How Sensitivity Analysis Uncovers Shadow Prices and Slack in Linear Programming

1 Sensitivity Analysis

Conducting sensitivity analysis is crucial for decision makers to interpret models. Simply building a model is insufficient; we must explain it from various perspectives to maximize its utility.

Sensitivity analysis studies how changes in features affect a linear programming model. In this method, we vary one feature (e.g., a constraint) while keeping others constant and observe the impact on the model output.

We can perform sensitivity analysis in three ways:

Impact of changes in objective function coefficient values

Impact of changes in the right‑hand side constants of constraints

Impact of changes in the coefficients of constraints

2 Shadow Prices and Slack Variables

A shadow price indicates how much the objective function value changes when the right‑hand side constant of a constraint increases by one unit. A slack variable represents the amount of unused resource; it defines the feasibility range of a constraint. If the slack variable equals zero, the constraint is binding—altering it changes the optimal solution. A non‑binding constraint means any change within its range does not affect the optimal objective value.

3 Example

3.1 Decision Variables

A glass manufacturing company produces two products, A and B. Let:

A = quantity of type‑A glass

B = quantity of type‑B glass

3.2 Objective Function

Maximize profit:

Profit = 60·A + 50·B

3.3 Constraints

Constraint 1: 4·A + 10·B ≤ 100

Constraint 2: 2·A + 1·B ≤ 22

Constraint 3: 3·A + 3·B ≤ 39

All constraints must be satisfied simultaneously.

3.4 Solving Code

<code>from pulp import *
import pandas as pd
# Initialize Class, Define Vars., and Objective
model = LpProblem("Glass_Manufacturing_Industries_Profits", LpMaximize)
# Define variables
A = LpVariable('A', lowBound=0)
B = LpVariable('B', lowBound=0)
# Define Objective Function: Profit on Product A and B
model += 60 * A + 50 * B
# Constraint 1
model += 4 * A + 10 * B <= 100
# Constraint 2
model += 2 * A + 1 * B <= 22
# Constraint 3
model += 3 * A + 3 * B <= 39
# Solve Model
model.solve()
print("Model Status:{}".format(LpStatus[model.status]))
print("Objective = ", value(model.objective))
for v in model.variables():
    print(v.name,"=", v.varValue)</code>

Output:

Model Status:Optimal
Objective =  740.0
A = 9.0
B = 4.0

<code>o = [{'name':name,'shadow price':c.pi,'slack': c.slack} for name, c in model.constraints.items()]
pd.DataFrame(o)</code>

3.5 Sensitivity Analysis: Calculating Shadow Prices and Slack Variables

The shadow price for constraint C2 is 10 and for C3 is 13.33, meaning a one‑unit increase in their right‑hand side constants would raise the objective function by those amounts respectively.

Constraint C1 has a slack value of 24, indicating it is a non‑binding (slack) constraint. Constraints C2 and C3 have slack of 0, making them binding; any change would affect the optimal objective.

4 Summary

This article introduced key concepts of linear programming sensitivity analysis—shadow prices and slack variables—and demonstrated a practical example solved with the PuLP library.

Reference

https://machinelearninggeek.com/sensitivity-analysis-in-python/