M. Tech.ED II SEMESTER Course Code: CED11T13 ADVANCED OPTIMIZATION TECHNIQUES LPC 3 - 3 SYLLABUS UNIT – I LINEAR PROGRAMMING: Two-phase simplex method, Big-M method, duality, interpretation, applications. UNIT – II ASSIGNMENT PROBLEM: Hungarian’s algorithm, Degeneracy, applications, unbalanced problems, Traveling salesman problem. UNIT – III CLASSICAL OPTIMIZATION TECHNIQUES: Single variable optimization with and without constraints, multi – variable optimization without constraints, multi - variable optimization with constraints - method of Lagrange multipliers, Kuhn-Tucker conditions. UNIT – IV NUMERICAL METHODS FOR OPTIMIZATION: Nelder Mead’s Simplex search method, Gradient of a function, Steepest descent method, Newton’s method, type s of penalty methods for handling constraints. UNIT – V GENETIC ALGORITHM (GA): Differences and similarities between conventional and evolutionary algorithms, working principle, reproduction, crossover, mutation, termination criteria, different reproduction and crossover operators, GA for constrained optimization, draw backs of GA. UNIT – VI GENETIC PROGRAMMING (GP): Principles of genetic programming, terminal sets, functional sets, differences between GA & GP, random population generation, solving differential equations using GP. UNIT – VII MULTI-OBJECTIVE GA: Pareto’s analysis, Non-dominated front, multi - objective GA, Nondominated sorted GA, convergence criterion, applications of multi-objective problems. UNIT – VIII APPLICATIONS OF OPTIMIZATION IN DESIGN AND MANUFACTURING SYSTEMS: Some typical applications like optimization of path synthesis of a four-bar mechanism, minimization of weight of a cantilever beam, optimization of springs and gears, general optimization model of a machining process, optimization of arc welding parameters, and general procedure in optimizing machining operations sequence. TEXT BOOKS: 1. Jasbir S. Arora (2007), Optimization of structural and mechanical systems, 1st Edition, World Scientific, Singapore. 2. Kalyanmoy Deb (2009), Optimization for Engineering Design: Algorithms and Examples , 1st Edition, Prentice Hall of India, New Delhi, India. 3. Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, 4th Edition, John Wiley & Sons, New Delhi, India. REFERENCE BOOKS: 1. D. E. Goldberg (2006), Genetic algorithms in Search, Optimization, and Machine learning, 28th Print, AddisonWesley Publishers, Boston, USA. 2. W. B. Langdon, Riccardo Poli (2010), Foundations of genetic programming, 1st Edition, Springer, New York. 3. R. Venkata Rao, Vimal J. Savsani (2012), Mechanical Design Optimization Using Advanced Optimization Techniques, 1st Edition, Springer, New York. Other Titles 1. Hamdy A Taha (2007), Operations Research – An introduction, 8th Ed, Pearson "Operations Research (OR) is the representation of real-world systems/problems/decisions by mathematical models together with the use of quantitative methods (algorithms) for solving such models. which are the unknowns to be determined by the solution to the model. “Linear Programming (LP)” is an optimization method applicable for the solution of problems in which the objective function and the constraints appear as linear functions of the decision variables. the organization’s attempt to achieve some objective (frequently maximizing profit/rate of return. and using a quantitative (explicit. duality. service levels. . Underlying OR is the philosophy that: decisions have to be made. Constraints to represent the physical limitations of the system An objective function An optimal solution to the model is the identification of a set of variable values which are feasible (satisfy all the constraints) and which lead to the optimal value of the objective function." A mathematical model of OR consists: Decision variables. Introduction The first formal application of Operations Research (OR) was initiated in England during World War II. with a view to optimizing. available machine time. capital). the ideas advanced in military operations were adapted to improve efficiency and productivity in the civilian sector (Taha 2007). After the war. capital. applications. In general terms. articulated) approach will lead to better decisions than using non-quantitative (implicit. minimizing costs) in view of limited or constrained resources (available capital/labor. one can regard OR as being the application of scientific methods / thinking to decision making. Indeed it can be argued that although OR is imperfect it offers the best available approach to making a particular decision in many instances (which is not to say that using OR will produce the right decision). The constraint equations in a linear programming problem may be in the form of equalities or inequalities (SS Rao 2009). LP is a mathematical technique designed to aid managers in allocating scarce resources (such as labor. Big-M method. unarticulated) approaches. or energy) among competing activities. when a team of British Scientists set out to make scientifically based decisions regarding the best utilization of war material.UNIT – I LINEAR PROGRAMMING Two-phase simplex method. in the form of a model. It reflects. interpretation. . bj . The objective function is of the minimization type. xn) = c1x1 + c2x2 + · · · + cnxn subject to the constraints a11x1 + a12x2 + · · · + a1nxn = b1 a21x1 + a22x2 + · · · + a2nxn = b2 .The objective of a decision maker in a linear programming problem is to maximize or minimize an objective function in consideration with resources subjected to some constraints.. n) are known constants. . and xj are the decision variables. . } { } . 