On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems

Transcription

1 Math. Program., Ser. A (2007) 110: DOI /s FULL LENGTH PAPER On the equivalence of the max-min transportation lower bound and the time-indexed lower bound for single-machine scheduling problems Yunpeng Pan Leyuan Shi Received: 4 April 2005 / Accepted: 12 May 2006 / Published online: 14 July 2006 Springer-Verlag 2006 Abstract New observations are made about two lower bound schemes for single-machine min-sum scheduling problems. We find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program. We show the equivalence of this strongest bound and the bound provided by the LP relaxation of the time-indexed integer programming formulation. These observations lead to a new lower bound scheme that yields fast approximation of the time-indexed bound. Several techniques are developed to facilitate the effective use of the new lower bound in branch-andbound. Numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from OR-Library. Results obtained with our new method are spectacular; we are able to solve all 125 open problems to optimality. Keywords Scheduling Combinatorial optimization Time-indexing 1 Introduction The application of linear programming (LP)-based lower bounds has fundamentally changed the landscapes of many areas of discrete optimization. Y. Pan (B) TomoTherapy Inc., 1240 Deming Way, Madison, WI , USA ypan@tomotherapy.com L. Shi ISyE Department, University of Wisconsin-Madison, Madison, WI 53706, USA leyuan@engr.wisc.edu

2 544 Y. Pan, L. Shi However, machine scheduling is one of the remaining strongholds where LPbased methods have not been able to compete with specialized algorithms. It is our hope that the results to be presented here would tip the balance of power to the favor of LP-based methods. First, let us describe the problem. Jobs 1,..., n need to be processed on a machine that can only process one job at a time without interruption. Job j takes time p j to complete; upon its completion at time t j, it incurs cost f j (t j ), which is an arbitrary function. The objective is to find a processing sequence such that the total cost, n j=1 f j (t j ), is minimized. Even for the most common cost functions, the problem may be intractable. The optimal solution of the problem hence entails clever enumeration, which in turn requires a tight lower bound. The classical scheduling approach assumes specific cost functions and develops lower bounds, typically without using LP. An apparent drawback is that specialized schemes become invalid with generalization of the problem. Within the mathematical programming community, there are two main schools of thought regarding how to formulate the problem. Integer programming (IP) formulations using continuous completion (or start) time variables along with binary variables have been proposed. Early work along this line includes that of Balas [6, 7]. The LP relaxations of these formulations are not very strong, despite various efforts to strengthen them using valid inequalities [4, 7] (see [18] for a comprehensive survey). On the other hand, time-indexed IP formulations have been studied. Dyer and Wolsey [12] show that a particular time-indexed formulation gives the strongest bound among several formulations, ranging from combinatorial lower bounds to LP-based ones. Polyhedral results are also proposed to further strengthen the LP relaxation of the timeindexed formulation [2, 21]. Despite its superior quality, one criticism of the time-indexed formulation is that time-indexing can result in a possibly huge number of time periods, as the number of time periods is not bounded by a polynomial function in the number of jobs. Therefore, model accuracy sometimes may have to be sacrificed with a coarse discretization of time, in an effort to curb the formulation size. A clear advantage of time-indexing is that the cost functions f j ( ) can be arbitrary, including nonlinear functions, as long as f j ( ) can be evaluated at each point. The biggest disadvantage of time-indexing, however, is that it involves a large number of variables and constraints, so much so that even the LP relaxation can be difficult to solve. To address this problem, Van den Akker et al. [3] exploit the sparsity in the LP solution through column generation. While column generation can speed up the solution of the LP relaxation as seen in [3], it has not overcome the breaking point where the large majority of instances with more than 30 jobs can be solved. More recent work includes [8] on the time-indexed formulation and column generation, and [5] on the time-indexed formulation and the Lagrangian relaxation. In this paper, the time-indexed formulation should be interpreted as the particular one studied in [2, 21]. The time-indexed lower bound refers to the bound provided by the LP relaxation of the time-indexed formulation. We report a somewhat unexpected result. It turns out that the time-indexed lower bound

