Bi-criteria scheduling against restrictive common due dates using a multi-objective differential evolution algorithm

Bi-criteria scheduling against restrictive common due dates using a multi-objective differential... Abstract Consideration is given to the single-machine early/tardy scheduling problem (SMETP). This type of scheduling sets costs (penalties) depending on whether a job finished before or after a specified common due date. Two optimization criteria are simultaneously considered for minimization: first, the total weighted earliness and tardiness penalty costs, and second the total flow time of the jobs. A multi-objective differential evolution algorithm is presented devoted to the search for Pareto-optimal solutions. This algorithm is an adaptation of an existing algorithm for the single-objective SMETP that has shown excellent performance in terms of solution quality and speed. Using results from existing benchmark problems, we test the performance of the proposed algorithm on various operating environments with up to 1000 jobs. Furthermore, extended experimental comparisons are performed against three well-known multi-objective evolutionary algorithms. The results demonstrate very satisfactory performance for our algorithm in terms of both solution time and quality. 1. Introduction Scheduling of a set of independent jobs on a single machine so that they are completed by a specified restricted common due date (CDD) and objective to minimize the jobs’ total earliness and tardiness is NP-hard (Baker & Scudder, 1990; French, 1990). This type of scheduling is very important in just-in-time production environments where the main effort is to have the jobs completed as close as possible to their due date, neither too early, nor too late. The reason is that early jobs result in inventory holding costs, while late jobs result to customers’ dissatisfaction and consequently to possible loss of orders, loss of reputation, etc. Therefore, both earliness and tardiness of jobs should be discouraged. This is usually achieved by setting penalty costs depending on whether a job finished before (earliness), or after (tardiness) the specified due date. Minimizing these costs pushes the completion time of each job as close as possible to the due date. The problem, referred to herein as the single-machine early/tardy scheduling problem (SMETP) is formally known in the literature as $$1\left| {d_{j} =d} \right|\sum {\left( {\alpha_{j} E_{j} +\beta_{j} T_{j} } \right)}$$. Research in CDD scheduling problems is generally distinguished into two main categories: the first category assumes that CDD is unrestrictive, that is, the due date is sufficiently large so that it does not constrain the scheduling of the jobs. This is known as the unrestricted version of the problem. Kanet’s study (Kanet, 1981) was the first attempt to solve a variant of this problem dealing exclusively with the special case when the earliness and tardiness penalties are equal to one. Later, many researchers addressed various versions of the basic problem through the means of heuristics. Among them, Bagchi et al. (1986) and Hall & Posner (1991) studied the problem with objective to minimize the sum of absolute deviations of the job completion times from a CDD. Bagchi et al. (1986) proposed an enumerative algorithm which is effective at solving problems up to 15 jobs in size that leads to an optimal solution under the assumption that the starting time of the schedule is zero. Hall & Posner (1991) showed that the symmetric (i.e. $$\alpha_{j} =\beta_{j} )$$ unrestricted SMETP is solvable in a pseudo-polynomial time. Moreover, the authors developed a dynamic programming approach to solve such a problem. De et al. (1994) proposed a greedy randomized adaptive heuristic for the unrestricted problem with different penalties to find both the optimal due date and the optimal sequence of the jobs. The procedure consists of two phases: an initial solution is constructed in the first phase through controlled randomization and is improved in the second phase through steepest descent neighbourhood search. Hoogeveen & Van de Velde (1997) addressed the unrestricted problem with ‘almost’ CDD using a dynamic programming algorithm. The due dates of the jobs are different but lie between a large constant $$d$$ and $$d+p_{j} $$ for each job $$j$$ (with $$p_{j} $$ being the processing time of job $$j)$$. Van den Akker et al. (2000) presented a combined column generation and Lagrangian relaxation algorithm that solves instances with up to 125 jobs to optimality. Comprehensive surveys on CDD scheduling problems can be found in Cheng & Gupta (1989), Baker & Scudder (1990) and Gordon et al. (2002). The second category, concerns the restricted version of the problem which is obviously harder since the CDD is small enough to constrain the scheduling process. Existing theoretical results showed that the restricted problem is NP-hard even if all the job penalty costs are equal to one (Hoogeveen & van de Velde, 1991). Moreover, for the problem with general penalty costs (which is the case of this study) no pseudo-polynomial algorithm has never been proposed (Feldmann & Biskup, 2003). Only for small-sized problems with up to 10 jobs was an optimal schedule found in a reasonable time applying the mixed-integer program formulation proposed by Biskup & Feldmann (2001) using a commercial optimizer. Due to these complexity results, the recent research effort in this field is spent on the development of robust approximation algorithms based on metaheuristics. The first attempt for solving the restricted problem by means of metaheuristics is due to (Lee & Kim, 1995). The authors developed a parallel genetic algorithm (GA) with the job schedules being coded into chromosomes via a binary representation scheme. Parallel subpopulations were constructed by considering only jobs that can be processed first in the schedule. Offspring solutions were generated by special operators reflecting the problem-specific properties. James (1997) proposed different forms of the tabu search (TS) algorithm including one based on a sequence of jobs solution space and another based on an early/tardy solution space. New solutions were generated using the swap neighbourhood scheme. Hao et al. (1996) also used TS to solve a similar but unrestricted CDD scheduling problem. The proposed algorithm starts by generating a feasible solution via a suitable simple greedy heuristic. Improved neighbour solutions are iteratively generated considering swap and insert moves. Feldmann & Biskup (2003) tackled the problem using three different metaheuristics namely evolution strategy, simulated annealing (SA) and threshold accepting (TA). To obtain feasible starting solutions the authors employed a greedy heuristic which schedules those jobs with relatively high tardiness costs compared to their earliness costs prior to the due date. Comparative analysis showed that TA is superior in terms of both solution time and quality. Hino et al. (2005) developed a TS and a GA for the problem. Hybridization was performed by combing TS and GA in a sequential form as in the following: the best solution obtained by the one algorithm passes in a second stage as input to the other for improvement. Two hybrid forms were examined TS followed by GA and vice versa. Liao & Cheng (2007) proposed a hybrid approach which combines TS with a variable neighbourhood search (VNS) heuristic. VNS uses insertion and swap schemes for generating new neighbourhood solutions. To prevent cycling back to previously visited solutions, two tabu lists are used to record the recent VNS moves one for each different search scheme. A hybrid algorithm was also proposed by M’Hallah (2007) to address a special case of the problem with unit penalties. This algorithm combines a GA with SA and some other local search heuristics including dispatching rules and hill climbing. Dispatching rules are used to construct the initial GA population. Hill climbing is applied to every child issued by crossover while SA is used as a mutant operator to the best members of every generation. Nearchou (2008) tackled the problem usingdifferential evolution (DE). DE is a population-based algorithm which evolves real-valued vectors (solutions) in an attempt to reach the global optimum. New vectors are generated by scaling the difference of two randomly selected population vectors. Each vector is mapped to a feasible schedule solution using an appropriate representation scheme which reflects all the properties of the problem. Within this general background, this paper considers a bi-criteria SMETP with general earliness and tardiness penalties. The first criterion is to minimize the total weighted earliness and tardiness penalties from a restricted CDD while the second criterion is to minimize the total flow time of the jobs. We are interested for a solution approach devoted to the determination of the Pareto-optimal solutions of the bi-criteria SMETP. In this context, we provide an adaptation of our previous DE algorithm (Nearchou, 2008) to this new environment, termed to herein as MODE (the multi-objective DE). Since its inception by Storn & Price (1997) as a global optimizer over continuous parameter spaces, DE has been applied with success in various engineering applications including electrical and power systems, robotics, machine scheduling, route planning, etc. (see Neri et al., 2010; Das et al., 2016; for recent surveys). As far as the restricted SMETP is concerned, DE showed a remarkable high solution quality performance over the wellknown Biskup & Feldmann (2001)’s benchmarks introducing new upper bounds to nearly 60% of the instances included in the related data set (see Nearchou, 2008 for details). To the best of our knowledge, this is the first study dealing with the multi-criteria SMETP. This observation stems from the literature review performed with the aim of positioning MODE within the relevant research field. This review allowed us to draw the following general conclusions: (a) All published work concern the solution of the single-objective SMETP. (b) Several of the previous works (e.g. Lee & Kim, 1995; James, 1997) have an underlying weakness connected to the properties of the problem. Particularly, they limit their search to schedules in which the first job starts at time zero although this (as will be discussed in Section 3) might exclude all optimal schedules a priori. (c) MODE overcomes this weakness using an effective problem representation scheme. The rest of the paper is organized as follows: Section 2 states the problem. Section 3 presents MODE (the proposed solution algorithm) for solving the bi-criteria SMETP. Computational results concerning the performance of MODE and comparisons with three well-known multi-objective evolutionary algorithms (MOEAs) namely, SPEA2, NSGA-II and L-NSGA respectively are provided in Section 4. It is highlighted that, SPEA2, NSGA-II and L-NSGA have never been applied before to the problem under consideration. Finally, conclusions and directions for future work are pointed out in Section 5. 2. Problem formulation This section presents and discusses a general formulation for the bi-criteria SMETP. We make use of the following notation: $$n$$  Number of jobs  $$j$$  Index for jobs  $$p_{j}$$  Processing time of job $$j$$  $$C_{j}$$  Completion time of job $$j$$  $$d$$  Due date for completion (common for all jobs)  $$E_{j} $$  Earliness of job $$j$$  $$T_{j}$$  Tardiness of job $$j$$  $$\alpha_{j} $$  Earliness penalty for job $$j_{\mathrm{\thinspace }}(\alpha_{j} > 0)$$  $$\beta_{j} $$  Tardiness penalty for job $$j_{\mathrm{\thinspace }}(\beta_{j} > 0)$$  $$n$$  Number of jobs  $$j$$  Index for jobs  $$p_{j}$$  Processing time of job $$j$$  $$C_{j}$$  Completion time of job $$j$$  $$d$$  Due date for completion (common for all jobs)  $$E_{j} $$  Earliness of job $$j$$  $$T_{j}$$  Tardiness of job $$j$$  $$\alpha_{j} $$  Earliness penalty for job $$j_{\mathrm{\thinspace }}(\alpha_{j} > 0)$$  $$\beta_{j} $$  Tardiness penalty for job $$j_{\mathrm{\thinspace }}(\beta_{j} > 0)$$  The problem can be formally defined as follows: a set of $$n$$ jobs must be processed without interruption on a single machine. Each job $$j$$$$j=1,2,\ldots ,n)$$ requires a positive processing time $$p_{j} $$ and ideally must be completed exactly on a specific (common for all jobs) due date $$d$$. Penalties ($$\alpha_{j} $$ or $$\beta_{j} )$$ are incurred whenever a job $$j$$ is completed before or after this due date. Therefore, an ideal schedule is one in which all jobs finish on the specific due date. Assuming that, $$C_{j} $$ is the completion time of job $$j$$, then the earliness and tardiness of job $$j$$ are given by the following relations,   Ej =max(0,d−Cj)Tj =max(0,Cj−d)for all j=1,2,…,n (1) Under these conditions, the bi-criteria SMETP seeks to determine a processing order $$\sigma$$ of the $$n $$ jobs (i.e. $$\sigma$$ is a permutation of the numbers $$1,2,\ldots ,n)$$ that minimizes the following two objectives:   Objective-1:minimize SumET=∑j=1n(αjEj+βjTj) (2)  Objective-2:minimize TFT=∑j=1nCj (3) The first objective denotes the total weighted earliness and tardiness penalty costs of the schedule while the second objective denotes the total flow time of the jobs. Penalties $$\alpha_{j} ,\beta_{j} $$ in Eq. (2) can be measured in different ways resulting in several variations of the basic SMETP. Note that, $$d$$ is called unrestrictive when $$d\geqslant \sum_{j=1}^n {p_{j} } $$ holds, otherwise is called restrictive. Moreover, $$d$$ is also called unrestrictive when it constitutes a decision variable for the problem. Consequently, one can refer to the problem as either unrestricted or restricted SMETP. In this study we examine the restricted case of this particular scheduling problem. The main assumptions for SMETP in relation to the jobs operations and machine availability are as follows: 1. Jobs are independent without precedence or other constraints. 2. Jobs are known in advance and are all available for processing at time zero. 3. The machine can handle only one job at a time. 4. The machine is continuously available and is never kept idle while work is waiting. 5. No job pre-emption is permitted. 