BUS 212 QA
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Lp models (Basic Features)
Dual and Primal
Graphical method of solution to LP models
Feasible region with an example
Types of kinds of models with example
Analytical method of solution LP model
Model description
Transportation modelling using North-West corner method and explain degeneracy
SOLUTIONS
1
Linear programming is a process of optimizing the problem which are subjected to certain
constraints. It means that it is the process of maximizing or minimizing the inner functions under
linear inequality constraint.
Every business has an objecttive function, every objective set will contenal limity constraint
1. Objective function; minimize cost or maximize profit
2. Constraints; constraint function is given as inequality function, minimization problem
(≥), maximization problem (≤).
3. Non negativity constraints; at which when set at zero (opposite of constraint at zero)
Examples
Minimize cost = 2A + 5B
\
Objective function
13A + 15B ≥71
21A + 12B≥100-----constraints
A,B≤0--------non negativity constraints
2
Dual and Primal
Every linear is a dual or primal and every dual has primal
Example
If the example given below is dual problem, find the primal
Minimize cost = 5x +8y
Subject to
6x-3y≥35
4x+8y≥90
x, y ≤0
The primal will be
Maximize cost = 35A+90B
6A+4B≤5
-3A+8B≤8
A,B≥0
3
Graphical method of solution to LP models
Example
2x+4y = 12
2x-y = -1
x-2y = -8
Solution. The use of intercept method
2x –y = -1 ----- equ(i)
x-2y= -8-------- equ(ii)
From equ(i) the x and y intercept
x- intercept when y = 0
2x-y = -1
2x-0 = -1
2x = -1
2
2
x = -1 or -0.5
2
Then the coordinate here is
(x,y) = (-0.5, 0)
Also y- intercept when x = 0
2x-y = -1
2(0) -y = -1
0-y = -1
-y = -1
-1 = -1
y=1
Then the coordinate here is
(x,y) = (0, 1)
From equ(ii) the x and y intercept
x- intercept when y = 0
x-2y = -8
x-2(0) = -8
x-0 = -8
x = -8
1
1
x = -8
Then the coordinate here is
(x,y) = (-8, 0)
Also y- intercept when x = 0
x-2y = -8
0-2y = -8
-2y = -8
-2
-2
y=4
The coordinate here is (x,y) = (0, 4)
Equ(i) coordinate
(x,y) (-0.5,0)
(x,y) (0,1)
Equ(ii) coordinate
(x,y) (-8,0)
(x,y) (0, 4)
4
Feasible region with an example
A feasible region is an area defined by a set of coordinates that satisfy a system of inequalities
Example
Consider this linear problem models form shown below
Maximize profit = 6x+3y
Subject to
3x-2y≤14
4x+8y≤24
x,y≥0
From the above it is clear that the objective function is maximization problem. Hence need to
compute the solution of the two variable x and y involved solving the problem requires. The use
of graphical method of solution
Solution
The use of intercept method
3x-2y = 14 -----equ(i)
4x+8y = 24 -----equ(ii)
Equ(i) x intercept when y = 0
3x-2(0) = 14
3x-0 = 14
3x = 14
2
3
x = 4.667
Coordinate is (x,y) = (4.667, 0)
y- Intercept when x = 0
3x-2y = 14
3(0)-2y = 14
0-2y = 14
-2y = 14
-2
-2
y = -7
The coordinate here is (x,y) = (0, -7)
Equ(ii)
4x+8y = 24
x- intercept when y = 0
4x + 8(0) = 24
4x+0 = 24
4x = 24
4
4
x=6
Coordinate here is (x,y) = (6,0)
y- Intercepts when x = 0
4x+8y = 24
4(0)=8y = 24
0+8y = 24
8y = 24
8y = 24
8
8
y=3
The coordinate here is (x,y) = (0,3)
5
Types of kinds of models with example
1. Linear regression model; this model is used to model the relationship between a
dependent variable and one or more independent variables. For example, a linear
regression model can be used to predict house prices based on variables such as square
footage, number of bedrooms and location
2. Logistic regression model; this model is used when the dependent variable is binary or
categorical. It estimates the probability of an event occurring based on one or more
independent variables, for example, a logistic regression model can be used to predict the
likelihood of a customer buying a product based on their demographic characteristics
3. Time series models; these models are used to analyze data collected over time, such as
stock prices or weather data. Examples of time series models include autoregressive
integrated moving average (ARIMA) models and exponential smoothing models
4. Decisiontrees; this model is used for classification and prediction problems. It use a tree
like graph to model decision based on the input variables. For example, a decision tree
can be used to predict whether a customer will buy a product based on their age, income
and other demographic factors.
5. Neural networks; these models are used to model complex relationship between
variables. They can be used for various purposes such as image recognition, speech
recognition and natural language processing.
6. Transportation models; these models help in optimizing transportation and logistics
operations. For example, a shopping company may use LP to determine the most efficient
routes and shipping schedules to minimize costs and maximize service levels
6
Analytical method of solution to LP model
Example
The LP model function given below is
Minimize cost = 30x+28y
Subject to
3x+4y≥3
5x+2y≥2
x,y≤0
Solution [crammar’s rule]
3x+4y=3 ------ Equ(i)
5x+2y=2-------Equ(ii)
A=
, B=
|A| =
|A| = (3.2) – (5.4)
|A| = 6-20
|A| = -14
X=
Check in equ(ii)
5x=2y = 3
5(6.1429) + 2(0.6429) = 2
X=
0.7145 + 1.2858 = 2
2.003 = 2
X=
X=
X = = or 0.1429
Y=
Check in equ(i)
3x+4y = 3
Y=
3(0.1429)+4(0.6429)
Y=
0.4287+2.5716
Y=
3.003 = 3
Y=
or 0.6429
Minimize cost = 30x+28y
= 30(0.1429)+28(0.6429)
= 4.287 + 18.0012
= 22.28712
7
Model description
Models are put together to show relationship between two or more variables making provision
for their error terms
Variables are identified characteristics that shows or explain behavior of any element, when an
element is seen too consistently demonstrate or exemplify similar and stable characteristics, it
therefore means that those characteristics features are consistent over time with the indentified
element.
In operational research, one basic feature of anything refer to as a model
It identical characteristics, there are physical model, iconic model, statistical model,
simulation model, time series model, decision trees model
Model variables are to take any value assigned to them discrete or continuous. Evidently
in models, there should be both dependent and independent
Y is the function of X = F (X) is a type of model that explain Y as the dependent variable while
X is the independent variable. The two variables, variable X and Y are representatives of real life
situations that are being considered for explanation
Example; “the impact of motivation on workers output” here let X be motivation and Y
be the workers output.
8
Transportation modelling using North-West corner method
This is a type and kind of linear programming model and is a cost minimization technique,
demand must be the same as supply and if it is not the same, you will use degeneracy method.
The source destination must be known and quantity available for supply must be ascertained.
Degeneracy: this is when number of rows is not equal to number of column i.e x+y a
domain variable/rows/column with value of zero
Destination
Source
Abia
Owerri
Cross River
Delta
250
50
-
Enugu
10
30
15
300
50
r+c-1
3+3-1 = 5 cells
Five cells
(1,1) 250×10 = 2,500
(2,1) 50×30 = 1,500
(2,2) 250×35 = 8,750
(3,2) 50×40 = 2,000
(3,3) 400×30 = 12,000
26,750
250
-
0
FCT
20
35
25
250
0
50
400
Supply
30
40
30
450
400
250 0
350 300
400 0
1000
0
50 0