Multi-objective Mathematical Models for Process Targeting S. O. Duffuaa, M. Darwish and A. Haroun...

Multi-objective Mathematical Models for Multi-objective Mathematical Models for Process TargetingProcess Targeting

Multi-objective Mathematical Models for Multi-objective Mathematical Models for Process TargetingProcess Targeting

S. O. Duffuaa, M. Darwish and A. HarounS. O. Duffuaa, M. Darwish and A. HarounSystems Engineering DepartmentSystems Engineering Department

King Fahd University Of Petroleum & MineralsKing Fahd University Of Petroleum & Minerals

OutlineOutlineOutlineOutline

• IntroductionIntroduction• Literature ReviewLiterature Review• Problem StatementProblem Statement• Project ObjectivesProject Objectives• Outline of Preliminary Modeling DirectionOutline of Preliminary Modeling Direction• Possible algorithms Possible algorithms • Conclusions and RemarksConclusions and Remarks

Introduction…Introduction…Introduction…Introduction…

L µ

Process Targeting (PT)Process Targeting (PT)• The problem of PT concerns with the The problem of PT concerns with the

determination of the optimum values of the process determination of the optimum values of the process parameters to optimize certain objective.parameters to optimize certain objective.

• The importance of PT is to ensure that a process The importance of PT is to ensure that a process produces products that not only satisfy customer produces products that not only satisfy customer needs, but also with a minimum production cost.needs, but also with a minimum production cost.

• Research in PT started in the early fifties with the Research in PT started in the early fifties with the CAN filling problems.CAN filling problems.

Cont…

IntroductionIntroductionIntroductionIntroduction

• A basic ‘CAN’ filling problem is as follows: A basic ‘CAN’ filling problem is as follows: • The quality characteristic is assumed to be the net weight of The quality characteristic is assumed to be the net weight of

the filled CAN. the filled CAN. • The value of this quality characteristic is a random variable The value of this quality characteristic is a random variable

X, and it has a lower specification limit (LSL)L.X, and it has a lower specification limit (LSL)L.• A 100% inspection is used for product quality control and it A 100% inspection is used for product quality control and it

is assumed to be error free.is assumed to be error free.• An item is accepted if X ≥ L and defective otherwise. An item is accepted if X ≥ L and defective otherwise.

Accepted items are sold at a fixed price Accepted items are sold at a fixed price aa, while rejected , while rejected items are sold at a reduced price items are sold at a reduced price rr..

IntroductionIntroduction IntroductionIntroduction

Cont…

Introduction…Introduction…Introduction…Introduction…

• Now, if the process mean is set higher, the chance of Now, if the process mean is set higher, the chance of producing defective items will reduce, however, this may producing defective items will reduce, however, this may result in a higher production cost.result in a higher production cost.

L µ

IntroductionIntroductionIntroductionIntroduction

• Y approximately follows a normal distribution with Y approximately follows a normal distribution with mean μ and standard deviation σ.mean μ and standard deviation σ.

• The objective is to find the target value μ so that the The objective is to find the target value μ so that the net income for the process is maximized.net income for the process is maximized.

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a g y L if y L

r if y L

($) ( ) ( ) ( ) ( )L

L L

E a f y dy g y L f y dy r f y dy

Literature ReviewLiterature ReviewLiterature ReviewLiterature Review

• C. Springer (1951): The problem was to find the mean for a canning process in order to minimize the cost. The price for producing under/over filled cans are assumed to be different.

• Hunter and Kartha (1977)Hunter and Kartha (1977) addressed the problem of addressed the problem of finding the optimal process mean (that maximizes the finding the optimal process mean (that maximizes the expected profit per item) with only a specified lower limit expected profit per item) with only a specified lower limit in which under-filled items are sold at reduced prices. in which under-filled items are sold at reduced prices. They also assumed that conforming items are sold at a They also assumed that conforming items are sold at a fixed price with a penalty cost due to excess in quality.fixed price with a penalty cost due to excess in quality.

• Golhar (1987)Golhar (1987) extended the model in Hunter and Kartha extended the model in Hunter and Kartha (1977) such that under-filled cans are reprocessed (emptied (1977) such that under-filled cans are reprocessed (emptied and refilled at a reprocessing cost).and refilled at a reprocessing cost).

