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Packaging strategies to counteract weight variability in extruded food products
Journal of Food Engineering 56 (2003) 353–360 www.elsevier.com/locate/jfoodeng
Packaging strategies to counteract weight variability in extruded food products K. Cronin *, J. Fitzpatrick, D. McCarthy Department of Process Engineering, University College Cork, Cork, Ireland Received 31 October 2001; accepted 14 April 2002
Abstract The effect of non-uniformity in the weights of an extruded food product on the efficiency of the subsequent packaging operation is analysed. The focus is on reducing the over-weights of the packs by better control over dispersion in product weight. Samples from an industrially extruded food product are analysed for variation in dimensional parameters and density. Variability in the product width dimension is shown to be the most significant source of weight variability. The control required over product width to obtain acceptable levels of pack over-weight is quantified. It is not practically possible to eliminate product weight variability. Hence alternative packing strategies that may be more successful in reducing the amount of product over-weight are examined. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Food extrusion; Packaging legislation; Weight control
1. Introduction A large number of consumer food products are formed in extrusion processes. Such products include breakfast cereals, snack foods and biscuits, confectionery and pasta products. These products are usually sold according to the rules of European average weight legislation. Because of the difficulties of ensuring consistent output from the extruder, non-uniformity in product weight can be a significant problem (Frame, 1994). This variability in the product can be due to random changes in the state of the raw material, such as the rheological consistency of dough and also because of unpredictable fluctuations in extruder operation with time (Quinlan & Butler, 1997). Variable product weight, irrespective of its source, imposes costs on the manufacturer in trying to satisfy the packaging rules; the most obvious manifestation of this is having to pack product with a deliberate over-weight included (Cronin, 1999). This paper outlines some strategies that may be adopted to counteract this problem based on the authors’ experience of industrial practice and some experimental work. It also attempts to show how relatively simple statistical analysis can *
Corresponding author. Tel.: +353-21-490-2644; fax: +353-21-4270249. E-mail address: [email protected] (K. Cronin).
help interpret industrial production data and suggest process improvements. Fig. 1 shows a schematic outline of the process producing a product similar in physical dimensions to liquorice sticks. The process will be treated as a generic example of extrusion and analysed as a case-study in extruded product weight control. The raw material is placed in the hopper and is processed in the single screw extruder. The die at the exit of the extruder is of square cross-section with the corners chamfered to a given radius. The continuously extruded rope, exiting the extruder, is moved forward by being dragged through two, contra-rotating, powered drums. Immediately after these drums, it passes over an idler wheel and pieces are cut sequentially from it into the required lengths by a cutting knife, rotating in the horizontal plane. The pieces (referred to as sticks) fall onto a conveyor and are carried to the packaging station. At the packaging station, the sticks are assembled into packs containing either 4, 6 or 8 individual pieces. The cross-section of each stick is a square with rounded corners as shown in Fig. 2. The nominal dimensions of the product are a width of 15 mm and length of 116 mm. The stated density of the material is 1150 kg/m3 , implying that the nominal weight of each stick is 30 g. Variability in the composition of the raw material entering the extruder (primarily in its rheological
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Nomenclature B L Ow Qn R TNE
stick width (mm) stick length (mm) pack overweight (–) nominal quantity (g) Stick corner radius (mm) Tolerable negative error (g)
lw lpw rw rpw q
mean stick weight (g) mean pack weight (g) standard deviation in stick weight (g) standard deviation in pack weight (g) product density (kg/m3 )
Fig. 1. Production line diagram.
at its set point. If extreme variations in extruder output occur, then the extruder speed itself can be adjusted.
2. Theory 2.1. Average weight legislation
Fig. 2. Product cross-section.
consistency) and extrusion process upsets lead to variability in the mass flow rate of product that leaves the extruder. This will reflect itself in changes to the dimensions and/or density of the extrudate rope. Variation in rope cross-sectional area can be counteracted by adjusting the rotational velocity of the powered drums; specifically a decrease in extruder output that manifests itself as a reduction in the rope cross-section can be compensated for by reducing the drum speed. Clearly production output will fall but the rope dimensions will be maintained at their prior value. The converse situation of an increase in rope dimensions coming from the extruder is counteracted by increasing the drum speed and stretching out the rope. The rotating cutter is synchronized with the powered drums to keep stick length
Variability in extruder performance produces a dispersion in individual stick weight which in turn causes a variability in pack weight. Because the packs are sold according to European average weight legislation (Anon., 1979), a distribution in pack weight has the consequence that the average pack weight, lpw must be greater than or equal to the declared pack weight, Qn (the nominal quantity) to satisfy the regulations of average weight legislation. The difference between average pack weight and declared weight is known as the overweight or giveaway weight, Ow and can obviously be directly related to process profitability. It can be defined in fractional terms as Ow ¼
lpw Qn Qn
ð1Þ
A recapitulation of the basis of the three rules of average weight legislation is needed to understand the precise relationship between product overweight and standard deviation in product weight (Murphy, 1985). The legislation assumes that the distribution in pack weights is characterized by the normal distribution and is thus completely determined by the mean and standard
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360 Table 1 Tolerable negative error for different nominal weights as set by average weight legislation Nominal quantity, Qn (g)
TNE % of Qn
5–50 50–100 100–200 200–300 300–500
critical i.e. if it is passed then the other two will automatically be satisfied. Eqs. (4)–(6) can be reformulated to determine the appropriate critical rule, depending upon the size of the standard deviation, as follows:
g
If rpw 6
9 4.5 4.5
355
TNE 1:96
Rule 1 is critical
