Dominance inequalities for scheduling around an unrestrictive common due date
aa r X i v : . [ c s . D M ] F e b Dominance inequalities for schedulingaround an unrestrictive common due date
Anne-Elisabeth Falq a, ∗ , Pierre Fouilhoux a, ∗ , Safia Kedad-Sidhoum b, ∗ a Sorbonne Université, CNRS, LIP6, 4 Place Jussieu, 75005 Paris, France b CNAM, CEDRIC, 292 rue Saint Martin, 75141 Paris Cedex 03, France
Abstract
The problem considered in this work consists in scheduling a set of tasks on a sin-gle machine, around an unrestrictive common due date to minimize the weightedsum of earliness and tardiness. This problem can be formulated as a compact mixedinteger program (MIP). In this article, we focus on neighborhood-based dominanceproperties, where the neighborhood is associated to insert and swap operations. Wederive from these properties a local search procedure providing a very good heuris-tic solution. The main contribution of this work stands in an exact solving context:we derive constraints eliminating the non locally optimal solutions with respect tothe insert and swap operations. We propose linear inequalities translating theseconstraints to strengthen the MIP compact formulation. These inequalities, calleddominance inequalities, are di ff erent from standard reinforcement inequalities. Weprovide a numerical analysis which shows that adding these inequalities signifi-cantly reduces the computation time required for solving the scheduling problemusing a standard solver. Keywords: scheduling, integer programming, common due date, dominanceproperties
1. Introduction
The scheduling problem studied in this work falls within the just-in-time schedul-ing field. In this framework, each task has a due date, and any deviation from this ∗ Corresponding author
Email addresses: [email protected] (Anne-Elisabeth Falq), [email protected] (Pierre Fouilhoux), [email protected] (SafiaKedad-Sidhoum)
Preprint submitted to EJOR February 16, 2021 ue date is penalized. From one hand, the tardiness is penalized to model the cus-tomer dissatisfaction, on the other hand, the earliness is penalized to model the in-duced storage costs . The reader can refer to the seminal surveys of Baker & Scudder(1990) and Kramer & Subramanian (2019) for the early results in this field.We consider a set J of n tasks with fixed processing times ( p j ) j ∈ J ∈ R J + , to benon-preemptively processed on a single machine. These tasks share a common duedate d for which tasks should be preferably completed. In this work, we assumethat the due date is unrestrictive , i.e. d > P p j .In a just-in-time framework, tasks completing before or after d will thereforeincur penalties according to unit earliness (resp. tardiness) penalties ( α j ) j ∈ J ∈ R J + (resp. ( β j ) j ∈ J ∈ R J + ). A schedule is defined by the task completion times denotedby ( C j ) j ∈ J . Using [ x ] + to denote the positive part of x ∈ R , the earliness (resp. tardiness ) of a task j ∈ J is given by [ d − C j ] + (resp. [ C j − d ] + ). Given parameters d , p , α and β , the Unrestrictive Common Due Date Problem (UCDDP) aims atfinding a schedule minimizing the total penalty : X j ∈ J α j [ d − C j ] + + β j [ C j − d ] + This criterion is non-regular, i.e. is not a non-increasing function of completiontime C j for any j ∈ J . Moreover, the penalty function is not a linear function of thecompletion times.If all task penalties are equal, i.e. α j = β j for any task j ∈ J , the UCDDP issolvable in polynomial time (Kanet, 1981). If task penalties are symmetric, i.e. α j = β j for all j ∈ J , the problem is NP-hard (Hall & Posner, 1991). Therefore, theproblem that we consider with arbitrary α and β coe ffi cients is also NP-hard.A wide range of variables can be used to formulate a single machine schedul-ing problem as a Mixed Integer Program (MIP): completion time variables, time-indexed variables, linear ordering variables, positional date and assignment vari-ables (Queyranne & Schulz, 1994). However, only few works based on linearprogramming are proposed for the UCDDP. Biskup & Feldmann (2001a) pro-pose a compact MIP formulation based on disjunctive variables, which is onlygiven to compare the proposed heuristic method to an exact method for small in-stances. van den Akker et al. (2002) propose a formulation based on an exponen-tial number of binary variables using column generation and Lagrangian relaxation.In Falq et al. (2021), we propose a MIP based on natural variables, similar to com-pletion time variables, together with a compact MIP based on partition variables.2urthermore, the UCDDP have been solved through several other dedicatedexact methods like branch-and-bound algorithms ( e.g. Sourd (2009)) and dynamicprogramming methods ( e.g.
