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An MILP Model for Oil Vessel at Refinery


Quse M.H. Shihab1 and V.H. Bajaj2

{1Research Student, 2Professor} Department of Statistics, Dr.B.A.M. University, Aurangabad-431004(MS) INDIA.

Corresponding Addresses:

1[email protected], 2[email protected]

Research Article



Abstract: Oil refinery is receiving its crude oil through a pipeline, which is linked to docking station where oil vessels or any other's tankers unloaded. The unloading schedule of these tankers is defend at corporate level cannot be changed easily. This paper is focused on the production scheduling optimization of operation modes concerning crude oil vessel unloading, storage, blending and feed to crude distillation units (CDUS). The model minimizes the operation cost for the total system shown. The model is formulated using general notation and MILP formulation, showing the possibility of using the model for a general problem of this matter.

Key Words: Oil Refinery, Oil Vessel, Crude oil, Unloading, Storage, Charging, MILP Model.


1. Introduction:


            This paper is focused on the production scheduling optimization of operation models concerning crude oil vessel unloading, storage, blending and feed to crude distillation units (CDUS) [4].

            The model proposes the strategic operation for the system the strategic operation for the system in accordance with the given condition and the optimal operational cost calculated [7].

            The strategic operation if it is feasible must follow the proposed suitable unloading days for vessels and the proposed flow rates among vessels and tanks, among storage and charging tanks and among tanks and plants for keeping the optimal production scheduling given by the model results [8].  This model can be used as a viable tool not only for supporting the shipment planning also for discovering system infeasibilities and for strategic decisions concerning investments in storage and pumping systems [5], [9].


    2. Problem Definition:


            The crude oil is unloaded into storage tanks at the docking station and the problem considers one pre-selected storage tank per vessel that manages the same crude oil composition of the vessel that manages the same crude oil composition of the vessel. Then, the crude oil is transferred from storage tanks to charging tanks. Each crude oil inside the charging tank must carry out to be within a range of blended crude oil composition determined at the planning level for the scheduling horizon. The fulfilment of this blended composition range is made per tanks having in account the material balance of the remaining volume and the different flows coming in or coming out the tank with their different compositions per time interval. Then, blended crude oils from charging tanks are charged into the CDUS and whenever that it is optimally required feed switches are done from one kind of blended crude oil to another for each CDU. Finally, it is important to mention that whether a charging tank is feeding a CDU; it must not be fed by any storage tank or vice versa [3], [4], [8].

            The problem will focus on determine the following operating variables to minimum costs:

a)                  Waiting time for each vessel in the sea after arriving.

b)                  Unloading duration time for each vessel.

c)                  Crude oil unloading rate from vessels to storage tanks.

d)                  Crude oil transfer and blending rates from storage tanks to charging tanks.

e)                  Inventory volumes of storage and charging tanks.

f)                   Crude distillation unit charging rates fulfilling the demand pee each CDU.

g)                  Sequence of type of blending crude oil to be charge in each CDU in accordance with the optimal model changeovers.

On the other hand these are the following operating constrains that must be met:

a.                   Equipment capacity limitations; Tank capacity and pumping rate.

b.                  Quality limitations of each blended crude oil; Range of component concentration in each blended crude oil.

c.                   Demand per interval of time (day) of each CDU.


 3. Nomenclature and Model Assumptions:



a.                   VE = {v = 1, 2, …V/ Crude Oil or Tankers}.

b.                  ST = {i=1, 2, ...I/ Storage Tanks}.

c.                   SCH ={t= 1, 2,…T/ Time intervals along the scheduling horizon}.

d.                  CT= {j, y = 1, 2, …J/ Charging tanks}.

e.                   CDU = {L= 1, 2,… L/ Crude distillation units}.

3-2. Parameters:

CUV     = Unloading cost of vessel v per unit time interval.

TARRV= Crude oil vessel arrival date to the docking station.

CSEAV= Sea waiting cost of vessel v per unit time interval.

CSINVi = Inventory cost of storage tank i per unit time interval.

CBINVj= Inventory cost of charging tank j per unit time interval.

CCHANGL=Changeover cost of CDUL.

TLEAV= Crude oil vessel maximum departure date from the decking station.                

VSimin= Storage tank minimum capacity.

VSimax= Storage tank maximum capacity.

VBjmin=  Charging tank minimum capacity.

VBjmax= Charging tank maximum capacity.

3.3 Variables:

3.3.1.Binary Variables

Zj,y,l,t= Variable to denote switch of the blended crude oil fed to                                

CDUL from charging tank j to charging tank y.

XFv,t= Variable to denote if vessel v starts unloading at time t.

XLv,t= Variable to denote if vessel v finishes unloading at time t.

Dj,l,t=  Variable to denote if the crude oil blended in charging tank

j charges CDUL at time t, otherwise charging tank j couldbe being fed by

Storage tanks.


3.3.2Integer Variables.                                                                      

TLv= Vessel v unloading Completion time.

TFv= Vessel v unloading initiation time.

3.3.3Continuous Variables.

VSi,t    = Volume of crude oil in storage tank i at time t.

VBj,t    = Volume of crude oil in charging tanks j at time t.

FVSv,i,t = Volumetric flow rate from vessel v to storage tank i at time t.        

FSBi,j,t  = Volumetric flow rater from storage tank i to charging tank j at time t.

FBCj,L,t = Volumetric flow rate from charging tank j to CDUL at time t.

The following are the assumptions for the proposed model:

1.Only one vessel docking station for crude oil unloading is considered.