3. The proposed MODE algorithm for the bi-criteria SMETP For the restricted SMETP with general earliness and tardiness penalties there is an optimal schedule with the following three properties. Note that, these properties are supported within MODE through the use of a suitable solution representation scheme aiming to obtain high quality schedules more efficiently. Property 1 No idle times are contained between consecutive jobs (Cheng & Kahlbacher, 1991). Property 2 The schedule is V-shaped around the CDD (Smith, 1956). That is, early jobs are sequenced in non-increasing order of $$\frac{p_{j} }{\alpha_{j} }$$ (called ‘$$\backslash $$-shaped’ format) and late jobs are sequenced in non-decreasing order of $$\frac{p_{j} }{\beta_{j} }$$ ($$j=1,2,\ldots ,n)$$ (called ‘/-shaped’ format). Property 3 The processing time of the first job either starts at time zero, or one job is completed at the due date (Hoogeveen & van de Velde, 1991). All the variations of the restricted SMETP result to an NP-hard combinatorial optimization problem (Gordon et al., 2002). Furthermore, as far as more than one optimization criteria are concerned, the problem still remains intractable since finding the global optimum to a general multi-objective optimization (MOO) problem is NP-complete (Bäck, 1996). Actually, an exact solution to MOO problems at which all decision variables satisfy the associated constraints and all the individual objective functions have reached their associated optimal values may not even exist. Hence, the various developed MOO algorithms are in fact in a situation where their attempt is to optimize each individual objective to the greatest possible extend. Usually, there is no single optimal solution to these problems but rather a set of optimal solutions. This set is known as Pareto-optimal solutions. The solutions in this set are such that they are non-dominated by any other solution of the search space of the given problem when all the objectives are considered, and moreover, they do not dominate each other in the set. In this section, we present MODE; a multi-objective DE algorithm for the solution of the bi-criteria SMETP. MODE effectively constitutes an adaptation of the DE algorithm proposed by Nearchou (2008) in the new bi-objective environment. MODE maintains two separate populations: a main evolving population (termed MAIN) and a secondary population of diverse Pareto-optimal solutions (termed PARETO) which is iteratively updated. Let $$N_{\mathrm{main}}$$ denotes the population size of the first population, and $$N_{\mathrm{pareto}}$$ the size of the second population. Let also $$k_{\max } $$ denotes the maximum permitted number of iterations of the algorithm. The pseudo-code of the developed MODE followed by a description of its main components is given below. The algorithm starts (Step 1) by generating randomly an initial population MAIN of $$N_{\mathrm{main}}$$ solutions $$x_{i,k} $$ ($$i=1,2,\ldots ,N_{\mathrm{main}})$$ ($$k$$ denotes the iteration number of the algorithm). Each component $$x_{i,k}^{l} $$, $$l=(1,2,\ldots ,n)$$ of a vector $$x_{i,k} $$ is a real-valued number in the range (0,1). After the creation of MAIN, an initially empty PARETO population is created. A cycle of $$k_{\max }$$ iterations is then applied to MAIN and PARETO (Steps 2--4). At each iteration all vectors in MAIN are targeted for replacement. Therefore, $$N_{\mathrm{main}}$$ competitions are held to determine the members of MAIN for the next iterations. This is achieved by using mutation (Step 2.1), crossover (Step 2.2) and acceptance operators (Step 2.3). In the mutation phase, for each target vector $$x_{i,k} $$ ($$i=1,2,\ldots,N_{\mathrm{main}})$$ a mutant vector $$\stackrel{\frown}{{x}} _{i,k} $$ is obtained by   x⌢i,k=xr1,k+F(xr2,k−xr3,k), (4) where $$r1\ne r2\ne r3\ne i\in \left\{ {1,2,\ldots ,N_{\mathrm{main}}}\right\}$$. $$x_{r1,k} $$ is known as the base vector and $$F>0$$ is a (user defined) scaling parameter. The crossover operator (Step 2.2) is then applied to obtain the trial vector $$\psi_{i,k} $$ from $$\stackrel{\frown}{{x}}_{i,k} $$ and $$x_{i,k} $$. The crossover is defined by   ψi,kl={x⌢i,kl if rnd ⩽CR or l=Iixi,kl if rnd>CR and  l≠Ii , (5) where $$I_{i} $$ is a randomly chosen integer in the set $$I$$, that is $$I_{i}\in I=(1,2,\ldots ,n)$$; the superscript $$l$$ represents the $$l\mbox{-th}$$component of respective vectors. $$\mbox{rnd}\in (0,1)$$ is drawn randomly for each $$l$$. The ultimate aim of the crossover rule is to obtain the trial vector $$\psi_{i,k} $$ with components coming from the components of the target vector $$x_{i,k} $$ and the mutated vector $$\stackrel{\frown}{{x}}_{i,k} $$. This is ensured by introducing CR and set $$I$$. In the acceptance phase (Step 2.3) the trial $$\psi_{i,k}$$ and target $$x_{i,k} $$ vectors are compared for domination. To do so, first the two real-valued vectors are mapped to actual SMETP schedule solutions using Decode function. Then, their costs (in regard to both of the optimization criteria given by Eqs (2) and (3)) are compared and the non-dominated solution between them survives reproducing its structure in the new MAIN. The core idea behind Decode is to build schedules that satisfy the three problem properties mentioned above. This is accomplished by the following scheme: let’s assume the solution $$x_{i,k} $$. Each component $$x_{_{i,k} }^{l} \in \left( {0,1} \right) l=(1,2,\ldots ,n)$$ of this vector is associated to a specific job $$1,2,\ldots,n$$ with that order. A value less than or equal to 0.5 in the vector indicates that the corresponding job is early otherwise the job is tardy. So, the jobs are distinguished into two sets namely, $$S_{E} $$ and $$ S_{T}$$ containing the early and the tardy jobs respectively. Following the V-shaped property, the jobs in $$S_{E}$$ are moved at the start of the schedule and sequenced in a ‘$$\backslash $$-shaped’ way. Late jobs are moved at the end of the schedule and sequenced in a ‘/-shaped’ manner. Let sump the total processing time of the early jobs in $$S_{E}$$. According to Feldmann & Biskup (2003), an optimal solution to SMETP can fall in one of the following three disjunctive cases: (A) The first job in $$S_{E}$$ starts at time zero and the last job in $$S_{E} $$ finishes exactly on due date $$d$$. (B) The first job in $$S_{E}$$ starts at time zero and the last job in $$S_{E} $$ is completed prior to $$d.$$ Further, a straddling job exists, that is a job starting executed before $$d $$ and ending after $$d$$. (C) The first job in $$S_{E}$$ does not start at time zero (i.e. it is delayed) and the last job in $$S_{E} $$ is finished exactly on the due date $$d$$. Case-(A) occurs when sump $$= d$$; Case-(B) occurs when sump > $$d$$, while Case-(C) occurs when sump < $$d$$. Therefore, according to the proposed scheme, for every candidate vector of the entire population, first the sets $$S_{E} $$ and $$S_{T}$$ are created. Second, the processing time of the jobs in $$S_{E}$$ are summed up into sump until the value of this variable surpassed $$d $$ or no other jobs are contained in $$S_{E}$$. Third, the starting time of the first job in $$S_{E} $$ is defined. That is, when sump $$\geqslant d$$ (Cases (A) and (B)), the first job starts at time zero, otherwise (Case (C)), the first job is delayed starting at time $$d$$-sump. Fourth, jobs in $$S_{E} $$ are ordered based on ‘$$\backslash $$-shaped’ property, while jobs in $$S_{T}$$ are ordered based on ‘/-shaped’ property. The operation of Decode is given below in algorithmic form. More details about this decoding scheme including application examples can be found in our previous work (Nearchou, 2008). Each iteration of MODE is completed with the updating of PARETO population (Step 3). First, a TEMP is built combining the members of MAIN and PARETO (Step 3.1). Then (Step 3.2), the weakness $$w_{z} $$ for each $$z\in \mbox{TEMP}$$ is computed which corresponds to the total number of the TEMP members dominating $$z$$. The more the value of $$w_{z} $$ the lesser the quality of the solution corresponding to $$z$$. The members in TEMP are then (Step 3.3) sorted in ascending order of their weakness value. The actual PARETO updating is accomplished by Update_Pareto function (Step 3.4). The pseudo-code of this function is given below. Its operation is simple: a non-dominated TEMP solution not already included in PARETO is copied into PARETO if the latter has available space or if the new TEMP solution dominates some existing PARETO solutions. In the latter case all the existing dominated solutions are removed from PARETO. 4. Experimental investigation and discussion In this section we present computational results obtained for assessing the quality of the MODE algorithm. Comparisons have been also performed against to three famous MOEAs namely, SPEA2, NSGA-II and L-NSGA. All experiments were performed on a PC with 3GHz Intel core Duo CPU, 4GB RAM and Windows XP operating system. The algorithms were implemented in Java programming language. The strength Pareto evolutionary algorithm-2 (SPEA2) introduced by Zitzler et al. (2002) and the non-dominated sorting GA-II (NSGA-II) proposed by Deb et al. (2002) are very popular in the Evolutionary Computation community as effective multi-objective optimizers and due to their efficiency often constitute test beds against any new multi-objective algorithm. The Lorenz non-dominated sorting GA (L-NSGA) is a recent MOEA developed by Dugardin et al. (2010). It is based on NSGA-II but uses the Lorenz dominance relationship instead of the Pareto dominance with the aim to provide a stronger selection of the non-dominated solutions. 4.1 Design of experiments Three versions of the proposed MODE algorithm were evaluated differ in the way the mutant vectors (Eq. (4)) are created. We will refer to these versions with the abbreviations MODE1, MODE2 and MODE3, respectively. MODE1 creates mutants using Eq. (4); while MODE2 and MODE3 create mutants using Eqs (6) and (7), respectively (see below). Details about DE variants can be found in the works of Storn & Price (1997) and Storn (1996).  x⌢i,k =xi,k+λ(xbest,k−xi,k)+F(xr2,k−xr3,k) (6)  x⌢i,k =xbest,k+F(xr1,k+xr2,k−xr3,k−xr4,k) (7) with, $$r1\ne r2\ne r3\ne r4\ne i\in \left\{ {1,2,\ldots ,N_{\mathrm{main}}} \right\}$$. $$x_{\mathrm{best,}k} $$ denotes the trial vector corresponding to the best schedule solution of the (current) iteration $$k$$. $$\lambda \in $$(0,1) (in Eq. (6)) is a parameter that controls the greediness of this particular DE scheme. Here, according to the indications of Storn & Price (1997) we set $$\lambda = 0.9$$. The three MODE algorithms as well as SPEA2, NSGA-II, and L-NSGA were tested and compared over a set of public benchmarks problems proposed by Biskup & Feldmann (2001). These benchmarks include test instances ranging from small size with 10 jobs to large size instances with 1000 jobs. Particularly, there are 7 problem classes with 10, 20, 50, 100, 200, 500 and 1000 jobs with each class containing 10 test instances in total. The value of a restrictive factor $$h = 0.2$$, 0.4, 0.6, 0.8 classifies the benchmarks test problems as less or more restricted against a CDD using the relation   d=⌊h∑j=1npj⌋. (8) With $$\left\lfloor\ y\ \right\rfloor $$ denoting the biggest integer smaller than or equal to $$y$$. That is, for each problem, $$d $$is estimated by multiplying the restrictive factor $$h$$ with the summation of the processing times of the $$n$$ jobs. The lower the value of $$h$$ the more restrictive is $$d$$ (i.e. the higher the expected percentage of the late jobs). In this study we present results concerning the most restricted CDD benchmarks of the Biskup & Feldmann (2001)’s data set (i.e. the test instances with $$h = 0.2$$). It is worth noting that, the existing upper bounds on the optimal objective function values for these benchmarks concern only the first optimization criterion (i.e. SumET objective given by Eq. (2)). The performance of the six algorithms was quantified through the use of the following indices: (1) The quality ratio (P*/P) as a percentage. The larger the value for this ratio for a given heuristic, the higher its performance. P* denotes the number of the different Pareto solutions generated by a heuristic over a specific test instance. While, P denotes the number of non-dominated solutions among all Pareto solutions obtained by all the heuristics. In particular, since some Pareto solutions obtained by one heuristic may be dominated by other heuristics, all the obtained solutions are compared to each other and the non-dominated among them are selected. (2) The Zitzler index (Zitzler & Thiele, 1999) here denoted as Z(x,y) which represents the percentage of the Pareto solutions in algorithm x dominated by at least one Pareto solution of algorithm y. Since this measure is not symmetrical, it is also necessary to calculate Z(y,x). Therefore, if Z(x,y) < Z(y,x) then x is better than y. To obtain the average performance of the heuristics, each one of them was ran 30 times over every test instance (starting each time from a different random number seed) and the solution quality was averaged. Hence, as there are 10 instances in each one of the 7 problem classes, this means that each heuristic was ran $$7 \times 10 \times 30 times = 2100$$ times in total. 