Problem 47
Question
Through its franchised stations, an oil company gives out 16,000 road maps per year. The cost of setting up a press to print the maps is \(\$ 100\) for each production run. In addition, production costs are 6 cents per map and storage costs are 20 cents per map per year. The maps are distributed at a uniform rate throughout the year and are printed in equal batches timed so that each arrives just as the preceding batch has been used up. How many maps should the oil company print in each batch to minimize cost?
Step-by-Step Solution
Verified Answer
The oil company should print 4,000 maps in each batch to minimize the total cost.
1Step 1: Define variables
Let \( Q \) be the number of maps printed in each batch. We need to calculate this value to minimize the total cost.
2Step 2: Calculate the number of batches
The number of batches produced in a year is \( \frac{D}{Q} \), where \( D \) is the total number of maps needed annually, which is 16,000 maps.
3Step 3: Compute setup cost
The setup cost per year is given by the number of batches multiplied by the cost of setting up each batch. It can be expressed as: \[ \text{Setup Cost} = \frac{D}{Q} \times 100 \]
4Step 4: Compute production cost
Production cost per year is the cost to produce all the maps. It is calculated as: \[ \text{Production Cost} = D \times 0.06 \]
5Step 5: Compute storage cost
The storage cost per year involves the cost to store the average number of maps throughout the year. This average is \( \frac{Q}{2} \) maps. The storage cost can be computed as: \[ \text{Storage Cost} = \frac{Q}{2} \times 0.20 \]
6Step 6: Formulate the total cost function
The total cost function (C) is the sum of setup cost, production cost, and storage cost: \[ C = \frac{D}{Q} \times 100 + D \times 0.06 + \frac{Q}{2} \times 0.20 \]
7Step 7: Substitute known values
Substitute \( D = 16,000 \) into the total cost function: \[ C = \frac{16000}{Q} \times 100 + 16000 \times 0.06 + \frac{Q}{2} \times 0.20 \] Simplified further: \[ C = 1600000 \times \frac{1}{Q} + 960 + 0.10Q \]
8Step 8: Differentiate the cost function
To find the optimal batch size \( Q \), differentiate the total cost function with respect to \( Q \) and set it equal to zero: \[ \frac{dC}{dQ} = -1600000 \times \frac{1}{Q^2} + 0.10 \]
9Step 9: Solve for Q
Set \( \frac{dC}{dQ} = 0 \) and solve for \( Q \): \[ -1600000 \times \frac{1}{Q^2} + 0.10 = 0 \] \[ 0.10 = 1600000 \times \frac{1}{Q^2} \] \[ Q^2 = \frac{1600000}{0.10} \] \[ Q^2 = 16000000 \] \[ Q = \sqrt{16000000} = 4000 \]
10Step 10: Conclusion
The oil company should print 4,000 maps in each batch to minimize the total cost.
Key Concepts
Cost MinimizationDifferential CalculusInventory Management
Cost Minimization
In business scenarios, cost minimization is crucial to maximize profits. In this exercise, we aim to minimize the cost associated with printing and storing maps. To achieve this, we must consider three main areas:
1. **Setup Costs:** These are the costs to set up the press for each production run. In this case, it's \(\text{\$100}\) per run. Simplifying these costs over the year requires understanding how many batches are produced: \(\frac{D}{Q} \), where \(\frac{D}{Q}\) is the annual production runs, and \({D}\) is the demand (16,000 maps).
2. **Production Costs:** These are the costs of producing one map, \(\text{\textcent 6}\), multiplied by the number of maps produced, which is constant each year, 16,000 maps. This can be calculated as \({16,000} \times \text{\cent 6} = \text{\$960} \).
3. **Storage Costs:** These cover the expenses related to storing the maps before they are distributed. We use the average inventory throughout the year, which is \(\frac{Q}{2}\), where Q is batch size. At \(\text{\textcent 20}\) per map annually, storage costs can be represented as \(\frac{Q}{2} \times \text{\textcent 20} = \text{\cent 0.10Q \).
By calculating and combining these costs, we can derive a total cost function to find the optimal batch size that minimizes overall expenditures.
1. **Setup Costs:** These are the costs to set up the press for each production run. In this case, it's \(\text{\$100}\) per run. Simplifying these costs over the year requires understanding how many batches are produced: \(\frac{D}{Q} \), where \(\frac{D}{Q}\) is the annual production runs, and \({D}\) is the demand (16,000 maps).
