This is as high as the line can be raised, i.e., as large a value as the right-hand side of that objective function can be, while some part of that function is in or on the boundary of the feasible region. If we change the 2 to a 4, we get the dash-dot line shown in that same figure. Changing the right-hand side moves the objective function, the dashed line, up and down but doesn’t change its slope. Since that function is to be maximized, our goal is to find the maximum value of its right-hand side while some part of that function is in the feasible region or on its boundary. To find the best combination of X and Y values in this feasible region, set the objective function equal to some value, such as X + Y = 2, and then plot that equation. Unbounded feasible regions result from one or more variables going to infinity as would be the case if there were no constraint 2 X + Y ≤ 4 or if the constraint had to be greater or equal to 4 or any other number. Optimization problems that do not have feasible regions have no feasible solutions, meaning that not all constraints can be satisfied. If a model is linear and has only two variables such asĪll the X Y pairs of values in the shaded region and its boundaries, called the feasible region, satisfy all the constraints. Hence, it seems reasonable to show how linear problems are solved, at least graphically, and when necessary, how some non-linear components of a model may be made linear to take advantage of linear optimization solution methods. Linear programming has found many applications in the military, in government agencies, industry and in agriculture, ecology, economics, engineering, public health, and urban planning to mention only a few subject areas. They are often employed just because of the efficiency and widespread availability of the solution methods for linear models. Thus, many tricks exist for making non-linear functions linear. This is not because the world is linear, but because the algorithms (solution methods) used to solve linear models are so efficient and are able to solve problems with many-even thousands-of variables and constraints, as long as they are linear. Developing and solving linear optimization models is often the first topic addressed in courses in systems analysis. Undoubtedly the most commonly used of all the mathematical programming (constrained optimization) methods is linear programming.
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