Improved heuristics for the single machine scheduling problem with linear early and quadratic tardy penalties

Transcription

1 Imroved heuristics for the single machine scheduling roblem with linear early and quadratic tardy enalties Jorge M. S. Valente* LIAAD INESC Porto LA, Faculdade de Economia, Universidade do Porto Postal Address: Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal Address: *Corresonding author Jeffrey E. Schaller Deartment of Business Administration, Eastern Connecticut State University Postal Address: Deartment of Business Administration, Eastern Connecticut State University, 83 Windham St., Willimantic, CT , USA Address: Abstract This aer considers the single machine scheduling roblem with linear earliness and quadratic tardiness costs. The research on the version with inserted idle time focused on an exact aroach, while several heuristics had already been roosed for the version with no idle time. These heuristics were then the basis for the develoment of new heuristic rocedures for the version with idle time. Some imrovement rocedures were also considered. The new heuristics outerformed the existing rocedures. A genetic algorithm rovides the best results in terms of solution quality, but is comutationally intensive. One of the backward scheduling disatching rules rovides results of similar quality, and can quickly solve even large instances. The new heuristics were also alied, with the aroriate modifications, to the version with no idle time. Again, the new rocedures rovided better results than the existing heuristics. Therefore, the rocedures develoed in this aer are the new heuristics of choice for both versions of the considered roblem. Keywords: roduction scheduling; heuristics; single machine; linear earliness; quadratic tardiness; disatching rules; genetic algorithms Biograhical Notes Jorge M. S. Valente is an Associate Professor of Oerations Research at the Faculty of Economics, University of Porto, Portugal. He holds a Ph.D. in Management Science and a M.S. in Economics from the University of Porto. His current research interests include roduction scheduling, combinatorial otimization and heuristic techniques. 1

2 Jeffrey E. Schaller is a Professor of Oerations Management at Eastern Connecticut State University. He holds a Ph.D. in Business Administration from the University of Florida, a Masters of Business Administration from the University of Michigan and a B.A. from Gettysburg College. His current research interests include roduction scheduling, cellular manufacturing and lean roduction. 1. Introduction This aer considers the single machine scheduling roblem with linear earliness and quadratic tardiness costs. Formally, the roblem can be stated as follows. A set of n indeendent obs {1, 2,, n} has to be scheduled on a single machine that can handle at most one ob at a time. The machine is assumed to be continuously available from time zero onwards, and reemtions are not allowed. Job, = 1, 2,, n, requires a rocessing time and should ideally be comleted on its due date d. For a given schedule, the earliness and tardiness of ob are resectively defined as E = max {0, d - C } and T = max {0, C - d }, where C is the comletion time of ob. The obective is then to find a schedule that minimizes the sum of the = + n linear earliness and quadratic tardiness costs ( E T ) Single machine scheduling environments actually occur in several ractical oerations; for a secific examle in the chemical industry, see Wagner, Davis and Kher (2002). Also, the erformance of many roduction systems is frequently determined by the quality of the schedules for a single bottleneck machine. Moreover, results and insights obtained for single machine roblems can often be alied to more comlex scheduling environments, such as flow shos or ob shos. Scheduling models with both earliness and tardiness costs have received considerable and increasing attention from the scheduling community, due to their ractical imortance. Indeed, early/tardy scheduling roblems are comatible with the concets of ust-in-time roduction and suly chain management. These roduction strategies, which have been adoted by many organisations, view both early and tardy deliveries as undesirable. The obective function considered in this aer includes linear earliness and quadratic tardiness costs. Indeed, early deliveries result in unnecessary inventory, and the costs incurred with this inventory tend to be roortional to the time held in stock. Late deliveries, on the other hand, can result in lost sales and loss of customer

3 goodwill. A quadratic enalty is then used for the tardy obs, since a customer's dissatisfaction tends to increase quadratically with the tardiness, as roosed in the loss function of Taguchi (1986). The considered roblem is of ractical relevance and interest when idle time may be inserted, as well as when the insertion of idle time is not allowed. Both of these versions of the roblem have been reviously studied. Schaller (2004) analysed the roblem with inserted idle time. He resented a timetabling algorithm and a lower bounding rocedure, as well as a branch-and-bound algorithm and simle heuristics. For the roblem with no idle time, both an exact rocedure and several heuristic techniques have been considered. Valente (2008) develoed a lower bounding rocedure and a branch-and-bound technique. Among the heuristics, disatching rules (Valente, 2007b), beam search rocedures (Valente, 2007a) and genetic algorithms (Valente and Gonçalves, 2008) have all been roosed. A large number of aers have been ublished on scheduling models with earliness and tardiness costs. Baker and Scudder (1990) rovide an excellent survey of the initial work on early/tardy scheduling. A recent survey of multicriteria scheduling which includes roblems with earliness and tardiness enalties is given in Hoogeveen (2005), while Kanet and Sridharan (2000) review scheduling models with inserted idle time. Previous research roosed a large number of heuristics for the roblem with no idle time. Among these heuristics, the EQTP_EXP disatching rule (henceforth denoted simly as EQTP) develoed in Valente (2007b) is the heuristic of choice for large instances, while the MA_IN genetic algorithm of Valente and Gonçalves (2008) rovides the best results for small and medium size roblems. This aer investigates heuristic rocedures for the version with inserted idle time, taking as its starting oint the existing heuristics for the roblem where idle time is not allowed. In this context, the erformance of the EQTP disatching rule, suitably adated in order to take into account the insertion of idle time, was first analysed. Since the results were not as good as desired, several additional disatching heuristics were then develoed, as well as a modification of the genetic algorithm MA_IN. Some imrovement rocedures were also considered, in order to imrove the schedule obtained by the heuristics. These new heuristics erformed quite well for the roblem with inserted idle time. Therefore, this raised the question of whether they would also erform well for the version with no idle time. Consequently, the new heuristics were also then 3

