Multi-objective scheduling on two dedicated processors

Abdelaziz Dammak; Adel Kacem

REVIEW 3 major objections 4 minor 43 references

Multi-objective scheduling on two dedicated processors

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Three genetic-algorithm variants (aggregative, Pareto, and NSGA-II) produce approximate Pareto fronts for the NP-hard problem of scheduling dedicated two-processor tasks under release dates, with lower bounds for makespan, total…

desk verdict Genuinely new tri-objective problem and a valid (if unproved) LBTC lower bound, but the 'effectiveness' claim is underdetermined by ratios to lower bounds of unknown tightness, especially for tardiness. read the letter →

arxiv 1908.04452 v2 pith:INU5NGDE submitted 2019-08-13 cs.DS

Adel Kacem , Abdelaziz Dammak This is my paper · ORCID

classification cs.DS MSC 90B3568M20

keywords schedulingdedicatedprocessorsmulti-objectiveoptimizationgeneticalgorithmNSGA-IIParetofrontlowerboundstotalcompletiontime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Every task in this scheduling problem is tied to one of two dedicated processors, or to both at once, and all tasks have release dates. The paper studies the simultaneous minimization of makespan, total tardiness, and total completion time, an NP-hard problem, and proposes three genetic-algorithm solvers: an aggregative version with Uniform Design weights, a Pareto-based version, and NSGA-II. To judge the quality of the approximations, the authors construct a lower bound for each criterion, including a new bound $\mathrm{LB}_{\mathrm{TC}}$ for total completion time obtained by splitting tasks and applying the shortest-remaining-processing-time (SRPT) theorem to underestimate completion times. The central claim, supported by experiments on instances with 10 and 20 tasks, is that the algorithms are effective: makespan and total completion time frequently land close to their lower bounds, while total tardiness remains the hardest criterion. If that claim holds, these are fast, practical solvers for a problem that arises in computer control and production systems.

What carries the argument

The load-bearing object is a single permutation-coded genetic algorithm whose selection pressure can be switched among three mechanisms. The aggregative mechanism draws weight vectors from a Uniform Design matrix, giving seven fitness functions that are weighted sums of the three normalized criteria. The Pareto mechanism keeps all non-dominated solutions in the population and fills the rest with the best individuals for each criterion. The NSGA-II mechanism merges parents and children, sorts into non-dominated fronts, and breaks ties by crowding distance. Alongside these, the paper installs three quality floors: $\mathrm{LB}_C$ from the two single-processor relaxations; $\mathrm{LB}_{\mathrm{TC}}$, built by splitting both kinds of tasks into half-length pieces, re-weighting them by $1/2$, adding the penalty $\sum_j p_j/4$ on each processor, and applying the SRPT-based theorem [10] to underestimate completion times; and $\mathrm{LB}_{\mathrm{TT}}$, built from the same splitting plus an assignment step solved by the Hungarian algorithm. The lower bounds do the work of turning the heuristic fronts into measurable claims about proximity to optimality.

What would settle it

Take a small instance (say five tasks with release dates), compute the paper's $\mathrm{LB}_{\mathrm{TC}}$ by hand, and compare it with the exact optimal total completion time from exhaustive enumeration. Any instance where $\mathrm{LB}_{\mathrm{TC}}$ exceeds the true optimum refutes the claimed bound and invalidates the $TC/\mathrm{LB}_{\mathrm{TC}}$ quality ratios.

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Extended reading notes

Core claim

The paper's central assertion is that the tri-objective problem $P2|fixj,rj|C_{\max},\sum T_j,\sum C_j$ can be tackled effectively by adapting genetic algorithms, and that the quality of the solutions can be certified by criterion-specific lower bounds. For makespan, the bound $\mathrm{LB}_C$ comes from relaxing the problem into two single-processor sub-problems and taking the maximum of their optimal values. For total completion time, the paper proposes $\mathrm{LB}_{\mathrm{TC}} = \mathrm{Lb}_1 + \mathrm{Lb}_2$: bi-processor tasks are split into two mono-processor subtasks, mono-processor tasks are halved with shifted release dates, a penalty $\sum_j p_j/4$ is added on each processor, and the SRPT theorem supplies per-task completion-time underestimates. On the five instance types tested, the reported ratios $C/\mathrm{LB}_C$ are between about $1.03$ and $1.36$, the ratios $TC/\mathrm{LB}_{\mathrm{TC}}$ mostly between about $1.1$ and $2.2$, and $TT/\mathrm{LB}_{\mathrm{TT}}$ varies much more, especially when release dates are spread widely. The conclusion the paper draws is that the three methods are effective and complementary: each one leads on at least one criterion or instance type, and the fronts are diverse enough to be useful.

