution quality versus CPU time for the various algorithms evaluated in our study, DGA consistently appeared on the efficient frontier.     

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.     

A search heuristic for the sequence-dependent economic lot scheduling problem

Almost all of the research on the economic lot scheduling problem (ELSP) has assumed that setup times are sequence-independent even though sequence-dependent problems are common in practice. Furthermore, most of the solution approaches that have been developed solve for a single optimal schedule when in practice it is more important to provide managers with a range of schedules of different length and complexity. In this paper, we develop a heuristic procedure to solve the ELSP problem with sequence-dependent setups. The heuristic provides a range of solutions from which a manager can choose, which should prove useful in an actual stochastic production environment. We show that our heuristic can outperform Dobson's heuristic when the utilization is high and the sequence-dependent setup times and costs are significant.     

Flowshop scheduling research after five decades

Since Johnson’s seminal paper in 1954, flowshop scheduling problems have received considerable research attention over the last fifty years. As a result, several optimization and heuristic solution procedures are available to solve a variety of flowshop scheduling problems. This paper provides a brief glimpse into the evolution of flowshop scheduling problems and possible approaches for their solution over the last fifty years. It briefly introduces the current flowshop problems being solved and the approaches being taken to solve (optimally or approximately) them. The paper concludes with some fruitful directions for future research.     

A genetic algorithm for service level based vehicle scheduling

In many practical applications, vehicle scheduling problems involve more complex evaluation criteria than simple distance or travel time minimization. Scheduling to minimize delays between the accumulation and delivery of correspondence represents a class of vehicle scheduling problems, where: the evaluation of candidate solutions is costly, there are no efficient schemes for evaluation of partial solutions or perturbations to existing solutions, and dimensionality is limiting even for problems with relatively few locations. Several features of genetic algorithms (GA's) suggest that they may have advantages relative to alternative heuristic solution algorithms for such problems. These include ease of implementation through efficient coding of solution alternatives, simultaneous emphasis on global as well as local search, and the use of randomization in the search process. In addition, a GA may realize advantages usually associated with interactive methods by replicating the positive attributes of existing solutions in the search process, without explicitly defining or measuring these attributes. This study investigates these potential advantages through application of a GA to a service level based vehicle scheduling problem. The procedure is demonstrated for a vehicle scheduling problem with 15 locations where the objective is to minimize the time between the accumulation of correspondence at each location and delivery to destination locations. The results suggest that genetic algorithms can be effective for finding good quality scheduling solutions with only limited search of the solution space.     

Aircraft and crew scheduling for fractional ownership programs

Fractional aircraft ownership programs, where individuals or corporations own a fraction of an aircraft, have revolutionized the corporate aviation industry. Fractional management companies (FMC) manage all aspects of aircraft operations enabling the owners to enjoy the benefits of private aviation without the associated responsibilities. We describe here the development of a scheduling decision support tool for a leading FMC. We present mathematical models, exact and heuristic solution methods. Our computational results using real and randomly generated data indicate that these models are quite effective in finding optimal or near-optimal solutions. The first phase of the implementation of one of these models at the FMC led to a significant improvement in effective utilization of the aircraft, reduction of costs due to reduced empty moves, and hence increased profits.