41. Linear Programming

Nature, and all of nature, human being, life, works with certain constraint at any given time, and consciously, subconsciously, unconsciously, tries to find the best or most optimal path forward from that situation. And life is a series of such path forwards, each moment a first step with available constraints, reality and exercise of will, and available resources to the optimal outcome.

One is more objective when one considers life like this perhaps. One's own as well as everyone and everything around. Eventually, the reckoning is only with the self, and the infinite inside, that whether you are training yourself enough to see well, to keep solving your linear programs optimally.

It somehow traverses the ground between mathematics, programming, life as in available reality at any point in time, and a self with its own shifting objective function - it is one of the most fascinating concepts in terms of philosophy/maths and very accessible to any being inherently, unconsciously. Perhaps more accessible to the pre-word subconscious than the shallow knowledge confused being.

Its use is mostly in operations planning in the real world, and only perhaps so because in the strange way art mimics life, because all life's underlying operations are managed through this function.

Mathematically it is fascinating because it uses maths for life in some way, somehow channels the abstraction to a more material purpose. Overall more contenment in each sphere.

A step towards win-win outcomes. The best possible solution to it.

*

Personal takes aside, Linear Programming: (from various sources)

Linear programming is the technique used for optimizing a particular scenario. Using linear programming provides us with the best possible outcome in a given situation. It uses all the available resources in a manner such that they produce the optimum result.

*

"Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists."

*

"Linear programming or Linear optimization is a technique that helps us to find the optimum solution for a given problem, an optimum solution is a solution that is the best possible outcome of a given particular problem.

In simple terms, it is the method to find out how to do something in the best possible way. With limited resources, you need to do the optimum utilization of resources and achieve the best possible result in a particular objective such as least cost, highest margin, or least time. 

The situation that requires a search for the best values of the variables subject to certain constraints is where we use linear programming problems. These situations cannot be handled by the usual calculus and numerical techniques."

*

"In operations research, linear programming (LP) is one of the mathematical techniques used to get an optimal solution to a given operational problem, considering resource scarcity and external and internal constraints.

To apply Linear Programming for process optimization, these requirements have to be met:

Problem statement: define the objective in clear mathematical terms

Decision variables: quantitative input variables impacting the objective

Constraints: quantitative and measurable conditions

Objective function: the relationship between the objective and the input variables has to be linear"

In terms of modern applications apart from the philosophical contemplation of life, nature and their operations:

Linear programming is a widely used field of optimization for several reasons. Many practical problems in operations research can be expressed as linear programming problems.[6] Certain special cases of linear programming, such as network flow problems and multicommodity flow problems, are considered important enough to have much research on specialized algorithms. A number of algorithms for other types of optimization problems work by solving linear programming problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as duality, decomposition, and the importance of convexity and its generalizations. Likewise, linear programming was heavily used in the early formation of microeconomics, and it is currently utilized in company management, such as planning, production, transportation, and technology. Although the modern management issues are ever-changing, most companies would like to maximize profits and minimize costs with limited resources. 

*

An optimal solution need not exist, for two reasons. First, if the constraints are inconsistent, then no feasible solution exists: For instance, the constraints x ≥ 2 and x ≤ 1 cannot be satisfied jointly; in this case, we say that the LP is infeasible.

Certain realities about the basic of fabric of cosmos - the equity of the planes, the either end of equation transference indifference. If it is not met, there is no conversation, understanding - it is all horizontal plane randomness. No enduring meaning can ever appear.

*

Linear programming, a powerful mathematical technique, is used to solve optimization problems in various industries. Here are some modern applications:

Supply Chain Optimization: Linear programming helps companies minimize costs and maximize efficiency in their supply chains. It’s used for determining the most cost-effective transportation routes, warehouse operations, and inventory management strategies.

Energy Management: In the energy sector, linear programming is utilized to optimize the mix of energy production methods. This includes balancing traditional energy sources with renewable ones to reduce costs and environmental impact while meeting demand.

Telecommunications Network Design: Linear programming aids in designing efficient telecommunications networks. It helps in allocating bandwidth, designing network layouts, and optimizing the flow of data to ensure high-speed communication at lower costs.

Financial Planning: Businesses and financial analysts use linear programming for portfolio optimization, risk management, and capital budgeting. It helps in making investment decisions that maximize returns while minimizing risk.

