Essay on linear programming


  1. Linear Programming Essay
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Thus, although the old upper limit of The product has been reset in fish meal from its lower limit of 8 to a new limit of zero. This is necessary because of the addition of meat protein and the setting of an RHS value for this new row element. SET-UP as outlined previously.

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However, it is suggested that if extensive modifications are to be made, some other approach to the problem should be used. Compared to the original solution in Figure 5, energy remains the limiting factor influencing cost. Despite a reduction in the cost of maize and replacement of the bound on fish meal usage by the more flexible meat-protein bound diet, cost is actually increased. This probably results from the removal of distillers grains from the ingredient list. We also see a further increase in the protein content which, in the original model, already exceeds the 30 percent minimum set.

This represents a waste and points to opportunities for research in the area of energy utilization in catfish. Some of these ramifications will be explored in the discussion of parametric methods in the next section. Numerous types of changes in the basic format of the original data deck can be made by proper use of the REVISE programme. One would not normally make as many changes as have been made in this example, but these are given to illustrate the different types of changes available.

It is pointed out, again, that if extensive modifications are to be made, the user might be advised to make changes in the original data deck rather than using the REVISE programme. The use of parametric linear programming PLP adds a new dimension to least-cost linear programming.

This technique is extremely versatile and in the hands of a creative and imaginative nutritionist can be valuable in many aspects of feed formulation. Some of the areas in which PLP can be used to advantage are: a planning and formulating diets for experimental purposes, b evaluating the potential sales value and usage level of new or improved feeds tuffs, c determining the economic benefit of improved feed processing techniques, and d determining costs and benefits of modifications in nutrient requirements.

Parametric programming is a technique for investigating the effects of a sequence of proportionate or incremental changes in the elements of a row, column, or right-hand-side. A selected factor is parametrized; i. Several different parametric procedures can be used in feed formulation.

First, the value of the objective function of the solution in this example the cost may be varied. These different methods will be discussed as to their application to feed formulation, and some examples will be given which will guide the reader to consider how PLP may be used to advantage under his own conditions. NONE 1. NONE By proper use of this programme one can quickly establish a series of linear changes in his nutritional specifications and examine their effects on the diet composition and costs. The changes may be either positive or negative; i. Changes in the level of more than one nutrient at a time may be examined simultaneously with the specified change in each nutrient independent of changes in other nutrients, either in direction or magnitude.

For example, one could increase calcium while at the same time decrease phosphorus requirements. One may increase lysine requirements by a factor of two while increasing methionine requirements by a factor of four, as another example. In the given initial model of a catfish feed, it was noted that DE was the only limiting nutrient in the optimal solution. Therefore, one will study the effects of reducing the DE requirement of the initial diet. To repeat, additional information must be included in both the data deck and the control programme.

First, add a new card to the RHS section of the data deck. This card, designated as RHS2, specifies the nutrient to be changed and the amount by which it is to be changed. It is to be punched as are other RHS cards in the original data deck. A negative sign immediately preceding the value denotes such a reduction.

This card must" be placed in the deck after the end of all the RHS1 cards. It must not be placed before any of the original RHS cards. These are outlined in Figure Since this is a post-optimal operation, the computer first solves the original solution and then proceeds to parametrize the right-hand-side factor indicated. This value can be changed by the user to fit his problem but must always be a positive number and contain a decimal point. The first column represents the solution of the initial model. As the minimum DE requirement is decreased by the PARARHS programme, a decrease in the cost of the diet is seen, as well as a progressive decrease in the protein level until a minimum for the nutrient 30 percent is reached.

The savings per unit decrease of DE requirement is constant up to the point where protein level becomes limiting; at this time the savings fall off sharply, reflecting the new cost of maintaining minimum protein level. The dual role of fish meal is also seen as a supplier of both protein and energy when energy cost is high. From the table, it can also be seen that as a protein source, copra meal is less expensive than distillers grains, whereas the latter ingredient is more cost effective in an energy-limiting diet.

A certain ingredient is selected only one ingredient at a time can be considered. One, several, or all of the row values within this column may be increased or decreased. It is often possible to change the nutritive value of an ingredient by some processing method, enzyme addition, etc. For example, it has been established that fully gelatinized maize has almost twice the DE value as uncooked maize for catfish.

