The Heartland Produce Company owns farms in the Midwest, where it grows and harvests potatoes. It then ships these potatoes to three processing plants in St. Louis, New Orleans, and Chicago, where different varieties of potato products, including potato chips, are produced. The company is experiencing increased demand, so it wants to buy one or more new farms to produce more potato products. The company is considering six new farms with the following annual fixed costs and projected harvest: The company currently has the following additional available production capacity (tons) at its three plants, which it wants to utilize: Available Capacity Plant (thousands of tons) St. Louis ……… 12 Dallas ………. 10 Chicago ………. 14 The shipping costs ($) per ton from the farms being considered for purchase to the plants are as follows: Which of the six farms should the company purchase to meet available production capacity at the minimum total cost (including annual fixed costs and shippingcosts)?
The Correct Answer and Explanation is :
To determine which farms Heartland Produce Company should purchase, the objective is to meet the available production capacity at the three processing plants (St. Louis, Dallas, and Chicago) while minimizing the total costs. The total costs include both the fixed costs for operating the new farms and the shipping costs from the farms to the plants.
To solve this, we can approach it using linear programming (LP) or optimization techniques. Here’s a simplified explanation:
Step 1: List the key information
- Available production capacity at each plant (in thousands of tons).
- Fixed costs and projected harvest (tons) for each of the six farms.
- Shipping costs from each farm to each plant.
Step 2: Set up the problem
The problem can be set up as a transportation problem where:
- The supply is the harvest from each of the farms.
- The demand is the available capacity at each plant.
- The costs are the fixed operating costs of the farm and the shipping costs.
You would need to minimize the sum of:
- Fixed costs for the selected farms.
- Shipping costs from the selected farms to the plants.
Step 3: Formulate constraints
For each farm, you must ensure that the total tons shipped from the farm meet or do not exceed its harvest capacity. For each plant, you must ensure that the total tons received do not exceed the available capacity.
Step 4: Solve the problem
By using a solver, such as a linear programming solver, you would calculate the most cost-efficient combination of farms to meet the plant capacity while minimizing the total costs. The solver would consider:
- Whether to buy each farm based on its fixed cost and harvest potential.
- How to allocate each farm’s harvest to the plants based on shipping costs and capacity limitations.
Conclusion:
The solution would specify which farms to purchase, how much to ship from each farm to each plant, and the total cost. The result would ensure that the company maximizes its production without exceeding the plant capacities and minimizes the costs in both fixed expenses and shipping. Without the specific values (costs, harvest, shipping rates), it’s impossible to give an exact farm selection, but the outlined method should guide you to the optimal decision.