4 alied, after the aroriate modifications, to the roblem without idle time. These new rocedures outerformed the existing heuristics, so the rocedures develoed in this aer are the new heuristics of choice for both versions of the considered roblem. The remainder of this aer is organized as follows. In section 2, the heuristic rocedures are described. The imrovement rocedures are resented in section 3. In section 4, the comutational results are reorted. Finally, some concluding remarks are rovided in section The heuristic rocedures In this section, the considered heuristic rocedures are described. First several disatching rules are roosed. These rules include the forward scheduling EQTP rocedure and several backward scheduling disatching rules. Finally, a modified version of the MA_IN genetic algorithm is also considered. Throughout this section, the focus will be on the roblem version with inserted idle time. Therefore, the seudo-codes and the algorithm descritions rovided are for this idle time version. These heuristic rocedures can also be imlemented for the roblem with no idle time by removing all the rocedures that insert idle time. 2.1 Disatching rules Disatching rules were roosed and tested in Valente (2007b) for the roblem without inserted idle time and the EQTP heuristic was found to be the best. In this section several new disatching rules are roosed. Since inserted idle time is allowed, a key consideration in develoing the disatching heuristics is how to incororate inserted idle time. Schaller (2004) develoed a rocedure that determines the otimal amount of idle time to insert before each ob in a given sequence of the obs, i.e. the amount of idle time that minimizes the obective function value for that sequence. In this aer, a slightly modified version of this rocedure is used. This modification consists only in a rerocessing hase that is erformed rior to alying the rocedure develoed in Schaller (2004). In this rerocessing hase, the lateness of all obs is first calculated assuming that no 4

5 idle time is used. Then, if all the obs in the sequence are early, the minimum earliness value is calculated. Finally, this value is inserted as idle time before the first ob. This rerocessing hase reduced considerably the comutation time required by the timing rocedure for the instances where most obs are early. This modified version of the otimal timing rocedure for the roblem with inserted idle time will be denoted by OPT_IIT. In the following, let S be the ordered set of the currently scheduled obs and U be the set of the yet unscheduled obs. Also, let idle[i] reresent the amount of idle time to be inserted before the ith osition in the sequence, i = 1, 2,, n. Finally, let LB_IIT denote the lower bounding rocedure roosed by Schaller (2004) for the version with inserted idle time. This rocedure, while calculating the lower bound, also comutes values for the idle times before each osition in the sequence. The seudo-code for the disatching heuristics for the version with inserted idle time is given in Algorithm 1. Algorithm 1. Disatching heuristics for the version with inserted idle time 1. Set S =, U = {1, 2,, n}. 2. Aly rocedure LB_IIT to the instance and set idle[i], i = 1, 2,, n, equal to the corresonding idle time comuted in rocedure LB_IIT. 3. Use a disatching rule to determine the sequence of obs. 4. Aly rocedure OPT_IIT to the sequence in set S and set idle[i], i = 1, 2,, n, equal to the corresonding otimal idle times comuted in rocedure OPT_IIT. 5. Calculate the obective function value of the schedule corresonding to the sequence in set S and the idle times in idle[i], i = 1, 2,, n. 6. Return sequence S, the idle times idle[i], i = 1, 2,, n and the obective function value of the corresonding schedule. In ste 2, the amount of idle time before each osition in the sequence is first set equal to the idle time values calculated in the lower bounding rocedure LB_IIT. This aroach roved greatly suerior to another strategy that was considered in reliminary exeriments, more secifically setting all idle times equal to 0. Ste 3 uses one of the disatching rules, to be described next, to generate a ob sequence. In ste 4, rocedure OPT_IIT is used to obtain the otimal idle times for the sequence 5

6 generated in ste 3. The obective function value of the corresonding schedule is calculated in ste 5, and these elements are then returned in ste The forward scheduling EQTP disatching rule The EQTP heuristic is the best of the existing disatching rules for the roblem with no idle time. The EQTP rule is a forward scheduling heuristic, i.e. the schedule is built from the front to the back, with obs being added to the end of the current artial sequence. When the machine becomes available, the EQTP heuristic calculates a riority index for each unscheduled ob, and the ob with the highest riority value is then selected to be rocessed next. Let I (t) denote the riority index of ob at the current time t (i.e. t is the time at which the next scheduled ob will start). The EQTP heuristic uses the following riority index I (t): I ( t) = [ ] ( 1/ ) + 2( t+ d ) if s 0 ( / ) ex[ ( + 1) s / k ] if 0< s < /( + 1) 2 3 ( 1/ ) [( / ) ( 1/ )( + 1) s / k ] if [ /( + 1) k ] s ( 1/ ) otherwise, [ k ] < k where s = d - t - is the slack of ob, is the average rocessing time of the remaining unscheduled obs and k is a lookahead arameter. The effectiveness of the EQTP rule deends on the value of the lookahead arameter k, which should reflect the number of cometing critical obs, i.e. the number of obs that may clash each time a sequencing decision is to be made. The following rocedure is used to calculate the value of k at each iteration. First, a critical slack value crit_slack is calculated. This critical slack value is calculated as crit_slack k = i + 1 n = slack_ro (n U + idle[ k] ), where n U is the number of unscheduled obs, and 0 slack_ro < 1 is a user-defined arameter. Then, each ob is classified as critical if 0 < s crit_slack, and non-critical otherwise. Therefore, a ob is considered critical if it is not already tardy (s > 0), but is about to become tardy (s crit_slack). Finally, the lookahead arameter k is set equal to the number of critical obs. Note that in this rocedure and other rocedures roosed in this aer for the roblem with inserted idle time, critical slack is defined as a roortion of the sum of 6

