Che-ming Yang, PhD, MD
Former Vice President of Scheduling and Staff Anesthesiologist
Providence Anesthesiology Associates, PA
Charlotte, North Carolina


Fairness in assignment scheduling is essential for the success of an anesthesiology group practice. Often the assignment schedule directly ties not only to income but also the lifestyle of individual members of the group. The inability to produce a fair assignment schedule is a common source of discord among members of anesthesiology groups. Fortunately, this can be readily solved by adopting a transparent and fair scheduling software.

Assignment schedules should be fair, and the scheduling software should be able to unfailingly distribute every assignment with different value and/or difficulty to eligible members at a predictable frequency. Moreover, scheduling software must be flexible in order to accommodate a host of complex conditions, rules, and restrictions mandated by the group, as well as special requests made by individual members or the group as a whole.

It doesn’t seem difficult to expect that a scheduling program be capable of fulfilling such seemingly obvious requirements. That being said, there is hardly any scheduling software available that can achieve these results. A common theme of scheduling programs is that statistics get out of whack over time, and the solution as recommended by the developers is the statistics should be reset to zero. Such a solution only perpetuates the problem. In this article, the root cause of such statistical problems is delineated and a solution is presented.

Another common deficiency in scheduling programs is the schedule that is generated often contains holes. A hole in a schedule means there is an assignment that no eligible member can do that day, because all the members eligible to do that assignment have been given other assignments; and, at the same time, a member is left with no assignment to do, because all the assignments that member is eligible to do have been delegated to other members. This situation, which should never happen, is undoubtedly a result of a poor scheduling strategy. A new strategy based on a totally different approach is introduced that effectively fixes the problem.

A Fair Schedule: Accurate Distribution of Assignment Probabilities

As previously noted, a fair assignment schedule is one in which every member has his or her fair and predictable share of eligible assignments over a period of time. To reach this goal, one must first figure out what the fair share is for each assignment for each member, and use it as the target in scheduling. In other words, the target needs to be defined first, because the scheduling algorithm is nothing more than a strategy developed to ensure the defined target is followed.

To illustrate, let’s assume one assignment is shared equally by 10 doctors (D1 to D0). Then, each doctor’s fair share is 0.1, which means over 100 days each doctor should be assigned exactly 10 times to this assignment. Or each doctor has a 0.1 probability of being assigned to this assignment each day.

Now, let’s look at a group practice setting where we have 10 doctors and 10 assignments (A1 to A0) to be filled. If every doctor does every assignment and does it equally, then every doctor’s fair share, or probability to be assigned to each assignment each day, is 0.100, as illustrated in Table 1.


Here, the assignment distribution is defined. The resulting schedule will show that over a period of 100 days, each doctor will do each assignment 10 times, and each assignment will be assigned to each doctor 10 times as well. The key in this example is not so much that each assignment is equally shared by every doctor, but that the probabilities for each doctor to do each assignment are correctly defined every day (in this case, they are all 0.1), or that the target is correctly defined for every doctor and every assignment. With a correctly defined target, the scheduling is feasible. It will become clear later that the requirement for correctly defined probabilities is the summation of probability for each column, and each row is exactly 1.0 in distribution Table 1. These constituent numbers can thus be treated as probability distributions for each assignment (in the column direction) as well as for each doctor (in the row direction).

Now, in a more realistic group setting, every doctor does not do every assignment and each assignment may not be equally shared among doctors (Table 2). The number in each cell represents the probability ratio among doctors, or how each assignment is distributed among doctors (the probability distribution being in the column direction). Note that these numbers do not represent the probability ratios in the row direction.


In this setting, how to distribute the probabilities among eligible assignments for each doctor (the probability distribution in the row direction) becomes a challenge. Most of the assignments can still be distributed following the defined probability ratios among eligible doctors regardless of how other assignments are distributed among eligible doctors. These assignments are named prospective assignments. It would be easy if all the assignments are distributed independently as if they are all prospective. However, in most group situations, not all assignments can be prospective unless the group agrees that everyone does everything equally, as illustrated in Table 1. It is not possible to treat every assignment as prospective because if all assignments are prospective, it will be unlikely that the summation of all assignment probabilities for each doctor(the row summation) will equal 1.0.