Cont…


• Golhar and Pollock (1988)Golhar and Pollock (1988) extended the model in extended the model in Golhar (1987) for the case where the ingredient Golhar (1987) for the case where the ingredient was assumed to be expensive. For this reason, the was assumed to be expensive. For this reason, the process mean and the USL were determined.process mean and the USL were determined.

• M. A. Rahim and P. K. Banerjee (1988) considered the process where the system has a linear drift (e.g., tool wear etc).

Cont…


• Taguchi, Elsayed and Hsiang (1989) proposed a loss Taguchi, Elsayed and Hsiang (1989) proposed a loss function approach as a measure of quality, and its use in function approach as a measure of quality, and its use in determining product specification, target values of product determining product specification, target values of product characteristics and desired tolerances relevant to target characteristics and desired tolerances relevant to target value.value.

• O. Carlsson (1989): determined, for the case of two quality

characteristics, the optimum process mean under acceptance variable sampling.

• R. Schmidt & P. Pfeifer (1989): investigated the effects on cost savings from variance reduction.

Cont…


• Boucher and Jafari (1991) extended the line of research by Boucher and Jafari (1991) extended the line of research by evaluating the problem of finding the optimum target value evaluating the problem of finding the optimum target value under a sampling plan as opposed to 100% inspection. Two under a sampling plan as opposed to 100% inspection. Two conditions were examined, (1) when sampling results in conditions were examined, (1) when sampling results in destructive testing and (2) when the testing is nondestructivedestructive testing and (2) when the testing is nondestructive ..

• Arcelus and Rahim (1994) developed a model for joint Arcelus and Rahim (1994) developed a model for joint determination of target value for variable and attribute determination of target value for variable and attribute quality characteristics under inspection sampling plan.quality characteristics under inspection sampling plan.

• Al-Sultan (1994) extended the model of Boucher and Jafari Al-Sultan (1994) extended the model of Boucher and Jafari (1991) to the case of two machines in series processing a (1991) to the case of two machines in series processing a product.product.

• Liu, Tang and Chun (1995) considered the case of a filling process with limited capacity constraint.

Cont…


• F. J. Arcelus (1996) introduced the consistency criteria in the targeting problem.

• J. Roan, L. Gong & K. Tang (1997) considered production decisions such as production setup and raw material procurement policies.

• Min Koo Lee & Joon Soon Jang (1997) developed the model for multi-class screening case.

• Arcelus (1997) developed a targeting model where he considered two objectives which are uniformity of the product and conformance to specifications.

• Sung Hoon Hong & E. A. Elsayed (1999) studied the effect of measurement error for targeting problem.

Cont…

• Wen and Mergen developed a process targeting cost model to determine otimal process target (mean).

• Al-Sultan and Pulak (2000Al-Sultan and Pulak (2000) extended Golhar’s (1987) ) extended Golhar’s (1987) model for the case of two-stage manufacturing model for the case of two-stage manufacturing process, also it is modified version of Al-Sultan’s process, also it is modified version of Al-Sultan’s (1994) model with 100% inspection.(1994) model with 100% inspection.

• Teeravaraprug and Cho (2002) studied the multivariate Teeravaraprug and Cho (2002) studied the multivariate quality loss function to incorporate the customer’s quality loss function to incorporate the customer’s overall perception of product quality into design.overall perception of product quality into design.

• Ferrell Chhoker (2002) presented a sequence of models Ferrell Chhoker (2002) presented a sequence of models that addressed 100% inspection and single sampling, that addressed 100% inspection and single sampling, with and without error when a Taguchi quadratic loss with and without error when a Taguchi quadratic loss function is used.function is used. Cont…


• Duffuaa and Siddiqui (2002) developed a process targeting model Duffuaa and Siddiqui (2002) developed a process targeting model for three class screening problem by incorporating product for three class screening problem by incorporating product uniformity, extended work in the literature by incorporating a uniformity, extended work in the literature by incorporating a measurement error present in inspection systems.measurement error present in inspection systems.

