TS Eng ROAD Intro prog Lineaire HR 14.ppt

Linear programming
Introduction

The Linear Programming (LP) has proven that it is a very
powerful method in the search for solutions to problems
arising from various areas,

The application of linear programming is extremely broad. It is
applied in industry, trade, finance, agriculture, marketing,
public service, etc.

The PL consists of two main parts:


Formulate the problem that concerns us,
Solving this problem.
2
Introduction
The following table shows some examples of actual use of Operations
Research by different organizations and profits and / or savings
achieved as a result by applying Linear Programming:
Organization
Application
Dutch Ministry of Infrastructure and National Water management policy
the Environment (Netherlands
development, adding new facilities,
Rijkswaterstaat)
operating and costs procedures
Year
Yearly savings
1985
$15 millions
Monsanto Corp.
Production's operations optimization to
obey goals with a minimum cost
1985
$2 millions
Weyerhaeuser Co.
Cutting trees optimization to maximize
wood products production
1986
$15 millions
Electrobras/CEPAL Brasil
Optimal allocation of hydraulic and thermic
resources in the national energy
1986
generation system
$43 millions
United Airlines
Shifts at book offices and airports
scheduling to accomplish with the
customer needs at minimal cost
$6 millions
1986
3
Introduction
CITGO Petroleum Corp.
Optimization of refinement, offer, distribution and
commercialization of products operations
1987
$70 millions
Santos, Ltd., Australia
Capital investment optimizing to produce natural gas
along 25 years in Australia
1987
$3 millions
1989
$59 millions
San Francisco Police
Department
Optimization programming and assignment of Patrol's
1989
officers with a computed system
$11 millions
Texaco Inc.
Optimizing the mixing of ingredients available in order
to obtain fuels which met with the quality requirements 1989
and sales
$30 millions
Integration of a national network of spare parts
inventory to improve support service
$20 millions
+ $250
millions in
minor
inventory
Administration of oil and coal inventories for the
Electric Power Research
electric service with the intention of balancing
Institute
inventory costs and risks of remaining
IBM
1990
4
Introduction
U.S. Military Airlift
Command
Rapidity in the airplanes, crew, load and
passengers coordination to drive the evacuation
1992
by air in the "Desert Storm" project in the Middle
Orient
Victory
American Airlines
Design of a pricing, overbooking and flight
coordination structure system to enhance
benefits
1992
$500 millions
of additional
revenue
Yellow Freight System,
Inc.
Optimizing the design of the national transport
network and the scheduling of shipping routes in 1992
the U.S.
$17.3 millions
New Haven Health Dept.
Design of an effective program of needles
change to combat the AIDS contagion
1993
33% less of
contagions
AT&T
Development of a computer system to design
call centers to guide customers
1993
$750 millions
5
Introduction
Delta Airlines
Maximizing profits from the allocation of aircraft
types in 2.500 national flights in the U.S.
1994
$100 millions
Restructuring of the whole supply chain among
Digital Equipment Corp. suppliers, plants, distribution centers, potential sites 1995
and market areas
$800 millions
Selection and optimum programming of mass
projects to obey with future energy needs of the
country
1995
$425 millions
Optimal restructuring of the size and form of the
South African National
South African National Defence Force and his
Defence Force (SANDF)
weapons system
1997
$1.100
millions
China
Procter & Gamble
Redesign of the North American production and
distribution system to reduce costs and to improve 1997
the incoming rapidity to the market
$200 millions
Taco Bell
Optimum employees programming to provide the
service to desired clients with a minimum cost
1998
$13 millions
Hewlett-Packard
Redesign of security inventories' size and location
at printer production line to obey the production
goals
1998
$280 millions
of additional
revenue
6
Introduction
General processes used to formulate each problem are:

Determine the essential elements constituting the treaty system
(such as capital, labor, materials, ...),

Set the goal (profit maximization or cost minimization),

Represent the problem mathematically (put the problem in
mathematical form) to find the optimal solution.
7
Introduction
Formulation or modeling: Some advice

