Definition Of Linear Program

In the activeness of production together with trade, both inwards large together with modest scale manufacture tin sack non endure separated from the lucre work that must endure obtained yesteryear the company. The master copy objective is to earn the maximum income yesteryear minimizing expenditures (raw fabric costs, production procedure costs, employee salaries, transportation, etc.)

For this purpose, commonly the company's administration makes several possibilities inwards determining the strategy that must endure taken to attain it. For event inwards producing ii kinds of goods amongst dissimilar costs together with benefits. The companionship tin sack calculate the profits that may endure obtained yesteryear magnitude amongst honour to the required materials, lucre per unit, shipping costs, together with others.

To solve the work used linear program. Linear programme is defined every bit a way to solve a work yesteryear using mathematical methods formulated inwards the cast of equations or linear inequalities.

To obtain the optimum solution, a graphical method applied to a elementary linear programme consisting of ii variables yesteryear agency of a corner exam or a work exam inwards the laid upwards expanse of completion.

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Mathematical Model Of Even Out Problem

Definition of Mathematical Model

The close of import thing inwards linear programming is to transform the verbal thing into a mathematical model that is a daily linguistic communication pronunciation into a simpler together with easier to empathize linguistic communication of mathematics. So the mathematical model is a formula derived from an interpretation when translating a verbal matter.

How To Convert Verbal Sentence To H5N1 Mathematical Model In The Form Of Inequality System

To arrive easier to alter the verbal problems inwards the shape of linear programme into the working capital missive of the alphabet mathematics used tabular array every bit follows:

Example:
To brand staff of life A required 200 grams of flour together with 25 grams of butter. As for staff of life B required 100 grams of flour together with 50 grams of butter. The available flour is exclusively 4kg together with the butter is 1.2kg. If the toll of staff of life A Rp. 400,00 together with staff of life B cost Rp. 500.00. Make a mathematical model!

Answer:
Suppose a lot of staff of life A = x together with lots of staff of life B = y, pregnant other variables are flour together with butter. So the tabular array is obtained every bit follows:

Flour together with butter available at close 4 kg = 4000 grams together with 1.2 = 1200 grams hence the sign of inequality is "<". From the tabular array tin last made inequality every bit follows:
200x + 100y < 4000, simplified to:
2x + y < 40

25x + 50y < 1200, simplified to:
x + 2y < 48 

Since x together with y are nonnegative integers then:
x > 0 
y > 0

The higher upwards 4 inequalities are requirements that must last met called the constraint function. the toll of staff of life A Rp.500,00 together with staff of life B Rp.400,00, hence the sales proceeds tin last formulated amongst Z = 400x + 500. y: Z is called objective component subdivision or target component subdivision that tin last maximized or minimized.

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Graphs Of 2 Variables Linear Inequality Set

The 2 linear inequalities of the 2 variables are inequalities containing 2 variables such every bit x in addition to y. The ready of inequalities short town tin endure presented inwards the cartisius field. The forms of linear inequality are every bit follows:

ax  +by < c
ax + past times < c
ax + past times > c
ax + past times > c

How to Complete The Two Variables Linear Inequality Set

Here are the steps taken to consummate the ready of 2 linear inequality sets:

• Draw ax + past times = c line of piece of job of cartesius past times searching. Cut points of graph alongside x axis (y = 0) and  y axis (x = 0).
• Take whatever betoken P (x₁ y₁) that is non located on the line, thence calculated the value of ax₁ + by₁. The value of ax₁ + by₁ is compared to the value of c.
• The short town portion for inequality ax + past times < c is determined every bit follows:
  If ax₁ + by₁ < c, thence the portion containing P is the short town region.
  f ax₁ + by₁ > c, thence the portion containing P is non a short town region.
• The short town portion for inequality ax + by > c is determined every bit follows:
  If ax₁ + by₁ > c, thence the portion containing P is the short town region.
  If ax₁ + by₁ < c, thence the portion containing betoken P is non a short town region.
• Areas that are non short town are given hatching, thence the short town expanse is an expanse alongside no shade. This is helpful when determining areas that regard closed to inequalities.
• Settlement areas for inequalities containing equal signals are drawn alongside total lines, whereas inequality settlements that produce non incorporate an equal sign are inwards the figure alongside dashed lines.

