The LP Procedure

Example 5.4 Additional Sensitivity Analysis

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 two-step 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 re-solve 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 $\phi$ times the change vector given in the SEN data set, where $\phi$ ranges from a minimum of -4.15891 to a maximum of 29.72973. At the minimum value of $\phi$ , the profit decreases to 1103.073. This value of $\phi$ corresponds to an increase in the cost of ARABIAN_HEAVY to 169.99 (namely, $-$ 175 + $\phi \, \times$ 1.2), ARABIAN_LIGHT to 179.16 ( $= -$ 175 + $\phi \, \times$ 1), and BREGA to 211.24 ( $= -$ 205 + $\phi \, \times$ 1.5), and corresponds to an increase in the price of JET_1 and JET_2 to 304.16 (= 300 + $\phi \, \times$ (-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

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
Col	Variable Name	Status	Activity	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 $\phi$ of 29.72973, the current optimal production plan yields a profit of 4695.95. Any increase or decrease in $\phi$ beyond the limits given results in a change in the production plan. More precisely, the columns that constitute the basis change.