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Sensitivity Analysis Sensitivity analysis examines how the optimal solution will be impacted by changes in the model coefficients due to uncertainty, error or human intervention. In particular, it can address: how a change in an objective function coefficient will affect the optimal solution how a change in the right hand side (RHS) of a constraint will affect the optimal solution how an alternate optimal solution can be identified

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Range of Optimality The range of optimality of an objective function coefficient specifies an interval for the objective function coefficient in which the original optimal solution remains optimal, while keeping all other coefficients in the problem constant. The value of the objective function may change in this range. If the coefficient is changed by more than the Allowable Increase or Allowable Decrease, another extreme point of the feasible region will be identified as the optimal solution.

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Alternate Optimal Solutions An alternate optimal solution exists if the Allowable Increase or Allowable Decrease for one or more variables are equal to zero and the solution is not degenerate. (Ragsdale pg.143) The solution is degenerate if the RHS values of any of the constraints have zero Allowable Increase or Allowable Decrease.

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Reduced Cost For each variable, the reduced cost is the per-unit amount the product contributes to the objective less the per-unit value of the resources it consumes. Reduced cost represents the amount that the variable is overpriced in minimization problems or underpriced in maximization problems. Reduced cost represents the amount that the objective function will be hurt if one unit of that variable is forced into the solution.

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Alternate Optimal Solutions An alternate optimal solution exists if the Final Value and the Reduced Cost entries for a variable are both zero. To identify the alternate optimal solution, change Solver’s target cell to maximize the identified zero variable value and add the constraint that the objective must equal the optimal value specified in the current solution.

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Binding Constraints The slack of a resource constraint is the amount of the corresponding resource that will not be used if the optimal solution is implemented. The surplus of a minimum requirement constraint is the amount by which the requirement will be exceeded if the optimal solution is implemented. A binding constraint implies that the corresponding resource or requirement is restricting the optimal solution and objective value. A binding constraint has slack or surplus of zero. A nonbinding constraint is one in which there is positive slack or surplus when evaluated at the optimal solution.

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Shadow Price A shadow price for a right hand side value (or resource limit) is the amount the objective function will change per unit increase in the right hand side value of a constraint. The shadow price for a nonbinding constraint is 0. Changing the right hand side of a binding constraint will result in a new optimal solution that will adjust to incorporate the new requirement. This new optimal solution must be found by resolving the model with Solver. When a variable is forced into the solution due to a lower bound constraint, the reduced cost represents shadow price of that lower bound constraint.

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Graphical Interpretation Graphically, a shadow price is determined by adding +1 to the right hand side value in question, enlarging the feasible region and then resolving for the optimal solution in terms of the same two binding constraints. The shadow price is equal to the difference in the values of the objective functions between the new and original problems. Excel snippit to illustrate shadow prices

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Range of Feasibility The range of feasibility for a change in the right hand side value is the Allowable Increase and Allowable Decrease for the coefficient in which the original shadow price remains constant.

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