Question 1210545
To help Gorimapa Nigeria PLC minimize its costs while meeting the order requirements, we can use **Linear Programming (LP)**. This mathematical approach allows us to find the most efficient combination of standard and deluxe cabinets within the given constraints.

### 1. Problem Formulation

First, we define our variables and the mathematical relationships between them.

**Decision Variables:**

* Let  = Number of **Standard** cabinets to produce.
* Let  = Number of **Deluxe** cabinets to produce.

**Objective Function:**
The goal is to minimize the total variable cost ():


**Constraints:**

1. **Total Order Quantity:** At least 200 cabinets in total.
* 


2. **Deluxe Requirement:** At least 60 deluxe cabinets must be made.
* 


3. **Assembly Time:** Total time cannot exceed 800 hours.
* 


4. **Non-negativity:** Production cannot be negative.
* 



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### 2. Graphical Method Analysis

To solve this graphically, we treat the inequalities as equations to find the boundary lines.

* **Line 1 (Total Order):** .
* If ; if .


* **Line 2 (Deluxe Min):** .
* A horizontal line at .


* **Line 3 (Time Limit):** .
* If ; if .



#### Identifying the Feasible Region

The feasible region is the area that satisfies all three conditions simultaneously. Looking at the intercepts:

* The **Time Limit** and **Total Order** lines actually meet at the x-axis at .
* However, we must have at least 60 Deluxe cabinets ().

By solving the intersection of  and the other constraints:

1. **Intersection of  and :** . Point: **(140, 60)**.
2. **Intersection of  and :** . Point: **(125, 60)**.

**Wait!** Note that at point (140, 60), the total hours needed would be  hours. This exceeds our 800-hour limit.

### 3. Finding the Optimal Solution

The feasible region is a small triangle bounded by the points where these constraints overlap. To minimize cost, we test the corner points of the feasible region:

| Corner Point  | Cost Calculation () | Total Cost |
| --- | --- | --- |
| **(125, 60)** |  | **#860,000** |
| **(0, 160)** |  | #960,000 |
| **(0, 200)** | *Not feasible (Exceeds 800 hours)* | - |

*Note: The point (200, 0) is also not feasible because  must be at least 60.*

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### 4. Expert Advice to the Manager

Based on the linear programming analysis, the best product mix to minimize costs while staying within the 800-hour limit and meeting the order requirements is:

* **Standard Cabinets:** 125 units
* **Deluxe Cabinets:** 60 units
* **Total Production:** 185 units