Why a Price Recommendation Can Be Infeasible
A pricing optimizer can return no recommendation even when its inputs and calculations appear reasonable. The cause may be a genuine conflict among requirements: customer-level movement limits may prevent the segment change a business rule requires, or segment limits may leave too little room to meet a portfolio target. Diagnosing that conflict means identifying which requirements cannot hold together, quantifying the size of the conflict, and communicating what would have to change.
Here, let \(x_j\) denote the proposed rate change for policy \(j\), measured in percentage points. For example, \(x_j=4\) means a proposed increase of 4 percentage points. A constraint is a required condition on one or more decision variables. The feasible set is the set of all decisions satisfying every constraint. If that set is empty, the optimization problem is infeasible, regardless of the objective being maximized or minimized.
The useful first question is not “Which constraint should be removed?” It is “What is the smallest change needed to make the constraints compatible, and what does that change mean?” That distinction keeps a mathematical diagnostic separate from a pricing decision.
Diagnose the Conflict Before Relaxing Anything
Write each requirement separately, with its population, weights, units, and direction made explicit. Common constraints include policy-specific floors and caps, a segment-average target or range, and a portfolio-level floor or cap. A segment average generally uses the applicable policy weights; a portfolio average generally uses portfolio weights. Substituting an unweighted average can create a false diagnosis.
A useful diagnostic is to derive the maximum or minimum outcome possible under the other hard constraints. For a weighted average with nonnegative weights summing to one, increasing a component cannot decrease the average. Therefore, an upper bound for each component produces an upper bound for the weighted average. If that maximum is below a required minimum, the system is infeasible. The same reasoning applies in the opposite direction to a required maximum.
Worked Example: Portfolio Floor Versus Segment Cap
A hypothetical portfolio has two segments. Segment S1 represents 80% of the portfolio and contains two equally weighted policies with proposed increases \(x_1\) and \(x_2\). Segment S2 represents 20% of the portfolio, with proposed increase \(y\). Each policy change must be between 0 and 8 percentage points. S1's weighted average increase may not exceed 3 points, while the portfolio average must be at least 4.5 points. Determine whether the requirements are feasible and quantify two possible relaxations.
Method. Find the highest portfolio average possible while respecting the policy bounds and S1 cap. The S1 average is \((x_1+x_2)/2\), so its largest allowed value is 3. S2 can reach 8. The greatest possible portfolio average is therefore
This upper bound is attainable: set \(x_1=x_2=3\) and \(y=8\). Since even the maximum, 4.00, is below the required 4.5, there is no feasible solution. The shortfall from the portfolio floor is \(4.5-4.0=0.5\) percentage points.
One option is to relax the S1 cap by \(s\) percentage points. Its average can then be at most \(3+s\), so the portfolio maximum becomes \(0.80(3+s)+0.20(8)=4.00+0.80s\). To reach 4.5 requires \(0.80s\geq0.50\), or \(s\geq0.625\). The S1 cap would need to increase from 3 to at least 3.625 points.
Alternatively, keep all segment and policy limits unchanged and reduce the portfolio floor by \(t\). The attainable maximum is 4.00, so the floor must be lowered by at least \(t=0.50\) point, from 4.5 to 4.0. These are different relaxations with different business meanings; the calculation alone does not select between them. Common mistake avoided: treating the two segment values as equally weighted would produce the incorrect maximum \((3+8)/2=5.5\). The portfolio shares are 80% and 20%, not 50% and 50%.
Slack Variables Record the Size and Direction of a Violation
For a requirement \(g(x)\leq b\), a nonnegative slack variable \(s\) can represent an allowed excess by writing \(g(x)\leq b+s\). For a requirement \(g(x)\geq b\), a nonnegative slack can represent a shortfall by writing \(g(x)+s\geq b\). In both cases, \(s=0\) means the original requirement is satisfied. A positive \(s\) reports how far the relaxed condition extends beyond the original requirement.
