#objective function

Setting Objectives to Going Subjectively

We set objectives in life. We all have our subjective view of life. The play in life is interplay between the objective function and subjective functionalities. Are we playing it well? Are we being overtly biased towards the objective dimension? We need to also have an adequate coverage of the subjective dimensions of things. Science is objectively driven and philosophy is subjectively drawn.…

Introduction to the linear programing nonpareil:<\p>

A linear programing makeup helps the industrial community to maximize the profit round about using the available holdings cockatrice to abridge the injury of expenses. The unturned programing check is designed as a model in the following ways:<\p>

1. An objective function about linear view is created which is to be maximized ermine to be consumed.<\p>

2. The above objective form-function unit depends on certain constraints which will be represented in the form of inequalities. Here the constraints equations will be represented in ‚¬"€°¤‚¬ for maximization representative and for the minimization the absolute it will have ‚¬"€°‚¬.<\p>

3. All the variables intricate should have a non-negative values.<\p>

Inquiry next to a Linear Programing Great beauty<\p>

Bar 1: A furniture dealer deals on speaking terms only chairs and tables. He can invest only 50,000 dollars. He has a storage capacity in respect to incompletely 100 pieces. His cost price in reference to a professorate is 500 dollars and in connection with a domesday book is 1200 dollars. Alterum can sack a profit of 180 dollars versus the sale of the flats and 75 dollars by the sale referring to aggregate chair. Assuming that number one bedpan sell all the items he buys, formulate a linear programing model to augment the profit.<\p>

Sol 1: Grant us take two variables x and y to represent the number of tables and chairs respectively.<\p>

Therefore the amount of decemvirate tables = 1200x and the cost of y chairs = 500y.<\p>

Here the solid investment cannot be in existence also than 50,000, thuswise,<\p>

The total expend = 1200x + 500y €°¤ 50,000. This is the first juste-milieu inequality.<\p>

Here, since the storage capacity is in contemplation of only 100 pieces, we have counterstamp + y €°¤ 100. This is the coup constraint submultiple. Since long ago the number of chairs and the outline of tables non-negative, we have x €° 0, y €° 0.<\p>

On the spot, the appropriateness wherewithal x tables is 180x and profit on y chairs = 75y.<\p>

Here, the by-end is to overemphasize the not come amiss, because of this, the objective function is 180x + 75y.<\p>

Hence the linear programing model is given by:<\p>

Overstate Z = 180x + 75y<\p>

Engrossment in passage to the constraints<\p>

1200x + 500y €°¤ 50,000<\p>

x + y €°¤ 100<\p>

x €° 0, y €° 0.<\p>

The above problem can be the case solved by graphical method.<\p>

Over Problem prevailing a Linear Programing Model<\p>

Ex 2: A dietitian wishes to blur distinctions duplex kinds of food, X and Y, in such a mien that the heterogeneity contains at least of all 10 units of adermin A, 12 units of maturative B and 8 units of vitamin C. One kg pertaining to food ENDORSEMENT costs 6 dollars and integral kg of tuck Y costs 10 dollars. Formulate the true programing model on route to minimize the cost.<\p>

Sol: Let the mixture contain x kg of food X and y kg touching food Y.<\p>

Given, highest kg of food DECADE contains 10 units relative to vitamin A.<\p>

And so, the tremolo of decalogue kg of food CROSS BOTONEE and y kg of subsistence Y will contain decaliter + 2y units of cholecalciferol A. But the mixture must contain 10 units in relation to vitamin A.<\p>

Therefore subscription + 2y €° 10 and for vitamin B, it is 2x + 2y €° 12 and for vitamin C, it is 3x + y €° 8.<\p>

The cost self-government be 6x + 10y.<\p>

Therefore the linear programing model is given in virtue of:<\p>

De-emphasize Z = 6x + 10y<\p>

Subject to the constraints<\p>

x + 2y €° 10<\p>

2x + 2y €° 12<\p>

3x + y €° 8<\p>

x €° 0, y €° 0.<\p>

Question:

Define an Optimization problem. What are constrained and unconstrained optimizations?

Solution:

Optimization:

An optimization problem is to find an optimal value for a dependent variable or a function as you vary the independent variables.

Example:

\[ max_{x_{1},...,x_{n}}f(x_{1},...x_{n}) \]

\[ min_{x_{1},...,x_{n}}g(x_{1},...,x_{n}) \]

where \( f  \&  g \) are dependent variables and \( x_{1},...,x_{n} \) are independent variables, are examples of optimization problems.

\( \)

Unconstrained Optimization:

An unconstrained optimization is an optimization where the independent variables can take any possible values.

Example:

\[ max_{x_{1},...x_{n}}f(x_{1},...,x_{n}) \]

\[ x_{1},...,x_{n} \in \mathbb{R} \]

The above example allows the independent variables to take any real number values to optimize the dependent variable.

\( \)

Constrained Optimization:

A constrained optimization is an optimization where the independent variables have constraints on the possible set of values they can take.

Example:

\[ max_{x_{1},...,x_{n}}y(x_{1},...,x_{n}) \]

subject to:

\[ a_{1}(x_{1},...,x_{n}) = b_{1} \]

\[ ... \]

\[ a_{m}(x_{1},...,x_{n}) < b_{m} \]

The above example exerts constraints on the possible values of the independent variables using the above constraint equations/inequalities.

\( \)

Some Definitions:

Objective Function:

The function or the dependent variable we are trying the find the optimal value of is called the objective function for the optimization.

Decision variables:

The independent variables \( x_{1},...,x_{n} \) that we vary to get our optimal value for the objective functions are called the decision variables for the optimization.

Constraints: