The LP Procedure

Example 4.5 Price Parametric Programming for the Oil Blending Problem

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 4.4. The analysis in the last example only accounted for an increase of a little over 4 units (because the minimum $\phi $ 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 right-hand-side 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 $\phi $ 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 $\times $ (change vector).

Output 4.5.1 shows the result of this analysis. The first page is the Price Sensitivity Analysis Summary , as discussed in Example 4.4. The next page is an accounting for the change in basis as a result of decreasing $\phi $ 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 $\phi $, 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 4.5.1: Price Parametric Programming for the Oil Blending Problem

The LP Procedure

Problem Summary
Objective Function Max profit
Rhs Variable _rhs_
Type Variable _type_
Problem Density (%) 45.00
   
Variables Number
   
Non-negative 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 1E-8
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 NON-NEG 0 77.3 0
5 jet_1 BASIC NON-NEG 300 60.65 0
6 jet_2 BASIC NON-NEG 300 63.33 0
7 naphtha_inter BASIC NON-NEG 0 21.8 0
8 naphtha_light BASIC NON-NEG 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

The LP Procedure

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

The LP Procedure

Price Parametric Programming Log
Sensitivity Vector change
Leaving Variable Entering Variable Objective Current Phi
brega brega 1103.0726 -4.158908

The LP Procedure

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 $\phi $ is exactly -30? Because the change in the objective is linear as a function of $\phi $, you can calculate the objective for any value of $\phi $ between those given by linear interpolation. For example, for any $\phi $ between -4.1589 and -30.6878, the optimal objective value is 
            

            

               
\[ \phi \times (1103.0726-0)/(-4.1589-30.6878)+ b \]

            

            
 where 

            

               
\[ b= 30.6878\times (1103.0726-0)/(-4.1589-30.6878) \]

            

            
 For $\phi = $-30, this is 28.5988.