A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheduling

Transcription

1 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling María I. Restrepo Bernar Genron Louis-Martin Rousseau November 2015

2 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling María I. Restrepo 1,2,*, Bernar Genron 1,3, Louis-Martin Rousseau 1,2 1 Interuniversity Research Centre on Enterprise Networks, Logistics an Transportation (CIRRELT) 2 Department of Mathematics an Inustrial Engineering, Polytechnique Montréal, P.O. Box 6079, Station Centre-Ville, Montréal, Canaa H3C 3A7 3 Department of Computer Science an Operations Research, Université e Montréal, P.O. Box 6128, Station Centre-Ville, Montréal, Canaa H3C 3J7 Abstract. This paper aresses a iscontinuous multi-activity tour scheuling problem uner eman uncertainty an when employees have ientical skills. The problem is formulate as a two-stage stochastic programming moel, where first-stage ecisions correspon to the assignment of employees to weekly tours, while secon-stage ecisions are relate to the allocation of work activities an breaks to aily shifts. A multi-cut L- shape metho is presente as a solution approach. Computational results on real an ranomly generate instances show that the use of the stochastic moel helps to prevent aitional costs, when compare with the expecte-value problem solutions. Keywors. Stochastic multi-activity tour scheuling problem, L-shape metho, column generation, context-free grammars. Acknowlegements. The authors woul like to thank the Fons e recherche u Québec - Nature et technologies (FRQNT) which supporte this work with a grant. Results an views expresse in this publication are the sole responsibility of the authors an o not necessarily reflect those of CIRRELT. Les résultats et opinions contenus ans cette publication ne reflètent pas nécessairement la position u CIRRELT et n'engagent pas sa responsabilité. * Corresponing author: Maria-Isabel.Restrepo@cirrelt.ca Dépôt légal Bibliothèque et Archives nationales u Québec Bibliothèque et Archives Canaa, 2015 Restrepo, Genron, Rousseau an CIRRELT, 2015

3 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling 1 Introuction The multi-activity tour scheuling problem (MATSP) is the integration of two problems, the multi-activity shift scheuling problem (MASSP) an the ays-off scheuling problem. In the MASSP, the planning horizon is usually one ay ivie into time perios of equal length. Since employees can perform ifferent work activities uring the same shift, the MASSP is concerne with choosing the work activities an the rest perios to assign to shifts to respon to a eman for service, that is translate into a eman for the number of employees require for each work activity an time interval. The ays-off scheuling problem eals with the selection of employee working ays an ays-off over a planning horizon of at least one week. In the MATSP, the constraints characterizing the feasibility of aily shifts an weekly tours, as well as the work rules for the allocation of work activities an rest breaks to the shifts, are usually efine by employee regulations an workplace agreements. The MATSP can be categorize into ifferent variants epening on the characteristics consiere. For instance, the personalize version of the MATSP appears when employees have iniviual preferences an skills. The anonymous version of the MATSP correspons to the case when employees have ientical skills. When shifts are allowe to span from one ay to another, the continuous version of the MATSP arises; otherwise, we have the iscontinuous version of the problem. In this paper, we consier the iscontinuous anonymous version of the MATSP. Realistic applications of the MATSP for companies that operate outsie the stanar 8-hours shift, 5-ays per week scheule an that face wie fluctuations in eman become challenging ue to several factors. First, complex large-scale moels result as a consequence of consiering multiple work activities an flexibility in the composition of aily shifts an weekly tours. Secon, since eman is typically unknown when scheuling ecisions nees to be taken, specialize solution techniques that allow to inclue this variability shoul be evelope. Specifically, such techniques allow to make a ecision on the employee scheule before a realization of the eman is known. Then, after eman becomes known, a recourse action shoul be implemente to compensate eficiencies in the previously mae scheules (e.g., unercovering an overcovering of eman). In this paper, we aress the iscontinuous stochastic multi-activity tour scheuling problem (SMATSP) for employees with ientical skills. In this problem, a long-term staffing ecision nees to be mae while heging for the short-term eman uncertainty. The problem is formulate as a two-stage stochastic programming moel, ecomposable by ays an by scenarios, in which first-stage ecisions correspon to the assignment of employees to weekly tours, while secon-stage ecisions (recourse actions) are relate with the allocation of work activities an breaks to aily shifts. The contribution of this paper lies in the proposal of an approach to efficiently solve practical instances of the problem. A heuristic multi-cut L-shape metho is implemente as a solution approach. Because the complete enumeration of weekly tours makes the problem intractable, the first-stage problem is solve via column generation. Aitionally, the secon-stage problems benefit from the use of context-free grammars to inclue work rules regaring the efinition of shifts an to efficiently hanle the multi-activity context. The paper is organize as follows. In Section 2, we review the relevant literature on shift scheuling an tour scheuling problems with multiple work activities an stochastic eman. Then, we present some backgroun material relate with the use of grammars for multi-activity shift scheuling problems. In Section 3, we escribe the two-stage moel 2

