Discrete representation of non-dominated sets in multi-objective linear programming

Shao, Lei; Ehrgott, Matthias

doi:10.1016/j.ejor.2016.05.001

Cited by 32 publications

(12 citation statements)

References 32 publications

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“…Model can be equivalently reformulated with bixel intensity variables x replaced by segment intensity variables

\overset{x}{true}

and dose deposition matrix for bixels A replaced by dose deposition matrix for segments

Ā

(where each column of

Ā

represents the dose deposited to the voxels for a particular segment under unit intensity). The reformulated model is then solved with column generation integrated in the revised normal boundary intersection (RNBI) procedure (Shao & Ehrgott, , ; Lin et al, ).…”

Section: Application To Radiotherapy Treatment Planningmentioning

confidence: 99%

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Multiobjective navigation of external radiotherapy plans based on clinical criteria

Lin

Ehrgott

2017

Multi Criteria Decision Anal

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This study considers a navigation method for finding the most preferable radiotherapy plan from a discrete set using planner-defined clinical criteria. The method is based on repeatedly solving an optimization model to identify a plan that best satisfies the aspiration values set by the planner.During navigation, the planner iteratively adjusts the aspiration values to match the preference information learned from previous plans until the most preferable plan is identified. The use of soft constraints to model aspiration values enables navigation amongst a discrete set and allows the planner to freely specify the aspiration values without producing an infeasible model. We demonstrate the use of the model by applying it to a prostate cancer case. This illustrates that improvements in optimization criteria do not necessarily lead to improvements in clinical criteria.Hence, the method obviates the need to simultaneously monitor both optimization and clinical criteria in current navigation systems. Instead, the direct use of clinical criteria for navigation aids the planner to quickly identify the most preferable plan. KEYWORDS data envelopment analysis, interactive multiobjective optimization, multiobjective decision making, navigation 1 J Multi-Crit Decis Anal. 2018;25: 31-41.wileyonlinelibrary.com/journal/mcda

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“…Model can be equivalently reformulated with bixel intensity variables x replaced by segment intensity variables

\overset{x}{true}

and dose deposition matrix for bixels A replaced by dose deposition matrix for segments

Ā

(where each column of

Ā

Section: Application To Radiotherapy Treatment Planningmentioning

confidence: 99%

“…The RNBI procedure computes a representative set of nondominated points for multiobjective linear programmes (MOLPs; Shao & Ehrgott, , ). The procedure constructs a set of equidistant reference points in the criterion space.…”

Section: Application To Radiotherapy Treatment Planningmentioning

confidence: 99%

Multiobjective navigation of external radiotherapy plans based on clinical criteria

Lin

Ehrgott

2017

Multi Criteria Decision Anal

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“…In bi-objective optimization, one solution cannot satisfy every objective, simultaneously. Previous research provides a variety of methods to generate the Pareto frontier [23], such as weighting method, ϵ -constraint method, and normalized normal constraint method. Normalized normal constraint (NNC) method, proposed by Messac et al [24], obtains an evenly spaced Pareto solution by constructing a series of single-objective optimization problems after normalized processing of the feasible region of multi-objective optimization [25].…”

Section: Introductionmentioning

confidence: 99%

Bi-objective inventory allocation planning problem with supplier selection and carbon trading under uncertainty

Kang

et al. 2018

PLoS ONE

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Concern is growing that business enterprises focus primarily on their economic activities while disregarding the adverse environmental and social effects of these activities. To contribute to the literature on this matter, this study investigates a novel bi-objective inventory allocation planning problem with supplier selection and carbon trading across multiple periods under uncertainty. The concepts of a carbon credit price and a carbon cap are proposed to demonstrate the effect of carbon emissions costs on inventory allocation network costs. Demands of manufacturers, transport price, and defect rate of materials that should be rejected are set as random variables. We combine normalized normal constraint method, differential evolution algorithm, and uncertainty simulation to deal with the complex model. One representative case shows the effectiveness and practicability of this model and proposed method. The Pareto frontier is generated by solving the bi-objective model. We extend the results of numerical examples in large scale problems, and compare the solution method results with exact solutions. The environmental objective across the inventory allocation network varies with changes of the carbon cap and the carbon credit price.

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“…Das and Denis propose the boundary intersection method for finding an evenly distributed set of solutions in the nondominated set [8]. Recently Shao and Ehrgott combine these methods to obtain well distributed nondominated solutions for MOLP [44]. Other methods for finding representations are due to Kim and De Weck for bicriteria nonconvex problems [27] and general MOPs [28], Sylva and Crema [46] and Masin and Bukchin [33] for MOMIP problems and Karasakal and Koksalan for MOLP [26].…”

Section: Introductionmentioning

confidence: 99%

Bilevel programming for generating discrete representations in multiobjective optimization

Kirlik

Sayın

2017

Math. Program.

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The solution to a multiobjective optimization problem (MOP) consists of the nondominated set that portrays all relevant trade-off information. The ultimate goal is to identify a Decision Maker's most preferred solution without generating the entire set of nondominated solutions. We propose a bilevel programming formulation that can be used to this end. The bilevel program is capable of delivering an efficient solution that maps into a given set, provided that one exits. If the Decision Maker's preferences are known a priori, they can be used to specify the given set. Alternatively, we propose a method to obtain a representation of the nondominated set when the Decision Maker's preferences are not available. This requires a thorough search of the outcome space. The search can be facilitated by a partitioning scheme similar to the ones used in global optimization. Since the bilevel programming formulation either finds a nondominated solution in a given partition element or determines that there is none, a representation with a specified coverage error level can be found in a finite number of iterations. While building a discrete representation, the algorithm also generates an approximation of the nondominated set within the specified error factor. We illustrate the algorithm on the multiobjective linear programming problem.

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Discrete representation of non-dominated sets in multi-objective linear programming

Cited by 32 publications

References 32 publications

Multiobjective navigation of external radiotherapy plans based on clinical criteria

Multiobjective navigation of external radiotherapy plans based on clinical criteria

Bi-objective inventory allocation planning problem with supplier selection and carbon trading under uncertainty

Bilevel programming for generating discrete representations in multiobjective optimization

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