Formulate A Linear Programming Model For The Rayhoon Restaur

Formulate a linear programming model for the Rayhoon Restaurant

Formulate a linear programming model for the Rayhoon Restaurant

The Rayhoon Restaurant: Brayden and Behrad were roommates who decided to open a restaurant in their small town, named Rayhoon. Their objective was to determine the optimal number of chicken and pork dinners to prepare each night to maximize profit while satisfying various operational constraints. The problem involves formulating a linear programming (LP) model based on the given constraints and then analyzing different scenarios, including advertising investment, staffing reliability, and pricing strategies.

Problem Description and Data

They plan to serve only two full-course meals nightly: one with pork and the other with chicken. The following key data are provided:

  • The maximum number of meals they expect to sell per night is initially 60 but could increase to 70 with advertising.
  • Each chicken dinner requires 15 minutes of preparation; each pork dinner takes twice as long, i.e., 30 minutes.
  • The total available kitchen staff labor time is 20 hours per day, which equals 1200 minutes.
  • The ratio of chicken to pork dinners sold should be at least 3:2, i.e., at least 1.5 chicken dinners for each pork dinner.
  • At least 10% of the customers will order pork dinners.
  • The profit for each chicken dinner is $12, which could increase to $14 if prices are raised; each pork dinner yields $16 profit.

Decision Variables

Let:

  • \( x_c \) = number of chicken dinners prepared per night
  • \( x_p \) = number of pork dinners prepared per night

Objective Function

Maximize total profit:

\( \text{Maximize } Z = 12x_c + 16x_p \)

(or, depending on scenario changes, profit for chicken could be increased to $14)

Constraints

  1. Supply constraint (maximum meals):
  • Initial scenario: \( x_c + x_p \leq 60 \)
  • With advertising: \( x_c + x_p \leq 70 \)
  • Labor time constraint: total preparation time must not exceed 1200 minutes:
    • Initial: \( 15x_c + 30x_p \leq 1200 \)
  • Demand ratio constraint: at least 1.5 chicken dinners for each pork dinner:
    • \( x_c \geq 1.5x_p \)
  • Minimum pork dinner sales: at least 10% of total sales:
    • \( x_p \geq 0.10(x_c + x_p) \)
  • Non-negativity constraints:
    • \( x_c \geq 0 \), \( x_p \geq 0 \)

    Scenario Analyses

    A. Advertising Investment

    Investing $30 per day in advertising increases the maximum meals from 60 to 70. The LP model remains the same with the updated maximum. Solving the LP with this new constraint reveals whether the increased capacity leads to higher profit, justifying the advertising expense.

    B. Staff Reliability and Reduced Working Hours

    Suppose staff availability decreases by 5 hours, reducing available labor time from 20 to 15 hours (900 minutes). The labor constraint adjusts accordingly to:

    \( 15x_c + 30x_p \leq 900 \)

    This reduction impacts the optimal production levels and profit, which the LP solution would demonstrate.

    C. Raising Chicken Dinner Prices

    Increasing the profit per chicken dinner to $14 alters the objective function to:

    \( Z = 14x_c + 16x_p \)

    This change's effect can be analyzed by updating the LP model and consulting the optimal solution to assess additional profit benefits and acceptability.

    Conclusion

    The formulation of this LP model allows Brayden and Behrad to make data-driven decisions about capacity investment, staffing, and product pricing. By solving the model under different scenarios, they can evaluate trade-offs and optimize their restaurant's profitability while adhering to operational constraints.

    References

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