Objectives

The surgical case sequencing problem at hand (SCSP) maximally comprises 6 objectives (|J | ≤ 6) that have to be optimized simultaneously. As such, we need to combine them into a multi-objective function that well balances their importance.

5.2.1.1 Description of the objectives

A first objective concerns the surgery scheduling of children (age ≤ 5 years). For medical reasons, patients need to be sober when the surgery is per- formed. Contrary to adults, children cannot easily cope with this obligation

5.2. Problem statement

Figure 5.3: Surgery sequence to clarify the calculation of α1(children as early as

possible)

and the lack of food can cause parents, surgeons, nurses or patients a lot of annoyance. Therefore, it is desirable to schedule these surgeries as early as possible. In particular, we want to minimize the sum of the starting times of surgeries performed on children. This objective is represented by the variable α1 and is expressed in periods. Figure 5.3 clarifies the result-

ing value for α1 based on the surgery sequence. When we assume that the

underlined surgeries represent children, the value for α1 is determined as

α1 = 2 + 9 = 11. Note that a switch of surgery B and surgery C would

decrease the objective value to α1 = 2 + 6 = 8 and hence result in a superior

sequence (at least for this objective).

The second objective is very similar to the first one, though this time we are concerned about prioritized patients. This category represents, for in- stance, patients who already had a canceled surgery once or surgeries that the surgeon preferably performs in the beginning of the slot. Similarly to the children, we want the surgeries of the prioritized patients to be sched- uled as early as possible, i.e. we want to minimize the sum of their starting times. We distinguish between the children and the prioritized patients as the weight that is assigned to the objectives can be different (see Section 5.2.1.2). The value for this objective is represented by the variable α2 and

is expressed in periods.

Third, we want to incorporate the travel distance between the patient’s residence and the day-care center while constructing the surgery schedule. Although the day-care center of the UZ Leuven Campus Gasthuisberg is centrally positioned in Belgium, it is possible that patients have to travel

Figure 5.4: Surgery sequence to clarify the calculation of α4 (minimization of

recovery overtime)

over 150 kilometers. The aim is to schedule these patients from a certain reference period (e.g. 11 a.m.) on. Penalty costs are incurred for patients whose surgery starts before the reference period. This cost is represented by the variable α3 and equals the number of patients with a surgery starting

time that precedes the reference period. The relevance of this objective is twofold. On the one hand, there is an increase in patient satisfaction if the patient’s effort to get in time to the day-care center is not too large. On the other hand, providing more time to enter the hospital reduces the probabil- ity to arrive late due to traffic uncertainty.

Fourth, we want to minimize the stay in recovery after the closure of the day-care center at 7 p.m., as this results in unplanned (and hence costly) hospitalizations or overtime for the nursing personnel. In particular, we minimize the number of periods in which recovery care has to be provided after closing time (=α4). Figure 5.4 clarifies the calculation of this variable.

In the example, the day-care center closes after period 12. However, the cur- rent surgery sequence results in a stay of patient B in PACU 2 from period 13 to period 15, so that α4= 3. When the closing time is advanced by one

period, the recovery overtime would increase with one more period, namely a fraction of the stay of patient B in PACU 1 (α4= 1 + 3 = 4).

Finally, we are interested in minimizing the peak number of bed spaces used in PACU 1 (=α5) and PACU 2 (=α6) in order to level the bed occupancy

5.2. Problem statement

already pointed at the different degree of monitoring between both recovery phases. Note that the leveling furthermore protects the flow of patients from the operating room to the PACU in case of schedule deviations as spare recovery capacity is more likely to be available. This should reduce the probability of bed blocking, in which the patients stays in the operating room until a recovery bed is idle. It should be clear that bed blocking is very expensive and hence has to be avoided. In practice, solutions to bed blocking are found in a quick transfer of patients from PACU 1 to PACU 2 when capacity is lacking in PACU 1, although this clearly decreases the degree of patient satisfaction and service quality (see Section 6.3.4). When we return to the example sequence of Figure 5.4, it should be clear that the peak use of recovery bed spaces in PACU 1 equals the peak of PACU 2, namely one single bed space (α5 = α6= 1).

5.2.1.2 Towards a multi-objective function

Intuitively, it seems necessary to take multiple objectives into account. When we would optimize the surgery schedule for one single objective, it is very likely that the schedule performs poorly with regard to some other objective. Question is, however, how we should combine the objectives into a well-balanced multi-objective function. One approach is to sum the values for each objective, i.e. P

j∈Jαj. However, since the objectives are expressed

in various units with a different granularity, this formula is not adequate. Moreover, it does not incorporate a notification of the (varying) importance that the scheduler assigns to each objective. One solution exists in the in- troduction of a weight wj for each objective j, determined by the human

scheduler, so that the multi-objective function is transformed toP

j∈Jwjαj.

The weights now incorporate both a trade-off between the units and an indi- cation of the objective’s importance. Setting these type of weights manually, however, is a very subjective decision, even for an experienced planner, and is difficult to argument. What is the trade-off between a period and a bed space? As such, the scheduling process may result in schedules that are not very favorable for the decision maker due to the mismanagement of the weights.

We suggest to combine the objectives as represented in Expression (5.1). Recall from Section 5.2.1.1 that α1 (α2) equals the sum of the surgery start-

ing times of children (prioritized patients), α3 equals the number of travel

patients that is scheduled before the reference period, α4 equals the total

amount of recovery overtime (expressed in periods) and α5 (α6) equals the

peak number of bed spaces used in PACU 1 (PACU 2). It should be clear that the expression has to be minimized in order to find an optimized surgery schedule. X j∈J wj· αj− bestvaluej worstvaluej− bestvaluej ! (5.1)

Expression (5.1) proposes a type of normalized objective function that orig- inates from the field of multiple criteria decision making (e.g. [209]). Since the patient population is known, we should be able to calculate for each single objective j, i.e. leaving all other objectives out of consideration, its best value (bestvaluej) and its worst value (worstvaluej). In other words,

each feasible schedule features values for αj that satisfy bestvaluej ≤ αj ≤

worstvaluej, which is easy to interpret. These extreme values are consecu-

tively used as indicated in Expression 5.1 to generate a relative measure of quality, i.e. the transformation discards the different units of the objectives. One could argue why we do not divide αj by bestvaluej and optimize this

kind of transformation as it would be a relative measure too. However, since the best value for an objective j possibly equals 0, this would result in a division by 0. One could argue again that the denominator of Expression 5.1 also equals 0 when bestvaluej equals worstvaluej. Then, however, we do

not take the optimization of objective j into account as it implies that the value of objective j is optimal for every feasible schedule that is obtained. This also implies that the set of objectives |J | for a particular surgery day not always equals 6 (0 ≤ |J | ≤ 6). The number of objectives that are even- tually incorporated in the problem setting depends on the constitution of the patient population of the specific day.

5.2. Problem statement

The stabilizing transformation or normalization ensures that all objectives j ∈ J will be gradually optimized to the same extent and that they will somehow be comparable to each other. It is unlikely, however, that the objectives are of equal importance to the human planner. Thus, we also incorporate a differentiator by assigning a weight wj to objective j. Note

that the weights now only indicate the preferences of the scheduler. When the sum of the weights equals 1, the multiple objective function has a value that is in the range [0, 1]. A value equal to 1 denotes that each αj is equal to

its worstvaluej, whereas a function value of 0 indicates that ∀j ∈ J : αj =

bestvaluej. We refer to Section 5.6.2 for a discussion on the calculation of

the extreme values.