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Would you recommend hiring an additional sewing worker at $15 an hour?

To answer whether hiring an additional sewing worker at $15 per hour is recommended, we must analyze the cost-effectiveness in the context of the linear programming problem.

### Key Information from the Problem:
- **Resources:** Leather, sewing hours, and finishing hours are constrained.
- **Product Prices:** Purses are sold for $24, bags for $22, and backpacks for $45.
- **Sewing hours:** Current available time is **40 hours**, and each product requires:
  - Purse: 2 hours
  - Bag: 1 hour
  - Backpack: 3 hours

#### Approach:
1. **Impact on Objective Function:**
   - Increasing sewing hours by hiring an additional worker allows more units of products to be produced, potentially increasing revenue.
   
2. **Cost of Hiring the Worker:**
   - If an additional worker is hired for $15 per hour, this cost needs to be compared with the increased revenue from additional products made possible by the extra sewing hours.

3. **Dual Price (Shadow Price):**
   - The shadow price (dual value) of the sewing hours constraint tells us how much the objective function (profit) increases per additional hour of sewing. If this value is higher than $15, hiring a worker is beneficial. If it is less than $15, hiring is not profitable.

4. **Optimal Solution:**
   - According to the problem, the company is already operating at an optimal point. By relaxing the sewing hour constraint through hiring, we need to check if the increase in profit justifies the cost.

### Steps:
- If the dual value of sewing hours is greater than or equal to $15, **hiring an additional worker** would increase profits.
- If the dual value is less than $15, the company should **not hire the worker** as it would lead to reduced profit.

Given that the problem asks for checking feasibility changes, the dual price of sewing hours can be used to determine whether to recommend hiring.

Would you like to analyze the linear program formulation or perform detailed calculations based on shadow prices to determine profitability?

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### Follow-up Questions:
1. What are the shadow prices of other constraints, such as leather or finishing hours?
2. What would be the new optimal solution with an additional worker?
3. How does the cost of hiring affect the maximum profit?
4. What is the sensitivity of the solution to changes in the resource availability?
5. Could hiring multiple workers provide better marginal benefits?

### Tip:
When evaluating hiring decisions, always compare the increase in profit (marginal revenue) with the cost per additional unit of resource (e.g., labor).

Evaluate whether hiring an additional sewing worker at $15 per hour is cost-effective in the context of a linear programming optimization problem. Learn about dual prices, shadow values, and resource constraints involving leather, sewing hours, and finishing hours.

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