3 Equivalence of max-min transportation bound and time-indexed bound 545 can be obtained in a completely different light. Specifically, we point out that the strongest bound from the family of lower bounds provided by transportation problem relaxations of the problem can be determined by solving a single linear program. Furthermore, this bound is shown to be equal to the time-indexed lower bound. We propose methods that offer a more cost-effective way to leverage the quality of the time-indexed lower bound in branch-and-bound. Although the initial work at the root node is no less than that required by the time-indexed lower bound, we are able to extract some crucial structural information that can be re-used at subsequent nodes. Computing lower bounds for nodes other than the root then amounts to solving transportation problems. The paper is organized as follows. Section 2 introduces the new lower bound and presents the equivalence proof. Section 3 addresses a number of issues regarding how to use the lower bound effectively in branch-and-bound. Section 4 reports on our computational experience in using the lower bound in branch-and-bound algorithms for solving the total weighted tardiness scheduling problem. Section 5 presents some concluding remarks. 2 Max-min transportation lower bound 2.1 General lower bound scheme The transportation problem relaxation of scheduling problems with min-sum objectives and general cost functions was first introduced by Lawler [14] in his 1964 seminal paper. Let us assume that jobs 1,..., n are to be processed during time window [0, T]. To discretize the time window, we refer to the time interval [t, t + 1) as period t; hence, t = 0,..., T 1. The following transportation problem can be constructed: (P(c)) min s.t. T 1 c jt x jt j=1 T 1 x jt = p j, j = 1,..., n, (1) x jt 1, t = 0,..., T 1, (2) j=1 x jt 0, j = 1,..., n; t = 0,..., T 1, (3) where x jt = 1ifjobj is assigned to period t, and x jt = 0 otherwise; c jt is the transportation cost. Lawler [14] gives the following sufficient condition under which (P(c)) will provide a lower bound for the scheduling problem:

4 546 Y. Pan, L. Shi Proposition 1 ([14]) A sufficient condition for (P(c)) to provide a lower bound for the scheduling problem is t+p j 1 s=t c js = f j (t + p j ), j = 1,..., n; t = 0,..., T p j. (4) A less stringent condition is obtained by changing = to in(4): Proposition 2 A sufficient condition for (P(c)) to provide a lower bound for the scheduling problem is t+p j 1 s=t c js f j (t + p j ), j = 1,..., n; t = 0,..., T p j. (5) The sufficiency of these conditions is obvious. Both (4) and (5) require that if all the p j units of processing are assigned to consecutive periods, the total transportation cost for this job cannot exceed the cost of job completion after the last of these periods. Since in (P(c)) jobs do not have to be assigned to consecutive periods, the minimum value of (P(c)) is therefore a lower bound. Further, we have found many examples indicating that the bounds produced by (5) are often strictly stronger than those produced by (4). Regardless of which of these conditions is adopted, it leaves an infinite number of choices for cost vector c. Lawler [14] seems to have left this question open. All previous work along this line focuses on particular job cost functions, and heuristically chooses cost structures that are plausible for their respective job cost functions; e.g., [12,19] using (4) and [9,13] using (5). We propose to determine an optimal cost structure that yields the strongest transportation problem relaxation. Our approach is fundamentally different than previous work in two respects. First, no assumption of a particular job cost function is made. Second, the bound provided by this strongest transportation problem relaxation has superior quality. In contrast, previous relaxations are viewed to have rather poor quality. The stark difference in bound quality is due to the fact that our optimal cost structure takes into account interactions among competing jobs, whereas previous heuristic cost structures have not been able to capture such interactions. 2.2 Optimal choice of c Let LB(c) denote the optimal objective value of the transportation problem (P(c)). Then, we define the max-min transportation lower bound as LB = max{lb(c) c satisfies (5)}. LB(c) is a concave function of c, and subgradient optimization may thus be used. However, it is important to note that the choice of c is constrained by (5). Although constrained subgradient optimization via projection can be applied, it makes the already slow convergence of subgradient