4 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling for the SMATSP. In Section 4, we introuce the solution approach to solve the problem. Computational experiments are presente an iscusse in Section 5. The concluing remarks are presente in Section 6. 2 Backgroun Material In this section, we review some literature on the moels an methos to solve the MASSP an the MATSP. We also present some references on workforce problems uner stochastic eman. Then, we finish with an introuction on the use of grammars for the MASSP. 2.1 Literature Review on Multi-Activity Shift an Tour Scheuling Although mono-activity shift scheuling an tour scheuling problems have been extensively stuie in the literature uring the last few ecaes [1, 15, 16, 32], only recently some attention has been given to the problem that eals with multiple work activities. Ritzman et al. [30] propose one of the first approaches to solve the MATSP. The metho is base on a heuristic solution approach that integrates a construction metho with a simulation component. Although employees are assigne to specific operations, breaks an rules relate with switching between work activities are not consiere. Heuristic approaches that use column generation (CG) [27] an tabu search [10] are also propose to solve multi-activity shift scheuling problems over multiple ays. Even though both approaches tackle long time horizons, the constraints characterizing the feasibility of weekly tours (e.g., total tour length) are not inclue in the formulation of the problem. In a similar way, Detienne et al. [12] an Lequy et al. [18] solve a multi-activity assignment problem by using ecomposition techniques an heuristics base on CG an branch-an-boun (B&B) as solution methos. Fixing the sequences of work, rest ays, shift types an breaks, might reuce the complexity of personnel scheuling problems, but it can also lea to sub-optimal solutions. Constraint programming (CP) techniques aim to solve that ifficulty by offering moeling languages to hanle complex optimization problems. Demassey et al. [11] present a CP-base CG algorithm to moel complex regulation constraints in a real-worl MASSP. Quimper an Rousseau [25] use formal languages to moel the work rules relate with the composition of shifts in a multi-activity context. Côté et al. [7] propose two approaches for the MASSP: the first one uses an automaton to erive a network flow moel, while the secon one takes avantage of context-free grammars to obtain a MIP moel in which an an/or graph structure is use. Côté et al. [8] present an implicit grammar-base moel for the MASSP that aresses symmetry issues by using general integer variables. Computational results show that, in the mono-activity case, the solution times of the moel are comparable an sometimes superior to the results presente in the literature an that, in the multi-activity case, the moel is able to solve to optimality instances with up to ten work activities. Côté et al. [9] an Boyer et al. [5] present grammar-base CG methos to solve the personalize MASSP an the personalize multi-activity multi-task shift scheuling problem, respectively. Both approaches use formal languages an ynamic programming to efficiently formulate an solve the pricing subproblems, but some limitations regaring long time horizons (e.g., one week) are present. To overcome these issues, Restrepo et al. [29] an Restrepo et al. [28] present approaches base on branch-an-price (B&P) an Beners ecomposition (BD), respectively. In the former approach [29], two B&P algorithms are presente for the personalize MATSP. 3

5 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling In the latter approach [28], a combine BD an CG metho is introuce for the anonymous MATSP. In both approaches, the work rules for the composition of multi-activity shifts are expresse with context-free grammars, while some constraints that guarantee the feasibility of weekly tours are embee into a irecte acyclic graph. 2.2 Literature Review on Stochastic Shift an Tour Scheuling Different moels an solution approaches have been propose in the literature to eal with stochastic eman in personnel scheuling problems. As an illustration, Easton an Rossin [14] an Easton an Mansour [13] evelop heuristic methos that aim to tackle problems where eman is uncertain. Easton an Rossin [14] propose a tabu search metho to solve a stochastic goal programming moel that integrates an optimizes labor eman an employee scheuling. Easton an Mansour [13] present a genetic algorithm to solve shift scheuling problems in which the recourse ecisions are relate with the unercovering an overcovering of eman. Although both approaches aim to solve problems over a one-week planning horizon, employee patterns are previously efine an only a small set of stochastic scenarios is consiere. Bar et al. [3] propose a heuristic two-stage moel that aresses tour scheuling problems over a one-week planning horizon. First-stage variables are relate with the number of full-time an part-time employees hire, while secon-stage ecisions correspon to the allocation of employees to specific shifts uring the week. Computational experiments on real instances that consier three stochastic scenarios (high, meium an low eman) show that significant savings are likely when the recourse problem is use. Some stuies that use ecomposition approaches have been recently propose as alternatives to solve workforce planning problems when eman is uncertain. Pacqueau an Soumis [21] propose a heuristic two-stage moel to solve a shift scheuling problem. The propose moel is base on a ecomposition of Aykin s [2] implicit moel, where first-stage variables are associate to the allocation of full-time shifts to the employees an recourse ecisions correspon to hiring part-time employees, using overtime for full-time shifts, the allocation of breaks an the allowance of unerstaffing. Punnakitikashem et al. [24] introuce a stochastic nurse scheuling problem that aims to minimize staffing costs an excess workloa. The authors present a BD approach, a Lagrangian relaxation with a BD approach an a neste BD approach as solution methos. Computational results suggest that simultaneously consiering nurse staffing an assignment is more esirable than oing them sequentially. Similarly, Kim an Mehrotra [17] present an integrate staffing an scheuling approach applie to nurse management when eman is uncertain. The problem is formulate as a two-stage stochastic integer program, where aily shifts an weekly patterns are previously enumerate. Firststage ecisions correspon to the number of employees assigne to aily shifts an to weekly patterns, while secon-stage ecisions correspon to: 1) the possibility of aing or canceling aily shifts for every working pattern; 2) allowing unercovering or overcovering of eman. A set of vali mixe-integer rouning inequalities that escribe the convex hull of feasible solutions in the secon-stage problem are inclue. Consequently, the integrality of the secon-stage ecision variables can be relaxe. Computational experiments show that the use of the stochastic moel prevents the hospital from being overstaffe. An L-shape metho is presente in Robbins an Harrison [31] to solve a combine server-sizing an staff scheuling problem for call centers in which a service level agreement must be satisfie. Firststage ecisions correspon to the employee staffing, while secon-stage ecisions correspon 4

