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REVIEW 3 major objections 5 minor 39 references

On Branch-and-Price for Project Scheduling

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper establishes why branch-and-price is ineffective for most project scheduling problems: single-mode pricing problems have the integrality property, so the master LP is as weak as the compact PDDT LP, while column generation is far…

desk verdict A useful, honest negative-result paper on branch-and-price for project scheduling, but the key bound-transfer claim rests on an unproved polytope identity. read the letter →

arxiv 2501.04563 v1 pith:JNAGJX6F submitted 2025-01-08 math.OC

Maximilian Kolter , Martin Grunow , Rainer Kolisch This is my paper · ORCID
classification math.OC MSC 90C1090C2790B35
keywords branch-and-pricecolumngenerationDantzig-Wolfedecompositionresource-constrainedprojectschedulingintegralitypropertylowerboundsdegeneracyRCPSP
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

Branch-and-price has been proposed for many resource-constrained project scheduling problems, but existing studies never show a clear win over solving the compact integer program directly. This paper explains why: for single-mode problems, the Dantzig-Wolfe pricing problem has the integrality property, so the master LP relaxation is exactly as strong as the compact PDDT relaxation, and solving it via column generation is far slower. For multi-mode problems the reformulation strengthens the bound only marginally, and the increase in runtime dominates. The paper also shows that branching on start times preserves the bound equality, while branching on resource demands can strengthen bounds but makes the pricing problem NP-hard. The conclusion is a recommendation against branch-and-price for classical RCPSP variants and a call for alternative decompositions with hard-but-efficiently-solvable pricing problems.

What carries the argument

The central object is the integrality property of the pricing problem: an integer program has this property when its linear relaxation has only integral vertices, so the convex hull of feasible integer solutions equals the LP polyhedron. Theorem 2 (attributed to Geoffrion's Lagrangian-relaxation analysis) then implies that the linear relaxation of the master problem achieves exactly the same bound as the compact formulation. The paper combines this with Proposition 1 on primal degeneracy of the restricted master problem, showing that many zero-valued basic variables mislead column generation and slow convergence. For the multi-mode case, the counterexample in Appendix A shows the integrality property fails, but the resulting bound gains are small. The branching analysis then shows start-time branching preserves the integrality property, while resource-demand branching adds one knapsack-style constraint that makes the pricing problem NP-hard.

What would settle it

Find a single-mode RCPSP instance where the linear relaxation of the master problem, solved exactly by column generation, has a strictly larger optimal value than the linear relaxation of the compact PDDT formulation; such an instance would contradict Propositions 2 and 5 and Theorem 2 as applied here. A complementary check is to test whether the aggregated-precedence pricing polytope ever has a fractional vertex that is not in the disaggregated PDDT polyhedron.

Watch

Extended reading notes

Core claim

For classical single-mode project scheduling, the Dantzig-Wolfe decomposition yields a pricing problem that can be solved as a linear program (Proposition 2), yet the linear relaxation of the master problem is as weak as the relaxation of the compact PDDT formulation (Theorem 2 and Proposition 2). Adding standard branching on start times does not change this: branch-and-price and branch-and-bound on the compact formulation generate the same branching trees (Proposition 5). The master problem is highly degenerate, so column generation needs many iterations and long runtimes for no bound gain. For multi-mode pricing problems, the integrality property fails, but computational experiments on MMLIB instances show only marginal lower-bound improvements that are offset by runtimes averaging 26 times longer than the compact LP; the compact formulation with solver-generated cuts matches or beats the master bound in a fraction of the time. The paper concludes that branch-and-price is not competitive for most project scheduling problems, with the possible exceptions of preemptive variants and dedicated-resource variants, where NP-hard pricing problems can yield genuinely stronger relaxations.

Load-bearing premise

The argument that literature pricing problems with aggregated precedence constraints give no bound gain relies on the assumption that the convex hull of the aggregated pricing polytope equals the feasible region of the disaggregated PDDT relaxation; if this fails for some instance class, the master bound could differ.