Healthcare Logistics: In healthcare, linear programming is applied to optimize the allocation of resources, such as hospital beds, medical staff, and equipment. It’s crucial for improving patient care, reducing wait times, and managing costs effectively.

Manufacturing Process Optimization: Linear programming is used to determine the optimal production levels for multiple products within a manufacturing facility, considering constraints like labor, materials, and machine availability.

Agricultural Planning: Farmers and agricultural planners use linear programming to decide on crop selection, land use, and resource allocation to maximize yields and profits while conserving resources.

Airline Crew Scheduling: Airlines employ linear programming to schedule crews efficiently, ensuring that flights are staffed in compliance with regulations and minimizing operational costs.

*

Can be solved through iterative process of Simplex method, or through graphing, to viusally chart the optimal area.  (For the straightforward Linear programming Problems. There are several branching thoughts and variations on linear programming - as a wikipedia dive reveals).

"Simplex Method

Optimization Algorithm: The Simplex Method is a powerful algorithm used in linear programming to find the optimal solution to linear inequalities.

Step-by-Step Approach: It iteratively moves towards the best solution by navigating the edges of the feasible region defined by constraints.

Efficiency: Known for its efficiency in solving large-scale linear programming problems.

Versatility: Applicable in various domains like diet planning, network flows, production scheduling, and more, showcasing its versatility.

*

One of the main challenges is to formulate your problem correctly and accurately, which requires a clear understanding of your operations, your constraints, your objectives, and your assumptions.

 *

The main benefit of optimization models is the ability to evaluate possible solutions in a quick, safe, and inexpensive way without actually constructing and experimenting with them. Other benefits are as  follows.

1. Structures the thought process. Constructing an optimization model of a problem forces a decision maker to think through the problem in a concise, organized fashion. The decision maker determines what factors he or she controls; that is, what the decision variables are. The decision maker specifies how the solution will be evaluated (the objective function). Finally, the decision maker describes the decision environment (the constraints). Modeling acts as a way of organizing and clarifying the problem.

2. Increases objectivity. Mathematical models are more objective since all assumptions and criteria are clearly specified. Although models reflect the experiences and biases of those who construct them, these biases can be identified by outside observers. By using a model as a point of reference, the parties can focus their discussion and disagreements on its assumptions and components. Once the model is agreed on, people tend to live by the results.

3. Makes complex problems more tractable. Many problems in managing an organization are large and complex and deal with subtle, but significant, interrelationships among organizational units. For example, in determining the optimal amounts of various products to ship from geographically dispersed warehouses to geographically dispersed customers and the routes that should be taken, the human mind cannot make the billions of simultaneous tradeoffs that are necessary. In these cases, the decision maker often uses simple rules of thumb, which can result in less than optimal solutions. Optimization models make it easier to solve complex organization-wide problems.

4. Make problems amenable to mathematical and computer solution. By representing a real problem as a mathematical model, we use mathematical solution and analysis techniques and computers in a way that is not otherwise possible.

5. Facilitates “what if” analysis. Mathematical models make it relatively easy to find the optimal solution for a specific model and scenario. They also make “what if” analysis easy. With “what if” analysis, we recognize that the prices, demands, and product availabilities assumed in constructing the model are simply estimates and may differ in practice. Therefore, we want to know how the optimal solution changes as the value of these parameters vary from the original estimates. That is, we want to know how sensitive the optimal solution is to the assumptions of the model. “What if” analysis is also called sensitivity or parametric analysis.

Although mathematical modeling has many advantages, there are also disadvantages. The actual formulation or construction of the model is the most crucial step in mathematical modeling. Since the problems tend to be very complex, it is possible to mismodel the real problem. Important decision variables or relationships may be omitted or the model may be inappropriate for the situation. The optimal solution to the wrong problem is of no value.

A second disadvantage is not understanding the role of modeling in the decision-making process. The optimal solution for a model is not necessarily the optimal solution for the real problem. Mathematical models are tools to help us make good decisions. However, they are not the only factor that should go into the final decision. Sometimes the model only evaluates solutions with regard to quantitative criteria. In these cases qualitative factors must also be considered when making the final decision. The bottom line for evaluating a model is whether or not it helps a decision maker identify and implement better solutions. The model should increase the decision maker’s confidence in the decision and the willingness to implement it.