A grocery store chain needs to transport of 3000m^3 of refrigerated goods and 4000m^3 of non-refrigerated goods. They plan to hire truck from a company that has two types of trucks for rent. A truck of type A has a refrigerated goods section of 20m^3 and a non-refrigerated goods section of 40m^3, while each truck of type B has both sections of the same volume of 30m^3. The cost per kilometer of a Type A is $30, and $40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?
To determine the minimum total cost, we need to find the number of trucks of each type that minimizes the cost while also satisfying the volume requirement.
Let's assume that the grocery store rents x trucks of type A and y trucks of type B.
The total refrigerated goods volume required is 3000m^3.
Each truck of type A has a refrigerated goods section of 20m^3, so the maximum capacity of refrigerated goods transport by type A trucks is 20m^3 * x trucks.
Each truck of type B has the same volume capacity as type A, so the maximum capacity of refrigerated goods transported by type B trucks is 30m^3 * y trucks.
Hence, the total refrigerated goods volume capacity is 20x + 30y.
Similarly, the total non-refrigerated goods volume capacity is 40x + 30y, which should be at least 4000m^3.
We can form the following constraints:
1. 20x + 30y >= 3000 (Refrigerated goods volume requirement)
2. 40x + 30y >= 4000 (Non-refrigerated goods volume requirement)
3. x, y >= 0 (Non-negativity constraint)
Now, let's calculate the cost for each truck type and find the objective function.
The cost per kilometer for type A trucks is $30, and the cost per kilometer for type B trucks is $40.
Let's assume the distance to be covered is d kilometers.
The cost of type A trucks = 30 * d * x
The cost of type B trucks = 40 * d * y
The objective function (total cost) is the summation of the costs for type A and type B trucks:
Total Cost = 30 * d * x + 40 * d * y
Now, we can formulate the linear programming problem as follows:
Minimize: Total Cost = 30 * d * x + 40 * d * y
Subject to:
20x + 30y >= 3000
40x + 30y >= 4000
x, y >= 0
Solving this linear programming problem will give us the values of x and y (the number of trucks of type A and type B) that minimize the total cost while satisfying the volume requirements.
You specify no distance, so let's work with the per/km cost. For x trucks of A and y trucks of B, you want to
minimize c=30x+40y subject to
20x+30y >= 3000
40x+30y >= 4000
Now crank out the vertices.