TNE TNE 6 rpw 6 Rule 2 is critical 1:96 1:76 TNE Rule 3 is critical If rpw P 1:76 For small values of the standard deviation, when rpw is approximately less than half the appropriate TNE, Rule 1 will be critical and no overweight will be required to meet the targets set by legislation. This obviously demonstrates the advantages of good process control. For large standard deviations, (in particular when rpw is greater than 0.57 times the TNE) the T2 lower weight limit (Rule 3) will be the determining rule and a significant amount of overweight will have to be included in the packs. The intermediate case is covered by Rule 2. For the process under study in this paper, the standard deviation in pack weights was generally of the same magnitude as the TNE itself. The associated minimum overweight necessary to satisfy the legislation (Rule 3) can then be defined from Eqs. (1) and (6) as If
9 3
deviation in pack weights, lpw and rpw respectively. The first rule states that average pack weight must be equal to or greater than the nominal weight. Furthermore the legislation lays down limits on the number of packs that can be below the declared weight using the concept of the tolerable negative error (TNE). Depending on the nominal weight of the pack, certain TNEs are recognized and for the pack weights of interest here are shown in Table 1. They are given as percentages of the nominal quantity or as absolute values to ensure equality of outcome at transition weights. Two lower limits on pack weight are assembled using the TNE. T1 ¼ Qn TNE
ð2Þ
T2 ¼ Qn 2TNE
ð3Þ
Rule 2 of the legislation states that not more than 2.5% of packs can contain an amount of product less than the T1 limit and Rule 3 states that no pack may contain less than the T2 limit. Any packs falling outside these limits are illegal. Check-weighers are installed on the line and such packs are rejected, unpacked and reassembled. This gives rise to significant extra labour costs. The proportion of packs that lie beyond these two limits in the lower tail of the pack weight distribution can be quantified in terms of the standard deviation in pack weight. Interpreting Rule 3 in a practical sense to mean that only 1 in 10,000 packs can be below the T2 limit, then the three rules can be mathematically defined as Rule 1
lpw P Qn
ð4Þ
Rule 2
lpw P Qn TNE þ 1:96rpw
ð5Þ
Rule 3
lpw P Qn 2TNE þ 3:72rpw
ð6Þ
To conform with the weight legislation, average pack weight lpw must satisfy the above three inequalities. Because there is dispersion in the pack weights, average pack weight must be greater than the declared weight. The task for the manufacturer is to select the optimum value for this average pack weight that limits the overweight as much as possible while avoiding the cost of repacking illegal underweight product. Examining the above equations, it can be seen that depending upon the magnitude of the standard deviation of the pack weights, one particular rule will be
Ow P
3:72rpw 2TNE Qn
ð7Þ
In summary, diminishing expensive overweight, requires action to reduce the standard deviation in pack weights. If rpw can be reduced below 0.51 times the TNE, then there will be no need for the manufacturer to include any overweight in the packs. 2.2. Stick weight variability Each pack consists of a certain number of sticks, n, where for the product in question n can be four, six or eight. The packs are assembled in three parallel packaging lines immediately after a stick marshalling area. Successively cut sticks do not exhibit a correlation in weight and in any case downstream processing operations, between the cutter and packaging station, introduce further randomization into stick weight. The individual stick weights are taken to be normally distributed. Forming a pack is then considered to be equivalent to randomly sampling (with a sample size of either four, six or eight) from an infinite normally distributed population. The standard deviation in pack weight rpw is solely due to the standard deviation in individual stick weight, rw assuming that standard deviation in tare weight (i.e. the weight of the packaging medium itself) is negligible. The mean and standard deviation in pack weight can be related to the mean and standard deviation in stick weight by
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lpw ¼ nlw pffiffiffi rpw ¼ nrw
ð8Þ ð9Þ
As is evident, lowering the standard deviation in pack weights must involve reducing the standard deviation in stick weights. To do this, a model is required for stick weight (or more precisely, stick mass) that gives its physical dependencies. The weight of a stick will be given as its volume times the average density. Volume is in turn fixed by the stick dimensional parameters. Assuming each stick has a homogeneous composition, each weight (actually mass) of the stick can be calculated by treating the stick as a prismatic bar, i.e. w ¼ B2 ð4 pÞR2 Lq ð10Þ The corner radius, R of the stick is 1.5 mm. Ignoring the rounded corners and treating the product as perfectly square will simplify the subsequent calculations and the corresponding error is less than 1%. Stick weight is then given as 2
ð11Þ
w ¼ B Lq
In reality the cross-section of the product is not perfectly square, the cross-section can vary along the length of the stick and the longitudinal axis of the stick is not always straight. These deviations from the assumed geometry will cause errors in the analysis and will subsequently be quantified. The variability in stick weight is considered to arise from variability in the geometric parameters B and L and in product density, q. The latter three quantities are taken to be random variables. If each of them is normally distributed and independent of the other two, then the technique of statistical differentials can be used to relate the mean and standard deviation of the three input variables to that of the output variable, weight (Kempthorne & Folks, 1971). Mean stick weight will thus be given as lw ffi l2B lL lq
which from Eq. (11) implies h i1=2 2 2 2 rw ffi ð2BLqÞ r2B þ ðB2 qÞ r2L þ ðB2 LÞ r2q
ð14Þ
where the quantities in round brackets are evaluated at their mean values. The approximate rather than full equality symbol is necessary in the above equations due to the non-linear relationship between stick weight and stick width. Knowing the standard deviation in stick width, length and density, Eq. (14) will allow the relative importance of each of these in producing variability in stick weight to be calculated. For instance the relative contribution of dispersion in stick width to variation in stick weight in percentage terms is given by ð2BLqÞ2 r2B 100 r2w 3. Materials and methods On three separate days, eighty sticks of product were extracted from the line, immediately after the cutter, at one minute intervals. The weight of each stick was measured on a Precisa top-pan balance that had a resolution of 0.01 g. The length, L of each stick was measured with a rule. From each group of eighty sticks, twenty sticks were randomly selected for further analysis of width and density. Measurement of these two parameters is more involved than weight and length and it was not though to be practical to measure the width and density of all eighty sticks. The perpendicular widths, B of the twenty sticks were measured at four points along the length of the stick with a remote laser micrometer, working on the shadow principle, with a resolution of 5 lm. The arithmetic mean of the eight recorded readings was taken as the width of the stick. The density of each of the twenty sticks was measured by the water displacement method, i.e. measuring the weight of each stick in air and then in water.
