Mohammad S. Aljohani*, Ahmad A. Moreb

Optimal Variance ReductionTechnique that Minimizes Filling Production Cost with Optimum Target Weight

MOHAMMAD S. ALJOHANI*, AHMAD A. MOREB**

*Department of Nuclear Engineering,

KingAbdulAzizUniversity,

SAUDI ARABIA

**Industrial & Management Systems Engineering,

Kuwait University, KUWAIT

Abstract:- The cost incurred in a filling production system is mainly due to variation in weights of products; resulting in an overfill cost, and a cost of non-satisfaction of customers because of the under-fill. Moreover, a cost is incurred for undertaking system improvement. Quality improvement reduces the variance, which subsequently affects the filling cost. This paper finds the optimal variance that minimizes total filling cost, including the cost of improvement (the cost of reducing filling variance). In the past, work involved in selecting a cost effective quality improvement effort is done separately from the efforts needed to minimize filling costs; thus resulting in suboptimal total cost. This paper however, combines the quality improvement cost with the filling cost; resulting in a global optimum total cost. The bases for data used in this paper are an empirical relationship between the quality improvement cost and the standard deviation of the process, which is subsequently fitted by an exponentially decaying function of the standard deviation. Historically, filling problems are difficult to minimize; this difficulty is due to the Gaussian distributed terms involved. This research offers an optimal solution that strikes a balance between overfill, under-fill and the variance reduction cost. In this paper, the methodology involves formulating the total cost function by an analytical semi-final solution, after which a numerical algorithm is used to arrive at the optimum. A numerical example is included to demonstrate how the techniques could be used.

Keywords: Optimization, Production system, Overfill cost, Under-fill cost,Variancereduction, Cost of Quality.

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1Introduction

The problem of finding the optimum process mean that minimizes the filling cost has been addressed for several decades by extensive research [1-5]. Their effort was centered around the theme of finding the lowest acceptable weight, below which the product is termed as under-fill and is liable to be charged a huge penalty cost. This penalty cost is a collection of many factors attributed to loss of customers known as “poor quality cost”. However, if the product has more than the lowest weight, it is similar to giving away free material. All of these researches had resorted to lookup tables in their optimization methodologies. A more recent effort to solve this type of problem was introduced by [6]. Their work had introduced a new turn into solving this problem; an analytical solution had replaced the lookup tables. Furthermore, the analytical technique was able to arrive at the final optimum solution in a one single problem. Moreover, two other techniques were also introduced, a statistical (numerical) method and a graphical one that proved to be of great convenience for practitioners in the field.

In many instances, organizations are continuously engaged in quality improvement, and many of these improvements are focused on ''six sigma'' approach that involves reducing product’s variance; thus resulting in high costs. Previous research has dealt with the decisions of choosing a cost effective quality improvement effort separate from the cost of filling cost; resulting in a suboptimal total cost. Schmidt and Pfeifer [7] had addressed the effect of quality improvement (variance reduction) on the total cost. Schmidt and Pfeifer [4] developed a linear relationship between cost reduction and reduction in standard deviation for pre-specified ranges of the standard deviation. This linear relationship is merely an approximation to the true nonlinear Gaussian relationship. Moreover, a separate linear relationship is needed for each range of standard deviation; subsequently, a separate problem must be solved for each range, followed by a ranking process for these options then the best option is selected.

This paper, however, develops a single (total) cost function and the problem is solved only once resulting in the best value for the standard deviation that minimizes total filling cost including quality improvement cost (the cost of reducing variance); the solution found is a global optimum. The quality improvement cost for the case under study was best fitted by an exponentially decaying function of the standard deviation. The methodology presented in this paper faces the same difficulty in minimizing the cost function, due to the Gaussian distributed terms involved. However, the optimal solution in this paper strikes a balance between overfill, under-fill and variance reduction costs. The methodology starts by formulating the total cost function with analytical semi-final solution, which is then terminated by a numerical algorithm. An example is included to demonstrate the methodology.

2Definition of variables

= The weight of a product.

= Total number of products.

= Number of products that has a weight of .

= Average weight of all products.

= Target weight (the lowest acceptable weight).

=Penalty cost incurred if a product is under-fill.

= The per unit cost ($ / unit weight).

= Standard deviation of the weight of products.

= Over fill cost (cost of material over the target weight).

= Under fill cost.

