A company produces two versions of a product. Each version is made from the same raw material that costs 10 per gram, and each version requires two different types of specialized labor to finish. 𝑈 is the higher priced version of the product. 𝑈 sells for 270 per unit and requires 10 grams of raw material, one hour of labor type 𝐴, two hours of labor type 𝐵. Due to the higher price, the market demand for 𝑈 is limited to 40 units per week. 𝑉 is the lower priced version of the product with unlimited demand that sells for 210 per unit and requires 9 grams of raw material, 1 hour of labor type 𝐴 and 1 hour of labor type 𝐵. This data is summarized in the following table:
| Version | Raw Material required | Labor A required | Labor B required | Market demand | Price | | --- | --- | --- | --- | --- | --- | | U | 10g | 1 hour | 2 hours | ≤ 40 units | 270 | | V | 9g | 1 hour | 1 hour | unlimited | 210 |
Weekly production at the company is limited by the availability of labor and the inventory of raw materials. Raw material has a shelf life of one week and must be ordered in advance. Any raw material left over at the end of the week is discarded. The following table details the cost and availability of raw material and labor.
| Resource | Amount Available | Cost |
|---|---|---|
| Raw Material | unlimited | 10/g |
| Labor A | 80 hours/week | 50/hour |
| Labor B | 100 hours/week | 40/hour |
The company wants to maximize its gross profits.
The problem statement above describes an optimization problem. Reformulating the problem statement as a mathematical model involves a few crucial elements:
The starting point is developing a mathematical model is to list decision variables relevant to the problem at hand. Decision variables are quantities that can be modified to achieve a desired outcome. While some decision variables introduced at this stage may prove redundant later, the goal at this point is to create a comprehensive list of variables that will be useful in expressing the problem’s objective and constraints.
For this problem statement, listed below are decision variables with symbols, descriptions, and any lower and upper bounds that are known from the problem data.
| Decision Variable | Description | lower bound | upper bound |
|---|---|---|---|
| 𝑥𝑀 | amount of raw material used | 0 | - |
| 𝑥𝐴 | amount of Labor A used | 0 | 80 |
| 𝑥𝐵 | amount of Labor B used | 0 | 100 |
| 𝑦𝑈 | number of 𝑈 units to produce | 0 | 40 |
| 𝑦𝑉 | number of 𝑉 units to product | 0 | - |
The next step is to formulate an objective function describing that describes how we will measure the value of candidate solutions to the problem. In this case, the value of the solution is measured by profit which is to be maximized.
max profit
Profit, in turn, is equal to the difference between revenue and cost of operations: