How do you create a linear programming problem?

"The variables must be non-negative, meaning they cannot take negative values." This is incorrect. Many modern implementations will assume your decision variables are non-negative by default, you usually can change this. Also, in order to perform the Simplex Algorithm the linear program is first transformed to "Standard Equality Form", in which the variables must be non-negative. However, the variables being non-negative is not inherent to the definition of linear programs.

Variable need to be continuous for problem to be linear programming problem. If its binary or integer then problem is non linear.

Creating a linear programming problem involves a balance between understanding the problem context, formulating mathematical relationships, and leveraging optimization techniques to find the best solution. 1. Identify decision variables 2.Formulate the objective function 3.Specify the constraints 4.Define variable bounds

Satish Kumar: What you are telling is right, but it is not a new modelling framework, but a conventional system existing since ever. In the real world problems, the number of variables, coefficients, constraints are very large. Hence that is a practical approach using indexes to represent variables and coefficients.

Identifying decision variables is the cornerstone of effective problem-solving. They represent what we aim to determine, like product quantities or routes. Symbolic representation, like x or descriptive names (x1, x2), brings clarity and enables mathematical modelling. Enforcing non-negativity ensures realistic solutions. These variables streamline complex problems, align with objectives, and aid in communication, making them an essential first step in decision-making processes.

Defining the objective function in linear programming offers clarity by quantifying the goal, aligns decision-making with strategic objectives, establishes a practical mathematical framework for efficient optimization techniques, and provides valuable insights into trade-offs between variables. However, it assumes linear relationships which might oversimplify complex real-world scenarios.

The included example brings up the point of clearly understanding your measures. Profit may be price times number o items sold, but only if costs are zero. Reminds me of a real world anecdote of a plant manager who did not care about plant operations costs because he only was concerned about profits!

System understanding is important. Consumption constraints (eg raw materials consumed must be less than stock) can be easy to determine. Logical, policy, or physics constraints may require more thought. If "both .. and" or "exclusive or" relationships exist, you may need integer variables (losing linearity). In military logistics, explosive compatibility may preclude certain product combinations. In a physical world example, some fragile products produced at higher altitude may not be intact at sea level. These relationships make the problem description more complete, but perhaps move out of the linear realm. Overlooking them may lead to elegant solutions that cannot be implemented. It is important to understand your decision space.

Setting constraints sets the limits for the possible values our variables can possibly take. In my experience I have observed to set hard constraints only if its required otherwise always use a soft constraint

You should also check your constraints for redundancy, which would lead you to a degenerate solution, as well as incompatibility, which would lead to an infeasible solution.

Formulating constraints in linear programming acts as a powerful translator, converting real-world intricacies into a language of optimization. This step not only grounds solutions within practical boundaries but also illuminates the delicate balance between various factors. For instance, in a production scenario with limited machine hours, constraints could be the linchpin that highlights the equilibrium between maximizing output and respecting operational constraints.

Should any assumption not hold, adjustments are pivotal; for instance, if variables are discrete, integer programming could be a fitting alternative, or stochastic programming may be warranted when coefficients lack certainty or vary. This step ensures the model's alignment with real-world intricacies and enhances its practical impact, which I like the most.

I love the notion of assumptions. I think we frequently assume the assumptions are known and are intact. Consider both endogenous (what is within your problem design), and exogenous (what is external). Simply put, what seems to optimize the use of your raw materials may simply result in excess inventory if your assumptions about demand do not hold.

'You can also use specialized software or tools, such as Excel Solver, MATLAB, or Python.' This is kind of vague and doesn't really express the right thing. You can't solve an LP with just Python. Surely, Python will help you write a code but you will need a solver e.g. Gurobi, Cplex that actually solves the LP.

The choice of solver can make a significant difference for larger or more complex problems, such as those with integer variables. For example, the Excel Solver is very limited on problem size and performance, and can only handle a few integer variables. On the other hand, commercial solvers such as Gurobi and CPLEX have advanced built-in techniques that can greatly enhance performance for linear, mixed-integer, and even nonlinear programming problems.

The fastest way to get started is to install the pyomo package for python to formulate the mathematical optimization problem and glpk to solve it. Both are open source and easy to install.

Turns out the world is frequently not a clean, point solution. Do not be afraid to test your assumptions with some sensitivity analysis. A robust situation is frequently preferable. Also, "the model" is frequently not the rationale for following your "optimal" result. Make sure it relates to the real world, and be able to explain and defend it in those terms. This may seem "overkill" for simple problems, but may be critical to convincing decisionmakers in complex situations.

Since we are specifically addressing “linear” programming, I think I would state the obvious “Step Zero” as being “Be sure you are working on a linear problem.” Approximating integer with linear may or may not “optimize” well. Missing non- linearity may lead to totally erroneous results. Of course, this could be addressed in the following steps, but here it emphasizes understanding of the problem space you will be working in.

Agree with Cyrus.. First we need to understand what are the conflicting business requirements and are they linear.. Are there more than one set of resources that are being consumed and hence can act as constraints