Solver tells you that the shadow price is 3 for a less than or equal to constraint in a maximization (profit) LP problem, this means:

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BlackTom AI · Accepted Answer

In a linear programming (LP) maximization problem, the shadow price (dual value) associated with a constraint indicates how much the objective function would improve if the right-hand side (RHS) of that constraint is increased by a small amount, typically 1 unit, while all else remains constant.

Option 1: 'if the coefficient of an objective function variable is increased by 1 then the optimal objective function value is increased by 3 dollars.' This statement confuses the effect of changing a primal constraint with changing a coefficient in the objective function. Shadow prices relate to RHS changes of constraints, not to coefficients in the objective.

Option 2: 'this constraint is nonbinding.' A shadow price of 3 implies the constraint is binding at the current optimum; otherwise, a change in RHS would not affect the objective value. Nonbinding (slack) constraints have shadow prices of zero.

Option 3: 'if the RHS for that constraint is increased by 1 then the optimal objective function value is increased by 3 dollars.' This matches the meaning of a shadow price of 3 for a ≤ constraint in a maximization problem: increasing the allowed limit by 1 unit yields a $3 improvement in the objective, assuming the basic feasible solution shifts accordingly and the model remains valid.

Option 4: 'if the RHS for that constraint is increased by 3 then the optimal objective function value is increased by 1 dollar.' This reverses the relationship and uses an incorrect magnification factor; the shadow price tells the change per 1 unit increase, not per 3 units, and the magnitude would not drop to 1 dollar for a 3-unit increase.

Option 5: 'if the coefficient of an objective function variable is increased by 3 then the optimal objective function value is increased by 1 dollar.' Again, this misattributes the effect to a constraint RHS and misstates the relationship between coefficient changes and the shadow price. The shadow price does not describe changes in objective coefficients in this way.

Taken together, the option that correctly interprets a shadow price of 3 for a ≤ constraint in a maximization LP is the one stating the objective increases by 3 for a 1-unit increase in the constraint's RHS, i.e., Option 3.

Business Modeling & Optimization - DAT-3659 - BMBAN2 A3. 1 In Class Quiz 01 - Business Modeling and Optimization Quiz 1 (Remotely Proctored)

Solver tells you that the shadow price is 3 for a less than or equal to constraint in a maximization (profit) LP problem, this means:

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