This can be partially realized by the pelleting process and completely realized by extruding maize under pressurized heat. Either process, however, involves some additional processing cost. Again, additional information should be added to the data deck and the control programme. First, RHS values must be added into the data deck to indicate the magnitude of the change in DE value of the maize and the additional cost associated with each increment of gelatinization.

These values will be designated as RHS3. The RHS3 values represent the new values added to the respective old values. Assume that 0. Assume, further, for simplicity, that each such step will produce 20 percent gelatinization, and that by the end of the fifth step, the gelatinization process is complete. Suppose each processing step results in a constant cost increment of Baht 0. Keep in mind that any or all of the attributes in the selected ingredient could be changed with the direction on magnitude of change for each attribute independent of the others.

For example, it might also be found that for each increase in the DE value due to the processing, there would be a decrease in lysine content. The user would then add another RHS card indicating this change. However, proceed on the assumption that the energy and cost values represent the only changes. Note that some familiar statements as well as some new ones are added. The change in the diets as a result of increases in the DE value of the maize is shown in Table 3.

As demonstrated in the preceding sections, DE constitutes not only the limiting nutrient in the initial diet but meeting its minimal requirement is at some cost to unwanted protein as well.

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The increasing DE values of maize, made available through gelatinization, more than off-set the added cost involved. Again, sharp drops in cost of the diet occur just before the lower limit for protein is reached. If diet cost is plotted against percent gelatinization, a sigmoid curve is obtained Figure 13 which shows the sharpest reduction in diet cost occurring at around 58 percent gelatinization coinciding with the highest level of maize usage. Incidentally, this is also the level at which the total cost of the diet is least sensitive to increase in cost of gelatinization.

In this programme, a specific row is selected and the value of this row, in one or more feed-stuffs, is altered by a specified amount. Assume, in this example, that the digestible energy content of feed grains, normally low for catfish, can be increased by processing. Assume, further, that stepwise processing of the grains will result in corresponding increases in digestible energy.

Linear Programming Essay

Sorghum is a new column which is added for the demonstration of this parametric technique it was absent in the initial model. Since there is a restriction to select only one row for ranging, it is not possible to consider increased cost with each increase in DE content as was rationalized in the earlier example of parametric column ranging.

This is one limitation of the method. This must be a restraint row and can be ranged. The values which will be given represent increments of the DE content in each of the five steps. It is noted that a total of 5 additions of the change row value are called for to the original DE value of the specified feedstuffs, with the solution to be printed out at each increment. Thus, at each increment, the original value will be replaced with the original value plus a multiple of the change row value.

Once again, the high cost of meeting the minimum requirement of digestible energy in the sample diet is demonstrated. At 40 percent grain starch gelatinization, the total DE needs can be met without sacrifice of unwanted excess of protein,. As further gelatinization of grain starch takes place, the cost of energy becomes subordinate to the cost of protein, and the choice of maize as the energy source gives way to sorghum - the latter having a higher protein content.

As a result of the higher cost of protein too, the greater utilization of feather meal at the expense of fishmeal is seen. While it is technically possible to range the price of several ingredients simultaneously, it would be difficult to interpret the output. The usage level will generally be a step-like function rather than a straight line with the width and height of the steps varying in relation to the competition from other feedstuffs.

The most common use is to estimate the economic or nutritive worth of an ingredient in relation to other ingredients and given an estimate of the market demand use level at different price ranges. One should be aware that the economic value of a particular feedstuff depends not only upon its nutrient content and biological availability, but also upon the type of diet in which it is fed, the limitation or bounds imposed upon the feedstuffs, and the nutrient composition and cost of other feed-stuffs. As an example, a parametric cost ranging study of soybean meal, an ingredient that was not selected for use by the computer in the second least-cost model, will be made.

This programme first solves the basic least-cost solution as described earlier, and then post-optimally considers the cost ranging of the chosen feedstuff. During the solving of the basic least-cost solution, the test ingredient is usually assigned a high artificial price so as to prevent its use in the diet. Certain additional information needs to be added to the data deck. Other objective functions may also be chosen.

Optimization and Linear Programming

This instructs the computer to increase the cost of the test ingredient by 1 each time it goes through the cycle. The intervals at which the solution is to be printed out may be specified in two ways.

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