Q$1.(20$ pts$)$ Suppose that a logistics company is trying to load $\backslash ($ n $\backslash )$ items to $\backslash ($ m $\backslash )$ trucks. Truck $\backslash ($ j $\backslash )$ has a weight capacity of $\backslash ($ C$\_\{$j$\}\backslash )$ tons and a volume capacity of $\backslash ($ K$\_\{$j$\}\backslash )$ units for $\backslash ($ j$=1,\backslash$ldots$,$ m $\backslash ).$ Each item $\backslash ($ i $\backslash )$ has a weight of $\backslash ($ w$\_\{$i$\}\backslash )$ tons and a volume of $\backslash ($ v$\_\{$i$\}\backslash )$ units for $\backslash ($ i$=1,\backslash$ldots$,$ n $\backslash ).$ The company will incur a profit of $\backslash ($ p$\_\{$i$\}\backslash )$ if item $\backslash ($ i $\backslash )$ is transported. On top of the weight and volume capacity restrictions, each truck can carry at most $15$ items. Item $1$ and item $2$ cannot be loaded in the same truck. Items $3$ and $4$ should be loaded in the same truck $($or they should not be loaded at all$).$ If items $5$ and $6$ are loaded to truck $1,$ then item $3$ should be loaded to truck $2.$ The total weight of items loaded to truck $3$ should be within $3$ tons of the total weight of the items loaded to truck $4.$

The company would like to find out which items to load to each of the trucks to maximize the total profit. Note that the company may not be able to transport all the items. Develop an integer programming model for this problem.