## 5 Problem Formulation
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**S. No.** & **Appliance** & **Category** & \begin{tabular}{l} **Power Rating** \\ \(\rho_{e,n}\) **(KW)** \\ \end{tabular} & \begin{tabular}{l} **Starting** \\ **Time (a)** \\ \end{tabular} & \begin{tabular}{l} **Ending** \\ **Time (\(\beta\))** \\ \end{tabular} & \
\begin{tabular}{l} **Time-Span** \\ **(\(\beta-a\)) (h)** \\ \end{tabular} &
\begin{tabular}{l} **LOT** \\ **(h)** \\ \end{tabular} \\ \hline
1 & Fridge-1 & Fixed & 0.3 & 00 & 24 & 24 & 24 \\
2 & Interior Lighting-1 & Fixed & 0.84 & 18 & 24 & 06 & 6.0 \\
3 & Dish Washer-1 & Elastic & 2.0 & 09 & 17 & 08 & 2.0 \\
4 & Washing Machine-1 & Elastic & 0.6 & 09 & 12 & 03 & 1.5 \\
5 & Spin Dryer-1 & Elastic & 2.5 & 13 & 18 & 05 & 1.0 \\
6 & Cook Top-1 & Elastic & 3.0 & 08 & 09 & 01 & 0.5 \\
7 & Over-1 & Elastic & 5.0 & 18 & 19 & 01 & 0.5 \\
8 & Microwave-1 & Elastic & 1.7 & 08 & 09 & 01 & 0.5 \\
9 & Laptop-1 & Elastic & 0.1 & 18 & 24 & 06 & 2.0 \\
10 & Desktop-1 & Elastic & 0.3 & 18 & 24 & 06 & 3.0 \\
11 & Vacuum Cleaner-1 & Elastic & 1.2 & 09 & 17 & 08 & 0.5 \\
12 & Electrical Car-1 & Elastic & 3.5 & 18 & 08 & 14 & 3.0 \\ \hline
1 & Fridge-2 & Fixed & 0.25 & 00 & 24 & 24 & 24 \\
2 & Interior Lighting-2 & Fixed & 0.9 & 19 & 24 & 07 & 7.0 \\
3 & Dish Washer-2 & Elastic & 1.9 & 11 & 15 & 04 & 2.0 \\
4 & Washing Machine-2 & Elastic & 0.5 & 10 & 14 & 04 & 2.0 \\
5 & Spin Dryer-2 & Elastic & 2.0 & 10 & 16 & 06 & 2.0 \\
6 & Cook Top-2 & Elastic & 3.5 & 09 & 10 & 01 & 0.5 \\
7 & Owen-2 & Elastic & 5.4 & 17 & 20 & 03 & 1.5 \\
8 & Microwave-2 & Elastic & 1.9 & 07 & 09 & 02 & 0.8 \\
9 & Laptop-2 & Elastic & 0.09 & 16 & 23 & 07 & 3.0 \\
10 & Desktop-2 & Elastic & 0.28 & 14 & 20 & 06 & 2.0 \\
11 & Vacuum Cleaner-2 & Elastic & 1.4 & 10 & 16 & 06 & 1.5 \\
12 & Electrical Car-2 & Elastic & 3.3 & 16 & 09 & 17 & 4.0 \\ \hline
1 & Fridge-3 & Fixed & 0.5 & 00 & 24 & 24 & 20 \\
2 & Interior Lighting-3 & Fixed & 0.62 & 17 & 06 & 13 & 13 \\
3 & Dish Washer-3 & Elastic & 2.5 & 10 & 16 & 06 & 2.5 \\
4 & Washing Machine-3 & Elastic & 0.8 & 08 & 14 & 06 & 1.8 \\
5 & Spin Dryer-3 & Elastic & 2.5 & 13 & 19 & 06 & 1.0 \\
6 & Cook Top-3 & Elastic & 3.2 & 07 & 09 & 02 & 0.5 \\
7 & Over-3 & Elastic & 5.3 & 16 & 18 & 02 & 1.5 \\
8 & Microwave-3 & Elastic & 1.9 & 10 & 14 & 04 & 1.0 \\
9 & Laptop-3 & Elastic & 5.3 & 16 & 18 & 02 & 1.5 \\
8 & Microwave-3 & Elastic & 1.9 & 10 & 14 & 04 & 1.0 \\
9 & Laptop-3 & Elastic & 0.2 & 16 & 24 & 08 & 2.5 \\
10 & Desktop-3 & Elastic & 0.4 & 18 & 20 & 02 & 1.0 \\
11 & Vacuum Cleaner-3 & Elastic & 1.3 & 11 & 12 & 01 & 0.5 \\
12 & Electrical Car-3 & Elastic & 3.4 & 16 & 07 & 11 & 5.0 \\ \hline
1 & Fridge-4 & Fixed & 0.4 & 00 & 24 & 24 & 18 \\
2 & Interior Lighting-4 & Fixed & 0.7 & 19 & 08 & 13 & 13 \\
3 & Dish Washer-4 & Elastic & 2.3 & 08 & 19 & 11 & 4.0 \\
4 & Washing Machine-4 & Elastic & 0.9 & 11 & 14 & 03 & 1.0 \\
5 & Spin Dryer-4 & Elastic & 2.0 & 14 & 20 & 06 & 1.0 \\
6 & Cook Top-4 & Elastic & 3.5 & 10 & 12 & 02 & 1.2 \\
7 & Over-4 & Elastic & 5.5 & 10 & 11 & 01 & 0.8 \\
8 & Microwave-4 & Elastic & 1.9 & 10 & 14 & 04 & 1.5 \\
9 & Laptop-4 & Elastic & 0.15 & 11 & 23 & 12 & 4.0 \\
10 & Desktop-4 & Elastic & 0.4 & 09 & 24 & 15 & 6.0 \\
11 & Vacuum Cleaner-4 & Elastic & 1.5 & 11 & 16 & 05 & 1.2 \\
12 & Electrical Car-4 & Elastic & 4.0 & 10 & 22 & 12 & 4.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Appliances and their running time constraints [31]. LOT, Length of Operational Time.