title: "Lecture 1: Introduction to Mathematical Modelling and LPs" author: "Junaid Hasan" date: "June 20, 2021"¶

A Real Life Scenario¶

  • Washington State Ferries do not carry enough lifeboats for all passengers.
  • They do carry enough life vests to make up the difference.
  • Reason:
  • Lifeboats take up too much space!
  • If we allow lifeboats for everyone, then capacity will go down by a lot.

A demo problem¶

  • Suppose a boat company wants to design a new ferry with the following:
  • A vest holds 1 person and requires $0.05 m^3$ of space to store.
  • A boat holds 20 people and requires $2.1 m^3$ of space to store.
  • The ferry must have a capacity of 1000.
  • The space devoted for emergency equipment is $85 m^3$.
  • Question: What to do?

Lets try out a few cases¶

  • Suppose all lifeboats then 50 boats need $50 \times 2.1 \approx 105 m^3$ space.
  • Suppose all lifevests then 1000 lifevests need $1000 \times 0.05 = 50 m^3$ space.
  • Therefore we must do a combination.

Computation¶

  • Let $x_1$ vests.
  • Let $x_2$ boats.
  • The capacity limit means $$x_1 \cdot 0.05 + x_2 \cdot 2.1 = 85.$$
  • The person limit means $$ x_1 \cdot 1 + x_2 \cdot 20 = 1000.$$
In [6]:
# you may need to install numpy
# do it via 'pip install numpy' or 'conda install numpy'
import numpy as np
A = np.array([[0.05, 2.1],[1, 20]])
b = np.array([85, 1000])
x = np.linalg.solve(A,b)
print(x)

[363.63636364  31.81818182]

Discussing the solution¶

  • We found out we need 363.64 vests and 31.82 boats.
  • Since fractional vests/boats are not practical we need to interpret this correctly:
  • How?
  • Note that the volume cannot be increased, so we must reduce boats by 1 and increase vests to accommodate to 1000.
  • If boats are 31, then 620 people are covered,
  • Therefore we need 380 vests.
  • The volume needed is $31 \cdot 2.1 + 380 \cdot 0.05 = 84.1 m^3$

Finishing touches¶

  • Now we write to the company with 31 boats and 380 vests.
  • The company replies: "You are crazy! Lifeboats are arranged on symmetric racks, therefore must be even! Fix it"
  • So we must have 30 boats and therefore the remaining 400 vests.

Reconsideration¶

  • Notice that we had $x_2 = 31.8$, so suppose we want 32 boats. But this means we need more space.
  • The space taken is $32 \cdot 2.1 + 360 \cdot 0.05 = 85.2 m^3$ slightly more than 85.
  • This means we write back to the company with "If the space were to be increased by $0.2 m^3$ we could accommodate 2 more boats means 40 more people in the boats!
  • A significant improvement indeed".
  • They respond: "Sure that works! The 85 figure was an approximation, we can easily find another $0.2 m^3$ of space if it means 2 more boats".

What do we learn from this¶

  • A modelling problem may not be explicitly stated.
  • There are multiple ways of modelling the same problem mathematically.
  • As we saw the equality condition may not hold, if inequalities hold then this problem is called
  • a linear programming problem, furthermore we want
  • the numbers to be integers so it was an integer linear programming problem.
  • Once we solve the problem we have to interpret the answer. More so we ask the question "Does it make sense?"

contd..¶

  • Often all the information may not be present and we might have to tweak the solution a little bit. For example the arrangement of the boats allowed us to change our solution.
  • Problem may be hard and solution may not exist or infeasible.
  • Therefore math modelling is the cycle
  • Model cycle
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