# Operations Research

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Operations Research (MTH601)
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SOLUTION TO LINEAR PROGRAMMING PROBLEMS
Having formulated a linear programming problem, we have to find the solution for the same. To find the
solution to a linear programming model, we have the graphical and the algebraic methods that can be applied
successfully.
If the problem involves only two decision variables, graphical method of solution is quite adequate. Even
when three decision variables are involved, a graphical solution can be resorted to. But this involves three-
dimensional representation. Therefore we can conveniently restrict the graphical method to problems involving two
decision variables.
The second method is the algebraic procedure or simplex method, which involves an iterative procedure.
This can be applied to linear programming problems involving two or more variables. This is the most versatile
method of solving linear programming models. In this method, an initial solution is assumed and this solution s
modified progressively through a well-defined iterative process until we get the optimal solution. The procedure is
simple and so mechanical that it requires time and patience to execute it manually. However, an electronic computer
with the readymade programmes can be used as a handy tool for solving LP problems involving many variables.
Graphical solution to LPP
We all know that a point can be represented by two co-ordinates namely and x and y in a graph. From the
origin we have to go x units along x axis and y units along y-axis. If we plot two points in a graph and join them we
get a segment of a straight line. A minimum of two points is needed to a draw a straight line. In a linear
programming problem, the objective function and constraints are linear, so that we can plot the straight lines
representing constraints and the objectives function.
Let us take an algebraic equation 3x + 5y =15. This is a first-degree equation in x and y. Geometrically this
represents a straight line. If we want to represent the above equation geometrically, we require a minimum of two
points to plot a line. Let x = 0, we get y = 3 and let y = 0 we get x = 5. Hence we have two points A(0, 3) and B(5, 0)
and connect them to get a straight line
Fig 1
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We normally come across, in linear programming problems, the linear constraints having inequalities either
'greater than equal to' or 'less than equal to' or strictly 'equal to'.
Consider the inequality
3x + 5y < 15
To plot the feasible area satisfying the above inequality, we consider the equation
3x + 5y = 15.
This has been plotted in the figure 1. This straight line has an area above the line and an area below the
line. To determine the appropriate area satisfying the inequality, 3x + 5y < 15,
Let us consider the origin (x = 0, y = 0) as a reference point. Left hand side for the reference point is 3.0 + 5.0 = 0.
The value '0' of the left hand side, is clearly less than 15 satisfying the constraint and therefore the origin
whose co-ordinates are (0, 0) is a point to be included in the feasible area. The origin is below the straight line and
the area A1 should also contain the trial point namely the origin. Hence the feasible area satisfying the constraint 3x
+ 5y < 15 is indicated as A1 in figure 2.
Fig. 2
Next consider the inequality
3x + 5y > 15
To find the feasible area satisfying the above inequality, we take once again the origin as a trial point and
see whether this has to be included in the area. Let x = 0 and y = 0, then we get the value of the left hand side is 0
and the right hand side is 15 and hence the area satisfying this space inequality 3x + 5y > 15, should not contain the
origin and hence the feasible solution space is as indicated by A2 in figure 3.
The constraints are linear in a linear programming problem, which can be plotted on a graph. The ingenuity
of graphical solution to a linear programming problem depends on fixing the proper solution space.
Consider the following inequality 4x - 5y < 20
To get the feasible space to satisfy the above constraint, first we plot the straight line 4 x - 5y = 20 as shown
in figure 4. The origin, which satisfies the constraint, should be included in the solution space.
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Fig. 3
Fig. 4
Hence the area above the straight line shows the feasible solution space satisfying the inequality 4x - 5y <
20. The area below the line will correspond to the constraint 4x - 5y > 20.
Consider the inequality
-4x + 5y < 20
We plot the straight line -4x + 5y = 20 as shown in figure 5
The origin satisfies the inequality -4x + 5y < 20 and this should be included in the feasible solution space. Hence the
area below the straight line should represent the feasible solution space
Consider the inequality
-4x - 5y < 20
We plot the straight line -4x - 5y = 20 as shown in figure 6
Fig 5
Fig. 6
Since the origin satisfies the inequality -4x - 5y = 20 it should be included in the feasible solution space. Hence the
area above the straight line should be the solution space.
