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literature review and problem DEFINITION
introduction
The purpose of this chapter is to present the literature review on the targeting problem. In addition, an outline of the basic model that is extended in this thesis is presented.
literature review
In this section, a brief literature review is given for the area of Process Targeting. The literature is presented in chronological order.
C. Springer (1951) firstly considered the problem of optimal Process Targeting. The problem was to determine the optimal mean for a canning process with specified upper and lower limits. He considered the price of producing under filled and over filled cans as fixed, but different.
D. Bettes (1962) proposed an empirical method, for the same model as in C. Springer (1951). His method was based on trial and error. The method is computationally tedious and does not give precise results.
W. Hunter and C. Kartha (1977) proposed a model in which under filled cans are sold in a secondary market at a reduced price. Cans above the lower specification limit are sold at a fixed price, which is an unrealistic assumption as cans barely over filled or nearly full are sold at a same price.
L. Nelson (1979) provided a nomograph for the model presented in C. Springer (1951).
An extension of W. Hunter and C. Kartha (1977) model was presented by S. Bisgaard W. Hunter and L. Pallensen (1984) where cans filled below specification limit are sold in a secondary market at a price proportional to the filled quantity.
D. Golhar (1987) studied the targeting problem with the assumption that under filled cans are to be emptied and refilled at the expense of a fixed reprocessing cost. The process is assumed to have known variance.
R. Vidal (1988) provided a simple graphical solution for the problem stated in S. Bisgaard W. Hunter and L. Pallensen (1984).
D. Golhar and S. Pollock (1988) extended the model presented by D. Golhar (1987) for the case where, the ingredient was assumed expensive. For that reason, upper specification limit of the quality characteristic, and the process mean, were determined. The model reduces to the model presented in D. Golhars [6] as upper limit tends to infinity.
M. A. Rahim and P. K. Banerjee (1988) firstly considered the process where the system has a linear drift (e.g., tool wear etc). A cost model for finished product is presented. A search algorithm as well as a graphical method is suggested to the optimal production run.
D. Golhar (1988) provided a computer program for the above model.
Another extension of the W. Hunter and C. Kartha (1977) is done by Carlsson (1989)(a). In this model the producer gives compensation for the under filled item, along with a benefit for the producer for the over filled items. The compensation penalty is a fixed amount plus an amount proportional to the difference between the lower and upper limit.
O. Carlsson (1989)(b) determined, for the case of two variable characteristics, the optimum process mean under acceptance variable sampling.
R. Schmidt and P. Pfeifer (1989)(a) investigated the effects on cost savings from variance reduction in a single level canning problem and an approximate simple linear relationship between percentage reduction in standard deviation and the cost reduction was presented.
An extension of D. Golhar (1987) model for the case of capacitated filling is presented in R. Schmidt and P. Pfeifer (1989)(b). In this work, a two level process control scheme to determine both process mean and the upper control limit is presented. The paper also compares the cost saving with that of a one level of process control.
In the extension of T. Boucher and M. Jafari (1991) the W. Hunter and C. karthas (1977) work was extended and single sampling inspection is introduced in the model.
D. Golhar and S. Pollock (1992) examined the effect of variance reduction on the cost for the D. Golhar and S. Pollock (1988) model and gave a close form approximate solution.
In Brain J. Melloy (1991), the problem of a uniform filling of an item is formulated under compliance testing. The objective was to minimize the noncompliance and give away cost.
Do sun Bai and Min Koo lee (1993) presented the problem of selecting the process mean and the cutoff value of a correlated variable for a filling process in which inspection is based on the correlated variable rather than the process mean itself.
In F. J. Arcelus and M. A. Rahim (1994) a model for simultaneously selecting the optimal target means for both the variable and attribute quality characteristic is presented. Optimality conditions are derived and a computational algorithm is given.
K. S. AlSultan (1994) addressed the problem of two machines in series where a sampling plan is used. An algorithm for finding optimal machine parameters for the two machines in series case, with single sampling inspection at each machine is given.
