TY - JOUR
AU - Lagodimos, A, G
AB - Abstract Consideration is given to a personnel planning problem occurring in multisite projects. In this problem a set of facilities (different sites) need to complete a given work load over a planning horizon of several days. The workforce is skilled and can work on any facility and it is employed to work on a basis of workday shifts. A fixed set-up cost associated with each facility is incurred each time a facility starts operating. The problem seeks to determine when each facility should be operated as well as the manpower needed at each shift to complete the workload targets of all the facilities within the planning horizon at a minimum cost. First, a formulation of the new problem as a mixed-integer linear programme is given. Then, since the problem is shown to be NP-hard, a novel genetic algorithm (GA) is presented for its solution which incorporates a problem-specific coding together with new special merging rules for creating offspring that exploits the structure of the problem. Using results from the standard CPLEX optimizer we test the performance of the GA for a variety of operating environments. The results demonstrate very satisfactory performance for the GA in terms of both solution time and quality. 1. Introduction In today’s business world it is well recognized that manpower should be considered as a resource that is as critical as capital. One of the reasons for this is that manpower directly affects available production capacity. Therefore, efficient manpower utilization involves the solution of several planning-related problems. Usually these problems include two main sets of decisions: firstly, how many workers to employ to achieve production targets; and, secondly, how to assign these workers to shifts in order to cover the demand for resources that vary over time. The former decision is known as manpower planning while the latter is known as manpower or personnel scheduling. Often the two decisions are faced sequentially in a multiphase activity to achieve tractability (Burch & Qiu, 1997). In an initial phase the number of employees needed for each timeslot of the planning horizon is determined. Then, the activity proceeds to determine possibly the number of shifts and the total number of employees needed to cover each shift. Finally, the assignment of the employees into shifts and, perhaps, days off to employees is performed. Several modern approaches, however, consider these decisions simultaneously aiming to complete all of the operations in a reduced amount of time or/and to complete them with a reduced number of workers. Examples of integrating manpower planning and scheduling can be found in many real-world applications including healthcare (Cheang et al., 2003), telephone operators in call centres (Wallace & Whitt, 2005), manufacturing (Lagodimos & Leopoulos, 2000; Nearchou et al., 2015) and constructions (Jun & El-Rayes, 2010). This work follows the latter approach and focuses on the solution of a new manpower planning problem referred to herein as manpower shift planning with set-up costs (MSPSC). Typical applications of this problem can be found in multisite construction projects, such as building development and highway construction. In such projects, a common requirement often encountered is the timely distribution of various groups of skilled personnel to different sites to perform prespecified work tasks within a desired time horizon. In highway construction projects for example, such skilled personnel include plumbers and/or electricians to carry out work related to the water and electricity supply networks, respectively: plumbers to lay and repair water mains and pipes, install new pipes and pumping stations, interrupt the operation of existing pumping stations to make necessary inspections at selected points of the network, install fire hydrants etc. and electricians to do electrical maintenance operations inside tunnels, on bridges or at busy exit points along the highways. This personnel is allocated to work on a particular workday shift (i.e. day, evening or night shift). Therefore, a planning problem faced by the management is to decide how many personnel of this category to allocate in every daily shift of some particular planning horizon so as to complete the total task of each construction site at the minimum cost. Additionally, since each construction site is based at a different location, a set-up cost is associated with each site and is incurred whenever a group of such skilled personnel is sent to that site. Hence, each site may be thought of as a facility that is opened whenever a crew starts working there. In order to avoid excessive transfers of workforce, it is desired that if a facility is opened, then it must operate for at least a certain number of periods. Since this opening cost may vary according to the location of the site and the time period, it makes sense to decide when to open each facility and how to distribute the manpower among the different shifts in order to minimize the total cost (TC). To the best of our knowledge, the MSPSC problem has never been formally addressed in the past. This observation stems from the literature review performed (Section 2) with the aim of positioning MSPSC within the relevant research field. Within this general background, a modified genetic algorithm (GA) is established for solving the MSPSC problem. The novelty of the developed GA is summarized below and constitutes the outcome of a comprehensive state-of-the-art review on existing GA-based methods for solving manpower planning problems (Section 4 analyses this issue in detail). In addition to the standard GA features, our GA (a) incorporates a problem-specific encoding of the solution structure. Individuals in the population on the level of the genotype are represented as two-dimensional binary matrices and not as vectors (binary, real-valued or integer) as traditional. Phenotypes, that is, solutions to the MSPSC problem are obtained through a decoder heuristic which performs the actual allocation of the jobs’ respective workforce to time periods to shifts. (b) Offspring solutions are created using special crossover and mutation operators that exploit the structure of the problem. These operators have been carefully designed to be effectively applied on the genotypes to always create feasible offspring solutions. Hence, no repairing routines are needed to transform infeasible solutions to feasible ones. This is an advantage over the existing GAs in the related literature which uses special procedures to repair infeasible solutions. These GAs are time consuming and may lead to poor material within the population as a result of not enough incentive for development (see Michalewicz & Fogel, 2000, for more about this issue). (c) It uses a dynamic control scheme for tuning mutation probability during its run. Particularly, the value of this parameter is readjusted in every iteration by taking some form of feedback from the genetic search related to the population’s diversity. Note that most of the existing GAs in the field of manpower planning use a fixed (user-defined) mutation probability; the value of which does not change during their run. The rest of the paper is organized as follows. Section 2 gives a survey of the related literature. Section 3 states the relevant assumptions and presents a mixed-integer linear programming (MILP) model for the problem. Section 4 introduces a novel GA for solving this problem. Experiments and comparisons between GA and results obtained by the CPLEX commercial optimizer are presented and analysed in Section 5. Finally, conclusions and indications for further research are given in Section 6. 2. Literature review There has been an abundance of research in the field of manpower planning and scheduling aimed at solving problems in functional areas within services and manufacturing (Bechtold et al., 1991; Jarrah et al., 1994; Alfares, 2000; Lai et al., 2002; Bard et al., 2003; Alfares, 2004; Ernst et al., 2004; Brecht et al., 2010; Van den Bergh et al., 2013; De Bruecker et al., 2015; Castillo-Salazar et al., 2016). Below we discuss these issues by considering direct services and manufacturing. In order to achieve satisfactory manpower utilization, service organizations deploy their personnel into so-called flexible shifts. These are work-time patterns that effectively distribute working hours over the workday while maintaining a given duration for each shift (usually 8 hours per workday). Hence, the major manpower planning problem in service environments concerns the determination of the personnel to be assigned to a number of flexible shifts so as to ensure the timely operation of all service points while optimizing some performance measure (usually manpower cost). One of the first attempts in the field was the pioneering work of Dantzig (1954) primarily concerned with the determination of the workforce needed by a given set of work-time patterns so as to minimize total labour costs. However, the problem today is quite different from that introduced by Dantzig in the 1950s. Companies now offer part-time contracts or flexible work hours and take into account employee preferences when creating work schedules. In its most general form where one practically considers any work-time pattern, the manpower planning problem in services is known as the tour scheduling problem. Tour scheduling integrates days off and shift scheduling. This integrated process involves choosing the days off for the workers and allocating shifts for each of their working days over the planning horizon. Tour scheduling reduces to shift scheduling when the planning horizon is one day and to days-off scheduling when only one shift exists on each day (see Ernst et al., 2004, for a survey on this topic). For over 60 years tour scheduling has been the focus of extensive research whose results are often summarized in the form of state-of-the-art reviews. The majority of the published works concern mainly applications in transportation, call centres and healthcare. See, for example, the earlier reviews of Bechtold et al. (1991), Jarrah et al. (1994) as well as the most recent reviews of Bard et al. (2003), Cheang et al. (2003), Alfares (2004), Ernst et al. (2004), Kohl & Karisch (2004), Van den Bergh et al. (2013), De Bruecker et al. (2015) and Castillo-Salazar et al. (2016). It is worth noting that any variation of tour scheduling