Linear Programming Examples And Solutions Pdf

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In business and in day-to-day living we know that we cannot simply choose to do something because it would make sense that it would unreasonably accomplish our goal. Instead, our hope is to maximize or minimize some quantity, given a set of constraints.

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research OR. They are now available for use by any students and teachers interested in OR subject to the following conditions. A full list of the topics available in OR-Notes can be found here. A company makes two products X and Y using two machines A and B. Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B.

Practice Problems

In business and in day-to-day living we know that we cannot simply choose to do something because it would make sense that it would unreasonably accomplish our goal. Instead, our hope is to maximize or minimize some quantity, given a set of constraints. Your hope is to get there in as little time as possible, hence aiming to minimize travel time. While we have only mentioned a few, these are all constraints —things that limit you in your goal to get to your destination in as little time as possible.

A linear programming problem involves constraints that contain inequalities. An airline offers coach and first-class tickets. For the airline to be profitable, it must sell a minimum of 25 first-class tickets and a minimum of 40 coach tickets. At most, the plane has a capacity of travelers.

How many of each ticket should be sold in order to maximize profits? The first step is to identify the unknown quantities. We are asked to find the number of each ticket that should be sold. Since there are coach and first-class tickets, we identify those as the unknowns. Next, we need to identify the objective function.

The question often helps us identify the objective function. Since the goal is the maximize profits, our objective is identified. If x coach tickets are sold, the total profit for these tickets is x.

We want to make the value of as large as possible, provided the constraints are met. In this case, we have the following constraints:. We will work to think about these constraints graphically and return to the objective function afterwards.

We will thus deal with the following graph:. We will first plot each of the inequalities as equations, and then worry about the inequality signs. That is, first plot,.

The first two equations are horizontal and vertical lines, respectively. Since this is a horizontal line running through a y -value of 25, anything above this line represents a value greater than We denote this by shading above the line:. We ask, when is the x -value larger than 40? Values to the left are smaller than 40, so we must shade to the right to get values larger than The blue area satisfies the second constraint, but since we must satisfy all constraints, only the region that is green and blue will suffice.

We have two options, either shade below or shade above. To help us better see that we will, in fact, need to shade below the line, let us consider an ordered pair in both regions.

Selecting an ordered pair above the line, such as 64, gives:. According to the graph, the point 64, 65 is one that falls below the graph. Putting this pair in yields the statement:. Therefore, we shade below the line:. The region in which the green, blue, and purple shadings intersect satisfies all three constraints.

We can verify that a point chosen in this region satisfies all three constraints. For example, choosing 64, 65 gives:. This gets us to a great point, but still does not answer the question: which point maximizes profit? Fortunately, there is a theorem discovered by mathematicians that allows us to answer this question. In our situation, we have three corner points, shown on the graph as the solid black dots:. That is, imagine you are looking at three fence posts connected by fencing black point and lines, respectively.

If you were to put your dog in the middle, you could be sure it would not escape assuming the fence is tall enough. If this is the case, then you have a bounded linear programming problem. If the dog could walk infinitely in any one direction, then the problem is unbounded. This means we have to choose among three corner points. To find the corner points as ordered pairs, we must solve three systems of two equations each:.

We could decide to solve by using matrix equations, but these equations are all simple enough to solve by hand:. The third point, ,25 maximizes profit. Therefore, we conclude that the airline should sell coach tickets and 25 first-class tickets in order to maximize profits. The above example was rather long and had many steps to complete. We will summarize the procedure below:.

Identify the corner points by solving systems of linear equations whose intersection represents a corner point. Test all corner points in the objective function. There is one instance in which we must take great caution. First, consider the true inequality,. Clearly, —5 is not larger than —3! To keep the statement true, we should change the direction of the inequality sign so that,. We can see by the number line below, that the two sets of numbers are symmetric about 0, except that the way in which we describe size is opposite.

This justifies that we should also use the opposite sign when we reflect values to the other side of 0. A health-food business would like to create a high-potassium blend of dried fruit in the form of a box of 10 fruit bars. It decides to use dried apricots, which have mg of potassium per serving, and dried dates, which have mg of potassium per serving SOURCE: www. The company can purchase its fruit through www. The company would like the box of bars to have at least the recommended daily potassium intake of about mg, but would like to keep it under twice the recommended daily intake.

In order to minimize cost, how many servings of each dried fruit should go into the box of bars? For apricots, there are 3 servings in one pound. The cost for x servings would thus be 3. For dates, there are 4 servings per pound. The cost for y servings would thus be 2. The feasible region is the green and blue shaded section between the two lines.

We see that there are four corner points that form an upside-down trapezoid, as shown in the graph below:. Again, we could solve by using matrix equations, but the systems are straightforward to solve by substitution. Since the problem is bounded, we now check to see which one minimizes cost:. The cheapest route for the company will be to create bars that contain no dried apricots and It is interesting to note that each of the corner points corresponds to either a horizontal or vertical intercept.

This is truly a case of real-world product creation! It makes complete sense to buy dates, since the same dollar amount yields a higher content of potassium. The question still remains: is it desirable to require a larger quantity of dates for a smaller price, or is it more desirable to require a smaller quantity of apricots for a larger price?

