Question

A company gets three byproducts A, B and C, in quantities of 6000, 10,000, and 5000 tons, respectively/month from the manufacturing process of their main product. The company can make a fertilizer consisting of 30% of A, 50% of B, and 20% of C and sell it at a profit $5/ton. They can also make a construction material consisting of 40% of A, 30% of B, and 30% of C and sell it at a profit $4/ton. Any leftover quantities of A,B, and C can be given away to a company dealing with bulk materials.

(1.1) Formulate a linear optimization model to find a plan that maximizes the profit of the company by using these three byproducts and use the primal simplex method to solve the LO model

(1.2) In a specific month, due to some disruption in the production process, the company finds that the amount of byproduct B reduces to 8000 tons. Find the optimal plan for the company in such a month.

(1.3) Based on recent market information, the company finds that the demand for the fertilizer is reduced to 15,000 tons. What is the new optimal plan for the company?

(1.4) The bulk material handling company (BMHC) developed a new plan to make fertilizer and construction material using the byproducts A, B and C. The BMHC wishes to force the manufacturing company to stop making fertilizer and construction material from their byproducts. Please find the optimal pricing strategy for BMHC to minimize its overall cost.

Answer 1

A linear optimization model is used to maximize profit from byproducts A, B, and C via the production of fertilizer and construction material, with constraints based on byproduct availability. Adjustments to this model are made depending on changes in byproduct amounts or product demand. For BMHC's optimal pricing strategy, a price is offered that is below the company's potential profit from production but above the market price.

Step-by-step explanation:

Linear Optimization Model for Profit Maximization

To maximize profit from the byproducts A, B, and C, we can set up a linear optimization model with two products: fertilizer and construction material. The objective function is to maximize profit, where Profit = $5 × (Tons of Fertilizer) + $4 × (Tons of Construction Material), subject to the constraints of byproduct availability and the percentage composition of each product. Without the specific demands for each of the byproducts in each product, an assumed percentage is utilized for the calculations:

• For fertilizer: 30% of A, 50% of B, and 20% of C
• For construction material: 40% of A, 30% of B, and 30% of C

The primal simplex method is used to solve this linear optimization model. The constraints are the quantities available of each byproduct, which limits how much of each final product can be produced. When the availability of byproduct B reduces to 8000 tons or the demand for fertilizer is reduced to 15,000 tons, the linear optimization model needs to be adjusted and resolved.

Optimal Pricing Strategy for BMHC

To minimize the overall cost for BMHC and deter the manufacturing company from making their own fertilizer and construction material, BMHC would need to develop a pricing strategy that makes it more cost-effective for the company to sell the byproducts A, B, and C directly to BMHC rather than producing the end products themselves.

This involves calculating the opportunity cost of the manufacturing company producing versus selling the byproducts and offering a price that is lower than the potential profit from production yet higher than the market price.