Slide 1

Optimizing Economic Return in egg productionSteve WilcoxSenior Mathematics MajorNorthern Kentucky University

April 29, 2011

The StoryRandy Weibe currently keeps his Shaver White Chickens for about 51 weeks at a time. The chickens are obtained at 19 weeks of age and the hen producers recommends that hens be disposed of at 70 weeks -- hence the 51 weeks.We suspect that they are productive beyond 70 weeks. How can we prove this?

A Shaver White ChickenThe DataRandy collects data over an entire cycle of egg production.Total eggs produced each day Gradeout information: breakdown of the size and quality of the eggs received in Winnipeg.

Randy noting the number of eggs for the day.Extra Large EggsLarge EggsMedium EggsSmall EggsPeeWee EggsGrade BGrade C / CrackedRejects

Talking about the data. Other data, Economic Data: cost to dispose of chickens, levy, etc.3The Egg Production CycleAn egg production cycle is in two parts: the production phase, when the chickens are laying eggs, and the very expensive transition phase when chickens are being replaced, barns are being cleaned, etc. Costs of the transition phase are well understood.

We used Randys data to estimate the average daily profit from his eggs during the egg production phase. As long as the average is high, we will keep these hens in production.

With this model, we want to predict the appropriate time to remove the old chickens and replace them with new ones.This model should help us decide whether or not the default of 70 weeks is really the appropriate cut-off time.4The Average Daily ReturnWe want to maximize the average daily return over a complete cycle.To do this, we created a model consistent with the following five factors:

Cost for daily operations (hen maintenance)Cost of downtime (time in-between old and new hens)Price of the eggsDaily egg productionGradeouts from the egg buyers

This will maximize Randys lifetime economic return.5Finding a Model EconomicsOne of the things we consider is the economics of the process.

The economics of the process consists of the return the farmer is gaining from selling eggs versus the costs required to purchase, maintain, and dispose of the chickens. (Profit=money in - money out.)

The farmer is gaining money from the eggs, but losing money based on several factors.Model EconomicsOn the negative side, the farmer is paying for the cost of maintenance on the chickens.e.g. feed for the chickens and cleaning supplies to keep the facility sanitary.

Another negative input to the economics is the downtime.This is the time interval in which old hens are disposed of, new hens are being purchased, and no eggs are being produced.Model EconomicsOn the positive side, the farmer is gaining money from the eggs being produced.For our purposes, the amount of money for each type of egg (grade) was set to a fixed price, the most common prices for this cycle.

The information provided to us is daily production numbers and the gradeouts the farmer receives weekly.

Model GradeoutsThe different gradeout types and set prices are as follows:Extra Large Eggs - $1.54 / dozenLarge Eggs - $1.54 / dozenMedium Eggs - $1.36 / dozenSmall Eggs - $0.94 / dozenPeewee Eggs - $0.2425 / dozenGrade B Eggs - $0.45 / dozenGrade C Eggs (Cracked) - $0.15 / dozenRejected Eggs - $0.00 / dozen

For the right to make this much money, you must pay a levy into the system.9Model GradeoutsThe gradeouts are essentially the invoices which lists how many eggs of each type were produced.

It also lists the current price for each grade and how much gross income was made off of each grade type.Again, for simplicity, we are setting the price for each egg grade to a constant.Modeling Daily Egg RevenueIf we look at the graph of the daily egg revenue over time (with all gradeouts combined), we see three phases: an increasing phase, a short stable plateau, and a decreasing phase.

Is there a better picture to use to capture the s-curve feature?11The ModelOur data suggests incorporating some S-curves (e.g. logistic models) to handle these three phases.A typical logistic function:11+e-c*tP(t) =

If we were to see our scatterplot before the 19 week mark, all of the points would be at 0.12The Model - Daily Egg Gross Return We decided on a model which consists of the product of two logistic functions.One is increasing (production ramping up) and the other is a decaying logistic (hens winding down).Our Daily Return Model during production:

Where: c is the height parameter of the first logistic.a1 is the central steepness of the first logistic.a2 is the central steepness of the second logistic.b1 is the center location for the first logistic.b2 is the center location for the second logistic.