6 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling to the computation of a telephone service shortfall. Computational results show that ignoring variability is a costly ecision, since the value of the stochastic solution for the moel is substantially high. Very limite literature is available on stochastic workforce planning for employees who have various skills to work on ifferent activities, tasks or unit epartments. Zhu an Sherali [34] aress a workforce planning problem for employees with multiple skills between service centers. A two-stage moel uner eman fluctuations is presente, where first-stage ecisions correspon to personnel recruiting an allocation of employees to multiple locations, while secon-stage ecisions consists in reassigning the workforce among the locations. The scheuling of cross-traine workers in a multi-epartment service environment with ranom eman is aresse in Campbell [6]. The author presents a two-stage moel ecomposable by ays an by scenarios, where first-stage ecisions are relate with the scheuling of ays-off an secon-stage ecisions correspon to the allocation of available employees at the beginning of each ay. In the approach, ays-off are previously efine an only a small number of scenarios is consiere (10 in total). Parisio an Jones [23] present a two-stage stochastic moel for a multi-skill tour scheuling problem in retail outlets, where first-stage variables are associate with the assignment of employees to weekly scheules, while recourse ecisions correspon to the allocation of overtime an to the unercovering an overcovering of eman. Although multiple work activities are inclue in the problem, the authors assume employees are allowe to work in only one activity per shift. Even though some authors have trie to tackle personnel scheuling problems uner stochastic eman, none of the previous stuies consier the integration of ays-off scheuling with shift scheuling in a multi-activity context. The metho propose in this paper aresses the iscontinuous MATSP when eman is uncertain an employee skills are ientical. Unlike the previous approaches, employee patterns an aily shifts are not previously fixe an a high egree of flexibility is inclue in their composition. Aitionally, the multi-activity context is efficiently hanle with context-free grammars, which are reviewe next. 2.3 Grammars for Multi-activity Shift Scheuling In shift scheuling, a context-free grammar (CFG) can be efine as a finite set of work rules that are use to generate vali sequences of work (shifts) for a given ay D, where D enotes the number of ays in the planning horizon. A CFG consists of a tuple G = Σ, N, S, P, where: Σ represents an alphabet of characters calle the terminal symbols for ay, which consists of work activities, breaks, lunch breaks, an rest stretches. N is a finite set of non-terminal symbols for ay. S N is the starting symbol for ay. P is a set of prouctions for ay, represente as A α, where A N is a nonterminal symbol an α is a sequence of terminal an non-terminal symbols. The work rules use to generate shifts are represente by the set of prouctions. The prouctions of a grammar can be use to generate new symbol sequences until only terminal symbols are part of the sequence. 5

7 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling A parse tree is a tree where each inner-noe is labele with a non-terminal symbol N an each leaf is labele with a terminal symbol Σ. A grammar recognizes a sequence if an only if there exists a parse tree where the leaves, when liste from left to right, reprouce the sequence. A DAG Γ is a irecte acyclic graph that embes all parse trees associate with wors (shifts) for ay of a given length n recognize by a grammar. The DAG Γ has an an/or structure where the an-noes represent prouctions (work rules) from P an the or-noes represent non-terminals from N an letters from Σ. An an-noe is true if all of its chilren are true. An or-noe is true if one of its chilren is true. The root noe is true if the grammar accepts the sequence encoe by the leaves. In Γ, Oil π enotes the or-noe associate with π N Σ, i.e., with non-terminals from N or letters from Σ, that generates a subsequence at position i of length l for ay. Note that if π Σ, the noe is a leaf an l is equal to one. On the contrary, if π N, the noe represents a non-terminal symbol an l > 1. A Π,k il is the kth an-noe representing prouction Π P that generates a subsequence from position i of length l at ay. There are as many A Π,k noes as there are ways of using P to generate il a sequence of length l from position i. In Γ, the root noe is escribe by O1n S an its chilren by A Π,k 1n ch(os 1n ). The chilren of or-noe Oπ il are represente by ch(oπ il ) an its parents by par(oil π ). Similarly, the chilren of an-noe AΠ,k il are represente by ch(a Π,k il ) an its parents by par(a Π,k il ). The sets of or-noes, an-noes an leaves in Γ are enote by O, A an L, respectively. The DAG Γ is built by a proceure propose in Quimper an Walsh [26]. Grammar G 1 presents an example on the use of context-free grammars for multi-activity shift scheuling. Two activities, w 1 an w 2, must be scheule, shifts have a length of n = 4 perios an shoul contain exactly one break, b, of one perio that can be place anywhere uring the shift except at the first or the last perio. For clarity, we o not inclue the subscript of the ay in the notation of grammar G 1 an noes from Γ 1. The grammar that efines the set of feasible shifts on this example follows: G 1 = (Σ = (w 1, w 2, b), N = (S, X, W, B), P, S), where prouctions P are: S XW, X W B, W W W w 1 w 2, B b, an symbol specifies the choice of prouction. In the previous example, prouctions W w 1, W w 2 an B b generate the terminal symbols associate with working on activity 1, working on activity 2, or having a break, respectively. Prouction W W W generates two non-terminal symbols, W, meaning that the shift will inclue a working subsequence. Prouction X W B means that the shift will inclue working time followe by a break. Finally, prouction S XW generates a sequence of length four (the aily shift), which inclues working time followe by a break to finish with more working time. Figure 1 represents the DAG Γ 1 associate with G 1. Observe that there are 16 parse trees (ifferent shifts) embee in Γ 1. As an illustration, we present a otte-parse tree that generates shift w 1 bw 1 w 2, an a ashe-line parse tree that generates shift w 2 w 2 bw 1. 6

