# Dynamic Relief Items Distribution Model with Sliding Time Window in the Post-Disaster Environment
## 3 Dynamic RID Model with Sliding Time Window
### Dynamic Path Calculation Using Fuzzy Logic  
In general, the shortest distance route is selected while making the selection from the supply point to the disaster region. Normally, it is assumed that the travel time is proportional to the distance between two locations. However, a disaster may damage the condition of the roads, which eventually impacts the humanitarian relief operations [52]. Therefore, due to the change in road conditions, the travel time and the selection of routes based on prior information in disaster environments may not be the optimal choice, as other factors such as road traffic condition, road geographical status, and road conditions are also crucial in defining travel time. An iterative best route selection strategy in each time slot is required based on updated information about the road conditions. In this dynamic distribution model, a fuzzified distance matrix has been used to identify travel time more realistically. The fuzzified distance matrix is a modification of the \begin{table}
\begin{tabular}{l l} \hline \hline
**Variables** & **Description** \\ \hline N\({}_{\text{c}}\) & Total demand for relief items in disaster regions. \\ \hline V\({}_{\text{t}}\) & Vehicle tours with multiple demand regions. \\ \hline r\({}_{\text{ij}}\) & Assigned DRs in resource (RS) with the jth tour of to the ith vehicle, along with attributes DR; d, dx, dy. \\ \hline f\(\delta\)d(r\({}_{\text{ij}}\)) & A function that gives the partial relief items at r\({}_{\text{ij}}\) demand regions. \\ \hline fdx, dy(r\({}_{\text{ij}}\)) & A function that returns the location of r\({}_{\text{ij}}\) demand region. \\ \hline Vt & The vehicle assigned for transportation. \\ \hline N\({}_{\text{v}}\) & Number of vehicles assigned in relief items scheduling. \\ \hline kmax & Maximum missions planned in resource scheduling. \\ \hline jxi & Executable missions of the assigned ith vehicle. \\ \hline \(\Phi_{\text{i}}\) & The velocity of the ith vehicle. \\ \hline \(\overline{\Psi}_{\text{i}}\) & Cost of the ith vehicle. \\ \hline T\({}_{\text{ij}}\) & Time spent between demand regions r\({}_{\text{ij}-1}\) and r\({}_{\text{ij}}\) of the ith vehicle. \\ \hline T\({}_{\text{offset}}\) & Offset time for the vehicle before starting the next journey. \\ \hline RS & Routing Schedule. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables and description used in the model. traditional distance matrix where the crisp distances between locations are changed into the fuzzified distance. Three fuzzy input variables (Road Condition, Traffic Condition and Number of Turns) and one fuzzy output variable (Weight Factor) are used in the dynamic RID model. There has not been any sufficient appropriate data available for defining the fuzzy memberships and rules for this paper. However, there has been evidence that in the absence of sufficient data, the appropriate knowledge base is useful in defining the fuzzy system [53]. In this model, based on research works on road traffic conditions after disaster environments [54; 55] and generalised knowledge from the literature, the three most appropriate fuzzy variables, values (listed in Table 2) and forty fuzzy rules are defined. These fuzzy variables reflect the generalised circumstances with different levels of travel route status after the disaster. Four fuzzy levels for each fuzzy variable have been applied to describe the road status. Each of these parameters has been fuzzified with different linguistic sets. The trapezoidal membership function is selected for each input and output parameter. For each fuzzy variable, the universe is defined in the range [0 100] and the membership value in the range [0 1]. Fuzzy values are defined by their corresponding range in the universe as shown in the plot in Figure 2. These ranges are defined in terms of the condition of the fuzzy variables.