7 the remaining rocessing time and idle time. This differs from the definition for critical slack used in the roblem without idle time, in which critical slack is defined as a roortion of the remaining rocessing time. Disatching Rule 1. EQTP disatching rule for the version with inserted idle time 1. Set t = For i = 1 to n 2.1. Set t = t + idle[i] Calculate the EQTP riority index I (t) for all obs U Let l be the ob with the largest riority index I (t) for U Add ob l to the end of set S and remove it from set U Set t = t+ l. Table 1 rovides the data for a numerical examle that will be used to illustrate the roosed heuristics. When alied to the instance in examle 1, the EQTP heuristic generates the sequence , with an obective function value of The details of the iterations erformed by the EQTP rocedure are rovided in Table Backward scheduling disatching rules Preliminary exeriments showed that the erformance of the EQTP heuristic fell short of that desired, articularly for instances with relatively loose due dates, and therefore with a higher roortion of early obs and larger amounts of inserted idle time. Comarisons with otimal sequences showed that the EQTP rule would sometimes erroneously select a ob with a large rocessing time early in the construction of the sequence. This is due to the fact that at the beginning of the EQTP rocedure, most or all obs could be early, and the riority index would then lead to choosing a ob with a large rocessing time, so as to reduce the earliness costs. However, this could cause a ob to become tardy later in the sequencing rocess. Since the earliness costs are linear, while the tardiness costs are quadratic, and 7

8 therefore more heavily enalized, the initial reduction in the earliness costs would be more than offset by the later increase in the tardiness of one or more obs. Due to this articular nature of the obective function, which enalizes tardiness much more heavily, backward scheduling rocedures were then considered. In backward scheduling heuristics, the schedule is built from the end to the front, with obs being added to the beginning of the current artial sequence. With backward scheduling, the tardy obs are considered first. Therefore, the backward heuristics consider right at the start the most imortant decisions, since they begin by scheduling the tardy obs, which contribute the most to the obective function value. In this subsection, four backward scheduling heuristics are resented. Two rules are first roosed that are standard in the sense that, at each iteration, they select the ob with the largest riority. Then exchange checks are incororated into these rules to create two additional rules. In these modified versions, the ob with the largest riority value may not actually be the ob selected for rocessing. Indeed, two tyes of exchange checks are erformed, and these may lead to choosing a different ob to be sequenced next. The two backward scheduling disatching rules without exchange checks are denoted by EQTP_Back and DR_Back. These rules share the same framework, and differ only in their riority index. The seudo-code for these two backward scheduling rules without exchange checks is given next. Disatching Rules 2 and 3. Backward scheduling disatching rules without exchange checks = 1 = n n 1. Set t = + idle[]. i i 1 2. Aly rocedure CT_Check. 3. For i = n to Aly rocedure Adust_T_IIT to ossibly adust the current comletion time t and the idle time before the revious ositions Calculate the riority index I (t) for all obs U Let l be the ob with the largest riority index I (t) for U Add ob l to the beginning of set S and remove it from set U Set t = t - l - idle[i]. 8

9 In ste 1, the comletion time t is initialized to the sum of the rocessing times of all obs and the idle times calculated by rocedure LB_IIT. In ste 3.1, before calculating the riority of the unscheduled obs, a rocedure denoted by Adust_T_IIT is alied. This rocedure may modify the current comletion time t, as well as the amount of idle time to be inserted before revious ositions in the schedule. In order to do this, the Adust_T_IIT rocedure uses information reviously obtained in ste 2 by the alication of rocedure CT_Check. The CT_Check and Adust_T_IIT rocedures were introduced after reliminary exeriments showed that it was sometimes ossible to detect that the existing inserted idle times were excessive. The two rocedures were therefore develoed to identify when LB_IIT inserted extra idle time, in order to then reduce the inserted idle times and the comletion times. The introduction of these two rocedures significantly imroved the erformance of the backward scheduling heuristics, articularly for instances with a higher roortion of early obs. The seudo-code for the CT_Check and Adust_T_IIT rocedures, as well as two roositions that identify the resence of excessive inserted idle time (and therefore rovide the theoretical basis for the two rocedures), are resented in the aendix. The aendix also contains an examle that is used to illustrate the CT_Check and Adust_T_IIT rocedures. As reviously mentioned, the EQTP_Back and DR_Back heuristics differ only on the riority index. The EQTP_Back disatching rule is basically a conversion of the EQTP heuristic to backward scheduling, and uses the following riority index I (t): I ( t) ( 1/ ) ( 1/ ) ex[ ( 1+ 1/ ( + 2k ) )( t d ) / k ] if 0< t d < [ 1/ ( k ) ] 2 3 [( 1/ )( + 2k ) ] [( 1/ ) ( 1/ )( k )( t d ) / k ] if [ 1/ ( k ) ] k t d ( 1/ )[ + 2( t d )] otherwise. = if t d As in the EQTP heuristic, the riority index of the EQTP_Back disatching rule deends on the value of a lookahead arameter k. This arameter is again set equal to the number of critical obs. However, a ob is now considered to be critical if 0 < t - d tardy_ro t, where 0 tardy_ro < 1 is a user-defined arameter. 0 k < k 9

