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Graphical Sensitivity Analysis for Variable Linear Programming Problems

Instructor: Giorgos-Nektarios Panayotidis

George-Nektarios has worked as a tutor and student consultant for five years and has a 4-year university degree in Applied Informatics.

In this lesson, we will be taking a look at a special type of analysis that is conducted on Variable Linear Programming Problems. It is called Graphical Sensitivity Analysis and is carried out after calculating the optimal solution to the problem.

Safety-Wise Optimal Solutions

Let's consider a situation where you are planning to paint a specific spot in your house, say, one wall in the living room. Normally, you would cover the floor in front of the wall and maybe tape the ceiling and edges of the wall. The rationale is pretty simple, if paint drops fall, they are expected to spatter in a certain manner. Some people, however, move out all of their furniture and cover the whole floor. These people have asked themselves the question, what if the spatter behaves differently than expected and hits the sofa? Or, what if I drop the whole can of paint? Total disaster! That's why, to exclude such possibilities, they cover the whole room.

A Sensitivity Analysis is analogous to taking extra precautions against paint spatters. Extra precautions are taken against uncertainty when estimating certain objective functions, such as profit. In a Graphical Sensitivity Analysis, two-dimensional graphs are used in order to depict changes when problem coefficients or constants have been altered because of better estimation.

Graphical Sensitivity Analysis

The Graphical Sensitivity Analysis involves three fundamental elements:

  • Linear problem variables - Linear problem variables or decision variables are quantities that have to be determined.
  • Objective function - An objective function is a function that needs to be either minimized or maximized in order to find an optimal solution. In a business context, this function is most often a profit function and the problem's optimal solution has to do with the profit's maximization.
  • Constraints - Constraints are linear inequalities which include the decision variables.

A sensitivity analysis allows us to discover what changes occur in the linear problem's optimal solution when certain aspects of the linear problem's data are subjected to changes. In other words, the optimal solution's sensitivity. In order for a graphical sensitivity analysis to occur, the optimal solution needs to be depicted in the plane. In this instance, the decision variables can only be two, as in Figure 1.

Figure 1: Feasible Region of Linear Programming Problem
Linear Problem Feasible Region

What types of data may change in the problem's elements? There are two fundamental changes that take place: change in the Objective Function Coefficient (OFC) and change in the Right-Hand Side (RHS) value within a constraint inequality.

Each one of these two basic change types will be analyzed separately.

Objective Function Coefficient Change

An objective function coefficient change means that a coefficient of the objective variable is subjected to a change. Specifically, this implies that a coefficient or one of the coefficients of the decision variables will either be increased or decreased. For example, a profit function may be altered from 260S + 245LX to 270S + 245LX. Another example (depicted in Figure 2) is that of an objective function having been revised from 7T + 5C = 4040 to 8T + 5C = 4360.

Figure 2: Objective Function Coefficient Change
Objective Function Coefficient Change

As is discernible in Figure 2, there are two basic characteristics of this particular type of sensitivity analysis:

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