Piraeus, Greece Published online: 03 Jan 2014.

Transcription

1 This article was downloaded by: [University of Patras] On: 09 February 2014, At: 09:46 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Engineering Optimization Publication details, including instructions for authors and subscription information: Efficient greedy algorithms for economic manpower shift planning A.C. Nearchou a, I.C. Giannikos a & A.G. Lagodimos b a Department of Business Administration, University of Patras, Rio, Greece b Department of Business Administration, University of Piraeus, Piraeus, Greece Published online: 03 Jan To cite this article: A.C. Nearchou, I.C. Giannikos & A.G. Lagodimos, Engineering Optimization (2014): Efficient greedy algorithms for economic manpower shift planning, Engineering Optimization, DOI: / X To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 Engineering Optimization, Efficient greedy algorithms for economic manpower shift planning A.C. Nearchou a, I.C. Giannikos a and A.G. Lagodimos b a Department of Business Administration, University of Patras, Rio, Greece; b Department of Business Administration, University of Piraeus, Piraeus, Greece (Received 21 December 2012; accepted 13 November 2013) Consideration is given to the economic manpower shift planning (EMSP) problem, an NP-hard capacity planning problem appearing in various industrial settings including the packing stage of production in process industries and maintenance operations. EMSP aims to determine the manpower needed in each available workday shift of a given planning horizon so as to complete a set of independent jobs at minimum cost. Three greedy heuristics are presented for the EMSP solution. These practically constitute adaptations of an existing algorithm for a simplified version of EMSP which had shown excellent performance in terms of solution quality and speed. Experimentation shows that the new algorithms perform very well in comparison to the results obtained by both the CPLEX optimizer and an existing metaheuristic. Statistical analysis is deployed to rank the algorithms in terms of their solution quality and to identify the effects that critical planning factors may have on their relative efficiency. Keywords: labour; resource allocation; assignment; combinatorial; workforce; personnel planning 1. Introduction Economic manpower shift planning (EMSP) is a combinatorial planning problem arising in industrial settings operating a predetermined number of fixed 8-hour workday shifts (i.e. day, evening and night shift) where a set of independent jobs needs be completed within a given time horizon. Each job has a known processing time (duration) and requires a fixed-size crew (manning) to be carried out. Workforce remuneration costs may differ between shifts. In this context, EMSP seeks the manpower to be deployed in each workday shift over the specific horizon so as to complete all jobs at minimum manpower cost. The problem was first identified (Lagodimos and Leopoulos 2000) as a real-world capacity planning problem for the packaging stage of process industries (e.g. processed foods, beverages, pharmaceuticals) where a number of product families need be packaged on independent packing lines. Viewing the production load of each packing line as a job, the respective EMSP solution allows managers to determine the semi-skilled operators needed to efficiently complete all production targets. Recently, EMSP was identified (Nearchou and Lagodimos 2013) in the context of maintenance planning and specifically in determining the engineers needed for carrying out the annual maintenance of a chemical plant during Corresponding author. nearchou@upatras.gr 2013 Taylor & Francis

3 2 A.C. Nearchou et al. seasonal shutdown. It should be noted that a very similar problem to EMSP for maintenance planning was previously identified in the steel industry and analysed by Chang et al. (1999). Special features of the problem are the inclusion of time windows and duration depending on manning. Pan et al. (2010) addressed a similar planning problem identified in the precision engineering industry. They developed a heuristic to assign operators to machines by considering skill requirements and operator expressed preferences for shifts and on/off-days. Considering research on the EMSP problem formulation and solution, Lagodimos and Leopoulos (2000) studied a simplified version of the problem (coined manpower shift planning or MSP), which assumes identical manpower remuneration costs between shifts. They provided an integer linear programming (ILP) formulation of the problem and showed (by analogy to makespan minimization on identical parallel machines) that the problem is NP-hard in the strong sense (Garey and Johnson 1979). In addition, they established a solution lower bound as well as a greedy heuristic (referred to hereafter as the LL algorithm) for the problem solution. This heuristic can generally be classified as a look-ahead greedy algorithm (see Arora, Jin, and Choi 2006; Lee, Ford, and Joglekar 2007 for a description of such algorithms in resource allocation problems) since it applies a trial-allocation operation (i.e. looks ahead in the solution space) to reserve those time periods that provide the lowest workforce cost at each step. Comparisons with the results obtained by a commercial optimizer (LINGO) revealed the excellent performance of the LL algorithm in terms of solution quality and speed. Aiming at improving this heuristic, Lagodimos and Paravantis (2006) focused on a particular part of its logic (the shift-selection procedure) and studied the effects of alternative shift priority rules on the resulting solution quality. Statistical analysis revealed one of these rules to clearly outperform the original one in the LL algorithm, leading to improved performance. Lagodimos and Mihiotis (2006) studied a different version of the problem, termed EMSP-O, which allows for different remuneration costs between shifts as well as overtime work. They formulated the problem as a mixed integer linear program (MILP) and explored its optimal solution structure using the LINGO optimizer for several different experimental settings. Their results demonstrated overtime as a useful management tool, primarily to increase manpower utilization and reduce labour cost. Similar conclusions were drawn by Bhatnagar, Saddikutti, and Rajgopalan (2007), who studied a closely related but more complex setting, allowing for multi-skilled manpower in an assembly environment. Later, Lagodimos and Mihiotis (2010) proposed a relaxed model for EMSP-O which imposes no constraints on the type of overtime used. They provided an algorithm for jointly determining manpower needs and overtime per workday shift for the case where manning for all jobs is identical. By comparison to results obtained using LINGO, this algorithm showed satisfactory performance. In a recent study, Nearchou and Lagodimos (2013) considered EMSP without including overtime but allowing for unequal remuneration costs between workday shifts and jobs with unequal manning. After introducing a lower bound to the EMSP optimal solution, they proposed three heuristics for its solution; namely, two somewhat simplistic greedy single-pass algorithms and a hybrid metaheuristic algorithm (termed μga/vns) combining a micro-genetic algorithm (μga) for fast global solutions generation with variable neighbourhood search (VNS) for μga solutions improvement. Computations showed that, despite being slower, the metaheuristic totally outperforms the greedy algorithms and leads to solutions of acceptable quality compared with results obtained using the CPLEX optimizer. Aiming to exploiting the inherent solution speed, which is particularly important in large combinatorial problems, this article presents efficient greedy algorithms for the EMSP solution. In this context, the EMSP problem is considered in its general form (with unequal remuneration costs between shifts) and three different adaptations are provided of the look-ahead LL algorithm to this new environment, termed the modified LL algorithm (mll), lowest cost increment algorithm (LCI) and block planning algorithm (BP). In brief, the mll heuristic is a direct adaptation of the improved LL algorithm (see above) to the cost-differentiated environments considered here. The

