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Mathematical optimization for decision making
By Mohammad
6 January 2023 • 4 min read
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A gentle introduction to operation research
Motivation
The demand for data analysts has exploded in recent years. Most companies are integrating data into their decision making process. However, in many practical applications, the company may not have data to answer the questions. An example of this scenario is when the managers want to start building a new product. In this case, many questions must be answered before the production. Some of these include
 How many units should we produce?
 Is this product profitable?
 How much budget should be allocated to marketing?
As you can see, these questions are directed toward areas where previous data may not help. So, should we just trust the intuition of the managers or is there a more systematic way to explore the answers? In this post, we explain the latter.
Mathematical Programming
Most of the decisions in a firm are either trying to maximize profit or minimize cost. In a way, the decisions are trying to achieve one of these extremes. You may remember this graph from high school; and, probably the “pointless” differentials.
Even though Y = X2 does not correspond to any particular thing, it can represent a whole range of possibilities. For example, if we substitute “Y” with “Production Cost”, things start to make sense. Now, trying to find the minimum value of “Y” is not a math exercise anymore; it is the process of minimizing the “Production Cost” given the parameter “X”. Parameters “X” and “Y” can represent anything. This is the simplest case of decision making where we control parameter “X” and the value of “Y” is derived from our decision. In this example, the relation between these parameters is quadratic. This is the essence of operations research and mathematical optimization.
An optimization task is defined by

Decision variables:
The parameters that you can control.

Objective function:
The thing we are trying to optimize (either by finding the minimum or the maximum) expressed in terms of decision variables.

Constraints:
A set of real world constraints. For example, the maximum budget. And, that’s it! Just by defining your problem and constraints in mathematical terms, you can have the solution using Microsoft Excel!
Real World Example
Suppose you want to start a business. You have 10000$ and want to build your awesome idea. You can buy 1 unit of material that is needed to build your idea for 2$. And, buying 100 units at once has a 10% discount. However, you can only work 4 hours a day because you have a full time job. But, you can always pay others to work for you. In this case you can pay someone 50$ per hour to work for you. Each hour of work produces 3 units of your awesome idea which you can sell each for 12$. And, each unit requires 2 materials.
Given these terms and a deadline of 30 days, how would you allocate your budget to maximize the profit?
Let’s translate this description into our framework.

Decision Variables:
There are several parameters that we control.
 Number of raw materials to buy?
 How many hours should we recruit someone to work for us?
 How many hours per day should we work?
 How many units to produce?

Objective function:
Maximizing profit in terms of the costs associated with each decision variable.

Constraints:
There are several constraints.
 Our budget.
 Each unit needs 2 materials to be produced.
 You can only work 4 hours per day.
 A 30 days deadline.
As you can see, there isn’t any formula or equation in the new form. But, the beauty of this framework is that you have to think about the situation explicitly. Every cost and its associated profit is now explicit. Even if you don’t want to enter the realm of optimization, you have a clear idea of what you can expect. The objective function tells us that we do not control the profit directly. It is our decisions in resource allocation that determine it. Furthermore, we can do some “WhatIf” analysis. For example, how much money can we make if we don't outsource the production?
Applications
As a company grows, the constraints for its decisions grow too. For example in the case of public transportation planning, hundreds of constraints must be taken into account.
As mentioned previously, even if you do not intend to use the mathematical framework for optimization, having the problem in a concise and explicit way is an invaluable tool. However, finding the optimal solution using computer softwares like Excel or IBM CPLEX opens the door for in depth analysis and can influence the decisions.
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