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# Linear Programming Theory

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Linear programming is used heavily in many different industries such as manufacturing, airlines, agriculture, investment banking, nonprofits, and the list goes on. Using linear programming will answer the question for the maximum or minimum product or service variable needed to achieve a specific outcome. The programming aspect is a complex algebraic calculation using constants and variables to arrive at the best solution. Because of the unlimited amount of variables that can be calculated projected outcomes can be optimized. The programming is described as linear because the components change in proportion to one another. Linear programming involves using a model, which is a mathematical calculation to determine an outcome for the problem identified. Businesses typically have plans or goals in mind for the most profitable outcome; incorporating linear programming as a way to plan appropriately in order to realize the most efficient plan of action will go a long way in maximizing resources.

The programming aspect is not referring to a computer language but instead it invents a solution to resolve an issue such as mitigating risk in a manufacturing product line expansion. Some refer to it as linear optimization. This system was developed nearly sixty-five years ago by 1940’s by George Dantzig in response to World War II on how best to disseminate manpower, supplies and artillery (Linear Programming, n.d.). Linear programming works well in situations where there are significant amounts of unpredictable occurrences, centered on an aspect that remains constant. The constant is called the coefficient. Since coefficient is not a variable, this aspect of the equation does not change. For example, if John Doe owns 5 KFC restaurants and restaurant A borrows chicken from restaurant C, but C is more profitable than A, the constant in this case is the number of restaurants. John Doe want to know how much chicken restaurant A is borrowing from C, and why is C making more money when A on a per week basis is cooking more chicken? When the total resources are stated such as the amount of chicken purchased per week, this economic tool is practical and accurate if given the correct formula.

Linear equations facilitate both the coefficient and variables. The variables are best labeled with what they represent versus using the letter X. A programming model is made up of decision variables (Linear Programming, n.d.). Determining these variables are the crux of precision planning in a linear program. If the variable are misstated, or the incorrect variable values are used this will result in a false outcome. The variable optimization is also impacted by the objective function (Hillier, 2010). This simply represents the expected product or results of the linear model, and is the calculated costs, profits, etc. based on projected variables. Constraints in linear problems involve the minimum and maximum capability of specific variables. There are usually several constraints in a linear problem including non-negative constraints. Examples of constraints are time, resources and finances.

When solving a linear equation, the geometric format is only one way to resolve a problem. Another non-geometric problem solving tool for linear equations is the Simplex Method (Linear Programming, n.d.). The modeling technique of the problems remains the same but uses a matrix to solve the problem versus drawing a graph and using geometric calculations. The constraints are exchanged for equations, and the problem is exchanged to a matrix equation, and the probable answer should match the matrix. Several companies such as IBM offer linear optimization products that are computer programs. These programs with the correct constraint variables and coefficients can process calculations without the element of human error. Linear problems seek to solve a sought after result using identified constraints. Whether a linear computer program or the Simplex Method is used the end result will be the same, one probably involving less mathematical expertise than the other.

Lowe’s home improvement stores contractor department undersells Home Depot’s contractor department by forty percent. On site product availability, inexperienced sales associates and budget constraints are known issues. Lowe’s seeks to determine what products are most requested by contractors, product availability and lead time, cost, consumer demand, and the production level needed to compete strategically with Home Depot, while optimizing the budget which is corporate directed. To write the linear equation the objective must be determined. Once that is understood, the next steps are to determine the coefficient or the constant of the equation and the constraint variables.

To completely incorporate the decision variables an in depth product review must be completed. For example what products are contractors requesting, compared to what is normally available and what is in stock. Products specific to contractors consist of the following departments: lumber and composites, concrete, cement and masonry; insulation; siding; molding and millwork (Shop by Department, n.d.). The team leader in each department is asked to keep a ledger of requested products that were either out of stock or were not carried for a period of 120 days. Sales reports by departments listing items sold and not returned over the same period will be analyzed. A budget for all departments is provided; these numbers are actual budget amounts and will not change. Lowe’s goal is to determine how best to use the current dollars by maximizing critical inventory and making cost adjustments if necessary to achieve the stated goal of increasing product availability and profit margin. Since Lowe’s computes projection based on regions, only the Midwest region is being considered, and specifically in the Chicago area. This doesn't take into account the adjustments needed for the continuing phenomenon of globalization, yet the same principles would apply.