2. The time applied for the changeovers are neglected and also the transient flows

Generated during either start up or shut down when a changeover is done.

3. Perfect blending is assume for each charging tank while it is being fed by different Crude oils, and additional blending time inside the tank is not required before it Chargesthe CDU.

4.The composition of the crude oil is decided by the amount of key Component presented in the crude oil or in the blended crude oil. Ingeneral, sulphur is at least one of the key components, for differentiating between crude oils.


 4. Model Mathematical formulation:


The model focus on minimizing the following operation cost of the system for the operations of crude oil vessel unloading, storage, blending and feeding to crude distillation units in an oil refinery [7], then this is the main objective equation that represents the total operation cost of the system:




The above equation is subject to the following constrains:


4.1 Vessel Arrival and Departure Operation Rules.

Each Vessel must arrive to the docking station for unloading only once:

Each vessel leaves the docking station only once:

        The unloading initiation time is:                      

The unloading completion time is:

Each vessel must start unloading either after or on the arrival time

established atthe planning level:

Each vessel must finish unloading up to one interval of time before the maximum

 departure time established at the planning level:

Except for the last vessel:

 Minimum duration of the vessel unloading is two time intervals:

The preceding vessel must finish unloading one time interval before the next

vessel in the sea arrives and start to unload:


4.2 Material Balance Equations for the Storage Tank.

                        The crude oil in storage tank i at time t+1 must be equal to the crude

oil in storage tank i at time t plus the crude oil transferred from vessel v to

storage tank i at time t taking a way the crude oil transferred to charging

tanks j at time t:

Volume capacity limitation for storage tank i:

4.3 Material Balance Equation for the Charging Tank.

            The crude oil blended in the charging tank j at timet+1 must beequal

 to the crude oil in the charging rank j at time t plus the crude oil transferred

from the storage tanks taking away the crude oil transferred to the CDUL at

time t:

Volume capacity limitation for charging tank j:


4.4 Operating Rules for Crude Oil Charging to Crude Distillation Units.

                        As it was stated above each CDULonly can be charged by one charging

               tank j at time t:

On the other hand, each charging tank j can charge at most one CDULat time t:

            If the CDUL is charged by crude oil blended j at time t and after is charged

by crude oil blended y at time t+1 then, changeover cost must be involved. The

following is the condition that confirms that changeover cost shall be charged:





5. Numerical Example:


                          Table1.Shows the system information for numerical example [2].


Scheduling Horizon (# of unit times : days)


No. of Vessel arrivals



Arrival time

Amount of crude Oil

Key component concentration

Vessel 1




Vessel 2




No. of Storage Tanks


Storage Tanks


Initial oil Amount

Key component concentration

Tank 1




Tank 2




No. of Charging Tanks


Charging Tanks


Initial oil Amount

Initial component concentration (Min-Max)

Tank 1



0.02 (0.015–0.025)

Tank 2



0.05 (0.045 – 0.055)

No. of CDUS



Unit Costs involved in vessel operation

Unloading Cost = 8,

Sea waiting Cost = 5

Tank Inventory Unit Cost

Storage Tank = 0.08,

Charging Tank = 0.05

Unit changeover cost for charged oil switch

50 (independent of sequence and  CDU)

Blended oils demand from charging tanks To CDUS

Blended oil 1 : 100,

Blended oil 2 : 100,

Blended oil 3: 100.

Maximum flow rate from vessel to one storage tank


Maximum flow from one storage tank to one charging tank


Maximum total flow rate from one storage tank to  charging tanks at any time




The given data for quantifying volumes and flow rates are given in barrels x 10,000 and barrels per time interval x 10,000 respectively, changeover costs are given in US$ x 1.000; sea waiting costs and unloading costs are given in US$ x1.000 per time interval (day) and tank inventory unit cost are given in US$ x 0.1 per oil barrel. Therefore, optimal value results will be in US$ x 1.000.


Table 2. Optimal Unloading Starting Date for Vessels.


Model A

Model B








Table 2. points out the comparisons of the optimal unloading starting up date results from both models for numerical example vessel 1 stars to unload on day (s) for  model'A'; that  is one day more than  model 'B'which considers this operation on day 2. Instead, for vessel 2 both coincide starting unloading on day7.

The optimization model Table 3.Involved36 discrete variables, 192 single variables, and 331 constraints [2]. The modelling system G.A.M.S. [1]. Was used for setting up the optimization model and the numbers of variables and constraints were reduced by considering the data structure of binary variables. The problem was solved by OSL, IBM [6]. An IBM RS- 6000 in 17.5 of CPU time.



       Table 3. Comparison of Optimal Result


Model A

Model B

Optimal value (US$ X 1.000)



Equations and constrains



Single Variables



Discrete Variables






Solving time (second)






The optimal results generated by model 'B' with No omission of any data given by model ‘A’ are shown and compared in advance with the optimal results of model 'A' in table 3.


6. Concluding Remarks:


                        1) The situations may be causing a big part of the difference between both optimal value since the optimal operational schedule of   Model 'A' should be paying more money in inventory total cost. On the other hand, Model 'A' results is paying one day more of sea waiting cost (US $ 5,000) than model 'B' results as is indicated in table (2).

                        2)The optimal results for numerical example are better for  model 'B' indicating a total operational cost saving for the problem conditions of US$ 10,717 (4.92%) cost reduction with respect to  model 'A' during the 8 day scheduling horizon in accordance with the results showed in table (1).




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