2. **Production Costs:** These are the costs of producing one map, \(\text{\textcent 6}\), multiplied by the number of maps produced, which is constant each year, 16,000 maps. This can be calculated as \({16,000} \times \text{\cent 6} = \text{\$960} \).
3. **Storage Costs:** These cover the expenses related to storing the maps before they are distributed. We use the average inventory throughout the year, which is \(\frac{Q}{2}\), where Q is batch size. At \(\text{\textcent 20}\) per map annually, storage costs can be represented as \(\frac{Q}{2} \times \text{\textcent 20} = \text{\cent 0.10Q \).
By calculating and combining these costs, we can derive a total cost function to find the optimal batch size that minimizes overall expenditures.
Differential Calculus
Differential calculus allows us to find the rate at which functions change. In this scenario, we use it to find the batch size that minimizes the total cost by following these steps:
1. **Build the Total Cost Function:** Combine setup, production, and storage costs:
\[ C = \frac{16000}{Q} \times 100 + 16000 \times 0.06 + \frac{Q}{2} \times 0.20 = 1600000 \times \frac{1}{Q} + 960 + 0.10Q \]
2. **Differentiate the Cost Function:** To find the minimum point, we differentiate the total cost function with respect to \(\text{Q}\) and set it to zero:
\[ \frac{dC}{dQ} = -1600000 \times \frac{1}{Q^2} + 0.10 \]
3. **Solve for Q:** Setting the derivative equal to zero helps find the critical point:
\[ 0 = -1600000 \times \frac{1}{Q^2} + 0.10 \] \[ 0.10 = 1600000 \times \frac{1}{Q^2} \]
This simplifies to: \[ Q^2 = \frac{1600000}{0.10} = 16000000 \] \[ Q = \sqrt{16000000} = 4000 \]
Thus, using differential calculus, we confirm that printing 4,000 maps per batch will minimize total costs.
1. **Build the Total Cost Function:** Combine setup, production, and storage costs:
\[ C = \frac{16000}{Q} \times 100 + 16000 \times 0.06 + \frac{Q}{2} \times 0.20 = 1600000 \times \frac{1}{Q} + 960 + 0.10Q \]
2. **Differentiate the Cost Function:** To find the minimum point, we differentiate the total cost function with respect to \(\text{Q}\) and set it to zero:
\[ \frac{dC}{dQ} = -1600000 \times \frac{1}{Q^2} + 0.10 \]
3. **Solve for Q:** Setting the derivative equal to zero helps find the critical point:
\[ 0 = -1600000 \times \frac{1}{Q^2} + 0.10 \] \[ 0.10 = 1600000 \times \frac{1}{Q^2} \]
This simplifies to: \[ Q^2 = \frac{1600000}{0.10} = 16000000 \] \[ Q = \sqrt{16000000} = 4000 \]
Thus, using differential calculus, we confirm that printing 4,000 maps per batch will minimize total costs.
Inventory Management
Effective inventory management ensures that an organization has the right products in the right quantities while minimizing cost. For the oil company in this problem, the aim is to synchronize map production with consumption to avoid excess inventory, which incurs storage costs.
Key strategies include:
1. **Economic Order Quantity (EOQ):** This model helps determine the optimal batch size to minimize total costs, considering setup, production, and storage. From our exercise, the EOQ for the oil company maps is 4,000 maps per batch.
2. **Just-In-Time (JIT):** Aims at reducing waste by receiving goods only as they are needed in the production process. It matches well with our solution since maps are printed in batches that arrive just as the previous batch runs out.
3. **Continuous Review System:** The company should continuously track the map inventory to ensure new orders are placed just in time. This balanced approach will mitigate the risks of stockouts and overstocking.
Understanding these inventory management principles can efficiently guide businesses in maintaining the balance between service levels and the cost constraints.
Key strategies include:
1. **Economic Order Quantity (EOQ):** This model helps determine the optimal batch size to minimize total costs, considering setup, production, and storage. From our exercise, the EOQ for the oil company maps is 4,000 maps per batch.
2. **Just-In-Time (JIT):** Aims at reducing waste by receiving goods only as they are needed in the production process. It matches well with our solution since maps are printed in batches that arrive just as the previous batch runs out.
3. **Continuous Review System:** The company should continuously track the map inventory to ensure new orders are placed just in time. This balanced approach will mitigate the risks of stockouts and overstocking.
Understanding these inventory management principles can efficiently guide businesses in maintaining the balance between service levels and the cost constraints.
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