Linear programming (LP) is used to find the optimal solution for functions operating under known constraints. Therefore, it is best suited for operating within a decision environment with certainty. Such situations are common in business and economics, where the costs are known, or when distributing limited resources. One example where both of these elements are true and an LP solution can be attempted is determining a schedule for public transportation that minimizes passenger waiting times and energy consumption by vehicles (Yin, Yang, Tang, Gao, & Ran, 2017). In this complex example, the energy costs and maximum passenger numbers are known, therefore, an LP-based solution is possible.
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Using the graphical solution approach
The graphical solution approach allows LP problems to be solved in an intuitive and visual way. It is generally used for problems with two decision variables because more than two results in a multi-dimensional graph that negates the approach’s advantage of being visual. To solve an LP problem with this method, one first has to define the problem’s feasible space by plotting all of its constraints, represented by inequalities, on a graph.
The area of the graph that falls within these constraints is then highlighted, showing all possible values for each variable. At this step, it is possible to discover that the areas defined by all the constraints have no intersections. This means that the LP problem has no feasible solution. Once the feasible area is determined, one assigns an arbitrary value to the objective function (i. e. the function one is trying to minimize or maximize) and plots it on the graph. Then one adjusts this value up (for maximization) or down (for minimization) until it intersects the furthest corner of the feasible area. One can simply use a ruler, moving it up or down parallel to the original line. Once this corner is found, one calculates all the variables at this point and the result is the optimized function.
Yin, J., Yang, L., Tang, T., Gao, Z., & Ran, B. (2017). Dynamic passenger demand oriented metro train scheduling with energy-efficiency and waiting time minimization: Mixed-integer linear programming approaches. Transportation Research Part B: Methodological, 97, 182-213. Web.