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Simplex Method: Practice Questions and Answers, Exams of Mathematics

Walden UniversityMathematics

A series of true/false and multiple-choice questions related to the simplex method in linear programming. It covers topics such as basic feasible solutions, simplex tableau setup, optimality conditions, and special cases like infeasibility and artificial variables. The questions are designed to test understanding of the algebraic overview of the simplex method and its application in solving optimization problems. It is useful for students studying operations research, management science, or related fields. Questions that assess comprehension of key concepts and procedures within the simplex method, such as identifying basic variables, understanding the role of unit columns, and interpreting the cj − zj row. It also covers topics like setting up the initial simplex tableau, improving solutions, and handling special cases like infeasibility and minimization problems. The questions are categorized by difficulty level and aligned with specific learning objectives and national standards.

Typology: Exams

2024/2025

Available from 08/28/2025

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CH 17 - Linear Programming: Simplex Method
Page 1
True / False
1. When a system of simultaneous equations has more variables than equations, there is a unique solution.
a.
True
b.
False
ANSWER:
POINTS:
DIFFICULTY:
LEARNING OBJECTIVES:
NATIONAL STANDARDS:
TOPICS:
KEYWORDS:
2. In order to determine a basic solution, set n − m of the variables equal to zero, and solve the m linear
constraint equations for the remaining m variables.
a.
True
b.
False
ANSWER:
POINTS:
DIFFICULTY:
LEARNING OBJECTIVES:
NATIONAL STANDARDS:
TOPICS:
KEYWORDS:
3. A basic feasible solution also satisfies the nonnegativity restriction.
a.
True
b.
False
ANSWER:
POINTS:
DIFFICULTY:
LEARNING OBJECTIVES:
NATIONAL STANDARDS:
TOPICS:
KEYWORDS:
4. The simplex method is an iterative procedure for moving from one basic feasible solution (an extreme point)
to another until the optimal solution is reached.
a.
True
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Download Simplex Method: Practice Questions and Answers and more Exams Mathematics in PDF only on Docsity!

True / False

  1. When a system of simultaneous equations has more variables than equations, there is a unique solution. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method KEYWORDS: Bloom's: Remember
  2. In order to determine a basic solution, set n − m of the variables equal to zero, and solve the m linear constraint equations for the remaining m variables. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method KEYWORDS: Bloom's: Understand
  3. A basic feasible solution also satisfies the nonnegativity restriction. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method KEYWORDS: Bloom's: Understand
  4. The simplex method is an iterative procedure for moving from one basic feasible solution (an extreme point) to another until the optimal solution is reached. a. True

b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method KEYWORDS: Bloom's: Remember

  1. In a simplex tableau, a variable is associated with each column and both a constraint and a basic variable are associated with each row. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.03 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.3 Setting up the Initial Simplex Tableau KEYWORDS: Bloom's: Remember
  2. At each iteration of the simplex procedure, a new variable becomes basic and a currently basic variable becomes nonbasic, preserving the same number of basic variables and improving the value of the objective function. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.04 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.4 Improving the Solution KEYWORDS: Bloom's: Understand
  3. Coefficients in a nonbasic column in a simplex tableau indicate the amount of decrease in the current basic variables when the value of the nonbasic variable is increased from 0 to 1. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Moderate

a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.06 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.6 Tableau Form: The General Case KEYWORDS: Bloom's: Understand

  1. A solution is optimal when all values in the cj − zj row of the simplex tableau are either zero or positive. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Understand
  2. The variable to enter into the basis is the variable with the largest positive cj − zj value. a. True b. False ANSWER: True POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.07 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.7 Solving a Minimization Problem KEYWORDS: Bloom's: Understand
  3. The variable to remove from the current basis is the variable with the smallest positive cj − zj value. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.07 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking

TOPICS: 17.7 Solving a Minimization Problem KEYWORDS: Bloom's: Understand

  1. The coefficient of an artificial variable in the objective function is zero. a. True b. False ANSWER: False POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.06 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.6 Tableau Form: The General Case KEYWORDS: Bloom's: Understand Multiple Choice
  2. Algebraic methods such as the simplex method are used to solve a. nonlinear programming problems. b. any size linear programming problem. c. programming problems under uncertainty. d. graphical models. ANSWER: b POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method KEYWORDS: Bloom's: Understand
  3. A basic feasible solution is a basic solution that a. also satisfies the nonnegativity conditions. b. contains 0 variables. c. corresponds to no extreme points. d. None of these are correct. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.01 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.1 An Algebraic Overview of the Simplex Method
  1. The values in the cj − zj , or net evaluation, row indicate the a. value of the objective function. b. decrease in value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. c. net change in the value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. d. values of the decision variables. ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.03 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.3 Setting up the Initial Simplex Tableau KEYWORDS: Bloom's: Understand
  2. The purpose of the tableau form is to provide a(n) a. infeasible solution. b. optimal infeasible solution. c. initial basic feasible solution. d. degenerate solution. ANSWER: c POINTS: 1 DIFFICULTY: Easy LEARNING OBJECTIVES: IMS.ASWC.19.17.03 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.3 Setting up the Initial Simplex Tableau KEYWORDS: Bloom's: Understand
  3. Which of the following is NOT a step that is necessary to prepare a linear programming problem for solution using the simplex method? a. Formulate the problem. b. Set up the standard form by adding slack and/or subtracting surplus variables. c. Perform elementary row and column operations. d. Set up the tableau form. ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.02 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.2 Tableau Form KEYWORDS: Bloom's: Understand
  1. In the simplex method, a tableau is optimal only if all the cj − zj values are a. zero or negative. b. zero. c. negative and nonzero. d. positive and nonzero. ANSWER: a POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Understand
  2. The way we guarantee that artificial variables will be eliminated before the optimal solution is reached is to assign each artificial variable the coefficient value of a. 0. b. 1. c. a very large negative number. d. a very large positive number. ANSWER: c POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.06 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.6 Tableau Form: The General Case KEYWORDS: Bloom's: Understand
  3. If one or more of the basic variables in a linear program have a value of zero, a. post-optimality analysis is required. b. their dual prices will be equal. c. converting the pivot element will break the tie. d. a condition of degeneracy is present. ANSWER: d POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.08 - 17. NATIONAL STANDARDS: United States - BUSPROG: Reflective Thinking TOPICS: 17.8 Special Cases KEYWORDS: Bloom's: Understand
  1. Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = 3x 1 + 8x 2 s.t. x 1 + x 2 ≤ 200 x 1 ≤ 80 x 2 ≤ 60 ANSWER: Min Z = 3x 1 + 8x 2 + Ma 1 + Ma 2 s.t. x 1 + x 2 + a 1 = 200 x 1 + s 1 = 80 x 2 + s 2 + a 2 = 60 The initial basis includes a 1 , a 2 , and s 2. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.06 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.6 Tableau Form: The General Case KEYWORDS: Bloom's: Create
  2. An initial simplex tableau is shown below. x 1 x 2 x 3 s 1 s 2 s 3 Basis cB 5 8 12 0 0 0 s 1 0 3 4 5 1 0 0 80 s 2 0 9 15 20 0 1 0 250 s 3 0 1 −1 2 0 0 0 20 zj 0 0 0 0 0 0 0 cj − zj 5 8 12 0 0 0 a. What variables form the basis? b. What are the current values of the decision variables? c. What is the current value of the objective function? d. Which variable will be made positive next, and what will its value be? e. Which variable that is currently positive will become 0? f. What value will the objective function have next? ANSWER: a. s 1 , s 2 , s 3 b. x 1 = 0, x 2 = 0, x 3 = 0, s 1 = 80, s 2 = 250, s 3 = 20 c. 0 d. x 3 , 10 e. s 3 f. z = 120 POINTS: 1 DIFFICULTY: Challenging

LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17.

NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Evaluate

  1. A simplex tableau is shown below. x 1 x 2 x 3 s 1 s 2 Basis cB 3 4 5 0 0 1/2 1 0 1/2 −1/2 6 0 0 1 −1/4 1 3 cj cj − zj a. What variables form the basis? b. What are the current values of the decision variables? c. What is the current value of the objective function? d. Which variable will be made positive next, and what will its value be? e. Which variable that is currently positive will become 0? f. What value will the objective function have next? ANSWER: a. x 2 , x 3 b. x 1 = 0, x 2 = 6, x 3 = 3 c. 39 d. x 1 , 12 e. x 2 f. 51 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Evaluate
  2. A simplex tableau is shown below. x 1 x 2 x 3 s 1 s 2 s 3 Basis cB 3 5 8 0 0 0 s 1 0 3 6 0 1 0 −9 126 s 2 0 −5/2 −1/2 0 0 1 −9/2 45 x 3 8 1/2 1/2 1 0 0 1/2 18 zj 4 4 8 0 0 4 144 cj − zj −1 1 0 0 0 − a. Perform one more iteration of the simplex procedure. b. What is the current complete solution?

