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What is mathematical optimization techniques?

A mathematical optimization technique is a method for solving problems through the use of numerical algorithms that attempt to find values which satisfy or work with constraints. Mathematical optimization techniques can be used in fields such as analytics, finance, biology, robotics, artificial intelligence and engineering. The size of available datasets frequently requires applications of optimization techniques for complex problems.

Mathematical optimization techniques are used to solve the following types of problems:

-finding locations for facilities with limited space, including airports, train stations, and shopping centers;

-assigning employees or sales people to different jobs or customers to minimize the total distance traveled or cost;

-planning the movement and scheduling of aircraft, ships, and trucks to minimize fuel consumption and reduce fleet costs;

-routing package delivery fleets to deliver packages while minimizing the total distance traveled.

Types of mathematical optimization techniques

There are many types of optimization problems, namely;

Linear programming problems

Linear programming problems are used to determine how much of each product to produce for maximizing profits while ensuring that the products meet the required specifications.

Integer linear programming

Integer linear programming problems deal with variables that can only be 0 or 1, while continuous linear programming problems permit fractional values such as 0.5 .Integer linear programming problems involve finding the minimum or maximum of a linear function subject to constraints that some or all variables be integers. Linear optimization may also deal with finding the best solution to a mathematical model, when no such optimal value exists and/or there is no certainty about the model.

Using the analogy of a traveler trying to reach his destination, linear optimization often concerns itself with problems like what is the shortest path or set of paths that passes through certain points (the airports), without passing through others (no backtracking). Formulating this type of problem typically requires special representation of the problem, such as a graph. Also see the branch of mathematics named combinatorial optimization which deals with finding shortest or longest paths in graphs without regard to constraints on where the path can go.

Nonlinear programming problems

Nonlinear programming problems are used to maximize or minimize values of functions subject to certain constraints. Nonlinear optimization problems are called constrained, bound-constrained or semi-infinite when the functions to be optimized are required to be nonnegative or, respectively, nonpositive. When the sign of the objective function is irrelevant (e.g., maximization/minimization), the problem is called unconstrained.

In physics, nonlinear programming problems are often solved by finding the extrema (critical points) from a set of auxiliary functions that measure or otherwise take account in a different way various aspects of a physical system. For example, in celestial mechanics problems involving the three-body problem are often solved using an auxiliary function that measures the distance between two fixed points in a coordinate system .

Constraint programming problems

Constraint programming problems are used to maximize or minimize values of functions where some variables are restricted to be integers, real numbers, binary, etc. Nonlinear optimization is used in a wide range of applications , such as scheduling problems, economic dispatch problems, optimal control problems, network traffic routing, and mathematical programming.

Stochastic programming problems

Stochastic programming is similar to linear optimization in that the solution space of the problem is described by a set of linear constraints. However, instead of maximizing or minimizing an objective function, the goal is to maximize or minimize the probability of a specific set of outcomes.

Dynamic programming

Dynamic programming is a mathematical optimization technique that solves problems by breaking them down into simpler subproblems . Dynamic programming can be used to find the best ways of doing things, to minimize or maximize some quantity. It was developed in the 1940s for applications in economics and business but now it is also used in many other areas

 

 

Applications of optimization techniques

The Stackelberg game model is used in economics and business studies to examine how players with different roles, e.g., leader and follower, interact with each other on their strategic moves.

The Lanchester square game model is used in business studies to determine a firm’s optimal pricing strategy, taking into account the competitor’s cost structure and its price setting response. The Lanchester square game model can be applied for two-side platforms, e.g., eBay and Uber .

Methods of solving non-linear programming problems..

Lagrangean multiplier method which involves introduction of an artificial variable in the initial problem . For details on how to solve constrained optimization problems using Lagrange’s multiplier methods visit the website assignmentguru.com.

Gradient search method involves calculating the gradient of the objective or cost function and then moving in the opposite direction by an infinitesimal amount multiplied by negative of the gradient. The method is iterative and it involves subsequent iterations until the value is optimized.

Kuhn-Tucker conditions involve setting the first-order conditions of the Lagrangean to zero and introducing an artificial variable in the constraint. Then solve the problem using Lagrange’s multiplier method.

methods of solving linear programming problems

They include linear programming networks, the simplex algorithm, and integer linear programming. These are described below. Other than these three there are hundreds of techniques for solving linear programs. Many of these methods find useful application in combinatorial optimization problems…

Simplex Algorithm

The simplex algorithm is an efficient technique for solving linear programs that meet certain conditions (see below). The critical feature of this method is that it can recover quite quickly from getting stuck in a local

pseudocode of the simplex algorithm is as follows:

The simplex method starts by putting an initial base vertex inside the feasible region. The next step is to identify, for each variable, whether it should take on an

equation always hold. This gives us

Converting the constraints into equalities means that they are all satisfied precisely when the optimal solution is achieved. The simplex algorithm carries out three basic steps at each iteration:

All these steps can be repeated until either a tolerance level has been reached or no improvement can be made to the current solution

The simplex method is very efficient, particularly when there are many variables and/or constraints. If the linear program does not have a feasible solution, then the algorithm cannot find one by moving from vertex to vertex on its graph. However, it can prove that no such solution .It uses a simple idea: at each iteration, choose an entering variable (one with the smallest index) and swap it with the leaving variable (one with the largest index).

5 benefits of using mathematical optimization techniques to improve your business results

If you’re familiar with the phrase “garbage in, garbage out”, then you’ll know that your results will be just as good – or bad – as the data that you enter into the system. The mathematics of optimization provides techniques to ensure that decision makers understand how their data is influencing their business outcomes. Let’s take a look at how mathematical optimization can help in business.

Handle data inconsistency and incompleteness

If your data is inconsistent or incomplete, then it will lead to incorrect results in your business. For example, if you are attempting to make an investment decision based on the number of users who visit your website, but your data doesn’t include the number of unique views you have for each page on your site, then it’s likely that you will make a poor business decision.

Gain insights from big data

Big Data is all about using predictive analytics to get the best possible results from your business.

Optimize business processes to increase profits

Mathematical optimization techniques allow you to identify the best production schedule for your materials, which can result in increased efficiency and output. It can also be used for other business processes such as transportation routing and scheduling problems.

 Make better decisions based on historical data

By using mathematical optimization techniques, you can make more accurate predictions about future events. These may be your customers’ response to a marketing campaign or the likely revenue from launching new products in specific markets.

 Integrate math into your business

By understanding how mathematical optimization can support you, you’ll be able to use it successfully in many different areas of your business. There are benefits for individuals who know how to apply optimization techniques to their role by looking at problems from a different angle. It also means that businesses will be able to make better decisions and improve their overall results.

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