Shaul P. Ladeny (1995) assumed the process in which oversized items and undersized items are repaired at different costs. The objective is to maximize the profit.
D. P. Mihalko and D. Golhar (1995) was the first to consider the process variance as an unknown parameter. In this paper, a method for the determination of the confidence interval for the optimal process setting for the case of an unknown process variance is proposed.
Liu, Tang and Chun (1995) considered the case of a filling process with limited capacity constraint. The optimal process parameters to be determined are process mean and upper specification limit.
F. J. Arcelus (1996) presented two models. The paper introduces the Taguchi quadratic loss function for the targeting problem. In the first model, the target mean is a trade off value between the target mean of the society and the mean that maximizes/minimizes profit/cost for the producer.
In F. J. Arcelus and M. A. Rahim (1996), four models are presented for different assumption related to finding a trade off between conflicting objectives of conformity and uniformity. A quadratic penalty is used for the uniformity of the product.
M. K. Lee and J. S. Jang (1997) introduced the case of the threeclass screening. In this paper, it is assumed that the products are sold in two different markets with different price structures. Two models were presented. In the first model, the objective is to find the optimal mean when inspection is based on the same quality characteristic. While in the second model it is assumed that the inspection is based on a correlated variable.
M. F. Pulak and K. S. ALSultan (1997). In this paper, a computer program is presented for nine different Process Targeting models.
K. S. ALSultan and M. A. AlFawzan (1997)(a) is an extension of M. A. Rahim and P. K. Banerjees (1988) model. The paper assumed a process with random linear drift and is assumed to have both upper and lower specification limits. The objective is to find the optimal initial mean and cycle length. Variance of the process is assumed known and constant.
K. S. AlSultan and M. F. Pulak (1997) presented a model for finding the optimal mean of a filling process under rectifying inspection. The effect of variance reduction is also considered for the case.
K. S. ALSultan and M. A. AlFawzan (1997)(b) studied the model of M. A. Rahim and P. K. Banerjee (1988) i.e., systems with linear drift for the case of variance reduction and optimal initial process mean and cycle time is given.
J. Roan, L. Gong, K. Tang (1997) considers other production decisions such as production setup and raw material procurement policies. Two discount rate policies for the inventory are adopted. The production rate is assumed to be the function of the mean of the process.
M. Cain and C. Janssen (1997) presented the model where the cost is asymmetric across the target. A linear cost below lower specification limit and a quadratic cost above specification limit are assumed.
S. Pollock, D. Golhar (1998). In this paper the canning process with constant demand and capacity constraint for the production process is considered. The model also assumes a penalty for producing a nonconforming item.
P. E. Pfeifer (1999) provided a general piecewise linear model for the canning model.
Sung Hoon Hong & E. A. Elsayed (1999) studied the effect of measurement error on the optimal mean settings for the case of two class screening situation.
Figure 21 shows the development of the targeting problem over the years. The literature has no model that incorporates Taguchi quadratic loss function and inspection error. This is the focus of the thesis.
Model 1 (EPM1)
As this thesis work extends the work done by Min Koo Lee and Joong Soon Jang (1997), their model (the model will be referred to as Model 1 or EPM1) is presented in this chapter. This model is used as a basis for the extension made in this thesis.
2.3.1 model DESCRIPTION
The model in their paper applies to the production processes that are filling a can or turning a metal bar etc. The quality characteristics Y could be the net weight of the can or the mean diameter of the turned metal bar. The quality characteristic, in this model, is assumed to be normally distributed. The product, out of the production process, is classified into three classes based on product specifications. Considering can filling example, class one (grade 1) is the product that has a net weight ( L1. The secondclass product (grade 2) is the product that has the net weight between L2 and L1. The third class is scrape, which has a net weight < L2.
The objective of the model is to find the optimal location of the process mean, to maximize the profit resulting from grade 1 and grade 2. The model assumptions are as follows
Figure 22: A production process with multi class screening
2.3.2 model ASSUMPTIONS
The assumptions of the models are:
A single item is to be sold in two different markets with different cost/profit structures.
