Week Six Homework Assignment

1. For the purposes of this assignment, partially manufactured air conditioners and fans are not acceptable.

The Electrocomp Corp manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours or wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach.

Electrocomp's management realizes that it forgot to include two critical constraints (see problem 14). In particular, management decides that there should be a minimum number of air conditioners produced in order to fulfill a contract. Also, due to an oversupply of fans in the preceding period, a limit should be placed on a total number of fans produced.

a. If Electrocomp decides that at least 20 air conditioners should be produced but no more than 80 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints?

b. If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, what would be the optimal solution? How much slack is there for each of the four constraints at the optimal solution?

2. A woodworking company manufactures and sells dining room tables and chairs. The owner has assumed that his customers are interested in buying tables and chairs individually rather than having to buy them in pre-defined sets, as is the case with most furniture manufacturers. The owner has established the following general guidelines for the company's production effort:

• The company's objective is to maximize profit during each production cycle.
• Fabrication of each table requires 4 units of wood and each chair requires 1 unit of wood.
• Fabrication of each table requires 20 units of fabrication labor and each chair requires 12 units of fabrication labor.
• Fabrication of each table requires 2 units of assembly labor and each chair requires 4 units of assembly labor.
• Fabrication of each table requires 12 units of finishing labor and each chair requires 8 units of finishing labor.
• Fabrication of each table requires 1 unit of packaging labor and each chair requires 1 unit of packaging labor.
• Producing partially manufactured (i.e., partially fabricated, assembled, finished and/or packaged) tables and/or chairs is acceptable during any given production period since they can be completed during the subsequent production cycle. However, potential profit for a given production cycle shall be calculated based only upon the number of complete tables and/or chairs produced during the production cycle, since partially manufactured tables or chairs cannot be sold.
• The company will earn a potential profit of $250 for each table sold and $65 for each chair sold.
• It is assumed that all complete tables and chairs manufactured during a given production period will also be sold during the same production period.

For the current production cycle, the owner anticipates having 750 units of wood, 5,000 units of fabrication labor, 1,225 units of assembly labor, 3,250 units of finishing labor and 500 units of packaging labor available. Create a linear programming model for the preceding scenario using the Excel Solver method in order to answer the following questions:

a. What is the optimal number of tables the company should produce during the current production cycle?
b. What is the optimal number of chairs the company should produce during the current production cycle?
c. What is the total amount of profit that the company would earn for producing the optimal number of tables and chairs during the current production cycle (keeping in mind that partially manufactured tables and/or chairs do not contribute to profit earned during the current production cycle)?
d. Which resources will be fully used in producing the optimal number of tables and chairs?

The following hints apply:

• Do not forget to consider whether or not a non-negativity constraint would be appropriate for inclusion in your model.
• Do not forget to consider whether or not an integer constraint would be appropriate for inclusion in your model.
• A given resource is fully consumed only if all of the 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 manufacture another complete or partial table or chair is irrelevant. Although you might not be able to use this surplus resource to make another complete or partial table or chair, it is nonetheless available and could 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 possibly be held in reserve for use in future production periods.

3. During the course of the current production cycle, the owner of the woodworking company discussed in the preceding problem discovers that his assumption regarding customers being interested in buying tables and chairs individually rather than in sets is incorrect. The vast majority of his customers are indicating that they are only interested in purchasing table and chair in sets, with each set consisting of one table and four chairs. Without revising the previously stated general guidelines or previously LP model for problem 2, answer the following questions based upon only being able to sell the tables and chairs produced in complete sets:

a. How many complete table and chair sets can the company assemble from the optimal number of tables and chairs produced during the current production cycle?
b. How many excess tables will the company have left in inventory at the end of the current production cycle?
c. How many excess chairs will the company have left in inventory at the end of the current production cycle?
d. What is the total amount of profit that the company would potentially earn, assuming that excess tables or chairs do not contribute to profit earned during the current production cycle?

4. Revise your linear programming model for the woodworking company discussed in problems 2 and 3 to take into account the following additional general guidelines for production:

• Tables and chairs shall only be sold in complete sets consisting of one table and four chairs (i.e., tables and chairs may not be sold individually).
• Profit for a given production period shall be calculated based upon the number of complete table and chair sets produced during the production period. Tables and chairs associated with partial table and chair sets shall not be considered when calculating profit.
• Partially manufactured tables and/or chairs do not contribute to profit earned during a given production cycle.
• The company must manufacture a minimum of 50 complete table and chair sets during any given production cycle in order to satisfy estimated customer demand.

Answer the following questions based upon your revised LP model:

a. What is the optimal number of tables the company should produce during the current production cycle in order to produce the optimal number of complete table and chair sets?
b. What is the optimal number of chairs the company should produce during the current production cycle in order to produce the optimal number of complete table and chair sets?
c. What is the total amount of profit that the company will potentially earn for producing the optimal number of complete table and chair sets?
d. Which resources will be fully used in producing the optimal number of tables and chairs?

In addition to the previously provided hints for problem 1, the following additional hints apply:

• You do not need to create a separate variable to represent a table and chair set. You can create a simple algebraic equation using only the two variables that represent the number of tables to be produced and the number of chairs to be produced that describes the ratio in which tables and chairs must be manufactured to produce only complete sets (i.e., no excess tables or chairs).

• Do not forget to account for the constraint requiring that you produce a specific minimum number of table and chair sets.