ð12Þ
Standard deviation in stick weight can be derived from the following general formula " 2 2 2 #1=2 ow ow ow 2 2 rw ffi rB þ rL þ r2q ð13Þ oB oL oq
4. Experimental results Table 2 contains the mean and standard deviation found in stick weight, width, length and density for the three trials. The mean stick weights for trials 1 and 2 are
Table 2 Experimentally measured mean and standard deviation in stick parameters Variable Weight, w Width, B Length, L Density, q
Unit g mm mm kg/m3
Trial 1
Trial 2
Trial 3
l
r
l
r
l
r
31.95 15.72 118.6 1139
3.06 0.63 1.2 42.2
30.25 15.68 116.0 1163
1.49 0.44 1.67 12.5
29.97 15.49 115.5 1150
1.42 0.57 1.35 16.7
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360
357
Fig. 3. (a) Frequency histogram of stick weight. (b) Frequency histogram of stick width. (c) Frequency histogram of stick length.
consistent with an average pack overweight of 6.5% and 0.8% respectively during this period. The mean stick weight for trial 3 is marginally below the nominal weight which implies that some of these packs would subsequently have been rejected at the check-weigher and that the line speed required adjustment. Fig. 3(a)–(c) illustrates in frequency histogram form the distribution in product weight, width and length for a selected trial (trial 1). Using the data in Table 2 and Eq. (14), the relative contributions to variability in stick weight by width, length and density can be assessed. Table 3 summarises the results. Examining the table, it can be seen that on
Table 3 Relative contributions to stick weight variability by stick physical parameters Parameter Width Length Density
Percentage contribution to weight variability Trial 1
Trial 2
Trial 3
81.4 1.2 17.4
88.4 5.8 5.8
94.0 2.4 3.6
average over 80% of the variability in stick weight is due to variability in stick width. The contributions of stick length and density to weight variability are an order of magnitude lower. This is not surprising bearing in mind that the coefficients of variation (standard deviation divided by mean value) for stick width for the three trials are greater than the coefficients of variation for the other two parameters. The effect of this greater relative variation in width is compounded by the fact that stick weight is quadratically related to width while only linearly related to length and density. Scatter plots of stick weight against stick width, length and density were assembled and are shown in Fig. 4(a)–(c), again for trial 1. In the case of weight versus width, stick weight is plotted against the square of stick width to linearise the relationship. Stick weight can be plotted directly against stick length and stick density. The coefficients of determination (R2 ) from each plot for all the trials are given in Table 4. Coefficients of determination of over 0.7 were obtained when plotting stick weight against the square of stick width. By contrast the coefficients between weight and length and density respectively are small.
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Fig. 4. (a) Scatter plot of stick weight versus the square of stick width. (b) Scatter plot of stick weight versus stick length. (c) Scatter plot of stick weight versus stick density.
Table 4 Coefficients of determination for stick weight versus stick width (squared), length and density Correlation coefficients Width Length Density
Trial 1
Trial 2
Trial 3
0.7 0.11 0.04
0.76 0.004 0.15
0.82 0.04 0.17
The methodology underlying the above analysis was checked by carrying out a number of statistical tests. The appropriateness of the stick weight model (Eq. (11))
was determined by comparing the actual measured stick weights against those predicted by the equation. Simultaneously the utility of the method of statistical differentials (Eq. (14)) was assessed by comparing the experimentally measured standard deviations in stick weight with those calculated by the equation. Table 5 contains the results with the percentage difference between theory (model prediction) and experiment given in the last row. It can be seen that with respect to mean weight, modelling each stick as a prismatic bar was accurate to within 10%, though the model consistently over-predicted weight. The measured standard deviations in stick weight are in line with the predictions of
Table 5 Comparison of experimental and predicted stick weight statistics Variable
Unit
Expt. weight, w Model weight, w Error
g g %
Trial 1
Trial 2
Trial 3
l
r
l
r
l
r
31.95 33.41 þ4.6
3.06 2.96 3.3
30.25 33.18 þ9.7
1.49 1.93 þ29.5
29.97 31.85 þ6.27
1.42 2.42 þ70.4
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360
Eq. (14); magnitudes of standard deviations are intrinsically more difficult to predict accurately than mean values. The histograms of weight, width, length and density were checked for normality by the Chi-Square test and the fit was acceptable. The independence of stick width, length and density was investigated by plotting scatter diagram of individual stick width, length and density against each other to check for any association. The coefficients of determination for all plots were less than 0.2; indeed for most of the plots the coefficients of determination were well below 0.1. Finally the weights of successive sticks that were produced on the line were checked for any association but the results were negative and no time dependent correlation was evident.
