Decentralization Cost in Two-Machine Job-shop Scheduling with Minimum Flow-time Objective

Transcription

1 Decentralization Cost in Two-Machine Job-shop Scheduling with Minimum Flow-time Objective Yossi Bukchin Eran Hanany May 07 Abstract Scheduling problems naturally arise in supply chains, ranging from manufacturing and business levels in a single firm through multi-organizational processes. In these environments, jobs are served by various resources. As scheduling choices are often decentralized, an inferior system solution may occur. In this paper we investigate a decentralized job-shop system, where jobs are processed utilizing two resources, each of which is minimizing their own flow-time objective. Analyzing the system as a non-cooperative game, we investigate the decentralization cost (DC), the ratio between the best Nash equilibrium and the centralized solution in terms of the system flow-time. We show that, even when both the system manager and each resource manager aim at minimizing the same (flow-time) objective, inefficiency can occur for small as well as large instances. For flow-shop settings we provide tight bounds on the maximal DC and insights regarding settings that lead to significant DCvalues.We namesuchsettingsasflow-shop misaligned. For job-shop settings, we present a typical structure of jobs that leads to large values of DC. We conclude that managers within such decentralized systems should implement strategies that overcome the inherent inefficiency. A simple mechanism is proposed to reduce the DC value, and its useful effectiveness is analyzed numerically. Introduction Scheduling problems deal with the allocation of scarce resources to tasks over time while optimizing one or more objectives (Pinedo 00). Such problems can be found in manufacturing and service Department of Industrial Engineering, Tel Aviv University, Tel Aviv 69978, Israel. bukchin@eng.tau.ac.il Department of Industrial Engineering, Tel Aviv University, Tel Aviv 69978, Israel. hananye@post.tau.ac.il

2 environments. Most traditional scheduling literature considers a centralized system with a single decision maker (DM) and suggests optimization models and heuristics to minimize objectives such as makespan, flow-time, tardiness and others. In reality, however, quite often several DMs are involved in the scheduling decision process, each aiming at maximizing their own utility. Decisions may be taken by local DMs since a central DM (system manager) either () does not exist; () does not have sufficient information; () is not able to provide an optimal solution due to problem scalability; and/or () does not see the necessity in enforcing a centralized solution. A decentralized process, whereby decisions made by some or all DMs are influenced by those made by other DMs, can be modeled using game theory approaches. The literature to date combining multiple DMs and scheduling decisions considers single or parallel (identical, related or unrelated) machine/resources, where DMs could be either jobs or machines (this literature is further discussed later in the Introduction). However, in many real world situations the scheduling problem is more general. In this paper we consider a decentralized setting of multiple DMs in a job-shop environment, whereby each DM manages a different resource serving multiple jobs, and each job passes along a predetermined route consisting of several resources. In doing so we provide the first model and analysis of decentralized job-shop settings. To address these issues we take a non-cooperative game theoretic approach and analyze a decentralized setting where the centralized objective is total flow-time minimization. The scheduling decisions on each of two resources are taken by the corresponding resource manager. The objective function of each resource manager is the same type as the objective function of the system, namely, the flow-time objective on their own resource. The flow-time for each resource is defined bythesumofflow-times over all jobs on this resource. The flow-time of a job on a resource is defined by the difference between the completion time of the job on this resource and the ready time (release time) of the job, i.e., the time the job was available at the resource site. We consider a static problem, in which all jobs are simultaneously ready at their respective initial machine at time zero. We also consider the special case of a flow-shop setting, in which all jobs have the same route. All the parameters of the model are commonly known to the DMs. As an example for such a setting, consider a construction project, in which multiple apartments (jobs) are built by several subcontractors (resources/machines). Each subcontractor is responsible for a specific operation, such as foundation & external activities, floors and walls, cladding, electric installation, etc. The project manager is interested in the earliest sum of completion times of all apartments to be delivered to the customers (sum of job flow-time). To this end, the manager

3 may urge each subcontractor, possibly by proposing proper incentives, to complete their part of the operation in each apartment as soon as possible (machine flow-time), where the duration is measured from the point in time where the apartment was available to the subcontractor (ready time). Also, in order to aid the subcontractors in making their decisions, the manager shares all the relevant information (job routing and processing times) among them. The research question then concerns the extent to which each subcontractor minimizing the flow-time would also minimize the total flow-time of the system. The objective of the paper is to analyze the decentralization cost (DC), i.e., the ratio between the best decentralized Nash equilibrium solution and the centralized optimal solution in the job-shop environment. We show that although the system manager and each of the resource managers aim in minimizing the same type of objective function, namely flow-time, the DC can be significant. We define the maximal DC over all possible processing durations for a given set of jobs. For flow-shop settings we provide tight bounds on the maximal DC and intuition for the sources of inefficiency. More specifically, we show that large DC values are obtained when the job processing durations are inversely related on the two machines, close to each other on machine and significantly different on machine. Wenamesuchsettingsasflow-shop misaligned, and provide insights for why they lead to inefficiency. The DC obtained may be as high as.. Flow-shop misaligned settings may occur in practice when variability of processing durations is low on machine andhighonmachine. For job-shop settings, we present a typical structure of jobs that leads to large values of DC. According to this structure half of the jobs start at machine and continue to machine, and the other half follow the opposite flow. The job durations are almost identical, however slightly higher on the second machine in the processing order. We show analytically that the obtained DC value may be as high as.5. Following this analysis we are able to draw managerial insights on the merit of intervention by a centralized DM, or the implementation of an improving mechanism. Managers within such decentralized systems may wish to implement strategies that overcome the inherent inefficiency. To this end, we suggest a simple improving mechanism and show numerically that it significantly reduces the DC values, possibly yielding the centralized solution in some inefficient cases that may occur when no intervention is applied. In the remainder of the Introduction we discuss the current literature on scheduling games. The Similar concepts named price of anarchy and price of stability are commonly used in the literature (see, e.g., Heydenreich et al. 007 and Anshelevich et al. 00), where the latter is the same as the DC, while the former refers to the worst Nash equilibrium rather than the best one as in the DC. For arguments justifying the DC definition see Bukchin and Hanany (007).

