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In fact, the field of unconstrained optimization is a large and important one for which a lot of algorithms and software are available. In practice, answers that make good sense about the underlying physical or economic problem, cannot often be obtained without putting constraints on the decision variables. Feasible and Optimal Solutions: A solution value for decision variables, where all of the constraints are satisfied, is called a feasible solution. Most solution algorithms proceed by first finding a feasible solution, then seeking to improve upon it, and finally changing the decision variables to move from one feasible solution to another feasible solution.

This process is repeated until the objective function has reached its maximum or minimum. This result is called an optimal solution. The basic goal of the optimization process is to find values of the variables that minimize or maximize the objective function while satisfying the constraints. There are well over solution algorithms for different kinds of optimization problems. The widely used solution algorithms are those developed for the following mathematical programs: convex programs, separable programs, quadratic programs and the geometric programs.

Linear Program Linear programming deals with a class of optimization problems, where both the objective function to be optimized and all the constraints, are linear in terms of the decision variables. A short history of Linear Programming: In , Lagrange solved tractable optimization problems with simple equality constraints. In , Gauss solved linear system of equations by what is now call Causssian elimination. In Wilhelm Jordan refinmened the method to finding least squared errors as ameasure of goodness-of-fit.

Now it is referred to as the Gauss-Jordan Method. In , Digital computer emerged. In , Dantzig invented the Simplex Methods. In , Karmarkar applied the Interior Method to solve Linear Programs adding his innovative analysis. Linear programming has proven to be an extremely powerful tool, both in modeling real-world problems and as a widely applicable mathematical theory. However, many interesting optimization problems are nonlinear. The study of such problems involves a diverse blend of linear algebra, multivariate calculus, numerical analysis, and computing techniques.

Important areas include the design of computational algorithms including interior point techniques for linear programming , the geometry and analysis of convex sets and functions, and the study of specially structured problems such as quadratic programming. Nonlinear optimization provides fundamental insights into mathematical analysis and is widely used in a variety of fields such as engineering design, regression analysis, inventory control, geophysical exploration, and economics. Quadratic Program Quadratic Program QP comprises an area of optimization whose broad range of applicability is second only to linear programs.

A wide variety of applications fall naturally into the form of QP. The kinetic energy of a projectile is a quadratic function of its velocity. The least-square regression with side constraints has been modeled as a QP. Certain problems in production planning, location analysis, econometrics, activation analysis in chemical mixtures problem, and in financial portfolio management and selection are often treated as QP.

There are numerous solution algorithms available for the case under the restricted additional condition, where the objective function is convex. Constraint Satisfaction Many industrial decision problems involving continuous constraints can be modeled as continuous constraint satisfaction and optimization problems.

Constraint Satisfaction problems are large in size and in most cases involve transcendental functions. They are widely used in chemical processes and cost restrictions modeling and optimization.

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When the objective function is convex and the feasible region is a convex set, both of these assumptions are enough to ensure that local minimum is a global minimum. Data Envelopment Analysis The Data Envelopment Analysis DEA is a performance metric that is grounded in the frontier analysis methods from the economics and finance literature. The strength of DEA relies partly on the fact that it is a non-parametric approach, which does not require specification of any functional form of relationships between the inputs and the outputs.

DEA output reduces multiple performance measures to a single one to use linear programming techniques. The weighting of performance measures reacts to the decision-maker's utility. Dynamic Programming Dynamic programming DP is essentially bottom-up recursion where you store the answers in a table starting from the base case s and building up to larger and larger parameters using the recursive rule s.

You would use this technique instead of recursion when you need to calculate the solutions to all the sub-problems and the recursive solution would solve some of the sub-problems repeatedly. While generally DP is capable of solving many diverse problems, it may require huge computer storage in most cases.

Separable Program Separable Program SP includes a special case of convex programs, where the objective function and the constraints are separable functions, i.

Geometric Program Geometric Program GP belongs to Nonconvex programming, and has many applications in particular in engineering design problems. Fractional Program In this class of problems, the objective function is in the form of a fraction i. Fractional Program FP arises, for example, when maximizing the ratio of profit capital to capital expended, or as a performance measure wastage ratio. Heuristic Optimization A heuristic is something "providing aid in the direction of the solution of a problem but otherwise unjustified or incapable of justification.