Editorial extensions

If this is right

  • Lower-bound computations for single-mode RCPSP that rely on Dantzig-Wolfe column generation do not deliver stronger bounds than solving the compact PDDT relaxation with a commercial solver.
  • A branch-and-price algorithm for single-mode problems that branches on start times will explore exactly the same branching tree as a standard branch-and-bound on the compact formulation, so it cannot be competitive unless its node solves are faster, which the runtime experiments rule out.
  • For multi-mode problems, the small bound improvement of the master LP can be recovered or even exceeded by adding cutting planes to the compact formulation, at a fraction of the runtime.
  • The only column-generation approaches that can promise stronger relaxations are those whose pricing problems are genuinely hard, such as preemptive antichain formulations or dedicated-resource decompositions; even there, the NP-hard pricing becomes the bottleneck.
  • Researchers should focus on alternative Dantzig-Wolfe decompositions whose pricing problems are hard but efficiently solvable by tailored algorithms.

Reading between the lines

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

  • The integrality-property argument likely extends to other scheduling or packing formulations whose pricing polytopes are known to be integral, meaning branch-and-price should not be expected to improve bounds in those settings either.
  • The degeneracy diagnosis suggests that dual stabilization could mitigate the runtime penalty of column generation for project scheduling, but since the underlying bound is the weak point in the single-mode case, stabilization alone would not make branch-and-price competitive.
  • The paper's negative result is scoped to the activity-subset decomposition; decompositions that change the pricing problem's structure, such as time-indexed or antichain-based formulations, are the direction most likely to escape the conclusion.
  • A testable extension would be to apply the same decomposition analysis to project scheduling variants with time-dependent resource capacities or calendars, where the pricing problems lose the integrality property and might behave like the multi-mode case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies Dantzig-Wolfe reformulations and branch-and-price for non-preemptive resource-constrained project scheduling. It proposes a general multi-mode multi-project model (MRCMPSP-GO) and a master/pricing framework that subsumes several decompositions from the literature. The main analytical claims are: the restricted master is prone to primal degeneracy (Proposition 1); single-mode resource-unconstrained pricing has the integrality property (Proposition 2), so by Theorem 2 the master relaxation is no stronger than the compact PDDT relaxation; multi-mode pricing is not integral (Proposition 3); resource-constrained pricing is not integral (Proposition 4); branching on start times preserves the bound equality (Proposition 5), while branching on resource demands can strengthen the bound (Proposition 6) but makes pricing NP-hard. Computational experiments on 480 PSPLIB j30 instances and 200 MMLIB instances quantify the runtime disadvantage of column generation. The paper concludes that branch-and-price is not competitive for classical single- and multi-mode RCPSP variants, with possible exceptions for dedicated-resource and preemptive settings.

Significance. The result, if established rigorously, is significant because it explains the poor performance of column generation for classical RCPSP and warns against routine Dantzig-Wolfe decomposition for this problem class. The paper's strengths are its systematic survey, the unifying MRCMPSP-GO model, the clean link between the integrality property of the pricing problem and the master bound (Theorem 2 with Proposition 2), and the direct PSPLIB/MMLIB comparisons. The single-mode equality z_LMP = z_PDDT-LP is a crisp, parameter-free theoretical statement, and the experiments confirm it. No free parameters are fitted anywhere in the analysis. However, two load-bearing arguments need repair: the unproved polytope identity in Section 6.1 and the logically inconsistent proof of Proposition 6 in Section 7.2. The multi-mode experiments also solve a different pricing problem than the one analyzed theoretically. The paper does not ship machine-checked proofs or code, and a reproducibility statement with instance selection and column-generation details would strengthen the experimental part.