4 first papers that combined game theory and scheduling were based on cooperative games, analyzing the possibility and benefit of cooperation among players competing on a common resource. Curiel et al. (989) initiated this line of research by proposing the single machine sequencing game, in which several jobs, each belonging to a different agent, must be processed on a single machine. The authors suggest that an allocation rule induced by a cooperative game can allocate the cost saving that results from coalitions decisions to move from the initial sequence to an optimal one. This model has been extended to consider ready times of the jobs (Hamers et al. 995), due dates (Borm et al. 00), multiple parallel identical machines (Hamers et al. 999), more general interchanges (Slikker 006) and outsourcing operations to a third party (Aydinliyim and Vairaktarakis 00a). Other cooperative game scheduling models are surveyed in Borm et al. (00), Curiel et al. (00), Chen and Hall (007), Hall and Liu (008) and Aydinliyim and Vairaktarakis (00b). There has also been a growing literature on non-cooperative games in supply chain (Cachon 00) and scheduling (Heydenreich et al. 007) settings, as it is natural to assume that DMs may not cooperate with each other when making decisions. In the majority of the literature, the jobs or customers (job owners) are modeled as players, and the makespan objective is considered in a complete information setting. Koutsoupias and Papadimitriou (999) and others (see e.g., Nisan et al. 007) analyzed scheduling environments by means of congestion games (Rosenthal 97), whereby each job is represented by a DM choosing a machine from a set of related parallel machines and all jobs are processed and released as a batch. Under the objective of minimum makespan and complete information they provided bounds for the price of anarchy, theratioof the worst equilibrium to the optimal solution. Analogue bounds were provided by Christodoulou et al. (00) and Immorlica et al. (005) and others for sequencing settings with unrelated/related machines where jobs are released immediately upon processing completion. Bukchin and Hanany (007) analyzed a different setting whereby each DM owns a set of jobs and may choose between processing on a common in-house resource or on a slower but uncapacitated subcontractor, i.e., a setting of related machines. Under the minimum individual and system total flow-time objective they investigated the DC, provided bounds on it, found empirically DC values of up to around 5 and proposed a coordinating mechanism based solely on scheduling decisions, i.e., with no monetary transfers between the players. Agussurja et al. (009) analyze the price of stability for identical machines. Correa and Queyranne (00) provide bounds for the price of anarchy under the weighted completion time objective in a setting where DMs, each having a single job, choose between related machines each only capable of processing a subset of jobs. Hamers et al. (0)

5 analyze the price of anarchy when agents owning sets of jobs process them on identical, parallel machines under the sum of completion times objective. Lee et al. (0) consider coordination mechanisms for scheduling on parallel machines of job agents, each holding one job, and provide bounds for the price of anarchy under various objectives. Vairaktarakis (0) consider a set of manufacturers, all of which can subcontract part of their workload to a third party, and develop pure Nash equilibria schedules under the makespan objective and three distinct protocols for production. Environments in which agents compete on a common resource were also investigated via auction games, in which the resource owner allocates time slots to the agents based on their bidding (see e.g., Kutanoglu and Wu 999, Wellman et al. 00, Dewan and Joshi 00 and Reeves et al. 005). For further discussion of literature on non-cooperative scheduling games see Hall and Liu (008) and Aydinliyim and Vairaktarakis (00b). Nisan and Ronen (00) initiated a line of research on mechanism design of centralized scheduling in settings where DMs have private information and the proposed mechanism uses money transfers between the DMs to induce truthfull information provision to the central manager. A mechanism design approach achieving truth revelation and/or non-costly money transfers to the central manager (i.e., money transfers that add up to zero) is not always possible in general (Jéhiel and Moldovanu 00), specifically when the private information is multi-dimensional such as when DMs own more than one job (Hain and Mitra 00 proposed a non-costly mechanism in a single machine setting with DMs each owning a single job). The paper is organized as follows. In Section we describe our model of decentralized job-shop scheduling, including several examples demonstrating the properties of Nash equilibrium outcomes compared to centralized optimal outcomes. In Section we analyze the DC for flow-shop settings. In Section we undertake a similar analysis for job-shop settings and provide a significant lower bound, suggesting that the DC should not be ignored. We complement the analysis with a numerical study of the DC. In Section 5 we provide a simple mechanism towards solving the problems raised in previous sections. Section 6 concludes. Modeling decentralized minimum flow-time scheduling Consider jobs { } to be processed in serial on two machines = { }, with processing durations ( ) 0. As in job-shop scheduling settings, each job may require its own different route and is available at the system at time zero to be processed on the initial 5

