Problem Statement

Suppose that you are throwing a pizza party for yourself and your closest friends, with a total of ten (10) people expected to attend. Further suppose that you are ordering large pizzas from your favorite deep-dish pizza chain, a large pizza can easily feed four adults, and that the only topping choices available are Cheese, Pepperoni, and Veggie. The goal here is to satisfy everyones preferences as best you can, without just buying everyone their own pizza of course. To this end, lets assume you asked your friends to rate their opinion of each pizza topping on a scale of 1 to 10 where 10 is the best, and you have recorded this data on the Part 2 sheet of the Module 12 Template workbook.

Armed with this information, you now need to make two decisions: (i) how many of each kind of pizza should you order, and (ii) who should receive each topping? Given that each pizza serves up to 4 people, so youll need to order three pizzas to have enough for all ten (10) people, but which three? One of each? Three of one kind? Fortunately, we can wait for Solver to tell us!

Part 1

Before we start building the what-if model, lets be sure we understand the problem itself; so, Part 1 tasks you with writing out the problem mathematically first. Follow the guidelines below to complete the has three text boxes on the Part 1 worksheet on the Module 12 Template, and for now, disregard any possible integer constraints you think we might need.

For the ease of typing the formulation into the text box, you can use Excel-speak e.g. SUMPRODUCT(x,d) in place of Summation/Product notation. You may also find that adding a brief verbal description is helpful; for example, rather than typing out the same constraint ten times (one for each guest), you can write it in general notation and describe which index (i or j) is being repeated. If that sounds more complicated than just spelling out each constraint, that is fine too.

Decision Variables

There are two main types of decision variables in this problem that coincide with the two decisions that you must make, so theres a group of decision variables for the pizzas that you order and another group of decision variables for how you distribute the servings. Since there are only three ordering variables (one for each pizza type), you can get away with an abbreviation-style variable name, but the pizza distribution variable has a lot more variations to consider (as can be seen on the Part 2 worksheet), so I encourage you to use the general LP notation introduced in Chapter 11.

Objective Function

This is the simplest box in that it only requires one function. Given that the objective is to satisfy everyones preferences according to their ratings on a scale from 1-10, we can think of this as trying to maximize what Im going to call the preference score for giving people the pizza they want. Dont forget to include either MAX or MIN in the Objective Function!

Constraints:

There are three constraints you must be aware of: the number of pizzas ordered shouldnt exceed three, everyone should receive exactly one serving of pizza (defined as of a pizza), and the number of servings distributed of each pizza cannot exceed the number of servings purchased (again assume 4 servings per pizza).

Decision Variables: Objective Function: Constraints