How do I know the exact allowable range for a right-hand-side (RHS) value before the current shadow price becomes invalid?
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To find the exact allowable range for a right-hand-side (RHS) value before its shadow price becomes invalid, check the point where changing that RHS forces any basic variable value down to zero.
Here is the exact step-by-step logic:
* Understand what invalidates the shadow price: A shadow price remains valid only as long as the current basis remains feasible (all basic variable values stay non-negative). The moment a RHS change forces any basic variable to drop below zero, the current basis becomes infeasible, forcing a basis change and invalidating the current shadow price.
* Track how the basic variables shift: Changing a single RHS value by an amount adjusts the values of your current basic variables proportionally. The rate at which each basic variable changes is determined by its corresponding column in the final optimal tableau.
* Calculate the Allowable Decrease:
* Look at the basic variables that decrease as the RHS decreases.
* Identify which basic variable hits zero first.
* The decrease amount that drives this first basic variable to zero establishes your exact Allowable Decrease.
* Calculate the Allowable Increase:
* Look at the basic variables that decrease as the RHS increases.
* Identify which basic variable hits zero first.
* The increase amount that drives this first basic variable to zero establishes your exact Allowable Increase.
Within this exact window—between your current RHS minus the Allowable Decrease and your current RHS plus the Allowable Increase—the current optimal basis remains valid, and the shadow price sta
ys constant.
You find this range by calculating the limits where current basic variables remain non-negative when the right-hand side changes.
Calculate the bounds where the basic variables remain non-negative, under the RHS change. The shadow price remains valid within this range, but exceeding its upper or lower limit forces a basic variable to become negative, invalidating the current optimal basis and its associated shadow price.
The exact allowable RHS range is given by the RHS sensitivity report, which shows the minimum and maximum values for which the current shadow price remains valid.
The exact allowable range for a right-hand-side value is determined by finding the minimum and maximum changes to that constant that keep all basic variables in the optimal simplex tableau non-negative when multiplied by the corresponding column of the inverse basis matrix.
The exact allowable RHS range is found in the Sensitivity Report under Allowable Increase and Allowable Decrease; within that range, the current shadow price remains valid.
You calculate the range by dividing the current basic variable values by the corresponding numbers in the matrix inverse column for that constraint. The smallest positive ratio gives the maximum allowable increase, while the smallest negative ratio gives the maximum allowable decrease. Exceeding these limits forces a basic variable to zero, changing the optimal basis and making the shadow price invalid.
You can determine the allowable RHS range from the Sensitivity Analysis report by checking the allowable increase and allowable decrease. The shadow price remains valid as long as the new RHS stays within that range.
To find the exact allowable range of an RHS value, use the allowable increase and allowable decrease shown in the sensitivity report; the range is from the current RHS minus the allowable decrease to the current RHS plus the allowable increase, and within this range the current shadow price remains valid.
he allowable range for a right-hand-side (RHS) value tells you how much a constraint's resource availability can increase or decrease while the current shadow price remains valid.
Check the sensitivity report for the allowable increase and decrease of the RHS value. The shadow price remains valid as long as the RHS stays within that range. If it goes beyond the range, the shadow price may change.
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