# Applications of Linear Programming

Linear Programming

Scenario # 1

A new woodworking company is going to manufacture and sell dining room tables and chairs. The company's owner has assumed that his customers are interested in buying tables and chairs individually, rather than having to buy them in predetermined sets, as is the case with his competitors. The owner has established the following general guidelines for the company's initial production cycle:

- The company's objective is to maximize profit earned for each production cycle.

- Fabrication of each table requires 12 units of wood, and fabrication of each chair requires 1.5 unit of wood.

- Fabrication of each table requires 20 units of fabrication labor. and fabrication of each chair requires 6 units of fabrication labor.

- Fabrication of each table requires 5 units of assembly labor, and fabrication of each chair requires 2 units of assembly labor.

- Fabrication of each table requires 12 units of finishing labor, and fabrication of each chair requires 3 units of finishing labor.

- Fabrication of each table requires 4 unit of packaging labor, and fabrication of each chair requires 1 unit of packaging labor.

- Producing partially manufactured (i.e., partially fabricated, assembled, finished and/or packaged) tables and/or chairs during a given production cycle is not permitted due to the fact that they cannot be sold.

- The company will earn a profit of $250 for each table sold, and a profit of $50 for each chair sold.

- Assume that the company will be able to sell all tables and/or chairs manufactured during a given production cycle.

- For the upcoming production cycle, the owner anticipates having 3,000 units of wood available; 6,000 units of fabrication labor available; 3,000 units of assembly labor available; 5,000 units of finishing labor available; and 1,500 units of packaging labor available.

Define the linear programming problem set-up (i.e., variables, objective function and constraints) for the preceding scenario, and then create an Excel model for the problem set-up. Next use the Excel Solver method in order to answer questions 1 through 4. Do not forget to consider whether or not a non-negativity constraint and/or an integer constraint would be appropriate for inclusion in your problem set-up and model.

1. What is the optimal number of tables the company should produce during the initial production cycle?

2. What is the optimal number of chairs the company should produce during the initial production cycle?

3. Which resources will be fully used in producing the optimal number of tables and chairs during the initial production cycle? (Note: A given resource is fully consumed only if all of the resource has been used. If even a fraction of a unit of a resource remains unused, then the resource has not been fully consumed.)

4. What is the total profit the company will earn for selling the optimal number of tables and chairs to be produced during the initial production cycle?

Scenario # 2

Assume that during the course of the initial production cycle the owner discovers that his assumption regarding customers wanting to buy tables and chairs individually, rather than in predetermined sets, was incorrect. Potential 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 LP problem set-up or Excel model you devised for Scenario # 1, or recalculating the optimal number of tables and chairs to produce, answer questions 5 through 8.

5. How many complete table and chair sets can the company assemble from the optimal number of tables and chairs being produced during the initial cycle?

6. If the company is only able to sell complete table and chair sets, how many excess tables will the company have left in inventory at the end of the initial production cycle?

7. If the company is only able to sell complete table and chair sets, how many excess chairs will the company have left in inventory at the end of the initial production cycle?

8. If the company is only able to sell complete table and chair sets, what is the adjusted total profit that the company will earn during the initial production cycle? (Note: Keep in mind that excess tables and chairs cannot be sold and thus will not count towards total profit earned.)

Scenario # 3

The owner has decided that for the next production cycle the company will only manufacture and sell tables and chairs in complete sets in response to the customer feedback. Revise the LP problem set-up and Excel model that you developed for Scenario # 1 to take into account the following additional general guidelines. Then use the Excel Solver method in order to answer questions 9 through 11.

- The optimal number of tables and chairs to produce during the production cycle shall be determined based upon only producing complete table and chair sets, with each set consisting of one table and four chairs. Excess tables and/or chairs shall not be produced due to the fact that they cannot be sold.

- Profit for shall be calculated based upon the number of complete table and chair sets produced during the production cycle. Excess tables and/or chairs cannot be sold, thus they do not contribute to profit earned.

- Assume that the company will be able to sell all complete table and chair sets manufactured during the production cycle.

- Unless otherwise indicated, all other Scenario # 1 guidelines remain the same.

- Disregard any excess tables or chairs that were left in inventory at the end of the initial production cycle.