8 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling O S 14 A S XW,1 14 A S XW,2 14 O X 13 A X WB,1 13 O X 12 O W 12 O W 32 A X WB,1 12 A W WW,1 12 A W WW,1 32 O W 11 O B 21 O W 21 O B 31 O W 31 O W 41 A W w1,1 11 A W w2,1 11 A B b,1 21 A W w1,1 21 A W w2,1 21 A B b,1 31 A W w1,1 31 A W w2,1 31 A W w1,1 41 A W w2,1 41 O w 1 11 O w 2 11 O21 b O w 1 21 O w 2 21 O31 b O w 1 31 O w 2 31 O w 1 41 O w 2 41 Figure 1 DAG Γ 1 on wors of length four an two work activities. Note that the chilren of the root noe ({A S XW,1 14, A S XW,2 14 } ch(o14 S )) can be seen as shift shells because they o not consier the allocation of specific work activities to the shifts, only the shift starting time an its length. Hence, an-noes A Π,k 1n are characterize by their starting time t Π,k 1n, working length wπ,k 1n, an length incluing breaks lπ,k 1n. In Γ 1, an-noe A S XW,1 14 generates shift wbww, while an-noe A S XW,2 14 generates shift wwbw. Both shifts have a working length of three time intervals, a total length of four time intervals an both start at time interval one (i = 1). Although the expressiveness of grammars allow to encoe a large number of work rules for the composition of aily shifts, some limitations regaring shift total length are present when long planning horizons are inclue in the problem (e.g., one week). To circumvent this problem, Restrepo et al. [28] present an approach that combines BD an CG to solve the eterministic iscontinuous MATSP for employees with ientical skills. The moel combines an explicit efinition of weekly tours with the implicit efinition of aily shifts from Côté et al. [8]. Since the moel presents a nice block structure ecomposable by ays, it is use in the formulation of the two-stage stochastic problem, presente next. 3 Two-Stage Stochastic Problem Stochastic shift an tour scheuling formulations exten an aapt eterministic moels to allow scheule moifications at a time closer to the actual eman realization. Two-stage stochastic programming moels give an example of such extensions. In these moels, some ecisions must be mae in the first-stage before values of ranom variables are observe. Then, in the secon-stage, a recourse action can be aopte after observing the actual values 7

9 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling of the ranom variables to ajust any ba ecision previously taken. In the moel propose, first-stage ecisions correspon to the number of employees assigne to each tour an to each aily shift shell, while secon-stage ecisions (recourse actions) correspon to the allocation of breaks an work activities to aily shifts an to the unercovering or overcovering of eman. The secon-stage problem is formulate with the implicit moel propose in Côté et al. [8]. In this approach, the authors translate the logical clauses associate with Γ, D, into linear constraints on integer variables, where the number of employees assigne to each an-noe (A ), each or-noe (O ) an each leaf (L ) in Γ are represente by an integer variable. In the first-stage problem, we efine a feasible tour as the integration of aily shift shells (chilren of root noes O1n S, D) an ays-off, over the set of ays in the planning horizon. Tours must meet the work rules relate with the total working length, with the number of working ays, with the rest time between consecutive shifts an with the allocation of aysoff. Figure 2 presents an example of three tours compose with the shifts presente in Γ 1. In this example, we assume that the DAG Γ for each ay D is the same. Aitionally, the planning horizon correspons to seven ays, the working length shoul fall between 15 an 18 time intervals, the number of working ays must fall between 5 an 6, an there are no rules for the allocation of ays-off an for the rest time between shifts. Finally, S 1 correspons to A S XW,1 14 wbww, S 2 correspons to A S XW,2 14 wwbw an DO correspons to a ay-off. Days S 1 DO DO S 1 S 2 S 1 S 2 Tours 2 S 1 S 1 S 2 S 2 S 2 DO S 2 3 DO S 1 S 2 S 2 DO S 2 S 1 Figure 2 Weekly tours compose of shifts from Γ 1. In efining a moel for the iscontinuous SMATSP, we assume that, at the moment we can act on secon-stage variables, the scenario for ay D is fully known. Hence, staffing ecisions (allocation of tours an aily shifts to employees) that are feasible to scheule without knowing in avance the eman, will be generate well ahea in time, while ajustments (allocation of work activities, position of breaks, overcovering an unercovering of eman) are mae once improve (aily) eman information is available. We also assume that the ranom vector ξ representing the stochastic perturbations of emans is non-negative an has a finite support. Henceforth, we efine Ω as the set of its possible realizations an p (w) > 0 as the probability of occurrence of scenario w Ω with w Ω p(w) = 1. The notation for the stochastic moel follows. Sets J: set of work activities; D: set of ays in the planning horizon; I : set of time intervals at ay D; 8

10 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling E: set of employees; T : set of feasible tours. First-stage problem Parameter δ Π,k t : parameter that takes value 1, if tour t inclues the kth shift shell built with prouction Π for ay (variable v Π,k 1n ), an assumes value 0 otherwise. Decision variables x t : integer variable that represents the number of employees assigne to tour t; v Π,k 1n : variable that represents the number of employees assigne to the kth an-noe built with prouction Π (chilren of the root noe O1n S from Γ ). Secon-stage problem Parameters b ij : eterministic eman for ay, time interval i an activity j; ξ (w) ij w; : stochastic perturbation of eman for ay, time interval i an activity j for scenario b (w) ij : stochastic eman for ay, time interval i an activity j for scenario w, b(w) ij = b ij + ξ (w) ij ; c ij : cost associate to one employee working on activity j, at time interval i, at ay ; c + ij, c ij : eman overcovering an unercovering costs for ay, time interval i an activity j, respectively. Decision variables y (w) ij : variable that enotes the number of employees assigne to activity j, at time interval i, for ay uner scenario w; v Π,k,(w) il : variable that enotes the number of employees assigne to the kth an-noe, representing prouction Π from Γ an that generates a sequence from i of length l < n, uner scenario w (this set of variables exclues the chilren of the root noe O1n S ); s +(w) ij, s (w) ij : slack variables representing overcovering an unercovering of eman of activity j, at time interval i, for ay uner scenario w, respectively. Aitionally, let V enote the set of variables corresponing to the union, over the set of ays D, of variables v Π,k 1n. Given that notation, the formulation for the first-stage moel, enote G T, is as follows. 9

