GS BUSA 424 CH 4 Sensitivity Analysis and the Simplex Method
The Sensitivity Report (continued) The original objective function coefficients associated with the variable cells are listed in the Objective Coefficient column in Figure 4.3. The next two columns show the allowable increases and decreases in these values. For example, the objective function value associated with Aqua-Spas (or variable X1) can increase by as much as $100 or decrease by as much as $50 without changing the optimal solution, assuming all other coefficients remain constant.
(You can verify this by changing the profit coefficient for Aqua-Spas to any value in the range from $300 to $450 and re-solving the model.) Similarly, the objective function value associated with Hydro-Luxes (or variable X2) can increase by $50 or decrease by approximately $66.67 without changing the optimal values of the decision variables, assuming all other coefficients remain constant. (Again, you can verify this by re-solving the model with different profit values for Hydro-Luxes.)
The Sensitivity Report his report summarizes information about the variable cells and constraints for our model. This information is useful in evaluating how sensitive the optimal solution is to changes in various coefficients in the model.
When solver is completed, select desired report from results window (see screenshot); creates on new tab. So before the manager decides to produce 122 Aqua-Spas and 78 Hydro-Luxes, he might want to determine how sensitive this solution is to the profit figures in the objective. That is, the manager might want to determine how much the profit figures could change before the optimal solution of X1 = 122, X2 = 78 would change. This information is provided in the Sensitivity Report shown in Figure 4.3.
The solution to an LP problem sometimes exhibits a mathematical anomaly known as degeneracy. The solution to an LP problem is degenerate if the RHS values of any of the constraints have an allowable increase or allowable decrease of zero. The presence of degeneracy impacts our interpretation of the values on the Sensitivity Report in a number of important ways: So before interpreting the results on a Sensitivity Report, you should always first check to see if the solution is degenerate because this has important ramifications on how the numbers on the report should be interpreted. However, degeneracy is sometimes caused by having redundant constraints in an LP model. Extreme caution (and perhaps consultation with an expert in mathematical programming) is in order if important business decisions are being made based on the Sensitivity Report for a degenerate LP problem.
1. When the solution is degenerate, the methods mentioned earlier for detecting alternate optimal solutions cannot be relied upon. 2. When a solution is degenerate, the reduced costs for the variable cells may not be unique. Additionally, in this case, the objective function coefficients for variable cells must change by at least as much as (and possibly more than) their respective reduced costs before the optimal solution would change. 3. When the solution is degenerate, the allowable increases and decreases for the objective function coefficients still hold, and, in fact, the coefficients may have to be changed substantially beyond the allowable increase and decrease limits before the optimal solution changes. 4. When the solution is degenerate, the given shadow prices and their ranges may still be interpreted in the usual way, but they may not be unique. That is, a different set of shadow prices and ranges may also apply to the problem (even if the optimal solution is unique).
Creating Spider Tables and Plots Recall that the optimal solution to the original Blue Ridge Hot Tubs problem involves producing 122 Aqua-Spas and 78 Hydro-Luxes for a total profit of $66,100. However, this solution assumes there will be exactly 200 pumps, 1,566 labor hours, and 2,880 feet of tubing available. In reality, pumps and tubing are sometimes defective, and workers sometimes call in sick. So, the owner of the company may wonder how sensitive the total profit is to changes in these parameters.
A Spider Table summarizes the optimal value for one output cell as individual changes are made to various model input cells (or parameters). In this case, the output cell of interest is cell D6 representing total profit. In this case, the output cell of interest is cell D6 representing total profit. The parameters of interest are cells E9, E10, and E 11 representing, respectively, the availability of pumps, labor, and tubing. The strategy in Figure 4.12 is to individually vary the availability of pumps, labor, and tubing between 90% and 110% of their original (or base case) values in 2% increments, as indicated by cells B15 through B25. The base case values for each of our three parameters are listed in cells F9 through F11. For each of these three parameters, we will create and optimize 11 different scenarios (while holding the other two parameters at their base case values) and record the corresponding optimal value for the objective function (cell D6). Therefore, we will use Risk Solver to solve a total of 33 (that is, 3x11) variations of our Blue Ridge Hot Tubs problem.