This indeed depends on the constraints. Perhaps the manufacturing and packaging costs could add constraints that alter the decision-making process.

A similar problem will be left as a homework exercise for the reader to think about. As a mathematical note, what we are seeing occurs as a result of having constraint lines that are parallel. A feasible region is said to be bounded if the constraints enclose the feasible region.

Both examples thus far have been examples of bounded linear programming problems, since the first feasible region was in the shape of a triangle and the second in the shape of a trapezoid. If the feasible region cannot be enclosed among the lines formed by constraints, it is said to be unbounded. An example of an unbounded linear programming problem would be:. A human resources office is working to implement an increase in starting salaries for new administrative secretaries and faculty at a community college.

The college would like to determine the percentage increase to allocate to each group, given that the college will be hiring 8 secretaries and 7 faculty in the upcoming academic year. What should the percentage increase be for each group? Our goal is to determine the percentage increase for administrative secretaries and faculty, so let. The college would like to minimize its total expenditures, so the objective function must include the total amount of money outflows.

To visualize the situation, we graph the constraint as an equation. To help us find points, we first find the intercepts:. This gives us three corner points, as shown above. We test each to verify which of the pairs of percentages gives the minimum cost:. Why did this happen, and what should we do to fix it? This outcome will occur anytime we are minimizing, have constraints with the le inequality sign, and when the origin is included in the feasible region.

To fix the problem, the company should make additional specifications, such as, what is the minimum percentage raise to give to each group? Is it desirable for one of the raises to be larger than the other? These are questions the analyst should discuss with human resources and administration.

Linear Programming Problems And Solutions Ppt

Linear programming LP , also called linear optimization is a method to achieve the best outcome such as maximum profit or lowest cost in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming also known as mathematical optimization. More formally, linear programming is a technique for the optimization of a linear objective function , subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality. Its objective function is a real -valued affine linear function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest or largest value if such a point exists. Linear programs are problems that can be expressed in canonical form as.

To address this issue, businesses have incorporated better strategies that will help identify transportation issues and develop the appropriate solutions. Based on Transportation Problems and Assignment Problems. There are many issues surrounding the low initial-cost provision of gravel roads. Unfortunately, although transportation planners recognize their potential benefits, they often treat them as last resort measures, to be used to address specific congestion and air pollution problems where conventional solutions prove to be ineffective. Solutions for First Order Equations Consider first the problem of finding the general solution for the equation tu x,t V x u x,t 0 for all x,t. Answer Table better about starting if that was the case. Our city has a big problem related to the aging fleet of buses, and a lack of technology needed for travel planning.

A mixed integer programming problem is a mathematical programming problem in which at least one, but not all, of the components of x 2 S are required to be integers. In linear programming problems of maximum profit solution or minimum cots solution always occurs at a corner point of the set of the feasible solution. Strictly speaking, this isn't necessary, but I think it makes the solution cleaner and easier to generalize. It will enormously ease you to see guide. It uses an iterative, algebraic procedure for simultaneously solving a set of linear equations. Integer Programming Model Formulation: Often, in linear programming problems, it is necessary that some or all of the variables have discrete values in the optimal solution. The one on the left has an optimal solution, but the one on the right does.


Determine how many dresses and trousers should be made to maximize profit and what the maximum profit is. Solution: Step 1: To solve the above problem we.


Linear Programming Questions And Answers Pdf

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems. After presenting the concepts of grey systems and grey numbers, this paper surveys existing approaches for solving grey linear programming problems and applications. Also, methods and approaches for solving grey linear programming are classified, and its advantages and disadvantages are expressed. The progress of grey programming has been expressed from past to present.

3.2a. Solving Linear Programming Problems Graphically

Optimization models are used extensively in almost all areas of decision-making, such as engineering design and financial portfolio selection. This site presents a focused and structured process for optimization problem formulation, design of optimal strategy, and quality-control tools that include validation, verification, and post-solution activities. Enter a word or phrase in the dialogue box, e. In deterministic models good decisions bring about good outcomes. You get that what you expect; therefore, the outcome is deterministic i.

 Танкадо отдал кольцо? - скептически отозвалась Сьюзан. - Да. Такое впечатление, что он его буквально всучил - канадцу показалось, будто бы он просил, чтобы кольцо взяли. Похоже, этот канадец рассмотрел его довольно внимательно.

Practice Problems

Кардинальное отличие Монокля заключалось не в его миниатюрном дисплее, а в системе ввода информации.

У вас есть возможность мгновенно получать информацию. Вы можете читать все, что пожелаете, - без всяких вопросов и запросов. Вы выиграли. - Почему бы не сказать - мы выиграли. Насколько мне известно, ты сотрудник АНБ.

Transportation problems and solutions pdf

 Нет. Это был шантаж. Все встало на свои места. - Ну конечно, - сказала она, все еще не в силах поверить в произошедшее.

2 Response
  1. Gemma B.

    The above stated optimisation problem is an example of linear programming problem. Graphical method of solving linear programming problems.

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