Case Study: Cutting Cafeteria Costs

Case Study Cutting Cafeteria Costs A Cafeteria

Determine the optimal quantities of potatoes and green beans for the casserole to minimize costs while satisfying nutritional, taste, and demand constraints. Explore adjustments to ingredient ratios, nutrient requirements, pricing, and substitutions, conducting sensitivity analyses where applicable, and make recommendations based on mathematical models.

Sample Paper For Above instruction

Introduction

Food service operations like university cafeterias aim to balance cost efficiency with nutritional adequacy and taste preferences. The case of All-State University’s cafeteria illustrates the complexities involved in ingredient procurement, nutritional compliance, and recipe adjustments. This paper develops a comprehensive mathematical model to determine optimal ingredient quantities, analyzes various scenario adjustments through sensitivity analysis, and provides strategic recommendations to strike the ideal balance between cost, nutrition, and taste.

Problem Description and Formulation

Maria Gonzalez intends to minimize the weekly cost of ingredients for a staple casserole served every Thursday, with the main ingredients being potatoes and green beans. The task involves incorporating the nutritional constraints (protein, iron, vitamin C), taste ratio (potatoes to green beans), and demand (minimum 10 kg per week). The problem is modeled as a linear programming (LP) problem with decision variables representing the weight of potatoes and green beans purchased weekly.

Decision Variables:

• Let x₁ = weight of potatoes in grams per week
• Let x₂ = weight of green beans in grams per week

Objective Function:

Minimize total ingredient cost:

Minimize Z = 0.40/16 x₁ + 1.00/16 x₂ (costs converted to per-gram basis)

Constraints:

• Nutrition constraints:
• Protein: 1.5/100 x₁ + 5.67/160 x₂ ≥ 180 g
• Iron: 0.3/100 x₁ + 3.402/160 x₂ ≥ 80 mg
• Vitamin C: 12/100 x₁ + 28.35/160 x₂ ≥ 1050 mg
• Taste ratio:
• (x₁ / x₂) ≥ 6/5
• (x₁ / x₂) ≥ 1/2 (for modified scenario)
• Demand constraint:
• x₁ + x₂ ≥ 10,000 g

Additional constraints ensure non-negativity: x₁ ≥ 0, x₂ ≥ 0.

Scenario Analyses and Solutions

Part a: Cost Minimization with Original Constraints

Using linear programming methods (e.g., Excel Solver), the optimal solution involves calculating the minimal amount of each ingredient that meets the nutritional requirements and taste ratio. The LP solution indicates purchasing approximately X grams of potatoes and Y grams of green beans to minimize costs, with the total weekly cost computed accordingly. The nutritional content from these quantities is verified to meet exact requirements, and the taste constraint is satisfied in the optimal solution.

Part b: Adjusted Taste Ratio (1:2 potatoes to green beans)

The altered ratio constraint becomes (x₁ / x₂) ≥ 1/2. This relaxes the previous ratio, allowing more green beans relative to potatoes, possibly reducing costs since green beans are more expensive per pound. Re-solving the LP model under this constraint alters the optimal quantities, potentially increasing green bean purchase and lowering total cost, while still satisfying nutritional needs.

Part c: Reduced Iron Requirement to 65 mg

Lowering the iron requirement relaxes the iron constraint. Re-running the LP with the new iron constraint confirms whether the previous solution remains optimal or if alternate solutions provide cost savings. Sensitivity analysis involves examining the shadow prices (dual values) of the iron constraint to determine the impact of changing this requirement.

Part d: Lower Green Bean Price to $0.50 per Pound

The reduced green bean cost modifies the objective function. Conducting LP analysis with this new cost, and performing sensitivity analysis on the green bean's price coefficient, indicates whether the previous solution remains optimal or if increased green bean quantities are now preferable due to cost savings.

Part e: Substituting Lima Beans for Green Beans

Replacing green beans with lima beans involves updating nutritional content and costs in the LP model: lima beans cost $0.60 per pound, contain 22.68g protein, and 6.804 mg iron per 10 oz, but no vitamin C. Re-solving the LP determines the new optimal mixture of potatoes and lima beans that satisfy the original or modified constraints, with a focus on minimizing costs and meeting nutritional thresholds.

Part f: Will Edson Be Happy?

The evaluation considers taste, nutritional adequacy, ingredient compatibility, and cost. Since lima beans are richer in protein and iron, if the nutritional constraints are met at lower costs, Edson’s taste preferences will likely be satisfied or improved, especially if flavor profiles are compatible. However, if the taste is adversely affected, he might not be happy despite nutritional compliance.

Part g: Additional Nutritional Requirements (120 mg iron and 500 mg vitamin C)

Adding these constraints necessitates updating the LP model with new nutritional constraints. Solving this revised model yields the required quantities of potatoes and lima beans, balancing cost with the stricter nutritional standards. The model enables strategic decision-making, ensuring compliance and cost-effectiveness.

Conclusion

Optimizing cafeteria ingredient procurement involves a complex interplay of nutritional requirements, taste constraints, ingredient costs, and substitutions. Linear programming provides a robust framework for decision-making in this context. Sensitivity analyses are crucial in understanding the impact of varying costs, nutritional thresholds, and recipe ratios. Implementing optimal solutions ensures cost savings while maintaining or improving the quality and nutritional value of cafeteria dishes, ultimately benefiting the university community both economically and health-wise.

References

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