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# 数学代写|运筹学代写Operations Research代考|MATH4202 Heuristics

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## 数学代写|运筹学代写Operations Research代考|Heuristics

The computational complexity of the integer programming model associated with the distribution problem rapidly surpasses any practical limit when $n(=$ the number of goods) is large. The problem cannot be solved in a time that is a polynomial function of the problem size; rather, the computation time grows exponentially in that size. As in most combinatorial optimization problems, it is practically impossible to find an optimal integer solution when the problem becomes very large. In such situations, we fall back on methods that restrict themselves to finding a reasonably good solution within an acceptable amount of computation time. Such methods are called heuristics. Often, one can develop different heuristics for a certain class of problems. When designing a heuristic, one must always weight the computation time and the quality of the calculated solution against each other. Ideally, a heuristic provides a near-optimal solution within a relatively short computation time. A good heuristic should have the following properties:

intuitively appealing,

easy to apply,

good solution in a relatively short computation time.
In Section 2.4, we already gave an example of a heuristic method for the knapsack problem. In this section and Sections $2.8$ and $2.9$, we give other examples of heuristics. Designing a heuristic is to a large extent an art that requires, in particular, creativity from the designer. The exact form of a heuristic strongly depends on the specific problem.

## 数学代写|运筹学代写Operations Research代考|A Partitioning Problem

A simple heuristic for the partitioning problem from Section $2.2 .5$ is as follows:

Order the goods in decreasing order of value.

Successively assign the goods to one of the two persons, where a product is assigned to the person who has the lowest total value of assets at that time.
Let us apply the heuristic to the numerical example from Section 2.2.5, where the numbers $a_i$ have already been ordered in the desired way. The heuristic provides the two groups
$$S_1={1,4,6,8,9,11,13,16,17,19}, S_2={2,3,5,7,10,12,14,15,18,20} .$$
In the heuristic solution, persons 1 and 2 receive goods worth 460 and 451 , with a difference of 9 , while in the optimal solution, the assigned goods have values 456 and 455, with a difference of 1.

## 数学代写|运筹学代写Operations Research代考|A Partitioning Problem

$$S_1=1,4,6,8,9,11,13,16,17,19, S_2=2,3,5,7,10,12,14,15,18,20 .$$

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