Max-Plus Linear Representation Research Articles

Two-level priority scheduling framework in a max-plus linear representation

A common type of scheduling policy includes first-in-first-out (FIFO) and earliest-outset bases. Among many approaches to this, max-plus linear representation is beneficial for event-driven discret...

Forward-compatible framework with critical-chain project management using a max-plus linear representation

Focusing on projects with uncertain task-duration times under limited resources, we develop a scheduling framework using a max-plus linear (MPL) representation. The base methodology is referred to as critical-chain project management (CCPM), designed to achieve a short lead time and observe the estimated due date. Our recent achievement called CCPM-MPL is an approach to handle the CCPM framework using an MPL representation; therein, no method for resolving resource contention is provided. We thus improve the existing CCPM-MPL framework for which it becomes compliant and forward compatible with the original CCPM. The forward compatibility yields additional benefits; it is able to list and compare all potential schedules with the same processing sequence. It can also handle multiple-input and/or multiple-output projects in a unified fashion. The former would be useful in instances when we are allowed to deliver multiple products separately, the latter when one or more resource is unavailable until a certain time.

Resolution of Resource Contentions in the CCPM-MPL Using Simulated Annealing and Genetic Algorithm

This research aims to plan a “good-enough” schedule with leveling of resource contentions. We use the existing critical chain project management-max-plus linear framework. Critical chain project management is known as a technique used to both shorten the makespan and observe the due date under limited resources; the max-plus linear representation is an approach for modeling discrete event systems as production systems and project scheduling. If a contention arises within a single resource, we must resolve it by appending precedence relations. Thus, the resolution framework is reduced to a combinatorial optimization. If we aim to obtain the exact optimal solution, the maximum computation time is longer than 10 hours for 20 jobs. We thus experiment with Simulated Annealing (SA) and Genetic Algorithm (GA) to obtain an approximate solution within a practical time. Comparing the two methods, the former was beneficial in computation time, whereas the latter was better in terms of the performance of the solution. If the number of tasks is 50, the solution using SA is better than that using GA.

Critical Chain Project Management Based on a Max-Plus Linear Representation for Determining Time Buffers in Multiple Projects

We propose a framework for describing the Critical Chain Project Management (CCPM) method based on a Max-Plus Linear (MPL) representation for multiple projects. The framework takes into account time buffers in both a single and multi-project environments which play a role for controlling undesirable state change and protecting the completion time for the entire system. The appropriate position and size of these buffers are determined for a large scale multi-project system. Furthermore, the MPL system with the CCPM method applied is redefined to easily comprehend the structure of the system. Finally, we confirm the proposed framework through a numerical example.

Resolution of Time and Worker Conflicts for a Single Project in a Max-Plus Linear Representation

This research develops a framework for resolving time and worker conflicts in the Critical Chain Project Management (CCPM) method, expressed in the form of a Max-Plus Linear (MPL) system. Our previous work proposed a method for resolving time conflicts. However, in practical cases, both time and worker conflicts may occur. Hence, we propose a method for resolving both time and worker conflicts for a single project. We first consider how to detect a resource conflict. Then, we define an adjacency matrix to resolve the detected conflicts. Using the proposed method, we confirm that the resource conflict can be resolved through a numerical example.