Let A and B be two events such that P(B) > 0. Then we can recall that the conditional probability of A given B is: or Hence, if we are given that event B occurred, the relevant sample space is reduced to B {P(B)=1 because we know B is true} and conditional probability becomes a … Read more
Independent event Two events are said to be independent of each other if the probability that one event occurs on any given trial of an experiment is not influenced by the occurrence of the other event under any condition.Alternatively, two events are considered disjoint if they have no outcomes in common and they can never … Read more
When we come across events that are not independent, ‘conditional’ events are plausible and account for the possible change in the multiplication of probability. Recall that if A and B are independent, then we can accept that : P(A ∩ B) = P(A) * P(B) However, if P(B) changes based on the occurrence of event … Read more
Conditional probabilities reflect how the probability of an event alters if we know that some other event has occurred. Consider an example involving manufactured parts as shown in the figure below. In this case, D denotes the event that a part is defective and F denotes the event that a part has a surface flaw. … Read more
Optimal feasible solutions (up to three non-trivial constraints) A feasible point on the optimal objective function line for an LP provides an acceptable optimal solution.The following Theorems are fundamental in solving linear programming problems to obtain an optimal solution: Theorem 1 When you consider R to be in the feasible region (convex polygon) and let … Read more
Feasible Solution A feasible solution for a linear program is a solution that satisfies all constraints that the program is subjected. It does not violate even a single constraint. Any x = (x1, xn) that satisfies all the constraints. Example x1 = 5 bowls x2 = 12 mugs Z = $40×1 + $50×2 = $700 … Read more
Feasible and infeasible regions (bounded and unbounded) For a standard maximum/minimum problem a range of values is said to be feasible if they satisfy the corresponding constraints.The set of feasible vectors is called the constraint set which lies on the feasible regions.So, if the constraint set is not empty, then the LP is feasible or … Read more
We have previously discussed word-problems translated into mathematical problems in the form of linear programs.The graphical method is applicable to solve the LPP involving two decision variables x1, and x2, however, more number of variables are difficult to optimize by graphical representation.The solution is a set of values for each variable: are consistent with the … Read more
The development of all LP models can be examined in a four-step process: (1) identification of the problem as solvable by LP (2) formulation of the mathematical model. (3) solution. (4) interpretation. The formulation of the mathematical problem involves translation of a problem scenario to a simple L.P framework containing a set of mathematical relationships. … Read more
Manufacturing problems In these problems, we determine the number of units of manufacturing products to be produced and sold by a firm. Each product requires fixed manpower, machine time, labour hour per unit of product, warehouse space per unit of the output etc., which can be optimized in order to make maximum profit. Example: NuGrow … Read more