1Properties of lower bounds for the RCPSP

1 Properties of lower bounds for the RCPSP Evgeny R. Gafarov1, Alexander A. Lazarev1 and Frank Werner2 1 2 Institute of Control Sciences of the Russian Academy of Sciences, Russia axel73@mail.ru, jobmath@mail.ru Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Germany frank.werner@mathematik.uni-magdeburg.de Keywords: Project scheduling, Makespan, Lower bounds 1 Abstract We show that the calculation of the well-known lower bound of Mingozzi for the RCPSP is an NP-hard problem and that the relative error of this lower bound can be equal to O(log n), where n is the number of jobs. 2 Introduction Problem RCPSP may be formulated as follows. Given a set N = {1, . . . , n} of jobs. A constant amount of Qk > 0 units of resource k, k = 1, . . . , K, is available at any time. Job j ∈ N has be processed for pj ≥ 0 time units without preemption. During this period, a constant amount of qjk ≥ 0 units of resource k is occupied. Furthermore, finish-start precedence relations i → j are defined between the jobs according to an acyclic directed graph G. The objective is to determine the starting times Sj for each job j = 1, . . . , n, in such a way that: at each time t, the total resource demand is less than or equal to the resource availability for each resource type; the given precedence constraints are fulfilled; the makespan Cmax = maxnj=1 Cj , where Cj = Sj + pj , is minimized. ∗ Let Cmax be the optimal value of the objective function for the problem when pre∗ (pmtn) be the optimal value when preemptions are emptions are not allowed and Cmax allowed. 3 Lower Bound of Mingozzi et al. We consider a linear programming formulation that partially relaxes the precedence constraints and allows preemption. The columns of this LP correspond to the so-called non-dominated feasible subsets. A feasible set X is a set of jobs that may be processed simultaneously, i.e., there are no precedence relations between any pair i, j ∈ X and all  qik ≤ Qk for k = 1, . . . , K). Such a set is called resource constraints are satisfied (i.e., i∈X non-dominated if it is not a proper subset X of another feasible set Y . We consider all non-dominated feasible sets and additionally the one-element sets {i} for all i = 1, . . . , n. We denote all these sets by X1 , X2 , . . . , Xf , where f is the number of such sets, and associate each set Xj with an incidence vector aj ∈ {0, 1}n defined by aji = 1 if i ∈ Xj , and aji = 0 otherwise, j = 1, . . . , f . Furthermore, let xj be a variable denoting the number of time units over which all the jobs in Xj are processed simultaneously. Then the following linear programming problem provides a lower bound LBM (Mingozzi A. et. al. 1998) for the RCP SP by relaxing the precedence constraints and allowing preemption: 2 f  xj −→ min (1) j=1 ⎧ f ⎨ aji xj ≥ pi , i = 1, . . . , n; ⎩ j=1 xj ≥ 0, j = 1, . . . , f. (2) It is known that the calculation of LBM is an N P -hard problem (by a reduction from the N P -hard Bin packing problem) (Lazarev A.A. and Gafarov E.R. 2008), and there are ∗ Cmax ≈ 2. instances for which LB M 4 Relative errors of well-known lower bounds for the problem ∗ In the paper (Lazarev A.A. and Gafarov E.R. 2008), there is a conjecture that Cmax < ∗ 2 · Cmax (pmtn). This conjecture is true for the special case of problem P m|prec|Cmax (Lawler E.L. et. al. 1998), for the special case of RCP SP with a constant amount of Q1 > 0 units of a single resource and without precedence constraints, and for the special case for which there are only one or two preempted job in an optimal schedule for RCPSP with preemptions. However, the conjecture is false for the general case. Theorem 1. There exists an instance of RCP SP for which ∗ Cmax = O(log n). ∗ (pmtn) Cmax Proof. We