different surgeon, the cleaning is again obligatory and it should be entirely performed in the slot of the infected patient.
5.3
Complexity analysis
In this section we will prove that the optimization of the SCSP is compu- tationally hard, i.e. NP-hard, by showing that it contains a problem, for which the optimization is already shown to be NP-hard, as a special case. This technique is referred to as a proof by restriction [105]. In particular, we will specify restrictions so that the restricted SCSP, which we will refer to as R-SCSP, is identical to the resource investment problem (RIP). The RIP is situated in the domain of the resource-constrained project scheduling prob- lems (RCPSP) and the optimization is shown in Neumann, Schwindt and Zimmerman [199] to be NP-hard. We may summarize the characteristics of the RIP as follows:
• PROBLEM: Resource investment problem (RIP)
• INSTANCE: A set of precedence-related activities that constitute a project. The project has to be finished before the project deadline. Each activity consumes resources during each period of its execution according to a particular resource consumption pattern. Each resource has a limited availability at each time instance p.
• GOAL: M IN P
ocosto·maxp consumptionop: minimize the costs that
are associated with the peak use of each resource during the course of a project by determining the activity starting times. In the ex- pression, costo denotes the procurement cost per unit of resource o
and consumptionopdenotes the number of units of resource o that are
needed in period p.
Theorem 1. Problem SCSP is NP-hard.
Proof of Theorem 1. In the R-SCSP, we only take objective 5 and 6 into account, i.e. minimizing the peak number of bed spaces used in PACU 1 and
PACU 2. We furthermore restrict the focus to a single surgery slot s and do not incorporate constraints concerning the medical equipment, incomplete pre-surgical tests or MRSA.
We cannot straightforwardly identify the RIP in the R-SCSP as there is a problem with the activity representation. We cannot define an activity for the RIP to be equal to an entire surgical process of a patient since this process is actually a sequence of three distinct activities. First, there is the surgery itself, which takes place in the operating room. Second, a recov- ery process is initiated in PACU 1. Finally, the patient is transferred for a second recovery process to PACU 2. The last two activities, though, con- sume resources when the surgery itself is already finished. This feature is not typical for the RIP and some modifications should hence be introduced. Instead of scheduling one activity that contains 3 processes (surgery, PACU 1 and PACU 2), we will schedule 3 precedence related (fictive) activities, namely n0, n00 and n000 in such a way that each activity now represents only one process. This substitution is depicted in Figure 5.5. In this figure, an activity-on-the-node representation is introduced. The duration of the activity is indicated above the node, whereas the resource consumption is indicated below using a vector. Only three resource types are represented in the R-SCSP, i.e. the operating room (o = 1), beds of PACU 1 (o = 2) and beds of PACU 2 (o = 3). The consumption of these resources by each activity is indicated in the respective entries of the vector: −→resn00= (0, 1, 0), for instance, denotes that only one resource is seized, namely a bed in PACU 1, when activity n00 is performed. The minimal and maximal zero time lags (F SM IN = 0 and F SM AX = 0) between the activities n0− n00 and n00− n000
in Figure 5.5 indicate that no time is allowed between the completion of the former and the start of the latter activity.
The equivalence between the RIP and the R-SCSP should now become trans- parent. We still have to introduce some modifications in order to complete the activity-on-the-node representation of the RIP. We have to define, for instance, a dummy start and a dummy end activity and add a F SM IN = 0
5.3. Complexity analysis
Figure 5.5: Representing a surgical process as a sequence of its constituent ac- tivities.
activity n0 of a substituting sequence and between the last activity n000 of a sequence and the dummy end activity. Moreover, a F FM AX = Msub−Mslb+1
precedence relation needs to be specified between the dummy start node and each activity n0 that represents a surgery in order to capture the project deadline. Recall from Section 5.2.2 that Mslb represents the starting period of slot s and that Msub equals the ending period of slot s. Note that, based on the precedence relation, the workload that has to be sequenced in a slot is equal to the capacity of the slot. This, however, does not imply that idle time cannot be incorporated as this is easily done by introducing a surgery that does not consume any resource, except for the operating room. The dummy start activity is completed at time p = Mslb. Since the surgical act inevitably needs an operating room to be performed in and the capacity of this resource is limited to 1 in the R-SCSP, we do not take the leveling of this resource into account (best value equals worst value in this case). Both the peak number of bed spaces in PACU 1 (maxp consumption2p = α5)
and PACU 2 (maxp consumption3p = α6), on the contrary, have to be
minimized. The procurement cost related to these resources is equal to cost2 = w5/(worstvalue5 − bestvalue5) for the use of one bed space in
PACU 1 and equal to cost3= w6/(worstvalue6− bestvalue6) for the use of
one bed space in PACU 2.
⇒ Assume that we have a solution to the RIP, i.e. we know for each patient n the start times vn0, vn00 and vn000 of the constituent activities, then we can construct a solution for the R-SCSP as follows: ∀n in the patient population: the surgery of patient n in slot s starts on period vn0.
⇐ Given a solution to the R-SCSP, we can construct a solution for the RIP as follows: ∀n in the patient population, we know that the surgery starts in
slot s on period vn ⇒ vn0 = vn, vn00 = vn+ surgery duration of patient n and vn000 = vn0+ stay of patient n in PACU 1.