6 machine. Note that for concreteness we refer throughout the paper to job processing on machines, but our analysis applies more generally also to service contexts, in which servers or capacitated resources operate instead of machines. We will refer to any job for which machine is first in the job s processing order as a job belonging to machine (or alternatively say that machine owns the job). Denote by the number of jobs belonging to machine, thus P =. Denote by ( ) the machine preceding in job s processing order, with the convention that ( ) =0when starts at (belongs to). Each machine determines its own processing sequence permutation independently from the other machine. Denote by ( ) { } the job sequenced to be processed in the th position on machine, and correspondingly { } denotes the position in the sequence on this machine for job, satisfying ( )=. Asequence ( ) for all and is denoted by. Given these sequences, each job starts its processing on machine as soon as it arrives to the machine and its preceding job on this machine, ( ), completed its processing. Upon processing completion on machine, the job immediately arrives to the next machine in its processing order or leaves the system. Therefore the starting time, ( ), ofjob on machine in sequence is defined recursively as ( ) max{ ( ( )) + ( ( )) ( ( ) )+ ( ( ) )}, () with initial conditions ( 0) = ( 0) = (0 )= (0 )= (0 )=0for all and. The first term within this maximum is the completion time of job on the machine preceding in this job s processing order (equal to zero when is the first machine) and the second term is the completion time of the job preceding job on machine (equal to zero when is the first job). Each machine determines its sequence with the objective of minimizing its own total flow-time, i.e., the sum of completion times minus ready times on machine, P = ( ( )+ ( )) P = ( ( ( )) + ( ( ))), () where each machine takes the other machine s sequence as given, thus generating a Nash equilibrium sequence. It follows that equilibrium corresponds to simultaneous scheduling of all jobs on each machine, with interdependent ready times, to minimize flow-time. For the special case of flow-shop scheduling settings, the machine processing order is identical for all jobs and can be assumed, without loss of generality, to be machine followed by machine, i.e., ( ) = 0 and ( ) = for all jobs. In this case, there is asymmetry in the sense that the 6

7 flow-time of machine, which depends on its ready times, is affected by the sequencing decision of machine, but not vice versa. Therefore an equilibrium outcome may be constructed as follows: an optimal sequence is first found for machine irrespective of machine, and then an optimal sequence is computed for machine given the ready times determined by the sequence on machine. Note that the second part of this algorithm is still a hard problem, and that finding the optimal centralized sequence is hard as well (Lenstra et al. 977). The expressions for the starting time ( ) of job on machine and machine s total flow-time, given by () and (), simplify to ( ) =max{ ( ) + ( ) ( ( ) )+ ( ( ) )} () and = P = ( ( )+ ( )) P = ( ( ) + ( )). () In general, we measure efficiency by the system flow-time, +,thusanoptimal solution is a two machine job-shop schedule minimizing total flow-time. The decentralization cost (DC) (Bukchin and Hanany 007) is the ratio between the minimal among all Nash equilibria and the minimal among all possible sequences. To illustrate, consider two jobs belonging to different machines, i.e., = =,implying ( ) = ( ) = 0, ( ) = and ( ) =. There are four possible sequences, each corresponding to a pair of job permutations. Note however that the sequence defined by ( ) =, ( ) =, ( ) = and ( ) = is infeasible because each job waits to start its processing on the first machine after the completion of the other job, which is waiting on the other machine, leading to a deadlock. The remaining three sequences are feasible. The upper two rows of Figure depicttheflow-times of each schedule and the Gantt chart of the unique equilibrium schedule for two instances of the problem, where the processing times are given in the two matrices on the left (the notation ( ) in the title of the figure refers to ( )). The rows and the columns in each matrix of flow-times correspond to the vectors [ ( ) ] and [ ( ) ], respectively, and each cell corresponds to the vector ( ). In the first row instance in Figure, each machine processes firstitsownjobandonlythen the other machine s job, while in the second row instance, machine prefers to process first the other machine s job, only then to process its own job. Note that for both instances, the unique equilibrium outcome, marked in the matrix of flow-times with bold letters, is also the optimal schedule, thus leading to DC value of. As a further illustration, consider two jobs belonging to 7

