# Solving Linear Programming Problems

Bananas cost 30 rupees per dozen (6 servings) and apples cost 80 rupees per kg (8 servings).Given: 1 banana contains 8.8 mg of Vitamin C and 100-125 g of apples i.e. Every person of the family would like to have at least 20 mg of Vitamin C daily but would like to keep the intake under 60 mg.

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Linear programming (LP) is a special form of multi-objective optimization, where the objectives and constraints that describe a decision are represented by linear equations, which are then used to find the best (optimal) solutions.

Linear programming can be divided into seven steps. Maybe your mother wants to be a named stakeholder in this decision because she likes to see you be productive.

How much fruit servings would the family have to consume on a daily basis per person to minimize their cost?

Solution: We begin step-wise with the formulation of the problem first.

Now begin from the far corner of the graph and tend to slide it towards the origin. Once you locate the optimum point, you’ll need to find its coordinates.

This can be done by drawing two perpendicular lines from the point onto the coordinate axes and noting down the coordinates.The constraint variables – ‘x’ = number of banana servings taken and ‘y’ = number of servings of apples taken. Total Cost C = 5x 10y Constraints: x ≥ 0; y ≥ 0 (non-negative number of servings) Total Vitamin C intake: 8.8x 5.2y ≥ 20 (1) 8.8x 5.2y ≤ 60 (2) To check for the validity of the equations, put x=0, y=0 in (1). Therefore, we must choose the side opposite to the origin as our valid region. Thus we should slide the ruler in such a way that a point is reached, which: 1) lies in the feasible region 2) is closer to the origin as compared to the other points This would be our Optimum Point. It is the one which you will get at the extreme right side of the feasible region here.Similarly, the side towards origin is the valid region for equation 2) Feasible Region: As per the analysis above, the feasible region for this problem would be the one in between the red and blue lines in the graph! I’ve also shown the position in which your ruler needs to be to get this point by the line in green.Most decisions require us to consider multiple, usually conflicting, objectives.For example, civil engineers face multi-objective decisions as they try to site a highway in order to balance efficient travel, noise reduction, air quality, cost, and proximity to residential areas.This is used to determine the domain of the available space, which can result in a feasible solution. A simple method is to put the coordinates of the origin (0,0) in the problem and determine whether the objective function takes on a physical solution or not.If yes, then the side of the constraint lines on which the origin lies is the valid side. The feasible solution region on the graph is the one which is satisfied by all the constraints.Choose the constant value in the equation of the objective function randomly, just to make it clearly distinguishable.An optimum point always lies on one of the corners of the feasible region. Place a ruler on the graph sheet, parallel to the objective function.Here we are going to concentrate on one of the most basic methods to handle a linear programming problem i.e. In principle, this method works for almost all different types of problems but gets more and more difficult to solve when the number of decision variables and the constraints increases. We will first discuss the steps of the algorithm: We have already understood the mathematical formulation of an LP problem in a previous section.Therefore, we’ll illustrate it in a simple case i.e. Note that this is the most crucial step as all the subsequent steps depend on our analysis here.

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