The polynomial solvability of selected bicriteria scheduling problems on parallel machines with equal length jobs and release dates

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1 The polynomial solvability of selected bicriteria scheduling problems on parallel machines with equal length jobs and release dates Hari Balasubramanian 1, John Fowler 2, and Ahmet Keha 2 1: Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, MA 2: Department of Industrial Engineering, Arizona State University, Tempe, AZ hbalasubraman@ecs.umass.edu, john.fowler@asu.edu, ahmet.keha@asu.edu Abstract We consider bicriteria scheduling problems involving several classical well-known scheduling objectives on parallel, identical machines with job release dates. The jobs are assumed to have equal processing times. Our bicriteria treatment of these problems includes lexicographic optimization, minimization of a composite linear function, generation of schedules on the efficient frontier and the generation of all Pareto optimal solutions. Using the framework provided by the dynamic program of Baptiste (2000), we show the polynomial solvability for several bicriteria problems when the number of machines is given. We also present algorithms that extend the linear programming approach of Brucker and Kravchenko (2008) and whose complexities are polynomial in the number of jobs and independent of the number of machines. The complexity status of the bicriteria problems considered in this paper was previously unknown. Keywords: bicriteria scheduling, parallel machines, equal processing time jobs. 1 Introduction Multicriteria scheduling stems from the need to address real-world problems that often involve conflicting objectives. A schedule that optimizes one criteria may in fact perform quite poorly for another. Decision makers must carefully evaluate the trade-offs involved in considering several criteria. In a production scheduling context, for instance, it may be important to the decision-maker to meet customer due-dates, while also keeping the work-in-process costs to a minimum. Much of the literature in multicriteria scheduling to date has been focussed on single machine scheduling. The recent survey by Hoogeoven (2004) is a comprehensive study of multicriteria scheduling research to date. The survey reveals the dearth of theoretical results in bicriteria problems involving parallel machines, mostly because very few single criteria problems have polynomial time complexity. We consider bicriteria scheduling of equal length jobs on identical parallel machines with job release dates. Our main goal in this paper is to advance the knowledge of computational complexity of bicriteria problems for this environment. Indeed, we view this as part of the effort to widen the existing frontiers of solvability in polynomial time (Baptiste et al. (2004)). The polynomial time algorithms provided in the paper may not always be of practical interest but they provide a 1

2 starting point for the development of more efficient approaches for complex problems encountered in practice. Additionally, while the assumption of equal processing times may seem to have limited applicability, situations in which processing times are unequal can be modeled as having a common processing time (the mean). Schedules obtained based on this assumption can then be used for the general problem, especially for cases where the processing times do not differ by much. The key reason for the use of a common processing time is the computational tractability that this assumption affords in the generation of the optimal or good solutions, as opposed to the NP-hard nature of most general parallel machine problems. Moreover, complex environments such as flexible flow shops and job shops are often decomposed into parallel machine problems. At several of these parallel machine workstations, it is possible the processing activity (assembly of parts, painting, cleaning etc.) has the same duration while the jobs differ in their priorities, arrival times and due-dates. Sarin and Prakash (2004), who also consider bicriteria parallel machine problems with identical processing times but no job release dates, point to cases in flexible flow shops where the assumption of identical processing times has been shown to be reasonable. 2 Problem Definition We consider the problem of scheduling n jobs J 1, J 2,..., J n on m parallel, identical machines. Each job, j J, has a release date r j 0, a due date d j 0 and weight w j > 0. All jobs have the same processing time p > 0. The goal is to find a completion time C j for each job j to solve selected bicriteria optimization problems. The completion times have to be feasible, that is 1) jobs start after their release date i.e. j, C j p r j and 2) no more than m machines are used at any time t,i.e t, {J j C j p t < C j } m. Additionally only non-preemptive schedules are considered. In the α β γ notation of Graham et al. (1979) we denote the α and β part this of environment as P m p j = p, r j. We leave the γ part blank for now since bicriteria optimization can be performed in many different ways. We also note here that the subscript m in P m is used to indicate that the number of machines in the environment is fixed and not a variable. In the context of computational complexity if a P β γ problem is polynomially solvable then the P m β γ problem, for the same β and γ, is also polynomially solvable. The reverse, however, is not necessarily true. We consider bicriteria problems that involve the classical scheduling objectives of makespan (C max ), sum of completion times ( C j ), sum of weighted completion times ( w j C j ), maximum lateness (L max, where L max = max j {C j d j }), total number of tardy jobs( U j, where U j = 1 if C j > d j and 0 otherwise) and total tardiness ( T j, where T j = max(c j d j, 0)). Since criteria are often conflicting, it is rare that a single solution is best for all of them. In any multicriteria problem it is important to point out the nature of optimization being performed given that such trade-offs exist. Let S be the set of feasible solutions for a bicriteria optimization problem and z 1 (x) and z 2 (x) be the objective values for criteria z 1 and z 2 (both of which need to be minimized) for a feasible solution x S. We follow, for the most part, the notation and terminology of T Kindt and Billaut (2002) and Hoogeoveen (2004). It may be of interest to minimize a secondary criterion z 2 given the optimal value of the primary criterion z 1. This is called lexicographic or hierarchial optimization represented as Lex (z 1, z 2 ). Sometimes the decision-maker may have a linear function in mind that he intends to minimize: a composite function of the form F l (z 1, z 2 ) = α z 1 + (1 α) z 2, where 0 α 1. But perhaps 2

