The document defines linear programming as a branch of mathematics used to find the optimal solution to problems with constraints. It provides examples of using linear programming to maximize profit or minimize costs in organizations. It also introduces drawing linear inequalities and solving simultaneous inequalities. The steps to formulate a linear programming problem are identified as defining variables and objectives, translating constraints, finding feasible solutions, and evaluating objectives to find optimal solutions.
* Presentation – Complete video for teachers and learners on Similarity
* GSCE, IGCSE, IB, PSAT, and AISL - Exam Style Questions which covers all the related concepts required for students to unravel any International Exam Style Similarity Questions
* Learner will be able to say authoritatively that:
I can apply similarity to model a real life situation and the various field of study: Engineering, Art and Design, Construction, etc..
I can solve any given question on Combined Similarity: Volume, Area, Standard Dimensions…
I can find the scale factor given any object or image parameter
I can use a given scale model to find unknown parameter of any similar shape and also apply the concepts in all field of studies: Construction, Cryptographer, Actuary, Astronomy, Physical Science, Biological Science, Astrophysics, etc….
* Presentation – Complete video for teachers and learners on Similarity
* GSCE, IGCSE, IB, PSAT, and AISL - Exam Style Questions which covers all the related concepts required for students to unravel any International Exam Style Similarity Questions
* Learner will be able to say authoritatively that:
I can apply similarity to model a real life situation and the various field of study: Engineering, Art and Design, Construction, etc..
I can solve any given question on Combined Similarity: Volume, Area, Standard Dimensions…
I can find the scale factor given any object or image parameter
I can use a given scale model to find unknown parameter of any similar shape and also apply the concepts in all field of studies: Construction, Cryptographer, Actuary, Astronomy, Physical Science, Biological Science, Astrophysics, etc….
This is an interactive presentation which contains the information about Algebra for student-teacher , who are going to teach maths. Further, it contains information about the curriculum alignment and objectives of algebraic teaching which are mentioned in Curriculum of Pakistan.
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
Friends if you found this helpful please click the like button. and share it :) thanks for watching
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
This is an interactive presentation which contains the information about Algebra for student-teacher , who are going to teach maths. Further, it contains information about the curriculum alignment and objectives of algebraic teaching which are mentioned in Curriculum of Pakistan.
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
Friends if you found this helpful please click the like button. and share it :) thanks for watching
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
A geometrical approach in Linear Programming ProblemsRaja Agrawal
Here you can see how we can solve any problems related to Linear Programming using Graphical method. You will also come to know about Simplex method as well as Dual method in LPP. All taken examples are easy to understand. Do give a try and share your valuable feedback.
This PowerPoint was created to help out graduating seniors who are taking the TAKS Mathematics Exit-Level test. It includes formulas, rules & things that they need to remember to pass the test.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
1. LINEAR PROGRAMING
Definition
Is a branch of mathematics which enables one to solve problems which either the greatest or
minimum/least value of a certain quantity is required under some given limitations or
constraints.
Example
In a big – organization, decision about distribution in order to realize maximum profit or
reduce costs of productionare done by use of linear programming.
Limitation/ constraints are translatedbylinear inequalities.