11 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling f(g T ) = min Q(V) (1) v Π,k 1n = δ Π,k t x t, D, A Π,k 1n ch(os 1n ), (2) t T x t = E, (3) t T x t 0 an integer, t T, (4) v Π,k 1n 0, D, AΠ,k 1n ch(os 1n ). (5) The objective of G T, (1), is to minimize the expecte recourse cost Q(V). Constraints (2) represent the link between aily shifts (chilren of root noes O1n S in Γ, D) an tours. Since a fixe number of employees is given an all the employees have the same skills, constraint (3) guarantees that exactly E employees are assigne to the tours. Finally, constraints (4)-(5) set the non-negativity an integrality of variables x t an the non-negativity of variables v Π,k 1n. The expecte recourse function is enote by Q(V) E ξ [Q(V, ξ)]. The recourse function Q(V, ξ(w)), for a given realization w of ξ, is represente by: Q(V, ξ(w)) = min c ij y (w) ij + D D A Π,k il i I j J y (w) ij s+(w) ij + s (w) ij Π,k,(w) vil = ch(oπ il ) i I j J (c + ij s+(w) ij + c ij s (w) ij ) (6) = b (w) ij, D, i I, j J, (7) A Π,k il Π,k v1n, par(oπ il ) D, Oil π ch(a π,k 1n ) \ L, (8) Π,k,(w) vil = A Π,t il ch(oπ il ) A Π,k il Π,k,(w) vil, par(oπ il ) D, Oil π O \ {O1n S L ch(a π,k 1n )}, (9) y (w) ij = Π,1,(w) i1, D, i I, j J, (10) v A Π,k i1 par(oj i1 ) v Π,k,(w) il 0, D, A Π,k il A \ ch(o1n S ), (11) s +(w) ij, s (w) ij 0, D, i I, j J, (12) y (w) ij 0 an integer, D, i I, j J. (13) Problem (6)-(13) is base on the implicit moel presente in Côté et al. [8]. The objective, (6), is to assign work activities to aily shifts in orer to minimize the staffing cost plus the unercovering an overcovering of eman. Constraints (7) ensure that the eman per ay, time interval i an work activity j is met for each eman realization w. Due to the structure of Γ, D, constraints (8)-(9) guarantee for every or-noe Oil π, excluing the root noe O1n S an the leaves L, that the summation of the value of its chilren, ch(oil π ), 10

12 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling is the same as the summation of the value of its parents, par(oil π ). Constraints (10) set the value of variables y (w) ij equal to the summation of the value of the parents of leaf noes O j i1. Constraints (11)-(13) set the non-negativity of variables vπ,k,(w) il, s +(w) ij, s (w) ij an the non-negativity an integrality of variables y (w) ij. Observe that problem (1)-(5) (first-stage problem) has complete recourse because for any realization of the ranom vector ξ an value of variables v Π,k 1n, problem (6)-(13) (seconstage problem) is always feasible ue to the allowance of unercovering an overcovering of eman. Aitionally, note that problem Q(V, ξ(w)) is ecomposable by ays ue to its particular block structure. Therefore, V, Ω an p (w) > 0 can also be ecompose by ays: V, Ω, p (w) > 0, D, an the expecte recourse function Q(V) can be represente as: Q(V) E ξ [ D Q(V, ξ )] D E ξ [Q(V, ξ )] (14) In the following, we present the solution metho propose to solve the SMATSP. 4 Heuristic Multi-cut L-shape Metho The basic iea behin the L-shape metho is to approximate the nonlinear term, Q(V), in the objective function of the two-stage stochastic problem (1)-(5). In particular, since the expecte recourse function involves solving all secon-stage recourse problems, the main principle of the L-shape metho is to avoi numerous function evaluations by using an outer linearization of Q(V), as in BD. Since ξ follows a non-negative istribution with a finite support, with Ω as the set of its possible realizations for each ay D, an p (w) > 0 as the probability of occurrence of scenario w Ω ( D w Ω p (w) = 1), Q(V) can be expresse as Q(V) = D w Ω p (w) Q(V, ξ (w)). By efining θ (w) as an aitional set of free variables, the two-stage stochastic problem G T can be reformulate as the following moel, enote as B T. f(b T ) = min D w Ω θ (w) (15) v Π,k 1n = δ Π,k t x t, D, A Π,k 1n ch(os 1n ), (16) t T θ (w) p (w) Q(V, ξ (w)), D, w Ω, (17) x t = E, (18) t T x t 0 an integer, t T, (19) v Π,k 1n 0, D, AΠ,k 1n ch(os 1n ). (20) Optimality cuts (17) ensure that the value of each variable θ (w) is larger than or equal to the optimal value of its corresponing secon-stage problem for each ay D an each 11