3 of most interest - and in most cases the most difficult to generate - is the set of Pareto optimal or non-dominated solutions. Definition 2.1. A solution x is Pareto optimal or non-dominated if there exists no other solution x S for which z 1 (x) z 1 (x ) and z 2 (x) z 2 (x ) where at least one of the inequalities is strict. Definition 2.2. A solution x is said to be weakly Pareto optimal if there exists no other solution x S for which z 1 (x) < z 1 (x ) and z 2 (x) < z 2 (x ). Let co(n D) represent the convex hull of all non-dominated solutions when each solution is plotted in the objective space (with each axis representing a criterion) based on the values the solution has for the two criteria. Hooegeveen (2004) defines the efficient frontier as the lower envelope of co(nd) and an extreme point as a Pareto optimal solution that is also a vertex of the efficient frontier. The set of extreme points is of interest as sometimes it is not possible to generate all the Pareto optimal solutions. It is worthwhile to note that there exists an extreme point that is an optimal solution to the linear combination F l (z 1, z 2 ) = α z 1 + (1 α) z 2, for an α such that 0 α 1. In other words, after solving optimally for the linear combination described above, we obtain an extreme point. All extreme points can be obtained by changing the value of α. The extreme points are also a subset of the supported solutions, defined by T Kindt and Billaut (2002) as the set of Pareto optimal points located on the efficient frontier. The set of non-supported solutions are the Pareto optimal points that do not lie on the efficient frontier. Figure 1 shows an example illustrating these ideas. Ạ B.. C. D.. I E. F.. G J. H Figure 1: Solutions in objective space. (Note that G is not a vertex of the efficient frontier and is therefore not an extreme point. But since it lies on the line joining F and H, it is a supported pareto optimal solution) It is possible to generate the set of Pareto optimal points for bicriteria problems, using the ɛ constraint approach (T Kindt and Billaut (2002)). The approach is written in the γ field of the α β γ scheduling notation as ɛ(z 1 z 2 ) (T Kindt and Billaut (2002)). The notation means that we seek for the best solution for z 1 given that the solution does not exceed a fixed z 2 value. If z 2 3

4 is always integral, it is possible to find all the Pareto optimal points by iterating over all values of z 2 in increments of 1 that lie between Lex(z 1, z 2 ) and Lex(z 2, z 1 ), and optimizing for each value of z 2 the value of z Contributions We look at bicriteria scheduling problems in terms of 1) Lexicographic optimization: P m r j, p j = p Lex(z 1, z 2 ) 2) Minimization of a composite linear function: P m r j, p j = p F l (z 1, z 2 ) 3) Generation of the extreme schedules, and 4) Generation of the set of all pareto optimal points: P m r j, p j = p ɛ(z 1 z 2 ). We show the polynomial time solvability of a number of bicriteria pairs of classical scheduling objectives in all the four types of bicriteria optimization listed above. The polynomial algorithms proposed are extensions of the single criteria algorithms known in the literature. Indeed, our main contribution is the illustration that these polynomial single criteria algorithms, with minor algorithmic changes - and hence almost no increase in their complexities - can be used for bicriteria optimization of several pairs of classical scheduling objectives. The remainder of the paper is organized as follows. In Section 3 we review the theoretical results in multicriteria parallel machine scheduling. We look at optimal single criteria algorithms in Section 4, with a detailed look at the dynamic program by Baptiste (2000). In Section 5 we consider lexicographic optimization of various bicriteria problems; we also include in this section an ɛ-constraint problem involving T j and L max. We exploit the structure provided by the dynamic program of Baptiste (2000) to prove complexity results for these problems. Sections 6 and 7 discuss how the linear programming approach of Brucker and Kravchenko (2008) can be extended to optimally minimize in polynomial time 1) a composite linear function involving w j C j and Tj, 2) to generate a subset of extreme non-dominated schedules 3) and to generate the set of all non-dominated schedules when one of the criteria is C max and the other is either w j C j or T j. 3 Related Work One of the earliest theoretical results in parallel machine bicriteria scheduling appears in Bruno et al. (1974) who show that P Lex( C j, C max ) is NP-hard. Tuzikov et al. (1998) consider bicriteria scheduling uniform processors: Q p i = p ɛ(g f max ), where f max = max i (Φ i (C i )) and g = i (Ψ i(c i )), where Φ i and Ψ i are strictly increasing functions. They provide polynomial algorithms for generating all Pareto optimal solutions. T kindt et al. (2001) consider a bicriteria scheduling problem on unrelated parallel machines applicable to the production of glass bottles. Angel et al.(2003) identify a class of multiobjective optimization problems (which include some parallel machine problems) possessing a fully polynomial time approximation scheme (FPTAS) for generating an ε-approximate Pareto curve. Baptiste and Brucker (2004) show that P m Lex( C j, U j ) is solvable in pseudopolynomial time. Gupta et al. (2003) prove further complexity results: they show that P Lex( C j, C max ) is strongly NP-hard and thus P Lex( C j, U j ) is strongly NP-hard as well; P Lex( C j, T max ) can be solved in pseudopolynomial time; and P m Lex( C j, U j ) can be solved in polynomial provided the processing times are all different. Sarin and Prakash (2004) propose polynomial algorithms for lexicographic bicriteria scheduling of various pairs of traditional scheduling objectives in an identical parallel machine environment with equal release dates assuming that all jobs have equal processing times (P p j = p Lex(z 1, z 2 )). Hoogeoven (2004) provides an excellent review of multicriteria scheduling. 4