5 Equivalence of max-min transportation bound and time-indexed bound 547 optimization even slower. To resolve the difficulty of this max-min problem, we consider the LP dual problem of (P(c)) instead: LB(c) = max j=1 T 1 p j u j + v t (DP(c)) s.t. u j + v t c jt, j = 1,..., n; t = 0,..., T 1, u j free, j = 1,..., n and v t 0, t = 0,..., T 1, where (u, v) are the dual variables associated with (1) (2). Hence, the original two-stage problem reduces to a linear program with variables (u, v, c): LB = max j=1 T 1 p j u j + v t (P1) s.t. (5), u j + v t c jt 0, j = 1,..., n; t = 0,..., T 1, (6) u j free, j = 1,..., n and v t 0, t = 0,..., T 1, (7) 2.3 Equivalence of LB and the time-indexed lower bound The max-min transportation lower bound LB, by definition, is the strongest lower bound from the family of lower bounds provided by transportation problem relaxations of the scheduling problem. For a particular c that satisfies (4), Dyer and Wolsey [12] prove that LB(c) is dominated by the time-indexed lower bound. Other than this result, LB seemingly has nothing to do with the timeindexed lower bound. However, in our preliminary experiments, we compared these two bounds, and to our great surprise, they were equal on all test problems. A closer examination revealed that it was no coincidence that the two bounds matched on those test problems. We recall the LP relaxation of the time-indexed IP formulation: (TI) min s.t. j=1 T p j T p j f j (t + p j )y jt y jt = 1, j = 1,..., n, (8) t j=1 s=t p j +1 y jt 1, t = 0,..., T 1, (9) y jt 0, j = 1,..., n; t = 0,..., T p j, where variables y jt = 1ifjobj is to start in period t, and y jt = 0 otherwise.

6 548 Y. Pan, L. Shi Theorem 1 The max-min transportation lower bound and the time-indexed lower bound are equivalent. Proof Consider problem (P1). We first project the polyhedron represented by constraints (5) (7) from (u, v, c)-space to (u, v)-space. This is straightforward since c variables are free and their coefficients in the objective function of (P1) are all zeros. Hence, (P1) is reduced to (P1 ): LB = max j=1 (P1 ) s.t. p j u j + T 1 p j u j + t+p j 1 s=t v t v s f j (t+p j ), j=1,..., n;,..., T p j, (10) u j free, j = 1,..., n and v t 0, t = 0,..., T 1. (11) If (u, v, c) satisfies (5) (7), then (u, v) satisfies (10) (11). Conversely, if (u, v) satisfies (10) (11), we define (u, v, c) where c jt = u j + v t for all j, t; such a (u, v, c) satisfies (5) (7). If we replace p j u j with a new variable u j, and v t with v t in (P1 ), it is not difficult to verify that (P1 ) is the LP dual problem of (TI). Hence, their optimal values are equal. Note that the variables u j = p ju j, j = 1,..., n and v t = v t, t = 0,..., T 1 (12) are the dual variables associated with constraints (8) (9) in (TI). 3 Using the new lower bound in branch-and-bound The time-indexed lower bound is known to be strong, typically, with an initial gap of less than 1 2% in our experience as well as those of others (see, e.g., [2]). As mentioned before, the computational expense of the bound is unfortunately too high for even moderate sized problems. In this section, we discuss how to use the proposed max-min transportation lower bound in branch-and-bound, which can be done much more efficiently due to the structure of transportation problems. The original contribution of our proposed method is that it eliminates the need to compute time-indexed lower bounds at all nodes except the root. Since the root and other nodes in the search tree are handled differently, we proceed to explain the treatment of each node type separately. 3.1 Root node The main task to be accomplished at the root is to compute the optimal transportation costs for later use at other nodes. While it suffices to solve (P1), it