Procedure for Graphical Solution to LPP
Having explained how to fix the feasible solution space satisfying the constraint, we now give various steps
involved in arriving at a solution to a linear programming problem.
STEP 1:Consider all the constraints. Taking the equality relationship, plot all the straight lines in a graph and get
feasible solution space satisfying the inequality. Usually we get a bounded solution space.
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STEP 2: Assign an arbitrary value for the objective function. Plot the straight line to represent the objective
function, with the arbitrary value for Z.
STEP 3: Move the objective line parallel in the appropriate direction in the solution space to maximize and in the
opposite direction to minimize the objective function under consideration.
STEP 4: In this process, the moving objective line may meet an extreme point (or corner point) beyond which we
cannot proceed as this violates the constraints. Note the co-ordinates of this extreme point, which will give
maximum or minimum value of the objective function.
The above steps are explained in the following example.
Example
Maximize Z = 4x + 7y
Subject to
x < 40, y < 30, x + y < 60,
x, y > 0
In the above example, we have three constraints and all of them are 'less than inequalities'. First step in the
graphical method of solution to the linear programming problem is to represent the constraints graphically. Consider
the first constraint
x < 40
This indicates that x can take a value 40 or less than 40, for all values of y. The equation x = 40 is a straight
line parallel to the y axis as in the figure 2.7 and the area to the left of this straight line is the solution space
satisfying the constraint x < 40.
Fig. 7
Fig. 8
Consider the second constraint y < 30. This demands that y can take a value of 30 or less than 30 for all the
values of x. The equation y = 30 is a line parallel to x-axis and is represented in figure 8.
Similarly consider the third constraint x + y < 60. To plot the straight line let x = 0, then y = 60 and let y =
0, then x = 60. Therefore the two points (0, 60) and (60, 0) represented in figure 9 will be connected, to get the
straight line x + y = 60. To get the solution space satisfying x + y < 60, let x = 0 and y = 0, then the left hand side
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reduces to 0 which is clearly less than 60. Therefore the area below the line including (0, 0) represented by the
shaded portion in figure 9 represents the solution space satisfying the constraint x + y = 60.
Fig. 9
Consider the non-negative constraints for x an y and the feasible solution space is obtained by combining
all the constraints in one figure as shown in figure 10.
The shaded portion of the figure 10 represents the solution space as feasible region. The solution space in
figure 2.10 is a bounded figure with five corner points (otherwise called extreme points) namely A, B, C, D and E.
The solution space is bounded by segments of five straight lines. The co-ordinates of A, B, C, D and E are
determined by solving the appropriate simultaneous constraint equations.
Thus we have A at the origin, having xo-ordinates x = 0, y = 0.
The co-ordinates for B are x = 0, y = 30.
The co-ordinates of C are found by solving equations:
y = 30
x + y = 60
Fig 10
Fig 11
We get the co-ordinates of C as x = 30, y = 30.
Similarly the co-ordinates of D are determined by solving equations:
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x = 40
x + y = 60
The co-ordinates of D are x = 40, y = 20
The co-ordinates of E are x = 40, y = 0.
The next step is to examine the value of the objective function at these points. This is done in the table
below:
Extreme point
Co-ordinates
Value of
or corner point
Z = 4x + 7y
A
(0, 0)
0
B
(0, 30)
210
C
(30, 30)
330*
D
(40, 20)
300
E
(40, 0)
160
From the above table we see that the objective function has a maximum value of 330 at C (30, 30). So, the
optimum Z denoted by Z* is given by Z* = 330 for x = 30 and y = 30.
Alternate Method:
Instead of evaluating the objective function at all the extreme points; a graphical optimum solution can be
obtained which is explained below.
Consider the objective function
Z = 4x + 7y
Assign a convenient arbitrary value for Z say Z = 140, then
Z = 4x + 7y = 140
This is a straight line, which can be represented in the solution space as shown in figure 11.
Consider Z = 4x + 7y = 140. If x = 0, we get y = 20 and if y = 0, we get x = 35. These two points are connected by a
straight line representing Z = 140.
Thus Z = 140 is called an iso-profit line since for all points in the straight line, Z = 140. With this value we
have obtained the slope and hence shape of the straight line.