5 For the parallel machine environment with no release dates and equal or unit processing times, bicriteria problems (linear combinations of criteria) involving classical non-decreasing scheduling criteria (i.e. regular measures) can be modeled as assignment problems with n jobs to be assigned to n possible completion time points (it can be verified easily only n completion time points need to be considered for non-decreasing criteria). The objectives C max and C j are especially trivial in this environment. Simons (1983) considers the problem of finding whether a feasible schedule exists in the P r j, p j = p environment where every job is scheduled before its deadline and the resulting schedule has the minimum C j and C max (both criteria are minimized simultaneously). She proposes a O(n 3 log(n)) algorithm for the problem. The P r j, p j = p ɛ( C j L max ) and P r j, p j = p ɛ( C max L max ) problems can be solved using the same algorithm in the following way. Let L max be the pre-specified value of maximum lateness not to be exceeded. To determine whether a feasible schedule (with optimum C j and C max ) exists such that maximum lateness is less than or equal to L max, the deadline D j for each job are set to D j = d j + L max. If the algorithm is run for every possible value of the maximum lateness, all the non-dominated points can be generated.brucker and Kravchenko (2008) provide an alternative linear programming approach to minimize C j in the presence of deadlines in the same environment. Their result too has the same bicriteria implication. To the best of our knowledge this result is the only known bicriteria result in parallel machine environment with release dates and equal processing times. However, in the last few years, a number of parallel machine single criterion scheduling problems with release dates, whose complexities had previously been unknown, were shown to be polynomial using dynamic programming(problems P m r j, p j = p w j C j and P m r j, p j = p T j by Baptiste (2000), and problem P m r j, p j = p w j U j by Baptiste et al. (2004)). More recently, Brucker and Kravchenko (2008) present a linear programming approach to solve the P r j, p j = p w j C j and P r j, p j = p T j problems. The complexity of the linear programming approach is independent of the number of machines and hence is an improvement over the dynamic program of Baptiste (2000). For our bicriteria optimization requirements, we use the the dynamic program of Baptiste (2000). For notational convenience we call this dynamic program DPB (Dynamic Program by Baptiste). Wherever possible we also extend the linear programming approach of Brucker and Kravchenko (2008). 4 Optimal single criterion algorithms We now present optimal single algorithms for several single criterion problems. We restrict ourselves only to algorithms that we extend to lexicographic bicriteria optimization in the next section. 4.1 P r j, p j = p C max and P r j, p j = p C j A feasible schedule is said to be non-delay if no machine is kept idle while an operation is available for processing. Since the machines are identical and the jobs have the same processing times, a non-delay schedule produces the optimum for both C max and C j objectives (for details on this see Brucker (1998)). Note, however, a schedule optimal in C max need not necessarily be non-delay. Simons (1983) considers the problem of obtaining a schedule, if one exists, in which each job is scheduled before its deadline and if so, solving the P r j, p j = p C max problem and P r j, p j = p C j. 5

6 For our purposes, suppose that the deadlines are equal to a common high value so they can be comfortably met. Minimizing C max and C j now reduces to the following algorithm, which produces a non-delay schedule. Let J be set of all jobs and let S i record the earliest time that machine i is available for processing. Completion Time Algorithm (CTA) Step 1 : Set S i = min{r j j J}, i Step 2 : Find machine i with the least S i. If more than one machine have the same minimum value, break ties arbitrarily. Step 3 : Find a job j J such that r j S i. Break ties arbitrarily if more than one such jobs are available. If at least one such job exists then, set S i = S i + p, C j = S i, and J := J \ j and go to Step 2. If no such job exists, then if J := STOP, else go to Step 1. We call the algorithm Completion Time Algorithm (CTA) (the idea for this comes from Simons (1983)). The steps of the algorithm are executed O(n) times, and each time the earliest available job or the the earliest available machine is to be determined; and Step 3 involves searching for a job for which r j S i and takes O(n) time. The complexity of the algorithm is therefore: O(n(nlog(n) + mlog(m) + n) = O(n 2 log(n)) for n > m. 4.2 P m r j, p j = p w j C j and P m r j, p j = p T j In this section we state the theorems from Baptiste (2000) that lead to his dynamic programming algorithm (DPB). For details regarding the proof of these theorems and ideas, refer to Baptiste (2000).We start with the following hypothesis that states the objective functions of the form i f i(c i )for which DPB produces the optimal solution. Hypothesis 4.1. The functions f i are non-decreasing, i.e., t 1, t 2 > t 1, f i (t 1 ) f i (t 2 ) and the functions f i f j are monotonous, i.e., either t 1, t 2 > t 1, (f i f j )(t 1 ) (f i f j )(t 2 ) or t 1, t 2 > t 1, (f i f j )(t 1 ) (f i f j )(t 2 ). It can be easily verified that the hypothesis holds for w j C j and T j. Note that it does not hold for U j and w j T j. A binary relation between jobs is then established. Given any pair of jobs (J i, J j ), J i J j if and only if f i f j is 1) either strictly increasing between some time points or 2) constant and i < j. For this definition, the following lemma holds: Lemma 4.1. The binary relation is a strict total order. Without any loss of generality, the jobs can now be sorted according to the strict total order, i.e., J 1 J 2... J n. For weighted sum of completion times this implies that jobs are indexed in the non-increasing order of their weights while for total tardiness and sum of tardy jobs the jobs are indexed in non-decreasing order of their due-dates. If S denotes the schedule, among optimal ones, that lexicographically minimizes the vector of completion times (C 1, C 2,..., C n ), then the following lemma characterizes the time points at which jobs start and end on the schedule S. (The lexicographic notion here has an entirely different context and should not be confused with bicriteria lexicographic optimization. Suppose 6

7 for example, that n = 3. Then the vector (5,3,2) is a lexicographically better vector than (5,3,3); the vectors are identical in their first two elements but the first better has a lower value for the third element). Lemma 4.2. The time points at which jobs start and end on the schedule S belong to T = {t : r i, l {0,..., n}, t = r i + lp}. It is easy to prove that this lemma holds for any form of bicriteria optimization as well, provided that both the criteria involved are regular measures, i.e., non-decreasing in the completion times. It follows from the lemma that the size of T has an upper bound of O(n 2 ). Definition 4.1. A resource profile ξ is a vector (ξ 1, ξ 2,..., ξ m ) such that ξ 1 ξ 2... ξ m and ξ m ξ 1 p. Let Ξ denote the set of resource profiles ξ such that ξ i T. For instance, consider a trivial 2-job example on 2 machines. Let r 1 = 0 and r 2 = 1. Let p = 3. For this example T := {0, 1, 3, 4, 6, 7}. Then the following are resource profiles: (0, 0), (0, 1), (0, 3), (1, 1), (1, 3), (1, 4), (3, 3), (3, 4), (3, 6), (4, 4), (4, 6), (4, 7), (6, 6), (6, 7), (7, 7). Note that the ith component of the resource profile does not necessarily correspond to the ith machine. Therefore, in the the resource profile (0,1), the component 0 may correspond to machine 2 and the component 1 may correspond to machine 1. See the note Intuitive meaning of resource profiles in section 4 of Baptiste (2000). Definition 4.2. Given two resource profiles σ and µ, σ µ if and only if for every index i in 1,..., m, σ i µ i. We now introduce the variables of DPB and the theorem that leads to it. Let U k (t 1, t 2 ) be the set of jobs whose index is lower than or equal to k and whose release date is in the interval [t 1, t 2 ). For any integer k n for any resource profiles σ and ε such that σ ε, let F k (σ, ε) be the minimum value that the function J i U k (σ m p,ε 1 ) f i(c i ) can take among the schedules of all the jobs in U k (σ m p, ε 1 ) such that 1) the starting times belong to T 2) the number of machines available at time t to schedule the jobs in the set U k (σ m p, ε 1 ) is m {i : t σ i } + {i : ε i t}. If no such schedule exists, F k (σ, ε) is equal to. By definition, the boundary condition, F 0 (σ, ε) is equal to 0. The following important theorem (Theorem 8 in Baptiste (2000)) leads us to DPB. Theorem 4.3. For any value of k in [1,n], for any resource profiles σ and ε such that σ ε, F k (σ, ε) is equal to F k 1 if r k / [σ m p, ε 1 ) and otherwise to min (F k 1 (σ, θ) + F k 1 (θ, ε) + f k (θ 1 + p)). θ Ξ,r k θ 1,σ θ θ =(θ 2,...,θ m,θ 1 +p) ε In essence, the theorem above states that the optimal solution at stage k between resource profiles σ and ε can be obtained by trying all possible resource profiles that are between σ and ε. The resource profile that yields the lowest value of the objective function is chosen and leads to a further decomposition of the problem. Since there exists an optimal solution in which starting times belong to T, the optimum for the problem is exactly: 7

