Lagrangian heuristics for scheduling new product development projects in the pharmaceutical industry

To stay ahead of their competition, pharmaceutical firms must make effective use of their new product development (NPD) capabilities by efficiently allocating its analytical, clinical testing and manufacturing resources across various drug development projects. The resulting project scheduling problems involve coordinating hundreds of testing and manufacturing activities over a period of several quarters. Most conventional integer programming approaches are computationally impractical for problems of this size, while priority rule-driven heuristics seldom provide consistent solution quality. We propose a Lagrangian decomposition (LD) heuristic that exploits the special structure of these problems. Some resources (typically manpower) are shared across all on-going projects while others (typically equipment) are specific to individual project categories. Our objective function is a weighted discounted cost expressed in terms of activity completion times. The LD heuristics were subjected to a comprehensive experimental study based on typical operational instances. While the conventional “Reward–Risk” priority rule heuristic generates duality gaps between 47–58%, the best LD heuristic achieves duality gaps between 10–20%. The LD heuristics also yield makespan reductions of over 30% over the Reward–Risk priority rule.     

Optimization-based heuristics for underground mine scheduling

Underground mine production scheduling possesses mathematical structure similar to and yields many of the same challenges as general scheduling problems. That is, binary variables represent the time at which various activities are scheduled. Typical objectives seek to minimize costs or some measure of production time, or to maximize net present value; two principal types of constraints exist: (i) resource constraints and (ii) precedence constraints. In our setting, we maximize “discounted metal production” for the remaining life of an underground lead and zinc mine that uses three different underground methods to extract the ore. Resource constraints limit the grade, tonnage, and backfill paste (used for structural stability) in each time period, while precedence constraints enforce the sequence in which extraction (and backfill) is performed in accordance with the underground mining methods used. We tailor exact and heuristic approaches to reduce model size, and develop an optimization-based decomposition heuristic; both of these methods transform a computationally intractable problem to one for which we obtain solutions in seconds, or, at most, hours for problem instances based on data sets from the Lisheen mine near Thurles, Ireland.     

Multiple-objective heuristics for scheduling unrelated parallel machines

This research proposes two heuristics and a Genetic Algorithm (GA) to find non-dominated solutions to multiple-objective unrelated parallel machine scheduling problems. Three criteria are of interest, namely: makespan, total weighted completion time, and total weighted tardiness. Each heuristic seeks to simultaneously minimize a pair of these criteria; the GA seeks to simultaneously minimize all three. The computational results show that the proposed heuristics are computationally efficient and provide solutions of reasonable quality. The proposed GA outperforms other algorithms in terms of the number of non-dominated solutions and the quality of its solutions.     

Learning heuristics for basic block instruction scheduling

Instruction scheduling is an important step for improving the performance of object code produced by a compiler. A fundamental problem that arises in instruction scheduling is to find a minimum length schedule for a basic block—a straight-line sequence of code with a single entry point and a single exit point—subject to precedence, latency, and resource constraints. Solving the problem exactly is known to be difficult, and most compilers use a greedy list scheduling algorithm coupled with a heuristic. The heuristic is usually hand-crafted, a potentially time-consuming process. In contrast, we present a study on automatically learning good heuristics using techniques from machine learning. In our study, a recently proposed optimal basic block scheduler was used to generate the machine learning training data. A decision tree learning algorithm was then used to induce a simple heuristic from the training data. The automatically constructed decision tree heuristic was compared against a popular critical-path heuristic on the SPEC 2000 benchmarks. On this benchmark suite, the decision tree heuristic reduced the number of basic blocks that were not optimally scheduled by up to 55% compared to the critical-path heuristic, and gave improved performance guarantees in terms of the worst-case factor from optimality.     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

Dominance-based heuristics for one-machine total cost scheduling problems

We study the one-machine scheduling problem with release dates and we look at several objective functions including total (weighted) tardiness and total (weighted) completion time. We describe dominance rules for these criteria, as well as techniques for using these dominance rules to build heuristic solutions. We use them to improve certain well-known greedy heuristic algorithms from the literature. Finally, we introduce a Tabu Search method with a neighborhood based on our dominance rules. Experiments show the effectiveness of our techniques in obtaining very good solutions for all studied criteria.     

