Optimisation and its applications An overview

Optimisation is the act of obtaining the best result under given circumstances.

In the design , construction and maintenance of any engineering system, engineers have to take many technological and managerial decisions at several stages.

The goal of such decisions is either to minimise the effort required or to maximise the desired benefit.

Optimisation can be defined as the process of finding the conditions that give the maximum value or minimum value of a function.

A number of optimisation methods have been developed for solving different types of optimsation problems

Historical Development

The existence of optimsation methods can be traced to the days of Newton, Lagrange, Cauchy, Leibnitz, Bernoulii, Euler etc.,

Very little progress was made until the middle of the twentieth century.

Then High speed digital computers made implementation of optimisation procedures possible.

Historical Developments (contd.,)

Von Neumann gave the foundations for game theory in the year 1928 Development of Simplex method by Dantzig in 1947 Kuhn and Tucker conditions for NLP in 1951

Bellmanns principle of optimality in 1957

Developments in Numerical methods of unconstrained optimisation have been made in the UK in the 1960s.

Historical Developments (contd.,)Duffin, Zener and Peterson developed Geometric Programming during1960s.

Gomory gave his cutting plane method for Integer programming.

Goal programming was proposed by Charnes and Cooper in 1961.Simulated annealing, genetic algorithms and neural network methods came during the last decade.

Engineering applications of optimisationSome typical applications from different engineering disciplines are given below:

1. Design of aircraft and aerospace structures for minimum weight

2.Finding the optimal trajectories of space vehicles

3.Design of civil engineering structures such as frames, foundations, bridges, towers, chimneys

Engineering applications of optimisation (contd.,)

4.Minimum-weight design of structures for earthquake, wind, and other types of random loading

5. Design of water resources systems for maximum benefit

6. Optimal design of structures

7. Optimum design of linkages, cams, gears, machine tools and other mechanical components

Engineering applications of optimisation (contd.,)

8. Selection of machining conditions in metal-cutting processes for minimum production cost

9. Design of material handling equipment such as conveyors, trucks, cranes for minimum cost

10.Design of pumps, turbines and heat transfer equipment for maximum efficiency

11.Optimum design of electrical machinery such as motors, generators, and transformers

12. Optimum design of electrical networks

Engineering applications of optimisation (contd.,)

13. Shortest route taken by a salesperson visiting various cities during one tour

14.Optimal production planning, controlling and scheduling

15. Optimum design of chemical processing equipment and plants

16.Design of optimum pipeline networks for process industries

Engineering applications of optimisation (contd.,)17. Selection of a site for industry

18.Planning of maintenance and replacement of equipment to reduce operating costs

19.Inventory control

20.Allocation of resources or services among several activities to maximise the benefit

Engineering applications of optimisation (contd.,)

21. Controlling the waiting and idle times and queuing in production lines to reduce costs

22. Planning the best strategy to obtain maximum profit in the presence of a competitor

23.Optimum design of control systems

Methods of Optimisation TechniquesMathematical Programming TechniquesStochastic Process TechniquesStatistical MethodsCalculus methodsStatistical Decision Theory Regression AnalysisCalculus of variationsMarkov ProcessesCluster Analysis, pattern recognitionNonlinear ProgrammingQueuing TheoryDesign of ExperimentsGeometric ProgrammingRenewal TheoryLinear ProgrammingSimulation methodsDynamic ProgrammingReliability TheoryInteger ProgrammingStochastic ProgrammingSeparable ProgrammingMulti objective ProgrammingNetwork methods: CPM and PERTGame TheorySimulated AnnealingGenetic AlgorithmsNeural Networks

Classification of optimisation problems

Based on the existence of constraints:

Constrained optimisation problem and unconstrained optimisation problem

(ii) Based on the nature of design variables:

Parameter or static optimisation problem and trajectory or dynamic optimisation problem

(iii) Based on physical structure of the problem:

Optimal control problem and nonoptimal control problem

Classification of Optimisation problems (contd.,)

(iv) Based on the nature of equations involved:

Linear Programming

Nonlinear Programming

Geometric programming

Quadratic programming

(v) Based on the permissible values of the design variables:Integer programming

Real value programming

Classification of Optimisation problems (contd.,)

(vi) Based on the deterministic nature of the variables:

Stochastic programming

(vii)Based on the separability of the objective functions andconstraint functions:

Separable programming Based on the number of objective functions to be optimised:

Single objective programmingMulti objective programming

Classical optimisation Techniques

To find the optimum solution of continuous and differential functions.

These methods make use of the techniques of differential calculus in locating optimum points. ( maxima and minima)

These methods have limited scope in practical applications because practical problems involve objective functions which are not continuous and/or differentiable.

Linear ProgrammingThis is applicable for the problems in which the objective function and the constraints are linear functions of the key variables. Applications: Product mix problem Optimal production planning in a manufacturing firmMake or buy decisions in metal working industriesMinimising trim loss in paper millOptimal routing of messages in communication network Routing of aircraft and shipTransportation problem and assignment problem

Methods to solve LP modelsGraphical MethodSimplex method- with slack variables only- Big M Method- Two Phase MethodDual Simplex methodRevised simplex methodDecomposition principleKarmarkars method (50 times faster than simplex)

Sensitivity Analysis/Post optimal analysisTo know the sensitivity of the optimality of a LP model with respect to the changes made in the parameters and characteristics of a LP model

Making changes in the RHS constants of the constraintsMaking changes in the objective function coefficientsAdding a new constraintAdding a new variable

-Also the resources used in the model may be classified as abundant and scarce and unit worth of a resource may also be calculated.