4.2 Choice of the control parameters’ settings All the algorithms were defined to evolve a fixed size population of $$N_{\mathrm{main}}=n$$ individual solutions (with $$n$$ being the number of the jobs) and run for a maximum time of 0.3$$n$$ CPU seconds. This means, a maximum running time equal to 3 s for 10-job problems, 6 s for 20-job, $$\ldots$$ 300 s for 1000-job problems. The maximum size of the Pareto population maintained by each heuristic was defined to be equal to $$N_{\mathrm{pareto}}=\left\lfloor {5+\sqrt n } \right\rfloor $$ solutions. To determine the final Pareto set solutions for each algorithm we applied the following procedure: the Pareto solutions generated in every different run are combined into a single union set with $$30N_{\mathrm{pareto}}$$ solutions. Then, a non-dominating sorting is performed and the first $$N_{\mathrm{pareto}}$$ solutions of this union set constitute the final Pareto set for the particular algorithm. To estimate the correct settings for CR and $$F$$ used within the three MODE algorithms a dynamic self-adapted control scheme was adopted previously proposed by Brest et al. (2006). According to this scheme for each new parent vector solution a new pair of CR and $$F$$ values is estimated using the following relations:   Fi,k+1 ={Fl+rand1×Fu,if rand2<r1Fi,k,otherwise  (9)  CRi,k+1 ={rand3,if rand4<r2CRi,k,otherwise  (10) where, $$F_{l} =0.1$$, $$F_{u} =0.9$$, $$r_{1} =r_{2} =0.1$$ and rand$$_{j}$$ ($$j\in \{ 1,2,3,4\} $$) are uniform random numbers in the range [0,1]. This dynamic scheme was found superior to various existing static control schemes examined in preliminary experimental studies. At the start, CR and $$F $$are set to 0.9 and 0.5, respectively for all the members of the initial population. SPEA2, NSGA-II and L-NSGA were implemented according to their description in the literature. The individuals are represented as floating-point vectors (as in the proposed MODE algorithms) while solutions encoding is performed using the developed procedure presented in Section 3.1. To determine the appropriate crossover and mutation rates for these algorithms we used the following experimental framework: we set the mutation rate to a fix value within the discrete range 0.0, 0.01, 0.1 and experimented with various crossover rates in the range 0.3, 0.6, 0.8, 0.9, 1.0. The best crossover and mutation rates obtained after the termination of these preliminary runs are given in Table 1. Note that, these rates were finally chosen according to the notion of non-dominance in relation to the metrics given in Section 4.1 above. Table 1 Control settings for the crossover and mutation rates of SPEA2, NSGA-II, L-NSGA     SPEA2  NSGA-II  L-NSGA  Crossover rate  0.8  0.8  0.8  Mutation rate  0.0  0.1  0.1     SPEA2  NSGA-II  L-NSGA  Crossover rate  0.8  0.8  0.8  Mutation rate  0.0  0.1  0.1  4.3 Comparative results and discussion The detailed results with the extreme Pareto solutions (i.e. best values for SumET and TFT objectives) obtained by the six heuristics for all test problems are given in the Appendix. The best objective value obtained in each test instance is indicated in bold. The cells in the third column ($$\mbox{UB}_{\mathrm{SumET}} )$$ of the Table A1 in the Appendix show the best known value in regard to SumET objective given by Nearchou (2008) when considering the related single-objective problem. One can rather easily recognize the high quality performance achieved by both MODE3 and SPEA2 heuristics. More important however, is the efficiency of the algorithms over the large-sized test problems (with $$n > 200$$). Hence, from these results we observe that MODE3 outperformed all the others in all but 2 of the 500-job instances, as well as, in all of the 1000-job test instances. Let us now discuss the performance of the six heuristics in regard to the quality ratio (P*/P). Table 2 displays the mean quality ratio (P*/P) obtained by each one of the heuristics (after the 30 runs) over the examined test problems. Note that, this ratio is averaged over the 10 test instances included in each class of problems. The higher the value of this ratio (maximizing at 1) for a particular algorithm the greater its performance. For example a quality ratio equal to 0.82 (see the first cell in Table 2 means that 82% of the solutions obtained by the specific algorithm (here MODE1) are non-dominated by any Pareto solution of the other algorithms. Hence, as one can see from Table 2, for test problems with up to 200 jobs, SPEA2 achieved higher quality ratio than the other heuristics. For larger size problems however (with 500 and 1000 jobs), MODE3 was found superior outperforming all the other heuristics. It is worth noting that, concerning the 1000-job problems, MODE3 attained a quality ratio equal to one. This means that all of the MODE3 Pareto solutions are non-dominated by any Pareto solution created by the remaining five algorithms. Note that, a zero ratio means that all of the solutions obtained by an algorithm are dominated by at least one solution of another algorithm. Table 2 Mean quality ratio over the seven test problem classes  $$n$$  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  10  0.82  0.85  0.89  0.77  0.84  0.85  20  0.61  0.30  0.60  0.78  0.81  0.86  50  0.00  0.03  0.59  0.10  0.43  0.95  100  0.00  0.00  0.12  0.00  0.00  0.99  200  0.00  0.46  0.43  0.00  0.00  0.60  500  0.00  0.13  0.97  0.00  0.00  0.10  1000  0.00  0.00  1.00  0.00  0.00  0.00  $$n$$  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  10  0.82  0.85  0.89  0.77  0.84  0.85  20  0.61  0.30  0.60  0.78  0.81  0.86  50  0.00  0.03  0.59  0.10  0.43  0.95  100  0.00  0.00  0.12  0.00  0.00  0.99  200  0.00  0.46  0.43  0.00  0.00  0.60  500  0.00  0.13  0.97  0.00  0.00  0.10  1000  0.00  0.00  1.00  0.00  0.00  0.00  Turning now to the Zitzler index, Table 3 shows the values of this performance metric obtained over the first test instance included in the 500-job problem class. Recall that, Zitzler index is estimated through an algorithm-to-algorithm comparison iterated procedure. That is, each one of the algorithms is compared against to all the others. Observe for example cell (x,y)$$=$$(MODE1, L-NSGA) in the first line of Table 3. The value 88.89% in this cell is the Zitzler index of MODE1 against L-NSGA and means that 88.89% of the solutions generated by MODE1 are dominated by at least one L-NSGA solution. The Zitzler index is not symmetric and therefore we must also examine the value in cell (y,x)$$=$$(L-NSGA,MODE1) in order to make a safe conclusion concerning the efficiency of the two algorithms. Z(L-NSGA,MODE1) $$=$$ 77.78%. Hence, since Z(L-NSGA,MODE1) < Z(MODE1,L-NSGA) we can conclude that L-NSGA is superior to MODE1 for this particular test instance. Counting the number of y’s dominated by each (algorithm) x gives us the overall performance of the latter. This information is reported in the last column of Table 3. As one can see, MODE3 outperforms all the other five heuristics attained the smallest Zitzler index, this is depicted with a score 5. The second best performance was achieved by MODE2 with an overall score equal to 4 (meaning that, MODE2 outperforms all the other heuristics except from MODE3). Next comes SPEA2 with a total score equal to 3. Table 3 The Zitzler index for the six algorithms over the first test instance 500-job SMETP        Y        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  No. Z(x,y) < Z(y,x)  X  MODE1  —  100.0%  100.0%  88.89%  88.85%  100.0%  0     MODE2  0.0%  —  66.67%  0.0%  0.0%  0.0%  4     MODE3  0.0%  22.22%  —  0.0%  0.0%  0.0%  5     L-NSGA  77.78%  100.0%  100.0%  —  85.19%  100.0%  2     NSGA-II  55.56%  100.0%  100.0%  88.89%  —  100.0%  1     SPEA2  0.0%  100.0%  100.0%  3.7%  3.7%  —  3        Y        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  No. Z(x,y) < Z(y,x)  X  MODE1  —  100.0%  100.0%  88.89%  88.85%  100.0%  0     MODE2  0.0%  —  66.67%  0.0%  0.0%  0.0%  4     MODE3  0.0%  22.22%  —  0.0%  0.0%  0.0%  5     L-NSGA  77.78%  100.0%  100.0%  —  85.19%  100.0%  2     NSGA-II  55.56%  100.0%  100.0%  88.89%  —  100.0%  1     SPEA2  0.0%  100.0%  100.0%  3.7%  3.7%  —  3  Obviously, we cannot give in a single table the detailed Zitzler values for the six algorithms over all the test problems. However, we can count the y’s dominated by each x (in terms of the Zitzler index) over the instances included in the examined data set. Table 4 depicts this information. From this table we can now safely conclude about the overall performance of the algorithms. As it is clear, SPEA2 attained the highest total score (equal to 31) meaning that in 31 out the 70 in total test instances showed superior performance than the other algorithms. The second best performance was attained by MODE3 with a total score equal to 28. The remaining algorithms showed fairly lower quality performance with L-NSGA being the weakest. Furthermore, Table 4 gives us another significant information: MODE3 efficiency increases as the size of the problem increases. As one can observe from Table 4, for large-sized problems ($$n\geqslant 200$$ jobs) MODE3 outperformed all the other heuristics in 21 out the 30 in total test instances included in these three problem classes. The next best performance is due to SPEA2 with a score equal to 6. Therefore, as a final remark, SPEA2 and MODE3 are both quite efficient multi-objective optimizers to tackle the bi-criteria SMETP examined in this paper; with the latter being clearly more efficient when addressing large-sized instances of the problem. Table 4 A synopsis of how many times each algorithm achieved the highest performance in regard to the Zitzler index over the examined problems  Test problems  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  $$n \leqslant 100$$  4  3  7  1  5  25  $$n $$ > 100  0  3  21  0  0  6  Total  4  6  28  1  5  31  Test problems  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  $$n \leqslant 100$$  4  3  7  1  5  25  $$n $$ > 100  0  3  21  0  0  6  Total  4  6  28  1  5  31  Turning to the small- and medium-sized instances ($$n \leqslant 100$$) (where MODE3 found inferior than SPEA2), further comparative experiments performed showed that, increasing the population size (for both algorithms) results to higher performance for MODE3. Table 5 displays these results for three different population sizes. Each cell of the Table presents the number of the instances an algorithm became superior to the other in regard to the Zitzler index. Remember that, there are 10 instances in each problem class. Hence, a value equal to 10 means absolute superiority of the corresponding algorithm against its opponent. As it is clear from Table 5, in all but one test problems MODE3 outperformed SPEA2. Moreover, MODE3’s superiority clearly increases with the population size. This observation comes in accordance to the indications of the literature for sizing DE populations. In particular, according to Storn & Price (1997) best DE performance is achieved using populations in which parental vectors are multiples of the problem parameters. However, an in-depth analysis is still required to formally establish the limiting performance of MODE3 under these general conditions. Different problems required different settings. Table 5 Comparative results between MODE3 and SPEA2 for different population sizes. How many times the two algorithms obtained the best performance in regard to Zitzler index     Population size ($$N_{\mathrm{main}}$$)     2$$n$$  3$$n$$  5$$n$$  Test problems ($$n)$$  MODE3  SPEA2  MODE3  SPEA2  MODE3  SPEA2  10  8  2  10  0  8  2  20  7  3  6  4  10  0  50  4  6  6  4  10  0  100  7  3  7  3  7  3  Total  23  14  29  11  35  5     Population size ($$N_{\mathrm{main}}$$)     2$$n$$  3$$n$$  5$$n$$  Test problems ($$n)$$  MODE3  SPEA2  MODE3  SPEA2  MODE3  SPEA2  10  8  2  10  0  8  2  20  7  3  6  4  10  0  50  4  6  6  4  10  0  100  7  3  7  3  7  3  Total  23  14  29  11  35  5  5. Conclusions The problem of scheduling a number of jobs on a single machine against a restrictive CDD considering multiple optimization criteria has been presented in this paper. Two criteria have been considered for minimization: the total earliness and tardiness penalties, and the total flow-time of the jobs. As the restrictive CDD problem is known to be intractable we decided to tackle the problem using metaheuristics. To that purpose, a new multi-objective differential evolution (MODE) algorithm devoted to the derivation of the Pareto set solutions has been developed to address the problem. Three variants of the proposed MODE have been implemented and tested on existing benchmark problems taken from the open literature. Moreover, extended comparisons have been performed against to three well-known MOEAs namely, SPEA2, NSGA-II and L-NSGA. The experiments showed that a particular MODE variant is quite efficient to address the problem under consideration since was found superior to all the other multi-objective algorithms especially when addressing large-sized problems. Future work will be focused on more realistic versions of this particular scheduling problem. The case where the optimization criteria are considered interact is an interesting direction of future work. Research along these lines will consider the discrete Choquet integral method (Grabisch & Labreuche, 2010; Abichou et al., 2015) as a means to aggregate the criteria in the fitness function of each individual solution. This technique seems to be quite promising as a way to model the interactions between the optimization criteria in a tangible way. Furthermore, in practice, many MOO problems have multiple conflicting objective functions expressed in differing units, and with an inverse, non-linear relationship among themselves. These objectives may be even imprecise or fuzzy in nature to be defined. In its present form the proposed MODE algorithm cannot address such problems. Hybridizing MODE with non-linear goal programming techniques (Tanino et al., 2003) as well as with other metaheuristics such as VNS (see e.g. Brito et al., 2016; Mjirda et al., 2016) may result to a promising optimization tool for these problems. 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Appendix Table A1 Best objective function values found by the heuristics for each test problem  $$n$$        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2        UB$$_{\mathrm{SumET}}$$  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  10  1  1936  1975  549  1992  546  1936  546  2091  546  1975  558  1936  558     2  1042  1042  594  1163  593  1042  593  1637  593  1042  593  1077  593     3  1586  1586  565  1714  565  1586  565  1763  565  1586  565  1586  565     4  2139  2214  463  2500  446  2139  459  2372  446  2205  478  2170  459     5  1187  1187  392  1199  391  1187  391  1362  391  1187  391  1187  392     6  1521  1523  386  1592  386  1521  386  1632  386  1521  386  1523  386     7  2170  2170  450  2365  450  2170  453  2612  450  2170  450  2170  451     8  1720  1720  291  1726  291  1720  291  1824  291  1726  291  1726  291     9  1574  1574  397  1574  397  1574  397  1746  397  1574  397  1574  401     10  1869  1888  546  1888  538  1869  539  2065  538  1869  546  1869  546  20  1  4394  4423  1656  4729  1645  4478  1678  4991  1645  4398  1656  4423  1648     2  8430  8555  1973  8914  1970  8496  1973  8936  1970  8460  1976  8443  1973     3  6210  6231  1974  6657  1947  6260  1982  7244  1947  6221  1989  6224  1956     4  9188  9357  1859  10869  1859  9289  2002  10656  1859  9332  1899  9192  1872     5  4215  4321  1361  4584  1342  4315  1352  4773  1342  4215  1377  4224  1343     6  6527  6586  1665  6983  1634  6578  1674  7259  1634  6541  1674  6550  1634     7  10455  10500  2108  11131  2080  10471  2125  11403  2080  10550  2107  10459  2080     8  3920  4252  1758  5793  1708  4126  1764  3948  1705  4390  1730  3954  1709     9  3465  3563  1015  3708  1003  3471  1011  3689  1003  3518  1010  3485  1003     10  4979  5000  1676  5870  1661  5051  1668  5997  1661  5122  1717  4995  1661  50  1  40697  43082  11009  42669  10287  43262  11061  43099  10343  41704  10688  41113  10441     2  30613  33367  9962  33131  9278  33433  9926  33533  9387  31586  9588  30748  9275     3  34435  38670  10992  37396  10282  37882  11300  38806  10453  35522  10581  34815  10341     4  27755  29964  9417  29206  8600  29489  9231  30802  8643  28588  8860  27907  8635     5  32307  38420  11592  39211  10857  36074  11701  42394  10892  33810  11339  32718  10992     6  34993  38536  11023  37581  10409  36880  11514  38802  10405  36121  10717  35601  10396     7  43136  47390  12126  49911  11436  46880  12032  52999  11609  45492  11680  43462  