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1 | 6218-6221 | With his capital of Rs 50,000,
he can buy 50000 ÷ 500, i e 100 chairs But he can store only 60 pieces |
1 | 6219-6222 | e 100 chairs But he can store only 60 pieces Therefore, he
is forced to buy only 60 chairs which will give him a total profit of Rs (60 × 75), i |
1 | 6220-6223 | 100 chairs But he can store only 60 pieces Therefore, he
is forced to buy only 60 chairs which will give him a total profit of Rs (60 × 75), i e |
1 | 6221-6224 | But he can store only 60 pieces Therefore, he
is forced to buy only 60 chairs which will give him a total profit of Rs (60 × 75), i e ,
Rs 4500 |
1 | 6222-6225 | Therefore, he
is forced to buy only 60 chairs which will give him a total profit of Rs (60 × 75), i e ,
Rs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables
and 50 chairs, as he can store only 60 pieces |
1 | 6223-6226 | e ,
Rs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables
and 50 chairs, as he can store only 60 pieces Total profit in this case would be
Rs (10 × 250 + 50 × 75), i |
1 | 6224-6227 | ,
Rs 4500 There are many other possibilities, for instance, he may choose to buy 10 tables
and 50 chairs, as he can store only 60 pieces Total profit in this case would be
Rs (10 × 250 + 50 × 75), i e |
1 | 6225-6228 | There are many other possibilities, for instance, he may choose to buy 10 tables
and 50 chairs, as he can store only 60 pieces Total profit in this case would be
Rs (10 × 250 + 50 × 75), i e , Rs 6250 and so on |
1 | 6226-6229 | Total profit in this case would be
Rs (10 × 250 + 50 × 75), i e , Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would
earn different profits by following different investment strategies |
1 | 6227-6230 | e , Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would
earn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum
profit |
1 | 6228-6231 | , Rs 6250 and so on We, thus, find that the dealer can invest his money in different ways and he would
earn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum
profit To answer this question, let us try to formulate the problem mathematically |
1 | 6229-6232 | We, thus, find that the dealer can invest his money in different ways and he would
earn different profits by following different investment strategies Now the problem is : How should he invest his money in order to get maximum
profit To answer this question, let us try to formulate the problem mathematically 12 |
1 | 6230-6233 | Now the problem is : How should he invest his money in order to get maximum
profit To answer this question, let us try to formulate the problem mathematically 12 2 |
1 | 6231-6234 | To answer this question, let us try to formulate the problem mathematically 12 2 1 Mathematical formulation of the problem
Let x be the number of tables and y be the number of chairs that the dealer buys |
1 | 6232-6235 | 12 2 1 Mathematical formulation of the problem
Let x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i |
1 | 6233-6236 | 2 1 Mathematical formulation of the problem
Let x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i e |
1 | 6234-6237 | 1 Mathematical formulation of the problem
Let x be the number of tables and y be the number of chairs that the dealer buys Obviously, x and y must be non-negative, i e ,
0 |
1 | 6235-6238 | Obviously, x and y must be non-negative, i e ,
0 (1)
(Non-negative constraints) |
1 | 6236-6239 | e ,
0 (1)
(Non-negative constraints) (2)
0
yx
The dealer is constrained by the maximum amount he can invest (Here it is
Rs 50,000) and by the maximum number of items he can store (Here it is 60) |
1 | 6237-6240 | ,
0 (1)
(Non-negative constraints) (2)
0
yx
The dealer is constrained by the maximum amount he can invest (Here it is
Rs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,
2500x + 500y ≤ 50000 (investment constraint)
or
5x + y ≤ 100 |
1 | 6238-6241 | (1)
(Non-negative constraints) (2)
0
yx
The dealer is constrained by the maximum amount he can invest (Here it is
Rs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,
2500x + 500y ≤ 50000 (investment constraint)
or
5x + y ≤ 100 (3)
and
x + y ≤ 60 (storage constraint) |
1 | 6239-6242 | (2)
0
yx
The dealer is constrained by the maximum amount he can invest (Here it is
Rs 50,000) and by the maximum number of items he can store (Here it is 60) Stated mathematically,
2500x + 500y ≤ 50000 (investment constraint)
or
5x + y ≤ 100 (3)
and
x + y ≤ 60 (storage constraint) (4)
© NCERT
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506
MATHEMATICS
The dealer wants to invest in such a way so as to maximise his profit, say, Z which
stated as a function of x and y is given by
Z = 250x + 75y (called objective function) |
1 | 6240-6243 | Stated mathematically,
2500x + 500y ≤ 50000 (investment constraint)
or
5x + y ≤ 100 (3)
and
x + y ≤ 60 (storage constraint) (4)
© NCERT
not to be republished
506
MATHEMATICS
The dealer wants to invest in such a way so as to maximise his profit, say, Z which
stated as a function of x and y is given by
Z = 250x + 75y (called objective function) (5)
Mathematically, the given problems now reduces to:
Maximise Z = 250x + 75y
