Management Science – Week 21
SOLVER SENSITIVITY, LIMITS AND ANSWER REPORTs ON LP SOLUTION
(Introduction to Management Science, Anderson et al, Ch. 3 and 6)
1. Sensitivity Analysis
One-off solution to a LP problem is of limited interest. In practical business situations, most
of the coefficients in the model can only be estimated values, and so it is of interest to
investigate what happens to the optimal solution if their values change.
We should consider how sensitive a model is to changes in:
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The objective function coefficients
•
The constraint coefficients
•
The RHS values for the constraints (available resources)
SOLVER provides some sensitivity analysis reports. Specifically it tells us:
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The range of values the objective function coefficients can assume without changing
the optimal solution.
•
The impact on the objective function value of increases or decreases in the availability
of various constrained resources.
•
The impact on the objective function 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.
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2. Excel Output
The next sheet to consider is the Answer Report (Figure 1). This report is just a summary of
objective function value (target cell), decision variables (adjustable cells) and constraints.
Figure 1. Answer Report.
Next, consider the Limits Report (Figure 2). This report gives information about the objective
function. The first row gives the value of the objective function evaluated at the optimal setting
for the decision variables (𝑥1 = 540 and 𝑥2 = 252). The bottom two rows give us what would
happen to the objective function if we were to set each of the variables independently to their
minimum value (zero). This basically, allows us to see the contribution to the objective
function of each variable separately. We can summarise from this information that Standard
bags contribute far more to the overall objective value (£5400) compared to Deluxe bags (£2268)
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Figure 2. Limits Report.
Finally, we have a Sensitivity Report. This gives us much useful information, but we need to
consider some of the values provided in more detail. For detailed explanation, please refer
Section 3.
Figure 3. Sensitivity Report.
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3. Extra Information about Sensitivity Report
3.1 Reduced Cost
Reduced cost for a decision variable is defined to be the negative of the objective function
coefficient increase necessary for the variable to be positive in the optimal solution. In this
example, both variables already have a positive value in the optimal solution so no
improvement is necessary in their coefficients to introduce them. However, suppose the profit
on standard bags was only £1. The objective function would be:
𝑃 = 𝑥1 + 9𝑥2
And the solution to this problem is presented in Figure 4.
Figure 4. Explanation of Reduced Cost.
That is, we are now manufacturing 540 deluxe and 0 standard bags. The reduced cost of –2.6
tells us that the objective function coefficient of standard bags needs to increase by £2.60 for it
to be worthwhile for us to manufacture any standard bags.
Alternatively, it is telling us that if we forced the manufacture of any standard bags at this
point, we would make a loss of £2.60 per unit produced compared with maximum profit.
We should note that the reduced cost gives the change in the objective function we will see
if we produce any items that are not currently part of the optimal solution i.e. the amount you
make the optimal solution “worse”. For maximise problems, that means reduced costs are
negative. For minimum problems, they will be positive.
These calculations are based on the assumption that there is no change in any of the other
coefficients.
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3.2 Allowable Increase / Allowable Decrease
The allowable increase for an objective function coefficient is defined to be the amount by
which that coefficient could be increased and the current optimal solution still to be valid
(assuming everything else remains the same). The allowable decrease is the change that could
be made in the current variable setting whilst not affecting the optimal solution. For example,
the profit per standard bag is £10. If that was increased to as much as £13.50, we now know
that the solution of x1 = 540, x2 = 252 would be valid. However, if it increased to more than
£13.50, then the solution would not be valid, and the problem would have to be resolved. In
the same way, we can see that profit per standard bag could be decreased to £6.30 without
affecting the optimal setting of the decision variables. Any further reduction would require
the problem to be resolved.
Note that these ranges are telling us that the variable values ( 𝑥1 and 𝑥2 ) would be
unchanged within these ranges for the coefficients. The profit value would, of course, be
changed because you are multiplying the variables by different coefficients.
3.3 Shadow Price
Consider the cutting and dyeing constraint. We can see that in the optimal solution all 630
hours available in the department are used. This constraint is binding, meaning that it is
actively restricting the solution. A question of interest is would it be worthwhile making
additional hours available in this department? The shadow price tells us the change in the
optimal solution that would be achieved if one extra unit of resource were made available.
Thus, the shadow price of 4.38 for the first constraint tells us that if we had one extra hour
available in the cutting and dyeing department, we would make an additional profit of £4.38.
Thus, we can see the maximum price we should be willing to pay to purchase the extra
resource. For constraints that are not binding, this means we are not using all available
resource as it is, so it would be pointless to purchase extra resources. Therefore, the shadow
price for non-binding constraints is zero. Shadow prices may be negative, indicating what
decrease in the objective function would be achieved by a one-unit increase in the constraint
resource. This is applicable to minimisation problems.
Note that the shadow price may only apply for a small increase in the constraint resource.
Clearly, we cannot keep buying extra resources and gaining benefit from it. We eventually get
to a situation where the additional resources are not used – this corresponds to the constraint
becoming non-binding. In this circumstance, we would need to resolve the LP. The allowable
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increase and allowable decrease figures for each constraint give the range of values for which
shadow costs are valid. Thus the shadow price of £4.38 for the first constraint is valid if the
hours available for cutting and dyeing are between 630 – 134.4 = 495.6 and 630 + 52.36 = 682.36
Note also that the shadow price also gives the change in the objective function if one unit
of resource is removed i.e. one hour removed from the total available in cutting and dyeing
would reduce the objective function by 4.38.
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