11476     8  43839  48414  14687  47286  13931  48074  14594  49274  14084  45334  14317  43903  13971     9  34228  37842  8992  37252  8321  36455  8986  39908  8430  35343  8841  34629  8356     10  32958  35056  10532  35445  9825  34587  10200  36862  9852  33547  10044  33191  9792  100  1  1..46E$$+$$5  1.97E$$+$$5  5.15E$$+$$4  1.64E$$+$$5  4.64E$$+$$4  1.82E$$+$$5  4.98E$$+$$4  1.85E$$+$$5  4.89E$$+$$4  1.68E$$+$$5  4.85E$$+$$4  1.54E$$+$$5  4.57E$$+$$4     2  1.25E$$+$$5  1.76E$$+$$5  4.87E$$+$$4  1.53E$$+$$5  4.43E$$+$$4  1.53E$$+$$5  4.77E$$+$$4  1.62E$$+$$5  4.62E$$+$$4  1.45E$$+$$5  4.70E$$+$$4  1.35E$$+$$5  4.49E$$+$$4     3  1.30E$$+$$5  1.67E$$+$$5  4.69E$$+$$4  1.44E$$+$$5  4.28E$$+$$4  1.54E$$+$$5  4.53E$$+$$4  1.49E$$+$$5  4.33E$$+$$4  1.53E$$+$$5  4.45E$$+$$4  1.36E$$+$$5  4.12E$$+$$4     4  1.30E$$+$$5  1.80E$$+$$5  4.55E$$+$$4  1.44E$$+$$5  4.19E$$+$$4  1.59E$$+$$5  4.41E$$+$$4  1.51E$$+$$5  4.39E$$+$$4  1.58E$$+$$5  4.40E$$+$$4  1.35E$$+$$5  4.19E$$+$$4     5  1.24E$$+$$5  1.81E$$+$$5  4.66E$$+$$4  1.43E$$+$$5  4.24E$$+$$4  1.53E$$+$$5  4.51E$$+$$4  1.51E$$+$$5  4.34E$$+$$4  1.43E$$+$$5  4.28E$$+$$4  1.32E$$+$$5  4.09E$$+$$4     6  1.39E$$+$$5  1.89E$$+$$5  4.73E$$+$$4  1.55E$$+$$5  4.24E$$+$$4  1.62E$$+$$5  4.44E$$+$$4  1.67E$$+$$5  4.34E$$+$$4  1.70E$$+$$5  4.53E$$+$$4  1.46E$$+$$5  4.20E$$+$$4     7  1.35E$$+$$5  1.78E$$+$$5  4.72E$$+$$4  1.51E$$+$$5  4.30E$$+$$4  1.62E$$+$$5  4.51E$$+$$4  1.59E$$+$$5  4.40E$$+$$4  1.58E$$+$$5  4.34E$$+$$4  1.40E$$+$$5  4$$+$$4     8  1.60E$$+$$5  2.07E$$+$$5  5.32E$$+$$4  1.87E$$+$$5  5.09E$$+$$4  1.93E$$+$$5  5.39E$$+$$4  1.90E$$+$$5  5.23E$$+$$4  1.93E$$+$$5  5.21E$$+$$4  1$$+$$5  5$$+$$4     9  1.17E$$+$$5  1.69E$$+$$5  4.79E$$+$$4  1.32E$$+$$5  4.38E$$+$$4  1.52E$$+$$5  4.68E$$+$$4  1.49E$$+$$5  4.60E$$+$$4  1.46E$$+$$5  4.53E$$+$$4  1$$+$$5  4$$+$$4     10  1.19E$$+$$5  1.67E$$+$$5  4.75E$$+$$4  1.34E$$+$$5  4.31E$$+$$4  1.46E$$+$$5  4.54E$$+$$4  1.48E$$+$$5  4.39E$$+$$4  1.38E$$+$$5  4.50E$$+$$4  1$$+$$5  4$$+$$4  200  1  4.99E$$+$$5  7.80E$$+$$5  1.96E$$+$$5  6.95E$$+$$5  1.88E$$+$$5  6.89E$$+$$5  1.88E$$+$$5  7.56E$$+$$5  1.94E$$+$$5  7.37E$$+$$5  1.94E$$+$$5  6$$+$$5  1$$+$$5     2  5.41E$$+$$5  8.38E$$+$$5  1.97E$$+$$5  7.00E$$+$$5  1$$+$$5  7.43E$$+$$5  1.89E$$+$$5  7.96E$$+$$5  1.95E$$+$$5  8.24E$$+$$5  1.93E$$+$$5  6$$+$$5  1.86E$$+$$5     3  4.89E$$+$$5  7.47E$$+$$5  1.84E$$+$$5  6.76E$$+$$5  1.74E$$+$$5  6$$+$$5  1.76E$$+$$5  7.17E$$+$$5  1.82E$$+$$5  7.25E$$+$$5  1.81E$$+$$5  6.47E$$+$$5  1$$+$$5     4  5.86E$$+$$5  8.56E$$+$$5  2.02E$$+$$5  7.71E$$+$$5  1.89E$$+$$5  7$$+$$5  1.92E$$+$$5  8.14E$$+$$5  1.97E$$+$$5  8.12E$$+$$5  1.99E$$+$$5  7.58E$$+$$5  1$$+$$5     5  5.13E$$+$$5  8.01E$$+$$5  2.02E$$+$$5  6.76E$$+$$5  1$$+$$5  6.99E$$+$$5  1.95E$$+$$5  7.62E$$+$$5  2.00E$$+$$5  7.71E$$+$$5  1.99E$$+$$5  6$$+$$5  1.96E$$+$$5     6  4.78E$$+$$5  7.30E$$+$$5  1.85E$$+$$5  6.80E$$+$$5  1$$+$$5  6.48E$$+$$5  1.81E$$+$$5  7.24E$$+$$5  1.88E$$+$$5  7.51E$$+$$5  1.88E$$+$$5  6$$+$$5  1.82E$$+$$5     7  4.55E$$+$$5  7.63E$$+$$5  1.89E$$+$$5  6.57E$$+$$5  1$$+$$5  6.49E$$+$$5  1.76E$$+$$5  7.07E$$+$$5  1.85E$$+$$5  6.72E$$+$$5  1.79E$$+$$5  6$$+$$5  1.79E$$+$$5     8  4.94E$$+$$5  7.98E$$+$$5  1.93E$$+$$5  7.04E$$+$$5  1.83E$$+$$5  6$$+$$5  1$$+$$5  7.30E$$+$$5  1.89E$$+$$5  7.37E$$+$$5  1.86E$$+$$5  6.87E$$+$$5  1.85E$$+$$5     9  5.29E$$+$$5  7.92E$$+$$5  1.92E$$+$$5  7.30E$$+$$5  1$$+$$5  6.92E$$+$$5  1.85E$$+$$5  7.73E$$+$$5  1.91E$$+$$5  7.54E$$+$$5  1.89E$$+$$5  6$$+$$5  1.84E$$+$$5     10  5.38E$$+$$5  8.12E$$+$$5  1.94E$$+$$5  7.16E$$+$$5  1$$+$$5  7$$+$$5  1.88E$$+$$5  8.11E$$+$$5  1.93E$$+$$5  8.11E$$+$$5  1.92E$$+$$5  7.37E$$+$$5  1.90E$$+$$5  500  1  2.95E$$+$$6  5.07E$$+$$6  1.24E$$+$$6  4.73E$$+$$6  1$$+$$6  4$$+$$6  1.19E$$+$$6  5.08E$$+$$6  1.23E$$+$$6  5.12E$$+$$6  1.24E$$+$$6  4.96E$$+$$6  1.22E$$+$$6     2  3.37E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  5.16E$$+$$6  1$$+$$6  4$$+$$6  1.25E$$+$$6  5.48E$$+$$6  1.29E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  4.99E$$+$$6  1.28E$$+$$6     3  3.10E$$+$$6  5.33E$$+$$6  1.26E$$+$$6  5.08E$$+$$6  1.21E$$+$$6  4$$+$$6  1$$+$$6  5.36E$$+$$6  1.26E$$+$$6  5.35E$$+$$6  1.26E$$+$$6  4.98E$$+$$6  1.22E$$+$$6     4  3.22E$$+$$6  5.41E$$+$$6  1.30E$$+$$6  5.20E$$+$$6  1.28E$$+$$6  4$$+$$6  1$$+$$6  5.39E$$+$$6  1.31E$$+$$6  5.38E$$+$$6  1.30E$$+$$6  5.23E$$+$$6  1.29E$$+$$6     5  3.11E$$+$$6  5.11E$$+$$6  1.20E$$+$$6  4.90E$$+$$6  1.17E$$+$$6  4$$+$$6  1$$+$$6  5.14E$$+$$6  1.20E$$+$$6  5.16E$$+$$6  1.20E$$+$$6  4.98E$$+$$6  1.20E$$+$$6     6  2.79E$$+$$6  4.95E$$+$$6  1.23E$$+$$6  4.57E$$+$$6  1.18E$$+$$6  4$$+$$6  1$$+$$6  4.75E$$+$$6  1.21E$$+$$6  4.88E$$+$$6  1.22E$$+$$6  4.68E$$+$$6  1.20E$$+$$6     7  3.17E$$+$$6  5.39E$$+$$6  1.27E$$+$$6  5.16E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.41E$$+$$6  1.27E$$+$$6  5.25E$$+$$6  1.27E$$+$$6  4.94E$$+$$6  1.24E$$+$$6     8  3.12E$$+$$6  5.41E$$+$$6  1.26E$$+$$6  5.09E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.30E$$+$$6  1.25E$$+$$6     9  3.36E$$+$$6  5.51E$$+$$6  1.29E$$+$$6  5.31E$$+$$6  1.26E$$+$$6  4$$+$$6  1$$+$$6  5.48E$$+$$6  1.28E$$+$$6  5.45E$$+$$6  1.27E$$+$$6  5.30E$$+$$6  1.27E$$+$$6     10  3.12E$$+$$6  5.37E$$+$$6  1.25E$$+$$6  5.09E$$+$$6  1.22E$$+$$6  4$$+$$6  1$$+$$6  5.28E$$+$$6  1.25E$$+$$6  5.30E$$+$$6  1.25E$$+$$6  5.18E$$+$$6  1.25E$$+$$6  1000  1  1.41E$$+$$7  2.32E$$+$$7  5.08E$$+$$6  2.26E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.35E$$+$$7  5.12E$$+$$6  2.34E$$+$$7  5.13E$$+$$6  2.19E$$+$$7  4.97E$$+$$6     2  1.23E$$+$$7  2.13E$$+$$7  4.88E$$+$$6  2.12E$$+$$7  4.88E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  4.93E$$+$$6  2.16E$$+$$7  4.93E$$+$$6  2.01E$$+$$7  4.84E$$+$$6     3  1.20E$$+$$7  2.20E$$+$$7  5.03E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  5.04E$$+$$6  2.21E$$+$$7  5.04E$$+$$6  2.11E$$+$$7  4.99E$$+$$6     4  1.18E$$+$$7  2.13E$$+$$7  4.94E$$+$$6  2.12E$$+$$7  4.94E$$+$$6  1$$+$$7  4$$+$$6  2.10E$$+$$7  4.92E$$+$$6  2.14E$$+$$7  4.93E$$+$$6  2.13E$$+$$7  4.93E$$+$$6     5  1.25E$$+$$7  2.24E$$+$$7  5.17E$$+$$6  2.23E$$+$$7  5.11E$$+$$6  1$$+$$7  4$$+$$6  2.27E$$+$$7  5.16E$$+$$6  2.20E$$+$$7  5.13E$$+$$6  2.22E$$+$$7  5.15E$$+$$6     6  1.17E$$+$$7  2.10E$$+$$7  5.10E$$+$$6  2.10E$$+$$7  5.06E$$+$$6  1$$+$$7  4$$+$$6  2.13E$$+$$7  5.09E$$+$$6  2.10E$$+$$7  5.10E$$+$$6  2.12E$$+$$7  5.07E$$+$$6     7  1.33E$$+$$7  2.31E$$+$$7  5.09E$$+$$6  2.27E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.31E$$+$$7  5.07E$$+$$6  2.30E$$+$$7  5.08E$$+$$6  2.28E$$+$$7  5.05E$$+$$6     8  1.23E$$+$$7  2.22E$$+$$7  4.99E$$+$$6  2.18E$$+$$7  4.95E$$+$$6  1$$+$$7  4$$+$$6  2.22E$$+$$7  4.98E$$+$$6  2.20E$$+$$7  4.96E$$+$$6  2.07E$$+$$7  4.87E$$+$$6     9  1.18E$$+$$7  2.16E$$+$$7  4.98E$$+$$6  2.10E$$+$$7  4.92E$$+$$6  1$$+$$7  4$$+$$6  2.14E$$+$$7  4.95E$$+$$6  2.14E$$+$$7  4.91E$$+$$6  2.12E$$+$$7  4.92E$$+$$6     10  1.24E$$+$$7  2.14E$$+$$7  5.10E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.19E$$+$$7  5.10E$$+$$6  2.14E$$+$$7  5.09E$$+$$6  2.19E$$+$$7  5.08E$$+$$6  $$n$$        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2        UB$$_{\mathrm{SumET}}$$  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  10  1  1936  1975  549  1992  546  1936  546  2091  546  1975  558  1936  558     2  1042  1042  594  1163  593  1042  593  1637  593  1042  593  1077  593     3  1586  1586  565  1714  565  1586  565  1763  565  1586  565  1586  565     4  2139  2214  463  2500  446  2139  459  2372  446  2205  478  2170  459     5  1187  1187  392  1199  391  1187  391  1362  391  1187  391  1187  392     6  1521  1523  386  1592  386  1521  386  1632  386  1521  386  1523  386     7  2170  2170  450  2365  450  2170  453  2612  450  2170  450  2170  451     8  1720  1720  291  1726  291  1720  291  1824  291  1726  291  1726  291     9  1574  1574  397  1574  397  1574  397  1746  397  1574  397  1574  401     10  1869  1888  546  1888  538  1869  539  2065  538  1869  546  1869  546  20  1  4394  4423  1656  4729  1645  4478  1678  4991  1645  4398  1656  4423  1648     2  8430  8555  1973  8914  1970  8496  1973  8936  1970  8460  1976  8443  1973     3  6210  6231  1974  6657  1947  6260  1982  7244  1947  6221  1989  6224  1956     4  9188  9357  1859  10869  1859  9289  2002  10656  1859  9332  1899  9192  1872     5  4215  4321  1361  4584  1342  4315  1352  4773  1342  4215  1377  4224  1343     6  6527  6586  1665  6983  1634  6578  1674  7259  1634  6541  1674  6550  1634     7  10455  10500  2108  11131  2080  10471  2125  11403  2080  10550  2107  10459  2080     8  3920  4252  1758  5793  1708  4126  1764  3948  1705  4390  1730  3954  1709     9  3465  3563  1015  3708  1003  3471  1011  3689  1003  3518  1010  3485  1003     10  4979  5000  1676  5870  1661  5051  1668  5997  1661  5122  1717  4995  1661  50  1  40697  43082  11009  42669  10287  43262  11061  43099  10343  41704  10688  41113  10441     2  30613  33367  9962  33131  9278  33433  9926  33533  9387  31586  9588  30748  9275     3  34435  38670  10992  37396  10282  37882  11300  38806  10453  35522  10581  34815  10341     4  27755  29964  9417  29206  8600  29489  9231  30802  8643  28588  8860  27907  8635     5  32307  38420  11592  39211  10857  36074  11701  42394  10892  33810  11339  32718  10992     6  34993  38536  11023  37581  10409  36880  11514  38802  10405  36121  10717  35601  10396     7  43136  47390  12126  49911  11436  46880  12032  52999  11609  45492  11680  43462  11476     8  43839  48414  14687  47286  13931  48074  14594  49274  14084  45334  14317  43903  13971     9  34228  37842  8992  37252  8321  36455  8986  39908  8430  35343  8841  34629  8356     10  32958  35056  10532  35445  9825  34587  10200  36862  9852  33547  10044  33191  9792  100  1  1..46E$$+$$5  1.97E$$+$$5  5.15E$$+$$4  1.64E$$+$$5  4.64E$$+$$4  1.82E$$+$$5  4.98E$$+$$4  1.85E$$+$$5  4.89E$$+$$4  1.68E$$+$$5  4.85E$$+$$4  1.54E$$+$$5  4.57E$$+$$4     2  1.25E$$+$$5  1.76E$$+$$5  4.87E$$+$$4  1.53E$$+$$5  4.43E$$+$$4  1.53E$$+$$5  4.77E$$+$$4  1.62E$$+$$5  4.62E$$+$$4  1.45E$$+$$5  4.70E$$+$$4  1.35E$$+$$5  4.49E$$+$$4     3  1.30E$$+$$5  1.67E$$+$$5  4.69E$$+$$4  1.44E$$+$$5  4.28E$$+$$4  1.54E$$+$$5  4.53E$$+$$4  1.49E$$+$$5  4.33E$$+$$4  1.53E$$+$$5  4.45E$$+$$4  1.36E$$+$$5  4.12E$$+$$4     4  1.30E$$+$$5  1.80E$$+$$5  4.55E$$+$$4  1.44E$$+$$5  4.19E$$+$$4  1.59E$$+$$5  4.41E$$+$$4  1.51E$$+$$5  4.39E$$+$$4  1.58E$$+$$5  4.40E$$+$$4  1.35E$$+$$5  4.19E$$+$$4     5  1.24E$$+$$5  1.81E$$+$$5  4.66E$$+$$4  1.43E$$+$$5  4.24E$$+$$4  1.53E$$+$$5  4.51E$$+$$4  1.51E$$+$$5  4.34E$$+$$4  1.43E$$+$$5  4.28E$$+$$4  1.32E$$+$$5  4.09E$$+$$4     6  1.39E$$+$$5  1.89E$$+$$5  4.73E$$+$$4  1.55E$$+$$5  4.24E$$+$$4  1.62E$$+$$5  4.44E$$+$$4  1.67E$$+$$5  4.34E$$+$$4  1.70E$$+$$5  4.53E$$+$$4  1.46E$$+$$5  4.20E$$+$$4     7  1.35E$$+$$5  1.78E$$+$$5  4.72E$$+$$4  1.51E$$+$$5  4.30E$$+$$4  1.62E$$+$$5  4.51E$$+$$4  1.59E$$+$$5  4.40E$$+$$4  1.58E$$+$$5  4.34E$$+$$4  1.40E$$+$$5  4$$+$$4     8  1.60E$$+$$5  2.07E$$+$$5  5.32E$$+$$4  1.87E$$+$$5  5.09E$$+$$4  1.93E$$+$$5  5.39E$$+$$4  1.90E$$+$$5  5.23E$$+$$4  1.93E$$+$$5  5.21E$$+$$4  1$$+$$5  5$$+$$4     9  1.17E$$+$$5  1.69E$$+$$5  4.79E$$+$$4  1.32E$$+$$5  4.38E$$+$$4  1.52E$$+$$5  4.68E$$+$$4  1.49E$$+$$5  4.60E$$+$$4  1.46E$$+$$5  4.53E$$+$$4  1$$+$$5  4$$+$$4     10  1.19E$$+$$5  1.67E$$+$$5  4.75E$$+$$4  1.34E$$+$$5  4.31E$$+$$4  1.46E$$+$$5  4.54E$$+$$4  1.48E$$+$$5  4.39E$$+$$4  1.38E$$+$$5  4.50E$$+$$4  1$$+$$5  4$$+$$4  200  1  4.99E$$+$$5  7.80E$$+$$5  1.96E$$+$$5  6.95E$$+$$5  1.88E$$+$$5  6.89E$$+$$5  1.88E$$+$$5  7.56E$$+$$5  1.94E$$+$$5  7.37E$$+$$5  1.94E$$+$$5  6$$+$$5  1$$+$$5     2  5.41E$$+$$5  8.38E$$+$$5  1.97E$$+$$5  7.00E$$+$$5  1$$+$$5  7.43E$$+$$5  1.89E$$+$$5  7.96E$$+$$5  1.95E$$+$$5  8.24E$$+$$5  1.93E$$+$$5  6$$+$$5  1.86E$$+$$5     3  4.89E$$+$$5  7.47E$$+$$5  1.84E$$+$$5  6.76E$$+$$5  1.74E$$+$$5  6$$+$$5  1.76E$$+$$5  7.17E$$+$$5  1.82E$$+$$5  7.25E$$+$$5  1.81E$$+$$5  6.47E$$+$$5  1$$+$$5     4  5.86E$$+$$5  8.56E$$+$$5  2.02E$$+$$5  7.71E$$+$$5  1.89E$$+$$5  7$$+$$5  1.92E$$+$$5  8.14E$$+$$5  1.97E$$+$$5  8.12E$$+$$5  1.99E$$+$$5  7.58E$$+$$5  1$$+$$5     5  5.13E$$+$$5  8.01E$$+$$5  2.02E$$+$$5  6.76E$$+$$5  1$$+$$5  6.99E$$+$$5  1.95E$$+$$5  7.62E$$+$$5  2.00E$$+$$5  7.71E$$+$$5  1.99E$$+$$5  6$$+$$5  1.96E$$+$$5     6  4.78E$$+$$5  7.30E$$+$$5  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4.92E$$+$$6     10  1.24E$$+$$7  2.14E$$+$$7  5.10E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.19E$$+$$7  5.10E$$+$$6  2.14E$$+$$7  5.09E$$+$$6  2.19E$$+$$7  5.08E$$+$$6  View Large © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

Bi-criteria scheduling against restrictive common due dates using a multi-objective differential evolution algorithm

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Oxford University Press
Copyright
© The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpw015
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Abstract

