## Custom «Linear and Mathematical Programming Use in the Petroleum Refineries» Essay Paper

Mathematical programming was used in the petroleum industry for the first time 40 years ago to manage various tasks and processes; moreover, it was actively used in management of such tasks in the 1990s. However, before the invention of computers its use was limited and it could only be used when there was the need to solve a few problems that involved hand applications (Pörn, Harjunkoski & Westerlund, 1999). The major events that culminated in the use of mathematical programming include the introduction of the simplex algorithm by Dantzig in the 20th century and the invention of electronic computers. As a result of introduction of the simplex algorithm in 1947, an area of programming called linear programming was created and it was used in a number of mathematical applications such as operations research. In 1949, other applications were developed and started to be used in management of tasks related to the petroleum industry.

Mixed-Integer Linear Programming Model for Gasoline Blending and Distribution scheduling

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During mixed integer programming, it is assumed that perfect mixing can be achieved at the blend header and that the changeover time between various products can be neglected. The scheduling model involves the use of a number of constraints such as material balance constraint (Pörn, Harjunkoski & Westerlund, 1999). This constraint states that the amount of product s in tank j at the previous event should not be higher than the amount of products stored in tank j. The following formula is used to obtain material balance:
Pst(s,j,+1) = Pst(s,j,n) + Blnd(s,j,n) -∑lift(I,j,n)

Another constraint used is the allocation constraint which states that when the amount of product j is not zero at point n, that is lift (I,j,n) =0 otherwise (Quesada & Grossmann, 1995). To ensure task splitting is not experienced, the constraint above is processed only once when the case of a small order is involved. In some cases, it can be processed three times.

Another constraint used is the demand constraint which is used to determine the amount of processed materials that can be contained in a particular tank at a particular time when the products are lifted from stock tanks (Ramage, 1998). In order to understand this constraint, the formula used is:
∑∑uv(I,j,n)=∑Prod-ord(I,s)

In addition, the sequence constraint is used. This constraint enables understanding of the sequences in which orders should be processed and the time at which processing should start and end. Viscosity in the tanks can be understood by applying the viscosity constraint which can be represented by the following formula:
VISCt= ∑∑(MIUV –XQC)

It should be understood that this equation is non-linear and thus hard to solve. However, when suitable mathematical treatments are provided to the equation, it is possible to derive an exact linear program model.

Application of Mathematical Programming Language during Supply of Oil, Blending, Product Distribution and Refinery Planning

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An example of mathematical programming language that can be used to describe the process of supply, blending and distribution of oil is the matrix method (Rigby, Lasdon & Waren, 1995). This is where matrices that represent supply and those that represent constraints are used so that an optimized problem solution can be obtained.

For instance, in the equation below, B1, B2 and B3 represent supply, blending and distribution constraints in the order in which they appear while A1, A2 and A3 represent the associated constraints of supply, blending and distribution in the order in which they are listed. As a result of the use of this mathematical program, it is possible to understand the dimensions of the resulting optimizations.

Z*= min c1X1 + C2X2 + c3X3
So that B1X1 > b1
B2X2>b2
A1x1 + A2x2 + A3x3 > 0
X1. x2 . x3 > 0

The above equation can be used during Liquid Petroleum gas (LPG) processing in a refinery when the raw material is fed into the heating tower so that the capacity of the heating tower can be understood and also during operation of the column such as during separation of the raw material into sub-components of petroleum such as butane and other byproducts of petroleum refinery.

Improving Oil Refinery Productivity Through Enhanced Crude Blending Using LP Model

In order to improve refinery productivity, it is important to ensure that optimum requirements during blending stage are established and the right mathematical and programming techniques are used to understand these requirements. For instance, programming languages such as Matlab can contribute towards understanding of the required resources so that there is no resource wastage (Shah, 1996). In addition, understanding of constraints such as the amount of a product to be stored in a tank can facilitate establishment of the right tank sizes to be used.

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It is recommended that linear programming tools are used in both long-term planning and daily scheduling of petroleum refining operations. However, there has been lack of consistency between the LP models and actual operations (Steinschorn & Hofferl, 1997). This is due to the effects of market forces and constraints in the internal environment. However, if linear programming tools are used it is possible to identify and quantify these variations. The use of LP tool ensures that optimum objectives are achieved in the refinery scheme and costs of production are reduced.

However, other planning techniques can contribute to the efficiency of oil refining activities such as planning and scheduling, receiving of products and finished products, designing processes aimed at controlling bottlenecks, coming up with optimum methods of operation that result into generation of optimum profits and coming up with a method of production that results into minimal  operating costs.