## 5 Pareto-Based Decision Making for Placement of Storage Containers  
The users of Cloud storage services usually need to consider multiple quality requirements, which often appear to be conflicting. Finding an optimal placement option for the necessary storage containers, that is, the deployment of the storage infrastructure is a complex task. Particularly, altering one parameter may strongly influence the other parameters. For instance, higher storage availability requires increased system redundancy, which could lead to higher operational costs. In order to facilitate the process, the proposed solution implements the concept of domination and Pareto optimality. This concept of domination in the field of multi-criteria decision making has been used and described in prior studies [54; 55]. Pareto front is an efficient tool in the decision-making process, because the shape and spread of the Pareto front can provide efficient exploration of the non-dominated solutions' space and investigate the trade-offs. In this section, we briefly introduce the reader to the multi-objective optimisation problem and to some of the related terms, and then finally express the investigated problem as a multi-objective optimisation problem. For a set of solutions \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) and a set of objective functions \(f_{i}:X\rightarrow\mathbb{R}\), \(i\in\{1,2,\ldots,m\}\), the efficient set is a set of non-dominated solutions. A solution \(x_{k}\) dominates solution \(x_{\ell}\) if and only if \(x_{k}\) is not worse than \(x_{\ell}\) in any of the objectives and is strictly better in at least one objective. The Pareto front is the objective space of non-dominated solutions. The efficient set is the solution to the corresponding multi-objective optimisation problem. In our problem, we are concerned with storage nodes selection in such a way that the SLOs are respected. This can be effectively translated to the constrained multi-objective optimisation problem, as follows. The set of solutions is represented by storage nodes. The objective functions characterise the storage nodes in terms of the SLA parameters as defined in Section 4.4; thus, the number of objectives is three. What means better depends on each of the SLOs. For the availability and throughput, higher values are better than smaller while for the cost, smaller values are better than higher. By negating the availability and throughput values, the dominance relation translates to minimisation in all the objectives. The method used to calculate the efficient set (and the Pareto front) is non-dominated sorting, such as the algorithm in NSGA-II [56] used for deriving the Pareto front of one generation of solutions. However, in our setting, no genetic operations of crossover and mutations are required, because all the solutions are known in advance. In other words, as we do not implement data partitioning (i.e., split a single file into data chunks and distribute them into several storages), no service composition is required. In terms of SLA, all the solutions presented in a Pareto set are feasible, i.e., they comply with the SLOs. Essentially, the Pareto set is important from the user perspective as it provides control over the optimisation step. It enables a user to express the QoS requirements and later on select the preferred storage option among the offered Paretooptimal storage infrastructures (deployed storage containers). After a point is selected, the point is translated to a decision vector, expressing the relative preference over multiple objectives. The decision vector is written into an SLA specification (see Section 4.3 for the format). Let \(x^{*}\in X^{*}\) denote a selected point in the Pareto set \(X^{*}\). Then, the decision vector \(w\) is obtained as  
\[w=\frac{x^{*}}{|x^{*}|},\]  
i.e., \(w\) equals to the selected point normalised by its second norm. Let \((c,r,p)\) respectively denote storage cost, reliability and performance objectives.