Time Allocation for News Program. The program manager for Channel $10$ would like to determine the best way to allocate the time for the $11$:$00-11$:$30$ evening news broadcast. Specifically, she would like to determine the number of minutes of broadcast time to devote to local news, national news, weather, and sports. Over the $30-$minute broadcast, $10$ minutes are set aside for advertising. The station's broadcast

policy states that at least $15\%$ of the time available should be devoted to local news coverage; the time devoted to the combination of local news and national news must be at least $50\%$ of the total broadcast time; the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment; the time devoted to the sports segment should be no longer than the total time spent on the local and national news; and at least $20\%$ of the time should be devoted to the weather segment. The production costs per minute are $$300$ for local news, $$200$ for national news, $$100$ for weather, and $$100$ for sports. LO $2,3,5$

a$.$ Formulate and solve a linear program that can determine how the $20$ available minutes should be used to minimize the total cost of producing the program.

b$.$ Interpret the dual value for the constraint corresponding to the available time. What advice would you give the station manager given this dual value?

c$.$ Interpret the dual value for the constraint corresponding to the requirement that at least $15\%$ of the available time should be devoted to local coverage. What advice would you give the station manager given this dual value?

d$.$ Interpret the dual value for the constraint corresponding to the requirement that the time devoted to the local and the national news must be at least $50\%$ of the total broadcast time. What advice would you give the station manager given this dual value?

e$.$ Interpret the dual value for the constraint corresponding to the requirement that the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment. What advice would you give the station manager given this dual value?