Exam 17: the Simplex Solution Method

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions. MAX    100x1 + 120x2 + 150x3 + 125x4 Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108       2. 3x1 + 5x2 + x4 ? 120       3. x1 + x3 ? 25       4. x2 + x3 + x4 > 50       x1, x2, x3, x4 ? 0 Optimal Solution: Objective Function Value = 7475.000 Variable Value Reduced Costs X1 8 0 X2 0 5 X3 17 0 X4 33 0 Constraint Slack/Surplus Dual Prices 1 0 75 2 63 0 3 0 25 4 0 -25 Objective Coefficient Ranges Variable Lower Limit Current Value Upper Limit X1 87.5 100 none X2 none 120 125 X3 125 150 162 X4 120 125 150 Right Hand Side Ranges Constraint Lower Limit Current Value Upper Limit 1 100 108 123.75 2 57 120 none 3 8 25 58 4 41.5 50 54 -What is the profit?

The linear programming problem whose output follows determines how many red nail polishes, blue nail polishes, green nail polishes, and pink nail polishes a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions. MAX    100x1 + 120x2 + 150x3 + 125x4 Subject to    1. x1 + 2x2 + 2x3 + 2x4 ? 108       2. 3x1 + 5x2 + x4 ? 120       3. x1 + x3 ? 25       4. x2 + x3 + x4 > 50       x1, x2, x3, x4 ? 0 Optimal Solution: Objective Function Value = 7475.000 Variable Value Reduced Costs X1 8 0 X2 0 5 X3 17 0 X4 33 0 Constraint Slack/Surplus Dual Prices 1 0 75 2 63 0 3 0 25 4 0 -25 Objective Coefficient Ranges Variable Lower Limit Current Value Upper Limit X1 87.5 100 none X2 none 120 125 X3 125 150 162 X4 120 125 150 Right Hand Side Ranges Constraint Lower Limit Current Value Upper Limit 1 100 108 123.75 2 57 120 none 3 8 25 58 4 41.5 50 54 -By how much can the profit on green nail polish increase before the solution would change?

Consider the following linear programming problem and the corresponding final tableau. MAX =3+5 s.t. \leq4 2\leq12 3+2\geq18 What is the sensitivity range for the first constraint?

The ________ form of a linear program is used to determine how much one should pay for additional resources.

In the simplex procedure, if all cj - zj ≤ 0 and one or more of the basic variables are artificial, this indicates that

Given the following linear programming problem: maximize 4+3 subject to 4+3\leq23 5-\leq5 ,\geq0 What is the (Cj - Zj) value for S1 at the initial solution?

The simplex method ________ be used to solve quadratic programming problems.

In using the simplex method, ________ optimal solutions are identified by cj - zj = 0 for a non-basic variable.

The leaving variable is determined by ________ the quantity values ________ the pivot column values and selecting the minimum possible value or zero.

A change in the objective function coefficient of a basic variable cannot change the value of zj for a non-basic variable in the final simplex tableau.

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

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

A primal maximization model with ≤ constraints converts to a ________ minimization model with constraints.

At the initial basic feasible solution at the origin, only slack variables have a value greater than 1.

A(n) ________ maximization linear programming problem has an artificial variable in the final simplex tableau where all cj - zj values are less than or equal to zero.

Given the following linear programming problem: maximize =\ 100+80 subject to +2\leq40 3+\leq60 ,\geq0 Using the simplex method, what is the value for S₁ in the final basic feasible solution?

The ________ variable allows for an initial basic feasible solution, but it has no meaning. Therefore, after we get the simplex tableau started, they are discarded in later iterations.

The ________ column is the column corresponding to the entering variable.

At the initial basic feasible solution at the origin, only slack variables have a value greater than:

Given the following linear programming problem: maximize =\ 100+80 subject to +2\leq40 3+\leq60 ,\geq0 Using the simplex method, what is the value for S₂ in the optimal tableau?