Local search metaheuristics for discrete–continuous scheduling problems

Problems of scheduling nonpreemptable jobs which require simultaneously a machine from a set of parallel, identical machines and a continuous, renewable resource are considered. For each job there are known: its processing speed as a continuous, concave function of a continuous resource allotted at a time and its processing demand. The optimization criterion is the schedule length. The problem can be decomposed into two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to find an optimal (continuous) resource allocation among jobs already sequenced. Problem (ii) can be formulated as a convex programming problem with linear constraints and solved using proper solvers. Thus, the problem remains to generate a set of all feasible sequences of jobs on machines (this guarantees finding an optimal schedule in the general case). However, the cardinality of this set grows exponentially with the number of jobs. Thus, we propose to use heuristic search methods defined on the space of feasible sequences. Three metaheuristics: tabu search (TS), simulated annealing (SA) and genetic algorithm (GA) have been implemented and compared computationally with a random sampling technique. The computational experiment has been carried out on an SGI PowerChallenge XL computer with 12 RISC R8000 processors. Some directions for further research have been pointed out.     

Tabu list management methods for a discrete–continuous scheduling problem

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. Processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimizes the makespan. The tabu search (TS) metaheuristic is presented to attack the problem. Three different tabu list management methods: the tabu navigation method (TNM), the cancellation sequence method (CSM) and the reverse elimination method (REM) are discussed and examined. A computational experiment is described and the results obtained for the methods tested are compared to optimal solutions. A few conclusions and final remarks are presented.     

On a methodology for discrete–continuous scheduling

Discrete–continuous problems of scheduling nonpreemptable jobs on parallel machines are considered. The problems arise e.g. when jobs are assigned to multiple parallel processors driven by a common electric, hydraulic or pneumatic power source. Existing models have assumed job processing rates as a function of the number of jobs currently being processed, or equivalently the number of machines currently in operation. In this paper a more general model is proposed in which processing rates of a job assigned to a machine depend on the amount of a continuous, i.e. continuously divisible resource (e.g. power) allotted to this job at a time. Thus the problem consists of two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to allocate the continuous resource among jobs already sequenced. We provide a comprehensive analysis of the problem. This includes properties of optimal schedules, efficiently (in particular analytically) solvable cases, formulations of the possibly simplest mathematical programming problems for finding optimal schedules in the general case, heuristics and the worst-case analysis. Although our objective function in this paper is to minimize makespan of a set of independent jobs, the presented methodology can be applied to other criteria, precedence-related jobs, and many resource types (apart from, or instead of machines).     

Applications of resource assignment and scheduling with Petri Nets and heuristic search

This paper introduces a new Petri Net based approach for resource allocation and scheduling. The goals are (i) minimize the number of required resources given a set of jobs, (ii) find both an assignment for all jobs in the span of a predefined shift and (iii) the sequence in which such jobs are executed. The studied problem was inspired from a complex real life manufacturing shop as described in this document. The modeling of the processes and jobs is carried out with Petri Nets due to their capability of representing dynamic, concurrent discrete-event dynamic systems. The resource assignment starts with an initial feasible solution (initial number of resources) and then follows with a re-optimization process aimed to further reduce the resource requirements. The algorithm is based on a modified Heuristic Search method previously presented. The algorithm was tested first on a number of instances from the literature and then on the aforementioned system (a car seat cover manufacturer). The proposed approach shows not only good results in terms of performance but also shows the potential of Petri Nets for modeling and optimizing real-life systems. An implementation phase at the first stages of the process is underway at the time of writing.     

Resource level minimization in the discrete–continuous scheduling

A discrete–continuous problem of non-preemptive task scheduling on identical parallel processors is considered. Tasks are described by means of a dynamic model, in which the speed of the task performance depends on the amount of a single continuously divisible renewable resource allotted to this task over time. An upper bound on the completion time of all the tasks is given. The criterion is to minimize the maximum resource consumption at each time instant, i.e., the resource level. This problem has been observed in many industrial applications, where a continuously divisible resource such as gas, fuel, electric, hydraulic or pneumatic power, etc., has to be distributed among the processing units over time, and it affects their productivity. The problem consists of two interrelated subproblems: task sequencing on processors (discrete subproblem) and resource allocation among the tasks (continuous subproblem). An optimal resource allocation algorithm for a given sequence of tasks is presented and computationally tested. Furthermore, approximation algorithms are proposed, and their theoretical and experimental worst-case performances are analyzed. Computer experiments confirmed the efficiency of all the algorithms.     

