## 3 The ZAPROS III-\(i\) Method  
The project manager has among his activities the role of making decisions. To come up with a solution, he has a set of alternatives to choose from. Each alternative has its own characteristics and, in some cases, these may be common to other alternatives. The use of methodologies that support the decision-maker can minimize possible negative impacts caused by wrong decisions due to poor choice of alternatives [20]. VDA aims to provide such support by presenting alternatives to the decision-maker in friendly language and in the most human way as possible. There are many methods within VDA that operate in such a manner. Among these is the ZAPROS III method [20], which provides a more consistent process of eliciting preferences compared with previous methods [20]. It structures preferences of values as the distances between the evaluations of two criteria, denominated quality variations (QV). In addition, it uses the formal index of quality (FIQ) to order established alternatives and minimize the number of pairs of alternatives to be compared to obtain a solution [21]. Some alternatives may be unmatched, and this leads to unsatisfactory results in decision-making models. Thus, the ZAPROS III-\(i\) method was devised, amending ZAPROS III slightly in the process of comparing alternatives to improve the decision method [21]. The use of the ZAPROS III-\(i\) methodology as a means to solve problems of ordering software requirements shows promise, since this method considers, in addition to the factors described in the previous section, the opinion of the project manager. ZAPROS III-\(i\) consists of a VDA method that aims at sorting alternatives in scenarios involving a reduced set of criteria and values and a large number of alternatives. The method relies on obtaining preferences around values that represent the distances between two criteria judgments. A preference scale can be structured, allowing the comparison of alternatives [17]. As explained in [17], the ZAPROS III-_i_ method is structured in three stages: problem formulation, elicitation of preferences, and comparison of alternatives. In the first step, criteria and their values relevant to the decision-making process are obtained. In the second step, the preference scale based on the preference of the decision-maker is generated. This process occurs in two stages: (i) elicitation of preferences for quality variation of the same criterion, and (ii) preference elicitation between pairs of criteria. In the last step, a comparison is made between the alternatives based on the preferences of the decision-maker. For details on the procedure, see [17]. The method follows the same formal statement of the problem proposed in [19; 20]:  
Given:  
1. K = 1, 2,..., N, representing a set of N criteria;  
2. n\({}_{q}\) represents the number of possible values on the scale of q-th criterion, (q \(\in\) K); for ill-structured problems, as in this case, usually n\({}_{q}\)\(\leq\) 4;  
3. \(X_{q}\) = \(\left\{x_{iq}\right\}\) represents a set of values for the q-th criterion, and this set is the scale of this criterion; \(\mid\)\(X_{q}\)\(\mid\) = n\({}_{q}\) (q \(\in\) K), where the values of the scale are ranked from best to worst, and this order does not depend on the values of other scales;  
4. Y = \(X_{1}\)\(\times\)\(X_{2}\)\(\times\)\(\ldots\)\(\times\)\(X_{n}\) represents a set of vectors y\({}_{i}\) (every possible alternative: hypothetical alternatives + real alternatives) in such a way that y\({}_{i}\) = (y\({}_{i1}\); y\({}_{i2}\); \(\ldots\); y\({}_{iQ}\)), and y\({}_{i}\)\(\in\) Y, y\({}_{iq}\)\(\in\)\(X_{q}\) and Q = \(\mid\)Y\(\mid\), such that \(\mid\)Y \(\mid\) = \(\prod^{Q}\)q=1 nq  
5. A = \(\left\{a_{i}\right\}\)\(\in\) Y, i = 1, 2,..., t such that the set of t vectors represents the description of the real alternatives. Required are the ranks of multi-criteria alternatives based on the decision-maker preferences.