### Linear Programming with Three Decision Variables

**Name:**

You decide to invite your friends to brunch. You go shopping and buy 12 eggs, 8 tablespoons (Tbsp) of butter, and 20 slices of bread.
During the brunch you will cook these ingredients into servings of French toast, toast, or scrambled eggs to feed to your friends.
If you run out of ingredients you'll have to buy more, so your objective is to cook as many servings of food as possible.
From your recipes, you see that you need the following amounts of ingredients per serving:

| **Eggs** | **Tbsp Butter** | **Slices of Bread** |

**French Toast** | 1 | 1 | 1 |

**Toast** | 0 | 1 | 2 |

**Scrambled Eggs** | 2 | 0 | 0 |

- Match the technical terms with the specific examples:
Constraint | | Number of servings of scrambled eggs |

Constraint Coefficient | | Total number of servings |

Objective Function | | Number of slices of bread available |

Decision Variable | | Amount of butter needed to make toast |

- Formulate a linear programming problem to determine how many servings of each type of food to cook. Please use the
following decision variables in your formulation:
**F** = number of servings of French toast

**T** = number of servings (2 slices) of toast

**S** = number of servings of scrambled eggs

- Use Microsoft Excel's Solver tool to solve the linear programming problem. Write the values you find for S, T and F below.

- Which constraints are binding?

- What is the "dual price" of eggs in this problem? What does "dual price" mean in this context?

- Your room mate ruined your non-stick pan; you must add 1 Tbsp butter to your recipe for scrambled eggs.
What are the values of F, T and S in the new optimal solution?

- Your friend Susan will bring you a new pan if you make her French toast. Once you've made Susan's French toast,
what is the maximum number of servings of brunch you can make using the original recipe for scrambled eggs? How many of those servings are French toast,
how many are toast, and how many are scrambled eggs?