Linear programming
Order ID |
53563633773 |
Type |
Essay |
Writer Level |
Masters |
Style |
APA |
Sources/References |
4 |
Perfect Number of Pages to Order |
5-10 Pages |
Description/Paper Instructions
Linear programming
Linear programming (LP) is a mathematical technique used to optimize a linear objective function subject to linear constraints. It is used to solve various problems in business, economics, engineering, and other fields. In this article, we will explain linear programming in 1000 words or less.
A linear program consists of an objective function and a set of constraints. The objective function is a linear function that needs to be maximized or minimized. The constraints are linear equations or inequalities that limit the values of the decision variables.
Decision variables are the variables that we want to determine. They represent the quantities that we need to optimize. For example, if we want to minimize the cost of producing a certain product, the decision variables could be the quantities of different raw materials that need to be used.
The general form of a linear program is as follows:
Maximize or Minimize: Z = c1x1 + c2x2 + … + cnxn
Subject to: a11x1 + a12x2 + … + a1nxn <= b1 a21x1 + a22x2 + … + a2nxn <= b2 … am1x1 + am2x2 + … + amnxn <= bm x1, x2, …, xn >= 0
In this formulation, Z is the objective function to be optimized, and x1, x2, …, xn are the decision variables. The coefficients c1, c2, …, cn represent the contribution of each variable to the objective function. The constraints a11x1 + a12x2 + … + a1nxn <= b1, a21x1 + a22x2 + … + a2nxn <= b2, …, am1x1 + am2x2 + … + amnxn <= bm, represent the limitations on the values of the decision variables. The non-negativity constraints x1, x2, …, xn >= 0, ensure that the decision variables cannot take negative values.
The solution to a linear program is a set of values for the decision variables that satisfy all the constraints and optimize the objective function. This set of values is called an optimal solution. There may be multiple optimal solutions, or no optimal solutions at all.
The simplex algorithm is the most widely used algorithm for solving linear programs. It is an iterative method that starts with an initial feasible solution and improves it at each step until an optimal solution is found. The algorithm works by moving from one corner point of the feasible region to an adjacent corner point that improves the objective function value. This process is repeated until no further improvement is possible.
The feasible region is the set of all feasible solutions that satisfy all the constraints. It is typically represented graphically as a polygon or polyhedron in two or three dimensions, respectively. The corner points of the feasible region are the points where the constraints intersect. These corner points are also called basic feasible solutions.
If the feasible region is unbounded, that is, it extends to infinity in one or more directions, then the linear program has no optimal solution. If the feasible region is empty, that is, there are no feasible solutions, then the linear program is said to be infeasible.
Linear programming has a wide range of applications. It is used to optimize production schedules, transportation routes, and inventory levels in manufacturing and logistics. It is also used in finance and investment to optimize portfolio selection and risk management. In addition, it is used in agriculture and resource management to optimize crop planting and water usage.