Example:
Find the settling portion of 2x + y < 4

Answer:
For how to discovery the intersection of the graph alongside the x-axis in addition to y-axis tin endure searched past times creating the next table:

Then the intersection points alongside the ten in addition to y axes are (2, 0) in addition to (0, 4)

Take the betoken P (0,0) every bit the examination betoken at 2x + y < 4 in addition to obtain 2.0 + 0 < 4.

because 2.0 + 0 < 4, thence the expanse containing P is the short town region.

Then the graph becomes:

The non-shaded expanse is the short town portion of 2x + y < 4.

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Target Purpose Optimum Value Of Regional Linear Inequality System

The of import affair inwards the linear programming occupation is to alter the occupation of variables into the cast of mathematical model which is the representation of everyday linguistic communication into a simpler as well as easier to empathize math language.

Steps to Get Optimum Value

Here are the steps that must endure taken to become the optimum value:

1. Convert variable issues into mathematical models.
2. Define Settlement Set.
3. Determine all the corner points inwards the faesible area.
4. Calculate the value of the objective cast for each corner indicate inwards the viable region.
5. From the lawsuit inwards measurement 4, the maximum or minimum value tin endure set.

Exmple:

An plane has a seating capacity of no to a greater extent than than 48 persons. Each principal plane rider tin comport lx kg baggage as well as economic scheme plane xx kg, spell the aircraft has a capacity of non to a greater extent than than 1440kg. If the ticket cost for the principal plane as well as economic scheme are Rp.1.000.000,00 as well as Rp.500.000,00 per person, decide the number of passengers per plane for maximum ticket sales result!

Answer:
The mathematical model is constructed past times supposing the number of principal plane passengers = x people as well as the number of economic scheme plane passengers = y people.

Maximum Z = 1.000.000x + 500.000y

Capacity requirements:
x + y < 48
60x + 20y < 1440
x > 0
y > 0

From the mathematical model inwards tin endure OBAC viable surface area alongside indicate B searched equally follows:

x + y = 48
y = 36

as well as thence the indicate B (12, 36)

Test the points of the corner, ie points O, A, B, as well as C.

The maximum value Z is Rp. 30.000.000,00 is fulfilled past times x = 12 as well as y = 36, or inwards other words ticket sales volition endure maximum if number of principal plane passengers are 12 people as well as economic scheme plane are 36 people.

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Probing Draw Of Piece Of Occupation Inwards Linear Program

The probing describe of piece of employment is a describe of piece of employment used to investigate the optimum value obtained from the objective function.

The optimum value of the objective shape of the fix of inequality arrangement resolutions other than past times using the corner signal method tin too hold upwards solved past times using the probing line.

Steps to Find the Optimum Value past times The Probing Line

Here are the steps to abide by the optimum value past times using the probing describe of piece of employment method:

1. Make the describe of piece of employment ax + past times = k, where ax + by is the objective shape that the optimum value is sought. To transcend away inwards easier to guide keep k = ab.
2. Make parallel lines ax + past times = k, ie past times taking dissimilar k or shifting ax + past times = k describe of piece of employment to the left or right.

• If ax + past times = k₁ is the leftmost describe of piece of employment inwards the short town portion through the signal (x₁, y₁), thence k₁ = ax₁ + bx₁ is the minimum value.
• If ax + past times = k₂ is the rightmost describe of piece of employment inwards the short town portion through the signal (x₂, y₂), thence k₂ = ax₂ + by₂ is the maximum value.

Example:
Using the probing describe of piece of employment method, cause upwards one's hear the maximum together with minimum values of the objective component Z = 2x + 3y inwards the viable portion inwards the figure below:

Answer:
To cause upwards one's hear the maximum together with minimum of the starting fourth dimension done is to brand the describe of piece of employment equation of the known objective component is 2x + 3y = half-dozen = k, together with named amongst the describe of piece of employment g.

Look at the pic above. Scroll the describe of piece of employment g thence equally to cutting the feasble portion at the leftmost point, ie the describe of piece of employment g₁ which is the describe of piece of employment parallel to the describe of piece of employment g together with correct through the signal (1, 2). Thus the minimum value Z is k₁ = ii (1) + iii (2) = 8. While g₂ is the almost correct together with correct describe of piece of employment through the signal (5, 4). Thus the maximum value of Z is k₂ = ii (5) + iii (4) = 22.

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