Abstract Consideration is given to the single-machine early/tardy scheduling problem (SMETP). This type of scheduling sets costs (penalties) depending on whether a job finished before or after a specified common due date. Two optimization criteria are simultaneously considered for minimization: first, the total weighted earliness and tardiness penalty costs, and second the total flow time of the jobs. A multi-objective differential evolution algorithm is presented devoted to the search for Pareto-optimal solutions. This algorithm is an adaptation of an existing algorithm for the single-objective SMETP that has shown excellent performance in terms of solution quality and speed. Using results from existing benchmark problems, we test the performance of the proposed algorithm on various operating environments with up to 1000 jobs. Furthermore, extended experimental comparisons are performed against three well-known multi-objective evolutionary algorithms. The results demonstrate very satisfactory performance for our algorithm in terms of both solution time and quality. 1. Introduction Scheduling of a set of independent jobs on a single machine so that they are completed by a specified restricted common due date (CDD) and objective to minimize the jobs’ total earliness and tardiness is NP-hard (Baker & Scudder, 1990; French, 1990). This type of scheduling is very important in just-in-time production environments where the main effort is to have the jobs completed as close as possible to their due date, neither too early, nor too late. The reason is that early jobs result in inventory holding costs, while late jobs result to customers’ dissatisfaction and consequently to possible loss of orders, loss of reputation, etc. Therefore, both earliness and tardiness of jobs should be discouraged. This is usually achieved by setting penalty costs depending on whether a job finished before (earliness), or after (tardiness) the specified due date. Minimizing these costs pushes the completion time of each job as close as possible to the due date. The problem, referred to herein as the single-machine early/tardy scheduling problem (SMETP) is formally known in the literature as $$1\left| {d_{j} =d} \right|\sum {\left( {\alpha_{j} E_{j} +\beta_{j} T_{j} } \right)}$$. Research in CDD scheduling problems is generally distinguished into two main categories: the first category assumes that CDD is unrestrictive, that is, the due date is sufficiently large so that it does not constrain the scheduling of the jobs. This is known as the unrestricted version of the problem. Kanet’s study (Kanet, 1981) was the first attempt to solve a variant of this problem dealing exclusively with the special case when the earliness and tardiness penalties are equal to one. Later, many researchers addressed various versions of the basic problem through the means of heuristics. Among them, Bagchi et al. (1986) and Hall & Posner (1991) studied the problem with objective to minimize the sum of absolute deviations of the job completion times from a CDD. Bagchi et al. (1986) proposed an enumerative algorithm which is effective at solving problems up to 15 jobs in size that leads to an optimal solution under the assumption that the starting time of the schedule is zero. Hall & Posner (1991) showed that the symmetric (i.e. $$\alpha_{j} =\beta_{j} )$$ unrestricted SMETP is solvable in a pseudo-polynomial time. Moreover, the authors developed a dynamic programming approach to solve such a problem. De et al. (1994) proposed a greedy randomized adaptive heuristic for the unrestricted problem with different penalties to find both the optimal due date and the optimal sequence of the jobs. The procedure consists of two phases: an initial solution is constructed in the first phase through controlled randomization and is improved in the second phase through steepest descent neighbourhood search. Hoogeveen & Van de Velde (1997) addressed the unrestricted problem with ‘almost’ CDD using a dynamic programming algorithm. The due dates of the jobs are different but lie between a large constant $$d$$ and $$d+p_{j} $$ for each job $$j$$ (with $$p_{j} $$ being the processing time of job $$j)$$. Van den Akker et al. (2000) presented a combined column generation and Lagrangian relaxation algorithm that solves instances with up to 125 jobs to optimality. Comprehensive surveys on CDD scheduling problems can be found in Cheng & Gupta (1989), Baker & Scudder (1990) and Gordon et al. (2002). The second category, concerns the restricted version of the problem which is obviously harder since the CDD is small enough to constrain the scheduling process. Existing theoretical results showed that the restricted problem is NP-hard even if all the job penalty costs are equal to one (Hoogeveen & van de Velde, 1991). Moreover, for the problem with general penalty costs (which is the case of this study) no pseudo-polynomial algorithm has never been proposed (Feldmann & Biskup, 2003). Only for small-sized problems with up to 10 jobs was an optimal schedule found in a reasonable time applying the mixed-integer program formulation proposed by Biskup & Feldmann (2001) using a commercial optimizer. Due to these complexity results, the recent research effort in this field is spent on the development of robust approximation algorithms based on metaheuristics. The first attempt for solving the restricted problem by means of metaheuristics is due to (Lee & Kim, 1995). The authors developed a parallel genetic algorithm (GA) with the job schedules being coded into chromosomes via a binary representation scheme. Parallel subpopulations were constructed by considering only jobs that can be processed first in the schedule. Offspring solutions were generated by special operators reflecting the problem-specific properties. James (1997) proposed different forms of the tabu search (TS) algorithm including one based on a sequence of jobs solution space and another based on an early/tardy solution space. New solutions were generated using the swap neighbourhood scheme. Hao et al. (1996) also used TS to solve a similar but unrestricted CDD scheduling problem. The proposed algorithm starts by generating a feasible solution via a suitable simple greedy heuristic. Improved neighbour solutions are iteratively generated considering swap and insert moves. Feldmann & Biskup (2003) tackled the problem using three different metaheuristics namely evolution strategy, simulated annealing (SA) and threshold accepting (TA). To obtain feasible starting solutions the authors employed a greedy heuristic which schedules those jobs with relatively high tardiness costs compared to their earliness costs prior to the due date. Comparative analysis showed that TA is superior in terms of both solution time and quality. Hino et al. (2005) developed a TS and a GA for the problem. Hybridization was performed by combing TS and GA in a sequential form as in the following: the best solution obtained by the one algorithm passes in a second stage as input to the other for improvement. Two hybrid forms were examined TS followed by GA and vice versa. Liao & Cheng (2007) proposed a hybrid approach which combines TS with a variable neighbourhood search (VNS) heuristic. VNS uses insertion and swap schemes for generating new neighbourhood solutions. To prevent cycling back to previously visited solutions, two tabu lists are used to record the recent VNS moves one for each different search scheme. A hybrid algorithm was also proposed by M’Hallah (2007) to address a special case of the problem with unit penalties. This algorithm combines a GA with SA and some other local search heuristics including dispatching rules and hill climbing. Dispatching rules are used to construct the initial GA population. Hill climbing is applied to every child issued by crossover while SA is used as a mutant operator to the best members of every generation. Nearchou (2008) tackled the problem usingdifferential evolution (DE). DE is a population-based algorithm which evolves real-valued vectors (solutions) in an attempt to reach the global optimum. New vectors are generated by scaling the difference of two randomly selected population vectors. Each vector is mapped to a feasible schedule solution using an appropriate representation scheme which reflects all the properties of the problem. Within this general background, this paper considers a bi-criteria SMETP with general earliness and tardiness penalties. The first criterion is to minimize the total weighted earliness and tardiness penalties from a restricted CDD while the second criterion is to minimize the total flow time of the jobs. We are interested for a solution approach devoted to the determination of the Pareto-optimal solutions of the bi-criteria SMETP. In this context, we provide an adaptation of our previous DE algorithm (Nearchou, 2008) to this new environment, termed to herein as MODE (the multi-objective DE). Since its inception by Storn & Price (1997) as a global optimizer over continuous parameter spaces, DE has been applied with success in various engineering applications including electrical and power systems, robotics, machine scheduling, route planning, etc. (see Neri et al., 2010; Das et al., 2016; for recent surveys). As far as the restricted SMETP is concerned, DE showed a remarkable high solution quality performance over the wellknown Biskup & Feldmann (2001)’s benchmarks introducing new upper bounds to nearly 60% of the instances included in the related data set (see Nearchou, 2008 for details). To the best of our knowledge, this is the first study dealing with the multi-criteria SMETP. This observation stems from the literature review performed with the aim of positioning MODE within the relevant research field. This review allowed us to draw the following general conclusions: (a) All published work concern the solution of the single-objective SMETP. (b) Several of the previous works (e.g. Lee & Kim, 1995; James, 1997) have an underlying weakness connected to the properties of the problem. Particularly, they limit their search to schedules in which the first job starts at time zero although this (as will be discussed in Section 3) might exclude all optimal schedules a priori. (c) MODE overcomes this weakness using an effective problem representation scheme. The rest of the paper is organized as follows: Section 2 states the problem. Section 3 presents MODE (the proposed solution algorithm) for solving the bi-criteria SMETP. Computational results concerning the performance of MODE and comparisons with three well-known multi-objective evolutionary algorithms (MOEAs) namely, SPEA2, NSGA-II and L-NSGA respectively are provided in Section 4. It is highlighted that, SPEA2, NSGA-II and L-NSGA have never been applied before to the problem under consideration. Finally, conclusions and directions for future work are pointed out in Section 5. 2. Problem formulation This section presents and discusses a general formulation for the bi-criteria SMETP. We make use of the following notation: $$n$$  Number of jobs  $$j$$  Index for jobs  $$p_{j}$$  Processing time of job $$j$$  $$C_{j}$$  Completion time of job $$j$$  $$d$$  Due date for completion (common for all jobs)  $$E_{j} $$  Earliness of job $$j$$  $$T_{j}$$  Tardiness of job $$j$$  $$\alpha_{j} $$  Earliness penalty for job $$j_{\mathrm{\thinspace }}(\alpha_{j} > 0)$$  $$\beta_{j} $$  Tardiness penalty for job $$j_{\mathrm{\thinspace }}(\beta_{j} > 0)$$  $$n$$  Number of jobs  $$j$$  Index for jobs  $$p_{j}$$  Processing time of job $$j$$  $$C_{j}$$  Completion time of job $$j$$  $$d$$  Due date for completion (common for all jobs)  $$E_{j} $$  Earliness of job $$j$$  $$T_{j}$$  Tardiness of job $$j$$  $$\alpha_{j} $$  Earliness penalty for job $$j_{\mathrm{\thinspace }}(\alpha_{j} > 0)$$  $$\beta_{j} $$  Tardiness penalty for job $$j_{\mathrm{\thinspace }}(\beta_{j} > 0)$$  The problem can be formally defined as follows: a set of $$n$$ jobs must be processed without interruption on a single machine. Each job $$j$$$$j=1,2,\ldots ,n)$$ requires a positive processing time $$p_{j} $$ and ideally must be completed exactly on a specific (common for all jobs) due date $$d$$. Penalties ($$\alpha_{j} $$ or $$\beta_{j} )$$ are incurred whenever a job $$j$$ is completed before or after this due date. Therefore, an ideal schedule is one in which all jobs finish on the specific due date. Assuming that, $$C_{j} $$ is the completion time of job $$j$$, then the earliness and tardiness of job $$j$$ are given by the following relations,   Ej =max(0,d−Cj)Tj =max(0,Cj−d)for all j=1,2,…,n (1) Under these conditions, the bi-criteria SMETP seeks to determine a processing order $$\sigma$$ of the $$n $$ jobs (i.e. $$\sigma$$ is a permutation of the numbers $$1,2,\ldots ,n)$$ that minimizes the following two objectives:   Objective-1:minimize SumET=∑j=1n(αjEj+βjTj) (2)  Objective-2:minimize TFT=∑j=1nCj (3) The first objective denotes the total weighted earliness and tardiness penalty costs of the schedule while the second objective denotes the total flow time of the jobs. Penalties $$\alpha_{j} ,\beta_{j} $$ in Eq. (2) can be measured in different ways resulting in several variations of the basic SMETP. Note that, $$d$$ is called unrestrictive when $$d\geqslant \sum_{j=1}^n {p_{j} } $$ holds, otherwise is called restrictive. Moreover, $$d$$ is also called unrestrictive when it constitutes a decision variable for the problem. Consequently, one can refer to the problem as either unrestricted or restricted SMETP. In this study we examine the restricted case of this particular scheduling problem. The main assumptions for SMETP in relation to the jobs operations and machine availability are as follows: 1. Jobs are independent without precedence or other constraints. 2. Jobs are known in advance and are all available for processing at time zero. 3. The machine can handle only one job at a time. 4. The machine is continuously available and is never kept idle while work is waiting. 5. No job pre-emption is permitted. 3. The proposed MODE algorithm for the bi-criteria SMETP For the restricted SMETP with general earliness and tardiness penalties there is an optimal schedule with the following three properties. Note that, these properties are supported within MODE through the use of a suitable solution representation scheme aiming to obtain high quality schedules more efficiently. Property 1 No idle times are contained between consecutive jobs (Cheng & Kahlbacher, 1991). Property 2 The schedule is V-shaped around the CDD (Smith, 1956). That is, early jobs are sequenced in non-increasing order of $$\frac{p_{j} }{\alpha_{j} }$$ (called ‘$$\backslash $$-shaped’ format) and late jobs are sequenced in non-decreasing order of $$\frac{p_{j} }{\beta_{j} }$$ ($$j=1,2,\ldots ,n)$$ (called ‘/-shaped’ format). Property 3 The processing time of the first job either starts at time zero, or one job is completed at the due date (Hoogeveen & van de Velde, 1991). All the variations of the restricted SMETP result to an NP-hard combinatorial optimization problem (Gordon et al., 2002). Furthermore, as far as more than one optimization criteria are concerned, the problem still remains intractable since finding the global optimum to a general multi-objective optimization (MOO) problem is NP-complete (Bäck, 1996). Actually, an exact solution to MOO problems at which all decision variables satisfy the associated constraints and all the individual objective functions have reached their associated optimal values may not even exist. Hence, the various developed MOO algorithms are in fact in a situation where their attempt is to optimize each individual objective to the greatest possible extend. Usually, there is no single optimal solution to these problems but rather a set of optimal solutions. This set is known as Pareto-optimal solutions. The solutions in this set are such that they are non-dominated by any other solution of the search space of the given problem when all the objectives are considered, and moreover, they do not dominate each other in the set. In this section, we present MODE; a multi-objective DE algorithm for the solution of the bi-criteria SMETP. MODE effectively constitutes an adaptation of the DE algorithm proposed by Nearchou (2008) in the new bi-objective environment. MODE maintains two separate populations: a main evolving population (termed MAIN) and a secondary population of diverse Pareto-optimal solutions (termed PARETO) which is iteratively updated. Let $$N_{\mathrm{main}}$$ denotes the population size of the