Hall & Posner (1991), Hoogeveen & van de Velde (1991),Tanaka & Araki (2013)). Moreover, a benchmark is provided in Biskup & Feldmann(2001a), together with a heuristic method. An important remark is that the e ffi -ciency of the latter approaches comes from the exploitation of dominance proper-ties. In particular, the dedicated Branch-and-Bound proposed in Sourd (2009) foran exact solving of UCDDP, solves up to 1000-tasks instances of the benchmarkprovided in Biskup & Feldmann (2001a).Among the dominance properties, we can distinguish the structural ones, whichallow to restrict the solution set to the set of the schedules having a specific struc-ture. These properties are already taken into account in the compact formulationthat we propose in Falq et al. (2021). Indeed, thanks to their structure, the dom-inant schedules are encoded by only n binary variables resulting in a formulationwhose size does not depend on the time horizon, in contrast with the time-indexedformulations. In the present work, we focus on another type of dominance propertycalled neighborhood based dominance properties. We propose to model them bylinear inequalities.One important contribution of this article is to propose inequalities strength-ening the linear formulation in a new way. They are di ff erent from standard re-inforcement and symmetry-breaking inequalities. The standard reinforcement in-equalities cut fractional points to improve the lower bound obtained from the linearrelaxation, the symmetry-breaking inequalities cut integer points, - which may beoptimal- to reduce the search space. In contrast, Dominance inequalities aims atcutting sub-optimal integer points.This article is organized as follows. In Section 2, we recall structural domi-nance properties leading to reformulate the UCDDP into a partition problem; thenwe adapt some neighborhood based dominance properties used on schedules forpartitions. Section 3 translates in a linear way the constraints proposed in Sec-tion 2, which leads to a new compact linear formulation, enriched with dominanceinequalities. Section 4 presents some experimental results to show the contributionof considering dominance in an exact solving process, as in a heuristic approach.
2. Dominance properties
A subset of solutions is said dominant if it contains at least one optimal solu-tion, and strictly dominant if it contains all the optimal solutions. For brevity, a3chedule will be said dominant (resp. strictly dominant) if it belongs to a dominant(resp. strictly dominant) set. In this article, we will only use dominance propertiesabout solutions, even if there also exist dominance properties about instances orproblems (Jouglet & Carlier, 2011).