• S. O. Duffuaa and A. W. Siddiqui (2003) developed a process S. O. Duffuaa and A. W. Siddiqui (2003) developed a process targeting model for a three-class screening problem in which targeting model for a three-class screening problem in which measurement errors exist. To reduce the effect of measurement errors exist. To reduce the effect of measurement errors, they introduced the concept of cut-off measurement errors, they introduced the concept of cut-off points. These cut-off points are considered to be the decision points. These cut-off points are considered to be the decision variables.variables.

• Bowling etc al (2003) developed the general form of a Markovian Bowling etc al (2003) developed the general form of a Markovian

model for optimum process target levels within the framework of a model for optimum process target levels within the framework of a multi-stage serial production system.multi-stage serial production system.


• Chung Chen (2003) modified Wen and Mergen Chung Chen (2003) modified Wen and Mergen model for determining the optimum process model for determining the optimum process mean for an indirect quality characteristics.mean for an indirect quality characteristics.

• Darwish and Duffuaa (2004) develop a modelto Darwish and Duffuaa (2004) develop a modelto determine simultaneously the optimal process determine simultaneously the optimal process target and inspection plan parameters.target and inspection plan parameters.

• Chung Chen (2005) modified Wen and Mergen Chung Chen (2005) modified Wen and Mergen model for determining the optimum process model for determining the optimum process mean for a process with a Log-normal mean for a process with a Log-normal distribution.distribution.


• Chung Chen (2006) modified Wen and Mergen Chung Chen (2006) modified Wen and Mergen model for determining the optimum process model for determining the optimum process mean using a mixed quality loss.mean using a mixed quality loss.

• Duffuaa, Kolus and Alturki (2006) extended Duffuaa, Kolus and Alturki (2006) extended process targeting models to processes in series process targeting models to processes in series with dependent quality characteristics.with dependent quality characteristics.


• Extensions of cost models.Extensions of cost models.

• Introduction of drift in the process.Introduction of drift in the process.

• Two quality characteristics ( Variable and attribute).Two quality characteristics ( Variable and attribute).

• Introduction of different quality control plans ( 100% Introduction of different quality control plans ( 100% inspection versus inspection plans).inspection versus inspection plans).

• Indirect measurement of quality characteristicIndirect measurement of quality characteristic

PTP DevelopmentPTP DevelopmentPTP DevelopmentPTP Development

• Error in measurement and inspectionError in measurement and inspection

• Two processes on two characteristics.Two processes on two characteristics.

• Uniformity criteria (Taguchi Quadratic loss function).Uniformity criteria (Taguchi Quadratic loss function).

• Simultaneous optimization of process parameters and Simultaneous optimization of process parameters and quality control schemes parameters.quality control schemes parameters.

• Multiple criteriaMultiple criteria

PTP DevelopmentPTP DevelopmentPTP DevelopmentPTP Development

Problem StatementProblem StatementProblem StatementProblem Statement

• Consider an industrial process in which items are Consider an industrial process in which items are produced continuously. An example is the can filling produced continuously. An example is the can filling problem. The quality characteristic is the net weight problem. The quality characteristic is the net weight of the material in the can and in a painting problem of the material in the can and in a painting problem the quality characteristic could be the thickness of the the quality characteristic could be the thickness of the paint.paint.

• Let Y be the measured quality characteristic of the Let Y be the measured quality characteristic of the product that has a lower specification limit L and a product that has a lower specification limit L and a target value T = L + δ. target value T = L + δ.

• The net selling price of a product that meets The net selling price of a product that meets specification is a $ and the selling price for a rejected specification is a $ and the selling price for a rejected item perhaps after processing is r $ (r < a). item perhaps after processing is r $ (r < a).

Cont…

• Let g be the excess quality measured for accepted Let g be the excess quality measured for accepted item ( g > 0). The problem under consideration is item ( g > 0). The problem under consideration is to find the optimal process parameters that to find the optimal process parameters that optimize the following three objectives:optimize the following three objectives:

• Maximizing net income.Maximizing net income.• Maximizing net profit. Maximizing net profit. • Maximize process yield or conformance to Maximize process yield or conformance to

specifications. specifications. • Maximizing product uniformity.Maximizing product uniformity.

Problem StatementProblem StatementProblem StatementProblem Statement

Cont…

Project ObjectivesProject ObjectivesProject ObjectivesProject Objectives

• Develop a multi-objective process targeting Develop a multi-objective process targeting model for the problem defined earlier using model for the problem defined earlier using 100% inspection as a mean for product quality 100% inspection as a mean for product quality control assuming perfect inspection. control assuming perfect inspection.

