This example continues to examine the effects of a change in the cost of crude and the selling price of jet fuel. Suppose that you know the cost of ARABIAN_LIGHT crude is likely to increase 30 units, with the effects on oil and fuel prices as described in Example 5.4. The analysis in the last example only accounted for an increase of a little over 4 units (because the minimum was 4.15891). Because an increase in the cost of ARABIAN_LIGHT beyond 4.15891 units requires a change in the optimal basis, it may also require a change in the optimal production strategy. This type of analysis, where you want to find how the solution changes with changes in the objective function coefficients or righthandside vector, is called parametric programming.
You can answer this question by using the PRICEPHI= option in the PROC LP statement. The following program instructs PROC LP to continually increase the cost of the crudes and the return from jet fuel using the ratios given previously, until the cost of ARABIAN_LIGHT increases at least 30 units.
proc lp sparsedata primalin=solution pricephi=30; run;
The PRICEPHI= option in the PROC LP statement tells PROC LP to perform parametric programming on any price change vectors specified in the problem data set. The value of the PRICEPHI= option tells PROC LP how far to change the value of and in what direction. A specification of PRICEPHI=30 tells PROC LP to continue pivoting until the problem has objective function equal to (original objective function value) 30 (change vector).
Output 5.5.1 shows the result of this analysis. The first page is the Price Sensitivity Analysis Summary, as discussed in Example 5.4. The next page is an accounting for the change in basis as a result of decreasing beyond 4.1589. It shows that BREGA left the basis at an upper bound and entered the basis at a lower bound. The interpretation of these basis changes can be difficult (Hadley 1962; Dantzig 1963).
The last page of output shows the optimal solution at the displayed value of , namely 30.6878. At an increase of 30.6878 units in the cost of ARABIAN_LIGHT and the related changes to the other crudes and the jet fuel, it is optimal to modify the production of jet fuel as shown in the activity column. Although this plan is optimal, it results in a profit of 0. This may suggest that the ratio of a unit increase in the price of jet fuel per unit increase in the cost of ARABIAN_LIGHT is lower than desirable.
Output 5.5.1: Price Parametric Programming for the Oil Blending Problem
Problem Summary  

Objective Function  Max profit 
Rhs Variable  _rhs_ 
Type Variable  _type_ 
Problem Density (%)  45.00 
Variables  Number 
Nonnegative  5 
Upper Bounded  3 
Total  8 
Constraints  Number 
EQ  5 
Objective  1 
Total  6 
Solution Summary  

Terminated Successfully 

Objective Value  1544 
Phase 1 Iterations  0 
Phase 2 Iterations  0 
Phase 3 Iterations  0 
Integer Iterations  0 
Integer Solutions  0 
Initial Basic Feasible Variables  7 
Time Used (seconds)  0 
Number of Inversions  2 
Epsilon  1E8 
Infinity  1.797693E308 
Maximum Phase 1 Iterations  100 
Maximum Phase 2 Iterations  100 
Maximum Phase 3 Iterations  99999999 
Maximum Integer Iterations  100 
Time Limit (seconds)  120 
Variable Summary  

Col  Variable Name  Status  Type  Price  Activity  Reduced Cost 
1  arabian_heavy  UPPERBD  165  0  21.45  
2  arabian_light  UPPBD  UPPERBD  175  110  11.6 
3  brega  UPPBD  UPPERBD  205  80  3.35 
4  heating_oil  BASIC  NONNEG  0  77.3  0 
5  jet_1  BASIC  NONNEG  300  60.65  0 
6  jet_2  BASIC  NONNEG  300  63.33  0 
7  naphtha_inter  BASIC  NONNEG  0  21.8  0 
8  naphtha_light  BASIC  NONNEG  0  7.45  0 
Constraint Summary  

Row  Constraint Name  Type  S/S Col  Rhs  Activity  Dual Activity 
1  profit  OBJECTVE  .  0  1544  . 
2  napha_l_conv  EQ  .  0  0  60 
3  napha_i_conv  EQ  .  0  0  90 
4  heating_oil_conv  EQ  .  0  0  450 
5  recipe_1  EQ  .  0  0  300 
6  recipe_2  EQ  .  0  0  300 
Price Sensitivity Analysis Summary  

Sensitivity Vector change  
Minimum Phi  4.158907511 
Entering Variable  brega 
Optimal Objective  1103.0726257 
Maximum Phi  29.72972973 
Entering Variable  arabian_heavy 
Optimal Objective  4695.9459459 
Col  Variable Name  Status  Activity  Minimum Phi  Maximum Phi  

Price  Reduced Cost  Price  Reduced Cost  
1  arabian_heavy  0  169.9907  24.45065  129.3243  0  
2  arabian_light  UPPBD  110  179.1589  10.027933  145.2703  22.837838 
3  brega  UPPBD  80  211.2384  0  160.4054  27.297297 
4  heating_oil  BASIC  77.3  0  0  0  0 
5  jet_1  BASIC  60.65  304.15891  0  270.27027  0 
6  jet_2  BASIC  63.33  304.15891  0  270.27027  0 
7  naphtha_inter  BASIC  21.8  0  0  0  0 
8  naphtha_light  BASIC  7.45  0  0  0  0 
Price Parametric Programming Log  

Sensitivity Vector change  
Leaving Variable  Entering Variable  Objective  Current Phi 
brega  brega  1103.0726  4.158908 
Price Sensitivity Analysis Summary  

Sensitivity Vector change  
Minimum Phi  30.68783069 
Entering Variable  arabian_light 
Optimal Objective  0 
Col  Variable Name  Status  Activity  Minimum Phi  

Price  Reduced Cost  
1  arabian_heavy  0  201.8254  43.59127  
2  arabian_light  ALTER  110  205.6878  0 
3  brega  0  251.0317  21.36905  
4  heating_oil  BASIC  42.9  0  0 
5  jet_1  BASIC  33.33  330.68783  0 
6  jet_2  BASIC  35.09  330.68783  0 
7  naphtha_inter  BASIC  11  0  0 
8  naphtha_light  BASIC  3.85  0  0 
What is the optimal return if is exactly 30? Because the change in the objective is linear as a function of , you can calculate the objective for any value of between those given by linear interpolation. For example, for any between 4.1589 and 30.6878, the optimal objective value is
where
For 30, this is 28.5988.