7 Equivalence of max-min transportation bound and time-indexed bound 549 is not a particularly efficient way, since this LP has (n + T + nt) columns and (2nT +n P) rows, where P = n j=1 p j is the sum of job processing times. Alternatively, we propose a method that first solves the projected problem (P1 )to determine optimal (u, v) values, and then computes an optimal c value for the fixed (u, v) values. Solution of problem (P1 ). In comparison with (P1), (P1 ) is a lot smaller in size, with only (n+t) columns and (nt +n P) rows. Furthermore, we find that it is conducive to transpose the coefficient matrix so that there are fewer rows than columns; then, fewer logical/slack variables would need to be added by an LP solver. We therefore solve, instead, the LP dual problem of (P1 ), which is (TI). After (TI) is solved, we extract the dual variable values associated with its constraints and use (12) to obtain (u, v) values. It is important to note that this is the only time when (TI) needs to be solved. A sparse reformulation of (TI). Although (TI) has fewer columns and rows than (P1), the size of (TI) remains the most serious challenge that we must face when trying to solve the LP. In fact, the size of (TI) as measured by the number of nonzeros in its coefficient matrix, roughly quadruples as the processing time range doubles. For example, with a processing time range of [1, 100], our 2.8 GHz PC equipped with 1 GB of memory would barely be able to solve the LP for n = 50; for n = 100, the initial loading of the LP data would activate virtual memory in addition to the 1 GB physical memory, thereby reducing solver execution to a crawl. To deal with this size issue, we introduce a simple, but useful sparse reformulation of (TI) whose size grows linearly with the expansion of the processing time range. As a result, an LP solver now can run faster and can solve problems that previously were too large to load into the memory. Let job 0 denote a dummy job with unit processing time, i.e., p 0 = 1. We expand the variable set of (TI) by adding variables y 0t (t = 0,..., T 1). When job 0 is started in period t, i.e., y 0t = 1, it indicates the idling of the machine in this period at cost f 0 (t + p 0 ) 0. In addition, job 0 is not subject to constraints (8), and can be started more than once. Hence, we have the following reformulation: min j=1 (TI ) s.t. (8), T p j j=0 s=t p j +1 f j (t + p j )y jt t y jt = 1, t = 0,..., T 1, (13) y jt 0, j = 0,..., n; t = 0,..., T p j, where the y 0t variables can also be viewed as slack variables. Furthermore, the constraints in (13) form an interval matrix, in which 1 s appear consecutively in each column [2, 15]. Hence, we can apply the following row

8 550 Y. Pan, L. Shi transformations to these constraints: The equation for t = 0 is left unchanged; for t = 1,..., T 1, we multiply both sides of the equation for t 1by 1 and then add it to the equation for t. This gives us a sparse reformulation: min j=1 T p j f j (t + p j )y jt (TIS) s.t. (8), y j0 = 1, (14) j=0 (y jt y j(t pj )) = 0, t = 1,..., T 1, (15) j=0 y jt 0, j = 0,..., n; t = 0,..., T p j. Note that the number of nonzeros in each equation of (15) does not exceed 2(n + 1), independent of processing time range. Now suppose that we have solved problem (TIS) to obtain an optimal primal solution y jt (j = 0,..., n; t = 0,..., T p j ) along with dual variable values u j (j = 1,..., n), v 0, and v t (t = 1,..., T 1) associated with constraints (8), (14), and (15), respectively. The restriction of the primal solution to index set (j = 1,..., n; t = 0,..., T p j ) is of course optimal to (TI). The dual variable values associated with constraints of (TI) are not explicit, but can be reconstituted as follows. Recall that (TI ) results from the addition of slack variables to (TI); thus, the dual variable values associated with the constraints are the same. Let A =[B; N] be the coefficient matrix of (TI ), where B is an (n + T) (n + T) matrix consisting of the basic columns associated with the optimal LP solution, and N consists of nonbasic columns. Also, let f =[..., f j (t + p j ),...] T be the vector of the objective function coefficients. From linear programming theory, we have at optimality, [u T, v T ]B = f T. (16) The row transformation involving consecutive periods t 1 and t can be written in matrix notation as L t = I I 2 (t + n)th row, (t n)th row, (17) where submatrices I 1 and I 2 are identity matrices. Similar to (16), the following also holds when optimality is attained for (TIS): [u T, v T ]L 1 L T 1 B = f T. (18)

