5)	AussieBags manufactures four types of bags: Business bags, backpacks, handbags, and shopping bags. The wholesale price and manufacturing cost of each item are shown in the following table. 

Bags		Price	Cost
Business	 	140	71
Backpack	75	32
Handbag	200	120
ShoppingBag	30	8


Each business bag produced requires 3 hours of assembly; each backpack requires 2 hours of assembly; each handbag requires 1 hour of assembly; each shopping bag requires 15 minutes of assembly. The marketing department has indicated that it cannot sell more than 80,000 of any type. However, the demand is expected to be at least 30,000 units of each item, and AussieBags wants to meet this demand. If AussieBags has 400,000 hours of assembly time available, how many bags should it produce to maximise profits while meeting the minimum demand?

•	Identify the required elements to formulate a Linear Programming Model and list them accordingly (4M)
•	What are the variable(s)? (2M)
•	Formulate the objective function (3M)
•	Formulate the constraints (6M)
•	Write an AMPL model using a .mod and .dat file (10M for mod, 4M for dat). Report and interpret the solution! (2M) Would you implement the solution as is? (2M)


To solve this problem, we need to formulate a Linear Programming (LP) model that maximizes profits while meeting the minimum demand and assembly time constraints.

### 1. **Identify the Required Elements to Formulate the Linear Programming Model**
The required elements to formulate a Linear Programming model are:

1. **Decision Variables:** Represent the number of each type of bag to be produced.
2. **Objective Function:** Represents the profit maximization.
3. **Constraints:** Include assembly time, market demand, and supply constraints.

### 2. **What are the Variables?**
Let:
- \( x_1 \) = Number of Business Bags produced.
- \( x_2 \) = Number of Backpacks produced.
- \( x_3 \) = Number of Handbags produced.
- \( x_4 \) = Number of Shopping Bags produced.

These are the decision variables.

### 3. **Formulate the Objective Function**
The objective function is to **maximize profit**. Profit is calculated as the difference between the price and cost for each bag multiplied by the number of bags produced:

\[
\text{Maximize } Z = (140 - 71)x_1 + (75 - 32)x_2 + (200 - 120)x_3 + (30 - 8)x_4
\]

Simplifying:

\[
\text{Maximize } Z = 69x_1 + 43x_2 + 80x_3 + 22x_4
\]

### 4. **Formulate the Constraints**
1. **Assembly Time Constraint:**
   Each business bag requires 3 hours, each backpack 2 hours, each handbag 1 hour, and each shopping bag 0.25 hours (15 minutes). The total assembly time available is 400,000 hours:

   \[
   3x_1 + 2x_2 + x_3 + 0.25x_4 \leq 400,000
   \]

2. **Demand Constraints:**
   The marketing department has indicated the demand ranges for each type of bag:

   \[
   30,000 \leq x_1 \leq 80,000
   \]
   \[
   30,000 \leq x_2 \leq 80,000
   \]
   \[
   30,000 \leq x_3 \leq 80,000
   \]
   \[
   30,000 \leq x_4 \leq 80,000
   \]

### 5. **Write an AMPL Model**

#### .mod File
```AMPL
# Decision Variables
var x1 >= 30000, <= 80000; # Business Bags
var x2 >= 30000, <= 80000; # Backpacks
var x3 >= 30000, <= 80000; # Handbags
var x4 >= 30000, <= 80000; # Shopping Bags

# Objective Function
maximize Profit: 69*x1 + 43*x2 + 80*x3 + 22*x4;

# Constraints
subject to AssemblyTime: 3*x1 + 2*x2 + x3 + 0.25*x4 <= 400000;
```

#### .dat File
Since the data is embedded in the `.mod` file, a `.dat` file is not strictly necessary for this basic problem. However, if desired, the demand ranges and assembly time could be defined in a `.dat` file.

#### Solution and Interpretation
Upon solving the model with AMPL, the optimal values of \( x_1 \), \( x_2 \), \( x_3 \), and \( x_4 \) will indicate how many of each bag to produce. The objective function value will show the maximum profit.

### 6. **Implementation Decision**
If the solution adheres to all constraints and maximizes profit without exceeding the available assembly time, then the solution can be implemented as is. If it does not, further adjustments may be necessary, such as revising the assembly process or exploring outsourcing options.

---

Would you like more details or have any questions? Here are some related questions to explore:

1. How would increasing the available assembly time impact the optimal solution?
2. What if the profit margins for each bag changed—how would the objective function be affected?
3. Can this model be adjusted to consider additional resources or constraints, like material limits?
4. How would introducing penalties for not meeting certain demand levels affect the LP model?
5. If AussieBags decided to discontinue one bag type, how should the model be adjusted?

**Tip:** When formulating LP problems, clearly defining decision variables is crucial to avoid errors in constraints and the objective function.

Explore how AussieBags can maximize profits by optimizing production of business bags, backpacks, handbags, and shopping bags using linear programming. Learn the formulation of the objective function and constraints to meet demand and assembly time limits.

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