Introduction

It is relatively easy to take the packer's kill sheets and critique the economic marketing results of someone's decision to sell a load of top hogs, as I did this morning. Everyone has 20/20 hind sight after the fact. Successful marketing means that one must be able to predict optimal market timing. The awareness of predictive strategies separates those who understand the process taking place in the finishing house from those who guess. Predictive market timing strategy will be generated by those who have the ability to understand the weight distribution of a population of hogs on a finishing floor and can make time oriented predictions about the hogs based on probabilities associated with that distribution.

This presentation is conceptual. You will be a better market planner if you understand the concepts. This presentation is not meant to make statisticians. It will introduce you to some statistical ideas that describe the topping hog in a population of hogs that occupy a floor. If you don't understand the math, don't be intimidated. Try to grasp the concept. After all, in actual use the math is simplified by computer spreadsheets and it is the concepts that are the primary tools of application. In application, the following material is more than conceptual. It is techniques that have been applied, some everyday, in the industry. However, there is also a portion of this material that is theoretical and has been only the subject of research.

The presentation will move from the less complex to the more complex. Don't feel that you are missing out because you struggle more as the topic unravels. The organization of this topic will present 5 facts and 5 predictive tools these facts imply. The prediction of market timing will be explained through the use of these tools.

Fact 1. Gain in a Finisher is Usually Straight Line

Growth curves are a plot of the pig weight over the days the pig spend in various facilities. Growth curves are used to describe the way hogs gain weight. The slope, or steepness, of this line (rise/run) is the average daily gain (pounds/days). The curve has little slope before the pig is 30 pounds, but soon bends up and stays straight line during finishing. As the hog matures, the slope again reduces after about 270 pounds to 300 pounds. The growth curve during the hogs time in the finishing house is in the middle area which is not very curved at all. It is more convenient to approximate this area of growth as a straight line. This is equivalent to stating that average daily gain (ADG) can be assumed to remain constant through the finishing while the pig is on a normal topping floor.

Tool 1. The Growth Line

The straight portion of the growth curve is convenient because the slope of the straight portion can be assumed to be the gain of a single topping floor and a predictive growth line can be made for that floor. This growth line is derived as a plot of average pig weight over calendar days. The total weight of all the pigs received pigs is usually known as well as the number of head received. Average entry weight is easily figured by dividing the total weight by the number of head. Using the average pig weight on entry and the date of entry , the initial entry point can be described on plot as the intersection from these two initial values. The growth line can be drawn onto the growth plot by using the straight slope from the growth curve and drawing a line originating from the initial point. This is the same as saying that these pigs will gain like most pigs have in the past. This means that the slope (ADG) from similar facilities were used to predictively align how the growth in this facility should appear. To use the growth line predictively is to extrapolate up the ADG line to find an intersection of a specified weight and a calendar date. For example, with the use of the growth line, one could claim that the finishing group will average 220 pounds on January 14th.

To get the best growth line, the variables affecting growth must be constant with those of the benchmarks that generated the growth curve. "Holding variables constant" simply means comparing apples to apples. If the growth curve was established in January on barrows fed with no fat addition in a tunnel ventilated 960, then the curve will only apply to barrows on the same feed in the same environment. The weakness of using growth curves is that one value of slope (ADG) is used to predict the growth within many finishers.

Fact 2. As a Group of Hogs Gains Weight, the Variation Between Individual Pigs Will Increase.

As hogs grow, individuals grow differently. Some have more potential to grow and they become larger at a time before their cohorts. Other pigs are set back by timid dispositions or disease. The longer that hogs are on a finishing floor, not only do their weights increase, but the more the individual weights of the pigs will vary one from another. Standard deviation is a measurement of this variation of weight. The more spread out the weight range becomes, the larger the standard deviation becomes.

This tendency for variation to increase among pigs over time could upset our ability to predict marketing dates, but luckily the variation occurs in a somewhat orderly manner. Body weight variation increases along a straight line over time like body weight.

Tool 2. The Variation Line

The amount of expected variation that will occur within a population of a finishing house can be plotted over calendar days as a variation line. This plot is very similar to the growth line. It is a graph of the standard deviation over time spent on the floor in calendar days. The plot is a straight line based on the slope of the change of variation per day in other finishing houses. The slope indicates the rate of change in weight variation, which is theoretically stable over time. This allows the prediction of weight variation on a topping floor over the feeding period.