5. Line control and optimisation strategies For the extrusion process studied, stick width has been identified as the dominant cause of stick weight and hence pack weight dispersion. Controlling stick width should permit improved control over stick weights and lead to lower pack over-weights. By combining Eqs. (9) and (14) and ignoring the contribution of stick length and density, a simple, approximate formula can be developed to relate pack weight standard deviation to stick width standard deviation pffiffiffi rpw n2BLqrB ð15Þ where mean stick width, length and density are taken to be at their nominal values. This formula can be used to quickly assess how reducing dispersion in stick width will improve packing statistics or alternatively to relate a target level of over-weight to the corresponding maximum level of allowable standard deviation in stick width. Table 6 contains illustrative results for a pack size of four items. The nominal quantity, Qn will be 120 g so by Table 1 the TNE for this case will be 5.4 g. An initial selected value of 0.6 mm for rB will result in a pack weight standard deviation of 4.8 g from Eq. (15) above. This in turn means that a minimum over-weight of 5.9% is necessary to satisfy the legislation (Rule 3) using Eq. (7). As the standard deviation in stick width is decreased in 0.1 mm increments, the required over-weight falls until the standard deviation in pack weight is less than Table 6 Relationship between standard deviation in stick width and pack overweight rB (mm)
rpw (g)
TNE (g)
Ow (%)
0.6 0.5 0.4 0.3 0.2
4.8 4.0 3.2 2.4 1.6
5.4 5.4 5.4 5.4 5.4
5.9 3.4 0.9 0 0
359
half the TNE and no over-weight is needed (Rule 1). Thus from the point of view of satisfying the packing rules, only a certain degree of improvement in process control is economically justifiable. The actual standard deviation in stick width for the process was of the order of 0.55 mm. Reducing this to less than 0.4 mm should bring substantial improvements to the over-weight situation. The above analysis demonstrates that stick width is the critical parameter in terms of achieving control over pack weights. In applying a control system to the extrusion and postextrusion process, rope width should be a good indicator of uniformity and on-line measurement of it should provide a useful control signal to the powered drums. In practice of course as dispersion in stick width falls, the relative importance of variability in stick length and stick density become more significant in the determination of stick weight variability. Also for this particular product, intrinsic difficulties in the extrusion process mean that a lower limit of about 1% on over-weight is found to be achievable. Given an inherent variability in individual stick weight, the advantage to the manufacturer in selling the product in larger pack sizes can be quantified. Table 7 gives the minimum over-weight that must be supplied for three different pack sizes, containing four, six and eight items respectively and assuming that the standard deviation in stick width, rB is constant at 0.5 mm. Dispersion in pack weight and over-weight are calculated as above (Eqs. (15) and (7)). In general, increasing the number of sticks in a pack enables lower over-weights to be achieved. The relationship is not monotonic due to the vagaries in how the TNE is calculated for different weight classes, i.e. in percentage terms or as an absolute quantity. Analysis like this could also be used in initial product design to assist in the selection of product dimensions that will minimize over-weight for a given retail nominal quantity; for instance it is clear that eight sticks with a nominal length of 58 mm are more profitable to pack than four sticks of 116 mm length though both packs would have the same declared weight. The potential benefits from sorting product by weight prior to packaging could be assessed as an aid in determining the cost-effectiveness of such a strategy. After being marshaled into straight lines prior to packaging, the sticks could be made to pass over an open grid to separate the light from the heavy product. Thus the sticks would be available in two different batches Table 7 Relationship between pack size and pack over-weight n (–)
Qn (g)
TNE (g)
rpw (g)
Ow (%)
4 6 8
120 180 240
5.4 8.1 9.0
4.0 4.9 5.7
3.4 1.13 1.34
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Table 8 Standard deviation in pack weights using sorted and unsorted pack assembly Unsorted Sorted
Qn (g)
TNE (g)
lpw (g)
rpw (g)
120 120
5.4 5.4
124.02 124
3.396 2.04
approximately classified as those with a weight below the mean weight and those with a weight above it. Assembling for example a four pack of product would then take place by randomly sampling two sticks from each batch and placing them in the pack. The weight distribution statistics of the packs become more complicated because the two separate underlying stick weight populations from which they are assembled are clearly no longer normally distributed. Each stick weight distribution will be a mirror image of the other, one being the left hand tail of the original normal distribution and the other being the right hand tail. Thus individually each of the two distributions will be highly non-symmetric. Taken in conjunction with the fact that pack sizes are small (either four, six or eight sticks in a pack), the distribution in pack weights itself will not be normally distributed as expected by the legislation. Statistically the process is equivalent to sampling from a non-normal distribution with a small sample size. A simple Monte Carlo numerical random sampling scheme can be used to simulate the assembly of packs (of four sticks) by the current unsorted method and by the proposed method with stick weights sorted into two batches, (Law & Kelton, 1991). Table 8 illustrates the results of the simulation. It gives the mean and standard deviation in pack weights that would result if the packs were assembled by the default system of randomly sampling from a single stick weight distribution and the corresponding pack statistics if they were assembled from the two sorted stick weight distributions. The mean and standard deviation in the original normal stick weight distribution are given values of 31 and 1.7 g respectively. The model simulates the assembly of 1000 packs of product. For the unsorted assembly the mean and standard deviation in pack weight (124.02 and 3.396 g respectively) are as expected very close to the values predicted by Eqs. (8) and (9). The slight discrepancies are solely due to the inherent random nature of the sampling and could be diminished by increasing the number of numerical trials. The mean pack weight using the method of sorting will obviously not change and can be seen to be 124 g. However the standard deviation is considerably lower at 2.04 g. This implies that the pack weight distribution that would be present if prior sorting by stick weight was introduced would be much tighter
than the distribution that exists with the current method. Hence lower over-weight targets are possible with this technique.
6. Conclusions The interaction between variability in product weight and the mechanics of European average weight legislation have been outlined, with a focus on the relationship between standard deviation in individual stick weight and a corresponding necessary pack overweight. A statistical analysis of industrial data has shown that for the particular product that was analysed, the dispersion in product width makes the dominant contribution to stick weight variation. Thus to achieve smaller pack overweights and improved control over the process, stick width should be the focus of attention. The potential benefits to packaging performance in reducing the standard deviation in stick width have been illustrated. Furthermore the advantages offered by other packing strategies such as packing product in larger size packs (i.e. containing more individual sticks) and sorting the sticks into weight classes prior to pack assembly have been investigated. The mathematical analysis underlying this approach is straightforward but has been shown to be adequate for the task.
Acknowledgement The authors would like to thank Mr. Ali Dogan Demir for help with the Monte Carlo simulation.
References Anonymous. (1979). EU Directive 79/1005/EEC. Cronin, K. (1999). A methodology for investigation into alternative packaging methods for column wrapped biscuits. Journal of Food Engineering, 39, 379–387. Frame, N. D. (Ed.). (1994). The technology of extrusion cooking. London: Blackie Academic & Professional. Kempthorne, O., & Folks, L. (1971). Probability, statistics, and data analysis. Ames, Iowa, USA: Iowa State University Press. Law, A. M., & Kelton, W. D. (1991). Simulation modelling and analysis. New York, USA: McGraw-Hill. Murphy, M. F., (1985). Aspects of the EC System of Average Quantity Control. Internal Report, Faculty of Food Science and Technology, University College Cork, Ireland. Quinlan, D., Butler, F., (1997). Introduction of a real time rheological dough test at Irish Biscuits. In: Recent Advances in Improving and Developing Bakery and Cereal Based Products, Non-Commissioned Food Research Programme Workshop. University College Dublin, Ireland.