= Variance reduction cost

= Total cost

3Problem Statement

Problem formulation starts by presenting a mathematical expression for the number of products that have weights falling in the interval :

where,

(1)

The number of products below a specific weight, say , can be found by integrating the above expression in equation (1) over the interval from to :

(2)

This integral is frequently evaluated using look up tables [8]. The overall total cost is composed of three components, the under-fill cost, the overfill cost and the variance reduction (quality improvement) cost, which can be expressed as:

(3)

The target weight is defined as the lowest acceptable weight; thus the number of units below the target can be found by integrating equation (1) from to . If this integral is multiplied by the penalty cost per unit, one gets the under-fill cost; that is equal to:

(4)

Similarly, the total weight of all units with weights above the target weight can be found by integrating equation (1) from to . Thus the overfill cost can be found by evaluating the integral:

(5)

The above equation can be rewritten as:

(6)

The variance reduction cost function is based on an empirical set of data that is best fitted by the following equation:

(7)

where , and are variance reduction cost parameters.

Therefore, the overall total cost is expressed by:

(8)

This equation can now be solved to arrive at a closed form for the total overall cost. The first two terms of equation (8) when integrated would result in the following error function terms (equations (9) and (10)):

(9)

And,

(10)

Before evaluating the integral of the third term of equation (8); it should be pointed out that the value of the error function at is equal to one. Moreover, the exponential component of this term is equal to zero at . Therefore, this term (the 3rd term of equation (8)) after integration will be:

(11)

Combining all four terms of the total cost in equation (8); namely, the solution terms in equations (9), (10), (11) and the variance reduction term, would result in:

(12)

After simplification, a closed form for the overall total cost can finally be written as:

(13)

The above equation can be solved numerically to arrive at an optimal and that give the minimum overall total cost of a filling production process including the quality improvement cost (variance reduction cost). Solving equation (13) can be done using the following algorithm:

4 Algorithm

A reasonable value for the process mean will be chosen, as a start, to cover all feasible values; the valuesand are chosen as the lower and upper feasible limits for the process mean. Similarly and are the limits chosen for .

The algorithm starts searching along both, the standard deviation and the mean of the process , in the predetermined ranges and , respectively. The steps of this algorithm are as follows:

1. Set =.
2. Set =.
3. Increment by and find the minimum cost by solving equation (13) until = ; record the values of in the array.
4. Find the minimum total cost from among the values of (in the array); let this minimum corresponds to and .
5. Increment by and go to step (2) until ; also record the values of in an array.
6. The smallest value among the recorded values of is the minimum total cost, ; identify the corresponding parameters and .
7. The algorithm terminates by finding and the corresponding and .

5 Example:

A synthetic data was used to illustrate the methodology presented above. The parameters in this example are as follows:

• The target weight is 1 kg per product.
• The cost of 1 kg of product is $ 120 / kg .
• The cost of poor quality, ; this includes the loss of customers due to selling one under-filled product and is estimated to be $ 70 / product.
• = 10, = 2, = 0.01 (best fitted parameters)

The solution found using the above algorithm is as follows:

The optimal = 1.12 kg.

The optimal = 0.057 kg

The minimum total filling cost = $ 19,414

The relationships, minimum total cost as a function of standard deviation at optimal mean, standard deviation versus optimal mean and the total cost as a function of mean at optimal standard deviation are plotted below for further demonstration; Figure 1, Figure 2 and Figure 3 respectively.

4Conclusion

The methodology adopted above is semi-analytical. The advantage of this semi-analytical solution stems from the ease of its use. Solving for the process mean and standard deviation that minimizes the total cost involves a simple algorithm that is a very straightforward.

References

[1] Springer, C.H., A Method of Determining the Most Economic Position of a Process Mean, Industrial Quality Control, Vol. 8, 1951, pp. 36-39.

[2] Bettes, D.C., Finding an Optimum Target Value in Relation to a Fixed Lower Limit and an Arbitrary Upper Limit, Applied Statistics, Vol. 11, 1962, pp. 202-210.

[3] Carlsson, O., Determining the Most Profitable Process Level for a Production Process under Different Sales Conditions, Journal of Quality Technology, Vol. 16, 1984, pp. 44-49.

[4] Schmidt, R. L. and Pfeifer, P. E., Economic Selection of the Mean and Upper Limit for a Canning Problem with Limited Capacity, Journal of Quality Technology, Vol. 23, No. 4, 1991, pp. 312-317.

[5] Tang, K. and Lo, J., Determination of the Optimal Process Mean when Inspection is based on a Correlated Variable,IIE Transactions, Vol. 25, No. 3, 1993, pp. 66-72.

[6] Moreb, A. A. and Aljohani, M. S., Minimizing Total Cost For A Gaussian Distributed Filling Production System, WSEAS Transactions on Mathematics, Vol. 2, No. 3, 2003, pp 214-218

[7] Schmidt, R. L. and P. E. Pfeifer, P. E., An Economic Evaluation of Improvements in Process Capability for a Single-Level Canning Problem, Journal of Quality Technology, Vol. 21, No. 1, 1989, pp. 16-19.

[8] Montgomery, D. C.,Introduction to Statistical Quality Control, 4th edition, John Wiley & Sons, 2001.

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