Greatest value or least value will be expressedas a function (calledthe objective function)
Introduction
Drawing of linear inequalities
Example 01
Draw and show the half plane representedby 8x+2y ≥16
Solution
For 8x +2y≥16; draw 8x+ 2y = 16
For x – intercept, y = 0
8x = 16
X = 2
For y – intercept, x = 0
2y = 16
Y = 8
Using (0, 0) as a test point
2. 8(0) +2(0)≥ 16
0 ≥ 16 (False)
Example 02
Determine the solutionset of the simultaneous inequalities
Solution
x + y ≥ 3 draw x + y = 3 (full line)
x + y = 3
At x – intercept y= 0 at y – intercept x= 0
x = 3 y = 3
For x – 2y ≤ 9, x – 2y = 9
3. At x – intercept y= 0 at y – intercept, x= 0
X =9 -2y= 9
Y =
Using (0, 0) as a test point Using (0,0) as a test point
x + y ≥ 3 x – 2y≤ 9
0 + 0 ≥ 3 0 – (2)(0) ≤ 9
0 ≥ 3 (F) 0 ≤ 9 (T)
The clear part is the solutionset
4. The solutionset is calledthe feasible region
Questions
Draw by shading unwanted regions of the half planes represented by following
simultaneous inequalities
(i) y≥ 2x – 1, y ≤ -1
(ii) y≤ 2x – 1, y ≥ x – 3, y ≥ -1
(iii) y < 2x – 1, y ≤ -1
(iv) 6x + 9y ≥ 12
0.4x+ 0.1y≥ 0.2
32x+ 10y≥ 20
Evaluationofa functionsatisfiedbythe givenset ofinequalities
→Example
Find the maximum and minimum value of c = 4x + 3y + 38 subjectedto
x + y ≥ 5
0 ≤ y ≤ 6 0 ≤ x ≤ 5
x ≥ 0, y ≥ 0
Solution
For x + y ≥ 5
x + y = 5
When x = 0, y = 5
y = 0, x = 5
For 0 ≤ y ≤ 6
0 = y = 6
Line y = 6
For 0 ≤ x ≤ 5
O = x = 5
Line x = 5
Test points x ≥ 0, shade left of x = 0
(0, 0) y ≥ 0, shade below of x = axis
6. B (5, 6) 4 (5) + 3 (6) + 38 = 76
C (0, 6) 4 (0) + 3 (6) +38 = 56
D (0, 5) 4 (0) + 3 (5) + 38 = 53
: . The maximum value of c = 76 and occurs at (5, 6)
The minimum value of c = 53 and occurs at (0, 5)
Questions
1. Find the maximum and minimum values of the given functions and the value
x and y where they occur
(i) Z = 4x+ 3y
Subject to
x + 2y ≤ 10
3x + y ≤ 5
x ≥ 0, y ≥ 0
(ii) P = 134x+ 20y
Subject to
x + y ≤ 160
10 ≤ x ≤ 60
0 ≤ y ≤ 120
(iii) T = 4x +7y
Subject to.
x + y ≤ 18
5 ≤ x ≤ 10
3 ≤ y ≤ 10
x ≥ 0, y ≥ 0
(iv) P = 2x + 4y
Subject to
2x + 3y ≥ 3
-5x+ 4y ≤ 0
3x + 4y ≤ 18
X ≥ 0, y ≥ 0.
FORMULATION OF A LINEAR PROGRAMMING PROBLEM.
Steps in formulatinga linear programming problem.
1. * Read the problem several times and assess what is known and what is to be determined.
7. 2. * Identify the unknown quantities and assign variables to them, be careful about the units.
3. * Determine the objective function;it involves the quantity to be maximized or minimized.
4. * Translate the constraints into linear inequalities,
* Constraints are limitation or restrictions to the problem for each constraints the units must
be same.
5. * Graph the constraints andfind the feasible solution
6. * Find the corner points of the feasible solution. These are points of intersectionof the graph
7. * Evaluate the objective function. Highest value of the objective function has to be
maximized or smallest value to be minimized.
Example
A student has 120 shillings to spend on exercise books. At a school shop an exercise
book costs 8 shillings, at stationery store an exercise book costs 12 shillings. The school
has only 6 exercise books and the student wants to obtain the greatest number of exercise
books possible usingthe money. Find the greatest number of exercise books he canbuy.
Solution
Let x be number of exercise books to be bought at school shop
Let y be number of exercise books to be bought at stationeryshop
→Objective function
Let f (x, y) = objective function
Then f (x, y) = x + y
→Constrains or linear inequalities
8x + 12y ≤ 120
x ≤ 6
Non – Negative constraints
x ≥ 0
y ≥ 0
→Equations
8x + 12y = 120
4x + 6y = 60
2x + 3y = 30
When x = 0, y = 10
y = 0, x = 15
x = 6, y = 0
8. x = 6
Corner points F (x, y) = x + y
A (0, 0) 0 + 0 = 0
B (6, 0) 6 + 0 = 6
C ( 6, 6) 6 + 6 = 12
D (0, 10) 0+10 = 10
9. : . The greatest numbers of exercise books he can buy are 12 books 6 from the school
shop and 6 from stationery.
