## 2 Materials and Methods
The index \(i\) is from the set of indices assigned to the analyzed product variants, the index \(j\) is from the set of indices reserved for the observed product characteristics. This concept can now be used in practice for the business objectives outlined earlier in the text as follows:  
1. Using interval \([0,1]\) for each weight that represents importance of a product feature, find weight combinations \(k_{i}\), \(i=1,\ldots,n\), where \(k_{i}\) is the ith vector of weights chosen from \([0,1]\times\ldots\times[0,1]\), which have the property that when used in the AHP model, the _currently_ top-selling product variant has the highest overall AHP-based value among the currently sold variants. Not all weight vectors from the multivariate interval will have this property. The weight vectors that lead to the best-selling product having the highest AHP value form a space potentially containing weight vectors used in the decision-making process by the clients who bought the best-selling variant, at least according to the AHP model. This is because, as described, the top seller can be expected to have the highest AHP value among the sold variants for those who bought it, or a value near the maximum. In practice, the interval \([0,1]\times\ldots\times[0,1]\), which contains an infinite number of weight vectors, should be reasonably discretized when finding the weights. 2. Within the technical capabilities of the company that seeks a good product design, generate new product variants \(p_{j}\), \(j=1,\ldots,m\), and calculate their AHP values \(A_{ij}\), using the sets of weights \(k_{i}\) from step 1. 3. For each generated product variant \(p_{j}\), form the set \(M_{j}\) of all those combinations of weights \(k_{i}\) for which \(A_{ij}\) is higher than the AHP value of all the currently sold product variants, or at least as high as the AHP value of the current top seller. 4. A suitable product variant is the variant \(p_{k}\) whose set \(M_{k}\) is the largest of all \(M_{j}\) sets. Step 1 works with the top-selling product variant, demanded by a potentially interesting number of clients, given that it is a top seller. Within the AHP approach, this variant should have the highest AHP value among all the variants currently sold, for those clients. Step 1 therefore finds various decision-making weights that lead to this expected result. The found weights encapsulate as an estimate the clients' weights, or their preferences regarding the product features. The knowledge of the clients' weights is important, for if they are known, a new variant, even more successful than the currently most successful variant, can be designed; the design should have such characteristics that, using the known weights, its AHP value will be higher than that of any existing variant. Such a product design could appeal to the market segment. But the weights are not known exactly and so step 1 creates a whole space of weights that is assumed to contain the unknown weights of the clients whose purchases gave rise to the top-selling product. In step 2, each newly generated product variant is evaluated with all the sets of weights obtained in step 1. This way, all these evaluations will contain, within the modelling, the \"true\" evaluations of the variant made by the clients from step 1. Now, since the desire is to create a variant potentially better than any of the existing variants, its AHP value should be the highest (step 3 of the procedure) and with respect to as many step 1 weights as possible (step 4 of the procedure) to increase the chance that it is indeed the best product variant for the discussed client segment, or at least its part. This is because it is unknown where exactly in the step 1 space of weights the clients' true weights are located, so the intent is to cover as much of that space as possible to capture the clients, or as many of them as possible.