Beaver Creek Pottery Company is a small crafts operation run by a Native American tribal council. The company employs skilled artisans to produce clay bowls and mugs with authentic Native American designs and colors. The two primary resources used by the company are special pottery clay and skilled labor. Given these limited resources, the company desires to know how many bowls and mugs to produce each day in order to maximize profit. This is generally referred to as a product mix problem type. This scenario is illustrated in Figure 2.1.
## Figure 2.1. Beaver Creek Pottery CompanyThe two products have the following resource requirements for production and profit per item produced (i.e., the model parameters):
There are 40 hours of labor and 120 pounds of clay available each day for production. We will formulate this problem as a linear programming model by defining each component of the model separately and then combining the components into a single model. The steps in this formulation process are summarized as follows :
A linear programming model consists of decision variables, an objective function, and constraints. ## Decision VariablesThe decision confronting management in this problem is how many bowls and mugs to produce. The two decision variables represent the number of bowls and mugs to be produced on a daily basis. The quantities to be produced can be represented symbolically as x x ## The Objective Function The objective of the company is to maximize total profit. The company's profit is the sum of the individual profits gained from each bowl and mug. Profit derived from bowls is determined by multiplying the unit profit of each bowl, $40, by the number of bowls produced, x maximize Z = $40 x where
## Model Constraints In this problem two resources are used for productionlabor and clayboth of which are limited. Production of bowls and mugs requires both labor and clay. For each bowl produced, 1 hour of labor is required. Therefore, the labor used for the production of bowls is 1 x 1 x However, the amount of labor represented by 1 x 1 x The "less than or equal to" ( ) inequality is employed instead of an equality (=) because the 40 hours of labor is a maximum limitation that can be used , not an amount that must be used . This constraint allows the company some flexibility; the company is not restricted to using exactly 40 hours but can use whatever amount is necessary to maximize profit, up to and including 40 hours. This means that it is possible to have idle, or excess, capacity (i.e., some of the 40 hours may not be used). The constraint for clay is formulated in the same way as the labor constraint. Because each bowl requires 4 pounds of clay, the amount of clay used daily for the production of bowls is 4 x 4 x A final restriction is that the number of bowls and mugs produced must be either zero or a positive value because it is impossible to produce negative items. These restrictions are referred to as nonnegativity constraints and are expressed mathematically as Nonnegativity constraints restrict the decision variables to zero or positive values. x The complete linear programming model for this problem can now be summarized as follows:
The solution of this model will result in numeric values for x
and
Because neither of the constraints is violated by this hypothetical solution, we say the solution is A feasible solution does not violate any of the constraints . Now consider a solution of x
Although this is certainly a better solution in terms of profit, it is
An infeasible problem violates at least one of the constraints. The solution to this problem must maximize profit without violating the constraints. The solution that achieves this objective is x |

Introduction to Management Science (10th Edition)

ISBN: 0136064361

EAN: 2147483647

EAN: 2147483647

Year: 2006

Pages: 358

Pages: 358

Authors: Bernard W. Taylor

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