The problem stated here is typical of the computing abilities of linear program optimization. The flexibility of linear equations are applicable to any business type, and this is one of the reason why it continues to be a much needed tool for professionals in the twenty-first century. After the 120 day test period, it is found that lumber and composites are the most requested products. Professional contractors use high quantities of lumber products which cannot be compared with individual consumer do-it-yourself (DIY) project usage. The test period revealed that Lowe’s only carries 55 sizes and varieties of pressure treated lumber, compared to 112 that their competitor Home Depot carries. In addition, the larger quantities are usually out of stock or required a special order. 12.5% of the pressure treated lumber involved sizes, color and types that were purchased infrequently and the stock was taking up needed real estate in stores. However, overall Lowe’s pressure treated lumber retail price was from 3% to 5% lower than Home Depot.

Although Home Depot offers more products for contractors to consider, the test period determined that contractors requested most frequently 15 common dimensions and type, however, only 7 of these are typically in stock in the Chicago stores and just 5 were available in the quantities most requested. Programming linear must have a define objective or outcome before a business model can be mapped, which has been explained previously. The programmer’s task is to decipher the correct variables and coefficients to calculate what changes Lowe’s should make to product availability, raw material cost, and retail real estate. Understanding these parameters will statistically reveal pressure lumber items to omit from inventory, and which ones to add that will result in the highest profit potential. However, while product availability is an essential component, another factor discovered during the test period and was employee knowledge. When the team lead was not available, the sales associates were unable to answer specific contractor questions about lumber type and size. This affected their ability to offer exceptional customer assistance and service, a corporate mandate and mission.

The flexibility of linear computations is only limited by the imagination. In this scenario, in addition to products specifics, cost, and retail pricing, is the people variable. This is not as easy to calculate, however many different programming scenarios can be worked based on productivity and its contribution to the overall retail sales. The Lowes model not only focused on pressure lumber, but other contractor items as well. And the same linear model can be used for the other four departments and each Chicago location arriving at an answer that is specific enough to generate the desired outcome.

At Lowe’s decision are made based on the available resources. Based on all factors, the goal is to maximize resources to achieve the highest maximum objective. These decisions cannot be made unless accurate assumptions are made. However, the program is only as great as the determined inputs. If the inputs are inaccurate the desired outcome cannot be achieved. Therefore checking data and documentation cannot be an assumption it must be actual based on historical activity.

Linear programming is an additional business tool if used efficiently and will provide a workable road map to make smarter business choices and decisions. Large and small businesses will benefit from this tool. In small businesses if staff availability is a problem, an outside firm can be enlisted whose expertise is developing business solutions using linear programming. Larger corporations have the flexibility to have in-house computer programming software to calculate the geometrical algorithms. However, in-house staff must not only understand the theories of linear programming but also be able to develop linear models to arrive at the desired destination. Using linear solutions in businesses will save significant dollars and the bonus is that current resources are working at high efficiency levels.

References

Hillier, F. L. (2010). Introduction to linear programming In Introduction to operations research. (3) Retrieved from http://www.mhhe.com/engcs/industrial/hillier/etext/PDF/chap03.pdf.

Linear programming. (n.d.). Cengage Learning. Retrieved from http://www.cengage.com/resource_uploads/downloads/0538495057_260715.pdf

Shop by department. (n.d.). Home Depot. Retrieved from http://www.homedepot.com/?cm_mmc=SEM|THD|G|BT1&gclid=CPfAyozw3bwCFe1aMgodw0cADg