Like the growth line, the slope of the variation line of one finisher could be used to predict the variation plot of another topping floor of similar environment, nutrition, and construction. However, when the growth line is used, a starting point was the intersection of the initial average weight and the day of entry. The use of the variation line also needs a starting point which is the measure of variation (standard deviation) of the weight of the pigs upon entry. This term is not as easily derived as the average weight. To determine the standard deviation of weight, individual pigs must be weighed to calculate this measurement. This calls for extra labor to derive a value for initial weight variation between the pigs on entry. Unlike calculating average entry weight, it is not feasible to include every pig in the calculation of the value of the standard deviation. This value can be estimated by weighing a random sample of pigs (usually about 20) and calculating the variation that exists within this sample. This is easier than finding the true variation, which would require weighing every pig.

If the variation within the group is known on entry, the variation line can be plotted from a known slope from previous finishing groups to predict the variation that will occur within the group on a future date. These two lines, the growth line and the variation line, allow the prediction of two parameters on a future date. These two parameters are a predicted future average weight of the finishing group and the future weight variation within the finishing group.

Fact 3. Weight is Distributed like a Bell Curve

There is a relationship between the average hog weight and the variation of hog weight. It is necessary to understand how the attribute of body weight is distributed amongst the pigs in an all-in all-out finishers to understand the relationship between the average weight and weight variation. Statisticians would say that weight is distributed among pigs within a finishing house in a normal fashion. This means that if a bar graph of the number of hogs categorized by the same body weights to the nearest 10 pound were made, a bell curve could be fit over the top of the bars.

The average weight and the variation of the weights among the pigs cause the bell to change shape, yet still retain the characteristic bell shape. The average describes the point that is the center of the bell and the sides of the bell are symmetric on either side of the center line. Variation describes how wide the bell spreads on either side of center. The more weights vary between pigs, the broader the bell spreads to take on wider ranges of weight. Also the spread affects the height of the bell at its central location. The more individual pigs become spread out among a greater variety of weights, the fewer weights there are to become concentrated around center. To sum this up, if average weight and weight variation are known, these two parameters will describe the shape of the weight distribution curve of a population of hogs on a topping floor. These 2 parameters are estimated by the growth and variation lines.

Pigs that "fall behind" can cause the distribution to become miss shaped or skewed, as these pigs create an asymmetric distribution with a long left tail. The bell curve is then lost to a degree and this type of distribution is harder to make marketing predictions from. Culling is helpful in this area as it serves to maintain a symmetric distribution that is similar on both sides of the average.

Tool 3. The Weight Distribution Curve

The importance of this shape is its ability to estimate the number of hogs in each weight category. The growth and variation lines are used to predict the average weight and variation of weight on the floor. If the average weight and the variation of weight can be closely estimated, the shape of the weight distribution curve can be assumed. With a distribution curve, the number of animals in each sequential weight category within the finishing house is accessible without the expense of weighing all the hogs.

This is possible because each section of the curve can be easily linked mathematically to the number of hogs in each weight category as represented by a section of the weight distribution curve. There is a known proportion of animals represented in every area of the curve as long as it maintains a bell shape. To calculate the number of animals that are in a certain weight category, it is as easy as taking the total number on the floor and multiplying by a proportion of the total. The proportion is available from a table. This idea is worked in reverse to find the light cut-off weight for a marketing group. For instance, lets say we want to market the first grade sort of 192 head from a 960 head topping floor. We can easily calculate the proportion; 192 is .20 of 960 (192/960), the proportion being .2000. The next step is to transform this proportion into a measure of variation. This is done with a Z table. The table tells us that .2000 is close to .2005 which corresponds with a point on the weight distribution curve that is .84 measures of variation to the right of the center of the distribution curve. As a definition, 1.00 measure of variation (Z) is enough movement from the center to delineate about 1/3 of the population represented by the whole curve. So, if we count the .84 measures of weight variation (standard deviation) to the right of the average body weight, it will stop at a place on the weight distribution curve that corresponds to a body weight given on the base axis of the curve, which could be the 236 pounds. This means that if we sorted all animals out of the barn that were 236 pounds and over, we would have very close to 192 head.

