A graph-theoretic approach to interval scheduling on dedicated unrelated parallel machines

Abstract

We study the problem of scheduling n non-preemptive jobs on m unrelated parallel machines. Each machine can process a specified subset of the jobs. If a job is assigned to a machine, then it occupies a specified time interval on the machine. Each assignment of a job to a machine yields a value. The objective is to find a subset of the jobs and their feasible assignments to the machines such that the total value is maximized. The problem is NP-hard in the strong sense. We reduce the problem to finding a maximum weight clique in a graph and survey available solution methods. Furthermore, based on the peculiar properties of graphs, we propose an exact solution algorithm and five heuristics. We conduct computer experiments to assess the performance of our and other existing heuristics. The computational results show that our heuristics outperform the existing heuristics.

Mobility Offer Allocations in Corporate Settings

“…Now, the aim of the tactical fixed interval scheduling problem is to minimize the number of needed machines given that all jobs have to be scheduled ( [15]). The goal of the operational fixed interval scheduling problem is to maximize the weight of the jobs that can be scheduled on a given set of machines ( [16]). In our application, machines correspond to vehicles, jobs to mobility demands, and the possible assignments of a job to a machine at a specific interval to mobility offers.…”

“…In our application, machines correspond to vehicles, jobs to mobility demands, and the possible assignments of a job to a machine at a specific interval to mobility offers. [16] present heuristics for a special case of the operational fixed interval scheduling problem, to which we compare our developed algorithms in Section 6.…”

“…The mobility offer allocation problem is equivalent to the interval scheduling on dedicated unrelated parallel machines (ISDU) introduced in [16]. A mapping between instances of ISDU and instances of MOAP is given by identifying machines with vehicles, jobs with mobility demands, and intervals with mobility offers.…”

The Application of Graph Theory for Vulnerability Assessment in Service Triads

“…The literature has not yet structured the potential vulnerabilities concerned with relational properties between the members of supply chain triads and has not provided a structured analytical approach towards the assessment of such vulnerabilities. We noticed that this issue could be effectively resolved using the modelling apparatus of graph theory, a mathematical approach widely adopted in operations research (e.g., Ahmadi-Javid, Ardestani-Jaafari, Foulds, Hojabri, & Farahani, 2015;Li, Kilgour, & Hipel, 2005;Ng, Cheng, Bandalouski, Kovalyov, & Lam, 2014) The aim of this study is two-fold: (i) using the toolset of graph theory, to model different cross-organizational pathways according to which risks can emerge and propagate in service triads seen as an elementary form of service supply networks; and (ii) to provide a method for vulnerability assessment of the suggested graph models of service triads, taking into consideration the typology and direction of the cross-organizational relationships within graph models of supply chain triads.…”

Multi-Objective Energy-Efficient Interval Scheduling in Hybrid Flow Shop Using Imperialist Competitive Algorithm

“…The lower bound and upper bound of interval are only required to indicate uncertain processing conditions, decision-maker prefers using interval number to indicate his expected performance and the obtained interval results can be understood easily, so it is a good choice to apply interval number for uncertainty. In the past decade, there are some works related to interval scheduling in parallel machines [30], flow shop [37] and job shop [27], [38], [39].…”