A soda producer makes and sells two products, Classic Cola and Diet Cola. During the planning period, if the producer spends x1 dollars on promotion of Classic Cola, it can sell 100x1^.5 cases of Classic Cola, and if it spends X^2 dollars on promotion of Diet Cola, it can sell 10x2^.75 cases of Diet Cola. Each case of Classic Cola sells for $12.00 and costs $0.95 to produce and ship to customers, while each case of Diet Cola sells for $12.50 and costs $1.00 to produce and ship to customers. A total of $7,500 is available for promotion during the planning period.
a) Formulate and solve a nonlinear optimization model to help this soda producer identify the best promotional strategies for its two products.
b) Suppose the producer can double the promotional budget. Formulate and solve a nonlinear optimization model to help this soda producer identify the best promotional strategies for its two products in that case. Does the change in profit justify the budget increase? Does the proportional amount spent promoting the two products remain the same?