8 the same machine, i.e. a flow-shop setting where ( ) = 0, ( ) = for each, and =, =0. Now all four sequences are feasible. The third row of Figure depicts one instance of this problem (again, the notation ( 0) in the title of the figure refers to ( )). Here the unique equilibrium outcome, marked again in the flow-time matrix with bold letters, is not an optimal schedule, thus leading to DC value of 9 8. ( )= 6 =\ = [ ] [ ] [ ] [ ] 6 M M ( )= 8 =\ = [ ] [ ] [ ] [ ] 6 6 M M ( )= 6 =\ = [ ] [ ] [ ] [ ] 7 7 Figure : Instances with two jobs: (-), (-) and (-0) Analysis of flow-shop settings M M In this section we provide tight bounds on the maximal DC in flow-shop settings. Consider the following intuition for sources of inefficiency: in equilibrium, machine always follows a Shortest Processing Time (SPT) order and machine responds to the resulting completion times on machine, which become its own ready times. The objective of machine is to minimize its own flow-time, or equivalently minimize the sum of starting times of the jobs on this machine. Keeping the same order as on machine reduces the starting time of the earlier sequenced jobs on machine, but ignores the processing durations on machine. To the contrary, taking into account the processing durations while ignoring the ready times would result in an SPT order on machine. Theoptimal sequence trade-offs these two considerations. In case the processing durations have the same order on both machines, these two extreme sequencing options coincide and lead to the optimal objective value, i.e. DC value of (this follows from a simple job interchange argument). However, if there is an inverse relation between the processing durations of the machines (i.e., increasing sequence 8

9 on machine and decreasing sequence on machine ), the above mentioned trade-off becomes significant: keeping the same order as on machine will yield additional flow-times due to the Longest Processing Time (LPT) order on machine, while the option of sequencing an SPT order on machine will result in relatively late starting times for the jobs on machine, as dictated by the completion times on machine. To conclude this part of the discussion, we see that an inverse order of the processing durations on both machines potentially leads to relatively large values of the DC. Having discussed the order of processing durations, let us now address more closely their structure on each machine. In an optimal solution, a large DC value implies a non-spt order on machine, otherwise the equilibrium is optimal in contradiction to the large DC value. Therefore, considering a shift from an equilibrium schedule to an optimal one, one may expect a reduced flow-time on machine in the expense of some additional flow-time on machine. A large value of the DC will be obtained when the flow-time reduction on machine is significantly larger than the flow-time increase on machine. A small additional flow-time on machine will be obtained when the processing durations on machine are almost identical, in which case all schedules yield almost the same flow-time. To the contrary, on machine one may expect significant differences among the processing durations for achieving a large flow-time reduction, as similar processing durations on machine imply similar flow-time values for all schedules on machine. To conclude, based on the above intuition, we may expect high DC values in the settings defined as follows: Definition Flow-shop processing durations are said to be flow-shop misaligned if the processing durations are inversely related on the two machines, close to each other on machine and significantly different on machine. To gain some insight on flow-shop misaligned settings and the corresponding DC values, let us start by considering the case of two jobs. For this case we have the following result: Theorem When flow-shop processing durations with two jobs are flow-shop misaligned such that there exists 0 for which ( ) = for { }, ( ) = and ( ) =, as 0, the DC approaches the maximal value of 9 7. Proof. All proofs are collected in the Appendix. 9

10 As shown in Figure, the maximal DC is generated in the limit with a unique optimal solution (Figure (a)) and two equilibria (Figures (b) and (c)). The processing durations and the resulting schedules follow the three intuitive principles defining flow-shop misaligned settings. Also, the first equilibrium follows an SPT order on machine while starting the first job at time, andthesecond equilibrium follows an LPT order on machine, however starts the first job earlier. In the optimal solution, machine chooses an LPT order, however the additional flow-time (as compared with the equilibria solution) is negligible, while the flow-time saving on the machine is relatively large (two time units). The question is then, why we should expect multiple equilibria at the maximal DC. To answer this question, let us assume that only one of the two equilibrium sequences in Figure istheuniqueequilibrium. Inthiscase,onecanshow(seetheproofofTheorem)thattheDC can be increased using local changes in the durations of the jobs, in particular, by decreasing the duration of the last job on machine in this unique equilibrium. However, when both Eq and Eq exist, any change in the processing durations which increases the flow-time ratio (equilibrium over optimal) of one sequence so that it is strictly higher than the ratio for the other sequence, leaves the latter as a single equilibrium, which, by the arguments above, cannot generate the maximal DC. M M M M M M (a) Optimal (b) Eq (c) Eq Figure : Gantt charts illustrating the maximal DC for two jobs The next theorem identifies more general flow-shop misaligned settings, for which we can characterize a closed form expression for the DC. Note that the misaligned processing durations of Theorem are the special case where =. Theorem When flow-shop processing durations are flow-shop misaligned such that there exist and 0 for which ( ) = for all, ( ) = for +,and ( ) = + P = + for +, as 0, the DC approaches ( ) ( + + )+ P = + ( + ( )) ( +)+P = + ( +)( +). To understand the principles of the misaligned flow-shop setting in Theorem, let us consider 0