FIGURE 4.15 Solver Table showing changes in the optimal solution, profit, and resource usage and the number of pumps changed
A number of interesting insights emerge from Figure 4.15. First, compare columns B and F: As the number of available pumps increases from 170 up to 205, they are always all used. With about 175 pumps, we also begin to use all the available labor. However, when the number of available pumps increases to 210 or more, only 207 pumps can be used because we run out of both tubing and labor at that point. This suggests that the company should not be interested in getting more than 7 additional pumps unless it can also increase the amount of tubing and/or labor available.
The Answer Report includes:
A summary of the original and final (optimal) value of the objective cell and the decision variables. To create this report, first solve the LP problem in the usual way, and then click Reports, Optimization, Answer on the Risk Solver Platform ribbon. OR when solver is completed, select desired report from results window (see screenshot); creates on new tab.
Changes in the RHS values Binding constraints prevent us from further improving (that is, maximizing or minimizing) the objective function. For example, the Answer Report in Figure 4.2 indicates that the constraints for the number of pumps and hours of labor available are binding, whereas the constraint on the amount of tubing available is nonbinding. This is also evident in Figure 4.3 by comparing the Final Value column with the Constraint R.H. Side column.
After solving an LP problem, you might want to determine how much better or worse the solution would be if we had more or less of a given resource. For example, Howie Jones might wonder how much more profit could be earned if additional pumps or labor hours were available. The Shadow Price column in Figure 4.3 provides the answers to such questions.
Analyzing Changes in Constraint Coefficients What would happen if the product could be produced in only seven hours? The reduced cost value for Typhoon-Lagoons is calculated as: $320 - $200 x 1 - $16.67 x 7 - $0 x 13 = $3.31 Because this new reduced cost value is positive, producing Typhoon-Lagoons would be profitable in this scenario, and the solution shown in Figure 4.8 would no longer be optimal.
As another example, suppose that we wanted to know the maximum amount of labor that is required to assemble a Typhoon-Lagoon while keeping its production economically justifiable. L3 represents the amount of labor required to produce a Typhoon-Lagoon, we want to find the maximum value of L3 that keeps the reduced cost for Typhoon-Lagoons greater than or equal to zero. $320- $200 x 1 - $16.67 x L3 - $0 x 13 >= 0 L3 <= 120/16.67 = 7.20
OTHER USES OF SHADOW PRICES Because shadow prices represent the marginal values of the resources in an LP problem, they can help us answer a number of other managerial questions that might arise. For example, suppose Blue Ridge Hot Tubs is considering introducing a new model of hot tub called the Typhoon-Lagoon. Suppose that each unit of this new model requires 1 pump, 8 hours of labor, and 13 feet of tubing, and can be sold to generate a marginal profit of $320. Would production of this new model be profitable? The shadow prices in Figure 4.3 indicate that each pump taken away from production of the current products will reduce profits by $200. Similarly, each labor hour taken away from the production of the current products will reduce profits by $16.67. The shadow price for the tubing constraint indicates that the supply of tubing can be reduced without adversely affecting profits. It is interesting to note that the shadow prices (marginal values) of the resources equate exactly with the marginal profits of the products that, at optimality, assume values between their simple lower and upper bounds. This will always be the case.
Because each Typhoon-Lagoon requires 1 pump, 8 hours of labor, and 13 feet of tubing, the diversion of resources required to produce one unit of this new model would cause a reduction in profit of $200 x 1 + $16.67 x 8 + $0 x 13 = $333.33. This reduction would be partially offset by the $320 increase in profit generated by each Typhoon-Lagoon. The net effect of producing each Typhoon-Lagoon would be a $13.33 reduction in profit ($320 - $333.33 = - $13.33). Therefore, the production of Typhoon-Lagoons would not be profitable (although the company might choose to produce a small number of Typhoon-Lagoons to enhance its product line for marketing purposes). The reduced cost for each variable is equal to the per-unit amount the product contributes to profits minus the per-unit value of the resources it consumes (where the consumed resources are priced at their shadow prices).
The Simplex Method - Creating Equality Constraints Using Slack Variable We have repeatedly mentioned that the simplex method is the preferred method for solving LP problems. This section provides an overview of the simplex method and shows how it relates to some of the items that appear on the Answer Report and the Sensitivity Report.