Example
Student in a certain class are about to take a certain test of BAM which has two sections
A and B; where in section A each question worth 10 marks while in section B; each
worth 25 marks. The student must do at least 3 questions of section A; but not more than
12. A student must also do 4 questions from section B but not more than 15. In addition
students cannot do more than 20 questions. How many questions of each type should the
student do to obtain the maximum scores?
Solution
Let x be number of questions to be done in sectionA
Let y be number of questions to be done in sectionB
→Objective function
f (x, y) = 10x+ 25y
→Constrains
3 ≤ x ≤ 12
4 ≤ y ≤ 15
x + y ≤ 20
x ≥ 0, y ≥ 0
Maximize f (x, y) = 10x+ 25ysubject to;
3 ≤ x ≤ 12
4 ≤ y ≤ 15
x + y ≤ 20
x ≥ 0, y ≥ 0
Equations
3 = x = 12
4 = y = 15
x + y = 20
x = 20, y = 0
x = 0, y = 20
x = 0
10. y = 0
Corner Points f(x, y) = 10x+ 25y
A (3, 4) 10 (3) + 25 (4) = 130
B (12, 4) 10 (12) + 25 (4) = 220
C (12, 8 ) 10 (12) + 25 (8) = 320
D (5, 15) 10 (5) + 25 (15) = 425
E (3, 15) 10 (3) + 25 (15) = 405
The student should do 5 questions from section A and 15 questions from section B to
obtain maximum score of 425.
Diet problems onlinear programming problem
Example 01
A doctor prescribes a special diet for patients containing the following number of units of
Vitamin A and B per kg of two types of foodF1 and F2
Type of Food Vitamin A Vitamin B
F1 20 units/kg 7 units/kg
F2 15 units /kg 14 units /kg
11. If the minimum daily intake required is 120 units of A and 70 units of B, what is the least
total mass of fooda patient must have so as to have enough of these vitamins?
EXAMPLE 02
Rice and beans provide maximum levels of protein, calories and vitamin B2. If used as a
staple diet. The food values per kg of uncooked rice and beans are as shown in the table
below.
Protein/kg Calories/kg Vitamin B2/kg Price kg
Rice 60g 3200 cal 0.4 400
Beans 90g 1000 cal 0.1 500
Min daily req. 120 2000 cal 0.2
What is the lowest cost of diet meeting, these specifications?
Solution.
Let x be number of kg of rice to be bought
Let y be number of kg of beans to be bought
→Objective function
400x+ 500y
Constrains
60x+ 90y≤ 120
3200x+ 1000y≤ 2000
0.4x + 0.1y≤ 0.2
Minimize f (x, y) = 400x+ 500ySubject to;
60x+ 90y≤ 120
3200x+ 1000y≤ 2000
12. 0.4x+ 0.1y≤ 0.2
x ≥ 0, y ≥ 0
For 60x+ 90y≥ 120
60x+ 90y= 120
2x + 3y = 4
When x = 0, y = 1.3
Y = 0, x = 2
For 3200x+ 1000y≥ 2000
32x+ 10y= 20
16x+ 5y = 10
When x = 0, y = 2
Y = 0, x = 0.63
For 0.4x + 0.1y ≤ 0.2
0.4x+ 0.1y= 0.2
When x = 0, y = 2
When y = 0, x = 0.5
13. Corner points F (x, y) = x + y
A (0, 8) 0 + 8 = 8
B (3.6, 3.2) 3.6 + 3.2 = 6.8
C (10,0) 10 + 0 = 10
: . The least total mass a patient should have is 6.8kg i.e. 3.6kg of food 1 and 3.2 kg of
food2.
Question
1. A doctor prescribes that in order to obtain adequate supply of vitamin A and C his patient
should have portions of food 1 and food 2. The number of units of vitamin A and C are given
in the followingtable
A C
14. Food1 3 2
Food2 1 7
The doctor prescribes a minimum of 14 units of vitamin A and 21 units of vitamin C. What
are the least portions of food1 and food2 that will fit the doctor’s prescriptions?