8 F n = ((min t T t,..., min t T t), (max t T t,..., max t T t)) The decomposition from Theorem 4.3 allows us to compute this value. The relevant values of σ and ε are in Ξ. Baptiste (2000) argues that for each of these resource profiles there are n 2 possible values for the first component and once it is fixed, there are n possible choices for the remaining m 1 components. Therefore there are O(n 2 n m 1 ) = O(n m+1 ) relevant resource profiles and O(n 2m+2 ) relevant pairs (σ, ε). The DPB works as follows: In the initialization phase F 0 (σ, ε) is set to 0 for any values σ,ε in Ξ such that σ ε. For k = 1 to n, based on Theorem 4.3, F k is computed for all possible values of the parameters in the theorem and also using all values of F k 1 computed at the previous step. The initialization takes O(n 2m+2 )steps. In the next step, for each value of k,o(n 2m+2 )values of F k have to be considered. In each consideration for F k, all possible θ values need to be checked, for which there are O(n m+1 ))possibilities (similar to the bounds for σ and ε). So the overall time complexity is O(n 2m+2 n m+1 n) = O(n 3m+4 ). 4.3 Results Summary Complexity results from this section are summarized in table 1. We also present in the table below the complexity result for the P m r j, p j = p w j U j problem shown by Baptiste et al. (2004). We do not describe the algorithm here as it is not used for the bicriteria optimization results discussed in this paper. Criterion Complexity Reference C max O(n 2 log(n)) Simons (1983) Cj O(n 2 log(n)) Simons (1983) wj C j O(n 3m+4 ) Baptiste (2000) Tj O(n 3m+4 ) Baptiste (2000) wj U j O(n 6m+1 ) Baptiste et al. (2004) wj T j open Table 1: Summary of complexity results for single criterion algorithms. Note that P r j, p j = p is also polynomially solvable by Brucker and Kravchenko (2008); we discuss and adapt their algorithm in section 6. 5 Lexicographic Bicriteria Optimization 5.1 P r j, p j = p Lex( C j, z 2 ) Algorithm CTA from Section 4.1 produces a non-delay schedule that is optimal for C j. We next show that the jobs in any schedule optimal in C j can finish only at the completion time points generated by CTA. Note here that jobs may be reassigned to different completion time points producing many alternately optimal schedules but the set of completion time points that produces the optimal C j is unique. 8

9 Lemma 5.1. Let κ := {κ 1, κ 2..., κ n } be the set of completion time points produced by a CTA schedule indexed in non-decreasing order. Then, the time points at which jobs end on any schedule optimal in total completion time are in κ. Proof. Let κ := {κ 1, κ 2,..., κ n} be a set of completion times from which at least one feasible schedule can be obtained. Let this vector be indexed in non-decreasing order. Suppose - as part of the contradiction argument - there exists at least one element of κ that is different from its corresponding element in κ and that the sum of the elements in κ produces the lowest total completion time. Let κ k be the lowest value in κ that is different from the elements of κ. If κ k > κ k any schedule realizing completion time points in C would not be non-delay which violates the non-delay nature of C. Hence κ k < κ k. Now consider n i=1 κ i and n i=1 κ i. Since k 1 i=1 κ i = k 1 i=1 κ i and κ k < κ k, n i=k+1 κ i > n i=k+1 κ i, otherwise optimality of the sum of time points in κ is not possible. In any schedule that realizes the completion times in the set κ, there are n k jobs scheduled at time points κ k+1, κ k+2,..., κ n. Next, we construct a new schedule as follows: schedule k jobs at time points κ 1, κ 2,..., κ k. There exists at least one such assignment as there exists a non-delay schedule that realizes the completion time points in κ; from this it also follows that there exists at least one job that can finish at κ k. The remaining n k jobs are scheduled such that they realize κ k+1, κ k+2,..., κ n. There existed such an assignment for n k jobs the set κ ; and since κ k < κ k such an assignment for n k jobs is still possible for the new schedule being constructed. Now consider the total completion time of this schedule: k i=1 κ i+ n i=k+1 κ i. Since n i=k+1 κ i > n i=k+1 κ i, n i=1 κ i > k i=1 κ i + n i=k+1 κ i, which contradicts the optimality of n i=1 κ i. The theorem tells us that there is a unique set of n time points to minimize the total completion time. This helps us to minimize other non-decreasing secondary criteria by solving an assignment problem. The minimization of non-decreasing sum criteria given the optimal value of total completion time can be done in polynomial time with the following two steps: 1) Run CTA 2) Use the assignment formulation to minimize the secondary criteria. The complexity of O(n 3 ) is determined by the second step. For total number of tardy jobs, total tardiness, and maximum lateness as secondary criteria, a minor modification in CTA can ensure minimization of any of these as a secondary criteria without the use of the assignment algorithm. Whenever a machine becomes available, and more than one job is available for processing, we choose the job with the lower due-date if the secondary criterion is total tardiness or maximum lateness. If the secondary criterion is the total number of tardy jobs, we follow an algorithm that closely resembles the algorithm for 1 U j by Moore (1968). Every time the machine becomes free and more than one job is available is available for processing, we first identify the set of jobs that can be finished before the due-date, and of the jobs in this set we pick the one with the earliest due-date. If no such set exists, the job to be scheduled is chosen arbitrarily (since all jobs have the same processing time). The following two lemmas state the above more formally and can be easily proved using contradiction arguments involving a pairwise exchange. Let N represent the set of unscheduled jobs that are available at time S i to be scheduled. Lemma 5.2. In Step 3 of CTA, If N > 1, then choosing the job with the least due-date produces a solution optimal in both secondary criteria L max and T j. 9