9 Equivalence of max-min transportation bound and time-indexed bound 551 By comparing (16) and (18), we obtain [u T, v T ]=[u T, v T ]L 1 L T 1. (19) Using (12), (17), and (19), the original dual variable values can be computed as u j = u j /p j = u j /p j, j = 1,..., n, and (20) { v t = v v t = t v, t = 0,..., T 2, t+1 v t, t = T 1. (21) This sparse reformulation is actually quite effective. For a processing time range of [1, 100], we are now able to solve problems with n = 40/50 within a few minutes and problems with n = 100 in less than an hour. Solving for optimal transportation costs c jt. Now suppose that optimal u and v values have been computed, taking into account (20) and (21). With u j and v t fixed, constraints (6) of (P1) reduce to lower bound constraints on the c jt variables. Any feasible vector c that satisfies (5) and the lower bound constraints, together with u and v, would optimally solve problem (P1). Hence, the task on hand is a trivial feasibility problem; e.g., one solution would be c jt u j + v t for all j, t. Moreover, we note that the interaction among jobs through the constraints are gone. Thus, we propose a further decomposition by job; this allows us to solve no more than n small feasibility problems instead of a very large one. Specifically, for each fixed j = 1,..., n, we consider a system of linear inequalities defined by t+p j 1 s=t c js f j (t + p j ), t = 0,..., T p j, (22) c jt u j + v t, t = 0,..., T 1. (23) Fixing of variables. Further savings can be achieved by noting that some of constraints (10) in problem (P1 ) hold at equality; i.e., for some t 0, p j u j + t 0 +p j 1 s=t 0 v s = f j (t 0 + p j ), which implies fixing of variables: c js u j + v s for all s = t 0,..., t 0 + p j 1. We can then drop the inequality in (22) that corresponds to t = t 0. Moreover, we show that at least one such t 0 exists. Consider variables y jt (t = 0,..., T p j ) of (TI), which are the dual variables associated with constraints (22). Since at least one of these variables takes on a nonzero value, the existence of t 0 follows from complementarity. Ironically, this is perhaps one of the few occasions when

10 552 Y. Pan, L. Shi we would wish that more of the decision variables y jt take on fractional values, thereby allowing more constraints to be dropped and more costs c jt to be fixed. Secondary considerations for choosing c jt. The u and v that we have computed, together with any c satisfying (22) (23), would attain the time-indexed bound. Therefore, the choice of c among all qualified choices does not change the lower bound for the root. However, this choice will affect bound quality at other nodes, since c is not re-optimized. This motivates secondary considerations in choosing c. The intuition behind our secondary criterion is to make the transportation costs as large as possible. It is then reasonable to require the following property be satisfied by a criterion: Property 1 No c jt1 can be increased without some other c jt2 being decreased. Here we propose a criterion that minimizes the total difference between the two sides of constraints (22), which clearly satisfies Property 1. More precisely, we minimize [ T p j f j (t + p j ) ] t+p j 1 s=t c js subject to (22) (23). 3.2 Subsequent nodes When a lower bound needs to be evaluated at a node other than the root, we solve a transportation problem of the form (P(c)) with the optimal cost structure found at the root. The transportation problem is solved using an algorithm proposed in [19], which is a specialized O(n 2 T) adaptation of the Hungarian method [15]. We refer to [19] for issues regarding warm start and early termination of the algorithm. The optimal transportation costs for the root are not necessarily optimal for the other nodes, but rather are an approximation. The other nodes represent subproblems of which schedules are partially determined. On the other hand, the transportation costs can still be used to compute valid lower bounds. Moreover, such lower bounds are most likely to be fairly good approximations of the time-indexed bounds when the subproblems do not deviate too much from the original problem. In other words, lower bounds should be tight for nodes that are not very deep in the tree, where effective pruning has the greatest impact. Although we do not re-optimize the transportation costs, some simple improvements can be carried out. Let J = {1,..., n} be the original set of jobs to be scheduled. Suppose that some jobs are already scheduled at a node other than the root and that the set of unscheduled jobs is J J. Also, suppose that for each j J, the feasible processing window is now narrowed to [a j, b j ] [0, T].Forjobj J, (5) is satisfied over time window [0, T]; therefore, (5) is obviously satisfied over time window [a j, b j ]. This ensures the validity of the lower bound. However, Property 1 may not hold with respect to the narrower time window. We give a heuristic for the adjustment of c jt (t [a j, b j ]) so that Property 1 is restored. The heuristic increases c jt for each j J and for each period t = b j 1,..., a j in a greedy fashion.

