Bakery ABC produces two products, butter cake, and chocolate cake. The sales volume for butter cake is at least 80% of the total sales of both butter cake and chocolate cake. However, the company cannot sell more than 100 pieces of butter cakes per day. Both products use flour, of which the maximum daily availability is 240 kg. The usage rates of the flour are 2 kg for every butter cake and 4 kg for every chocolate cake. The profit per piece of butter cake and chocolate cake are RM20 and RM50 respectively. Solve this case study using the graphical method.
a) Construct the objective function of the given case-study.
b) Construct the constrains of the given case-study.
c) Propose the quantity of butter cake and chocolate cakes to be produced daily in order to maximize profit using graphical method approach.
To solve this case study using the graphical method, we need to follow these steps:
a) Construct the objective function:
The objective function represents the function we want to maximize or minimize. In this case, we want to maximize the profit.
Let's define:
x = number of butter cakes produced daily
y = number of chocolate cakes produced daily
The profit function can be defined as:
P = 20x + 50y
b) Construct the constraints:
The constraints define the limitations or restrictions in the problem. Based on the information given, we have the following constraints:
1. Sales volume constraint:
The sales volume for butter cake should be at least 80% of the total sales of both butter cake and chocolate cake.
This can be represented as:
x ≥ 0.8(x + y)
=> x ≥ 0.8x + 0.8y
=> -0.8x + 0.2y ≥ 0
2. Maximum daily limit of butter cakes:
The company cannot sell more than 100 pieces of butter cakes per day.
This can be represented as:
x ≤ 100
3. Flour usage constraint:
The maximum daily availability of flour is 240 kg, and the usage rates are 2 kg for every butter cake and 4 kg for every chocolate cake.
This can be represented as:
2x + 4y ≤ 240
4. Non-negativity constraint:
The number of cakes cannot be negative.
This can be represented as:
x ≥ 0
y ≥ 0
c) Propose the quantity of butter cake and chocolate cakes to be produced daily:
To use the graphical method, we need to plot the constraints on a graph and find the feasible region. In this case, since we have more constraints than variables, we will have a bounded feasible region.
1. Plot the sales volume constraint: -0.8x + 0.2y ≥ 0
For simplicity, let's convert it to the equality: -0.8x + 0.2y = 0
Plot this line on the graph.
2. Plot the maximum daily limit of butter cakes: x ≤ 100
This will be a vertical line at x = 100.
3. Plot the flour usage constraint: 2x + 4y ≤ 240
Rearranging, we have: y ≤ 60 - 0.5x
Plot this line on the graph.
4. Plot the non-negativity constraints: x ≥ 0 and y ≥ 0
This will create a shaded area in the first quadrant of the graph.
5. Find the feasible region:
The feasible region is the region where all constraints are satisfied. It will be the intersection of all the shaded areas from the constraints.
6. Identify the corner points of the feasible region:
The corner points of the feasible region represent the potential solutions. Find the coordinates of these points.
7. Calculate the profit for each corner point:
Substitute the coordinates of each corner point into the profit function P = 20x + 50y to find the profit for each combination of butter cakes and chocolate cakes.
8. Select the combination with the highest profit:
Identify the combination of butter cakes and chocolate cakes that gives the highest profit. This will be the optimal solution.
By following these steps, you can propose the quantity of butter cake and chocolate cakes to be produced daily in order to maximize profit using the graphical method approach.