4.7 Ranging the Simple Oil Blending Model

Let us now return to our simple oil blending example and analyze the solution. The optimal solution and sensitivity analysis information from our optimiser are shown in Figure 11.

Figure 11: Ranging the Oil Blending Model

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       Problem Title  :  Simple_Oil_Blending_Mode Date  :     May-11-1995
                                                  Time  :            7:28

       Input File     :  RANGN2.SIM                               
       Output File    :  RANGN2.LIS                               
     ----------------------------------------------------------------------
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       Optimal Solution  : 

        MIN  OBJ               =            16.0000

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            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             .   

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

        1)   SGRAV             `<'          0.0250             .   
        2)   VBI               `<'             .               -0.5455
        3)   SULPHUR           `<'          0.1000             .   
        4)   MBALANCE          `='             .               36.1818
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            Ranges objective         Lower        Current        Upper
     ======================================
        1)   LGO                    28.0000       30.0000         -   
        2)   HGO                    10.0000       22.0000       22.2222
        3)   WAXD                   19.8182       20.0000         -   
        4)   ARES                   14.3636       15.0000         -   
        5)   VRES                    6.0000       10.0000       11.0000

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            Ranges RHS-values        Lower        Current        Upper
     ======================================
        1)   SGRAV                   0.9550        0.9800         -   
        2)   VBI                    26.0000       37.0000       37.7857
        3)   SULPHUR                 3.6000        3.7000         -   
        4)   MBALANCE                0.9098        1.0000        1.0356
     ----------------------------------------------------------------------

The first question we asked involves the sensitivity of our solution to changes in the cost coefficients of the chosen components, HGO and VRES. Looking at the Ranges Objective section, note that for HGO its current cost coefficient is 22.0 units. The Lower Cost figure of 10.0 indicates that the cost of HGO can reduce to "just above" 10.0 units before a change of basis occurs. The Upper Cost of 22.2222 indicates that the cost of HGO can increase to 22.2222 units before a change of basis occurs. Therefore, if our cost data is forecasted or estimated in some way, we know, for instance that as long as our actual cost for HGO does not rise by more than 0.2222 units above the estimated cost or fall more than 12.0 units below it we will have an accurate solution.

Similarly, for VRES, the cost coefficient may fall from the current cost of 10.0 units to 6.0 units and rise to 11.0 units before a change of basis occurs. Remember, though, that sensitivity analysis information considers individual changes to the data only.

Next, we asked how much cheaper Light Gas Oil would need to be in order to be included in the solution. This information can be read directly from the reduced cost information given for each variable. We noted that the reduced cost of a non basic variable indicated how much the cost coefficient of that variable would have to change by in order for it to be made basic (i.e. included in the solution). From the solution file, the reduced cost for Light Gas Oil, LGO, is 2.0. This indicates that the cost of Light Gas Oil would have to fall by 2.0 units from 30.0 units to 28.0 units before it will be included in the solution.

We then considered the Viscosity Blend Index constraint, noting that, since the constraint is binding, the model is trying to maximise the VBI of the blend to minimize costs. To find the potential cost savings if we were permitted to relax the upper limit on the VBI we look at the shadow price of the VBI constraint. This indicates the change in the objective function value per unit relaxation of the constraint. From the solution file, this is given as -0.5455 indicating that the objective function will decrease by 0.5455 units for each unit we raise our upper limit on the VBI of the blend.

As shown earlier, as we relax a binding constraint further the cost savings will increase from those indicated by the shadow price. To find how far the saving occurs at the shadow price rate we look at the Ranges RHS_values section of the solution file. The Upper Activity figure for the VBI constraint indicates that the right hand side of the constraint may be relaxed from 37.0 to 37.7857 at the shadow price rate. The cost saving from raising the upper limit on VBI occurs at a rate of 0.5455 / unit for 0.7857 units at total saving of:

0.3787 * 0.5455 = 0.2065 units

Our last question addressed the effect of forcing an amount of Waxy Distillate to be used in the blend. Note that the reduced cost information for non basic variables indicates the unit cost of forcing a non basic variable off its lower bound. The reduced cost information shows that Waxy Distillate has a reduced cost of 0.1818. Therefore, it will cost us 0.1818 units to raise the value of WAXD by a unit, i.e. since we are minimising cost the objective function value will increase by 0.1818 units for each unit of Waxy Distillate we use.

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