What is linear programming?
Linear programming is an optimization method for a system of linear constraints and a linear objective function. What we mean by objective function is the quantity to be optimized and the goal of linear programming is to find the values of the variables that minimize or maximize the objective function.
The application of linear programming is basically everywhere you go. You use linear programming at professional and personal fronts. For example, you are using linear programming when you are driving from a destination to another and want to take a shortcut.
Linear programming and optimization are applied in several industries. The manufactory and service industry uses linear programming on a regular basis.
Where do we use Linear Programming?
-Applications in Engineering
Engineers use linear programming to help, design, solve, and manufacturing problems. For instance, in airfoils meshes engineers seek aerodynamic shape optimization. This allows for the loss of the drag coefficient of the airfoil.
Limitations may include life coefficient relative maximum thickness nose radius and trailing edge angle. Shape optimization seeks to make a shock-free airfoil with a possible shape. So linear programming provides engineers with essential tools in shape optimization.
Modern energy grid systems incorporate not only traditional electrical systems but also renewables like solar photovoltaics and wind. In order to optimize the electric load requirement generators, transmission and distribution lines, and storage must be taken into account. However, the cost must remain sustainable for profits.
Linear programming provides a method to optimize the electrical power system design. It allows for coordinating the electrical load in the lowest total distance between the generation of the electrical and its request overtime. Linear programming can be applied to optimize load-matching or to optimize price giving a valuable tool to the energy industry.
-Food and Agriculture
Farmers also apply linear programming techniques to their work. By determining what crops they should increase the quantity of it and how to use it efficiently farmers can progress their revenue. In nutrition linear programming provides healthy, low-cost food baskets for needy families, nutritionists can use line programming.
Mathematical modeling provides assistance to calculate the foods needed to provide nutrition at a low cost in order to prevent non-communicable diseases.
Unprocessed food data and prices are required for such calculations while considering the cultural aspects of the food types. The objective function is the complete cost of the food basket. Linear programming also allows time variations for the frequency of performing this kind of food baskets.
Well, the applications of Linear programming do not end here there were just a few examples. There are many more purposes of linear programming in real life like applied by Stock Markets, Sports, Shareholders, etc. Go on and explore further.
Example of a linear programming problem
We will give you an example so you can understand what linear programming is.
For example, a delivery man has 6 packages to deliver in a day. The warehouse is located at point A. The 6 delivery destinations are given by V, U, W, X, Z, and Y. The number of lines indicates the distance between the cities. To save on time and fuel the delivery person wants to take the shortest way.
The delivery person will calculate different ways of going to all the 6 destinations and then come up with the shortest path. This technique of determining the shortest path is called linear programming. In this case, the goal of the delivery person is to deliver the package on time at all 6 destinations. The method of deciding on the best route is called Operation Research.
Operation research is an approach to decision-making, which includes a set of methods to operate a system. Linear programming is used for obtaining the most optimal solution for a problem with given limitations. In linear programming, we formulate our real-life problem into a mathematical form. It includes an objective purpose, linear inequalities with subject to constraints.
Is the linear design 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 route. It would likely have multiple turns, signals, U-turns, and traffic jams. But with a simple theory, we have overcome the complexity of the problem drastically and are creating a solution that should work in most situations.
ADVANTAGES OF LINEAR PROGRAMMING
Following are certain advantages of linear programming:
- Linear programming helps in accomplishing the optimum use of productive resources. It also indicates how a decision-maker can apply his productive factors effectively by selecting and distributing these resources.
- Linear programming techniques improve the quality of decisions. The decision-making method of the user of this technique shifts more objective and less subjective.
- Linear programming methods provide possible and practical solutions considering that there might be other limitations operating outside of the problem which must be taken into the statement. Just because we can produce so many units this doesn’t mean they can be sold. Thus, the required adjustment of its mathematical solution is required for the sake of convenience to the decision-maker.
- Highlighting of bottlenecks in the production processes is the most essential benefit of this method. For example, when a bottleneck occurs, some devices cannot meet demand while others remains idle for some time.
- Linear programming also serves in the re-evaluation of a basic plan for improving conditions. If conditions change when the plan is partly carried out, they can be determined to adjust the remainder of the plan for the best outcomes.
LIMITATIONS OF LINEAR PROGRAMMING
- There should be an objective that should be clearly identifiable and measured in quantitative terms. It could be, for example, maximization of sales, profit, minimization of cost, and so on, which is not possible in real life.
- The activities to be included should be distinctly identifiable and measurable in quantitative terms, for example, the products included in a production planning problem and all the activities can’t be covered in quantitative terms for instance if labor is sick, which will lower his performance which can’t be measured.
- The resources of the system which are to be allocated for the achievement of the goal should also be identifiable and measurable quantitatively. They must be in limited quantity. The technique would include the allocation of these resources in a manner that would trade off the returns on the investment of the resources for the attainment of the objective.
- The relationships representing the objective as also the resource restriction considerations, represented by the objective function and the constraint equations or inequalities, respectively must be linear in nature, which is not possible.
- There should be a series of possible options courses of action possible to the decision-makers, which are determined by the resource constraints.
What is Nonlinear Programming?
Nonlinear programming is the process of solving optimization problems that affect some of the nonlinear constraints or nonlinear objective functions. It includes minimizing or maximizing a nonlinear objective function subject to bound constraints, linear constraints, nonlinear constraints, etc.
These constraints can inequalities or equalities. In addition, nonlinear programming helps in analyzing design tradeoffs, computing optimal trajectories and portfolio optimization, selecting optimal designs, and model calibration in computation finance.
What are some distinctions between linear and nonlinear programming?
The main distinctions between linear and nonlinear programming are that linear programming helps to find the best solution from a set of parameters or conditions that have a linear relationship while on the other hand, nonlinear programming helps to find the best solution from a set of parameters or requirements that have a nonlinear relationship.
Linear programming is an outstanding concept in optimization techniques in mathematics because it helps to find the most optimized solution to a given problem. On the other hand, nonlinear programming is the mathematical method of finding the optimized answer by considering restrictions or objective functions that are nonlinear.