Shadow Price Calculation In Linear Programming

Understanding shadow prices is crucial in linear programming for effective decision-making. In this article, we'll explore how to calculate the shadow price of a variable based on changes in its coefficient and the resulting impact on the objective function.

Understanding Shadow Prices

Let's dive into shadow prices, guys! In linear programming, the shadow price of a constraint represents the change in the optimal objective function value resulting from a one-unit increase in the right-hand side of that constraint. It essentially tells us how much better (or worse) our objective function would become if we could relax or tighten a constraint by a small amount. Shadow prices are also known as dual values or marginal values.

Why are shadow prices important? Well, they provide valuable insights into the sensitivity of the optimal solution to changes in the constraints. This information is incredibly useful for resource allocation, cost-benefit analysis, and identifying bottlenecks in the system. Imagine you're running a factory. Knowing the shadow price of a particular resource (like labor hours or raw materials) can help you determine whether it's worth investing in more of that resource. If the shadow price is high, it means that increasing the availability of that resource would significantly improve your profits. Conversely, if the shadow price is low, it might not be worth the investment.

How do we find shadow prices? Shadow prices are typically obtained from the optimal solution of the linear programming problem. Specifically, they are found in the dual solution, which is a byproduct of solving the primal problem. Most linear programming solvers (like those in Excel, Python libraries like SciPy, or dedicated optimization software) will provide the shadow prices along with the optimal variable values and objective function value. To get a better handle on this, consider a simple example. Suppose we have a constraint that limits the amount of labor hours available. The shadow price of this constraint might tell us that each additional labor hour would increase our profit by, say, $10. This would be super useful when deciding whether to hire more workers or invest in overtime. So, next time you're dealing with a linear programming problem, pay close attention to those shadow prices. They can be your secret weapon for making smart decisions and optimizing your outcomes!

Calculating Shadow Price from Coefficient Change

Now, let's tackle the main question: how do we calculate the shadow price when we know the percentage change in a decision variable's coefficient and the resulting percentage change in the objective function? Here's the scenario: We increase the coefficient of a decision variable by 20%, and this leads to a 15% increase in the objective function. What's the shadow price of this variable?

To solve this, we need to understand the relationship between the change in the coefficient and the change in the objective function. Let's use some variables to represent the key values:

• Let Δc be the change in the coefficient of the decision variable.
• Let ΔZ be the change in the objective function.
• Let x be the value of the decision variable at the optimal solution.
• Let π be the shadow price of the variable.

The change in the objective function can be approximated as:

ΔZ ≈ π * Δx

However, we are given the percentage changes, not the absolute changes. So, let's denote:

• %Δc = 20% = 0.20 (percentage change in the coefficient)
• %ΔZ = 15% = 0.15 (percentage change in the objective function)

We can express the changes as:

• Δc = 0.20 * c (where c is the original coefficient)
• ΔZ = 0.15 * Z (where Z is the original objective function value)

The relationship between the change in the objective function and the change in the coefficient can also be expressed as:

ΔZ ≈ Δc * x

Since we want to find the shadow price π, we need to relate ΔZ to Δc. Given that the objective function changes by 15% when the coefficient changes by 20%, we can set up the following relationship:

0.15 * Z ≈ π * (0.20 * c) * x

However, this formula does not directly give us the shadow price π. Instead, we should consider the ratio of the percentage change in the objective function to the percentage change in the coefficient. This ratio gives us an elasticity measure, which can be related to the shadow price.

Let's think about this a bit differently. If we increase the coefficient by 20% and the objective function increases by 15%, it means that for every 1% increase in the coefficient, the objective function increases by 0.75%. Mathematically:

(0.15 * Z) / (0.20 * c) = 0.75 * (Z / c)

This value, 0.75, represents the sensitivity of the objective function to changes in the coefficient. In the context of shadow prices, it indicates that for a small unit increase in the coefficient, the objective function increases by 0.75 units. Therefore, the shadow price of the variable is 0.75.

Detailed Explanation

Alright, let's break this down even further to make sure we're all on the same page. When we talk about linear programming, we're essentially trying to find the best possible outcome (maximizing profit or minimizing cost) given a set of constraints. The objective function is the mathematical expression that represents what we're trying to optimize. The decision variables are the values we can control to achieve this optimization. Now, imagine we tweak one of these variables. Specifically, we're looking at a variable's coefficient in the objective function. This coefficient tells us how much that variable contributes to the overall objective. If we increase this coefficient, we'd expect the objective function to change, right?

What we're trying to find is the shadow price. The shadow price tells us how sensitive the optimal value of the objective function is to changes in the constraints or, in this case, the coefficients of the decision variables. It's like asking: "If I could make a small change to this constraint (or coefficient), how much better or worse would my optimal solution be?" In our problem, we're told that increasing the coefficient by 20% results in a 15% increase in the objective function. This is a direct relationship that we can use to calculate the shadow price. Let's denote the original coefficient as c and the original objective function value as Z. The change in the coefficient Δc is 20% of c, which is 0.20c. The change in the objective function ΔZ is 15% of Z, which is 0.15Z. The shadow price π is the ratio of the change in the objective function to the change in the coefficient. In other words:

π = ΔZ / Δc = (0.15Z) / (0.20c) = 0.75 (Z/c)

This means that for every unit increase in the coefficient c, the objective function Z increases by 0.75(Z/c). Now, let's consider a scenario where Z/c = 1. In this case, π = 0.75. Therefore, the shadow price of the variable is 0.75. This aligns with the alternatives provided, making option A the correct choice. It's important to note that this is a simplified explanation, and in more complex linear programming problems, the shadow prices may vary depending on the specific problem and its constraints. However, for the given scenario, this calculation provides a clear and concise solution.

Answer

Therefore, the shadow price of the variable is:

A) 0.75

Understanding and calculating shadow prices is vital for anyone working with linear programming, as it provides critical insights for optimization and decision-making. Keep practicing, and you'll become a pro in no time!