Explanation The given data is: Time in hours per week Component A Component B Component C Maximum available time Fabrication 2 3 2 1000 Assembly 1 1 2 800 Profit $7 $8 $10 Note that the company must decide on the number of components each week that should be manufacture. So, the decision variables are; x 1 = Number of components A x 2 = Number of components B x 3 = Number of components C Total number of components produced each week cannot exceed 420. x 1 + x 2 + x 3 ≤ 420 The manufacturer's objective is to maximize profit: P = 7 x 1 + 8 x 2 + 10 x 3 The manufacturer is limited by the number of hours available for fabrication and assembly. These limitations lead to the following constraints: 2 x 1 + 3x 2 + 2x 3 ≤ 1000 Fabrication time constraint x 1 + x 2 + 2 x 3 ≤ 800 Assembly time constraint Adding the nonnegative constraints, the model for the linear programming problem is: Maximize P = 7 x 1 + 8 x 2 + 10 x 3 subject to 2 x 1 + 3x 2 + 2x 3 ≤ 1000 x 1 + x 2 + 2 x 3 ≤ 800 x 1 + x 2 + x 3 ≤ 420 x 1 , x 2 , x 3 ≥ 0 To convert the problem constraints into a system of equations slack variables is used. Since there are m = 2 problem constraints, so it needs two slack variables s 1 , s 2 to convert the inequality into equality. 2 x 1 + 3x 2 + 2x 3 + s 1 = 1000 x 1 + x 2 + 2 x 3 +s 2 = 800 x 1 + x 2 + x 3 +s 3 = 420 − 7 x 1 − 8 x 2 − 10 x 3 + P = 0 Form the simplex tableau: x 1 x 2 x 3 s 1 s 2 s 3 P s 1 s 2 s 3 P [ 2 3 2 1 0 0 0 1000 1 1 2 0 1 0 0 800 1 1 1 0 0 1 0 420 − 7 − 8 − 10 0 0 0 1 0 ] It is to be noted that a variable can be selected as a basic variable only if it corresponds to a column in the tableau that has exactly one nonzero element (usually 1) and the nonzero element in the column is not in the same row as the nonzero element in the column of another basic variable. So, the basic variables are s 1 , s 2 , s 3 , P . And non-basic variables are x 1 , x 2 , x 3 . To select the pivot column, select the most negative number in the bottom row of the tableau and to the left of the P column. The column containing this element is the pivot column . If there is a tie for the most negative indicator, choose either column. Hence, the third column is the pivot column. To select the pivot row, check for the smallest quotient for each row above the dashed line. If there is a tie for the smallest quotient, choose either row. 1000 2 = 500 800 2 = 400 420 1 = 420 So, the second row is selected as the pivot row. The element at the intersection of the pivot column and the pivot row is called the pivot element. x 1 x 2 x 3 s 1 s 2 s 3 P s 1 s 2 s 3 P [ 2 3 2 1 0 0 0 1000 1 1 2 0 1 0 0 800 1 1 1 0 0 1 0 420 − 7 − 8 − 10 0 0 0 1 0 ] So, the element enclosed in rectangle 2 is the pivot element. The variable on the top of the pivot column is called as the entering variable. the variable to the left of the pivot row is the exiting variable. So, x 3 is the entering variable and s 2 is the exiting variable. x 1 x 2 x 3 s 1 s 2 s 3 P s 1 s 2 s 3 P [ 2 3 2 1 0 0 0 1000 1 1 2 0 1 0 0 800 1 1 1 0 0 1 0 420 − 7 − 8 − 10 0 0 0 1 0 ] R 2 ÷ 2 → R 2 ~ [ 2 3 2 1 0 0 0 1000 0

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 6.2, Problem 42E

To determine

The number of components should the company manufacture each week in order to maximize its profit for the given fabrication and assembly constraints.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.2, Problem 41E

