Rafał Różycki

Simulated Annealing for Multi-Mode Resource-Constrained Project Scheduling

In this paper the resource-constrained project scheduling problem with multiple execution modes for each activity and the makespan as the minimization criterion is considered. A simulated annealing approach to solve this problem is presented. The feasible solution representation is based on a precedence feasible list of activities and a mode assignment. A comprehensive computational experiment is described, performed on a set of standard test problems constructed by the ProGen project generator. The results are analyzed and discussed and some final remarks are included.

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.

Power-aware scheduling of preemptable jobs on identical parallel processors to meet deadlines

This paper deals with a power-aware scheduling of preemptable independent jobs on identical parallel processors where ready time for each job is given and its completion time has to meet a given deadline. Jobs are described by (different) continuous, strictly concave functions relating their processing speeds at a time to the amount of power allotted at the moment. Power is a continuous, doubly constrained resource, i.e. both: its availability at each time instant and consumption over scheduling horizon are constrained. A methodology based on properties of minimum-length schedules is utilized to determine the existence of a feasible schedule for given amounts of energy and power. The question about minimum levels of power and energy ensuring the existence of a feasible schedule for a given set of jobs is also studied.

Solving the discrete-continuous project scheduling problem via its discretization

Abstract. In this paper a discrete-continuous project scheduling problem is considered. In this problem activities simultaneously require discrete and continuous resources. The processing rate of each activity depends on the amount of the continuous resource allotted to this activity at a time. All the resources are renewable ones. The activities are nonpreemtable and the objective is to minimize the makespan. Discretization of this problem leading to a classical (i.e. discrete) project scheduling problem in the multi-mode version is presented. A simulated annealing (SA) approach to solving this problem is described and tested computationally in two versions: with and without finding an optimal continuous resource allocation for the final schedule. In the former case a nonlinear solver is used for solving a corresponding convex programming problem. The results are compared with the results obtained using SA for the discrete-continuous project scheduling problem where the nonlinear solver is used for exact solving the continuous part in each iteration. The results of a computational experiment are analyzed and some conclusions are included.

Power-aware scheduling of preemptable jobs on identical parallel processors to minimize makespan

This paper deals with power-aware scheduling of preemptable jobs on identical parallel processors to minimize schedule length when jobs are described by continuous, strictly concave functions relating their processing speed at time t to the amount of power allotted at the moment. Power is a continuous, doubly constrained resource, i.e. both: its availability at time t and consumption over scheduling horizon are constrained. Precedence constraints among jobs are represented by a task-on-arc graph. A methodology based on properties of optimal schedules is presented for solving the problem optimally for a given ordering of nodes in the graph. Heuristics for finding an ordering which leads to possibly short schedules are proposed and examined experimentally.

Solving a power-aware scheduling problem by grouping jobs with the same processing characteristic

In the paper a power-aware problem of scheduling preemptable jobs on parallel identical machines to minimize the schedule length is considered. Exact approaches utilizing the idea of grouping jobs are presented and compared from the viewpoint of the size of an appropriate non-linear programming problem.

Discrete-continuous scheduling to minimize the makespan for power processing rates of jobs

Abstract A class of discrete-continuous scheduling problems is considered when each nonpreemptable, independent job simultaneously requires for its processing a machine from a set of m identical, parallel machines and an amount, arbitrary within interval [0,1], of a continuously divisible, renewable resource available in amount 1. Job processing rates are described by power functions of the resource amount allotted at a time. Some properties of optimal schedules are proved which allow to find such schedules analytically in some cases and to construct efficient heuristics in the general case. Results of a computational experiment are described.

An Almost Optimal Heuristic for Preemptive C max Scheduling of Dependent Tasks on Parallel Identical Machines

We consider the problem of scheduling preemptable, dependent tasks on parallel, identical machines to minimize the makespan. The computational complexity of this problem remains open if the number of machines is fixed and larger than 2. The aim of this paper is to compare two heuristic algorithms on a basis of a computational experiment. The solutions generated by the heuristics are compared with optimal solutions obtained by a branch-and-bound algorithm. Computational results show that the heuristic based on node ordering finds optimal schedules for 99.9% of instances with the maximum relative deviation from optimum of 4.8%.

Heuristic solving some discrete-continuous project scheduling problems with discounted cash flows

In this paper a discrete-continuous project scheduling problem with discounted cash flows is considered. Each activity requires for its processing discrete resources and an amount of a continuous resource. Processing rate of an activity is a concave function of the amount of the continuous resource allotted to this activity at a time. A positive cash flow is associated with the completion of each activity. The objective is the maximization of the net present value of all cash flows of the project. Two heuristics for allocating the continuous resource are proposed, and compared to optimum on a basis of a computational experiment. Some conclusions as well as directions for future research are given.

A Genetic Algorithm for Some Discrete-Continuous Scheduling Problems

In this paper a Genetic Algorithm (GA) for solving some discrete-continuous scheduling problems is proposed. In this algorithm, each genotype represents a solution which is derived directly from the idea of a feasible sequence rather than being coded as a binary string. Applying suitable genetic operators allows to conduct the searching process in a feasible region only.