Packaging strategies to counteract weight variability in extruded food products K. Cronin *, J. Fitzpatrick, D. McCarthy Department of Process Engineering, University College Cork, Cork, Ireland Received 31 October 2001; accepted 14 April 2002
Abstract The effect of non-uniformity in the weights of an extruded food product on the efficiency of the subsequent packaging operation is analysed. The focus is on reducing the over-weights of the packs by better control over dispersion in product weight. Samples from an industrially extruded food product are analysed for variation in dimensional parameters and density. Variability in the product width dimension is shown to be the most significant source of weight variability. The control required over product width to obtain acceptable levels of pack over-weight is quantified. It is not practically possible to eliminate product weight variability. Hence alternative packing strategies that may be more successful in reducing the amount of product over-weight are examined. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Food extrusion; Packaging legislation; Weight control
1. Introduction A large number of consumer food products are formed in extrusion processes. Such products include breakfast cereals, snack foods and biscuits, confectionery and pasta products. These products are usually sold according to the rules of European average weight legislation. Because of the difficulties of ensuring consistent output from the extruder, non-uniformity in product weight can be a significant problem (Frame, 1994). This variability in the product can be due to random changes in the state of the raw material, such as the rheological consistency of dough and also because of unpredictable fluctuations in extruder operation with time (Quinlan & Butler, 1997). Variable product weight, irrespective of its source, imposes costs on the manufacturer in trying to satisfy the packaging rules; the most obvious manifestation of this is having to pack product with a deliberate over-weight included (Cronin, 1999). This paper outlines some strategies that may be adopted to counteract this problem based on the authors’ experience of industrial practice and some experimental work. It also attempts to show how relatively simple statistical analysis can *
Corresponding author. Tel.: +353-21-490-2644; fax: +353-21-4270249. E-mail address: [email protected] (K. Cronin).
help interpret industrial production data and suggest process improvements. Fig. 1 shows a schematic outline of the process producing a product similar in physical dimensions to liquorice sticks. The process will be treated as a generic example of extrusion and analysed as a case-study in extruded product weight control. The raw material is placed in the hopper and is processed in the single screw extruder. The die at the exit of the extruder is of square cross-section with the corners chamfered to a given radius. The continuously extruded rope, exiting the extruder, is moved forward by being dragged through two, contra-rotating, powered drums. Immediately after these drums, it passes over an idler wheel and pieces are cut sequentially from it into the required lengths by a cutting knife, rotating in the horizontal plane. The pieces (referred to as sticks) fall onto a conveyor and are carried to the packaging station. At the packaging station, the sticks are assembled into packs containing either 4, 6 or 8 individual pieces. The cross-section of each stick is a square with rounded corners as shown in Fig. 2. The nominal dimensions of the product are a width of 15 mm and length of 116 mm. The stated density of the material is 1150 kg/m3 , implying that the nominal weight of each stick is 30 g. Variability in the composition of the raw material entering the extruder (primarily in its rheological
0260-8774/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 2 ) 0 0 1 6 1 - 9
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K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360
Nomenclature B L Ow Qn R TNE
stick width (mm) stick length (mm) pack overweight (–) nominal quantity (g) Stick corner radius (mm) Tolerable negative error (g)
lw lpw rw rpw q
mean stick weight (g) mean pack weight (g) standard deviation in stick weight (g) standard deviation in pack weight (g) product density (kg/m3 )
Fig. 1. Production line diagram.
at its set point. If extreme variations in extruder output occur, then the extruder speed itself can be adjusted.
2. Theory 2.1. Average weight legislation
Fig. 2. Product cross-section.
consistency) and extrusion process upsets lead to variability in the mass flow rate of product that leaves the extruder. This will reflect itself in changes to the dimensions and/or density of the extrudate rope. Variation in rope cross-sectional area can be counteracted by adjusting the rotational velocity of the powered drums; specifically a decrease in extruder output that manifests itself as a reduction in the rope cross-section can be compensated for by reducing the drum speed. Clearly production output will fall but the rope dimensions will be maintained at their prior value. The converse situation of an increase in rope dimensions coming from the extruder is counteracted by increasing the drum speed and stretching out the rope. The rotating cutter is synchronized with the powered drums to keep stick length
Variability in extruder performance produces a dispersion in individual stick weight which in turn causes a variability in pack weight. Because the packs are sold according to European average weight legislation (Anon., 1979), a distribution in pack weight has the consequence that the average pack weight, lpw must be greater than or equal to the declared pack weight, Qn (the nominal quantity) to satisfy the regulations of average weight legislation. The difference between average pack weight and declared weight is known as the overweight or giveaway weight, Ow and can obviously be directly related to process profitability. It can be defined in fractional terms as Ow ¼
lpw Qn Qn
ð1Þ
A recapitulation of the basis of the three rules of average weight legislation is needed to understand the precise relationship between product overweight and standard deviation in product weight (Murphy, 1985). The legislation assumes that the distribution in pack weights is characterized by the normal distribution and is thus completely determined by the mean and standard
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360 Table 1 Tolerable negative error for different nominal weights as set by average weight legislation Nominal quantity, Qn (g)
TNE % of Qn
5–50 50–100 100–200 200–300 300–500
critical i.e. if it is passed then the other two will automatically be satisfied. Eqs. (4)–(6) can be reformulated to determine the appropriate critical rule, depending upon the size of the standard deviation, as follows:
g
If rpw 6
9 4.5 4.5
355
TNE 1:96
Rule 1 is critical