In order to describe dominant schedules, we first provide some useful defini-tions. A task j ∈ J is said early (resp. tardy ) if it completes at time C j d ( resp. C j > d ). Among the early tasks, the one completing at d (if it exists), is called the on-time task. The early-tardy partition of a schedule is the pair ( E , T ) where E (resp. T ) is the early (resp. tardy) task subset. Moreover, we will use α -ratio (resp. β -ratio ) to designate α j / p j (resp. β j / p j ). We define a block as a feasible schedulewithout idle time between task execution, a d-schedule as a feasible schedule withan on-time task, and a d-block as a d -schedule which is also a block. A schedule issaid V-shaped if early tasks are scheduled in non-decreasing order of their α -ratiosand the tardy ones in non-increasing order of their β -ratios.The following lemma gives dominance properties already known for the un-restrictive common due date problem with symmetric penalties (Hall & Posner,1991). These results have been extended to asymmetric penalties in Falq et al.(2021), using the same task shifting and exchange proof arguments. Lemma 1 (
Falq et al. (2021) ) The setof V-shaped schedules isstrictly dominant forthe UCDDP.The set of d -blocks is dominant for the UCDDP. Moreover, if unit earlinessandtardiness penaltiesarepositive, i.e. ( α, β ) ∈ R ∗ + J × R ∗ + J ,thenthesetofblocksis strictly dominant.Thanks to the d -block dominance, we can make some assumptions on param-eters p , α, β without loss of generality. A task having a zero processing time canbe inserted between two tasks of a schedule without impacting other tasks. In par-ticular in a d -block, such a task can be inserted at the due date, which incurs nopenalty variation. Hence, we assume that ∀ j ∈ J , p j ∈ R ∗ + .Since the due date d is large enough, a task having a zero earliness (resp. tardiness)penalty can be added at the beginning (resp. at the end) of a d -block without incur-ring any cost variation. Hence, we assume that ∀ j ∈ J , ( α j , β j ) ∈ ( R ∗ + ) .Since they form a dominant set, we only consider V-shaped d -blocks in thefollowing. Note that in such schedules, tardy tasks are completely processed after d , since there is no straddling task, i.e. there is no task starting before d and com-pleting after d . Therefore, the early (resp. tardy) task set can be referred to the left4resp. right) side of d .In general, a V-shaped d -block cannot be encoded by its early-tardy partition.Indeed, if two early (resp. tardy) tasks have the same α -ratio (resp. β -ratio), onecannot determine which one is processed first, they can be sequenced arbitrarily.However, swapping two such tasks in a V-shaped d -block results in another V-shaped d -block having the same penalty. Furthermore, all the V-shaped d -blockhaving the same early-tardy partition have the same penalty. Based on this remark,two V-shaped d -blocks having the same early-tardy partition will be said equiva-lent . We denote by ∼ this relation.We define an ordered bi-partition of a set A as a couple ( A , A ) where { A , A } is a partition of A . This is not only a partition into two subsets since the two subsetsare not symmetric, i.e. ( A , A ) , ( A , A ). Note that the early-tardy partition ofa schedule is an ordered bi-partitions. More precisely, there is a one-to-one corre-spondence between equivalence classes for ∼ and ordered bi-partitions ( E , T ) of J where E , ∅ . Indeed, the set of the early tasks of a d -block cannot be empty sinceit contains at least the on-time task. Let ~ P ∗ ( J ) denote the set of such ( E , T ), and ~ P ( J ) denote the set of any ordered bi-partition of J . In the sequel, we will onlysay partition to refer to an ordered bi-partition or to an early-tardy partition, whenthere is no ambiguity.For any partition ( E , T ) ∈ ~ P ∗ ( J ), let f ( E , T ) denote the penalty of the equiva-lence class ( E , T ), that is the penalty of any V-shaped d -block of partition ( E , T ).For sake of consistency, we extend the definition of f to ~ P ( J ) by setting f ( ∅ , J ) asthe penalty of any V-shaped block starting at time d . From now on, our aim is tofind a partition of J minimizing f , since the UCDDP can be formulated as follows.