R(t)=What was the reasoning behind this model?13Using the ModelNow that we have the model, it is possible for us to calculate the total area underneath the logistic curve. This will give us the total number of eggs produced over the course of the chickens lives.

However, this is not what we were interested in. Even though it does help us get to where we want to go, we still have a few more things to get what we are looking for: the total economic value of these eggs.Using the Gradeout DataThe gradeouts provide us data that tells us the percentage and total number of eggs (in dozens) produced in a particular week.

From this, we can calculate an economic value for this particular week, based on these percentages given to us and the cost for each type of egg.Using the GradeoutsFor example, the 46th week had the following data:Extra Large 14.56% 1704Large 57.15% 6687Medium 24.12% 2822Small 0.65% 76Peewee0.00% 0B0.36% 42Cracked1.82% 213Rejects1.33% 156

Note: On this particular day, 11,700 dozen eggs were produced in total.Egg Type Percentage Total (dozens)Egg Type Percentage Total (dozens) We then multiply the total number of eggs by the value of the egg types ($1.54 for the extra large eggs and the large eggs, $1.36 for the medium eggs, etc). This gives us the total economic value for that weeks worth of egg production. The problem with this is that we need daily gradeouts, not just weekly gradeouts. This way we can assign a particular economic value to each day instead of each week. How could we possibly achieve this?InterpolationIn order to obtain the daily values, we need to use interpolation on the data.

Interpolation is when you estimate intermediate data values based on known data points. In our case, we know the grades from week to week. By using interpolation, we are able to estimate daily gradeouts based on adjacent weeks of data.More future work, consider different splines to use (ours leads to negative values)We adjust for any nonsensical extrapolations (like negative values)17InterpolationWe chose a conservative method: linear splines, a standard type of interpolation (effectively we are just connecting the dots to estimate between).

.63.59Week 24Week 25Proportion of Large EggsCalculating the Economic ValueFrom here, we can now integrate the model over a span of time in order to estimate the total return that Randy can expect from one egg production cycle.

After we obtain the total value of the eggs, we must then subtract all fees for maintenance during production, and the costs of the transition period, such as the daily living cost of the chickens (e.g. chicken feed), the cost for disposal of the chickens, the cost of cleaning the barns, etc.Economic Model Average Daily Return

ADR(D)=D = duration of the egg production phase.CPD = Cost per Day during egg production (Randy provided $1,712.91).TTC = Total Transition Cost (Randy provided $139,091.20).TTT = Total Transition Time (Randy provided 10 days).Looking at the graph of the Economic Function

This is what the product of two logistics looks like.The downtime (in red) corresponds to TTT in the formula, it is the amount of time for the transition from old chickens to new ones (in days)The green region indicates the total value of the egg production over the duration (then all the daily costs are taken out).

21ProblemsIf you remember back to the beginning, we said that chickens must be raised to the age of 19 weeks before they even start producing eggs.

But what does this mean for our model?

ProblemsThis means that we must be able to predict at least 19 weeks in advance when the appropriate time is to call for the next batch of hens.

ProblemsIf we have a good model, we should have well enough time to predict when the cut-off time period should be.

This cut-off period should include a grace period in which Randy will actually be able to make preparations for new chickens. He wouldnt reasonably be able to expect new chickens to be started the very next day.Calculating the best cut-off dayTo optimize the number of days to keep the chickens in production, we take the derivative of the economic model and then set this equal to 0.

