This document provides an overview of linear programming and the simplex method for solving linear programming problems. It begins with defining the basic linear programming problem as having an objective function and a set of constraints. It then describes how to formulate a sample linear programming problem as a set of equations in standard form. The document explains how to find the feasible region and optimal solution graphically. It introduces the concept of basic feasible solutions and shows how the simplex method works by iteratively moving from one basic feasible solution to an adjacent better solution until the optimal solution is found. Key steps like choosing entering and leaving variables are demonstrated.
For a good business plan creative thinking is important. A business plan is very important and strategic tool for entrepreneurs. A good business plan not only helps entrepreneurs focus on specific steps necessary for them to make business ideas succeed, but it also helps them to achieve short-term and long-term objectives. As an inspiring entrepreneur who is looking towards starting a business, one of the businesses you can successfully start without much stress is book servicing café.
Importance:
Nowadays, network plays an important role in people’s life. In the process of the improvement of the people’s living standard, people’s demand of the life’s quality and efficiency is more higher, the traditional bookstore’s inconvenience gradually emerge, and the online book store has gradually be used in public. The online book store system based on the principle of providing convenience and service to people.
With the online book servicing café, college student do not need to blindly go to various places to find their own books, but only in a computer connected to the internet log on online book servicing café in the search box, type u want to find of the book information retrieval, you can efficiently know whether a site has its own books, if you can online direct purchase, if not u can change the home book store to continue to search or provide advice to the seller in order to supply. This greatly facilitates every college student saving time.
The online book servicing café’s main users are divided into two categories, one is the front user, and one is the background user. The main business model for Book Servicing Café relies on college students providing textbooks, auctions, classifieds teacher evaluations available on website. Therefore, our focus will be on the marketing strategy to increase student traffic and usage. In turn, visitor volume and transactions will maintain the inventory of products and services offered.
Online bookstore system i.e. Book Servicing Café not only can easily find the information and purchase books, and the operating conditions are simple, user-friendly, to a large extent to solve real-life problems in the purchase of the books.
When you shop in online book servicing cafe, you have the chance of accessing and going through customers who have shopped at book servicing café and review about the book you intend to buy. This will give you beforehand information about that book.
While purchasing or selling books at the book servicing café, you save money, energy and time for your favorite book online. The book servicing café will offer discount coupons which help college students save money or make money on their purchases or selling. Shopping for books online is economical too because of the low shipping price.
Book servicing café tend to work with multiple suppliers, which allows them to offer a wider variety of books than a traditional retail store without accruing a large, costly inventory which will help colle
A machine learning method for efficient design optimization in nano-opticsJCMwave
Explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradiant-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A machine learning method for efficient design optimization in nano-optics JCMwave
The slideshow contains a brief explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradient-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A brief study on linear programming solving methodsMayurjyotiNeog
This small presentation includes a brief study on various linear programming solving methods. These methods (graphical & simplex) are used to solve industrial engineering related problems in practical use.
Simple Linear Regression: Step-By-StepDan Wellisch
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Crystallization, or crystallisation, is the process of atoms or molecules arr...Mayurkumarpatil1
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For a good business plan creative thinking is important. A business plan is very important and strategic tool for entrepreneurs. A good business plan not only helps entrepreneurs focus on specific steps necessary for them to make business ideas succeed, but it also helps them to achieve short-term and long-term objectives. As an inspiring entrepreneur who is looking towards starting a business, one of the businesses you can successfully start without much stress is book servicing café.
Importance:
Nowadays, network plays an important role in people’s life. In the process of the improvement of the people’s living standard, people’s demand of the life’s quality and efficiency is more higher, the traditional bookstore’s inconvenience gradually emerge, and the online book store has gradually be used in public. The online book store system based on the principle of providing convenience and service to people.
With the online book servicing café, college student do not need to blindly go to various places to find their own books, but only in a computer connected to the internet log on online book servicing café in the search box, type u want to find of the book information retrieval, you can efficiently know whether a site has its own books, if you can online direct purchase, if not u can change the home book store to continue to search or provide advice to the seller in order to supply. This greatly facilitates every college student saving time.