This is shown in the last column of Table 3, where the probability summations in the row direction are not all 1.0. In this example, doctor D1 will be assigned, out of 100 days, 10 times to assignment A1, 9.4 times to A2, 11.5 times to A3, etc, for a total of 103.6 assignments in 100 days, as indicated by D1’s total assignment probability of 1.036 in the last column. Since doctor D1 is expected to do 103.6 assignments in 100 days as the set target, he or she is always behind in statistics for some or all assignments. And this statistical derangement will be amplified over time, which eventually will necessitate a reset of statistics.


Obviously, fair scheduling is not possible with such an incorrect probability distribution as the target. In other words, even though these numbers are correct probability distributions for each assignment (in the column direction, all summations are 1.0), they cannot be viewed as correct probability distributions for each doctor (in the row direction, the summations are not all 1.0) unless the row summation is exactly 1.0. This example demonstrates that without a correct probability distribution to define the correct target, the correct scheduling will not be possible and the schedule thus generated will not be fair.

To fulfill the requirement that every doctor’s probability summation (the last column of Table 3) is exactly 1.0, some assignments need to be distributed differently. Such assignment’s probability distribution will depend on how much probability is already occupied by the combined probabilities of prospective assignments for each doctor. This is illustrated in Table 4, where only assignments A1 through A7 are prospective. The probabilities of assignments A8, A9, and A0 are strategically distributed among doctors (in the column direction) so the row summation of probabilities for each doctor is exactly 1.0, are shown in the last column of Table 4.


In addition to the requirement that every row summation of probabilities equals 1.0, the simultaneous requirement that every column summation of probabilities equals 1.0 for every assignment is also fulfilled, as shown in the last row of Table 4. Assignments like A8, A9, and A0, for which the probability distributions depend on each doctor’s prospective assignment probabilities, are called retrospective assignments. So essentially, the retrospective assignment probability is a “make-up” or “catch-all” probability for each doctor. As illustrated in Table 4, doctor D3 has a high total prospective probability (sum of A1 through A7 = 0.850), which means D3 will spend a lot of time in prospective assignments. As a result, D3 can only have limited time for retrospective assignments as indicated by D3’s relatively small combined retrospective probability (sum of A8, A9, and A0 = 0.150).

The importance of having the correct distribution of assignment probabilities for each doctor, through the aid of retrospective assignment probabilities, cannot be overemphasized in searching for a fair schedule. Now that the correct assignment probability distribution is in place, a correct target for scheduling is thus defined and the correct scheduling process now becomes possible, as is demonstrated later.

The probability for each assignment, generated daily, can be viewed as an earned “credit” (for being available to do that assignment), which can be saved and used toward acquiring that very assignment. By treating the probability as a credit, it is easier to accept the idea that this number can be used in calculation. So by being available to work, the doctor earns a certain credit (equals the probability) for each assignment he or she is eligible to do. In addition, the total credit each doctor received (or earned) for being available to work is 1.0 for the day (because the row summation is 1.0).

This earned credit is cumulative. When the doctor is given a certain assignment, his or her accumulated credit for that assignment is reduced by 1.0, as the doctor redeems the accumulated credit for that assignment. Earning credit puts the doctor more behind or less ahead (thus a greater negative or less positive number) in that assignment and redeeming the accumulated credit for the respective assignment makes the doctor less behind or more ahead (thus a less negative or more positive number) in that assignment. For example, as illustrated in Table 5, if a doctor has accumulated credit of –0.1 for an assignment (ie, behind by 0.1 for that assignment) and is assigned to that assignment that day, the resulting accumulated credit for that assignment will be 0.9 (–0.1+1.0=0.9, so ahead by 0.9 for that assignment). Since each doctor receives a combined total of 1.0 credit for all the assignments he or she is eligible to do (the row direction) and redeems 1.0 of credit for an assignment, the net total accumulated credit for all eligible assignments (the row direction) always remains zero for each doctor. By the same token, the net total combined accumulated credit among all eligible doctors (the column direction) always remains zero for each assignment as well.


This is also illustrated in Table 5. The group issues a total of 1.0 credit to eligible doctors for each assignment (the column direction in Table 4), and receives back 1.0 of credit from the doctor who is given that assignment (like D9 to A1, D3 to A2, D1 to A3, etc, in Table 5), resulting in a net of 0.0 in the column summation. The accumulated credit (which is actually the residual credit because it is the amount after redemption or reception) thus derived can be viewed as the assignment statistics because it represents how much the doctor is ahead (positive number) or behind (negative number) in the respective assignment he or she is eligible to do.