Load-bearing premise

The load-bearing premise is that the split-and-penalty relaxation, including the added term $\sum_j p_j/4$, gives a value at or below the true optimal total completion time; if that inequality ever fails, the $TC/\mathrm{LB}_{\mathrm{TC}}$ columns in the result tables are not valid quality measures.

Editorial extensions

If this is right

For instances in the tested range ($n=10$ and $n=20$, five load-balance types), the algorithms produce useful approximate Pareto fronts in under a third of a second, so the approach is fast enough for interactive or repeated use.
The total-completion-time lower bound $\mathrm{LB}_{\mathrm{TC}}$ gives future heuristics a concrete quality floor: any heuristic can be reported as a ratio $TC/\mathrm{LB}_{\mathrm{TC}}$, and a ratio near 1 indicates a schedule near the bound.
No single selection mechanism dominates: the results imply that the right choice among aggregative, Pareto, and NSGA-II depends on the objective being prioritized and on the task-load distribution.
Total tardiness is the criterion with the largest gap to its bound, especially at wide release-date spreads, identifying the place where better algorithms or tighter bounds would pay off.
The hypervolume ratio and the number of non-dominated solutions give a concrete way to rank the three methods on each instance group.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural test not run in the paper is varying the split weight $w$ in $\mathrm{LB}_{\mathrm{TC}}$; the authors fix $w=1/2$, and checking whether other weights strengthen the bound would show whether that choice is essential.
The same splitting idea could be extended to more than two dedicated processors, but the penalty argument would need a new proof, since the two-processor split and the added penalty term do not transfer automatically.
Because the reported values are averages over ten random instances, reporting the spread across those instances would sharpen the effectiveness claim, a step the paper leaves implicit.
The three selection mechanisms could be transferred to other multi-objective scheduling problems with fixed job-to-machine assignments, such as the parallel-machine case the conclusion names as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The genuinely new piece here is the tri-objective problem P2|fixj,rj|Cmax,ΣTj,ΣCj, obtained by adding total completion time to the authors' earlier bi-objective work. That formulation is new, and so is the lower bound LBTC. The reader's worry about LBTC doesn't survive contact with the construction: splitting each mono-processor job into two half-jobs of length pj/2 and later adding pj/4 per original mono-job is a valid relaxation. In any feasible original schedule you can schedule the two halves contiguously, and the weighted objective of the relaxed schedule sits exactly pj/4 below the original ΣC for that job, so the penalty restores the original objective. It is a genuine lower bound. The presentation gap is that the text asserts it in two lines instead of proving it; that's a fixable defect, not a correctness flaw.

What the paper does well: the problem is described cleanly, the three genetic variants (aggregative, Pareto, NSGA-II) are standard but correctly adapted, and the test bed across five instance types and three α levels is reasonable. Makespan ratios are close to the lower bound, which suggests the bound is informative and the algorithms are decent on that criterion. The relative ranking among algorithms is not circular—it is based on the same instances and direct comparisons.

The soft spots are real. First, the 'effectiveness' claim is underdetermined. Every quality number is a ratio to a lower bound of unknown tightness, and total tardiness ratios reach 99.9. If LBTT is tight, the algorithms simply fail on tardiness; if LBTT is loose, the ratio tells you nothing. No exact MIP baseline is provided even though n=10 and 20 are small enough for exact solving. Second, the hypervolume ratio in Eq. (20) is not properly defined: HVLB is never specified, and the text says the reference point comes from initial solutions rather than a lower bound. As written, HVr is not reproducible. Third, the experimental section omits basic details: population size, crossover and mutation rates, number of runs per instance, variance. Ten instances per group with averages and no spread cannot support the fine-grained comparisons in Tables 6–11.

Citation pattern: the authors lean heavily on their own prior papers, but that is acceptable—LBTT comes from earlier work and is correctly credited. No sign of fabricated data.

Verdict: the paper deserves a serious referee but needs major revision. The new problem is legitimate, LBTC is valid, and the heuristics are plausible. A good referee should push for a proof of LBTC, a spelled-out HVr, an exact baseline for small n, and a rewrite of the abstract that doesn't claim uniform 'effectiveness' across all three criteria.

Referee Report

3 major / 4 minor

Summary. The paper studies the tri-objective scheduling problem P2|fixj,rj|Cmax,ΣTj,ΣCj on two dedicated processors. It proposes three genetic algorithm variants: an aggregative method using uniform-design weights, a Pareto-based method, and NSGA-II. It also presents lower bounds for the three criteria (LBC for makespan, LBTC for total completion time, LBTT for total tardiness) and evaluates the algorithms on randomly generated instances with n=10 and n=20 across five problem types and three release-date ranges. The evaluation uses ratios of each objective to its lower bound, the number of non-dominated solutions, and a hypervolume ratio. The abstract and conclusion claim that the proposed algorithms are effective.

