The formulation of this problem has given below:

Suppose a company has the option of advertising its product on certain months with a campaign cost of CH/month. If a campaign is on, we expect sales to go up by 15% on that month. But, in the next month we see a decrease of 8% in sales. Suppose we have forecasted demand of D(t) for next 6 months. A monthly capacity of CAP units per month is available for production. If inventory cost is h/unit/month and production cost (p(t)) is variable over the planning horizon due to inflation, formulate the problem to maximize profits, given a fixed sales price of P.

QUESTION1: Find a way to maximize NET PRESENT VALUE OF PROFITS in this formulation

QUESTION2: Also, add multiple products to this formulation and also add a monthly budget for multiple product campaigns.

The formulation of this problem:

Answer: Maximize sum(t, P*S(t)-CH*X(t)-p(t)*PROD(t)+h I(t))

s.t. S(t)= D(t)+0.15 *D(t) *X(t)-0.08* D(t)* X(t-1) for all t

I(t) = I(t-1)+PROD(t)-S(t) for all t

PROD(t) ≤ CAP for all t

Where X(t): binary variable indicating if there is campaign in month tPROD(t): production quantity in month tS(t): Sales quantity in month t