TNE TNE 6 rpw 6 Rule 2 is critical 1:96 1:76 TNE Rule 3 is critical If rpw P 1:76 For small values of the standard deviation, when rpw is approximately less than half the appropriate TNE, Rule 1 will be critical and no overweight will be required to meet the targets set by legislation. This obviously demonstrates the advantages of good process control. For large standard deviations, (in particular when rpw is greater than 0.57 times the TNE) the T2 lower weight limit (Rule 3) will be the determining rule and a significant amount of overweight will have to be included in the packs. The intermediate case is covered by Rule 2. For the process under study in this paper, the standard deviation in pack weights was generally of the same magnitude as the TNE itself. The associated minimum overweight necessary to satisfy the legislation (Rule 3) can then be defined from Eqs. (1) and (6) as If
9 3
deviation in pack weights, lpw and rpw respectively. The first rule states that average pack weight must be equal to or greater than the nominal weight. Furthermore the legislation lays down limits on the number of packs that can be below the declared weight using the concept of the tolerable negative error (TNE). Depending on the nominal weight of the pack, certain TNEs are recognized and for the pack weights of interest here are shown in Table 1. They are given as percentages of the nominal quantity or as absolute values to ensure equality of outcome at transition weights. Two lower limits on pack weight are assembled using the TNE. T1 ¼ Qn TNE
ð2Þ
T2 ¼ Qn 2TNE
ð3Þ
Rule 2 of the legislation states that not more than 2.5% of packs can contain an amount of product less than the T1 limit and Rule 3 states that no pack may contain less than the T2 limit. Any packs falling outside these limits are illegal. Check-weighers are installed on the line and such packs are rejected, unpacked and reassembled. This gives rise to significant extra labour costs. The proportion of packs that lie beyond these two limits in the lower tail of the pack weight distribution can be quantified in terms of the standard deviation in pack weight. Interpreting Rule 3 in a practical sense to mean that only 1 in 10,000 packs can be below the T2 limit, then the three rules can be mathematically defined as Rule 1
lpw P Qn
ð4Þ
Rule 2
lpw P Qn TNE þ 1:96rpw
ð5Þ
Rule 3
lpw P Qn 2TNE þ 3:72rpw
ð6Þ
To conform with the weight legislation, average pack weight lpw must satisfy the above three inequalities. Because there is dispersion in the pack weights, average pack weight must be greater than the declared weight. The task for the manufacturer is to select the optimum value for this average pack weight that limits the overweight as much as possible while avoiding the cost of repacking illegal underweight product. Examining the above equations, it can be seen that depending upon the magnitude of the standard deviation of the pack weights, one particular rule will be
Ow P
3:72rpw 2TNE Qn
ð7Þ
In summary, diminishing expensive overweight, requires action to reduce the standard deviation in pack weights. If rpw can be reduced below 0.51 times the TNE, then there will be no need for the manufacturer to include any overweight in the packs. 2.2. Stick weight variability Each pack consists of a certain number of sticks, n, where for the product in question n can be four, six or eight. The packs are assembled in three parallel packaging lines immediately after a stick marshalling area. Successively cut sticks do not exhibit a correlation in weight and in any case downstream processing operations, between the cutter and packaging station, introduce further randomization into stick weight. The individual stick weights are taken to be normally distributed. Forming a pack is then considered to be equivalent to randomly sampling (with a sample size of either four, six or eight) from an infinite normally distributed population. The standard deviation in pack weight rpw is solely due to the standard deviation in individual stick weight, rw assuming that standard deviation in tare weight (i.e. the weight of the packaging medium itself) is negligible. The mean and standard deviation in pack weight can be related to the mean and standard deviation in stick weight by
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lpw ¼ nlw pffiffiffi rpw ¼ nrw
ð8Þ ð9Þ
As is evident, lowering the standard deviation in pack weights must involve reducing the standard deviation in stick weights. To do this, a model is required for stick weight (or more precisely, stick mass) that gives its physical dependencies. The weight of a stick will be given as its volume times the average density. Volume is in turn fixed by the stick dimensional parameters. Assuming each stick has a homogeneous composition, each weight (actually mass) of the stick can be calculated by treating the stick as a prismatic bar, i.e. w ¼ B2 ð4 pÞR2 Lq ð10Þ The corner radius, R of the stick is 1.5 mm. Ignoring the rounded corners and treating the product as perfectly square will simplify the subsequent calculations and the corresponding error is less than 1%. Stick weight is then given as 2
ð11Þ
w ¼ B Lq
In reality the cross-section of the product is not perfectly square, the cross-section can vary along the length of the stick and the longitudinal axis of the stick is not always straight. These deviations from the assumed geometry will cause errors in the analysis and will subsequently be quantified. The variability in stick weight is considered to arise from variability in the geometric parameters B and L and in product density, q. The latter three quantities are taken to be random variables. If each of them is normally distributed and independent of the other two, then the technique of statistical differentials can be used to relate the mean and standard deviation of the three input variables to that of the output variable, weight (Kempthorne & Folks, 1971). Mean stick weight will thus be given as lw ffi l2B lL lq
which from Eq. (11) implies h i1=2 2 2 2 rw ffi ð2BLqÞ r2B þ ðB2 qÞ r2L þ ðB2 LÞ r2q
ð14Þ
where the quantities in round brackets are evaluated at their mean values. The approximate rather than full equality symbol is necessary in the above equations due to the non-linear relationship between stick weight and stick width. Knowing the standard deviation in stick width, length and density, Eq. (14) will allow the relative importance of each of these in producing variability in stick weight to be calculated. For instance the relative contribution of dispersion in stick width to variation in stick weight in percentage terms is given by ð2BLqÞ2 r2B 100 r2w 3. Materials and methods On three separate days, eighty sticks of product were extracted from the line, immediately after the cutter, at one minute intervals. The weight of each stick was measured on a Precisa top-pan balance that had a resolution of 0.01 g. The length, L of each stick was measured with a rule. From each group of eighty sticks, twenty sticks were randomly selected for further analysis of width and density. Measurement of these two parameters is more involved than weight and length and it was not though to be practical to measure the width and density of all eighty sticks. The perpendicular widths, B of the twenty sticks were measured at four points along the length of the stick with a remote laser micrometer, working on the shadow principle, with a resolution of 5 lm. The arithmetic mean of the eight recorded readings was taken as the width of the stick. The density of each of the twenty sticks was measured by the water displacement method, i.e. measuring the weight of each stick in air and then in water.
