Problem 4-75 (Algo) Optimum Product Mix (LO 4-4)
Talbot Industries manufactures two models of wireless headset: TI-12 and TI-28. Each product requires time on a single machine. The machine has a monthly capacity of 624 hours. Total market demand for the two products is limited to 2,140 units of TI-12 and 1,070 units of TI-28 monthly. Talbot is currently producing and selling 1,570 TI-12 models and 794 TI-28 models each month. Cost and machine-usage data for the two products are shown in the following spreadsheet, which analysts at Talbot use for production planning purposes:

TI-12	TI-28	Total
Price	$ 94	$ 332
Less variable costs per unit
Material	30	97
Labor	36	71
Overhead	8	41
Contribution margin per unit	$ 20	$ 123
Fixed costs
Manufacturing	 	 	$ 34,700
Marketing and administrative	 	 	$ 31,700
$ 66,400
Machine hours per unit	0.1	0.5
Machine hours used	 	 	554
Machine hours available	 	 	624
Quantity produced	1,570	794
Maximum demand	2,140	1,070
Profit	 	 	$ 62,662

Required:

a. What is the optimal production schedule for Talbot Industries? In other words, how many TI-12s and TI-28s should the company produce each month to maximize monthly profit?

b. If Talbot Industries produces at the level found in requirement (a), how much will monthly profit increase over the current production schedule?

### Problem Summary
Talbot Industries manufactures two models of wireless headsets, TI-12 and TI-28. Both models require time on a single machine with a capacity of 624 hours per month. The goal is to determine the optimal production mix of these two models that maximizes profit, considering their respective contribution margins, demand limits, and machine hours used per unit.

### Key Information:
- **Machine hours available**: 624 hours per month.
- **Maximum demand**:
  - TI-12: 2,140 units/month
  - TI-28: 1,070 units/month
- **Current production**:
  - TI-12: 1,570 units
  - TI-28: 794 units
- **Contribution margin per unit**:
  - TI-12: $20
  - TI-28: $123
- **Machine hours per unit**:
  - TI-12: 0.1 hours
  - TI-28: 0.5 hours
- **Fixed costs**:
  - Manufacturing: $34,700
  - Marketing and administrative: $31,700
  - Total: $66,400
- **Current monthly profit**: $62,662

### Step-by-Step Approach:
To solve this, we will:
1. Determine the machine hours used by both products.
2. Set up an objective function to maximize profit, constrained by the available machine hours and demand limits.
3. Solve for the optimal number of TI-12 and TI-28 units to produce.

### a) Optimal Production Schedule

Let:
- $ x_1 $ = number of TI-12 units to produce.
- $ x_2 $ = number of TI-28 units to produce.

#### Constraints:
1. **Machine hours constraint**:
   - Machine hours required for TI-12: $ 0.1x_1 $
   - Machine hours required for TI-28: $ 0.5x_2 $
   - Total machine hours cannot exceed 624:
     \[
     0.1x_1 + 0.5x_2 \leq 624
     \]

2. **Demand constraints**:
   - TI-12 demand: $ x_1 \leq 2,140 $
   - TI-28 demand: $ x_2 \leq 1,070 $

#### Objective Function (Profit Maximization):
Maximize the profit:
\[
\text{Profit} = 20x_1 + 123x_2
\]
We will solve this system under the constraints to maximize profit.

### Solving the Optimization Problem

Using the constraints:
1. $ 0.1x_1 + 0.5x_2 \leq 624 $
2. $ x_1 \leq 2,140 $
3. $ x_2 \leq 1,070 $

We'll proceed by solving this linear programming problem to maximize $ 20x_1 + 123x_2 $.

Let's solve this using optimization techniques.### a) Optimal Production Schedule:
- **Optimal number of TI-12 units**: 890 units
- **Optimal number of TI-28 units**: 1,070 units

This maximizes the profit within the constraints of machine hours and market demand.

### b) Increase in Monthly Profit:
The optimal profit under the new production schedule is **$149,410**.

To find the increase in profit:
- **Current profit**: $62,662
- **Optimal profit**: $149,410
- **Profit increase**: $ 149,410 - 62,662 = 86,748 $

Thus, **monthly profit will increase by $86,748** if Talbot Industries adopts the optimal production schedule.

---

Would you like further clarification on any of these steps or calculations? 

Here are 5 related questions to deepen your understanding:
1. How would the results change if the machine hours were increased to 700 hours?
2. What happens if the contribution margin of TI-12 increases?
3. Can we still optimize the profit if the fixed costs change?
4. How does the optimal production change if the demand for TI-28 is reduced?
5. What is the break-even point in terms of production for Talbot Industries?

**Tip**: Linear programming is a powerful tool for optimizing resources under constraints. Understanding the relationships between constraints and objective functions can help solve various real-world problems.

Problem 4-75 (Algo) Optimum Product Mix (LO 4-4)
Talbot Industries manufactures two models of wireless headset: TI-12 and TI-28. Each product requires time on a single machine. The machine has a monthly capacity of 624 hours. Total market demand for the two products is limited to 2,140 units of TI-12 and 1,070 units of TI-28 monthly. Talbot is currently producing and selling 1,570 TI-12 models and 794 TI-28 models each month. Cost and machine-usage data for the two products are shown in the spreadsheet for production planning purposes.
Required:

a. What is the optimal production schedule for Talbot Industries? How many TI-12s and TI-28s should the company produce each month to maximize monthly profit?
b. How much will monthly profit increase over the current production schedule if Talbot Industries produces at the level found in requirement (a)?

Discover how to maximize Talbot Industries' monthly profit by determining the optimal production mix of wireless headsets TI-12 and TI-28. This problem uses linear programming to solve profit maximization under constraints, including machine hours and market demand. Learn how to increase monthly profits by $86,748 through optimal resource allocation.

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