Fact 4. Consecutive Marketings are Sequential Divisions Placed on the Weight Distribution Curve

Assume the 960 place house in the example is on a 4 finisher site that is filled on a continuous basis by buildings as they become available. The policy of the producer is to market in lots that are evenly divisible by about 200 head, the capacity of a trailer load. If 192 head was just marketed, there will be about 4 trailer loads left in the house to finish and sort (960/5=192, 5 equally sized loads with 1 shipped leaves 4). To see these loads on the weight distribution will be just a matter of following the same steps that we did above to cut the first load. The first load pulled a proportion of .20. The next load will pull a cumulative proportion of .40 (192 x 2 / 960). For the same reasons, the next three loads will be divided out by cumulative proportions .60 and .80 of the population of the building. There are 4 divisions cutting 5 possible sorted loads. These loads can be referred to as A, B, C, D, and E; with A being the first load cut from the right of the weight distribution curve and the others following in order to the left.

To see where these proportional cuts fall on the weight distribution curve requires transformation to Z values. Using the table again; we can see that the second division will be .25 measures of variation to the right of the center, .25 measures of variation to the left of the center, and .84 measures to the left of center. This means the divisions will all occur close to the mean, since all are less than 1 variation measurement away on either side, which represents 2/3 of the population of the floor.

The proximity of these lines on the weight distribution curve suggest some marketing strategy that will be applicable to this example as well as other schemes of marketing. First, assume that the 5 loads were all sorted separately by the 4 divisional lines. Now access the weight variation in the consecutive loads. Which loads have the greatest variation? Obviously the first and the last (A and E). In fact, there is about a 35 pound weight range on both of these two proposed loads.

Which loads have the least variation of body weight? Obviously the middle three loads. The third load (C) has a weight range of less than 15 pounds. The second and fourth loads (B and D) both have about a 16 pound spread. Therefore, the loads coming from the middle of the weight distribution curve are the most uniform and contain the most hogs in a narrow spread as compared to the first and the last.

Now that we know where the variation is, wouldn't it make sense to market the first load with a 35 pound spread and then market the next two loads together with 1 cut? This would make trailer two loads with only about a 30 pound spread.

This would get 60 per cent of the hogs out and we are left with 2 loads, one with much variation and the other with much less variation. We know that we would wait a long time for the last load to finish if both were marketed separately. The wait would probably be so long for so few hogs, that the savings in sort loss would be surpassed by the opportunity costs of not having the facility full with a new compliment of hogs. This means that the close out marketing might as well be both loads. The distribution also shows the variation in the close-out marketing that is composed of the last two loads (D and E). There will be at least a 50 pound weight range between these. In fact, 80 per cent of the hogs in the last sort will be within 30 pounds of each other and the remaining 20 per cent will be lightweights that have an additional 20 pounds of weight range. Can you understand why packers, who desire consistency, dread clean-out marketings?

What we have done is mathematically proven the old marketing system of two grade sorts and a clean-out. The actual sorts took place on the weight distribution curve at +.84 and -.25 measures of variation off the center.

Tool 4. The Combination of Tools:

The Growth Line, Variation Line, and Weight Distribution Curve Can Be Linked Together to Build a Marketing Device.

It is easier to understand how to use these three tools together when one considers their contribution to prediction and their relationship to each other. The growth line is used to predict the average weight of the house at a point in time, usually defined by a certain calendar day. The average weight of the house on a particular day also describes the center of the weight distribution curve. The spread and height of that distribution curve on that same day can be estimated by predicting from the variation line. The three tools together create the model of the weight distribution divided into marketed segments and relates the marketings to time to allow planning.

As time passes and days go by, the distribution curve changes by spreading out and becoming shorter at the center. This is because growth increases the amount of variation. However, the center of the weight distribution curve can always be positioned on one day by using the prediction of the growth line. The variation line will reset the spread and height of the curve on that the same day. Marketing can be related to the same positions on the curve and lined up to new days on the growth line. In this way, future marketing days and weights can be predicted.

In fact, the loss model that was discussed in this morning's presentation could be installed within the marketing groups. Since the weight distribution curve allows one to predict the number of head in each weight category, the predicted weights of individuals could be placed into the loss model of each marketing to be analyzed for an estimated optimal sale date. This would turn the range of dates between the marketing lines into a specific date of marketing.

Fact 5. The Growth Line and The Variation Line are not Representative of the Process Performance on Every Topping Floor.

The marketing device is a workable blend of concepts and mathematics. There is an obvious inaccuracy that can be addressed. The growth line and the variation line serve as a likely outcome, but in truth their slopes represent the performance outcomes of other finishers and may or may not be applicable to the finisher on which market timings are being made. Not only do individual hogs perform differently, but individual floors do also. This is especially true of the variation line, as it can change more dramatically in short time period than the growth line. To improve on the marketing device, there needs to be another tool utilized that directly estimates the rate of growth and the variation of pig weights within a finisher.