Parametric Linear ProgrammingTo find out the range for objective function coefficients so that the optimality remains unaffected.

To find out the range for right hand side constants of constraints so that the optimality remains unaffected.

Nonlinear Programming

- It is an extension of linear programming.

The objective function may be nonlinear and the constraints may be linear or nonlinear.

Lagrangean methodLagrangean method is used to solve a NLP where the objective function is nonlinear and the constraint is in linear equation form.

Kuhn-Tucker conditionsKT conditions are used to solve a NLP where the Objective function is in nonlinear form and constraint is in nonlinear inequality form.

Quadratic ProgrammingQuadratic Programming is a NLP problem in which the highest order of the polynomial of the objective function is restricted to 2.

Such quadratic programming problems are solved using Wolfes method.

The QPP is converted into a LPP and solved using suitable simplex method.

Separable ProgrammingSeparable programming problem is a NLP where the objective function consists of linear and nonlinear term as well as the constraint will also consist of linear and nonlinear term.

In such problems the nonlinear terms are converted into equivalent sum of linear terms using piece wise linearisation and the problem becomes a LPP and it is solved using simplex with restricted basis method.

Stochastic programming or Chance-constrained programmingSuch problems deal with situations where some or all of the parameters of the optimisation problem are described by stochastic (random or probabilistic) variables. The stochastic model is converted into an equivalent deterministic model and solved.

Integer ProgrammingInteger Linear programming is a LPP where the optimal solution for the key variables are to be obtained in terms of integers.

Methods used:Gomoris cutting plane methodBranch and Bound method

Dynamic programming-Also known multistage programming-Bellmanns optimality condition is used.Terms used in DP:StageState VariableRecursive function (Forward/Backward)More amount of computations to be made

Applications of Dynamic ProgrammingCapital Budgeting problemReliability improvement problemShortest route problemInventory control/production controlLinear programmingDesign of a gear trainMinimum cost pipe network

Goal Programming or Multiobjective programming-It is an extension of Linear programming.-Suitably formulated into a LPP and solved.Three types:(i) Single goal programming (ii) Multi goal programming with equal priorities(iii)Multi goal programming with different priorities

Network TechniquesCPM and PERT Project networks

Shortest path model

Minimum spanning tree problem Kruskals Algorithm

(iv) Maximal Flow Problem- MFP Algorithm

SequencingOptimal sequence of a few jobs to be processed under a few machines so that elapsed time and the idle times are minimised.Types:N Jobs x 2 machinesN jobs x 3 machinesN Jobs x m machinesm jobs x 2 machines

Game Theory-Two person Zero sum game-Maxi-min and Mini-max criteria-Pure strategy and Mixed strategy-Value of the game and the optimal strategies are found out.Types:Games with saddle pointMethod of sub gamesGraphical method(iv) Matrix method

Queuing ModelsCharacteristics of a queuing model Arrival patternService patternNo. of service channelService orderNo. of customers servedPopulation

Applications of queuing modelsQueuing model with customers will be studied.-Customers dont refer only to persons.-Customers may also be machines to be repaired/programs waiting to be executed by computers/jobs waiting to be processed etc.- Idle time or the waiting time of the customers and the service personnel will be identified

Monte Carlo simulation-Used for queuing models with random distribution.-Random numbers are generated.-Using random numbers, frequency distribution of arrival and service the queue will be simulated.-Idle time of the customers and the service counter will be identified to take further decision.

Design of Experiments or Experimental DesignTerms: FactorsLevelsReplicationInteractionResponseTreatment conditionDegrees of freedom

DOE is used for parameter optimisation of a process or a productThe significant factors can be identified.The levels of the factors may be fixed to have an optimum response value.

It is known as parameter design

It is also known as Design for Robustness

DOE for friction welding The reduction of RPN with respect to low tensile strength and weld crack can be achieved by optimising the process parameters. In order to optimise the process parameters, Design of Experiments (DOE) was used as explained below.Upset force, friction force, burn off and spindle speed are the major causes of weld crack and low tensile strength. These parameters were optimised using 2 level DOE involving multiple factors.The experimental factors of the DOE are upset force, friction force, burn off length and spindle speed.

Factors and levels of DOE

Factors/LevelsUpset force(Tons)Friction force(Tons)Burn off(mm)Spindle speed(rpm)ABCD10.7120.35164210021.030.4154.52300

Initial RPN

Sl.No.Failure ModeRPN1Low Tensile strength1602Weld crack120

Due to the FMEA that was carried out for friction welding in the manufacture of engine valves, the RPN for the different failure modes were reduced and are shown below.

RPN Values

Sl.No.Failure ModeInitial RPNImproved RPN1Low Tensile strength160322Weld crack12040

One factor at a time DOE Full factorial DOE

Taguchis DOE

Steps in Taguchis DOE-Decide on the number of factors and levels-Consider interaction if anyFind out the degrees of freedom based on factors, levels and interaction.Choose the proper Orthogonal Array which should be greater than or equal to the degrees of freedom.

Conclusion-The optimisation techniques available are plenty.A given problem must be properly understood to identify the function to be optimised, key variables to be optimised, constraints if any and accordingly the problem has to be formulated.- A suitable optimisation technique/model must be selected accordingly to derive the maximum benefit from these available techniques.

Thank you

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