How do you use linear programming to benchmark performance?

2 How to formulate a linear programming problem?

In order to formulate a LP problem, you must identify and define decision variables, an objective function, constraints, and non-negativity. Decision variables are the variables that you can control and that affect the outcome of the problem, such as the number of units of each product that you produce and sell. The objective function is the function that you want to optimize, either by maximizing or minimizing it; for example, the total profit from selling both products. Constraints are conditions that limit the values of the decision variables; for instance, the available materials, labor, and market demand for each product. Non-negativity is the assumption that decision variables cannot take negative values. After defining these elements, you can write them as a system of equations or inequalities. For example, a LP problem of maximizing profit from selling two products could be expressed as: Maximize Z = 10x1 + 15x2 Subject to: x1 + 2x2 <= 40 (material constraint); 2x1 + x2 <= 30 (labor constraint); x1 <= 12 (demand constraint for product 1); x2 <= 10 (demand constraint for product 2); x1 >= 0, x2 >= 0 (non-negativity). Here Z is the objective function, x1 and x2 are the decision variables, and the coefficients are the unit prices, requirements, and limits.

3 How to solve a linear programming problem?

To solve a LP problem, you need to find the values of the decision variables that optimize the objective function, while satisfying the constraints. There are different methods to solve a LP problem, such as graphical, simplex, or interior point methods. However, the most common and convenient way is to use a software tool or a programming language that can handle LP problems, such as Excel, MATLAB, Python, or R. These tools can take your LP problem as an input and output the optimal solution, as well as the sensitivity analysis, which shows how the solution changes with respect to changes in the parameters. For example, using Excel Solver, you can enter your LP problem in a spreadsheet, specify the objective function, the decision variables, and the constraints, and then click Solve to get the optimal solution and the sensitivity report.

4 How to use linear programming to benchmark performance?

Using LP to benchmark your performance requires comparing your optimal solution with a reference value or target value that represents the performance of competitors, industry standards, or best practices. This can help in identifying gaps and opportunities for improvement in process, strategy, or system. For example, if you want to benchmark your profit from selling two products against an industry average of $200, you can solve your LP problem and get the optimal solution and objective function value. Then, calculate the percentage difference between your profit and the industry average profit. After interpreting the results and drawing conclusions, you can analyze the sensitivity report to see how your profit would change if parameters like prices, demands, or costs change.

5 How to use linear programming to improve performance?

To use LP to improve your performance, you need to use the sensitivity analysis to identify the most influential parameters and variables in your problem, and then adjust them to increase your objective function value or decrease your gap with the reference value or the target value. For instance, suppose you want to improve your profit from selling two products by increasing the demand for product 1, which has a higher unit price. In this case, you should analyze the sensitivity report and find the shadow price and the allowable increase or decrease for each constraint. Then, adjust the parameter or variable that has the highest impact on your objective function value or gap with the reference/target value. Finally, solve the LP problem again to get the new optimal solution and objective function value. For example, if you increase the demand for product 1 by 8 units, it would increase your profit by 80 units and reduce your gap with industry average profit to 0%.

6 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?