The online book servicing café’s main users are divided into two categories, one is the front user, and one is the background user. The main business model for Book Servicing Café relies on college students providing textbooks, auctions, classifieds teacher evaluations available on website. Therefore, our focus will be on the marketing strategy to increase student traffic and usage. In turn, visitor volume and transactions will maintain the inventory of products and services offered.
Online bookstore system i.e. Book Servicing Café not only can easily find the information and purchase books, and the operating conditions are simple, user-friendly, to a large extent to solve real-life problems in the purchase of the books.
When you shop in online book servicing cafe, you have the chance of accessing and going through customers who have shopped at book servicing café and review about the book you intend to buy. This will give you beforehand information about that book.
While purchasing or selling books at the book servicing café, you save money, energy and time for your favorite book online. The book servicing café will offer discount coupons which help college students save money or make money on their purchases or selling. Shopping for books online is economical too because of the low shipping price.
Book servicing café tend to work with multiple suppliers, which allows them to offer a wider variety of books than a traditional retail store without accruing a large, costly inventory which will help colle
A machine learning method for efficient design optimization in nano-opticsJCMwave
Explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradiant-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A machine learning method for efficient design optimization in nano-optics JCMwave
The slideshow contains a brief explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradient-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.
A brief study on linear programming solving methodsMayurjyotiNeog
This small presentation includes a brief study on various linear programming solving methods. These methods (graphical & simplex) are used to solve industrial engineering related problems in practical use.
Simple Linear Regression: Step-By-StepDan Wellisch
This presentation was made to our meetup group found here.: https://www.meetup.com/Chicago-Technology-For-Value-Based-Healthcare-Meetup/ on 9/26/2017. Our group is focused on technology applied to healthcare in order to create better healthcare.
Crystallization, or crystallisation, is the process of atoms or molecules arr...Mayurkumarpatil1
Crystallization, or crystallisation, is the process of atoms or molecules arranging into a well-defined, rigid crystal lattice in order to minimize their energetic state
LA HUG - Video Testimonials with Chynna Morgan - June 2024Lital Barkan
Have you ever heard that user-generated content or video testimonials can take your brand to the next level? We will explore how you can effectively use video testimonials to leverage and boost your sales, content strategy, and increase your CRM data.🤯
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1. How to capture video testimonials that convert from your audience 🎥
2. How to leverage your testimonials to boost your sales 💲
3. How you can capture more CRM data to understand your audience better through video testimonials. 📊
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Affordable Stationery Printing Services in Jaipur | Navpack n PrintNavpack & Print
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At Techbox Square, in Singapore, we're not just creative web designers and developers, we're the driving force behind your brand identity. Contact us today.
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"𝑩𝑬𝑮𝑼𝑵 𝑾𝑰𝑻𝑯 𝑻𝑱 𝑰𝑺 𝑯𝑨𝑳𝑭 𝑫𝑶𝑵𝑬"
𝐓𝐉 𝐂𝐨𝐦𝐬 (𝐓𝐉 𝐂𝐨𝐦𝐦𝐮𝐧𝐢𝐜𝐚𝐭𝐢𝐨𝐧𝐬) is a professional event agency that includes experts in the event-organizing market in Vietnam, Korea, and ASEAN countries. We provide unlimited types of events from Music concerts, Fan meetings, and Culture festivals to Corporate events, Internal company events, Golf tournaments, MICE events, and Exhibitions.
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➢ Korean President visits Samsung Electronics R&D Center
➢ Vietnam Food Expo with Lotte Wellfood
"𝐄𝐯𝐞𝐫𝐲 𝐞𝐯𝐞𝐧𝐭 𝐢𝐬 𝐚 𝐬𝐭𝐨𝐫𝐲, 𝐚 𝐬𝐩𝐞𝐜𝐢𝐚𝐥 𝐣𝐨𝐮𝐫𝐧𝐞𝐲. 𝐖𝐞 𝐚𝐥𝐰𝐚𝐲𝐬 𝐛𝐞𝐥𝐢𝐞𝐯𝐞 𝐭𝐡𝐚𝐭 𝐬𝐡𝐨𝐫𝐭𝐥𝐲 𝐲𝐨𝐮 𝐰𝐢𝐥𝐥 𝐛𝐞 𝐚 𝐩𝐚𝐫𝐭 𝐨𝐟 𝐨𝐮𝐫 𝐬𝐭𝐨𝐫𝐢𝐞𝐬."
