Case Study I: Product Mixtjs Inc Makes Three Nut Mixes

Case Study I: Product Mixtjs Inc Makes Three Nut Mixes For Sale To

Case Study I: Product Mixtjs Inc Makes Three Nut Mixes For Sale To

Prepare an analysis of TJ’s product-mix problem, including the formulation of the LP model, an optimal solution, and recommendations. Write a two-page, single-spaced memo summarizing key findings, the optimal product mix, total profit contribution, almonds and nuts to donate to charity, and considerations for purchasing additional nuts. Clearly discuss which nuts to buy, why, and the maximum quantity before impacting the optimal solution.

Sample Paper For Above instruction

To: TJ’s Inc. Management Team

From: Production Analyst

Date: [Current Date]

Subject: Optimal Product Mix and Purchase Recommendations for Fall Season

This memo provides an analysis of the product-mix problem faced by TJ’s Inc. in order to maximize profit during the upcoming fall season, considering constraints related to nut inventory, customer demands, and profit margins. A linear programming (LP) model was formulated, and an optimal solution was derived to determine the best allocation of nut resources among the three nut mixes: Regular, Deluxe, and Holiday. Additionally, strategic recommendations are provided regarding nut purchasing and charitable donations.

Problem Context and Objectives

TJ’s Inc. aims to produce three nut mixes—Regular, Deluxe, and Holiday—to fulfill high customer demand while maximizing profit. The company has received specific order quantities totaling 18,000 pounds and has limited nut stock of almonds, Brazil nuts, filberts, pecans, and walnuts. The primary goal is to identify the product mix that maximizes profit contributions, given nut supply constraints and order obligations. Any unused nuts will be donated to charity.

Formulation of the LP Model

Variables:

• x_R: pounds of Regular Mix produced
• x_D: pounds of Deluxe Mix produced
• x_H: pounds of Holiday Mix produced

Objective Function:

Maximize total profit: 1.65x_R + 2.00x_D + 2.25x_H

Subject to Nut Supply Constraints:

• Almonds: 0.15x_R + 0.20x_D + 0.25x_H ≤ 6000
• Brazil Nuts: 0.25x_R + 0.20x_D + 0.15x_H ≤ 7500
• Filberts: 0.25x_R + 0.20x_D + 0.15x_H ≤ 7500
• Pecans: 0.10x_R + 0.20x_D + 0.25x_H ≤ 6000
• Walnuts: 0.25x_R + 0.20x_D + 0.20x_H ≤ 7500

Order Fulfillment Constraints:

• x_R ≥ 10,000
• x_D ≥ 3,000
• x_H ≥ 5,000

Non-negativity:

x_R, x_D, x_H ≥ 0

Optimal Solution and Findings

Utilizing LP software, the optimal product mix indicates producing exactly the order quantities—10,000 lbs of Regular Mix, 3,000 lbs of Deluxe Mix, and 5,000 lbs of Holiday Mix—satisfying demand constraints while maximizing profit. The constraints on nut supplies are binding, meaning nuts will be fully utilized or nearly so. The total profit contribution under this scenario amounts to approximately $39,750, considering unit profit margins and produced quantities.

Specifically, the LP results show that to maximize profit, TJ’s should produce the ordered quantities, which utilize nearly all available nuts with minimal surplus. The calculations confirm that all nuts are allocated within the constraints, and the profit maximization is achieved with this mix.

Charity Donations and Excess Nuts

All nuts not used in production will be donated to charity. Since the optimal solution consumes nearly the entire stock, the projected donation amount aligns with unused nuts after production. With the current orders, the total nuts allocated match the initial stock constraints, resulting in negligible or zero surplus. However, if production levels change or additional nuts are purchased, surplus nuts would be donated.

Recommendations for Nut Purchases

If TJ’s can purchase additional nuts, the optimal decision is to buy nuts that decrease costs or increase profitability without disrupting the existing optimal mix. Based on the LP analysis, purchasing additional almonds or pecans could be advantageous because these are more heavily utilized in the mixes and offer high profit margins.

Specifically, the company should prioritize buying almonds or pecans up to the maximum quantities that satisfy the binding constraints in the LP model. The maximum purchase quantity before affecting the current optimal solution is dictated by the point at which additional nuts are no longer cost-effective or cause constraint violations. Careful incremental purchasing, monitored via LP re-optimization, would allow maximization of profit potential without compromising the current optimal mix.

Conclusion

In summary, TJ’s should produce the exactly ordered quantities of 10,000 lbs of Regular Mix, 3,000 lbs of Deluxe Mix, and 5,000 lbs of Holiday Mix. The total profit contribution is approximately $39,750, with all nuts used efficiently under the current constraints. Buying additional nuts should be focused on almonds and pecans, which are most beneficial for sustaining or improving profit margins without exceeding resource constraints. Finally, surplus nuts, if any, should be donated to charity, aligning with TJ’s corporate social responsibility objectives.

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