has been shown to be NP-complete (Bartholdi, 1981) and therefore most of the recent associated research concentrated in developing efficient heuristic solutions. One of the first studies of manpower planning in manufacturing was Lagodimos & Leopoulos (2000). This problem, termed manpower shift planning (MSP), formed part of the master production planning activity of a food and beverage producer and concerned the monthly staffing of the packing shop. Assuming identical manpower remuneration costs between shifts, MSP seeks for the minimum workforce needed to work in each shift over a given planning horizon in order to complete predetermined production objectives related to the packing lines. Interestingly, Chang et al. (1999) and Pan et al. (2010) identified similar problems in steel industry and precision engineering industry, respectively. The aim by Chang et al.’s (1999) study was to minimize the workforce level required to complete a set of maintenance jobs within predetermined time windows. Each job has a known total work content (given in man-days) and its duration may vary according to the exact manning used. Special features of the problem studied by Pan et al. (2010) were the assignment of operators to machines by considering skill requirements and operator-expressed preferences for shifts and on/off days. Focussing on a special setting where all packing lines have identical manning, Lagodimos & Mihiotis (2006) modified MSP allowing for different shift-related costs and overtime thus defining the economic MSP problem (EMSP). Both MSP and EMSP have been shown to be NP-hard. The solution of the EMSP problem in the general case, which is allowing for different remuneration shift costs and unequal manning, was studied by Nearchou et al. (2015) through the means of three very fast greedy heuristics. An adaptation of one of these greedy heuristics, namely, the lowest-cost-increment (LCI) algorithm, was also used in accordance with a GA to tackle EMSP with even higher success in terms of solution quality (Nearchou et al., 2014). This previous GA encodes the decision variables of the problem as an integer vector separated into two parts: the first part represents a job list while the second part represents a shift list for starting jobs’ operation assignment to periods. Solution decoding (i.e. the actual allocation of the jobs’ workforce to time periods) is performed by the LCI algorithm. Offspring solutions are generated using a modified one-point crossover and a shift-mutation operator. Recently, some researchers turned their attention to workforce scheduling in environments involving the mobilization of personnel at different locations in order to perform work activities (see Castillo-Salazar et al., 2016, for a survey on this issue). Typical examples of these situations are nurses visiting patients at their homes to administer medication or provide treatment, care workers aiding members of the community to perform difficult tasks (Eveborn et al., 2006), technicians carrying out repairs and installations (Cordeau et al., 2010) as well as security guards performing night rounds on several premises. These problems combine features from both workforce scheduling and vehicle routing since it is assumed that employees have to travel long distances between locations to do the jobs. Therefore, reducing the travel time is of major importance since it could potentially increase productivity. The MSPSC problem can be classified as a manpower planning problem involving allocation of skilled personnel at different sites where special tasks need to be performed. Every time a group of skilled personnel arrives at a site to do the job, a set-up cost is incurred. This cost is related to preparatory operations needed to be carried out at the particular site before these personnel start working there. In terms of highway construction projects, the preparations may include cleaning and setting machines and tools, transferring and arrangement of materials, setting up the required equipment for communicating with the headquarters, fuelling of vehicles etc. These operations require time that is essentially non-productive in the sense that it is not related to the particular tasks that must be carried out at each site. In order to reduce this non-productive time, a limit is imposed on the number of times this set-up process is performed. We know of no previous research that has studied this problem within either manufacturing or services context. 3. Problem formulation In this section we present a formal statement of MSPSC and develop the corresponding optimization model. We also discuss its principal features together with its computational complexity. 3.1. Formal problem definition A set of |$M$| facilities should operate for a number of |$T$| time periods in a particular time horizon of |$D$| days and |$S$| daily shifts (i.e. |$T=S.D$|⁠), in order to offer a service or cover a demand of some sort. Each open facility |$j$| requires a number of |${a}_j$| skilled workers for its operation and has a workload |${w}_j$|⁠, i.e. it must operate for a total of |${w}_j$| time periods in the planning horizon. Opening a facility |$j$| at time period |$t$| implies a fixed cost equal to |${f}_{tj}$|⁠. This fixed cost may involve the cost of fuel for vehicles, machinery spare parts or other cost elements which may vary according to the location of the facility as well as the time period. The fixed cost |${f}_{tj}$| can be thought of as a set-up cost and is incurred only in the initial time period whenever the facility is opened. As mentioned earlier, the set-up process at each location |$j$| usually implies non-productive time |${q}_j$| which must be kept to a minimum. Since it may not be practical to open and close facilities without any restrictions, we assume that a facility may open at most |$k$| times within the specified time horizon. The requirement that each facility |$j$| may open at most |$k$| times, in conjunction with its workload |${w}_j$|⁠, effectively implies that if the facility opens at some time period, then it must remain open for a minimum number of |${n}_j$| successive periods. In general, |${n}_j=\lfloor \frac{w_j}{k}\rfloor$|⁠. For instance, if |${w}_j=23$| and |$k=5$|⁠, then each time facility |$j$| opens, it must remain open for at least four periods. The only exception is when |$\operatorname{mod}\ ({w}_j,k)=\lfloor \frac{w_j}{k}\rfloor$| where mod(•) denotes the remainder of the division. In this case, facility |$j$| must remain open for at least |${n}_j=\lfloor \frac{w_j}{k}\rfloor +1$| periods each time it opens. For instance, if |${w}_j=24$| and |$k=5$|⁠, then the facility must remain open for at least five periods at a time, so that the total number of times that it opens does not exceed five. Finally, each shift |$s$| is associated with a remuneration cost |${c}_s$|⁠. This may differ for each shift since the shifts may refer to different periods of the day (morning, afternoon or evening) with different manpower costs. The MSPSC problem consists of determining both the set of facilities to open and the minimum number of workers to be assigned in each shift (⁠|$s\in S$|⁠) in order to complete the required work on all the facilities within the predetermined time horizon of the |$D$| days at the lowest cost (total opening cost plus manpower cost). Throughout the analysis, the following operational assumptions are used: Workforce is skilled and flexible so that it can serve any facility. Workforce is assigned to a particular shift and cannot be moved to another shift. Workload |${w}_j$| of any facility |$j$| covers an integer number of periods (day-shift combinations) and can be completed within the available productive time, excluding the time that may be required for set-up; that is, |${w}_j\le S\cdot D-k\cdot{q}_j$|⁠. Workforce remuneration rates |${c}_v$| strictly satisfy |${c}_v\ge{c}_u$| for |$v>u$|⁠. Facilities can open (and close) at most |$k$| times. All facilities are unrelated. No constraints or operation dependencies exist between the facilities. 3.2. The mathematical model The necessary notations and parameters for the problem are summarized as follows: Indices |$j=1,\ldots,M$| facilities index |$t=1,\ldots,T$| periods index |$s=1,\ldots,S$| shifts index. Parameters |$D$| means total number of days to work. |$S$| means number of shifts per day. |$T$| means total number of time periods, namely, |$T=S\cdot D$|⁠. |${f}_{tj}$| means fixed cost of opening facility |$j$| at time period |$t$|⁠. |${w}_j$| means workload of facility |$j$|⁠, i.e. number of periods that facility |$j$| must operate. |${a}_j$| means staffing requirements at facility |$j$|⁠. |${c}_s$| means staffing cost for shift |$s$|⁠. |$k$| means maximum number of times permitted for opening each facility over |$T$|⁠. |${n}_j$| $$=\begin{cases}\left\lfloor \frac{w_j}{k}\right\rfloor +1,\quad \mathrm{if} \operatorname{mod}\big({{n}}_{{j}},k\big)=\left\lfloor \frac{w_j}{k}\right\rfloor \\{}\\ \left\lfloor \frac{w_j}{k}\right\rfloor\!, \qquad\;\;\, \mathrm{otherwise} \end{cases}$$ |${p}_{ts}$| $$=\begin{cases}1,\kern0.5em \mathrm{if}\ \mathrm{time}\ \mathrm{period}\ t\ \mathrm{is}\ \mathrm{in}\mathrm{cluded}\ \mathrm{in}\ \mathrm{shift}\ s\\{}0,\,\,\, \mathrm{otherwise.}\kern10.25em \end{cases}$$ Decision variables |${x}_{tj}$| $$ =\begin{cases}1,\kern0.5em \mathrm{if}\ \mathrm{facility}\ j\ \mathrm{is}\ \mathrm{operating}\ \mathrm{during}\ \mathrm{period}\ t\ \\{}0,\,\,\, \mathrm{otherwise}\kern11.75em \end{cases}$$ |${z}_s$||$=\mathrm{number}\ \mathrm{of}\ \mathrm{workers}\ \mathrm{employed}\ \mathrm{in}\ \mathrm{shift}\ s$| |${\delta}_{tj}$| $$=\begin{cases}1,\kern0.5em \mathrm{if}\ \mathrm{facility}\ j\ \mathrm{opens}\ \mathrm{at}\ \mathrm{period}\ t \\{}0,\,\,\, \mathrm{otherwise.}\kern7.7em \end{cases}$$ According to the above definitions, MSPSC can be formulated by the MILP shown below: $$ \operatorname{Minimize}\kern0.5em TC=\sum \limits_s{c}_s\cdot{z}_s+\sum \limits_t\sum \limits_j{f}_{tj}\cdot{\delta}_{tj}.$$ Subject to $$\begin{equation} \hskip-143pt\sum \limits_t{x}_{tj}={w}_j\qquad\qquad\qquad\qquad\qquad \mathrm{for}\ \mathrm{all}\kern0.5em j \end{equation}$$ (1) $$\begin{equation} \hskip-130pt{z}_s\ge \sum \limits_j{a}_j\cdot{p}_{ts}\cdot{x}_{tj}\quad\quad\quad\qquad\qquad\!