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Articles de revues sur le sujet « Multiple solution »

Auteur : Grafiati

Publié le 9 mars 2023

Mis à jour le 10 mars 2023

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1

Casari, Stefano. "Multiple Display Solution." Industrial Vehicle Technology International 27, no. 1 (2019): 157. http://dx.doi.org/10.12968/s1471-115x(23)70456-x.

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INDUSTRIAL VEHICLE OEMs CAN STREAMLINE MACHINE DESIGNS BY INSTALLING A SINGLE COMPUTER THAT IS ABLE TO HANDLE MULTIDISPLAY SYSTEMS, INCLUDING 360° SURROUND VIEW, ON A CHOICE OF TWO OPERATING PLATFORMS

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Karakaya, G., and M. Köksalan. "Evaluating solutions and solution sets under multiple objectives." European Journal of Operational Research 294, no. 1 (2021): 16–28. http://dx.doi.org/10.1016/j.ejor.2021.01.021.

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Koutitas, G., and C. Tzaras. "Multiple cylinder UTD solution." Electronics Letters 41, no. 9 (2005): 515. http://dx.doi.org/10.1049/el:20058308.

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ZHU, K., X. H. MO, C. Z. YUAN, and P. WANG. "A MATHEMATICAL REVIEW ON THE MULTIPLE-SOLUTION PROBLEM." International Journal of Modern Physics A 26, no. 25 (2011): 4511–20. http://dx.doi.org/10.1142/s0217751x11054589.

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The recent multiple-solution problem in extracting physics information from a fit to the experimental data in high energy physics is reviewed from a mathematical viewpoint. All these multiple solutions were previously found via a fit process, while in this paper we prove that if the sum of two coherent Breit–Wigner functions is used to fit the measured distribution, there should be two and only two nontrivial solutions, and they are related to each other by analytical formulae. For real experimental measurements in more complicated situations, we also provide a numerical method to derive the other solution from the already obtained one. The excellent consistency between the exact solution obtained this way and the fit process justifies the method. From our results it is clear that the physics interpretation should be very different depending on which solution is selected. So we suggest that all the experimental measurements with potential multiple solutions be re-analyzed to find the other solution because the result is not complete if only one solution is reported.

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Younas, Waqar, Gauhar Ali, Naveed Ahmad, et al. "Improving Convergence Speed of Bat Algorithm Using Multiple Pulse Emissions along Multiple Directions." Sensors 22, no. 23 (2022): 9513. http://dx.doi.org/10.3390/s22239513.

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Metaheuristic algorithms are effectively used in searching some optical solution space. for optical solution. It is basically the type of local search generalization that can provide useful solutions for issues related to optimization. Several benefits are associated with this type of algorithms due to that such algorithms can be better to solve many issues in an effective way. To provide fast and accurate solutions to huge range of complex issues is one main benefit metaheuristic algorithms. Some metaheuristic algorithms are effectively used to classify the problems and BAT Algorithm (BA) is one of them is more popular in use to sort out issues related to optimization of theoretical and realistic. Sometimes BA fails to find global optima and gets stuck in local optima because of the absence of investigation and manipulation. We have improved the BA to boost its local searching ability and diminish the premature problem. An improved equation of search with more necessary information through the search is set for the generation of the solution. Test set of benchmark functions are utilized to verify the proposed method’s performance. The results of simulation showed that proposed methods are best optimal solution as compare to others.

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Berezin, Alexander A. "Multiple Submissions: An Abstract Solution." Physics Today 41, no. 1 (1988): 128. http://dx.doi.org/10.1063/1.2811305.

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Jackson, Sophie E. "The Solution to Multiple Structures." Structure 16, no. 5 (2008): 659–61. http://dx.doi.org/10.1016/j.str.2008.04.004.

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Soleimani–Damaneh, Majid, and Moslem Zamani. "On compromise solutions in multiple objective programming." RAIRO - Operations Research 52, no. 2 (2018): 383–90. http://dx.doi.org/10.1051/ro/2017071.

