Manufacturing Problem

To tackle this manufacturing problem, we typically want to approach it as a graph traversal or dynamic programming problem. We'll break down the challenge into manageable steps, which will allow us to design a solution that identifies the cheapest way to produce the target product through a series of possible inputs or combinations.

## Problem Breakdown

1. **Inputs**:
- A target product that you want to produce.
- A set of products along with their costs and the components required to create them.

2. **Output**:
- The cheapest way (cost) to obtain the target product.

3. **Define the dependency structure**:
- Each product might depend on several other products or raw materials.
- We need to consider both costs of purchasing materials and costs of assembling them.

4. **Create a recursive strategy or use dynamic programming**:
- We'll either directly cache results (dynamic programming) or recursively calculate while storing intermediate results.
- We need to account for scenarios where a product can be both bought and built (like along the tree of dependencies).

## Implementation in Python

Here's how you could implement a solution for this problem:

```python
from collections import defaultdict
import math

def parse_input(target_product, product_data):
costs = {}
dependencies = defaultdict(list)

for line in product_data:
parts = line.split(' ')
product = parts[0]
cost = int(parts[1])
if product not in costs or costs[product] > cost: # only store the cheaper option
costs[product] = cost
dependencies[product] = parts[2:]

return costs, dependencies

def find_cheapest_product(product, costs, dependencies, memo):
if product in memo:
return memo[product]

if product not in costs:
# If the product is not purchasable directly, we should return infinity
return math.inf

# Minimum cost starts with cost of direct purchase
min_cost = costs[product]

# Check each dependency/component
for dep in dependencies[product]:
dep_cost = find_cheapest_product(dep, costs, dependencies, memo)
min_cost = min(min_cost, dep_cost)

memo[product] = min_cost
return min_cost

def cheapest_way_to_make_product(target_product, product_data):
costs, dependencies = parse_input(target_product, product_data)
memo = {}
cheapest_cost = find_cheapest_product(target_product, costs, dependencies, memo)

# Check if product can actually be built or bought
return cheapest_cost if cheapest_cost != math.inf else "Cannot obtain the target product"

# Example Usage
target_product = "TeddyBear"
product_data = [
"PaintedEye 3",
"TinyShirt 2",
"FakeCloth 5",
"SewingThread 1",
"TeddyBear 10 PaintedEye TinyShirt FakeCloth SewingThread",
"GlassEye 6",
"Paint 4",
"PaintedEye GlassEye Paint"
]

print(cheapest_way_to_make_product(target_product, product_data))
```

## Explanation of the Code
- **Data Parsing**: We read the input to create two dictionaries: one for costs and another for the dependencies of each product.
- **Recursion with Memoization**:
- We build a recursive function to calculate the cheapest cost for each product by checking if it's cheaper to build it from its components or buy it outright.
- The results of previous calculations are stored in a memo dictionary to avoid redundant calculations, thus optimizing the process.
- **Final Output**: If the product cannot be obtained (cost remains infinity), we handle that case appropriately.

This approach efficiently finds the minimum cost to produce the target product by considering all available options.