LINEAR PROGRAMMING PROBLEMS
1. Two printers N and T produce three types of books. N produces 80 types I books
per day, 10 type II books per day and 20 types III books per day, while T produces
20 types I books per day 10 type II books per day and 70 types III books per day.
The orders placed are 1600 type I, 500 type II and 2100 type III books. The daily
operating costs for N shs. 10,000/=, for T shs, 20,000/= how many days should
each printer operate to meet the orders at aminimum cost.
2. A small textile company manufactures three different size of shirts, Large (L),
medium (M) and small (S) at two different plants A and B. The number of shirts
of each size producedand the cost of productionper day are as follows!
A B Monthly demand
Large size per day 50 60 2500
Medium size per day 100 70 3500
Small size per day 100 200 7000
Production Cost per day
T. shs.
2500 3500 _
(i) How many days per month should each factory operate in order to minimize
total cost.
(ii) What is the minimum cost of production
Solution01
Let x be number of days printer N shouldoperate
15. Let Y be number of days printer T should operate
→Objective function(f(x, y))
10000x+ 20000y
→Constrains
80x + 20y≥ 1600
10x + 10y≥ 500
20x + 70y≥ 2100
X ≥ 0, y ≥ 0
→Equations
80x + 20y≥ 1600
80x+ 20y= 1600
8x + 2y = 160
When x = 0, y = 80
y = 0, x = 20
10x + 10y≥ 500
10x+ 10y= 500
x + y = 50
When x = 0, y = 50
y = 0, x = 50
20x + 70y≥ 2100
20x+ 70y= 2100
2x + 7y = 210
When x = 0, y = 30
16. y = 0, x = 105
Corner points F (x, y) = 10000x+ 20000y
A (0, 80) 10000 (0)+ 20000 (80) = 1,600,000
B (28, 22) 10000 (28) + 20000(22) = 720,000
C (105, 0) 10,000 (105)+ 20000 (0)= 1,050,000
Printer N should be operated for 28 days and printer T should work for 22 days to meet the
orders at minimum cost.
Solution02
Let x be number of days per month factoryA shouldoperate
Let y be number of days per month factoryB shouldoperate
→Objective function
17. F (x, y) = 2500x+ 3500y
→Constrains
50x + 60y≥ 2500
100x+ 70y ≥ 3500
100x+ 200y≥ 7000
x ≥ 0, y ≥ 0
Minimize f (x, y) = 2500x+ 3500y
Subject to 50x+ 60y≥ 2500
100x+ 70y≥ 3500
100x+ 200y≥ 7000
X ≥ 0, y ≥ 0
→Equations
50x + 60y= 2500
When y = 0, x = 50
X = 0, y = 41.7
100x+ 70y = 3500
When y = 0, x = 35
x = 0, y = 50
100x+ 200y= 7000
When y = 0, x = 70
X = 0, y = 35
18. Corner points F (x, y) = 2500x+ 3500y
A (0,50) 2500 (0) + 3500(50)= 175,000
B (15, 30) 2500 (15) + 3500 (30) = 142,500
C (20,25) 2500 (20) + 3500 (25) = 137,500
D (70, 0) 2500 (70) + 3500 (0)= 175,000
Factory A should operate for 20 days and factory B should operate for 25 days in order to
minimize total cost.
→Minimum cost of productionis 137,500
3. In a certain garage the manager had the following facts floor space required for a
saloon is 2m2 and for a lorry is 3m2. Four technicians are required to service a
saloon car and three technicians for a lorry per day. He has a maximum of 24m2 of
19. a floor space and a maximum of 36 technicians available; in addition he is not
allowed to service more Lorries than saloon cars. The profit for serving a saloon
car is 40,000/= and a lorry is 60,000/=. How many motor vehicles of each type
should be serviceddaily in order to maximize the profit?