subject to the constraints:
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y ≥ 0
So, we have to maximise the linear function Z subject to certain conditions determined
by a set of linear inequalities with variables as non-negative |
1 | 6241-6244 | (3)
and
x + y ≤ 60 (storage constraint) (4)
© NCERT
not to be republished
506
MATHEMATICS
The dealer wants to invest in such a way so as to maximise his profit, say, Z which
stated as a function of x and y is given by
Z = 250x + 75y (called objective function) (5)
Mathematically, the given problems now reduces to:
Maximise Z = 250x + 75y
subject to the constraints:
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y ≥ 0
So, we have to maximise the linear function Z subject to certain conditions determined
by a set of linear inequalities with variables as non-negative There are also some other
problems where we have to minimise a linear function subject to certain conditions
determined by a set of linear inequalities with variables as non-negative |
1 | 6242-6245 | (4)
© NCERT
not to be republished
506
MATHEMATICS
The dealer wants to invest in such a way so as to maximise his profit, say, Z which
stated as a function of x and y is given by
Z = 250x + 75y (called objective function) (5)
Mathematically, the given problems now reduces to:
Maximise Z = 250x + 75y
subject to the constraints:
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y ≥ 0
So, we have to maximise the linear function Z subject to certain conditions determined
by a set of linear inequalities with variables as non-negative There are also some other
problems where we have to minimise a linear function subject to certain conditions
determined by a set of linear inequalities with variables as non-negative Such problems
are called Linear Programming Problems |
1 | 6243-6246 | (5)
Mathematically, the given problems now reduces to:
Maximise Z = 250x + 75y
subject to the constraints:
5x + y ≤ 100
x + y ≤ 60
x ≥ 0, y ≥ 0
So, we have to maximise the linear function Z subject to certain conditions determined
by a set of linear inequalities with variables as non-negative There are also some other
problems where we have to minimise a linear function subject to certain conditions
determined by a set of linear inequalities with variables as non-negative Such problems
are called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the
optimal value (maximum or minimum value) of a linear function (called objective
function) of several variables (say x and y), subject to the conditions that the variables
are non-negative and satisfy a set of linear inequalities (called linear constraints) |
1 | 6244-6247 | There are also some other
problems where we have to minimise a linear function subject to certain conditions
determined by a set of linear inequalities with variables as non-negative Such problems
are called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the
optimal value (maximum or minimum value) of a linear function (called objective
function) of several variables (say x and y), subject to the conditions that the variables
are non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are
linear relations while the term programming refers to the method of determining a
particular programme or plan of action |
1 | 6245-6248 | Such problems
are called Linear Programming Problems Thus, a Linear Programming Problem is one that is concerned with finding the
optimal value (maximum or minimum value) of a linear function (called objective
function) of several variables (say x and y), subject to the conditions that the variables
are non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are
linear relations while the term programming refers to the method of determining a
particular programme or plan of action Before we proceed further, we now formally define some terms (which have been
used above) which we shall be using in the linear programming problems:
Objective function Linear function Z = ax + by, where a, b are constants, which has
to be maximised or minimized is called a linear objective function |
1 | 6246-6249 | Thus, a Linear Programming Problem is one that is concerned with finding the
optimal value (maximum or minimum value) of a linear function (called objective
function) of several variables (say x and y), subject to the conditions that the variables
are non-negative and satisfy a set of linear inequalities (called linear constraints) The term linear implies that all the mathematical relations used in the problem are
linear relations while the term programming refers to the method of determining a
particular programme or plan of action Before we proceed further, we now formally define some terms (which have been
used above) which we shall be using in the linear programming problems:
Objective function Linear function Z = ax + by, where a, b are constants, which has
to be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function |
1 | 6247-6250 | The term linear implies that all the mathematical relations used in the problem are
linear relations while the term programming refers to the method of determining a
