2.5 Minimising the Cost of the Blend

Now let us consider some other possible objectives. The most obvious one is cost. I appreciate that there is no immediately obvious cost which can be used for components produced in the refinery. There is in fact a set of costs which it is natural to use, but they are only obtainable from a LP model of the complete refinery.


First Set of Costs

Fig 3.1 shows the matrix in which the costs of the materials have been set as:

		Cost

LGO 25.0

HGO 22.0

WAXD 20.0

ARES 15.0

VRES 10.0

Fig 3.1: LP Matrix - Minimise Cost of Blend using 25, 22, 20, 15, 10

{ Simple Oil Blending Model - Version 3 }
{ In this model we are attaching costs to the components of the blend }

TITLE Simple_Oil_Blending_Model_V3 ;

MIN  OBJ =   25.0 LGO + 22.0 HGO + 20.0 WAXD + 15.0 ARES
           + 10.0 VRES ;

SUBJECT TO

SGRAV :       0.83 LGO  + 0.88 HGO  + 0.92 WAXD
            + 0.97 ARES + 1.03 VRES - 1.0 ACTUALSG
            = 0.0;

VBI :         15.0 LGO  + 26.0 HGO  + 30.0 WAXD
            + 40.0 ARES + 48.0 VRES - 1.0 ACTUALVI
            = 0.0 ;

SULPHUR:      1.0 LGO  + 2.2 HGO  + 2.8 WAXD
            + 4.1 ARES + 5.0 VRES - 1.0 ACTUALSU
            = 0.0 ;

MBALANCE:     1.0 LGO  + 1.0 HGO + 1.0 WAXD
            + 1.0 ARES + 1.0 VRES
            = 1.0

BOUNDS

            ACTUALSG <= 0.98 ;
            ACTUALVI <= 37.0 ;
            ACTUALSU <= 3.7 ;

END

The objective function which we are minimising is

OBJ  :  25.0 LGO + 22.0 HGO + 20.0 WAXD + 15.0 ARES + 10.0 VRES

Note that the coefficients of ACTUALSG, ACTUALVI and ACTUALSU are zero, so we are completely indifferent to the qualities of the HFO blend which we make.

Fig 3.2 shows the solution.

Fig 3.2: LP Solution - Minimised Cost of Blend using 25, 22, 20, 15, 10

     ----------------------------------------------------------------------
       Problem Title  :  Simple_Oil_Blending_Mode Date  :     Apr-29-1995
                                                  Time  :           16:11

       Input File     :  SIMBLN3.SIM                              
       Output File    :  SIMBLN3.LIS                              
     ----------------------------------------------------------------------
     ----------------------------------------------------------------------

       Optimal Solution  : 


        MIN  OBJ               =            15.0000

     ----------------------------------------------------------------------


            Decision variables            Values           Reduced Cost
     ======================================================================

        1)   LGO               =             0.3333             .   
        2)   HGO               =              .                2.0000
        3)   WAXD              =              .                1.8182
        4)   ARES              =              .                1.3636
        5)   VRES              =             0.6667             .   
        6)   ACTUALSG          =             0.9633             .   
        7)   ACTUALVI'         =            37.0000            0.4545
        8)   ACTUALSU          =             3.6667             .   

     ----------------------------------------------------------------------



            Constraints       Type         Slack          Shadow prices
     ======================================================================

        1)   SGRAV            '='             .                 .   
        2)   VBI              '='             .               -0.4545
        3)   SULPHUR          '='             .                 .   
        4)   MBALANCE         '='             .               31.8182

     ----------------------------------------------------------------------

Given our indifference to the quality of the blend, it may come as something of a surprise that the optimum solution is

LGO = 0.333

VRES = 0.667

with these properties

ACTUALSG = 0.9633

ACTUALVI  = 37.0

ACTUALSU = 3.667

This is exactly the same as when our objective was to maximise specific gravity. Why should this be so? There is an inverse correlation between cost and specific gravity. The high cost components have low specific gravity and the low cost components high specific gravity. Thus minimisation of cost is very similar to maximisation of specific gravity.


Second Set of Costs

What if we had used a different set of costs? Figs 4.1 and 4.2 show what happens if we increase the cost of VRES to 12.0, i.e.

Cost

LGO 25.0

HGO 22.0

WAXD 20.0

ARES 15.0

VRES 12.0

Fig 4.1: LP Matrix - Minimise Cost of Blend using 25, 22, 20, 15, 12

{ Simple Oil Blending Model - Version 4 }
{ In this model we are increasing the cost of VRES by 2 to 12 }

TITLE Simple_Oil_Blending_Model_V4 ;

MIN  OBJ =   25.0 LGO + 22.0 HGO + 20.0 WAXD + 15.0 ARES
           + 12.0 VRES ;

SUBJECT TO

SGRAV :       0.83 LGO  + 0.88 HGO  + 0.92 WAXD
            + 0.97 ARES + 1.03 VRES - 1.0 ACTUALSG
            = 0.0;

VBI :         15.0 LGO  + 26.0 HGO  + 30.0 WAXD
            + 40.0 ARES + 48.0 VRES - 1.0 ACTUALVI
            = 0.0 ;

SULPHUR:      1.0 LGO  + 2.2 HGO  + 2.8 WAXD
            + 4.1 ARES + 5.0 VRES - 1.0 ACTUALSU
            = 0.0 ;

MBALANCE:     1.0 LGO  + 1.0 HGO + 1.0 WAXD
            + 1.0 ARES + 1.0 VRES
            = 1.0

BOUNDS

            ACTUALSG <= 0.98 ;
            ACTUALVI <= 37.0 ;
            ACTUALSU <= 3.7 ;