11 an example of four jobs. For this case, the processing durations of Theorem are presented in Figure. The top Gantt chart presents the system optimal schedule, and the remaining charts correspond to each value of, the number of distinct job processing duration values on machine. Clearly, each set of durations provides a possibly different value for the DC, while the maximal value (in this case, =) provides a lower bound on the maximal DC. We can see that the three intuitive principles defining flow-shop misaligned settings are applied in all sets of durations. WhenlookingattheGanttchartofeach we observe multiple sequences, each having a different initial job on machine. The proof of Theorem establishes that each such sequence is optimal for machine among all sequences starting with the given initial job. Consequently, any best response of machine must be one of these sequences. Furthermore, we show that there are multiple best responses, each corresponding to a different equilibrium sequence (as depicted in Figure ). The different schedules demonstrate the intuitive arguments stated above, as in some of the schedules an early start of the first job is preferred, resulting in an inefficient non-spt order of the jobs on machine (see for example the schedule of = 0 =), while in others, the SPT order is preferred, leading to a relatively late start of the first job (see for example the schedule of = 0 =). As in the discussion following Theorem for the case of two jobs, we should expect multiple equilibria for the durations of the maximal DC for any given number of jobs,. When considering local changes in job durations with more than one equilibrium, the system flow-time may need to increase simultaneously for all equilibrium sequences so that all of them will remain equilibria after the increase. When the number of equilibria is large enough, such a simultaneous increase may become impossible, which would result in the maximal DC. This maximal number of equilibria depends on the number of job durations that are allowed to be locally changed. Figure demonstrates the structure of such equilibria, based on the processing durations in the theorem. In particular, when the duration of only one job can be changed (job on machine, and =), two equilibria generate the DC; when two job durations can be changed (jobs and on machine and =), three equilibria are observed; and when three job durations can be changed (jobs and on machine and =), four equilibria are observed. The structure of processing durations and their resulting equilibrium schedules are very informative for understanding the potential inefficiency that may generate in the system. In fact, we have the following: Conjecture The maximal DC among all possible instances of jobs in a flow-shop setting is achieved using the flow-shop misaligned processing durations in Theorem. With these durations

12 the maximal DC approaches max ( ). M opt M k= M k= M k= M k=, t(j,)=, t(j,)={,,,+/} M j'= M j'= M j'= (Eq.) M j'= (Eq.) M k=, t(j,)=, t(j,)={,,+/,+/+/} M j'= M j'= (Eq.) M j'= (Eq.) M j'= (Eq.) M k=, t(j,)=, t(j,)={,+,++/,++/+/} M j'= (Eq.) M j'= (Eq.) M j'= (Eq.) M j'= (Eq.) Lower bound =max{ = = = = =max{ 58 6 = = = = = = } = 8 = 75 = = = = = = = Figure : Gantt charts illustrating lower bound on maximal DC = }

13 Further to the results and intuitive arguments presented above, our conjecture may be justified with numerical experimentation. To this end, we applied a simulation based, meta-heuristic optimization method, specifically the Cross Entropy (CE) method (Rubinstein and Kroese, 00). Taking the processing durations as variables for maximizing the DC, each iteration of this heuristic generates a sample of processing durations according to Normal distributions with job-machine specific expectations and variances. For each instance in this sample, the DC value is computed by solving two mixed-integer linear-programming (MILP) formulations (using the CPLEX solver), one for the optimal system flow-time and the other for the system flow-time in equilibrium. The first formulation minimizes the flow-time of a two machines flow-shop system, while the second solves thesinglemachineflow-time minimization problem for given ready times determined by an SPT order on machine. At the end of the iteration, the parameters of the Normal distributions to be used for generating the processing durations in the next iteration are updated to be the mean and variance coming from a sub-sample with the highest DC values. This allowed obtaining the highest DC values and corresponding processing durations for up to 8 jobs. 6 5 M M 5.5 M M.5.5 M M t jm.5 t jm Job index (j) Job index (j) 5 Job index (j) DC CE =.066, DC Th =.077 (a) Three jobs DC CE =.685, DC Th =.750 (b) Four jobs DC CE =.70, DC Th =.80 (c) Five jobs M M M M M M Job index (j) Job index (j) Job index (j) DC CE =.860, DC Th =.07 (d) Six jobs DC CE =.76, DC Th =.8 (e) Seven jobs DC CE =.00, DC Th =.086 (f) Eight jobs Figure : Cross Entropy (CE) processing duration results and DC vs. DC according to Theorem The numeric results are reported in Figure, with bars showing the processing durations sorted in a descending order according machine, and DC values below the graphs comparing the results of the CE method to the results of Theorem. Evidently, the resulting processing durations

14 are flow-shop misaligned, i.e., inversely related on the two machines, close to each other on machine and significantly different on machine. The DC values are very close to those obtained in Theorem. Moreover, no solution was found to contradict Conjecture, according to which the processing durations of Theorem are those of the maximal DC. Although the conjectured maximal DC in Theorem approaches when goes to infinity, the value of the DC is quite high even for large values of the number of jobs,. For example, numeric calculation shows that it is equal to 0876 for =5000. Figure 5 illustrates the DC as a function of for 00, and of the ratio within the DC as a function of for = 5000 and. Figure 5: The conjectured maximal DC as a function of (a) the number of jobs, and (b) the parameter for = 5000 jobs. Analysis of job-shop settings In this section we analyze general job-shop settings. Considering first the case of two jobs, recall that the instances depicted in the first two rows of Figure in Section possess an optimal equilibrium, thus leading to DC value of. This property is not special to the processing durations chosen in these examples, as shown in the next theorem. Theorem For all job-shop settings with two jobs, where =for each, thedcequals. As shown in Theorem, the conclusion of Theorem does not hold for a flow-shop setting with two jobs. Also, Theorem fails if we add a single job to the setting of the theorem, as demonstrated in the example in Figure 6. Here machine owns jobs and machine owns job. The top and bottom Gantt charts show the unique optimal schedule and the unique equilibrium schedule, respectively (the corresponding flow-times are marked in the matrix below the charts with bold letters). The DC obtained for this example is equal to. We now present a general structure of jobs that leads to large values of DC in job-shop settings. This analysis provides a lower bound on the maximal DC. The general structure involves jobs with