particular programme or plan of action Before we proceed further, we now formally define some terms (which have been
used above) which we shall be using in the linear programming problems:
Objective function Linear function Z = ax + by, where a, b are constants, which has
to be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function Variables x and
y are called decision variables |
1 | 6248-6251 | Before we proceed further, we now formally define some terms (which have been
used above) which we shall be using in the linear programming problems:
Objective function Linear function Z = ax + by, where a, b are constants, which has
to be maximised or minimized is called a linear objective function In the above example, Z = 250x + 75y is a linear objective function Variables x and
y are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a
linear programming problem are called constraints |
1 | 6249-6252 | In the above example, Z = 250x + 75y is a linear objective function Variables x and
y are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a
linear programming problem are called constraints The conditions x ≥ 0, y ≥ 0 are
called non-negative restrictions |
1 | 6250-6253 | Variables x and
y are called decision variables Constraints The linear inequalities or equations or restrictions on the variables of a
linear programming problem are called constraints The conditions x ≥ 0, y ≥ 0 are
called non-negative restrictions In the above example, the set of inequalities (1) to (4)
are constraints |
1 | 6251-6254 | Constraints The linear inequalities or equations or restrictions on the variables of a
linear programming problem are called constraints The conditions x ≥ 0, y ≥ 0 are
called non-negative restrictions In the above example, the set of inequalities (1) to (4)
are constraints Optimisation problem A problem which seeks to maximise or minimise a linear
function (say of two variables x and y) subject to certain constraints as determined by
a set of linear inequalities is called an optimisation problem |
1 | 6252-6255 | The conditions x ≥ 0, y ≥ 0 are
called non-negative restrictions In the above example, the set of inequalities (1) to (4)
are constraints Optimisation problem A problem which seeks to maximise or minimise a linear
function (say of two variables x and y) subject to certain constraints as determined by
a set of linear inequalities is called an optimisation problem Linear programming
problems are special type of optimisation problems |
1 | 6253-6256 | In the above example, the set of inequalities (1) to (4)
are constraints Optimisation problem A problem which seeks to maximise or minimise a linear
function (say of two variables x and y) subject to certain constraints as determined by
a set of linear inequalities is called an optimisation problem Linear programming
problems are special type of optimisation problems The above problem of investing a
© NCERT
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LINEAR PROGRAMMING 507
given sum by the dealer in purchasing chairs and tables is an example of an optimisation
problem as well as of a linear programming problem |
1 | 6254-6257 | Optimisation problem A problem which seeks to maximise or minimise a linear
function (say of two variables x and y) subject to certain constraints as determined by
a set of linear inequalities is called an optimisation problem Linear programming
problems are special type of optimisation problems The above problem of investing a
© NCERT
not to be republished
LINEAR PROGRAMMING 507
given sum by the dealer in purchasing chairs and tables is an example of an optimisation
problem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem |
1 | 6255-6258 | Linear programming
problems are special type of optimisation problems The above problem of investing a
© NCERT
not to be republished
LINEAR PROGRAMMING 507
given sum by the dealer in purchasing chairs and tables is an example of an optimisation
problem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem In this
chapter, we will be concerned only with the graphical method |
1 | 6256-6259 | The above problem of investing a
© NCERT
not to be republished
LINEAR PROGRAMMING 507
given sum by the dealer in purchasing chairs and tables is an example of an optimisation
problem as well as of a linear programming problem We will now discuss how to find solutions to a linear programming problem In this
chapter, we will be concerned only with the graphical method 12 |
1 | 6257-6260 | We will now discuss how to find solutions to a linear programming problem In this
chapter, we will be concerned only with the graphical method 12 2 |
1 | 6258-6261 | In this
chapter, we will be concerned only with the graphical method 12 2 2 Graphical method of solving linear programming problems
In Class XI, we have learnt how to graph a system of linear inequalities involving two
variables x and y and to find its solutions graphically |
1 | 6259-6262 | 12 2 2 Graphical method of solving linear programming problems
In Class XI, we have learnt how to graph a system of linear inequalities involving two
variables x and y and to find its solutions graphically Let us refer to the problem of
