A generalized job-shop problem with more than one resource demand per task

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5 R 2 P 2 W 2,3 W 2,2 W 2,1 0 0 W 1,1 W 1,2 P 1 R 1 Figure 1: Plane with obstacles for two robots finishing it. The y-coordinate is interpreted similarly. In job-shop for each pair of task O 1,j, O 2,j with µ 1,j = µ 2,j a rectangle is drawn (see figure 1) in order to forbid parallel processing. A solution to this job-shop problem corresponds to a path in the plane from (0, 0) to (P 1, P 2 ) which avoids the interior of all obstacles and uses only horizontal (R 1 working), vertical (R 2 working) or diagonal (both working) segments. Consequently, the length of a horizontal or vertical segment is the euclidean length, i. e. x respective y. For the diagonal parts it is defined as the euclidean length of the projection on the x axis (or equivalently on the y axis). For every such path from (0, 0) to (P 1, P 2 ) the total length is the makespan of the associated schedule. Brucker showed that this problem can be transformed into an ordinary shortest path problem in a directed acyclic graph G = (V, A) which can then be solved in O( V + A ) by dynamic programming. For this purpose he introduced one vertex for the point (0, 0), one for (P 1, P 2 ) and two vertices for every obstacle, representing the north west and the south east corner. From every vertex v he started a line sweep moving diagonal upwards until he hits another obstacle or one of the border lines x = P 1 respective y = P 2. In the first case the vertex will be connected with the two vertices of the hit obstacle by two arcs (see figure 2), in the second one v will be connected with the vertex (P 1, P 2 ). The weight of the arcs is set to the length of the path from v to the corresponding corner. Because every obstacle has 2 vertices and every vertex at most two arcs, the shortest path can be solved in O(n) where n is the number of obstacles, i. e. n = n 1 n 2 in the case of LSP-T. Brucker also showed the G can be constructed using O(n log n) operations. 5

6 NW linesweep SO v Figure 2: Construction of Arcs 3.2 Integrating Collision Avoidance For handling line line and line point collisions the geometric algorithm needs three modifications: First, instead of drawing a rectangle for every pair of tasks on the same machine, we just demand that they share at least one common resource. Thus, we draw a rectangle between O 1,j and O 2,j if both are welding jobs and there is only one laser source or if (O 1,j, O 2,j ) C ll. Second, let (O i,j, q i,j ) C lp where q i,j d i, i.e. R i is not allowed to wait at q i,j while R i is processing O i,j. Remember: we have(o i,j, O i,j ), (O i,j, O i,j +1) C ll and thus two neighboring rectangles. Taking care of these line point collision can therefore be simply done by unifying these two rectangles. For line point collisions of the form (O 1,j, q 2,1 ) involving the depot of robot R 2 we modify the network constructed by Brucker s algorithm. We must force R 2 to leave d 2 before R 1 can start the processing of O 1,j. This means forbidding the horizontal line segment O 1,j and can be done by removing the vertex associated with the south eastern corner of the obstacle defined by O 1,j and O 2,1. In the case of (O 1,j, q 2,n2+2) C lp the task O 1,j has to be processed before O 2,n2+1. So we stretch the rectangle on the left-hand side to the y-axis (see figure 3) and remove the vertex of the north west corner to forbid the vertical strip afterwards. R 2 P 2 O i,j R 1 Figure 3: Line Point collision with depot 6

7 3.3 Handling the latency time Now let τ l > 0: Tuchscherer et al. [5] observed for the case C lp = C ll = that LSP-T can be transformed into an equivalent problem with τ l = 0 if τ l is sufficiently small, i. e. smaller then all driving times. Their modification increases the welding time of all jobs by τ l and reduces the following driving time accordingly. This transformation is not valid anymore when collisions are present: Consider the example in Figure 4. Here we are given two welding tasks and two driving tasks which have a line-line collision. In the transformed problem the no wait schedule is feasible while it induces a collision in the original one. Transformed Problem Original Problem Figure 4: τ l cannot be removed Let W 1,j, W 2,j be two welding tasks. We have to take care that at least τ l time units have past after the finishing of W 1,j before the processing of W 2,j can start, or vice versa. In the following we will concentrate on the first situation the other is analogous. If the next driving move D 1,j+1 does not induce a line-line collision with W 2,j we can locally increase the welding time of W 1,j stretching the rectangle to the right. If D 1,j+1 τ l we are done, otherwise we can only increase it by D 1,j+1. But now the two obstacles (W 1,j, W 2,j ) and (W 1,j+1, W 2,j ) are side-by-side. If we apply the linesweep algorithm we get a vertical arc from the south east corner of (W 1,j, W 2,j ) to the north west corner of (W 1,j+1, W 2,j ) with weight p w 2,j. This weight only needs to be increased by τ l p d 1,j+1 (see figure 5). If (D 1,j+1, W 2,j ) C ll then we already have adjacent rectangles and thus their size does not need to be changed. The linesweep algorithm will produce two vertical arcs of weight p w 2,j : one from the south east corner of (W 1,j, W 2,j ) to the north west corner of (D 1,j+1, W 2,j ) and one from the south east corner of (D 1,j+1, W 2,j ) to the north west corner of (W 1,j+1, W 2,j ). The weight of the first arc will be increase by τ l and the weight of the second one by max{τ l p d 1,j+1, 0}. Note, if some of the rectangles have been unified by line-point collision handling before, then the weight changes of the non existent arcs will not be carried out. We finally state the main result of this section: Proposition 1. LSP-T with R = 2 can be solved in O(n log n) where n is the number task pairs sharing a common resource, i. e. a laser source or a line-line collision. 7

