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You are going to start a small business to sell premium cupcakes.  The demand for cupcakes will depend on your selling price.  If the selling price is p , then the monthly demand for your cupcakes (in units) is:

monthly demand=50000-1500×p.

You have a budget of $25,000 for production cost to start up your business in the first month.  Due to economy of scales, the average production cost of each cupcake decreases as production increases.  The cost of producing x  cupcakes is:

cost=6x+ √15 x

You should only produce as many cupcakes as you can sell, and you cannot sell more cupcakes than you produce.  Your goal is to maximize your revenue = monthly demand×price  for the first month.

Questions:

1.     Write a non-linear program for the optimization problem.  Allow non-integer solutions!

2.     Set up an Excel spreadsheet.  Compute the optimal production and pricing plan using Solver with central differencing.

3.     Suppose you want to run Excel with multistart.  To save some searching time, list some reasonable bounds to impose on your decision variables.  Show/explain your solution.

·       Hint:  You should not use your solution from Q2!  What is the largest price without making demand negative?  What is the minimum cost per cupcake?

·       You do not need to set up Solver for this question.

4.     Make a new tab and copy your NLP from Question 2.  Suppose you do not have a production budget and can produce as many cupcakes as you want.  Remove the appropriate constraints and re-run Solver to obtain the optimal price.  Why is it not optimal to produce an infinite number of cupcakes?

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