11 Equivalence of max-min transportation bound and time-indexed bound Effect of node depth on bound quality Normally, the quality of the lower bound improves as branch-and-bound explores deep into the tree. Our lower bound behaves somewhat differently. At a node other than the root, it approximates the time-indexed bound. As the search goes deeper and deeper into the tree, the transportation costs c jt deviate further and further from the optimal costs for the individual nodes. Therefore, the quality of this approximation deteriorates and may reach a point where the tree size is noticeably enlarged. We further illustrate this phenomenon using two numerical examples of 40-job total weighted tardiness problems (denoted by 1 w j T j in the scheduling literature). In the two examples, we assume that positional branching is used; therefore, the depth of a node is measured by the number of jobs fixed. In Fig. 1a, b the horizontal axis indicates a certain depth of the tree, and the vertical axis shows the node count at this depth (the number of nodes that cannot be fathomed). To eliminate the influence of the upper bound, we initially set it to the known optimal value. In the legend, TI and MMT refer to the curves for the time-indexed bound and for our max-min transportation bound, respectively. In both Fig. 1a and b, it can be seen that our bound approximates the time-indexed bound fairly well until around 20 jobs are fixed. After that, our bound deviates noticeably from the time-indexed bound, as indicated by the larger node counts. All the node counts eventually come down when nearly all jobs are fixed. There is also an important distinction between (a) and (b). The degradation of the transportation lower bound and its effect seen in (a) are reasonable; the node count curve for MMT follows an up-and-down pattern similar to that for TI. In comparison, the situation in (b) indicates poor approximation of the timeindexed bound by our bound when roughly n > 20. We develop two methods for alleviating such pathological occurrences. First, we may recompute c jt from time to time as the search progresses. But, recomputation cannot be too frequent, since that would conflict with our initial intention to approximate the hard LP by the easy transportation problem. A second method is to employ one or more combinatorial lower bounds in conjunction with our lower bound, provided that such bounds can be derived. The running time of a combinatorial lower bound is typically negligible, while No. nodes at depth Fig (a) TI MMT MMT/DPD No. jobs fixed (depth) No. nodes at depth No. jobs fixed (depth) (b) Comparison of max-min transportation bound and time-indexed bound TI MMT MMT/DPD

12 554 Y. Pan, L. Shi the quality may not be satisfactory. However, it seems that some combinatorial lower bounds can complement our lower bound. For example, we have observed that the pathological behavior occurs mostly on instances of 1 w j T j that are considered to be easy for a combinatorial lower bound in [17]. Further, we note that the pathological behavior tends to occur when initial lower bounds are poor, but nodes can be fathomed effectively using problem-specific dominance rules or general-purpose dynamic programming dominance techniques. The curves (MMT/DPD) in Fig. 1a, b show the effectiveness of a partial dynamic programming (PDP) dominance technique [16] in helping tame the erratic tree growth when it is used alongside MMT. The solution time is relatively small for this type of instance. In our experiments, we encountered few occasions when timeconsuming measures like recomputing the transportation costs are justified. 4 Branch-and-bound algorithms for 1 w j T j We incorporate the max-min transportation lower bound in a branch-andbound algorithm to solve the problem 1 w j T j, which is unary NP-hard. Now the job cost function takes the form f j (t j ) = w j max{0, t j d j }, where d j is a given due date. 4.1 Algorithms Our proposed lower bound (MMT) and three previous bounds are wrapped in a branch-and-bound framework presented in [1] and then compared computationally. The three existing bounds are the time-indexed LP bound (TI), Potts and Van Wassenhove s combinatorial bound (PVW) [17], and Lawler s transportation lower bound (LT) with the costs c jt determined in accordance with (4) and the free variable values defined as c jt = 0, t = 0,..., p 2; c jt = c j(t pj ) + f j (t + 1) f j (t), t = p 1,..., T 1 (for all j). By varying the configuration of branch-and-bound, we obtain a number of different algorithms that are meant to reveal various characteristics of the lower bounds. All the algorithms use a standard positional node representation and branching scheme. The strategy for selecting a node to branch on is a new variation of the depth-first strategy: When more than one node have the same lower bound, select the node whose most recently fixed job has a due date that is closest to the due date of the job occupying the same position in the incumbent solution. In addition to the upper bound heuristics discussed in [1], we also compute the best-α schedule [10] and improve upon it using the standard adjacent pairwise interchange heuristic. The algorithms were coded in C++ and tested on a Pentium IV 2.8 GHz Linux PC with 1 GB RAM. LPs are solved with QSopt. 4.2 Experimental design We consider four sets of instances. Let us begin with the last set, which is taken from OR-Library. There are three control parameters in the process of

