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All rights reserved. This paper describes a simple enhancement of this heuristic. On benchmark test problems, the modified method does not result in much larger computation times and almost always produces lower solution values than does the Esau—Williams algorithm.
The improvements are often significant. Journal of the Operational Research Society 53, — DOI: Vertex n is called the root. With each vertex i 2 V nfng is associated a nonnegative weight or demand qi, and with each edge fi; jg is associated a length or cost cij. Given a spanning tree, any subtree linked to the root by a single edge is called a main subtree. Given a vertex j 2 V nfng, the main subtree containing j is simply called the subtree of j and the edge linking that subtree to the root is called the gate of j.
The CMSTP consists of determining a minimum spanning tree on G such that the total demand of every main subtree does not exceed a given capacity Q. The CMSTP arises in several telecommunications and energy network applications in which it is desired to control the flow passing through edges incident to the terminal vertex.
It is rather simple to implement and uses the concept of savings, as in the Clarke and Wright19 algorithm for the vehicle routing problem. While the more recent heuristics typically provide better solution values than does the EW heuristic, the improvement in solution quality is often obtained at the expense of computational efficiency and ease of implementation.
The purpose of this paper is to provide a simple modification to the EW heuristic. The proposed change is easy to implement and the resulting algorithm is only slightly slower than the original method.
However, as computational results will show, solution quality can be significantly improved. The modified algorithm will be described in the following section, followed by computational results and the conclusion. Starting with a star tree in which each vertex of V nfng is linked to the root by an edge, the EW heuristic iteratively repeats the following operations: Step 1. Identify the two vertices i and j belonging to two different main subtrees and yielding the largest Journal of the Operational Research Society Vol.
Step 2. Figure 1 Identify two vertices i and j belonging to two different main subtrees and yielding the largest positive modified saving s0ij , and such that the total weight of the subtree of i and the modified subtree of j does not exceed Q. If no such saving exists, stop. Go to Step 1. A limitation of the EW algorithm is that, in Step 2, vertex j is always disconnected from the root by deleting its gate while it may have been more advantageous to remove the longest edge on the path linking j to the root.
We use this idea in the MEW. Define the modified gate of j as the longest edge on its path to the root and let gj0 denote its length. Also define the modified subtree of j as the subtree containing j obtained if the modified gate of j is deleted see Figure 2. The letter c means that the root is centrally located whereas the letter e means that it is eccentric. The latter instances tend to be more difficult. We also solved cma b instances with non-unit demands.
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ESAU WILLIAMS ALGORITHM PDF
MEWA: Gewijzigde algoritme besproken Esau-Williams