Example 3.3: Analyzing the Sensitivity of the Solution to Changes in the Objective Coefficients

The LP Procedure

Example 3.3: Analyzing the Sensitivity of the Solution to Changes in the Objective Coefficients

Simple solution of a linear program is often not enough. A manager needs to evaluate how sensitive the solution is to changing assumptions. The LP procedure provides several tools that are useful for "what if," or sensitivity, analysis. One tool studies the effects of changes in the objective coefficients.

For example, in the oil blending problem, the cost of crude and the selling price of jet fuel can be highly variable. If you want to know the range over which each objective coefficient can vary without changing the variables in the basis, you can use the RANGEPRICE option in the PROC LP statement.

   proc lp sparsedata RANGEPRICE PRIMALOUT=SOLUTION;
   run;

In addition to the Problem and Solution summaries, the LP procedure produces a Price Range Summary, shown in Output 3.3.1.

For each structural variable, the upper and lower ranges of the price (objective function coefficient) and the objective value are shown. The blocking variables, those variables that would enter the basis if the objective coefficient were perturbed further, are also given. For example, the output shows that if the cost of ARABIAN_LIGHT crude were to increase from 175 to 186.6 per unit (remember that you are maximizing profit so the ARABIAN_LIGHT objective coefficient would decrease from -175 to -186.6), then it would become optimal to use less of this crude for any fractional increase in its cost. Increasing the unit cost to 186.6 would drive its reduced cost to zero. Any additional increase would drive its reduced cost negative and would destroy the optimality conditions; thus, you would want to use less of it in your processing. The output shows that, at the point where the reduced cost is zero, you would only be realizing a profit of 268=1544-(110*11.6) and that ARABIAN_LIGHT enters the basis, that is, leaves its upper bound. On the other hand, if the cost of ARABIAN_HEAVY were to decrease to 143.55, you would want to stop using the formulation of 110 units of ARABIAN_LIGHT and 80 units of BREGA and switch to a production scheme that included ARABIAN_HEAVY, in which case the profit would increase from the 1544 level.

Output 3.3.1: Price Range Summary for the Oil Blending Problem

The LP Procedure

Price Range Analysis
Col	Variable Name	Minimum Phi			Maximum Phi
Col	Variable Name	Price	Entering	Objective	Price	Entering	Objective
1	arabian_heavy	-INFINITY	.	1544	-143.55	arabian_heavy	1544
2	arabian_light	-186.6	arabian_light	268	INFINITY	.	INFINITY
3	brega	-208.35	brega	1276	INFINITY	.	INFINITY
4	heating_oil	-7.790698	brega	941.77907	71.5	arabian_heavy	7070.95
5	jet_1	290.19034	brega	949.04392	392.25806	arabian_heavy	7139.4516
6	jet_2	290.50992	brega	942.99292	387.19512	arabian_heavy	7066.0671
7	naphtha_inter	-24.81481	brega	1003.037	286	arabian_heavy	7778.8
8	naphtha_light	-74.44444	brega	989.38889	715	arabian_heavy	6870.75

Note that in the PROC LP statement, the PRIMALOUT=SOLUTION option was given. This caused the procedure to save the optimal solution in a SAS data set named SOLUTION. This data set can be used to perform further analysis on the problem without having to restart the solution process. Example 3.4 shows how this is done. A display of the data follows in Output 3.3.2.

Output 3.3.2: The PRIMALOUT= Data Set for the Oil Blending Problem

Obs	_OBJ_ID_	_RHS_ID_	_VAR_	_TYPE_	_STATUS_	_LBOUND_	_VALUE_	_UBOUND_	_PRICE_	_R_COST_
1	profit	_rhs_	arabian_heavy	UPPERBD		0	0.00	165	-165	-21.45
2	profit	_rhs_	arabian_light	UPPERBD	_UPPER_	0	110.00	110	-175	11.60
3	profit	_rhs_	brega	UPPERBD	_UPPER_	0	80.00	80	-205	3.35
4	profit	_rhs_	heating_oil	NON-NEG	_BASIC_	0	77.30	1.7977E308	0	0.00
5	profit	_rhs_	jet_1	NON-NEG	_BASIC_	0	60.65	1.7977E308	300	0.00
6	profit	_rhs_	jet_2	NON-NEG	_BASIC_	0	63.33	1.7977E308	300	0.00
7	profit	_rhs_	naphtha_inter	NON-NEG	_BASIC_	0	21.80	1.7977E308	0	0.00
8	profit	_rhs_	naphtha_light	NON-NEG	_BASIC_	0	7.45	1.7977E308	0	0.00
9	profit	_rhs_	PHASE_1_OBJECTIV	OBJECT	_DEGEN_	0	0.00	0	0	0.00
10	profit	_rhs_	profit	OBJECT	_BASIC_	0	1544.00	1.7977E308	0	0.00

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