Submitted manuscript to International Journal of Operations Research _ June 2008
On the Oil Refinery’s Linear Programming Model
Gholam Reza Amin  and Farshid Alasvand
 Department
of Computer Engineering, Postgraduate Engineering Centre, Azad University of South Tehran Branch, Tehran-Iran

Department of Petroleum Engineering, Research and Science Branch, Azad University, Tehran-Iran
Abstract:
This paper proves some significant properties of the linear programming model
used in oil refinery. The paper demonstrates an infeasible case of the LP model
and then introduced a necessary and sufficient condition for feasibility of the
refinery’s LP model. Furthermore the paper detects the redundant constraints of
the oil refinery’s model. Therefore the computational effort needed for solving
the oil refinery LP model is reduced.
Keywords: Linear programming; Oil refinery; Feasibility; Redundancy.
1. Introduction
Linear Programming (LP) deals with the problem of minimizing or maximizing a
linear function in the presence of linear equality and/or inequality constraints or
restrictions, ([1], [9]). Linear programming has proven to be an extremely
powerful tool, both in modeling real-world problems and as a widely applicable
mathematical theory. Thirteen of the Nobel laureates in Economics from 1969 to

Corresponding author. Email: [email protected]
1992 were authors or co-authors of papers or books on linear programming, [4].
Even today it still saved millions of dollars annually for business throughout the
world, [3]. In oil industry, linear programming is widely used by multiproduct
oil-refining firms, which minimize a refinery's variable cost under a set of
constraints. The blending of gasoline was among the first popular applications of
LP in refineries [2]. Also Esso Standard Oil Company was probably the first to
publish a book in 1955 entitled “Linear Programming—The Solution of Refinery
Problems” [8]. Several authors have used linear programming to operating costs
associated with the refinery's CO2 emissions, [6,7]. Recently Tehrani Nejad [7]
used the optimal solutions of the oil refinery’s primal and dual LP models and
presented a novel economic interpretation on the allocation of CO2 emissions.
This paper introduces some properties of the refinery’s LP model. The paper
shows when the refinery's LP model is feasible. Also the paper detects the
redundant restrictions corresponding to the refinery’s LP model used in Tehrani
Nejad [7]. The presence of redundant constraints does not change the optimal
solution(s), but may consume extra computational effort, [5]. Therefore the
contribution of this paper is that it reveals further properties of the oil refinery’s
LP model which are important practically. The rest of this paper is organized as
follows:
Section 2 gives a brief explanation of the refinery’s LP model. It also shows an
infeasible case of the model. A necessary and sufficient condition for feasibility of
the LP model is given in Section 3. It also detects the redundant constraints of the
refinery’s LP model. Section 4 shows that how a decision maker can use the
obtained theoretical results of this paper practically. Section 5 concludes the
paper.
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2. The oil refinery’s LP model
The oil refinery’s LP model can be explained as follows:
Consider an oil refinery cost minimization model in which the refinery processes
n different types of crude oils (denoted by x  ( x1 ,, xn ) ) to produce m main
type of oil products, for instance gasoline, diesel and fuel oil. The refiner's
objective is to satisfy its output target for each of the oil products (denoted
by b  ( b1 , , bm ) t ) at minimum cost subject to the market crude prices (denoted
by c  ( c1 , , c n ) ), respectively, and its prevailing linear technology. The
availability of the crude oils is limited to x  ( x1 , , x n ) t , respectively. It is also
assumed that the quantities of CO2 released by processing each of the crude oils
are E  (e1 ,, e n ) , respectively. Therefore the cost minimization LP model of this
refinery can be formulated as shown below, [7]
min cx
s.t.
Ax  b
(1.1)
Ex    0
(1.2)
xx
(1.3)
x  0n ,   0
(1.4)
(1)
Where, A  (a ij ) for each i  1,  , m and j  1,, n , are the technical coefficients
represent the average productivity of crudes in oil products and variable 
corresponds to the total carbon dioxide emissions generated from the burning of
fuel gas (ethane and propane), liquefied fuel (e.g., vacuum residue) and the coke
of the catalytic cracker in the refinery, each of which being assigned a specific
CO2 emission coefficient, E  (e1 ,, e n ) , [7]. The constraints or restrictions of the
model can be described as follows:
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(1.1): The product demand constraints (m constraints), (1.2): the CO2 balance
equation, (1.3) the capacity constraints (n restrictions) and the nonnegativity of
decision variables is given in (1.4).
