Simplex Method - Minimize the Cost of Equine Nutrient Requirements in the United States
This paper deals with optimization theory in the cost of equine nutrient requirements. To reach a state of optimization in the supply of nutrients for horses, this paper will utilize analyses through simplex methods, in solving the maximum flow problem, which in theory has already been solved in the works of Ford and Fulkerson in the form of their famed FFA algorithm. What remains now is the direct application of this algorithm into the subject at hand as well as the inclusion of theory developed by George Dantzig, who developed the simplex algorithm of solving linear problems. 
The question at hand is one of amounts of nutrients to be supplied to horses, but there’s a deeper mathematical formula to be worked out. Basically, there’s need to figure out an algorithm that would work in minimizing the costs of equine maintenance in nutrient supply, but without sacrificing quality of life for the horses in question. This might sound rather ghastly, considering this paper aims to minimize the costs of feeding the horses, but there is no intention of actually underperforming, rather, the point is to minimize the bottle neck effect in feeding the horses during various stages of their lives.
Horses require a very specific amount of food with well measured nutrients within. These numbers have been gauged and well defined a very long time ago, will refining procedures every few years or so. Either way, the numbers are very trustworthy and are in use globally. This means that we know exactly how much of certain foods horses need, for every different stage of development. The NRC has carefully gauged values of protein, energy, macrominerals, microminerals and vitamins, during maintenance, pregnancy, lactation, performance and growth.  They acknowledged that it is nigh impossible to narrow down on a concrete number for amounts of these elements, so they instead focused on effective ranges on daily intakes. Furthermore, they defined a daily nutrient intake in grams per day over several categories such as calcium, magnesium, potassium, vitamin D, vitamin C and many others.
The problem with simply meeting the requirements for these amounts lies in that there isn’t a clear amount that can be administered at all times, and there is no way to gauge the nutrients across all the foods equally. In other words, there is no single food that fits all the necessary criteria that can be administered on average. Horses require varied amounts of countless nutrients and these nutrients could be taken to represent the various sinks for the model of this scenario. The sources, of course, would be the various foods available.
Researchers in Central India have already written on a similar topic in using simplex algorithms to optimize the efficiency of dairy farmers in feeding large amounts of cows. They maintain that this was a necessary step in optimizing for cost efficiency during a particularly dry economic season. Had they not decided to optimize, it is very likely that their inefficiency could have driven many farms to bankruptcy.
Unlike a general flow network, this one’s flow isn’t constricted on its own. In other words, we can supply the horses with as much food as we like, however, the idea is to find the most cost efficient way of doing so. To apply an optimization algorithm to this case, we would have to firstly acknowledge all the factors that would go into the formula. Currently, the process is as follows: There are various foods, which are sources of nutrients. Different foods offer different amounts of nutrients in certain random overlaps. In other words, many different foods offer Vitamin C, but not all of them offer Vitamin D as well, etc. These foods all have a certain price, therefore the cost efficiency would have to stem from best choice between these various foods, keeping in mind that the horses’ needs must be met without compromise. There is also the case of varying scenarios for different types of horses in different stages of development, but this does not need to be analyzed piece by piece since a functional algorithm should be applicable to every situation. Therefore, it is sufficient to simply develop an algorithm that maxes out efficiency regardless of the situation.  The following chart represents the workflow in an overly simplified manner, but what matters is the logic behind it:
The first notable aspect of this graph is that it assumes components from different food groups, as opposed to simply tying the foods to the final nutrients. This might seem like a logistic waste, but is actually a necessary component to the optimization because, for instance, farmers often purchase food in bulk for their horses. One cannot expect them to purchase every bit of food individually. So they purchase treats for instance, which include carrots, apples and beets in a single “food” group, which further contains other foods, here defined as components. But it is still unclear as to why a division is necessary. Well, let’s say a farmer receives an offer from a merchant that offers to supply him apples, beets, oranges and carrots in a package, as those are the products that the merchant produces. That merchant does not sell his products in any other combination. So for instance, the farmer in question has the choice of purchasing 10kg of apples and beets and 12kg of oranges and carrots in a package, and then of course, it’s up to him to decide how many packages he wants to buy, if any. The problem here is that receiving all food from a single merchant will invariably lead to an unnecessary surplus, every single time. This surplus will not go away, since, if the horses flat out don’t need 12kg of oranges, rather, they need only 2, then with every purchased package, the farmer has to literally throw away 10kg worth of oranges, which is a gigantic waste of funds.
In a real life scenario, merchants would not be this picky, but they would still have conditions. It is fact that no bulk supplier measures their merchandise in grams. They would most likely supply in kilograms or tons, depending on the size of the ranch. That is why it is necessary to define every source as a group of its own that branches into different subgroups. This way the algorithm will have the chance to sift through the different combinations and find the most efficient one.
In developing the actual algorithm we would have to firstly define the variables that we are dealing with. For the sake of argument, let’s say we’re working with three different foods a,b and c, which all contain some amount of nutrients w,x,y and z.
A = 2w + 32x +0.1z
B = 16w + 2x + 20y+2.5z
C = 52w + 0.2x + 5y
These are the three possible food sources. We know from research that to maintain his farm, example farmer needs 120w, 70x, 22y and 17z. This means the formula we’re trying to solve, or at least optimize for is lA+mB+nC≥120w+70x+22y+17z
It is evident upon first glance that it is impossible to find a combination that will satisfy the requirements perfectly, however, that is the point of workflow optimization. We could find the values that would yield the least waste, thereby reaching the point of highest cost efficiency. To this end we would need to apply the simplex algorithm, since this is a simple linear problem. In doing so we would come to a set of values for l m and n, which in a real world scenario would tell us exactly how much food to order from the various available sources, to feed a given amount of horses while minimizing the amount of waste. There would always be waste though, which is unfortunate. It is in fact, due to this conclusion that most ranchers don’t go out of their way to optimize through a simplex algorithm, rather they “ball-park” their amounts as close as possible, and always have a surplus that needs to be thrown away. This might seem cost efficient in a single month, but after a few years, the waste would be extremely significant and absolutely unacceptable by any standards. To that end, an efficient model is pivotal.
 G. B. Dantzig, Linear Programming and Extensions, Princeton.
 J. D. Pagan, "Nutrient Requirements: Applying the Science."
 A. C. S. C. N. G. P. T. K. S. T. N. H. a. R. S. G. S. N. Goswami*, "Least cost diet plan of cows for small dairy farmers of Central India."
 M. Trick, "The Simplex Method."