10 Lemma 5.3. In Step 3 of CTA, 1) If N > 1, then choosing among {j j N, d j (S i + p) 0} the job with the least due-date produces a solution optimal in secondary measure of total number of tardy jobs. If no such jobs exist, the job to be scheduled can be chosen arbitrarily. The above lemmas imply that the minimization of the secondary criterion can be achieved by using simple sorting algorithms (O(n(log(n)) time) each time a machine is free. Since machines get freed up to n times during the course of the algorithm, the total running time of the algorithms is O(n 2 (log(n))) - this is in addition to the n 2 time to run the CTA algorithm. Thus, for C j as the primary measure, we now can solve for the following objectives as the secondary measure: 1) C max in O(n 2 log(n)) time (CTA) 2) L max in O(n 2 log(n) + n 2 log(n)) = O(n 2 log(n)) time (Modified CTA) 3) T j in O(n 2 log(n) + n 2 log(n)) = O(n 2 log(n))time (Modified CTA) 4) U j in O(n 2 log(n) + n 2 log(n)) = O(n 2 log(n))time (Modified CTA) 5) w j U j in O(n 2 log(n) + n 3 ) = O(n 3 ) time(cta with assignment) 6) w j T j in O(n 2 log(n) + n 3 ) = O(n 3 ) time (CTA with assignment) 5.2 P m r j, p j = p Lex( w j C j, z 2 ) and P m r j, p j = p Lex( T j, z 2 ) We now extend DPB to solve lexicographic optimization problems for bicriteria combinations that satisfy a crucial lemma (lemma 5.4 below). We identify 5 such combinations: 2 with w j C j as the primary criterion and 3 with T j as the primary criterion.we consider the following bicriteria problems with w j C j as the primary criteria. 1) Lex ( w j C j, L max ) 2) Lex ( w j C j, w j T j ) For these 2 problems, we index the jobs in decreasing order of their weights (since w j C j is the primary measure). In case the weights are equal, jobs are ordered in the non-decreasing order of their due-dates. We consider the following bicriteria problems with T j as the primary criteria. 1) Lex ( T j, C j ) 2) Lex ( T j, L max ) 3) Lex ( T j, w j U j ) For these problems, we index the jobs in the increasing order of their due-dates; in case of due-dates are equal, we index jobs in the non-decreasing order of their weights. Definition 5.1. We say a job k is scheduled between resource profiles σ and ε if the following conditions are met: 1) there exists a resource profile, τ Ξ such that σ τ ε 2) τ 1 r k 3) k is scheduled at τ 1 and 4) τ = (τ 2, τ 3...τ m, τ 1 + p) is also such that σ τ ε. Since τ Ξ, all time points at which jobs are scheduled belong to T. We do not need to consider schedules in which jobs do not finish at points in T as it can be easily shown that for any bicriteria problem with non-decreasing criteria, there exists a schedule with better or same value of the two criteria and in which all jobs finish at time points that belong to T. For any integer k n and for any resource profiles σ and ε such that σ ε, let L be the set of all feasible schedules of jobs in U k (σ m p, ε 1 ) such that in each schedule all jobs are scheduled between resource profiles σ and ε. 10

11 Let ζ L be any such feasible schedule. C i (ζ) denotes the completion time of job i U k (σ m p, ε 1 ) on ζ. We denote primary and secondary criteria of this job i, given its completion time is C i, as fi I(C i) and fi (C i ) (the primary and secondary criteria values of job i on the schedule ζ are written as fi I ( Ci (ζ) ) and fi ( Ci (ζ) ).) We now define Fk I (σ, ε), the optimal primary criteria value for schedules in L. More formally, { Fk I (σ, ε) = min fi I (C i (ζ)) } ζ L i U k (σ m p,ε 1 ) If no feasible schedule is possible, Fk I(σ, ε) = and by definition F 0 I (σ, ε) = 0. Next, we define Fk (σ, ε), the optimal secondary criteria value for all schedules in L that realize F k I (σ, ε). If the secondary criterion is a sum function, then F k (σ, ε) = min { i U k (σ m p,ε 1 ) f i (C i (ζ)) ζ L, i U k (σ m p,ε 1 ) f I i (C i (ζ)) = F I k (σ, ε) } If the secondary criterion is a minimax function we first define objective value for a given schedule in L and then value Fk (σ, ε) for all schedules that realize F k I (σ, ε). { Fmax(ζ) = max fi } (C i (ζ)) i U k(σ m p, ε 1 ), ζ L, { Fk (σ, ε) = min Fmax(ζ) ζ L, j U k (σ m p,ε 1 ) f I i (C i (ζ)) = F I k (σ, ε) } If no feasible schedule is possible, Fk (σ, ε) = (for both sum and minimax functions ) and by definition F0 (σ, ε) = 0 (for both sum and minimax functions). We restrict our attention to fi I and fi functions that correspond to the primary and secondary criteria of the five lexicographic problems listed at the beginning of the section. Our next steps are to show an important technical lemma that will ultimately allow us to carry out a decomposition identical to that of DPB (Theorem 4.3). Suppose that Fk (σ, ε) is finite (this means that at least one feasible schedule exists which in turn implies that Fk I(σ, ε) is finite) and r k [σ m p, ε 1 ). There are many schedules that could be optimal for Fk (σ, ε). Among these schedules that realize F k (σ, ε) (and therefore F k I (σ, ε)) let H be the schedule on which the vector made of completion times, taken in increasing order of index, is lexicographically minimum. (Note that the lexicographic notion mentioned here is different from bicriteria lexicographic minimization. For example, suppose there are 3 jobs in total with indexes 1, 4 and 5 in the set U 5 (σ m p, ε 1 ). Then the vector (C 1, C 4, C 5 ) = (4, 7, 8) is better for the lexicographical order than (C 1, C 4, C 5 ) = (4, 8, 7)). Let t k be the starting time of job k on H ( job k is on H as r k [σ m p, ε 1 )). Lemma 5.4 below is required to prove Theorem 5.5 which leads us to a decomposition structure identical to Theorem 4.3; and this in turn allows us to follow the DPB algorithm for lexicographic bicriteria optimization with only minor modifications. We note our claim in lemma 6.4 is restricted 11