4 Engineering Optimization 3 LCI heuristic effectively constitutes a reduced (myopic) version of mll. In each step, instead of a look-ahead search covering the required time periods to complete a job, this heuristic considers much shorter look-ahead paths, completing the step when the entire job has been scheduled. Finally, the BP heuristic practically combines the logic underlying the EMSP solution lower bound (Nearchou and Lagodimos 2013) with the LCI heuristic above. All new algorithms are easily implemented. Moreover, as presented here, they are clearly found to be superior in terms of both solution time and quality to the best existing EMSP heuristic (i.e. the μga/vns metaheuristic described above). The remainder of the article is organized as follows. Section 2 provides a formal ILP formulation of the EMSP problem. Section 3 presents the three new algorithms for solving EMSP after discussing their underlying logic. It also demonstrates their application via a common numerical example. Section 4 presents and discusses computational results for all algorithms using a large set of test problems representing a variety of operating environments. It also provides comparisons, together with statistical analysis, of the algorithms performance with that of the best existing heuristic (i.e. the μga/vns algorithm) and the solutions obtained via the standard CPLEX commercial optimizer. Finally, Section 5 highlights the article s contributions and concludes with directions for future research. 2. Mathematical formulation This section presents and discusses a general ILP formulation for the EMSP problem. The following notation is used: i j k N D S w i a i c k C z k index for jobs index for days index for shifts number of jobs number of days in planning horizon number of shifts per day duration of job i manning (crew size) required for work on job i remuneration cost for work in shift k; c k c l for k > l total workforce cost number of workers assigned to shift k There are N jobs which must be completed within a predetermined planning horizon of D days. Each day comprises S distinct and each shift has D time periods. Each job i (i = 1,..., N) requires a i workers (manning) to be carried out and has a duration w i measured in time periods. There is a fixed remuneration cost c k for working at any workday shift k (k = 1,..., S) which may differ between shifts (e.g. work in the morning shift may cost less than in the night shift). Under these conditions, the EMSP problem seeks to determine the workforce needed in each workday shift in order for all jobs to be completed within the planning horizon of D days at minimum cost. The main assumptions for the EMSP problem in relation to jobs operations and workforce availability are as follows. (1) Jobs are independent without precedence or other constraints. (2) Jobs are all available at time zero (that is, processing of any job may start at the first shift of the first day of the planning horizon). In addition, processing of any job need not be performed in consecutive periods (i.e. preemption is allowed). (3) Workforce is flexible and can operate on any job.

5 4 A.C. Nearchou et al. (4) Workforce is employed for a particular shift and cannot be moved to another shift. (5) Duration w i of any job i covers an integer number of periods and must be smaller than the total available periods in the planning horizon; w i SD. (6) Workforce remuneration costs c k strictly satisfy: c k c l for k > l. (7) No overtime is allowed. Although these assumptions may appear restrictive at first, they are quite realistic since they characterize most of the industrial environments where EMSP is found to apply (see Section 1 and references therein). It should be noted that even the assumption concerning the independence of jobs (Assumption 1 above), which is totally realistic in packing shop planning, also holds in maintenance planning provided the respective plan is designed at a somewhat aggregate level, as happens when time is measured in 8-hour units. Introducing the decision variable, { 1 if job i is assigned to shift k of day j x ijk = 0, otherwise EMSP can be represented by the following ILP formulation: subject to: z k Minimize : C = S c k z k k=1 N a i x ijk 0 (k = 1,..., S and j = 1,..., D) (1) i=1 D j=1 k=1 S x ijk = w i (i = 1,..., N) (2) x ijk {0, 1} i, j, k (3) The objective function of EMSP represents the total workforce cost C, being the sum of the respective costs incurred in all shifts. Constraint set (1) essentially defines z k as the workforce level assigned to shift k, ensuring that it is sufficient to cover the workforce needs for all jobs assigned to each period of this shift. Notice that owing to constraint set (1), z k = max j ( N i=1 a ix ijk ), which corresponds to the maximum manpower assignment among the periods of the respective shift. Constraint set (2) ensures that each job is assigned to a number of periods equal to its duration. So, all jobs can be completed within the available horizon and all jobs may be assigned to any period once (at most). Finally, constraint set (3) ensures that decision variables x ijk are binary. As formulated, EMSP is clearly an assignment problem where a solution corresponds to a complete assignment of the given set of jobs to available periods (day-shift combinations). This particular model effectively constitutes a special case of the model in Lagodimos and Mihiotis (2006) and is obtained by equating all overtime variables therein to zero. The model may also be considered as an extension to the MSP model in Lagodimos and Leopoulos (2000) by simply allowing for different remuneration costs between shifts. Therefore, since MSP has been found to be NP-hard in the strong sense, the same holds for EMSP as formulated above. This section closes with an association of EMSP with other known combinatorial problems. As shown in Lagodimos and Leopoulos (2000), for the special case where w i = 1 (for i = 1,..., N)

