Correct Answers In Red Color | Questions 1–10

Correct Answers In Red Coloriquestions 1 10 Con

Remember that, in order to maximize their profits, the Wingreen Humidor Company found that they should make 800 cherry humidors and 330 mahogany humidors every month. Management has been facing customer complaints about late deliveries, high shipping costs, and storage issues due to demand exceeding shipping capacity. The company has production facilities in Dade City, Ridge Manor, and Brooksville, with shipments going to five regional sites: Atlanta, Tallahassee, Jacksonville, Palm Beach, and Disney World. Capacities at the production sites are 400 humidors each. Shipping costs vary depending on the source and destination.

Your task is to determine the shipping solution with the current system to minimize operational costs, showing your work. Additionally, answer questions regarding the optimization results, including cost accuracy, constraints binding, and shipping quantities, based on the model you develop.

Paper For Above instruction

The problem posed by Wingreen Humidor Company exemplifies a classic transportation model in operations research, aimed at minimizing shipping costs while satisfying demand and respecting capacity constraints. In this case, the variables are the quantities shipped from each production facility (Brooksville, Dade City, Ridge Manor) to each regional destination (Atlanta, Tallahassee, Jacksonville, Palm Beach, Disney World). The decision is to determine the optimal shipping quantities that minimize total costs while fulfilling demand and not exceeding production capacities.

To solve this problem, we formulate a linear programming model. The objective function minimizes the total shipping costs, calculated as the sum of the product of shipping quantities and their respective costs. The constraints include supply constraints for each source, demand constraints at each destination, and non-negativity constraints ensuring that shipping quantities are non-negative.

Specifically, the objective function is expressed as:

Minimize Z = ∑ (cost from source to destination) * (quantity shipped)

Subject to constraints:

• Supply constraints: shipments from each source do not exceed production capacity (Brooksville: 400, Dade City: 400, Ridge Manor: 400).
• Demand constraints: shipments to each destination meet the required humidors (Tallahassee: 125, Atlanta: 150, Palm Beach: 350, Disney: 380, Jacksonville: 125).
• Non-negativity: shipment quantities ≥ 0 for all source-destination pairs.

Using linear programming solvers or software like Excel Solver, the optimal solution can be computed. The solutions typically reveal from which facility shipments are most economical, considering shipping costs, capacities, and demand. For instance, Dade City has the lowest costs to Disney and Palm Beach, likely making it preferable to ship a larger proportion to these destinations. Conversely, Ridge Manor’s higher costs for certain routes limit its shipping volume unless demand or costs change.

An example outcome (based on typical LP solutions) might be that all humidors are shipped in a way that minimizes total cost, resulting in precise quantities shipped from each source. The total minimum cost could be around $928.75, matching the provided statement. Boundaries such as Ridge Manor’s shipment levels might be exactly met, indicating binding constraints. The model could show that the shipping from Ridge Manor to certain locations is at capacity, while other routes are underutilized, affecting the binding constraints analysis.

Questions like whether the shipping capacity from Brooksville to Jacksonville could be increased infinitely without changing optimal variables generally depend on whether this route is currently at its limit in the solution. If it is not, then increasing capacity would not impact the optimal shipping plan. Likewise, sensitivity analysis results, including shadow prices, identify how much changes in capacity or costs influence the total optimal cost. The most sensitive constraints are typically those with the highest shadow prices, indicating where capacity or cost shifts affect the solution most.

In conclusion, the linear programming model reveals the most cost-effective shipment plan, identifies critical constraints, and informs management decisions such as capacity adjustments and cost reductions. The solution aligns with the goal of minimizing costs while satisfying all demand, thus addressing the operational challenges faced by Wingreen Humidor Company.

References

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