3/2-approximation algorithm for a single machine scheduling problem

The problem of minimizing the maximum delivery times while scheduling tasks on a single processor is a classical combinatorial optimization problem. Each task ui must be processed without interruption for t(ui) time units on the machine, which can process at most one task at time. Each task uw; has a release time r(ui), when the task is ready for processing, and a delivery time g(ui). Its delivery begins immediately after processing has been completed. The objective is to minimize the time, by which all jobs are delivered. In the Graham notation this problem is denoted by 1|rj,qi|Cmax, it has many applications and it is NP-hard in a strong sense. The problem is useful in solving owshop and jobshop scheduling problems. The goal of this article is to propose a new 3/2-approximation algorithm, which runs in O(n log n) times for scheduling problem 1|rj.qi|Cmax. An example is provided which shows that the bound of 3/2 is accurate. To compare the effectiveness of proposed algorithms, random generated problems of up to 5000 tasks were tested.

Effective Heuristic Algorithms Solving the Jobshop Scheduling Problem with Release Dates

Manufacturing industry reflects a country’s productivity level and occupies an important share in the national economy of developed countries in the world. Jobshop scheduling (JSS) model originates from modern manufacturing, in which a number of tasks are executed individually on a series of processors following their preset processing routes. This study addresses a JSS problem with the criterion of minimizing total quadratic completion time (TQCT), where each task is available at its own release date. Constructive heuristic and meta-heuristic algorithms are introduced to handle different scale instances as the problem is NP-hard. Given that the shortest-processing-time (SPT)-based heuristic and dense scheduling rule are effective for the TQCT criterion and the JSS problem, respectively, an innovative heuristic combining SPT and dense scheduling rule is put forward to provide feasible solutions for large-scale instances. A preemptive single-machine-based lower bound is designed to estimate the optimal schedule and reveal the performance of the heuristic. Differential evolution algorithm is a global search algorithm on the basis of population, which has the advantages of simple structure, strong robustness, fast convergence, and easy implementation. Therefore, a hybrid discrete differential evolution (HDDE) algorithm is presented to obtain near-optimal solutions for medium-scale instances, where multi-point insertion and a local search scheme enhance the quality of final solutions. The superiority of the HDDE algorithm is highlighted by contrast experiments with population-based meta-heuristics, i.e., ant colony optimization (ACO), particle swarm optimization (PSO) and genetic algorithm (GA). Average gaps 45.62, 63.38 and 188.46 between HDDE with ACO, PSO and GA, respectively, are demonstrated by the numerical results with benchmark data, which reveals the domination of the proposed HDDE algorithm.