Significance. If the computational results were fully supported, the paper would make a useful contribution: a practical multi-objective heuristic study for an NP-hard scheduling problem, together with a new lower bound LBTC for the total completion time criterion. A strength is that the relative ranking of the three algorithms is internally coherent, since all comparisons use the same lower-bound baselines, and the LBTC construction is, on inspection, a valid relaxation: scheduling the two halves of each mono-processor task contiguously in any feasible original schedule yields a relaxed schedule whose weighted objective is ΣC_j − Σp_j/4 for the mono-processor tasks, so adding the p_j/4 penalty restores a valid lower bound. However, the manuscript does not provide this proof, and more importantly, it offers no validation against exact optima or against existing benchmarks. The problem sizes considered (n=10 and n=20) are small enough that exact MIP or branch-and-bound results should be obtainable for at least a subset of instances.

major comments (3)

[§5.2, Tables 6–11] The absolute quality of the solutions is measured only by ratios to lower bounds whose tightness is never established. For example, Table 9 reports TT/LBTT = 99.889 for Type 2, α=1.5 with the aggregative method; this could mean either that the solutions are very poor for the total tardiness criterion or that LBTT is very weak. Similarly, TC/LBTC values above 2.0 in Tables 9–11 could reflect a loose bound rather than a poor front. Since the instances have only n=10 or n=20 jobs, exact methods are feasible: one could solve the single-criterion problems exactly (e.g., a MIP solver or the branch-and-bound of Manaa and Chu [33] for makespan, and analogous exact methods for the other criteria) and report true optimality gaps on at least a subset of instances. Without this, the abstract's statement that the obtained results show 'the effectiveness of the proposed algorithms' is not supported, even though the relative ordering of the three algorithms is largely unaffected by bound tightness.
[Eq. (20), §5.2] The definition of the hypervolume ratio HVr is incomplete and, as written, appears incoherent. The formula HVr = 1 − (HV_LB − HV_Algorithm)/HV_LB requires an object HV_LB, which is never defined in the text. In the usual definition, the hypervolume of a single lower-bound point is zero, and if the lower bound is instead used as the reference point, the denominator vanishes. The text earlier states that the reference point is the worst value in each criterion among the initial solutions, which is inconsistent with Eq. (20). All HVr-based comparisons in Tables 6–11 and Figures 4 and 6 depend on this quantity, so the definition must be made precise, with a clear statement of how HV_LB is computed and why the resulting ratio measures what the paper claims.
[§3.4] The lower bound LBTC is asserted without proof. The phrase 'with Σ_{j∈P1} p_j/4 is a penalty to be added according to Webster formula' is not a justification, and no reference to a 'Webster formula' is provided. The construction is in fact valid: for any feasible original schedule, scheduling each split mono-processor task's two halves contiguously yields a feasible relaxed schedule whose weighted objective equals ΣC_j − Σ_{j∈P1} p_j/4 on processor P1 (and analogously on P2), so adding the penalty over the mono-processor tasks gives a lower bound on the original total completion time. This argument should be written out explicitly in the paper, along with a clarification that the penalty applies only to mono-processor tasks and not to the bi-processor sub-tasks. As it stands, the validity of LBTC, and hence all TC/LBTC ratios in Tables 6–11, is not established in the manuscript itself.

minor comments (4)

[§5.1, Table 5] The text reads '0, 300 seconds' where '0.300' is clearly intended; the comma should be a decimal point.
[§5.2, discussion of Tables 6–8] The sentence 'For the problem of Type 2 with (α = 1.5, α = 1), Type 3 with (α = 1.5, α = 1) and Type 4 with (α = 1.5, α = 1)' is garbled; it appears that separate α values are meant for each problem type, and the repeated pairs should be corrected.
[References] Several references contain typographical errors, for example [10] 'Naval Researech Logistics' should be 'Naval Research Logistics', and [15] is titled 'Some New Uniform Design' but appears to be an Emmons scheduling paper; these should be checked and corrected.
[§4.2] The text says the aggregative method 'consists in generating weights for each sequence', but the uniform-design weights are generated once independently of the sequences; the wording should be clarified to avoid implying per-individual weight generation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the algorithms and lower bounds are independent; the effectiveness claim is under-supported but not circular.

