Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

In the paper we consider the problem of scheduling $n$ identical jobs on 4 uniform machines with speeds $s_1 \geq s_2 \geq s_3 \geq s_4,$ respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree $\Delta$, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if $s_1=s_2=s_3$. If, however, $\Delta \leq 4$ and $s_1 \geq 12 s_2$, $s_2=s_3=s_4$, then the problem can be solved to optimality in time $O(n^{1.5})$. The same algorithm returns a solution of value at most 2 times optimal provided that $s_1 \geq 2s_2$. Finally, we study the case $s_1 \geq s_2 \geq s_3=s_4$ and give an $O(n^{1.5})$-time $32/15$-approximation algorithm in all such situations

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are

SOLVING PROCESS PLANNING AND SCHEDULING PROBLEMS USING THE CONCEPT OF MAXIMUM WEIGHTED INDEPENDENT SET

Process planning and scheduling (PPS) is an essential and practical topic but a very intractable problem in manufacturing systems. Many research studies use iterative methods to solve such problems; however, they cannot achieve satisfactory results in both quality and computational speed. Other studies formulate scheduling problems as a graph coloring problem (GCP) or its extensions, but these formulations are limited to certain types of scheduling problems. In this dissertation, we propose a novel approach to formulate a general type of the PPS problem with resource allocation and process planning integrated towards a typical objective, minimizing the makespan. The PPS problem is formulated into an undirected weighted conflicting graph, where nodes represent operations and their resources; edges represent constraints, and weight factors are guidelines for the node selection at each time slot. Then, the Maximum Weighted Independent Set (MWIS) problem, which considers a graph with weights assigned to nodes and seeks to discover the “heaviest” independent set, that is, a set of nodes with maximum total weight so that no two nodes in the set are connected by an edge, can be solved to find the best set of operations with their desired resources for each discrete time slot. This proposed approach solves the PPS problem directly (a direct method in computational mathematics context). We establish that the proposed approach always returns a feasible optimum or near-optimum solution to the PPS problem. The performance of the proposed approach for the PPS problem depends on the accuracy and computational speed of solving the MWIS problem. We propose a divide-and-conquer algorithm structure with relatively low complexity for solving the MWIS problem. An exact MWIS algorithm and an All Maximal Independent Set Listing (AMISL) algorithm are developed based on this algorithm structure. The proposed algorithm structure can also be used to compose the exact MWIS algorithm with existing approximation MWIS algorithms. This is an effective way to improve the accuracy of existing approximation MWIS algorithms or improve the computational speed of the exact MWIS algorithm. All eight algorithms for the MWIS problem, the exact MWIS algorithm, the AMISL algorithm, two approximation algorithms from the literature, and four composed algorithms, are tested on the test instances based on the PPS application environment. The different configurations of the proposed approach for solving the PPS problem are tested on a real-world PPS example and further designated test instances to evaluate the scalability, accuracy, and robustness