Operations research is a particular way of mathematical modeling (a subset of it if you will), using mostly linear algebra techniques, that seek to restrict the search domain for suitable answers, or even to directly optimize parameters of a problem. Mathematical models need not use only linear algebra, or not at all, which would be the case of stochastic modeling, or using ordinary differential equations.
A very common example of operations research problem would be something along the lines of the following:
A furniture company produces two types of chairs: wooden chairs and metal chairs. The company wants to determine the optimal production quantities for each type of chair to maximize profit while considering various constraints.
Objective: Maximize total profit
Decision Variables: x = number of wooden chairs produced y = number of metal chairs produced
Parameters:
- Profit per wooden chair:
- Profit per metal chair:
- Production time for wooden chair: 2 hours
- Production time for metal chair: 1.5 hours
- Available production time: 300 hours per week
- Maximum storage capacity: 200 chairs
- Minimum demand for wooden chairs: 30 units
- Minimum demand for metal chairs: 40 units
Objective Function: Maximize Z = 50x + 40y
Subject to the following constraints:
The goal is to find the values of x and y that maximize the total profit while satisfying all the given constraints."
This example includes the following key components of an operations research problem:
An example of mathematical modeling, using, say, probability techniques, as opposed of using linear algebra for operations research, could be the following, also very classical example:
Problem: A whale can be seen in a given zone of the Ocean once every 8 hours. What is the probability of not seeing a whale after 24 hours?
Solution:
Therefore, the probability of not seeing a whale after 24 hours is approximately 0.0498 or about 4.98%.
Operation research models are used for decision making, improving efficacy and optimization of the complex systems whereas mathematical models represent real-world systems using mathematical techniques and widely used in physics, biology, economics etc.
Operations Research models are generally decision-making models where we need to make the best decisions under the restricted domain. Whereas mathematical models are generally the real world models to understand the effect of variables on each other and find all possible solutions in the restricted domain.
Operations research uses pure (both combinatorics and combinatorial) maths for modeling complex systems. Differential geometry plays a key role in solving such optimization problems.
Refer to :
- R. B. Misra, "Differential Geometry, Field Theory and Operations Research", 2011 -
Book Differential Geometry, Field Theory and Operations Research.
- R. B. Misra, "Differentiel Geometry: Its Past and Future", 2024 -
Article DIFFERENTIAL GEOMETRY: ITS PAST AND FUTURE
Related topic by following https://www.researchgate.net/post/For_all_specialists_in_the_field_of_pure_mathematics_and_statistics
Network models in Operations research are a perfect illustration of the successful combination of different fields of mathematics: differential geometry, graph theory (decision tree, decision forest), combinatorial optimization, linear algebra, probabilities and statistics, etc.
Refer to Chapter 6-Network Models in Eiselt et al., "Operations Research - A Model-Based Approach", 3rd Edition, Springer, 2022 - https://link.springer.com/chapter/10.1007/978-3-030-97162-5_6#citeas
The difference between operations research models and mathematical models is of the same order as that between applied mathematics and pure mathematics:
- Follow Pure vs. Applied Mathematics: https://www.geeksforgeeks.org/pure-vs-applied-mathematics/
More specifically, operations research models are real-life applications of discrete (applied) mathematics:
- Follow Real-Life Applications of Discrete Mathematics: https://www.geeksforgeeks.org/real-life-applications-of-discrete-mathematics/
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