[Note: This is a partial preview. To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
Leading companies such as Nike, Toyota, and Siemens are prioritizing sustainable innovation in their business models, setting an example for others to follow. In this Sustainability training presentation, you will learn key concepts, principles, and practices of sustainability applicable across industries. This training aims to create awareness and educate employees, senior executives, consultants, and other key stakeholders, including investors, policymakers, and supply chain partners, on the importance and implementation of sustainability.
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1. Introduction and Key Concepts of Sustainability
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4. Sustainability Implementation & Best Practices
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2. 2
Optimization Methods
One-Dimensional Unconstrained Optimization
Golden-Section Search
Quadratic Interpolation
Newton's Method
Multi-Dimensional Unconstrained Optimization
Non-gradient or direct methods
Gradient methods
Linear Programming (Constrained)
Graphical Solution
Simplex Method
3. 3
• Basic linear programming (LP) problem consists
of two major parts:
– The objective function
– A set of constraints
• For maximization problem, the objective function
is generally expressed as
n
n x
c
x
c
x
c
2
2
1
1
Z
Maximize
cj= payoff of each unit of the jth activity that is undertaken
xj= magnitude of the jth activity
Z= total payoff due to the total number of activities
Standard Form Linear Programming Problem
4. 4
• The constraints can be represented generally as
where
aij= amount of the ith resource that is consumed for each
unit of the jth activity
bi = amount of the ith resource that is available
• The general second type of constraints specifies
that all activities must have a positive value. i.e,
i
n
in
i
i b
x
a
x
a
x
a
2
2
1
1
Standard Form Linear Programming Problem
0
i
x
5. 5
Problem
Products
Resource Regular Premium Resource
Availability
Raw gas 7 m3/tonne 11 m3/tonne 77 m3/week
Production
time
10 hr/tonne 8 hr/tonne 80 hr/week
Storage 9 tonne 6 tonne
Profit $150/tonne $175/tonne
How many tonnes of regular and premium gas to
produce in order to maximize weekly profit?
6. 6
Developing LP Formulation
Let x1 = 1 tonne of regular gas
x2 = 1 tonne of premium gas
Maximize Z = 150x1 + 175x2
subject to
7x1 + 11x2 ≤ 77 (material constraint) (1)
10x1 + 8x2 ≤ 80 (time constraint) (2)
x1 ≤ 9 (regular gas storage) (3)
x2 ≤ 6 (premium gas storage) (4)
x1, x2 ≥ 0 (positivity) (5, 6)
7. 7
Graphical Solution
Feasible solution space
(the shaded area) contains
all points that obey the
constraints or points that
represent the feasible
solutions.
Points located outside of
this area are known as
infeasible solution.
8. 8
Graphical Solution
We can adjust the line
representing the objective
function Z = 150x2 + 175x2
to locate the optimum
point.
One of the corner points
(points where two or more
lines intersect) will be an
optimum point.
– Corner points can be
feasible or infeasible
9. 9
Graphical Solution
Some insights:
• Increasing storages (
constraints (3) and (4))
does not improve profit.
• Raising either materials or
production time can
improve profit.
10. 10
Possible outcomes that can generally be
obtained in a LP problem
1. Unique solution. The maximum objective
function intersects a single point.
2. Alternate solutions. Problem has an infinite
number of optima corresponding to a line
segment.
3. No feasible solution.
4. Unbounded problems. Problem is under-
constrained and therefore open-ended.
11. 11
(a)Unique solution – represented by a corner point
Alternate solutions – represented by points on one
of the constraint line
(b) No feasible solution
(c) Unbounded problems
12. 12
Key properties of LP that drives the design of
Simplex Method (An algorithm for solving LP
problems)
1. The optimum point is always at a feasible corner
point. (Why?)