Using a set of mathematical relationships to reflect as
accurately as possible a real situation

Compromise between the fit with the reality and easy to solve
the model
8
Mathematical model
Three entities to identify
i)
ii)
iii)
all actions (activities) available to the decision maker
(variables)
the objective is expressed as a (objective function)
mathematical function
Constraints defining the nature of the system under study
expressed in terms of mathematics (constraints) relationship
9
Resolution

Three entities to identify
i)
all actions (activities) available to
the decision maker (variables)
the objective is expressed as a
(objective function) mathematical
function
Constraints defining the nature of
the system under study expressed
in terms of mathematics
(constraints) relationship
ii)
iii)

Model resolution
Using a procedure (algorithm)
method to determine
1. variable values representing the
amplitude of the use of various
actions
2. to optimize the economic function
(reach the goal)
3.
within the constraints imposed
10
Linear model
Three specific properties :
1.
Additivity of variables: the global effect of actions
(variables) is equal to the sum of the particular effects of
each action (variables). There is therefore no cross effect of
actions (independent variables)
2.
The variables are always non-negative values
3.
The order of power is equal to one for each variable
11
Example: Problem of diet





3 types of grain available to feed the flock : g1, g2, g3
Each kg of grain contains 4 nutrients : ENA, ENB, ENC, END
Weekly required quantity of each nutrient is specified
The price per kg of each type of grain is specified.
Problem: Determine the quantity of each grain (kg) used to establish a diet
at minimum cost respecting the required quantity of each nutrient
12
Given of Problem
quantity





3 types of grain available to feed
the flock : g1, g2, g3
Each kg of grain contains 4
nutrients : ENA, ENB, ENC, END
Weekly required quantity of each
nutrient is specified
The price per kg of each type of
grain is specified.
Problem: Determine the quantity
of each grain (kg) used to
establish a diet at minimum cost
respecting the required quantity of
each nutrient
g1
g2
g3
Weekly
1250
250
900
232.5
ENA
ENB
ENC
END
2
3
1
1
5
3
0.6 0.25
7
0
0
1
$/kg
41
96
35
13
Variables of problem





3 types of grain available to feed
the flock : g1, g2, g3
Each kg of grain contains 4
nutrients : ENA, ENB, ENC, END
Weekly required quantity of each
nutrient is specified
The price per kg of each type of
grain is specified.
Entities of model
variables
i) Actions
# kg of g1
x1
# kg of g2
x2
# kg of g3
x3

Problem: Determine the quantity
of each grain (kg) used to
establish a diet at minimum cost
respecting the required quantity of
each nutrient
14
Economic Function and constraints
Weekly
quantity
ii) Economic Function
Cost of diet per week =
41x1 + 35x2 + 96x3
To minimize
ENA
ENB
ENC
END
iii) Constraints
ENA:
ENB:
ENC:
END:
2x1 + 3x2
+7x3
1x1 + 1x2
5x1 + 3x2
0.6x1 + 0.25x2 + x3
≥ 1250
≥ 250
≥ 900
≥ 232.5
$/kg
g1
g2
2
3
1
1
5
3
0.6 0.25
41
35
g3
7
0
0
1
1250
250
900
232.5
96
No negativity of variables :
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
15
Mathematical model
min z = 41x1 + 35x2 + 96x3
With Constraints:
2x1 + 3x2
+7x3 ≥ 1250
1x1
+ 1x2
≥ 250
5x1
+ 3x2
≥ 900
0.6x1 + 0.25x2 + x3 ≥ 232.5
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
16
Problem of restorer

Availability of restorer:
30 urchins
24 shrimp
18 oysters

Two types of seafood dishes offered by the restaurant :
first type is composed from 5 urchins, 2 shrimp and one oyster with profit 8 $,
 Second type is composed from 3 urchins, 3 shrimp and 3 oysters with profit 6 $


Problem: determine the number of plates of each type to provide for the
restorer maximizes its income in accordance with the availability of
seafood
17
Variables of problem

