What is Shadow Price in Linear Programming?
Linear programming is a mathematical method used to find the best outcome in a given set of conditions. It is widely used in various fields, such as economics, engineering, and logistics. One of the most important concepts in linear programming is the shadow price, which provides valuable insights into the optimization process. In this article, we will explore what shadow price is and its significance in linear programming.
Shadow price, also known as dual price or marginal value, represents the rate of change in the objective function value due to a unit change in the right-hand side of the constraint. In simpler terms, it indicates how much the objective function will improve if the constraint is relaxed by one unit.
Understanding Shadow Price in Linear Programming
To understand shadow price, let’s consider a linear programming problem with the following constraints:
1. x + y ≤ 10
2. 2x + 3y ≥ 20
3. x, y ≥ 0
The objective function is to maximize z = 5x + 3y.
In this problem, we have two constraints: the first one is a resource constraint (x + y ≤ 10), and the second one is a demand constraint (2x + 3y ≥ 20). The shadow price of each constraint will help us understand the impact of relaxing or tightening these constraints on the objective function.
Calculating Shadow Price
To calculate the shadow price, we first need to convert the constraints into their dual form. The dual form of the constraints is obtained by taking the reciprocal of the coefficients of the variables in the constraints. In our example, the dual form of the constraints is:
1. -x + y ≥ -10
2. -2x – 3y ≤ -20
Next, we solve the dual problem, which is a minimization problem. The objective function of the dual problem is to minimize w = 10y + 20x. The shadow price of each constraint is the coefficient of the variable in the dual objective function.
For the first constraint (-x + y ≥ -10), the shadow price is the coefficient of y, which is 1. This means that if we increase the available resources (x + y) by one unit, the objective function (z) will improve by 1 unit.
For the second constraint (-2x – 3y ≤ -20), the shadow price is the coefficient of x, which is 20. This indicates that if we increase the demand (2x + 3y) by one unit, the objective function (z) will improve by 20 units.
Significance of Shadow Price in Linear Programming
The shadow price has several important implications in linear programming:
1. Resource Allocation: The shadow price helps in determining the optimal allocation of resources. If the shadow price of a resource is high, it indicates that the resource is scarce and should be allocated efficiently.
2. Decision Making: The shadow price provides valuable information for decision-making. If the shadow price of a constraint is positive, it means that relaxing the constraint will improve the objective function. Conversely, if the shadow price is negative, it suggests that tightening the constraint will improve the objective function.
3. Sensitivity Analysis: Shadow price is an essential tool for sensitivity analysis. It helps in assessing the impact of changes in the parameters of the problem on the optimal solution.
In conclusion, shadow price is a crucial concept in linear programming that provides valuable insights into the optimization process. By understanding the shadow price, decision-makers can make informed decisions regarding resource allocation and problem-solving.
