Staff Rostering Problem: Choice of a mathematical model ´ Edith Naudin, Peter Chan, Michael Hiroux, Georges Weil, Tahar Zemmouri Equitime, 1 all´ee de Cert`eze, 38610 Gi`eres, France, {enaudin,pchan,mhiroux,tzemmouri,gweil}@equitime.fr

1

Introduction

Since the first use of the Dantzig-Wolfe decomposition ([2]) by Gilmore and Gomory ([5]) on the cutting stock problem, models based on column generation became more and more popular. In this context, we propose to compare the computational behavior of several mathematical models for the same problem: the staff rostering problem. The first model, considered to be the most conventional, uses decision variables containing the least information. The other models are obtained through a Dantzig-Wolfe decomposition of the first. The variables being more complex, these models often require column generation to be solved (see [1], [4] and [6]). The staff rostering problem solved here consists of assigning workers to tasks whose starting and ending times are fixed, and subject to constraints defined over different horizons. The first model uses assignment variables; the other models use variables representing rosters with different horizons. Let I be the set of workers to be assigned over the set of tasks T , defined over a set of consecutive days D = {1, . . . , |D|}. Each task t ∈ T is characterised by: • a day dayt ∈ D • a starting time bt ∈ IN • an ending time et ∈ IN • a duration dt = et − bt ∈ IN∗ • a staffing requirement rt ∈ IN∗ • a positive cost in case of understaffing: ct ≥ 0 per person missing A solution to this problem must respect the following constraints: The staffing requirement of each task is to be covered as completly as possible The set of assignments of each worker must verify: • At all instants, each worker is assigned to at most one task;

(1) (2)

• Two consecutive working days are separated by a rest period of which the minimal duration is DayR. (3) • The working time per day is limited to a maximum of DayW ;

(4)

• The total working time is limited to a maximum of W eekW ;

(5)

As we can see, the problem has constraints expressed over different horizons: • Short-horizon constraints applied over a task: (1); • Medium-horizon constraints applied over one day: (2) and (4); • Long-horizon constraints applied over several days being scheduled: (3) and (5).

2

The Models

2.1

First Model: Short-term variables

The first model uses the following boolean assignment variables: ∀i ∈ I, ∀t ∈ T, xi,t ∈ {0; 1}, xi,t = 1 if the person i is assigned to the task t, otherwise, 0 . The following variables are used: ∀t ∈ T, st ∈ IN is the understaffing of the task t. We define the following sets: • IN C0 (T ) ⊆ T × T the list of pairs of incompatible tasks for the constraint (2): IN C0 (T ) = {(t1 , t2 ), t1 , t2 ∈ T, dayt1 = dayt2 , bt1 ≤ bt2 and bt2 < et1 } • IN C1 (T ) ⊆ T × T the list of pairs of incompatible tasks for the constraint (3): IN C1 (T ) = {(t1 , t2 ), t1 , t2 ∈ T, dayt1 + 1 = dayt2 , bt2 < et1 + DayR} • ∀d ∈ D, LDd ⊆ T the list of tasks on each day d: ∀d ∈ D, LDd = {t ∈ T, dayt = d}  X  min ct .st     t∈T     s.t. X      xi,t + st ≥ rt ,     i∈I     xi,t1 + xi,t2 ≤ 1, 

(P )

                      

∀t ∈ T

(1)

∀i ∈ I, ∀(t1 , t2 ) ∈ IN C0 (T ) (2) xX ∀i ∈ I, ∀(t1 , t2 ) ∈ IN C1 (T ) (3) i,t1 + xi,t2 ≤ 1, dt .xi,t ≤ DayW, ∀i ∈ I, ∀d ∈ D (4)

t∈LD X d

dt .xi,t ≤ W eekW,

∀i ∈ I

(5)

t∈T

xi,t ∈ {0; 1}, ∀i ∈ I, ∀t ∈ T st ∈ IN, ∀t ∈ T The set of constraints (1) computes the understaffing st of tasks. Each task t induces a linear cost ct when it is understaffed. (P ) computes a planning which satisfy the sets of contraints (2) - (5) and which minimize the weighted understaffing of tasks.

2.2 2.2.1

Other Models Second model: mid-term variables

The second model can be obtained by applying a Dantzig-Wolfe decomposition on the first model. It uses assignment variables that represent valid daily rosters. The resource constraints over workers (2) and the limited daily working hours constraints (4) are respected implicitly by the variables. The other problem constraints are handled as linear constraints in the master problem (MP). Given the huge number of possible valid daily rosters, the resolution of this model relies on the column generation technique. For each day d ∈ D, the sub problem is a constrained shortest path problem in a graph Gd where the nodes represent the tasks of the day and the arcs link compatible tasks: Gd = (Nd , Ad ), where Nd = LDd } and Ad = IN C0 (LDd ) = {(t1 , t2 ), t1 , t2 ∈ LDd , et1 ≤ bt2 }. The resource constraint (2) is expressed by the graph construction . The daily maximum work time constraint (4) is processed like a constraint of capacity in the

constrained shortest path problem and considered during the construction of rosters. To find rosters satisfying the constraints (4) in the graphs Gd , a dynamic programming algorithm is used. This form of decomposition is interesting when the constraints which are transferred in the variables are so difficult to put into a linear form in the first model that one can gain in the quality of the lower bound obtained from a linear relaxation. For instance, the constraint (2), expressed with the inequality xi,t1 +xi,t2 ≤ 1 in (P ) is a linear form of the following quadratic constraint: xi,t1 .xi,t2 = 0. The quadratic form gives a better lower bound, but is not easy to use: with the linear form, we can use the simplex algorithm to compute the lower bound. The lost of quality with the linear form can be recovered by using the second model: each roster verify the constraint (2). Unfortunately, this model has a cost: the sub-problem (SP) for column generation is NP-hard (Dror, [3]). 2.2.2

Third model: long term variables

The third model - also obtained by a Dantzig-Wolfe decomposition - uses variables that represent valid rosters over the whole horizon. The master problem is a set covering problem with the staffing requirements (1). The other constraints are expressed by the validity of rosters. Here the sub-problem for column generation is a constrained shortest path problem in a graph G where the nodes represent the tasks over the whole horizon and the arcs join compatible tasks: G = (N, A), where N = T and A = IN C0 (T ) ∪ IN C1 (T ) = {(t1 , t2 ), t1 , t2 ∈ T, et1 ≤ bt2 } \ {(t1 , t2 ), t1 , t2 ∈ T, dayt1 + 1 = dayt2 , bt2 < et1 + DayR}. The resource constraints (2) and the minimum daily rest duration constraint (3) are satisfied by construction of the graph. The other constraints (maximum daily working time (4) and over the whole horizon (5)) are handled as constraints of capacity and considered during the column generation. The following table resumes the processing of each constraint in the three models: Contrainte Model 1 Model 2 Model 3 Staffing requirements (1) Master Problem Master Problem Master Problem Resource Constraint (2) Master Problem SP: Graph SP: Graph Daily rest duration (3) Master Problem Master Problem SP: Graph Daily work duration (4) Master Problem SP: Capacity SP: Capacity Total work duration (5) Master Problem Master Problem SP: Capacity In our presentation, we will compare the computational behavior of these models in different scheduling scenarios.

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