Tehrani Nejad [7] assumed that model (1) and the corresponding dual have a
feasible solution. The following example shows an infeasible case of the
refinery’s LP model (1).
min 55 x1  50 x 2  72 x 3  85 x 4  60 x 5
s.t.
0.29 x1  0.19 x 2  0.39 x 3  0.35 x 4  0.49 x 5  100
0.34 x1  0.29 x 2  0.35 x 3  0.29 x 4  0.29 x 5  87
0.34 x1  0.48 x 2  0.25 x 3  0.34 x 4  0.19 x 5  72
(2)
0.05 x1  0.04 x 2  0.03x 3  0.06 x 4  0.05 x 5    0
x1  45 , x 2  40 , x 3  50 , x 4  55 , x 5  52
x j  0 j  1,2 , 3, 4 , 5   0
The above model shows an infeasible oil refinery production model. The coming
section gives a necessary and sufficient condition for feasibility of the oil refinery
model (1). Therefore the proposed feasibility condition helps the decision maker
to save the computational time of solving an infeasible LP model.
3. A necessary and sufficient condition for feasibility
The following theorem provides a necessary and sufficient condition for
feasibility of the refinery’s LP model (1).
Theorem 1: The refinery’s LP model (1) is feasible if and only if Ax  b .
Proof: To show the necessary condition, assume the refinery’s LP model (1) is
feasible and therefore has a solution, say, ( x 0 ,  0 ) . So
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Ax 0  b
Ex 0 -  0  0
x0  x
x0  0n
0 0
Hence b  Ax 0  Ax and it concludes the necessary side. Now suppose Ax  b .
To show the sufficient condition let Ax  b . It is enough to show that the model
is feasible. Note that the nonnegative vector defined by ( x 0 ,  0 )  ( x, E x ) is a
feasible solution of model (1) because it satisfies in all of the constraints of the
refinery’s LP model (1). This completes the proof. 
Corollary: Assume that there is an index k for which x k  M and a ik  0 for
each i  1,  , m , where A  (a ij ) and M is enough large. Then the refinery’s LP
model (1) is feasible.
Proof: Due to the assumption there is an index k for which x k  M , where M is
arbitrary large. This implies Ax  b (since a ik  0 for each i  1,  , m ). Therefore
the necessary and sufficient condition given in Theorem 1 holds. 
The next theorem shows that the CO2 balance constraint, Ex    0 , in model (1)
is redundant. In fact the necessary and sufficient condition provided by Theorem
(1) only depends on the parameters A , b and x . This shows the parameter E
does not affect the feasibility of model (1) which sounds the redundancy of the
CO2 balance constraint (1.2). Theorem (2) proves this claim.
Theorem 2: The CO2 balance constraint (1.2) in model (1) is redundant.
Proof: To show the redundancy of the CO2 balance equation in model (1) it is
enough to show that z1*  z *2 , where z1* is the optimal value of model (1) and z *2 is
the optimal value of model (1) by ignoring constraint (1.2). That is
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z *2  min cx
s.t.
Ax  b
(3)
xx
x  0n
Let ( x * ,  * ) and x̂ * denote the optimal solutions of models (1) and (3),
respectively. From the linear programming theory we know that z1*  z *2 , (Bazaraa
et al., 2005). If z1*  z *2 we are done, otherwise assume that z1*  z *2 . Therefore
c x *  c x̂ * . Now consider the vector ( xˆ * , ˆ ) , where ˆ  E xˆ * . Note that ( xˆ * , ˆ ) is a
feasible solution of model (1) with the objective value c x̂ * . But
c xˆ *  z 2*  z1*  c x *
This contradicts to the optimality of ( x * ,  * ) for model (1). So the assumption is
incorrect and therefore z1*  z 2* . 
In the oil refining industry, due to various factors including evaporation and
leakage, the quantity, weight or volume of oil products are less than that of crude
oil inputs and feedstocks. In cost accounting systems, this is referred to as normal
process loss and is usually expressed as a percentage of the input activity volume.
In fact, this constraint already exists in most operational LP refinery models to
capture the quantity of process losses. Introducing the material balance
constraint for losses, the new LP model takes on the following specifications, [7].
min cx
s.t.
