A company produces two products, A and B, which are mutaully substitutable to a certain degree. The
Question:
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 QA and QB denote the quantity sold (demand) of product A and product B respective while PA and PB denote the unit price of product A and product B respectively.
QA = 3600 - 11PA + 1PB
QB = 1800 - 9PB + 3PA
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 PA and PB 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 PA and PB. You can start with a draft formulation which also includes QA and QB. Then, substitute QA and QB with PA and PB using the two functions. [hint: total revenue = PA*QA+PB*QB] 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. PA + PB + PA2+ PB2 + PAPB
PA + PB ≤
PA + PB ≤
PA, PB ≥ 0
Expert Answer:
