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Introduction to Management Science, 11e (Taylor) Module A: The Simplex Solution Method 1) The simplex method cannot be used to solve quadratic programming problems. Answer: TRUE Diff: 3 Section Heading: Converting the Model into Standard Form Keywords: simplex method 2) The simplex method is a general mathematical solution technique for solving linear programming problems. Answer: TRUE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simplex method 3) In the simplex method, the model is put into the form of a table, and then a number of mathematical steps are performed on the table. Answer: TRUE Diff: 3 Section Heading: Converting the Model into Standard Form Keywords: simplex method 4) The simplex method can be used to solve quadratic programming problems. Answer: FALSE Diff: 3 Section Heading: Converting the Model into Standard Form Keywords: simplex method 5) The simplex method is a general mathematical solution technique for solving nonlinear programming problems. Answer: FALSE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simplex method 6) The simplex method moves from one better solution to another until the best one is found, and then it stops. Answer: TRUE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simplex method 7) The mathematical steps in the simplex method replicate the process in graphical analysis of moving from one extreme point on the solution boundary to another. Answer: TRUE Diff: 1 Section Heading: Converting the Model into Standard Form 1 Copyright Š 2013 Pearson Higher Education, Inc. Publishing as Prentice Hall


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Keywords: simplex method 8) The first step in solving a linear programming model manually with the simplex method is to convert the model into standard form. Answer: TRUE Diff: 1 Section Heading: Converting the Model into Standard Form Keywords: standard form, simplex method 9) The last step in solving a linear programming model manually with the simplex method is to convert the model into standard form. Answer: FALSE Diff: 1 Section Heading: Converting the Model into Standard Form Keywords: standard form, simplex method 10) Slack variables are added to constraints and represent unused resources. Answer: TRUE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: slack variables, simplex method 11) Artificial variables are added to constraints and represent unused resources. Answer: FALSE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: artificial variables, slack variables, simplex method 12) A basic feasible solution satisfies the model constraints and has the same number of variables with non-negative values as there are constraints. Answer: TRUE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: basic feasible solution 13) A basic feasible solution satisfies the model constraints and has the same number of variables with negative values as there are constraints. Answer: FALSE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: basic feasible solution 14) Row operations are used to solve simultaneous equations where equations are multiplied by constants and added or subtracted from each other. Answer: TRUE Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simultaneous equations, row operations 2 Copyright Š 2013 Pearson Higher Education, Inc. Publishing as Prentice Hall


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15) The basic feasible solution in the initial simplex tableau is the origin where all decision variables equal zero. Answer: TRUE Diff: 1 Section Heading: The Simplex Method Keywords: basic feasible solution, initial simplex tableau 16) At the initial basic feasible solution at the origin, only slack variables have a value greater than zero. Answer: TRUE Diff: 2 Section Heading: The Simplex Method Keywords: basic/initial basic feasible solution, slack variables 17) At the initial basic feasible solution at the origin, only slack variables have a value greater than 1. Answer: FALSE Diff: 2 Section Heading: The Simplex Method Keywords: basic/initial basic feasible solution, slack variables 18) In using the simplex method, the number of basic variables is equal to the number of constraints. Answer: TRUE Diff: 2 Section Heading: The Simplex Method Keywords: basic feasible solution, constraints 19) The simplex method does not guarantee an integer solution. Answer: TRUE Diff: 1 Section Heading: The Simplex Method Keywords: simplex method 20) 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 Answer: TRUE Diff: 2 Section Heading: The Simplex Method Keywords: simplex method 21) 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. Answer: FALSE Diff: 3 Section Heading: The Simplex Method Keywords: simplex method

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22) 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. Answer: TRUE Diff: 3 Section Heading: The Simplex Method Keywords: simplex method 23) Final tableaus cannot be used to conduct sensitivity analysis. Answer: FALSE Diff: 1 Section Heading: The Simplex Method Keywords: simplex method 24) The dual form of a linear program is used to determine how much one should pay for additional resources. Answer: TRUE Diff: 1 Section Heading: The Simplex Method Keywords: dual 25) Multiple optimal solutions cannot be determined from the simplex method. Answer: FALSE Diff: 1 Section Heading: The Simplex Method Keywords: simplex method 26) The theoretical limit on the number of decision variables that can be handled by the simplex method is 50. Answer: FALSE Diff: 1 Section Heading: The Simplex Method Keywords: simplex method 27) The ________ column is the column corresponding to the entering variable. Answer: pivot Diff: 2 Keywords: pivot column 28) The variable with the largest positive cj - zj is the ________ variable. Answer: entering Diff: 2 Keywords: entering variable 29) ________ variables are added to constraints and represent unused resources. Answer: Slack Diff: 2 Keywords: slack variables

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30) The first step in solving a linear programming model manually with the simplex method is to convert the model into ________ form. Answer: standard Diff: 2 Keywords: standard form 31) The ________ values are contribution to profit for each variable. Answer: cj Diff: 2 Keywords: cj values, contribution to profit. 32) The ________ values are computed by multiplying the cj column values by the variable column values and summing. Answer: zj Diff: 2 Keywords: zj values 33) 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. Answer: artificial Diff: 2 Keywords: artificial variables 34) In solving a minimization problem, artificial variables are assigned a ________ in the objective function to eliminate them from the final solution. Answer: large cost Diff: 2 Keywords: artificial variables 35) 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. Answer: infeasible Diff: 2 Keywords: infeasible problem, infeasible solution 36) In using the simplex method, ________ optimal solutions are identified by cj - zj = 0 for a non-basic variable. Answer: multiple or alternative Diff: 2 Keywords: alternative optimal solutions, multiple optimal solutions 37) A primal maximization model with ≤ constraints converts to a ________ minimization model with constraints. Answer: dual Diff: 2 Keywords: dual model

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38) The quantity values on the right-hand side of the primal inequality constraints are the ________ coefficients in the dual. Answer: objective function Diff: 2 Keywords: dual model 39) If the primal problem has three constraints, then the corresponding dual problem will have three ________. Answer: decision variables Diff: 2 Keywords: dual model 40) Whereas the maximization primal model has ≤ constraints, the ________ dual model has ≥ constraints. Answer: minimization Diff: 1 Keywords: dual model 41) ________ 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). Answer: Degeneracy Diff: 2 Keywords: degeneracy 42) In a ________ problem, artificial variables are assigned a very high cost. Answer: minimization Diff: 1 Keywords: artificial variables 43) A(n) ________ problem can be identified in the simplex procedure when it is not possible to select a pivot row. Answer: unbounded Diff: 2 Keywords: simplex irregularity, unbounded solution 44) The ________ form of a linear program is used to determine how much one should pay for additional resources. Answer: dual Diff: 2 Keywords: dual 45) To determine the sensitivity range for the coefficient of a variable in the objective function, calculations are performed such that all values in the cj - zj row are ________. Answer: less than or equal to zero Diff: 2 Keywords: sensitivity analysis

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46) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What are the basic variables in the initial tableau? Answer: S1, S2 Diff: 1 Section Heading: The Simplex Method Keywords: basic variables, initial tableau 47) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What are the Cj values for the basic variables? Answer: 0, 0 Diff: 1 Section Heading: The Simplex Method Keywords: basic variables, objective function coefficients 48) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What is the (Cj - Zj) value for S1 at the initial solution? Answer: 0 Diff: 1 Section Heading: The Simplex Method Keywords: Cj - Zj values 49) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What is the (Ci- Zi) value for S2 at the initial solution? Answer: 0 Diff: 1 Section Heading: The Simplex Method Keywords: Cj - Zj values

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50) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What is the value of X1 in the final tableau? Answer: 0 or 4.25 Diff: 2 Section Heading: The Simplex Method Keywords: simplex method, simplex tableaus 51) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What is the value of x2 in the final tableau? Answer: 2 or 7.667 Diff: 2 Section Heading: The Simplex Method Keywords: simplex method, simplex tableaus 52) Solve the following problem using the simplex method. Minimize Z = 3x1 + 4x2 + 8x3 Subject to: 2x1 + x2 ≥ 6 x2 + 2x3 ≥ 4 x1, x2 ≥ 0 Answer: x1 = 1, x2 = 4, x3 = 0 and Z = 19 Diff: 3 Section Heading: The Simplex Method Keywords: simplex method, simplex tableaus 53) Solve the following problem using the simplex method. Minimize Z = 2x1 + 6x2 Subject to: 2x1 + 4x2 ≤ 12 3x1 + 2x2 ≥ 9 x1, x2 ≥ 0 Answer: x1 = 1.5, x2 = 2.25, and Z = 16.5 Diff: 3 Section Heading: The Simplex Method Keywords: simplex method, simplex tableaus

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54) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 What is the optimal value of this objective function? Answer: 23 Diff: 2 Section Heading: The Simplex Method Keywords: objective function value, simplex tableaus 55) Given the following linear programming problem: maximize 4x1 + 3x2 subject to 4x1 + 3x2 ≤ 23 5x1 - x2 ≤ 5 x1, x2 ≥ 0 How many iterations did we have to perform before reaching the final tableau? Answer: 3 Diff: 3 Section Heading: The Simplex Method Keywords: simplex tableaus, simplex iterations 56) Given the following linear programming problem: maximize Z = $100x1 + 80x2 subject to x1 + 2x2 ≤ 40 3x1 + x2 ≤ 60 x1, x2 ≥ 0 Using the simplex method, what is the optimal value for X1? Answer: 16 Diff: 2 Section Heading: The Simplex Method Keywords: simplex tableaus, simplex iterations 57) Given the following linear programming problem: maximize Z = $100x1 + 80x2 subject to x1 + 2x2 ≤ 40 3x1 + x2 ≤ 60 x1, x2 ≥ 0 Using the simplex method, what is the optimal value for X2? Answer: 12 Diff: 2 Section Heading: The Simplex Method Keywords: simplex tableaus, simplex iterations

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58) Given the following linear programming problem: maximize Z = $100x1 + 80x2 subject to x1 + 2x2 ≤ 40 3x1 + x2 ≤ 60 x1, x2 ≥ 0 Using the simplex method, what is the value for S2 in the optimal tableau? Answer: 0 Diff: 2 Section Heading: The Simplex Method Keywords: simplex tableaus, simplex iterations 59) Given the following linear programming problem: maximize Z = $100x1 + 80x2 subject to x1 + 2x2 ≤ 40 3x1 + x2 ≤ 60 x1, x2 ≥ 0 Using the simplex method, what is the optimal value for the objective function? Answer: $2560 Diff: 2 Section Heading: The Simplex Method Keywords: objective function value 60) Given the following linear programming problem: maximize Z = $100x1 + 80x2 subject to x1 + 2x2 ≤ 40 3x1 + x2 ≤ 60 x1, x2 ≥ 0 Using the simplex method, what is the value for S1 in the final basic feasible solution? Answer: 0 Diff: 2 Section Heading: The Simplex Method Keywords: slack variables, simplex iterations

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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

Objective Coefficient Ranges

Right Hand Side Ranges

61) How much space will be left unused? Answer: 0 Diff: 1 Section Heading: The Simplex Method 11 Copyright © 2013 Pearson Higher Education, Inc. Publishing as Prentice Hall


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Keywords: computer output of linear programming method, slack variables

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62) How much time will be used? Answer: 57 Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear programming problems, slack variables 63) By how much will the second marketing restriction be exceeded? Answer: 0 Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear programming problems, slack variables 64) What is the profit? Answer: 7475 Diff: 2 Section Heading: The Simplex Method Keywords: comp output of linear prog problems, objective function value 65) To what value can the profit on red nail polish drop before the solution would change? Answer: 87.5 Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear prog problems, sensitivity analysis 66) By how much can the profit on green nail polish increase before the solution would change? Answer: 12.5 Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear prog problems, sensitivity analysis 67) By how much can the amount of space decrease before there is a change in the profit? Answer: 0 Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear prog problems, sensitivity analysis 68) You are offered the chance to obtain more space. The offer is for 15 units and the total price is 1500. What should you do? Answer: Reject the offer. Diff: 2 Section Heading: The Simplex Method Keywords: computer output of linear prog problems, sensitivity analysis

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69) Consider the following linear programming problem: MAX s.t.

Z = 10x1 + 30x2 4x1 + 6x2 ≤ 12 8x1 + 4x2 ≤ 16

Use the two tables below to create the initial tableau and perform 1 pivot.

Answer:

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Diff: 2 Section Heading: The Simplex Method Keywords: simplex procedure 70) Consider the following linear programming problem and the corresponding final tableau. MAX s.t.

Z = 3x1 + 5x2 x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≥ 18

What is the shadow price for each constraint? Answer: constraint 1, 3; constraint 2, 2.5; constraint 3, 0 Diff: 2 Section Heading: The Simplex Method Keywords: sensitivity analysis, shadow price

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71) Consider the following linear programming problem and the corresponding final tableau. MAX s.t.

Z = 3x1 + 5x2 x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≥ 18

What is the sensitivity range for the first constraint? Answer: maximum decrease of 2, and an infinite increase Diff: 2 Section Heading: The Simplex Method Keywords: sensitivity analysis, quantity ranges for constraints 72) Write the dual form of the following linear program. MAX s.t.

