Food Product Orientation

In the canning industry, the sorting, orienting, and feeding of food products can present more complex problems of analysis due to the random variables, which cannot be avoided. Automation research by Kwei at the University of Arkansas has led to the development of an optimum in-feeder design for the chopping of vegetables in a frozen food process (see Figure 3.18). At the point where the carrot was attached to the top of the plant is a tough portion that is considered undesirable. Mechanical choppers are capable of chopping off the tops if the proper end is fed to the chopper. The picture shows an in-feed sorting and conveying mechanism, the objective of which is to advance the carrot towards the mechanical choppers. The carrots are dumped from bulk conveyors onto a “shaker table,” which is full of holes. The top end is heavier and will likely fall through the holes first regardless of carrot orientation on the shaker table. What happens after the carrots fall depends principally upon the relationship between carrot length and the size of the gap between the in-feed conveyor and the shaker table. Other factors are the speed of the in-feed conveyor and the downward slant of the entire assembly. If the carrots could be counted upon to be of constant length, the problem would be much easier. Analysis of the feed system design must consider the random variables of carrot size. Case Study 3.5 illustrates the computations for a simplified case.

Design of a Frozen Vegetable In-Feeder

Figure 3.18 shows part of an automated system for preparing frozen carrots. In the diagram, a carrot is falling through a hole in the shaker pan onto the in-feed conveyor with a correct orientation. However, if the carrot is too short for the gap between shaker pan and in-feed conveyor, it is clear that the carrot might fall such that the point of the carrot will be forward and the cap to the wrong end. On the other hand, if the gap is too small, the carrot will not feed and will clog the system. Clogging causes disruption of the production line and is considered twice as serious a difficulty as “wrong end” orientation. As an approximation to the true situation, assume that clogging always will occur if the gap is one inch less than carrot length and that clogging will never occur if the gap is larger than one inch less than carrot length. Also, as an approximation, assume that a carrot will always fall incorrectly if the gap is greater than the carrot length. Assume that carrot length is normally distributed with mean 3 in. and standard deviation of 2 in.

1. What size gap should be specified?
2. What percent of the incoming carrots would clog the in-feeder?
3. What percent of the carrots would be chopped incorrectly?

Solution

Let L length of carrot (normally distributed, t = 3 in., u = ‘/ in.)

F[L] = cumulative distribution function of L

G = gap in inches — optimum gap

Q(G) = penalty function = Prob (carrot will be chopped wrong end] + 2 Prob [carrot will clog]

Q(G*) = min Q(G)

Q(G) Prob[G  L] + 2(Prob[G  L — 1])

The function can be minimized either by standard search techniques or differential calculus. Intuitively, the gap should be greater than 2 in. because clogging is a more serious problem than wrong end chopping. The minimum for the penalty function is found to lie in the vicinity of a 2.7 in. gap.

The carrot processing case study required some understanding of random variables and statistical theory for analysis but even for those readers without a background in these techniques, the similarities between such a problem and machine parts feeding problems should be evident. Just as in the slot feeding problem there is a working range to be established. For a nominal 3-in, carrot the in-feed conveyor can be set anywhere from 2 in. to 3 in. with no problem. The complication is that carrots vary in size, and a working range for a 3-in, carrot is not a working range for a 5-in, carrot. Fortunately, most real world statistical distributions have a central tendency, and automation system designs can be aimed for the bulk in the mid-range of the distribution. This was done in Case study 3.5, except that the upper tail of the distribution was favored somewhat over the lower tail because of the higher penalty associated with clogging the in feeder.