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Compromise solutions, as feasible points as close as possible to the ideal (utopia) point, are important solutions in multiple objective programming. It is known in the literature that each compromise solution is a properly efficient solution if the sum of the image set and conical ordering cone is closed. In this paper, we prove the same result in a general setting without any assumption.

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Lin, Huei-li. "Multiple Solutions of Quasilinear Elliptic Equations in." International Journal of Differential Equations 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/673526.

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Assume that is a positive continuous function in and satisfies some suitable conditions. We prove that the quasilinear elliptic equation in admits at least two solutions in (one is a positive ground-state solution and the other is a sign-changing solution).

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Kim, Ki Sung, Kyung Su Kim, and Ki Sup Hong. "Grillage Optimization with Multiple Objectives." Key Engineering Materials 306-308 (March 2006): 517–22. http://dx.doi.org/10.4028/www.scientific.net/kem.306-308.517.

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The structural design problems are acknowledged to be commonly multicriteria in nature. The various multicriteria optimization methods are reviewed and the most efficient and easy-to-use Pareto optimal solution methods are applied to structural optimization of grillages under lateral uniform load. The result of the study shows that Pareto optimal solution methods can easily be applied to structural optimization with multiple objectives, and the designer can have a choice from those Pareto optimal solutions to meet an appropriate design environment.

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YUAN, C. Z., X. H. MO, and P. WANG. "MULTIPLE SOLUTIONS IN EXTRACTING PHYSICS INFORMATION FROM EXPERIMENTAL DATA." International Journal of Modern Physics A 25, no. 32 (2010): 5963–72. http://dx.doi.org/10.1142/s0217751x10051153.

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Multiple solutions exist in many experimental situations where several interfering amplitudes are summed to fit experimentally measured distributions, such as cross sections, mass spectra, and/or angular distributions. We show a few examples where multiple solutions are found, but only one solution is reported in publications. Since there is no standard rule for choosing one among the solutions as the physics one, we propose a simple rule that agrees with what has been adopted in previous literature: the solution corresponding to the minimal magnitudes of the amplitudes must be the physical solution. We suggest testing this rule in future analyses.

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WANG, PEI. "REFERENCE CLASSES AND MULTIPLE INHERITANCES." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 03, no. 01 (1995): 79–91. http://dx.doi.org/10.1142/s0218488595000086.

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The reference class problem in probability theory and the multiple inheritances (extensions) problem in non-monotonic logics are special cases of conflicting beliefs. One popular solution accepted in the two domains is the specificity principle. By analyzing an example, several factors beyond specificity are found to be relevant to the priority of a reference class. A new approach, Non-Axiomatic Reasoning System (NARS), is discussed, where these factors are all taken into account. It is argued that the solution provided by NARS is better than the solutions provided by probability theory and non-monotonic logics.

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Troy, William C. "Multiple solutions of a nonlinear boundary value problem." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 113, no. 3-4 (1989): 191–209. http://dx.doi.org/10.1017/s0308210500024069.

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SynopsisWe consider a system of nonlinear differential equations derived from a model for the self-trapping of light. We prove that there is a parameter range for which there is a fundamental mode solution. Further, given any positive integer N, there also exists a parameter range over which the boundary value problem has N solutions. The first of these is a fundamental mode solution while the next N — 1 solutions represent additional modes.

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de Araujo, Anderson L. A., and Kennedy Martins Pedroso. "Multiple Periodic solutions and Positive Homoclinic Solution for a differential equation." Bulletin of the Belgian Mathematical Society - Simon Stevin 20, no. 3 (2013): 535–46. http://dx.doi.org/10.36045/bbms/1378314514.

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Renzo, Francesco Di. "Simulating lattice field theories on multiple thimbles." EPJ Web of Conferences 175 (2018): 07015. http://dx.doi.org/10.1051/epjconf/201817507015.

