How to Calculate Shadow Price in Linear Programming
Linear programming is a mathematical method used to optimize a linear objective function, subject to linear equality and inequality constraints. It is widely applied in various fields, such as economics, engineering, and logistics. One of the critical concepts in linear programming is the shadow price, which provides valuable insights into the sensitivity of the optimal solution to changes in the constraints. In this article, we will discuss how to calculate the shadow price in linear programming.
Firstly, let’s define the shadow price. The shadow price, also known as the dual variable, represents the rate of change in the objective function value per unit increase in the right-hand side of a constraint. In other words, it indicates the economic value of relaxing a constraint. To calculate the shadow price, we need to solve the dual problem of the given linear programming problem.
Assume we have a linear programming problem in the following form:
Maximize Z = c1x1 + c2x2 + … + cnxn
Subject to:
a11x1 + a12x2 + … + a1nxn <= b1
a21x1 + a22x2 + ... + a2nxn <= b2
...
am1x1 + am2x2 + ... + amnxn <= bm
where x1, x2, ..., xn are decision variables, c1, c2, ..., cn are coefficients of the objective function, aij are coefficients of the constraints, and b1, b2, ..., bm are constants.
The dual problem of the given linear programming problem is as follows:
Minimize W = b1y1 + b2y2 + ... + bmym
Subject to:
a11y1 + a21y2 + ... + am1ym >= c1
a12y1 + a22y2 + … + am2ym >= c2
…
an1y1 + an2y2 + … + annym >= cn
where y1, y2, …, ym are dual variables.
To calculate the shadow price, we need to find the optimal solution of the dual problem. This can be done using various methods, such as the simplex method or the interior-point method. Once we have the optimal solution, the shadow price for each constraint can be determined as follows:
1. For a constraint of the form ai1x1 + ai2x2 + … + ainxn <= bi, the shadow price is equal to the corresponding dual variable, yi. 2. For a constraint of the form ai1x1 + ai2x2 + ... + ainxn >= bi, the shadow price is equal to the negative of the corresponding dual variable, -yi.
In summary, to calculate the shadow price in linear programming:
1. Formulate the dual problem of the given linear programming problem.
2. Solve the dual problem using an appropriate method to find the optimal solution.
3. Determine the shadow price for each constraint based on the corresponding dual variable.
Understanding the shadow price can help decision-makers assess the impact of relaxing or tightening constraints on the optimal solution. This information can be valuable in making strategic decisions and optimizing resource allocation in various applications.