ð12Þ
Standard deviation in stick weight can be derived from the following general formula " 2 2 2 #1=2 ow ow ow 2 2 rw ffi rB þ rL þ r2q ð13Þ oB oL oq
4. Experimental results Table 2 contains the mean and standard deviation found in stick weight, width, length and density for the three trials. The mean stick weights for trials 1 and 2 are
Table 2 Experimentally measured mean and standard deviation in stick parameters Variable Weight, w Width, B Length, L Density, q
Unit g mm mm kg/m3
Trial 1
Trial 2
Trial 3
l
r
l
r
l
r
31.95 15.72 118.6 1139
3.06 0.63 1.2 42.2
30.25 15.68 116.0 1163
1.49 0.44 1.67 12.5
29.97 15.49 115.5 1150
1.42 0.57 1.35 16.7
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360
357
Fig. 3. (a) Frequency histogram of stick weight. (b) Frequency histogram of stick width. (c) Frequency histogram of stick length.
consistent with an average pack overweight of 6.5% and 0.8% respectively during this period. The mean stick weight for trial 3 is marginally below the nominal weight which implies that some of these packs would subsequently have been rejected at the check-weigher and that the line speed required adjustment. Fig. 3(a)–(c) illustrates in frequency histogram form the distribution in product weight, width and length for a selected trial (trial 1). Using the data in Table 2 and Eq. (14), the relative contributions to variability in stick weight by width, length and density can be assessed. Table 3 summarises the results. Examining the table, it can be seen that on
Table 3 Relative contributions to stick weight variability by stick physical parameters Parameter Width Length Density
Percentage contribution to weight variability Trial 1
Trial 2
Trial 3
81.4 1.2 17.4
88.4 5.8 5.8
94.0 2.4 3.6
average over 80% of the variability in stick weight is due to variability in stick width. The contributions of stick length and density to weight variability are an order of magnitude lower. This is not surprising bearing in mind that the coefficients of variation (standard deviation divided by mean value) for stick width for the three trials are greater than the coefficients of variation for the other two parameters. The effect of this greater relative variation in width is compounded by the fact that stick weight is quadratically related to width while only linearly related to length and density. Scatter plots of stick weight against stick width, length and density were assembled and are shown in Fig. 4(a)–(c), again for trial 1. In the case of weight versus width, stick weight is plotted against the square of stick width to linearise the relationship. Stick weight can be plotted directly against stick length and stick density. The coefficients of determination (R2 ) from each plot for all the trials are given in Table 4. Coefficients of determination of over 0.7 were obtained when plotting stick weight against the square of stick width. By contrast the coefficients between weight and length and density respectively are small.
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Fig. 4. (a) Scatter plot of stick weight versus the square of stick width. (b) Scatter plot of stick weight versus stick length. (c) Scatter plot of stick weight versus stick density.
Table 4 Coefficients of determination for stick weight versus stick width (squared), length and density Correlation coefficients Width Length Density
Trial 1
Trial 2
Trial 3
0.7 0.11 0.04
0.76 0.004 0.15
0.82 0.04 0.17
The methodology underlying the above analysis was checked by carrying out a number of statistical tests. The appropriateness of the stick weight model (Eq. (11))
was determined by comparing the actual measured stick weights against those predicted by the equation. Simultaneously the utility of the method of statistical differentials (Eq. (14)) was assessed by comparing the experimentally measured standard deviations in stick weight with those calculated by the equation. Table 5 contains the results with the percentage difference between theory (model prediction) and experiment given in the last row. It can be seen that with respect to mean weight, modelling each stick as a prismatic bar was accurate to within 10%, though the model consistently over-predicted weight. The measured standard deviations in stick weight are in line with the predictions of
Table 5 Comparison of experimental and predicted stick weight statistics Variable
Unit
Expt. weight, w Model weight, w Error
g g %
Trial 1
Trial 2
Trial 3
l
r
l
r
l
r
31.95 33.41 þ4.6
3.06 2.96 3.3
30.25 33.18 þ9.7
1.49 1.93 þ29.5
29.97 31.85 þ6.27
1.42 2.42 þ70.4
K. Cronin et al. / Journal of Food Engineering 56 (2003) 353–360
Eq. (14); magnitudes of standard deviations are intrinsically more difficult to predict accurately than mean values. The histograms of weight, width, length and density were checked for normality by the Chi-Square test and the fit was acceptable. The independence of stick width, length and density was investigated by plotting scatter diagram of individual stick width, length and density against each other to check for any association. The coefficients of determination for all plots were less than 0.2; indeed for most of the plots the coefficients of determination were well below 0.1. Finally the weights of successive sticks that were produced on the line were checked for any association but the results were negative and no time dependent correlation was evident.
