Banquet Seating A banquet hall offers two types of tables...

Banquet Seating A banquet hall offers two types of tables for rent: 6 -person rectangular tables at a cost of 28 dollar each and 10 -person round tables at a cost of 52 dollar each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The hall can have a maximum of 35 tables, and the hall has only 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Key Concepts

Linear Programming

Linear programming is an optimization method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. In such problems, a linear objective function is optimized, subject to a set of linear equality and inequality constraints.

Objective Function

The objective function is a formula that defines the goal of the optimization, typically to minimize or maximize a certain quantity – in this instance, the total cost. It aggregates the contributions of different decision variables weighted by their respective coefficients.

Constraints

Constraints represent the limits or requirements of the problem in the form of linear inequalities or equations. They restrict the solution space by imposing conditions such as seating capacity, maximum number of tables, and table availability.

Feasible Region

The feasible region is the set of all solutions that satisfy all the imposed constraints. It provides the candidate solutions for the optimization problem, from which the optimum is selected by evaluating the objective function.

Integer Programming

Integer programming is a branch of optimization where some or all decision variables are constrained to be integers. This is especially important in real-life problems, such as table allocation, where fractional values are not meaningful.

Transcript

00:01 So i've started off this problem by writing down everything i know.

00:04 So i know that rectangular tables can seat six people each and cost $28.

00:10 Round tables can seat 10 people each and cost $52.

00:14 We need to seat 250 people total.

00:17 We only have 15 rectangular tables and we need 35 tables total at most.

00:23 So to begin, i'm going to write up some variables.

00:28 I want to set x equal to the number.

00:30 Of rectangular tables, y equal to the number of round tables, and z equal to my total cost.

00:41 And in this case, i'm looking for the minimum cost.

00:45 Now i'm going to have to write my objective function, which starts with z equals.

00:50 Now i multiply the cost of the rectangular tables, which would be 28 by the number of rectangular tables, which is x, plus the same thing for roundtables, 52y.

01:03 And now we need to write down my constraints.

01:06 So we know that the number of tables can't be negative, so x is going to be greater than zero, and so is y.

01:13 Now looking up here, i know i need to seat 250 people, so the number of people seated at each rectangular table plus the number of people seated at each round table must be greater than or equal to 250.

01:32 Now looking here, we only have 15 rectangular tables, so i limit x to be less than or equal to 15.

01:42 And now only 35 tables fit in the banquet hall, so x plus y must be less than or equal to 35.

01:52 Now that i have all of my constraints, i want to start graphing this.

01:57 So since x and y are both greater than or equal to 0, we know that this will be in our first quadrant.

02:03 And i'm going to start by simplifying this equation here.

02:07 So i'm going to subtract 6x from both sides, and i end up with 10y is greater than or equal to negative 6x plus 250.

02:17 I divide both sides by 10, and i isolate y to get y is greater than or equal to negative three -fifths x plus 25.

02:27 And now i can graph this with my y intercept at 25, and my x intercept is approximately at 41 .7...

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