A company installs different POS computer systems. POS system A requires two hours to configure and one hour for assemble. POS B requires three hours to configure and one hour to assemble. POS C requires two hours to configure and two hours to assemble. The company has up to 100 labor hours for configure and 800 labor hours for assemble each week. If the profit of POS systems A, B, and C is $700, $ 800, and $1000 respectively how many of each system should be installed each week to maximize profit? What is the maximum profit each week? this has to be set up using the simplex method

To solve this problem using the simplex method, we need to formulate the problem into a linear programming (LP) model. Let's define the decision variables as follows:

Let x1 = number of POS system A installed per week
Let x2 = number of POS system B installed per week
Let x3 = number of POS system C installed per week

Objective Function:
Our goal is to maximize the profit, so we can create the objective function as follows:

Maximize: Z = 700x1 + 800x2 + 1000x3

Constraints:
1. Configuration constraint:
The total hours spent on configuration should not exceed the available labor hours for configuration (100 hours):

2x1 + 3x2 + 2x3 ≤ 100

2. Assembly constraint:
The total hours spent on assembly should not exceed the available labor hours for assembly (800 hours):

x1 + x2 + 2x3 ≤ 800

3. Non-negativity constraint:
The number of systems installed cannot be negative:

x1, x2, x3 ≥ 0

Now, we can set up the simplex matrix and apply the simplex method to solve the LP model. The simplex method involves multiple iterations of finding the pivot column and pivot row until an optimal solution is reached.

I'll calculate the optimal solution and maximum profit using the simplex method.