5. Line control and optimisation strategies For the extrusion process studied, stick width has been identified as the dominant cause of stick weight and hence pack weight dispersion. Controlling stick width should permit improved control over stick weights and lead to lower pack over-weights. By combining Eqs. (9) and (14) and ignoring the contribution of stick length and density, a simple, approximate formula can be developed to relate pack weight standard deviation to stick width standard deviation pffiffiffi rpw n2BLqrB ð15Þ where mean stick width, length and density are taken to be at their nominal values. This formula can be used to quickly assess how reducing dispersion in stick width will improve packing statistics or alternatively to relate a target level of over-weight to the corresponding maximum level of allowable standard deviation in stick width. Table 6 contains illustrative results for a pack size of four items. The nominal quantity, Qn will be 120 g so by Table 1 the TNE for this case will be 5.4 g. An initial selected value of 0.6 mm for rB will result in a pack weight standard deviation of 4.8 g from Eq. (15) above. This in turn means that a minimum over-weight of 5.9% is necessary to satisfy the legislation (Rule 3) using Eq. (7). As the standard deviation in stick width is decreased in 0.1 mm increments, the required over-weight falls until the standard deviation in pack weight is less than Table 6 Relationship between standard deviation in stick width and pack overweight rB (mm)
rpw (g)
TNE (g)
Ow (%)
0.6 0.5 0.4 0.3 0.2
4.8 4.0 3.2 2.4 1.6
5.4 5.4 5.4 5.4 5.4
5.9 3.4 0.9 0 0
359
half the TNE and no over-weight is needed (Rule 1). Thus from the point of view of satisfying the packing rules, only a certain degree of improvement in process control is economically justifiable. The actual standard deviation in stick width for the process was of the order of 0.55 mm. Reducing this to less than 0.4 mm should bring substantial improvements to the over-weight situation. The above analysis demonstrates that stick width is the critical parameter in terms of achieving control over pack weights. In applying a control system to the extrusion and postextrusion process, rope width should be a good indicator of uniformity and on-line measurement of it should provide a useful control signal to the powered drums. In practice of course as dispersion in stick width falls, the relative importance of variability in stick length and stick density become more significant in the determination of stick weight variability. Also for this particular product, intrinsic difficulties in the extrusion process mean that a lower limit of about 1% on over-weight is found to be achievable. Given an inherent variability in individual stick weight, the advantage to the manufacturer in selling the product in larger pack sizes can be quantified. Table 7 gives the minimum over-weight that must be supplied for three different pack sizes, containing four, six and eight items respectively and assuming that the standard deviation in stick width, rB is constant at 0.5 mm. Dispersion in pack weight and over-weight are calculated as above (Eqs. (15) and (7)). In general, increasing the number of sticks in a pack enables lower over-weights to be achieved. The relationship is not monotonic due to the vagaries in how the TNE is calculated for different weight classes, i.e. in percentage terms or as an absolute quantity. Analysis like this could also be used in initial product design to assist in the selection of product dimensions that will minimize over-weight for a given retail nominal quantity; for instance it is clear that eight sticks with a nominal length of 58 mm are more profitable to pack than four sticks of 116 mm length though both packs would have the same declared weight. The potential benefits from sorting product by weight prior to packaging could be assessed as an aid in determining the cost-effectiveness of such a strategy. After being marshaled into straight lines prior to packaging, the sticks could be made to pass over an open grid to separate the light from the heavy product. Thus the sticks would be available in two different batches Table 7 Relationship between pack size and pack over-weight n (–)
Qn (g)
TNE (g)
rpw (g)
Ow (%)
4 6 8
120 180 240
5.4 8.1 9.0
4.0 4.9 5.7
3.4 1.13 1.34
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Table 8 Standard deviation in pack weights using sorted and unsorted pack assembly Unsorted Sorted
Qn (g)
TNE (g)
lpw (g)
rpw (g)
120 120
5.4 5.4
124.02 124
3.396 2.04
approximately classified as those with a weight below the mean weight and those with a weight above it. Assembling for example a four pack of product would then take place by randomly sampling two sticks from each batch and placing them in the pack. The weight distribution statistics of the packs become more complicated because the two separate underlying stick weight populations from which they are assembled are clearly no longer normally distributed. Each stick weight distribution will be a mirror image of the other, one being the left hand tail of the original normal distribution and the other being the right hand tail. Thus individually each of the two distributions will be highly non-symmetric. Taken in conjunction with the fact that pack sizes are small (either four, six or eight sticks in a pack), the distribution in pack weights itself will not be normally distributed as expected by the legislation. Statistically the process is equivalent to sampling from a non-normal distribution with a small sample size. A simple Monte Carlo numerical random sampling scheme can be used to simulate the assembly of packs (of four sticks) by the current unsorted method and by the proposed method with stick weights sorted into two batches, (Law & Kelton, 1991). Table 8 illustrates the results of the simulation. It gives the mean and standard deviation in pack weights that would result if the packs were assembled by the default system of randomly sampling from a single stick weight distribution and the corresponding pack statistics if they were assembled from the two sorted stick weight distributions. The mean and standard deviation in the original normal stick weight distribution are given values of 31 and 1.7 g respectively. The model simulates the assembly of 1000 packs of product. For the unsorted assembly the mean and standard deviation in pack weight (124.02 and 3.396 g respectively) are as expected very close to the values predicted by Eqs. (8) and (9). The slight discrepancies are solely due to the inherent random nature of the sampling and could be diminished by increasing the number of numerical trials. The mean pack weight using the method of sorting will obviously not change and can be seen to be 124 g. However the standard deviation is considerably lower at 2.04 g. This implies that the pack weight distribution that would be present if prior sorting by stick weight was introduced would be much tighter
than the distribution that exists with the current method. Hence lower over-weight targets are possible with this technique.
6. Conclusions The interaction between variability in product weight and the mechanics of European average weight legislation have been outlined, with a focus on the relationship between standard deviation in individual stick weight and a corresponding necessary pack overweight. A statistical analysis of industrial data has shown that for the particular product that was analysed, the dispersion in product width makes the dominant contribution to stick weight variation. Thus to achieve smaller pack overweights and improved control over the process, stick width should be the focus of attention. The potential benefits to packaging performance in reducing the standard deviation in stick width have been illustrated. Furthermore the advantages offered by other packing strategies such as packing product in larger size packs (i.e. containing more individual sticks) and sorting the sticks into weight classes prior to pack assembly have been investigated. The mathematical analysis underlying this approach is straightforward but has been shown to be adequate for the task.
Acknowledgement The authors would like to thank Mr. Ali Dogan Demir for help with the Monte Carlo simulation.
References Anonymous. (1979). EU Directive 79/1005/EEC. Cronin, K. (1999). A methodology for investigation into alternative packaging methods for column wrapped biscuits. Journal of Food Engineering, 39, 379–387. Frame, N. D. (Ed.). (1994). The technology of extrusion cooking. London: Blackie Academic & Professional. Kempthorne, O., & Folks, L. (1971). Probability, statistics, and data analysis. Ames, Iowa, USA: Iowa State University Press. Law, A. M., & Kelton, W. D. (1991). Simulation modelling and analysis. New York, USA: McGraw-Hill. Murphy, M. F., (1985). Aspects of the EC System of Average Quantity Control. Internal Report, Faculty of Food Science and Technology, University College Cork, Ireland. Quinlan, D., Butler, F., (1997). Introduction of a real time rheological dough test at Irish Biscuits. In: Recent Advances in Improving and Developing Bakery and Cereal Based Products, Non-Commissioned Food Research Programme Workshop. University College Dublin, Ireland.