12 to: 1) Lex ( w j C j, L max ); 2) Lex ( w j C j, w j T j ); 3) Lex ( T j, C j ); 4) Lex ( T j, L max ); 5) Lex ( T j, w j U j ). Lemma 5.4. On H, jobs with a release date lower than or equal to t k start before or at t k. Proof. Suppose that there is a job i with r i t k that starts at time t i > t k on H. Let H be the schedule obtained by exchanging jobs i and k. Let δ I and δ be the difference in the primary and secondary criteria values respectively (value of the schedule H is subtracted from value of the schedule H ) due-to this exchange. Indeed, since processing times are equal, we need to consider the difference in criteria values of these two jobs only. Our proof proceeds as follows: we first consider the implications of this exchange on the five lexicographic problems listed at the beginning of the section. We show that for each of these five cases, δ I = 0 and δ = 0. This implies that H and H have the same values for both the primary criterion and the secondary criterion. But since k is the job in U k (σ m p, ε 1 ) with the highest index value (its index is therefore greater than that of job i), H is better for the lexicographic order. This contradicts our claim that H is the schedule on which the vector made of completion times, taken in increasing order of index, is lexicographically minimum. Lex ( w j C j, L max ) δ I = w k (t k + p) + w i (t i + p) w i (t k + p) + w k (t i + p) = w k (t k t i ) w i (t k t i ) = (w k w i )(t k t i ) Since w k w i, δ I 0. But since H achieves Fk (σ, ε) it achieves F k I (σ, ε) as well, and hence δ I = 0, which means w i = w k. Since the weights are equal, based on our indexing of jobs, d k d i. δ = max(t k + p d k, t i + p d i ) max(t k + p d i, t i + p d k ) Since d k d i and t k < t i, max(t k + p d k, t i + p d i ) = t i + p d i. Also, t i + p d i t i + p d k and t i + p d i t k + p d i. Hence, δ 0 but since H achieves F k (σ, ε), δ = 0. Lex ( w j C j, w j T j ) Just as in the previous case, δ I = 0, implying w i = w k and d i d k Jobs i and j are either tardy or early depending on whether they are scheduled at t k + p or t i + p. Given that we know d i d k, we list all possible cases and the δ value for each case in table 2. These values are easily verifiable through straightforward calculations. δ > 0 contradicts our assumption that H realizes F k (σ, ε). Therefore, it must be that δ = 0. Lex ( T j, C j ) The analysis for δ I is the same as the analysis for δ for the previous problem. It must be, therefore, that δ I = 0.Though jobs are swapped, the finish times are merely exchanged and it is easy to see that that the δ = 0, when the secondary criteria is C j. 12

13 Job k Job i t k + p t i + p t k + p t i + p δ tardy tardy tardy tardy = 0 early tardy early tardy 0 early tardy tardy tardy > 0 early early early tardy 0 early early tardy tardy > 0 early early early early = 0 Table 2: Different cases for job i and k when scheduled at t i or t k Lex ( T j, L max ) As before δ I = 0. The analysis of δ is the same as that for Lex( w j C j, L max ). Therefore, δ = 0. Lex ( T j, w j U j ) Again δ I = 0. Jobs i and j are either tardy or early depending on whether they are scheduled at t k + p or t i + p. In Table 3, we first consider the case d i < d k, list all possibilities and the corresponding δ I, δ values for each possibility. These values are easily verifiable through straightforward calculations. The cases with δ I > 0 are a contradiction on our assumption on H. The only possible cases are δ I = 0 and δ = 0. For case d i = d k, we know based on our indexing of jobs that w i w k. Clearly, δ I = 0 and for all the cases listed in table 3, δ 0, but since H realizes F k (σ, ε), δ = 0. Job k Job i t k + p t i + p t k + p t i + p δ I δ tardy tardy tardy tardy 0 0 early tardy early tardy > 0 0 early tardy tardy tardy > 0 0 early early early tardy > 0 > 0 early early tardy tardy > 0 0 early early early early 0 0 Table 3: Different cases for job i and k when scheduled at t i or t k, given that d k > d i Conclusion: For all the five problems considered above, the schedules H and H both realize Fk (σ, ε) but H is better for the lexicographical order, which contradicts our assumption on H. The above lemma does not work for a number of bicriteria combinations - most notably the Lex( T j, w j C j ) case. This is because the both jobs i and k could be early (or tardy) and remain early (or tardy) after the exchange; in this situation if w i < w k then H would have a 13

