4.4 Sensitivity Analysis

4.1 The Oil Blending Model

Once the solution to an LP problem has been found, it is good practice to investigate the "sensitivity" of the solution to small changes in the model data before taking any action on recommendations given by the model.

Consider one of the simple oil blending models used earlier. The objective is to find a least-cost mix of components subject to the quality constraints. The problem can be expressed as in Figure 2 below:

Figure 2: Simple Oil Blending Model

{ Simple Oil Blending Model - Version 4b }
{ In this model we constrain the qualities directly }

TITLE Simple_Oil_Blending_Model_V4b ;

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
            <= 0.98 ;

VBI :         15.0 LGO  + 26.0 HGO  + 30.0 WAXD
            + 40.0 ARES + 48.0 VRES
            <= 37.0 ;

SULPHUR:      1.0 LGO  + 2.2 HGO  + 2.8 WAXD
            + 4.1 ARES + 5.0 VRES
            <= 3.7 ;

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

END

The optimal solution to the above blending problem is as follows:

Objective function value

OBJ = 16.0

Components

HGO = 0.5  ( Basic Variable )

VRES = 0.5 ( Basic variable )

LGO = 0     ( Non-basic Variable )

WAXD = 0   ( Non-basic Variable )

ARES = 0    ( Non-basic Variable )

Qualities

SGRAV = 0.955 ( Basic Row )

VBI = 37.0        ( Non-basic Row )

SULPHUR = 3.6 ( Basic Row )

Material Balance

MBAL = 1        ( Non-basic Row )


Drawing Inferences from the Solution

Note that the non-basic rows in the solution represent constraints that are binding. We can use Sensitivity Analysis to address the following types of question:

  1. Many LP models are built with data that is not known exactly. For instance it may be forecasted or estimated by some other means. Although the model has chosen to make the blend up from Vacuum Residue and Heavy Gas Oil what would happen if the costs we attached to them were slightly wrong? Would they still be included in the blend? How much can the cost of each of our chosen components vary and the optimal blend remain the same?
  2. Presumably, since Light Gas Oil is unused in the blend, it is too expensive compared to the other materials chosen ie. Heavy Gas Oil and Vacuum Residue. How much cheaper should Light Gas Oil be before it is worth using it in the blend?
  3. The Viscosity Blend Index constraint is binding, indicating that the model is trying to make the VBI content of the blend as high as possible. We could therefore reduce the cost of the blend if we were allowed to relax the viscosity constraint. How much would the cost be reduced if we were permitted to raise the limit on the Viscosity Blend Index? How much can we raise this limit and guarantee cost savings?
  4. No Waxy Distillate is currently being used in the blend. Suppose some Waxy Distillate has to be used; the cost of the blend must increase (otherwise the model would have used some initially); how much will the cost increase if we force an amount of Waxy Distillate to be included?

The meaning of the information provided by the sensitivity analysis of optimisation codes is best explained graphically using two variable problems. These types of problem are atypical of most LP applications but are easiest to visualise since they can be drawn in two dimensions.

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