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Chapter 6: Problem 9

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. $$ \begin{array}{ccrcrc|c} x & y & z & u & v & P & \text { Constant } \\ \hline 1 & 0 & \frac{3}{5} & 0 & \frac{1}{5} & 0 & 30 \\ 0 & 1 & -\frac{19}{5} & 1 & -\frac{3}{5} & 0 & 10 \\ \hline 0 & 0 & \frac{26}{5} & 0 & 0 & 1 & 60 \end{array} $$

Short Answer

Expert verified
The given simplex tableau is in its final form. The solution to the associated regular linear programming problem is \((x, y, z, u, v) = (30, 10, 0, 0, 0)\).

Step by step solution

01

Check if the tableau is in final form.

In order for the tableau to be in final form, all entries of the bottom row except the last one (i.e., the constant column) must be non-negative, since maximum value is sought in a linear programming problem. For the given tableau: $$ \begin{array}{ccrcrc|c} x & y & z & u & v & P & \text { Constant } \\\ \hline 1 & 0 & \frac{3}{5} & 0 & \frac{1}{5} & 0 & 30 \\\ 0 & 1 & -\frac{19}{5} & 1 & -\frac{3}{5} & 0 & 10 \\\ \hline 0 & 0 & \frac{26}{5} & 0 & 0 & 1 & 60 \end{array} $$ We observe that all entries in the bottom row (ignoring the constant column) are non-negative. Hence, the tableau is in final form.
02

Obtain the variable values and find the solution.

Since the tableau is in final form, we can obtain the variable values for the associated linear programming problem directly from the tableau. Variables that have a 1 in the row corresponding to a basic variable (which are \(x\) and \(y\) in this case) and 0 in the other rows of the same column are assigned the value in the constant column of that row. All other variables get assigned a value of 0. From the tableau, we get the following values for the variables: - \(x = 30\) - \(y = 10\) - \(z = 0\) - \(u = 0\) - \(v = 0\) Therefore, the solution to the regular linear programming problem is: \((x, y, z, u, v) = (30, 10, 0, 0, 0)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Programming
Linear programming (LP) is a mathematical method for determining the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. Its typical function is to maximize or minimize a linear objective function, subject to a set of linear equality or inequality constraints.

LP is widely used in various fields, including economics, business, engineering, and military applications, for various optimization problems. To solve an LP problem, one needs to establish an objective function, define the constraints, and then find the optimal solution that either maximizes or minimizes the objective function while satisfying the constraints.
Pivot Element
The pivot element plays a crucial role in the implementation of the simplex method, an algorithm used to solve linear programming problems. During each iteration of this method, the pivot element is used to perform a series of operations that move the solution closer to the optimal.

A pivot is selected from the non-basic variable columns with positive coefficients in the objective function row. The chosen pivot should be in a column which would increase the objective function value when increased from zero. Then, to maintain the feasibility of the solution, the pivot row is chosen such that the ratios of the constants to the pivot column's positive entries are minimized. Once the pivot is selected, row operations are used to transform the pivot element into 1 and all other elements in its column to 0, leading to the next iteration of the tableau.
Tableau Final Form
In the context of the simplex method, the tableau represents the system of equations corresponding to the linear programming problem. The final form of the simplex tableau signifies that the optimal solution has been reached.

A tableau is in its final form when all the entries in the bottom row (objective function row), except the last one (which holds the value of the objective function), are non-negative. This indicates there's no other adjacent feasible solution that could provide a better value of the objective function. At this point, if we are aiming to maximize our objective, we can read off the values for the variables directly from the columns with unit coefficients and proceed to interpret the solution.
Optimization
Optimization involves finding the 'best available' values of a function within a given domain, which in the case of linear programming, means identifying the maximum or minimum value of the objective function subject to constraints.

The simplex method is a systematic approach used to solve optimization problems within LP. It iteratively moves through feasible solutions to find the optimal one. The method starts with a feasible solution, generally at a corner or boundary of the feasible region, and walks along the edges of the feasible region to the vertex that optimizes the objective function. When the tableau is in its final form, indicating that there are no adjacent vertices that will improve the value of the objective function, it is concluded that the optimal solution has been reached.

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Most popular questions from this chapter

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{r} x+y \leq 6 \\ 0 \leq x \leq 3 \\ y \geq 0 \end{array} $$

A company manufactures two products, \(\mathrm{A}\) and \(\mathrm{B}\), on two machines, 1 and II. It has been determined that the company will realize a profit of $$\$ 3 $$ unit of product \(A\) and a profit of $$\$ 4 $$ unit of product \(\mathrm{B}\). To manufacture a unit of product A requires 6 min on machine \(\mathrm{I}\) and 5 min on machine II. To manufacture a unit of product \(\mathrm{B}\) requires \(9 \mathrm{~min}\) on machine \(\mathrm{I}\) and 4 min on machine II. There are \(5 \mathrm{hr}\) of machine time available on machine I and \(3 \mathrm{hr}\) of machine time available on machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit? What is the optimal profit?

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. $$ \begin{array}{rrrrr|c} x & y & u & v & P & \text { Constant } \\ \hline 1 & 1 & 1 & 0 & 0 & 6 \\ 1 & 0 & -1 & 1 & 0 & 2 \\ \hline 3 & 0 & 5 & 0 & 1 & 30 \end{array} $$

The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gallons of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $$\$ 300$$ and that for pipeline water is $$\$ 500$$, how much water should the manager get from each source to minimize daily water costs for the city?

Construct the dual problem associated with the primal problem. Solve the primal problem. $$ \begin{aligned} \text { Minimize } & C=3 x+2 y \\ \text { subject to } & 2 x+3 y \geq 90 \\ & 3 x+2 y \geq 120 \\ & x \geq 0, y \geq 0 \end{aligned} $$
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