DIFFICULTY: Challenging

NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Evaluate

  • POINTS:
  • LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17.
  • Max 14x 1 + 14.5x 2 + 18x 35. Solve the following problem by the simplex method.
  • s.t. x 1 + 2x 2 + 2.5x 3 ≤
    • x 1 + x 2 + 1.5x 3 ≤
      • x 1 , x 2 , x 3 ≥ - x 1 x 2 x 3 s 1 s ANSWER:
        • Basis cB 14 14.5
          • s 1 0 1 2 2.5
          • s 2 0 1 1 1.5 - zj
            • cj − zj 14 14.5 - x 1 x 2 x 3 s 1 s
        • Basis cB 14 14.5
          • x 3 18 0.4 0.8 1 0.4
          • s 2 0 0.4 −0.2 0 −0.6 - zj 7.20 14.4 18 7.2
            • cj − zj 6.8 0.1 0 −7.2 - x 1 x 2 x 3 s 1 s
        • Basis cB 14 14.5
          • x 3 18 0 1 1 1 −1
          • x 1 14 1 −0.5 0 −1.5 2.5 - zj 14 11 18 −3
            • cj − zj 0 3.5 0 −3 − - x 1 x 2 x 3 s 1 s
        • Basis cB 14 14.5
          • x 2 14.5 0 1 1 1 −1
          • x 1 14 1 0 0.5 −1 - zj 14 14.5 21.5 0.5 13.5
            • cj − zj 0 0 −3.5 −0.5 −13.
  • POINTS:
  • LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. DIFFICULTY: Moderate

NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Evaluate

  1. Solve the following problem by the simplex method. Max 100x 1 + 120x 2 + 85x 3 s.t. 3x 1 + 1x 2 + 6x 3 ≤ 120 5x 1 + 8x 2 + 2x 3 ≤ 160 x 1 , x 2 , x 3 ≥ 0 ANSWER: x 1 x 2 x 3 s 1 s 2 Basis cB 100 120 85 0 0 s 1 0 3 1 6 1 0 120 s 2 0 5 8 2 0 1 160 zj 0 0 0 0 0 0 cj − zj 100 120 85 0 0 x 1 x 2 x 3 s 1 s 2 Basis cB 100 120 85 0 0 s 1 0 2.375 0 5.75 1 −0.125 100 x 2 120 0.625 1 0.25 0 0.125 20 zj 75 120 30 0 15 2400 cj − zj 25 0 55 0 − x 1 x 2 x 3 s 1 s 2 Basis cB 100 120 85 0 0 x 3 85 0.413 0 1 0.174 −0.0217 17. x 2 120 0.522 1 0 −0.043 0.1304 15. zj 97.745 120 85 9.63 13.8035 3356. cj − zj 2.283 0 0 −9.565 −13. x 1 x 2 x 3 s 1 s 2 Basis cB 100 120 85 0 0 x 3 85 0 −0.792 1 0.2083 −0.125 5 x 1 100 1 1.917 0 −0.0833 0.250 30 zj 100 124.38 85 9.3755 14.375 3425 cj − zj 0 −4.375 0 −9.375 −14. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Evaluate
  2. Determine from a review of the following tableau whether the linear programming problem has multiple

s.t. x 1 − 2x 2 + x 3 ≤ 11 −4 x 1 + x 2 + 2x 3 ≥ 3 2x 1 − x 3 ≥ − ANSWER: Min Z = −3x 1 + x 2 + x 3 + Ma 1 + Ma 2 s.t. x 1 − 2x 2 + x 3 + s 1 = 11 −4x 1 + x 2 + 2 x 3 − s 2 + a 1 = 3 −2x 1 + x 3 + a 2 = 1 The initial basis includes s 1 , a 1 , and a2. 2x 1 + 3.5x 2 − s 2 + a 1 = 60 2x 1 − 1x 2 + a 2 = 50 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.06 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.6 Tableau Form: The General Case KEYWORDS: Bloom's: Analyze

  1. Identify the type of solution shown in this simplex tableau. x 1 x 2 x 3 s 1 s 2 a 1 Basis cB 1 2 5 0 0 −M a 1 −M −3 −1 0 −1 −2 1 4 x 3 5 1 1/2 1 0 1/2 0 4 zj 5 + 3M 2.5 + M 5 M 2.5 + 2M −M −4M + 20 cj − zj −4 − 3M −0.5 − M 0 −M −2.5 − 2M 0 ANSWER: The tableau indicates an infeasible solution. POINTS: 1 DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.08 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.8 Special Cases KEYWORDS: Bloom's: Analyze
  2. What type of solution is shown in this simplex tableau? x 1 x 2 x 3 s 1 s 2 Basis cB 4 6 5 0 0 s 1 0 3 0 −1 1 −1 26 x 2 6 1 1 1 0 −1 10 zj 6 6 6 0 −6 60 cj − zj −2 0 −1 0 6 ANSWER: The tableau indicates an unbounded solution. POINTS: 1