– We only need to check the corner points.
2. If a corner point feasible solution has an objective
function value that is better that or equal to all
adjacent corner point feasible solutions, then it is
optimal.
– We don't necessarily need to search all corner
points.
13. 13
Key properties of LP that drives the design
of Simplex Method
3. There are finite number of corner point feasible
solutions.
–Any method that checks only corner points will
terminate eventually.
14. 14
Basic idea behind the Simplex method
1. Starting at a feasible corner point solution.
2. Repeatedly move to a better adjacent corner point
feasible solution until an optimum point is found.
15. 15
Finding Corner Points Algebraically
• How to find corner points algebraically?
• How to locate adjacent corner points?
• How to decide which adjacent point to
move to next?
• Simplex method offers a neat solution
(involves the use of slack variables) to
address these questions.
16. 16
First, constraint equations are reformulated as
equalities by introducing slack variables – variables
that measures how much of a constrained resource is
available.
0
,
,
,
,
,
6
9
80
8
10
77
11
7
to
subject
175
150
Maximize
4
3
2
1
2
1
4
2
3
1
2
2
1
1
2
1
2
1
S
S
S
S
x
x
S
x
S
x
S
x
x
S
x
x
x
x
Z
0
,
6
9
80
8
10
77
11
7
to
subject
175
150
Maximize
2
1
2
1
2
1
2
1
2
1
x
x
x
x
x
x
x
x
x
x
Z
Slack variables
Si > 0 means the corresponding resource is not fully consumed.
Si < 0 means the corresponding resource is over-consumed.
17. 17
• Adding slack variables results in an augmented
system of linear equations which is under specified
(has more unknowns than equations).
– 6 unknowns (= 2 original variables + 4 slack variables)
– 4 equations
• In general, if a LP problem has n variables and m
constraints, the resulting augmented system will
have a total of (n original variables + m slack
variables) and m equations.
Augmented System
Continue …
18. 18
• To solve an augmented system with
– 6 unknowns (= 2 original variables + 4 slack variables)
– 4 equations
We need to set 2 of the variables constants before
we can solve the system of equations.
e.g., Setting x1 = 0, x2 = 0 yields
Augmented System
Continue …
6
9
80
77
4
3
2
1
S
S
S
S
6
9
80
8
10
77
11
7
4
2
3
1
2
2
1
1
2
1
S
x
S
x
S
x
x
S
x
x
19. 19
Solving the resulting system yields x1=0, x2=0, S1=77, S2=80,
S3=9, S4=6, which represent one of the augmented solutions –
values of all (original + slack) variables are given.
For this example, the values tell us that if we don't produce any regular (x1)
or premium (x2) gas, we would have the following amount of unconsumed
resources:
– 77 m3 of raw gas,
– 80 production hours
– 9 tonne of regular gas storage
– 6 tonne of premium gas storage
6
9
80
77
4
3
2
1
S
S
S
S
Augmented System
20. 20
• An augmented solution which corresponds to a
corner point is also known as a basic solution.
– The augmented solution x1=0, x2=0, S1=77, S2=80, S3=9,
S4=6 is a basic solution. It corresponds to corner point A.
– In addition, it is a basic feasible solution because all
variables ≥ 0.
• So far, you know how to find an augmented
solution.
– How should we set the variables so that the augmented
solution is a basic solution?
Augmented System
Continue …
21. 21
• The equations (or constraints) have a one-to-one
relationship with the slack variables.
– For example, S1 relates to the 1st equation, S4 relates to
the 4th equation.
• If a point is on the line 7x1 + 11x2 = 77 (line labeled
"1"), what's the value of S1?
• If a point is on the line 10x1 + 8x2 = 80 (line labeled
"2"), which variable has the value zero?
• At the corner point D, which variables have the
value zero?
Characteristics of Slack Variables
Show Graph & Equations
22. 22
• Setting two variables to zero has the effect of
"selecting two lines", and solving the resulting
system means finding the intersecting point (a
corner point) of the selected lines.