10 The riority index of the DR_Back heuristic utilizes a value denoted by min_tardy which equals the minimum value of the tardiness among all obs that are currently tardy. This riority index is given by the following exression: I ( t) = ( 1/ ) 2( 1/ ' )( t d ) if t d 0 otherwise, where ' (a modified rocessing time) = min{, min_tardy}. Both the EQTP_Back and DR_Back heuristics select an early ob whenever such a ob exists. This not only revents the earliness of this ob from increasing, but also decreases the tardiness of the unscheduled obs that are currently still late. Whenever several early obs exist, the exression of the riority index when t - d 0 assures that those obs are scheduled in longest rocessing time order (when looking at the schedule from the front to the back), since this reduces the earliness costs of those obs. When all obs are tardy (or quite tardy, in the case of the EQTP_Back heuristic), the riority index favours obs with a low tardiness and a high rocessing time. Indeed, choosing a ob with a low tardiness means that the increase in the obective function will be small. Also, selecting a ob with a large rocessing time allows for a larger reduction in the tardiness of the remaining unscheduled obs. As reviously mentioned, the reason for favouring obs with larger rocessing times is the fact that this allows a larger decrease in the tardiness of the other obs. However, if a ob has a rocessing time that is equal to min_tardy, choosing this ob now will allow the selection, at the next iteration, of a ob that will then be right on- time. Giving more imortance to obs with rocessing times that are larger than min_tardy does not rovide a benefit in this situation, since it would actually increase, in the next iteration, the earliness of the ob that has currently the minimum tardiness min_tardy. Therefore, the original rocessing time was then relaced by the modified rocessing time ' in the second branch of the riority index of the DR_Back heuristic, which imroved the results. Table 3 and Table 4 illustrate the alication of the EQTP_Back and DR_Back heuristics, resectively, to the instance in examle 1. The EQTP_Back rocedure generates the sequence , with an obective function value of 10

11 2544. The DR_Back rocedure, on the other hand, obtains the sequence , with a cost of The following two heuristics, denoted by EQTP_Back_Ex and DR_Back_Ex, are modified versions of the EQTP_Back and DR_Back heuristics, resectively. These modified versions were obtained by adding exchange checks to the original rocedures. The exchange checks, as will be described below, may lead to choosing a ob different from the one with the largest riority value. Disatching Rules 4 and 5. Backward scheduling disatching rules with exchange checks = 1 i= n n 1. Set t = + idle[]. i 1 2. Aly rocedure CT_Check. 3. For i = n to Aly rocedure Adust_T_IIT to ossibly adust the current comletion time t and the idle time before the revious ositions Calculate the riority index I (t) for all obs U Let l be the ob with the largest riority index I (t) for U Aly rocedure Exchange_Checks Add ob l to the beginning of set S and remove it from set U Set t = t - l - idle[i]. This rocedure erforms two sets of checks that may lead to selecting a ob different from the one with the largest riority index I (t). The seudo-code for Exchange_Checks is given in Procedure 1. Procedure 1. Exchange_Checks 1. Consider, in decreasing order of due dates, all obs h U \ {l} such that d k > d l : 1.1. Let k be the ob currently being considered Calculate the cost C k,l of scheduling ob l in osition i and ob k in osition i

12 1.3. Calculate the cost C l,k of scheduling ob k in osition i and ob l in osition i If C l,k < C k,l, set l = k and move to ste 2. Otherwise, consider the next ob h. 2. Consider, in increasing order of rocessing times, all obs h U \ {l} such that k < l : 2.1. Let k be the ob currently being considered Calculate the cost C k,l of scheduling ob l in osition i and ob k in osition i Calculate the cost C l,k of scheduling ob k in osition i and ob l in osition i If C l,k < C k,l, set l = k and sto. Otherwise, consider the next ob h. Procedure Exchange_Checks erforms two sets of checks that may change the ob that will be chosen for rocessing at the current iteration. These checks were motivated by the several reliminary tests that were erformed, as well as by the comarison of the EQTP_Back and DR_Back results with otimal sequences. The first set of exchange checks is erformed in ste 1. In this ste, the ob currently selected is comared, in decreasing order of due dates, with the remaining unscheduled obs that have a larger due date. For each candidate ob the cost of that ob and the currently selected ob is then calculated when those obs are scheduled in the next two ositions, in the two ossible orders. If the cost when the candidate ob is scheduled at the current osition is lower, that candidate ob becomes the currently selected ob, and the rocedure then skis to ste 2. Otherwise, the next candidate ob is considered. The reason for including this set of checks is to avoid large squared tardiness enalties. Checking obs with later due dates may reduce the squared tardiness enough to offset the benefits of scheduling the ob with the largest riority index. In ste 2, the second set of exchange checks is erformed. This second ste is similar to the first, since it also calculates the cost of scheduling the currently selected ob and the candidate ob in the next two ositions in the sequence, in the two ossible orders. However, the candidate obs are now the remaining unscheduled obs that have a smaller rocessing time than the currently selected ob. Furthermore, these candidate obs are considered in increasing order of their rocessing times. The reason 12