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Simulating thimble regularization of lattice field theory can be tricky when more than one thimble is to be taken into account. A couple of years ago we proposed a solution for this problem. More recently this solution proved to be effective in the case of 0+1 dimensional QCD. A few lessons we can learnt, including the role of symmetries and general hints on algorithmic solutions.

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Gething, P. "Optimum solution for multiple emitter location." IEEE Transactions on Antennas and Propagation 35, no. 7 (1987): 876. http://dx.doi.org/10.1109/tap.1987.1144189.

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Spyker, Daniel A. "Multiple Chemical Sensitivities–Syndrome and Solution." Journal of Toxicology: Clinical Toxicology 33, no. 2 (1995): 95–99. http://dx.doi.org/10.3109/15563659509000456.

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Saville, D. J. "Multiple Comparison Procedures: The Practical Solution." American Statistician 44, no. 2 (1990): 174. http://dx.doi.org/10.2307/2684163.

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Lipovetsky, Stan. "Multiple Regression in Pair Correlation Solution." Journal of Modern Applied Statistical Methods 8, no. 1 (2009): 122–31. http://dx.doi.org/10.22237/jmasm/1241136600.

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Galybin, Konstantin, Leon Dahlhaus, and Sergei Tcherkashnev. "Multiple Analysis – A Borehole Seismic Solution." ASEG Extended Abstracts 2010, no. 1 (2010): 1–3. http://dx.doi.org/10.1081/22020586.2010.12041940.

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Saville, D. J. "Multiple Comparison Procedures: The Practical Solution." American Statistician 44, no. 2 (1990): 174–80. http://dx.doi.org/10.1080/00031305.1990.10475712.

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Achiron, A. "The multiple sclerosis labyrinth—Poirot's solution." Autoimmunity Reviews 5, no. 8 (2006): 509–10. http://dx.doi.org/10.1016/j.autrev.2006.02.013.

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Nissen, Phillip. "A Geometry Solution from Multiple Perspectives." Mathematics Teacher 93, no. 4 (2000): 324–27. http://dx.doi.org/10.5951/mt.93.4.0324.

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It is perhaps informative for students to see that no one approach to geometry is the best. Students should be encouraged to try a variety of approaches when attempting a solution to any problem, experimenting and discussing which method is helping them find an answer. For some problems, students could develop a hybrid method, which would allow them to experience the creative aspects of mathematics.

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Nowak, Ethan J., and Matthias G. Imhof. "Amplitude preservation of Radon-based multiple-removal filters." GEOPHYSICS 71, no. 5 (2006): V123—V126. http://dx.doi.org/10.1190/1.2243711.

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This study examines the effect of filtering in the Radon transform domain on reflection amplitudes. Radon filters are often used for removal of multiple reflections from normal moveout-corrected seismic data. The unweighted solution to the Radon transform reduces reflection amplitudes at both near and far offsets due to a truncation effect. However, the weighted solutions to the transform produce localized events in the transform domain, which minimizes this truncation effect. Synthetic examples suggest that filters designed in the Radon domain based on a weighted solution to the linear, parabolic, or hyperbolic transforms preserve the near- and far-offset reflection amplitudes while removing the multiples; whereas the unweighted solutions diminish reflection amplitudes which may distort subsequent amplitude-versus-offset (AVO) analysis.

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Zhao, Yongdong, and Suhada Jayasuriya. "A New Formulation of Multiple-Input Multiple-Output Quantitative Feedback Theory." Journal of Dynamic Systems, Measurement, and Control 118, no. 4 (1996): 748–52. http://dx.doi.org/10.1115/1.2802352.

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The QFT robust performance problem in its entirety may be reduced to an H∞ problem by casting each specification as a frequency domain constraint on the nominal sensitivity function and the complementary sensitivity function. It is shown that the existence of a solution to a standard Ha. problem guarantees a solution to the QFT problem whereas the existence of a QFT solution does not necessarily guarantee an H∞ solution. A solution obtained via this formulation for the QFT problem is in general more conservative when compared to what may be obtained from classical QFT loopshaping. However, one does not have to restrict the QFT controller to be diagonal as is usually done in MIMO-QFT. In addition, a simple constructive approach is provided for the design of a prefilter matrix for MIMO systems. In the standard QFT approach, the synthesis of a prefilter matrix for the MIMO case is much more involved than that of the SISO case.