Solution
Let x be number of salooncars to be serviceddaily
Let y be number of Lorries to be serviceddaily
→Objective function
F (x, y) = 40000x+ 60000y
→Constrains
2x + 3y ≤ 24
4x + 3y ≤ 36
x ≥ y
x ≥ 0 and y ≥ 0
Maximize f (x, y) = 40000x+ 60000y
Subject to 2x + 3y ≤ 24
4x + 3y ≤ 36
x ≥ y
x ≥ 0 and y ≥ 0
→Equations
2x + 3y ≤ 24
When x = 0, y = 8
20. Y = 0, x = 12
4x + 3y ≤ 36
When x = 0, y = 12
y = 0, x = 9
x = y
Corner points F (x, y) = 40000x+ 60000y
A (0, 0) 40000 (0) + 60000(0) = 0
B (4.8, 4.8) 40000 (4.8) + 60000 (4.8) = 480,000
C (6,4) 40000 (6)+ 60000 (4) = 480,000
D (9, 0) 40000 (9) + 60000 (0) = 360,000
6 salooncars and 4 Lorries shouldbe serviceddaily to maximize profit to 480,000/=
21. More example
A builder has two stores, one at S1 and the other at S2. He is building houses at P1, P2, and P3.
He needs 5 tons of bricks at P1, 6 tons of bricks at P2 and 4 tons of bricks at P3. The stores
contain 9 tons of bricks at S1 and 6 tons of bricks at S2. The transport cost per ton are shown
in the diagram
To From P1 P2 P3
S1 6/= 3/= 4/=
S2 4/= 2/= 6/=
How does the builder send his bricks at a minimum cost?What is the minimum overall cost?
Solution
Let the builder send x tons of bricks from S1 to P1 and y tons of bricks from S1 to P2
Then the transportationof bricks to P1, P2 and P3 will be as follow: -
To
From P1 P2 P3
S1 X Y 9 – (x +
y)
S2 5 – x 6 – y 4 – [9 –
(x + y)]
The constrains are obtainedas follows
x ≥ 0, y ≥ 0
9 – (x + y) ≥ 0 i.e. x + y ≤ 9
5 – x ≥ 0 i.e. x ≤ 5
6 – y ≥ 0 i.e. y ≤ 6
4 – [9 - (x + y)} ≥ 0 i.e. x + y ≥ 5
22. Objective function
F (x, y) = 6x + 3y + 4(9 - (x + y)) + 4 (5 – x) + 2(6 – y) +6 [4 – (9 - (x + y)]
= 6x + 3y +36 – 4x+ 4y + 20 – 4x+ 12 – 2y + 24 – 54 + 6x+ 6y
F (x, y) = 4x – 3y+ 38
Minimize f (x, y) = 4x – 3y + 38
Subject to x + y ≤ 9
x + y ≥ 5
x ≤ 5, y ≥ 0
y ≤ 6, y ≥ 0
→Equation
x + y = 9
When x = 0, y = 9
y = 0, x = 9
x + y = 5
x = 0, y = 5
y = 0, x = 9
x = 5
y = 6
23. Corner points F (x, y) = 4x + 3y + 38
A (0, 5) 4 (0) + 3 (5) + 38 = 53
B (0, 6) 4 (0) + 3 (6) + 38 = 56
C (3, 6) 4 (3) + 3 (6) + 38 = 68
D (5, 4) 4 (5) + 3 (4) + 38 = 70
E (5, 0) 4 (5) + 3 (0) + 38 = 58
The builder should send the bricks of tons as follows
To
From P1 P2 P3
S1 0 5 4
S2 5 1 0
24. The overall minimum cost is 53/=
EXERCISE
There is a factory located at each of the places P and Q. From these location a certain
commodity is delivered to each of the three deports situated at A, B and C. The weekly
requirements of the deports are respectively 5,5 and 4 unit of the commodity while the
production capacity of the factories P and Q are 8 and 6 units respectively, just sufficient for
requirement of deports. The cost of transportationper unit is given.
To
From A B C
P 16 10 15
Q 10 12 10
Formulate this linear programming problem and how the commodities can be transported at
minimum cost. What is the overall minimum cost?