• Develop a multi-objective process targeting Develop a multi-objective process targeting model for the problem defined above using model for the problem defined above using acceptance sampling as a mean for product acceptance sampling as a mean for product quality control assuming perfect inspection. quality control assuming perfect inspection.

Cont…


• Generalize the two models developed in objectives Generalize the two models developed in objectives 1 and 2 to situations where inspection error is 1 and 2 to situations where inspection error is present.present.

• Study the effect of the inspection errors on the Study the effect of the inspection errors on the

optimal parameters of the models optimal parameters of the models


• Assumptions:Assumptions:• Same assumptions as Hartha and Kartha ModelSame assumptions as Hartha and Kartha Model

• One quality characteristicsOne quality characteristics

• Approximately normally distributed with mean Approximately normally distributed with mean µ µ and known variance and known variance 2.2.

• Characteristics has a lower specification limit L.Characteristics has a lower specification limit L.

• All items meeting specification are sold at price All items meeting specification are sold at price aa

• Items not meeting specification Items not meeting specification are sold at a are sold at a reduced price reduced price rr..

• No drift in the process at the beginning this could be No drift in the process at the beginning this could be

relaxedrelaxed..

Multi Objective OptimizationMulti Objective OptimizationMulti Objective OptimizationMulti Objective Optimization

• Min ( fMin ( f11(X),f(X),f22(X), …, f(X), …, fnn(X))(X))

Subject toSubject to: X : X εε S S

• Optimality conditionOptimality condition

• Pareto optimality. (Vilferdo Pareto Pareto optimality. (Vilferdo Pareto 1848- 1923)1848- 1923)

• Non-inferior setNon-inferior set

• Efficient setEfficient set

• Edgeworth (1881)Edgeworth (1881)

Solution ApproachSolution ApproachSolution ApproachSolution Approach

• Multi-objective model requires a decent approach Multi-objective model requires a decent approach to achieve Pareto optimalityto achieve Pareto optimality

• Weighted sum approachWeighted sum approach• Constraints approachConstraints approach• Goal programming.Goal programming.• Value (Utility) function approach.Value (Utility) function approach.

Solution ApproachSolution ApproachSolution ApproachSolution Approach

• Multi-objective model requires a decent approach Multi-objective model requires a decent approach to achieve Pareto optimalityto achieve Pareto optimality

• Non-preference methodsNon-preference methods• Posterior MethodsPosterior Methods• A prior methodsA prior methods• Interactive methods.Interactive methods.

Non-preference methodsNon-preference methods Non-preference methodsNon-preference methods

• Methods of global criteriaMethods of global criteria• Minimize the distance from an ideal reference Minimize the distance from an ideal reference

number using an L-p norm.number using an L-p norm.

• Proximal Bundle MethodProximal Bundle Method• Move in the direction where all objectives Move in the direction where all objectives

decrease. Termination is done when certain decrease. Termination is done when certain level of accuracy reached. level of accuracy reached.

A Posteriori methodsA Posteriori methods A Posteriori methodsA Posteriori methods

• Weighting methodWeighting method

• εε- constraint method- constraint method

• Hybrid methodsHybrid methods• Weighted metricWeighted metric

A Priori methodsA Priori methods A Priori methodsA Priori methods

• Value ( Utility) function value.Value ( Utility) function value.

• Lexicographic ordering.Lexicographic ordering.

• Goal programming.Goal programming.

Concluding RemarksConcluding RemarksConcluding RemarksConcluding Remarks

• PTP is an active area of research in quality for PTP is an active area of research in quality for over a half century.over a half century.

• No serious work has been done in modeling the No serious work has been done in modeling the PTP as a multi objective optimization model up to PTP as a multi objective optimization model up to the project team members knowledge.the project team members knowledge.

• Multi-objective optimization is expected to Multi-objective optimization is expected to reveal new insights in this problem and reveal new insights in this problem and open new doors of research.open new doors of research.

Questions & CommentsQuestions & CommentsQuestions & CommentsQuestions & Comments

Multi-objective Mathematical Models for Process Targeting S. O. Duffuaa, M. Darwish and A. Haroun...

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