Because our original formulation of the LP model for the Blue Ridge Hot Tubs problem has only two decision variables (X1 and X2), you might be surprised to learn that Solver actually used five variables to solve this problem. To solve an LP problem using the simplex method, Solver temporarily turns all inequality constraints into equality constraints by adding one new variable to each less than or equal to constraint and subtracting one new variable from each greater than or equal to constraint. The new variables used to create equality constraints are called slack variables. We will refer to X1 and X2 as the structural variables in the model to distinguish them from the slack variables. Recall that we did not set up slack variables in the spreadsheet or include them in the formulas in the constraint cells. Solver automatically sets up the slack variables it needs to solve a particular problem.
Simultaneous Changes in Objective Function Coefficients A technique known as The 100% Rule determines whether the current solution remains optimal when more than one objective function coefficient changes. The following two situations could arise when applying this rule: Case 1. All variables whose objective function coefficients change have nonzero reduced costs. In case 1, the current solution remains optimal provided that the objective function coefficient of each changed variable remains within the limits indicated in the Allowable Increase and Allowable Decrease columns of the Sensitivity Report.
Case 2. At least one variable whose objective function coefficient changes has a reduced cost of zero. Case 2 is a bit trickier. In case 2, we must perform the following analysis where: (see image) Notice that rj measures the ratio of the planned change in cj to the maximum allowable change for which the current solution remains optimal. If only one objective function coefficient changed, the current solution remains optimal provided that rj <= 1 (or, if rj is expressed as a percentage, it must be less than or equal to 100%). Similarly, if more than one objective function coefficient changes, the current solution will remain optimal provided that rj <= 1. (Note that if rj > 1, the current solution might remain optimal, but this is not guaranteed.)
FINDING THE BEST SOLUTION The simplex method operates by first identifying any basic feasible solution (or extreme point) for an LP problem, and then moving to an adjacent extreme point, if such a move improves the value of the objective function. When no adjacent extreme point has a better objective function value, the current extreme point is optimal, and the simplex method terminates. The process of moving from one extreme point to an adjacent one is accomplished by switching one of the basic variables with one of the nonbasic variables to create a new basic feasible solution that corresponds to the adjacent extreme point.
For example, in Figure 4.20, moving from the first basic feasible solution (point 1) to the second basic feasible solution (point 2) involves making X1 a basic variable and S2 a nonbasic variable. Similarly, we can move from point 2 to point 3 by switching basic variables with nonbasic variables. So, starting at point 1 in Figure 4.20, the simplex method could move to point 2 and then to the optimal solution at point 3. Alternatively, the simplex method could move from point 1 through points 5 and 4 to reach the optimal solution at point 3. Thus, although there is no guarantee that the simplex method will take the shortest route to the optimal solution of an LP problem, it will find the optimal solution eventually.
shadow price The shadow price for a constraint indicates the amount by which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant. Amount by which the value of the objective function would change with a one-unit change in the RHS value of a constraint If a shadow price is positive, a unit increase in the RHS value of the associated constraint results in an increase in the optimal objective function value. The shadow price values apply provided that the increase or decrease in the RHS value falls within the allowable increase or allowable decrease limits in the Sensitivity Report for each constraint.
If a shadow price is negative, a unit increase in the RHS value of the associated constraint results in a decrease in the optimal objective function value. To analyze the effects of decreases in the RHS values, you reverse the sign on the shadow price. That is, the negated shadow price for a constraint indicates the amount by which the optimal objective function value changes given a unit decrease in the RHS value of the constraint, assuming all other coefficients remain constant.
The Simplex Method - Basic Feasible Solutions After all the inequality constraints in an LP problem have been converted into equalities (by adding or subtracting appropriate slack variables), the constraints in the LP model represent a system (or collection) of linear equations. If there are a total of n variables in a system of m equations, one strategy for finding a solution to the system of equations is to select any m variables and try to find values for them that solve the system, assuming all other variables are set equal to their lower bounds (which are usually zero). This strategy requires more variables than constraints in the system of equations—or that n >= m. The m variables selected to solve the system of equations in an LP model are sometimes called basic variables, while the remaining variables are called nonbasic variables.