15 small differences in processing durations (i.e., the variance in processing durations is small relative to their mean) such that the processing duration is smaller on the initial machine than on the later machine, possibly justified due to some setup or handling time. Under the additional assumption that the processing durations are identical on the initial machine and identical on the later machine, the following theorem provides the resulting lower bound on the maximal DC. ( )= M M M M =\ = [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Figure 6: Instance with three jobs ( ) and = 5 =. Theorem A lower bound on the maximal DC among all possible instances of a job-shop setting with jobs approaches b c + (b c +), where each machine owns (almost) half of the jobs, i.e., = b c, = b c, and the processing durations are slightly smaller on the initial machine than on the later machine but are otherwise identical, i.e., for all and and some 0 and 0 +, ( ) = if is the initial machine for job, otherwise ( ) =. The proof establishes the result by considering two sequence with a typical structure, one for the equilibrium (Figure 7) and one for the optimum (Figures 8). In the equilibrium sequence each machine sequences the jobs in a SPT order, starting with their own jobs followed by the other 5

16 machine s jobs. This creates large delays to the each job due to the sequencing by the machine processing it last. By comparison, the optimum is much more efficient because each job is processed immediately by the second machine. Although this creates a small increase in flow-times for the machine processing first, but decreases substantially the flow-time for the other machine, thus reducing the overall system flow-time. M n- n n+ n+ n M n+ n+ n- n n Figure 7: Type sequence M n+ n+ n n M n+ n+ n+ n n Figure 8: Type sequence The value of the lower bound increases with the number jobs, attains relatively high values even for a small number of jobs and asymptotes to an upper bound of 5, as illustrated in Figure 9. Figure 9: Two instances with two jobs ( ) One can show using simulation that similar results to that of Theorem are obtained when the processing durations are not identical but are instead distributed with some variance while still being smaller on the initial machine than on the later machine. Next, the DC is analyzed empirically via a wide set of experiments. The purpose of the experimentation is to study some characteristics of the DC, such as its average value and how it is affected by the problem parameters. For each instance, the centralized optimal solution(s) as well 6

17 as all pure Nash equilibria were calculated via exhaustive search. Due to scalability limitations, only small sized instances were studied (for example, in a 6-job instance, the number of cells in the game matrix is (6!), so the number of comparisons needed for computing all Nash equilibria is (6!) ' ). Still such settings may be found within real life organizations. The analysis includes instances with three, four and five jobs. We distinguish between various types of instances according to the symbol ( ), where denote the number of jobs belonging to machine, respectively (i.e., jobs starting processing on the respective machine, then proceeding to the other machine). The following five types of instances were examined: ( ), ( ), ( ), ( ) and ( ). Note that due to symmetry, these types contain all possible partitions for three to five jobs problem. For each type, 500 instances were generated with random processing durations taken from a uniform distribution between 0 and. The results were analyzed according to two factors: the size of the instance in terms of the total number of jobs, and the degree of symmetry in the instance in terms of the number of jobs each machine owns. In asymmetric instance types one machine owns significantly more jobs than the other, hence is expected to have more power in the game, while in symmetric instance types both machines own about the same number of jobs, hence we expect both to have similar amount of power. We accordingly consider - and5-job instances of types ( ) and ( ) as asymmetric, and types ( ) and ( ) as symmetric. We found that around 60% of the instances are inefficient in the sense that the best decentralized solution (equilibrium) gives larger flow-time than that of the optimal solution (DC ). This percentage value increases with the instance size (see Figure 0(a)), as 8% of the 5-job instances were inefficient. Consequently, we may conjecture that as the instance size increases, the optimal centralized solution will not likely be an equilibrium. The proportion of inefficient instances also increases as the problem becomes more symmetric, as demonstrated in Figure 0(b). This may be explained by the fact that when one machine has much more power than the other, it will have higher interest in the system global objective, leading to a solution minimizing simultaneously their own cost and the system cost. Among the inefficient problems, the average DC is 0, with a maximal value of. The average and maximal DC values increase as instances become more symmetric, as demonstrated in Figure 0(c) and 8(d). Along with the previous result, we may conclude that symmetric instances, in which players are equally strong, lead to a higher likelihood of inefficiency and a higher DC. 7

18 (a) (b) (c) (d) (e) (f) Figure 0: Graphical demonstration of experiment results Additional results indicate that a unique optimal solution was obtained in 79% of the instances, and this value decreases with the instance size, as demonstrated in Figure 0(e). This result is quite expected since the feasible set increases with the instance size, thus there is higher likelihood of finding multiple optima. When analyzing the existence of pure equilibria, the results show that 8% of the instances have a single unique equilibrium, 7% have multiple equilibria and % have 8

