# What is Operations Research?

## 1. Definition

Operations research is, as Statistics, a branch of Applied Mathematics. Its purpose is, for a given problem, to help make the best possible decision, or even the optimal decision (maximization of gains, minimization of costs). To do this, it relies on a specific methodology and mathematical tools. The methodology is broken down into three phases:

1. Problem modeling: writing of the problem in the form of mathematical expressions involving variables (quantities which will be modifiable during the second phase), constraints that variables must respect and the objective function which is evaluated for the set of values taken by the variables;

2. Calculation of the optimum (minimization or maximization of the objective function);

3. Decision making or possibly reformulation of the problem and return to phase 1 if the solution is not satisfactory.

## 2. A quick history of operations research

Gaspard Monge (1746-1818), a French mathematician, who in 1817 was the first director of the Ecole Polytechnique, published in 1781 a "Mémoire sur la théorie des déblais et des remblais". Without completely solving the problem, but asking himself the question of how to transport from a surface A to a surface B (he then studies the problem in three dimensions) some "molecules" while minimizing the cost of transport, which he considers proportional to the weight, it lays the basis for a discipline of mathematics, optimal transport, whose applications are numerous and which has been a subject of study for many mathematicians including Cédric Villani, one of the most famous living mathematicians (Topics in Optimal Transportation, 2003, American Mathematical Association).

The term Operations Research was introduced in 1937 by AP Rowe, a british physicist, in a military context. The few years before and during the Second World War, the discipline experienced a rapid development: the minimization of the costs of production, of the times of routing, of decision, the efforts of logistics and the maximization of the efficiency of the collecting of intelligence have had a decisive impact (see British Operational Research in World War II and O.R. in World War 2: operational research against the U-boat). The Americans adopted the discipline from 1942. After the war, Operations Research was applied in industry, economical sciences, management and in what gradually became the decision sciences.

## 4. A key discipline within the data science ecosystem

Operations Research takes full advantage of the development of computer science and data science. For its use in business, it can be placed as shown below among the different processes involving data. Optimization engines, also called solvers, take as input structured data, forecasts, or simulations. The proposed solution itself generates data (for example the best scheduling for the tasks to be accomplished) which can in turn feed the information system or, more often, lead to an operational decision.

## 5. Practicing Operations Research

Here are some examples of common OR projects in companies:

• Planning of gas or gasoline trucks
• Planning of television advertising
• Optimization of construction equipment orders
• Planning of construction for motorway building sites
• Planning of the maintenance stops of nuclear thermal power stations
• Planning of the maintenance of the oars on trains
• Replanning of trains in the event of incidents
• Optimization of public bike rental stocks
• Sizing and allocation of storage servers
• Calculation of flight staff schedules

## 6. Carry out your Operations Research project ## 7. Some definitions:

### What is an Optimization problem?

An optimization search problem is the search for the optimal solution or a solution respecting a set of constraints applied to a set of variables, while maximizing or minimizing a criterion materialized by an objective function. For that, we will model a real or theoretical problem using decision variables and constraints. Then we will solve it with a solver.

### What is Modeling?

A modeling of a problem consists of initially finding the facts of the case, then to define indices (temporal, resource,…), then to write the constraints for finally writing the objective function of the problem. We will adopt this methodology in 5 steps in order to model some problems in the form of Optimization Problem: find the dimensions of the problem, then identify the datan then write the decision variables, then define the constraints and the objective function.

### What is an Objective function?

The objective function indicates how much each variable contributes to the value to be optimized in the problem. The objective function takes the following general form:

The objective function allows to calculate the criterion we want to maximize or minimize that is usually the essence of the operations research problem: minimizing costs, maximizing a gain, minimizing an energy consumption, maximizing the availabilty of a system. All the decision variables are included in the calculation of the objective function. Sometimes, the objective function can be a combination of several of several criteria.

### What is a Decision Variable?

A decision variable represents a value of the problem that can vary during the search for a solution. There can be one or several decision variables (sometimes tens of thousands) in an optimization problem and these variables are constrained.

### What is a Constraint?

The constraints make it possible to bound the search space for the solution. They make it possible, on the one hand, to make search possible and, on the other hand, when they are well defined, to accelerate the search for the solution. Constraints apply to decision variables, as well as sometimes to the objective function. For example, the constraints will indicate that a pH must be between 0 and 14 or that the number of paths is a positive integer less than a given value. The constraints remain fixed, unlike the decision variables.

### What is a Solution?

To each set (n-tuple) of values of the decision variables that respect the constraints corresponds a solution. The solution results in a single value of the objective function. It is possible that several solutions correspond to identical values of the objective function. In the case of very complex problems, one will be able to be satisfied with a solution without being perfectly sure that it is the optimal solution (too long computation time, or too complex problem).

### What is an Optimal Solution?

An optimal solution is the one or one of the solutions to which corresponds the best possible value of the objective function. It is materialized by the set (n-tuple) of values which respect the constraints while optimizing the objective function (minimum or maximum depending on the criterion). ### What is a Solver?

A solver is a computer program or a software, which will allow you to model your problem (variables, constraints, objective function) in order to then find either a solution or an optimal solution.

There are different solvers on the market and not all of them rely on the same methodologies / approaches to find the solution (simplex, local search, heuristics, etc.).

After analyzing the different solutions on the market, Addinsoft® chose the powerful LocalSolver® engine to develop XLOPTIM®, integrated with Microsoft Excel. You will find a series of use cases based on XLOPTIM on our website.