investment in tables and chairs discussed in Section 12 |
1 | 6260-6263 | 2 2 Graphical method of solving linear programming problems
In Class XI, we have learnt how to graph a system of linear inequalities involving two
variables x and y and to find its solutions graphically Let us refer to the problem of
investment in tables and chairs discussed in Section 12 2 |
1 | 6261-6264 | 2 Graphical method of solving linear programming problems
In Class XI, we have learnt how to graph a system of linear inequalities involving two
variables x and y and to find its solutions graphically Let us refer to the problem of
investment in tables and chairs discussed in Section 12 2 We will now solve this problem
graphically |
1 | 6262-6265 | Let us refer to the problem of
investment in tables and chairs discussed in Section 12 2 We will now solve this problem
graphically Let us graph the constraints stated as linear inequalities:
5x + y ≤ 100 |
1 | 6263-6266 | 2 We will now solve this problem
graphically Let us graph the constraints stated as linear inequalities:
5x + y ≤ 100 (1)
x + y ≤ 60 |
1 | 6264-6267 | We will now solve this problem
graphically Let us graph the constraints stated as linear inequalities:
5x + y ≤ 100 (1)
x + y ≤ 60 (2)
x ≥ 0 |
1 | 6265-6268 | Let us graph the constraints stated as linear inequalities:
5x + y ≤ 100 (1)
x + y ≤ 60 (2)
x ≥ 0 (3)
y ≥ 0 |
1 | 6266-6269 | (1)
x + y ≤ 60 (2)
x ≥ 0 (3)
y ≥ 0 (4)
The graph of this system (shaded region) consists of the points common to all half
planes determined by the inequalities (1) to (4) (Fig 12 |
1 | 6267-6270 | (2)
x ≥ 0 (3)
y ≥ 0 (4)
The graph of this system (shaded region) consists of the points common to all half
planes determined by the inequalities (1) to (4) (Fig 12 1) |
1 | 6268-6271 | (3)
y ≥ 0 (4)
The graph of this system (shaded region) consists of the points common to all half
planes determined by the inequalities (1) to (4) (Fig 12 1) Each point in this region
represents a feasible choice open to the dealer for investing in tables and chairs |
1 | 6269-6272 | (4)
The graph of this system (shaded region) consists of the points common to all half
planes determined by the inequalities (1) to (4) (Fig 12 1) Each point in this region
represents a feasible choice open to the dealer for investing in tables and chairs The
region, therefore, is called the feasible region for the problem |
1 | 6270-6273 | 1) Each point in this region
represents a feasible choice open to the dealer for investing in tables and chairs The
region, therefore, is called the feasible region for the problem Every point of this
region is called a feasible solution to the problem |
1 | 6271-6274 | Each point in this region
represents a feasible choice open to the dealer for investing in tables and chairs The
region, therefore, is called the feasible region for the problem Every point of this
region is called a feasible solution to the problem Thus, we have,
Feasible region The common region determined by all the constraints including
non-negative constraints x, y ≥ 0 of a linear programming problem is called the feasible
region (or solution region) for the problem |
1 | 6272-6275 | The
region, therefore, is called the feasible region for the problem Every point of this
region is called a feasible solution to the problem Thus, we have,
Feasible region The common region determined by all the constraints including
non-negative constraints x, y ≥ 0 of a linear programming problem is called the feasible
region (or solution region) for the problem In Fig 12 |
1 | 6273-6276 | Every point of this
region is called a feasible solution to the problem Thus, we have,
Feasible region The common region determined by all the constraints including
non-negative constraints x, y ≥ 0 of a linear programming problem is called the feasible
region (or solution region) for the problem In Fig 12 1, the region OABC (shaded) is
the feasible region for the problem |
1 | 6274-6277 | Thus, we have,
Feasible region The common region determined by all the constraints including
non-negative constraints x, y ≥ 0 of a linear programming problem is called the feasible
region (or solution region) for the problem In Fig 12 1, the region OABC (shaded) is
the feasible region for the problem The region other than feasible region is called an
infeasible region |
1 | 6275-6278 | In Fig 12 1, the region OABC (shaded) is
the feasible region for the problem The region other than feasible region is called an
infeasible region Feasible solutions Points within and on the
boundary of the feasible region represent
feasible solutions of the constraints |
1 | 6276-6279 | 1, the region OABC (shaded) is
the feasible region for the problem The region other than feasible region is called an
infeasible region Feasible solutions Points within and on the
boundary of the feasible region represent
feasible solutions of the constraints In
Fig 12 |
1 | 6277-6280 | The region other than feasible region is called an
infeasible region Feasible solutions Points within and on the
boundary of the feasible region represent
feasible solutions of the constraints In
Fig 12 1, every point within and on the
boundary of the feasible region OABC