Next step is to draw straight lines parallel to 4x + 7y = 140.
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This results in moving the objective line with various values in the solution space in a proper direction
(upwards in this example for maximization and downwards for minimization). In this process, the objective line may
cross a corner point indicating the maximum or minimum value.
Drawing parallel lines we see that a line passing through C (30, 30) gives the maximum value for Z and
value of Z* = 330.
Example
Minimize
Z = 4x + 5y
Subject to
x + y > 10
2x + 5y > 35
x, y > 0
Solution
STEP 1 Represent the constraints graphically
Consider
x + y = 10
If x = 0, then y = 10
If y = 0, then x = 10
Consider
2x + 5y = 35
If x = 0, then y = 7
If y = 0, then x = 17.5
The above constraints are represented in figure 12.
STEP 2:
Mark the solution space in the graph satisfying all the constraints.
Fig. 12
Choose an arbitrary value of Z, say 100. Then
STEP 3
Z = 4x + 5y = 100
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If x = 0, then y = 20
If y = 0, then x = 25
Represent Z = 100 in the graph.
Move down the line Z = 100 to minimize the total cost Z. Thus we get the minimum value of Z at the point B, which
is the intersection of two lines:
x + y = 10
2x + 5y = 35
Solving the two equations, we get x = 5, y = 5
The minimum value of Z = 4 x 5 + 5 x 5 = 45,
Therefore, Z* = 45, x = 5, y = 5
RESULT:
Certain Definitions:
1.
A feasible solution is the value of all points for which all constraints are satisfied.
2.
An optimal solution is a feasible solution, which maximizes or minimizes the objective function.
These terms can be interpreted in terms of the graphical representation. The feasible solution includes all
the points within the permissible region (or solution space) including the boundary points.
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REVIEW QUESTIONS
6.
The standard weight of a brick is 5 kg, and it
Solve Graphically
contains two basic ingredients B1 and B2.B1 costs Rs.
5/kg and B2 costs Rs. 8/kg Strength considerations
1.
Maximize Z = 3x + 7y
dictate that the brick contains not more than 4 kg of
B1 and a minimum of 2 kg of B2. Since the demand
Subject to
for the product is likely to be related to the price of
x + 4y < 20
the brick, find out graphically the minimum cost of
2x + y < 30
the brick satisfying the above conditions.
x+y<8
and x, y > 0
7.
Solve by graphical method the following
problem.
2.
Maximize Z = 2x + 3y
Minimize
Subject to
-x + 2y < 16
Z = 5x1 + 4x2
Subject to
4x1 + x2 > 40
x + y < 24
x + 3y > 45
2x1 + 3x2 > 90
-4x + 10y > 20
and
x, y > 0
x1 > 0, x2 > 0
3.
Maximize Z = 3x1 + 5x2
b) Food A contains 20 units of vitamin x and 40 units
Subject to
-3x1 + 4x2 < 12
of vitamin y per gram. Food B contains 30 units of
2x1 - x2 > -2
each of vitamins x and y. The daily minimum human
2x1 + 3x2 > 12
requirements of vitamin x and y are 900 units and
1200 units respectively. How many grams of each
x1 < 4
type of food should be consumed so as to minimize
x2 > 2
the cost, if food A costs 60 paise per gram and food B
x1 > 0
80 paise per gram.
4.
Maximize
Z = 4x1 + 6x2
8.
A company produces two types of products
say type A and type B. Product A is of superior
Subject to
x1 < 2
quality and product B is of a lower quality.
x2 < 4
Respective profits for the two types of products are
x1 + x2 > 3
Rs. 40/- and 30/-. The data on the resource required,
and
x1, x2 > 0
availability of resources are given below:
5.
What are the four major types of allocation
Capacity
Requirements
problems, which could be solved using linear
Product A
Product B
available
programming technique? Briefly explain each one of
Per month
them with an example.
Raw materials
120
60
12000
(kg)
b)
Minimize
Z = 4x1 + x2
Machining time
5
8
600
(hrs/piece)
Subject to
3x1 + 4x2 > 20
Assembly
4
3
500
-x1 - 5x2 < -15
(man-hour)
and
x1, x2 > 0
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