19 no equilibrium. The percentage of instances with a unique equilibrium decreases with the instance size, as demonstrated in Figure 0(f). Interestingly, most of this decrease is compensated by an increase in the percentage of instances with no pure equilibrium. Another interesting result relates to the equilibrium characteristics. In around 9% of the inefficient instances, exactly one of the machines improves its utility versus the optimal solution. However, in the remaining instances, a lose-lose situation is obtained in the sense that both machines worsen their utility compared to the optimal solution. This percentage increases with the instance size, as demonstrated in Figure (a). Another observation, as demonstrated in Figure (b), indicates that when some machine owns significantly more jobs than the other, this machine improves its utility equilibrium significantly more often than the other. This result supports the reasoning above regarding the relative advantage of having more power by owning more jobs (at the expense of the other machine). (a) (b) Figure : Additional demonstration of experiment results 5 A simple mechanism for coordination In this section we investigate a simple mechanism for improving the system efficiency and reducing the DC. In discussing possible mechanisms, one should note that cost sharing based on the system flow-time may be potentially used to eliminate the inefficiency. However, such a mechanism requires a binding agreement that involves monetary transfers between the DMs after any sequencing decision they might make. Such an agreement may not be easy to implement in some environments. Therefore we propose instead a mechanism based on penalizing a machine found to have a simple 9

20 possible change within its chosen sequence, leading to improvement for the system. By a simple change we mean a swap of only two jobs in the machine s sequence. We call it a simple change because finding whether such a change exists is a computation task of polynomial complexity as a function of the number of jobs in the system. Whenever such a swap is found, the machine is penalized. Suchpenaltiescanbeachievedbypostponing the departure of the jobs from the machine, resulting in a very high flow-time for this machine. Alternatively, monetary penalties may be imposed on the manager responsible for the machine. Note that this penalty method has the advantage that it is easy to implement, and moreover only takes the role of a threat which is never actually used. Despite the heuristic nature of this mechanism, it can be shown that it completely solves the inefficiency existing in the settings investigated in Sections and. Although these settings lead to significantly high DC values, the mechanism reduces the DC to for any number of jobs possible in the system. We now examine the performance of the proposed simple mechanism on various instances, relying on the same structure of numeric experimenation conducted in Section. The mechanism was applied to 00 instances of each type. As shown in Figure, the percentage of inefficient instances increases up to 78% without the mechanism, and up to only 9% when the mechanism is applied. Namely, the proportion of inefficient instances increases both with and without the mechanism, however, the increase rate diminishes significantly when the mechanism is applied. Thus in general, the mechanism performs well in reducing the DC. Moreover, although one may expect that the mechanism increases the DC for some instances, this was not observed in the numerical study, suggesting that it is not likely to happen. We may conclude that the state of the system can be significantly improved when using a simple polynomial mechanism. 6 Concluding remarks This paper investigates the multiple jobs, two machines job-shop problem modeled as a noncooperative game, in which the sequencing decisions for each machine are taken independently by the respective machine owner. Such situations may be found in the operational level of manufacturing and service systems, where jobs or projects are processed by different machines or organization resources/functions, and each resource manager can determine the job processing sequences within their territory. Decentralized decisions are expected to provide inferior solutions compared to the 0

21 Proportion of inefficient instances No mechanism Mechanism number of jobs Figure : Figure : Percentage of inefficient instances centralized environment, mostly because the incentives of each decision maker are not necessarily aligned with the centralized objective. We examined the potential system loss due to the decentralized decision making process, which is expressed by the decentralization cost (DC). We analyzed a setting where single players as well as the system aim at minimizing the same flow-time objective and showed that the system loss can be still significant. After presenting our model and demonstrating the optimal (centralized) and equilibrium (decentralized) schedules using small examples, we provided tight bounds on the maximal DC for flow-shop settings and insights into the sources of significant DC values. Our bounds on the maximal DC show high values for small instances as well as for large scale systems (above 08 for a job instance), with a maximum value of ' 06 for 0 jobs. For job-shop settings we showed that system efficiency can be guaranteed only for the smallest instance of two jobs that start their processing on different machines and proceed to the other machine. For more than two jobs, we provided a lower bound on the maximal DC using a symmetric system with an arbitrary even number of jobs with equal processing time, half belonging to machine and the other half belonging to machine. The DC obtained for this system, which is a lower bound on the maximal DC expected for general instances, approaches 5 asymptotically as the number of jobs increases to infinity, with relatively high values for small instances (value of for 8 jobs). We then developed a simple mechanism based on pairwise exchange for improving the efficiency of the system. The mechanism coordinates the job-shop and the flow-shop settings for which our bounds were developed, and is shown numerically to be highly efficient in general job-shop problems. A wide experimentation of relatively small problems provided some insights regarding the DC value. We found that most problems are inefficient and the DC value increases with the instance size. An average DC of 0 was obtained, with a maximal value of. When comparing