14 worse secondary criterion value. The decomposition (similar to the DPB decomposition) that we propose next will therefore not work. The decomposition also does not work if the w j T j criterion were being minimized as a total ordering of jobs (or indexing) cannot be established. Let Θ(σ, ε) be the set of resource profiles such that for every θ Θ(σ, ε), θ 1 r k, σ θ ε and θ Θ(σ, ε) where θ = (θ 2,..., θ m, θ 1 + p). We define Ω(σ, ε) Θ(σ, ε) as follows: Ω(σ, ε) := { θ Θ(σ, ε) F I k 1 (σ, θ) + Fk 1 I (θ, ε) + fk I (θ 1 + p) = Fk I (σ, θ)} Ω(σ, ε) is therefore the set of resource profiles that are optimal for the primary criterion. The single criterion DPB decomposition described in Theorem 4.3 searches for the resource profile in the set Θ(σ, ε) (the set of resource profiles between σ and ε) and chooses the one that gives the minimum value of the criterion being considered. Here we follow the same procedure; only now we search for a resource profile in Ω(σ, ε) ( the set of resource profiles between σ and ε and that are also optimal for the primary criterion), and choose the one that gives the minimum value of the secondary criterion. Due to the similarity of the lexicographic bicriteria case to the single criterion case, the theorem follows the same steps presented in Baptsite (2000). We note again our claim is restricted to the following problems: 1) Lex ( w j C j, L max ); 2) Lex ( w j C j, w j T j ); 3) Lex ( T j, C j ; 4) Lex ( T j, L max ); 5) Lex ( T j, w j U j ). Theorem 5.5. For any value of k in [1,n], for any resource profiles σ and ε such that σ ε, Fk (σ, ε) is equal to F k 1 (σ, ε) if r k / [σ m p, ε 1 ) and otherwise to. min (Fk 1(σ, θ) + Fk 1 θ Ω(σ,ε) (θ, ε) + fk (θ 1 + p)) Proof. Let F be the expression above. We describe only the sketch of the proof here, as it follows the same course as that of Theorem 8 in Baptiste (2000). If r k / U k (σ m p, ε 1 ) the proposition holds as U k (σ m p, ε 1 ) = U k 1 (σ m p, ε 1 ). We now consider the case where r k U k (σ m p, ε 1 ). The following are the steps of the proof: 1) F Fk (σ, ε): Assume F is finite (as otherwise the theorem holds). Let θ Ξ be the resource profile that realizes F. There is a schedule H 1 that realizes Fk 1 (σ, θ); and a schedule and a schedule H 2 that realizes Fk 1 (θ, ε). Notice that any job in U k 1[σ m p, ε 1 ) is either scheduled on H 1 or on H 2. Consider the schedule H build as follows: schedule J k at time t k and all other jobs in U k (σ m p, ε) at the time they were scheduled on H 1 or on H 2. The next steps showing the feasibility of this schedule are identical to the proof in theorem 8 of Baptiste (2000). F Fk (σ, ε) follows from the proof of feasibility. 2) F Fk (σ, ε): Assume finiteness of F k (σ, ε) (as the proposition holds otherwise). Among the schedules that realize Fk (σ, ε), let H be the one on which the vector made of completion times of the jobs in U k (σ m p, ε 1 ), taken in increasing order of index, is lexicographically minimum. Let t k be the starting time of job k on H (job k is on H as r k U k (σ m p, ε 1 ). From lemma 5.4, we know for all our 5 bicriteria lexicographic problems that on H jobs that arrive before or at t k start at t k or before. Next a resource profile θ is exhibited such that (1) θ 1 = t k (2)σ θ (3) θ = (θ 2, θ 3,..., θ m, θ 1 + p) ε (4) Jobs in the set U k 1 (σ, θ ) are scheduled between σ and θ (5) Jobs in the set U k 1 (θ, ε) are scheduled between θ and ε. The steps here are identical to the proof 14

15 of Theorem 8 in Baptiste (2000). We now calculate Fk (σ, ε) for the schedule H. F k (σ, ε) = = i U k (σ m p,ε) i U k 1 (σ m p,θ 1 ) fi ( Ci (H ) ) fi ( Ci (H ) ) + i U k 1 (θ 1,ε 1 ) fi ( Ci (H ) ) + fk (θ 1 + p) By definition, Fk 1 (σ, θ) i U k 1 (σ m p,θ 1 ) fi ( Ci (H ) ) and Fk 1 (θ, ε) i U k 1 (θ 1,ε 1 ) fi ( Ci (H ) ) Therefore, Fk (σ, ε) Fk 1 (σ, θ) + Fk 1 (θ, ε) + fk (θ 1 + p) Fk (σ, ε) F When the secondary criterion is the minimax function (like L max or T max ), we have the following theorem. Theorem 5.6. For any value of k in [1,n], for any resource profiles σ and ε such that σ ε, Fk (σ, ε) is equal to F k 1 (σ, ε) if r k / [σ m p, ε 1 ) and otherwise to min θ Ω max(fk 1 (σ, θ), Fk 1 (θ, ε), fk (θ 1 + p)) Proof. Let F be the expression above. The steps are identical. We show only the calculations for the Fk (σ, ε) F part. Fk (σ, ε) = Fmax(H ) 15

16 By definition, k 1 (σ, θ) max{ fi ( Ci (H ) ) i U k 1 (σ m p, θ 1 ) } F and k 1 (θ, ε) max{ fi ( Ci (H ) ) i U k 1 (θ 1, ε 1 ) } F Therefore, Fk (σ, ε) max(fk 1 (σ, θ), Fk 1 (θ, ε), fk (θ 1 + p)) Fk (σ, ε) F A dynamic programming algorithm Let a be the resource profile such that a 1 = a 2 =... = a m = min t T t and b be the resource profile such that b 1 = b 2 =... = b m = min t T t. Let S be the schedule, among optimal ones (optimal for one of 5 bicriteria problems listed at the beginning of the section) on which the vector of completion times, taken in increasing order of index is lexicographically optimum. Then we know that S realizes Fn (a, b). 1) In the initialization phase F0 I (θ, ε) and F 0 (θ, ε) are set to zero for any values σ, ε in Ξ such that σ ε 2) We then iterate from k = 1 to n. Each time, Fk is calculated for all possible values of the parameters due to the formulas in theorems 5.5 and 5.6, and the values of Fk 1 calculated at the previous step. The complexity of the algorithm is identical to the complexity of DPB. The only extra feature is to keep track of the secondary criterion values in addition to the primary criterion values. Its time complexity therefore remains O(n 3m+4 ). We call this algorithm DPB-Lex(z 1, z 2 ), where z 1 and z 2 correspond to the primary and secondary criteria of the 5 problems listed at the beginning of the section. 5.3 Results Summary Table 4 lists the results obtained in this section. 5.4 P m r j, p j = p ɛ( T j L max ) It is also possible to use DPB to generate the set of all non-dominated points when one of the criteria is T j and the other is L max. We use the ɛ-constraint approach in conjunction with DPB. To minimize T j such that the solution does not exceed given a value of L max, we change the definition of a feasible solution in DPB. Recall that in any state of DPB, Θ is the set of resource profiles such that for every θ Θ, θ 1 r k, and σ θ ε and θ Θ where θ = (θ 2,..., θ m, θ 1 +p). 16

