Question 1210545: Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including at least 60 of the deluxe kind. The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 . There are 400 hours available for assembling time. The equipment can be used to assemble either kind of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize the company's cost of this special order. You are required as an expert to formulate this problem in a linear programming form and using the graphical method advise the manager on the best product that will enable his firm to minimize it's cost
Found 3 solutions by ikleyn, CPhill, n2:
Answer by ikleyn(53575)   (Show Source):
You can put this solution on YOUR website! .
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This problem, as it is presented in the post, has no solution, at all.
Indeed, to make 200 cabinets, it requires at least 4*200 = 800 hours of work,
counting the minimum required time of 4 hours for a standard cabinet.
It is just more than 400 hours available, so the problem does not have a solution.
You do not need to deploy the heavy artillery of the Linear Programming to get this conclusion.
Answer by CPhill(2189)  (Show Source):
You can put this solution on YOUR website! 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.
*
---
### 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.*
---
### 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
Answer by n2(49)  (Show Source):
You can put this solution on YOUR website! .
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~
The answer in the post by @CPhill (125 standard cabinets, 60 Deluxe cabinets) does not satisfy
the restriction of 400 hours: 125*4 + 60*5 = 800 hours, which greatly exceeds the restriction of 400 hours.
It also does not satisfy the requirement "at least 200 bathroom cabinets", since 125 + 60 = 185 is less than 200.
So, @CPhill solved DIFFERENT problem from what is posed in the post.
He made it even without explicit announcement/declaration about changing the problem,
which is inappropriate practice and can confuse a reader.
For the correct treatment of the problem, see the post by @ikleyn at this spot,
where it was shown that the problem, as posed in the post, has no solution, at all.
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