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A fuzzy multi-objective programming based approach is presented to solve the small and medium size problems to optimality.
We presented two methods to solve the problem and we concluded that their performance was acceptable for small and medium size problems, but not for large problems.
Using optimality conditions, we reformulate the non-linear bi-level model as a single-level mixed integer linear program, which is computationally tractable for medium size problems using a commercial solver.
The results show that for small and medium size problems whether or not a computationally more demanding algorithm is employed to solve the level 1 or level 2 problem does not have any bearing on the makespan evaluated.
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Now, a randomly generated medium size problem should be considered.
In this section, a complete randomly generated medium size problem is considered as a numerical example.
To facilitate the synthesis of medium-size problems, the following synthesis/analysis scheme is proposed.
However, the node-based formulation can yield competitive lower bounds in a reasonable amount of time on medium-size problems.
For small- and medium-size problems, the technique proposed in this article will prove to be very effective for the design of heat exchanger networks.
The last column in Table 5 confirms above claim and shows that in large-size problems, the difference between 2-VRPTW and VRPCDTWS is more than medium-size problems.
Indeed, the Peters et al. matrices only represent medium-size problems for RAxML, which has successfully analysed a data set of nearly 10,000 aligned sites for over 55,000 flowering plant species [ 11].
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com