Relocation methods

It should be noted that a resequence or repipe does not involve zero capital cost, even though no new heat exchanger equipment might be purchased. The pipework modifications for a resequence or repipe might be very expensive. Also, equipment might need to be relocated. Methods for capital cost estimation for retrofit were discussed in Chapter 2. 

The best-known relocation method is the k-means method, for which there exist many variants and different algorithms for its implementation. The k-means algorithm minimizes the sum of the squared Euclidean distances between each item in a cluster and the cluster centroid. The basic method used most frequently in chemical applications proceeds as follows ... 

The most widely used and most effective general technique for estimating the mixture model parameters is the expectation maximization (EM) algorithm. " It finds (possibly suboptimally) values of the parameters using an iterative refinement approach similar to that given above for the k-means relocation method. The basic EM method proceeds as follows ... 

Note that the k-means relocation method is really a special case of EM that assumes (1) each cluster is modeled by a spherical Gaussian distribution, (2) each data item is assigned to a single cluster, and (3) the mixture weights... 

Nonhierarchical relocation methods, such as the k-means method, start with a user-defined number of seed clusters and iteratively reassign molecules between clusters to see if better clusters result, where the success of the solution is measured by some parameter such as the ratio of the mean intercluster similarity to the mean intracluster similarity, The methods can be quite... 

Anchoring. Bismuth alloys that expand on solidification are particularly useful for aligning and setting punches in a die plate. It is much easier to melt and pour an ahoy around a punch than to machine the entire die plate and punch at the same time. This method also makes it easier to relocate parts or change dies. The low temperatures involved do not cause distortion. 

Economic and safety considerations encourage minimal stockpiling of chemicals and avoiding transportation of hazardous substances. These increasing demands offer many opportunities for microwave chemistry in the development of environmentally benign methods for the preparation of intermediates, specialty chemicals and pharmaceuticals [6[. It appears likely that within the next few years, individual chemical reactors will be required for diverse tasks and will need to be easily relocated... 

We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point trim times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2. 

Relocation algorithms are widely used outside of chemical applications, largely because of their simplicity and speed. The original k-means noncombi-natorial methods, such as that by Forgy, and the combinatorial methods, such as that by MacQueen, have been modified into different versions for use in many disciplines, a few of which are mentioned here. Efficient implementations of k-means include those by Hartigan and Wong ° and Spaeth. 

Specialization of labor, the placement of individuals into categories based on job fxmctions, leads to increased productivity. This increased productivity is seen by (1) an improvement in a worker s ability to do a task, resulting from a worker s concentration on one or a few tasks (2) time saving due to minimized need for physical relocation or from adapting, or orienting, a worker to the new task and (3) concentration on one or a few tasks increasing the likelihood of discovering easier and better methods. 

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