Characteristics of Slack Variables
6
9
80
8
10
77
11
7
4
2
3
1
2
1
2
1
S
x
S
x
x
x
x
x
The value of x1 and x2 of this system is the coordinate of
the corner point C (the point where line "1" and line "2"
intersect.)
For example, let S1=0, S2=0. Then the system becomes
23. 23
Finding Corner Points Algebraically
We now know how to find the corner points.
Next
• How to find an adjacent corner point?
• How to decide which adjacent corner point
to move to?
24. 24
Basic and Non-Basic Variables
Corner
Point
Non-Basic
Variables
Basic
Variables
Some of the
Adjacent
Corner Points
A x1, x2 S1, S2, S3, S4 B, E
B x2, S2 x1, S1, S3, S4 A, C
C S1, S2 x1, x2, S3, S4 B, D
D S1, S4 x1, x2, S2, S3 C, E
E x1, S4 x2, S1, S2, S3 A, D
F x1, S2 x2, S1, S3, S4
Variables in which their values are set to zero are
called non-basic variables.
Remember: Non-basic, variable set to zero,
corresponding constraint is active
Do you see a
pattern between
any pair of
adjacent corner
points?
25. 25
Basic and Non-Basic Variables
• If two points are adjacent corner points, then
– Their basic sets are identical except for one member.
– Their non-basic sets are identical except for one
member.
• The reverse of the above statement is not always
true. (e.g., F and A)
• Questions:
– To find an adjacent corner point, can we just switch one
variable between the basic and non-basic sets?
– How should we decide which corner point to move to
next?
26. 26
Basic and Non-Basic Variables
How do we make sure the following
conditions are satisfied?
– The corner points are adjacent.
– The corner points are feasible.
– The new corner point gives better value
for the objective function than the current
corner point.
27. 27
Which variables to switch?
All three conditions from the previous slide can be
satisfied if the following steps are taken:
• Selecting an entering basic variable – a non-
basic variable that will increase the objective
function value the most if allowed to take on a
positive value. Move this variable from the non-
basic set to the basic set.
• Finding the leaving basic variable – a basic
variable that is forced to a value zero first after
allowing the entering basic variable to increase.
Move the variable that is forced to zero from the
basic set to the non-basic set.
28. 28
Which variables to switch?
Suppose we are currently at point A.
Basic set = { S1, S2, S3, S4 }, non-basic set = { x1, x2 }
Which variable should be the entering basic variable?
Which variable should be the leaving basic variable?
0
,
,
,
,
,
6
9
80
8
10
77
11
7
to
subject
175
150
Maximize
4
3
2
1
2
1
4
2
3
1
2
2
1
1
2
1
2
1
S
S
S
S
x
x
S
x
S
x
S
x
x
S
x
x
x
x
Z
29. 29
Which variables to switch?
x2 has a larger coefficient in the objective function, so
we should make x2 the entering basic variable. (x1
remains a non-basic variable and has the value zero)
If we keep increasing x2, other basic variables will
decrease. Which basic variable will become zero
first?
0
,
,
,
,
,
6
9
80
8
77
11
to
subject
175
150
Maximize
4
3
2
1
2
1
4
2
3
2
2
1
2
2
1
S
S
S
S
x
x
S
x
S
S
x
S
x
x
x
Z
30. 30
Which variables to switch?
If we take the ratio of the RHS value to the coefficient
of the entering basic variable (x2), the smallest non-
negative ratio can tell us which basic variable will
reach zero first if we keep increasing the value of x2.
So S4 should be made the leaving basic variable.
0
,
,
,
,
,
)
6
1
/
6
(
6
N/A
9
)
10
8
/
80
(
80
8
)
7
11
/
77
(
77
11
to
subject
175
150
Maximize
4
3
2
1
2
1
4
2
3
2
2
1
2
2
1
S
S
S
S
x
x
S
x
S
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Show Graph & Equations
Minimum
ratio test
31. 31
Which variables to switch?
So setting x1=0, S4=0 and solve the system of
equations would yield an augmented solution
correspond to point E.
Solving the system of equations at every corner point
is not very efficient. Simplex method offers an
efficient way to compute the value of each variable
by updating the equations incrementally.