`=>` If R is unbounded, then a maximum or a minimum value of the objective function may not exist. However, if it exists, it must occur at a corner point of R. (By Theorem 1).
`=>` In the above example, the corner points (vertices) of the bounded (feasible) region are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and (0, 60) respectively. Let us now compute the values of Z at these points.
We have
`=>` We observe that the maximum profit to the dealer results from the investment strategy (10, 50), i.e. buying 10 tables and 50 chairs.
`=>` This method of solving linear programming problem is referred as Corner Point Method. The method comprises of the following steps:
`color{blue} 1`. Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point.
`color{blue} 2`. Evaluate the objective function` Z = ax + by` at each corner point. Let M and m, respectively denote the largest and smallest values of these points.
`color{blue} 3`. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.
(ii) In case, the feasible region is unbounded, we have:
`color{blue} 4`. (a) M is the maximum value of Z, if the open half plane determined by `ax + by > M `has no point in common with the feasible region. Otherwise, Z has no maximum value.
(b) Similarly, m is the minimum value of Z, if the open half plane determined by `ax + by < m` has no point in common with the feasible region. Otherwise, Z has no minimum value.
We will now illustrate these steps of Corner Point Method by considering some examples:
`=>` If R is unbounded, then a maximum or a minimum value of the objective function may not exist. However, if it exists, it must occur at a corner point of R. (By Theorem 1).
`=>` In the above example, the corner points (vertices) of the bounded (feasible) region are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and (0, 60) respectively. Let us now compute the values of Z at these points.
We have
`=>` We observe that the maximum profit to the dealer results from the investment strategy (10, 50), i.e. buying 10 tables and 50 chairs.
`=>` This method of solving linear programming problem is referred as Corner Point Method. The method comprises of the following steps:
`color{blue} 1`. Find the feasible region of the linear programming problem and determine its corner points (vertices) either by inspection or by solving the two equations of the lines intersecting at that point.
`color{blue} 2`. Evaluate the objective function` Z = ax + by` at each corner point. Let M and m, respectively denote the largest and smallest values of these points.
`color{blue} 3`. (i) When the feasible region is bounded, M and m are the maximum and minimum values of Z.
(ii) In case, the feasible region is unbounded, we have:
`color{blue} 4`. (a) M is the maximum value of Z, if the open half plane determined by `ax + by > M `has no point in common with the feasible region. Otherwise, Z has no maximum value.
(b) Similarly, m is the minimum value of Z, if the open half plane determined by `ax + by < m` has no point in common with the feasible region. Otherwise, Z has no minimum value.
We will now illustrate these steps of Corner Point Method by considering some examples: