Fractional 0–1 programs: links between mixed-integer linear and conic quadratic formulations

Mehmanchi, Erfan; Gómez, Andrés; Prokopyev, Oleg A.

doi:10.1007/s10898-019-00817-7

Cited by 14 publications

(11 citation statements)

References 40 publications

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“…To summarize these results, we observe that MILP 1 [8 ab ] tends to have the best continuous relaxation bound. This observation is consistent with the earlier observations in the literature that the corresponding nominal reformulation FP 1 typically has the best relaxation quality (Borrero et al 2016, Mehmanchi et al 2019. Nonetheless, this does not always (or even often) lead to superior solution times mainly because of the large size of the reformulation.…”

Section: Disjoint Reformulationssupporting

confidence: 91%

“…It is also possible to develop a binary expansion reformulation of RFP 1 [8 ab ]. However, based on our experiments, such a formulation performs poorly in computations; also, refer to Borrero et al 2016and Mehmanchi et al (2019) for an analogous comparison regarding deterministic FP. Hence, we omit this formulation for brevity.…”

Section: Disjoint Uncertainty Setmentioning

confidence: 64%

“…Finally, it is worth mentioning that conic quadratic programming-based reformulations can be used as an alternative solution approach to tackle deterministic fractional 0-1 programs (see, e.g., Ş en et al 2018, Mehmanchi et al 2019, Atamtürk and Gómez 2020 because they lead to strong convex relaxations. However, solving large-scale mixed-integer conic quadratic programs is still challenging in practice despite the recent advances in off-the-shelf commercial optimization software.…”

Section: Resultsmentioning

confidence: 99%

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On Robust Fractional 0-1 Programming

Mehmanchi

Gillen

Gómez

et al. 2020

INFORMS Journal on Optimization

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We study single- and multiple-ratio robust fractional 0-1 programming problems (RFPs). In particular, this work considers RFPs under a wide range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator and the latter accounts for different forms of interrelatedness between them. We first demonstrate that unlike the deterministic case, a single-ratio RFP is nondeterministic polynomial-time hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure, variants of the well-known budgeted uncertainty, the disjoint and joint single-ratio RFPs are polynomially solvable when the deterministic counterpart is. We also propose mixed-integer linear programming (MILP) formulations for multiple-ratio RFPs. We conduct extensive computational experiments using test instances based on real and synthetic data sets to evaluate the performance of our MILP reformulations as well as to compare the disjoint and joint uncertainty sets. Finally, we demonstrate the value of the robust approach by examining the robust solution in a deterministic setting and vice versa.

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Section: Disjoint Reformulationssupporting

confidence: 91%

Section: Disjoint Uncertainty Setmentioning

confidence: 64%

Section: Resultsmentioning

confidence: 99%

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On Robust Fractional 0-1 Programming

Mehmanchi

Gillen

Gómez

et al. 2020

INFORMS Journal on Optimization

Self Cite

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“…The model presented in the previous section is a multiple-ratio quadratic 0-1 fractional programming model as the highest order of the product of binary variables in the objective function is two, and the objective function is expressed as the sum of G ratios [41][42][43][44]. A quadratic 0-1 programming model can be converted into a linear 0-1 programming one by introducing a new set of 0-1 variables (one for each quadratic term included in the base model) and additional constraints [28].…”

Section: Model Conversionmentioning

confidence: 99%

The Machine-Part Cell Formation Problem with Non-Binary Values: A MILP Model and a Case of Study in the Accounting Profession

et al. 2021

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The traditional machine-part cell formation problem simultaneously clusters machines and parts in different production cells from a zero–one incidence matrix that describes the existing interactions between the elements. This manuscript explores a novel alternative for the well-known machine-part cell formation problem in which the incidence matrix is composed of non-binary values. The model is presented as multiple-ratio fractional programming with binary variables in quadratic terms. A simple reformulation is also implemented in the manuscript to express the model as a mixed-integer linear programming optimization problem. The performance of the proposed model is shown through two types of empirical experiments. In the first group of experiments, the model is tested with a set of randomized matrices, and its performance is compared to the one obtained with a standard greedy algorithm. These experiments showed that the proposed model achieves higher fitness values in all matrices considered than the greedy algorithm. In the second type of experiment, the optimization model is evaluated with a real-world problem belonging to Human Resource Management. The results obtained were in line with previous findings described in the literature about the case study.

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“…Assumption A1 is standard in fractional optimization [10,11,34]. In particular, it ensures that the objective function is well defined.…”

Section: Introductionmentioning

confidence: 99%

Fractional 0-1 programming and submodularity

Han¹,

Gómez²,

Prokopyev³

2020

Preprint

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In this note we study multiple-ratio fractional 0-1 programs, a broad class of N P-hard combinatorial optimization problems. In particular, under some relatively mild assumptions we provide a complete characterization of the conditions, which ensure that a single-ratio function is submodular. Then we illustrate our theoretical results with the assortment optimization and facility location problems, and discuss practical situations that guarantee submodularity in the considered application settings. In such cases, near-optimal solutions for multiple-ratio fractional 0-1 programs can be found via simple greedy algorithms.

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Fractional 0–1 programs: links between mixed-integer linear and conic quadratic formulations

Cited by 14 publications

References 40 publications

On Robust Fractional 0-1 Programming

On Robust Fractional 0-1 Programming

The Machine-Part Cell Formation Problem with Non-Binary Values: A MILP Model and a Case of Study in the Accounting Profession

Fractional 0-1 programming and submodularity

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