The inspection process is error free.
100% sampling is used.
The quality characteristic Y is assumed normally distributed with unknown process mean EMBED Equation.3 and known variance EMBED Equation.3 .
The inspection is based on Y.
EMBED Equation.3 .
EMBED Equation.3
The production cost per item is EMBED Equation.3
Specification limits on different grade are:
Specification limits on Y for grade 1 areY ( L1,Specification limits on Y for grade 2 areL2(Y< L1,Specification limits on Y for scrape areY< L2,
2.3.3 model formulation
Now the profit (per unit e.g., per kilogram etc.) function EMBED Equation.3 can be expressed as:
EMBED Equation.3
Now, The per unit expected profit can be written as:
EMBED Equation.3 (31)
where EMBED Equation.3 is the distribution of the quality characteristic Y. Equation (31) can be simplified as:
(
EMBED Equation.3
(
EMBED Equation.3
(
EMBED Equation.3 (32)Using the normality assumption i.e.,
EMBED Equation.3
and letting:
EMBED Equation.3
For standard normal the limits can be redefined as
at y = L1 EMBED Equation.3
at y = L2 EMBED Equation.3
Now, assuming EMBED Equation.3 , EMBED Equation.3 and EMBED Equation.3 , EMBED Equation.3 be the p.d.f. and c.d.f. of the normal and standard normal distributions i.e.,
EMBED Equation.3
EMBED Equation.3
Using the relationship EMBED Equation.3 and the relationships for standard normal distribution, also, letting EMBED Equation.3 we can rewrite the equation (32) as:
EMBED Equation.3 (33)
This is the form presented in M. K. Lee and J. S. Jang (1997). The model is used to determine the process mean.
conclusion
In this chapter, the literature in the area of process targeting is reviewed. Then the model of Min Koo Lee and Joong Soon Jang (1997) is presented. The work in this thesis extends the model of Min Koo Lee and Joog Soon Jang (1997) in two directions. First, the model is extended by incorporating errors in the measurement systems and second the concept of product consistency is incorporated through Taguchi loss function.
The next chapter extends the model by incorporating measurement errors. In case of measurement error, the observed quality characteristic is not the true value and this may regime modifying inspection criteria to counter the effect of the error. This issue is addressed in the next chapter in detail.
L2 L1
Scrap
Grade 1
Grade 2
Given: USL, USL, cost of overfilled and under filled items that are different, 1) Process normally distributed with known variance (2 2) Pearson type III
Objective: Minimize the cost [1]
Rectifying inspection is used instead of 100% inspection. Effect of variance reduction on the profit is studied [28]
Process variance is assumed to be unknown [21]
Two machines in series [19]
Measurement error considered [35]
Asymmetric cost function for undefiled and overfilled items. Linear and quadratic cost functions are assumed [31]
Three class screening considered.
Same model when inspection is based on correlated variable [25]
Inspection based on correlated variable [17]
USL also a decision variable. Both overfilled and under filled items are reprocessed [8]
Under filled items reprocessed at a fixed cost. [6]
Figure 21: Process Targeting Problem Development Over The Years
Problem is to determine optimal mean when considering cost of inventory such that the production rate is a function of mean [30]
A general piecewise linear model presented [34]
Multiclass screening problem with limits also unknown [33]
Effect of variance reduction [29]
USL and LSL given [27]
System with linear drift. Optimal cycle length to be determined [9]
A penalty for the uniformity of the product is added. (4 models presented) [24]
Problem under capacity constraint [22]
Oversized and under sized items are repaired at different costs [20]
Simultaneous selection of means for variable and attribute quality characteristics. [18]
Same problem under compliance testing. Both USL and LSL to be determined [16]
Single sampling plan is used instead of 100% inspection [14]
Effect of variance reduction on the profit. [12]
Over filled items sold at a price proportional to overfill [5]
Above problem solved for the maximization of profit case. Overfilled items sold at a fixed price. [3]
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