This paper considers the problem of optimally choosing the mean of a filling process for a number of model variations. Optimality is defined as that setting which maximizes expected profit. Issues considered include the costs of waste, over-fill, and "top-up". An industrial example is discussed with both numerical as well as graphical solutions provided. The effects of change of the process variance on the optimal solution as well as on the expected profit are also discussed. Implications to "Weights and Measures" requirements of following this optimality path are provided with particular reference to loss in expected profit.

Keywords: filling process, optimal mean

*by* **Violetta I. Misiorek, CSIRO-CMIS, South Clayton
MDC, Victoria 3169, Australia **and** Neil S. Barnett,
Victoria University, MCMC, Melbourne, Victoria 8001, Australia**

*INTRODUCTION*

This paper considers a problem concerned with the selection of the mean setting of a filling process, where the focus is on maximizing the expected profit. Other authors have addressed a similar problem but have considered situations where containers, when under/over-filled are not recovered and the cost of an overflow is constant regardless of the amount of material in the container. This paper is an extension to some of the models that have already appeared in the literature where the cost of the container is included in the profit structure. Some additional cost associated with emptying out under-filled containers and putting the material back into the process as well as that of recapturing over-flow, are also considered. Moreover, the profit from over-filled containers is presented as being proportional to the amount of ingredient in the container. In addition, the optimization of the mean setting is considered with the likelihood of meeting "Weights and Measures" requirements. The monetary penalties associated with breaching these requirements are usually high and are discussed in more detail later in the paper.

In the many and varied industrial filling processes that exist in practice, overfilling frequently occurs and in extreme circumstances product actually overflows the container. Both over-filling and overflowing contribute additional costs to the producer. There are various circumstances, some of which result in over-flowing product being retrieved and yet others where over-flowing material is lost. Underfilling containers can lead to breaches of Weights and Measures Legislation, over-filling containers, on the other hand, may lead to a different confrontation with the legal system because, for example, for wines and spirits, there are penalties for over-filling associated with the excise tax. Other problems can arise through over-filling as is the case when containers need to be hermetically sealed, since this cannot be done if they are filled to capacity.

Under-filled containers and their contents may be discarded, which is usually the case in the food industry. In certain other industries, however, containers are either "topped-up" or emptied out and the material re-used. There are many different situations that involve different cost parameters, these lead to the consideration of various models. In this paper, special attention is paid to providing the potential user with a simple method that will provide a ready numerical solution for the initial mean process setting for the filling process.

Early work in the area was presented in Burr (1949), Springer (1951), and Bettes (1962). These all considered related problems; the latter two took a number of economic aspects into account and determined the optimal location of the mean, for minimization of total cost. The work in Burr (1949) was later extended in Nelson (1979) with the development of a graphical method to select the optimal setting.

Hunter and Kartha (1977) developed
a method for determining the optimal target value of an industrial
process to maximize the expected profit taking process variability
and production costs into account. The problem they considered,
which related to a filling process, revolved around the situation
where product above a certain dimensional threshold attracts
a fixed selling price and product below the threshold attracts
a reduced, yet fixed, selling price. In addition, product
above the threshold implies *give-away* which diminishes
the net profit per item. Besides successfully formulating
the problem, they provided a graphical method of solution.
The authors considered the problem under the assumption that
once the initial process mean setting is made, no other control
actions are subsequently necessary. The assumed conditions
in their model do not facilitate provision of an explicit
optimal solution, however, Nelson (1978) found an appropriate
function, which allowed for a close approximation to a solution
to be obtained. He also included a plot of errors.

A generalization of this model was presented by Bisgaard, Hunter, and Pallesen (1987), where the authors, again making the connection to a filling problem, developed a procedure for selecting optimal values for the process mean as well as the variance. They eliminated the assumption made in Hunter and Kartha (1977), that all under-filled items are sold at a fixed price. They considered a situation where under-filled items are sold for a price that is proportional to the amount of ingredient in the container. This assumption, however, is also not realistic since due to government regulations, containers with the amount of content below a certain proportion of the nominal weight or volume cannot be sold. A solution for a filling process that is approximately normal was given together with a table that provides the optimal process setting. Situations in which the distribution has a log-normal or Poisson distribution were also discussed. The objective was to maximize the expected profit.

A model also similar to that of Hunter and Kartha (1977), was studied by Carlsson (1984). In this paper the author analysed the choice of the process mean as well as the net expected income taking production costs and selling price into account. The discussion was based on an example from the steel construction industry, where rejected items are either sold at a reduced price or reprocessed. The net income function was represented as a piecewise linear function of the main quality characteristic. The situations where the customer is willing to pay extra for good quality as well as when the producer may have to compensate the customer for bad quality, were both discussed.

Gohlar (1987), also addressed the issue of finding the most economic setting of a process mean for a canning problem. He modelled a situation where the over-filled product could only be sold in a regular market and under-filled cans emptied and re-filled, with the penalty of extra cost. The capacity of the container was implicitly assumed infinite.

The canning problem analysed by Gohlar (1987) was later discussed by Schmidt and Pfeifer (1989). They explored the cost reductions achievable through a reduction in the process standard deviation. A linear relationship was found between the percentage reduction in both cost and process standard deviation. Revenue per can was constant, implying that minimization of expected cost is equivalent to maximization of expected profit.

Gohlar and Pollock (1988) have extended their previous model by including the upper limit in the cost structure. They considered a case where cans filled above the optimum upper limit are reprocessed. Their model is suitable for processes where the container's capacity is considerably bigger than the anticipated volume/weight. It was further extended by Schmidt and Pfeifer (1991) who altered the formulation of profit from rejected containers so that the model is also suitable for the capacitated case.

Bai and Lee (1993), concentrated on a process where a lower specification limit of the quality characteristic is given and where each container is inspected on a surrogate variable correlated with the primary variable. The method of selection of the optimal target values for the mean of the quantity of material in the container as well as the cut-off value on the surrogate variable were presented. The authors assumed that over-filled containers, no matter how much the over-fill, are sold for a fixed price. The same situation was discussed by Tang and Lo (1993) who, besides formulating the problem, performed sensitivity analyses.

Lee and Kim (1994) extended the model discussed in Bai and Lee (1993) by including a controllable upper limit. The under-filled as well as the over-filled containers were both assumed to be emptied and refilled. The developed profit model took into consideration selling price as well as the costs of filling, inspection, rework and also penalty costs.

Misiorek and Barnett (1995), focused on a model where production between two dimensional values can be re-processed at a cost, but where items produced below the lower of these is unsaleable. Items initially produced above the upper threshold attract a fixed selling price but involve give-away product. The objective was also to maximize the expected profit.

The purpose of this current
paper is to present various filling process models for which
it is desired to select the optimal process mean. Extra cost
parameters are considered. Filling, where overflowing material
is either recaptured or lost are both considered, together
with issues related to the cost of the containers. In relation
to under-filled containers two situations are considered,
one where the containers are emptied-out and material re-used
as well as one where the containers are "topped-up".
A case where overflow is not an issue but under-filled containers
are emptied-out and material re-used is also discussed. The
models considered here are an extension to models described
by Gohlar and Pollock as well as by Schmidt and Pfeifer. The
objective, having allowed for a fixed filling cost, is to
maximize the expected profit. Following analysis, implications
to Australian Weights and Measures legislation requirements
are considered. An industrial example is given to illustrate
possible use of the models described. Both graphical illustrations
and numerical solutions are provided, via both *Mathematica*
and *S-Plus*.

A different approach to somewhat similar problem is discussed by Pfeifer (1999).

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