END

Fig 4.2: LP Solution - Minimised Cost of Blend using 25, 22, 20, 15, 12

     ----------------------------------------------------------------------
       Problem Title  :  Simple_Oil_Blending_Mode Date  :     Apr-29-1995
                                                  Time  :           16:11

       Input File     :  SIMBLN4.SIM                              
       Output File    :  SIMBLN4.LIS                              
     ----------------------------------------------------------------------
 
     ----------------------------------------------------------------------

       Optimal Solution  : 


        MIN  OBJ               =            16.2609

     ----------------------------------------------------------------------
            Decision variables            Values           Reduced Cost
     ======================================================================

        1)   LGO               =             0.2174             .   
        2)   HGO               =              .                1.0435
        3)   WAXD              =              .                0.8696
        4)   ARES              =             0.4783             .   
        5)   VRES              =             0.3043             .   
        6)   ACTUALSG          =             0.9578             .   
        7)   ACTUALVI'         =            37.0000            0.1304
        8)   ACTUALSU'         =             3.7000            2.1739


     ----------------------------------------------------------------------

            Constraints       Type         Slack          Shadow prices
     ======================================================================

        1)   SGRAV            '='             .                 .   
        2)   VBI              '='             .               -0.1304
        3)   SULPHUR          '='             .               -2.1739
        4)   MBALANCE         '='             .               29.1304
     ----------------------------------------------------------------------

The solution is

LGO = 0.217

ARES = 0.478

VRES = 0.304

with these properties

ACTUALSG = 0.9578

ACTUALVI  = 37.0

ACTUALSU = 3.7

So we have now returned to our first solution, in which our objective function was

OBJ : - ( ACTUALSG + ACTUALVI + ACTUALSU )


Third Set of Costs

Are we forever obliged to oscillate between these two solutions? The answer is `No', as is shown in Figs. 5.1 and 5.2. Here the costs are

Cost

LGO 30.0

HGO 22.0

WAXD 20.0

ARES 15.0

VRES 10.0

i.e. we have increased the cost of LGO to 30.0 but left the cost of VRES as 10.0.

Fig 5.1: LP Matrix - Minimise Cost of Blend using 30, 22, 20, 15, 10

{ Simple Oil Blending Model - Version 5 }
{ In this model we are leaving the cost of VRES as 10.0 }
{ We are now increasing the cost of LGO by 5 to 30.0 }

TITLE Simple_Oil_Blending_Model_V5 ;

MIN  OBJ =   30.0 LGO + 22.0 HGO + 20.0 WAXD + 15.0 ARES
           + 10.0 VRES ;

SUBJECT TO

SGRAV :       0.83 LGO  + 0.88 HGO  + 0.92 WAXD
            + 0.97 ARES + 1.03 VRES - 1.0 ACTUALSG
            = 0.0;

VBI :         15.0 LGO  + 26.0 HGO  + 30.0 WAXD
            + 40.0 ARES + 48.0 VRES - 1.0 ACTUALVI
            = 0.0 ;

SULPHUR:      1.0 LGO  + 2.2 HGO  + 2.8 WAXD
            + 4.1 ARES + 5.0 VRES - 1.0 ACTUALSU
            = 0.0 ;

MBALANCE:     1.0 LGO  + 1.0 HGO + 1.0 WAXD
            + 1.0 ARES + 1.0 VRES
            = 1.0

BOUNDS

            ACTUALSG <= 0.98 ;
            ACTUALVI <= 37.0 ;
            ACTUALSU <= 3.7 ;

Fig 5.2: LP Solution - Minimise Cost of Blend using 30, 22, 20, 15, 10

     ----------------------------------------------------------------------
       Problem Title  :  Simple_Oil_Blending_Mode Date  :     Apr-29-1995
                                                  Time  :           16:12

       Input File     :  SIMBLN5.SIM                              
       Output File    :  SIMBLN5.LIS                              
     ----------------------------------------------------------------------


     ----------------------------------------------------------------------

       Optimal Solution  : 


        MIN  OBJ               =            16.0000

     ----------------------------------------------------------------------




            Decision variables            Values           Reduced Cost
     ======================================================================

        1)   LGO               =              .                2.0000
        2)   HGO               =             0.5000             .   
        3)   WAXD              =              .                0.1818
        4)   ARES              =              .                0.6364
        5)   VRES              =             0.5000             .   
        6)   ACTUALSG          =             0.9550             .   
        7)   ACTUALVI'         =            37.0000            0.5455
        8)   ACTUALSU          =             3.6000             .   

     ----------------------------------------------------------------------



            Constraints       Type         Slack          Shadow prices
     ======================================================================

        1)   SGRAV            '='             .                 .   
        2)   VBI              '='             .               -0.5455
        3)   SULPHUR          '='             .                 .   
        4)   MBALANCE         '='             .               36.1818

     ----------------------------------------------------------------------

The solution is

HGO = 0.5

VRES = 0.5

with these properties
ACTUALSG	=  0.955
ACTUALVI  = 37.0

ACTUALSU = 3.6

We should not be surprised that different objective functions may give rise to the same solutions. In a subtle way this is closely connected to the fact that LP works at all. There are infinitely many possible solutions to a problem even as simple as those studied here. To be able to find the optimum solution in a fraction of a second of CPU is very remarkable. It is explained by the fact that the use exclusively of linear constraints reduces the number of potentially optimum solutions to a finite number defined by the intersection of the constraints. The optimum solution can only lie at one of these 'extreme' point should be best by reference to criteria which are similar. The costs that I have used are related to the qualities. VRES is a less valuable material then WAXD or LGO precisely because its qualities are worse.

One other message which should be clear is that there is no unique optimum solution. A particular solution is optimum only with respect to some objective function (although several objective functions may give rise to the same optimum solution, as here). Optimisation is meaningful only with objectives and an optimum solution is only as good as the objective function.

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