VIPSolutionsGive step-by-step solution with explanation and final answer:1) A production manager is planning the scheduling of four products on three machines. Each product cn be manufactured on any of the machines. The unt production costs (008) are summarized below. The production rate (nuts per hour) for ech product on exch of the machines s summarized below Clee]
Question:
Give step-by-step solution with explanation and final answer:
1) A production manager is planning the scheduling of four products on three machines.
Each product cn be manufactured on any of the machines. The unt production costs
(008) are summarized below.
The production rate (nuts per hour) for ech product on exch of the machines s
summarized below
Clee]
This is AI generated Response.
Asked by: Test User 100147
Created at: 2025-05-07 13:42:59
Asked by: Test User 100147
Created at: 2025-05-07 13:42:59
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Let's break down the problem **step-by-step**:
### **Step 1: Understanding the Problem**
You have 4 products and 3 machines.
- **Each product can be manufactured on any machine.**
- **Unit production costs** (in $) for each product-machine combination are given.
- **Production rates** (units/hour) for each product-machine combination are also given.
Let's summarize the tables:
#### **Unit Production Cost Table ($):**
| Product | Machine 1 | Machine 2 | Machine 3 |
|---------|-----------|-----------|-----------|
| 1 | 4 | 5 | 3 |
| 2 | 8 | 6 | 4 |
| 3 | 10 | 7 | 8 |
| 4 | 6 | 3 | 12 |
#### **Production Rate Table (units/hour):**
| Product | Machine 1 | Machine 2 | Machine 3 |
|---------|-----------|-----------|-----------|
| 1 | 4 | 1.75 | 3 |
| 2 | 2.5 | 3 | 2 |
| 3 | 1.75 | 4 | 2.5 |
| 4 | 2.5 | 1.5 | 3.25 |
---
### **Step 2: Calculating Cost per Unit Time**
To find the most efficient machine for each product, **calculate the cost per unit time** (cost per hour) for each product on each machine:
\[
\text{Cost per hour} = \text{Unit cost} \times \text{Production rate}
\]
#### Let's calculate for each cell:
---
#### **For Product 1:**
- Machine 1: \( 4 \times 4 = 16 \)
- Machine 2: \( 5 \times 1.75 = 8.75 \)
- Machine 3: \( 3 \times 3 = 9 \)
#### **For Product 2:**
- Machine 1: \( 8 \times 2.5 = 20 \)
- Machine 2: \( 6 \times 3 = 18 \)
- Machine 3: \( 4 \times 2 = 8 \)
#### **For Product 3:**
- Machine 1: \( 10 \times 1.75 = 17.5 \)
- Machine 2: \( 7 \times 4 = 28 \)
- Machine 3: \( 8 \times 2.5 = 20 \)
#### **For Product 4:**
- Machine 1: \( 6 \times 2.5 = 15 \)
- Machine 2: \( 3 \times 1.5 = 4.5 \)
- Machine 3: \( 12 \times 3.25 = 39 \)
---
### **Step 3: Summary Table (Cost per Hour)**
| Product | Machine 1 | Machine 2 | Machine 3 |
|---------|-----------|-----------|-----------|
| 1 | 16 | 8.75 | 9 |
| 2 | 20 | 18 | 8 |
| 3 | 17.5 | 28 | 20 |
| 4 | 15 | 4.5 | 39 |
---
### **Step 4: Choosing the Best Machine for Each Product**
Pick the **minimum cost per hour** for each product:
- **Product 1:** Machine 2 ($8.75)
- **Product 2:** Machine 3 ($8)
- **Product 3:** Machine 1 ($17.5)
- **Product 4:** Machine 2 ($4.5)
---
### **Step 5: Final Answers**
**The optimum assignment of products to machines to minimize cost per hour is:**
| Product | Best Machine | Minimum Cost per Hour |
|---------|-------------|----------------------|
| 1 | 2 | $8.75 |
| 2 | 3 | $8.00 |
| 3 | 1 | $17.50 |
| 4 | 2 | $4.50 |
---
## **Summary Table**
| Product | Assigned Machine | Cost per Hour |
|---------|------------------|--------------|
| 1 | 2 | $8.75 |
| 2 | 3 | $8.00 |
| 3 | 1 | $17.50 |
| 4 | 2 | $4.50 |
---
**If you have any specific constraints (e.g., each machine can do only one product), let me know!** Otherwise, this is the step-by-step optimal allocation based on minimizing cost per hour.
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