represents feasible solution to the problem |
1 | 6278-6281 | Feasible solutions Points within and on the
boundary of the feasible region represent
feasible solutions of the constraints In
Fig 12 1, every point within and on the
boundary of the feasible region OABC
represents feasible solution to the problem For example, the point (10, 50) is a feasible
solution of the problem and so are the points
(0, 60), (20, 0) etc |
1 | 6279-6282 | In
Fig 12 1, every point within and on the
boundary of the feasible region OABC
represents feasible solution to the problem For example, the point (10, 50) is a feasible
solution of the problem and so are the points
(0, 60), (20, 0) etc Any point outside the feasible region is
called an infeasible solution |
1 | 6280-6283 | 1, every point within and on the
boundary of the feasible region OABC
represents feasible solution to the problem For example, the point (10, 50) is a feasible
solution of the problem and so are the points
(0, 60), (20, 0) etc Any point outside the feasible region is
called an infeasible solution For example,
the point (25, 40) is an infeasible solution of
the problem |
1 | 6281-6284 | For example, the point (10, 50) is a feasible
solution of the problem and so are the points
(0, 60), (20, 0) etc Any point outside the feasible region is
called an infeasible solution For example,
the point (25, 40) is an infeasible solution of
the problem Fig 12 |
1 | 6282-6285 | Any point outside the feasible region is
called an infeasible solution For example,
the point (25, 40) is an infeasible solution of
the problem Fig 12 1
© NCERT
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508
MATHEMATICS
Optimal (feasible) solution: Any point in the feasible region that gives the optimal
value (maximum or minimum) of the objective function is called an optimal solution |
1 | 6283-6286 | For example,
the point (25, 40) is an infeasible solution of
the problem Fig 12 1
© NCERT
not to be republished
508
MATHEMATICS
Optimal (feasible) solution: Any point in the feasible region that gives the optimal
value (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints
as given in (1) to (4), and since there are infinitely many points, it is not evident how
we should go about finding a point that gives a maximum value of the objective function
Z = 250x + 75y |
1 | 6284-6287 | Fig 12 1
© NCERT
not to be republished
508
MATHEMATICS
Optimal (feasible) solution: Any point in the feasible region that gives the optimal
value (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints
as given in (1) to (4), and since there are infinitely many points, it is not evident how
we should go about finding a point that gives a maximum value of the objective function
Z = 250x + 75y To handle this situation, we use the following theorems which are
fundamental in solving linear programming problems |
1 | 6285-6288 | 1
© NCERT
not to be republished
508
MATHEMATICS
Optimal (feasible) solution: Any point in the feasible region that gives the optimal
value (maximum or minimum) of the objective function is called an optimal solution Now, we see that every point in the feasible region OABC satisfies all the constraints
as given in (1) to (4), and since there are infinitely many points, it is not evident how
we should go about finding a point that gives a maximum value of the objective function
Z = 250x + 75y To handle this situation, we use the following theorems which are
fundamental in solving linear programming problems The proofs of these theorems
are beyond the scope of the book |
1 | 6286-6289 | Now, we see that every point in the feasible region OABC satisfies all the constraints
as given in (1) to (4), and since there are infinitely many points, it is not evident how
we should go about finding a point that gives a maximum value of the objective function
Z = 250x + 75y To handle this situation, we use the following theorems which are
fundamental in solving linear programming problems The proofs of these theorems
are beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming
problem and let Z = ax + by be the objective function |
1 | 6287-6290 | To handle this situation, we use the following theorems which are
fundamental in solving linear programming problems The proofs of these theorems
are beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming
problem and let Z = ax + by be the objective function When Z has an optimal value
(maximum or minimum), where the variables x and y are subject to constraints described
by linear inequalities, this optimal value must occur at a corner point* (vertex) of the
feasible region |
1 | 6288-6291 | The proofs of these theorems
are beyond the scope of the book Theorem 1 Let R be the feasible region (convex polygon) for a linear programming
problem and let Z = ax + by be the objective function When Z has an optimal value
(maximum or minimum), where the variables x and y are subject to constraints described
by linear inequalities, this optimal value must occur at a corner point* (vertex) of the
feasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let
Z = ax + by be the objective function |
