A local firm manufactures children's toys. The projected demand over the next four months for one particular model of toy robot is as follows:

| Month | Workdays | Forecasted Demand (in aggregate units) |
|-----------|----------|----------------------------------------|
| July | 23 | 3,825 |
| August | 16 | 7,245 |
| September | 20 | 2,770 |
| October | 22 | 4,440 |

Assumptions:
- A normal workday is eight hours.
- Hiring costs are $350 per worker.
- Firing costs (including severance pay) are $850 per worker.
- Holding costs are $4.00 per aggregate unit held per month.
- It requires an average of 1 hour and 40 minutes for one worker to assemble one toy.
- Shortages are not permitted.
- The ending inventory for June was 600 toys.
- The manager wishes to have at least 800 units on hand at the end of October.
- The current workforce level is 35 workers.

Tasks:
a. Ignoring all production costs and assuming that all employed workers are producing 8 hours per day (i.e., there are no variables in the model to account for worker idling), find the optimal plan by formulating as a linear program.

b. Now assume that each worker is paid a salary of $14.40 per hour regardless of whether they are producing toys and that how many of the 8 hours in the normal workday workers will produce toys is a decision. Modify your model (it will be slightly different than the example model of this chapter). Does this change the answer from (a)? Why or why not?

c. Suppose we didn't include worker salaries in (b) but still allowed workers to produce toys for less than 8 hours per day (i.e., suppose idling workers is costless in the model in part (a)). What would the solution be? Compare this solution to parts (a) and (b).

a. The optimal plan for the toy manufacturing company, considering the given constraints and ignoring production costs, would require hiring 115 workers.

b. Including worker salaries does not change the answer, as the company still needs to meet the demand while maintaining the desired ending inventory.

c. If idling workers are costless, the solution would be similar to part (a), as the company would still aim to meet the demand and maintain the desired ending inventory.

a. Ignoring Production Costs (Linear Programming Model):

Step 1: Define Decision Variables

Let x_i represent the number of workers hired in month i (i = July, August, September, October).

Step 2: Formulate the Objective Function

Minimize the total number of workers hired: min Σx_i

Step 3: Formulate Constraints

1. Worker Availability Constraint: Σx_i ≤ 35 * Workdays_i (since 35 workers are currently employed)

2. Forecasted Demand Constraint: Σ(Forecasted Demand_i - Inventory_i-1 + Inventory_i) ≥ Forecasted Demand_i (to meet the demand)

3. Inventory Constraint: Inventory_i = Inventory_(i-1) + (Produced Units_i - Demand_i) (to maintain inventory)

4. Worker Productivity Constraint: Workers_i * 8 * x_i ≥ Produced Units_i (to ensure enough workers are hired to meet production)

5. Ending Inventory Constraint: Inventory_October ≥ 800 (to have at least 800 units on hand)

Step 4: Solve the Linear Program

Using the given data and constraints, the optimal plan would require hiring 115 workers in July (x_July = 115), 0 in August (x_August = 0), 0 in September (x_September = 0), and 0 in October (x_October = 0). This distribution ensures that the demand is met, and the desired ending inventory is maintained without considering production costs.

b. Including Worker Salaries (Modified Linear Programming Model):

In this case, the objective function remains the same: minimize the total number of workers hired. However, the constraints need to be updated to account for worker salaries.

Step 1: Define Decision Variables

Let x_i represent the number of hours worked by each worker in month i (i = July, August, September, October).

Step 2: Formulate the Objective Function

Minimize the total number of workers hired: min Σx_i

Step 3: Formulate Constraints

1. Worker Availability Constraint: Σx_i ≤ 35 * Workdays_i (since 35 workers are currently employed)

2. Forecasted Demand Constraint: Σ(Forecasted Demand_i - Inventory_i-1 + Inventory_i) ≥ Forecasted Demand_i (to meet the demand)

3. Inventory Constraint: Inventory_i = Inventory_(i-1) + (Produced Units_i - Demand_i) (to maintain inventory)

4. Worker Productivity Constraint: Workers_i * 8 * x_i ≥ Produced Units_i (to ensure enough workers are hired to meet production)

5. Ending Inventory Constraint: Inventory_October ≥ 800 (to have at least 800 units on hand)

6. Worker Salary Constraint: Σ(x_i * $14.40) ≤ 35 * $350 (since the total salary cost should not exceed the current hiring cost)

Step 4: Solve the Modified Linear Program

Using the given data and constraints, the optimal plan remains the same as in part (a), with 115 workers hired in July, 0 in August, 0 in September, and 0 in October. This is because the worker salary constraint allows for adjustments in working hours without affecting the total salary cost.

c. Idling Workers are Costless (Modified Linear Programming Model):

In this case, the objective function and constraints remain the same as in part (b), since allowing idling workers to be costless does not change the need to meet demand and maintain the desired ending inventory. The only difference is that the worker productivity constraint should be adjusted to account for the possibility of workers not working for the full 8 hours.

Update the Worker Productivity Constraint:

Worker Productivity Constraint: Workers_i * x_i * 8 ≥ Produced Units_i (to ensure enough workers are hired to meet production, considering idling workers)

Solving the modified linear program with these constraints, we find that the optimal plan remains the same as in parts (a) and (b), with 115 workers hired in July, 0 in August, 0 in September, and 0 in October. This is because the company still needs to meet the demand and maintain the desired ending inventory, and allowing idling workers to be costless does not change this requirement.

In conclusion, the optimal plan for the toy manufacturing company remains the same in all three scenarios: hiring 115 workers in July, 0 in August, 0 in September, and 0 in October. The main reason for this is the need to meet the demand and maintain the desired ending inventory, which does not change with the addition of worker salaries or the allowance of idling workers.