Multi-objective Mathematical Models for Process Targeting

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Multi-objective Mathematical Models for Process
Targeting

• S. O. Duffuaa, M. Darwish and A. Haroun
• Systems Engineering Department
• King Fahd University Of Petroleum Minerals

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Outline

• Introduction
• Literature Review
• Problem Statement
• Project Objectives
• Outline of Preliminary Modeling Direction
• Possible algorithms
• Conclusions and Remarks

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Introduction
L µ
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Introduction

• Process Targeting (PT)
• The problem of PT concerns with the
  determination of the optimum values of the
  process parameters to optimize certain objective.
• The importance of PT is to ensure that a process
  produces products that not only satisfy customer
  needs, but also with a minimum production cost.
• Research in PT started in the early fifties with
  the CAN filling problems.

Cont
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Introduction

• A basic CAN filling problem is as follows
• The quality characteristic is assumed to be the
  net weight of the filled CAN.
• The value of this quality characteristic is a
  random variable X, and it has a lower
  specification limit (LSL)L.
• A 100 inspection is used for product quality
  control and it is assumed to be error free.
• An item is accepted if X L and defective
  otherwise. Accepted items are sold at a fixed
  price a, while rejected items are sold at a
  reduced price r.

Cont
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Introduction

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

L µ
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Introduction

• Y approximately follows a normal distribution
  with mean µ and standard deviation s.
• The objective is to find the target value µ so
  that the net income for the process is maximized.

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Literature 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) addressed the problem of
  finding the optimal process mean (that maximizes
  the expected profit per item) with only a
  specified lower limit in which under-filled items
  are sold at reduced prices. They also assumed
  that conforming items are sold at a fixed price
  with a penalty cost due to excess in quality.
• Golhar (1987) extended the model in Hunter and
  Kartha (1977) such that under-filled cans are
  reprocessed (emptied and refilled at a
  reprocessing cost).

Cont
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Literature Review

• Golhar and Pollock (1988) extended the model in
  Golhar (1987) for the case where the ingredient
  was assumed to be expensive. For this reason, the
  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
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Literature Review

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

• Boucher and Jafari (1991) extended the line of
  research by evaluating the problem of finding the
  optimum target value under a sampling plan as
  opposed to 100 inspection. Two conditions were
  examined, (1) when sampling results in
  destructive testing and (2) when the testing is
  nondestructive.
• Arcelus and Rahim (1994) developed a model for
  joint determination of target value for variable
  and attribute quality characteristics under
  inspection sampling plan.
• Al-Sultan (1994) extended the model of Boucher
  and Jafari (1991) to the case of two machines in
  series processing a product.
• Liu, Tang and Chun (1995) considered the case of
  a filling process with limited capacity
  constraint.

Cont
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Literature Review

• 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
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Literature Review

• Wen and Mergen developed a process targeting cost
  model to determine otimal process target (mean).
• Al-Sultan and Pulak (2000) extended Golhars
  (1987) model for the case of two-stage
  manufacturing process, also it is modified
  version of Al-Sultans (1994) model with 100
  inspection.
• Teeravaraprug and Cho (2002) studied the
  multivariate quality loss function to incorporate
  the customers overall perception of product
  quality into design.
• Ferrell Chhoker (2002) presented a sequence of
  models that addressed 100 inspection and single
  sampling, with and without error when a Taguchi
  quadratic loss function is used.

Cont
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Literature Review

• Duffuaa and Siddiqui (2002) developed a process
  targeting model for three class screening problem
  by incorporating product uniformity, extended
  work in the literature by incorporating a
  measurement error present in inspection systems.
• S. O. Duffuaa and A. W. Siddiqui (2003)
  developed a process targeting model for a
  three-class screening problem in which
  measurement errors exist. To reduce the effect of
  measurement errors, they introduced the concept
  of cut-off points. These cut-off points are
  considered to be the decision variables.
• Bowling etc al (2003) developed the general form
  of a Markovian model for optimum process target
  levels within the framework of a multi-stage
  serial production system.