Optimal resource allocation and scheduling for?the?CELL BE platform

Resource allocation and scheduling for multicore platforms is one of the most critical challenges in today??s embedded computing. In this paper we focus on a well-known multicore platform, namely the Cell BE processor, and we address the problem of allocating and scheduling its processors, communication channels and memories, with the goal of minimizing execution time for complex data streaming applications. We propose three complete approaches that optimally solve the problem and prove optimality. The first is based on the recursive application of the Logic Based Benders decomposition, resulting in a three stage algorithm. The second is a pure CP approach while the third is a hybrid approach integrating the first two. Extensive experimental evaluation shows the features of each approach and its effectiveness on a specific instance structure.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

Tabu search for the linear ordering problem with cumulative costs

Given a matrix of weights, the Linear Ordering Problem (LOP) consists of finding a permutation of the columns and rows in order to maximize the sum of the weights in the upper triangle. This well known NP-complete problem can also be formulated on a complete weighted graph, where the objective is to find an acyclic tournament that maximizes the sum of arc weights. The variant of the LOP that we target here was recently introduced and adds a cumulative non-linear propagation of the costs to the sum of the arc weights. We first review the previous methods for the LOP and for this variant with cumulative costs (LOPCC) and then propose a heuristic algorithm for the LOPCC, which is based on the Tabu Search (TS) methodology. Our method achieves search intensification and diversification through the implementation of both short and long term memory structures. Our extensive experimentation with 224 instances shows that the proposed procedure outperforms existing methods in terms of solution quality and has reasonable computing-time requirements.     

Nonlinear discrete Sturm–Liouville problems with global boundary conditions

This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm–Liouville problem.     

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C ¹ continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized Green’s functions for the second-order discrete problems with nonlocal conditions

In this paper, we investigate a generalized discrete Green’s function that describes the general least squares solution of every second-order discrete problem with two nonlocal conditions. We develop the problem where the necessary and sufficient existence condition of ordinary discrete Green’s function is not satisfied. Some examples are also presented.     

Competitive exclusion in a discrete juvenile–adult model with continuous and seasonal reproduction

We develop a general discrete juvenile–adult population model that describes two competing species. We consider species in which the juveniles only compete with other juveniles, and the adults only compete with other adults, i.e. juveniles and adults of either species do not compete. This is typical of amphibians where juveniles (tadpoles) live in water and adults (frogs) live on land. Assuming competition efficiencies of the two species are similar, we analyse the cases where reproduction is either continuous or seasonal. In both cases, we develop conditions on the invasion net reproductive numbers of the two species that will lead to competitive exclusion. We show using numerical simulations that coexistence and bistability are possible outcomes when competition efficiencies of the two species are different.     

Corrigendum to “Resource-monotonicity for house allocation problems”

Ehlers and Klaus (Int J Game Theory 32:545–560, 2003) study so-called allocation problems and claim to characterize all rules satisfying efficiency, independence of irrelevant objects, and resource-monotonicity on two preference domains (Ehlers and Klaus 2003, Theorem 1). They explicitly prove Theorem 1 for preference domain R₀{mathcal{R}_0} which requires that the null object is always the worst object and mention that the corresponding proofs for the larger domain R{mathcal{R}} of unrestricted preferences “are completely analogous.” In Example 1 and Lemma 1, this corrigendum provides a counterexample to Ehlers and Klaus (2003, Theorem 1) on the general domain R{mathcal{R}} . We also propose a way of correcting the result on the general domain R{mathcal{R}} by strengthening independence of irrelevant objects: in addition to requiring that the chosen allocation should depend only on preferences over the set of available objects (which always includes the null object), we add a situation in which the allocation should also be invariant when preferences over the null object change. Finally, we offer a short proof of the corrected result that uses the established result of Theorem 1 for the restricted domain R₀{mathcal{R}_0}.     

Green’s functions for discrete mTH-order problems*

We investigate an mth-order discrete problem with additional conditions, described by m linearly independent linear functionals. We find the solution to this problem and present a formula and the existence condition of Green??s function if the general solution of a homogeneous equation is known. We obtain a relation between Green??s functions of two nonhomogeneous problems. It allows us to find Green??s function for the same equation, but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.     

Metaheuristic methods based on Tabu search for assigning judges to competitions

Two metaheuristic methods based on Tabu search are introduced to assign judges to individual competitions in a tournament. The complexity of the mathematical formulation accounting for the assignment rules, leads us to use such an approach. The first metaheuristic includes two different Tabu searches that are combined with a diversification strategy. The second metaheuristic is applied to a penalized version of the original model formulated as an assignment problem. This metaheuristic is also based on a Tabu search procedure including a diversification strategy driven by the constraints violated. Numerical results are provided to indicate the efficiency of the methods to generate very good solutions.     

Optimized Runge–Kutta pairs for problems with oscillating solutions

Three types of methods for integrating periodic initial value problems are presented. These methods are (i) phase-fitted, (ii) zero dissipation (iii) both zero dissipative and phase fitted. Some particular modifications of well-known explicit Runge–Kutta pairs of orders five and four are constructed. Numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems.