# total amount of profit

Willis Eckley owns a fine woodworking company that manufactures and sells dining room table and chair sets. A table and chair set consists of one table and four chairs. Willis has established the following guidelines for production:

1. Tables and chairs shall not be sold individually, they shall only be sold in sets.

2. Tables and chairs shall only be manufactured if sufficient resources are available to manufacture a complete set (i.e., partial sets shall not be manufactured).

3. A given table or chair shall only be manufactured if sufficient resources are available to manufacture a complete table or chair (i.e., partially fabricated, assembled, finished or packaged tables or chairs are not allowed).

Each table requires 4 units of wood and each chair requires 1 unit of wood. Each table requires 20 hours of fabrication labor and each chair requires 12 hours of fabrication labor. Each table requires 2 hours of assembly labor and each chair requires 4 hours of assembly labor. Each table requires 12 hours of finishing labor and each chair requires 8 hours of finishing labor. Each table requires 1 hour of packaging labor and each chair requires 1 hour of packaging labor. The company earns a unit profit of $200 for each table sold and $65 for each chair sold. Historical sales records indicate that the company must manufacture a minimum of 100 table and chair sets during each production cycle in order to satisfy customer demand. At the start of the current production cycle, Willis anticipated having 1,500 units of wood, 10,000 hours of fabrication labor, 3,000 hours of assembly labor, 6,500 hours of finishing labor and 1,000 hours of packaging labor available.

Create a linear programming model for the preceding scenario in order to answer the following questions.

Hint: You do not need to create a separate variable to represent a table and chair set. You can represent the number of table and chairs sets manufactured as either a function of the total number of tables produced or the total number chairs produced.

Hint: Some students believe that a resource has been fully consumed if the remaining portion is insufficient to make either a complete table, chair or table and chair set. While this is not necessarily an unreasonable interpretation, it is actually incorrect. A given resource is fully consumed only if all of the available resource has been used. If even a fraction of the resource remains unused (e.g., a fraction of a unit of wood or a fraction of a unit of labor), then the resource has not been fully consumed. The fact that the portion remaining may be insufficient to make another complete table, chair or set is irrelevant. Although you might not be able to use this surplus resource to make another complete table, chair or set, it is nonetheless available and could thus be reallocated elsewhere within the company where it could possibly be put to use during the current production period to manufacture other products or could be held in reserve for use in future production periods.

1. Can Willis produce the required minimum number of table and chair sets?

2. What is the optimal number of tables Willis should produce if he wants to maximize his profits during the production cycle?

3. What is the optimal number of chairs Willis should produce if he wants to maximize his profits during the production cycle?

4. What is the total amount of profit that Baxter Fine Woodworking will earn for producing the optimal number of table and chair sets?

5. Which of the resources will be fully used in producing the optimal number of table and chair sets?

As fate would have it, during the next production cycle one of the trucks that was supposed to deliver a shipment of wood to the Baxter Fine Woodworking plant was involved in an accident and its cargo was destroyed in the ensuing fire. Willis will now only have 750 units of wood available for the production cycle. All other resource constraints remain unchanged from the original production cycle.

6. Can Willis produce the required minimum number of table and chair sets?

Given the availability of only 750 units of wood, assume that Willis decides to reduce the required minimum number of table and chair sets to 90. All other resource constraints remain unchanged from the original production cycle.

7. Can Willis produce the revised minimum number of table and chair sets?

8. What is the optimal number of tables that Willis can produce if he wants to maximize his profits during the production cycle?

9. What is the optimal number of chairs that Willis can produce if he wants to maximize his profits during the production cycle?

10. What is the total amount of profit that Willis can earn for producing the optimal number of table and chair sets during the production cycle?

11. Which of the resources will be fully used in producing the optimal number of table and chair sets during the production cycle?

During the following production cycle, Willis receives the full shipment of 1,500 units of wood. However, a large number of employees have been stricken with the flu and have called in sick during the first few weeks of the production cycle. Willis estimates that he will only have 8,000 hours of fabrication labor, 1,700 hours of assembly labor, 4,000 hours of finishing labor and 750 hours of packaging labor available for the production cycle. All other resource constraints remain unchanged from the original production cycle.

12. Can Willis produce the required minimum number of table and chair sets?

Given the limitations on the amount of labor available, Willis once again decides to reduce the required minimum number of table and chair sets to 90. All other constraints remain the same.

13. Can Willis produce the revised required minimum number of table and chair sets?

14. What is the optimal number of tables that Willis can produce given the revised resource constraints if he wants to maximize his profits during the production cycle?

15. What is the optimal number of chairs that Willis can produce given the revised resource constraints if he wants to maximize his profits during the production cycle?

16. What is the total amount of profit that Baxter Fine Woodworking can earn for producing the optimal number of table and chair sets given the revised resource constraints?

17. Which of the resources will be fully used in producing the optimal number of table and chair sets?

A number of customers have indicated that they would potentially be interested in purchasing additional chairs for their table and chair set. Willis has estimated that he could sell an additional 40 chairs during each production cycle based upon customer inquiries. All other constraints remain unchanged from the original production cycle.

18. Can Willis still produce the required minimum number of table and chair sets?

19. What is the optimal number of tables Willis should produce if he wants to maximize his profits during the production cycle?

20. What is the optimal number of chairs Willis should produce if he wants to maximize his profits during the production cycle?

21. What is the total amount of profit that Willis will earn for producing the optimal number of table and chair sets?

22. Which of the resources will be fully used in producing the optimal number of table and chair sets?

23. Would Willis earn more profit by producing the optimal number of table and chair sets plus the extra chairs instead of simply producing the optimal number of table and chair sets?

#### Solution Summary

Profit function is integrated in this solution.