Ax  b
lxl  0
Ex    0
xx
x  0n ,   0
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(4)
Where the n-vector l  (l1 ,, ln ) represents the loss coefficients for each input
activity, and the variable l measures the total losses inherent in the production
process. The following numerical example gives an infeasible case of the
refinery’s LP model in the presence of material balance constraint.
min 55 x1  50 x 2  72 x 3  85 x 4  60 x 5
s.t.
0.29 x1  0.19 x 2  0.39 x 3  0.35 x 4  0.49 x 5  100
0.34 x1  0.29 x 2  0.35 x 3  0.29 x 4  0.29 x 5  87
0.34 x1  0.48 x 2  0.25 x 3  0.34 x 4  0.19 x 5  72
(5)
0.03 x1  0.04 x 2  0.01x 3  0.02 x 4  0.03 x 5  l  0
0.05 x1  0.04 x 2  0.03 x 3  0.06 x 4  0.05 x 5    0
x1  45 , x 2  35 , x 3  50 , x 4  70 , x 5  52
x j  0 j  1,2 , 3, 4 , 5   0 , l  0
It is easy to verify that the refinery’s LP model (5) is feasible if and only if Ax  b .
Furthermore, in the cost minimization model (5) the material balance constraint
is redundant. A similar discussion applied for Theorem 2 reveals the redundancy
of the constraint. Therefore we have the following theorem.
Theorem 3: Model (4) is feasible if and only if Ax  b . Besides, the material and
CO2 balance constraints are redundant.
4. How to use the results practically?
This section uses the obtained theoretical properties of the oil refinery’s linear
programming model to show how a decision maker can use the results
practically. In other words, how he/she can modify an infeasible oil refinery’s LP
model to a feasible one. Assume that the decision maker encounters with an
infeasible model. According to the result of Theorem 1 (or Theorem 3) it is
enough to change the parameters b or x to meet the sufficient condition. Note
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that he/she is not allowed to change the technical coefficient A  (a ij ) , since it is
given from the current technology. So at least one of the following modifications
should be done.
1) the output target bi for some oil product, i  1,  , m should be decreased,
or
2) the availability of the jth crude oil x j , for some j  1,, n should be increased.
Since the feasibility condition of oil refinery LP model discussed in this paper is
both necessary and sufficient to make an infeasible oil refinery’s LP model to a
feasible one, the only possible sensitivity analysis must be done as explained
above.
5. Conclusion
This paper proved some significant properties of the linear programming model
used in the oil refinery. The paper demonstrated that the feasibility of the oil
refinery LP model is depending upon only some of the refinery’s data. Also a
necessary and sufficient condition for feasibility of the refinery’s LP model is
introduced. In addition, two redundant constraints of the LP model are detected.
Therefore the contribution of this paper is that it rectified the significant
problems of the existing refinery’s LP model which has not already discussed.
Besides the computational effort needed for solving the oil refinery LP model is
reduced.
Reference:
[1] Bazarra, M., Jarvis, J. J. and Sherali, H., 2005, Linear Programming and Network
Flows, John Wiley & Sons, Third edition, USA.
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[2] Charnes, A., Cooper, W.W., Mellon, B., 1952, Blending aviation gasoline—a
study in programming interdependent activities in an integrated oil company.
Econometrica, 20, 135–139.
[3] Caixeta-Filho, J. V., van Swaay-Neto, J. M. and Wagemaker, A. DP., 2002,
Optimization of the Production Planning and Trade of Lily Flowers at Jan de
Wit Company. Interfaces, 32, 35–46.
[4] Orden, A., 1993, LP from the 40s to the 90s, Interface, 23, 2-12.
[5] Paulraj, S., Chellappan, C. and T. R. Natesan, T. R., 2006, A heuristic approach
for identification of redundant constraints in linear programming models.
International Journal of Computer Mathematics, 83, 675–683.
[6] Pierru, A., 2007, Allocating the CO2 emissions of an oil refinery with
Aumann–Shapley prices. Energy Economics, 29, 563-577.
[7] Tehrani Nejad M., A., 2007, Allocation of CO2 emissions in petroleum
refineries to petroleum joint products: A linear programming model for
practical applications. Energy Economics, 29, 974-997.
[8] Symonds, G.H., 1955, Linear Programming—The Solution of Refinery Problems,
ESSO Standard Oil Company, New York.
[9] Vanderbei, R. J., 1997, Linear Programming: Foundation and Extensions, Kluwer
Academic Publisher, Second edition, USA.
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