When there is no solution to an LP problem that satisfies all of the constraints given, then no feasible solution exists. Graphically, it means that no feasible solution region exists—a situation that might occur if the problem was formulated with conflicting constraints. This, by the way, is a frequent occurrence in real-life, large-scale LP problems that involve hundreds of constraints. For example, if one constraint is supplied by the marketing manager, who states that at least 300 tables must be produced (namely, $X_1\mathrm{ \ge }300$) to meet sales demand, and a second restriction is supplied by the production manager, who insists that no more than 220 tables be produced (namely, $X_1\le 220$) because of a lumber shortage, no feasible solution region results. When the operations research analyst coordinating the LP problem points out this conflict, one manager or the other must revise his or her input. Perhaps more raw materials could be procured from a new source, or perhaps sales demand for that table could be lowered by substituting a different model table to meet customer needs.

As a further graphic illustration of this, let us consider the following three constraints:

As seen in Figure 7.12, there is no feasible solution region for this LP problem because of the presence of conflicting constraints.

Sometimes a linear program will not have a finite solution. This means that in a maximization problem, for example, one or more solution variables, and the profit, can be made infinitely large without violating any constraints. If we try to solve such a problem graphically, we will note that the feasible region is open ended.

Let us consider a simple example to illustrate the situation. A firm has formulated the following LP problem:

As you see in Figure 7.13, because this is a maximization problem and the feasible region extends infinitely to the right, there is unboundedness, or an unbounded solution. This implies that the problem has been formulated improperly. It would indeed be wonderful for the company to be able to produce an infinite number of units of $X_1$ (at a profit of $3 each!), but obviously no firm has infinite resources available or infinite product demand.

The presence of redundant constraints is another common situation that occurs in large LP formulations. Redundancy causes no major difficulties in solving LP problems graphically, but you should be able to identify its occurrence. A redundant constraint is simply one that does not affect the feasible solution region. In other words, other constraints may be more binding or restrictive than the redundant constraint.

Let’s look at the following example of an LP problem with three constraints:

The third constraint, $X_1\le 25,$ is less binding, but the first two constraints are indeed more restrictive constraints (see Figure 7.14).

An LP problem may, on occasion, have two or more alternate optimal solutions. Graphically, this is the case when the objective function’s isoprofit or isocost line runs perfectly parallel to one of the problem’s constraints—in other words, when they have the same slope.

Managers of a firm noticed the presence of more than one optimal solution when they formulated this simple LP problem:

As we see in Figure 7.15, our first isoprofit line of $8 runs parallel to the first constraint equation. At a profit level of $12, the isoprofit line will rest directly on top of the segment of the first constraint line. This means that any point along the line between A and B provides an optimal $X_1$ and $X_2$ combination. Far from causing problems, the existence of more than one optimal solution allows managers great flexibility in deciding which combination to select. The profit remains the same at each alternate solution.