Problem 2: Product Pricing for Remington Manufacturing
Remington Manufacturing is planning its next production cycle. The company can produce three products, each of which must undergo machining, grinding, and assembly operations. The company needs to determine the number of each product to produce to maximize its profit in the next quarter. The following table summarizes the hours of machining, grinding, and assembly required by each unit of each product, and the total hours of capacity available for each operation.
Operation
Machining Grinding Assembly
Product 1 2
4
5
Hours Required By Product 2
3 3 5
Product 3 6
4
2
Total Hours Available per Quarter
500
300
400
The costs of manufacturing products 1, 2 and 3 are $500, $600, and $700 per unit, respectively.
Having established a reputation for high quality and reliability, the company believes it can increase profits by increasing the prices of the products. However, a price increase might have a detrimental effect on demand, so the company has engaged a marketing research firm to estimate the level of demand for its products at various prices. The marketing research firm used the technique of regression analysis to develop a model of the relationship between the prices and demand for the products. After analyzing the situation, the marketing research firm concluded that the company can expect the demand for the products in the next quarter to vary with price in the following way:
Demand for Product 1 = 400 – 0.22 × price of Product 1 Demand for Product 2 = 350 – 0.15 × price of Product 2 Demand for Product 3 = 550 – 0.18 × price of Product 3
The company also needs to determine the number of each product to produce. Apparently, he cannot sell more than the anticipated demand for a product. For example, suppose Product 1 is priced at $920 each, the demand will be 400-(0.22)(920)=197.6. That means the company can sell at most 197.6 products. Let’s assume the demand can be fractional. How many products should the company produce and how much should they be priced for the company to maximize its quarterly profit? Remember the company can only produce integer number of products.
Please provide a mathematical formulation for this problem (defining decision variables, objective function and constraints). You do NOT need to solve it.
Suppose there is a setup cost of $1,000 to produce any number of product 1. That is, the cost will be incurred as long as any product 1 is produced, and the cost is $1,000 regardless of the number of products produced. How would you revise your formulation to incorporate this change?
(Not a continuum of part 2) If the company chooses to produce Product 2, it needs to produce at least 300 of it. How would you revise your formulation to incorporate this change?
(Not a continuum of parts 2 and 3) The production manager of the factory realized that their facility is not designed for production of large volume. As a result, they can produce more than 300 units for at most one of the three products. How would you revise your formulation to incorporate this change?