8 weight = p w 2,j + τ l p d 1,j+1 W 2,j W 1,j W 1,j+1 D 1,j+1 Figure 5: Integrating Latency Proof. Define an obstacle for every task pair sharing a common resource with all modification subject to collisions and latency as described above. This requires O(n) operations. Apply Brucker s algorithm to construct the network. This works in O(n log n) (see [3]) and creates 2n vertices and 4n arcs. Adjust the weights of the arcs for handling latency and remove vertices as needed for the line point collisions. This can be done in O(n). Finally, solve a shortest path problem in the network. Applying topological sort and dynamic programming this requires O(n). 4 NP-hardness of three robots 4.1 The N P-hardness result We will reduce the even-odd partition problem to an instance of LSP-T that is described by three robots and one laser source. Furthermore the instance we use for the reduction is without any collisions and latency. Therefore this instance can also be seen as a special instance of the job-shop scheduling problem with three jobs and four machines. However three of these machines are solely used by one robot/job each. Therefore all the tasks of a robot/job scheduled on his own machine are conflict free with respect to the other two robots/jobs. Only the forth machine (corresponding to the laser source) has tasks of all of the three jobs assigned. So this subproblem of job-shop has a very simple structure and the hardness result of Sotskov and Shakhlevich [14] for (J3 3 C max ) does not apply to the instances we use. Their result heavily relies on the fact that all three machines have to process tasks of all three jobs. However since the instances we use is a job-shop instances with makespan objective we can concentrate on active schedules when asking for optimal solutions (cf. Pinedo [9]). Definition 1. Even-odd partition 8

10 e1 R1 H e2 3H H + δi 1 2 H + 2H 2H e1 R2 H e1 4H 2H 2H H 2H e1 R3 6H + 2e1 H H 4H e1 δi 12H e1 Figure 6: Block Structure B11 R1 e1 H e2 3H H + δi 1 2 H + 2H 2H e1 R2 H e1 4H 2H 2H H 2H e1 R3 6H + 2e1 H H 4H e1 δi 12H e1 Figure 7: Block Structure B21 get the correspondence between even-odd partition and LSP-T. Note that even tough the basic idea relies on Sotskov and Shakhlevich [14], the construction we present differentiates in significant points from their approach, moreover the proof we present is simpler. Scheduling the welding tasks of T i 1, T i 2 and T i 3 as the sequence (W i 11, W i 21, W i 12, W i 22, W i 23, W i 13, W i 31, W i 32, W i 24, W i 14) on the laser source is called block structure B 1i (cf. figure 6). Scheduling them as sequence (W i 21, W i 11, W i 22, W i 12, W i 23, W i 31, W i 13, W i 32, W i 24, W i 14) is called B 2i (cf. figure 7). If all blocks T1, i T2 i and T3 i are scheduled as B 1i or B 2i and if they are finished before the welding tasks of T1 i+1, T2 i+1 and T3 i+1 start, we call such a schedule useful. As we will show for getting the connection between even-odd partition and LSP- T it suffices to consider only active schedules with C max = 12 n H + E/2. On the other hand for instance J we can immediately get a sharp lower bound of 12 n H + n i=1 e 2i by looking at the minimal total time for all tasks of R 3 to be finished. By considering just the tasks of R 1 and R 2 and scheduling them according to B 2i for all i we get the same lower bound, moreover this bound is obviously sharp. Lemma 1. Scheduling all tasks of R 2 and R 3 in the trivial order can be done feasibly without causing any task to wait (not taking R 1 into account). 10