They are, at best, educated guesses. Several heuristic tools have evolved in the last decade that facilitate solving optimization problems that were previously difficult or impossible to solve. These tools include evolutionary computation, simulated annealing, tabu search, particle swarm, etc. Common approaches include, but are not limited to: comparing solution quality to optimum on benchmark problems with known optima, average difference from optimum, frequency with which the heuristic finds the optimum.

Global Optimization The aim of Global Optimization GO is to find the best solution of decision models, in presence of the multiple local solutions. While constrained optimization is dealing with finding the optimum of the objective function subject to constraints on its decision variables, in contrast, unconstrained optimization seeks the global maximum or minimum of a function over its entire domain space, without any restrictions on decision variables. Nonconvex Program A Nonconvex Program NC encompasses all nonlinear programming problems that do not satisfy the convexity assumptions.

However, even if you are successful at finding a local minimum, there is no assurance that it will also be a global minimum. Therefore, there is no algorithm that will guarantee finding an optimal solution for all such problem. NSP are arising in several important applications of science and engineering, including contact phenomena in statics and dynamics or delamination effects in composites.

These applications require the consideration of nonsmoothness and nonconvexity. Metaheuristics Most metaheuristics have been created for solving discrete combinatorial optimization problems. Practical applications in engineering, however, usually require techniques, which handle continuous variables, or miscellaneous continuous and discrete variables. Theoretical and experimental studies on metaheuristics adapted to continuous optimization, e.

Software implementations and algorithms for metaheuristics adapted to continuous optimization. Real applications of discrete metaheuristics adapted to continuous optimization. Performance comparisons of discrete metaheuristics adapted to continuous optimization with that of competitive approaches, e. Multilevel Optimization In many decision processes there is a hierarchy of decision makers and decisions are taken at different levels in thishierarchy.

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Multilevel Optimization focuses on the whole hierarchy structure. The field of multilevel optimization has become a well known and important research field. Hierarchical structures can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, supply chain, game theory and economics. Moreover, new applications are constantly being introduced. Multiobjective Program Multiobjective Program MP known also as Goal Program, is where a single objective characteristic of an optimization problem is replaced by several goals.

In solving MP, one may represent some of the goals as constraints to be satisfied, while the other objectives can be weighted to make a composite single objective function. Multiple objective optimization differs from the single objective case in several ways: The usual meaning of the optimum makes no sense in the multiple objective case because the solution optimizing all objectives simultaneously is, in general, impractical; instead, a search is launched for a feasible solution yielding the best compromise among objectives on a set of, so called, efficient solutions; The identification of a best compromise solution requires taking into account the preferences expressed by the decision-maker; The multiple objectives encountered in real-life problems are often mathematical functions of contrasting forms.

A key element of a goal programming model is the achievement function; that is, the function that measures the degree of minimisation of the unwanted deviation variables of the goals considered in the model. A Business Application: In credit card portfolio management, predicting the cardholder's spending behavior is a key to reduce the risk of bankruptcy. Given a set of attributes for major aspects of credit cardholders and predefined classes for spending behaviors, one might construct a classification model by using multiple criteria linear programming to discover behavior patterns of credit cardholders.

Non-Binary Constraints Program Over the years, the constraint programming community has paid considerable attention to modeling and solving problems by using binary constraints. Only recently has non-binary constraints captured attention, due to growing number of real-life applications.

A non-binary constraint is a constraint that is defined on k variables, where k is normally greater than two. A non-binary constraint can be seen as a more global constraint. Modeling a problem as a non-binary constraint has two main advantages: It facilitates the expression of the problem; and it enables more powerful constraint propagation as more global information becomes available. Success in timetabling, scheduling, and routing, has proven that the use of non-binary constraints is a promising direction of research. Bilevel Optimization Most of the mathematical programming models deal with decision-making with a single objective function.

The bilevel programming on the other hand is developed for applications in decentralized planning systems in which the first level is termed as the leader and the second level pertains to the objective of the follower. In the bilevel programming problem, each decision maker tries to optimize its own objective function without considering the objective of the other party, but the decision of each party affects the objective value of the other party as well as the decision space. Bilevel programming problems are hierarchical optimization problems where the constraints of one problem are defined in part by a second parametric optimization problem.

If the second problem has a unique optimal solution for all parameter values, this problem is equivalent to usual optimization problem having an implicitly defined objective function. However, when the problem has non-unique optimal solutions, the optimistic or weak and the pessimistic or strong approaches are being applied. Combinatorial Optimization Combinatorial generally means that the state space is discrete e.