\! \mathrm{for}\ \mathrm{all}\kern0.5em s,t \end{equation}$$ (2) $$\begin{equation} \hskip10pt{x}_{t^{\prime }j}\ge{\delta}_{tj}\qquad\qquad\quad\qquad\qquad\quad\quad\ \! \mathrm{for}\ \mathrm{all}\kern0.5em j,t,{t}^{\prime}\kern0.5em \mathrm{with}\kern0.5em {t}^{\prime}\ge t\kern0.5em \mathrm{and}\kern0.5em {t}^{\prime}\le t+{n}_j-1 \end{equation}$$ (3) $$\begin{equation} {\hskip-70pt\delta}_{tj}\ge{x}_{tj}-{x}_{\left(t-1\right)j}\qquad\qquad\quad\qquad\quad\!\!\! \mathrm{for}\ \mathrm{all}\kern0.5em j,t\kern0.5em \mathrm{with}\kern0.5em t\ge 2\kern1.25em \end{equation}$$ (4) $$\begin{equation} \hskip-136pt{\delta}_{1j}\ge{x}_{1j}\qquad\qquad\qquad\qquad\qquad\; \; \mathrm{for}\ \mathrm{all}\kern0.5em j \end{equation}$$ (5) $$\begin{equation} \hskip-135pt{x}_{tj}\kern0.5em \in \left\{0,1\right\}\ \mathrm{and}\kern0.5em {\delta}_{tj} \in \left\{0,1\right\}\kern2.75em\,\, \mathrm{for}\ \mathrm{all}\kern0.5em t, j \end{equation}$$ (6) $$\begin{equation} {\hskip-130ptz}_s\ge 0\qquad\quad\qquad\qquad\quad\qquad\;\;\;\;\mathrm{for}\kern0.3em \mathrm{all}\kern0.5em s. \end{equation}$$ (7) The objective function expresses the minimization of TC. The first term expresses the staffing cost of all shifts, whereas the second term corresponds to the total opening cost of all facilities. Constraint set (1) specifies that each facility |$j$| must operate for a total of |${w}_j$| periods. Constraint set (2) defines the total number of workers employed during shift |$s$|⁠. Note that parameter |${p}_{ts}$| simply indicates whether or not time period |$t$| is included in shift |$s$|⁠. Constraint set (3) ensures that if a facility is opened at period |$t$|⁠, then it must operate for at least a certain number of |${n}_j$| periods. For instance, if |$k=2$| in constraint set (3), then the facility must remain open for at least half its specified workload. In conjunction with constraint set (1), this implies that the facility may open at most |$k$| times during the planned time horizon. These constraint sets are necessary to ensure that even when the set-up costs for opening the facilities are sufficiently small, facilities may not open more than |$k$| times which would be impractical. Without loss of generality, this study investigates the case where |$k=2$|⁠. Constraint set (4) essentially forces variable |${\delta}_{tj}$| to take a value of 1 only when |${x}_{tj}$| = 1 and |${x}_{(t-1)j}$| = 0, namely, when facility |$j$| is opened at period |$t$|⁠. In all other cases, |${\delta}_{tj}$| is equal to 0 due to the minimization of the objective function. Constraint set (5) defines these variables for the first time period. Finally, constraint sets (6) and (7) refer to the nature of the decision variables. A final issue concerns the computational complexity of the problem. For the special case where |${f}_{tj}$| = 0 for all |$t$| and |$j$|⁠, |${c}_s$| = 1 for all |$s$|⁠, and no restrictions in opening/closing facilities exist, the developed model reduces to the MSP model. Since MSP has been shown to be NP-hard in the strong sense (Lagodimos & Leopoulos, 2000), by reduction to makespan minimization on identical parallel machines known to be such (Garey & Johnson, 1979), the same holds for the MSPSC problem. Hence, because of this complexity result in the next section a novel GA for solving the MSPSC problem is introduced. 4. The GA solution heuristic We now present and discuss the developed GA for solving the MSPSC problem. Throughout the presentation we explain its main components and their operation in detail. We also give the layout of the algorithm in a pseudo-code format. To position our GA within the relevant research field a comprehensive literature review was performed. We concentrated our bibliographic search in journal papers and conference proceedings. The outcome of this review is presented in tabular form in the Appendix. Each paper is categorized according to its classification (the type of the problem considered) and its application area. This review allowed us to draw the following general observations: (a) the majority of the research papers addressed problems in the context of services (mainly in hospitals and airlines sectors) and (b) nurse scheduling is by far the most studied problem. Any GA undergoes the following five steps (for an introduction into GAs, we refer to Michalewicz & Fogel, 2000): (a) Create randomly a population of individuals that represent potential solutions to the physical optimization problem. (b) Evaluate the quality of each individual in the population by computing their fitness function. (c) Promote individuals of higher quality by introducing selective pressure on the entire population. (d) Generate new individuals by applying variation operators on the population. (e) Repeat steps (b)–(d) several times until the satisfaction of a suitable termination criterion. The above steps can be implemented in a number of possible ways. However, the most important considerations are the representation mechanism (a way of encoding the problem solutions to artificial chromosomes), the evaluation mechanism (the computation of the fitness function), the genetic operators (methods to create new offspring from selected parent solutions) and appropriate settings for the control parameters that control how the components of GA combine and operate. Following these indications the main components of the developed GA are as in the following: 4.1. Chromosome syntax The chromosome (or genotype) is a two-dimensional |$M\times T$| binary matrix, with |$M$| being the number of the facilities and |$T=S\cdot D$| being the permitted time horizon for serving demand. Hence, each cell |$(\,j,t)$| (⁠|$\,j\in [1\dots M]$|⁠, |$t\in [1\dots T]$|⁠) of the chromosome has a binary value denoting the status of the corresponding facility |$j$| at the specified period |$t$|⁠. A zero value denotes that facility |$j$| is closed at period |$t$|⁠; otherwise, it is opened (meaning that it is operating). Figure 1 displays the syntax of the chromosome designed to be used within the proposed GA. Fig. 1. Open in new tabDownload slide Chromosome syntax for the MSPSC problem. Fig. 1. Open in new tabDownload slide Chromosome syntax for the MSPSC problem. 4.1.1. Application example Let us assume an MSPSC problem with the following settings: M = 3; D = 5; S = 2; |${c}_1$| = 1; |${c}_2$| = 2. The characteristics of the facilities are given in Table 1. The opening costs associated with these facilities are reported in Table 2. Each period corresponds to a specific shift of a particular day. For example, period 10 corresponds to the second shift of Day 5. The costs of opening the three facilities at that period are 2, 2 and 3. Table 1. Data for a three-facility MSPSC application example Facility . Workload . . Workforce (manning) . 1 5 10 2 3 9 3 8 7 Facility . Workload . . Workforce (manning) . 1 5 10 2 3 9 3 8 7 Open in new tab Table 1. Data for a three-facility MSPSC application example Facility . Workload . . Workforce (manning) . 1 5 10 2 3 9 3 8 7 Facility . Workload . . Workforce (manning) . 1 5 10 2 3 9 3 8 7 Open in new tab Table 2. Facilities opening costs . Periods . Facility . 1 2 3 4 5 6 7 8 9 10 1 3 6 7 5 6 5 4 6 5 2 2 3 5 8 5 5 6 3 5 6 2 3 2 4 8 5 4 6 4 6 4 3 Period 1 2 3 4 5 6 7 8 9 10 Day/shift 1/1 1/2 2/1 2/2 3/1 3/2 4/1 4/2 5/1 5/2 . Periods . Facility . 1 2 3 4 5 6 7 8 9 10 1 3 6 7 5 6 5 4 6 5 2 2 3 5 8 5 5 6 3 5 6 2 3 2 4 8 5 4 6 4 6 4 3 Period 1 2 3 4 5 6 7 8 9 10 Day/shift 1/1 1/2 2/1 2/2 3/1 3/2 4/1 4/2 5/1 5/2 Open in new tab Table 2. Facilities opening costs . Periods . Facility . 1 2 3 4 5 6 7 8 9 10 1 3 6 7 5 6 5 4 6 5 2 2 3 5 8 5 5 6 3 5 6 2 3 2 4 8 5 4 6 4 6 4 3 Period 1 2 3 4 5 6 7 8 9 10 Day/shift 1/1 1/2 2/1 2/2 3/1 3/2 4/1 4/2 5/1 5/2 . Periods . Facility . 1 2 3 4 5 6 7 8 9 10 1 3 6 7 5 6 5 4 6 5 2 2 3 5 8 5 5 6 3 5 6 2 3 2 4 8 5 4 6 4 6 4 3 Period 1 2 3 4 5 6 7 8 9 10 Day/shift 1/1 1/2 2/1 2/2 3/1 3/2 4/1 4/2 5/1 5/2 Open in new tab Fig. 2. Open in new tabDownload slide The MSPSC solution corresponding to genotype |$\Psi$|. Fig. 2. Open in new tabDownload slide The MSPSC solution corresponding to genotype |$\Psi$|. Let us also assume that at some time instance of the algorithm’s run the following chromosome |$\Psi$| was generated: $$ \Psi =\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline 0&0&1&1&1&1&1&0&0&0\\\hline 0&0&0&0&0&1&1&1&0&0\\\hline1&1&1&1&0&0&1&1&1&1\\\hline\end{array} $$ |$\Psi$| is interpreted as follows: facility 1 opens at the third period and remains open until the seventh period (Day 4, shift 1). Facility 2 opens at the sixth period and closes at the ninth period. Finally, facility 3 opens at the first period, closes at the fifth and reopens at the seventh period. The MSPSC solution corresponding to |$\Psi$| is illustrated in Fig. 2. As one can see, 26 workers are needed in shift 1 and 19 workers in shift 2 per day, which means a staffing cost equal to |${c}_1$| × 26 + |${c}_2$| × 19 = 64. The fixed cost associated with the three facilities is estimated as facility 1 opens at period 3 which means a cost equal to 7. Facility 2 opens at period 6 with a cost equal to 6, while facility 3 opens and closes twice with a cost equal to 2 + 4 = 6. Hence, the total fixed cost is equal to 7 + 6 + 6 = 19. Summarizing, the MSPSC solution corresponding to |$\Psi$| has a cost equal to 64 + 19 = 83. 4.2. Initial population Initially, population |$\Phi$| consists of a set of |${POP}_{size}$| individuals, each one representing a potential MSPSC solution. |$\Phi$| is created using the algorithm INIT given below. INIT starts by generating (for each individual |$l$|⁠) a random permutation of the integers 1,…|$M$|⁠. This permutation corresponds to the facility list. Then, it selects (and considers) each one of the facilities in the order appeared in the list. For each facility |$j$|⁠, INIT reserves time periods always starting from the first period and continues until all time periods needed have been covered (that is, until the number of periods reserved exactly equals its workload needs |${w}_j$|⁠). 4.3. Mapping genotypes to phenotypes A phenotype |${\mathbf{PT}}_l$| (l = 1,…,M) of an individual solution is a unique time period table (a two-dimensional matrix with M lines and SD columns) which corresponds to an actual MSPSC solution. |${\mathbf{PT}}_l$| can be generated from |${\boldsymbol{\Psi}}_l$| using ALLOC algorithm (see below). Each non-zero cell in |${\boldsymbol{\Psi}}_l$| indicates a reserved period. Hence, ALLOC scans cell by cell every line of |${\boldsymbol{\Psi}}_l$| and when this cell has a non-zero value it allocates the respective workforce |${a}_j$| for the operation of facility |$j$|⁠. Figure 2 illustrates the MSPSC solution corresponding to chromosome |$\Psi$| mentioned above. 