Hint: You do not need to create a new variable to represent a table and chair set. In order to create only complete sets, you must produce exactly four chairs for every table produced. In order to represent this constraint in your problem set-up and model, simply create an additional constraint equation, using only the two variables you previously identified to 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 in order to produce only complete table and chair sets (i.e., avoid producing excess tables or excess chairs).

9. What is the optimal number of complete table and chair sets the company should produce during the production cycle?

10. Which resource will be fully used in producing the optimal number of set table and chair sets? (Note: A given resource is fully consumed only if all of the resource has been used. If even a fraction of a unit of a resource remains unused, then the resource has not been fully consumed.)

11. Which resource(s) restrict the number of complete table and chairs sets that can be produced? (Note: A resource would be considered restrictive if there is insufficient slack for this resource remaining after producing the optimal number of tables and chair sets to produce an additional complete table and chair set.)

12. What is the total profit that the company will earn for producing the optimal number of complete table and chair sets during the production cycle?

Scenario # 4

Customers have advised the owner that they would like to purchase a matching china hutch along with their table and chair set in order to have a matched dining room set. Accordingly, the owner has decided to focus the next production effort on manufacturing dining room sets. Revise the LP problem set-up and Excel model you developed for Scenario # 3 to take into account the following additional general guidelines and use the Excel Solver method in order to answer questions 13 through 15:

- Tables, chairs and china hutches shall only be sold as a complete dining room set consisting of one table, four chairs and one china hutch. Excess tables, chairs, and/or hutches shall not be produced due to the fact that they cannot be sold.

- Producing partially manufactured (i.e., partially fabricated, assembled, finished and/or packaged) tables, chairs, and/or china hutches is not permitted due to the fact that they cannot be sold and thus do not contribute to profit earned.

- Fabrication of each china hutch will require 15 units of wood; 28 units of fabrication labor; 10 units of assembly labor; 12 units of finishing labor; and 5 units of packaging labor.

- The company will earn a unit profit of $300 for each china hutch sold.

- All other guidelines for scenarios # 1 and 3 guidelines remain the same.

- Profit for a given production period shall be calculated based upon the number of complete dining room sets produced during the production cycle.

- Assume that all complete dining room sets manufactured during the production cycle will be sold during the production cycle.

- The owner anticipates needing to manufacture a minimum of 75 complete dining room sets during the production cycle in order to satisfy customer demand.

Hint: You do not need to create a new variable to represent a complete dining room set. You can represent the requirement to only manufacture complete dining room sets by creating two simple constraints. First, in order to produce only complete dining sets, you must produce exactly four chairs for every table produced. In order to represent this constraint, simply create a constraint equation, using the two variables you previously identified to 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. Second, in order to produce only complete dining sets, you must also produce exactly one china hutch for every table produced. In order to represent this constraint, simply create a constraint equation, using the two variables you previously identified to represent the number of tables to be produced and the number of china hutches to be produced, that describes the ratio in which tables and china hutches must be manufactured. Simultaneously satisfying both of these constraints will ensure that only complete dining room sets are produced (i.e., ensure no excess tables, chairs or china hutches are produced).

Hint: Do not forget to account for the constraint requiring that you produce a specific minimum number of complete dining room sets. Once again, you do not need a separate variable to represent a complete dining room set. You can represent this constraint requirement by selecting one of your three variables and specifying the minimum number of units that must be produced in order to satisfy the requirement to produce a specific minimum number of complete dining room sets..

13. What is the optimal number of complete dining room sets the company should produce during the production cycle?

14. Which resource will be fully used in producing the optimal number of complete dining room sets? (Note: A given resource is fully consumed only if all of the resource has been used. If even a fraction of a unit of a resource remains unused, then the resource has not been fully consumed.)

15. What is the total profit the company will earn for producing the optimal number of complete dining room sets?

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#### Solution Summary

The following posting discusses various applications of linear programming.

Maximizing or Minimizing and Application to Linear Programming

(See attached file for full problem description)

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1) Discuss from your own experience a case where you had to maximize or minimize something and also how you think linear programming techniques might have helped you arrive at your final decision.

2) Read the short, well-written article on the Simplex Method which appears here:

http://www.sce.carleton.ca/faculty/chinneck/po/Chapter4.pdf

Discuss what is going on geometrically inside the Simplex Method.

3) Do a search using an Internet search engine and discuss some of the many computer algorithms and programs that exist to solve linear programming problems.

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(See attached file for full problem description)

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