Huanfa Chen

Huanfa Chen

he/him

Associate Professor in Spatial Data Science, UCL CASA

Notes on multiple objective combinatorial optimisation problems

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Updates 2024/01/31

This post records some notes on multi-objective combinatorial optimization (MOCO) problems.

Categorisation of strategies

Ehrgott and Gandibleux [7] categorised solution methods for addressing multi-objective optimization problems into three groups, based on the information provided by decision-makers. These groups are as follows:

  • Priori mode: In this strategy, decision-makers express their preferences at the beginning of the process. An example of this approach is goal programming, where decision-makers specify their objectives upfront.
  • Posteriori mode: The preferences of decision-makers are analyzed after obtaining all the non-dominated points. This mode focuses on evaluating and comparing different solutions based on the decision-makers’ preferences. Some algorithms of this approach aim to obtain the entire non-dominated frontier. These algorithms involve reformulating the multiple-objective problem as a collection of single-objective models. Two effective approaches are the two-phase method and $\epsilon$-constraint method.
  • Interactive mode: This strategy involves active involvement and collaboration between decision-makers and the decision-making process. Decision-makers play a significant role, providing feedback and refining their preferences during the computation steps.

Goal programming

Here, for each of the objectives a target value (goal) is specified by the decision maker. The overall aim is to minimize the deviation from the specified goals. This approach is sometimes considered a different field from multiobjective optimization.

A good example of goal programming can be found here.

Other references on GP

  • Ignizio JP (1976) Goal programming and extensions. Lexington Books,Lexington, KY

  • Lee SM (1972) Goal Programming for decision analysis. Auerbach,Philadelphia,PA

Efficient solution and supported solution

Lexicographic optimization

Max-ordering problem

As far as the other definitions of optimality in (MOCO) are concerned,we note that the max-ordering problem with sum objectives is INP -hard in general (see [22]), but can be reduced to a single objective problem in the case of bottleneck objectives.

For lexicographic optimization it is known that a lexicographically optimal solution is always efficient,and even a supported efficient solution

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Problems of MS Word in academic writing

  1. There are obvious differences between the desktop software of MS Word and the MS Word 365 on a browser, especially the formulas and figures.
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Benefits of using Overleaf

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Problems of using Overleaf

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Tutorials of Overleaf

  1. UCL webpage and overleaf course at UCL: see link.
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