major comments (3)
  1. [Section 6.1] Section 6.1 states, without proof or citation, that 'the convex hull of the pricing problem with aggregated precedence constraints coincides with the feasible region of the linear relaxation of the stronger formulation with disaggregated precedence constraints.' This identity is the bridge that transfers the no-bound-improvement result from the disaggregated pricing problem min{(18):(19)-(21)} to the actual algorithms of Deckro et al. (1991), Drexl and Kimms (2001), and Volland et al. (2017), which use aggregated precedence in pricing. Without a proof, the central conclusion in Section 8 that these branch-and-price approaches offer no bound advantage is not supported. The identity is in fact true and follows from Proposition 2 together with the observation that, for single-mode binary schedules, the integer feasible sets of the two precedence formulations coincide; the authors should include this argument or provide a precise citation.
  2. [Section 7.2, Proposition 6] The proof of Proposition 6 is logically inconsistent. It says that NP-hardness of the pricing problem 'implies' that 'the feasible region of the linear relaxation of the compact formulation LP is a proper subset of the linear relaxed master problem LMP,' and then concludes 'This implies the existence of a point x* in LP and x* not in LMP.' These two statements are contradictory: if LP were a subset of LMP, a point in LP could not fail to be in LMP. The intended inclusion is that LMP is a proper subset of LP. Moreover, NP-hardness alone does not establish the required separation; one must exhibit a fractional vertex of the pricing polytope and an objective for which that fractional point is the unique LP optimum. Please rewrite the proof with a concrete instance or an explicit construction, and correct the inclusion direction.
  3. [Section 6.2, Table 8] The multi-mode LMP is computed using aggregated precedence constraints in the pricing problem as a speed-up, while Proposition 3 and the surrounding theoretical analysis concern the disaggregated pricing problem (19)-(21). The reported 0.42% average improvement over PDDT-LP is therefore not a direct measurement of the bound gain of the theoretically analyzed Dantzig-Wolfe formulation. The practical conclusion that column generation is uncompetitive may survive this issue, but the claim that the stronger relaxations 'translate only into marginal lower-bound improvements' should be either tested with the disaggregated pricing problem or explicitly qualified as applying to the aggregated-price LMP.
minor comments (5)
  1. [Section 6.1] The statement that the violated disaggregated precedence constraints 'are likely to be generated by commercial solvers automatically' is speculative and not directly supported by the experiments; please soften or verify.
  2. [Appendix B] In constraint (B.3), the sum over all times other than t' is written with j as the summation index ('sum over j in T\{t'}'); it should be sum over t in T\{t'}.
  3. [Section 7.2, Eq. (25)] The right-hand side of the up-branch contains an apparent typo, 'rkntr*knt'; it should likely be just 'r*knt'.
  4. [References] The reference to Geoffrion (1974) lists the journal as 'Mathemtical Programming' instead of 'Mathematical Programming'.
  5. [Section 5, computational setup] The experimental section does not describe how the initial column set for the LMP was generated, how the column-generation runs were terminated, or how the 200 MMLIB instances were sampled (no seed or list is given); please add a reproducibility statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central negative claim about branch-and-price for project scheduling is derived from external theorems and independent experiments, with no fitted input renamed as a prediction.

full rationale

The derivation chain is self-contained and non-circular. The central bound-equivalence results, Theorem 2 and Proposition 2, are imported from external prior work, namely Geoffrion (1974) and Möhring et al. (2001), neither of which is authored by the present paper's authors. Proposition 3 is proved with an explicit counterexample, and Proposition 5 is proved by a vertex-subset argument showing that the integrality property is preserved under the considered branching constraints. The claim in Section 6.1 that the convex hull of the pricing problem with aggregated precedence constraints coincides with the feasible region of the linear relaxation of the disaggregated formulation is asserted without a detailed proof, and the paper's negative conclusion for the Deckro, Drexl-Kimms, and Volland approaches relies on it. However, this identity is not circular: it is a derived polytope claim, and it follows from Proposition 2 because the integer feasible schedules are the same under the aggregated and disaggregated precedence constraints. The computational experiments compare independent formulations on standard PSPLIB and MMLIB benchmarks and do not fit parameters to the conclusion. No fitted input is renamed as a prediction, and no load-bearing premise rests on a self-citation; the PSPLIB citation is a standard benchmark dataset, not an argument. The identified risk is a completeness or correctness gap concerning the polytope identity, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All free parameters and invented entities lists are empty; the paper introduces no fitted constants and no new postulated objects. The central claims rest on standard results in polyhedral combinatorics and on a representativeness assumption about the decompositions covered.