If a solution to the system of equations can be obtained using a given set of basic variables (while the nonbasic variables are all set equal to zero), that solution is called a basic feasible solution. Every basic feasible solution corresponds to one of the extreme points of the feasible region for the LP problem, and we know that the optimal solution to the LP problem also occurs at an extreme point. So, the challenge in LP is to find the set of basic variables (and their optimal values) that produce the basic feasible solution corresponding to the optimal extreme point of the feasible region. The first five solutions in Figure 4.19 are feasible and, therefore, represent basic feasible solutions to this problem. The remaining solutions are infeasible because they violate the nonnegativity conditions.
shadow price (continued) For example, Figure 4.3 indicates that the shadow price for the labor constraint is 16.67. Therefore, if the number of available labor hours increased by any amount in the range from 0 to 234 hours, the optimal objective function value changes (increases) by $16.67 for each additional labor hour.
If the number of available labor hours decreased by any amount in the range from 0 to 126 hours, the optimal objective function value changes (decreases) by $16.67 for each lost labor hour. A similar interpretation holds for the shadow price for the constraint on the number of pumps. (It is coincidental that the shadow price for the pump constraint (200) is the same as that constraint's RHS and Final Values.)
Creating Spider Tables and Plots (continued) Note that the second argument in this PsiOptValue( ) simply computes the appropriate optimization number (from 1 to 33) for each of the cells in the range from C15 through E25. (It returns the appropriate value from A15 through A25 plus 0, 11, or 22 if the formula is in column C, D, or E, respectively.)
In Figure 4.13, cells C15 through E25 show the optimal objective function values for each of our 33 optimization runs. As indicated, we can also graph this data to create an aptly named Spider Plot. The center point in the graph corresponds to the optimal solution to the original model with 100% of the pumps, labor, and tubing available. Each of the points in the graph shows the impact on total profit of varying the original resource levels by the indicated percentage. It is clear from Figure 4.13 that total profit is relatively insensitive to modest decreases or large increases in the availability of tubing.
A note about shadow prices To illustrate this point, let's suppose that the RHS value of the labor constraint for our example problem increases by 162 hours (from 1,566 to 1,728) due to the addition of new workers. Because this increase is within the allowable increase listed for the labor constraint, you might expect that the optimal objective function value would increase by $16.67 x 162 = $2,700. That is, the new optimal objective function value would be approximately $68,800 = ($66,100 + $16.67 x 162 = $68,800). Figure 4.6 shows the re-solved model after increasing the RHS value for the labor constraint by 162 labor hours to 1,728.
In Figure 4.6, the new optimal objective function value is $68,800, as expected. But this solution involves producing 176 Aqua-Spas and 24 Hydro-Luxes. That is, the optimal solution to the revised problem is different from the solution to the original problem shown in Figure 4.1. This is not surprising because changing the RHS of a constraint also changes the feasible region for the problem. The effect of increasing the RHS of the labor constraint is shown graphically in Figure 4.7. So, although shadow prices indicate how the objective function value changes if a given RHS value changes, they do not tell you which values the decision variables need to assume in order to achieve this new objective function value. Determining the new optimal values for the decision variables requires that you make the appropriate changes in the RHS value and re-solve the model.
Alternate Optimal Solutions Sometimes, the allowable increase or allowable decrease for the objective function coefficient for one or more variables will equal zero.
In the absence of degeneracy (to be described later), this indicates that alternate optimal solutions exist. You can usually get Solver to produce an alternate optimal solution (when they exist) by (1) adding a constraint to your model that holds the objective function at the current optimal value, and then (2) attempting to maximize or minimize the value of one of the decision variables that had an objective function coefficient with an allowable increase or decrease of zero. This approach sometimes involves some trial and error in step 2 but should cause Solver to produce an alternate optimal solution to your problem.
Shadow prices and the value of additional resources In the previous example, an additional 162 hours of labor allowed us to increase profits by $2,700. A question might then arise as to how much we should be willing to pay to acquire these additional 162 hours of labor. The answer to this question is, "It depends. . . ." If labor is a variable cost that was subtracted (along with other variable costs) from the selling price of the hot tubs to determine the marginal profits associated with each type of tub, we should be willing to pay up to $2,700 above and beyond what we would ordinarily pay to acquire 162 hours of labor. In this case, notice that both the original and revised profit figures of $66,100 and $68,800, respectively, represent the profit earned after the normal labor charge has been paid. Therefore, we could pay a premium of up to $2,700 to acquire the additional 162 hours of labor (or an extra $16.67 per additional labor hour) and still earn at least as much profit as we would have without the additional 162 hours of labor. Thus, if the normal labor rate is $12 per hour, we could pay up to $28.67 per hour to acquire each of the additional 162 hours of labor.