full rationale

The derivation chain is not circular. The three genetic algorithms produce schedules from random initial populations via crossover, mutation, and selection; no parameter is fitted to the reported quality metrics or to the lower bounds. The lower-bound ratios are computed after the fact as evaluation floors. The new LBTC in Section 3.4 is not an input-renaming: for any feasible original schedule, splitting each mono-processor job into two p/2 sub-jobs with shifted release dates and weight 1/2 gives a relaxed per-processor schedule whose weighted completion sum is sum(C_j) minus sum over mono jobs of p_j/4; adding the p_j/4 penalty restores the original objective contribution. Hence LBTC is a genuine relaxation lower bound, and its use as a denominator is not a self-justifying fit. The LBTT lower bound is imported from the authors' own prior work [24], but it is used as a published evaluation floor rather than as a step that defines the algorithms' outputs; no uniqueness theorem or ansatz is smuggled in. The paper does assert LBTC's validity without a full proof, which is an omitted-proof rigor gap rather than circularity. The real weakness is evidential: ratios such as C/LBC, TT/LBTT, and TC/LBTC are relative to bounds of unmeasured tightness, and no comparison to exact optima is provided even for n=10-20, so the blanket 'effectiveness' claim is underdetermined. That is a validation weakness, not a circular derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its main postulates are the validity of the LBTC relaxation and the unstated Webster penalty formula, both load-bearing for the quality assessment. The algorithm parameters are tuned but not reported, which is an additional hidden degree of freedom.

free parameters (2)

weight w in LBTC = 0.5
Section 3.4 sets w = 1/2 to split bi-processor tasks between the two processors; no justification is given for the choice, and it affects the strength and validity of the total completion time lower bound.
genetic algorithm parameters (population size, crossover/mutation rates) = not reported
Section 5 notes 'some preliminary tests have motivated our choices' but never lists the parameter values; these are tuned quantities required to reproduce the experiments.

assumptions (4)

standard math Chu's theorem: the SRPT rule yields pointwise minimum completion times for the single-machine preemptive problem 1|rj,pre|ΣCj
Invoked in Sections 3.4 and 3.5 to compute lower bounds on completion times; it is cited as Theorem 1 from Chu [10].
ad hoc to paper Splitting bi-processor tasks into weighted mono-processor sub-tasks yields a valid relaxation (lower bound)
Sections 3.4 and 3.5 assert that partitioning P12 tasks into two weighted independent problems on each processor gives lower bounds on the original total completion time and total tardiness, without proof.
ad hoc to paper The 'Webster formula' penalty of p_j/4 correctly converts the split-task completion time bound back to the original task completion time bound
Section 3.4 adds the penalty with 'a penalty to be added according to Webster formula' but provides no statement, proof, or citation of the formula.
domain assumption Instance generation distributions (processing times uniform in {0,...,50}, release dates in {0,...,L}, due dates in {r_j+p_j,...,r_j+p_j+L})
Section 5 defines the random instance model; results are only meaningful for this model and are averages over 10 instances per group.

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Pith. "Pith review of Multi-objective scheduling on two dedicated processors." pith.science (2026). https://pith.science/paper/INU5NGDE

@misc{pith2026190804452,
  author       = {Pith},
  title        = {Pith review of: Multi-objective scheduling on two dedicated processors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INU5NGDE}},
  note         = {Machine review of arXiv:1908.04452}
}

read the original abstract

We study a multi-objective scheduling problem on two dedicated processors. The aim is to minimize simultaneously the makespan, the total tardiness and the total completion time. This NP-hard problem requires the use of well-adapted methods. For this, we adapted genetic algorithms to multi-objective case. Three methods are presented to solve this problem. The first is aggregative, the second is Pareto and the third is non-dominated sorting genetic algorithm II (NSGA-II). We proposed some adapted lower bounds for each criterion to evaluate the quality of the found results on a large set of instances. Indeed, these bounds also make it possible to determine the dominance of one algorithm over another based on the different results found by each of them. We used two metrics to measure the quality of the Pareto front: the hypervolume indicator (HV) and the number of solutions in the optimal front (ND). The obtained results show the effectiveness of the proposed algorithms.

Figures

Figures reproduced from arXiv: 1908.04452 by the authors.

**Figure 2.** Example hypervolume in three dimensions (reproduced from Bradstreet [ [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Average number of non-dominated solutions for n =10 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Hypervolume ratios for n =10 (NSGA-II and Pareto) are almost identical. The aggregative selection method is less effective for problems of T ype1 with (α = 1.5), T ype2 with (α = 1.5), T ype3 with (α = 1.5) and T ype5 with (α = 1.5). The numerical results of the total …

**Figure 5.** Figure 5: Average number of non-dominated solutions for n =20 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Hypervolume ratios for n =20 the solutions found by NSGA-II algorithm, Pareto and aggregative methods proposed for solving the multi-objective problem. To generate the weight for the aggregative approach, we used the method of Uniform Design (proposed by Leung and Wang…

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