Maximizing Revenue from Coffee Sales: Linear Programming Approach

QUESTION

a) You are considering producing two types of drink: Original and Venti, both of which consist solely of milk, cream cheese, and coffee

At present, you have in stock

  1. 100 oz of coffee
  2. 20 oz of cream cheese, and
  3. 30 oz of milk

The mixture used to make Venti must contain at least 20% cream cheese.

The mixture used to make Original must have at least 10% cream cheese and 10% milk.

Both types of coffee must have at least 50% coffee.

Each ounce of Venti can be sold for 25¢ , and each ounce of Original for 20¢. Formulate an LP that will enable you to maximize your revenue from coffee sales.

b) Your partner Jane suggests that the selling price of your product should change according to the demand at different times of the year. She approximates that the selling price should increase by 10% in the fourth quarter of the year, and decrease by 20% in the second quarter of the year. Assume her approximations are accurate and you can solve the LP as often as you want, explain how would you change your current approach in part (a) to accommodate her suggestions. How much would these adjustments change the optimal solution and why?

ANSWER

Maximizing Revenue from Coffee Sales: Linear Programming Approach

To maximize revenue from coffee sales, we can use linear programming (LP) to find the optimal quantities of Original and Venti drinks to produce. Let’s define the decision variables:

  • Let � be the number of ounces of Venti produced.
  • Let � be the number of ounces of Original produced.

The objective is to maximize the revenue, which is the sum of the revenues from Venti and Original: Maximize 0.25�+0.20�

Now, let’s set up the constraints based on the available resources and the composition requirements:

  1. Coffee Constraint: The coffee used in both drinks must not exceed the available coffee. �+�≤100
  2. Cream Cheese Constraint for Venti: Venti must contain at least 20% cream cheese. 0.20�≤20
  3. Cream Cheese Constraint for Original: Original must have at least 10% cream cheese. 0.10�≤20
  4. Milk Constraint for Original: Original must have at least 10% milk. 0.10�≤30
  5. Coffee Composition Constraint: Both drinks must have at least 50% coffee. �≥0.50�+0.50� �−0.50�−0.50�≥0

The final LP can be summarized as follows: Maximize 0.25�+0.20� Subject to: �+�≤100 0.20�≤20 0.10�≤20 0.10�≤30 �−0.50�−0.50�≥0

Adjusting Prices According to Seasonal Demand

Incorporating Jane’s suggestion to adjust prices based on the demand at different times of the year requires modifying the objective function to account for varying prices. Let � represent the price multiplier:

1 & \text{First and Third Quarters} \\ 1.1 & \text{Fourth Quarter} \\ 0.8 & \text{Second Quarter} \end{cases} \] The objective function becomes: \[ \text{Maximize } (0.25P)x + (0.20P)y \] The rest of the constraints remain the same. The linear programming problem now aims to maximize revenue considering the price adjustments based on seasonal demand. **Effect of Seasonal Price Adjustments on Optimal Solution** These price adjustments will impact the optimal solution by affecting the balance between producing Venti and Original drinks. In the fourth quarter, with a 10% price increase, the LP might lean more towards producing Venti to capitalize on the higher selling price. Conversely, the 20% price decrease in the second quarter might shift the production balance towards Original to maximize revenue despite the lower individual selling price. The optimal solution would change because the price adjustments alter the trade-off between selling price and production quantity for each type of drink. The LP solver, when considering these adjustments, will determine the quantities of Venti and Original drinks that yield the highest overall revenue, considering both the composition constraints and the variable prices. In conclusion, incorporating seasonal price adjustments into the linear programming approach allows the business to adapt to changing demand patterns. The optimal solution will vary based on the price multipliers, impacting the production quantities of different drink types to maximize revenue. This dynamic approach optimizes revenue under varying demand scenarios, offering flexibility and responsiveness to market conditions.

 

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