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Formulate a linear programming model for Rayhoon Restaurant that helps estimate the optimal number of meals to prepare each night, considering constraints such as maximum sales, preparation times, labor hours, and customer preferences. The problem involves at least three constraints and two decision variables, is bounded and feasible with a single optimal solution, and includes sensitivity analysis and shadow price discussion. Use Excel Solver to find the solution, and prepare a short writeup describing the problem, the LP model, the optimal solution, and insights from sensitivity analysis. Submit both the writeup and the Excel spreadsheet showing your setup and results.

Paper For Above instruction

The Rayhoon Restaurant case presents a classic linear programming problem to maximize profitability while respecting operational constraints. The decision variables are the number of chicken dinners (x) and pork dinners (y) served each night. The primary goal is to maximize total profit, given the profitability of each meal type and various resource limitations, such as kitchen preparation time, customer demand, and customer preferences. The model's constraints stem from kitchen labor hours, maximum meal sales, and customer preferences, along with non-negativity conditions for the decision variables.

This problem is a maximization linear programming model because the objective is to maximize profit, with the profit per meal being positive ($12 for chicken and $16 for pork). The constraints include:

- Total meals served cannot exceed maximum demand (initially 60, potentially 70 with advertising),

- Total preparation time must not exceed 20 hours (or 1,200 minutes),

- The ratio of chicken to pork dinners must be at least 3:2,

- Pork dinners must constitute at least 10% of total sales.

Furthermore, the model can be augmented with sensitivity analysis to evaluate the effects of changes in profit margins, resource availability, and market strategies like advertising spending adjustments. Shadow prices give insight into the marginal value of constraints, such as additional labor hours or improved customer demand.

The LP formulation begins with defining variables:

- x = number of chicken dinners,

- y = number of pork dinners.

The objective function to maximize profit:

Maximize Z = 12x + 16y.

Subject to the constraints:

1. Meal quantity limit: x + y

2. Time constraint: 15x + 30y

3. Customer preference: x >= (3/2) y (at least 3 chicken for every 2 pork).

4. Pork customer share: y >= 0.10(x + y).

5. Non-negativity: x >= 0, y >= 0.

Once the LP model is set up in Excel, Solver can be employed to find the optimal number of meals. The results tell us the combination of chicken and pork dinners that maximizes profit under the given constraints.

The sensitivity analysis component involves examining the shadow prices for each constraint—indicating how much profit would increase with a one-unit increase in resource availability—and considering how changes in advertising, staff reliability, and pricing affect profitability.

In terms of policy recommendations:

- Investing in advertising may increase maximum capacity from 60 to 70 meals, which possibly raises profit if demand exceeds initial capacity.

- Staff reductions of 5 hours diminish available labor hours, which could lower profit, especially if the constraint becomes binding.

- Raising the chicken dinner price to $14 increases profit per unit and could improve overall profitability without adversely affecting demand if customers accept the higher price.

This comprehensive LP model, combined with sensitivity analysis, provides strategic insights for Rayhoon Restaurant's decision-making regarding capacity, staffing, and pricing strategies.

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