# Formulating a Linear Programming Model

The Barron Textile Mill in High Point, North Carolina was checked by inspectors enforcing the Occupational Safety and Health Act (OSHA). The inspectors found violations in four categories: hazardous materials, fire protection, hand-powered tools, and machine safety guards. In each category the mill was not in 100% compliance. Each percentage point of increase in the compliance level in each category will reduce the frequency of accidents, decrease the accident cost per worker, and constitute progress toward satisfying the OSHA compliance level. The following table shows the benefits (in accident frequency and accident cost per worker) and the costs of a percentage point increase in compliance in each category.

(see chart in attached file)

To achieve 100% compliance in all four categories, the mill would have to increase compliance in hazardous materials by 60 percentage points, in fire protection by 28 percentage points, in hand-powered tools by 35 percentage points, and in machine guarding by 17 percentage points. Only $52,000 is available to spend on safety at this time. Any larger expenditure could jeopardize the financial standing of the mill. Therefore, management hopes to achieve a level of accident reduction and compliance that is within the company's budget limitation sufficient to temporarily delay any punitive action from OSHA. Management at Barron has established the following prioritized goals to accomplish this objective.

? The company needs to remain within a total safety budget of $52,000.

? The company wants to achieve the percentage increases necessary to achieve 100% compliance in each category.

? The company wants to achieve total accident frequency reduction of 20 accidents/105 hours of exposure even if 100% compliance cannot be achieved in all categories.

? The company wishes to reduce the total accident cost per worker by $115.

a. Formulate a linear goal programming model to determine the percentage points of compliance needed in each category that will best satisfy the company's goals.

b. Determine the solution that will best achieve the company's goals in minimizing accident frequency and constitute progress toward satisfying OSHA compliance levels. Interpret the solution results including the levels of goal achievement.

#### Solution Preview

Please see the attached files.

a. Let xi be the percentage point change in category i

Let ci be the cost/percentage point change for category i

Let ai be the accident cost per worker reduction for category i

Let fi be accident frequency ...

#### Solution Summary

The solution provides detailed explanations and calculations, including Excel Solver code in the attached files.