An Illustrated Guide to Linear Programming
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There is also some objective to be optimized which enables us to select at least one of the feasible solutions to be the optimum. The finding of the feasible solutions and the determination of an optimal one represents the computational aspects of programming problems which are discussed in later sections.
As we shall concern ourselves only with linear-programming problems, it should be stressed that linear programs are a special subset of general programming problems usually called mathematical programming in that the mathematical description of linear programs can be written in terms of linear or straight-line relationships. On the other hand, if the seller allowed 10 cents off for the second pound purchased, 20 for the third, etc. In sum, linear-programming problems are those programming problems whose relationships—the constraints of the problem and the objective function—can be represented mathematically as linear relationships.
Although this appears to be quite restrictive, many important problems have this simplifying feature. Programming problems, especially linear-programming problems, arise in a wide variety of settings. There are standard applications of programming techniques found in the areas of agriculture, petrochemicals, paper manufacturing, transportation, production scheduling and inventory control, military and defense, engineering, economics, to cite just a few. A bibliography of applications is given in the Appendix.
An Illustrated Guide to Linear Programming - Saul I. Gass - Google книги
We shall deal with a few such applications in order to develop the fundamentals of linear programming. But first—as a slight digression—we must discuss the place of linear programming within the scheme of modern-day scientific decision-making, i. Definitions of operations research OR , like advice to the lovelorn, have always been in demand and are plentiful. They range from the detailed one of operations research is the application of scientific techniques, tools, and methodology to problems involving the operations of a system so as to provide those in control of the system with optimum solutions to the problems, to the succinct ones of operations research is quantitative common sense or research into operations.
For our needs, we consider OR to be the application of scientific techniques to decision problems.
What interests us most, however, is not what OR claims to be, but its methodology. The phases of an OR project can be delineated into six overlapping and somewhat ill-defined stages. Formulating the problem. Developing a mathematical model to represent the system under study. Deriving a solution from the model.
Testing the model and solution. Establishing controls over the solution.
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Putting the solution to work. Here we have introduced the concept of a mathematical model, which we shall see is central to the methodology of OR. These phases of an OR project can be viewed as accomplishing the following: For any problem we need to define the broad objectives and goals of the system—examine the environment we are working in—become familiar with the jargon, the people and things associated with the problem—determine the alternative courses of action available to the decision-maker—develop some statement, verbal or otherwise, of the problem to be investigated; translate the problem into a suitable logical or mathematical model which relates the variables of the problem by realistic constraints and a measure of effectiveness; find a solution which optimizes the measure of effectiveness, i.
As our ability to develop precise mathematical models of operational problems is not a highly developed science—most people believe it to be an art—we must be sensitive to discrepancies in the solution and feed back to the model refinements that will cause future solutions to be more realistic and accurate. This action might not be possible to undo. Are you sure you want to continue? Upload Sign In Join. Home Books Science. Save For Later.
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An illustrated guide to linear programming
A representative example of my problem might be:. My final project is involving a more complicated problem similar to the one above. The structure of the problems won't change- only the coefficients and inputs will change. So with the example above, I would be looking for a set of functions that might allow a C program to quickly determine x , then y , then z like:.
If your problem is that you want to solve linear programming problems efficiently, you can use Cassowary.