Here’s a detailed analysis of the given linear programming problem:

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(a) Which two constraints give rise to the optimal solution?

The constraints that give rise to the optimal solution are those with non-zero shadow prices, as these indicate that these constraints are binding (i.e., they limit the feasible region of the solution).

From the table:

• Skilled labour has a shadow price of $12/hour.
• Materials has a shadow price of $3/kg.

Thus, the two constraints giving rise to the optimal solution are Skilled labour and Materials.

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(b) Is it worth paying overtime to help relax constraints?

Overtime is paid at "time-and-a-half," meaning:

• Skilled labour overtime costs: $1.5 \times 20 = 30 \, \text{per hour}$.
• Unskilled labour overtime costs: $1.5 \times 10 = 15 \, \text{per hour}$.

Analysis:

1. Skilled labour:

   • Shadow price: $12/hour.
   • Overtime cost: $30/hour.
   • Since the overtime cost ($30) exceeds the shadow price ($12), it is not worth paying overtime for skilled labour.

2. Unskilled labour:

   • Shadow price: $0/hour.
   • Overtime cost: $15/hour.
   • Since the shadow price is $0, unskilled labour is not a limiting factor, so overtime has no value in relaxing this constraint.

Conclusion: It is not worth paying overtime for either type of labour to relax the constraints.

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(c) Should the new product be manufactured?

The proposed product has the following details:

• Selling price: $80
• Costs:
  • Skilled labour: 2 hours at $20/hour → $2 \times 20 = 40$.
  • Unskilled labour: 1 hour at $10/hour → $1 \times 10 = 10$.
  • Materials: 3 kg at $5/kg → $3 \times 5 = 15$.
• Profit per unit: $15.

Adjusting for shadow prices:

The shadow price for binding constraints reflects the opportunity cost of resource usage. For this product:

1. Skilled labour:

   • Used: 2 hours.
   • Shadow price: $12/hour.
   • Opportunity cost: $2 \times 12 = 24$.

2. Materials:

   • Used: 3 kg.
   • Shadow price: $3/kg.
   • Opportunity cost: $3 \times 3 = 9$.

Effective profit per unit: $\text{Profit per unit} = 15 - 24 (\text{Skilled labour}) - 9 (\text{Materials}) = -18.$

Since the adjusted profit is negative, manufacturing the product would lead to a loss. Thus, the new product should not be manufactured.

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Summary of Answers:

(a) The two constraints giving rise to the optimal solution are Skilled labour and Materials.
(b) It is not worth paying overtime to relax constraints for either type of labour.
(c) The new product should not be manufactured as it would result in a loss after accounting for opportunity costs.

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Do you want further clarification or help with similar problems? Here are some questions to expand your understanding:

1. What are shadow prices and how do they influence decision-making in linear programming?
2. How does opportunity cost affect profitability in constrained optimization problems?
3. Can you derive how to calculate effective profit considering shadow prices for other cases?
4. What are the implications of zero shadow prices in non-binding constraints?
5. How might relaxing non-binding constraints impact overall feasibility in linear programming?

Tip: Always consider shadow prices as a critical tool to assess the economic value of constrained resources in optimization problems.