/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Determine whether the given simp... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 6

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{rrrrrrr|c} x & y & z & u & v & w & P & \text { Constant } \\ \hline \frac{1}{2} & 0 & \frac{1}{4} & 1 & -\frac{1}{4} & 0 & 0 & \frac{19}{2} \\\ \frac{1}{2} & 1 & \frac{3}{4} & 0 & \frac{1}{4} & 0 & 0 & \frac{21}{2} \\ 2 & 0 & 3 & 0 & 0 & 1 & 0 & 30 \\ \hline-1 & 0 & -\frac{1}{2} & 6 & \frac{3}{2} & 0 & 1 & 63 \end{array} $$

Short Answer

Expert verified
The given simplex tableau is not in final form. The pivot element for the next iteration of the simplex method is the element 3 in Row 3, at the intersection of the entering variable column \(z\) and the leaving variable row \(w\).

Step by step solution

01

Check if the tableau is in final form

The stopping condition for the tableau is that all coefficients in the P-row (last row) must be nonnegative. Inspecting the P-row, we find that the coefficient for \(z\) is negative (i.e., \(-\frac{1}{2}\)). This indicates that the tableau is not in final form.
02

Identify the entering variable

Next, we find the entering variable which will be the one with the most negative coefficient in the P-row. We only have one negative coefficient in P-row, which is for the \(z\) variable (i.e., \(-\frac{1}{2}\)). Therefore, \(z\) will be our entering variable.
03

Identify the leaving variable

Now, we should identify the leaving variable by performing the minimum ratio test. To perform the minimum ratio test, we must create ratios between the constants in the tableau and the values of the entering variable column. We must discard any negative values or values equal to zero in this step. Ratios for rows: - Row 1: \(\frac{19/2}{1/4} = 38\) - Row 2: \(\frac{21/2}{3/4} = 14\) - Row 3: \(\frac{30}{3} = 10\) The minimum positive ratio is 10, corresponding to Row 3. Therefore, the leaving variable is \(w\).
04

Identify the pivot element

The pivot element is the intersection of the entering variable column and the leaving variable row, so in this case, the pivot element is 3 located in Row 3, Column of the variable \(z\). In conclusion, the given simplex tableau is not in final form, and the pivot element for the next iteration of the simplex method is the element 3 in Row 3, at the intersection of the entering variable column \(z\) and the leaving variable row \(w\).

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Most popular questions from this chapter

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{array}{ll} \text { Minimize } & C=40 x+30 y+11 z \\ \text { subjcct to } & 2 x+y+z \geq 8 \\ & x+y-z \geq 6 \\ x & x+y \geq 0, z \geq 0 \end{array} $$

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows: $$ \begin{array}{lccc} \hline & \begin{array}{c} \text { Standard } \\ \text { Model } \end{array} & \begin{array}{c} \text { Deluxe } \\ \text { Model } \end{array} & \begin{array}{c} \text { Luxury } \\ \text { Model } \end{array} \\ \hline \text { Material } & \$ 6,000 & \$ 8,000 & \$ 10,000 \\ \hline \text { Factory Labor (hr) } & 240 & 220 & 200 \\ \hline \text { On-site Labor (hr) } & 180 & 210 & 300 \\ \hline \text { Profit } & \$ 3,400 & \$ 4,000 & \$ 5,000 \\ \hline \end{array} $$ For the first year's production, a sum of \(\$ 8,200,000\) is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed \(218,000 \mathrm{hr} ;\) and the amount of labor for on-site work is to be less than or equal to 237,000 labor-hours. Determine how many houses of each type Boise should produce (market research has confirmed that there should be no problems with sales) to maximize its profit from this new venture.

Solve each linear programming problem by the simplex method. $$ \begin{aligned} \text { Maximize } & P=4 x+6 y+5 z \\ \text { subject to } & x+y+z \leq 20 \\ & 2 x+4 y+3 z \leq 42 \\ & 2 x+3 z \leq 30 \\ & x \geq 0, y \geq 0, z & \geq 0 \end{aligned} $$

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{rrrrr|r} x & y & u & v & P & \text { Constant } \\ \hline 1 & 1 & 1 & 0 & 0 & 6 \\ 1 & 0 & -1 & 1 & 0 & 2 \\ \hline 3 & 0 & 5 & 0 & 1 & 30 \end{array} $$

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Maximize } & P=x-2 y+z \\ \text { subject to } & 2 x+3 y+2 z \leq 12 \\ & x+2 y-3 z \geq 6 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$
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