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 ModelHere is a picture of what the egg production data looks like over the course of the 71 weeks the hens were housed.We need to model the economic value of these eggs.

Steve: this is WEEKLY egg productionFinding a ModelWe are given two different pieces of information to compute egg value: daily egg production, and the gradeouts that the farmer receives weekly from the egg processor.

We seek a model that incorporates both of these different data types.

We must also keep in mind that the farmer is winning money from the eggs, but losing money based on several factors which will be incorporated into the model.Finding a Model - EconomicsOn the negative side, the farmer is paying to maintain the chickens.e.g. feed, medicine, insurance, cleaning supplies to keep the facility sanitary.Another negative input to the economics is the transition phase.This is the time interval in which old hens are disposed of, new hens are purchased, and no eggs are being produced.On the positive side, the farmer is gaining money from the eggs being produced.

For our purposes, the value of each type of egg (grade) was set to a fixed price, the most common prices for this cycle. Prices do float in reality, however.Finding a Model - GradeoutsThe gradeouts are invoices that list the percentage, total number of eggs (in dozens) produced that week, and the current price for each grade.For simplicity, we set the price for each egg grade to a constant.From this, we can calculate an economic value for any particular week, based on these percentages given and the cost for each type of egg.

Finding a Model - GradeoutsThe different grades and set prices are:

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.10Example - Using the GradeoutsFor example, the 46th week gradeout shows:Extra Large 14.56% 1704Large 57.15% 6687Medium 24.12% 2822Small 0.65% 76Peewee0.00% 0B0.36% 42Cracked1.82% 213Rejects1.33% 156

11,700 dozen total eggs were produced during the week:Egg Grade Percentage Total (dozens)Egg Grade Percentage Total (dozens) We multiply the total proportion of each grade by the value of the grade (e.g. .1456 times $1.54 for the extra large eggs), and add them to get the average dozen value. This times total production (11,700) gives total economic value for that week. The problem with this is that we would prefer daily gradeouts, not just weekly gradeouts, so that we can assign economic value to each days production instead. How can we possibly achieve this?

InterpolationWe used interpolation to create intermediate estimates of gradeout data.

Interpolation is when you estimate intermediate data values based on known data points (and if you estimate at a data location, you get the data value back).

In our case, we know the grades from week to week. By using interpolation, we are able to estimate daily grades, based on nearby 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)We use linear algebra for our interpolation12InterpolationWe 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 EggsWeek 24.5Derived Data: Daily Return Daily Return data is created the same way the Weekly Return Data was calcu- lated, only using the interpolated gradeout data.

Calculation or Graph inserted here (probably graph)Modeling Daily Egg RevenueIf we look at the graph of the daily egg revenue over time, we see three phases: an increasing phase, a short stable plateau,a decreasing phase.We want a model that will respect these phases.

Is there a better picture to use to capture the s-curve feature?15The ModelOur data suggests incorporating some S-curves (e.g. logistic models) to handle these three phases.Where: c is the asymptotic height,b is the shift of center, anda determines the steepness.

A typical logistic function:

If we were to see our scatterplot before the 19 week mark, all of the points would be at 0.16The Model - Daily Egg Return We chose a model which consists of the product of two logistic functions:One is increasing (production ramping up), andOne is decaying (hens winding down).The Daily Return model for daily egg value:

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.

What was the reasoning behind this model?17Calculating the Economic ValueFrom here, we can now integrate the Daily Return model over a given duration of time in order to estimate the total return expected from one egg production cycle:

However, there is more work to be done: we need to incorporate the transition phase, with its associated costs, in order to compute an average daily return over a cycle.

Economic Model Average Daily ReturnD = duration of the egg production phase. This is the crucial parameter whose value we seek to determine.CPD = Cost per Day during egg production (Randy estimated this as $1,712.91 on a per-day basis)TTC = Total Transition Cost (Randy: $139,091.20)TTT = Total Transition Time (Randy: 10 days)

Elements of theADR FunctionThe 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 egg production cycle.Not represented: Maintenance costs during egg production.

20ProblemsWe must be able to predict at least 19 weeks in advance when we need the next batch of hens.

If 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?If 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.

The ResultsThis is Average Daily Return for our model:

We want to maximize the Average Daily Return if we maximize this, well maximize the farmers yearly revenue.

It appears that this graph has a maximum at around day 436 or so (which is a little over 62 weeks)Calculating the best cut-off dayTo maximize the ADR for the optimal duration D, we take the derivative of ADR with respect to D and set it equal to 0.

The derivative of this function is a little ugly. We differentiate it, then use Newtons Method to obtain our optimal duration D.

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 Method

The ADR (Average Daily Return) model gives the amount of money made throughout a complete production cycle as a function of the duration of egg production. We can use this to compare how the duration will affect Randys profits.

23The Results

After using Newtons method to obtain the maximum, we summarize our results in the following graph:If the duration were 357 days (as hen producer suggests) then ADR(357)= $275.87 per day.By keeping them for 371 days, Randy made an average of $285.18 dollars a day.If he would have kept them for 436 days, Randy would have made up to $303.76 a day. This is about $18.00 more dollars per day. Over the course of a year, that adds up to about $6500.00 over one cycle of hens.Should we believe the model?This assumes that the model is believable to 436 days: if so, we can say that it would be beneficial to Randy to keep the chickens for 11 weeks longer.

We dont know for sure that these hens would continue producing asour model suggests:Randy has to decide whether its worth the riskof extending the egg production cycle, basedon our results.

This model assumes a consistent trend after our last set of data points.

That is, we have no way to know whether or not the chickens would just suddenly stop producing after day 371.

Should we believe the model?This model of hen data with more than 71 weeks of egg production comes from Institut de Slection Animale (ISA):

This shows us that hens are capable of producing eggs long after the age of 81 weeks, even up to 100 or 105 weeks. Theres no dramatic change at 70 weeks (or any other moment in time).

This picture comes from a paper intent on expanding the life of the Shaver White chickens.Shaver White Chicken26Back-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 logistic function is modeled.Backing up and using the model at day 171 gives us the following graph:This tells us that the hens have not begun to drop in number of eggs produced.

Real Time Prediction of DHow the Farmer will use our resultsBack-TrackingHere is what the data looks like if we start at day 240.By day 240, the egg production has started to drop (and the second logistic function becomes significant in the model). At this point, Randy gets his first estimates of duration D for this group of hens.

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.

28Back-Tracking

This graph illustrates the successive projected times for the optimal duration for the flock as time goes on.At around day 280, we can see that it appear the data seems to be converging on a particular optimal day, around day 436. If we were to keep the chickens to day 436, then at day 280, we would still have about 22 weeks before the hens will need to be replaced. If Randy starts making preparations at this point, he will have a little over 3 weeks to prepare.29The ResultsAccording to our model, we predict that Randys chickens could have produced for up to 62 weeks after they began production, or to around week 81 of their lives, thus maximizing Randys profits.

These chickens never got the chance to reach their maximum economic value: they were killed 9 weeks prior to this time period.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 hens age and die.

This has obvious negative impact on the economic value of the hens housed.

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.Mortality 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.