4.4. The genetic operators GA attempts to improve the solutions in |$\Phi$| using three operators, namely, selection, crossover and mutation. The role of selection is to distinguish among individuals in |$\Phi$| based on their quality and allow the ‘better’ of them to become parents. Each parent undergoes variation through crossover and mutation in order to create offspring. Crossover merges information from two parents into one or two offsprings. Mutation is applied on a single parent and creates a slightly modified offspring. In the developed GA, selection is performed using binary tournament selection strategy (Michalewicz & Fogel, 2000) while parent variation is performed using new developed problem-specific crossover and mutation operators. These variation operators are described below. Binary tournament selection works as follows: two individual solutions are randomly selected from Φ and the one with the highest fitness value reproduces its structure in the population of the next generation. As explained below the highest population fitness individual corresponds to the lowest cost MSPSC solution in the population. 4.4.1. Variation operators Figure 3 displays the developed crossover operator. This operator works on a pair of selected parent genotypes as follows: two crossing lines are randomly selected along the genotypes and the contents between these lines are exchanged between the two mated parents. In Fig. 3, the arrows indicate the crossover points. Then, the contents between these points are exchanged between the parents to produce new offspring for mating in the next generation. Fig. 3. Open in new tabDownload slide A two-point crossover applied on the binary matrix chromosome. Fig. 3. Open in new tabDownload slide A two-point crossover applied on the binary matrix chromosome. Considering mutation, two different forms of this operator were developed. Mutation is applied along a randomly selected line (a facility) of a particular genotype. The first mutation form concerns the case where the selected facility opens/closes only once. On the other hand the second mutation form concerns the case where the selected facility opens and closes exactly twice (when k = 2). Below we make a detailed description of the developed mutation forms. Mutation first form: the selected facility opens/closes only once. This operator consists of the following two steps: (a) split the reserved periods block into two equal sub-parts and (b) shift either the first part to the left r periods or shift the second part r periods to the right. r is selected randomly and should correspond to a valid number of periods in the available time horizon. The two sub-parts are selected with equal probability. An example of this operator is illustrated in Fig. 4. Fig. 4. Open in new tabDownload slide First mutation form. Fig. 4. Open in new tabDownload slide First mutation form. Mutation second form: the selected facility opens and closes exactly twice. Mutation consists of shifting left or right r periods either the first or the second sub-part with the reserved periods. The operator always keeps track to create legal offspring by choosing r from the permitted range of the free periods in the time periods table. An example of this mutation is shown in Fig. 5 with the second sub-part chosen to be moved two periods to the left. Fig. 5. Open in new tabDownload slide Second mutation form. Fig. 5. Open in new tabDownload slide Second mutation form. 4.5. The evaluation mechanism This mechanism concerns the computation of the fitness function for each candidate solution. Fitness reflects the quality of the MSPSC solution corresponding to a particular chromosome. The higher the fitness value of a chromosome, the better the quality of the corresponding solution is. Fitness in this study is formulated as $$\begin{equation} fitness=\frac{1}{1+\mathit{TC}}. \end{equation}$$ (8) Obviously, high values of the fitness function correspond to low values of the TC. 4.6. The pseudo-code of the developed GA We now present and discuss the general layout and operation of the algorithm. For presenting the algorithm the following additional notation is necessary. Φ Population of individual solutions |$l$| Index for individual solutions in Φ |${POP}_{size}$| Number of individual solutions in Φ (population size) |${\boldsymbol{\Psi}}_l$| Genotype (chromosome) of individual |$l$|⁠. A two-dimensional binary matrix with |$M$| lines and |$SD$| columns |${\Psi}_l[\,j,t]$| Entry in |${\boldsymbol{\Psi}}_l$| concerning the operation of facility |$j$| at period |$t$| (1 ≤ |$t$| ≤ |$SD$|⁠) |${\boldsymbol{\pi}}_l$| A facility list (a permutation of the integers 1,2,…,|$M$|⁠) to individual |$l$| |${\pi}_l[\,j]$| Facility in the jth position of permutation |${\boldsymbol{\pi}}_l$| |${\mathbf{PT}}_l$| Phenotype of individual |$l$|⁠; |${\mathbf{PT}}_l$| is a two-dimensional matrix of integers with |$M$| lines and |$SD$| column |${PT}_l[\,j,t]$| Entry in |${\mathbf{PT}}_l$| concerning the operation of facility |$j$| at period |$t$| (1 ≤ |$t$| ≤ |$SD$|⁠) |${\mathbf{PT}}_{best}$| Phenotype of the best population individual |${\mathbf{PT}}_{bsf}$| Phenotype of the best-so-far individual |$CR$| Crossover probability |$MUT$| Mutation probability |${MUT}_{init}$| Initial value of the mutation probability |$\Theta$| Decrease rate for estimating the mutation probability The algorithm starts (step 1) by generating randomly an initial population |$\Phi$| of individuals and continues with their evaluation (step 2). Each individual represents a potential MSPSC solution. Initialization of |$\Phi$| is performed using procedure INIT. Evaluating an individual solution corresponds to the computation of its cost. To that purpose, ALLOC procedure is called (see subsection 3.3) to generate the time periods table corresponding to a particular (genotype) solution (step 2.1a). Then, the cost of this solution is computed (step 2.1b). At the end of the evaluation phase the population best as well as the best-so-far individual are determined (steps 2.2 and 2.3). A cycle of iterations is then applied to |$\Phi$| (steps 3–8). At each iteration, individuals first undergo evolution by means of suitable genetic operators (step 3). The individuals of the new population (step 4) are then evaluated (step 5). Next (step 6), the mutation rate decreases slowly by a factor |$\Theta$| and is conditionally reinitialized to its original |${MUT}_{init}$| value (step 6.2) with the diversity of |$\Phi$|⁠. This control scheme will be explained in the experimental section of the paper. At the end of each iteration, the best individual solution in |$\Phi$| is compared to the best solution attained so far (step 7). The latter will finally constitute the actual MSPSC solution (step 9). 5. Experiments and discussion Multiple computations were performed in order to test the performance of the developed GA. In the absence of a definite knowledge of the optimal solution, GA results were tested against the results provided by the CPLEX optimizer (version 12.6/2014). GA was coded and compiled in Delphi 5.0 and run on a PC with the following specifications: an AMD Athlon 64_2 Dual-Core 2.11 GHz processor, 2.0 GB RAM and Windows XP. The CPLEX solutions were obtained using the standard settings of the package with a stopping condition at 3600 s (1 h) of computer run time. 5.1. Test problems There were 120 test instances involving 3 classes of problems (40 instances per class) with 20, 50 and 100 facilities were generated. For all instances we kept the same planning horizon of 30 days (⁠|$D$| = 30) and 3 daily shifts (⁠|$S$| = 3) and varied the respective manning and workload. The remuneration rates of the three workday shifts were set to c1 = 1, c2 = 2 and c3 = 3. The test problems were constructed as the result of combining five manning (M1–M5) with eight workload (P1–P8) profiles. Specifically, job manning profiles vary in the range (1,10) and were chosen to represent several different operating environments. Hence, profiles M1 and M2 represent cases with fairly homogeneous manning requirements for all jobs, while profiles M3–M5 allow for more extreme manning distributions. Workload profiles vary in the range (3,25) and include approximately even (P1), entirely random (P2), parabolic (P3–P4) and skewed (P5–P8) profiles. Table 3. Settings investigated for the GA’s control parameters Static control scheme . |$CR\in \{0.6,0.8,0.9\}$| . |$MUT\in \{0.001,0.01,0.1\}$| . Adaptive control scheme |$CR=1$| |$MUT$| Adaptable using equation (9) Population size |${POP}_{size}\in \{M,3M,5M,10M\}$| Static control scheme . |$CR\in \{0.6,0.8,0.9\}$| . |$MUT\in \{0.001,0.01,0.1\}$| . Adaptive control scheme |$CR=1$| |$MUT$| Adaptable using equation (9) Population size |${POP}_{size}\in \{M,3M,5M,10M\}$| Open in new tab Table 3. Settings investigated for the GA’s control parameters Static control scheme . |$CR\in \{0.6,0.8,0.9\}$| . |$MUT\in \{0.001,0.01,0.1\}$| . Adaptive control scheme |$CR=1$| |$MUT$| Adaptable using equation (9) Population size |${POP}_{size}\in \{M,3M,5M,10M\}$| Static control scheme . |$CR\in \{0.6,0.8,0.9\}$| . |$MUT\in \{0.001,0.01,0.1\}$| . Adaptive control scheme |$CR=1$| |$MUT$| Adaptable using equation (9) Population size |${POP}_{size}\in \{M,3M,5M,10M\}$| Open in new tab The fixed costs for opening the facilities were generated randomly for each facility and each time period. In particular, each set-up cost |${f}_{tj}$| was drawn from a discrete uniform distribution U(a,b) where a = 2 and b = 5 for periods in the first shift, a = 4 and b = 6 for periods in the second shift and a = 5 and b = 8 for periods in the third shift. All the test problems were first solved with GA and then with CPLEX. The full data set can be downloaded from http://www.bma.upatras.gr/staff/nearchou/benchmarks_MSPSC.rar. Table 4. Parameter tuning: average dev% from CPLEX solution. Experiments over a 20-facility MSPSC problem Workload profile . |$CR$| . |$MUT$| . |${POP}_{size}$| . |$M$| . |$3M$| . |$5M$| . |$10M$| . (1) . (2) . (3) . (4) . (5) . (6) . (7) . P1 0.6 0.001 −1.43 −2.18 −2.21 −2.33 0.01 −2.74 −2.89 −2.92 −2.97 0.1 −3.30 −3.28 −3.31 −3.07 0.8 0.001 −1.53 −2.34 −2.18 −2.33 0.01 −2.88 −2.76 −2.89 −2.89 0.1 −3.29 −3.31 −3.07 −3.30 0.9 0.001 −1.70 −2.19 −2.25 −2.26 0.01 −2.79 −2.79 −2.85 −2.87 