![]() ![]() Therefore as x increases without bound, z increases without bound and there is no maximum value of z. Since the feasible region is unbounded there may be no maximum value of z. Maximize z=2 x+ y subject to 3 x+ y≥6, x+ y≥4, x≥0, and y≥0. ▸ Objective =x has a minimum, reached at both corners, and between the two corners. Then the feasible set is unbounded and has two corners. Suppose the constraints are y≥0, x≥0, y≤2. ▸ Objective = y has a minimum, reached along the ray starting at the corner and moving to the right. ▸ Objective = x+ y has a minimum, reached uniquely at the corner. Then theįeasible set is unbounded and has one corner. ▸ Objective = x– y has no minimum, and no maximum ▸ Objective =- y has no minimum, but has a maximum It has a minimum, reached along the entire x-axis. Then the feasible set is unbounded and has no corners. This is a little more nuanced than the Theorem stated on page ( How can we Maximize an Objective Function Using Search-Line Method to the Constraints?) of the text (which is not really a true theorem ☻) … ▸ at a corner or along an infinite ray leaving that corner, or ▸ at two adjacent corners and at all points in between, ![]() If there is a maximum/minimum, it can happen ▸ if it is a minimization problem, there might be a minimum, or it might be possible to make the objective arbitrarily small (big and negative) inside the feasible set. ▸ if it is a maximization problem, there might be a maximum, or it might be possible to make the objective arbitrarily large inside the feasible set, and If the feasible set of a linear programming problem is not bounded (there is a direction in which you can travel indefinitely while staying in the feasible set) then a particular objective may or may not have an optimum:
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