Cases

Linear programming is a powerful tool, that when used can be a very valuable asset for any company. Ever since it was invented, companies have been using it to help them with the various problems they have, whether it be trying to minimize costs that their projects are incurring, or trying to maximize profit (as is the goal of every company out there). The following will present some select cases from companies that have used linear programming and had proven successful to them.



Cases

Picture of the Maharashtra State in India [2]

India_Maharashtra.jpg

Aundha Minor Irrigation Project

Not all the companies that use linear programming use it for strict commercial purposed and model it out for an office environment. In this case, India (specifically the state of Maharashtra) had been undergoing a massive development of their irrigation systems. As time went on, more and more minor irrigation project were popping up around the country, and they all needed to have an efficient water management system for them, as water resources were becoming very expensive. [1]

Being that this whole planning process was very complex and could end up with several alternate plans, the need for making an optimal decision for these projects was required. Thus, the need to solve this problem with linear programming, as well as the fuzzy technique was seen. One of the main needs for irrigation is to supply farmer's with ample water for their crops, and for the farmers to get efficient crop production, thus trying to maximize their production. Some of the constraints the farmers faced was limited access to water, limited soil, and the economic limitations of the farmer's themselves. A few models were produced for this situation, including single objective models such as maximizing net benefit and production, and minimizing investment and labor. For alternate plans, they used multi objective models while using fuzzy techniques to explore even more possibilities.[1]

Once all of these factors were considered, the Aundha minor irrigation project could finally begin. For this project, they had 2 specific objectives; to find the optimal location of the irrigation ditch itself, and to consider the economical benefits of the alternate plans they had. The model began to slowly develop then, as different numbers were found to be put into the model. Fertilizer availability was found, as well as what optimal irrigation efficiency would be. Some of the cropping constraints they found made them realize that they could only cultivate in certain areas. Different levels of their net benefit vs. fertilizer availability was also found, such as a higher level of fertilizer availability would only increase net benefit by 0.12%. Here are some additional results they found, as quoted from the source:[1]

It is clear from the results that the cropping intensity of 1.75 became a limiting factor in case of benefit maximization, investment minimization and for both labour maximization and minimization. However, production maximization and benefit maximization without considering the poultry industry plans gave the crop intensity of 200 per cent.[1]

This shows that they used a comparison of several different objectives in order to solve the model. The constraints they had also played a critical role as it showed the limitations of their goals and how they could then go out and achieve it. In the end, it was found that their main plan would give a net profit of 31 million, while the other plans would give less than half of that amount. Linear Programming allowed them to achieve the maximum amount of profit given the constraints that they had to abide to. [1]


Shell International Petroleum

Although linear programming has somewhat of a knack for helping out farmers in their cause for maximum profit, it has many other uses in the modern world as well. In one case, Shell International Petroleum has used it to solve some of their problems they have been having. Using the technique, they have been able to more efficiently allocate crude oil to their refineries so that they can be processed in the most efficient way. Once this oil was fully processed, the company would use linear programming again to figure out the most optimal way to transport the oil to the market. This type of example would be called a transportation problem, and is under the category of a network model.

Illustrated Figure of the problem [3]

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Shell has stated that while linear programming has been around for half a decade and has consumed much computer time in its calculations, around half of that computer time was used in some way for oil companies. This shows the heavy need that many of the world's oil companies have on decision modeling and quantitative analysis to solve their problems. As for Shell itself, it had to figure out a way to get its oil from the oil drills out to the refineries, and then to the market. They started by drawing a figure to illustrate what the problem actually entailed, and this is shown in the figure to the left. [3]

As you can see, the process starts from the top of the figure at the crude oil fields, and then slowly makes its way down to the market. As seen in the figure, there are many ways in which the oil can move both to the refineries, and to the market. If there were to just choose an arbitrary path for this oil, then their maximum profit would not be realized and they would be losing valuable time in both lost production capacity and lost sales. Thus the need for a linear programming was sought after to find it, and they would end up solving this model using the transportation model. [3]

There are many constraints and factors which they had to deal with as well. When the oil goes from the oil fields to the refineries, they had to take into account the differences in freight costs and the difference in quality that the crude oil has. Once the entire model was mapped out, the company then needed to start to solve it. Once of the restrictions they faced here was a limit on computer speeds and computer techniques in relation to how complex they could make the problem. They had to size down some of the variables as the computers then could not handle a problem as massive as it was at first. [3]


References

1. Gore, K.P. July 31, 2002. "Linear programming and Multi objective allocation model to
Watershed management." <http://asae.frymulti.com/request.asp?JID=5&AID=9329&CID=cil2002&T=2>
3. Catchpole, A.R. "Application of LP to Integrated Supply Problems in the Oil Industry." <http://www.jstor.org/stable/view/3007567?seq=1>
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