0.1 −2.34 −3.07 −3.29 −3.30 1 Adapt −3.28 −3.31 −3.31 −3.07 P2 0.6 0.001 0.81 0.97 0.94 0.79 0.01 2.98 2.43 2.23 2.27 0.1 1.46 1.51 1.40 1.46 0.8 0.001 0.99 0.94 0.94 0.94 0.01 0.99 0.94 0.94 0.94 0.1 1.41 1.44 1.34 1.41 0.9 0.001 0.96 0.93 0.95 0.94 0.01 2.91 2.31 2.23 2.23 0.1 1.78 1.32 1.25 1.30 1 Adapt 0.94 0.78 0.78 0.78 P3 0.6 0.001 0.98 0.95 0.95 0.95 0.01 3.27 2.78 2.59 2.56 0.1 1.66 1.43 1.59 1.49 0.8 0.001 0.82 0.83 0.83 0.83 0.01 3.00 2.85 2.50 2.69 0.1 1.54 1.58 1.51 1.46 0.9 0.001 0.82 0.80 0.80 0.84 0.01 3.15 2.58 2.71 2.56 0.1 1.76 1.73 1.61 1.63 1 Adapt 0.82 0.80 0.80 0.80 P4 0.6 0.001 1.66 1.60 1.60 1.60 0.01 2.50 3.26 2.47 2.47 0.1 2.88 2.74 2.67 2.83 0.8 0.001 1.62 1.60 1.61 1.66 0.01 3.61 3.98 3.02 3.02 0.1 2.81 2.86 2.73 2.60 0.9 0.001 1.60 1.61 1.61 1.64 0.01 2.63 2.26 2.08 3.99 0.1 2.66 2.81 2.39 2.08 1 Adapt 1.61 1.61 1.59 1.61 P5 0.6 0.001 0.05 0.01 0.01 0.02 0.01 2.19 1.85 1.90 1.88 0.1 1.47 1.68 1.64 1.59 0.8 0.001 0.46 0.45 0.46 0.46 0.01 2.02 1.69 1.43 1.40 0.1 1.40 1.25 1.75 1.24 0.9 0.001 0.45 0.45 0.46 0.46 0.01 2.12 2.88 2.78 2.70 0.1 1.56 1.48 1.17 2.18 1 Adapt 0.07 0.00 0.00 0.01 P6 0.6 0.001 1.34 1.32 1.32 1.33 0.01 2.40 2.25 2.06 2.06 0.1 2.11 2.32 2.33 2.38 0.8 0.001 1.02 1.06 1.02 1.02 0.01 2.69 2.05 2.21 2.03 0.1 2.31 1.95 2.24 2.08 0.9 0.001 1.38 1.36 1.36 1.38 0.01 1.29 1.19 1.19 1.19 0.1 1.92 1.19 2.08 2.29 1 Adapt 1.06 1.06 0.88 0.88 P7 0.6 0.001 3.20 3.07 3.07 3.07 0.01 3.63 3.57 3.75 3.24 0.1 3.80 3.04 3.04 3.10 0.8 0.001 3.51 3.55 3.53 3.53 0.01 4.03 4.66 4.63 4.66 0.1 4.44 4.97 4.47 3.91 0.9 0.001 4.49 4.52 4.49 4.57 0.01 3.53 3.81 3.51 3.40 0.1 4.22 3.49 3.83 3.11 1 Adapt 2.79 2.70 2.70 2.70 P8 0.6 0.001 3.96 3.95 3.94 3.97 0.01 2.97 2.34 2.93 2.88 0.1 2.48 2.59 3.24 3.82 0.8 0.001 3.95 3.96 3.96 2.91 0.01 3.40 3.01 3.09 3.14 0.1 3.24 3.16 3.16 3.16 0.9 0.001 4.04 4.01 4.01 3.97 0.01 3.93 4.01 3.09 3.14 0.1 2.54 3.97 3.09 3.09 1 Adapt 2.19 2.19 2.19 2.19 Workload profile . |$CR$| . |$MUT$| . |${POP}_{size}$| . |$M$| . |$3M$| . |$5M$| . |$10M$| . (1) . (2) . (3) . (4) . (5) . (6) . (7) . P1 0.6 0.001 −1.43 −2.18 −2.21 −2.33 0.01 −2.74 −2.89 −2.92 −2.97 0.1 −3.30 −3.28 −3.31 −3.07 0.8 0.001 −1.53 −2.34 −2.18 −2.33 0.01 −2.88 −2.76 −2.89 −2.89 0.1 −3.29 −3.31 −3.07 −3.30 0.9 0.001 −1.70 −2.19 −2.25 −2.26 0.01 −2.79 −2.79 −2.85 −2.87 0.1 −2.34 −3.07 −3.29 −3.30 1 Adapt −3.28 −3.31 −3.31 −3.07 P2 0.6 0.001 0.81 0.97 0.94 0.79 0.01 2.98 2.43 2.23 2.27 0.1 1.46 1.51 1.40 1.46 0.8 0.001 0.99 0.94 0.94 0.94 0.01 0.99 0.94 0.94 0.94 0.1 1.41 1.44 1.34 1.41 0.9 0.001 0.96 0.93 0.95 0.94 0.01 2.91 2.31 2.23 2.23 0.1 1.78 1.32 1.25 1.30 1 Adapt 0.94 0.78 0.78 0.78 P3 0.6 0.001 0.98 0.95 0.95 0.95 0.01 3.27 2.78 2.59 2.56 0.1 1.66 1.43 1.59 1.49 0.8 0.001 0.82 0.83 0.83 0.83 0.01 3.00 2.85 2.50 2.69 0.1 1.54 1.58 1.51 1.46 0.9 0.001 0.82 0.80 0.80 0.84 0.01 3.15 2.58 2.71 2.56 0.1 1.76 1.73 1.61 1.63 1 Adapt 0.82 0.80 0.80 0.80 P4 0.6 0.001 1.66 1.60 1.60 1.60 0.01 2.50 3.26 2.47 2.47 0.1 2.88 2.74 2.67 2.83 0.8 0.001 1.62 1.60 1.61 1.66 0.01 3.61 3.98 3.02 3.02 0.1 2.81 2.86 2.73 2.60 0.9 0.001 1.60 1.61 1.61 1.64 0.01 2.63 2.26 2.08 3.99 0.1 2.66 2.81 2.39 2.08 1 Adapt 1.61 1.61 1.59 1.61 P5 0.6 0.001 0.05 0.01 0.01 0.02 0.01 2.19 1.85 1.90 1.88 0.1 1.47 1.68 1.64 1.59 0.8 0.001 0.46 0.45 0.46 0.46 0.01 2.02 1.69 1.43 1.40 0.1 1.40 1.25 1.75 1.24 0.9 0.001 0.45 0.45 0.46 0.46 0.01 2.12 2.88 2.78 2.70 0.1 1.56 1.48 1.17 2.18 1 Adapt 0.07 0.00 0.00 0.01 P6 0.6 0.001 1.34 1.32 1.32 1.33 0.01 2.40 2.25 2.06 2.06 0.1 2.11 2.32 2.33 2.38 0.8 0.001 1.02 1.06 1.02 1.02 0.01 2.69 2.05 2.21 2.03 0.1 2.31 1.95 2.24 2.08 0.9 0.001 1.38 1.36 1.36 1.38 0.01 1.29 1.19 1.19 1.19 0.1 1.92 1.19 2.08 2.29 1 Adapt 1.06 1.06 0.88 0.88 P7 0.6 0.001 3.20 3.07 3.07 3.07 0.01 3.63 3.57 3.75 3.24 0.1 3.80 3.04 3.04 3.10 0.8 0.001 3.51 3.55 3.53 3.53 0.01 4.03 4.66 4.63 4.66 0.1 4.44 4.97 4.47 3.91 0.9 0.001 4.49 4.52 4.49 4.57 0.01 3.53 3.81 3.51 3.40 0.1 4.22 3.49 3.83 3.11 1 Adapt 2.79 2.70 2.70 2.70 P8 0.6 0.001 3.96 3.95 3.94 3.97 0.01 2.97 2.34 2.93 2.88 0.1 2.48 2.59 3.24 3.82 0.8 0.001 3.95 3.96 3.96 2.91 0.01 3.40 3.01 3.09 3.14 0.1 3.24 3.16 3.16 3.16 0.9 0.001 4.04 4.01 4.01 3.97 0.01 3.93 4.01 3.09 3.14 0.1 2.54 3.97 3.09 3.09 1 Adapt 2.19 2.19 2.19 2.19 Open in new tab Table 4. Parameter tuning: average dev% from CPLEX solution. Experiments over a 20-facility MSPSC problem Workload profile . |$CR$| . |$MUT$| . |${POP}_{size}$| . |$M$| . |$3M$| . |$5M$| . |$10M$| . (1) . (2) . (3) . (4) . (5) . (6) . (7) . P1 0.6 0.001 −1.43 −2.18 −2.21 −2.33 0.01 −2.74 −2.89 −2.92 −2.97 0.1 −3.30 −3.28 −3.31 −3.07 0.8 0.001 −1.53 −2.34 −2.18 −2.33 0.01 −2.88 −2.76 −2.89 −2.89 0.1 −3.29 −3.31 −3.07 −3.30 0.9 0.001 −1.70 −2.19 −2.25 −2.26 0.01 −2.79 −2.79 −2.85 −2.87 0.1 −2.34 −3.07 −3.29 −3.30 1 Adapt −3.28 −3.31 −3.31 −3.07 P2 0.6 0.001 0.81 0.97 0.94 0.79 0.01 2.98 2.43 2.23 2.27 0.1 1.46 1.51 1.40 1.46 0.8 0.001 0.99 0.94 0.94 0.94 0.01 0.99 0.94 0.94 0.94 0.1 1.41 1.44 1.34 1.41 0.9 0.001 0.96 0.93 0.95 0.94 0.01 2.91 2.31 2.23 2.23 0.1 1.78 1.32 1.25 1.30 1 Adapt 0.94 0.78 0.78 0.78 P3 0.6 0.001 0.98 0.95 0.95 0.95 0.01 3.27 2.78 2.59 2.56 0.1 1.66 1.43 1.59 1.49 0.8 0.001 0.82 0.83 0.83 0.83 0.01 3.00 2.85 2.50 2.69 0.1 1.54 1.58 1.51 1.46 0.9 0.001 0.82 0.80 0.80 0.84 0.01 3.15 2.58 2.71 2.56 0.1 1.76 1.73 1.61 1.63 1 Adapt 0.82 0.80 0.80 0.80 P4 0.6 0.001 1.66 1.60 1.60 1.60 0.01 2.50 3.26 2.47 2.47 0.1 2.88 2.74 2.67 2.83 0.8 0.001 1.62 1.60 1.61 1.66 0.01 3.61 3.98 3.02 3.02 0.1 2.81 2.86 2.73 2.60 0.9 0.001 1.60 1.61 1.61 1.64 0.01 2.63 2.26 2.08 3.99 0.1 2.66 2.81 2.39 2.08 1 Adapt 1.61 1.61 1.59 1.61 P5 0.6 0.001 0.05 0.01 0.01 0.02 0.01 2.19 1.85 1.90 1.88 0.1 1.47 1.68 1.64 1.59 0.8 0.001 0.46 0.45 0.46 0.46 0.01 2.02 1.69 1.43 1.40 0.1 1.40 1.25 1.75 1.24 0.9 0.001 0.45 0.45 0.46 0.46 0.01 2.12 2.88 2.78 2.70 0.1 1.56 1.48 1.17 2.18 1 Adapt 0.07 0.00 0.00 0.01 P6 0.6 0.001 1.34 1.32 1.32 1.33 0.01 2.40 2.25 2.06 2.06 0.1 2.11 2.32 2.33 2.38 0.8 0.001 1.02 1.06 1.02 1.02 0.01 2.69 2.05 2.21 2.03 0.1 2.31 1.95 2.24 2.08 0.9 0.001 1.38 1.36 1.36 1.38 0.01 1.29 1.19 1.19 1.19 0.1 1.92 1.19 2.08 2.29 1 Adapt 1.06 1.06 0.88 0.88 P7 0.6 0.001 3.20 3.07 3.07 3.07 0.01 3.63 3.57 3.75 3.24 0.1 3.80 3.04 3.04 3.10 0.8 0.001 3.51 3.55 3.53 3.53 0.01 4.03 4.66 4.63 4.66 0.1 4.44 4.97 4.47 3.91 0.9 0.001 4.49 4.52 4.49 4.57 0.01 3.53 3.81 3.51 3.40 0.1 4.22 3.49 3.83 3.11 1 Adapt 2.79 2.70 2.70 2.70 P8 0.6 0.001 3.96 3.95 3.94 3.97 0.01 2.97 2.34 2.93 2.88 0.1 2.48 2.59 3.24 3.82 0.8 0.001 3.95 3.96 3.96 2.91 0.01 3.40 3.01 3.09 3.14 0.1 3.24 3.16 3.16 3.16 0.9 0.001 4.04 4.01 4.01 3.97 0.01 3.93 4.01 3.09 3.14 0.1 2.54 3.97 3.09 3.09 1 Adapt 2.19 2.19 2.19 2.19 Workload profile . |$CR$| . |$MUT$| . |${POP}_{size}$| . |$M$| . |$3M$| . |$5M$| . |$10M$| . (1) . (2) . (3) . (4) . (5) . (6) . (7) . P1 0.6 0.001 −1.43 −2.18 −2.21 −2.33 0.01 −2.74 −2.89 −2.92 −2.97 0.1 −3.30 −3.28 −3.31 −3.07 0.8 0.001 −1.53 −2.34 −2.18 −2.33 0.01 −2.88 −2.76 −2.89 −2.89 0.1 −3.29 −3.31 −3.07 −3.30 0.9 0.001 −1.70 −2.19 −2.25 −2.26 0.01 −2.79 −2.79 −2.85 −2.87 0.1 −2.34 −3.07 −3.29 −3.30 1 Adapt −3.28 −3.31 −3.31 −3.07 P2 0.6 0.001 0.81 0.97 0.94 0.79 0.01 2.98 2.43 2.23 2.27 0.1 1.46 1.51 1.40 1.46 0.8 0.001 0.99 0.94 0.94 0.94 0.01 0.99 0.94 0.94 0.94 0.1 1.41 1.44 1.34 1.41 0.9 0.001 0.96 0.93 0.95 0.94 0.01 2.91 2.31 2.23 2.23 0.1 1.78 1.32 1.25 1.30 1 Adapt 0.94 0.78 0.78 0.78 P3 0.6 0.001 0.98 0.95 0.95 0.95 0.01 3.27 2.78 2.59 2.56 0.1 1.66 1.43 1.59 1.49 0.8 0.001 0.82 0.83 0.83 0.83 0.01 3.00 2.85 2.50 2.69 0.1 1.54 1.58 1.51 1.46 0.9 0.001 0.82 0.80 0.80 0.84 0.01 3.15 2.58 2.71 2.56 0.1 1.76 1.73 1.61 1.63 1 Adapt 0.82 0.80 0.80 0.80 P4 0.6 0.001 1.66 1.60 1.60 1.60 0.01 2.50 3.26 2.47 2.47 0.1 2.88 2.74 2.67 2.83 0.8 0.001 1.62 1.60 1.61 1.66 0.01 3.61 3.98 3.02 3.02 0.1 2.81 2.86 2.73 2.60 0.9 0.001 1.60 1.61 1.61 1.64 0.01 2.63 2.26 2.08 3.99 0.1 2.66 2.81 2.39 2.08 1 Adapt 1.61 1.61 1.59 1.61 P5 0.6 0.001 0.05 0.01 0.01 0.02 0.01 2.19 1.85 1.90 1.88 0.1 1.47 1.68 1.64 1.59 0.8 0.001 0.46 0.45 0.46 0.46 0.01 2.02 1.69 1.43 1.40 0.1 1.40 1.25 1.75 1.24 0.9 0.001 0.45 0.45 0.46 0.46 0.01 2.12 2.88 2.78 2.70 0.1 1.56 1.48 1.17 2.18 1 Adapt 0.07 0.00 0.00 0.01 P6 0.6 0.001 1.34 1.32 1.32 1.33 0.01 2.40 2.25 2.06 2.06 0.1 2.11 2.32 2.33 2.38 0.8 0.001 1.02 1.06 1.02 1.02 0.01 2.69 2.05 2.21 2.03 0.1 2.31 1.95 2.24 2.08 0.9 0.001 1.38 1.36 1.36 1.38 0.01 1.29 1.19 1.19 1.19 0.1 1.92 1.19 2.08 2.29 1 Adapt 1.06 1.06 0.88 0.88 P7 0.6 0.001 3.20 3.07 3.07 3.07 0.01 3.63 3.57 3.75 3.24 0.1 3.80 3.04 3.04 3.10 0.8 0.001 3.51 3.55 3.53 3.53 0.01 4.03 4.66 4.63 4.66 0.1 4.44 4.97 4.47 3.91 0.9 0.001 4.49 4.52 4.49 4.57 0.01 3.53 3.81 3.51 3.40 0.1 4.22 3.49 3.83 3.11 1 Adapt 2.79 2.70 2.70 2.70 P8 0.6 0.001 3.96 3.95 3.94 3.97 0.01 2.97 2.34 2.93 2.88 0.1 2.48 2.59 3.24 3.82 0.8 0.001 3.95 3.96 3.96 2.91 0.01 3.40 3.01 3.09 3.14 0.1 3.24 3.16 3.16 3.16 0.9 0.001 4.04 4.01 4.01 3.97 0.01 3.93 4.01 3.09 3.14 0.1 2.54 3.97 3.09 3.09 1 Adapt 2.19 2.19 2.19 2.19 Open in new tab 5.2. Tuning the GA’s control parameters To determine the correct settings for GA’s control parameters (⁠|${POP}_{size}$|⁠, |$CR$| and |$MUT$|⁠), multiple experiments were performed in a preliminary investigation. Two control schemes were studied: (a) a static scheme, in which |$CR$| was defined to take values in the discrete set {0.6, 0.8, 0.9} and |$MUT$| in {0.001, 0.01, 0.1}; (b) a dynamic scheme, in which |$CR$| was fixed to a high value and |$MUT$| was allowed to be self-adapted during the algorithm’s evolution. Particularly, |$CR$| was set equal to 1 and |$MUT$| was defined to be high at the beginning and decreasing slowly by a factor |$\Theta$| = 0.95 using the linear relation |$MUT=\Theta \times MUT$|⁠. Mutation rate was further defined to be adapted by the relation $$\begin{equation} MUT=\left\{\!\!\begin{array}{l}\Theta \times MUT,\kern0.5em \mathrm{if}\ \mathrm{avg}.\mathrm{pop}<0.95\times \max\!.