6 Engineering Optimization 5 and S = 1, the ILP formulation of EMSP is identical to the problem of scheduling a set of N independent tasks i (each with a processing time a i ) on D identical parallel machines aiming at minimizing makespan. We refer to Lagodimos and Mihiotis (2010) for an association of EMSP with the two-dimensional strip-packing problem. It is through such associations that bounds for the EMSP solution have been developed. 3. Heuristic solution algorithms This section, after discussing the general logic underlying their operation, presents three new greedy heuristic algorithms (termed mll, LCI and BP) for the EMSP solution Background and logic In order to explain the logic underlying the new algorithms, some background information is necessary. Consider an N-job EMSP problem and consider any feasible solution having workforce levels z k with k = 1,..., S. Clearly, by constraints (1), the total available capacity (in manperiods) is D S k=1 z k. Since the total capacity needed for completing all jobs (irrespective of their assignment to shifts) is N i=1 a iw i, then U = D S z k k=1 N a i w i (4) gives the total unutilized capacity pertaining from the particular solution. Notice that, for any EMSP setting, the workforce levels z k are the only variables entering unutilized capacity U. This implies that for the special EMSP setting without remuneration cost differences between shifts (i.e. c 1 = =c S ): (a) minimizing total cost C (see ILP formulation) is equivalent to minimizing unutilized capacity U in Equation (4); and (b) any solutions having identical total workforce levels Z = S k=1 z k are equivalent, irrespective of their assignment to shifts. Unfortunately, these properties do not hold in the general EMSP setting (i.e. c 1 c S ), since not total workforce Z but its individual elements z k (for all k = 1,..., S) determine total cost C. Therefore, while capacity utilization remains important, efficient workforce deployment in shifts with relatively small remuneration costs becomes critical. As already stated, the LL algorithm was developed for the solution of the special EMSP problem setting (with c 1 = =c S ). Under this algorithm, jobs are ordered in decreasing manning (hence its greedy characterization) and assigned to periods one job at a time covering their required duration (until all jobs are assigned). To keep job dispersion to shifts small, available shifts are first prioritized (using some dynamic shift-priority index) and are covered in this order. Only if a job s duration exceeds the feasible periods of the selected shift (due to constraints 2) is the next priority-shift used for assignment, and so on. Note that the assignment of any job to the periods of the selected shift aims at keeping workforce variability between periods minimal, so minimizing unutilized workforce. This is achieved by simply choosing the periods with the least assigned workforce (so far). A key factor in the operation of the LL algorithm is the shift-priority index used. As originally developed, shift-priority is decided on the basis of some capacity utilization-related criterion (see Lagodimos and Paravantis 2006 for a discussion of alternative such criteria). While such criteria are well suited for the special EMSP setting without remuneration cost differences between shifts, based on the previous discussion, they are clearly inappropriate for the general EMSP setting with remuneration cost differences between shifts. Provided, however, that shift-priorities were based i=1

7 6 A.C. Nearchou et al. Table 1. Data for an eight-job EMSP application example. Job Duration Manning on some cost-based criterion giving preference to shifts with low remuneration costs, then the LL algorithm could in principle be adapted for solving the general EMSP problem. This is the main idea underlying the three heuristic algorithms presented here. Although these heuristics are fully described below, it is worth noting that the two of them (mll and LCI heuristics) constitute direct applications of the LL algorithm logic, while the third (BP heuristic) applies this logic after decomposing the duration of all jobs and forcing all components consisting of D-period blocks to be assigned at the earliest possible shifts. In the presentation that follows, to facilitate comparisons, the presentation of each algorithm is followed by the solution of a common application example. This represents an eight-job EMSP problem, using the job-related data (i.e. manning and duration) shown in Table 1. It also assumes three workday shifts (S = 3 with remunerations c 1 = 1, c 2 = 2 and c 3 = 3) and a planning horizon of 10 days (D = 10) The modified LL (mll) algorithm mll carries out the assignment of jobs operation to periods in shifts in two successive stages, namely initial and shift assignment. The difference between these two stages relates to the set of periods used as candidates for assignment. In particular, during initial assignment, mll considers those periods for which possible assignment of the job under consideration cannot increase overall manpower needs. During shift assignment, the algorithm assigns job operation to periods in shifts, prioritized according to a dynamically evaluated priority index. The basic inputs for this algorithm, which are identical for all the developed algorithms, are: a list of N jobs, manning a i and duration w i required for each job i (i = 1,..., N) in the list, number of shifts S, number of days D in the planning horizon, and remuneration costs c k for each shift k (k = 1,..., S). Algorithm 1: The mll algorithm Input: a list of N jobs, a i and w i ( i = 1,..., N); S, D, c k ( k = 1,..., S) Output: a period list PL over the planning horizon of D days. begin Initialize PL (i.e. all periods in PL are initially empty); Sort the N jobs in decreasing order of manning; Set the jobs counter, i = 1; for all t = 1,..., SD do Set f t = 0 (counters for workforce assigned so far to periods); for all k = 1,..., S do Set z k = 0 (workforce assigned so far to shift k); repeat // 1 st stage: initial assignment // Set the periods counter, t = 0; Set the reserved periods counter, p = 0;