DIFFICULTY: Moderate LEARNING OBJECTIVES: IMS.ASWC.19.17.08 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.8 Special Cases KEYWORDS: Bloom's: Analyze

  1. Joe Forrester, an operations analyst for a manufacturing company, developed the following LP formulation. From it, create an initial simplex tableau. Max 40xl + 30x 2 + 50x 3 s.t. 2xl + 3x 2 + 4x 3 < 200 x 1 + 2x 2 + 2x 3 < 300 3xl + x 2 + 5x 3 < 500 ANSWER: x 1 x 2 x 3 s 1 s 2 s 3 Basis cB 40 30 50 0 0 0 s 1 0 3 4 5 1 0 0 200 s 2 0 9 15 20 0 1 0 300 s 3 0 1 – 1 2 0 0 1 500 zj 0 0 0 0 0 0 0 cj – zj 40 30 50 0 0 0 POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.03 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.3 Setting up the Initial Simplex Tableau KEYWORDS: Bloom's: Create
  2. Jackie Quinn developed the following LP formulation for a problem she is working on and now needs to create an initial simplex tableau. Create the initial simplex tableau. Min 75xl + 45x 2 s.t. 3xl + 2x 2 > 10 xl + 6x 2 > 15 ANSWER: x 1 x 2 a 1 a 2 s 1 s 2 Basis cB – 75 – 45 – M – M 0 0 a 1 0 3 2 1 0 – 1 0 10 a 2 0 1 6 0 1 0 – 1 15 zj – 4M – 8M – M – M M M – 25M

Basis cB 550 350 0 0 0 s 1 0 0 1 1 – 1 0 500 x 1 550 1 1/2 0 1/2 0 750 s 3 0 0 3/2 0 – 1/2 1 2250 zj 550 275 0 275 0 412, cj – zj 0 75 0 – 275 0 ANSWER: Third tableau x 1 x 2 s 1 s 2 s 3 Basis cB 550 350 0 0 0 x 2 350 0 1 1 – 1 0 500 x 1 550 1 0 – 1/2 1 0 500 s 3 0 0 0 – 3/2 1 1 1500 zj 550 350 75 200 0 450, cj – zj 0 0 – 75 – 200 0 Solution is optimal. There are no positive cj – zj values. x 1 = 550, x 2 = 350, and objective function value is 450,000. POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Create

  1. An operations research analyst for a communications company has the following LP problem and wants to solve it using the simplex method. Max 50x 1 + 20x 2 s.t. 2x 1 + x 2 < 200 x 1 + x 2 < 350 xl + 2x 2 < 275 ANSWER: Initial tableau x 1 x 2 s 1 s 2 s 3 Basis cB 50 20 0 0 0 s 1 0 2 1 1 0 0 200 s 2 0 1 1 0 1 0 350 s 3 0 1 2 0 0 1 275 zj 0 0 0 0 0 0 cj – zj 50 20 0 0 0

Second tableau x 1 x 2 s 1 s 2 s 3 Basis cB 50 20 0 0 0 x 1 50 1 1/2 1/2 0 0 100 s 2 0 0 1/2 – 1/2 1 0 250 s 3 0 0 3/2 – 1/2 0 1 175 zj 50 25 25 0 0 5000 cj – zj 0 – 5 – 25 0 0 Tableau is optimal. Optimal solution is x 1 = 100, and the objective function =

POINTS: 1 DIFFICULTY: Challenging LEARNING OBJECTIVES: IMS.ASWC.19.17.05 - 17. NATIONAL STANDARDS: United States - BUSPROG: Analytic TOPICS: 17.5 Calculating the Next Tableau KEYWORDS: Bloom's: Create

  1. The operations research analyst for a big manufacturing firm in Oregon developed the following variable definitions for an LP maximization problem she was working on. The company was trying to determine how many consoles of each model to produce next week given each console had to go through three production departments. She obtained the following optimal simplex tableau for the problem and wanted to interpret its meaning. Variable definitions: xl = number of model 1 consoles produced x 2 = number of model 2 consoles produced s 1 = unused personnel hours in department 1 s 2 = unused personnel hours in department 2 s 3 = unused personnel hours in department 3 objective function = total profit on model 1 and model 2 consoles produced in the coming week Optimal tableau x 1 x 2 s 1 s 2 s 3 Basis cB 180 350 0 0 0 x 1 180 1 0 1 0 – 1 15 s 2 0 0 0 – 1 1 1 10 x 2 350 0 1 0 0 1 20 zj 180 350 180 0 170 9700 cj – zj 0 0 – 180 0 – 170 ANSWER: Optimal solution: Produce 15 model 1 consoles, produce 20 model 2 consoles, all personnel hours will be used in departments 1 and 3, 10 personnel hours will be unused in department 2, and