The director of advertising for a retail chain is considering how to allocate her $$200,000$ budget for television advertising among four programs $($A$,$ B$,$ C$,$ D$)$ on three channels $(1,2,3).$ Market research studies have shown that the chain$$s customers can be broken down into two groups: $$High$-$end Achievers$$ and $$Aspiring Achievers.$$ High$-$end Achievers spend twice as much in the store as do Aspiring Achievers. The director wants to maximize the total in$-$store spending potential of the audience but must have at least three ads on each program, and cannot spend more than $50\%$ of the budget on any one channel. The audience by program and the ad costs are below.