K-median clustering

P-median or k-median clustering problems are well studied in the literature. The IP model has the objective of minimizing total weighted distance from F facilities to stores satisfying all C customer demands and assuming that each facility can satisfy customer demands. The model can be extended to have capacity constraints. Minimizing total weighted costs can be employed as an objective function. 

Notice that our method provides not only a sales forecast for the chain, but for each cluster. The cluster forecasts were used to guide allocation of product to stores for th k-median clustering based on sales method. In our prediction formula, otpSpj represents the forecast of product j in the cluster of stores corresponding to test store p, and hence this quantity is the ideal amount to send to this cluster. Total sales voliunes at individual stores were used as a basis to determine allocations to stores within a cluster through the formula apSpjWj/ Swj. For other methods, we used the existing approach... 

Recall that a test period of 3 weeks was used in developing the data reported in Table 4. To determine the impaet of the length of the test period on eost and foreeast error we applied k-median clustering based on sales with test periods of varying lengths. Results are reported in Table 4 and suggest the industry practice of a 3-week test period is not unreasonable. 

Table 4. Impact of test period length on forecast error and cost for k-median clustering based on sales...

Fig. 2 The procedure for colour image segmentation, (a) Original ZN-stained tissue slide image and its result after applying (b) the C-Y based colour filter (c) the k-mean clustering (d) the median filter (e) the region...

We first describe the optimization model used to form clusters and select a test store within each cluster. This model is a specialized integer program known as the k-median problem, which we solve with the highly efficient algorithm given in Comuejols et al. (1977). 

In order to check the results of the analysis, K-Nearest Neighbor distances were computed for the scaled data set Including the cadmium results. The median of the distances from a given laboratory to the three nearest neighbors ranged from 0.26 to 1.24 with the median distance between members of the cluster (1,2,3,5,6,7) equal to 0.79. The median distances of Laboratories 4 and 8 from members of this cluster were 1.24 and 1.22, respectively, supporting the view that these laboratories are outliers. 

The set of PRISRI rivers has been subdivided into 12 classes of runoff for which the median elemental concentrations have been determined (Figure 3). The Q25 and Qis quantiles and, to some extend the Qio to Qgo quantiles follow the same patterns for all elements. On the basis of Na" ", K" ", Cl, and two different clusters can be... 

To compare the two criteria we examined 32 data sets where each data set has between 150 and 600 objects, split into 20 true clusters on the average. We applied simulated annealing partitional clustering 5 times to each set with both B(p) and W(p). We then compared the best test results for W(p) and B(p) over the 32 data sets. Table 1 shows for each data set the best Jaccard score for each of the criteria. In addition, for W(p) it shows the best number of clusters parameter k in 18, 19, 20, 21, 22. Similarly, for B(p) the table shows the best median distance parameter v in the set 2.0, 2.5, 3.0, 3.5, 4.0 and the associated number of clusters in the final partitioning. At the bottom of the table is the minimum, maximum, and mean of each column. 

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