13 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling scenario w Ω. Observe that the structure of problem (15)-(20) allows the L-shape metho to be extene to inclue multiple cuts at each iteration, i.e., one per ay an per scenario, instea of aing one aggregate cut. Birge an Louveaux [4] showe that in an iterative algorithm, aing multiple cuts at the same iteration may spee up convergence an reuce the number of iterations. Since the secon-stage problems (6)-(13) are MILP moels that o not possess the integrality property, we relax integrality constraints (13) on variables y (w) ij because 1) the ual to the LP relaxation of each secon-stage problem will prouce a cut that forces θ (w) to be at least as great as the objective value of the relaxation, which is a vali lower boun for the actual recourse function value; 2) Restrepo et al. [28] showe that, in practice, problems (6)-(13) o not exhibit a large integrality gap an that optimal or near-optimal solutions can be foun by solving the LP relaxation of the secon-stage problems. Let Q(V, ξ (w)) enote the LP relaxation of problem Q(V, ξ (w)). Let ρ (w) ij, γπ,(w) il be the ual variables associate with constraints (7) an (8) from Q(V, ξ (w)), respectively. Let (w) be the projection over the space of variables ρ (w) ij, γπ,(w) il of the polyheron efine by the constraints associate with the ual of moel Q (V, ξ (w)). Note that (w) polyheron [33]. Let E (w) be the set of extreme points of (w) B T are replace by the following ones, efining formulation B T : is itself a. Inequalities (17) in moel θ (w) ( p (w) i I j J b (w) ij ρ(w) ij + O π il ch(aπ,k 1n )\L γ π,(w) il D, w Ω, (ρ, γ ) E (w) v A Π,k 1n par(oπ il ) ) Π,k 1n, (21) Since these new optimality cuts are using linear approximations of Q(V, ξ (w)), moel B T is a MILP relaxation of B T i.e., f(b T ) f(b T ). Optimality cuts (21) o not nee to be exhaustively generate, since only a subset of them are active in the optimal solution of the problem. Hence, an iterative algorithm can be use to generate only the subset of cuts that will represent the optimal solution. The algorithm consists in a multi-cut version of the L-shape metho where, at each iteration l 1, a relaxation of the first-stage problem is solve. Such relaxation is obtaine by replacing the set of extreme points at each ay D an each scenario w Ω, by subsets E l E (w) (w). Note that in the first-stage moel, B T, it is assume that the complete set of tours T is known. However, with the incorporation of shift an tour flexibility, the complete enumeration of the set of feasible tours might be intractable. To aress this issue, we propose a heuristic CG approach in which a master problem B LP, is efine as the LP relaxation of B T over a restricte set of tours T. We also efine the MILP associate to B LP as B MILP, where for a given subset of columns T, the integrality constraints on x t variables are impose to obtain a heuristic integer solution (i.e., B MILP is solve by a state-of-the-art B&B metho, using only the columns corresponing to ). The algorithm for the L-shape metho is then ivie into two parts: multi-cut generation an CG. Two algorithm enhancements were implemente to spee-up the convergence of the multi- 12

14 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling cut L-shape metho. First, we aopte the strategy propose in McDaniel an Devine [20], which consists in initially solving the LP relaxation of the first-stage problem to generate, in a fast way, a number of vali cuts. Then, when some criterion is met (i.e., the relative gap between the upper boun an the lower boun of the problem is smaller than a certain value), the metho then solves the MILP of the first-stage problem. Secon, we implemente the metho presente in Papaakos [22] for the generation of strong optimality cuts. The author proposes an alternative to eliminate the necessity of solving the extra auxiliary subproblem introuce in Magnanti an Wong [19]. Aitionally, since fining a core point for the problem is a ifficult task, the author suggests to use an approximation of the core point that consists of a convex linear combination of the previously generate core point an the current solution for the first-stage problem. Let ɛ 1 be the tolerance that efines if an optimality cut is ae or not to the first-stage problem an let ɛ 2 be the tolerance we use to stop solving B LP, i.e., stop the McDaniel an Devine [20] strategy. Let Int be a boolean variable that inicates whether the MILP (B MILP ) of the first-stage problem is solve (Int=true) or not (Int=false). Let θ (w) enote l w Ω θ (w) l the optimal value of variables θ (w) from B LP, at iteration l. Note that θ l = D enotes a lower boun on f(g T ) at iteration l. Since we are using a heuristic approach to (w) fin integer solutions for the first-stage moel, we also enote θ l as the optimal values of variables θ (w) when B MILP is solve at iteration l. To simplify the algorithm escription, we use the same notation when solving B LP (w), even though in that case, we have θ(w) l = θl for each D, w Ω, since the CG algorithm is performe until all columns have non-negative reuce costs. Let s (w) l, sl (w) be the optimal value of secon-stage problem (6)-(13) for ay, uner scenario w at iteration l, when integrality constraints on variables y (w) ij are impose an relaxe, respectively. Note that θ l = D w Ω p (w) s (w) l is an upper boun on f(gt ) at iteration l. Similarly, θ l = D w Ω p (w) s l (w) is an upper boun on f(b T ) f(g T ) at iteration l, which we call the approximate upper boun. Let v π,k 1nl, v π,k 0 1nl enote an optimal solution an the core point approximation, respectively of first-stage variables v π,k 1n from moel B T at iteration l. The flow iagram of the algorithm is presente in Figure 3. The escription of the multi-cut L-shape metho follows. 13

15 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling Begin Initialization Get primal sol. from B LP no Int=true? yes Solve B MILP get primal sol. Solve LP rel. secon-stage prob. get ual sol. Multi-cut generation Solve B LP get ual sol. Column generation A columns to B LP no yes Opt. cuts? yes A opt. cuts Solve pricing subp. c t < 0? no Solve MILP secon-stage prob. Compute the value of the stochastic sol. En Figure 3 Flow chart for the multi-cut L-shape metho. Initialization: The multi-cut L-shape algorithm starts with an empty set of optimality cuts, E l =, D, w Ω (w). In this step, the number of iterations l is set to zero, the lower an upper bouns of the problem are initialize as θ l =, θ l =, θ l =, the Boolean variable Int is set to false, an an initial set of columns, generate with the proceure shown in the Column generation step, is ae to B LP. Int=true?: In this step of the algorithm, we verify if the Boolean variable Int is true or false. If Int = false, we continue with the step Get primal sol. from B LP. If Int = true we continue with the step Solve B MILP get primal sol. The value of Int is change from false to true when ( θ l θ l )/ θ l < ɛ 2 an Int = false. Get primal sol. from B LP : In this step of the algorithm, we get the primal solution v π,k (w) 1nl, θ l from B LP. Then, we calculate the approximation of the core point as: 14