22 symmetric versus asymmetric instances in terms of the number of jobs owned by each machine, we found that symmetry leads to higher likelihood of inefficiency and higher DC values. Therefore a symmetric system in which machines have roughly the same power will perform in lower efficiency than a system with a dominant player. The reason may be that under asymmetry, the interest of the dominant player will be more aligned with the system objective in the expense of the other player. When comparing the equilibrium solution to the optimal one, we found that in 9% of the cases exactly one of the machines improves its utility in equilibrium as compared to the optimal solution, where the improving machine is likely to be the one with the larger number of jobs in asymmetric instances. The remaining instances show a lose-lose situation, where both machines worsen their utility. Appendix Proof of Theorem. ( ) max Given any job durations ( ), let max ( ) denote the maximal ratio among all pairs of Nash equilibrium and optimum. Let max max be the maximal over all job durations ( ). We first find max max andthenuseittodetermine the limit maximal value of DC. Since inefficiency is possible (see the example in the third row of Figure ), max max must be strictly greater than. For the remainder of the proof, fix job durations ( ) that generate max max. If each had a sequence on machine as in some, then the best response of machine in each would generate,implyingmax max =, a contradiction. Since only two sequences are possible for machine, all must have the same sequence on machine, which is different from the sequence on machine in some. Suppose ( ) 6= ( ). Then machine has an SPT order in all and an LPT order in all. Therefore, for the long job on machine, thecoefficient of its duration in for all is for that duration value or near lower values, which is greater than or equal to the corresponding coefficient in for all (as is the maximal possible value for any coefficient). Thus, decreasing the duration of the long job on machine increases, a contradiction. Therefore, the durations on machine must be equal. Without loss of generality assume that = =and, thus =for all. Suppose that a single generates ( ) max. Then, for the last job on machine, thecoefficient of its duration in is for that duration value or near lower values, which is less than or equal to the corresponding coefficient in (because the coefficients of job durations on machine

23 are always positive). Thus, decreasing the duration of the last job on machine increases, contradicting max ( ) =max max. Therefore there are exactly two equilibria generating max max,namely =[[ ] [ ]] and =[[ ] [ ]]. By the definition of a best response, =.Consequently = because =. If, then =and = +, and these are the minimal possible values for and, respectively, so is optimal and max max =, a contradiction. Therefore, and = +(+ + )= =( + )+( ).Thisimplies = +. It follows that = =6+ and =(+ ) + ( +max(0 ) + )=5+ +max(0 ). Maximizing s.t. 0 results uniquely in =,consequently = = =, = and max max = 9 7. It remains to show that this is exactly the value to which the maximal DC approaches when ( ) = ( ) =, ( ) = for some 0 and ( ) =, as 0. To see this, note that for small 0, the equilibria obtained above continue to hold, while the uniquely efficient equilibrium [[ ] [ ]] is eliminated. Since the existence of efficient equilibria is the only difference between the maximal DC and max max,as 0, the maximal DC approaches max max. ProofofTheorem. Machine strictly prefers an SPT sequencing of all jobs to minimize its flow-time,, leading to the sequence defined by ( ) = +for. For simplicity of presentation we establish the remainder of the proof with = 0, as similar arguments are valid when 0 + (the only role of 0 is to eliminate Nash equilibria that are different from those obtained below, with almost no change in the system flow-time). Note that for machine the starting time is ( ) = for all. Taking this sequence imposed by machine as given, we would like to obtain an optimal sequence for machine. To this end we augment with particular sequences for this machine, denoted 0 for 0, such that 0 starts with job 0 and all other jobs follow SPT. We will prove for each 0 that 0 is optimal for machine among all sequences starting with job 0.Toprovethis,itwillbesufficient to show that all jobs after 0 are scheduled with no idle times. This is true because the sum of ready times on machine, P = ( ( ) + ( )), is the same for all sequences, so minimizing machine s total flow-time, (see eq. ()), is equivalent to minimizing the sum of completion times on machine, P = ( ( ) + ( )). Since all sequences starting with job 0 on machine have the same

24 ( 0 ), SPT sequencing of all other jobs minimizes the sum of completion times on machine. Consider 0 defined for 0 +by ( ) = 0 +, and for + 0 defined by 0 ( ) = 0 = and 0 +, where the lower two row cases are relevant only for some values of and 0. The Gantt charts in Figure illustrate these sequences for =and. Consider first 0 for 0 +. Notethat 0 ( 0 ) = 0 +. For the job 0 + placed in the th position for =, the starting time is 0 ( 0 ( ) ) = max{ 0 ( 0 + )+ ( 0 + ) 0 ( 0 ( ) )+ ( 0 ( ) )} =max{( ( 0 +))+ ( 0 + )+} = 0 +, i.e., with no idle time. By induction, this holds also for all jobs placed in 0. In particular, job in the ( 0 ) th position starts at 0 ( ) =, therefore all later jobs have already arrived to machine, thus can be processed with no idle times. Therefore, by the argument above, 0 is optimal for machine among all sequences starting with job 0. We can now compare the 0 for 0 +. Since all of these sequences follow SPT on machine for all jobs, including job 0, minimizing the total flow-time 0 over 0 +is equivalent to minimizing the starting time, 0 ( ) = 0 +. Thus the minimum is attained at the maximum value of 0, i.e., +. For this 0, since the starting time on machine is 0 ( 0 ) = 0 +and all other jobs are scheduled with no idle times, the system flow-time sums for all jobs the starting time 0 +plus the remaining flow-time on machine (as demonstrated in Figure