## 4 Framework Implementation
### Implementation of Linear Programming  
To identify the best placement in the warehouse, optimizing the transportation or minimizing the pallet movement, data were collected from the logistics department of Decathlon. Table 2 highlights the dimensions of the warehouse, rack, shelf, box, and pallet. Table 3 presents the box quantities based on the percentage and additional information based on the number of products that can be stored in the box. Table 4 shows the weekly product demand from July 2020 to October 2020 based on the information provided by the supplier. The steps from Section 3.1 were followed to formulate the LP model:  
Step A1: Highlight the essential key variables that are needed for the model development. Step A2: Identify the decision variables that need to be optimized, and include the given data:  
The decision variables in this warehouse study are the number of boxes that can be stored on each pallet and placed on the shelves. _Quantity of Box 1 that can be stored on the rack = x_ \begin{table}
\begin{tabular}{c c c c} \hline
**Week** & **Average Demand (Units)** & **Week** & **Average Demand (Units)** \\ \hline Week 27 & 265,127 & Week 35 & 349,649 \\ Week 28 & 403,777 & Week 36 & 462,614 \\ Week 29 & 421,158 & Week 37 & 562,452 \\ Week 30 & 449,974 & Week 38 & 277,279 \\ Week 31 & 368,098 & Week 39 & 209,425 \\ Week 32 & 292,351 & Week 40 & 271,177 \\ Week 33 & 313,496 & Week 41 & 177,547 \\ Week 34 & 316,759 & Week 42 & 134,029 \\ \hline \end{tabular}
\end{table} 
Table 4: Weekly product demand from July to October. \begin{table}
\begin{tabular}{c c} \hline
**Key Information** & **Dimension** \\ \hline \multirow{4}{*}{Warehouse} & Total Storage Area & 557.42 m\({}^{2}\) (6000 Square feet) \\  & Storage Capacity & 11,000 to 12,000 Cartons \\  & Height of the Warehouse & 4 m \\  & Length of the Warehouse & 110 m \\ \hline \multirow{4}{*}{Rack} & Maximum Racking Height of the Boxes & 2.150 m \\  & Number of Racks & 15 \\  & Number of Shelves in a Rack & 3 \\  & (Length * Width * Height) of Rack 1 & 22 m \(\times\) 1.3 m \(\times\) 6.6 m \\  & (Length * Width * Height) of Rack 2 & 19.8 m \(\times\) 1.3 m \(\times\)6.6 m \\  & Total Volume of 15 Rack (13R1 + 2R2) & 2793.65 m\({}^{3}\) \\ \hline \multirow{4}{*}{Shelves} & Length of Shelves & 22 m \\  & Width of Shelves & 1.3 m \\  & Height of Shelves & 2.2 m \\ \hline \multirow{4}{*}{Box} & Maximum Weight & 20 kg \\  & (Length * Width * Height) of Box 1 & 0.6 m \(\times\) 0.4 m \(\times\) 0.4 m \\  & Volume of Box 1 & 0.048 m\({}^{3}\) \\  & (Length * Width * Height) of Box 2 & 0.6 m \(\times\) 0.2 m \(\times\) 0.2 m \\  & Volume of Box 2 & 0.024 m\({}^{3}\) \\  & (Length * Width * Height) of Box 3 & 0.6 m \(\times\) 0.3 m \(\times\) 0.2 m \\  & Volume of Box 3 & 0.036 m\({}^{3}\) \\ \hline \multirow{4}{*}{Pallet} & Maximum Weight & 700 Kg \\  & Length * Width * Height & 1.20 m \(\times\) 0.80 m \(\times\) 0.150 m \\ \cline{1-1}  & Volume & 0.144 m\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Warehouse, rack, shelves, box, and pallet dimensions. \begin{table}
\begin{tabular}{c c c c} \hline
**Boxes** & **Maximum Quantity** & **Percentage Considered for** & **Max Cartons Based on** & **Average Quantities of** \\  & **in a Single Pallet** & **Storing in Warehouse** & **the Percentage** & **Products** \\ \hline Box 1 & 20 & 55–60 & 7200 & 35 to 40 \\ Box 2 & 80 & 20–30 & 3600 & 20 to 25 \\ Box 3 & 40 & 25–10 & 1200 & 15 to 20 \\ \hline \end{tabular}
\end{table} 
Table 3: Box quantities and information based on the percentage and number of products. Quantity of Box 2 that can be stored on the rack = \(x_{2}\) Quantity of Box 3 that can be stored on the rack = \(x_{3}\)_  
Step A3: Formulate the objective function. The objective of this optimization model is to maximize space utilization by placing the maximum number of boxes on the pallets at their respective rack in the warehouse.