For an equality \(g(x)=b\), use two one-sided deviations if the direction matters: \(g(x)-b\leq s^+\) and \(b-g(x)\leq s^-\), where \(s^+\) and \(s^-\) are nonnegative. Alternatively, an absolute-deviation measure \(s\) can be written as \(-s\leq g(x)-b\leq s\). The latter measures the greatest deviation in either direction, but does not distinguish over-target from under-target.
Slack units must match the underlying constraint. A shortfall of 0.5 percentage point in an average increase is not the same quantity as a 0.5% relative change or a dollar premium shortfall. If several constraints use different units, comparing raw slack values is not meaningful without a stated scaling or priority rule.
Worked Example: An Exact Segment Target Conflicts With Policy Bounds
A hypothetical two-policy segment requires an exact weighted average increase of 3.8 percentage points. Policy A has weight 60% and a maximum permitted increase of 2 points. Policy B has weight 40% and a permitted range from 4 to 6 points. Find the maximum attainable segment average, then calculate how much the target or one policy cap would need to change.
Method. To maximize the weighted average under these upper bounds, set both policies to their respective maximum values. The result is \(0.60(2)+0.40(6)=1.20+2.40=3.60\). Since 3.60 is less than 3.8, the exact target is infeasible.
If the target is allowed to move downward, its closest attainable value is 3.60, a reduction of \(3.80-3.60=0.20\) percentage point. If instead only Policy B's cap is relaxed, let its new cap be \(6+s\). Even with Policy A at 2, meeting the target requires \(0.60(2)+0.40(6+s)=3.8\). Thus \(1.20+2.40+0.40s=3.80\), so \(s=0.50\) point. Policy B would need a cap of 6.50.
For comparison, keep Policy B at 6 and relax Policy A's cap to \(2+r\). Then \(0.60(2+r)+0.40(6)=3.8\), giving \(1.20+0.60r+2.40=3.80\), so \(r=1/3\) point. The three calculations describe different potential policy changes: lower the target by 0.20 point, raise B's cap by 0.50 point, or raise A's cap by one-third of a point. None should be presented as interchangeable without explaining which rule is being changed. Check: each proposed change makes the weighted average \(3.80\), as required.
Comparing Relaxations Without Hiding Their Trade-Offs
When several constraints conflict, assign a separate slack to each candidate relaxation rather than combining them into one unexplained “feasibility” allowance. A diagnostic objective might minimize a weighted sum of slacks, but the weights encode preferences. If a one-point customer-limit violation is assigned a much larger penalty than a one-point segment-target shortfall, the optimizer will tend to use segment slack instead. That is a consequence of the chosen weights, not a neutral finding.
A more transparent comparison reports the alternatives and their implications directly. If a weighted slack objective is used, disclose the slack definitions, units, weights, and resulting values. A lexicographic approach can also be used: first minimize violations of the highest-priority constraint class, then, among those solutions, minimize the next class. This avoids implying that unlike constraints are naturally interchangeable. It still depends on an explicitly approved priority order.
Record each policy, segment, and portfolio constraint with its direction, weight basis, and units before adding any slack.
Use attainable bounds or inspect the mutually inconsistent requirements. An objective change cannot repair an empty feasible set.
Add only the relevant nonnegative slacks, calculate their minimum values, and state the business rule each slack would relax.
Show the original violation, each proposed relaxation, its units, and any priority or penalty assumptions used to choose among them.
Worked Example: How Penalties Choose Which Rule to Relax
Two equally weighted policies have changes \(x_A\) and \(x_B\). Their original constraints are \(0\leq x_A\leq1\) and \(0\leq x_B\leq6\), in percentage points, and the segment average must be at least 4. Introduce \(s_A\geq0\) to relax A's cap and \(s_S\geq0\) to allow a segment shortfall:
For fixed \(s_A\), the greatest average is obtained at \(x_A=1+s_A\) and \(x_B=6\). It equals \((1+s_A+6)/2=3.5+0.5s_A\). Therefore feasibility requires \(s_S\geq\max(0,\,0.5-0.5s_A)\). This relation makes the trade-off explicit: increasing the customer-cap relaxation can reduce the required segment shortfall.