Chapter 6.2, Problem 43E

Students have asked these similar questions

Pidgeonhole Principle 1. The floor of x, written [x], also called the integral part, integer part, or greatest integer, is defined as the greatest integer less than or equal to x. Similarly the ceiling of x, written [x], is the smallest integer greater than or equal to x. Try figuring out the answers to the following: (a) [2.1] (b) [2] (c) [2.9] (d) [2.1] (e) [2] (f) [2.9] 2. The simple pidgeonhole principle states that, if you have N places and k items (k> N), then at least one hole must have more than one item in it. We tried this with chairs and students: Assume you have N = 12 chairs and k = 18 students. Then at least one chair must have more than one student on it. 3. The general pidgeonhole principle states that, if you have N places and k items, then at least one hole must have [] items or more in it. Try this out with (a) n = 10 chairs and k = 15 students (b) n = 10 chairs and k = 23 students (c) n = 10 chairs and k = 20 students 4. There are 34 problems on these pages, and we…

Determine if the set of vectors is linearly independent or linearly dependent. linearly independent O linearly dependent Save Answer Q2.2 1 Point Determine if the set of vectors spans R³. they span R³ they do not span R³ Save Answer 23 Q2.3 1 Point Determine if the set of vectors is linearly independent or linearly dependent. linearly independent O linearly dependent Save Answer 1111 1110 Q2.4 1 Point Determine if the set of vectors spans R4. O they span R4 they do not span IR4 1000; 111O'

The everything combined problem Suppose that a computer science laboratory has 15 workstations and 10 servers. A cable can be used to directly connect a workstation to a server. For each server, only one direct connection to that server can be active at any time. 1. How many cables would you need to connect each station to each server? 2. How many stations can be used at one time? 3. How many stations can not be used at any one time? 4. How many ways are there to pick 10 stations out of 15? 5. (This one is tricky) We want to guarantee that at any time any set of 10 or fewer workstations can simultaneously access different servers via direct connections. What is the minimum number of direct connections needed to achieve this goal?