13 for including this set of checks was that it was found that sometimes scheduling the ob with the largest riority index (if it had a large rocessing time) caused the remaining obs to be quite early. It was found that incororating this set of checks sometimes scheduled a ob with a lower rocessing time and a higher tardiness than the original ob chosen, which resulted in a lower obective when the entire schedule was comleted. Again, examle 1 is used to illustrate the alication of the EQTP_Back_Ex and DR_Back_Ex heuristics. When alied to this instance, both of these rocedures generate the sequence , with an obective function value of Although the two rocedures give the same final sequence, they erform different exchange checks. The details of the iterations erformed by the EQTP_Back_Ex and DR_Back_Ex heuristics are rovided in Table 5 and Table 6, resectively Genetic algorithm The MA_IN genetic algorithm is the best erforming heuristic for the roblem with no idle time. However, due to its comutational requirements, this rocedure can only be alied to small and medium size instances. In this rocedure, a so-called initialization of the initial oulation is erformed. This initialization consists in introducing in the initial oulation chromosomes that corresond to the sequences of four of the disatching rules analysed in Valente (2007b). One of these four disatching rules is recisely the EQTP heuristic that has been reviously resented. Also, an adacent airwise interchange rocedure is used to imrove each chromosome in the successive oulations. Therefore, the MA_IN rocedure can be seen as a memetic algorithm, since it incororates a local search rocedure. In this section, a modified version of this rocedure, denoted as MA_IN_New, is roosed. The MA_IN_New heuristic is essentially identical to the MA_IN rocedure, with the excetion that, in the generation of the initial oulation, the chromosome corresonding to the solution of the EQTP heuristic has been relaced by a chromosome that contains the solution of the DR_Back_Ex heuristic. Indeed, the DR_Back_Ex rocedure rovided the best results among the disatching rules that were considered, and its solution then relaced the EQTP sequence in the initial oulation of MA_IN_New. Furthermore, in the calculation of the fitness value of each chromosome, rocedure OPT_IIT is also necessarily alied in order to 13

14 otimally insert idle time when calculating the obective function value of the sequence corresonding to that chromosome. The MA_IN_New genetic algorithm was also alied to the version with no inserted idle time. For this version the rocedure OPT_IIT is not required. However, and in addition to this usual difference between the rocedures for the versions with and without idle time, there is one other distinction between the MA_IN_New algorithms for the two versions of the roblem. This difference is in the way the local search rocedure is alied. The discussion of this difference, however, will be deferred to the next section, where the imrovement rocedures are resented. 3. The imrovement rocedures In this section, the roosed imrovement rocedures are described. Four imrovement rocedures have been considered. The first three are quite common in scheduling roblems, and will only be briefly described. These three methods are the adacent airwise interchanges (API), 3-swas (3SW) and first-imrove interchanges (INTER) imrovement rocedures. The fourth rocedure, even though it uses insertions, which are also common in scheduling roblems, is more roblem-secific. Therefore, this rocedure, which will be denoted as INS, will be described in more detail. The API rocedure considers adacent ob ositions. A air of adacent obs is then swaed if such an interchange imroves the obective function value. This rocess is reeated until no further imrovement is found, i.e. until no additional imrovement is ossible through an adacent swa. The 3SW rocedure is similar to the API method, but it considers three consecutive ob ositions, instead of only an adacent air of obs. All ossible ermutations of these three obs are then analysed, and the best configuration is selected. Once more, the rocedure is alied reeatedly until no further imrovement is ossible. The INTER rocedure starts at the first osition in the sequence, and then considers all the successive ositions, until the end of the sequence is reached. For each osition in the sequence, the INTER method considers interchanging the ob that is currently scheduled in that osition with the obs that are scheduled in the following ositions. Whenever such an interchange imroves the obective function value, that 14

15 interchange is erformed. If any interchange is erformed before the end of the sequence is reached, the rocedure starts again at the beginning of the schedule. Otherwise, no further imrovement is ossible, and the rocedure terminates. The alication of imrovement rocedures to the version where idle time is allowed is made more comlicated recisely due to that inserted idle time. Indeed, when an interchange is erformed, there is a change in the sequence, and consequently the otimal amount of inserted idle time may therefore also change. In Schaller (2004), an INTER imrovement rocedure had already been roosed. This method alied the OPT_IIT rocedure to re-otimize the inserted idle times each time an interchange was erformed. However, this made the rocedure too comutationally intensive, and therefore it could only be alied to small instances. Alying the OPT_IIT rocedure each time an interchange was erformed would still make the imrovement rocedures too comutationally demanding, and incaable of handling medium or large instances. Therefore, in this aer the API, 3SW and INTER rocedures are first alied while keeing constant the idle times from the original schedule. Once no further imrovement is ossible, the OPT_IIT rocedure is then used to re-otimize the inserted idle time. If the new idle times are different from the original ones, the imrovement rocedure is again alied (keeing the new idle times constant). This is reeated until there is no change in the idle times after the alication of the imrovement rocedure. This aroach was not only much more comutationally efficient, but also erformed well in terms of the imrovement in the obective function value. In the revious section, it was mentioned that there was a difference in the way the API local search rocedure was alied in the MA_IN_New genetic algorithms for the two versions of the considered roblem. For the version with no idle time, the API rocedure is alied until no further imrovement is ossible. In the version with inserted idle time, however, only a single ass of the API rocedure is erformed. Therefore, in the MA_IN_New algorithm for the version with idle time the API rocedure is not alied as described above. That is, while for the disatching rules the API rocedure is alied reeatedly until there is no change in the idle times, for the MA_IN_New genetic algorithm only a single ass of this rocedure is erformed. This is due to comutational efficiency concerns. Indeed, the MA_IN_New algorithm alies the local search rocedure to each chromosome. Emloying only a single ass 15