assumptions (5)
  • standard math Theorem 2 (Geoffrion 1974): if the pricing problem has the integrality property, the Dantzig-Wolfe relaxation bound equals the compact LP bound.
    Used in Section 6 to translate integrality of the pricing problem into absence of bound improvement; the paper cites Desrosiers et al. (2024) for the proof.
  • standard math The single-mode resource-unconstrained PSP polytope with disaggregated precedence constraints is integral (Proposition 2).
    Quoted from Möhring et al. (2001); it is the basis for the no-improvement result for single-mode decompositions.
  • standard math The convex hull of the aggregated PDT pricing problem coincides with the LP feasible region of the disaggregated PDDT formulation.
    Invoked in Section 6.1 to extend the no-improvement conclusion to the aggregated pricing problems actually used by Deckro et al. (1991), Drexl and Kimms (2001), and Volland et al. (2017).
  • domain assumption The activity-subset decomposition of Section 4.3 covers the decompositions used by most branch-and-price papers for project scheduling.
    The paper's negative conclusion is scoped by this representativeness claim; Section 6.4 concedes that the MSRCPSP and PRCPSP formulations fall outside the decomposition.
  • domain assumption Computational results on PSPLIB and MMLIB instances are representative of the difficulty of the problem class.
    The single-mode experiments use all 480 j30 instances; the multi-mode experiments use 200 randomly sampled MMLIB50 instances. Randomness is not seeded, so the sample is not fully reproducible.

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Pith. "Pith review of On Branch-and-Price for Project Scheduling." pith.science (2026). https://pith.science/paper/JNAGJX6F

@misc{pith2026250104563,
  author       = {Pith},
  title        = {Pith review of: On Branch-and-Price for Project Scheduling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNAGJX6F}},
  note         = {Machine review of arXiv:2501.04563}
}
read the original abstract

Integer programs for resource-constrained project scheduling problems are notoriously hard to solve due to their weak linear relaxations. Several papers have proposed reformulating project scheduling problems via Dantzig-Wolfe decomposition to strengthen their linear relaxation and decompose large problem instances. The reformulation gives rise to a master problem that has a large number of variables. Therefore, the master problem is solved by a column generation procedure embedded in a branching framework, also known as branch-and-price. While branch-and-price has been successfully applied to many problem classes, it turns out to be ineffective for most project scheduling problems. This paper identifies drivers of the ineffectiveness by analyzing the structure of the reformulated problem and the strength of different branching schemes. Our analysis shows that the reformulated problem has an unfavorable structure for column generation: It is highly degenerate, slowing down the convergence of column generation, and for many project scheduling problems, it yields the same or only slightly stronger linear relaxations as classical formulations at the expense of large increases in runtime. Our computational experiments complement our theoretical findings.

Figures

Figures reproduced from arXiv: 2501.04563 by the authors.

Figure 1
Figure 1. Convergence for solving the linear relaxation of t [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

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Works this paper leans on

39 extracted references · 38 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

  2. [2]

    , author Drexl, A

    author Akkan, C. , author Drexl, A. , author Kimms, A. , year 2005 . title Network decomposition-based benchmark results for the discrete time-cost tradeoff problem . journal European Journal of Operational Research volume 165 , pages 339--358

    work page 2005

  3. [3]

    , year 2017

    author Artigues, C. , year 2017 . title On the strength of time-indexed formulations for the resource-constrained project scheduling problem . journal Operations Research Letters volume 45 , pages 154--159

    work page 2017

  4. [4]

    , author Johnson, E.L

    author Barnhart, C. , author Johnson, E.L. , author Nemhauser, G.L. , author Savelsbergh, M.W. , author Vance, P.H. , year 1998 . title Branch-and-price: Column generation for solving huge integer programs . journal Operations Research volume 46 , pages 316--329

    work page 1998

  5. [5]

    , author Lenstra, J.K

    author Blazewicz, J. , author Lenstra, J.K. , author Kan, A.R. , year 1983 . title Scheduling subject to resource constraints: classification and complexity . journal Discrete Applied Mathematics volume 5 , pages 11--24

    work page 1983

  6. [6]

    , author Vanhoucke, M

    author Bredael, D. , author Vanhoucke, M. , year 2023 . title Multi-project scheduling: A benchmark analysis of metaheuristic algorithms on various optimisation criteria and due dates . journal European Journal of Operational Research volume 308 , pages 54--75

    work page 2023

  7. [7]

    , author Knust, S

    author Brucker, P. , author Knust, S. , year 2000 . title A linear programming and constraint propagation-based lower bound for the RCPSP . journal European Journal of Operational Research volume 127 , pages 355--362

    work page 2000

  8. [8]