The fact that the summation of all the residual credits or assignment statistics for each doctor is always zero (called the zero sum rule) is very helpful for scheduling. The zero sum rule applies to every doctor (in the row direction) and every assignment (in the column direction) simultaneously every day. So if a doctor is ahead in some assignments, he or she must be behind in some other assignments with the same total magnitude. At the same time, if there are some doctors who are ahead in an assignment, there must be some other doctors who are behind in this assignment with the same total magnitude, regardless of the prospective or retrospective assignment.

In other words, all the residual credits or assignment statistics for a doctor should always “scatter” around the zero baseline. By giving a different assignment to the doctor, it will alter the scattering pattern or change the assignment statistics distribution. For example, refer to Table 5: If doctor D9 is assigned to A8 instead of A1, the resulting distribution of assignment statistics will be (–0.100, –0.094, –0.115, –0.091, –0.143, –0.125, –0.125, 0.932, –0.139, 0.000), which is different from the assignment statistics distribution for D9 in Table 5. The resulting statistics distribution has a different scattering pattern. The distribution pattern of assignment statistics for a doctor can thus be a gauge of how good an assignment is given to that doctor that day.

Generally speaking, an assignment, if appropriately delegated to a doctor, would make the doctor’s distribution pattern of assignment statistics closer to the zero baseline (or less scattering around the zero baseline). Thus, this defines the target or the objective of the scheduling process—to minimize the overall scattering of assignment statistics for all doctors and assignments as a whole.

A Flexible Schedule: A Powerful Scheduling Strategy

Now that the target of a fair schedule is defined, the next thing to do is to find a strategy to reach that target. As mentioned earlier, the schedule made must fulfill the requirements of all the conditions, restrictions, and rules agreed upon by the group and special requests made by the group and doctors from the group. A flexible scheduling software is one that can accommodate all those conditions and requests. To that end, the best way to make a schedule (and often the only way, as noted below) is to do it day by day as opposed to assignment by assignment over a predetermined time period. That is, the software has to finish assigning all the assignments of the day, and update the complete assignment statistics before proceeding to make the next day’s schedule.

The reason why the schedule has to be made day by day is there are always some conditions to be fulfilled that call for all the previous day’s assignment information, and assignment-by-assignment scheduling cannot guarantee that such information is already available. Besides, assignment statistics need to be updated daily so the target can be refreshed and be ready for use in searching for the next day’s schedule.

To make an assignment schedule for the day, the strategy is to evaluate all possible assignment permutations of the day (eg, there are 6 possible permutations for a group of 3 doctors to fill 3 assignments), exclude permutations that violate any conditions (restrictions, rules, or special requests), and select the permutation that makes the assignment distribution profile closest to the target for the day or that minimizes the overall scattering pattern of the assignment statistics of the day. The resulting best permutation is likely the one that gives most doctors their respective assignments who are most behind in statistics.

Exploring all the possible permutations of the day makes it more likely than any other strategy to reach a schedule that accommodates all the imposed conditions, restrictions, rules, and special requests. In fact, unless the valid schedule or permutation doesn’t exist under the imposed conditions (eg, there is one in-house call assignment to be filled but all the doctors are either off after call or requesting no calls on that day), a schedule can always be discovered with this strategy. With this understanding, the above outlined strategy is undoubtedly the most powerful and flexible one in assignment scheduling.

Statistics Tracking: An Example

Scheduling statistics reveal not only how each doctor’s assignments are distributed over a certain time period, but also how each assignment is distributed among doctors in the group. To illustrate this, let’s use the probabilities distribution table (or the time allocation table) depicted in Table 4. Note that Table 4 is only for weekday assignments. (Treatment for the weekend assignments is simpler and will not be dealt with in this article.)

Table 6 shows the resulting weekday assignment schedule for January 2018. (This scheduling software automatically recognizes that January 1, 2018, is Monday and is a holiday by default, therefore no assignment is assigned, or on weekends.) Table 7 shows the assignment statistics at month’s end (January 31, 2018). Note that for every assignment, whether prospective or retrospective, statistics center around zero, indicating that all assignments are neither ahead nor behind for any doctor.


Table 8 shows the number of times each assignment is delegated to each doctor during the month. The ratios of these numbers closely follow the probability distribution depicted in Table 4 for every assignment, regardless of the prospective or retrospective one. As predicted in Table 4, the retrospective assignments (A8, A9, and A0) are distributed more “unevenly.” Since doctors D4 and D8 do prospective assignments less frequently, they are given retrospective assignments more often as expected.


