On the other hand, the point (x1 15, x2 70) is not in the feasible region, because even though x1 15 and x2 70 satisfy (2), (4), (5), and (6), they fail to satisfy
On the other hand, the point (x1 15, x2 70) is not in the feasible region, because even though x1 15 and x2 70 satisfy (2), (4), (5), and (6), they fail to satisfy (3): 15 70 is not less than or equal to 80. Any point that is not in an LP’s feasible region is said to be an infeasible point. As another example of an infeasible point, consider (x1 40, x2 20). Although this point satisfies all the constraints and the sign restriction (5), it is infeasible because it fails to satisfy the sign restriction (6), x2 0. The feasible region for the Giapetto problem is the set of possible production plans that Giapetto must consider in searching for the optimal production plan. DEFINITION ■ For a maximization problem, an optimal solution to an LP is a point in the feasible region with the largest objective function value. Similarly, for a minimization problem, an optimal solution is a point in the feasible region with the smallest objective function value. ■ Most LPs have only one optimal solution. However, some LPs have no optimal solution, and some LPs have an infinite number of solutions (these situations are discussed in Section 3.3). In Section 3.2, we show that the unique optimal solution to the Giapetto problem is (x1 20, x2 60). This solution yields an objective function value of z 3×1 2×2 3(20) 2(60) $180 When we say that (x1 20, x2 60) is the optimal solution to the Giapetto problem, we are saying that no point in the feasible region has an objective function value that exceeds 180. Giapetto can maximize profit by building 20 soldiers and 60 trains each week. If Giapetto were to produce 20 soldiers and 60 trains each week, the weekly profit would be $180 less weekly fixed costs. For example, if Giapetto’s only fixed cost were rent of $100 per week, then weekly profit would be 180 100 $80 per week.
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