[Paper Review] The Single Machine Total Weighted Tardiness Problem - Is it (for Metaheuristics) a Solved Problem ?

Martin Josef Geiger|ArXiv.org|Jul 17, 2009

Scheduling and Optimization Algorithms20 references3 citations

TL;DR

This paper investigates whether the single machine total weighted tardiness problem (SMTWTP) is effectively 'solved' for metaheuristics using existing benchmark instances. It evaluates hillclimbing and Variable Neighborhood Search (VNS), finding that simple local search methods achieve near-optimal results on standard instances, suggesting current benchmarks are too easy. The key contribution is identifying that existing instances lack sufficient difficulty, especially for advanced metaheuristics, and proposing new, more challenging benchmarks with larger sizes and specific parameter settings.

ABSTRACT

The article presents a study of rather simple local search heuristics for the single machine total weighted tardiness problem (SMTWTP), namely hillclimbing and Variable Neighborhood Search. In particular, we revisit these approaches for the SMTWTP as there appears to be a lack of appropriate/challenging benchmark instances in this case. The obtained results are impressive indeed. Only few instances remain unsolved, and even those are approximated within 1% of the optimal/best known solutions. Our experiments support the claim that metaheuristics for the SMTWTP are very likely to lead to good results, and that, before refining search strategies, more work must be done with regard to the proposition of benchmark data. Some recommendations for the construction of such data sets are derived from our investigations.

Motivation & Objective

• To assess whether current benchmark instances for the SMTWTP are sufficiently challenging for modern metaheuristics.
• To evaluate the performance of simple local search methods (hillclimbing and VNS) on standard SMTWTP instances.
• To identify characteristics of 'difficult' instances that challenge metaheuristic performance.
• To propose new, more challenging benchmark instances to support future research in SMTWTP metaheuristics.
• To encourage theoretical grounding of empirical findings in SMTWTP metaheuristic design.

Proposed method

• Employs hillclimbing and Variable Neighborhood Search (VNS) with multiple neighborhood operators (exchange and shift) on standard SMTWTP instances.
• Uses the standard benchmark set from Potts and Van Wassenhove (1991), with n=40, 50, and 100 jobs, and parameters RDD ∈ {0.2, 0.4, 0.6, 0.8, 1.0} and TF ∈ {0.2, 0.4, 0.6, 0.8, 1.0}.
• Measures solution quality via relative deviation from best-known or optimal solutions, and analyzes solution diversity using entropy of precedence relations in solution pools.
• Applies entropy analysis to assess solution similarity, computing E_ijk = (-1/log√2) * (ω_ijk/μ) * log(ω_ijk/μ) for precedence relations between jobs.
• Compares results across different TF and RDD values to identify parameter combinations that yield harder instances.
• Analyzes the impact of neighborhood operators by measuring their relative performance in local search.

Experimental results

Research questions

• RQ1 Are current benchmark instances for the SMTWTP sufficiently challenging for modern metaheuristics?
• RQ2 Which neighborhood operators (e.g., exchange vs. shift) yield better performance in local search for SMTWTP?
• RQ3 What parameter settings (TF and RDD) produce the most difficult SMTWTP instances?
• RQ4 How does the diversity of optimal solutions (measured via entropy) relate to problem difficulty?
• RQ5 Why do simple local search methods perform so well on existing benchmarks, and what does this imply for future algorithm development?

Key findings

• Simple hillclimbing with exchange neighborhood operators significantly outperforms shift operators on SMTWTP benchmarks.
• Variable Neighborhood Search (VNS) achieves near-optimal solutions on all tested instances, with only a few unsolved cases, especially for n=100.
• Instances with TF ∈ {0.6, 0.8} and higher RDD values are more challenging, while TF ∈ {0.2, 1.0} leads to easier instances.
• A large number of distinct global optima exist for difficult instances, and their codings show very low entropy (close to 0), indicating high similarity among optimal solutions.
• The entropy of solution pools is extremely low (approaching 0), suggesting that optimal solutions are 'micro-mutations' of each other, even when distinct.
• The current benchmark set is insufficiently challenging, as even simple metaheuristics achieve near-optimal results, implying that recent advances may not be as impactful as claimed.