## 4. Specified Problem and Multi-Objective Function Derived for Reliable Cloud Storage  
### Problem Definition  
Other than the above issues, cloud data storage mainly focuses on resolving the general and specific constraints in the components where the cloud data are stored effectively [15]. To define the issues in the reliable storage process, let us discuss the components and VMs in the cloud network. Here, the components are represented by \(N=\{N_{1},N_{2},\ldots,N_{C}\}\) and VMs are denoted as \(R=\{R_{1},R_{2},\ldots,R_{M}\}\). Every component is defined with requirement sets along with hardware traits. Similarly, each machine is composed of software or hardware features. These all come under the structural constraints among the components. Hence, the problems are listed below:  
1. An assignment matrix is constructed with the components and VMs, which are interpreted by each other. Hence, the entities in the matrix are either 0 or 1. This is given by Equation (1). \[\text{A}_{xy}= \begin{cases}1,&\text{if}\quad N_{x}\text{ is given to }R_{y}\\ 0,&\text{if}\quad N_{x}\text{ is not given to }R_{y}\end{cases} \] (1)
2. The type of every machine is denoted by its selection vector as \(v_{y}\), where \(y\in 1\) to \(M\)  
Finally, all the constraints are: (i) the structural constraints; (ii) the hardware requirements (capacity constraints) of all components are satisfied; (iii) the purchasing/leasing price is minimized. The assigned matrix and its type selection are elaborated in Equations (2) and (3). \[\text{A}=\left( \begin{array}{cccccc}0&0&0&0&1&0\\ 0&0&1&1&0&0\\ 0&1&0&0&0&0\\ 1&0&0&0&0&0\\ 0&1&1&1&0&0\end{array} \right) \tag{2}\]  
\[v=\left[12,12,13,13,15,0\right] \tag{3}\]  
Both the general and structural constraints are explored below. General constraints:  
While considering every component for allocating the data into any VMs, the allocation rules are specified concerning exclusive deployment. This is shown in Equation (4). \[\sum_{y=1}^{M}\text{A}_{xy}\geq 1,\quad\text{where }x=1\text{ to C} \tag{4}\]  
The capacity constraints consider solving the type of resource presented in the VM, which is derived using Equation (5). \[\sum_{x=1}^{N}\text{A}_{xy}\cdot\text{I}_{x}^{p}\leq\text{I}_{y_{y}}^{p},\quad \text{where }x=1\text{ to C} \tag{5}\]One VM is linked with the data occupied by another VM. The term \(\wedge\) is denoted as the wedge operator, which helps estimate two events' occurrence. Hence, the VM offers or links to store the cloud data are represented using Equation (6). \[u_{y}=1\wedge v_{y}=0\Rightarrow Q_{y}^{p}=J_{v_{y}}^{p}\forall p\in(1,P)\wedge r _{y}=R_{v_{y}} \tag{6}\]  
Furthermore, several VMs are used for deployment and estimation purposes. Here, \(M\) represents the number of VMs used for deployment. Moreover, the variable \(u\) determines the vector of the binary occupancy. It is defined by the vector value and given in Equation (7). \[ \begin{split}\sum_{x}^{N}\text{A}_{xy}=0\Rightarrow v_{y}=0, \quad\text{where $y=1$ to M}\\ \sum_{x}^{N}\text{A}_{xy}\geq 1\Rightarrow u_{y}=1,\quad\text{ where $y=1$ to M}\end{split}  \tag{7}\]  
Structural or application-oriented constraints:  
These two major constraints are taken as interactions among the components and instances, which are explained below:  
* Conflict: When using the same VM for allocating the data, such conflicts are raised. The variable \(\Im_{xy}\) is denoted as the conflicts among the two components encoded in the matrix \(\Im\). The conflicts are in the matrix format shown in Equation (8). \[\text{A}_{xy}+\text{A}_{zy}\leq 1,\quad\text{where $\Im_{xy}=1$}\] (8)
* Co-location: This relation is given in the matrix when utilizing the same VM to define the constraints. Hence, \(\Re_{xy}\) represents the co-location relation stored in matrix \(\aleph\). It is described in Equation (9). \[\text{A}_{xy}=\text{A}_{zy},\quad\text{where $\Re_{xy}=1$}\] (9)
* Exclusive deploying: From the different sets of components, only one is used for storage purposes. In that scenario, the constraint is defined by Equation (10).