16 of the API rocedure allowed for a reduction in the comutation time of the MA_IN_New algorithm, without comromising the solution quality. The INS imrovement rocedure, as reviously mentioned, is more roblemsecific, and uses ob insertions. In addition to insertions, this rocedure also erforms a rescheduling of the obs between the insertion osition and the original osition of the ob that was moved. The seudo-code of the INS imrovement method is given in rocedure 2. In the following, let index[i] denote the index of the ob in osition i, i = 1, 2,, n. Also, let os() be the osition of ob in the current schedule, = 1, 2,, n. Procedure 2. INS imrovement rocedure 1. Determine the longest rocessing time (LPT) ordering of the obs. Let LPT k denote the index of the kth ob, k = 1, 2,, n, when the obs are in LPT order (i.e. sequenced in non-increasing order of rocessing times). 2. For = LPT 1 to LPT n : 2.1. Determine os(). Set ins_os = os() For i = 1 to os() - 1: If index[i] >, continue to the next osition If ob index[i] is not tardy if ob is inserted before it: Set ins_os = i and go to ste Else: Calculate the cost C index[i], of scheduling ob index[i] in osition i and ob in osition i Calculate the cost C,index[i] of scheduling ob in osition i and ob index[i] in osition i If C,index[i] < C index[i],, set ins_os = i and go to ste If ins_os < os(): Calculate the cost C current of the current schedule Create the following trial schedule from the current schedule: Move ob to osition ins_os Reschedule the obs in ositions ins_os + 1 to os() using the DR_Back_Ex rule Calculate the cost C trial of the trial schedule. 16

17 If C trial < C current, relace the current schedule with the trial schedule. The INS imrovement rocedure was motivated by the comarison of the sequences generated by the disatching rules with otimal sequences. Indeed, these comarisons showed that in some cases a ob with a large rocessing time should have been scheduled earlier in the sequence. Therefore, in ste 2, the INS rocedure considers the obs in non-increasing order of their rocessing times. Let the ob being currently considered for insertion earlier in the sequence be denoted as the insertion ob. For each insertion ob, its osition in the sequence is first determined in ste 2.1. Then, in ste 2.2, a search is erformed, in revious ositions, for a candidate osition for inserting the insertion ob. In this search, and as can be seen by ste 2.2.1, only ositions that contain a ob with a rocessing time that is not larger than the rocessing time of the insertion ob are considered. If one of those ositions contains a ob that will not be tardy even if the insertion ob is inserted before it, then that osition becomes the candidate osition. Otherwise, the cost of the insertion ob and the ob in the current osition is then calculated, when those two obs are scheduled in the current and the next ositions, in the two ossible orders. If the cost when the insertion ob is scheduled first is lower, then the current osition becomes the candidate osition. If a candidate osition is found, ste 2.3 is then executed. In this ste, the cost of the current schedule is first calculated. Then, a trial schedule is generated in ste This trial schedule is obtained by moving the insertion ob to the candidate osition, and rescheduling all the obs between the next osition and the original osition of the insertion ob using the DR_Back_Ex rule. The cost of this trial schedule is then calculated. If this cost is lower than that of the current schedule, the current schedule is relaced by the trial schedule. Table 7 rovides the data for an examle that will be used to illustrate the INS rocedure. The DR_Back_EX heuristic, when alied to the instance in examle 2, will generate the sequence , with an obective function value of After the alication of the INS imrovement rocedure, the sequence has been modified to , with a lower cost of The details of the iterations erformed by the INS rocedure are given in Table Comutational results 17