Availability of restorer:
30 urchins
24 shrimp
18 oysters

Two types of seafood dishes offered
by the restaurant :




Entities of model
variables
i) Actions
# plates $8
x
# plates $6
y
first type is composed from 5 urchins, 2
shrimp and one oyster with profit 8 $,
Second type is composed from 3
urchins, 3 shrimp and 3 oysters with
profit 6 $
Problem: determine the number of plates of
each type to provide for the restorer
maximizes its income in accordance with the
availability of seafood
18
Economic function and constraints


Availability of restorer:
30 urchins
24 shrimp
18 oysters
Two types of seafood dishes offered
by the restaurant :



first type is composed from 5 urchins, 2
shrimp and one oyster with profit 8 $,
Second type is composed from 3
urchins, 3 shrimp and 3 oysters with
profit 6 $
Problem: determine the number of plates of
each type to provide for the restorer
maximizes its income in accordance with the
availability of seafood

Entities of model
ii) Economic Function
restorer income =
8x + 6y
To maximize
iii) Constraints
urchins : 5x + 3y ≤ 30
shrimp : 2x + 3y ≤ 24
oysters : 1x + 3y ≤ 18
non-negativity of variables:
x,y≥0
19
Mathematical model
max 8x + 6y
With constraints:
5x + 3y ≤ 30
2x + 3y ≤ 24
1x + 3y ≤ 18
x,y≥0
20
Exercise
Sam wants to invest $ 5,000 next year in the two types of
investment: A and B.
The investment brings in A 5% while the investment B earns
him 8%.
A study of market suggests that it must invest at least 25% in
A and at most 50% in B.
In addition, the amount of investment in A must be at least
half the amount of investment in B.
Establish a Model of this problem as a linear program.
21
Exercise
A bank plans its operations for the next year. the bank provides three types of
loans whose rates of return are the following:
loans
rates of return
personal
14%
household
10%
Auto (car)
12%
the policy of the bank imposes certain restrictions on the distribution of
amounts in the different categories. Personal loans should not exceed 25%
of budget of the bank while the personal and household loans should not
exceed 45% of the budget. The amount allocated to auto loans should not
exceed 70% of the budget but should represent at least 80% of the amount
allocated personal and household loans. The bank has a budget of $
500,000 to be divided into different categories.
Write a linear program to maximize returns of this bank.
22
Exercise
Top Toys prepares new advertising campaign on radio and TV.
Each radio advertising costs him $ 300, each television advertising costs him $
2,000.
Top Toys has a budget of $ 20,000 for his campaign. However, to ensure that there
is at least one radio and one TV commercial advertising, it was decided that no
more than 80% of the total budget can be allocated to each type of media.
A study predicts that the first radio advertising will be followed by 5000 listeners,
and only 2,000 listeners for the following.
The same study states that the first commercial TV viewers will be followed by
4500, and 3000 only for the following viewers.
Write a linear program for the company's Top Toys.
23
Resolution Graphic Method

Methods for problem with only two variables

Back to the problem of the restorer
max z = 8x + 6y
With following Constraints:
5x + 3y ≤ 30
2x + 3y ≤ 24
1x + 3y ≤ 18
x, y ≥0
24
Domain of possible solutions
Let us draw the straight line that has
the following equation:
5x + 3y = 30
The set of points that satisfy the
constraint
5x + 3y ≤ 30
are under this line because all points
satisfy this relationship
25
Domain of possible solutions
Let us draw the straight line that has
the following equation:
2x + 3y = 24
The set of points that satisfy the
constraint
2x + 3y ≤ 24
are under this line because all
points satisfy this relationship
26
Domain of possible solutions

Now Let us draw the straight line
that has the following equation:
1x + 3y = 18
The set of points that satisfy the
constraint
1x + 3y ≤ 18
are under this line because all points
satisfy this relationship
27
Domain of possible solutions
The set of points that satisfy all
constraints:
5x + 3y ≤ 30
2x + 3y ≤ 24
1x + 3y ≤ 18
x,y≥0
28
Optimal solution