16 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling v π,k 0 1nl = 1 2 vπ,k 0 1n l vπ,k 1nl, D, A Π,k 1n boun of the problem as: θ l = D to the secon-stage problems. w Ω θ (w) l ch(os 1n ) an we upate the lower. The values of v π,k 1nl, v π,k 0 1nl are sent Solve B MILP get primal sol.: In this step of the algorithm, we get the primal solution θ (w) l from B LP to upate the lower boun of the problem as: θ l = D w Ω θ (w) l. Then we solve B MILP (w) to get the primal solution θ l, v π,k 1nl an to calculate the approximation of the core point as: v π,k 0 1nl = 1 2 vπ,k 0 1n l vπ,k 1nl, D, A Π,k 1n ch(o1n S ). The values of vπ,k 1nl, v π,k 0 1nl are sent to the secon-stage problems. Multi-cut generation: The objective of the multi-cut step is to generate optimality cuts (21) in orer to approximate the recourse function Q(V, ξ). The proceure to generate an to a new optimality cuts to the first-stage problem follows. Solve LP rel. secon-stage prob.: In this step of the algorithm, we solve the LP relaxation of the secon-stage problems twice. First, we fix variables v π,k 1n with the value of core point v π,k 0 1nl to get a ual solution (ρ, γ ). Secon, we fix variables v π,k 1n with the value of point vπ,k 1nl to recover the real objective value of the seconstage problems an to upate the approximate upper boun of the problem as: θ l = min{ θ l, D w Ω p (w) s l (w) }. Opt. cuts? an a opt. cuts: In orer to a optimality cuts to the first-stage problem, we verify if (s l (w) θ l (w) )/s l (w) > ɛ 1, in which case a new optimality cut is ae for scenario w an ay. After aing the optimality cuts, we increase by one the number of iterations l. Column generation: The CG metho consists of a master problem B LP an a pricing subproblem. The former problem, as mentione before, is the LP relaxation of moel B T over a reuce set of tours T. The latter problem is responsible for fining tours with negative reuce cost that will be ae to B LP in an iterative way. Let λ Π,k 1n an δ be the ual variables associate with the constraints (16) an (18) from B LP, respectively. Let S = D ch(os 1n ) be a set of shift shells, efine as the union, over the set of ays in the planning horizon, of all the chilren of root noes O1n S, D. Let G(N, A) be a irecte acyclic graph, compose of a set of noes N = {v s s S {v b, v e }}, where v s correspons to shift s an v b, v e are the source an sink noes, respectively. Each shift s S hols, besies a set of attributes inherite from its corresponing an-noe (start perio, working time, an length incluing breaks), a reuce cost contribution corresponing to value of the ual variable λ Π,k 1n. The set of arcs A represents the connection between noes epening on the work rules for the allocation of ays-off an rest time between consecutive shifts. New columns for B LP correspon to resource-constraine shortest paths over G(N, A). More specifically, each feasible tour t must meet the work rules relate with the minimum an maximum number of working ays in a tour, with the minimum an maximum tour length in time intervals, with the maximum number of ays-off, an 15

17 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling with the minimum rest time between two consecutive aily shifts. reuce cost c t of tour t is given by Aitionally, the ( c t = λ D A Π,k 1n ch(os 1n ) ) Π,k 1n δπ,k t σ. (22) The proceure to generate an a new columns for B LP is as follows: Solve B LP, get ual sol.: In this step of the algorithm, we solve problem BLP get the ual solution (λ, σ) that will be sent to the pricing subproblem. Solve pricing subp.: New variables (tours) for the B LP are generate by using a label setting algorithm for the resource-constraine shortest-path problem over graph G(N, A). In the algorithm, the total length of the tour an the number of working ays represent global resources that are consume by the labels while they are extene. c t < 0? an a columns to B LP : In this step of the algorithm we evaluate if negative reuce cost columns were foun by the pricing subproblem. If yes, such columns are sent to B LP which is re-optimize to start a new iteration. Solve MILP secon-stage prob.: In this step of the algorithm, we solve the MILP of the secon-stage problems when v π,k 1n variables are fixe with the value of vπ,k 1nl. In this step, we also compute the value of the upper boun as θ l = w Ω D p(w) s (w) l an the gap with respect to the approximate upper boun: 100 ( θ l θ l )/ θ l. This gap helps to measure the quality of the solution obtaine when the integrality constraints on secon-stage variables y (w) ij are relaxe. Compute the value of the stochastic sol.: The multi-cut L-shape algorithm ens with the computation of the value of the stochastic solution (VSS) which is efine as V SS = EEV HN. HN correspons to the value of the two-stage stochastic programming problem an EEV correspons to the expecte value of the expecte-value problem (EV). Recall that, since secon-stage problems are MILP moels that o not possess the integrality property, the final (heuristic) solution of the two-stage stochastic problem might not be optimal an HN = θ l θ l. 5 Computational Experiments In this section, we test the propose multi-cut L-shape metho on real an ranomly generate instances of the SMATSP. First, we escribe the generation an the characteristics of the set of instances use. Secon, we present the problem efinition an the grammar built for the composition of aily shifts. Thir, we report an analyze the computational results. The computational experiments were performe on a 64-bit GNU/Linux operating system, 96 GB of RAM an 1 processor Intel Xeon X5675 running at 3.07GHz. The multi-cut L-shape algorithm was implemente in C++. The LP relaxation of both the first-stage problem an to 16