first population, and $$N_{\mathrm{pareto}}$$ the size of the second population. Let also $$k_{\max } $$ denotes the maximum permitted number of iterations of the algorithm. The pseudo-code of the developed MODE followed by a description of its main components is given below. The algorithm starts (Step 1) by generating randomly an initial population MAIN of $$N_{\mathrm{main}}$$ solutions $$x_{i,k} $$ ($$i=1,2,\ldots ,N_{\mathrm{main}})$$ ($$k$$ denotes the iteration number of the algorithm). Each component $$x_{i,k}^{l} $$, $$l=(1,2,\ldots ,n)$$ of a vector $$x_{i,k} $$ is a real-valued number in the range (0,1). After the creation of MAIN, an initially empty PARETO population is created. A cycle of $$k_{\max }$$ iterations is then applied to MAIN and PARETO (Steps 2--4). At each iteration all vectors in MAIN are targeted for replacement. Therefore, $$N_{\mathrm{main}}$$ competitions are held to determine the members of MAIN for the next iterations. This is achieved by using mutation (Step 2.1), crossover (Step 2.2) and acceptance operators (Step 2.3). In the mutation phase, for each target vector $$x_{i,k} $$ ($$i=1,2,\ldots,N_{\mathrm{main}})$$ a mutant vector $$\stackrel{\frown}{{x}} _{i,k} $$ is obtained by   x⌢i,k=xr1,k+F(xr2,k−xr3,k), (4) where $$r1\ne r2\ne r3\ne i\in \left\{ {1,2,\ldots ,N_{\mathrm{main}}}\right\}$$. $$x_{r1,k} $$ is known as the base vector and $$F>0$$ is a (user defined) scaling parameter. The crossover operator (Step 2.2) is then applied to obtain the trial vector $$\psi_{i,k} $$ from $$\stackrel{\frown}{{x}}_{i,k} $$ and $$x_{i,k} $$. The crossover is defined by   ψi,kl={x⌢i,kl if rnd ⩽CR or l=Iixi,kl if rnd>CR and  l≠Ii , (5) where $$I_{i} $$ is a randomly chosen integer in the set $$I$$, that is $$I_{i}\in I=(1,2,\ldots ,n)$$; the superscript $$l$$ represents the $$l\mbox{-th}$$component of respective vectors. $$\mbox{rnd}\in (0,1)$$ is drawn randomly for each $$l$$. The ultimate aim of the crossover rule is to obtain the trial vector $$\psi_{i,k} $$ with components coming from the components of the target vector $$x_{i,k} $$ and the mutated vector $$\stackrel{\frown}{{x}}_{i,k} $$. This is ensured by introducing CR and set $$I$$. In the acceptance phase (Step 2.3) the trial $$\psi_{i,k}$$ and target $$x_{i,k} $$ vectors are compared for domination. To do so, first the two real-valued vectors are mapped to actual SMETP schedule solutions using Decode function. Then, their costs (in regard to both of the optimization criteria given by Eqs (2) and (3)) are compared and the non-dominated solution between them survives reproducing its structure in the new MAIN. The core idea behind Decode is to build schedules that satisfy the three problem properties mentioned above. This is accomplished by the following scheme: let’s assume the solution $$x_{i,k} $$. Each component $$x_{_{i,k} }^{l} \in \left( {0,1} \right) l=(1,2,\ldots ,n)$$ of this vector is associated to a specific job $$1,2,\ldots,n$$ with that order. A value less than or equal to 0.5 in the vector indicates that the corresponding job is early otherwise the job is tardy. So, the jobs are distinguished into two sets namely, $$S_{E} $$ and $$ S_{T}$$ containing the early and the tardy jobs respectively. Following the V-shaped property, the jobs in $$S_{E}$$ are moved at the start of the schedule and sequenced in a ‘$$\backslash $$-shaped’ way. Late jobs are moved at the end of the schedule and sequenced in a ‘/-shaped’ manner. Let sump the total processing time of the early jobs in $$S_{E}$$. According to Feldmann & Biskup (2003), an optimal solution to SMETP can fall in one of the following three disjunctive cases: (A) The first job in $$S_{E}$$ starts at time zero and the last job in $$S_{E} $$ finishes exactly on due date $$d$$. (B) The first job in $$S_{E}$$ starts at time zero and the last job in $$S_{E} $$ is completed prior to $$d.$$ Further, a straddling job exists, that is a job starting executed before $$d $$ and ending after $$d$$. (C) The first job in $$S_{E}$$ does not start at time zero (i.e. it is delayed) and the last job in $$S_{E} $$ is finished exactly on the due date $$d$$. Case-(A) occurs when sump $$= d$$; Case-(B) occurs when sump > $$d$$, while Case-(C) occurs when sump < $$d$$. Therefore, according to the proposed scheme, for every candidate vector of the entire population, first the sets $$S_{E} $$ and $$S_{T}$$ are created. Second, the processing time of the jobs in $$S_{E}$$ are summed up into sump until the value of this variable surpassed $$d $$ or no other jobs are contained in $$S_{E}$$. Third, the starting time of the first job in $$S_{E} $$ is defined. That is, when sump $$\geqslant d$$ (Cases (A) and (B)), the first job starts at time zero, otherwise (Case (C)), the first job is delayed starting at time $$d$$-sump. Fourth, jobs in $$S_{E} $$ are ordered based on ‘$$\backslash $$-shaped’ property, while jobs in $$S_{T}$$ are ordered based on ‘/-shaped’ property. The operation of Decode is given below in algorithmic form. More details about this decoding scheme including application examples can be found in our previous work (Nearchou, 2008). Each iteration of MODE is completed with the updating of PARETO population (Step 3). First, a TEMP is built combining the members of MAIN and PARETO (Step 3.1). Then (Step 3.2), the weakness $$w_{z} $$ for each $$z\in \mbox{TEMP}$$ is computed which corresponds to the total number of the TEMP members dominating $$z$$. The more the value of $$w_{z} $$ the lesser the quality of the solution corresponding to $$z$$. The members in TEMP are then (Step 3.3) sorted in ascending order of their weakness value. The actual PARETO updating is accomplished by Update_Pareto function (Step 3.4). The pseudo-code of this function is given below. Its operation is simple: a non-dominated TEMP solution not already included in PARETO is copied into PARETO if the latter has available space or if the new TEMP solution dominates some existing PARETO solutions. In the latter case all the existing dominated solutions are removed from PARETO. 4. Experimental investigation and discussion In this section we present computational results obtained for assessing the quality of the MODE algorithm. Comparisons have been also performed against to three famous MOEAs namely, SPEA2, NSGA-II and L-NSGA. All experiments were performed on a PC with 3GHz Intel core Duo CPU, 4GB RAM and Windows XP operating system. The algorithms were implemented in Java programming language. The strength Pareto evolutionary algorithm-2 (SPEA2) introduced by Zitzler et al. (2002) and the non-dominated sorting GA-II (NSGA-II) proposed by Deb et al. (2002) are very popular in the Evolutionary Computation community as effective multi-objective optimizers and due to their efficiency often constitute test beds against any new multi-objective algorithm. The Lorenz non-dominated sorting GA (L-NSGA) is a recent MOEA developed by Dugardin et al. (2010). It is based on NSGA-II but uses the Lorenz dominance relationship instead of the Pareto dominance with the aim to provide a stronger selection of the non-dominated solutions. 4.1 Design of experiments Three versions of the proposed MODE algorithm were evaluated differ in the way the mutant vectors (Eq. (4)) are created. We will refer to these versions with the abbreviations MODE1, MODE2 and MODE3, respectively. MODE1 creates mutants using Eq. (4); while MODE2 and MODE3 create mutants using Eqs (6) and (7), respectively (see below). Details about DE variants can be found in the works of Storn & Price (1997) and Storn (1996).  x⌢i,k =xi,k+λ(xbest,k−xi,k)+F(xr2,k−xr3,k) (6)  x⌢i,k =xbest,k+F(xr1,k+xr2,k−xr3,k−xr4,k) (7) with, $$r1\ne r2\ne r3\ne r4\ne i\in \left\{ {1,2,\ldots ,N_{\mathrm{main}}} \right\}$$. $$x_{\mathrm{best,}k} $$ denotes the trial vector corresponding to the best schedule solution of the (current) iteration $$k$$. $$\lambda \in $$(0,1) (in Eq. (6)) is a parameter that controls the greediness of this particular DE scheme. Here, according to the indications of Storn & Price (1997) we set $$\lambda = 0.9$$. The three MODE algorithms as well as SPEA2, NSGA-II, and L-NSGA were tested and compared over a set of public benchmarks problems proposed by Biskup & Feldmann (2001). These benchmarks include test instances ranging from small size with 10 jobs to large size instances with 1000 jobs. Particularly, there are 7 problem classes with 10, 20, 50, 100, 200, 500 and 1000 jobs with each class containing 10 test instances in total. The value of a restrictive factor $$h = 0.2$$, 0.4, 0.6, 0.8 classifies the benchmarks test problems as less or more restricted against a CDD using the relation   d=⌊h∑j=1npj⌋. (8) With $$\left\lfloor\ y\ \right\rfloor $$ denoting the biggest integer smaller than or equal to $$y$$. That is, for each problem, $$d $$is estimated by multiplying the restrictive factor $$h$$ with the summation of the processing times of the $$n$$ jobs. The lower the value of $$h$$ the more restrictive is $$d$$ (i.e. the higher the expected percentage of the late jobs). In this study we present results concerning the most restricted CDD benchmarks of the Biskup & Feldmann (2001)’s data set (i.e. the test instances with $$h = 0.2$$). It is worth noting that, the existing upper bounds on the optimal objective function values for these benchmarks concern only the first optimization criterion (i.e. SumET objective given by Eq. (2)). The performance of the six algorithms was quantified through the use of the following indices: (1) The quality ratio (P*/P) as a percentage. The larger the value for this ratio for a given heuristic, the higher its performance. P* denotes the number of the different Pareto solutions generated by a heuristic over a specific test instance. While, P denotes the number of non-dominated solutions among all Pareto solutions obtained by all the heuristics. In particular, since some Pareto solutions obtained by one heuristic may be dominated by other heuristics, all the obtained solutions are compared to each other and the non-dominated among them are selected. (2) The Zitzler index (Zitzler & Thiele, 1999) here denoted as Z(x,y) which represents the percentage of the Pareto solutions in algorithm x dominated by at least one Pareto solution of algorithm y. Since this measure is not symmetrical, it is also necessary to calculate Z(y,x). Therefore, if Z(x,y) < Z(y,x) then x is better than y. To obtain the average performance of the heuristics, each one of them was ran 30 times over every test instance (starting each time from a different random number seed) and the solution quality was averaged. Hence, as there are 10 instances in each one of the 7 problem classes, this means that each heuristic was ran $$7 \times 10 \times 30 times = 2100$$ times in total. 4.2 Choice of the control parameters’ settings All the algorithms were defined to evolve a fixed size population of $$N_{\mathrm{main}}=n$$ individual solutions (with $$n$$ being the number of the jobs) and run for a maximum time of 0.3$$n$$ CPU seconds. This means, a maximum running time equal to 3 s for 10-job problems, 6 s for 20-job, $$\ldots$$ 300 s for 1000-job problems. The maximum size of the Pareto population maintained by each heuristic was defined to be equal to $$N_{\mathrm{pareto}}=\left\lfloor {5+\sqrt n } \right\rfloor $$ solutions. To determine the final Pareto set solutions for each algorithm we applied the following procedure: the Pareto solutions generated in every different run are combined into a single union set with $$30N_{\mathrm{pareto}}$$ solutions. Then, a non-dominating sorting is performed and the first $$N_{\mathrm{pareto}}$$ solutions of this union set constitute the final Pareto set for the particular algorithm. To estimate the correct settings for CR and $$F$$ used within the three MODE algorithms a dynamic self-adapted control scheme was adopted previously proposed by Brest et al. (2006). According to this scheme for each new parent vector solution a new pair of CR and $$F$$ values is estimated using the following relations:   Fi,k+1 ={Fl+rand1×Fu,if rand2<r1Fi,k,otherwise  (9)  CRi,k+1 ={rand3,if rand4<r2CRi,k,otherwise  (10) where, $$F_{l} =0.1$$, $$F_{u} =0.9$$, $$r_{1} =r_{2} =0.1$$ and rand$$_{j}$$ ($$j\in \{ 1,2,3,4\} $$) are uniform random numbers in the range [0,1]. This dynamic scheme was found superior to various existing static control schemes examined in preliminary experimental studies. At the start, CR and $$F $$are set to 0.9 and 0.5, respectively for all the members of the initial population. SPEA2, NSGA-II and L-NSGA were implemented according to their description in the literature. The individuals are represented as floating-point vectors (as in the proposed MODE algorithms) while solutions encoding is performed using the developed procedure presented in Section 3.1. To determine the appropriate crossover and mutation rates for these algorithms we used the following experimental framework: we set the mutation rate to a fix value within the discrete range 0.0, 0.01, 0.1 and experimented with various crossover rates in the range 0.3, 0.6, 0.8, 0.9, 1.0. The best crossover and mutation rates obtained after the termination of these preliminary runs are given in Table 1. Note that, these rates were finally chosen according to the notion of non-dominance in relation to the metrics given in Section 4.1 above. Table 1 Control settings for the crossover and mutation rates of SPEA2, NSGA-II, L-NSGA     SPEA2  NSGA-II  L-NSGA  Crossover rate  0.8  0.8  0.8  Mutation rate  0.0  0.1  0.1     SPEA2  NSGA-II  L-NSGA  Crossover rate  0.8  0.8  0.8  Mutation rate  0.0  0.1  0.1  4.3 Comparative results and discussion The detailed results with the extreme Pareto solutions (i.e. best values for SumET and TFT objectives) obtained by the six heuristics for all test problems are given in the Appendix. The best objective value obtained in each test instance is indicated in bold. The cells in the third column ($$\mbox{UB}_{\mathrm{SumET}} )$$ of the Table A1 in the Appendix show the best known value in regard to SumET objective given by Nearchou (2008) when considering the related single-objective problem. One can rather easily recognize the high quality performance achieved by both MODE3 and SPEA2 heuristics. More important however, is the efficiency of the algorithms over the large-sized test problems (with $$n > 200$$). Hence, from these results we observe that MODE3 outperformed all the others in all but 2 of the 500-job instances, as well as, in all of the 1000-job test instances. Let us now discuss the performance of the six heuristics in regard to the quality ratio (P*/P). Table 2 displays the mean quality ratio (P*/P) obtained by each one of the heuristics (after the 30 runs) over the examined test problems. Note