Consider the economic function :
z = 8x + 6y.
More we move away from the
origin, Mor the value increases :
x = 0 et y = 0 => z = 0
29
Optimal solution


Consider the economic function :
z = 8x + 6y.
More we move away from the
origin, Mor the value increases :
x = 0 et y = 0 => z = 0
x = 0 et y = 6 => z = 36
30
Optimal solution


Consider the economic function :
z = 8x + 6y.
More we move away from the
origin, Mor the value increases :
x = 0 et y = 0 => z = 0
x = 0 et y = 6 => z = 36
x = 6 et y = 0 => z = 48
31
Optimal solution



Consider the economic function :
z = 8x + 6y.
More we move away from the
origin, Mor the value increases :
x = 0 et y = 0 => z = 0
x = 0 et y = 6 => z = 36
x = 6 et y = 0 => z = 48
x = 3 et y = 5 => z = 54.
Impossible to go further without
leaving the possible domain.
Optimal Solution :
x = 3 et y = 5
Optimal Value :
z = 54
32
Another Example
A company wants to make two different kinds of drinks.
The technical characteristics of the production and the placing
of labels for each 12 bottles are given in the following table :
Type of drink
production
Labeling
12 bottles of A
2 hours
1 hour
12 bottles of B
3 hours
4 hours
33
Another Example
The company makes a profit of $ 10 on a dozen bottles of A
and $ 20 on a dozen bottles of B.
Given that the Department of production has 20 hours
available and the department of labeling has 15 hours
available ,
Find the number of dozens of drink A and B that the
company needs to do for maximize its profit.
34
Solution
Mathematical model :
2x1 + 3x2  20
x1 + 4x2  15
x1, x2  0
x1 = 7, x2 = 2
Z = 110
Max Z = 10x1 + 20x2
Method 1: the straight line of Z must be moved
in parallel direction of itself to farthest point
of the polygon for the maximum value
35
Solution
Mathematical model :
2x1 + 3x2  20
x1 + 4x2  15
x1, x2  0
x1 = 7, x2 = 2
Z = 110
Max Z = 10x1 + 20x2
2nd Method: Calculate the corners of the
polygon and put their values in Z to find the
maximum value
36
Exercise
A customer asks a company to manufacture monthly her
products for two types A and B. This client is able to pay the
following prizes::


138$ per set of 100 pieces of A,
136$ per set of 100 pieces of B.
The Manufacture of A requires their passage through three
workshops:
37
Exercise
number of hours
required for the
production of a series
of 100 pieces of A
number of hours
required for the
production of a series
of 100 pieces of B
Cost/hour of work
Workshop 1
2
1
10$
Workshop 2
1
4.5
12$
Workshop 3
4
3
14$
However, the company has a limited number of hours in each workshop
as it has:
 200 hours for Workshop 1
540 hours for Workshop 2
480 hours for the workshop 3
38
Exercise
What are the quantities to make A and B within the time
limit required and the resources of the company; all
seeking to maximize profit?
Give a graphical solution.
39
Type of solutions
In general we can identify three types of solutions in the
linear programming:



A unique optimal solution,
No optimal solution,
A set of optimal solutions
40
Type of solutions
Example: no optimal solution
Min Z = x + 2y
With constraints:
x+y1
x2
x, y  0
41
Type of solutions
Example: A set of optimal solutions
Max Z = x + y
With constraints:
x + 3y  18
x+y8
2x + y  14
x, y  0
42
Linear programming in three variables
The space geometry is used to solve LP problems with three
variables.
For example:
x1  1000
x2  500
x3  1500
3x1 + 6x2 + 2x3  6750
x1, x2, x3  0
Max z = 4x1 + 12x2 + 3x3
43
Difficulties of generalization
The geometric solution can not of course extend to the
case of a number greater than three variables.
For this reason we need methods that are able to solve LP
problems with n variables of activities such that n> 3.
There is a method called a simplex algorithm to resolve this
kind of problems.
44