A printing company orders paper from two different
suppliers. Supplier charges per case.
Supplier charges per case. The company the constraints on the system.
needs to order at least 45 cases per day to meet
demand and can order no more than 30 cases
from Supplier . The company needs no more
than 2 times as many cases from Supplier as
from Supplier . Let the number of cases from
Supplier and the number of cases from
Supplier .

Ask by Logan Vaughan. in the United States

Mar 22,2025

Question

A printing company orders paper from two different
suppliers. Supplier charges per case.
Supplier charges per case. The company the constraints on the system.
needs to order at least 45 cases per day to meet
demand and can order no more than 30 cases
from Supplier . The company needs no more
than 2 times as many cases from Supplier as
from Supplier . Let the number of cases from
Supplier and the number of cases from
Supplier .

Ask by Logan Vaughan. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

**Step 1. Define the Variables**

Let  
$$
X = \text{number of cases ordered from Supplier } X,
$$  
$$
Y = \text{number of cases ordered from Supplier } Y.
$$

**Step 2. Write Down the Constraints**

1. **Demand Constraint:**  
   The company needs to order at least 45 cases per day, so:  
   $$
   X + Y \geq 45.
   $$

2. **Supplier $$ X $$ Constraint:**  
   The company can order no more than 30 cases from Supplier $$ X $$, so:  
   $$
   X \leq 30.
   $$

3. **Supplier $$ Y $$ Constraint:**  
   The company needs no more than 2 times as many cases from Supplier $$ Y $$ as from Supplier $$ X $$, so:  
   $$
   Y \leq 2X.
   $$

4. **Non-negativity Constraints:**  
   Negative orders are not possible, so:  
   $$
   X \geq 0 \quad \text{and} \quad Y \geq 0.
   $$

**Step 3. Write the Cost Function (if needed)**

Since Supplier $$ X $$ charges $$\$25$$ per case and Supplier $$ Y $$ charges $$\$20$$ per case, the total cost $$ C $$ is:  
$$
C = 25X + 20Y.
$$

**Final System of Constraints**

$$
\begin{aligned}
X + Y &\geq 45, \
X &\leq 30, \
Y &\leq 2X, \
X &\geq 0, \
Y &\geq 0.
\end{aligned}
$$
**Constraints for the Printing Company's Paper Orders:** - **Total Cases:** At least 45 cases must be ordered each day. - **Supplier $$ X $$:** No more than 30 cases can be ordered from Supplier $$ X $$. - **Supplier $$ Y $$:** No more than twice as many cases can be ordered from Supplier $$ Y $$ as from Supplier $$ X $$. - **Non-negativity:** The number of cases ordered from each supplier must be zero or more. Mathematically, these constraints can be expressed as: $$ \begin{aligned} X + Y &\geq 45, \ X &\leq 30, \ Y &\leq 2X, \ X &\geq 0, \ Y &\geq 0. \end{aligned} $$

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