8 Engineering Optimization 7 repeat Advance periods counter, t = t + 1; Set L t = z k f t (calculate non-utilized workforce in period t of shift k); if a i L t then Set PL[i, t]=a i ; // i.e. reserve period t for job i Set f t = f t + a i (update workforce counter for period t); Set z k = max(k 1)D < t kd, k t{f t } (update workforce for shift kt); Advance reserved periods counter, p = p + 1; end if until (p = w i ) or (t = SD); Set R li = w i p;//residual load (i.e. remaining periods) to complete job i // 2 nd stage: shift assignment // if (R li = 0) then SetPL2 = PL;//make a replica of the original period list PL //trial allocation phase: work with the temporary list PL2 // for each shift k = 1 to S do Determine the number fp k of the free periods in shift k; Set fp k = min(rl i, fp k ); Compute C b which is the cost so far for shift k in PL2; Call LCI on PL2 to reserve fp k periods in shift k for job i; Compute the cost C a for shift k after this reservation; Set MCI k = (Ca Cb)/fp k ;//marginal cost increment for shift k; end for // shift prioritization and assignment phase // Sort the S shifts in increasing order of their index MCI k (k = 1,..., S); Set the shifts counter, k = 1; repeat // Keep in PL the same periods allocation for shift k as in PL2 // for all periods t k do Set PL[i, t] = PL2[i, t]; Set Rli = Rl i _ fp k ;//update the remaining load for job i Advance the shifts counter, k = k + 1; until (Rl i = 0) or (k > S); end if //end of 2 nd stage for job i // Advance jobs counter, i = i + 1; until i > N; return (PL); end; After sequencing the jobs in decreasing order of their resource requirements (manning), mll reserves (initial assignment) all possible periods (irrespective of the shift they belong to) that can accommodate the operation of the selected job without any workforce increase. In particular, an available (free) period t [1, SD] is selected for assignment for the operation of job i if the available (non-utilized) workforce for this period (L t ) is enough to cover the workforce needs of job i (i.e. a i L t ). The initial assignment stage is performed once for each job and terminated when either the required periods to complete the job have been covered or no other period from the set of the available periods can be assigned to the particular job. Shift assignment follows initial assignment and is activated only if there is residual load (Rl i = 0) for the job under consideration not assigned during the initial assignment stage of the algorithm. That is, shift assignment considers all the available periods of one particular shift, which is selected according to a performance criterion. This stage is accomplished in two successive phases, namely

9 8 A.C. Nearchou et al. Figure 1. Results obtained for the application example using the mll algorithm. the trial allocation phase and the shift prioritization phase. During trial allocation, jobs operation is assigned to periods using the logic of the LCI algorithm (see Subsection 3.3, below). The assignment is performed to each shift which has available (free) periods as if it is actually selected. Each time a period is allocated for the operation of a job, this period is considered as reserved (not available) for the remaining operation of the job. Thus, the aim with the trial allocation phase is to obtain all the necessary information regarding the actual implications of selecting each shift for assignment. This information is then used by the shift prioritization phase to establish the final order (priority) for considering each shift for the actual assignment. To that purpose, the magnitude of a specific performance criterion (priority index) for each shift is evaluated. The shift selection criterion adopted with the mll algorithm is to select that shift for which the job assignment provides the smallest marginal increment to the shift cost, i.e. select first the shift with minimum MCI k (k = 1,..., S). In case of ties, priority is given to the earliest shift. Marginal cost increment (MCI k ) denotes the cost increment in shift k per period due to trial allocation and computed as MCI k = (C a C b )/fp k, with C b and C a denoting the total labour cost of shift k so far before and after the trial allocation phase, respectively. Figure 1 shows the period list PL (i.e. the EMSP solution) generated by applying mll on the example in Table 1. Each PL line represents a particular allocation decision for the operation of the corresponding job. The entries in PL represent the workforce assigned to each period after following the allocation logic of mll. The final four lines in PL represent the total workforce (Total) allocated in each period (by summing all corresponding column entries), the total workforce (z k ) allocated in each shift (i.e. the maximum workforce allocated in all periods of the shift), the cost (c k z k ) of this allocation for each shift (taking into account the shift remuneration cost), and the total cost (C) for the obtained EMSP solution, respectively. For clarity of the allocation decisions illustrated in Figure 1, workforce assigned during the initial assignment stage of mll is shown in bold and italics. Observe, for example, the lines in PL associated with jobs 1 and 2. All the required workforce concerning job 1 has been allocated to periods during the shift assignment (second stage of the algorithm), while the required workforce concerning job 2 has been allocated to periods during the initial assignment stage (this is why they are given in bold and italics). As a result of this assignment, z 1 = z 2 = z 3 = 10. That is, 10 workers have been assigned so far to each one of the three shifts to carry out jobs 1 and 2. With a more careful observation of this assignment, one can easily realize that no workforce has been allocated so far to the last two periods of shift 3 (i.e. days 9 and 10 of shift 3). Therefore, the non-utilized workforce for these periods is L 9 = L 10 = 10. This permits mll to reserve (in the next step) days 9 and 10 of shift 3 for the needs of job 3 during the initial assignment stage (since a 3 > L 9, a 3 < L 10 ). The residual load (operation) of job 3 (Rl 3 = 9) is accomplished by allocating the first Rl 3 periods of shift 1 during the shift assignment stage of the algorithm (since, after the trial allocation phase for job 3,

