Linear Programming Essays - Mathematical Optimization,

Linear Programming Linear programming is a nonstatistical mathematical technique whereby the maximization or minimization of a linear expresion of variables, call the objective function, is determined in the presence of known or assumed restrictions, call constraint. In essence, it's a procedure for solving the problems in which there are more variables than simultaneous equations in which the variables are expressed. No probability or statistics are needed to study linear programming. The mathematics involved in linear programming is relatively easy to understand and to manipulate in contrast to calculus. Linear equations and inequations form the mathematical skeleton around which linear programming is built. A linear function called the object function is to be maximized or minimized in some sense, like optimzed. Most real world problems have many possible solutions. The purpose of optimization is to choose from among many possible solutions the "best" possible solutions. Some example of "best" are highest profit, lowest cost, largest sales, lowest production time, etc. The optimization of the objective functions take place in teh presence of known or assumed restriction. The technical term constraints is used to describe the restrictions present in linear programming problem. The constraints are expressed mathemically as inequalities. In a practical real-world situation, the constraints are generated by the presence of limited resources or commodities such as capital manpower and raw material. Mathematically, inequations can be converted to equations by the introduction of slack variables. Linear programming can be dated from the year 1947 when G.B. Dantzing evolved an efficent technique call the Simplex Method, for solving linear programming problems. The following decades, the rapid development of both the theory and applications of linear programming which were aided by the simultaneous introduction of the electronic computers. One of the first probelm to be solved by the simplex method was Stigler's diet problem (1945). Here is the diet problem Protein Fat Carbohydrate Cost 100g bread 40 5 205 2.2p 100g cheese 60 380 60 12p Minimal daily requirement 300 790 1350 The problem is determine how much bread and cheese Mrs. Jones should buy each day in order to minimize the cost of the diet, whilst fulfilling the calorie requirements. Suppose shy buys x' * 100g of cheese and x'' * 100g of cheese, then the mathematical problem, known as a linear programme is as follows. Minimize z=2.2x' + 12x'' (Cost Of Diet) Subject To 40x' + 60x'' > or = 300 (At least 300 cal of protein) 5x' + 380x'' > or = 790 (At least 790 cal of fat) 205x' + 60x'' > or = 1350 (At least 1350 cal of carbo.) x' > or = 0, x2 > = 0 (quantites must be non-negative) The easiest and most illustrative method of solving problems in two unknowns is the graphical method. The value of x' and x'' satisfying 40x' + 60x'' > or = 300 lies in the upper half-plane bounded by the straight line 40x' + 60x'' = 300, so the x' and x'' satistying all the above inequalities lie in the intersection of their respective half - plane. Interger Solutions Provided the supplies and demands are positive intergers, the matrix minimum method always leads to and optimal solution with integer values as the method only involves operations on integers which results in integers. Obviously a non-integer optimal solution would be useless. Uniqueness It can happen that two or more differnet allocations of ships between ports give rise to thesame minimum cost. However, if v'=u'

5 Types of Redundancy

5 Types of Redundancy 5 Types of Redundancy 5 Types of Redundancy By Mark Nichol Various words and phrases serve to communicate what a word or phrase refers to or herald to a reader that additional information is forthcoming or a comparison is being made, but writers sometimes make the mistake of unnecessarily employing more than one of these indicators at once. The following five sentences illustrate an array of redundancies. Discussions and revisions follow each erroneous sentence. 1. Many of them are between the ages of 15 and 35 years old. â€Å"The ages of† and â€Å"years old† serve the same purpose- to identify what the numbers 15 and 35 signify- so use one or the other: â€Å"Many of them are between 15 and 35 years old† or â€Å"Many of them are between the ages of 15 and 35.† 2. Despite differences between millennials and Generation Z, both share the same workplace learning needs. Both and share have the same function; they indicate that there is something in common. Either write â€Å"Despite differences between millennials and Generation Z, they share the same workplace learning needs,† or start the main clause with â€Å"both have† and detail the specifics. 3. In addition, financial institutions should also consider having teams that provide oversight at regional or global levels. Also is redundant to â€Å"in addition,† so use one or the other: â€Å"In addition, financial institutions should consider having teams that provide oversight at regional or global level† or â€Å"Financial institutions should also consider having teams that provide oversight at regional or global levels.† 4. He hosted an educational television program from 1993 to 1998, and he also wrote several books on scientific topics. And indicates that additional information is forthcoming, so also is extraneous: â€Å"He hosted an educational television program from 1993 to 1998, and he wrote several books on scientific topics.† (Another option is to slightly simplify the sentence: â€Å"He hosted an educational television program from 1993 to 1998 and wrote several books on scientific topics.† Alternatively, also can be retained if and is deleted, which requires, however, that a semicolon supplant the comma: â€Å"He hosted an educational television program from 1993 to 1998; he also wrote several books on scientific topics.†) 5. His account was bolstered by Jones, who, like Smith, also relies on communications with a network of family and acquaintances still in the city. Like indicates that Jones shares a characteristic with Smith, so also is superfluous: â€Å"His account was bolstered by Jones, who, like Smith, relies on communications with a network of family and acquaintances still in the city.† Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! Browse the Style category, check our popular posts, or choose a related post below:Yours faithfully or Yours sincerely?8 Writing Tips for Beginners10 Varieties of Syntax to Improve Your Writing

Speech Essay Example | Topics and Well Written Essays - 250 words - 6

Speech - Essay Example have added advantage as they inherited with richness and fair complexity that generates dominating behavior and attitudes towards Hispanics (Macartney et al., 2014). In addition, older people will show that as they have spent much period of life and have assembled considerable experiences (Population Division, n.d.). b. There exist variations in framing and evaluating the data and information based on the categories of audiences. It is equally important that where there are more women than men, or more Asian Americans than Hispanics, or older people than younger people, the speech delivered to them need to be highly intellectual and presented in such a manner that addresses past, present and future aspects (Pearson Education Ltd, 2014). The specific purpose statement defines about the scope, objective and purpose of the research study in relation to the relevant topic. On the other hand, thesis statement states about main objective, significance and researcher opinion about the topic (SBCC, n.d.). In order to critically evaluate the Internet sources, it is crucial to keep certain key aspect in mind. In this regard, the evaluator should consider authorship, publishing body, accuracy, authority, currency and coverage among others (Western Kentucky University, 2014). A proper credit to sources can be given, if the internet sources meet the required information needs. Besides, appropriateness of format, integrity, authenticity and reliability of the internet source can be measured to provide order proper credit to the sources (American Sociological Association,