13 Equivalence of max-min transportation bound and time-indexed bound 555 generating these instances: n {40, 50, 100}, RDD {0.2, 0.4, 0.6, 0.8, 1.0}, and TF {0.2, 0.4, 0.6, 0.8, 1.0}. There are five instances for each combination of the values. All the 40/50-job instances can be solved optimally using the algorithm in [1] that employs PVW [17]. According to [11], there were no attempts to solve the 100-job instances, since CPU times were expected to be prohibitively large. We derived the other three sets of instances from the above instances through time scaling. Let SF be a scaling factor. For each of the 375 instances with p j U[1, 100] for all j, we modify the instance by setting p j p j /SF and d j d j /SF. The other input of an instance stays unchanged. The above set of instances corresponds to the case where SF = 1. Via this scheme, three more sets of instances were obtained with SF = 10, 5, 2. Thus, the processing time ranges, [1, p max ], p max = 10, 20, 50, 100, are all represented. 4.3 Results Our initial set of experiments was designed to evaluate the relative strength of MMT in relation to the other bounds. Let z be the optimal value and let z LB be a lower bound. Then, the percentage gap is defined as g LB = (z z LB )/z 100 if z > 0. Instances with z = 0 are trivial; in that event, we define g LB = 0. g avg LB and glb max stand for the average and maximum values. We first compare the quality of PVW, LT, and MMT at the root. TI and MMT are equal at the root. The fourth test set (p max = 100) is used in this experiment. For the same combination (n, TF) we report g avg LB and gmax LB computed over 25 instances. TF is the dominant factor governing the difficulty of instances of the same n. In Table 1, g avg MMT never exceeds 2%. Large gmax MMT values are observed for TF = 0.4 across all n. Nevertheless, the instances with TF = 0.4 are actually easy, despite large initial gaps. Table 1 Lower bound quality at root node (p max = 100) n TF g avg PVW g max PVW g avg LT g max LT g avg MMT g max MMT

14 556 Y. Pan, L. Shi Next, we examine the quality of LT, MMT, and TI as search progresses. We take the tree size generated by each lower bound as an indicator of bound strength. In this experiment, each lower bound was used independently, and all pruning devices were turned off except trivial ones. The initial upper bound was set to the optimal value. The set of instances with (n = 40, p max = 10) is used since it is impractical for TI to deal with larger problems. Also, results for PVW were omitted because its tree size is orders of magnitude larger. In Table 2, n avg LB and nmax LB denote the average and maximum numbers of nodes in the tree for lower bound LB. We can see that MMT generates markedly fewer nodes than LT. The extent of reduction is most spectacular in the parameter range 0.6 RDD 1, 0.6 TF 1. Within the same range, MMT generates 2 4 times more nodes than TI. This is a strong indication that on average MMT is a fairly good approximation of TI. The results thus far indicate that the max-min transportation lower bound can be effectively used in branch-and-bound in a stand-alone manner. The idea of re-using optimal transportation cost structure found at the root node is indeed viable computationally. Now we present a specialized algorithm that employs the best combination of MMT and other existing techniques. Results on the fourth (and the most difficult) set of instances are reported. In Table 3, the CPU times in seconds Table 2 Tree sizes of using a single lower bound (n = 40, p max = 10) RDD TF n avg LT n max LT n avg MMT n max MMT n avg TI n max TI ,369 5, ,483 1, ,142 10, ,418 2, ,718 10, ,087 34, ,455 10, ,863 46, ,721 15, ,752 47, ,347 1,082, , , ,326 7,

15 Equivalence of max-min transportation bound and time-indexed bound 557 Table 3 CPU time and node counts of specialized algorithm (p max = 100) n TF NO t avg t max t avg 2 t max 2 n avg n max required by the algorithm are denoted by t avg and t max, and further broken down into t avg 2 and t2 max after the root node times have been subtracted. n avg and n max are the node counts. NO shows the numbers of instances (out of 25) solved optimally. Within the 4-hour per instance time limit, our algorithm solved to optimality all but one of the 125 open instances with 100 jobs. The remaining instance (#44) was also solved optimally after 9 hours of computation. It turns out that for all these instances, their best known upper bounds are in fact optimal. Additionally, in our other experiments, we found that the proposed bound demonstrates good scalability with increase of n and p max. 5 Conclusion We presented new results as to finding the strongest bound from the family of lower bounds provided by transportation problem relaxations of the singlemachine scheduling problem. As an application of the theory, we proposed a new scheme to approximate the time-indexed bound during branch-and-bound. The theory behind our results should bring due attention to the study of the LP dual problem (P1 ) associated with (TI) the LP relaxation of the time-indexed formulation. (P1 ) is related to the transportation problem (P(c)) via (P1). An immediate impact of recognizing these relations has been unambiguously illustrated through our computational results. The proposed lower bound is based on the family of transportation problem relaxations. Interestingly, an alternative assignment problem relaxation has been suggested in [20], where constraints (1) of (P(c)) are replaced by (T s)/p j k=0 x j(s+kpj ) = 1, j = 1,..., n; s = 0,..., p j 1. This relaxation is at least as strong as (P(c)) because the above constraints imply (1). Our lower bound idea can again be applied while condition (5) is imposed as before. We

