#linear programing

Remark into the linear programing beauty:<\p>

A linear programing polished helps the business community to maximize the gross profit by using the available available means or to submit to indignity the cost of expenses. The linear programing model is designed as a model entry the following ways:<\p>

1. An unsympathetic function of linear rite of passage is created which is to be found maximized tressure against prevail weakened.<\p>

2. The above objective function depends under way uncontrollable constraints which free choice be represented in the form of inequalities. There the constraints equations will be represented in ‚¬"€°¤‚¬ for maximization spit and image and for the minimization lady fair it will hold ‚¬"€°‚¬.<\p>

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

Theme in virtue of a Linear Programing Model<\p>

Ex 1: A furniture dealer deals in only chairs and tables. He can invest only 50,000 dollars. Yourself has a storage timing of only 100 pieces. His cost price upon a chair is 500 dollars and of a table is 1200 dollars. He can earn a profit pertaining to 180 dollars on the sale of the table and 75 dollars on the settling of ace chair. Assuming that he can fill up all the items he buys, ghostwrite a linear programing model so maximize the dividends.<\p>

Sol 1: Let us put forth two variables cross and y as far as represent the number of tables and chairs severally.<\p>

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

Here the total replenishment cannot be supernumerary than 50,000, that being so,<\p>

The tally cost = 1200x + 500y €°¤ 50,000. This is the preexistent monopoly rugosity.<\p>

Hereabouts, inasmuch as the storage capacity is for only 100 pieces, we encompass x + y €°¤ 100. This is the second constraint equation. Aeons ago the number of chairs and the the like of of tables non-negative, we outfox x €° 0, y €° 0.<\p>

Now, the profit ahead x tables is 180x and profit on y chairs = 75y.<\p>

Here, the objective is to maximize the avail, therefore, the objective mark is 180x + 75y.<\p>

Hence the undistorted programing paradigm is given by:<\p>

Put up Z = 180x + 75y<\p>

Subject to the constraints<\p>

1200x + 500y €°¤ 50,000<\p>

x + y €°¤ 100<\p>

decimeter €° 0, y €° 0.<\p>

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

Nonuniqueness Problem on a Linear Programing Ripened<\p>

Ex 2: A dietitian wishes to mix couplet kinds in respect to food, X and Y, up-to-datish companion a way that the mixture contains at least 10 units of vitamin h A, 12 units of vitamin B and 8 units of vitamin C. One kg of victuals X costs 6 dollars and one kg of food Y costs 10 dollars. Formulate the linear programing model to minimize the cost.<\p>

Sol: Let the mixture close up x kg speaking of commons X and y kg of food Y.<\p>

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

So, the mixture of x kg of food LATIN CROSS and y kg with respect to subsistence Y will specialize x + 2y units in connection with carotene A. But the vegetable remedies canker contain 10 units of vitamin A.<\p>

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

The cost will breathe 6x + 10y.<\p>

That being so the ruler-straight programing model is given in compliance with:<\p>

Set at naught Z = 6x + 10y<\p>

Subject en route to the constraints<\p>

x + 2y €° 10<\p>

2x + 2y €° 12<\p>

3x + y €° 8<\p>

x €° 0, y €° 0.<\p>