What are shadow prices in linear programming?
Shadow prices, also known as dual prices or marginal values, are an essential concept in linear programming (LP) that provides valuable insights into the economic implications of the decision variables. In this article, we will explore what shadow prices are, how they are calculated, and their significance in solving linear programming problems.
The concept of shadow prices arises from the dual problem of a linear programming problem. In linear programming, we have a primal problem that seeks to maximize or minimize an objective function subject to a set of linear constraints. The dual problem, on the other hand, seeks to maximize or minimize the sum of the products of the shadow prices and the right-hand side values of the constraints.
Understanding the Dual Problem
To understand shadow prices, it is crucial to first understand the dual problem. The dual problem is derived from the primal problem by introducing a set of variables, known as dual variables, for each constraint. These dual variables represent the rate of change of the objective function with respect to a change in the right-hand side value of the constraint.
The dual problem can be formulated as follows:
Maximize/Minimize Z’ = Σ(c_j x_j)
Subject to:
A x <= b where A is the coefficient matrix, b is the right-hand side vector, c is the objective function vector, and x is the decision variable vector.
Calculating Shadow Prices
Shadow prices are the optimal values of the dual variables. To calculate the shadow prices, we need to solve the dual problem. The shadow prices can be obtained by solving the dual problem and interpreting the optimal values of the dual variables.
In general, the shadow price for a constraint is the rate at which the objective function changes with respect to a unit change in the right-hand side value of that constraint, while holding all other constraints and the objective function constant.
Significance of Shadow Prices
Shadow prices have several important applications in linear programming:
1. Sensitivity Analysis: Shadow prices can be used to analyze the sensitivity of the optimal solution to changes in the right-hand side values of the constraints. This information is valuable for decision-making and risk assessment.
2. Economic Interpretation: Shadow prices provide an economic interpretation of the constraints. They indicate the maximum amount that the objective function can be increased (or decreased) for each constraint, without affecting the feasibility of the solution.
3. Resource Allocation: Shadow prices help in determining the optimal allocation of resources by indicating the value of additional units of a resource. A higher shadow price suggests that the resource is more valuable and should be allocated accordingly.
4. Constraint Relaxation: Shadow prices can be used to decide which constraints can be relaxed or removed without affecting the optimal solution. A shadow price of zero indicates that the constraint is inactive and can be ignored.
In conclusion, shadow prices are an essential concept in linear programming that provides valuable insights into the economic implications of decision variables. By understanding shadow prices, we can make more informed decisions, conduct sensitivity analysis, and optimize resource allocation.
