Question

A businessman is considering opening a small, specialized trucking firm. To make the firm profitable, it is estimated that it must have a daily trucking capacity of at least 84,000 cu. ft. Two types of trucks are appropriate for specialized operation. Their characteristics and costs are summarized in the table below. Note that truck 2 requires 3 drivers for long haul trips. There are 41 potential drivers available and there are facilities for at most 40 trucks. The businessman's objective is to minimize the total cost outlay for trucks.

Truck Cost Capacity (Cu. Ft.) Drivers Needed
Small $18,000 2,400 1
Large $45,000 6,000 3
Solve the problem graphically and note there are alternate optimal solutions. Which optimal solution:
a) uses only one type of truck?
b) utilizes the minimum total number of trucks?
c) uses the same number of small and large trucks?

Answer 1

Explanation:

To solve this problem graphically, we can create a graph with the number of small trucks (x-axis) and the number of large trucks (y-axis). The objective is to minimize the total cost outlay for trucks.

a) To find the optimal solution that uses only one type of truck, we need to consider the cost and capacity of each truck. We can calculate the cost of using only small trucks and only large trucks separately and choose the option with the lowest cost.

For example, if we use only small trucks:

Cost = (Number of small trucks) * (Cost per small truck)

Capacity = (Number of small trucks) * (Capacity per small truck)

If we use only large trucks:

Cost = (Number of large trucks) * (Cost per large truck)

Capacity = (Number of large trucks) * (Capacity per large truck)

By comparing the costs and capacities of using only small trucks and only large trucks, we can determine which option is the optimal solution that uses only one type of truck.

b) To find the optimal solution that utilizes the minimum total number of trucks, we need to minimize the number of trucks used while still meeting the daily trucking capacity of at least 84,000 cu. ft. This means we need to find the combination of small and large trucks that minimizes the total number of trucks used.

We can calculate the total capacity of each combination of small and large trucks and check if it meets the minimum requirement. Then, we can determine the combination with the minimum total number of trucks used.

For example, if we use x small trucks and y large trucks:

Total capacity = (x * Capacity per small truck) + (y * Capacity per large truck)

If the total capacity is at least 84,000 cu. ft., we can calculate the total number of trucks used:

Total trucks used = x + y

By trying different combinations of small and large trucks and calculating the total number of trucks used, we can find the optimal solution that utilizes the minimum total number of trucks.

c) To find the optimal solution that uses the same number of small and large trucks, we need to find the combination that balances the capacity and cost. This means the total capacity and the total cost should be as close as possible.

We can start by assuming an equal number of small and large trucks (let's call it n). Then, we can calculate the total capacity and the total cost for this combination.

Total capacity = (n * Capacity per small truck) + (n * Capacity per large truck)

Total cost = (n * Cost per small truck) + (n * Cost per large truck)

By varying the value of n and calculating the total capacity and total cost for each combination, we can find the optimal solution that uses the same number of small and large trucks.