Question

A company produces three products, A, B, and C. The sales volume for A is at least 50% of the total sales of all three products. However, the company cannot sell more than 75 units of A per day. The three products use one raw material, of which the maximum daily availability is 240lb. The usage rates of the raw material are 2lb per unit of A,4 lb per unit of B, and 3lb per unit of C. The unit prices for A,B, and C are $20,$50, and $35, respectively. a) Determine the optimal product mix for the company. b) Determine the dual price of the raw material resource and its allowable range. If available raw material is increased by 120lb, determine the optimal solution and the change in total revenue using the dual price. c) Use the dual price to determine the effect of changing the maximum demand for product A by ±10 units.

Answer 1

To determine the optimal product mix for the company, we calculate the total sales volume of all three products and set up relevant constraints. By maximizing the revenue and considering the dual price of the raw material resource, we can determine the optimal solution.

Step-by-step explanation:

To determine the optimal product mix for the company, we need to consider the sales volume, resource availability, and unit prices.

Let's calculate the total sales volume of all three products: A, B, and C. Since the sales volume for A should be at least 50% of the total sales, we can assume A at 50% and divide the remaining 50% equally between B and C.

This means A = 0.5x, B = 0.25x, and C = 0.25x, where x is the total sales volume.

We have a constraint on the maximum sales of A, which is 75 units per day. The usage rates of the raw material are 2lb per unit of A, 4lb per unit of B, and 3lb per unit of C.

Assuming the maximum daily availability of the raw material is 240lb, we can set up the following equations: 2A + 4B + 3C ≤ 240 and A ≤ 75.

To optimize the product mix, we need to maximize the revenue. The revenue for each product is calculated by multiplying the unit price with the sales volume.

The total revenue is the sum of the revenue from each product. By solving these equations and considering the allowable range of the dual price for the raw material resource, we can determine the optimal product mix and the dual price of the raw material resource.

Answer 2

To find the optimal product mix and the dual price of the raw material for the company, one must consider the sales volume of product A and the raw material constraints. For the exercise regarding Doggies Paradise Inc., we calculate total revenue, marginal revenue, total cost, and marginal cost for different output levels and identify the profit maximizing quantity where marginal cost equals marginal revenue.

Step-by-step explanation:

To determine the optimal product mix for the company manufacturing products A, B, and C, we must consider several constraints. The sales volume of product A should be at least 50% of the total sales, and the maximum units of A sold per day cannot exceed 75 units. Additionally, there's a constraint on the raw materials, with a maximum daily availability of 240lb. The raw material usage rates are 2lb, 4lb, and 3lb per unit for products A, B, and C respectively. With unit prices at $20, $50, and $35 for A, B, and C, the objective is to maximize revenue.

To solve part b), determining the dual price of the raw material resource, we would usually rely on linear programming methods such as the simplex algorithm to find the shadow prices or dual values. As the availability of the raw material increases by 120lb, the optimal solution and the change in total revenue can be recalculated using the dual price derived from the original problem.

For part c), the change in the maximum demand for product A by ±10 units, we would adjust the constraints for product A's production and recalculate the optimal mix, using the dual price to assess the impact on total revenue due to this change in constraint.

Profit Maximizing Quantity for Doggies Paradise Inc.

Considering the information given for Doggies Paradise Inc., we can compute the total revenue, marginal revenue, total cost, and marginal cost for each output level. By plotting the total revenue and total cost curves on one diagram, and the marginal revenue and marginal cost curves on another, we can identify the profit maximizing quantity where marginal cost equals marginal revenue.