Hi everyone. David from rondofiniti here. I completed my MBA from Deakin University in 2018 which included a specialisation in Business Analytics. In this specialisation I undertook a subject called Decision Modelling for Business Analytics, which fits into the prescriptive modelling space. We did a lot of work on linear and non-linear programming, probability modelling, stochastic modelling and monte-carlo simulations. To be totally honest, I personally haven’t seen these fantastic tools used to their full potential in corporate and non-corporate business, which is a shame. Therefore, I thought I’d write this brief article on one of these aspects, namely linear programming.
What is linear programming (LP)?
Firstly, LP is not computer programming – it’s a specific problem mathematically modelled with boundary conditions (constraints) which define a feasible solution zone. The feasible solution zone is used to extract a solution to the problem based on an objective function. Often the objective is to maximise profit, or minimise cost, or minimise material usage – it can be anything of that nature.
To illustrate, consider the simple example below:
A store sells two types of figurines, X and Y. The owner pays $7.5 and $13 for each unit of X and Y respectively. Each unit of figurine X yields a profit of $1.8 while a unit of figurine Y yields a profit of $2.5. The store owner estimates that no more than 2200 figurines will be sold every month and does not plan to invest more than $20,000 in stored inventory. How many units of each type of figurine should be stocked in order to maximize the monthly total profit?
This problem can be set up mathematically as follows:
X = Number of X figurines in inventory per month
Y = Number of Y figurines in inventory per month
MAX 1.8 X + 2.5 Y (Profit – we want to maximise this)
X + Y ≤ 2200 (Sold estimate)
7.5 X + 13 Y ≤ 20000 (Total inventory)
X, Y = Integers (We need whole figurines)
X, Y ≥ 0 (Figurines can’t be negative)
Image 1 tells us that if the owner wishes to maximise profit, they would need to ensure that the monthly stored inventory consists of 1564 and 636 X and Y figurines respectively. We could also talk about slack variables, sensitivities, shadow prices and several different linear programming methods, but I wanted to keep this pretty simple. There are many applications of this method as follows (with an example listed):
Finance (determining how to optimally split your investment in a diversity share fund)
Manufacturing (optimising the least amount of material required)
Distribution (optimising which vehicles to use for different styles of delivery)
Apartment rental (optimising what to charge each room to maximise profit)
Process sequencing (determining in what order to run each process to reduce lost time)
Retail goods (optimising price to maximise profits)
Inventory (optimising how much of each product to store to minimise costs)
This list can be greatly expanded but will suffice for the purpose of this article. Also, to note, Excel has a solver feature for this style of programming, which makes it simple to solve such problems where there are many decision variables. Hope you enjoyed the article.