The Milton Lumber Company sells boards of various lengths (3-foot, 8-foot, and 10-foot wood boards). This company's customers demand 50 3-foot boards, 15 8-foot boards, and 70 10-foot boards per week. Milton Lumber cuts up boards 14 feet in length to meet this demand and wants to determine how to satisfy its customers' demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model. a) Define the decision variables. b) Define the objective function. c) Define the constraints.
The Milton Lumber Company sells boards of various lengths (3-foot, 8-foot, and 10-foot wood boards). This company's customers demand 50 3-foot boards, 15 8-foot boards, and 70 10-foot boards per week. Milton Lumber cuts up boards 14 feet in length to meet this demand and wants to determine how to satisfy its customers' demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model. a) Define the decision variables. b) Define the objective function. c) Define the constraints.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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![**Formulating and Solving an Integer Programming (IP) Model for the Milton Lumber Company**
The Milton Lumber Company sells boards of various lengths (3-foot, 8-foot, and 10-foot wood boards). This company’s customers demand 50 3-foot boards, 15 8-foot boards, and 70 10-foot boards per week. Milton Lumber cuts up boards 14 feet in length to meet this demand and wants to determine how to satisfy its customers’ demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model.
**a) Define the decision variables.**
Let:
\( x_1 \) = number of 14-foot boards cut into 3-foot pieces
\( x_2 \) = number of 14-foot boards cut into 8-foot pieces
\( x_3 \) = number of 14-foot boards cut into 10-foot pieces
**b) Define the objective function.**
Minimize waste:
\[ \text{Waste} = 14(x_1 + x_2 + x_3) - (3 \cdot x_1 + 8 \cdot x_2 + 10 \cdot x_3) \]
**c) Define the constraints.**
The constraints are based on the customer demands:
\( 3 \cdot x_1 \geq 50 \) (to meet the demand for 3-foot boards)
\( 8 \cdot x_2 \geq 15 \) (to meet the demand for 8-foot boards)
\( 10 \cdot x_3 \geq 70 \) (to meet the demand for 10-foot boards)
Additionally, we need the integer constraints for \( x_1 \), \( x_2 \), and \( x_3 \):
\( x_1, x_2, x_3 \in \mathbb{Z}_{\geq 0} \)
By solving the above IP model, Milton Lumber Company can determine the optimal way to cut 14-foot boards to minimize waste while satisfying the customer demands for different lengths of boards.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b9d4688-d7d2-4904-9fe3-877af97ac226%2Fd85da059-62e5-419f-b0ee-0a49b07371b2%2Ftot9rl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Formulating and Solving an Integer Programming (IP) Model for the Milton Lumber Company**
The Milton Lumber Company sells boards of various lengths (3-foot, 8-foot, and 10-foot wood boards). This company’s customers demand 50 3-foot boards, 15 8-foot boards, and 70 10-foot boards per week. Milton Lumber cuts up boards 14 feet in length to meet this demand and wants to determine how to satisfy its customers’ demands with a minimal amount of waste. Assume that all boards share the same width and thickness. Formulate and solve an IP model.
**a) Define the decision variables.**
Let:
\( x_1 \) = number of 14-foot boards cut into 3-foot pieces
\( x_2 \) = number of 14-foot boards cut into 8-foot pieces
\( x_3 \) = number of 14-foot boards cut into 10-foot pieces
**b) Define the objective function.**
Minimize waste:
\[ \text{Waste} = 14(x_1 + x_2 + x_3) - (3 \cdot x_1 + 8 \cdot x_2 + 10 \cdot x_3) \]
**c) Define the constraints.**
The constraints are based on the customer demands:
\( 3 \cdot x_1 \geq 50 \) (to meet the demand for 3-foot boards)
\( 8 \cdot x_2 \geq 15 \) (to meet the demand for 8-foot boards)
\( 10 \cdot x_3 \geq 70 \) (to meet the demand for 10-foot boards)
Additionally, we need the integer constraints for \( x_1 \), \( x_2 \), and \( x_3 \):
\( x_1, x_2, x_3 \in \mathbb{Z}_{\geq 0} \)
By solving the above IP model, Milton Lumber Company can determine the optimal way to cut 14-foot boards to minimize waste while satisfying the customer demands for different lengths of boards.
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