10 Engineering Optimization 9 we have MCI 1 < MCI 2 < MCI 3, which means that shift 1 has the lowest marginal cost increment index, thus gaining priority for assignment over the other two shifts). Turning now to the solution derived by mll, Figure 1 shows that the total cost of the mll solution is C = 96, corresponding to a total workforce of 56 (obtained as the sum of the workforce allocated in all shifts); namely, 28 workers in the morning shift, 16 in the evening shift and 12 in the night shift. Solving the same problem with the CPLEX commercial optimizer (with a maximum time limit of 1 hour on a personal computer (PC) with a 2.11 GHz processor) the optimum solution C = 90 was obtained. It is worth noting that the lower bound of the EMSP solution for this example is C = 87, a result indicating the efficiency of this bound as a tool for assessing the quality of a solution (when optimality is unknown) (see Nearchou and Lagodimos 2013 for details) The Lowest Cost Increment (LCI) algorithm LCI constitutes a reduced version of mll. The only difference between the two algorithms is that, in each step, LCI considers periods one at a time according to the lowest cost increment rule until the required periods for the entire job have been covered. In contrast, as explained above, mll considers all the available periods of the selected shift (prioritized by the MCI index) and continues by considering the periods of the next priority shift only if there is a residual job load. In particular, for each job i (i = 1,..., N) in the jobs list, LCI reserves (by allocating the respective workforce a i ) the next available time period t [1, SD] in the period list PL for which possible assignment of a i workers results to the lowest (shift) cost increment CI ik (k = 1,..., S). The process of periods reservation for job i continues until all periods needed have been covered (i.e. until the periods reserved exactly equal its duration w i ). Decisions concerning the allocation of periods to jobs are made once and are not revised or altered during the execution of the algorithm. In this respect, LCI constitutes a single-pass heuristic. a characteristic greatly affecting its speed. This issue will be discussed in more detail in the experimental section of the article. Algorithm 2: The LCI algorithm Input: a list of N jobs, a i and w i ( i = 1,,N); S, D, c k ( k = 1,..., S) Output: a period list PL over the planning horizon of D days. begin Initialize PL; Sort the N jobs in decreasing order of manning; Set the jobs counter, i =1; for all t = 1,,SD do Set f t = 0 (counters for workforce assigned so far to periods); for all k = 1,,S do Set z k = 0 (workforce assigned so far to shift k); repeat Set the reserved periods counter, p =0; Repeat Determine the next available period t* for job i such that CI ik = min{c k ((f t + a i ) z k )} (k = 1,,S) (t = 1,,D), where CI ik denotes the cost increment of shift k when a i workers are assigned to period t for job i; Set PL[i,t*] = a i (i.e. reserve period t* for job i); Set f t = f t + a i (update workforce counter for period t*); Set z k = max (k 1)D<t kd, k t {f t } (update workforce for shift k t ); Advance reserved periods counter, p = p +1; until p = w i ;

11 10 A.C. Nearchou et al. Figure 2. Results obtained for the application example using the LCI algorithm. Advance jobs counter, i = i +1; until i > N; return (PL); end; The application of LCI to the example of Table 1 gave the solution depicted in Figure 2. As shown in this figure, the LCI heuristic arrived at a solution with objective function value C = 100 (deploying a workforce of 55), which is worse than the mll solution. However, these are just the results for a single test problem. The overall performance of the two algorithms will be analysed and discussed in the experimental section of the article (Section 4) through extended comparisons on a plethora of test problems The Block Planning (BP) algorithm BP combines the logic underlying the EMSP solution lower bound (Nearchou and Lagodimos 2013) with the LCI heuristic. The algorithm starts by considering jobs in decreasing order of their workforce requirements (see the pseudo-code below). For each job i in the job list, BP assigns the job s operation to periods by allocating the relevant workforce a i. The assignment logic used by this algorithm is based on the following two-stage mechanism: (1) Cover all the D-period blocks (i.e. all the D days) of the first w i /D shifts (where y denotes the biggest integer y). Since by assumption 6 (see Section 2) these are the cheapest shifts, this corresponds to minimum cost assignment for the particular subset of the required job s operation. (2) Assign the residual job s operation to the periods of the remaining shifts using a suitable assignment algorithm. In this respect, the required duration w i for each job i is decomposed into two subsets, q i and r i. The former consists of w i /D D period blocks which can be assigned to cover all days of the first w i /D shifts. The latter (r i ) denotes the residual job s operation. The BP heuristic assigns the residual operation of the job under consideration to the available (free) periods using the LCI algorithm. Algorithm 3: The BP algorithm Input: a list of N jobs, a i and w i ( i =1,,N);S, D, c k ( k = 1,,S) Output: a period list PL over the planning horizon of D days. begin

