Complete the model. Min s.t. + 50 Q₁ Q₂ Q3 ≥ 0. 2,000 + + 1,000 + • Q₁ + Q₁ Q₂ 2,000 2,000 - 1,000 Q₂ + + 0.30. + 0.30. ON Li + 0.30. . Q3 ≤ 9,000 4 Solve the model. (Round Q₁, Q₂, and Q₂ to three decimal places and total cost (in $) to two decimal places.)
Complete the model. Min s.t. + 50 Q₁ Q₂ Q3 ≥ 0. 2,000 + + 1,000 + • Q₁ + Q₁ Q₂ 2,000 2,000 - 1,000 Q₂ + + 0.30. + 0.30. ON Li + 0.30. . Q3 ≤ 9,000 4 Solve the model. (Round Q₁, Q₂, and Q₂ to three decimal places and total cost (in $) to two decimal places.)
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Question
![Complete the model.
Minimize:
\[
\left[ \text{[ ]} \right) \cdot 2,000 + \left( \text{[ ]} \right) \cdot 2,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_1}{2}
\]
\[
+ 50 \cdot \left[ \left( \text{[ ]} \right) \cdot 2,000 + \left( \text{[ ]} \right) \cdot 2,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_2}{2}
\]
\[
+ \left[ \left( \text{[ ]} \right) \cdot 1,000 + \left( \text{[ ]} \right) \cdot 1,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_3}{2}
\]
Subject to:
\[
\left[ \left( \text{[ ]} \right) \cdot Q_1 + \left( \text{[ ]} \right) \cdot Q_2 + \left( \text{[ ]} \right) \cdot Q_3 \right] \le 9,000
\]
\[
Q_1, Q_2, Q_3 \ge 0.
\]
Solve the model. (Round \(Q_1\), \(Q_2\), and \(Q_3\) to three decimal places and total cost (in $) to two decimal places.)
\[
Q_1 = \_\_\_\_\_\_\_
\]
\[
Q_2 = \_\_\_\_\_\_\_
\]
\[
Q_3 = \_\_\_\_\_\_\_
\]
\[
\text{Total cost} = \$\_\_\_\_\_\_\_\_
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2F911e385d-3bfd-4a19-b74e-2152deb6eb29%2F4q3sn33_processed.png&w=3840&q=75)
Transcribed Image Text:Complete the model.
Minimize:
\[
\left[ \text{[ ]} \right) \cdot 2,000 + \left( \text{[ ]} \right) \cdot 2,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_1}{2}
\]
\[
+ 50 \cdot \left[ \left( \text{[ ]} \right) \cdot 2,000 + \left( \text{[ ]} \right) \cdot 2,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_2}{2}
\]
\[
+ \left[ \left( \text{[ ]} \right) \cdot 1,000 + \left( \text{[ ]} \right) \cdot 1,000 \right] + 0.30 \cdot \left( \text{[ ]} \right) \cdot \frac{Q_3}{2}
\]
Subject to:
\[
\left[ \left( \text{[ ]} \right) \cdot Q_1 + \left( \text{[ ]} \right) \cdot Q_2 + \left( \text{[ ]} \right) \cdot Q_3 \right] \le 9,000
\]
\[
Q_1, Q_2, Q_3 \ge 0.
\]
Solve the model. (Round \(Q_1\), \(Q_2\), and \(Q_3\) to three decimal places and total cost (in $) to two decimal places.)
\[
Q_1 = \_\_\_\_\_\_\_
\]
\[
Q_2 = \_\_\_\_\_\_\_
\]
\[
Q_3 = \_\_\_\_\_\_\_
\]
\[
\text{Total cost} = \$\_\_\_\_\_\_\_\_
\]
![The Economic Order Quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for this model are that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed.
Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation.
- Let:
- \( D_j \) = annual demand for item \( j \)
- \( c_j^p \) = unit cost of item \( j \)
- \( S_j \) = cost per order placed for item \( j \)
- \( w_j \) = space required per item \( j \)
- \( W \) = maximum amount of space available for all goods
- \( i \) = inventory carrying charge as a percentage of the cost per unit
The decision variables are \( Q_j \), the amount of item \( j \) to order. The model is:
Minimize
\[
\sum_{j=1}^n \left[ c_j^p D_j + S_j \frac{D_j}{Q_j} + \frac{i c_j^p Q_j}{2} \right]
\]
subject to
\[
\sum_{j=1}^n w_j Q_j \leq W
\]
\[
Q_j \geq 0
\]
In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (\( D_j / Q_j \) is the number of orders), and the last term is the annual inventory holding cost (\( Q_j / 2 \) is the average amount of inventory).
Construct and solve a nonlinear optimization model for the following data:
| | Item 1 | Item 2 | Item 3 |
|-------|--------|--------|--------|
| **Annual Demand** | 2,000 | 2,000 | 1,000 |
| **Item Cost (\$)** | 100 | 50 | 80 |
| **Order Cost (\$)** | 150 | 135 |](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862c5bc2-e09a-4feb-bd28-f7638a835094%2F911e385d-3bfd-4a19-b74e-2152deb6eb29%2Fwb3otcl_processed.png&w=3840&q=75)
Transcribed Image Text:The Economic Order Quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for this model are that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed.
Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation.
- Let:
- \( D_j \) = annual demand for item \( j \)
- \( c_j^p \) = unit cost of item \( j \)
- \( S_j \) = cost per order placed for item \( j \)
- \( w_j \) = space required per item \( j \)
- \( W \) = maximum amount of space available for all goods
- \( i \) = inventory carrying charge as a percentage of the cost per unit
The decision variables are \( Q_j \), the amount of item \( j \) to order. The model is:
Minimize
\[
\sum_{j=1}^n \left[ c_j^p D_j + S_j \frac{D_j}{Q_j} + \frac{i c_j^p Q_j}{2} \right]
\]
subject to
\[
\sum_{j=1}^n w_j Q_j \leq W
\]
\[
Q_j \geq 0
\]
In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (\( D_j / Q_j \) is the number of orders), and the last term is the annual inventory holding cost (\( Q_j / 2 \) is the average amount of inventory).
Construct and solve a nonlinear optimization model for the following data:
| | Item 1 | Item 2 | Item 3 |
|-------|--------|--------|--------|
| **Annual Demand** | 2,000 | 2,000 | 1,000 |
| **Item Cost (\$)** | 100 | 50 | 80 |
| **Order Cost (\$)** | 150 | 135 |
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