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To solve this aggregate planning problem, let's go through each step required to develop the plan and calculate the total cost based on the requirements.

### Problem Breakdown and Solution Outline

1. **Employee Production Capacity**:
   - Each of the 20 full-time employees produces 10 units per period, resulting in a regular output capacity of:
     \[
     20 \text{ employees} \times 10 \text{ units/employee} = 200 \text{ units per period}
     \]
   
2. **Demand and Regular Production**:
   - Compare the regular production capacity of 200 units to the forecasted demand for each period.
   - If demand exceeds 200 units, use overtime (up to 25 additional units per period at $9 per unit) or allow backlogs (costing $10 per unit per period, limited to 60 units).

3. **Inventory and Backlog Handling**:
   - Inventory carrying cost is $5 per unit per period.
   - The ending inventory at period 9 should be zero, and the backlog is limited to 60 units per period.

4. **Cost Components**:
   - Regular production cost: $6 per unit.
   - Overtime production cost: $9 per unit (up to 25 units).
   - Inventory carrying cost: $5 per unit per period.
   - Backlog cost: $10 per unit per period.

5. **Objective**:
   - Minimize costs by balancing regular production, overtime, inventory, and backlogs to meet the demand over nine periods with no leftover inventory in period 9.

### Step-by-Step Calculations

We'll now calculate the production, inventory, backlog, and cost for each period based on these rules. This approach involves:
- Calculating the difference between demand and regular production.
- Deciding whether to use overtime, inventory, or backlog to meet demand.
- Computing costs for each approach and summing them up for the total cost.

Let’s proceed with a detailed table showing these values for each period.

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I’ll create this table to show and compute the costs.I have displayed the table for the aggregate plan, showing the calculations of production, overtime, inventory, backlog, and the costs associated with each period. This table allows us to see the total cost for each period and how different components like overtime, inventory, and backlog costs affect the overall cost.

Would you like further details on the calculations for each component or have any questions?

Here are five additional questions to expand on this topic:

1. How does the maximum overtime limit impact the overall cost efficiency of the aggregate plan?
2. What would be the effect of increasing the backlog limit on total costs?
3. How would a change in regular production capacity per employee affect the plan?
4. What strategies can be used to minimize inventory holding and backlog costs?
5. How would fluctuating inventory or backlog costs per period influence decision-making in aggregate planning?

**Tip:** Reviewing how each cost component contributes to the total can help in finding areas to optimize and reduce overall costs.

A manager is attempting to put together an aggregate plan for the coming nine months based on a forecast of expected demand. The department has 20 employees who produce 10 units each per period at $6 per unit, with additional costs for overtime, inventory, and backlog. The goal is to meet demand while minimizing backlogs and ensuring zero inventory at the end of period 9.

Explore how to develop an aggregate plan for a nine-month forecast with a focus on minimizing costs related to regular production, overtime, inventory, and backlog. This problem covers key topics in production scheduling and cost minimization in operations management, suitable for college-level students.

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