Linear Programming In Operations Research

Introduction

Optimization is the way of life. We all have finite resources and time and we want to make the most of them. From using your time productively to solving supply chain problems for your company – everything uses optimization. It’s an especially interesting and relevant topic in data science. menfashdesign

It is also a very interesting topic – it starts with simple problems, but it can get very complex. For example, sharing a bar of chocolate between siblings is a simple optimization problem. We don’t think in mathematical terms while solving it. On the other hand, devising inventory and warehousing strategies for an e-tailer can be very complex. Millions of SKUs with different popularity in different regions are to be delivered in defined time and resources – you see what I mean!

Linear programming (LP) is one of the simplest ways to perform optimization. It helps you solve some very complex LP problems and linear optimization problems by making a few simplifying assumptions. As an analyst, you are bound to come across applications and problems to be solved by Linear Programming solutions.

For some reason, LP doesn’t get as much attention as it deserves while learning data science. So, I thought let me do justice to this awesome technique. I decided to write an article that explains Linear programming examples in simple English. I have kept the content as simple as possible. The idea is to get you started and excited about Linear Programming. ethicmenvoguee

Note- If you want to learn this in a course format, we have curated this free course for you- Linear Programming for Data Science Professionals

Table of contents

What is Linear Programming?

Now, what is linear programming? Linear programming is a simple technique where we depict complex relationships through linear functions and then find the optimum points. The important word in the previous sentence is depicted. The real relationships might be much more complex – but we can simplify them to linear relationships.

Linear programming is a mathematical modeling technique that involves maximizing or minimizing a linear function while taking into account various constraints. This approach has proven useful in guiding quantitative decisions across different fields such as business planning, industrial engineering, and to some extent, in the social and physical sciences. Linear programming, also known as linear optimization, is a method for achieving the best possible outcome in a mathematical model where the requirements are defined by linear relationships. businessdirectorypc

Moreover, it’s important to note that linear programming is a special case of mathematical programming. It involves finding the optimal solution to a problem that involves linear relationships between the decision variables and the constraints. This method has applications in various industries and fields, from production planning to logistics optimization, and is a powerful tool in making data-driven decisions that can lead to improved efficiency and cost savings.

Applications of linear programming are everywhere around you. You use linear programming at personal and professional fronts. You are using linear programming when you are driving from home to work and want to take the shortest route. Or when you have a project delivery you make strategies to make your team work efficiently for on-time delivery.

Example of a Linear Programming Problem (LPP)

Let’s say a FedEx delivery man has 6 packages to deliver in a day. The warehouse is located at point A. The 6 delivery destinations are given by U, V, W, X, Y, and Z. The numbers on the lines indicate the distance between the cities. To save on fuel and time the delivery person wants to take the shortest route.

So, the delivery person will calculate different routes for going to all 6 destinations and then come up with the shortest route. This technique of choosing the shortest route is called linear programming.

In this case, the objective of the delivery person is to deliver the parcel on time at all 6 destinations. The process of choosing the best route is called Operation Research. Operation research is an approach to decision-making, which involves a set of methods to operate a system. In the above example, my system was the Delivery model.

Linear programming is used for obtaining the most optimal solution for a problem with given constraints. In linear programming, we formulate our real-life problem into a mathematical model. It involves an objective function, and linear inequalities with subject to constraints.  allinternetbuziness

Is the linear representation of the 6 points above representative of the real world? Yes and No. It is an oversimplification as the real route would not be a straight line. It would likely have multiple turns, U-turns, signals and traffic jams. But with a simple assumption, we have reduced the complexity of the problem drastically and are creating a solution that should work in most scenarios.

Formulating a Problem

Let’s manufacture some chocolates…Example: Consider a chocolate manufacturing company that produces only two types of chocolate – A and B. Both the chocolates require Milk and Choco only.  To manufacture each unit of A and B, the following quantities are required:

The company kitchen has a total of 5 units of Milk and 12 units of Choco. On each sale, the company makes a profit of

Now, the company wishes to maximize its profit. How many units of A and B should it produce respectively

Solution: The first thing I’m gonna do is represent the problem in a tabular form for better understanding.

Let the total number of units produced by A be = X

Let the total number of units produced by B be = Y

Now, the total profit is represented by Z

The total profit the company makes is given by the total number of units of A and B produced multiplied by its per-unit profit of Rs 6 and Rs 5 respectively.

which means we have to maximize Z.

The company will try to produce as many units of A and B to maximize the profit. But the resources Milk and Choco are available in a limited amount.