week3 type ups-2 - LINEAR PROGRAMMING: Minimization...

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LINEAR PROGRAMMING: Minimization Introduction In the previous lesson we have discussed what linear programming problems are and how they can be formulated. We have also learned how to find the optimal solution to a maximization problem using the graphical solution method for a two-variable case or the simplex procedure that can handle any number of variables. In many business settings, however, an analyst is interested in minimizing the value of his objective function. For instance, in a diet problem, dietician’s maybe to minimize the cost of foods subject to minimum daily requirements. The computational procedures discussed in the preceding lesson are readily applicable to a minimization problem. In this lesson we will show how the simplex method can be used to find the optimal solution to a problem whose main objective is to minimize costs. Minimization Problem As we pointed out earlier, the criterion for optimality of an objective function of a linear program may be its minimization rather than maximization. Usually a cost minimization objective function is associated with a set of greater-than-or-equal-to constraints or equality constraints. Such problems need to be modified and put into an appropriate form before the simplex method can be applied for a solution. However, if less-than-or-equal-to constraints occur with a minimization problem the application of the simplex computational procedure is rather straight forward. We will first consider the latter format of the problem for illustration and then discuss the former. Suppose we are given the following LP problem to solve: M i n i m i z e Z = - 4 x 1 + 5 x 2 Subject to x 1 + 3 x 2 5 2 x 1 - 3 x 2 8 x 1, x 2 0 There are two ways in which we can solve this problem using the simplex method. The first requires that we reserve the rule of introducing a new variable into the basis. Recall, for maximization, the simplex rule requires that we select a variable with the most negative objective function coefficient to enter the new solution. This is because the most negative coefficient indicates the highest contribution to profit per unit of that variable. An optimal solution is available when all the objective function coefficients are non- negative (zero or positive). For the same reason, a positive objective function coefficient in a minimization problem indicates reduction in the cost per unit of that variable. In order to minimize the cost, we would therefore select a variable for the next solution that
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has the most positive coefficient. The process will terminate when all the objective function coefficients are nonpositive (zero or negative). When this condition occurs we have a solution to the minimization problem. Thus, letting s
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week3 type ups-2 - LINEAR PROGRAMMING: Minimization...

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