Chapter 6 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 6.1 - Refer to Example 1. Find the basic solution for...Ch. 6.1 - Construct the table of basic solutions and use it...Ch. 6.1 - Prob. 3MP Ch. 6.1 - Prob. 4MP Ch. 6.1 - Prob. 1ED Ch. 6.1 - The following linear programming problem has only...Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - In how many ways can two variables be chosen from...Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Problems 9–12 refer to the system 9. Find the...Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Problems 31–40 refer to the partially completed...Ch. 6.1 - Prob. 40E Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - Prob. 55E Ch. 6.1 - Prob. 56E Ch. 6.1 - Prob. 57E Ch. 6.1 - Prob. 58E Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - Prob. 60E Ch. 6.1 - Prob. 61E Ch. 6.1 - Prob. 62E Ch. 6.1 - Prob. 63E Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - Prob. 66E Ch. 6.1 - Prob. 67E Ch. 6.1 - Prob. 68E Ch. 6.1 - Prob. 69E Ch. 6.1 - Prob. 70E Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - Prob. 72E Ch. 6.1 - Prob. 73E Ch. 6.1 - Prob. 74E Ch. 6.1 - Prob. 75E Ch. 6.1 - Prob. 76E Ch. 6.2 - Solve the following linear programming problem...Ch. 6.2 - Solve using the simplex method: Ch. 6.2 - Prob. 3MP Ch. 6.2 - Prob. 1ED Ch. 6.2 - Prob. 1E Ch. 6.2 - Prob. 2E Ch. 6.2 - Prob. 3E Ch. 6.2 - Prob. 4E Ch. 6.2 - Prob. 5E Ch. 6.2 - Prob. 6E Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - In Problems 9–12, Using slack variables, write the...Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 14E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Repeat Problem 17 with P = x1 + 2x2. 17. Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 24E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 28E Ch. 6.2 - Prob. 29E Ch. 6.2 - Prob. 30E Ch. 6.2 - Prob. 31E Ch. 6.2 - Prob. 32E Ch. 6.2 - Prob. 33E Ch. 6.2 - Prob. 34E Ch. 6.2 - Prob. 35E Ch. 6.2 - Prob. 36E Ch. 6.2 - Prob. 37E Ch. 6.2 - Prob. 38E Ch. 6.2 - In Problems 37–40, there is a tie for the choice...Ch. 6.2 - Prob. 40E Ch. 6.2 - Prob. 41E Ch. 6.2 - Prob. 42E Ch. 6.2 - Prob. 43E Ch. 6.2 - Prob. 44E Ch. 6.2 - Prob. 45E Ch. 6.2 - Prob. 46E Ch. 6.2 - Prob. 47E Ch. 6.2 - Prob. 48E Ch. 6.2 - Prob. 49E Ch. 6.2 - Prob. 50E Ch. 6.2 - Prob. 51E Ch. 6.2 - Prob. 52E Ch. 6.2 - Prob. 53E Ch. 6.2 - Prob. 54E Ch. 6.2 - Prob. 55E Ch. 6.2 - Prob. 56E Ch. 6.3 - Form the dual problem: Ch. 6.3 - Prob. 2MP Ch. 6.3 - Prob. 3MP Ch. 6.3 - Repeat Example 4 if the shipping charge from plant...Ch. 6.3 - Prob. 1ED Ch. 6.3 - Prob. 2ED Ch. 6.3 - Prob. 1E Ch. 6.3 - Prob. 2E Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 1–8, find the transpose of each...Ch. 6.3 - In Problems 9 and 10, Form the dual problem. Write...Ch. 6.3 - Prob. 10E Ch. 6.3 - In Problems 11 and 12, a minimization problem, the...Ch. 6.3 - Prob. 12E Ch. 6.3 - Prob. 13E Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - Prob. 16E Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Solve the linear programming problems in Problems...Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 34E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.3 - Prob. 38E Ch. 6.3 - Prob. 39E Ch. 6.3 - Prob. 40E Ch. 6.3 - Prob. 41E Ch. 6.3 - Prob. 42E Ch. 6.3 - Prob. 43E Ch. 6.3 - Prob. 44E Ch. 6.3 - Solve the linear programming problems in Problems...Ch. 6.3 - Prob. 46E Ch. 6.3 - Prob. 47E Ch. 6.3 - Prob. 48E Ch. 6.3 - Prob. 49E Ch. 6.3 - Mining. A mining company operates two mines, each...Ch. 6.3 - Prob. 51E Ch. 6.3 - Prob. 52E Ch. 6.3 - Prob. 53E Ch. 6.3 - Prob. 54E Ch. 6.3 - Prob. 55E Ch. 6.3 - Prob. 56E Ch. 6.3 - Prob. 57E Ch. 6.3 - Prob. 58E Ch. 6.4 - Repeat Example 1 for EXAMPLE 1 Finding the...Ch. 6.4 - Prob. 2MP Ch. 6.4 - Prob. 3MP Ch. 6.4 - Prob. 4MP Ch. 6.4 - Prob. 5MP Ch. 6.4 - Interpret the values of the slack and surplus...Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Prob. 4E Ch. 6.4 - Prob. 5E Ch. 6.4 - Prob. 6E Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Use the big M method to solve Problems 9–22. 9. Ch. 6.4 - Prob. 10E Ch. 6.4 - Prob. 11E Ch. 6.4 - Prob. 12E Ch. 6.4 - Use the big M method to solve Problems 9–22. 13. Ch. 6.4 - Prob. 14E Ch. 6.4 - Use the big M method to solve Problems 9–22. 15. Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - Prob. 20E Ch. 6.4 - Use the big M method to solve Problems 9-22. 21....Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - Problems 25–32 are mixed. Some can be solved by...Ch. 6.4 - Prob. 26E Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Prob. 32E Ch. 6.4 - In Problems 33–38, construct a mathematical model...Ch. 6.4 - In Problems 33–38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - Prob. 36E Ch. 6.4 - Prob. 37E Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - Prob. 40E Ch. 6.4 - Prob. 41E Ch. 6.4 - Prob. 42E Ch. 6.4 - Prob. 43E Ch. 6.4 - Prob. 44E Ch. 6.4 - Prob. 45E Ch. 6.4 - Prob. 46E Ch. 6.4 - Prob. 47E Ch. 6 - Prob. 1RE Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Prob. 26RE Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE Ch. 6 - In Problems 28 and 29, Introduce slack, surplus,...Ch. 6 - Prob. 30RE Ch. 6 - Prob. 31RE Ch. 6 - Prob. 32RE Ch. 6 - Prob. 33RE Ch. 6 - Prob. 34RE Ch. 6 - Prob. 35RE Ch. 6 - Prob. 36RE Ch. 6 - Prob. 37RE Ch. 6 - Prob. 38RE Ch. 6 - Prob. 39RE Ch. 6 - Prob. 40RE Ch. 6 - Prob. 41RE

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.