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Galán-García, José Luis, Pedro Rodríguez-Cielos, Yolanda Padilla-Domínguez, et al. "SMIS: A Stepwise Multiple Integration Solver Using a CAS." Mathematics 9, no. 22 (2021): 2866. http://dx.doi.org/10.3390/math9222866.

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Multiple Integration is a very important topic in different applications in Engineering and other Sciences. Using numerical software to get an approximation to the solution is a normal procedure. Another approach is working in an algebraic form to obtain an exact solution or to get general solutions depending on different parameters. Computer Algebra Systems (CAS) are needed for this last approach. In this paper, we introduce SMIS, a new stepwise solver for multiple integration developed in a CAS. The two main objectives of SMIS are: (1) to increase the capabilities of CAS to help the user to deal with this topic and (2) to be used in Math Education providing an important tool for helping with the teaching and learning process of this topic. SMIS can provide just the final solution or an optional stepwise solution (even including some theoretical comments). The optional stepwise solutions provided by SMIS are of great help for (2). Although SMIS has been developed in the specific CAS Derive, since the code is provided, it can be easily migrated to any CAS which deals with integrals and text management that allow us to display comments for intermediate steps.

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Yermachenko, I. "MULTIPLE SOLUTIONS OF THE FOURTH‐ORDER EMDEN‐FOWLER EQUATION." Mathematical Modelling and Analysis 11, no. 3 (2006): 347–56. http://dx.doi.org/10.3846/13926292.2006.9637322.

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Two-point boundary value problems for the fourth-order Emden-Fowler equation are considered. If the given equation can be reduced to a quasi‐linear one with a non‐resonant linear part so that both equations are equivalent in some domain D, and if solution of the quasi‐linear problem is located in D, then the original problem has a solution. We show that a quasi‐linear problem has a solution of definite type which corresponds to the type of the linear part. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions.

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Ding, Wei. "The Problem of Scheduling Multiple Groups of Jobs on Multiple Processors at Different Speeds." Advances in Operations Research 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/748597.

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We mainly study the problem of schedulingngroups of jobs onnspecial-purpose processors andmgeneral-purpose processors at different speeds provided that the setup time of each job is less than times of its processing time. We first propose an improved LS algorithm. Then, by applying this new algorithm, we obtain two bounds for the ratio of the approximate solution to the optimal solutionT* under two different conditions.

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KADZIŃSKI, MIŁOSZ, and ROMAN SŁOWIŃSKI. "INTERACTIVE ROBUST CONE CONTRACTION METHOD FOR MULTIPLE OBJECTIVE OPTIMIZATION PROBLEMS." International Journal of Information Technology & Decision Making 11, no. 02 (2012): 327–57. http://dx.doi.org/10.1142/s0219622012400056.

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We introduce a new interactive procedure for multiple objective optimization problems. The identification of the most preferred solution is achieved by means of a systematic dialogue with the decision maker (DM) during which (s)he specifies pairwise comparisons of nondominated solutions from a current sample. We represent this preference information by a compatible form of the achievement scalarizing function, i.e., we are searching for weights of objectives which ensure that the reference solutions are compared by the function in the same way as by the DM. Directions of the isoquants of all compatible achievement scalarizing functions create a cone in the evaluation space, with the origin in a reference point. In successive iterations, each new pairwise comparison of solutions contracts the cone which is zooming on a subregion of nondominated points of greatest interest for the DM. The procedure ends when at least one satisfactory solution is selected or when the DM comes to conclusion that there is no such solution for the current problem setting.

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Hsu, Tsing-San. "Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains." Abstract and Applied Analysis 2007 (2007): 1–19. http://dx.doi.org/10.1155/2007/43018.

