Simplex Method Paper and Presentation
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Simplex Method Paper and Presentation
TAB Furniture Company manufactures high quality wooden office and residential furniture. Since 1998, this privately owned, family operated company employs 85 people, operating from one location in Houston, TX. Sustaining the company for 13 years, business could continue without interruption. However, imparting lean operations proves to create profit maximization leading to strengthening the companys market position and longevity.
Currently, TABs management expresses concern over the cost of labor. A directive to optimize the level of production to maximize profit comes as no surprise to the companys laborers. One problem area arises with the production of chairs and tables. Evidencing a failure to meet labor limitations necessitates pinpointing the problem and its constraints as vital components to move forward with lean production measures. Identifying the optimal method to maximize profits occurs with the essential labor hours for the production of chairs and tables. Therefore, applying the Simplex Method to obtain fundamental data supports managements decision to detect ways in reducing labor costs.
Linear Programming
Todays competitive environment compels firms to develop strategies and methods to minimize costs. However, with the complex nature of production, increasing costs, and various other considerations, multiple constraints emerge. In this regard, to maximize profits given specified constraints, formulating methods of operations maintain optimal use of scarce resources to maximize profit for the company.
Linear programming is an effective method used to help an organization allocate scarce resources and maximize profits. A linear program model consists of the following components:
Objective function: this function provides the relationship of decision variables. The objective function is the equation subject to optimization.
Constraints: this refers to the availability of resources; it defines the limitations or constraints facing the firm. Constraints are inequality problems and therefore equal to or less than the amount of available resources.
Slack variables: these variables capture the unused resources in every constraint. Slacks are included in the objective function but given a coefficient of zero.
Linear programming takes into account technology or the criteria of selecting the best value for decision variables given the constraints a firm faces. Linear programming is a very important operations research model. In business analysis, linear programming model by construction relates market prices, output (decision variables), technology, and cost equation in the form of an objective function. It integrates resource constraints to provide a complete model used to obtain the optimal level of production given certain limiting conditions called constraints.
Simplex Method
One of the most practical methods used to solve linear programs is the simplex method. This method is practical because unlike the graphical method, it applies when there are more than two decision variables. This method proves to be more realistic because in actual situations, decision variables more than likely generate more than two types of products with different margins of profit using the same method.
Second, upon establishing the optimal production quantities, the simplex method uncovers unused resources generally known as slack variables in maximization problems. This might be very useful to reducing costs and with decision-making for a profit maximizing firm. The Simplex Method is a powerful tool because it evaluates all possible solutions to obtain the best outcome. The following problems illustrate the use of the simplex method in solving a problem for profit maximization.
Maximization problem facing the firm
A firm produces chairs and tables; each of the products undergoes three stages, namely carpentry, upholstery, and finishing. Management has concerns over the rising cost of labor reflecting more than 40% of total revenue. Determining the optimal level of production to maximize profit is integral to the companys long-term survival.
Currently, manufacturing a chair requires three hours of carpentry, nine hours of upholstery, and two hours of finishing work. Similarly, table production requires two hours of carpentry, four hours of upholstery, and ten hours of finishing work. The factory allocated production limits of 66 labor hours for carpentry, 180 labor hours for upholstery, and 200 labor hours for finishing. The profit per chair and table equates to $90 and $75, respectively. The requirements to create a maximization problem include determining the level of production for chairs and tables that maximizes profit and investigating the slack, if any, to advise management accordingly upon producing the results.
The objective function and constraints
Utilizing this information, the aim of the firm is to maximize profit. To achieve this, the construction of a linear programming problem requires decision variables; in this case, five are selected. These variables include the number of chairs (X), the number of tables to be produced (Y), the unused labor hours or slack in carpentry department (S1), the unused labor hours or slack in upholstery department (S2), and the unused labor hours or slack in finishing department (S3).
The linear program model generates an objective function given by:
The above objective function is subject to the following constraints:
Solution to the Problem
Using the simplex method to solve this problem, the next step is to obtain an initial basic feasible solution. This