    , author Knust, S

    author Brucker, P. , author Knust, S. , year 2003 . title Lower bounds for resource-constrained project scheduling problems . journal European Journal of Operational Research volume 149 , pages 302--313

    work page 2003

Show all 39 references
  1. [9]

    , author Alvarez-Vald \'e s, R

    author Christofides, N. , author Alvarez-Vald \'e s, R. , author Tamarit, J.M. , year 1987 . title Project scheduling with resource constraints: A branch and bound approach . journal European Journal of Operational Research volume 29 , pages 262--273

    1987

  2. [10]

    , author Contardo, C

    author Costa, L. , author Contardo, C. , author Desaulniers, G. , year 2019 . title Exact branch-price-and-cut algorithms for vehicle routing . journal Transportation Science volume 53 , pages 946--985

    2019

  3. [11]

    , author L \"u bbecke, M.E

    author Coughlan, E.T. , author L \"u bbecke, M.E. , author Schulz, J. , year 2015 . title A branch-price-and-cut algorithm for multi-mode resource leveling . journal European Journal of Operational Research volume 245 , pages 70--80

    2015

  4. [12]

    , author Wolfe, P

    author Dantzig, G.B. , author Wolfe, P. , year 1960 . title Decomposition principle for linear programs . journal Operations Research volume 8 , pages 101--111

    1960

  5. [13]

    , author Winkofsky, E

    author Deckro, R.F. , author Winkofsky, E. , author Hebert, J.E. , author Gagnon, R. , year 1991 . title A decomposition approach to multi-project scheduling . journal European Journal of Operational Research volume 51 , pages 110--118

    1991

  6. [14]

    , author L \"u bbecke, M

    author Desrosiers, J. , author L \"u bbecke, M. , author Desaulniers, G. , author Gauthier, J. , year 2024 . title Branch-and-Price . number G-2024-36 , publisher Les Cahiers du GERAD , address GERAD, Montréal QC H3T 2A7, Canada

    2024

  7. [15]

    , author Kimms, A

    author Drexl, A. , author Kimms, A. , year 2001 . title Optimization guided lower and upper bounds for the resource investment problem . journal Journal of the Operational Research Society volume 52 , pages 340--351

    2001

  8. [16]

    , year 1974

    author Geoffrion, A.M. , year 1974 . title Lagrangean relaxation for integer programming . journal Mathemtical Programming Study volume 2 , pages 82 --1974

    1974

  9. [17]

    , year 2003

    author Guignard, M. , year 2003 . title Lagrangean relaxation . journal Top volume 11 , pages 151--200

    2003

  10. [18]

    , author Briskorn, D

    author Hartmann, S. , author Briskorn, D. , year 2010 . title A survey of variants and extensions of the resource-constrained project scheduling problem . journal European Journal of Operational Research volume 207 , pages 1--14

    2010

  11. [19]

    , author Briskorn, D

    author Hartmann, S. , author Briskorn, D. , year 2022 . title An updated survey of variants and extensions of the resource-constrained project scheduling problem . journal European Journal of Operational Research volume 297 , pages 1--14

    2022

  12. [20]

    , author Sprecher, A

    author Kolisch, R. , author Sprecher, A. , year 1997 . title PSPLIB-a project scheduling problem library: OR software-ORSEP operations research software exchange program . journal European Journal of Operational Research volume 96 , pages 205--216

    1997

  13. [21]

    , author Maniezzo, V

    author Mingozzi, A. , author Maniezzo, V. , author Ricciardelli, S. , author Bianco, L. , year 1998 . title An exact algorithm for project scheduling with resource constraints based on a new mathematical formulation . journal Management Science volume 44 , pages 714--729

    1998

  14. [22]

    , year 1984

    author M \"o hring, R.H. , year 1984 . title Minimizing costs of resource requirements in project networks subject to a fixed completion time . journal Operations Research volume 32 , pages 89--120

    1984

  15. [23]

    , author Schulz, A.S

    author M \"o hring, R.H. , author Schulz, A.S. , author Stork, F. , author Uetz, M. , year 2001 . title On project scheduling with irregular starting time costs . journal Operations Research Letters volume 28 , pages 149--154

    2001

  16. [24]