Z = 3x1 + 5x2 x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≥ 18

Answer: MIN Zd = 4y1 + 12y2 + 18y3 s.t. y1 + 3y3 ≥ 3 2y2 + 2y3 ≥ 5 Diff: 2 Section Heading: The Simplex Method Keywords: dual form

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73) The simplex method ________ be used to solve quadratic programming problems. A) can B) cannot C) may D) should Answer: B Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simplex method 74) The simplex method is a general mathematical solution technique for solving ________ programming problems. A) integer B) non-linear C) linear D) A, B, and C Answer: C Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: simplex method 75) Slack variables are added to ________ constraints and represent unused resources. A) ≤ B) < C) ≥ D) > E) = Answer: A Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: slack variables 76) The ________ step in solving a linear programming model manually with the simplex method is to convert the model into standard form. A) first B) second C) last D) only Answer: A Diff: 2 Section Heading: Converting the Model into Standard Form Keywords: standard form, simplex method

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77) Row operations are used to solve simultaneous equations where equations are ________ by constants and added to or subtracted from each other. A) converted B) restrained C) divided D) multiplied Answer: D Diff: 3 Section Heading: Converting the Model into Standard Form Keywords: row operations, simultaneous equations 78) The basic feasible solution in the initial simplex tableau is the origin where all decision variables equal: A) 0 B) 1 C) -1 D) 1 or -1 Answer: A Diff: 3 Section Heading: The Simplex Method Keywords: basic feasible solution, initial simplex tableau 79) At the initial basic feasible solution at the origin, only slack variables have a value greater than: A) 0 B) 1 C) -1 D) 1 or -1 Answer: A Diff: 3 Section Heading: The Simplex Method Keywords: basic feasible solution, initial simplex tableau 80) At the initial basic feasible solution at the origin, only ________ variables have a value greater than zero. A) linear B) slack C) non-linear D) integer Answer: B Diff: 1 Section Heading: The Simplex Method Keywords: basic feasible solution

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81) The leaving variable is determined by ________ the quantity values ________ the pivot column values and selecting the minimum possible value or zero. A) adding, to B) multiplying, by C) dividing, by D) subtracting, from Answer: C Diff: 2 Section Heading: The Simplex Method Keywords: simplex tableau, leaving variable 82) The leaving variable is determined by dividing the quantity values by the pivot column values and selecting the A) maximum positive value. B) minimum negative value. C) minimum positive value. D) maximum negative value. Answer: C Diff: 3 Section Heading: The Simplex Method Keywords: simplex tableau, leaving variable 83) The simplex method ________ guarantee integer solutions. A) sometimes does B) does C) does not D) may Answer: C Diff: 2 Section Heading: The Simplex Method Keywords: simplex method 84) For a maximization linear programming problem, a(n) ________ is ________ for a less-than-orequal-to constraint. A) surplus, subtracted B) slack, added C) artificial, added D) artificial, subtracted E) surplus, added Answer: B Diff: 2 Section Heading: The Simplex Method Keywords: slack variables

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85) The objective function coefficient of an artificial variable for a minimization linear programming problem is: A) +M B) -M C) 0 D) 1 E) an arbitrary value between 0 and positive infinity Answer: A Diff: 2 Section Heading: The Simplex Method Keywords: artificial variables 86) If a slack variable has a positive value (is basic) in the optimal solution to a linear programming problem, then the shadow price of the associated constraint A) is always zero. B) is always greater than zero. C) is always less than zero. D) could be any value (i.e., zero greater than zero or less than zero). Answer: A Diff: 2 Section Heading: The Simplex Method Keywords: slack variable, shadow price 87) In the simplex procedure, if cj - zj = 0 for a non-basic variable, this indicates that A) the solution is infeasible. B) the solution is unbounded. C) there are multiple optimal solutions. D) the formulation is incorrect. Answer: C Diff: 2 Section Heading: The Simplex Method Keywords: simplex irregularity, multiple optimal solutions 88) In the simplex procedure, if it is not possible to select a pivot row, this indicates that A) the solution is infeasible. B) the solution is unbounded. C) there are multiple optimal solutions. D) the formulation is incorrect. Answer: B Diff: 2 Section Heading: The Simplex Method Keywords: simplex irregularity, unbounded solution

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89) In the simplex procedure, if all cj - zj â&#x2030;¤ 0 and one or more of the basic variables are artificial, this indicates that A) the solution is infeasible. B) the solution is unbounded. C) there are multiple optimal solutions. D) the formulation is incorrect. Answer: A Diff: 2 Section Heading: The Simplex Method Keywords: simplex irregularity, infeasible solution 90) The ________ form of a linear program is used to determine how much one should pay for additional resources. A) standard B) primal C) feasible D) dual E) simplex Answer: D Diff: 2 Section Heading: The Simplex Method Keywords: dual

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