College of Veterinary Medicine

North Carolina State University

Raleigh, NC 27606

**Quality, Weight, Time, and Loss**

After the smoke clears around negotiations between the producer and the processor concerning bid pricing of finished hogs, what is the most important quality characteristic used to determine price per pound? All else aside, the most important quality characteristic is the final carcass weight of the animal.

What control does the producer have on carcass weight? The only control the producer can exercise on final carcass weights of a group of feeder pigs placed on a topping floor is the control the producer utilizes in determining final weight of the marketed animals. The main determinant of final individual hog weight is simply the time the producer allows the hogs to stay on the finishing floor. It all seems elementary, except the situation is complicated by the fact that there is variation among the weights of the individual animals and, therefore, they all don't reach market weight at the same time.

In fact, body weight is usually distributed like a bell curve within an all-in all-out feeding group. The bell shape distribution is maintained during the finishing period from the effects of culling, which keeps the form of the distribution from being skewed or pulled to the side. The question the producer must answer concerning market timing relates to how many times the bell curve must be divided into consecutive grade sorted marketings before the final close-out. This will allot each sequential division more time to develop added weight prior to their marketing.

Even though the main topic of discussion during finishing is the average weight of the pigs on the floor, the average has less to do with the choice of marketing dates than does the spread of the base of this distribution. Look at this another way. The broader the base of the weight distribution, the longer the producer will wait between marketings of the same number of hogs.

Excluding price fluctuation and carcass traits, marketing success is a matter of proper timing. It is an attempt to most accurately schedule grade and close-out sorts. If these sorts are properly timed, the losses associated with the sales of underweight and overweight animals will be minimized. This is what weight defined quality is about, the reduction of losses to a minimum, which also creates a more consistent product being delivered to the packer on grade sorted loads.

**Loss Functions**

A loss functions is a graphic tool that describes the manner in which economic loss occur to a producers when their products are completed with end point measurements that are not exactly what their customer specified. This graph is a plot of the different financial losses which can happen as related to the degree that the final product is out of specification. The range of measured end point quality values is depicted on the horizontal scale and the extent of the associated dollar amount of loss is expressed on the vertical scale. In other words, if we look at this in the framework of selling a grade sort of hogs, this would be a plot of dollars lost as compared to optimum against the range of days this sorted group could be sold. In other industries, the shape of the loss function graphs are controversial. The Japanese have a firm idea of the form this plot should take.

**The "Quadratic" Loss Function**

The Japanese have proven to our manufacturing industries how shrewd control of product quality can pay off with better products and reduced per unit production costs. These people view the loss function as it is proposed by a well known quality control engineer named Taguchi. In this view, losses other than the lost market values of the product should included in the total losses. To examine this idea in the context of finishing hogs, we need to redefine sort loss. In other words, we have to think of the total loss that occurs from selling an overweight hog late and not just the reduction of the payment rate. Add to the lost payment the wasted production costs of feeding, housing, and maintaining the animal past the optimal selling date and you would closely derive the total loss. In the same thought, the loss from marketing an underweight hog early includes the loss from a reduced payment rate and the lost opportunity of adding more weight (profit) to the carcass, but there is a savings on feed, housing, and maintenance that wasn't used.

Taguchi not only suggested the shape of the loss function, but he went as far as to describe the function with a mathematical formula, L(x)=k(x-T)2. To relate this in terms of properly timing the marketing of a sorted group, L(x) would represent the dollars lost by marketing on a specified day, k is a constant value defined by the process of finishing hogs, and (x-T) is the difference between the specified day and the day that is targeted to yield the highest return. Notice that the (x-T) value is squared. This means that two situations exist. First, as the days increase or decrease from an optimal marketing day, the loss increases much more quickly than a straight line rate. It accelerates in multiples of itself, buffered only by k. This is why the loss is known as a quadratic loss. Second, the use of the square implies that the results to the dollar value of loss will always be positive and the loss will increase on either the previous or the future marketing days as compared to the optimal marketing day.

The formula can be worked for various timings of marketing by days, which are the various values of x. The formula then predicts what the losses will be on certain marketing days compared to the optimum day. If these losses are plotted against time in days, the loss function appears as a prediction of losses to be expected from suboptimal timing of marketing. The shape of the plot expected from this formula is like that of figure 1. It has a point of no marketing losses occurring on the best day to market with the arms of the plot rising vertically on both sides (before and after) of that optimal date. The arms accelerate upward and are not straight lines.

**Application**

The idea of the quadratic loss function of Taguchi's strikes one as being interesting. However, it doesn't seem logical that a "catch all" formula could describe the losses of market timing. To try this formula out, we found an actual marketing situation and adapted it to the computer to make determinations of the increased or reduced income that would be realized if a sorted group of hogs would have been marketed earlier or later days. The group studied was a 189 head marketing that was the first sort from a 460 head topping floor in Sampson County. All hogs in the building had been weighed before the sort and the distribution of weight was known to be shaped like a bell curve. We had the payment grid, individually identified carcass weights, individually identified live weights, the per cent lean on each animal, and the payment rate with or without bonus given to each carcass.

The computer simulation involved the placement of this data on a spreadsheet that allowed the animals to increase weight by 1.5 lbs. for added days and reduce body weight by the same for subtracted days. The animals were assumed to stay the same per cent lean on forward and reverse marketing days. The total of feed expense, yardage, average mortality, and maintenance for each hog was added or deducted, depending if the market timing was budgeting for greater or lesser days than the actual marketing. Payment for the hogs was figured on the increased or reduced carcass weight by the grid. A total group payment was found, plus or minus listed variable expenses as calculated for the group. The tendency of prices to change over time was not included.

The results are shown in figure 2. One of the lines describes the loss function as predicted by the Taguchi formula. The other line is the computer simulation of the outcome of the loss function if the marketing had occurred over the range of days. The two are very similar and suggests that the loss associated with suboptimal marketing of this load of hogs was quadratic in nature. It appears that if this load would have been marketed 6 days earlier, the profit side would have been improved by about $400 or about $2/head.

**Take-Home Message**

- There are optimal days to market groups of hogs.
- Total loss for not marketing on the correct day causes losses that are not straight line, but are quadratic. Losses accelerate (as compared to optimum) over time before and after the best marketing day.

**Suggested Reading**

Evans, J.R. and Lindsay,W.M., 1993. The Management and Control of Quality. St. Paul: West Publishing Co.