    , author Schulz, A.S

    author M \"o hring, R.H. , author Schulz, A.S. , author Stork, F. , author Uetz, M. , year 2003 . title Solving project scheduling problems by minimum cut computations . journal Management Science volume 49 , pages 330--350

    2003

  17. [25]

    , author Bellenguez-Morineau, O

    author Montoya, C. , author Bellenguez-Morineau, O. , author Pinson, E. , author Rivreau, D. , year 2014 . title Branch-and-price approach for the multi-skill project scheduling problem . journal Optimization Letters volume 8 , pages 1721--1734

    2014

  18. [26]

    , author Quilliot, A

    author Moukrim, A. , author Quilliot, A. , author Toussaint, H. , year 2015 . title An effective branch-and-price algorithm for the preemptive resource constrained project scheduling problem based on minimal interval order enumeration . journal European Journal of Operational ...

    2015

  19. [27]

    , author Sadykov, R

    author Pessoa, A. , author Sadykov, R. , author Uchoa, E. , author Vanderbeck, F. , year 2018 . title Automation and combination of linear-programming based stabilization techniques in column generation . journal INFORMS Journal on Computing volume 30 , pages 339--360

    2018

  20. [28]

    , author Larsson, T

    author R \"o nnberg, E. , author Larsson, T. , year 2014 . title All-integer column generation for set partitioning: Basic principles and extensions . journal European Journal of Operational Research volume 233 , pages 529--538

    2014

  21. [29]

    , author Vanderbeck, F

    author Sadykov, R. , author Vanderbeck, F. , year 2013 . title Bin packing with conflicts: a generic branch-and-price algorithm . journal INFORMS Journal on Computing volume 25 , pages 244--255

    2013

  22. [30]

    , author Vanderbeck, F

    author Sadykov, R. , author Vanderbeck, F. , author Pessoa, A. , author Tahiri, I. , author Uchoa, E. , year 2019 . title Primal heuristics for branch and price: The assets of diving methods . journal INFORMS Journal on Computing volume 31 , pages 251--267

    2019

  23. [31]

    , author Zimmermann, J

    author Schwindt, C. , author Zimmermann, J. , year 2015 . title Handbook on Project Management and Scheduling Vol. 1 . publisher Springer , address Cham, Switzerland

    2015

  24. [32]

    , author Sherali, H.D

    author Subramanian, S. , author Sherali, H.D. , year 2008 . title An effective deflected subgradient optimization scheme for implementing column generation for large-scale airline crew scheduling problems . journal INFORMS Journal on Computing volume 20 , pages 565--578

    2008

  25. [33]

    , year 1982

    author Talbot, F.B. , year 1982 . title Resource-constrained project scheduling with time-resource tradeoffs: The nonpreemptive case . journal Management Science volume 28 , pages 1197--1210

    1982

  26. [34]

    , author Vanhoucke, M

    author Van Den Eeckhout, M. , author Vanhoucke, M. , author Maenhout, B. , year 2020 . title A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems . journal European Journal of Operational Research volume 280 , pages 845--859

    2020

  27. [35]

    , author Vanhoucke, M

    author Van Peteghem, V. , author Vanhoucke, M. , year 2014 . title An experimental investigation of metaheuristics for the multi-mode resource-constrained project scheduling problem on new dataset instances . journal European Journal of Operational Research volume 235 , pages 62--72

    2014

  28. [36]

    , year 2000

    author Vanderbeck, F. , year 2000 . title On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm . journal Operations Research volume 48 , pages 111--128

    2000

  29. [37]

    , year 2005

    author Vanderbeck, F. , year 2005 . title Implementing mixed integer column generation , in: booktitle Column generation . publisher Springer , pp. pages 331--358

    2005

  30. [38]

    , author F \"u gener, A

    author Volland, J. , author F \"u gener, A. , author Brunner, J.O. , year 2017 . title A column generation approach for the integrated shift and task scheduling problem of logistics assistants in hospitals . journal European Journal of Operational Research volume 260 , pages 316--334

    2017

  31. [39]

    , author Liu, C

    author Wang, Q. , author Liu, C. , author Zheng, L. , year 2019 . title A column-generation-based algorithm for a resource-constrained project scheduling problem with a fractional shared resource . journal Engineering Optimization volume 52 , pages 798--816

    2019

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