A Small Lumber Company Produces Two Types of Pine Boards

question 19

Essay

A small lumber company produces two types of pine boards used in home construction: 2x4s and 2x6s (dimensions in inches). It is attempting to determine how many of each to produce so as to minimize its costs on a per-minute basis. It has sales commitments to produce four 2x4s and two 2x6s per minute, but the management thinks they should not produce any more than eight 2x6s because of market demand. The company is also trying to support the community by employing people. Thus, it wants to keep at least 12 men employed, but only needs 2 men to produce each 2x4 and 1 person to produce each 2x6 per minute. It costs the company $.50 to produce a 2x4 and $.80 to produce a 2x6 per minute.
a. What is the objective function for this LP problem?
b. What is the constraint for the employment issue, assuming x₁ corresponds to 2x4s?
c. What is the optimal solution point for this problem?
d. What is the optimal value of the objective function?

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Definitions:

Herfindahl Index

A measure of market concentration to assess the level of competition, calculated by squaring each firm's market share and then summing the total.

Market Shares

The portion of a market controlled by a particular company, often expressed as a percentage of total sales in the industry.

Government Subsidies

Financial assistance provided by the government to individuals, organizations, or industries to encourage certain activities or reduce the burden on the recipients.

Repeated Game

A strategic situation where the same game is played multiple times by the same participants, allowing for strategy development based on previous outcomes.