The objective coefficient ranging analysis, discussed in the last example, is useful for assessing the effects of changing costs and returns on the optimal solution if each objective function coefficient is modified in isolation. However, this is often not the case.
Suppose you anticipate that the cost of crude will be increasing and you want to examine how that will affect your optimal production plans. Furthermore, you estimate that if the price of ARABIAN_LIGHT goes up by 1 unit, then the price of ARABIAN_HEAVY will rise by 1.2 units and the price of BREGA will increase by 1.5 units. However, you plan on passing some of your increased overhead on to your jet fuel customers, and you decide to increase the price of jet fuel 1 unit for each unit of increased cost of ARABIAN_LIGHT.
An examination of the solution sensitivity to changes in the cost of crude is a twostep process. First, add the information on the proportional rates of change in the crude costs and the jet fuel price to the problem data set. Then, invoke the LP procedure. The following program accomplishes this. First, it adds a new row, named CHANGE, to the model. It gives this row a type of PRICESEN. That tells PROC LP to perform objective function coefficient sensitivity analysis using the given rates of change. The program then invokes PROC LP to perform the analysis. Notice that the PRIMALIN= SOLUTION option is used in the PROC LP statement. This tells the LP procedure to use the saved solution. Although it is not necessary to do this, it will eliminate the need for PROC LP to resolve the problem and can save computing time.
data sen; format _type_ $8. _col_ $14. _row_ $6.; input _type_ $ _col_ $ _row_ $ _coef_; datalines; pricesen . change . . arabian_light change 1 . arabian_heavy change 1.2 . brega change 1.5 . jet_1 change 1 . jet_2 change 1 ; data; set oil sen; run;
proc lp sparsedata primalin=solution; run;
Output 5.4.1 shows the range over which the current basic solution remains optimal so that the current production plan need not change. The objective coefficients are modified by adding times the change vector given in the SEN data set, where ranges from a minimum of 4.15891 to a maximum of 29.72973. At the minimum value of , the profit decreases to 1103.073. This value of corresponds to an increase in the cost of ARABIAN_HEAVY to 169.99 (namely, 175 + 1.2), ARABIAN_LIGHT to 179.16 (175 + 1), and BREGA to 211.24 (205 + 1.5), and corresponds to an increase in the price of JET_1 and JET_2 to 304.16 (= 300 + (1)). These values can be found in the Price column under the section labeled Minimum Phi.
Output 5.4.1: The Price Sensitivity Analysis Summary 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 
The Price Sensitivity Analysis Summary also shows the effects of lowering the cost of crude and lowering the price of jet fuel. In particular, at the maximum of 29.72973, the current optimal production plan yields a profit of 4695.95. Any increase or decrease in beyond the limits given results in a change in the production plan. More precisely, the columns that constitute the basis change.