Optimizing Economic Return in egg production

1. Optimizing Economic Return in egg production Steve Wilcox Senior Mathematics Major Northern Kentucky University April 29, 2011

2. The Story A Shaver White Chicken Randy 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?

3. The Data Randy noting the number of eggs for the day. Extra Large Eggs Large Eggs Medium Eggs Small Eggs PeeWee Eggs Grade B Grade C / Cracked Rejects Randy 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.

4. The Egg Production Cycle We used Randy’s 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. An 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.

5. The Average Daily Return • Cost for daily operations (hen maintenance) • Cost of downtime (time in-between old and new hens) • Price of the eggs • Daily egg production • Gradeouts from the egg buyers We want to maximize the average daily return over a complete cycle. To do this, we created a model consistent with the following five factors:

6. Finding a Model Here 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.

7. Finding a Model We 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 seeka 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.

8. Finding a Model - Economics On 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.

9. Finding a Model - Gradeouts The “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.

10. Finding a Model - Gradeouts The different grades and set prices are: • Extra Large Eggs - $1.54 / dozen • Large Eggs - $1.54 / dozen • Medium Eggs - $1.36 / dozen • Small Eggs - $0.94 / dozen • Peewee Eggs - $0.2425 / dozen • Grade “B” Eggs - $0.45 / dozen • Grade “C” Eggs (Cracked) - $0.15 / dozen • Rejected Eggs - $0.00 / dozen

11. Example - Using the Gradeouts 11,700 dozen total eggs were produced during the week: Egg Grade Percentage Total (dozens) Egg Grade Percentage Total (dozens) Extra Large 14.56% 1704 Large 57.15% 6687 Medium 24.12% 2822 Small 0.65% 76 • Peewee 0.00% 0 • B 0.36% 42 • Cracked 1.82% 213 • Rejects 1.33% 156 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 day’s production instead. For example, the 46th week gradeout shows:

12. Interpolation We 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.

13. Interpolation Proportion of Large Eggs .63 .59 Week 24.5 Week 25 Week 24 We chose a conservative method: linear splines, a standard type of interpolation (effectively we are just “connecting the dots” to estimate between).

14. Derived Data: Daily Return Daily Return data is created the same way the Weekly Return Data was calcu- lated, only using the interpolated gradeoutdata.

15. Modeling Daily Egg Revenue If 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.

16. The Model A typical logistic function: Where: • c is the asymptotic height, • b is the shift of center, and • a determines the steepness. Our data suggests incorporating some “S-curves” (e.g. “logistic” models) to handle these three phases.

17. The Model - Daily Egg Return 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. We chose a model which consists of the product of two logistic functions: • One is increasing (production ramping up), and • One is decaying (hens winding down). The Daily Return model for daily egg value:

18. Calculating the Economic Value From 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.

19. Economic Model – Average Daily Return D = 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)

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

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

22. The Results We want to maximize the Average Daily Return – if we maximize this, we’ll maximize the farmer’s yearly revenue. It appears that this graph hasa maximum at around day 436 or so (which is a little over 62 weeks) This is Average Daily Return for our model:

23. Calculating the best cut-off day To 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 Newton’s Method to obtainour optimal duration D.

24. The Results After using Newton’s 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.

25. 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 don’t know for sure that these hens would continue producing as our model suggests: Randy has to decide whether it’s worth the risk of extending the egg production cycle, based on our results.

26. Should we believe the model? This shows us that hens are capable of producing eggs long after the age of 81 weeks, even up to 100 or 105 weeks. There’s no dramatic change at 70 weeks (or any other moment in time). This model of hen data with more than 71 weeks of egg production comes from Institut de Sélection Animale (ISA):

27. Back-Tracking to calculate the day “D” in our economic model 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. Using 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.

28. Back-Tracking 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. Here is what the data looks like if we start at day 240.

29. Back-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 ona 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.

30. The Results According to our model, we predict that Randy’s chickens could have produced for up to 62 weeks after they began production, or to around week 81 of their lives, thus maximizing Randy’s profits. These chickens never got the chance to reach their maximum economic value: they were killed 9 weeks prior to this time period.

31. Fine Tuning the model – Mortality Perhaps 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.

32. Acknowledgments Thanks 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.gif • Picture for egg sizes http://www.worldpoultry.net/public/image/ISA%20graph.JPG • Picture for egg production after 52 weeks.