\mathrm{pop}\\{}{MUT}_{init},\kern0.5em \mathrm{otherwise},\end{array}\right. \end{equation}$$ (9) where avg.pop and max.pop denote the average and maximum population fitness, respectively. A too small diversity of the population is encountered when avg.pop is almost identical to max.pop. In this case, |$MUT$| is reinitialized to |${MUT}_{init}$|⁠. Regarding population size, four possible values were examined within the discrete set {|$M$|⁠, |$3M$|⁠, |$5M$|⁠, |$10M$|}, with |$M$| being the number of facilities. GA stops after performing a maximum number of 30,000 evaluations, where an evaluation corresponds to the computation of the fitness of an individual solution. Table 3 summarizes the parameter control schemes examined in this work. Table 4 displays the results of the GA runs over a particular 20-facility MSPSC problem. Column (1) of Table 4 displays the workload profiles. Columns (2) and (3) show the various |$CR$| and |$MUT$| rates examined, respectively. For each combination (⁠|$CR$|⁠, |$MUT$| and |${POP}_{size}$|⁠) GA was run five times to reduce the impact of the initial solution. Columns (4)–(7) report the average percentage deviation (dev%) of the GA solution from CPLEX solution over each different |${POP}_{size}$| setting. The best solutions (lowest average dev%) are given in bold and underlined in the table. As one can see from the table, the best results were obtained by the dynamic control scheme (equation (9)). Considering the population size, it seems that the best results are encountered when |$5M$| individuals are maintained in the populations. In order to confirm these results from a statistical point of view, we initially investigated the effect of population size on the results. Specifically, we performed the well-known Friedman test (see, for example, Sprent & Smeeton, 2007), a powerful non-parametric test for detecting statistically significant rank differences across samples. We tested the null hypothesis that the four alternative population sizes produce results with the same median versus the alternative hypothesis that at least one median is different from the rest (i.e. non-directional test mode). After the null hypothesis was rejected at 5% level of significance (p-value = 0.000), we proceeded with a post hoc analysis to determine where the differences actually occur. The statistics indicated that the results obtained from a population size of 5M were statistically better than the ones of M (p-value = 0.000) and 3M (p-value = 0.002) and did not differ significantly from the results of size 10M (p-value = 0.261). Consequently, a population size of 5M individuals was preferred. We then investigated the effect of CR and MUT following the same procedure as the one described above. The application of the Friedman test confirmed that there were statistically significant differences across the different parameter settings (p-value = 0.000) and the results of the post hoc analysis verified that the dynamic control scheme produced significantly better solutions (p-value = 0.000 for all pairwise comparisons). Hence, the dynamic control scheme for estimating |$CR$| and |$MUT$| was employed. 5.3. Numerical results The results of all test problems are given in Table 5. The first column (Profile) of the table indicates the characteristics of each test instance as a combination of manning (M1–M5) and workload (P1–P8) settings. For each class of problem (20, 50 and 100 facilities) there are 4 columns in the table showing the linear programming lower bound (LP bound) of the best solution of the relaxed problem, the TC of the solution obtained by CPLEX and GA for the various combination of manning and workload profiles and the percentage deviation (column dev%) of GA solution from CPLEX solution. It is underlined that CPLEX was not able to determine the optimal solution to any of the 120 test instances examined within the allowed CPU time of 1 h. Hence, CPLEX solutions will serve as benchmark solutions that were provided by an exact method for comparison with the results of the GA. Observe that many of the dev% are negative; these are cases where GA outperformed CPLEX. Table 5. Comparative results between GA and CPLEX for 20-, 50- and 100-facility MSPSC problems . . 20-Facility MSPSC . . 50-Facility MSPSC . . 100-Facility MSPSC . Profile . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . M1P1 304.29 362 350 −3.31 768.50 3174 888 −72.02 1355.93 6292 1766 −71.93 P2 225.64 258 260 0.78 534.53 655 612 −6.56 956.67 3598 1222 −66.04 P3 214.16 251 253 0.80 545.14 2658 651 −75.51 1166.56 3640 1324 −63.63 P4 214.16 251 255 1.59 548.30 629 629 0.00 1026.32 3739 1263 −66.22 P5 221.87 280 280 0.00 548.49 699 643 −8.01 1081.76 3595 1270 −64.67 P6 209.69 226 228 0.88 512.16 597 594 −0.50 997.56 5950 1228 −79.36 P7 203.52 222 228 2.70 433.92 1364 525 −61.51 802.78 1970 1165 −40.86 P8 203.16 228 233 2.19 555.83 636 636 0.00 711.44 2171 1187 −45.32 M2P1 258.48 312 311 −0.32 626.16 2557 716 −72.00 1241.99 3270 1473 −54.95 P2 186.68 242 235 −2.89 418.78 2280 509 −77.68 829.62 2862 1045 −63.49 P3 207.08 258 248 −3.88 443.57 2380 548 −76.97 994.13 4427 1116 −74.79 P4 186.58 231 232 0.43 439.96 601 529 −11.98 728.01 3045 1066 −64.99 P5 191.61 225 230 2.22 439.38 2358 540 −77.10 906.94 3830 1069 −72.09 P6 177.00 207 209 0.97 414.76 528 503 −4.73 840.05 4834 1037 −78.55 P7 190.09 233 225 −3.43 403.93 461 461 0.00 723.66 1688 956 −43.36 P8 157.81 196 195 −0.51 425.17 502 502 0.00 829.07 2752 995 −63.84 M3P1 185.90 229 217 −5.24 469.99 1828 541 −70.40 937.48 3596 1089 −69.72 P2 139.92 176 171 −2.84 347.01 443 392 −11.51 693.51 2055 781 −62.00 P3 151.27 190 182 −4.21 323.79 439 381 −13.21 630.53 2054 814 −60.37 P4 136.21 171 177 3.51 377.56 433 432 −0.23 655.84 2138 820 −61.65 P5 137.85 159 163 2.52 320.65 1772 390 −77.99 732.32 2138 826 −61.37 P6 129.98 155 155 0.00 341.13 430 396 −7.91 652.12 2436 767 −68.51 P7 129.68 152 153 0.66 280.87 331 331 0.00 628.35 1589 756 −52.42 P8 129.39 158 158 0.00 349.98 449 411 −8.46 647.07 1954 745 −61.87 M4P1 195.11 233 225 −3.43 452.12 1754 527 −69.95 847.61 2376 1099 −53.75 P2 148.89 183 181 −1.09 328.59 418 377 −9.81 644.29 2151 834 −61.23 P3 170.34 203 207 1.97 349.57 406 400 −1.48 651.01 2170 840 −61.29 P4 129.21 159 162 1.89 330.12 881 375 −57.43 670.87 2178 798 −63.36 P5 142.66 177 176 −0.56 315.99 385 380 −1.30 674.09 2172 821 −62.20 P6 141.97 169 173 2.37 315.46 382 380 −0.52 588.53 1739 807 −53.59 P7 130.86 156 160 2.56 282.58 388 330 −14.95 595.96 1551 790 −49.07 P8 137.51 180 170 −5.56 334.44 412 393 −4.61 617.82 2051 723 −64.75 M5P1 178.72 213 206 −3.29 378.97 1397 435 −68.86 791.47 2872 908 −68.38 P2 134.16 162 166 2.47 286.32 1268 326 −74.29 529.45 1648 657 −60.13 P3 156.53 181 183 1.10 295.70 369 343 −7.05 542.27 1670 693 −58.50 P4 119.60 153 154 0.65 264.77 338 325 −3.85 554.18 1698 690 −59.36 P5 130.45 156 158 1.28 268.07 347 329 −5.19 591.76 1754 699 −60.15 P6 126.40 145 147 1.38 261.27 860 316 −63.26 547.50 1638 644 −60.68 P7 120.65 147 149 1.36 243.17 289 280 −3.11 388.53 1092 626 −42.67 P8 127.81 157 154 −1.91 287.46 861 327 −62.02 534.57 1550 630 −59.35 . . 20-Facility MSPSC . . 50-Facility MSPSC . . 100-Facility MSPSC . Profile . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . M1P1 304.29 362 350 −3.31 768.50 3174 888 −72.02 1355.93 6292 1766 −71.93 P2 225.64 258 260 0.78 534.53 655 612 −6.56 956.67 3598 1222 −66.04 P3 214.16 251 253 0.80 545.14 2658 651 −75.51 1166.56 3640 1324 −63.63 P4 214.16 251 255 1.59 548.30 629 629 0.00 1026.32 3739 1263 −66.22 P5 221.87 280 280 0.00 548.49 699 643 −8.01 1081.76 3595 1270 −64.67 P6 209.69 226 228 0.88 512.16 597 594 −0.50 997.56 5950 1228 −79.36 P7 203.52 222 228 2.70 433.92 1364 525 −61.51 802.78 1970 1165 −40.86 P8 203.16 228 233 2.19 555.83 636 636 0.00 711.44 2171 1187 −45.32 M2P1 258.48 312 311 −0.32 626.16 2557 716 −72.00 1241.99 3270 1473 −54.95 P2 186.68 242 235 −2.89 418.78 2280 509 −77.68 829.62 2862 1045 −63.49 P3 207.08 258 248 −3.88 443.57 2380 548 −76.97 994.13 4427 1116 −74.79 P4 186.58 231 232 0.43 439.96 601 529 −11.98 728.01 3045 1066 −64.99 P5 191.61 225 230 2.22 439.38 2358 540 −77.10 906.94 3830 1069 −72.09 P6 177.00 207 209 0.97 414.76 528 503 −4.73 840.05 4834 1037 −78.55 P7 190.09 233 225 −3.43 403.93 461 461 0.00 723.66 1688 956 −43.36 P8 157.81 196 195 −0.51 425.17 502 502 0.00 829.07 2752 995 −63.84 M3P1 185.90 229 217 −5.24 469.99 1828 541 −70.40 937.48 3596 1089 −69.72 P2 139.92 176 171 −2.84 347.01 443 392 −11.51 693.51 2055 781 −62.00 P3 151.27 190 182 −4.21 323.79 439 381 −13.21 630.53 2054 814 −60.37 P4 136.21 171 177 3.51 377.56 433 432 −0.23 655.84 2138 820 −61.65 P5 137.85 159 163 2.52 320.65 1772 390 −77.99 732.32 2138 826 −61.37 P6 129.98 155 155 0.00 341.13 430 396 −7.91 652.12 2436 767 −68.51 P7 129.68 152 153 0.66 280.87 331 331 0.00 628.35 1589 756 −52.42 P8 129.39 158 158 0.00 349.98 449 411 −8.46 647.07 1954 745 −61.87 M4P1 195.11 233 225 −3.43 452.12 1754 527 −69.95 847.61 2376 1099 −53.75 P2 148.89 183 181 −1.09 328.59 418 377 −9.81 644.29 2151 834 −61.23 P3 170.34 203 207 1.97 349.57 406 400 −1.48 651.01 2170 840 −61.29 P4 129.21 159 162 1.89 330.12 881 375 −57.43 670.87 2178 798 −63.36 P5 142.66 177 176 −0.56 315.99 385 380 −1.30 674.09 2172 821 −62.20 P6 141.97 169 173 2.37 315.46 382 380 −0.52 588.53 1739 807 −53.59 P7 130.86 156 160 2.56 282.58 388 330 −14.95 595.96 1551 790 −49.07 P8 137.51 180 170 −5.56 334.44 412 393 −4.61 617.82 2051 723 −64.75 M5P1 178.72 213 206 −3.29 378.97 1397 435 −68.86 791.47 2872 908 −68.38 P2 134.16 162 166 2.47 286.32 1268 326 −74.29 529.45 1648 657 −60.13 P3 156.53 181 183 1.10 295.70 369 343 −7.05 542.27 1670 693 −58.50 P4 119.60 153 154 0.65 264.77 338 325 −3.85 554.18 1698 690 −59.36 P5 130.45 156 158 1.28 268.07 347 329 −5.19 591.76 1754 699 −60.15 P6 126.40 145 147 1.38 261.27 860 316 −63.26 547.50 1638 644 −60.68 P7 120.65 147 149 1.36 243.17 289 280 −3.11 388.53 1092 626 −42.67 P8 127.81 157 154 −1.91 287.46 861 327 −62.02 534.57 1550 630 −59.35 Open in new tab Table 5. Comparative results between GA and CPLEX for 20-, 50- and 100-facility MSPSC problems . . 20-Facility MSPSC . . 50-Facility MSPSC . . 100-Facility MSPSC . Profile . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . M1P1 304.29 362 350 −3.31 768.50 3174 888 −72.02 1355.93 6292 1766 −71.93 P2 225.64 258 260 0.78 534.53 655 612 −6.56 956.67 3598 1222 −66.04 P3 214.16 251 253 0.80 545.14 2658 651 −75.51 1166.56 3640 1324 −63.63 P4 214.16 251 255 1.59 548.30 629 629 0.00 1026.32 3739 1263 −66.22 P5 221.87 280 280 0.00 548.49 699 643 −8.01 1081.76 3595 1270 −64.67 P6 209.69 226 228 0.88 512.16 597 594 −0.50 997.56 5950 1228 −79.36 P7 203.52 222 228 2.70 433.92 1364 525 −61.51 802.78 1970 1165 −40.86 P8 203.16 228 233 2.19 555.83 636 636 0.00 711.44 2171 1187 −45.32 M2P1 258.48 312 311 −0.32 626.16 2557 716 −72.00 1241.99 3270 1473 −54.95 P2 186.68 242 235 −2.89 418.78 2280 509 −77.68 829.62 2862 1045 −63.49 P3 207.08 258 248 −3.88 443.57 2380 548 −76.97 994.13 4427 1116 −74.79 P4 186.58 231 232 0.43 439.96 601 529 −11.98 728.01 3045 1066 −64.99 P5 191.61 225 230 2.22 439.38 2358 540 −77.10 906.94 3830 1069 −72.09 P6 177.00 207 209 0.97 414.76 528 503 −4.73 840.05 4834 1037 −78.55 P7 190.09 233 225 −3.43 403.93 461 461 0.00 723.66 1688 956 −43.36 P8 157.81 196 195 −0.51 425.17 502 502 0.00 829.07 2752 995 −63.84 M3P1 185.90 229 217 −5.24 469.99 1828 541 −70.40 937.48 3596 1089 −69.72 P2 139.92 176 171 −2.84 347.01 443 392 −11.51 693.51 2055 781 −62.00 P3 151.27 190 182 −4.21 323.79 439 381 −13.21 630.53 2054 814 −60.37 P4 136.21 171 177 3.51 377.56 433 432 −0.23 655.84 2138 820 −61.65 P5 137.85 159 163 2.52 320.65 1772 390 −77.99 732.32 2138 826 −61.37 P6 129.98 155 155 0.00 341.13 430 396 −7.91 652.12 2436 767 −68.51 P7 129.68 152 153 0.66 280.87 331 331 0.00 628.35 1589 756 −52.42 P8 129.39 158 158 0.00 349.98 449 411 −8.46 647.07 1954 745 −61.87 M4P1 195.11 233 225 −3.43 452.12 1754 527 −69.95 847.61 2376 1099 −53.75 P2 148.89 183 181 −1.09 328.59 418 377 −9.81 644.29 2151 834 −61.23 P3 170.34 203 207 1.97 349.57 406 400 −1.48 651.01 2170 840 −61.29 P4 129.21 159 162 1.89 330.12 881 375 −57.43 670.87 2178 798 −63.36 P5 142.66 177 176 −0.56 315.99 385 380 −1.30 674.09 2172 821 −62.20 P6 141.97 169 173 2.37 315.46 382 380 −0.52 588.53 1739 807 −53.59 P7 130.86 156 160 2.56 282.58 388 330 −14.95 595.96 1551 790 −49.07 P8 137.51 180 170 −5.56 334.44 412 393 −4.61 617.82 2051 723 −64.75 M5P1 178.72 213 206 −3.29 378.97 1397 435 −68.86 791.47 2872 908 −68.38 P2 134.16 162 166 2.47 286.32 1268 326 −74.29 529.45 1648 657 −60.13 P3 156.53 181 183 1.10 295.70 369 343 −7.05 542.27 1670 693 −58.50 P4 119.60 153 154 0.65 264.77 338 325 −3.85 554.18 1698 690 −59.36 P5 130.45 156 158 1.28 268.07 347 329 −5.19 591.76 1754 699 −60.15 P6 126.40 145 147 1.38 261.27 860 316 −63.26 547.50 1638 644 −60.68 P7 120.65 147 149 1.36 243.17 289 280 −3.11 388.53 1092 626 −42.67 P8 127.81 157 154 −1.91 287.46 861 327 −62.02 534.57 1550 630 −59.35 . . 20-Facility MSPSC . . 50-Facility MSPSC . . 100-Facility MSPSC . Profile . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . LP bound . CPLEX . GA . dev% . M1P1 304.29 362 350 −3.31 768.50 3174 888 −72.02 1355.93 6292 1766 −71.93 P2 225.64 258 260 0.78 534.53 655 612 −6.56 956.67 3598 1222 −66.04 P3 214.16 251 253 0.80 545.14 2658 651 −75.51 1166.56 3640 1324 −63.63 P4 214.16 251 255 1.59 548.30 629 629 0.00 1026.32 3739 1263 −66.22 P5 221.87 280 280 0.00 548.49 699 643 −8.01 1081.76 3595 1270 −64.67 P6 209.69 226 228 0.88 512.16 597 594 −0.50 997.56 5950 1228 −79.36 P7 203.52 222 228 2.70 433.92 1364 525 −61.51 802.78 1970 1165 −40.86 P8 203.16 228 233 2.19 555.83 636 636 0.00 711.44 2171 1187 −45.32 M2P1 258.48 312 311 −0.32 626.16 2557 716 −72.00 1241.99 3270 1473 −54.95 P2 186.68 242 235 −2.89 418.78 2280 509 −77.68 829.62 2862 1045 −63.49 P3 207.08 258 248 −3.88 443.57 2380 548 −76.97 994.13 4427 1116 −74.79 P4 186.58 231 232 0.43 439.96 601 529 −11.98 728.01 3045 1066 −64.99 P5 191.61 225 230 2.22 439.38 2358 540 −77.10 906.94 3830 1069 −72.09 P6 177.00 207 209 0.97 414.76 528 503 −4.73 840.05 4834 1037 −78.55 P7 190.09 233 225 −3.43 403.93 461 461 0.00 723.66 1688 956 −43.36 P8 157.81 196 195 −0.51 425.17 502 502 0.00 829.07 2752 995 −63.84 M3P1 185.90 229 217 −5.24 469.99 1828 541 −70.40 937.48 3596 1089 −69.72 P2 139.92 176 171 −2.84 347.01 443 392 −11.51 693.51 2055 781 −62.00 P3 151.27 190 182 −4.21 323.79 439 381 −13.21 630.53 2054 814 −60.37 P4 136.21 171 177 3.51 377.56 433 432 −0.23 655.84 2138 820 −61.65 P5 137.85 159 163 2.52 320.65 1772 390 −77.99 732.32 2138 826 −61.37 P6 129.98 155 155 0.00 341.13 430 396 −7.91 652.12 2436 767 −68.51 P7 129.68 152 153 0.66 280.87 331 331 0.00 628.35 1589 756 −52.42 P8 129.39 158 158 0.00 349.98 449 411 −8.46 647.07 1954 745 −61.87 M4P1 195.11 233 225 −3.43 452.12 1754 527 −69.95 847.61 2376 1099 −53.75 P2 148.89 183 181 −1.09 328.59 418 377 −9.81 644.29 2151 834 −61.23 P3 170.34 203 207 1.97 349.57 406 400 −1.48 651.01 2170 840 −61.29 P4 129.21 159 162 1.89 330.12 881 375 −57.43 670.87 2178 798 −63.36 P5 142.66 177 176 −0.56 315.99 385 380 −1.30 674.09 2172 821 −62.20 P6 141.97 169 173 2.37 315.46 382 380 −0.52 588.53 1739 807 −53.59 P7 130.86 156 160 2.56 282.58 388 330 −14.95 595.96 1551 790 −49.07 P8 137.51 180 170 −5.56 334.44 412 393 −4.61 617.82 2051 723 −64.75 M5P1 178.72 213 206 −3.29 378.97 1397 435 −68.86 791.47 2872 908 −68.38 P2 134.16 162 166 2.47 286.32 1268 326 −74.29 529.45 1648 657 −60.13 P3 156.53 181 183 1.10 295.70 369 343 −7.05 542.27 1670 693 −58.50 P4 119.60 153 154 0.65 264.77 338 325 −3.85 554.18 1698 690 −59.36 P5 130.45 156 158 1.28 268.07 347 329 −5.19 591.76 1754 699 −60.15 P6 126.40 145 147 1.38 261.27 860 316 −63.26 547.50 1638 644 −60.68 P7 120.65 147 149 1.36 243.17 289 280 −3.11 388.53 1092 626 −42.67 P8 127.81 157 154 −1.91 287.46 861 327 −62.02 534.57 1550 630 −59.35 Open in new tab To facilitate the comparisons between GA and CPLEX we formed Table 6 which summarizes the results of all tests. Each row of Table 6 indicates a specific performance metric. Particularly, the first line of the table shows the number of occasions GA gave a solution equal to CPLEX. The second line reports the number of occasions GA outperformed CPLEX. The third line shows the average dev% of the GA solution relative to the CPLEX solution and the fourth line presents the most extreme dev% over the 40 instances of each problem class. Finally, the last line of the table reports the average CPU time spent by GA to achieve the best solution. As one can see from Table 6, GA solution surpasses CPLEX solution in 90 problem settings (i.e. in 75% of the tests). In eight problem settings GA found the CPLEX solution (i.e. for circa 6% of the tests). More important, however, is the fact that CPLEX performed better than GA only in 22 problem settings (i.e. 18% of the tests) all belonging in the small-sized class of problems (that with 20 facilities). In these cases, the GA solution deviated less than 3.51% from the CPLEX solution. However, as the number of facilities increases, the superiority of the GA over CPLEX becomes more obvious. In fact, in the most difficult class of problems (that with 100 facilities), GA outperformed CPLEX in all 40 test instances examined, generating solutions of much higher quality. As one can clearly see from Table 6, in the case of the 100-facility problems the cost of the GA solutions is on average 61% lower than the cost of the CPLEX solutions. Given the nature of the problem this indicates a very satisfactory overall quality of the GA solution. Table 6. Summary of GA results over the 120 MSPSC test problems . 20-Facility . 50-Facility . 100-Facility . No. C = C(CPLEX) 3 5 0 No. C < C(CPLEX) 15 35 40 Average % deviation from C(CPLEX) −0.15% −29.55% −61.26% % deviation from C(CPLEX) with maximum absolute value −5.56% −77.99% −79.36% GA’s average computation time (second) 2.91 3.38 9.79 . 20-Facility . 50-Facility . 100-Facility . No. C = C(CPLEX) 3 5 0 No. C < C(CPLEX) 15 35 40 Average % deviation from C(CPLEX) −0.15% −29.55% −61.26% % deviation from C(CPLEX) with maximum absolute value −5.56% −77.99% −79.36% GA’s average computation time (second) 2.91 3.38 9.79 Open in new tab Table 6. Summary of GA results over the 120 MSPSC test problems . 20-Facility . 50-Facility . 100-Facility . No. C = C(CPLEX) 3 5 0 No. C < C(CPLEX) 15 35 40 Average % deviation from C(CPLEX) −0.15% −29.55% −61.26% % deviation from C(CPLEX) with maximum absolute value −5.56% −77.99% −79.36% GA’s average computation time (second) 2.91 3.38 9.79 . 20-Facility . 50-Facility . 100-Facility . No. C = C(CPLEX) 3 5 0 No. C < C(CPLEX) 15 35 40 Average % deviation from C(CPLEX) −0.15% −29.55% −61.26% % deviation from C(CPLEX) with maximum absolute value −5.56% −77.99% −79.36% GA’s average computation time (second) 2.91 3.38 9.79 Open in new tab Turning to the computer run time required for arriving at a GA solution; from Table 6 (last line) we can see that the average GA solution time varied between circa 3 and 10 s computer run time. This is considered as quite fast for a population-based heuristic which in our case maintains and evolves a population of |$5M$| individuals. This means, for example, a population of 500 individuals for the case of 100-facility problems. Finally, to explore the effects of the workload and manning specifications on GA solution quality we build Table 7. Each cell (i.e. workload-manning profile combinations) in this table shows three entries concerning the percent deviation of GA solutions from CPLEX over the 20-, 50- and 100-facility instances. For instance, for the case of M1:P1 profile combination, the deviation of GA solutions from CPLEX was −3.31% for the 20-facility, −72.02% for 50-facility and −71.93% for 100-facility problems. Notice that Table 7 also presents averages overall production workloads (PAV; last column) and manning profiles (MAV; last line). Table 7 The effect of manning and workload profiles on the GA’s performance Profile . . P1 . . . P2 . . . P3 . . . P4 . . . . . M1 −3.31 −72.02 −71.93 0.78 −6.56 −66.04 0.80 −75.51 −63.63 1.59 0.00 −66.22 M2 −0.32 −72.00 −54.95 −2.89 −77.68 −63.49 −3.88 −76.97 −74.79 0.43 −11.98 −64.99 M3 −5.24 −70.40 −69.72 −2.84 −11.51 −62.00 −4.21 −13.21 −60.37 3.51 −0.23 −61.65 M4 −3.43 −69.95 −53.75 −1.09 −9.81 −61.23 1.97 −1.48 −61.29 1.89 −57.43 −63.36 M5 −3.29 −68.86 −68.38 2.47 −74.29 −60.13 1.10 −7.05 −58.50 0.65 −3.85 −59.36 MAV −3.12 −70.65 −63.75 −0.71 −35.97 −62.58 −0.84 −34.84 −63.72 1.61 −14.70 −63.12 Profile P5 P6 P7 P8 PAV M1 0.00 −8.01 −64.67 0.88 −0.50 −79.36 2.70 −61.51 −40.86 2.19 0.00 −45.32 0.70 −28.01 −62.25 M2 2.22 −77.10 −72.09 0.97 −4.73 −78.55 −3.43 0.00 −43.36 −0.51 0.00 −63.84 −0.93 −40.06 −64.51 M3 2.52 −77.99 −61.37 0.00 −7.91 −68.51 0.66 0.00 −52.42 0.00 −8.46 −61.87 −0.70 −23.71 −62.24 M4 −0.56 −1.30 −62.20 2.37 −0.52 −53.59 2.56 −14.95 −49.07 −5.56 −4.61 −64.75 −0.23 −20.01 −58.66 M5 1.28 −5.19 −60.15 1.38 −63.26 −60.68 1.36 −3.11 −42.67 −1.91 −62.02 −59.35 0.38 −35.95 −58.65 MAV 1.09 −33.92 −64.10 1.12 −15.38 −68.14 0.77 −15.91 −45.68 −1.16 −15.02 −59.03 −0.15 −29.55 −61.26 Profile . . P1 . . . P2 . . . P3 . . . P4 . . . . . M1 −3.31 −72.02 −71.93 0.78 −6.56 −66.04 0.80 −75.51 −63.63 1.59 0.00 −66.22 M2 −0.32 −72.00 −54.95 −2.89 −77.68 −63.49 −3.88 −76.97 −74.79 0.43 −11.98 −64.99 M3 −5.24 −70.40 −69.72 −2.84 −11.51 −62.00 −4.21 −13.21 −60.37 3.51 −0.23 −61.65 M4 −3.43 −69.95 −53.75 −1.09 −9.81 −61.23 1.97 −1.48 −61.29 1.89 −57.43 −63.36 M5 −3.29 −68.86 −68.38 2.47 −74.29 −60.13 1.10 −7.05 −58.50 0.65 −3.85 −59.36 MAV −3.12 −70.65 −63.75 −0.71 −35.97 −62.58 −0.84 −34.84 −63.72 1.61 −14.70 −63.12 Profile P5 P6 P7 P8 PAV M1 0.00 −8.01 −64.67 0.88 −0.50 −79.36 2.70 −61.51 −40.86 2.19 0.00 −45.32 0.70 −28.01 −62.25 M2 2.22 −77.10 −72.09 0.97 −4.73 −78.55 −3.43 0.00 −43.36 −0.51 0.00 −63.84 −0.93 −40.06 −64.51 M3 2.52 −77.99 −61.37 0.00 −7.91 −68.51 0.66 0.00 −52.42 0.00 −8.46 −61.87 −0.70 −23.71 −62.24 M4 −0.56 −1.30 −62.20 2.37 −0.52 −53.59 2.56 −14.95 −49.07 −5.56 −4.61 −64.75 −0.23 −20.01 −58.66 M5 1.28 −5.19 −60.15 1.38 −63.26 −60.68 1.36 −3.11 −42.67 −1.91 −62.02 −59.35 0.38 −35.95 −58.65 MAV 1.09 −33.92 −64.10 1.12 −15.38 −68.14 0.77 −15.91 −45.68 −1.16 −15.02 −59.03 −0.15 −29.55 −61.26 Open in new tab Table 7 The effect of manning and workload profiles on the GA’s performance Profile . . P1 . . . P2 . . . P3 . . . P4 . . . . . M1 −3.31 −72.02 −71.93 0.78 −6.56 −66.04 0.80 −75.51 −63.63 1.59 0.00 −66.22 M2 −0.32 −72.00 −54.95 −2.89 −77.68 −63.49 −3.88 −76.97 −74.79 0.43 −11.98 −64.99 M3 −5.24 −70.40 −69.72 −2.84 −11.51 −62.00 −4.21 −13.21 −60.37 3.51 −0.23 −61.65 M4 −3.43 −69.95 −53.75 −1.09 −9.81 −61.23 1.97 −1.48 −61.29 1.89 −57.43 −63.36 M5 −3.29 −68.86 −68.38 2.47 −74.29 −60.13 1.10 −7.05 −58.50 0.65 −3.85 −59.36 MAV −3.12 −70.65 −63.75 −0.71 −35.97 −62.58 −0.84 −34.84 −63.72 1.61 −14.70 −63.12 Profile P5 P6 P7 P8 PAV M1 0.00 −8.01 −64.67 0.88 −0.50 −79.36 2.70 −61.51 −40.86 2.19 0.00 −45.32 0.70 −28.01 −62.25 M2 2.22 −77.10 −72.09 0.97 −4.73 −78.55 −3.43 0.00 −43.36 −0.51 0.00 −63.84 −0.93 −40.06 −64.51 M3 2.52 −77.99 −61.37 0.00 −7.91 −68.51 0.66 0.00 −52.42 0.00 −8.46 −61.87 −0.70 −23.71 −62.24 M4 −0.56 −1.30 −62.20 2.37 −0.52 −53.59 2.56 −14.95 −49.07 −5.56 −4.61 −64.75 −0.23 −20.01 −58.66 M5 1.28 −5.19 −60.15 1.38 −63.26 −60.68 1.36 −3.11 −42.67 −1.91 −62.02 −59.35 0.38 −35.95 −58.65 MAV 1.09 −33.92 −64.10 1.12 −15.38 −68.14 0.77 −15.91 −45.68 −1.16 −15.02 −59.03 −0.15 −29.55 −61.26 Profile . . P1 . . . P2 . . . P3 . . . P4 . . . . . M1 −3.31 −72.02 −71.93 0.78 −6.56 −66.04 0.80 −75.51 −63.63 1.59 0.00 −66.22 M2 −0.32 −72.00 −54.95 −2.89 −77.68 −63.49 −3.88 −76.97 −74.79 0.43 −11.98 −64.99 M3 −5.24 −70.40 −69.72 −2.84 −11.51 −62.00 −4.21 −13.21 −60.37 3.51 −0.23 −61.65 M4 −3.43 −69.95 −53.75 −1.09 −9.81 −61.23 1.97 −1.48 −61.29 1.89 −57.43 −63.36 M5 −3.29 −68.86 −68.38 2.47 −74.29 −60.13 1.10 −7.05 −58.50 0.65 −3.85 −59.36 MAV −3.12 −70.65 −63.75 −0.71 −35.97 −62.58 −0.84 −34.84 −63.72 1.61 −14.70 −63.12 Profile P5 P6 P7 P8 PAV M1 0.00 −8.01 −64.67 0.88 −0.50 −79.36 2.70 −61.51 −40.86 2.19 0.00 −45.32 0.70 −28.01 −62.25 M2 2.22 −77.10 −72.09 0.97 −4.73 −78.55 −3.43 0.00 −43.36 −0.51 0.00 −63.84 −0.93 −40.06 −64.51 M3 2.52 −77.99 −61.37 0.00 −7.91 −68.51 0.66 0.00 −52.42 0.00 −8.46 −61.87 −0.70 −23.71 −62.24 M4 −0.56 −1.30 −62.20 2.37 −0.52 −53.59 2.56 −14.95 −49.07 −5.56 −4.61 −64.75 −0.23 −20.01 −58.66 M5 1.28 −5.19 −60.15 1.38 −63.26 −60.68 1.36 −3.11 −42.67 −1.91 −62.02 −59.35 0.38 −35.95 −58.65 MAV 1.09 −33.92 −64.10 1.12 −15.38 −68.14 0.77 −15.91 −45.68 −1.16 −15.02 −59.03 −0.15 −29.55 −61.26 Open in new tab First, considering the effects of problem size, the results of Table 7 indicate that the GA solution quality improves with increased problem size. This happens for all aggregate metrics (i.e. MAV and PAV entries). As far as the manning profile is concerned, solution quality is generally better when manning requirements are homogeneous for all facilities (as for profiles M1 and M2), whereas quality seem to deteriorate for manning requirements not evenly distributed across facilities (see profiles M3 and M4). On the other hand, quality seems to improve when workload is distributed evenly (profile P1) or follows a parabolic distribution (profile P3) across facilities. 6. Conclusions This paper has dealt with the formulation and the heuristic solution of a new manpower planning problem termed MSPSC aiming at exploring the cost effectiveness of manpower planning decisions in situations where flexible and similarly skilled workforce need to perform tasks at different sites (locations) within a specified time horizon. Such applications have been identified in highway construction projects. Even in its simplest single-shift variation, MSPSC is shown to be NP-hard. After formulating the MSPSC problem as an MILP model, a heuristic method based on a modified GA was introduced for its solution. The developed GA incorporates a problem-specific coding of a solution structure that automatically satisfies the problem constraints, a set of special variation genetic operators for creating legal offspring solutions and an adaptive control scheme for the self-tuning of the GA’s control parameters. The GA performance was evaluated on multiple benchmarks test problems of various sizes ranging from 20 to 100 facilities. The results showed that the GA heuristic is quite fast and robust, showing a remarkable high performance over all the experiments. Since this is the first formal study of the MSPSC problem there is still much ground for further research with respect to both the problem formulation and the improvement of the GA-based solution heuristic. Considering first the MSPSC formulation, we may attempt to incorporate special features such as the use of less flexible workforce, limits to the workforce employed in each shift, constraints restricting particular facilities to operate simultaneously or daily shifts differing in duration. Furthermore, in our model the fixed opening costs and the staffing cost are added in the objective function to determine the TC to be minimized. Alternatively, one may wish to treat these two cost elements as separate objectives and formulate a multiobjective version of the model which may include other, non-cost-related elements as well. Considering possible improvements of the heuristic solution, it is very interesting to experiment with different hybrid models which might improve the efficiency of GA. Ongoing research attempts to hybridize the developed GA with extended variable neighbourhood search (VNS) schemes such as the skewed VNS method. 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(2010) Services general Valls et al. (2009) Hospitals (nurse) Burke et al. (2001), Aickelin & Dowsland (2004), Aickelin & White (2004), Maenhout & Vanhoucke (2008), Tsai & Li (2009) and Li et al. (2009) Airlines Levine (1996) Software projects Luna et al. (2012) Shift planning Manufacturing (packing shops) Nearchou & Lagodimos (2013) Manufacturing (maintenance planning in chemical plants) Nearchou et al. (2014) Job rotation Manufacturing general Asensio-Cuesta et al. (2012) Staff allocation Multiple R&D projects Wu & Sun (2006) Workforce planning Manufacturing general Fowler et al. (2008) Problem . Application area . Paper . Crew rostering Airlines Lucic & Teodorovic (2007) Hospitals (doctor rostering) Puente et al. (2009) Hospitals (nurse rostering) Moz & Pato (2007), Burke et al. (2008), Pato & Moz (2008), Bai et al. (2010), Maenhout & Vanhoucke (2011) and Beddoe & Petrovic (2006) Transportations (railway) Monfroglio (1996) Crew scheduling Transportations (underground passengers) Elizondo et al. (2010) Retail sector Zolfaghari et al. (2010) Services general Valls et al. (2009) Hospitals (nurse) Burke et al. (2001), Aickelin & Dowsland (2004), Aickelin & White (2004), Maenhout & Vanhoucke (2008), Tsai & Li (2009) and Li et al. (2009) Airlines Levine (1996) Software projects Luna et al. (2012) Shift planning Manufacturing (packing shops) Nearchou & Lagodimos (2013) Manufacturing (maintenance planning in chemical plants) Nearchou et al. (2014) Job rotation Manufacturing general Asensio-Cuesta et al. (2012) Staff allocation Multiple R&D projects Wu & Sun (2006) Workforce planning Manufacturing general Fowler et al. (2008) Open in new tab Table A1. List of papers per problem and application area employed GAs for solving manpower planning problems Problem . Application area . Paper . Crew rostering Airlines Lucic & Teodorovic (2007) Hospitals (doctor rostering) Puente et al. (2009) Hospitals (nurse rostering) Moz & Pato (2007), Burke et al. (2008), Pato & Moz (2008), Bai et al. (2010), Maenhout & Vanhoucke (2011) and Beddoe & Petrovic (2006) Transportations (railway) Monfroglio (1996) Crew scheduling Transportations (underground passengers) Elizondo et al. (2010) Retail sector Zolfaghari et al. (2010) Services general Valls et al. (2009) Hospitals (nurse) Burke et al. (2001), Aickelin & Dowsland (2004), Aickelin & White (2004), Maenhout & Vanhoucke (2008), Tsai & Li (2009) and Li et al. (2009) Airlines Levine (1996) Software projects Luna et al. (2012) Shift planning Manufacturing (packing shops) Nearchou & Lagodimos (2013) Manufacturing (maintenance planning in chemical plants) Nearchou et al. (2014) Job rotation Manufacturing general Asensio-Cuesta et al. (2012) Staff allocation Multiple R&D projects Wu & Sun (2006) Workforce planning Manufacturing general Fowler et al. (2008) Problem . Application area . Paper . Crew rostering Airlines Lucic & Teodorovic (2007) Hospitals (doctor rostering) Puente et al. (2009) Hospitals (nurse rostering) Moz & Pato (2007), Burke et al. (2008), Pato & Moz (2008), Bai et al. (2010), Maenhout & Vanhoucke (2011) and Beddoe & Petrovic (2006) Transportations (railway) Monfroglio (1996) Crew scheduling Transportations (underground passengers) Elizondo et al. (2010) Retail sector Zolfaghari et al. (2010) Services general Valls et al. (2009) Hospitals (nurse) Burke et al. (2001), Aickelin & Dowsland (2004), Aickelin & White (2004), Maenhout & Vanhoucke (2008), Tsai & Li (2009) and Li et al. (2009) Airlines Levine (1996) Software projects Luna et al. (2012) Shift planning Manufacturing (packing shops) Nearchou & Lagodimos (2013) Manufacturing (maintenance planning in chemical plants) Nearchou et al. (2014) Job rotation Manufacturing general Asensio-Cuesta et al. (2012) Staff allocation Multiple R&D projects Wu & Sun (2006) Workforce planning Manufacturing general Fowler et al. (2008) Open in new tab © The Author(s) 2019. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Multisite and multishift personnel planning with set-up costs
JF - IMA Journal of Management Mathematics
DO - 10.1093/imaman/dpy017
DA - 2019-12-17
UR - https://www.deepdyve.com/lp/oxford-university-press/multisite-and-multishift-personnel-planning-with-set-up-costs-zURK9weNr9
SP - 5
VL - 31
IS - 1
DP - DeepDyve
ER -