What Will Happen To The Solution If The Objective Function Coefficient For Variable 1 Decreases By 20?

Whether the solution is affected when the objective function coefficient for variable 1 decreases by 20 depends entirely on whether this change remains within the variable's allowable decrease range. The analysis is a core part of linear programming known as sensitivity analysis.

Scenario 1: The change is within the allowable decrease range

If the 20-unit decrease is less than or equal to the allowable decrease for variable 1, the optimal solution—meaning the set of values for all decision variables—will not change.

• Optimal Variable Values: The quantities of each variable in the optimal solution will remain exactly the same. The set of "basic" variables (those with positive values) and "non-basic" variables (those with zero values) will not change.
• Objective Function Value: The total objective function value (e.g., total profit) will change. The new value will be the original objective value minus 20 times the optimal value of variable 1.
• Reduced Costs and Shadow Prices: The reduced costs for non-basic variables and the shadow prices for the constraints will remain unchanged.

Example:Suppose the current objective coefficient for variable 1 is 50, and the sensitivity report shows an allowable decrease of 30. A decrease of 20 (from 50 to 30) is within this range. The solution's structure is stable, but the total profit will decrease.

Scenario 2: The change exceeds the allowable decrease range

If the 20-unit decrease is greater than the allowable decrease for variable 1, the current optimal solution will change.

• Optimal Variable Values: A new variable will enter the basis, and one of the current basic variables will leave. This means the combination of decision variables that defines the optimal production plan or resource allocation will be different.
• Objective Function Value: A new optimal objective value must be calculated by re-solving the linear program. The new value will be lower than the old value minus 20 times the old value of variable 1.
• Reduced Costs and Shadow Prices: Because the set of basic variables has changed, the reduced costs and shadow prices will also change. The dual prices will reflect the new set of binding constraints.
• Graphical Interpretation: In a two-variable problem, changing an objective function coefficient alters the slope of the objective function line. If the change is too large, the line will pass through a different corner point of the feasible region, which represents a new optimal solution.

Key concepts in sensitivity analysis

This analysis relies on key concepts from linear programming:

• **Objective Function Coefficients (cjc sub j

  𝑐𝑗

  )**: These represent the per-unit contribution of each variable to the objective (e.g., profit per unit of a product). A change in a cjc sub j

  𝑐𝑗

  alters the slope of the objective function.

• Range of Optimality: For each variable in the optimal solution, this range specifies how much its objective coefficient can change without changing the optimal combination of decision variables.

  • Allowable Increase: The maximum increase in the coefficient.
  • Allowable Decrease: The maximum decrease in the coefficient.

• Reduced Cost: For a non-basic variable (a variable with a value of zero in the optimal solution), the reduced cost indicates how much its objective coefficient must improve (increase for maximization, decrease for minimization) before it becomes profitable to include that variable in the solution.

• Shadow Price (Dual Price): This is the change in the optimal objective value per unit increase in a constraint's right-hand side. The shadow prices are only valid within a certain range of the right-hand side value and remain constant as long as the same set of constraints is binding.

Practical application and summary

In practice, a sensitivity report from a solver like Excel provides the allowable increase and decrease for each objective function coefficient. By referencing this report, a manager can immediately determine the impact of a cost or price change without re-running the entire optimization model.