The Extremum Point Theorem

When seeking to minimize the objective function, the graphical solution involves moving the objective function up from below the feasible region until it first encounters the feasible region.

The optimum solution for the revised problem is for the farmer to buy two bags of high grade and 4.8 bags of low grade per hectare. This will cost him $43.20 per hectare while still giving a yield of 75 tonnes.
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The simplest situation where no maximum or minimum can be found is where there is no feasible region. Some of the constraints are mutually incompatible. For example, consider our original problem. If the farmer had already committed himself to buy six bags per hectare of high grade (instead of the two we have been considering) there would be no feasible region, and hence no optimum would be possible. You could think of this as being too little feasible region.
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In some situations the feasible region is not bounded so that there is no maximum, not because there is no possible solution (as when there is no feasible region) but because for any solution there is always a better one. One such situation is illustrated in the applet. You could think of this as being too much feasible region.
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Similarly, the feasible region can be open in such a way that there is no minimum. The usual non-negativity constraints usually take care of this problem, although if the graph of the objective function has a positive slope then this situation can also occur even when there are non-negativity constraints. (This doesn't happen very often in real world problems.) The applet illustrates a situation that has no minimum unless non-negativity constraints are applied. Once the non-negative constraints are applied it has a solution of 42 at (4.2,0).
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Sometimes as you are moving the objective line towards the feasible region it will hit two vertices at the same time. This means that the graph of the objective function is parallel to that of the constraint that runs between these two vertices. In this situation, the to vertices and every point between them are equally optimal solutions.

The applet illustrates one such situation. As you move the objective line towards the feasible region, it encounters the line of constraint 2 all at once.

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