Dorian Auto has a $$20,000$ advertising budget. Dorian can purchase full$-$page ads in two magazines: Inside Jocks $($IJ$)$ and Family Square $($FS$).$ An exposure occurs when a person reads a Dorian Auto ad for the first time.

The number of exposures generated by each ad in IJ is as follows: ads $1-6,2,500$ exposures; ads $7-10,3,000$ exposures; ads $11-15,10,000$ exposures. For example, $8$ ads in IJ would generate $6(2,500)+2(3,000)=21,000$ exposures.

The number of exposures generated by each ad in FS is as follows: ads $1-4,2,000$ exposures; ads $5-12,6,000$ exposures; ads $13-15,8,000$ exposures. Thus, $13$ ads in FS would generate $4(2,000)+8(6,000)+1(8,000)=64,000$ exposures.

Each full$-$page ad in either magazine costs $$1,000.$ Assume there is no overlap in the readership of the two magazines. Formulate an IP to maximize the number of exposures that Dorian can obtain with limited advertising funds.

$3$

IE $311$

Let $x$ and $y$ respectively represent the number of adds in IJ and FS$.$ The number of exposures in IJ is given as

$E_{IJ}(x)=\{\begin{array}{ccc}2500x& if& 0x6\\ 15000+3000(x-6)& if& 7x10\\ 27000+10000(x-10)& if& 11x15\end{array}$

while the number of exposures in FS is given as

$E_{FS}(y)=\{\begin{array}{ccc}2000y& if& 0y4\\ 8000+6000(y-4)& if& 5y12\\ 56000+8000(y-12)& if& 13y15\end{array}$

Then, the model becomes

$max{E}_{IJ}(x)+E_{FS}(y)(maximize$ the exposure$)$

$s.t.x+y20(budget)$

$0x,y15$ and $x,y$ are integer. $(variable$ domain restrictions$)$

Notice that both $E_{IJ}(x)$ and $E_{FS}(y)$ are nonconcave functions. In order to linearize the above model, we introduce three "copies" of $x$ and $y$ variables each, and control them via newly defined binary variables. The integer linear programming model is given as below:

max,$[2500x_1+3000x_2-3000u_2+10000x_3-73000u_3],$

$+,[2000y_1+6000y_2-16000v_2+8000y_3-40000v_3](maximize$ the exposure$)$

$s.t.(x_1+x_2+x_3)+(y_1+y_2+y_3)20(budget)$

$,0x_{1}6u_1,7u_2{x}_{2}10u_2,11u_3{x}_{3}15u_3(piecewise$ linearization$-E_{IJ})$

$,0y_{1}4v_1,5v_2{y}_{2}12v_2,13v_3{y}_{3}15v_3(piecewise$ linearization$-E_{FS})$

$,u_1+u_2+u_3=1,v_1+v_2+v_3=1(piecewise$ linearization$)$

$,u_1,u_2,u_3,v_1,v_2,v_{3}in\{0,1\}$ and $x_1,x_2,x_3,y_1,y_2,y_3$ are integer. $(variable$ domain restrictions$)$ I couldn't really understand the solution can you explain please$$