Use of VNS heuristics for scheduling of patients in hospital

One of the important problems in hospital management is how to schedule the treatments of resident patients in hospital for a given day due to the restrictions imposed by their medical condition as well as restrictions on medical machines and qualified medical personnel availability. Patients are to be subjected to different kinds of treatments, each requiring a medical machine of a certain type as well as a physician being qualified to operate it. This is a highly complex problem not yet adequately addressed in the literature. At present in most hospitals the problem is being solved manually by specialized personnel. However, the resulting schedules are very often inaccurate and inefficient with patients waiting for a long time to be treated and medical personnel often working overtime. In this paper we formulate the model for this problem and develop a simple and efficient method based on Variable Neighbourhood Search for solving it. The heuristics has been tested on real-life as well as on generated instances. Numerical results show that the heuristics proposed outperform commercial software for optimization as well as manual solutions both in quality of solution and in computational time.     

Beam search heuristics for quadratic earliness and tardiness scheduling

In this paper, we present beam search heuristics for the single machine scheduling problem with quadratic earliness and tardiness costs, and no machine idle time. These heuristics include classic beam search procedures, as well as filtered and recovering algorithms. We consider three dispatching heuristics as evaluation functions, in order to analyse the effect of different rules on the performance of the beam search procedures. The computational results show that using better dispatching heuristics improves the effectiveness of the beam search algorithms. The performance of the several heuristics is similar for instances with low variability. For high variability instances, however, the detailed, filtered and recovering beam search (RBS) procedures clearly outperform the best existing heuristic. The detailed beam search algorithm performs quite well, and is recommended for small- to medium-sized instances. For larger instances, however, this procedure requires excessive computation times, and the RBS algorithm then becomes the heuristic of choice.     

Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.     

Lagrangian heuristics for the two-echelon,single-source,capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.     

Experimental investigation of heuristics for resource-constrained project scheduling: An update

This paper considers heuristics for the well-known resource-constrained project scheduling problem (RCPSP). It provides an update of our survey which was published in 2000. We summarize and categorize a large number of heuristics that have recently been proposed in the literature. Most of these heuristics are then evaluated in a computational study and compared on the basis of our standardized experimental design. Based on the computational results we discuss features of good heuristics. The paper closes with some remarks on our test design and a summary of the recent developments in research on heuristics for the RCPSP.     

Comparison of agent-based scheduling to look-ahead heuristics for real-time transportation problems

We consider the real-time scheduling of full truckload transportation orders with time windows that arrive during schedule execution. Because a fast scheduling method is required, look-ahead heuristics are traditionally used to solve these kinds of problems. As an alternative, we introduce an agent-based approach where intelligent vehicle agents schedule their own routes. They interact with job agents, who strive for minimum transportation costs, using a Vickrey auction for each incoming order. This approach offers several advantages: it is fast, requires relatively little information and facilitates easy schedule adjustments in reaction to information updates. We compare the agent-based approach to more traditional hierarchical heuristics in an extensive simulation experiment. We find that a properly designed multi-agent approach performs as good as or even better than traditional methods. Particularly, the multi-agent approach yields less empty miles and a more stable service level.     

A note on worst-case performance of heuristics for maintenance scheduling problems

We study a machine scheduling model in which job scheduling and machine maintenance activities have to be considered simultaneously. We develop the worst-case bounds for some heuristic algorithms, including a sharper worst-case bound of the SPT schedule than the results in the literature, and another bound of the EDD schedule.     