consider an instance of the type given in Fig. 1 (a). For this instance, we have m + 1 levels of jobs. At the highest level, we have a job j11 with processing time M p, at the second highest level, we have two jobs j12 and j22 with processing times M 2 p, and so m+1 → e2 → on. At the lowest level, we have a chain of short and very short jobs e1 → j1 m+1 → eM+1 , where pei = ε, qei = m + 1, i = 1, . . . , M + 1, j2m+1 → . . . → eM → jM and pj m+1 = p, qj m+1 = 1, i = 1, . . . , M . For each job jik from level k, k = 2, . . . , m, i i M p, qjik = 1. At each level k, k = 2, . . . , m, we have a chain of jobs we have pjik = 2(k−1) k e1 → j1 → e M +1 → j2k → e2 M +1 → . . . → j2kk−1 → eM+1 . Some of these precedence 2(k−1) 2(k−1) relations are illustrated in Fig. 1 (a). For this instance, we have (M + 1)ε ≪ p. By dotted lines, we mark all jobs which will be processed in parallel in an optimal schedule for the non-preemptive problem. In Fig. 1 (b), for an instance with only m + 1 = 6 levels, we give the resulting optimal ∗ ∗ = 112p + 33ε. (pmtn) = 32p + 33ε and Cmax solution. For this instance, we obtain Cmax ∗ ∗ For the general case, we have Cmax (pmtn) = M p + (M + 1)ε and Cmax = M p + m 2 Mp+ (M + 1)ε. Hence, we obtain ∗ m+2 Cmax ≈ . ∗ Cmax (pmtn) 2 Let us now express M by means of m. We have 2m = M . Then n = 2M + 1 + 1 + 2 + . . . + 2m−1 = 2M + 1 + 2m − 1 = 2 · 2m + 2m = 3 · 2m from which we obtain m = log n . 3 3 Therefore, we get ∗ Cmax ∗ Cmax (pmtn) ≈ m+2 log n − log 3 + 2 = . 2 2  As a consequence, there exists an instance of RCP SP for which we obtain the following result: ∗ Cmax LBM = O(log n), and Theorem 2. There exists a type of instances of RCP SP for which ∗ Cmax = O(log n), LBM and the calculation of LBM is an NP-hard problem. The idea of constructing such instances is not difficult. The instance contains two subsets of jobs N1 and N2 . The jobs from the first subset correspond to the instance illustrated in Fig. 1, where Q1 = m+1. In the set N2 , we have n independent jobs with unit processing times pj = 1 and j∈N2 qj = 2m + 2. Additionally, we have a dummy job o1 such that j → o1 → l for all j ∈ N1 , l ∈ N2 . It is obvious that we can give a reduction from the partition problem to the problem of calculating LBM for this type of instances. Additionally, let us consider the relaxation of the problem in which we do not take into consideration non-preemptive jobs, or different processing times, or different values qi , ′ the optimal value of the or we do not consider the precedence relations. Denote by Cmax objective function for the relaxed instance. Then there exist instances for which ∗ Cmax = O(log n), ′ Cmax ′ i.e., we have bad approximation ratio for the lower bound Cmax . Acknowledgements Partially supported by DAAD (Deutscher Akademischer Austauschdienst; A/08/80442/Ref. 325). References Mingozzi A., Maniezzo V., Ricciardelli S. and Bianco L., 1998, “An exact algorithm for project scheduling with resource constraints based on new mathematical formulation", Management Science, Vol. 44, pp. 714–729. Lawler E.L., Lenstra J.K., Rinnooy Kan A.H.G. and Shmoys D.B., 1998, “Sequencing and Scheduling: Algorithms and Complexity.", Report BS-R8909, Centre for Mathematics and Computer Science, Amsterdam, The Netherlands. Lazarev A.A. and Gafarov E.R. , 2008, “On project scheduling problem.", Automation and Remote Control. Vol. 69, N 12, pp. 2070-2087 Contribution to C max { +MP +1|2 Mp +1|2 Mp pi = ½ Mp p = 1/4 Mp i pi = 1/8 Mp qi1 = m+1, p = +1|2 Mp +1|2 Mp precedence relations i (a) qi1 = 1, p = p i +1|2 Mp 2M+1 jobs, total processing time Mp+(M+1) (b) Fig. 1. An instance for illustrating Theorem 1 m levels pi = Mp pi = ½ Mp 32p p 2p p 4p p 2p p 8p p 2p p 4p p 2p p 2p p 4p p 2p p 8p p 2p p 4p p 2p p 16p p 112p+33 4