Sensitivity analysis in PuLP

Introduction to Optimization in Python

Jasmin Ludolf

Content Developer

Sensitivity analysis in linear programming

Sensitivity analysis changes one parameter and to see changes in the output/outcome
Linear programming: how much the objective changes
Three approaches:
1. Change a coefficient in the objective
2. Change a coefficient in a constraint
3. Change the constant term in a constraint
Solve original and changed model
Use PuLP!

Gauge indicator with left green sector vector icon to represent sensitivity

Definitions

Constraint:
- Holds with equality: active, binding
- Holds with strict inequality: non active, non-binding, slack (adj.)
Slack (noun):
- Amount to add to the left of constraint to make it binding
Shadow price:
- Marginal increase in the objective with respect to the right of the constraint

The diet problem in PuLP

Food	Cost ($/lb)	Protein (%)	Fat (%)
corn	0.11	10	2.5
soybean	0.28	40	1

Recommendation: at least 17% protein, 2% fat, 7lb food
Objective: $0.11C + 0.28S$
Protein: $10C+40S \geq 17(C+S)$
Fat: $2.5C + S \geq 2(C+S)$
Weight: $C + S \geq 7$

from pulp import *

model = LpProblem('diet', LpMinimize)

C = LpVariable('C', lowBound=0)
S = LpVariable('S', lowBound=0)

model += 0.11*C + 0.28*S

model += 10*C + 40*S >= 17 * (C+S), 'Protein' 
model += 2.5*C + S >= 2 * (C+S), 'Fat'
model += C + S >= 7, 'Weight' 

model.solve()

Slack and shadow price in PuLP

print(f"Status: {LpStatus[model.status]}\n")

for name, c in model.constraints.items():
    print(f"{name}: slack = {c.slack:.2f}, shadow price = {c.pi:.2f}")

Status: Optimal

Protein: slack = -0.00, shadow price = 0.01

Fat: slack = -1.05, shadow price = 0.00

Weight: slack = -0.00, shadow price = 0.15

Let's practice!

Introduction to Optimization in Python