that, this ratio is averaged over the 10 test instances included in each class of problems. The higher the value of this ratio (maximizing at 1) for a particular algorithm the greater its performance. For example a quality ratio equal to 0.82 (see the first cell in Table 2 means that 82% of the solutions obtained by the specific algorithm (here MODE1) are non-dominated by any Pareto solution of the other algorithms. Hence, as one can see from Table 2, for test problems with up to 200 jobs, SPEA2 achieved higher quality ratio than the other heuristics. For larger size problems however (with 500 and 1000 jobs), MODE3 was found superior outperforming all the other heuristics. It is worth noting that, concerning the 1000-job problems, MODE3 attained a quality ratio equal to one. This means that all of the MODE3 Pareto solutions are non-dominated by any Pareto solution created by the remaining five algorithms. Note that, a zero ratio means that all of the solutions obtained by an algorithm are dominated by at least one solution of another algorithm. Table 2 Mean quality ratio over the seven test problem classes  $$n$$  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  10  0.82  0.85  0.89  0.77  0.84  0.85  20  0.61  0.30  0.60  0.78  0.81  0.86  50  0.00  0.03  0.59  0.10  0.43  0.95  100  0.00  0.00  0.12  0.00  0.00  0.99  200  0.00  0.46  0.43  0.00  0.00  0.60  500  0.00  0.13  0.97  0.00  0.00  0.10  1000  0.00  0.00  1.00  0.00  0.00  0.00  $$n$$  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  10  0.82  0.85  0.89  0.77  0.84  0.85  20  0.61  0.30  0.60  0.78  0.81  0.86  50  0.00  0.03  0.59  0.10  0.43  0.95  100  0.00  0.00  0.12  0.00  0.00  0.99  200  0.00  0.46  0.43  0.00  0.00  0.60  500  0.00  0.13  0.97  0.00  0.00  0.10  1000  0.00  0.00  1.00  0.00  0.00  0.00  Turning now to the Zitzler index, Table 3 shows the values of this performance metric obtained over the first test instance included in the 500-job problem class. Recall that, Zitzler index is estimated through an algorithm-to-algorithm comparison iterated procedure. That is, each one of the algorithms is compared against to all the others. Observe for example cell (x,y)$$=$$(MODE1, L-NSGA) in the first line of Table 3. The value 88.89% in this cell is the Zitzler index of MODE1 against L-NSGA and means that 88.89% of the solutions generated by MODE1 are dominated by at least one L-NSGA solution. The Zitzler index is not symmetric and therefore we must also examine the value in cell (y,x)$$=$$(L-NSGA,MODE1) in order to make a safe conclusion concerning the efficiency of the two algorithms. Z(L-NSGA,MODE1) $$=$$ 77.78%. Hence, since Z(L-NSGA,MODE1) < Z(MODE1,L-NSGA) we can conclude that L-NSGA is superior to MODE1 for this particular test instance. Counting the number of y’s dominated by each (algorithm) x gives us the overall performance of the latter. This information is reported in the last column of Table 3. As one can see, MODE3 outperforms all the other five heuristics attained the smallest Zitzler index, this is depicted with a score 5. The second best performance was achieved by MODE2 with an overall score equal to 4 (meaning that, MODE2 outperforms all the other heuristics except from MODE3). Next comes SPEA2 with a total score equal to 3. Table 3 The Zitzler index for the six algorithms over the first test instance 500-job SMETP        Y        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  No. Z(x,y) < Z(y,x)  X  MODE1  —  100.0%  100.0%  88.89%  88.85%  100.0%  0     MODE2  0.0%  —  66.67%  0.0%  0.0%  0.0%  4     MODE3  0.0%  22.22%  —  0.0%  0.0%  0.0%  5     L-NSGA  77.78%  100.0%  100.0%  —  85.19%  100.0%  2     NSGA-II  55.56%  100.0%  100.0%  88.89%  —  100.0%  1     SPEA2  0.0%  100.0%  100.0%  3.7%  3.7%  —  3        Y        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  No. Z(x,y) < Z(y,x)  X  MODE1  —  100.0%  100.0%  88.89%  88.85%  100.0%  0     MODE2  0.0%  —  66.67%  0.0%  0.0%  0.0%  4     MODE3  0.0%  22.22%  —  0.0%  0.0%  0.0%  5     L-NSGA  77.78%  100.0%  100.0%  —  85.19%  100.0%  2     NSGA-II  55.56%  100.0%  100.0%  88.89%  —  100.0%  1     SPEA2  0.0%  100.0%  100.0%  3.7%  3.7%  —  3  Obviously, we cannot give in a single table the detailed Zitzler values for the six algorithms over all the test problems. However, we can count the y’s dominated by each x (in terms of the Zitzler index) over the instances included in the examined data set. Table 4 depicts this information. From this table we can now safely conclude about the overall performance of the algorithms. As it is clear, SPEA2 attained the highest total score (equal to 31) meaning that in 31 out the 70 in total test instances showed superior performance than the other algorithms. The second best performance was attained by MODE3 with a total score equal to 28. The remaining algorithms showed fairly lower quality performance with L-NSGA being the weakest. Furthermore, Table 4 gives us another significant information: MODE3 efficiency increases as the size of the problem increases. As one can observe from Table 4, for large-sized problems ($$n\geqslant 200$$ jobs) MODE3 outperformed all the other heuristics in 21 out the 30 in total test instances included in these three problem classes. The next best performance is due to SPEA2 with a score equal to 6. Therefore, as a final remark, SPEA2 and MODE3 are both quite efficient multi-objective optimizers to tackle the bi-criteria SMETP examined in this paper; with the latter being clearly more efficient when addressing large-sized instances of the problem. Table 4 A synopsis of how many times each algorithm achieved the highest performance in regard to the Zitzler index over the examined problems  Test problems  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  $$n \leqslant 100$$  4  3  7  1  5  25  $$n $$ > 100  0  3  21  0  0  6  Total  4  6  28  1  5  31  Test problems  MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2  $$n \leqslant 100$$  4  3  7  1  5  25  $$n $$ > 100  0  3  21  0  0  6  Total  4  6  28  1  5  31  Turning to the small- and medium-sized instances ($$n \leqslant 100$$) (where MODE3 found inferior than SPEA2), further comparative experiments performed showed that, increasing the population size (for both algorithms) results to higher performance for MODE3. Table 5 displays these results for three different population sizes. Each cell of the Table presents the number of the instances an algorithm became superior to the other in regard to the Zitzler index. Remember that, there are 10 instances in each problem class. Hence, a value equal to 10 means absolute superiority of the corresponding algorithm against its opponent. As it is clear from Table 5, in all but one test problems MODE3 outperformed SPEA2. Moreover, MODE3’s superiority clearly increases with the population size. This observation comes in accordance to the indications of the literature for sizing DE populations. In particular, according to Storn & Price (1997) best DE performance is achieved using populations in which parental vectors are multiples of the problem parameters. However, an in-depth analysis is still required to formally establish the limiting performance of MODE3 under these general conditions. Different problems required different settings. Table 5 Comparative results between MODE3 and SPEA2 for different population sizes. How many times the two algorithms obtained the best performance in regard to Zitzler index     Population size ($$N_{\mathrm{main}}$$)     2$$n$$  3$$n$$  5$$n$$  Test problems ($$n)$$  MODE3  SPEA2  MODE3  SPEA2  MODE3  SPEA2  10  8  2  10  0  8  2  20  7  3  6  4  10  0  50  4  6  6  4  10  0  100  7  3  7  3  7  3  Total  23  14  29  11  35  5     Population size ($$N_{\mathrm{main}}$$)     2$$n$$  3$$n$$  5$$n$$  Test problems ($$n)$$  MODE3  SPEA2  MODE3  SPEA2  MODE3  SPEA2  10  8  2  10  0  8  2  20  7  3  6  4  10  0  50  4  6  6  4  10  0  100  7  3  7  3  7  3  Total  23  14  29  11  35  5  5. Conclusions The problem of scheduling a number of jobs on a single machine against a restrictive CDD considering multiple optimization criteria has been presented in this paper. Two criteria have been considered for minimization: the total earliness and tardiness penalties, and the total flow-time of the jobs. As the restrictive CDD problem is known to be intractable we decided to tackle the problem using metaheuristics. To that purpose, a new multi-objective differential evolution (MODE) algorithm devoted to the derivation of the Pareto set solutions has been developed to address the problem. Three variants of the proposed MODE have been implemented and tested on existing benchmark problems taken from the open literature. Moreover, extended comparisons have been performed against to three well-known MOEAs namely, SPEA2, NSGA-II and L-NSGA. The experiments showed that a particular MODE variant is quite efficient to address the problem under consideration since was found superior to all the other multi-objective algorithms especially when addressing large-sized problems. Future work will be focused on more realistic versions of this particular scheduling problem. The case where the optimization criteria are considered interact is an interesting direction of future work. Research along these lines will consider the discrete Choquet integral method (Grabisch & Labreuche, 2010; Abichou et al., 2015) as a means to aggregate the criteria in the fitness function of each individual solution. This technique seems to be quite promising as a way to model the interactions between the optimization criteria in a tangible way. Furthermore, in practice, many MOO problems have multiple conflicting objective functions expressed in differing units, and with an inverse, non-linear relationship among themselves. These objectives may be even imprecise or fuzzy in nature to be defined. In its present form the proposed MODE algorithm cannot address such problems. Hybridizing MODE with non-linear goal programming techniques (Tanino et al., 2003) as well as with other metaheuristics such as VNS (see e.g. Brito et al., 2016; Mjirda et al., 2016) may result to a promising optimization tool for these problems. 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Appendix Table A1 Best objective function values found by the heuristics for each test problem  $$n$$        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2        UB$$_{\mathrm{SumET}}$$  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  10  1  1936  1975  549  1992  546  1936  546  2091  546  1975  558  1936  558     2  1042  1042  594  1163  593  1042  593  1637  593  1042  593  1077  593     3  1586  1586  565  1714  565  1586  565  1763  565  1586  565  1586  565     4  2139  2214  463  2500  446  2139  459  2372  446  2205  478  2170  459     5  1187  1187  392  1199  391  1187  391  1362  391  1187  391  1187  392     6  1521  1523  386  1592  386  1521  386  1632  386  1521  386  1523  386     7  2170  2170  450  2365  450  2170  453  2612  450  2170  450  2170  451     8  1720  1720  291  1726  291  1720  291  1824  291  1726  291  1726  291     9  1574  1574  397  1574  397  1574  397  1746  397  1574  397  1574  401     10  1869  1888  546  1888  538  1869  539  2065  538  1869  546  1869  546  20  1  4394  4423  1656  4729  1645  4478  1678  4991  1645  4398  1656  4423  1648     2  8430  8555  1973  8914  1970  8496  1973  8936  1970  8460  1976  8443  1973     3  6210  6231  1974  6657  1947  6260  1982  7244  1947  6221  1989  6224  1956     4  9188  9357  1859  10869  1859  9289  2002  10656  1859  9332  1899  9192  1872     5  4215  4321  1361  4584  1342  4315  1352  4773  1342  4215  1377  4224  1343     6  6527  6586  1665  6983  1634  6578  1674  7259  1634  6541  1674  6550  1634     7  10455  10500  2108  11131  2080  10471  2125  11403  2080  10550  2107  10459  2080     8  3920  4252  1758  5793  1708  4126  1764  3948  1705  4390  1730  3954  1709     9  3465  3563  1015  3708  1003  3471  1011  3689  1003  3518  1010  3485  1003     10  4979  5000  1676  5870  1661  5051  1668  5997  1661  5122  1717  4995  1661  50  1  40697  43082  11009  42669  10287  43262  11061  43099  10343  41704  10688  41113  10441     2  30613  33367  9962  33131  9278  33433  9926  33533  9387  31586  9588  30748  9275     3  34435  38670  10992  37396  10282  37882  11300  38806  10453  35522  10581  34815  10341     4  27755  29964  9417  29206  8600  29489  9231  30802  8643  28588  8860  27907  8635     5  32307  38420  11592  39211  10857  36074  11701  42394  10892  33810  11339  32718  10992     6  34993  38536  11023  37581  10409  36880  11514  38802  10405  36121  10717  35601  10396     7  43136  47390  12126  49911  11436  46880  12032  52999  11609  45492  11680  43462  11476     8  43839  48414  14687  47286  13931  48074  14594  49274  14084  45334  14317  43903  13971     9  34228  37842  8992  37252  8321  36455  8986  39908  8430  35343  8841  34629  8356     10  32958  35056  10532  35445  9825  34587  10200  36862  9852  33547  10044  33191  9792  100  1  1..46E$$+$$5  1.97E$$+$$5  5.15E$$+$$4  1.64E$$+$$5  4.64E$$+$$4  1.82E$$+$$5  4.98E$$+$$4  1.85E$$+$$5  4.89E$$+$$4  1.68E$$+$$5  4.85E$$+$$4  1.54E$$+$$5  4.57E$$+$$4     2  1.25E$$+$$5  1.76E$$+$$5  4.87E$$+$$4  1.53E$$+$$5  4.43E$$+$$4  1.53E$$+$$5  4.77E$$+$$4  1.62E$$+$$5  4.62E$$+$$4  1.45E$$+$$5  4.70E$$+$$4  1.35E$$+$$5  4.49E$$+$$4     3  1.30E$$+$$5  1.67E$$+$$5  4.69E$$+$$4  1.44E$$+$$5  4.28E$$+$$4  1.54E$$+$$5  4.53E$$+$$4  1.49E$$+$$5  4.33E$$+$$4  1.53E$$+$$5  4.45E$$+$$4  1.36E$$+$$5  4.12E$$+$$4     4  1.30E$$+$$5  1.80E$$+$$5  4.55E$$+$$4  1.44E$$+$$5  4.19E$$+$$4  1.59E$$+$$5  4.41E$$+$$4  1.51E$$+$$5  4.39E$$+$$4  1.58E$$+$$5  4.40E$$+$$4  1.35E$$+$$5  4.19E$$+$$4     5  1.24E$$+$$5  1.81E$$+$$5  4.66E$$+$$4  1.43E$$+$$5  4.24E$$+$$4  1.53E$$+$$5  4.51E$$+$$4  1.51E$$+$$5  4.34E$$+$$4  1.43E$$+$$5  4.28E$$+$$4  1.32E$$+$$5  4.09E$$+$$4     6  1.39E$$+$$5  1.89E$$+$$5  4.73E$$+$$4  1.55E$$+$$5  4.24E$$+$$4  1.62E$$+$$5  4.44E$$+$$4  1.67E$$+$$5  4.34E$$+$$4  1.70E$$+$$5  4.53E$$+$$4  1.46E$$+$$5  4.20E$$+$$4     7  1.35E$$+$$5  1.78E$$+$$5  4.72E$$+$$4  1.51E$$+$$5  4.30E$$+$$4  1.62E$$+$$5  4.51E$$+$$4  1.59E$$+$$5  4.40E$$+$$4  1.58E$$+$$5  4.34E$$+$$4  1.40E$$+$$5  4$$+$$4     8  1.60E$$+$$5  2.07E$$+$$5  5.32E$$+$$4  1.87E$$+$$5  5.09E$$+$$4  1.93E$$+$$5  5.39E$$+$$4  1.90E$$+$$5  5.23E$$+$$4  1.93E$$+$$5  5.21E$$+$$4  1$$+$$5  5$$+$$4     9  1.17E$$+$$5  1.69E$$+$$5  4.79E$$+$$4  1.32E$$+$$5  4.38E$$+$$4  1.52E$$+$$5  4.68E$$+$$4  1.49E$$+$$5  4.60E$$+$$4  1.46E$$+$$5  4.53E$$+$$4  1$$+$$5  4$$+$$4     10  1.19E$$+$$5  1.67E$$+$$5  4.75E$$+$$4  1.34E$$+$$5  4.31E$$+$$4  1.46E$$+$$5  4.54E$$+$$4  1.48E$$+$$5  4.39E$$+$$4  1.38E$$+$$5  4.50E$$+$$4  1$$+$$5  4$$+$$4  