To determine To calculate: The maximization problem. Explanation Given: In a gift shop, there are 3 items named as A , B , C . Each unit of item A costs $ 2 to buy, $ 1 to sell and $ 2 to deliver, further each unit of item B costs $ 3 to buy, $ 2 to sell and $ 2 to deliver, further each unit of item C costs $ 6 to buy, $ 2 to sell and $ 4 to deliver. The profit on A , B , C are $ 4 , $ 3 , $ 3 respectively and if $ 1200 , $ 800 , $ 500 is spent to buy, sell and deliver respectively. Formula used: Calculation: Consider the maximization problem, first formulate the problem in the system of linear equations. Take, x 1 be the number of the item A then take, x 2 be the number of item B further, take, x 3 be the number of the item C . Take, z be the objective function to maximize as the profit is given A , B , C are $ 4 , $ 3 , $ 3 respectively. So, the objective function is: z = 4 x 1 + 3 x 2 + 3 x 3 From all three items spend in buy is $ 1200 and the cost for A , B , C are $ 2 , $ 3 , $ 6 respectively, so the constraints is: 2 x 1 + 3 x 2 + 6 x 3 ≤ 1200 From all three items spent in sell is $ 800 and the cost for A , B , C are $ 1 , $ 2 , $ 2 respectively, so the constraints is: x 1 + 2 x 2 + 2 x 3 ≤ 800 From all three items spent on deliver is $ 500 and the cost for A , B , C are $ 2 , $ 2 , $ 4 respectively, so the constraints are: 2 x 1 + 2 x 2 + 4 x 3 ≤ 500 Therefore, formulate the problem that is: Maximize, z = 4 x 1 + 3 x 2 + 3 x 3 Subject to, 2 x 1 + 3 x 2 + 6 x 3 ≤ 1200 , x 1 + 2 x 2 + 2 x 3 ≤ 800 , 2 x 1 + 2 x 2 + 4 x 3 ≤ 500 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 Use two stages method to solve the maximization problem, then there are various steps which must follow: Step (1): Modify each equation constraint by an equivalent pair of inequality constraint Step (2): Write each constraint with a positive constant. Step (3): If minimization problem, then converts into maximization by replacing z = − w . Step (4): Add or subtract surplus variables according to the need of the constraints into equations. Step (5): Write initial simplex table. Step (6): Find a basic feasible solution if a solution exists, which is stage I. Step (7): If basic feasible solution found then do the simplex method further which is a stage and get the optimal solution. Further, to convert linear inequality into the linear equation then take a slack variable and surplus variable which is done by adding and subtracting a non-negative variable to each constraint. If linear equality has less than or equal to sign then use slack variable by add as the equation is less than or equal to the value if it is equal then the slack variable is zero. If linear equality has greater than or equal to sign then use surplus variable by subtracting as the equation is greater than or equal to the value, if it is equal then the surplus variable is zero which is: 2 x 1 + 3 x 2 + 6 x 3 + s 1 = 1200 , x 1 + 2 x 2 + 2 x 3 + s 2 = 800 , 2 x 1 + 2 x 2 + 4 x 3 + s 3 = 500 where s 1 , s 2 , s 3 are greater than or equal to zero Then, rewrite constraints by adding or subtracting the non-negative slack and surplus variables which are: Maximize, z = 4 x 1 + 3 x 2 + 3 x 3 Subject to, 2 x 1 + 3 x 2 + 6 x 3 + s 1 = 1200 , x 1 + 2 x 2 + 2 x 3 + s 2 = 800 , 2 x 1 + 2 x 2 + 4 x 3 + s 3 = 500 , x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0 Then, convert the linear programming into a system of linear equations in which all the coefficient of variables are written on the left of the equal sign and the constants are on the right then convert the objective function into linear equations which are: − 4 x 1 − 3 x 2 − 3 x 3 + z = 0 Then, write the constraint and objective function into the augmented matrix which is in the form of a simplex table: x 1 x 2 x 3 s 1 s 2 s 3 [ 2 3 6 1 0 0 1200 1 2 2 0 1 0 800 2 2 4 0 0 1 500 − 4 − 3 − 3 0 0 0 0 ] Here, the last row shows the coefficients of the objective function and last columns shows the constants. The basic variables correspond to columns which have an entry of 1 and rest is zero its value corresponds to the last column which is constant, and the non-basic variables correspond to the other columns and its value is zero. In the above nonbasic variables are x 1 , x 2 , x 3 and the basic variables are s 1 , s 2 , s 3 . Here, x 1 = 0 , x 2 = 0 , x 3 = 0 , s 1 = 1200 , s 2 = 800 , s 3 = 500 Above simplex, the table has a basic feasible solution as the basic variable has no negative value so it is said to be basic feasible solution and stage I is completed. In the last row, all the entries except the last and second last entries are called indicators. The simplex table represents a system of four linear equations in seven variables which is x 1 , x 2 , x 3 , s 1 , s 2 , s 3 , z as there are more variable than equations then the system is dependent, and it has infinitely many solutions

MATH W/APPLICAT.W/NOTES GDE +ACCESS CODE

11th Edition

ISBN: 9781323751671

Author: Lial

Publisher: PEARSON C

See similar textbooks

Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear re…

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Videos

Question

Chapter 7, Problem 57RE

To determine

To calculate: The maximization problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7, Problem 56RE

Chapter 7, Problem 58RE

Students have asked these similar questions

Suppose an oil spill covers a circular area and the radius, r, increases according to the graph shown below where t represents the number of minutes since the spill was first observed. Radius (feet) 80 70 60 50 40 30 20 10 0 r 0 10 20 30 40 50 60 70 80 90 Time (minutes) (a) How large is the circular area of the spill 30 minutes after it was first observed? Give your answer in terms of π. square feet (b) If the cost to clean the oil spill is proportional to the square of the diameter of the spill, express the cost, C, as a function of the radius of the spill, r. Use a lower case k as the proportionality constant. C(r) = (c) Which of the following expressions could be used to represent the amount of time it took for the radius of the spill to increase from 20 feet to 60 feet? r(60) - r(20) Or¹(80-30) r(80) - r(30) r-1(80) - r−1(30) r-1(60) - r¹(20)

6. Graph the function f(x)=log3x. Label three points on the graph (one should be the intercept) with corresponding ordered pairs and label the asymptote with its equation. Write the domain and range of the function in interval notation. Make your graph big enough to see all important features.

Find the average value gave of the function g on the given interval. gave = g(x) = 8√√x, [8,64] Need Help? Read It Watch It

Chapter 7 Solutions

MATH W/APPLICAT.W/NOTES GDE +ACCESS CODE

Ch. 7.1 - Checkpoint 1 Graph the given...Ch. 7.1 - Prob. 2CP Ch. 7.1 - Prob. 3CP Ch. 7.1 - Prob. 4CP Ch. 7.1 - Prob. 5CP Ch. 7.1 - Checkpoint 6 Graph the feasible region of the...Ch. 7.1 - Prob. 7CP Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...

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.