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We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ>λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

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Sharma, Abhinav Kumar, and Indrajit Mukherjee. "Enhanced multi-objective solution approach for multiple quality characteristics optimisation problems considering predictive uncertainties." International Journal of Quality & Reliability Management 37, no. 1 (2019): 112–44. http://dx.doi.org/10.1108/ijqrm-01-2019-0011.

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Purpose The purpose of this paper is to address three key objectives. The first is the proposal of an enhanced multiobjective optimisation (MOO) solution approach for the mean and mean-variance optimisation of multiple “quality characteristics” (or “responses”), considering predictive uncertainties. The second objective is comparing the solution qualities of the proposed approach with those of existing approaches. The third objective is the proposal of a modified non-dominated sorting genetic algorithm-II (NSGA-II), which improves the solution quality for multiple response optimisation (MRO) problems. Design/methodology/approach The proposed solution approach integrates empirical response surface (RS) models, a simultaneous prediction interval-based MOO iterative search, and the multi-criteria decision-making (MCDM) technique to select the best implementable efficient solutions. Findings Implementation of the proposed approach in varied MRO problems demonstrates a significant improvement in the solution quality in worst-case scenarios. Moreover, the results indicate that the solution quality of the modified NSGA-II largely outperforms those of two existing MOO solution strategies. Research limitations/implications The enhanced MOO solution approach is limited to parametric RS prediction models and continuous search spaces. Practical implications The best-ranked solutions according to the proposed approach are derived considering the model predictive uncertainties and MCDM technique. These solutions (or process setting conditions) are expected to be more reliable for satisfying customer specification compared to point estimate-based MOO solutions in real-life implementation. Originality/value No evidence exists of earlier research that has demonstrated the suitability and superiority of an MOO solution approach for both mean and mean-variance MRO problems, considering RS uncertainties. Furthermore, this work illustrates the step-by-step implementation results of the proposed approach for the six selected MRO problems.

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Dasgupta, Sajib, Richard Golden, and Vincent Ng. "Clustering Documents Along Multiple Dimensions." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (2021): 879–85. http://dx.doi.org/10.1609/aaai.v26i1.8325.

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Traditional clustering algorithms are designed to search for a single clustering solution despite the fact that multiple alternative solutions might exist for a particular dataset. For example, a set of news articles might be clustered by topic or by the author's gender or age. Similarly, book reviews might be clustered by sentiment or comprehensiveness. In this paper, we address the problem of identifying alternative clustering solutions by developing a Probabilistic Multi-Clustering (PMC) model that discovers multiple, maximally different clusterings of a data sample. Empirical results on six datasets representative of real-world applications show that our PMC model exhibits superior performance to comparable multi-clustering algorithms.

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Tang, Hao, and Mingwei Hu. "Dynamic Vehicle Routing Problem with Multiple Objectives." Transportation Research Record: Journal of the Transportation Research Board 1923, no. 1 (2005): 199–207. http://dx.doi.org/10.1177/0361198105192300121.

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This paper addresses a dynamic vehicle routing problem related to courier mail services with explicit consideration of multiple objectives. This problem is modeled as a dynamic traveling repairman problem with time windows, and with three objectives considered: maximization of the number of serviced customers, minimization of customer waiting, and minimization of total travel time. A solution (simulation) framework, extending techniques developed for single-objective dynamic routing problems in the literature, is proposed to tackle this multiobjective problem. An efficient chained local search heuristic is embedded in this solution framework to solve the underlying static problem, the traveling salesman problem with time windows. Comprehensive simulation experiments conducted on data sets derived from benchmark problems show that the proposed algorithmic approach is able to provide high-quality solutions with limited computing effort. These experiments also indicate that the multiobjective model proposed in this study can offer much more attractive solutions to decision makers than single-objective models.