1 | 6289-6292 | Theorem 1 Let R be the feasible region (convex polygon) for a linear programming
problem and let Z = ax + by be the objective function When Z has an optimal value
(maximum or minimum), where the variables x and y are subject to constraints described
by linear inequalities, this optimal value must occur at a corner point* (vertex) of the
feasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let
Z = ax + by be the objective function If R is bounded**, then the objective function
Z has both a maximum and a minimum value on R and each of these occurs at a
corner point (vertex) of R |
1 | 6290-6293 | When Z has an optimal value
(maximum or minimum), where the variables x and y are subject to constraints described
by linear inequalities, this optimal value must occur at a corner point* (vertex) of the
feasible region Theorem 2 Let R be the feasible region for a linear programming problem, and let
Z = ax + by be the objective function If R is bounded**, then the objective function
Z has both a maximum and a minimum value on R and each of these occurs at a
corner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective
function may not exist |
1 | 6291-6294 | Theorem 2 Let R be the feasible region for a linear programming problem, and let
Z = ax + by be the objective function If R is bounded**, then the objective function
Z has both a maximum and a minimum value on R and each of these occurs at a
corner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective
function may not exist However, if it exists, it must occur at a corner point of R |
1 | 6292-6295 | If R is bounded**, then the objective function
Z has both a maximum and a minimum value on R and each of these occurs at a
corner point (vertex) of R Remark If R is unbounded, then a maximum or a minimum value of the objective
function may not exist However, if it exists, it must occur at a corner point of R (By Theorem 1) |
1 | 6293-6296 | Remark If R is unbounded, then a maximum or a minimum value of the objective
function may not exist However, if it exists, it must occur at a corner point of R (By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region
are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and
(0, 60) respectively |
1 | 6294-6297 | However, if it exists, it must occur at a corner point of R (By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region
are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and
(0, 60) respectively Let us now compute the values of Z at these points |
1 | 6295-6298 | (By Theorem 1) In the above example, the corner points (vertices) of the bounded (feasible) region
are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and
(0, 60) respectively Let us now compute the values of Z at these points We have
*
A corner point of a feasible region is a point in the region which is the intersection of two boundary lines |
1 | 6296-6299 | In the above example, the corner points (vertices) of the bounded (feasible) region
are: O, A, B and C and it is easy to find their coordinates as (0, 0), (20, 0), (10, 50) and
(0, 60) respectively Let us now compute the values of Z at these points We have
*
A corner point of a feasible region is a point in the region which is the intersection of two boundary lines **
A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a
circle |
1 | 6297-6300 | Let us now compute the values of Z at these points We have
*
A corner point of a feasible region is a point in the region which is the intersection of two boundary lines **
A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a
circle Otherwise, it is called unbounded |
1 | 6298-6301 | We have
*
A corner point of a feasible region is a point in the region which is the intersection of two boundary lines **
A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a
circle Otherwise, it is called unbounded Unbounded means that the feasible region does extend
indefinitely in any direction |
1 | 6299-6302 | **
A feasible region of a system of linear inequalities is said to be bounded if it can be enclosed within a
circle Otherwise, it is called unbounded Unbounded means that the feasible region does extend
indefinitely in any direction Vertex of the
Corresponding value
Feasible Region
of Z (in Rs)
O (0,0)
0
C (0,60)
4500
B (10,50)
6250
A (20,0)
5000
Maximum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 509
We observe that the maximum profit to the dealer results from the investment
strategy (10, 50), i |
1 | 6300-6303 | Otherwise, it is called unbounded Unbounded means that the feasible region does extend
indefinitely in any direction Vertex of the
Corresponding value
Feasible Region
of Z (in Rs)
O (0,0)
0
C (0,60)
4500
B (10,50)
6250
A (20,0)
5000
Maximum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 509
We observe that the maximum profit to the dealer results from the investment
strategy (10, 50), i e |
1 | 6301-6304 | Unbounded means that the feasible region does extend
indefinitely in any direction Vertex of the
Corresponding value
Feasible Region
of Z (in Rs)
O (0,0)
0
C (0,60)
4500
B (10,50)
6250
A (20,0)
5000
Maximum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 509
