Capacitated plant location - case study P3

Supply Chain Analytics in Python

Aaren Stubberfield

Supply Chain Analytics Mgr.

Capacitated plant location model

Modeling

Production at regional facilities
- Two plant sizes (low / high)
Exporting production to other regions
Production facilities open / close

image of globe with regional production

Expected ranges

What should we expected for values of our decision variables?

Production Quantities:

High production in regions with low variable production and shipping costs
Maxed production in regions that also have relatively low fixed production costs

Production Plant Open Or Closed:

High capacity production plant in regions with high demand
High capacity production plant in regions with relatively low fixed costs

Sensitivity analysis of constraints

Total Production = Total Demand:

shadow prices = Represent changes in total cost per increase in demand for a region
slack = Should be zero

Total Production ≤ Total Production Capacity:

shadow prices = Represent changes in total costs per increase in production capacity
slack = Regions which have excess production capacity

from pulp import *
import pandas as pd

# Initialize Class
model = 
    LpProblem("Capacitated Plant Location Model",
               LpMinimize)

# Define Decision Variables
loc = ['A', 'B', 'C', 'D', 'E']
size = ['Low_Cap','High_Cap']
x = LpVariable.dicts(
                "production_", 
                [(i,j) for i in loc for j in loc],
                 lowBound=0, upBound=None, 
                 cat='Continuous')

y = LpVariable.dicts(
               "plant_", 
               [(i,s) for s in size for i in loc], 
                cat='Binary')
# Define Objective Function
model += 
  (lpSum([fix_cost.loc[i,s]*y[(i,s)] 
          for s in size for i in loc])
 + lpSum([var_cost.loc[i,j]*x[(i,j)] 
          for i in loc for j in loc]))

# Define the Constraints
for j in loc: model += 
  lpSum([x[(i, j)] 
        for i in loc]) == demand.loc[j,'Dmd']
for i in loc: model += 
  lpSum([x[(i, j)] for j in loc]) <= lpSum(
             [cap.loc[i,s]*y[(i,s)]for s in size])

# Solve
model.solve()

# Print Decision Variables and Objective Value
print(LpStatus[model.status])
o = [{'prod':"{} to {}".format(i,j), 'quant':x[(i,j)].varValue} 
     for i in loc for j in loc]
print(pd.DataFrame(o))
o = [{'loc':i, 'lc':y[(i,size[0])].varValue, 'hc':y[(i,size[1])].varValue}
     for i in loc]
print(pd.DataFrame(o))
print("Objective = ", value(model.objective))

# Print Shadow Price and Slack
o = [{'name':name, 'shadow price':c.pi, 'slack': c.slack} 
     for name, c in model.constraints.items()]
print(pd.DataFrame(o))

Business questions

Likely Questions:

What is the expected cost of this supply chain network model?
If demand increases in a region how much profit is needed to cover the costs of production and shipping to that region?
Which regions still have production capacity for future demand increase?

Summary

Reviewed:

Expected ranges for decision variables
Interpreted the output of sensitivity analysis (shadow prices and slack)
Code to solve and output results
Likely business related question

Great work! Your turn

Supply Chain Analytics in Python