# Linear Programming and Profit Reduction: Finding New Constraint Hours

## What is the new available hours for this constraint after a profit reduction of $1920?

Original R.H.S: 108 hours, 120 hours

## Answer:

The new available hours for the constraint with an original R.H.S of 108, after a profit reduction of $1920 and with a Shadow Price of 15.5555556, cannot be practically calculated as it results in a negative value, which implies that the constraints need to be reevaluated.

## Explanation:

The question deals with the concept of linear programming in the context of business and operations research, specifically the changes in the right-hand side (R.H.S) values of constraints when a profit reduction occurs. Given a profit reduction of $1920 and the original R.H.S of the constraints being 108 and 120, we need to determine the new R.H.S.

To do this, one must understand the effect of the 'Shadow Price' on the constraint's R.H.S value. The Shadow Price indicates how much the objective function's value (profit in this case) will change with a unit increase in the R.H.S of a constraint.

According to the provided information, the 'Inspection' constraint has a Shadow Price of 15.5555556. The profit has been reduced by $1920, so we divide the profit reduction by the Shadow Price to find the change in hours, which gives us:

1920 / 15.5555556 ≈ 123.5 hours

Since we want to find the reduction in hours, we subtract this value from the original R.H.S of 108 hours:

108 - 123.5 ≈ -15.5

As the result cannot be negative in practical situations, it suggests that the new R.H.S should be 0, but this would mean that the constraint is no longer binding or the situation demands a reevaluation of the constraint, as the profit cannot be reduced by $1920 without breaking existing constraints.

Conclusion: Based on the given Shadow Price and the amount of profit reduction, the inspection constraint would theoretically require a reduction of hours that would make the new R.H.S negative, which is impractical. The situation indicates a need to reassess the constraints and their relationship with the profitability of the operations.