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This space could be finite or denumerable sets. For example, a discrete problem is combinatorial. Problems where the state space is totally ordered can often be solved by mapping them to the integers and applying "numerical" methods. If the state space is unordered or only partially ordered, these methods fail. This means that the heuristics methods becomes necessary, such as simulated annealing. Combinatorial optimization is the study of packing, covering, and partitioning, which are applications of integer programs. They are the principle mathematical topics in the interface between combinatorics and optimization.

These problems deal with the classification of integer programming problems according to the complexity of known algorithms, and the design of good algorithms for solving special subclasses. In particular, problems of network flows, matching, and their matroid generalizations are studied. This subject is one of the unifying elements of combinatorics, optimization, operations research, and computer science. Evolutionary Techniques Nature is a robust optimizer. By analyzing nature's optimization mechanism we may find acceptable solution techniques to intractable problems. Two concepts that have most promise are simulated annealing and the genetic techniques.

Scheduling and timetabling are amongst the most successful applications of evolutionary techniques. Genetic Algorithms GAs have become a highly effective tool for solving hard optimization problems. However, its theoretical foundation is still rather fragmented. Such systems are typically made up of a population of simple interacting agents without any centralized control, and inspired by cases that can be found in nature, such as ant colonies, bird flocking, animal herding, bacteria molding, fish schooling, etc.

There are many variants of PSO including constrained, multiobjective, and discrete or combinatorial versions, and applications have been developed using PSO in many fields. Swarm Intelligence Biologists studied the behavior of social insects for a long time. After millions of years of evolution all these species have developed incredible solutions for a wide range of problems. The intelligent solutions to problems naturally emerge from the self-organization and indirect communication of these individuals. Indirect interactions occur between two individuals when one of them modifies the environment and the other responds to the new environment at a later time.

Swarm Intelligence is an innovative distributed intelligent paradigm for solving optimization problems that originally took its inspiration from the biological examples by swarming, flocking and herding phenomena in vertebrates. Online Optimization Whether costs are to be reduced, profits to be maximized, or scarce resources to be used wisely, optimization methods are available to guide decision-making.

In online optimization, the main issue is incomplete data and the scientific challenge: how well can an online algorithm perform? Can one guarantee solution quality, even without knowing all data in advance? In real-time optimization there is an additional requirement: decisions have to be computed very fast in relation to the time frame we are considering. Further Readings: Abraham A. Grosan and V.

Ramos, Swarm Intelligence , Springer Verlag, It deals with the applications of swarm intelligence in data mining, using different intelligent approaches. Charnes A. Dempe S. Diwekar U. Covers almost all the above techniques. Liu B. Luenberger D. Miller R. Reeves C. Rodin R. For more books and journal articles on optimization visit the Web site Decision Making Resources Linear Programming Linear programming is often a favorite topic for both professors and students.

The ability to introduce LP using a graphical approach, the relative ease of the solution method, the widespread availability of LP software packages, and the wide range of applications make LP accessible even to students with relatively weak mathematical backgrounds. Additionally, LP provides an excellent opportunity to introduce the idea of "what-if" analysis, due to the powerful tools for post-optimality analysis developed for the LP model. Linear Programming LP is a mathematical procedure for determining optimal allocation of scarce resources. LP is a procedure that has found practical application in almost all facets of business, from advertising to production planning.

Transportation, distribution, and aggregate production planning problems are the most typical objects of LP analysis. In the petroleum industry, for example a data processing manager at a large oil company recently estimated that from 5 to 10 percent of the firm's computer time was devoted to the processing of LP and LP-like models.

Pdf Linear Programming In Industry Theory And Applications An Introduction 1965

Linear programming deals with a class of programming problems where both the objective function to be optimized is linear and all relations among the variables corresponding to resources are linear. This problem was first formulated and solved in the late 's. Rarely has a new mathematical technique found such a wide range of practical business, commerce, and industrial applications and simultaneously received so thorough a theoretical development, in such a short period of time.

Today, this theory is being successfully applied to problems of capital budgeting, design of diets, conservation of resources, games of strategy, economic growth prediction, and transportation systems. In very recent times, linear programming theory has also helped resolve and unify many outstanding applications. It is important for the reader to appreciate, at the outset, that the "programming" in Linear Programming is of a different flavor than the "programming" in Computer Programming.