To demonstrate that the integrity of assignment statistics is well maintained over a longer time period, the scheduling process is carried out for all of 2018, and the resulting statistics are summarized in Tables 9 and 10. Table 9 shows the zero sum rule is preserved after a year of scheduling and the assignment statistics are still tightly embracing the zero baseline for every assignment, indicating that no assignment is ahead or behind for any doctor. Table 10 shows the number of times each assignment is delegated to each doctor. The distribution ratios of these numbers clearly follow the assignment distribution probabilities depicted in Table 4.


Conclusion and Comments

A truly fair and flexible assignment schedule that accommodates all conditions, rules, restrictions, and special requests is ideal for any anesthesiology group. Such a schedule can only be generated by giving all assignments simultaneously, day by day. The strategy is to evaluate all possible permutations of the doctor-by-assignment combinations each day, and select the one that fulfills all requirements and most closely follows the fair assignment distribution target. With the aid of retrospective assignments, a true assignment probability distribution is generated daily (as in Table 4). Such a probability distribution, through which the target of scheduling is derived, is ultimately the key in the scheduling process. By evaluating all possible permutations daily, this strategy ensures each doctor gets his or her fairest possible share of each assignment, regardless of the prospective or retrospective one.

It is worth reiterating that the summations of assignment statistics always remain zero for every doctor and every assignment (the zero sum rule). This rule allows for a new doctor being added to the group or an established doctor retiring at any time without causing appreciable disturbance to the rest of the scheduling statistic. Furthermore, a doctor will not be overwhelmed with “banked” assignments (such as call jobs) after an extended vacation because the doctor does not accrue any credit for any assignment when he or she is not available to work. This convenience in scheduling is a result of having true assignment probability distributions.

Besides being the key in scheduling, the assignment probability distributions can be used to shed some light on other important information for the group. An example is an earning power analysis for group doctors. The earning potential of each doctor depends on which assignment he or she is given. By having a higher probability of being delegated high-value assignments, the doctor will have greater earning power. The earning potential for each doctor is mathematically simply the product of the doctor’s assignment probability distribution and the respective assignment values. This analysis can be applied to evaluate quality of assignments, as long as the quality can be quantified. For example, the assignment value can be related to the degree of assignment difficulty, associated work hours, predictability of early off or next day off, etc, in addition to the financial compensation for the assignment. This analysis tool can be further used to even out the earning potential among doctors, by altering the probability distribution profiles (through adjusting the ratios in Table 2). Without a doubt, tremendous useful information can be derived from the assignment probability distribution data.

The importance of having a correct probability distribution in scheduling cannot be overemphasized. If the probability distribution is not correctly defined (as in Table 3), the error in assignment statistics will accumulate over time due to violation of the zero sum rule. The quick runaway error in assignment statistics necessitates a frequent re-zeroing of all or part of the assignment statistics so scheduling can proceed. By manipulating statistics, the schedule generated is undeniably incorrect and unfair. Unfortunately, this is the most common defect in other scheduling programs (ie, not using a correct assignment probability distribution).

Another common difficulty in other scheduling programs is that the more conditions (rules, restrictions, and special requests) are imposed on the schedule, the harder it is for scheduling, which frequently appears as “holes” in the resulting schedule. It is worth mentioning that by adopting the strategy of evaluating all possible permutations, the more strenuous the conditions are, the easier it is in scheduling, simply because the strenuous conditions effectively limit the number of qualified permutations that require further evaluation.

Ironically, the fewer the number of qualified permutations, the less likely they will be discovered by schedulers who give assignment by assignment over a predetermined time period. In the search for the best doctor to fill a sequence of assignments one by one, an assignment in the middle of that sequence may have no eligible doctor available because all eligible doctors have been delegated to assignments earlier in the sequence. This situation is more likely to occur if the imposed conditions are strenuous. With an assignment that no eligible doctor can do, a hole will appear in the generated schedule. To patch up the holes, doctors will need to switch assignments, thereby ignoring statistics and bending rules. This manipulation undoubtedly perpetuates unfairness and lowers the quality of schedules.

This article is an introduction to a truly fair and flexible scheduling software. The scheduling program utilizing the strategy presented in this article has been devised and thoroughly tested for over 20 consecutive years in our group, with highly strenuous and demanding rules and restrictions imposed. The concept and strategy worked well and as anticipated. No apparent deviation from statistics has ever occurred and no re-zeroing of assignment statistics has ever taken place, and no hole in the schedule has occurred.