On the other hand, if labor is a sunk cost, which must be paid regardless of how many hot tubs are produced, it would not (or should not) have been subtracted from the selling price of the hot tubs in determining the marginal profit coefficients for each tub produced. In this case, we should be willing to pay a maximum of $16.67 per hour to acquire each of the additional 162 hours of labor.
Creating a Solver Table The Spider Plot in Figure 4.13 suggests that the total profit earned is most sensitive to changes in the available supply of pumps. We can create a Solver Table to study in greater detail the impact of changes in the available number of pumps.
Recall that a Solver Table summarizes the optimal value of multiple output cells as changes are made to a single input cell. In this case, the single input we want to change is cell E9 representing the number of pumps available. We might want to track what happens to several output cells, including the optimal number of Aqua-Spas and Hydro-Luxes (cells B5 and C5), the total profit (cell D6), and the total amount of pumps, labor, and tubing used (cells D9, D10, and D11). Figure 4.14 (and file Fig4-14.xlsm that accompanies this book) shows how to set up the Solver Table for this problem.
Ad Hoc Sensitivity Analysis Risk Solver Platform provides a number of powerful features that we can use to address ad hoc sensitivity analysis questions when they arise. In this section, we will consider two such ad hoc techniques:
Spider Tables/Plots and A Spider Table/Plot summarizes the optimal value for one output cell as individual changes are made to various input cells. Solver Tables A Solver Table summarizes the optimal value for multiple output cells as changes are made to a single input cell.
The Answer Report includes: (continued) The first section of the report summarizes the original and final (optimal) value of the objective cell. The next section summarizes the original and final (optimal) values of the decision variable cells. The final section of this report provides information about the constraints. In particular, the Cell Value column shows the final (optimal) value assumed by each constraint cell. Note that these values correspond to the final value assumed by the LHS formula of each constraint. The Formula column indicates the upper or lower bounds that apply to each constraint cell.
The Status column indicates which constraints are binding and which are nonbinding. A constraint is binding if it is satisfied as a strict equality in the optimal solution; otherwise, it is nonbinding. Notice that the constraints for the number of pumps and amount of labor used are both binding, meaning that all the available pumps and labor hours will be used if this solution is implemented. Therefore, these constraints are preventing Blue Ridge Hot Tubs from achieving a higher level of profit.
Robust Optimization the optimal solution to an LP problem can be somewhat fragile and could actually become infeasible (and a costly mistake) if any of the coefficients in an LP model are incorrect or differ from the real-world phenomena being modeled. This reality has led a number of researchers and practitioners to consider (and often prefer) robust solutions to optimization problems. A robust solution to an LP problem is a solution in the interior of the feasible region (rather than on the boundary of the feasible region) that has a reasonably good objective function value. In Figure 4.16, notice that the constraint on the number of pumps used was defined in the usual way and appears as a "normal" constraint in the Risk Solver task pane Model tab. When defining the labor and tubing constraints, we must select the USet constraint type and specify a value for the Set Size. (Notice that these constraints appear as Chance constraints in the Risk Solver task pane Model tab.) The Set Size is sometimes referred to as the budget of uncertainty for the constraint. There is no prescribed way of determining the Set Size value. Generally speaking, as the Set Size value increases, the solution obtained becomes more conservative (or more robust).
The actual amount of labor and tubing might vary a bit from the values assumed earlier. Similarly, the amount of tubing required per Aqua-Spa is uncertain but can reasonably be expected to vary uniformly. In Figure 4.16, note that the previous numeric constants in cells B10 through C11 have been replaced by the following random number generators: • Formula for cell B10: =PsiUniform(8.75, 9.25) • Formula for cell C10: =PsiUniform(5.75, 6.25) • Formula for cell B11: =PsiUniform(11.5, 12.5) • Formula for cell C11: =PsiUniform(15.5, 16.5)
The Limits Report This report lists the optimal value of the set cell. It then summarizes the optimal values for each variable cell and indicates what values the set cell assumes if each variable cell is set to its upper or lower limits.
The values in the Lower Limits column indicate the smallest value each variable cell can assume while the values of all other variable cells remain constant and all the constraints are satisfied. The values in the Upper Limits column indicate the largest value each variable cell can assume while the values of all other variable cells remain constant and all the constraints are satisfied.