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KANG, Kun-peng. "Core-based multidimensional multiple-choice knapsack solution." Journal of Computer Applications 32, no. 8 (2013): 2168–71. http://dx.doi.org/10.3724/sp.j.1087.2012.02168.

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Lim, Woong, Hongjoong Kim, Lynn Stallings, and Ji-Won Son. "Celebrating Diversity by Sharing Multiple Solution Methods." Mathematics Teacher 109, no. 4 (2015): 290–97. http://dx.doi.org/10.5951/mathteacher.109.4.0290.

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Chung, Y. N., D. L. Gustafson, and E. Emre. "Extended solution to multiple maneuvering target tracking." IEEE Transactions on Aerospace and Electronic Systems 26, no. 5 (1990): 876–87. http://dx.doi.org/10.1109/7.102720.

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Janko, Balazs, Chris P. Turner, Kim S. Cave Ayland, Martyn K. Pittuck, Sebastian O. H. Madgwick, and William S. Harwin. "A multiple robot solution for urban reconnaissance." International Journal of Intelligent Defence Support Systems 4, no. 1 (2011): 70. http://dx.doi.org/10.1504/ijidss.2011.037808.

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Koutitas, G., and C. Tzaras. "A UTD Solution for Multiple Rounded Surfaces." IEEE Transactions on Antennas and Propagation 54, no. 4 (2006): 1277–83. http://dx.doi.org/10.1109/tap.2006.872675.

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Panjikar, Santosh, Manfred Weiss, and Victor Lamzin. "Automated structure solution from multiple data sets." Acta Crystallographica Section A Foundations and Advances 71, a1 (2015): s18. http://dx.doi.org/10.1107/s2053273315099696.

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Mirrazavi, S. K., D. F. Jones, and M. Tamiz. "MultiGen: an integrated multiple-objective solution system." Decision Support Systems 36, no. 2 (2003): 177–87. http://dx.doi.org/10.1016/s0167-9236(02)00135-5.

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Bhatta, Ambika, Miroslava Raspopvic, Max Denis, and Charles Thompson. "Exact image solution approach for multiple reflections." Journal of the Acoustical Society of America 126, no. 4 (2009): 2160. http://dx.doi.org/10.1121/1.3248399.

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McCallum, Bennett T. "Multiple-solution indeterminacies in monetary policy analysis." Journal of Monetary Economics 50, no. 5 (2003): 1153–75. http://dx.doi.org/10.1016/s0304-3932(03)00061-8.

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Bernstein, Joseph B., Moti Gabbay, and Ofir Delly. "Reliability matrix solution to multiple mechanism prediction." Microelectronics Reliability 54, no. 12 (2014): 2951–55. http://dx.doi.org/10.1016/j.microrel.2014.07.115.

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Tong, Lee-Ing, Chung-Ho Wang, Chih-Chien Chen, and Chun-Tzu Chen. "Dynamic multiple responses by ideal solution analysis." European Journal of Operational Research 156, no. 2 (2004): 433–44. http://dx.doi.org/10.1016/s0377-2217(03)00017-1.

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Ogawa, Tadayuki, and Nobutaka Hirokawa. "Multiple analyses of protein dynamics in solution." Biophysical Reviews 10, no. 2 (2017): 299–306. http://dx.doi.org/10.1007/s12551-017-0354-7.

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Luccio, F., and M. C. Pinotti. "Suboptimal solution for PLA multiple column folding." Computer-Aided Design 22, no. 8 (1990): 515–20. http://dx.doi.org/10.1016/0010-4485(90)90068-n.

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Srivastava, H. M., and R. K. Raina. "Solution of a certain multiple integral equation." Journal of Mathematical Analysis and Applications 143, no. 2 (1989): 512–16. http://dx.doi.org/10.1016/0022-247x(89)90056-5.

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BAKRI, TAOUFIK, YURI A. KUZNETSOV, FERDINAND VERHULST, and EUSEBIUS DOEDEL. "MULTIPLE SOLUTIONS OF A GENERALIZED SINGULAR PERTURBED BRATU PROBLEM." International Journal of Bifurcation and Chaos 22, no. 04 (2012): 1250095. http://dx.doi.org/10.1142/s0218127412500952.

