. A food industry produces chips at three factories, located in Birmingham, Glasgow and London. Since the customer should be supplied with fresh products, these are not stored in the factories. The monthly production ability of the factory in London is 750 tones, while of the two are 500 tones. Every day 300 tones are given to the five warehouses in order to subsequently be transferred to customers. The profit per sale tone from the first warehouse is0.4 pounds if produced in London, 0.6 if produced in Glasgow and 2.2 pounds if produced in Birmingham. The corresponding gains for the second warehouse are1.1,1.2 and2 pounds. The profit per tone of production in London is 1.7, 1.3, and 2.5 pounds when sold by the third, fourth and fifth warehouse respectively. The corresponding gains for Birmingham are 1.6,1 and 0.5 pounds, while for Glasgow are 1.1,0.8 and 2.1 pounds. What is the maximum monthly profit that can be achieved by the industry? Interpret your result.
To find the maximum monthly profit that can be achieved by the industry, we need to determine the optimal allocation of production between the three factories and the five warehouses.
Let's define the decision variables:
x1 = Monthly production in London (in tones)
x2 = Monthly production in Glasgow (in tones)
x3 = Monthly production in Birmingham (in tones)
Now, we can set up the objective function to maximize the profit. The profit is calculated by multiplying the production quantity by the profit per sale tone, summing up the profits from all factories and warehouses.
The objective function can be formulated as follows:
Maximize Profit =
0.4x1 + 0.6x2 + 2.2x3 + 1.1x1 + 1.2x2 + 2x3 + 1.7x1 + 1.3x2 + 2.5x3 + 1.6x1 + 1x2 + 0.5x3 + 1.1x1 + 0.8x2 + 2.1x3
subject to the following constraints:
x1 + x2 + x3 ≤ 750 (Production limit in London)
x1 + x2 ≤ 500 (Production limit in Glasgow and Birmingham)
x1 + x1 + x1 + x1 + x1 ≤ 300 (Supply to the warehouses)
The first constraint ensures that the total production from all factories does not exceed the monthly production limits. The second constraint ensures that the production from Glasgow and Birmingham does not exceed their respective production limits. The third constraint ensures that the total supply to the warehouses does not exceed the daily supply limit.
Solving this linear programming problem will give us the optimal production quantities for each factory and the corresponding maximum monthly profit.
To find the maximum monthly profit that can be achieved by the industry, we need to determine the optimal production and distribution plan.
Let's define the decision variables:
x1 = tons produced in London and supplied to the first warehouse
x2 = tons produced in London and supplied to the second warehouse
x3 = tons produced in Glasgow and supplied to the first warehouse
x4 = tons produced in Glasgow and supplied to the second warehouse
x5 = tons produced in Birmingham and supplied to the first warehouse
x6 = tons produced in Birmingham and supplied to the second warehouse
We need to find the maximum value of the objective function:
Profit = (0.4 * x1 + 0.6 * x3 + 2.2 * x5) + (1.1 * x2 + 1.2 * x4 + 2 * x6) + (1.7 * x1 + 1.3 * x2) + (1.6 * x5 + 1 * x6) + (1.1 * x3 + 0.8 * x4 + 2.1 * x5)
subject to the following constraints:
x1 + x3 + x5 <= 750 (production constraint for London)
x2 + x4 + x6 <= 500 (production constraint for Glasgow and Birmingham)
x1 + x2 <= 300 (supply constraint for the first warehouse)
x3 + x4 <= 300 (supply constraint for the second warehouse)
x5 + x6 <= 150 (supply constraint for the third, fourth, and fifth warehouses)
Solving this linear programming problem will give us the optimal production and distribution plan that maximizes the monthly profit.