14 it is equal to 12 n H + e j B e j). By the same argument as in the proof of Lemma 6 we get that the sum of completion times of robots R 1 and R 2 on the one hand and of robot R 3 equals: 24 n H + n (e 2i 1 + e 2i ) = 24 n H + E i=1 5 A pseudo polynomial algorithm and an FP- TAS In this section we will show that LSP-T for a fixed number of robots can be solved in pseudo polynomial time. By complete enumeration we can also assume w. l. o. g. that the assignments of laser sources to robots is fixed. 5.1 The pseudo polynomial algorithm Our algorithm uses a transversal graph approach as introduced to scheduling problems by Middendorf and Timkovsky [7]. We will construct a directed acyclic graph G = (V, A) with unit weigh arcs where a shortest path coincides with an optimal solution to LSP-T. For i = 1,..., k let O i,0 with p i,0 be an artificial start task. Our vertex set V consists of a terminal vertex T together with tuples (i 1, t 1,..., i k, t k, j 1, d 1,..., j h, d h ) with the following meaning: i r {0,..., 2n r + 1} is the index of the task currently being processed by robot R r. t r {0,..., p r,ir } denote how long R r has already worked on O r,ir. j l { 1, 1,..., k} is the index of the robot who used laser source L l last. Here j l = 1 means, that the laser source has not be used so far. d l {0,..., τ l } is the time laser source L l has been idle since its last usage. Let v = (i 1, t 1,..., i k, t k, j 1, d 1,..., j h, d h ) T be a vertex. Denote by I c := {r t r < p r,ir } the set of robots currently working and by I n := {r i r < 2n r + 1, t r = p r,ir and if O r,ir+1 is a welding task additional j l(rr) = r j l(rr) = 1 d l(rr) = τ l(rr) } 14

15 the set of robots which can start their next task. Here l(r r ) denotes the index of the laser source robot R r is attached to. Denote by I the set of all subsets of I c I n which can work in parallel. This means, I induces no line-line collision, includes at most one welding task per laser source and induces no line-point collisions with any robot r I. For every I I we create an arc of weight one from v to w = (i 1, t 1,..., t k, i k, j 1, d 1,..., j h, d h ) defined by i r = i r + 1, r I I n, i r = i r otherwise t r = t r + 1, r I I c, t r = 1, r I I n, t r = t r otherwise j l = r if r I I n with R r starts a welding task on L l, j l = j l otherwise d l = 0 if r I I N with R r starts a welding task on L l, d l = d l + 1 if no robot in I with l r = l processes a welding task So every robot in I processes a task for one time unit. Every vertex with i r = n r for all robots will be connected with the terminal vertex T by an arc of zero weight. Observation 1. Every path in G from (0,..., 0) to T corresponds to a feasible schedule of LSP-T where the makespan is the sum over the arcs weights of P. Moreover every non preemptive active schedule can be written in this form. Given such a path P we can extract the start times of the k-th task of robot r by summing up the arc weights of P between vertices with i r < k. Observation 2. The number of vertices in G can be bounded by V k max r=1 (2n r + 2) k k n r max max r=1 j=1 pk r,j h max l=1 (τ l ) h and there are at most 2 k V arcs. Because h k and k is constant, the size of G is pseudo polynomial in the size of LSP-T. Theorem 2. LSP-T can be solved in pseudo-polynomial time when the number of robots is constant.. In section 4 we proved that LSP-T with three robots, no latency and no collisions is already N P-hard. However, there is a polynomial solveable special case. Proposition 2. Let C ll = C lp = and τ l = 0, l = 1,..., h. If k n i max max i=1 j=1 pd i,j min k n i i=1 then LSP-T can be solved in polynomial time. min j=1 pw i,j (3) 15

16 Proof. In the absense of collisions LSP-T decomposes into h problems with a single laser source. We can therefore assume h = 1. We will construct a transversal graph similar to the above but it does not contain any time index in the vertices. V consists of a source vertex S, a terminal vertex T and vertices of the form v = (i 1,..., i k, j) with i r {0,..., n r }, r = 1,..., k and j {1,..., k}. Here i r is the index of the last performed welding task of robot R r and j is the index of the robot worked last. Let v = (i 1,..., i k, j) be a vertex. For every r with i r < n r we add an arc to the node w = (i 1,..., i r 1, i r + 1, i r+1,..., i k, r). The weight of this arc will be set as follows: If r = j then R r has to move to its next welding task and thus the weight is p d r,i + r+1 pw r,i. Otherweise by (3) robot R r+1 r has already done the move D r,ir+1 while R j welded. Thus the arc weight is only p w r,i. r+1 For every r we connect S and w = (i 1,..., i k, r) with i r = 1, i r = 0, r r by an arc of weight p d r,1 + p w r,1. Every vertex v = (n 1,..., n k, j) has an arc of weight p d j,n j+1 to the terminal vertex T. Now it is easy to see that a shortest path from S to T in the directed acycled graph G yields an optimal solution to LSP-T. Futhermore the size of G is polynomial in the input size of LSP-T. 5.2 A fully polynomial approximation scheme for LSP-T We close this section by providing an FPTAS for LSP-T with a constant number of robots: Theorem 3. There is a fully polynomial approximation scheme (FPTAS) for LSP-T. Proof. We will use standard rounding argument which has been introduce by Ibarra and Kim [6] for the knapsack problem and is widely used in scheduling (see e. g. Agnetis et al. [1]). Let := max{max h l=1 τ l, max k r=1 max nr j=1 p r,j} be the maximum time distance and N := k r=1 2n r + 1, the total number of tasks. Define Z := ɛ N. We construct a modified instance of LSP-T with τ l := τ l Z and p r,j := pr,j Z. By construction we have τ l, p r,j 1 + N 1 ɛ. Thus all time distances of the modified problem are polynomially bounded in N and 1 ɛ. Therefore the size of the transversal graph described above is also polynomial bounded in N and 1 ɛ. Solving the shortest path problem yields a solution S with makespan T ( S). If we multiply all task start times by Z we obtain a feasible solution S of the original problem with makespan T (S) = Z T ( S). Now let S be an optimal solution for the original problem with makespan T (S ). We have to show T (S) (1 + ɛ)t (S ). To see this, transform S to the modified problem by forming an active schedule S keeping the order of the tasks. Because rounding up enlarges the processing time of each task by at most 16