We observe that the maximum profit to the dealer results from the investment
strategy (10, 50), i e buying 10 tables and 50 chairs |
1 | 6302-6305 | Vertex of the
Corresponding value
Feasible Region
of Z (in Rs)
O (0,0)
0
C (0,60)
4500
B (10,50)
6250
A (20,0)
5000
Maximum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 509
We observe that the maximum profit to the dealer results from the investment
strategy (10, 50), i e buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point
Method |
1 | 6303-6306 | e buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point
Method The method comprises of the following steps:
1 |
1 | 6304-6307 | buying 10 tables and 50 chairs This method of solving linear programming problem is referred as Corner Point
Method The method comprises of the following steps:
1 Find the feasible region of the linear programming problem and determine its
corner points (vertices) either by inspection or by solving the two equations of
the lines intersecting at that point |
1 | 6305-6308 | This method of solving linear programming problem is referred as Corner Point
Method The method comprises of the following steps:
1 Find the feasible region of the linear programming problem and determine its
corner points (vertices) either by inspection or by solving the two equations of
the lines intersecting at that point 2 |
1 | 6306-6309 | The method comprises of the following steps:
1 Find the feasible region of the linear programming problem and determine its
corner points (vertices) either by inspection or by solving the two equations of
the lines intersecting at that point 2 Evaluate the objective function Z = ax + by at each corner point |
1 | 6307-6310 | Find the feasible region of the linear programming problem and determine its
corner points (vertices) either by inspection or by solving the two equations of
the lines intersecting at that point 2 Evaluate the objective function Z = ax + by at each corner point Let M and m,
respectively denote the largest and smallest values of these points |
1 | 6308-6311 | 2 Evaluate the objective function Z = ax + by at each corner point Let M and m,
respectively denote the largest and smallest values of these points 3 |
1 | 6309-6312 | Evaluate the objective function Z = ax + by at each corner point Let M and m,
respectively denote the largest and smallest values of these points 3 (i)
When the feasible region is bounded, M and m are the maximum and
minimum values of Z |
1 | 6310-6313 | Let M and m,
respectively denote the largest and smallest values of these points 3 (i)
When the feasible region is bounded, M and m are the maximum and
minimum values of Z (ii) In case, the feasible region is unbounded, we have:
4 |
1 | 6311-6314 | 3 (i)
When the feasible region is bounded, M and m are the maximum and
minimum values of Z (ii) In case, the feasible region is unbounded, we have:
4 (a) M is the maximum value of Z, if the open half plane determined by
ax + by > M has no point in common with the feasible region |
1 | 6312-6315 | (i)
When the feasible region is bounded, M and m are the maximum and
minimum values of Z (ii) In case, the feasible region is unbounded, we have:
4 (a) M is the maximum value of Z, if the open half plane determined by
ax + by > M has no point in common with the feasible region Otherwise, Z
has no maximum value |
1 | 6313-6316 | (ii) In case, the feasible region is unbounded, we have:
4 (a) M is the maximum value of Z, if the open half plane determined by
ax + by > M has no point in common with the feasible region Otherwise, Z
has no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by
ax + by < m has no point in common with the feasible region |
1 | 6314-6317 | (a) M is the maximum value of Z, if the open half plane determined by
ax + by > M has no point in common with the feasible region Otherwise, Z
has no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by
ax + by < m has no point in common with the feasible region Otherwise, Z
has no minimum value |
1 | 6315-6318 | Otherwise, Z
has no maximum value (b) Similarly, m is the minimum value of Z, if the open half plane determined by
ax + by < m has no point in common with the feasible region Otherwise, Z
has no minimum value We will now illustrate these steps of Corner Point Method by considering some
examples:
Example 1 Solve the following linear programming problem graphically:
Maximise Z = 4x + y |
1 | 6316-6319 | (b) Similarly, m is the minimum value of Z, if the open half plane determined by
ax + by < m has no point in common with the feasible region Otherwise, Z
has no minimum value We will now illustrate these steps of Corner Point Method by considering some
examples:
Example 1 Solve the following linear programming problem graphically:
Maximise Z = 4x + y (1)
subject to the constraints:
x + y ≤ 50 |
1 | 6317-6320 | Otherwise, Z
has no minimum value We will now illustrate these steps of Corner Point Method by considering some
examples:
Example 1 Solve the following linear programming problem graphically:
Maximise Z = 4x + y (1)
subject to the constraints:
x + y ≤ 50 (2)
3x + y ≤ 90 |
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