18 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling Employees Day 1 Day 2 (a) Unimoal Employees Day 1 Day 2 (b) Level Figure 4 Deterministic eman profiles. the secon-stage problems was solve by using the barrier metho of CPLEX version We set a time limit of 3 hours to solve each instance. Aitionally, a relative gap tolerance of 0.01 was set as a stopping criterion for solving the MILPs with CPLEX. The value of tolerances ɛ 1, ɛ 2 were set to an 0.01, respectively. 5.1 Instances Generation The set of instances use to test our metho is ivie into two groups: ranomly generate instances an real instances from a small retail shop. The eterministic eman profiles for the set of ranom instances were generate such that they follow a unimoal behavior. The eterministic eman profiles for the real instances, are presente in Côté et al. [8]. These eman profiles present a constant (uniform) eman across hours (level behavior). Figure 4 shows an illustration, over two ays, of the eman profiles use for the computational experiments. Stochastic instances were create by aing to the eterministic eman profile, a ranom perturbation that follows a iscrete uniform istribution. We create instances with 11, 49, 81 an 125 scenarios. Their escription follows. Instances with 11 scenarios: In this group of instances, one large perturbation is generate for the complete week. Such perturbation follows a iscrete uniform istribution between -5 an 5. Instances with 49 scenarios: In this group of instances, two perturbations that follow a iscrete uniform istribution between -3 an 3 are generate. The first perturbation 17

19 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling will affect the first 12 hours of each ay uring the week, while the secon perturbation will affect the last 12 hours of each ay uring the week. Instances with 81 scenarios: In this group of instances, we generate perturbations for each ay of the week every two hours between 10am an 6pm. Such perturbations follow a iscrete uniform istribution between -1 an 1. Instances with 125 scenarios: In this group of instances, we generate perturbations for each ay of the week every two hours between 11am an 5pm. The perturbations follow a iscrete uniform istribution between -2 an 2. It is important to highlight that the eman is not perturbe when b (w) ij = b ij + ξ (w) ij 0. However, this case only applies when b ij = 0 because when the eman takes a positive value, this value is always higher than the value of the lower realization of the stochastic perturbation ξ (w) ij. 5.2 Problem Definition an Grammar The work rules for shift an tour generation, as well as the grammar use in the problem are as follows. Tour generation 1. The planning horizon is seven ays, where each ay is ivie into 96 time intervals of 15 minutes. 2. Shifts are not allowe to span from one ay to another (iscontinuous problem). 3. The tour working length shoul fall between 35 an 40 hours per week. 4. The number of working ays in the tour shoul fall between five an six. 5. There must be a minimum rest time of twelve hours between consecutive shifts. Daily shift generation 1. Shifts can start at any time interval uring any ay, allowing enough time to complete their uration in ay. 2. Three types of shifts are consiere: 8-hour shifts with 1-hour lunch break in the mile an two 15-minute breaks. 6-hour shifts with one 15-minute break an no lunch, an 4-hour shifts with one 15-minute break an no lunch. 3. If performe, the uration of a work activity is at least one hour an at most five hours. 4. A break (or lunch) is necessary between two ifferent work activities. 5. Work activities must be inserte between breaks, lunch an rest stretches. 6. A fixe number of employees E is given, therefore unercovering an overcovering of eman is allowe. Let a j be a terminal symbol that efines a time interval of work activity j J. Let b, l an r be the terminal symbols that represent break, lunch an rest perios, respectively. In prouctions Π P, Π [min, max] restricts the subsequences generate by a given prouction to a length between a minimum an maximum number of perios. The grammar an the prouctions that efine the multi-activity shifts are as follows: 18

20 A Two-Stage Stochastic Programming Approach for Multi-Activity Tour Scheuling G =(Σ = (a j j J, b, l, r), N = (S, F, Q, N, W, A j j J, B, L, R), P, S), S RF R F R RF RQR QR RQ RNR NR RN, B b, L llll, F [38,38] NLN, Q [25,25] W BW, N [17,17] W BW, R Rr r, W [4,20] A j j J, A j A j a j a j j J. 5.3 Computational Results Tables 1-2 present the computational results on stochastic weekly instances ealing with up to five work activities. Ten ifferent emans were teste for each activity (Nb.Act) an for each version on the number of scenarios (Scen.). We present the average CPU time in secons to solve the problem (T. time), the average CPU time spent in the CG approach (Time CG), which inclues the time to solve the pricing subproblems an the time to solve the LP relaxation when new columns are ae, the average CPU time to solve the first-stage problem (Time F-S), an the average CPU time to solve the secon-stage problems (Time S-S). The average gap between the best upper boun an best lower boun is presente in Gap1. This gap is compute as: Gap1 = 100 ( θ θ)/ θ. Since the secon-stage problems are MILPs that o not possess the integrality property an we are relaxing integrality constraints on variables y (w) ij, we also calculate the average gap between the upper boun θ an the approximate upper boun θ: Gap2 = 100 ( θ θ)/ θ. Conv. presents the number of instances that converge to a near-optimal solution, i.e., the algorithm stoppe when no more optimality cuts are ae to the first-stage moel. The average value of the stochastic solution (VSS), in percentage, is presente in the last column. This value is compute as: V SS = 100 (EEV HN)/EEV an it is only calculate for instances that converge (Conv.=1). Scen. Nb. Act T. time Time CG Time F-S Time S-S Gap1 Gap2 Conv. VSS % 0.00% % 2 1, , % 0.00% % , , % 0.00% % 4 2, , % 0.00% % 5 3, , % 0.02% % % 0.00% % 2 4, , % 0.00% % , , % 0.00% % 4 8, , % 0.01% % 5 10, , % 0.02% 0-1 2, , % 0.00% % , , % 0.00% % 3 9, , % 0.01% % 4 10, , % 0.01% % 1 2, , % 0.00% % , , % 0.00% % 3 10, , % 0.01% % 4 10, , % 0.01% % Table 1 Results on stochastic weekly instances with unimoal eman shape. 19