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### Lecture-2

Course: COMP 323, Fall 2009
School: Allan Hancock College
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Word Count: 571

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Solutions Given + + Feasible/Infeasible an LP problem and a vector X = (x1, . . . , xn) We say that X is feasible if it satisfies all the constraints of the LP problem (including non-negativity X 0); X is infeasible otherwise. The set of all feasible solutions is called the feasible region of the LP problem. Note: The above definitions do not depend on the objective function. X is an optimal solution if...

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Solutions Given + + Feasible/Infeasible an LP problem and a vector X = (x1, . . . , xn) We say that X is feasible if it satisfies all the constraints of the LP problem (including non-negativity X 0); X is infeasible otherwise. The set of all feasible solutions is called the feasible region of the LP problem. Note: The above definitions do not depend on the objective function. X is an optimal solution if X is feasible and the objective function obtains its optimal value (maximal or minimal) at X. + 1 + + Example Consider the following constraints 6 x1 + x2 5 1 x1 + x2 2 3 x1 + 2x2 2 x1 , x2 72 50 120 0 We have the following feasible/infeasible solutions Solutions (-1, 0) (5, 60) (20, 20) (0, 0) (40, 20) + feasible No No (3rd constraint) Yes Yes Yes (2nd constr. is tight) 2 + + Graphical LP Solutions We will show how a two-variable LP model can be solved graphically. It should be noted that two-variable models rarely occur in practice (a typical LP model may include thousands of variables and constraints), but the ideas of the graphical procedure lay the foundation for the development of the general solution technique (the simplex method). The graphical procedure includes two basic steps: 1. The determination of the solution spaces that defines the feasible solutions that satisfy all the constraints of the model (feasible region). 2. The determination of the optimal solution from among all the points in the feasible solution space. + 3 + + Determination of Feasible Region For each constraint a1x1 + a2x2 () b plot the straight line a1x1 + a2x2 = b. Decide which half-plane we need: take any test point T = (, ) outside of the line (1) and compare a1 + a2 with b. If (1) is satisfied then we take the half-plane which T belongs to; otherwise the other half plane. (1) Overlap all half-planes and take their intersection. + 4 + + Optimal Solution Take any point z0 and plot c1x1 + c2x2 = z0 (2) where z = c1x1 c2x2 + is the objective function. (2) is a level curve (line). All level curves are parallel. Slide (2) parallel to itself until it still hits the feasible region and obtains its corresponding to the smallest possible z0 for a Minimal problem; corresponding to the largest possible z0 for a Maximal problem. The point of the feasible region in which the above level curve hits the feasible region is an optimal solution. + 5 + + Example 1 Determine the feasible space graphically for the following constraints of an LP problem. 6 x1 + x2 5 1 x1 + x2 2 3 x1 + 2x2 2 x1 , x2 72 50 120 0 + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + Example 2 Give the graphic solution for the following LP problem: maximise z subject to x1 - 2x2 2x1 + x2 x2 x1 , x2 = 2x1 + 3x2 4 18 10 0 + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + Example 3 Give the graphic solution for the following LP problem: minimise subject to z = x1 - 2x2 -x1 + x2 x2 x1 , x2 x1 + 3x2 4 3 2 0 + 19 + + + 20 + + Example 4 + 21 + + Example 5 Give the graphic solution for the following LP problem: Maximise z subject ...

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