The Answer Report includes: (continued) Finally, the values in the Slack column indicate the difference between the LHS and RHS of each constraint. By definition, binding constraints have zero slack, and nonbinding constraints have some positive level of slack.
The values in the Slack column indicate that if this solution is implemented, all the available pumps and labor hours will be used, but 168 feet of tubing will be left over. The slack values for the nonnegativity conditions indicate the amounts by which the decision variables exceed their respective lower bounds of zero.
Shadow prices for nonbinding constraints Now, let's consider the shadow price for the nonbinding tubing constraint. The tubing constraint has a shadow price of zero with an allowable increase of infinity and an allowable decrease of 168.
Therefore, if the RHS value for the tubing constraint increases by any amount, the objective function value does not change (or changes by zero). This result is not surprising. Because the optimal solution to this problem leaves 168 feet of tubing unused, additional tubing will not produce a better solution. Furthermore, because the optimal solution includes 168 feet of unused tubing, we can reduce the RHS value of this constraint by 168 without affecting the optimal solution.
Robust Optimization When we solve this problem, Risk Solver actually formulates a larger LP problem with several different coefficient values for the USet constraints. It then solves this problem to obtain a solution that satisfies all of these possible constraint configurations. Figure 4.17 shows the first solution Risk Solver found for this problem.
This solution involves producing about 105 Aqua-Spas and 94 Hydro-Luxes for a profit of $64,945. If you evaluate this solution under our original deterministic assumptions—where Aqua-Spas require 9 hours of labor and 12 feet of tubing and Hydro-Luxes require 6 hours of labor and 16 feet of tubing—you will see that this solution uses about 1,509 of the available 1,566 hours of labor and about 2,764 feet of the available 2,880 feet of tubing. So as indicated on the Output tab in the Risk Solver task pane, this is a "conservative solution" that does not make use of all the available resources. If you click the smaller green arrow on the Output tab, Risk Solver will display a less conservative solution to the problem as shown in Figure 4.18
However, another advantage of using the simplex method is that it provides more sensitivity analysis information than the other techniques. In particular, the simplex method provides us with information about the following:
• The range of values the objective function coefficients can assume without changing the optimal solution • The impact on the optimal objective function value of increases or decreases in the availability of various constrained resources • The impact on the optimal objective function value of forcing changes in the values of certain decision variables away from their optimal values • The impact that changes in constraint coefficients will have on the optimal solution to the problem
Our discussion of Solver's Sensitivity Report highlights some key points concerning shadow prices and their relationship to reduced costs. These key points are summarized as:
• The shadow prices of resources equate the marginal value of the resources consumed with the marginal benefit of the goods being produced. • Resources in excess supply have a shadow price (or marginal value) of zero. • The reduced cost of a product is the difference between its marginal profit and the marginal value of the resources it consumes. • Products whose marginal profits are less than the marginal value of the goods required for their production will not be produced in an optimal solution.
Variables with optimal values equal to their simple upper bounds have reduced cost values that are greater than or equal to zero for maximization problems, or
less than or equal to zero for minimization problems. Figure 4.10 summarizes these relationships. Generally, at optimality, a variable assumes its largest possible value (or is set equal to its simple upper bound) if this variable helps improve the objective function value. In a maximization problem, the variable's reduced cost must be nonnegative to indicate that if the variable's value increased, the objective value would increase (improve). In a minimization problem, the variable's reduced cost must be nonpositive to indicate that if the variable's value increased, the objective value would decrease (improve).
The Sensitivity Report (continued) A note about constancy: The phrase "assuming all other coefficients remain constant" in the previous paragraph underscores the fact that the allowable increases and decreases shown in the Sensitivity Report apply
only if all the other coefficients in the LP model do not change.
All the coefficients in this model (the cj, aij, and bi) represent numeric constants. So, when we formulate and solve an LP problem, we implicitly assume
that we can specify the exact values for these coefficients. However, in the real world, these coefficients might change from day to day or minute to minute. Realizing that such uncertainties exist, a manager should consider how sensitive an LP model's solution is to changes or estimation errors that might occur in (1) the objective function coefficients (the cj), (2) the constraint coefficients (the aij), and (3) the RHS values for the constraints (the bi).