Tool 5. Statistical Process Control

There is an alternative tool that can be used instead the growth curve and the variation line. That tool has been successfully used in industrial manufacturing for years. This technique is called on-line process control or statistical process control (SPC). SPC can be applied to discover the average daily gain and the variation that is occurring during the finishing process.

The basis of SPC uses sampling to make assumptions about individual pigs, as well as the group of finishers as a whole. Sampling will create another expenditure for extra labor to sort, gather, and weigh pig weights to derive data to generate descriptive statistics. If 17 to 20 pigs are selected randomly from a 960 head floor and weighed weekly, these samples will be significant enough to use in SPC.

The descriptive statistics are essentially plotted over time as the samples are taken. The plot is used to study the stability of a parameter and estimate its real value in the finisher being studied. These plots are called control charts. We have established that gain is straight line and should be a stable value during finishing. If we plot the average daily gain of the samples over time, we create a control chart. To assure that the information is stable and applicable to our marketing device, we can draw 2 horizontal lines, both 3 standard deviations off of center, that describe the stability of ADG during the part of the finishing period when samples were taken.

If the plot of ADG stays between the two horizontal lines, the ADG of the finishing process is said to be stable. It would then be safe to use the sequentially weighted average of the ADG to estimate the slope of the growth line. This will make a growth line that is directly defined by the floor that is being marketed. A sequentially weighted term we is one that gives more strength to recent values than past values during calculation; so the sequentially weighted average of gain uses more information from recent samples of gain than from samples that came from the house 4 weeks ago.

If outside forces, like disease, come to bear on the finishing process; the ADG will change and become unstable. The plot will move outside the horizontal lines. The chart not only warns of a damaging occurrence in the process, but it also tells us that the sequential average is not a good estimate of slope for the growth line estimation. If the unstable chart returns to a stable state, a revised estimate of the growth line slope can be made that will be predictively accurate.

The same method can be applied to a control chart of weight variation. The best term to plot is not standard deviation. Standard deviation increases over time during growth, so standard deviation of weight within a finisher is unstable by definition. To get around this the standard deviations of the sampled weights are divided by the average weights of the samples. This expresses the weight variation in terms of average weight, which is also increasing and causes a dampening of the variation shift. This expression of variation divided by the average results in a stable parameter that can be plotted on a SPC chart.

This new parameter is known as a coefficient of variation (CV) and will allow us to estimate the variation of the finishing process at a time in the future. Just like the charting of gain, CV can be plotted by the time of the sample and the stability of the CV during the finishing process can be determined with the two horizontal lines. When the CV chart is stable, the sequentially weighted average of the CV values is the estimate of what the future CV will be on a possible marketing date in the future. From the growth line we will also predict the average body weight at the future time. The pooled CV value from the SPC chart is multiplied by the average weight to get the future variation of body weight. With the average weight and the weight variation being predicted at a future time, the shape of the distribution curve can be modeled. With the shape of the distribution curve available, the marketing device becomes functional. The interpretation of samplings with the use of SPC will allow more accurate prediction of the shape of the distribution curve and, therefore, market timings.

To sum up the implementation of SPC into our predictive market timing device; if weekly sampling were to begin on week 8 of the finishing period and the process was stable, the chart of ADG would give an estimate of slope to set the growth line by week 13. The growth line would be drawn from a point originating at average initial entry weight. The growth line would allow one to pinpoint the center of the weight distribution curve by predicting the average weight. The CV chart will generate an estimate of the CV of the finishing process. The predicted average weight when multiplied by the CV gives an estimate of the weight variation among the pigs at that future date. The predicted variation sets the spread on the weight distribution curve. When the distribution curve is placed over the time scale of the growth curve, the divisional market group lines of the weight distribution curve can be related through the growth line to a range of dates. A computer model, as in the morning presentation, could select the optimal predicted date from this range of dates.

Take-Home Message

Pig weight in an all-in all-out topping floor is distributed like a bell curve. The shape of this curve at a future time can be predicted by estimating what the average pig weight will be and what the variation between pig weights will be. Marketings divide up the bell shaped weight distribution curve. The divisions can be related to dates on the growth line of a floor. A spreadsheet can choose the optimal marketing date to remove that group.