12 Engineering Optimization 11 Figure 3. Results obtained for the application example using the BP algorithm. Initialize PL; Sort the N jobs in decreasing order of manning; Set the jobs counter, i =1; for all t = 1,,SD do Set f t = 0 (counters for workforce assigned so far to periods); for all k = 1,,S do Set z k = 0 (workforce assigned so far to shift k); repeat // separate w i into two subsets q i and r i Set q i = w i /D D;//where y denotes the biggest integer y Set r i = w i q i ; if w i /D = 0 then // Reserve all the periods of the first w i /D shifts for job i for each period t =1to q i do Set PL[i, t] =a i Set f t = f t + a i (update workforce counter for period t); end for for each shift k =1 to w i /D do Set z k = max (k 1)D<t kd {f t } (update workforce for shift k); end for end if if r i = 0 then Call LCI algorithm to reserve the remaining r i periods for job i; end if Advance jobs counter, i = i + 1; until i > N; return (PL); end; Application of the BP algorithm to the numerical example in Table 1 obtained the solution depicted in Figure 3. The cost of this solution is C = 92 (deploying a workforce of 55 as by the LCI heuristic), which outperforms the solutions obtained by the mll and the LCI heuristics. 4. Computational experiments This section presents computational results obtained for assessing the quality of the proposed heuristics. All experiments were performed on a 2.11 GHz PC with 2 GB RAM and the Windows

13 12 A.C. Nearchou et al. XP operating system. The new algorithms (mll, LCI and BP) were implemented in Borland Delphi Pascal programming language Design of experiments A two-phase numerical investigation was carried out. The first phase aimed at evaluating the performance of the three new algorithms relative to the performance of the best existing EMSP heuristic (the μga/vns algorithm) in Nearchou and Lagodimos (2013) as well as the solutions from a popular optimization package, the CPLEX optimizer (version 12.2/2010). The second phase aimed at a formal ranking of the algorithms on the basis of their efficiency through a comprehensive statistical analysis. The first investigation phase used the benchmarks data set introduced in Nearchou and Lagodimos(2013) for testing μga/vns. This data set includes 120 test instances involving three classes of problem (40 instances per class) with eight, 20 and 50 jobs, respectively. All correspond to a shop operating three workday shifts over a planning horizon of 10 days (i.e. S = 3 and D = 10) and respective remuneration costs per shift equal to c 1 = 1, c 2 = 2 and c 3 = 3. Each test instance is formed as a combination of particular manning and duration specifications. These specifications are given by a particular (manning and duration) profile selected to cover many different operating environments. The 120 instances used here prevail as the result of all possible combinations of five manning (M 1 M 5 ) with eight duration (P 1 P 8 ) profiles (see Nearchou and Lagodimos 2013 for full specifications 1 ). All the test problems were first solved with the proposed algorithms and then with CPLEX. For the latter, a stopping condition was set at 3600 seconds (1 hour) in case no optimal solution was reached before then. CPLEX was used in its standard setting without any alteration of its branch-and-bound implicit enumeration algorithm. The second investigation phase considered shops with 50 and 100 jobs also operating three daily shifts over a horizon of 10 days and remuneration costs per shift equal to c 1 = 1, c 2 = 2 and c 3 = 3. Real-world applications typically involve no more than 20 lines for packing shops and 100 jobs in the maintenance industry, hence the selection of these particular problem sizes in the overall experimentation. Multiple test instances were generated varying the two EMSP variables as follows: job duration was kept unchanged (at one level) and manning varied at three levels to form three different EMSP settings. In particular, duration was sampled from a U(1, 29) distribution, ensuring the consistent generation of three-shift problems. Manning was sampled from three different distributions, thus covering a variety of shop configurations; the distributions used were U(1, 2), U(1, 5) and U(1, 10). Within this experimental framework, 1000 independent EMSP test instances were generated for each setting, and solved by the three new heuristics (mll, LCI and BP) and the μga/vns heuristic. So, for each test instance, four different solutions were obtained, one for each algorithm. All computations involving μga/vns were carried out using the control parameter settings in Nearchou and Lagodimos (2013). In addition, to consider the stochastic behaviour of μga/vns fairly, this algorithm was run five times on each test problem (starting each time from a different random seed) and used the best solution obtained. The results of the second investigation phase were tested for statistical significance. Considering the solution (cost) of each algorithm as a random variable, it was observed that these variables violated the required conditions for normality. So, in order to rank the algorithms, the Friedman test (e.g. Sprent and Smeeton 2007) was chosen, a powerful non-parametric test for detecting statistically significant rank differences across samples. Using this, the null hypothesis was tested that the different samples were drawn from distributions with the same median versus the alternative hypothesis that at least one median was different from the rest (i.e. non-directional test mode).