200  1  4.99E$$+$$5  7.80E$$+$$5  1.96E$$+$$5  6.95E$$+$$5  1.88E$$+$$5  6.89E$$+$$5  1.88E$$+$$5  7.56E$$+$$5  1.94E$$+$$5  7.37E$$+$$5  1.94E$$+$$5  6$$+$$5  1$$+$$5     2  5.41E$$+$$5  8.38E$$+$$5  1.97E$$+$$5  7.00E$$+$$5  1$$+$$5  7.43E$$+$$5  1.89E$$+$$5  7.96E$$+$$5  1.95E$$+$$5  8.24E$$+$$5  1.93E$$+$$5  6$$+$$5  1.86E$$+$$5     3  4.89E$$+$$5  7.47E$$+$$5  1.84E$$+$$5  6.76E$$+$$5  1.74E$$+$$5  6$$+$$5  1.76E$$+$$5  7.17E$$+$$5  1.82E$$+$$5  7.25E$$+$$5  1.81E$$+$$5  6.47E$$+$$5  1$$+$$5     4  5.86E$$+$$5  8.56E$$+$$5  2.02E$$+$$5  7.71E$$+$$5  1.89E$$+$$5  7$$+$$5  1.92E$$+$$5  8.14E$$+$$5  1.97E$$+$$5  8.12E$$+$$5  1.99E$$+$$5  7.58E$$+$$5  1$$+$$5     5  5.13E$$+$$5  8.01E$$+$$5  2.02E$$+$$5  6.76E$$+$$5  1$$+$$5  6.99E$$+$$5  1.95E$$+$$5  7.62E$$+$$5  2.00E$$+$$5  7.71E$$+$$5  1.99E$$+$$5  6$$+$$5  1.96E$$+$$5     6  4.78E$$+$$5  7.30E$$+$$5  1.85E$$+$$5  6.80E$$+$$5  1$$+$$5  6.48E$$+$$5  1.81E$$+$$5  7.24E$$+$$5  1.88E$$+$$5  7.51E$$+$$5  1.88E$$+$$5  6$$+$$5  1.82E$$+$$5     7  4.55E$$+$$5  7.63E$$+$$5  1.89E$$+$$5  6.57E$$+$$5  1$$+$$5  6.49E$$+$$5  1.76E$$+$$5  7.07E$$+$$5  1.85E$$+$$5  6.72E$$+$$5  1.79E$$+$$5  6$$+$$5  1.79E$$+$$5     8  4.94E$$+$$5  7.98E$$+$$5  1.93E$$+$$5  7.04E$$+$$5  1.83E$$+$$5  6$$+$$5  1$$+$$5  7.30E$$+$$5  1.89E$$+$$5  7.37E$$+$$5  1.86E$$+$$5  6.87E$$+$$5  1.85E$$+$$5     9  5.29E$$+$$5  7.92E$$+$$5  1.92E$$+$$5  7.30E$$+$$5  1$$+$$5  6.92E$$+$$5  1.85E$$+$$5  7.73E$$+$$5  1.91E$$+$$5  7.54E$$+$$5  1.89E$$+$$5  6$$+$$5  1.84E$$+$$5     10  5.38E$$+$$5  8.12E$$+$$5  1.94E$$+$$5  7.16E$$+$$5  1$$+$$5  7$$+$$5  1.88E$$+$$5  8.11E$$+$$5  1.93E$$+$$5  8.11E$$+$$5  1.92E$$+$$5  7.37E$$+$$5  1.90E$$+$$5  500  1  2.95E$$+$$6  5.07E$$+$$6  1.24E$$+$$6  4.73E$$+$$6  1$$+$$6  4$$+$$6  1.19E$$+$$6  5.08E$$+$$6  1.23E$$+$$6  5.12E$$+$$6  1.24E$$+$$6  4.96E$$+$$6  1.22E$$+$$6     2  3.37E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  5.16E$$+$$6  1$$+$$6  4$$+$$6  1.25E$$+$$6  5.48E$$+$$6  1.29E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  4.99E$$+$$6  1.28E$$+$$6     3  3.10E$$+$$6  5.33E$$+$$6  1.26E$$+$$6  5.08E$$+$$6  1.21E$$+$$6  4$$+$$6  1$$+$$6  5.36E$$+$$6  1.26E$$+$$6  5.35E$$+$$6  1.26E$$+$$6  4.98E$$+$$6  1.22E$$+$$6     4  3.22E$$+$$6  5.41E$$+$$6  1.30E$$+$$6  5.20E$$+$$6  1.28E$$+$$6  4$$+$$6  1$$+$$6  5.39E$$+$$6  1.31E$$+$$6  5.38E$$+$$6  1.30E$$+$$6  5.23E$$+$$6  1.29E$$+$$6     5  3.11E$$+$$6  5.11E$$+$$6  1.20E$$+$$6  4.90E$$+$$6  1.17E$$+$$6  4$$+$$6  1$$+$$6  5.14E$$+$$6  1.20E$$+$$6  5.16E$$+$$6  1.20E$$+$$6  4.98E$$+$$6  1.20E$$+$$6     6  2.79E$$+$$6  4.95E$$+$$6  1.23E$$+$$6  4.57E$$+$$6  1.18E$$+$$6  4$$+$$6  1$$+$$6  4.75E$$+$$6  1.21E$$+$$6  4.88E$$+$$6  1.22E$$+$$6  4.68E$$+$$6  1.20E$$+$$6     7  3.17E$$+$$6  5.39E$$+$$6  1.27E$$+$$6  5.16E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.41E$$+$$6  1.27E$$+$$6  5.25E$$+$$6  1.27E$$+$$6  4.94E$$+$$6  1.24E$$+$$6     8  3.12E$$+$$6  5.41E$$+$$6  1.26E$$+$$6  5.09E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.30E$$+$$6  1.25E$$+$$6     9  3.36E$$+$$6  5.51E$$+$$6  1.29E$$+$$6  5.31E$$+$$6  1.26E$$+$$6  4$$+$$6  1$$+$$6  5.48E$$+$$6  1.28E$$+$$6  5.45E$$+$$6  1.27E$$+$$6  5.30E$$+$$6  1.27E$$+$$6     10  3.12E$$+$$6  5.37E$$+$$6  1.25E$$+$$6  5.09E$$+$$6  1.22E$$+$$6  4$$+$$6  1$$+$$6  5.28E$$+$$6  1.25E$$+$$6  5.30E$$+$$6  1.25E$$+$$6  5.18E$$+$$6  1.25E$$+$$6  1000  1  1.41E$$+$$7  2.32E$$+$$7  5.08E$$+$$6  2.26E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.35E$$+$$7  5.12E$$+$$6  2.34E$$+$$7  5.13E$$+$$6  2.19E$$+$$7  4.97E$$+$$6     2  1.23E$$+$$7  2.13E$$+$$7  4.88E$$+$$6  2.12E$$+$$7  4.88E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  4.93E$$+$$6  2.16E$$+$$7  4.93E$$+$$6  2.01E$$+$$7  4.84E$$+$$6     3  1.20E$$+$$7  2.20E$$+$$7  5.03E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  5.04E$$+$$6  2.21E$$+$$7  5.04E$$+$$6  2.11E$$+$$7  4.99E$$+$$6     4  1.18E$$+$$7  2.13E$$+$$7  4.94E$$+$$6  2.12E$$+$$7  4.94E$$+$$6  1$$+$$7  4$$+$$6  2.10E$$+$$7  4.92E$$+$$6  2.14E$$+$$7  4.93E$$+$$6  2.13E$$+$$7  4.93E$$+$$6     5  1.25E$$+$$7  2.24E$$+$$7  5.17E$$+$$6  2.23E$$+$$7  5.11E$$+$$6  1$$+$$7  4$$+$$6  2.27E$$+$$7  5.16E$$+$$6  2.20E$$+$$7  5.13E$$+$$6  2.22E$$+$$7  5.15E$$+$$6     6  1.17E$$+$$7  2.10E$$+$$7  5.10E$$+$$6  2.10E$$+$$7  5.06E$$+$$6  1$$+$$7  4$$+$$6  2.13E$$+$$7  5.09E$$+$$6  2.10E$$+$$7  5.10E$$+$$6  2.12E$$+$$7  5.07E$$+$$6     7  1.33E$$+$$7  2.31E$$+$$7  5.09E$$+$$6  2.27E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.31E$$+$$7  5.07E$$+$$6  2.30E$$+$$7  5.08E$$+$$6  2.28E$$+$$7  5.05E$$+$$6     8  1.23E$$+$$7  2.22E$$+$$7  4.99E$$+$$6  2.18E$$+$$7  4.95E$$+$$6  1$$+$$7  4$$+$$6  2.22E$$+$$7  4.98E$$+$$6  2.20E$$+$$7  4.96E$$+$$6  2.07E$$+$$7  4.87E$$+$$6     9  1.18E$$+$$7  2.16E$$+$$7  4.98E$$+$$6  2.10E$$+$$7  4.92E$$+$$6  1$$+$$7  4$$+$$6  2.14E$$+$$7  4.95E$$+$$6  2.14E$$+$$7  4.91E$$+$$6  2.12E$$+$$7  4.92E$$+$$6     10  1.24E$$+$$7  2.14E$$+$$7  5.10E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.19E$$+$$7  5.10E$$+$$6  2.14E$$+$$7  5.09E$$+$$6  2.19E$$+$$7  5.08E$$+$$6  $$n$$        MODE1  MODE2  MODE3  L-NSGA  NSGA-II  SPEA2        UB$$_{\mathrm{SumET}}$$  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  SumET  TFT  10  1  1936  1975  549  1992  546  1936  546  2091  546  1975  558  1936  558     2  1042  1042  594  1163  593  1042  593  1637  593  1042  593  1077  593     3  1586  1586  565  1714  565  1586  565  1763  565  1586  565  1586  565     4  2139  2214  463  2500  446  2139  459  2372  446  2205  478  2170  459     5  1187  1187  392  1199  391  1187  391  1362  391  1187  391  1187  392     6  1521  1523  386  1592  386  1521  386  1632  386  1521  386  1523  386     7  2170  2170  450  2365  450  2170  453  2612  450  2170  450  2170  451     8  1720  1720  291  1726  291  1720  291  1824  291  1726  291  1726  291     9  1574  1574  397  1574  397  1574  397  1746  397  1574  397  1574  401     10  1869  1888  546  1888  538  1869  539  2065  538  1869  546  1869  546  20  1  4394  4423  1656  4729  1645  4478  1678  4991  1645  4398  1656  4423  1648     2  8430  8555  1973  8914  1970  8496  1973  8936  1970  8460  1976  8443  1973     3  6210  6231  1974  6657  1947  6260  1982  7244  1947  6221  1989  6224  1956     4  9188  9357  1859  10869  1859  9289  2002  10656  1859  9332  1899  9192  1872     5  4215  4321  1361  4584  1342  4315  1352  4773  1342  4215  1377  4224  1343     6  6527  6586  1665  6983  1634  6578  1674  7259  1634  6541  1674  6550  1634     7  10455  10500  2108  11131  2080  10471  2125  11403  2080  10550  2107  10459  2080     8  3920  4252  1758  5793  1708  4126  1764  3948  1705  4390  1730  3954  1709     9  3465  3563  1015  3708  1003  3471  1011  3689  1003  3518  1010  3485  1003     10  4979  5000  1676  5870  1661  5051  1668  5997  1661  5122  1717  4995  1661  50  1  40697  43082  11009  42669  10287  43262  11061  43099  10343  41704  10688  41113  10441     2  30613  33367  9962  33131  9278  33433  9926  33533  9387  31586  9588  30748  9275     3  34435  38670  10992  37396  10282  37882  11300  38806  10453  35522  10581  34815  10341     4  27755  29964  9417  29206  8600  29489  9231  30802  8643  28588  8860  27907  8635     5  32307  38420  11592  39211  10857  36074  11701  42394  10892  33810  11339  32718  10992     6  34993  38536  11023  37581  10409  36880  11514  38802  10405  36121  10717  35601  10396     7  43136  47390  12126  49911  11436  46880  12032  52999  11609  45492  11680  43462  11476     8  43839  48414  14687  47286  13931  48074  14594  49274  14084  45334  14317  43903  13971     9  34228  37842  8992  37252  8321  36455  8986  39908  8430  35343  8841  34629  8356     10  32958  35056  10532  35445  9825  34587  10200  36862  9852  33547  10044  33191  9792  100  1  1..46E$$+$$5  1.97E$$+$$5  5.15E$$+$$4  1.64E$$+$$5  4.64E$$+$$4  1.82E$$+$$5  4.98E$$+$$4  1.85E$$+$$5  4.89E$$+$$4  1.68E$$+$$5  4.85E$$+$$4  1.54E$$+$$5  4.57E$$+$$4     2  1.25E$$+$$5  1.76E$$+$$5  4.87E$$+$$4  1.53E$$+$$5  4.43E$$+$$4  1.53E$$+$$5  4.77E$$+$$4  1.62E$$+$$5  4.62E$$+$$4  1.45E$$+$$5  4.70E$$+$$4  1.35E$$+$$5  4.49E$$+$$4     3  1.30E$$+$$5  1.67E$$+$$5  4.69E$$+$$4  1.44E$$+$$5  4.28E$$+$$4  1.54E$$+$$5  4.53E$$+$$4  1.49E$$+$$5  4.33E$$+$$4  1.53E$$+$$5  4.45E$$+$$4  1.36E$$+$$5  4.12E$$+$$4     4  1.30E$$+$$5  1.80E$$+$$5  4.55E$$+$$4  1.44E$$+$$5  4.19E$$+$$4  1.59E$$+$$5  4.41E$$+$$4  1.51E$$+$$5  4.39E$$+$$4  1.58E$$+$$5  4.40E$$+$$4  1.35E$$+$$5  4.19E$$+$$4     5  1.24E$$+$$5  1.81E$$+$$5  4.66E$$+$$4  1.43E$$+$$5  4.24E$$+$$4  1.53E$$+$$5  4.51E$$+$$4  1.51E$$+$$5  4.34E$$+$$4  1.43E$$+$$5  4.28E$$+$$4  1.32E$$+$$5  4.09E$$+$$4     6  1.39E$$+$$5  1.89E$$+$$5  4.73E$$+$$4  1.55E$$+$$5  4.24E$$+$$4  1.62E$$+$$5  4.44E$$+$$4  1.67E$$+$$5  4.34E$$+$$4  1.70E$$+$$5  4.53E$$+$$4  1.46E$$+$$5  4.20E$$+$$4     7  1.35E$$+$$5  1.78E$$+$$5  4.72E$$+$$4  1.51E$$+$$5  4.30E$$+$$4  1.62E$$+$$5  4.51E$$+$$4  1.59E$$+$$5  4.40E$$+$$4  1.58E$$+$$5  4.34E$$+$$4  1.40E$$+$$5  4$$+$$4     8  1.60E$$+$$5  2.07E$$+$$5  5.32E$$+$$4  1.87E$$+$$5  5.09E$$+$$4  1.93E$$+$$5  5.39E$$+$$4  1.90E$$+$$5  5.23E$$+$$4  1.93E$$+$$5  5.21E$$+$$4  1$$+$$5  5$$+$$4     9  1.17E$$+$$5  1.69E$$+$$5  4.79E$$+$$4  1.32E$$+$$5  4.38E$$+$$4  1.52E$$+$$5  4.68E$$+$$4  1.49E$$+$$5  4.60E$$+$$4  1.46E$$+$$5  4.53E$$+$$4  1$$+$$5  4$$+$$4     10  1.19E$$+$$5  1.67E$$+$$5  4.75E$$+$$4  1.34E$$+$$5  4.31E$$+$$4  1.46E$$+$$5  4.54E$$+$$4  1.48E$$+$$5  4.39E$$+$$4  1.38E$$+$$5  4.50E$$+$$4  1$$+$$5  4$$+$$4  200  1  4.99E$$+$$5  7.80E$$+$$5  1.96E$$+$$5  6.95E$$+$$5  1.88E$$+$$5  6.89E$$+$$5  1.88E$$+$$5  7.56E$$+$$5  1.94E$$+$$5  7.37E$$+$$5  1.94E$$+$$5  6$$+$$5  1$$+$$5     2  5.41E$$+$$5  8.38E$$+$$5  1.97E$$+$$5  7.00E$$+$$5  1$$+$$5  7.43E$$+$$5  1.89E$$+$$5  7.96E$$+$$5  1.95E$$+$$5  8.24E$$+$$5  1.93E$$+$$5  6$$+$$5  1.86E$$+$$5     3  4.89E$$+$$5  7.47E$$+$$5  1.84E$$+$$5  6.76E$$+$$5  1.74E$$+$$5  6$$+$$5  1.76E$$+$$5  7.17E$$+$$5  1.82E$$+$$5  7.25E$$+$$5  1.81E$$+$$5  6.47E$$+$$5  1$$+$$5     4  5.86E$$+$$5  8.56E$$+$$5  2.02E$$+$$5  7.71E$$+$$5  1.89E$$+$$5  7$$+$$5  1.92E$$+$$5  8.14E$$+$$5  1.97E$$+$$5  8.12E$$+$$5  1.99E$$+$$5  7.58E$$+$$5  1$$+$$5     5  5.13E$$+$$5  8.01E$$+$$5  2.02E$$+$$5  6.76E$$+$$5  1$$+$$5  6.99E$$+$$5  1.95E$$+$$5  7.62E$$+$$5  2.00E$$+$$5  7.71E$$+$$5  1.99E$$+$$5  6$$+$$5  1.96E$$+$$5     6  4.78E$$+$$5  7.30E$$+$$5  1.85E$$+$$5  6.80E$$+$$5  1$$+$$5  6.48E$$+$$5  1.81E$$+$$5  7.24E$$+$$5  1.88E$$+$$5  7.51E$$+$$5  1.88E$$+$$5  6$$+$$5  1.82E$$+$$5     7  4.55E$$+$$5  7.63E$$+$$5  1.89E$$+$$5  6.57E$$+$$5  1$$+$$5  6.49E$$+$$5  1.76E$$+$$5  7.07E$$+$$5  1.85E$$+$$5  6.72E$$+$$5  1.79E$$+$$5  6$$+$$5  1.79E$$+$$5     8  4.94E$$+$$5  7.98E$$+$$5  1.93E$$+$$5  7.04E$$+$$5  1.83E$$+$$5  6$$+$$5  1$$+$$5  7.30E$$+$$5  1.89E$$+$$5  7.37E$$+$$5  1.86E$$+$$5  6.87E$$+$$5  1.85E$$+$$5     9  5.29E$$+$$5  7.92E$$+$$5  1.92E$$+$$5  7.30E$$+$$5  1$$+$$5  6.92E$$+$$5  1.85E$$+$$5  7.73E$$+$$5  1.91E$$+$$5  7.54E$$+$$5  1.89E$$+$$5  6$$+$$5  1.84E$$+$$5     10  5.38E$$+$$5  8.12E$$+$$5  1.94E$$+$$5  7.16E$$+$$5  1$$+$$5  7$$+$$5  1.88E$$+$$5  8.11E$$+$$5  1.93E$$+$$5  8.11E$$+$$5  1.92E$$+$$5  7.37E$$+$$5  1.90E$$+$$5  500  1  2.95E$$+$$6  5.07E$$+$$6  1.24E$$+$$6  4.73E$$+$$6  1$$+$$6  4$$+$$6  1.19E$$+$$6  5.08E$$+$$6  1.23E$$+$$6  5.12E$$+$$6  1.24E$$+$$6  4.96E$$+$$6  1.22E$$+$$6     2  3.37E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  5.16E$$+$$6  1$$+$$6  4$$+$$6  1.25E$$+$$6  5.48E$$+$$6  1.29E$$+$$6  5.52E$$+$$6  1.29E$$+$$6  4.99E$$+$$6  1.28E$$+$$6     3  3.10E$$+$$6  5.33E$$+$$6  1.26E$$+$$6  5.08E$$+$$6  1.21E$$+$$6  4$$+$$6  1$$+$$6  5.36E$$+$$6  1.26E$$+$$6  5.35E$$+$$6  1.26E$$+$$6  4.98E$$+$$6  1.22E$$+$$6     4  3.22E$$+$$6  5.41E$$+$$6  1.30E$$+$$6  5.20E$$+$$6  1.28E$$+$$6  4$$+$$6  1$$+$$6  5.39E$$+$$6  1.31E$$+$$6  5.38E$$+$$6  1.30E$$+$$6  5.23E$$+$$6  1.29E$$+$$6     5  3.11E$$+$$6  5.11E$$+$$6  1.20E$$+$$6  4.90E$$+$$6  1.17E$$+$$6  4$$+$$6  1$$+$$6  5.14E$$+$$6  1.20E$$+$$6  5.16E$$+$$6  1.20E$$+$$6  4.98E$$+$$6  1.20E$$+$$6     6  2.79E$$+$$6  4.95E$$+$$6  1.23E$$+$$6  4.57E$$+$$6  1.18E$$+$$6  4$$+$$6  1$$+$$6  4.75E$$+$$6  1.21E$$+$$6  4.88E$$+$$6  1.22E$$+$$6  4.68E$$+$$6  1.20E$$+$$6     7  3.17E$$+$$6  5.39E$$+$$6  1.27E$$+$$6  5.16E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.41E$$+$$6  1.27E$$+$$6  5.25E$$+$$6  1.27E$$+$$6  4.94E$$+$$6  1.24E$$+$$6     8  3.12E$$+$$6  5.41E$$+$$6  1.26E$$+$$6  5.09E$$+$$6  1.24E$$+$$6  4$$+$$6  1$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.43E$$+$$6  1.26E$$+$$6  5.30E$$+$$6  1.25E$$+$$6     9  3.36E$$+$$6  5.51E$$+$$6  1.29E$$+$$6  5.31E$$+$$6  1.26E$$+$$6  4$$+$$6  1$$+$$6  5.48E$$+$$6  1.28E$$+$$6  5.45E$$+$$6  1.27E$$+$$6  5.30E$$+$$6  1.27E$$+$$6     10  3.12E$$+$$6  5.37E$$+$$6  1.25E$$+$$6  5.09E$$+$$6  1.22E$$+$$6  4$$+$$6  1$$+$$6  5.28E$$+$$6  1.25E$$+$$6  5.30E$$+$$6  1.25E$$+$$6  5.18E$$+$$6  1.25E$$+$$6  1000  1  1.41E$$+$$7  2.32E$$+$$7  5.08E$$+$$6  2.26E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.35E$$+$$7  5.12E$$+$$6  2.34E$$+$$7  5.13E$$+$$6  2.19E$$+$$7  4.97E$$+$$6     2  1.23E$$+$$7  2.13E$$+$$7  4.88E$$+$$6  2.12E$$+$$7  4.88E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  4.93E$$+$$6  2.16E$$+$$7  4.93E$$+$$6  2.01E$$+$$7  4.84E$$+$$6     3  1.20E$$+$$7  2.20E$$+$$7  5.03E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.18E$$+$$7  5.04E$$+$$6  2.21E$$+$$7  5.04E$$+$$6  2.11E$$+$$7  4.99E$$+$$6     4  1.18E$$+$$7  2.13E$$+$$7  4.94E$$+$$6  2.12E$$+$$7  4.94E$$+$$6  1$$+$$7  4$$+$$6  2.10E$$+$$7  4.92E$$+$$6  2.14E$$+$$7  4.93E$$+$$6  2.13E$$+$$7  4.93E$$+$$6     5  1.25E$$+$$7  2.24E$$+$$7  5.17E$$+$$6  2.23E$$+$$7  5.11E$$+$$6  1$$+$$7  4$$+$$6  2.27E$$+$$7  5.16E$$+$$6  2.20E$$+$$7  5.13E$$+$$6  2.22E$$+$$7  5.15E$$+$$6     6  1.17E$$+$$7  2.10E$$+$$7  5.10E$$+$$6  2.10E$$+$$7  5.06E$$+$$6  1$$+$$7  4$$+$$6  2.13E$$+$$7  5.09E$$+$$6  2.10E$$+$$7  5.10E$$+$$6  2.12E$$+$$7  5.07E$$+$$6     7  1.33E$$+$$7  2.31E$$+$$7  5.09E$$+$$6  2.27E$$+$$7  5.06E$$+$$6  2$$+$$7  4$$+$$6  2.31E$$+$$7  5.07E$$+$$6  2.30E$$+$$7  5.08E$$+$$6  2.28E$$+$$7  5.05E$$+$$6     8  1.23E$$+$$7  2.22E$$+$$7  4.99E$$+$$6  2.18E$$+$$7  4.95E$$+$$6  1$$+$$7  4$$+$$6  2.22E$$+$$7  4.98E$$+$$6  2.20E$$+$$7  4.96E$$+$$6  2.07E$$+$$7  4.87E$$+$$6     9  1.18E$$+$$7  2.16E$$+$$7  4.98E$$+$$6  2.10E$$+$$7  4.92E$$+$$6  1$$+$$7  4$$+$$6  2.14E$$+$$7  4.95E$$+$$6  2.14E$$+$$7  4.91E$$+$$6  2.12E$$+$$7  4.92E$$+$$6     10  1.24E$$+$$7  2.14E$$+$$7  5.10E$$+$$6  2.19E$$+$$7  5.04E$$+$$6  1$$+$$7  4$$+$$6  2.19E$$+$$7  5.10E$$+$$6  2.14E$$+$$7  5.09E$$+$$6  2.19E$$+$$7  5.08E$$+$$6  View Large © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Journal

IMA Journal of Management MathematicsOxford University Press

Published: Jan 1, 2018

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