In the former case, it means to plan and organize as in "Get with the program! While in the latter case, it means to write codes for performing calculations. Training in one kind of programming has very little direct relevance to the other. In fact, the term "linear programming" was coined before the word "programming" became closely associated with computer software. This confusion is sometimes avoided by using the term linear optimization as a synonym for linear programming.

Any LP problem consists of an objective function and a set of constraints. In most cases, constraints come from the environment in which you work to achieve your objective. When you want to achieve the desirable objective, you will realize that the environment is setting some constraints i. This is why religions such as Buddhism, among others, prescribe living an abstemious life. No desire, no pain. Can you take this advice with respect to your business objective? What is a function: A function is a thing that does something. For example, a coffee grinding machine is a function that transform the coffee beans into powder.

The objective function maps and translates the input domain called the feasible region into output range, with the two end-values called the maximum and the minimum values. When you formulate a decision-making problem as a linear program, you must check the following conditions: The objective function must be linear. That is, check if all variables have power of 1 and they are added or subtracted not divided or multiplied The objective must be either maximization or minimization of a linear function.

The objective must represent the goal of the decision-maker The constraints must also be linear. This very simple problem has no solution. As always, one must be careful in categorizing an optimization problem as an LP problem. Here is a question for you. Is the following problem an LP problem?

Therefore, the above problem is indeed an LP problem. For most LP problems one can think of two important classes of objects: The first is limited resources such as land, plant capacity, or sales force size; the second, is activities such as "produce low carbon steel", "produce stainless steel", and "produce high carbon steel". Each activity consumes or possibly contributes additional amounts of the resources. There must be an objective function, i. The problem is to determine the best combination of activity levels, which do not use more resources than are actually available.

Many managers are faced with this task everyday. Fortunately, when a well-formulated model is input, linear programming software helps to determine the best combination. The Simplex method is a widely used solution algorithm for solving linear programs.

An algorithm is a series of steps that will accomplish a certain task. Any linear program consists of four parts: a set of decision variables, the parameters, the objective function, and a set of constraints. In formulating a given decision problem in mathematical form, you should practice understanding the problem i. While trying to understand the problem, ask yourself the following general questions: What are the decision variables?

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That is, what are controllable inputs? Define the decision variables precisely, using descriptive names. Remember that the controllable inputs are also known as controllable activities, decision variables, and decision activities. What are the parameters? That is, what are the uncontrollable inputs? These are usually the given constant numerical values. Define the parameters precisely, using descriptive names.

What is the objective? What is the objective function? Also, what does the owner of the problem want? How the objective is related to his decision variables? Is it a maximization or minimization problem? The objective represents the goal of the decision-maker. What are the constraints? That is, what requirements must be met? Should I use inequality or equality type of constraint? What are the connections among variables? Write them out in words before putting them in mathematical form. Learn that the feasible region has nothing or little to do with the objective function min or max.

These two parts in any LP formulation come mostly from two distinct and different sources. The following is a very simple illustrative problem. However, the way we approach the problem is the same for a wide variety of decision-making problems, and the size and complexity may differ. The first example is a product-mix problem. The Carpenter's Problem: Allocating Scarce Resources Among Competitive Means During a couple of brain-storming sessions with a carpenter our client , he told us that he, solely, makes tables and chairs, sells all tables and chairs he makes at a market place, however, does not have a stable income, and wishes to do his best.

The objective is to find out how many tables and chairs he should make to maximize net income. We begin by focusing on a time frame, i. To learn more about his problem, we must go to his shop and observe what is going on and measure what we need to formulate i.

Linear Programming in Industry: Theory and Application; An Introduction | UVA Library | Virgo

We must confirm that his objective is to maximize net income. We must communicate with the client. The carpenter's problem deals with finding out how many tables and chairs to make per week; but first an objective function must be established: Since the total cost is the sum of the fixed cost F and the variable cost per unit multiplied by the number of units produced. F1 and F2 are the fixed costs for the two products respectively.

Theory and Applications. An Introduction

The constraining factors which, usually come from outside , are the limitations on labors this limitation comes from his family and raw material resources this limitation comes from scheduled delivery. Production times required for a table and a chair are measured at different times of day, and estimated to be 2 hours and 1 hour, respectively. Honorable prefrontal Sir William R. Good particular Sir William R. Two Ways Of Dying for a Husband.

Mathematical optimization

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