( F ) : min ( E , T ) ∈ ~ P ∗ ( J ) f ( E , T ) Let us recall some definitions used in local search context (Aarts & Lenstra, 2003).A neighborhood function N is a function which associates to any solution S a sub-set of solutions N ( S ), called the neighborhood of S . A solution of N ( S ) is called a neighbor of S . Moreover, a solution S is locally optimal with respect to minimiz-ing function φ if φ ( S ) φ ( S ′ ), for any neighbor S ′ ∈ N ( S ). If, on the contrary,there exists S ′ ∈ N ( S ) such that φ ( S ′ ) < φ ( S ), S is dominated (by S ′ ).5iven a neighborhood function, the set of locally optimal solutions always con-tains all optimal solutions, and is therefore a strictly dominant set. This statementcan be seen as a generic dominance property denoted G . This kind of dominanceproperty is commonly used in local search. Indeed, a step of a local search proce-dure consists in enumerating some of the neighbors of a given solution S , comput-ing their values and, if a better solution is found, then moving to the best solutionfound at the current iteration, which is equivalent to discard S since it is dominated. • Operation-based neighborhoods
We call operation any (eventually partial) function, which maps a solution toanother solution. In this work, we will consider neighborhood functions based on aset of operations. The neighborhood of a solution S is then the set of the solutionsobtained by applying to S any operation defined on S . This kind of neighborhoodfunctions allows to use the generic dominance property G in a di ff erent way. In-stead of considering sequentially the neighborhood of each solution, we will con-sider sequentially each operation. For each one, any solution is compared to itsneighbor obtained using this operation. This di ff erentiates the solution-centeredand the operation-centered point of view, as developed in the next section / point. • The solution-centered and the operation-centered points of view
Figures 1 and 2 illustrate these two points of view on the same set of solutions { A , B , . . . , N } represented by the blue points. We consider a given set of operationsand the associated neighborhood function. An arrow is set from a solution X to asolution Y if X is compared to Y in order to determine whether X is dominated by Y .In Figure 1, we focus on one solution, J , which is compared to all the solutionsobtained by applying to J an operation defined on J . Since J is compared to all itsneighborhood, i.e. { C , F , G , I , M , N } , one can determine if J is locally optimal.In Figure 2, we focus on one operation. All the solutions where this operationcan be applied are compared with the obtained neighbors. Solutions A , B , C , G and N are not compared to others solutions, since the considered operation cannot beapplied on these solutions. Conversely, since solutions D , E , F , H , I , J , K , L and M are compared to one of their neighbors, they might be discarded. For exampleif E is better than H , H can be discarded, no matter whether H is better than K .However, we cannot say that the non-discarded solutions are locally optimal, sinceonly one neighbor is taken into account. To this end, all the operations must beconsidered. 6 A • B • C • D • E • F • G • H • J • K • L • M • N • I Figure 1: Illustration of the solution-centered point of view • A • B • C • D • E • F • G • H • I • J • K • L • M • N Figure 2: Illustration of the operation-centered point of view
Let us now present the main ideas of a generic method to handle operation-based dominance property. First, find operations inducing an objective functionvariation that can be explicitly expressed. Then derive, for each operation, a con-straint ensuring that any solution is not dominated by applying this operation. Us-ing all the obtained constraints, only the dominant set of property G subsists as allthe non-locally optimal solutions have been removed (where local is understoodwith respect to the neighborhood defined by the operations).In Section 2.3, we introduce two families of operations over the partitions, and fol-low this method, resulting in constraints ( I u ), ( I ′ u ) and ( S u , v ) translating Property 3. In the case of problems where the solutions are partitions, two operations arecommonly considered to define a neighborhood: the insertion , which consists inmoving two elements, one in each subset and the swap which consists in swappingtwo elements of each subset. Let us define solutions which are locally optimal inthe neighborhoods associated with these operations.