18 In this section, the comutational exeriments and results are resented. First, the set of test roblems used in the comutational tests, and the arameter values that were selected for several heuristics, are described. Then, the comutational results for the version with inserted idle time are resented. Finally, the results for the version where idle time is not allowed are analysed. More extensive results are resented for the version with inserted idle time, since this is the main focus of this aer. For this version, the heuristics are first comared with the EQTP rule. Then, the effectiveness of the imrovement rocedures is analysed. Finally, the heuristic results are comared with otimum results. For the version with no idle time, only a summary of the main results is resented. Throughout this section, and in order to avoid excessively large tables, results will sometimes be resented only for some reresentative cases Exerimental design and arameter values The comutational tests were erformed on a set of roblems with 10, 15, 20, 25, 30, 40, 50, 75, 100, 250, 500, 750 and 1000 obs. These roblems were randomly generated as follows. For each ob, an integer rocessing time was generated from one of two uniform distributions [45, 55] and [1, 100], to create low (L) and high (H) variability, resectively. For each ob, an integer due date d is generated from the uniform distribution [P(1 T R/2), P(1 T + R/2)], where P is the sum of the rocessing times of all obs, T is the tardiness factor and R is the range of due dates. The range of due dates arameter was set at 0.2, 0.4, 0.6 and 0.8 for both the idle time and no idle time versions of the roblem. The tardiness factor T was set at 0.0, 0.2, 0.4 and 0.6 for the version with inserted idle time. When idle time is not allowed, the additional values of 0.8 and 1.0 were also considered for this arameter. These two values were not used for the version with idle time, since for these high values of T most obs are tardy, and the insertion of idle time does not imrove the obective function. For each combination of roblem size n, rocessing time variability (var), T and R, 10 instances were randomly generated. Therefore, a total of 160 (240 for the roblem without idle time) instances were generated for each combination of roblem size and variability. The rocedures for the version with inserted idle time were coded 18

19 in Turbo Pascal, and executed on a HP GHz ersonal comuter. For the version with no idle time, all the algorithms were coded in Visual C++ 6.0, and executed on a Pentium IV 2.8 GHz ersonal comuter. The EQTP, EQTP_Back and EQTP_Back_Ex heuristics, require a value for the slack_ro and the tardy_ro arameters, resectively. For the EQTP heuristic on the roblem with no idle time, a value of 0.60 was used, as recommended in Valente (2007b). Preliminary exeriments were then erformed to determine an aroriate value for slack_ro for the EQTP rule on the version with inserted idle time, as well as for the tardy_ro arameter for the EQTP_Back and EQTP_Back_Ex heuristics, for both versions of the roblem. The values {0.00, 0.05, 0.10,,0.95} were considered, and the obective function value was comuted for each slack_ro or tardy_ro value and each instance. An analysis of these results then showed that slack_ro = 0.60 also roved adequate for the EQTP heuristic when idle time is allowed. For the EQTP_Back and EQTP_Back_Ex heuristics, and for both versions of the roblem, the best results were obtained by setting tardy_ro at The MA_IN_New genetic algorithm requires values for several arameters. These arameters were set at the values that were reviously used in Valente and Gonçalves (2008) for the MA_IN rocedure for the version with no idle time Results for the version with inserted idle time In this section, the comutational results are resented for the version with inserted idle time. The heuristics are first comared with the EQTP rule. Then, the effectiveness of the imrovement rocedures is analysed. Finally, the heuristic results are comared with otimum results Comarison with the EQTP heuristic In this section, the backward scheduling disatching rules and the MA_IN_New genetic algorithm are comared with the forward scheduling EQTP heuristic. Table 9 rovides the mean relative imrovement in obective function value over the EQTP rule. The relative imrovement is calculated as (eqt_ofv heur_ofv) / 19

20 eqt_ofv 100, where eqt_ofv and heur_ofv are the obective function values of the EQTP rule and the aroriate heuristic, resectively. The backward scheduling rules and the MA_IN_New genetic algorithm clearly outerform the EQTP heuristic. The best erformance, as exected, is rovided by the MA_IN_New genetic rocedure. However, the best erforming of the backward disatching heuristics gives results that are extremely close to those of the genetic algorithm. The erformance of the backward scheduling rules is somewhat similar. However, slightly better results are obtained when the DR riority index and / or the exchange checks rocedure are used. In fact, the DR_Back(_Ex) heuristic is usually marginally suerior to its EQTP_Back(_Ex) counterart. Also, the EQTP_Back_Ex and DR_Back_Ex rocedures are suerior to the resective versions that do not incororate the exchange checks. The rocessing time variability has a significant imact on the imrovement rovided by the heuristic rocedures. As can be seen by the results in Table 9, the imrovement over the EQTP rule is much higher for the instances with high variability. Indeed, the relative imrovement is about 4% when the variability is low, but it usually ranges from 15% to 19% for high variability instances. Table 10 gives the effect of the T and R arameters on the relative imrovement over the EQTP rule, for instances with 100 obs. The heuristic results are quite close when T = 0.6. For these instances, there are a larger roortion of tardy obs, and little or no inserted idle time is required. The relative imrovement is also quite minor when T = 0.4 and the due dates are not widely sread. However, the relative imrovement is quite high (larger than 40% for some arameter combinations) for the lower values of the tardiness factor (as well as for the T = 0.4 and R = 0.8 arameter combination). Therefore, a larger imrovement is obtained for instances with a higher roortion of early obs (T = 0.0 or 0.2), where larger amounts of inserted idle time are needed. The heuristic runtimes (in seconds) are rovided in Table 11. The MA_IN_New genetic algorithm is comutationally quite demanding, and can only be alied to small and medium size instances. The disatching rules, however, are very efficient, and can quickly solve even quite large roblems. The versions that incororate the exchange checks do require a little additional time when comared 20

