Enumeration methods 1: Unterschied zwischen den Versionen

Version vom 1. Juli 2013, 20:07 Uhr

There are two solution sets for difficult combinatorical optimization problems available: complete and incomplete enumeration methods. These solution sets also apply on integer or mixed-integer optimization problems.

Theory

Complete enumeration

A complete enumeration method guarantees, that from all possible solutions the optimum is chosen. Algorithms that always provide the optimal solution are called exact methods.

Explicit complete enumeration

One possibility to solve a problem is explicit complete enumeration. The algorithm generates consecutively a enumeration tree with all possible solutions and memorizes the current best solution of the examined problem.

Implicit complete enumeration

Another possibility to exactly solve a combinatorical optimization problem is implicit complete enumeration. The algorithm consecutively cuts of all subsets of the solution space in which, under predetermined conditions, no optimal solution could be expected. The algorithm generates consecutively a enumeration tree of the remaining solution space.

Incomplete enumeration

In case of large scale optimization problems the computing time for complete enumeration techniques often exceeds the economic reasonable time. Therefore a use of incomplete enumeration (heuristics) is advised. Incomplete enumeration techniques just search in a subset of the valid solution space. Hence the best generated solution is not necessarily a optimal solution. It is also possible, that a heuristic stops without a valid solution even if there is one.

Example: Traveling Salesman Problem (TSP)

In the following we discuss one examples which we solve with the three enumeration methods mentioned before. The following table shows the cost a salesman has to travel from a location i to a location j.

Table 1:

Solution

Solution: Explicit complete enumeration

First you change every undesirable cost coefficient ${\displaystyle c_{ij}}$ from "i to j" with ∞ (=infinite), e.g. the elements of the main diagonal and elements with unusual high transportation costs (e.g. ${\displaystyle c_{32}}$). To reduce the complexity of the explicit complete enumeration process you subtract in every row the lowest cost from all the other costs (compare table 2): Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c_{ij}^{*} = c_{ij} - min\ c_{ij}

Table 2:

With this table you know look for every possible enumeration.

Table 3:

From table 3 you see that the optimal solution has the order 1 - 2 - 4 - 5 - 3 - 1 with minimal cost of 4.

Solution: Implicit complete enumeration

Solution: Incomplete enumeration

Sources

Neumann, K.; Morlock, M.: Operations Research, 2.Auflage, Carl Hanser Verlag München Wien, 2002

Zimmermann, W.:Operations Research, Quantitative Methoden zur Entscheidungsvorbereitung, 6.Auflage, R.Oldenbourg Verlag München Wien, 1992