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# Solving TWT Problems Math

Categories: MathProblems

### 2.1. Objective to Minimize TWT

Monch (2008) presented an efficient method to solve unrelated parallel machine total weighted tardiness (TWT) scheduling problems. He applied an ant colony optimization (ACO) approach as a heuristic to solve this NP hard problem. A colony of artificial ants is used to construct iteratively solutions of the scheduling problem using artificial pheromone trails and heuristic information. For the computation of the heuristic information, he used the Apparent Tardiness Cost (ATC) dispatching rule. He additionally improved the TWT value by applying a decomposition heuristic that solves a sequence of smaller scheduling problems optimally.

A multi-criteria scheduling problem with the goal of minimizing the maximum completion time, so called makespan as well as earliness and tardiness penalties simultaneously on unrelated parallel machines is studied in the research conducted by Vahid et al., (2014) in which jobs are sequence dependent setup times (SDST) and due dates are distinct.

### 2.2. Dispatching rules

Monch (2008) applied the ATC dispatching rule to solve the scheduling problem. The ATC heuristic is a composite dispatching rule that tends to schedule jobs with higher priority indices first, given by the combination of two other rules: MS, Minimum slack (time until due date) and WSPT, Weighted Shortest Processing Time.

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It is well known that ATC type dispatching rules are very efficient heuristics with respect to obtain small values for the performance measure TWT.

### 2.3. Metaheuristics

Monch (2008) presents first a very simple heuristic that is based on the ATC dispatching rule. The second heuristic uses a decision theory approach to improve the first heuristic.

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The third heuristic is based on an ACO approach. The necessary heuristic information is determined by using techniques from the second heuristic. The ACO approach is hybridized with a decomposition heuristic that can be used to improve the TWT value of the schedules. In the paper published by Rabadi et al., (2006), Meta-RaPS construction and improvement heuristic was used to find all optimal solutions. Meta-RaPS (Meta-heuristic for Randomized Priority Search) is a master strategy that uses both construction and improvement heuristics to generate high quality solutions. In Meta-RaPS, as with other meta-heuristics, randomness is a mechanism used to prevent the algorithm from getting trapped into local optima. A heuristic called Initial Sequence based on Earliness- Tardiness criterion on Parallel machines (ISETP) was presented by Vahid et al., (2014) to acquire the jobs sequence on parallel machines regarding minimizing total weighted tardiness and earliness.

### 2.4. Mathematical models and exact algorithms

A Mixed Integer Program (MIP) was formulated to find optimal solutions for the problem at hand by Rabadi et al., (2006). During Meta-RaPS execution, the number of iterations determines the number of feasible solutions constructed. In general, a construction heuristic builds a solution by systematically adding feasible jobs to the current schedule. The job with the best priority value is added to the current schedule. the best solution from all iterations is reported. Mixed Integer Programming (MIP) mathematical model was proposed by Vahid et al., (2014) to minimize total earliness and tardiness in addition to makespan, simultaneously, on unrelated parallel machines considering SDST assumption. In this context, a set of N jobs denoted by 1, 2,.., n has to be processed on a set of M unrelated parallel machines denoted by M1, M2, ·, Mm. The setups are assumed to be simultaneously machine and job dependent, consequently a setup time Skim is incurred, when a given job k is processed immediately after job i on machine j.