14 Engineering Optimization 13 Deviation from CPLEX (%) LCI mll BP µga/vns 12% 10% 8% 6% 4% 2% 0% -2% 8 jobs 20 jobs 50 jobs Figure 4. Table 2. Percentage deviation of each algorithm from CPLEX solution. Summary results of the first investigation phase. LCI mll BP μga/vns No. best C (for 8/20/50 jobs) 4/13/2 5/11/6 11/28/36 40/7/0 No. worst C 33/14/8 28/13/4 21/2/0 2/21/31 No. C = C(CPLEX) 4/1/0 5/1/0 12/5/4 38/0/0 No. known optimal 3/1/0 3/1/0 8/5/4 38/0/0 No. C < C(CPLEX) 0/1/0 0/0/0 0/1/0 2/0/0 % Average deviation from C(CPLEX) 8.3/8.5/ /8.6/ /5.2/ /8.5/ Results and discussion The detailed results of the experiments for each combination of manning and duration settings are given in Appendix. Figure 4 summarizes the results showing the average percentage deviation of the solutions given by each algorithm with respect to the corresponding CPLEX solutions. As can be observed from Figure 4, μga/vns outperforms all new algorithms in the case of the eight-job instances. In fact, μga/vns finds strictly better solutions than CPLEX in two eight-job instances and the exact CPLEX solution in the remaining eight-job instances, which explains the negative deviation of μga/vns from CPLEX in this case (-0.06%). In the 20- and 50-job instances, however, the comparison is not directly obvious. To facilitate this, Table 2 was formed, which summarizes the comparative results corresponding to all test problems considered in the first investigation phase. Each row of Table 2 indicates a specific performance metric while columns correspond to the four algorithms considered. Each entry in the table includes three numerical values (separated by / ) corresponding to the three problem classes, i.e. eight-job/20-job/50-job problems. The first and second lines of the table show the occasions on which an algorithm achieved the best solution (of all algorithms) and the worst, respectively. The third line shows the occasions on which an algorithm gave a solution equal to CPLEX. The fourth and fifth lines report, respectively, the occasions on which an algorithm achieved the optimal solution, and when it outperformed CPLEX in the sense that it generated solutions of lower cost. Finally, the last line displays the average percentage deviation of the solution of each algorithm relative to the CPLEX solution, calculated over the 40 instances of each problem class. The metrics in Table 2 allow a number of observations. For small-size problems (eight-job instances), LCI and mll display similar results for all metrics while BP appears superior. For this class of problem, μga/vns clearly outperforms all new heuristics in all metrics. As problem size increases, however, the relative performance of the new heuristics drastically improves.

15 14 A.C. Nearchou et al. Specifically, for medium-size problems (20-job instances), LCI and mll appear to perform identically to μga/vns (with an average deviation from CPLEX of approximately 8.5%), while BP clearly outperforms all others in nearly all metrics. Noticeably, BP has the worst solution in only two cases, whereas each of the other three algorithms is the worst in at least 13 cases. Finally, for large-size problems (50-job instances), all new heuristics clearly outperform μga/vns in all metrics; in fact, BP remains the best algorithm, followed by mll and LCI. What is remarkable about BP is that its performance seems to improve with increased problem size, showing the smallest deviation compared with CPLEX for large-size problems. It should be noted that, considering the processing time required for arriving at a solution, the three new heuristics are substantially faster than μga/vns, requiring (on average) less than 0.4 ms of CPU time for a solution for eight- and 20-job problems, and less than 1 ms for 50-job problems. The respective time for μga/vns varied between 3 seconds (s) (eight -jobs) and 34 s (50 jobs) (see Nearchou and Lagodimos 2013 for more details). Turning to the CPLEX solution times, it is worth noting that for the test instances where CPLEX reached the exact optimal solution, these times fluctuate significantly depending on each particular data set, ranging from 0.01 s to 2484 s run-time. In fact, the maximum CPU time of 2484 s was observed in an eightjob test instance! This is in line with the work of Cornuéjols, Karamanov, and Li (2006), which showed that in integer programming, problem size and CPU requirements are not linearly related. For the instances where no optimal solution was found, the CPLEX solution was retained after the stopping condition was activated at the set run-time limit (3600 s). The effects of the duration and manning specifications on heuristic solution quality have also been explored. Since μga/vns is the best performing algorithm in the eight-job problems, while BP is the best in 20- and 50-job problems, only these two algorithms were studied. Table 3(a) (c) shows the effectiveness of BP and μga/vns relative to CPLEX (percentage deviation of heuristic solutions from CPLEX) over eight-job, 20-job and 50-job instances, respectively. For each cell (i.e. duration manning profile combinations), each table shows two entries reported in two lines. The value of the first line corresponds to the BP result over the relevant problem size, while the value of the second line (given in italics) corresponds to the μga/vns result. For instance, for the case of M 1 :P 1 profile combination, the deviation of the BP solutions from CPLEX was 3.7% for the eight-job (Table 3a), 3.23% for the 20-job (Table 3b) and 3.13% for the 50-job problem (Table 3c). The respective deviation of the μga/vns solutions for the same problems was 0%, 6.45% and 7.5%. Each table also presents averages over all durations (DAV) (last column) and manning profiles (MAV) (last line). First considering the effects of problem size, the results of Table 3(a) (c) indicate that the μga/vns solution quality deteriorates with increased problem size. This happens for all aggregate metrics (i.e. MAV and DAV entries). As far as the manning profile is concerned, solution quality is generally better when manning requirements are not evenly distributed across jobs (as for profiles M 3 and M 4 ), whereas quality deteriorates significantly for even manning (see profile M 1 ). Quality seems to improve when durations are distributed evenly (profile P 1 ) or follow a parabolic distribution (profile P 4 ). To a large extent, these findings are consistent with those in Nearchou and Lagodimos (2013), where the μga/vns was assessed relative to an optimal solution lower bound. Unlike μga/vns, the quality of the BP solutions does not seem to deteriorate with increased problem size. In fact, there are cases where solution quality over 50-job instances is superior to that achieved over 20-job instances. See, for example, the results concerning profiles P 2, P 5 and P 7. Considering manning profiles, there does not seem to be a significant effect on BP solution quality, and the same appears to hold for the duration profiles (see MAV and DAV entries in Table 3). In conclusion, BP appears to be better suited to large-size instances and more capable of coping with different duration and manning requirements in comparison with μga/vns.

