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Linear Programming: Introduction (page 1 of 5)
Sections: Optimizing linear systems, Setting up word problems
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Linear programming is the process of taking various linear inequalities relating to some situation, and finding the 'best' value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the 'best' production levels for maximal profits under those conditions.
In 'real life', linear programming is part of a very important area of mathematics called 'optimization techniques'. Daisydisk 4 1040. This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These 'real life' systems can have dozens or hundreds of variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable linear case.
The general process for solving linear-programming exercises is to graph the inequalities (called the 'constraints') to form a walled-off area on the x,y-plane (called the 'feasibility region'). Then you figure out the coordinates of the corners of this feasibility region (that is, you find the intersection points of the various pairs of lines), and test these corner points in the formula (called the 'optimization equation') for which you're trying to find the highest or lowest value.
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- Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:
= 0, x - y <= 2'>
The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula 'z = 3x + 4y' is the optimization equation. I need to find the (x, y) corner points of the feasibility region that return the largest and smallest values of z.
My first step is to solve each inequality for the more-easily graphed equivalent forms: Film wizard 2 7 0 6.
It's easy to graph the system: Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
To find the corner points -- which aren't always clear from the graph -- I'll pair the lines (thus forming a system of linear equations) and solve:
y = –( 1/2 )x + 7 | y = –( 1/2 )x + 7 | y = 3x |
–( 1/2 )x + 7 = 3x y = 3(2) = 6 | –( 1/2 )x + 7 = x – 2 y = (6) – 2 = 4 | 3x = x – 2 y = 3(–1) = –3 |
corner point at (2, 6) | corner point at (6, 4) | corner pt. at (–1, –3) |
So the corner points are (2, 6), (6, 4), and (–1, –3).
Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. So, to find the solution to this exercise, I only need to plug these three points into 'z = 3x + 4y'.
(2, 6): z = 3(2) + 4(6) = 6 + 24 = 30
(6, 4): z = 3(6) + 4(4) = 18 + 16 = 34
(–1, –3): z = 3(–1) + 4(–3) = –3 – 12 = –15
(6, 4): z = 3(6) + 4(4) = 18 + 16 = 34
(–1, –3): z = 3(–1) + 4(–3) = –3 – 12 = –15
Then the maximum of z = 34 occurs at (6, 4),
and the minimum of z = –15 occurs at (–1, –3).
and the minimum of z = –15 occurs at (–1, –3).
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Cite this article as: | Stapel, Elizabeth. 'Linear Programming: Introduction.' Purplemath. Available from https://www.purplemath.com/modules/linprog.htm. Accessed [Date] [Month] 2016 |
The Color Purple Plot Summary
The book opens as young Celie address God with her fears about the future. She is raped numerous times by who she believes to be her father, sees her mother beaten, and fears for her sister Nettie's life. She and her mother give birth on the same day to 'His' children; 'He' immediately kills the mother's baby and sells her own crying newborn. Soon enough, 'He' sells her into marriage with an older widow with four children named Mr. Albert ______. Mr._____ beats her, taunts her, and forces her to clean, cook, and look after his dirty and ungrateful children. Meanwhile, Mr.______ lusts after Nettie until she runs away to the Reverend Mr.______'s home (with Celie's protective orders).
As her life with Mr._____ progresses, the children grow. Harpo, the eldest son, falls in love with a young girl named Sofia, and introduces her to the family already swelling and pregnant. She speaks her mind and flees to live with her sister Odessa. The two eventually marry and move in next door to Celie and Mr.____. Celie and Sofia develop a strong friendship, as one accepts her beatings and the other beats her husband.
Celie continues to write to God and Nettie, explaining her miserable life with Mr._____ and hoping to reunite as soon as possible. Suddenly, Mr._____ brings his mistress, Shug Avery, home with him, for she is ill with venereal disease, and has no other salvation. Celie immediately finds herself attracted to the celebrated singer, speaking with her, bathing her, and worshipping her. After an initial introduction of terror, the two become close friends.
Eventually, Sofia becomes bored with her marriage and leaves Harpo and their four children alone to stay with her sister, Odessa. Harpo learns how to live alone, and builds a juke joint in their old home. Shug Avery sings there and brings it big business. Celie and Shug continue to discuss sexuality, life, love, and family. One evening at Harpo's, Sofia returns with a new boyfriend. Harpo brings his new girlfriend, Squeak, to the joint, and everyone is reunited. However, when the dancing begins, Sofia punches Squeak in the mouth, knocking out her front teeth, and flees town again. The family learns that later that week, Sofia found herself in an argument with the mayor's wife, punched the mayor, and was thrown in jail after a brutal beating by the police. She is left there for years, while the family tries to help. Squeak discovers that she is related to the warden and tries to help Sofia by visiting him. When he sees her, he rapes her and leaves the family helpless.
Shug and Celie fall in love, as she teaches Celie about her own body, femininity, and sexuality. They discover that Mr.______ has been hiding all of the letters that Nettie has sent to Celie. She is so angry that she develops violent feelings towards him. They begin to read the letters and learn all about Nettie's new and adventurous life. When she ran away from Mr._____ years ago, she found Corrine and Samuel, two Christian missionaries who took her in. Their two adopted children, Adam and Olivia, are in fact Celie's biological children taken at birth. They travel to England and Africa to teach and help the youth of Africa. Starry night pro plus 8 0 2. In Africa, they live in the village of Olinka, where they find a new home, the children grow and learn, and they discover new theories on life and religion. Corrine falls ill with the African fever and eventually dies. Years later, Nettie and Samuel fall in love, marry, and are forced to leave their post in Africa. At this point, they decide to return home to Nettie's family. Adam and Olivia have become young adults, and Adam falls in love with an African girl named Tashi, whom he marries and brings to America.
Shug takes Celie, her new husband Grady, and Squeak to Memphis with her to escape the violent and cruel wrath of Albert. Celie is in heaven as she lives in a beautiful house with her beloved Shug, and discovers a method of making pants. Her pants become a runaway success and she opens Folkspants, Unlimited. Furthermore, she learns that her cruel father (who she learns is not her biological father) has died, leaving her the old house.
Unfortunately, Mr._____ hands Celie a letter stating that the ship on which Nettie and her family sailed has sunk. Celie is miserable, for she feels completely alone. While at home fixing her new house, Shug informs her that she is in love with someone else and just wants a six-month affair with the young man. Celie is brokenhearted for both grave losses, and spends the remainder of her time with Mr.______. He has grown, changed, and found religion, making him a bearable companion.
Eventually, Shug returns to Celie's home wanting her love forever and Nettie arrives on their doorstep with their children. Celie's life is now complete, as the two families join as one, despite blood relations and years of separation.