## 3 Formalization of Dima Optimization
### Objectives  
The objectives of DIMA optimization can be considered from quality measures in an avionics system, which can be classified into two categories, that is, effectiveness and cost. Effectiveness refers to the capacity of the system, such as safety, reliability, availability, and scalability. They are summarized as follows:  
* Safety: it can be measured by the number of harmful events during the flight, that is, \(\frac{N_{h}}{T}\), where \(N_{h}\) is the number of harmful events and \(T\) is the number of flight hours. * Reliability: it can be modeled as an exponential function of failure rate, that is, \(R(t)=e^{-\frac{t}{t}}\), where \(r=\frac{N_{f}}{T}\) with \(N_{f}\) as the number of failures during the flight and \(t\) is the elapsed time. * Availability: it can be modeled as the ratio between Mean Time To Failure (MTTF) \(\overline{T_{f}^{\prime\rho}}\) and Mean Time Between Failures (MTBF) \(\overline{T_{f}^{\prime\rho}}\), that is, \(A=\frac{\overline{T_{f}^{\prime\rho}}}{T_{f}^{\prime\rho}}\). MTTF is the averaged elapsed time from normal operation to a failure, which indicates the averaged operation time without failures for system availability. MTBF is the averaged elapsed time between two failures. The gap between MTBF and MTTF is MTTR, which is short for Mean Time To Repair that turns the system back to normal condition from failures. Ideally, we seek to achieve low MTTR and high MTTF for high system availability. * Scalability: it can be considered as the percentage of spare resources, that is, \(\frac{R_{spare}}{R_{total}}\), where \(R_{spare}\) refers to as the spare resources and \(R_{total}\) refers to as the total resources. The cost refers to the operating cost and acquisition cost of the system, such as the complexity, weight, power consumption, maintainability and mass from an economic perspective. They are summarized as follows:  
* Complexity: the ratio between the number of DIMA devices \(m\) and the number of device types \(m_{T}\), \(m/m_{T}\). * Weight: the mass of all the DIMA devices. * Operation: the costs for operation include Ship Set Costs (SSC), Operational Interruption Costs (OIC), and Initial Provisioning Costs (IPC). SSC are the recurring costs for installing all the devices and wires onto the avionics system. OIC are the average costs due to flight delays and cancelations for tackling unscheduled maintenance events, which depends on the reliability of avionics devices and the criticality of functions hosted by the devices. IPC are the costs for provisioning spare parts to guarantee an interruption-free operation; it is increased by a higher number of devices or lower MTBF. * Power consumption: power consumed by DIMA devices. * Maintainability: MTTR is used to measure the maintainability of items that need to be repaired. Long MTTR indicates that it is difficult to perform maintenance. Particularly, the operational costs, that is, OIC and IPC are affected by MTBF from the performance aspect. The quality measures can be evaluated through a discrete event simulation method [36; 37]. As an example, we can consider the mass of devices as follows. The total mass of devices can be written as  
\[\sum_{D\in\mathcal{D}}M_{D}\cdot\mathscr{K}(D), \tag{13}\]  
where \(M_{D}\) is the mass of device \(D\). The total mass of cables is expressed as  
\[\sum_{ \begin{subarray}{c}D,D^{\prime}\in\mathcal{D}\\ L,L^{\prime}\in\mathcal{L}\end{subarray} }\rho\cdot d_{L,L^{\prime}}\cdot \mathscr{K}(D,D^{\prime})\cdot\mathscr{K}(D,L)\cdot\mathscr{K}(D^{\prime},L^{ \prime}), \tag{14}\]  
where \(d_{L,L^{\prime}}\) is the distance between locations \(L\) and \(L^{\prime}\), and \(\rho\) is the mass of the cable per unit length. Note that the objective involves a product term \(\mathscr{K}(D,D^{\prime})\cdot\mathscr{K}(D,L)\cdot\mathscr{K}(D^{\prime},L^ {\prime})\). However, since all the variables are binary, standard tricks can be applied so that the term could be transformed into a set of linear constraints.