Suppose the diagnostic objective is to minimize \(2s_A+s_S\). For \(0\leq s_A\leq1\), the least permissible segment slack is \(0.5-0.5s_A\), and the objective is \(2s_A+0.5-0.5s_A=0.5+1.5s_A\). It is smallest at \(s_A=0\), giving \(s_S=0.5\) and objective value 0.5. For \(s_A\geq1\), the least \(s_S\) is 0 and the objective is \(2s_A\), which is at least 2. Thus the overall minimum is indeed the first solution.
Now change the objective to \(0.2s_A+s_S\). For \(0\leq s_A\leq1\), it is \(0.2s_A+0.5-0.5s_A=0.5-0.3s_A\), which decreases to 0.2 at \(s_A=1\). For \(s_A\geq1\), the objective is \(0.2s_A\), increasing from 0.2. The new minimum is \(s_A=1,\ s_S=0\). It permits \(x_A=2,\ x_B=6\), whose average is \((2+6)/2=4\). The first objective prefers a segment shortfall; the second prefers a larger customer-cap relaxation. Common mistake avoided: treating the selected slacks as inevitable. They depend on the penalty weights, so the weights and the implied trade-off must be disclosed.
Common Diagnostic Traps
- Changing the objective instead of the constraints. A different profit or contribution objective cannot make incompatible hard constraints feasible.
- Using the wrong weights. Calculate segment and portfolio outcomes with the weights specified for those quantities, not a convenient unweighted average.
- Putting slack on the wrong side. For a minimum requirement, a shortfall slack belongs on the left, as \(g(x)+s\geq b\). Adding it to the required minimum would make the condition stricter, not relax it.
- Adding incomparable slacks without scaling. A dollar premium deviation and a percentage-point rate deviation cannot be compared directly as if one unit of each had equal meaning.
- Calling a relaxed result compliant. A positive slack means the original constraint is violated. It should be labeled and reported as such.
A clear infeasibility report identifies the conflicting requirements, demonstrates why they cannot all hold, and gives the smallest quantified alternatives under stated assumptions. That is more useful than silently dropping a constraint or presenting a relaxed recommendation as though it satisfied the original rules. This approach is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each hypothetical situation, identify the feasibility result or the minimum required relaxation. All rate changes are in percentage points.
- Two portfolio groups have shares 70% and 30%. Their respective maximum increases are 2 and 7, while the portfolio minimum is 4. Calculate the maximum attainable portfolio increase and the minimum shortfall from the required floor. (Answer: Maximum 3.5; minimum portfolio-floor relaxation 0.5 point.)
- A segment requires an average of at least 5. Policy A has weight 25% and a maximum increase of 4; Policy B has weight 75% and a maximum increase of 5. What is the maximum attainable average and the minimum segment shortfall? (Answer: Maximum 4.75; minimum shortfall 0.25 point.)
- Two equally weighted policies must average exactly 5. Their caps are 3 and 6. What is the maximum attainable average, and how much must the first policy's cap increase if the second remains at its cap? (Answer: Maximum 4.5; the first cap must increase by 1 point, to 4.)
- Two equally weighted policies can each take only an increase of 0 or 4. The segment requires an exact average of 3. Is the requirement feasible, and what is the smallest absolute deviation from the target among allowed choices? (Answer: Infeasible; allowed averages are 0, 2, and 4, so the smallest absolute deviation is 1 point.)
- A portfolio floor is \(0.8\) times a segment average plus \(0.2\) times another segment average. The first segment is capped at 3 and the second at 8; the portfolio floor is 4.3. Find the maximum portfolio average and the minimum floor reduction. (Answer: Maximum \(0.8(3)+0.2(8)=4.0\); minimum floor reduction 0.3 point.)