21 with their EQTP_Back and DR_Back counterarts, but are nevertheless still quite efficient. The DR_Back_Ex disatching rule then seems to be the heuristic rocedure of choice. Indeed, its erformance in terms of solution quality is virtually identical to that of the MA_IN_New genetic algorithm, which rovides the best results. Also, the DR_Back_Ex rule is comutationally quite efficient, and can quickly solve even quite large instances The imrovement rocedures In this section, the comutational results for the imrovement rocedures are resented. As mentioned in the revious section, the DR_Back_Ex disatching rule is the most effective of the considered heuristics, since it not only erforms well in terms of solution quality, but is also comutationally quite efficient. Therefore, in this section the imrovement rocedures are only alied to the DR_Back_Ex disatching rule, in order to see if it is ossible to further imrove the schedules generated by this heuristic. Due to the large comutational times that would be required, the imrovement rocedures were only alied on instances with u to 500 obs. Table 12 rovides the mean relative imrovement in obective function value rovided by the imrovement rocedures. The imrovement rocedures only rovide a minor imrovement. Nevertheless, the rocessing time variability has a maor imact on the effect of the imrovement rocedures. Indeed, when the variability is low, the imrovement rocedures are nearly ineffective. For instances with high variability, however, the relative imrovement increases significantly, although it is still usually less than 0.5%. The INTER and INS rocedures usually rovide a larger imrovement than the API and 3SW methods. Also, the relative imrovement given by the 3SW rocedure is higher than that rovided by the API method. Table 13 gives the effect of the T and R arameters on the relative imrovement rovided by the imrovement rocedures, for instances with 100 obs. The relative imrovement is quite minor for instances with T = 0.4 or 0.6, where little or no inserted idle time is required. The only excetion occurs when the due dates are quite close (R = 0.2), since in this case the INTER and INS rocedures do rovide a larger imrovement. The relative imrovement is visibly higher for the instances with 21

22 a higher roortion of early obs (T = 0.0 or 0.2), where larger amounts of inserted idle time are needed. The runtimes required by the imrovement rocedures (in seconds) are rovided in Table 14. The INTER rocedure is comutationally quite intensive, and can therefore only be alied to small or medium size instances. The INS rocedure is also somewhat demanding, but it is much faster than the INTER rocedure, and can be alied to large instances. The 3SW and API rocedures are quite fast, and can then be used even for quite large roblems. The INS and INTER rocedures are then recommended for small and medium size instances, while the INS method can be used even for large roblems. For extremely large instances, the 3SW rocedure can still be alied efficiently Comarison with otimum results In this section, the heuristic results are comared with the otimum obective function values, for instances with u to 20 obs. Table 15 gives the mean relative deviation from the otimum (%dev), calculated as (H O) / O 100, where H and O are the heuristic and the otimum obective function values, resectively. The ercentage number of times each heuristic generates an otimum schedule (%ot) is also rovided. In the following, let DR_Back_Ex + INS denote the DR_Back_Ex rule, followed by the alication of the INS imrovement rocedure. The MA_IN_New and DR_Back_Ex + INS heuristics erform extremely well. Indeed, the relative deviation from the otimum is quite small for these heuristics. Also, these rocedures rovide an otimum solution for a quite large number of the test instances. The DR_Back_Ex and EQTP_Back_Ex heuristics also rovide quite good results. As reviously mentioned, the incororation of the exchange checks imroved the heuristic results. This can also be clearly seen in Table 15, since the DR_Back_Ex and EQTP_Back_Ex rocedures outerform their counterarts that do not erform the exchange checks. Nevertheless, even the DR_Back and EQTP_Back heuristics noticeably outerform the forward scheduling EQTP rule. From Table 15, it can be seen that the heuristics are much closer to the otimum when the rocessing time variability is low. This is quite clear for the worst erforming heuristics, articularly for the EQTP rule. Indeed, for the low variability 22

23 instances, the EQTP heuristic is 3-7% above the otimum. When the variability is high, however, the deviation from the otimum is over 25%. This is in line with the results reviously resented in Table 9 and Table 12. Indeed, the imrovement over the EQTP rule rovided by the heuristic rocedures was much higher for the instances with high variability. Similarly, the imrovement rocedures rovided a larger imrovement when the rocessing time variability was high. This is in accordance with the results given in Table 15, since there is indeed more room for imrovement when the variability is high. Table 16 gives the effect of the T and R arameters on the relative deviation from the otimum, for instances with 20 obs. The relative deviation from the otimum is usually higher for instances with a low tardiness factor, where larger amounts of inserted idle time are required, articularly for the worst erforming heuristics. Again, this is in accordance with the results resented in Table 10 and Table 13. In fact, both the imrovement over the EQTP rule and the imrovement rovided by the imrovement rocedures were higher for instances with T = 0.0 or Results for the version with no idle time In this section, a summary of the main comutational results is resented for the version with no idle time. Table 17 rovides the mean relative imrovement in obective function value over the EQTP rule. The results in Table 17 are similar to those reviously resented for the version with inserted idle time. Indeed, the backward scheduling rules and the genetic algorithms clearly outerform the EQTP heuristic. The best erformance is again rovided by the genetic algorithms and the DR_Back_Ex rule. Moreover, the use of the DR riority index and / or the exchange checks rocedure lead to slightly better results. The rocessing time variability also once more has a significant effect on the imrovement given by the heuristic rocedures. In fact, the imrovement over the EQTP rule is again much higher for the high variability instances. Table 18 rovides the mean relative deviation from the otimum (%dev), as well as the ercentage number of times each heuristic generates an otimum schedule (%ot). Once more, the results in Table 18 are similar to those reviously given for the inserted idle time version. The genetic algorithms and the DR_Back_Ex + INS 23