16 Engineering Optimization 15 Table 3. Effectiveness of BP and μga/vns relative to CPLEX for (a) eight-job, (b) 20-job and (c) 50-job test instances. P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 DAV (a) M M M M M MAV (b) M M M M M MAV (c) M M M M M MAV Note: DAV = averages over all durations; MAV = manning profile. The results of the second investigation phase, aimed at ranking the existing EMSP solution algorithms in terms of their efficiency, are now examined. As already stated (Section 4.1), 1000 repetitions were performed and the classical Friedman non-parametric test for detecting differences across multiple samples was employed (see Derrac et al for a recent application of the test in algorithms ranking). Like many non-parametric tests, the Friedman test uses data ranks as opposed to absolute data values. Table 4 summarizes the comparative results of the second investigation phase over the 1000 test instances of each setting and each problem class. Each entry in the first four rows in this table represents the average ranking of the corresponding algorithm with respect to the solution cost for each manning setting examined. The table also shows the Friedman statistic (fifth row) and the corresponding p-value (last row), namely the probability of the differences across the four algorithms being due to chance.

17 16 A.C. Nearchou et al. Table 4. Second investigation phase: average ranking of the algorithms performance. Job manning 50-job 100-job Algorithm U(1, 2) U(1, 5) U(1, 10) U(1, 2) U(1, 5) U(1, 10) LCI mll BP μga/vns Friedman statistic p-value <0.001 <0.001 <0.001 <0.001 <0.001 <0.001 As indicated in Table 4, there are highly statistically significant differences between all algorithms in both problem-size classes (see p-value in the last line of the table). Although on the basis of the average ranks BP appeared to be the best, it is not possible to safely determine the full ranking of the algorithms unless all possible pairs are compared. Such multiple comparisons, however, result in an accumulated statistical error known as the family-wise error rate (FWER), defined as the probability of drawing one or more false conclusions among all necessary comparisons. A post hoc analysis was therefore carried out to determine which algorithms were statistically different from each other, based on their mean rank differences. This analysis used Shaffer s static procedure (Shaffer 1986), a well-known step-down process for testing the statistical significance of multiple hypotheses. At any stage s of this process, two algorithms are compared under the null hypothesis that they are identical versus the alternative that they differ. The null hypothesis H s is rejected if p s α/t s, where p s is the p-value obtained for the sth hypothesis and t s is the maximum number of hypotheses that can be true given that all previous hypotheses are false. Note that Shaffer s static procedure was chosen, rather than its dynamic counterpart, because of its relative simplicity and power (Derrac et al. 2011). The results of the post hoc analysis over the various settings are illustrated in Table 5. Each row of the table shows the stage s of the Shaffer procedure, the hypothesis being tested (i.e. the pair of the algorithms compared), the corresponding p-value for each setting, the t s value and the value of the ratio α/t s. Note that, in each stage of the test, the calculated value of the ratio α/t s was identical for all three settings of both classes of problem and therefore is given once (final column of Table 5). Consider, for example, the second line of the table. Here, stage 5 of the process (comparison of BP versus mll) gave t s = 3 and α/t s = This means that at most three of the remaining hypotheses may be true given that the previous hypothesis (that of stage 6) is rejected. Moreover, since the p-value in all entries of this line is smaller than the ratio α/t s, the hypothesis of stage 5 is rejected for all settings. That is, BP is statistically different from mll. Considering the 50-job problems, the only hypothesis not rejected concerns the comparison of mll and μga/vns for the U(1, 2) manning distribution. In the cases of U(1, 5) and U(1, 10) settings (profiles with increased variability), mll performs significantly better than μga/vns (as indicated by the rankings in Table 4 above). In all other comparisons, statistically significant differences prevail, BP being by far the best algorithm, followed by LCI, mll and μga/vns (see average rankings in Table 4). Turning now to the 100-job problems, based on Table 5, all pairwise differences are statistically significant. In fact, under all manning settings, the ranking of the algorithms remains the same; namely, BP is found to be best, followed by LCI, mll and μga/vns. Furthermore, the average ranks of the algorithms (Table 4) indicate that the superiority of BP is even more pronounced in the 100-job instances than in the 50-job instances. This implies that as the size of the problem increases, BP displays an even better performance than the other algorithms.