16 558 Y. Pan, L. Shi find that it is counter-intuitive, but nevertheless true, that the resulting bound is also equal to the time-indexed lower bound. Some issues are yet to be fully explored. First, we should be able to speed up the root node computation. A second issue aries from the fact that more than one set of optimal (u, v, c) can attain the same lower bound at the root. This issue has been partially addressed, but needs to be further examined. In parallel with our work, there are other efforts to leverage the bound quality of the time-indexed formulation while reducing its computational expense. For example, Bigras et al. [8] suggest temporal decomposition for the column generation algorithm for (TI). On a similar computer, they were able to solve 89 of the 125 open instances within a 12-h time limit per instance. We are looking into incorporating the idea in our method. Acknowledgements The authors would like to thank Professor Nemhauser for an insightful discussion, Professor Jünger for pointing out the alternative assignment formulation, the associated editor for the encouraging comments, and two anonymous referees for their helpful suggestions. This research was supported in part by NSF (DMI , DMI , and DMI ), by AFOSR (FA ), and by John Deere Horicon Works. References 1. Abdul-Razaq, T.S., Potts, C.N., van Wassenhove, L.N.: (1990) A survey of algorithms for the single machine total weighted tardiness scheduling problem. Discrete Appl. Math. 26, van den Akker, J.M., van Hoesel, C.P.M., Savelsbergh, M.W.P.: (1999) A polyhedral approach to single-machine scheduling problems. Math. Program. 85, van den Akker, J.M., Hurkens, C.A.J., Savelsbergh, M.W.P.: (2000) Time-indexed formulations for machine scheduling problems: Column generation. INFORMS J. Comput. 12(2), Applegate, D., Cook, W.: (1991) A computational study of the job-shop scheduling problem. ORSA J. Comput. 3, Avella, P., Boccia, M., D Auria, B.: (2005) Near-optimal solutions of large-scale single-machine scheduling problems. INFORMS J. Comput. 17(2), Balas, E.: (1969) Machine sequencing via disjunctive graphs: an implicit enumeration algorithm. Oper. Res. 17, Balas, E.: (1979) Disjunctive programming. Ann. Discrete Math. 5, Bigras, L., Gamache, M., Savard, G.: (2005) Time-indexed formulations and the total weighted tardiness problem. G , Les Cahiers du GERAD 9. Bülbül, K., Kaminsky, P., Yano, C.: (2005) Preemption in single machine earliness/tardiness scheduling. J. Sched. Forthcoming 10. Chekuri, C., Motwani, R., Natarajan, B., Stein, C.: (2001) Approximation techniques for average completion time scheduling. SIAM J. Comput. 31, Congram, R.K., Potts, C.N., van de Velde, S.L.: (2002) An iterated dynasearch algorithm for the single-machine total weighted tardiness scheduling problem. INFORMS J. Comput. 13(1), Dyer, M.E., Wolsey, L.A.: (1990) Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Appl. Math. 26, Gelders, L., Kleindorfer, P.R.: (1974) Coordinnating aggregate and detailed scheduling in the one-machine job shop: Part I. Theory. Oper. Res. 22, Lawler, E.L.: (1964) On scheduling problems with deferral costs. Management Sci. 11(2), Nemhauser, G.L., Wolsey, L.A.: (1988) Integer and Combinatorial Optimization. John Wiley, NY 16. Pan, Y.: (2003) An improved branch and bound algorithm for single machine scheduling with deadlines to minimize total weighted completion time. Oper. Res. Lett. 31(6),

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