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Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu″ + 2u′ + f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.

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Cezikturk, Ozlem. "Multiple solution problems and mathematical thinking: Wasan geometry example." Academic Perspective Procedia 2, no. 1 (2019): 37–46. http://dx.doi.org/10.33793/acperpro.02.01.10.

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Wasan geometry flourished in Edo period of Japan as arty, multiple solutioned and systematic. Teachers are not open and papers are not evaluated. Multiple solutions are seen in parallel with advanced mathematics thinking. 80 middle school and 37 high school pre service math teachers were given a Wasan problem and grouped according to correct answers, different answers, and originality. 24 different answers (equality of diagonals of a square, symmetry, equality of tangent lines, auxiliary lines and auxiliary shapes, cosines’ theorem, tan (67.5), similarities, types of triangles etc.) were detected. Answers were also coded for composite answers (who include two different rules), or simple one ways, first answers, similar solution ways and 5-6 different solutions.

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Rossen, William Richard, and Johannes Bruining. "Foam Displacements with Multiple Steady States." SPE Journal 12, no. 01 (2007): 5–18. http://dx.doi.org/10.2118/89397-pa.

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Summary A number of experimental and theoretical studies suggest that the fractional-flow function for foam can be either multivalued in water saturation or else can comprise distinct fractional-flow curves for two or more foam regimes, with jumps between them where each regime reaches its limiting condition. We construct fractional-flow solutions for these cases. When such a foam is employed in a surfactant-alternating-gas (SAG) process, the usual "tangency" condition is modified and the foam can be considerably weaker than the foam formed at what appears to be the point of tangency of the multivalued fractional-flow function. If the capillary-pressure function Pc (Sw) differs between foam regimes, that difference can substantially change the nature of the displacement. This alters the effective fractional-flow function and hence the global solution of the equations. It is therefore important to determine how capillary pressure varies in foam displacements, by direct measurement in situ if possible. Special care is needed in numerical simulation of processes using fixed grids if capillary pressure depends directly on foam regime. Using gridblocks that are too large can weaken the effect of capillary pressure that would enforce the correct shock on the small scale. Using an upscaled fractional-flow function appears to eliminate this problem, however. Introduction Foams are injected into geological formations for gas diversion in improved oil recovery (IOR) (Schramm 1994; Rossen 1996), acid diversion in matrix acid well stimulation (Gdanski 1993), and environmental remediation (Hirasaki et al. 2000). In IOR and environmental remediation, it is often useful to inject gas and surfactant solution in alternating slugs, a process called SAG injection. SAG injection holds several advantages over continuous coinjection of gas and liquid, as described elsewhere (Shi and Rossen 1998; Shan and Rossen 2004). Method of Characteristics and Fractional-Flow Theory. Many problems involving conservation equations can be formulated in the so-called hyperbolic framework. The ensuing equations can then be solved using the method of characteristics (see, for example, Smoller (1980), after page 266). The solution consists of rarefactions or spreading waves, constant states, and shocks. Additional conditions are also required to obtain a unique solution. Numerical solutions of the equations can, in effect, pick out the wrong uniqueness conditions and give inaccurate results. A complete analysis requires the traveling-wave representation of a shock. Bruining and Van Duijn (2000, 2007) present an example in which the conditions on the traveling wave are essential to identifying the correct solution of the macroscopic equations. Bruining et al. (2002, 2004) give a regularization procedure for constructing such a traveling wave solution for an application of steam injection. The conditions on the traveling wave must be kept in mind when using a graphical procedure for finding the solution with the method of characteristics, a technique introduced by Buckley and Leverett (1941). It is widely used for petroleum engineering applications [see also Pope (1980), who generalized fractional-flow theory to deal with more complex problems]. In this formulation, the uniqueness condition is called the Welge tangent condition (1952).

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