17 1, we obtain T ( S ) T (S ) Z + N. Thus we have T (S) = Z T ( S) Z T ( S ) T (S ) + Z N = T (S ) + ɛ }{{} T (S ) (1 + ɛ)t (S ) 6 Conclusion In this article we fully settled the approximability status of LSP-T with a constant number of robots by deriving the exact boundary between polynomial time solvable and weakly N P-hard together wit an FPTAS. This situation is relevant for the industrial application of our problem since practical instances are characterized by the fact that the number of robots involved is small ( 5). From a theoretical point of view the N P-hardness of Section 4 refines the boundary between polynomial time solvable and N P-hard for job-shop scheduling by considering a a very special subproblem of job-shop scheduling. Still interesting questions concerning LSP-T remain: are there special graph classes in the conflict graph representation of the problem with a non-constant number of robots that guarantee a similar approximability (resp. inapproximability) behavior as in standard job-shop scheduling. References [1] Alessandro Agnetis, Marta Flamini, Gaia Nicosia, and Andrea Pacifici. A job-shop problem with one additional resource type. Journal of Scheduling, pages 1 13, /s [2] Sheldon B. Akers Jr. A graphical approach to production scheduling problems. Operations Research, 4(2): , [3] P. Brucker. An efficient algorithm for the job-shop problem with two jobs. Computing, 40(4): , [4] M.R. Garey and D.S. Johnson. Computers and intractability: a guide to NP-completeness. WH Freeman and Company, San Francisco, [5] M. Grötschel, H. Hinrichs, K. Schröer, and A. Tuchscherer. Ein gemischtganzzahliges lineares Optimierungsproblem für ein Laserschweißproblem im Karosseriebau. Zeitschrift für wissenschaftlichen Fabrikbetrieb, 5: ,

18 [6] Oscar H. Ibarra and Chul E. Kim. Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM, 22: , October [7] Martin Middendorf and Vadim G. Timkovsky. Transversal graphs for partially ordered sets: Sequencing, merging and scheduling problems. Journal of Combinatorial Optimization, 3: , [8] Alix Munier and Francis Sourd. Scheduling chains on a single machine with non-negative time lags. Math. Methods Oper. Res., 57(1): , [9] M. Pinedo. Scheduling: theory, algorithms, and systems. Springer Verlag, [10] J. Rambau and C. Schwarz. On the benefits of using NP-hard problems in branch & bound. In Operations Research Proceedings 2008, pages Springer, [11] J. Rambau and C. Schwarz. Exploiting combinatorial relaxations to solve a routing & scheduling problem in car body manufacturing. Preprint, Universität Bayreuth, [12] J. Rambau and C. Schwarz. How to avoid collisions in scheduling industrial robots? Preprint, Universität Bayreuth, [13] T. Schneider. Ressourcenbeschränktes Projektscheduling zur optimierten Auslastung von Laserquellen im Automobilkarosseriebau. Diplomarbeit, University of Bayreuth, [14] Y.N. Sotskov and NV Shakhlevich. NP-hardness of shop-scheduling problems with three jobs. Discrete Applied Mathematics, 59(3): , [15] E.D. Wikum. One-Machine Generalized Precedence Constrained Scheduling. PhD thesis, School of Industrial and Systems Enigeering, Georgia Institute of Technology, [16] E.D. Wikum, D.C. Llewellyn, and G.L. Nemhauser. One-machine generalized precedence constrained scheduling problems. Operations Research Letters, 16:87 99,

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