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Literature Review

• Chung Chen (2003) modified Wen and Mergen model
  for determining the optimum process mean for an
  indirect quality characteristics.
• Darwish and Duffuaa (2004) develop a modelto
  determine simultaneously the optimal process
  target and inspection plan parameters.
• Chung Chen (2005) modified Wen and Mergen model
  for determining the optimum process mean for a
  process with a Log-normal distribution.

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Literature Review

• Chung Chen (2006) modified Wen and Mergen model
  for determining the optimum process mean using a
  mixed quality loss.
• Duffuaa, Kolus and Alturki (2006) extended
  process targeting models to processes in series
  with dependent quality characteristics.

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PTP Development

• Extensions of cost models.
• Introduction of drift in the process.
• Two quality characteristics ( Variable and
  attribute).
• Introduction of different quality control plans
  ( 100 inspection versus inspection plans).
• Indirect measurement of quality characteristic

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PTP Development

• Error in measurement and inspection
• Two processes on two characteristics.
• Uniformity criteria (Taguchi Quadratic loss
  function).
• Simultaneous optimization of process parameters
  and quality control schemes parameters.
• Multiple criteria

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Problem Statement

• Consider an industrial process in which items are
  produced continuously. An example is the can
  filling problem. The quality characteristic is
  the net weight of the material in the can and in
  a painting problem the quality characteristic
  could be the thickness of the paint.
• Let Y be the measured quality characteristic of
  the product that has a lower specification limit
  L and a target value T L d.
• The net selling price of a product that meets
  specification is a and the selling price for a
  rejected item perhaps after processing is r (r
  lt a).

Cont
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Problem Statement

• Let g be the excess quality measured for accepted
  item ( g gt 0). The problem under consideration is
  to find the optimal process parameters that
  optimize the following three objectives
• Maximizing net income.
• Maximizing net profit.
• Maximize process yield or conformance to
  specifications.
• Maximizing product uniformity.

Cont
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Project Objectives

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

Cont
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Project Objectives

• Generalize the two models developed in
  objectives 1 and 2 to situations where inspection
  error is present.
• Study the effect of the inspection errors on the
  optimal parameters of the models

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Project Objectives

• Assumptions
• Same assumptions as Hartha and Kartha Model
• One quality characteristics
• Approximately normally distributed with mean µ
  and known variance ?2.
• Characteristics has a lower specification limit
  L.
• All items meeting specification are sold at
  price a
• Items not meeting specification are sold at a
  reduced price r.
• No drift in the process at the beginning this
  could be relaxed.

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Multi Objective Optimization

• Min ( f1(X),f2(X), , fn(X))
• Subject to X e S
• Optimality condition
• Pareto optimality. (Vilferdo Pareto 1848- 1923)
• Non-inferior set
• Efficient set
• Edgeworth (1881)

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Solution Approach

• Multi-objective model requires a decent approach
  to achieve Pareto optimality
• Weighted sum approach
• Constraints approach
• Goal programming.
• Value (Utility) function approach.

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Solution Approach

• Multi-objective model requires a decent approach
  to achieve Pareto optimality
• Non-preference methods
• Posterior Methods
• A prior methods
• Interactive methods.

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Non-preference methods

• Methods of global criteria
• Minimize the distance from an ideal reference
  number using an L-p norm.
• Proximal Bundle Method
• Move in the direction where all objectives
  decrease. Termination is done when certain level
  of accuracy reached.

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A Posteriori methods

• Weighting method
• e- constraint method
• Hybrid methods
• Weighted metric

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A Priori methods

• Value ( Utility) function value.
• Lexicographic ordering.
• Goal programming.

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Concluding Remarks

• PTP is an active area of research in quality for
  over a half century.
• No serious work has been done in modeling the PTP
  as a multi objective optimization model up to the
  project team members knowledge.
• Multi-objective optimization is expected to
  reveal new insights in this problem and open new
  doors of research.

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Questions Comments
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