Answer :
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.