In conclusion, linear programming is a powerful mathematical technique for optimizing linear objective functions subject to linear constraints. It is widely used in various fields to solve a wide range of optimization problems. The simplex algorithm is the most widely used algorithm for solving
Linear programming
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Content (worth a maximum of 50% of the total points) |
Zero points: Student failed to submit the final paper. |
20 points out of 50: The essay illustrates poor understanding of the relevant material by failing to address or incorrectly addressing the relevant content; failing to identify or inaccurately explaining/defining key concepts/ideas; ignoring or incorrectly explaining key points/claims and the reasoning behind them; and/or incorrectly or inappropriately using terminology; and elements of the response are lacking. |
30 points out of 50: The essay illustrates a rudimentary understanding of the relevant material by mentioning but not full explaining the relevant content; identifying some of the key concepts/ideas though failing to fully or accurately explain many of them; using terminology, though sometimes inaccurately or inappropriately; and/or incorporating some key claims/points but failing to explain the reasoning behind them or doing so inaccurately. Elements of the required response may also be lacking. |
40 points out of 50: The essay illustrates solid understanding of the relevant material by correctly addressing most of the relevant content; identifying and explaining most of the key concepts/ideas; using correct terminology; explaining the reasoning behind most of the key points/claims; and/or where necessary or useful, substantiating some points with accurate examples. The answer is complete. |
50 points: The essay illustrates exemplary understanding of the relevant material by thoroughly and correctly addressing the relevant content; identifying and explaining all of the key concepts/ideas; using correct terminology explaining the reasoning behind key points/claims and substantiating, as necessary/useful, points with several accurate and illuminating examples. No aspects of the required answer are missing. |
Use of Sources (worth a maximum of 20% of the total points). |
Zero points: Student failed to include citations and/or references. Or the student failed to submit a final paper. |
5 out 20 points: Sources are seldom cited to support statements and/or format of citations are not recognizable as APA 6th Edition format. There are major errors in the formation of the references and citations. And/or there is a major reliance on highly questionable. The Student fails to provide an adequate synthesis of research collected for the paper. |
10 out 20 points: References to scholarly sources are occasionally given; many statements seem unsubstantiated. Frequent errors in APA 6th Edition format, leaving the reader confused about the source of the information. There are significant errors of the formation in the references and citations. And/or there is a significant use of highly questionable sources. |
15 out 20 points: Credible Scholarly sources are used effectively support claims and are, for the most part, clear and fairly represented. APA 6th Edition is used with only a few minor errors. There are minor errors in reference and/or citations. And/or there is some use of questionable sources. |
20 points: Credible scholarly sources are used to give compelling evidence to support claims and are clearly and fairly represented. APA 6th Edition format is used accurately and consistently. The student uses above the maximum required references in the development of the assignment. |
Grammar (worth maximum of 20% of total points) |
Zero points: Student failed to submit the final paper. |
5 points out of 20: The paper does not communicate ideas/points clearly due to inappropriate use of terminology and vague language; thoughts and sentences are disjointed or incomprehensible; organization lacking; and/or numerous grammatical, spelling/punctuation errors |
10 points out 20: The paper is often unclear and difficult to follow due to some inappropriate terminology and/or vague language; ideas may be fragmented, wandering and/or repetitive; poor organization; and/or some grammatical, spelling, punctuation errors |
15 points out of 20: The paper is mostly clear as a result of appropriate use of terminology and minimal vagueness; no tangents and no repetition; fairly good organization; almost perfect grammar, spelling, punctuation, and word usage. |
20 points: The paper is clear, concise, and a pleasure to read as a result of appropriate and precise use of terminology; total coherence of thoughts and presentation and logical organization; and the essay is error free. |
Structure of the Paper (worth 10% of total points) |
Zero points: Student failed to submit the final paper. |
3 points out of 10: Student needs to develop better formatting skills. The paper omits significant structural elements required for and APA 6th edition paper. Formatting of the paper has major flaws. The paper does not conform to APA 6th edition requirements whatsoever. |
5 points out of 10: Appearance of final paper demonstrates the student’s limited ability to format the paper. There are significant errors in formatting and/or the total omission of major components of an APA 6th edition paper. They can include the omission of the cover page, abstract, and page numbers. Additionally the page has major formatting issues with spacing or paragraph formation. Font size might not conform to size requirements. The student also significantly writes too large or too short of and paper |
7 points out of 10: Research paper presents an above-average use of formatting skills. The paper has slight errors within the paper. This can include small errors or omissions with the cover page, abstract, page number, and headers. There could be also slight formatting issues with the document spacing or the font Additionally the paper might slightly exceed or undershoot the specific number of required written pages for the assignment. |
10 points: Student provides a high-caliber, formatted paper. This includes an APA 6th edition cover page, abstract, page number, headers and is double spaced in 12’ Times Roman Font. Additionally, the paper conforms to the specific number of required written pages and neither goes over or under the specified length of the paper. |
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