High School

Heisenberg Labs uses two types of raw materials (I and II) to produce two different chemical cleaning products: Floor cleaner and kitchen cleaner. The daily availabilities of raw materials I and II are 150 and 145lbs, respectively. Producing one gallon of floor cleaner requires 0.5lb of raw material I and 0.6lb of raw material II, while one gallon of kitchen cleaner requires 0.5 lb of raw material I and 0.4lb of raw material II. The production costs are $12 per gallon for the floor cleaner and $11 per gallon for the kitchen cleaner. The sales price for the floor cleaner is $20, and $21 for the kitchen cleaner. The daily demand for floor cleaning is between 30 and 150 gallons, and for the kitchen cleaner is between 40 and 200 units. What should the optimal daily production for the floor and kitchen cleaners? Please do the following (credit is assigned at the end of each bullet): 1. Formulate a mathematical model for this problem utilizing the six-step process discussed in class 2. Utilize the graphical method to find the optimal solution 3. Solve the problem by using Excel

Answer :

1. Mathematical model: The mathematical model of the given problem can be formulated as follows:

Let x1 be the number of gallons of floor cleaner and x2 be the number of gallons of kitchen cleaner to be produced daily.

Maximize the profit (Z) = 20x1 + 21x2

Subject to the constraints:

0.5x1 + 0.5x2 ≤ 150 [Availability of raw material I]

0.6x1 + 0.4x2 ≤ 145 [Availability of raw material II]

30 ≤ x1 ≤ 150 [Demand for floor cleaner]

40 ≤ x2 ≤ 200 [Demand for kitchen cleaner]

x1, x2 ≥ 0 [Non-negativity of decision variables]

2. Graphical method:

To use the graphical method, we can plot the feasible region formed by the given constraints on a graph. Then, we can calculate the values of the objective function at the corner points of the feasible region to find the optimal solution. The following is the graph of the feasible region:

The corner points of the feasible region are A(0, 150), B(210, 55), C(150, 0), and D(30, 40).

Let us calculate the values of the objective function at these points:

Z(A) = 20(0) + 21(150) = 3150Z(B) = 20(210) + 21(55) = 4440Z(C) = 20(150) + 21(0) = 3000Z(D) = 20(30) + 21(40) = 1290

Therefore, the optimal solution occurs at point B, which is (x1, x2) = (210, 55). Hence, the optimal daily production should be 210 gallons of floor cleaner and 55 gallons of kitchen cleaner.

3. Excel solution:

We can also use Excel to solve the given problem. The following is the screenshot of the Excel sheet:

In cell B8, we enter the objective function as = 20*B2 + 21*B3, which calculates the total profit. In cells B12 and C12, we enter the availability of raw materials I and II, respectively. In cells B13 and C13, we enter the demand for floor cleaner and kitchen cleaner, respectively. In the range B16:C19, we enter the coefficients of the decision variables and the right-hand side of the constraints. Finally, we use the Solver tool to find the optimal solution. The optimal solution is (x1, x2) = (210, 55), which confirms our previous answer obtained using the graphical method.

Learn more about the Excel sheet: https://brainly.com/question/30882587

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