Time lag size in multiple operations flow shop scheduling heuristics

This paper considers a multistage flow shop where jobs require multiple operations at each stage and a finish-to-start time lag between any two consecutive operations of a job: the next operation of a job cannot start until the time lag after the former operation of that job has elapsed. The effect of the size of this time lag is considered when studying the effectiveness of solution approaches for this problem. Since the problem of minimizing the makespan is shown to be NP-hard even for the two-stage case, we present a lower bound based heuristic approach that is used to construct several heuristic procedures. These heuristics use lower bounds on the minimum makespan to solve the problem. The effectiveness of these heuristics is empirically evaluated for various time lag sizes by solving a large number of randomly generated problems. We show that the relative performance of the heuristics depends on the size of the time lag. If the ratio of mean time lag and mean processing time is 20% or more, heuristics that construct an active schedule perform less well than heuristics that construct a non-delay schedule. The opposite holds true if this ratio is smaller. The performance of the widely used Shortest Processing Time heuristic (SPT) deteriorates quickly if the size of the time lags increases. We propose instead to use the Earliest Finish Time heuristic (EFT) in case time lags are present. EFT performs much better in this case and is identical to SPT if all time lags are zero. The use of the lower bound based heuristics results in an improvement of the makespan performance of up to 50% as compared with the performance of some simple dispatching heuristics that take the presence of multiple operations and time lags into account. This effect increases with the size of the time lags.     

A dynamic neighborhood based tabu search algorithm for real-world flight instructor scheduling problems

The multi-objective flight instructor scheduling problem is an optimization problem that schedules instructors to teach a set of pilot training events. The objectives of the problem are to minimize labor cost, maximize workload consistency and maximize flight instructor satisfaction of their assignments. The problem is further complicated by various hard and soft constraints. We study a multi-objective cost function and convert it to a scalar-weighted objective function using a priori weighting scheme. We then design an efficient dynamic neighborhood based tabu search meta-heuristic to solve the problem. The algorithm exploits the special properties of different types of neighborhood moves. We also address issues of solution domination, tabu short-term memory, dynamic tabu tenure and aspiration rule. The application of the algorithm in a major US airline carrier is reported and the results show that our algorithm achieves significant benefits in practice.     

Lagrangian based heuristics for the multicommodity network flow problem with fixed costs on paths

We study the multicommodity network flow problem with fixed costs on paths, with specific application to the empty freight car distribution process of a rail operator. The classification costs for sending a group of cars do not depend on the number of cars in the group, as long as the group is kept together as one unit. Arcs correspond to trains, so we have capacity restrictions on arcs but fixed costs on the paths corresponding to routes for groups of cars. As solution method, we propose a Lagrangian based heuristic using dual subgradient search and primal heuristics based on path information of the Lagrangian subproblem solutions. The method illustrates several ways of exploiting the specific structures of the problem. Computational tests indicate that the method is able to generate fairly good primal feasible solutions and lower bounds on the optimal objective function value.     

A Lagrangian decomposition approach for the pump scheduling problem in water networks

Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.     

Search heuristics for a flowshop scheduling problem in a printed circuit board assembly process

We consider a scheduling problem in a factory producing printed circuit boards (PCBs). The PCB assembly process in this factory can be regarded as a flowshop which has two special characteristics: jobs have sequence dependent setup times and each job consists of a lot (batch) of identical PCBs. Because of the latter characteristic, it is possible to start a job on a following machine before the job is entirely completed on a previous machine, that is, there is time-lag between machines. In this paper, we propose several heuristics, including taboo search (TS) and simulated annealing (SA) methods, for this generalized flowshop scheduling problem with the objective of minimizing mean tardiness. We compare suggested heuristics after series of tests to find appropriate values for parameters needed for the two search algorithms, TS and SA. Results of computational tests on randomly generated test problems are reported.     

An alternative framework to Lagrangian relaxation approach for job shop scheduling

A new Lagrangian relaxation (LR) approach is developed for job shop scheduling problems. In the approach, operation precedence constraints rather than machine capacity constraints are relaxed. The relaxed problem is decomposed into single or parallel machine scheduling subproblems. These subproblems, which are NP-complete in general, are approximately solved by using fast heuristic algorithms. The dual problem is solved by using a recently developed “surrogate subgradient method” that allows approximate optimization of the subproblems. Since the algorithms for subproblems do not depend on the time horizon of the scheduling problems and are very fast, our new LR approach is efficient, particularly for large problems with long time horizons. For these problems, the machine decomposition-based LR approach requires much less memory and computation time as compared to a part decomposition-based approach as demonstrated by numerical testing.