Simplex Method Minimization Examples Pdf
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Simplex Method Minimization Examples Using
Simplex Method For Function Minimization

4.2 The Simplex Method: Standard Minimization Problems

Drum extract vst free. The simplex algorithm assumes we are on the feasible region, in which case the test ratios assure we pivot to the nearest intersection with a constraint, which is a vertex of the feasible region. If we’re not inside the feasible region, the test ratio method unnecessarily requires more pivots. The other approach to Phase 1 is the following. An example can help us explain the procedure of minimizing cost using linear programming simplex method. Example: Assume that a pharmaceutical firm is to produce exactly 40 gallons of mixture in which the basic ingredients, x and y, cost $8 per gallon and $15 per gallon, respectively, No more than 12 gallons of x can be used, and at least 10.

Learning Objectives

Use the Simplex Method to solve standard minimization problems.

Notes

This section is an optional read. This material will not appear on the exam.

We can also use the Simplex Method to solve some minimization problems, but only in very specific circumstances. The simplest case is where we have what looks like a standard maximization problem, but instead we are asked to minimize the objective function.

Minimize	$C=-2x-3y$
subject to	$leftlbracebegin{align}5x+4y&leq 32x+2y&leq 10xgeq 0,&~ygeq 0end{align}right.$

We notice that minimizing $C$ is the same as maximizing $P=-C$. Thus the solution to the minimization problem can be found by solving the standard maximization problem below with the techniques learned in Section 4.1.

Simplex Method Minimization Examples Pdf

Maximize	$P=2x+3y$
subject to	$leftlbracebegin{align}5x+4y&leq 32x+2y&leq 10xgeq 0,&~ygeq 0end{align}right.$

The solution to this example is left as an exercise. The other important class of minimization problems we encounter are called standard minimization problems.

Definition. A standard minimization problem is a linear programming problem in which we seek to minimize an objective function $C=c_1x_1+ldots+c_nx_n$ subject to the constraints$$begin{align}a_{11}x_1 + ldots + a_{1n}x_n & geq b_1& vdotsa_{m1}x_1 + ldots + a_{mn}x_n & geq b_mend{align}$$where all $x_igeq 0$.

Definition. Each maximization problem in linear programming is associated with a counterpart minimization problem, and vice versa. For the purposes of identification, the given problem will be referred to as the primal problem, and the counterpart to this problem is called the dual problem.

Example. Consider the following standard minimization problem.

Minimize	$C=4x+2y$
subject to	$leftlbracebegin{align}5x+y&geq 55x+3y&geq 10xgeq 0,&~ygeq 0end{align}right.$

We first write down the following tableau for the given primal problem:$$begin{array}{cc|c}x & y &hline5 & 1 & 55 & 3 & 10hline4 & 2end{array}$$Note that this is not a typical Simplex tableau. Free version of inssider. There are no slack variables, and we don't rewrite the objective function for the last row. Next we transpose.$$begin{array}{cc|c}u & v &hline5 & 5 & 41 & 3 & 2hline5 & 10end{array}$$Then we construct the dual problem:

Simplex Method Minimization Examples Worksheet

Maximize	$P=5u+10v$
subject to	$leftlbracebegin{align}5u+5v&leq 4u+3v&leq 2ugeq 0,&~vgeq 0end{align}right.$

Then we set up the Simplex tableau and proceed as the Section 4.1, using $x$ and $y$ as our slack variables.$$begin{array}{ccccc|c}u & v & x & y & P & text{Constant}hline5 & 5 & 1 & 0 & 0 & 41 & 3 & 0 & 1 & 0 & 2hline-5 & -10 & 0 & 0 & 1 & 0end{array}$$Solving this tableau is left to the reader as an exercise. The final tableau looks like this.$$begin{array}{ccccc|c}u & v & x & y & P & text{Constant}hline1 & 0 & 3/10 & -1/2 & 0 & 1/50 & 1 & -1/10 & 1/2 & 0 & 3/5hline0 & 0 & 1/2 & 5/2 & 1 & 7end{array}$$Interpreting this in the usual manner, the solution of the dual problem is $u=1/5$, $v=3/5$, and $P=7$. The solution to the primal problem appears under the respective slack variables in the last row of the final tableau: $x=1/2$ and $5/2$.

Simplex Method Minimization Calculator

The graphical method of solution illustrated by the example in the preceding section is useful only for systems of inequalities involving two variables. In practice, problems often involve hundreds of equations with thousands of variables, which can result in an astronomical number of extreme points. In 1947 George Dantzig, a mathematical adviser for the U.S. Air Force, devised the simplex method to restrict the number of extreme points that have to be examined. The simplex method is one of the most useful and efficient algorithms ever invented, and it is still the standard method employed on computers to solve optimization problems. First, the method assumes that an extreme point is known. (If no extreme point is given, a variant of the simplex method, called Phase I, is used to find one or to determine that there are no feasible solutions.) Next, using an algebraic specification of the problem, a test determines whether that extreme point is optimal. If the test for optimality is not passed, an adjacent extreme point is sought along an edge in the direction for which the value of the objective function increases at the fastest rate. Sometimes one can move along an edge and make the objective function value increase without bound. If this occurs, the procedure terminates with a prescription of the edge along which the objective goes to positive infinity. If not, a new extreme point is reached having at least as high an objective function value as its predecessor. The sequence described is then repeated. Termination occurs when an optimal extreme point is found or the unbounded case occurs. Although in principle the necessary steps may grow exponentially with the number of extreme points, in practice the method typically converges on the optimal solution in a number of steps that is only a small multiple of the number of extreme points.

Simplex Method Minimization Examples Using

To illustrate the simplex method, the example from the preceding section will be solved again. The problem is first put into canonical form by converting the linear inequalities into equalities by introducing “slack variables” x₃ ≥ 0 (so that x₁ + x₃ = 8), x₄ ≥ 0 (so that x₂ + x₄ = 5), x₅ ≥ 0 (so that x₁ + x₂ + x₅ = 10), and the variable x₀ for the value of the objective function (so that x₁ + 2x₂ − x₀ = 0). The problem may then be restated as that of finding nonnegative quantities x₁, …, x₅ and the largest possible x₀ satisfying the resulting equations. One obvious solution is to set the objective variables x₁