Question

Bangs Leisure Chairs produces three hand-crafted outdoor chairs famous for beaches, pools, and patios: sling, Adirondack, and hammocks. The unit profit for these products is $40, $70, and $80, respectively. Each type of chair requires cutting, assembling, and finishing. The owner is retired and willing to work six hours per day for five days per week, so he has 120 hours available each month. He does not want to spend more than 50 hours each month on any one activity (that is, cutting, assembling, and finishing). The retailer he works with ensures that all products he makes can quickly be sold. Sling chairs are made up of ten wood pieces for the frame and one piece of cloth. The actual cutting of the wood takes 30 minutes. Assembling includes sewing the fabric and the attachment of rivets, screws, fabric, and dowel rods, which takes 45 minutes. The finishing stage involves sanding, staining, and varnishing the various parts, which takes one hour. Adirondack chairs take one and a half an hour to cut and one hour to assemble, and finishing takes one hour. For hammocks, cutting takes 0.4 hours, assembly takes 1.5 hours, and finishing takes one hour. How many of each type of chair should he produce each month to maximize profit?

Answer 1

To maximize monthly profit, the owner should set up a linear programming problem based on the time required for cutting, assembling, and finishing sling, Adirondack, and hammocks. By introducing variables for the number of each type of chair made and applying the time constraints, a solution can be found to determine the optimal number of chairs to produce within the 120-hour monthly work limit.

Step-by-step explanation:

To maximize profit, the owner of Bangs Leisure Chairs needs to determine how many of each type of chair - sling, Adirondack, and hammocks - to produce each month given the time constraints and unit profits for each product. We must take into account the time available, which is 120 hours per month, with no more than 50 hours dedicated to any one activity (cutting, assembling, and finishing). Based on the time each chair takes for each activity, we can set up a linear programming problem to find the optimal production mix.

The time required for each chair is:

• Sling chairs: Cutting - 0.5 hours, Assembling - 0.75 hours, Finishing - 1 hour
• Adirondack chairs: Cutting - 1.5 hours, Assembling - 1 hour, Finishing - 1 hour
• Hammocks: Cutting - 0.4 hours, Assembling - 1.5 hours, Finishing - 1 hour

Let's denote the number of sling chairs, Adirondack chairs, and hammocks produced per month as S, A, and H respectively. The total profit P can be calculated as:

P = 40S + 70A + 80H

The constraints based on the time limitations are:

Cutting time: 0.5S + 1.5A + 0.4H ≤ 50
Assembling time: 0.75S + 1A + 1.5H ≤ 50
Finishing time: 1S + 1A + 1H ≤ 50

The owner should use linear programming methods to solve this problem, which could include graphical methods or the simplex algorithm, to find the values of S, A, and H that maximize profit while staying within the constraints of the available time for each activity.