2.. 2. are 1. . . . a= [ ] . hence the name "linear" for the problem in analysis. . stated in standard form.. 2. The constraints take the form of linear inequalities. . All the constraints are of the equality type. . { } { . All the decision variables are nonnegative. . (1) (2) (3) The characteristics of a linear programming problem. and aij (i = 1.m. The general linear programming problem can be stated in the following standard forms: SCALAR FORM Minimize f (x1. (4) (5) (6) . am1x1 + am2x2 + · · · + amnxn = bm x1 ≥ 0 x2 ≥ 0 . 3. xn ≥ 0 where cj . j = 1. . . x2.. MATRIX FORM Minimize f (X) = cTX subject to the constraints aX = b X≥0 Where. In such cases. e. Such a variable is numerically equal to the difference between the right. It represents the part of unutilized resource of type . negative. Similarly. an unrestricted variable (which can take a positive. depending on whether x′′ j is greater than. or positive. a variable may be unrestricted in sign in some problems. nonnegative variable to the left-hand side of the inequality. However. The maximization of a function f (x1. . it can be written as xj = x′j − x′′j .Any linear programming problem can be expressed in standard form by using the following transformations. or less than x′j 3. zero. or zero value) can be written as the difference of two nonnegative variables. if the constraint is in the form of a “greater than or equal to” type of inequality as ak1x1 + ak2x2 + · · · + aknxn ≥ bk it can be converted into the equality form by subtracting a variable as ak1x1 + ak2x2 + · · · + aknxn − S1= bk where S1 is a nonnegative variable known as a surplus variable. In most engineering optimization problems.g: The objective function minimize f = c1x1 + c2x2 + · · · + cnxn is equivalent to maximize f ′ = −f = −c1x1 − c2x2 − · · · − cnxn Consequently. If a constraint appears in the form of a “less than or equal to” type of inequality as ak1x1 + ak2x2 + · · · + aknxn ≤ bk it can be converted into the equality form by adding a nonnegative slack variable S1 as follows: ak1x1 + ak2x2 + · · · + aknxn + S1= bk A linear constraint of the form ∑ can be converted into equality by adding a new. 1. . Such a variable is numerically equal to the difference between the right. xn) is equivalent to the minimization of the negative of the same function. 2. nonnegative variable to the left-hand side of the inequality. and hence the variables xj will be nonnegative. equal to. A linear constraint of the form ∑ can be converted into equality by adding a new. x2. the objective function can be stated in the minimization form in any linear programming problem. where x′j ≥ 0 and x′′j ≥ 0 It can be seen that xj will be negative. . the decision variables represent some physical dimensions.and left-hand sides of the inequality and is known as surplus .and left-hand sides of the inequality and is known as slack variable. Thus if xj is unrestricted in sign. . PS: Please refer . The case m < n corresponds to an underdetermined set of linear equations. 2. Two-phase simplex method . for if m > n. It has no feasible solutions (The feasible region contains no points). one of the following four cases will occur: 1. there would be m – n redundant equations that could be eliminated. We can assume that m < n.3 or 3. (3. for then there is either a unique solution X that satisfies Eqs. The case n = m is of no interest.SSRao (2009) for more fundamental concepts. The LP has a unique optimal solution. if they have one solution. (3. It has more than one (actually an infinite number of) optimal solutions 3. It represents the part of resource required in excess of the minimum limit of type .2 or 3.5) and (3.5) and (3. in which case the constraints are inconsistent. 4. The LP is infeasible. have an infinite number of solutions. In the feasible region there are points with arbitrarily large (in a max problem) objective function values.2 or 3.variable. LP Solutions: When an LP is solved.6) and yields the minimum value of f (Objective function in consideration for optimization). which.6) (in which case there can be no optimization) or no solution.3 or 3. The LP has alternative (multiple) optimal solutions. It can be seen that there are m equations in n decision variables in a linear programming problem. The problem of linear programming is to find one of these solutions that satisfies Eqs. The LP is unbounded.