With this data, we would be able to calculate the amount of money he would have made on a daily basis based on the day the chickens were kept alive and producing. We can then use these to compare how much of a difference keeping the chickens longer will affect Randys profits.Best way to show the derivative? Should I simply show this derivative? Or should I exclude it and simply say that taking the derivative and setting it equal to 0 is how we find the maximum of a curve? Newtons Method25Back-Tracking to calculate the day D in our economic modelUsing our data, we back-tracked to determine if this model was able to accurately predict a good cut-off time 19 weeks in advance. At this point, only the first logistics function is affect the data.Backing up and using the model at day 171 gives us the following graph:This tells us that the chickens have not even started to drop in number of eggs produced.

Back-TrackingHere is what the data looks like if we start at day 240.At this point, the egg production has started to drop (and the second logistic function can be seen). At this point, Randy should be watching his data to see when the economic value of the eggs will drop as well.

I want to use more examples like the previous slide to give an idea of exactly what the model will do when the farmer is entering in his data.

27Back-Tracking

If we do this enough, we can look at the projected time for the optimal duration that the flock should be kept to obtain maximum economic value.At around day 280, we can see that it appear the data seems to be convergingto a particular optimal day, in this case the data appears to be converging to around day 436. If we were to keep the chickens to day 436, as the graph appears to be saying thus far, then at day 280, we still have 22 weeks before the old chickens will need to be replaced. If Randy starts making preparations at this point, he will have about 3 weeks to make preparations for the new batch of chickens.28The ResultsThis is what the Average Daily Return looks like:

We want to take the derivative of this function to find the maximum

So it looks like from this graph that the maximum economic value of these chickens peaks at around day 436 or so (which is a little over 62 weeks)The Results

After taking a derivative and setting the function equal to 0, then solving, we get the following data points:If he did keep the chickens for 357 days (as the manufacture suggested) then Randy would have made a maximum of $275.87 dollars a day.By keeping them for 371 days, Randy only made a maximum of $285.18 dollars a day.If he would have kept them for 436 days instead, Randy would have made up to $303.76 dollars a day. This is about 18 dollars a day more than what he actually made. Over the course of a year, that adds up to about $6500.00 more he could have made.The ResultsSo, according to the model that we have found, we predict that Randys chickens could have lived up to 62 weeks after the began ovulation, or to around week 81 of their lives.

What does this mean? These chickens never got the chance to reach their maximum economic value! They were killed 11 weeks prior to this time period.Should we believe the model?Assuming the trend is believable, we can say that it would be beneficial to Randy to keep the chickens for 11 weeks longer.However, this model assumes a consistent trend after our last set of data points. We have no way to know for sure that these chickens would continue producing as before. That is, we have no way to knowwhether or not the chickens would just suddenly stop producing after day 371.

Should we believe the model?Here is a picture of an example of Shaver White Chicken data with more than 71 weeks of egg production.

This shows us that these chickens are capable of producing eggs long after the age of 81 weeks, even up to 100 or 105 weeks.

So yes, I believe the model is believable.

This picture comes from a paper intent on expanding the life of the Shaver White chickens.33Future Works Naturally, the model isnt perfect and its possible that we did not look at this model from as many angles as we could have to capture the whole picture.

The following slides consists of suggestions I have that may help improve this model in the future.Fine Tuning the model MortalityPerhaps another factor we could have included to the model is the mortality of the chickens.Naturally, as time goes on and chickens get older, exhausted, or otherwise worn down; some of the chickens will start to die off.This has another negative impact on the economic value of the current chickens that are in house.Fine Tuning the model MortalityMortality Effect:Even if chickens are producing eggs as well as ever, a smaller number of chickens still means a smaller number of eggs.

This, of course, means less gradeouts; which in turn is hurting the economic value of the current chickens.AcknowledgmentsThanks to Randy Weibe for providing the data and answers to several chicken-related questions.

Thanks to Dr. Long for his willingness to help me with this capstone and all the help he provided along the way.

http://www.hotelmule.com/attachments/2009/08/26_20090824062802439gB.gifPicture for egg sizeshttp://www.worldpoultry.net/public/image/ISA%20graph.JPGPicture for egg production after 52 weeks.