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Exam 17: the Simplex Solution Method

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If the primal problem has three constraints, then the corresponding dual problem will have three ________.

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Row operations are used to solve simultaneous equations where equations are multiplied by constants and added or subtracted from each other.

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Given the following linear programming problem: $\begin{array} { l l } \operatorname { maximize } & \mathrm { Z } = \$ 100 x _ { 1 } + 80 x _ { 2 } \\\text { subject to } & x _ { 1 } + 2 x _ { 2 } \leq 40 \\& 3 x _ { 1 } + x _ { 2 } \leq 60 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$ Using the simplex method, what is the optimal value for the objective function?

Write the dual form of the following linear program. $\begin{array} { l l } \text { MAX } & \mathrm { Z } = 3 x _ { 1 } + 5 x _ { 2 } \\\text { s.t. } & x _ { 1 } \leq 4 \\& 2 x _ { 2 } \leq 12 \\& 3 x _ { 1 } + 2 x _ { 2 } \geq 18\end{array}$

Solve the following problem using the simplex method. $\begin{array} { l } \text { Minimize } \mathrm { Z } = 2 x _ { 1 } + 6 x _ { 2 } \\\text { Subject to: } \quad 2 x _ { 1 } + 4 x _ { 2 } \leq 12 \\\qquad 3 x _ { 1 } + 2 x _ { 2 } \geq 9 \\x _ { 1 } , x _ { 2 } \geq 0\end{array}$

When solving a linear programming problem, a decision variable that leaves the basis in one iteration of the simplex method can return to the basis on a later iteration.

The leaving variable is determined by dividing the quantity values by the pivot column values and selecting the

In solving a linear programming problem with simplex method, the number of basic variables is the same as the number of constraints in the original problem

The simplex method is a general mathematical solution technique for solving linear programming problems.

Consider the following linear programming problem: $\begin{array} { l l } \text { MAX } & \mathrm { Z } = 10 x _ { 1 } + 30 x _ { 2 } \\\text { s.t. } & 4 x _ { 1 } + 6 x _ { 2 } \leq 12 \\& 8 x _ { 1 } + 4 x _ { 2 } \leq 16\end{array}$ Use the two tables below to create the initial tableau and perform 1 pivot.

Consider the following linear programming problem and the corresponding final tableau. $\begin{array} { l l } \text { MAX } & Z = 3 x _ { 1 } + 5 x _ { 2 } \\\text { s.t. } & x _ { 1 } \leq 4 \\& 2 x _ { 2 } \leq 12 \\& 3 x _ { 1 } + 2 x _ { 2 } \geq 18\end{array}$ What is the shadow price for each constraint?

The simplex method does not guarantee an integer solution.

Given the following linear programming problem: $\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$ What is the value of x2 in the final tableau?

The simplex method is a general mathematical solution technique for solving ________ programming problems.

________ in linear programming is when a basic variable takes on a value of zero (i.e., a zero in the right-hand side of the constraints of the tableau).

Given the following linear programming problem: $\begin{array} { l l } \operatorname { maximize } & 4 x _ { 1 } + 3 x _ { 2 } \\\text { subject to } & 4 x _ { 1 } + 3 x _ { 2 } \leq 23 \\& 5 x _ { 1 } - x _ { 2 } \leq 5 \\& x _ { 1 } , x _ { 2 } \geq 0\end{array}$ What is the (Ci- Zi) value for S2 at the initial solution?

Slack variables are added to constraints and represent unused resources.

The simplex method cannot be used to solve quadratic programming problems.

The ________ step in solving a linear programming model manually with the simplex method is to convert the model into standard form.

Artificial variables are added to constraints and represent unused resources.