Model and applications

We study a class of two-stage robust optimization problems with the following structure. (P1) min x∈X  c|₁x + max u∈U  min y∈Y c | 2y  (6.1) Subject to : Ax + Bu + Cy ≤ b (6.2)

In the above formulation, x and y are the vector of decision variables in the first and the second stage respectively. u is the vector of uncertain parameters, also referred to as adversarial variables, that are restricted to the uncertainty set U . c1 and c2 are given cost vectors, A,

B and C are known matrices with appropriate dimensions and b denotes the known vector

of right-hand side values. x ∈ X and y ∈ Y represent the integrality and bound constraints that we may have for variables in the first and second stages. Objective (6.1) minimizes the sum of the first- and second-stage costs. In this model, x ∈ X must be selected such that for all realizations u ∈ U there is y ∈ Y such that constraint (6.2) is satisfied. We assume that U is a finite uncertainty set. This assumption is necessary for the convergence proofs of the proposed Benders algorithms discussed in Section 4.1. The first- and second-stage variables can be continuous, integer or mixed, but all adversarial variables are supposed to be integer. We also assume that C is a block-diagonal matrix. The main focus of this research is to exploit this block-diagonal structure and develop algorithms to solve the reformulated problem efficiently. In the rest of this paper we sometimes include constraints x ∈ X , u ∈ U and y ∈ Y together with constraint (6.2). We also point out that in the literature of two- stage robust optimization the inner max(min(.)) problem in the above problem setting is sometimes referred to as the “adversarial problem”.

Let us now explain how Model (P1) can fit a wide range of two-stage robust resource planning problems where a set of tasks denoted by T with uncertain arrivals and durations are expected within a given planning horizon. Let D denote the set of days in the planning horizon. To perform each task, a set of resources is needed. Let R denote the set of all resources. We

suppose that the daily required amount of resource r ∈ R to perform task t ∈ T , denoted by

htr, is known and discrete. Also let Ptdenote the set of all discrete scenarios for the uncertain

arrival and duration of task t and let atp and ltp be the arrival day and the duration of task t in scenario p ∈ Pt respectively. In the first stage, the decision maker must decide on the

amount of resources made available over the planning horizon, paying a daily cost of c1r per unit of resource r ∈ R over that period. In the adversarial problem, considering the first-stage decisions, the worst possible combination of scenarios about tasks is assumed to take place. Because of this worst-case realization, the amount of resource r ∈ R supplied for day d ∈ D in the first stage may not be enough. In this case, in the second stage, the decision maker needs to get some additional amounts of resource r ∈ R on day d to meet the demand level. In the second stage, the daily cost of resource r ∈ R is c2r > c1r and there may be some additional restrictions on the maximum daily amounts of extra resources available.

The above problem can arise in various contexts such as project resource planning, main- tenance planning and workforce planning where the durations of activities and their start times are uncertain. To formulate this problem as an example of Model (P1) we define the following variables and sets.

Variables:

xdr : The amount of resource r supplied on day d in the first stage.

utp : 1 if scenario p ∈ Pt (previously explained) realizes for task t ; 0 otherwise. (adversa-

rial variables)

ydr : The additional amount of resource r supplied on day d in the second stage. Sets:

Ptd : The subset of scenarios in Pt where task t is in process on day d, i.e., Ptd =

{p ∈ Pt: atp ≤ d, atp+ ltp > d} (note that atp and ltp are defined above as the arrival

day and the duration of task t in scenario p ∈ Pt respectively).

The two-stage robust resource planning problem is formulated as follows :

minx∈X P d∈D

P

r∈R

c1rxdr+ maxu miny∈Y P d∈D P r∈R (c2rydr) !! (6.3) Subject to : xdr+ ydr ≥ P t∈T P p∈Ptd htrutp r ∈ R, d ∈ D (6.4) P p∈Pt utp= 1 t ∈ T (6.5) utp ∈ {0, 1} t ∈ T , p ∈ Pt (6.6)

In the above model, constraint (6.4) corresponds to the daily demand constraints over the planning horizon and constraints (6.5) and (6.6) define the uncertainty set. x ∈ X and y ∈ Y enforce non-negativity and integrality constraints of the first- and second-stage variables. In constraint (6.4) for each (r, d) ∈ R×D we have a block in the technology matrix corresponding to vector y.

As an illustrative instance of the problem defined above, we will concentrate in this paper on a nurse planning problem. In this problem, we plan wards’ nurses of a hospital for a medium term. The daily workloads of nurses depend on the number of patients brought from operating rooms to wards. Patients are already scheduled in operating rooms over the planning horizon. Before transferring patients from operating rooms to wards they may stay in ICUs for some days. The lengths of stays in ICUs and wards are uncertain and discrete. For each patient a number of scenarios about the lengths of stays in ICUs and wards are available. In the first stage of this problem, we assign some nurses to wards over the planning horizon. In the second stage if the nurses’ workload on a day is more than the service capacity of nurses assigned to that day, some extra nurses are hired. Nurses hired in the second-stage are paid more than those hired in the first-stage. The problem is formulated as follows :

Parameters:

c1 : The daily cost of a nurse hired in the first stage.

c2 : The daily cost of a nurse hired in the second stage.

Md : The maximum number of nurses available for hiring on day d in the second-stage. δ : The amount of service time provided by a first- or second-stage nurse per day (in

hours).

ρ : The average of required service time for each patient per day (in hours). lICU

tp : The length of stay in ICUs for patient t in scenario p ∈ Pt. lW ard

tp : The length of stay in wards for patient t in scenario p ∈ Pt. d0_t : The surgery day for patient t.

Set:

D : The set of days in the planning horizon.

T : The set of patients (tasks) already scheduled in operating rooms over the planning horizon.

Td : The set of patients scheduled on day d.

Pt : The set of scenarios for patient t. Each scenario gives information on the lengths of

Ptd : The subset of scenarios in Pt where patient t is in wards on day d, i.e., Ptd = n p ∈ Pt : d0t≤ d, d 0 t+ ltpICU + lW ardtp > d o . Variables:

xd : The number of nurses assigned to day d in the first stage.

utp : 1 if patient t follows scenario p after its surgery, 0 otherwise. (adversarial variable) yd : The number of nurses hired on day d in the second stage.

min x P d∈D c1xd+ max_u min_y P d∈D (c2yd) !! (6.7) Subject to : δxd+ δyd≥ ρ P t∈T P p∈Ptd utp d ∈ D (6.8) xd≥ 0, integer d ∈ D (6.9) P p∈Pt utp= 1 t ∈ T (6.10) utp ∈ {0, 1} t ∈ T , p ∈ Pt (6.11) 0 ≤ yd≤ Md, integer d ∈ D (6.12)

Constraint (6.8) is the daily demand constraints over the planning horizon. In this constraint, the reason that we have used coefficient ρ as the average workload of each patient rather than the actual workloads is that we do not know patients’ workloads in advance and therefore we cannot adjust the level of second-stage nurses accordingly in the morning. Constraint (6.10) and (6.11) define the discrete uncertainty set. Constraints (6.9) and (6.12) represent the bounds and integrality constraints for first- and second-stage variables respectively. In the next section, we use the nurse planning problem to illustrate the proposed reformulation. We also present extensive computational results on this problem in Section 6.5.