Abstract: In this paper we formulate a mathematical programming model for solving a class of man- power scheduling problems. The objective is to assign ...
ZOR - Methods and Models of Operations Research (1992) 36:93- 105
A Class of Manpower Scheduling Problems By R. Cerulli 1, M. Gaudioso 2, and R. Mautone 1
Abstract: In this paper we formulate a mathematical programming model for solving a class of manpower scheduling problems. The objective is to assign working schedules to a fixed number of employees in order to meet the workforce demand, assumed to be constant over the planning period. Necessary and sufficient conditions of existence of the solution are stated, heuristic methods are presented and the results of computational experiences are reported.
1 Introduction
A relevant number of algorithms have been developed in the aim of supporting managerial decisions in the area of workforce scheduling. Feasible schedules are those satisfying the workforce demand over the time, while meeting the legal constraints on the employment. Manpower scheduling algorithms are particularly needed whenever the systems are in operation 24 hours a day, seven days a week and backlogging operations is not allowed. An extensive survey on this class of problems may be found in [8]. In this paper we describe some methods for assigning the working schedules to a fixed number of employees in presence of workforce demand assumed to be constant over the time. In such problems it often happens that feasible individual working schedules are characterized by consecutive working days and this fact results in formal representations involving circular matrices. The related properties have been investigated in [2], [9]. In general it is assumed that a certain number of feasible individual working schedules are given. Each of them consists of a sequence of working and free 1 2
R. Cerulli and R. Mautone, Dip. Informatica ed Appl., Univ. Salerno, 84100 Fisciano, Italia. M. Gaudioso, Dip. Sistemi, Univ. Calabria, 87036 Rende (CS), Italia.
0340-9422/92/1/93-105 $2.50 9 1992 Physica-Verlag, Heidelberg
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shifts that span the planning horizon, and each employee is to be assigned to exactly one of them. The models presented in this paper could be applied in particular to the health system. Specific treatment of the related issues may be found in [5], [6], [7]. The paper is organized as follows. In Section 2 notations are introduced and the model is stated. Heuristic procedures are presented in Section 3. Necessary an sufficient conditions of existence are discussed in Section 4 and, finally, computational experience is reported in Section 5.
2 The Model
The model described in this section is aimed to determine a feasible assignment of schedules to a fixed number of employees in order to satisfy the workforce demand. It is a variant of the well known staffing problem analyzed in literature for example in [10], [11] and [12]. In particular we assume that:
n= m = l= x =
number of employees; number of planning periods; number of feasible work schedules; integer/-vector, xj = number of employees assigned to the j-th work schedule; e = vector of ones of appropriate dimension; A = (m, l) binary matrix defined as follows
I10 i f p e r i o d i i s a w o r k i n g o n e f o r s c h e d u l e j a/j = otherwise r = constant workforce demand over the planning horizon.
Thus the problem is to find a feasible solution to the system of linear equalities and inequalities:
Ax>_re (P)
e T x = El
x_O ,
integer .
A Class of Manpower Scheduling Problems
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It is easy to verify that (P) is feasible if and only if the optimal value of the following program is equal to 0 min v Ax+
ve>_re
(P1) eTx = n
x,v_>O ,
integer ,
where v has the obvious meaning of maximum violation in fulfilling the workforce demand.
3 Heuristic Algorithms
We describe now some heuristic algorithms for solving problem (P1). We observe that problem (P1) evokes the well known class of set covering problem for which an extensive literature is available, in particular on the heuristic techniques [1], [3], [4]. Such techniques may be fruitfully taken into account in devising algorithms for our problem. In particular we refer to the greedy heuristic [3]. We assign sequentially a feasible schedule to each employee, in the attempt o f keeping as balanced as possible the workforce allocated to the m periods. We note that finding a feasible solution to (P1) amounts to selecting exactly n columns of A, not necessarily distinct. Hence, we assign to the ( k + l)-th employee, k = 0 . . . . . n - 1 , the schedule which results in the best "smoothing" of the vector S, sum of the k columns already selected. Obviously at the initial step S is set equal to zero. To find such a schedule we define a merit function ~ ( j ) which allows us to associate a "value" to each column j, j = 1 . . . . . l. We consider two alternative definitions of function r ~l(J) and ~2(J) respectively: 1) We define the average value of S m i=I
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and the sets
L = {i I S i < s a }
U = {ilS~> s.I . Function r
is defined as follows
q~l(J) = ~ aij+ ~ ( 1 - a i j ) i~L
j=
1. . . . .
l .
iEU
It contributes "utility one" to the "ones" corresponding to the components of S below the average as well as to the "zeroes" corresponding to the components of S above the average. Thus the ( k + 1)-th employee is assigned the schedule j * such that j * = argmax r (J) 9 1
