A company produces two products, A and B, which are mutaully substitutable to a certain degree.  The expected quantity sold (demand) for each product is dependent on the price of both products dictated by the following two functions where Q_A and Q_B denote the quantity sold (demand) of product A and product B respective while P_A and P_B denote the unit price of product A and product B respectively.

Q_A = 3600 - 11P_A + 1P_B

Q_B = 1800 - 9P_B + 3P_A

The two functions provide the relationship between the demand and the price.  For example, demand of product A drops as the price of A increases, but increases as the price of product B rises, i.e. some customers buying product A as a substitute of B when B becomes expensive. [The interpretation is based on the sign of P_A and P_B in the function.]

Resources usage per unit of products A and B as well as total amount of resources available are given in the table below.

	Product A	Product B	Total Resources
Labor	2	5	4500
Machine	3	3	4000

(For example, each unit of product A requires 2 labor hours and 3 machine hours. There are 4500 labor hours and 4000 machine hours available.)

 

Formulate a quadratic program (quadratic objective function with linear constraints) to maximize total revenue subject to the two resources contraints (labor and machine) using decision variables P_A and P_B.  You can start with a draft formulation which also includes  Q_A and  Q_B.  Then, substitute Q_A and  Q_B with P_A and P_B using the two functions.  [hint: total revenue = P_A*Q_A+P_B*Q_B]  Hence, the purpose of this quadratic program is to find the best pricing policy to maximize total revenuel

[Rearrange the terms so that the constant appears on the right hand side.  Due to the variable substitution and rearrangement of terms, there could be a few negative coefficients or right hand sides.  Make sure you include a negative sign for a negative value when filling in your answer.]

Max.  P_A +  P_B +  P_A²+  P_B² +  P_AP_B

 P_A +  P_B  ≤  

 P_A +  P_B  ≤  

P_A, P_B ≥ 0