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Exam 17: the Simplex Solution Method

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Question 1

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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?

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The simplex method does not guarantee an integer solution.

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Artificial variables are added to constraints and represent unused resources.

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

________ 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).

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

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

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

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

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

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.

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.

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

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.

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

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

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

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?