17 Primary C max Cj wj C j Tj Secondary C max - O(n 2 log(n)) O(n 3m+4 ) O(n 3m+4 ) Cj O(n 2 log(n)) - O(n 3m+4 ) O(n 3m+4 ) wj C j O(n 3m+4 ) O(n 2 log(n)) - open L max O(n 6 log(n)) O(n 2 log(n)) O(n 3m+4 ) O(n 3m+4 ) Uj open O(n 2 log(n)) open O(n 3m+4 ) Tj O(n 3m+4 ) O(n 2 log(n)) O(n 3m+4 ) - wj U j open O(n 3 ) open O(n 3m+4 ) wj T j open O(n 3 ) O(n 3m+4 ) open Table 4: Complexity results for lexicographic optimization. Results for C max as primary criteria are from Balasubramanian et al. (2005); improved complexity results for C max are illustrated in Section 7.2 We now redefine it as : Θ := {θ θ 1 r k, θ 1 + p d k L max, σ θ ε, σ θ ε}, where L max is the given value of maximum lateness not to be exceeded. Since our criterion is T j, F k (σ, ε) is the optimal value of that the function j U k (σ m p,ε 1 ) T j can take among jobs in U k (σ m p, ε 1 ) such that 1) All jobs are scheduled between σ and ε 2)None of the jobs exceed the prefixed value of L max. If no feasible schedule is possible then F k (σ, ε) = and by definition the boundary condition F 0 (σ, ε) is 0. Theorem 5.7. For any value of k in [1,n], for any resource profiles σ and ε such that σ ε, F k (σ, ε) is equal to F k 1 if r k / [σ m p, ε 1 ) and otherwise to min (F k 1(σ, θ) + F k 1 (θ, ε) + T k (θ 1 + p)) θ Θ Proof. It is clear that in any feasible schedule the given L max value is not exceeded by any job. The rest of the proof is identical to theorem 8 in Baptiste (2000). The theorem directly leads to a dynamic programming algorithm identical to DPB. We now discuss the complexity of the procedure to generate all the Pareto optimal points (a discussion on generating the set of Pareto optimal solutions for other bicriteria pairs is given in section 7). Let L max represent the maximum lateness value for the P m r j, p j = p Lex( T j, L max ) problem. Since jobs can finish at only O(n 2 ) time points, there are only O(n 2 ) values that L max can take for each job. Therefore, the total number of possible L max values are O(n 2 n) = O(n 3 ). It is sufficient to consider the L max values in this set of O(n 3 ) possible values that are L max or higher. The complexity of DPB is O(n 3m+4 ), and the DPB will be run O(n 3 ) times to generate all Pareto optimal solutions. Enumerating all the Pareto optimal solutions therefore takes O(n 3 n 3m+4 ) = O(n 3m+7 ) time. 17

18 6 Minimizing a Composite Linear Function: LP approach Balasubramanian et al. (2005) showed that DPB could be used to solve a composite bicriteria linear combination of the form α w j C j + (1 α) T j. DPB works for any sum function that is monotonous (Hypothesis 5.1), and Balasubramanian et al. (2005) show that the function α w j C j + (1 α) T j is monotonous given that 1 α 1/(1 + w min ), where w min = min { } w i w j J i, J j, w i > w j, d i > d j. We now describe a linear programming approach to minimize the composite linear function α w j C j + (1 α) T j. The linear programming approach presented is an extension of the approach proposed by Brucker and Kravchenko (2008) for the P r j, p j = p w j C j and P r j, p j = p T j problems. Our contribution is the demonstration that the composite bicriteria linear function of the form α w j C j + (1 α)t j can be solved in polynomial time, given that 1 α 1/(1 + w min ), where w min = min { } w i w j J i, J j, w i > w j, d i > d j. The condition on the range of α is identical to the one established in Balasubramanian et al. (2005), based on DPB. But the LP-approach is an improvement over DPB (recall that DPB runs in O(n 3m+4 ) time) as the complexity of the LP based approach is independent of the number of machines. Brucker and Kravchenko (2006) propose an LP formulation for the P r j, p j = p w j C j problem. The formulation assigns job processing times to the possible time intervals in which jobs can be processed. The time intervals are in a sense analogous to the time points considered in DPB. This equivalence implies that the number of possible time intervals is O(n 2 ). First a D j value is defined for each job J j by setting D j = D = max j {r j } + np, j = 1,..., n. Let r = min j {r j }, j = 1...n. The {[ intervals considered are: rj + kp, r j + kp + p ] k Z, rj + kp r, r j + (k + 1)p D } In the above expression, note that k is an integer that can also be negative. All the intervals from the above set are enumerated in the increasing order of their left endpoints. The set obtained is denoted by {I i i {1...z}}. The left endpoint of each interval I i is denoted by r(i i ) and the right endpoint by D(I i ). Brucker and Kravchenko (2008) state that there exists some l such that I i+1... I i+l for any i 0,..., z l. The variable y is then set to the maximum of such l, i.e, y = max { l I i+1... I i+l, i {0,..., z l} }. In the LP formulation, the decision variable x ji is equal to the amount of job J j processed in the interval I i. While Brucker and Kravchenko (2008) define their objective function as the total weighted completion time, the objective function in our case is a bicriteria composite linear function. We model it as z n i=1 j=1 B jix ji, where B ji is the value α w j C j + (1 α)t j when job J j is scheduled in interval I i.the formulation for the problem is given below: such that z n min B ji x ji i=1 j=1 z x ji = p, j = 1,..., n i=1 n n x j,i x j,i+y mp, i = 0,...z y j=1 j=1 18