What are some best practices and tips for setting up and checking systems of equations in linear programming?

1 Define the variables

The first step is to define the variables that you will use to represent the decision variables, the constraints, and the objective function of the problem. You should choose clear and consistent symbols for the variables, such as x, y, z, or a, b, c. You should also assign units and meanings to the variables, such as x = number of units of product A, or y = hours of labor required. This will help you to keep track of what each variable represents and avoid confusion.

2 Write the objective function

The second step is to write the objective function, which is the expression that you want to maximize or minimize in the problem. The objective function should be a linear combination of the decision variables, such as z = 3x + 2y, where z is the profit and x and y are the units of products A and B. You should also specify whether you want to maximize or minimize the objective function, such as max z or min z.

3 Write the constraints

The third step is to write the constraints, which are the conditions that limit the values of the decision variables. The constraints should also be linear equations or inequalities involving the decision variables, such as x + y <= 100, where x and y are the units of products A and B and 100 is the limit of production capacity. You should write all the constraints that apply to the problem, such as resource availability, demand, quality standards, or budget.

4 Check the feasibility

The fourth step is to check the feasibility of the system of equations, which means to verify that there is at least one solution that satisfies all the constraints and the objective function. You can do this by graphing the system of equations on a coordinate plane and finding the feasible region, which is the area where all the constraints are met. You can also use algebraic methods, such as substitution or elimination, to solve the system of equations and find the feasible solutions. If there is no feasible solution, then the problem is infeasible and cannot be solved by linear programming.

5 Check the optimality

The fifth step is to check the optimality of the system of equations, which means to find the best solution that maximizes or minimizes the objective function within the feasible region. You can do this by finding the vertices or corners of the feasible region and evaluating the objective function at each vertex. The vertex that gives the highest or lowest value of the objective function is the optimal solution. You can also use methods such as the simplex algorithm or the graphical method to find the optimal solution more efficiently.

6 Check the sensitivity

The sixth step is to check the sensitivity of the system of equations, which means to analyze how the optimal solution changes when some parameters of the problem change. For example, you may want to know how the optimal solution changes when the profit per unit of product A increases or decreases, or when the production capacity increases or decreases. You can do this by using methods such as the sensitivity analysis or the shadow prices, which measure how much the objective function value changes when a constraint value changes by one unit.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?