Definition 2
Let ( E , T ) ∈ ~ P ∗ ( J ).(i) ( E , T ) isan insert local optimum if ∀ v ∈ T , f (cid:0) E , T (cid:1) f (cid:0) E ∪{ v } , T \{ v } (cid:1) , ∀ u ∈ E , f (cid:0) E , T (cid:1) f (cid:0) E \{ u } , T ∪{ u } (cid:1) .(ii) ( E , T ) isa swap local optimum if ∀ u ∈ E , ∀ v ∈ T , f (cid:0) E , T (cid:1) f (cid:0) E \{ u }∪{ v } , T ∪{ u }\{ v } (cid:1) . Property 3
The set of insert locally optimal partitions, as well as the set of swap locallyoptimal partitions, isstrictly dominant whenminimizing f over ~ P ∗ ( J ).7n terms of scheduling, the insert operation consists in removing a task j ∈ J from the early (resp. tardy) side and inserting j on the tardy (resp. early) side,as early (resp. as tardy) as possible according to its β -ratio (resp. α -ratio). Thetasks scheduled before (resp. after) j are right-shifted (resp. left-shifted) by p j timeunits. The swap operation consists in sequentially inserting an early task on thetardy side and inserting another tardy task on the early side, or vice-versa, as de-scribed above. Let us denote by insert u ( S ) (resp. swap u , v ( S )) the schedule obtainedfrom a schedule S by the insert (resp. swap) operation on task u (resp. on tasks u , v ).Since the insert and swap operations are fundamentally defined on partitions( i.e. over equivalence classes of V-shaped d -blocks), these operations preserve theequivalence. Therefore we have ∀ u ∈ J , S ∼ S ′ ⇒ insert u ( S ) ∼ insert u ( S ′ ) and ∀ ( u , v ) ∈ J , S ∼ S ′ ⇒ swap u , v ( S ) ∼ swap u , v ( S ′ ), assuming that u and v are not onthe same side in these schedules.In order to provide an expression of the penalty variation induced by such anoperation, it is convenient to choose a specific V-shaped d -block as representativeof an equivalence class. More precisely, the chosen representative will depend onthe inserted task (resp. on the swapped tasks). Let us introduce some notations,related to a given task u ∈ J . To describe the early side of a V-shaped d -block withregard to u , the set of remaining tasks J \{ u } can be split according to their α -ratiointo two subsets A ( u ) and ¯ A ( u ) defined as follows. A ( u ) = ( i ∈ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i p i > α u p u ) and ¯ A ( u ) = ( i ∈ J \{ u } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i p i α u p u ) Similarly, we introduce the two following subsets to describe the tardy side. B ( u ) = ( i ∈ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β i p i > β u p u ) and ¯ B ( u ) = ( i ∈ J \{ u } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β i p i β u p u ) Note that if u is early in a V-shaped d -block, early tasks belonging to A ( u ) arenecessarily scheduled after u , because of their α -ratio. To illustrate, we can observein Figure 3 that the tasks of A ( u ) ∩ E are scheduled after u in the schedule S .Conversely, an early task of ¯ A ( u ) is not necessarily scheduled before u , when its α -ratio is the same as u . However, we will consider a representative where u isplaced after all the early tasks of ¯ A ( u ), that is as tardy as possible according to its α -ratio. Similarly, in case where u is tardy, we will consider a representative where u is scheduled before all tardy tasks of ¯ B ( u ). Such a representative will be calleda u - canonical V-shaped d -block. The schedule S (resp. S ′ ) in Figure 3 representsthe shape of a u -canonical V-shaped d -block when u is early (resp. tardy).8 Penalty variation induced by an insert operation
Given u ∈ J , let ( E , T ) be a partition such that u ∈ E . We aim to express the variationof f induced by the insertion of u in T, i.e. between ( E , T ) and ( E \{ u } , T ∪{ u } ). Tothis end, we consider a u -canonical representative S of ( E , T ), and the V-shaped d -block S ′ obtained from S by inserting u in T , as early as possible, that is justafter tardy tasks of B ( u ). Note that S ′ is thus a u -canonical representative of (cid:0) E \ { u } , T ∪ { u } (cid:1) . Let ( e , t ) (resp. ( e ′ , t ′ )) denote the earliness and tardiness vec-tor of tasks in S (resp. S ′ ).As we can observe in Figure 3, early tasks of A ( u ) and tardy tasks of B ( u ) areidentically scheduled in S and S ′ . The penalty variation induced by the insertionof u is then only due to the move of u and to the right-shifting of early tasks of ¯ A ( u )and tardy tasks of ¯ B ( u ). S ¯ A ( u ) ∩ E A ( u ) ∩ E B ( u ) ∩ T ¯ B ( u ) ∩ Tdu S ′ ¯ A ( u ) ∩ E A ( u ) ∩ E B ( u ) ∩ T ¯ B ( u ) ∩ Td up u p u Figure 3: Illustration of the insert operation of an early task u on the tardy side Each task j ∈ ¯ A ( u ) ∩ E of S is postponed p u time units later in S ′ , while stayingearly. We then have e ′ j = e j − p u . The earliness penalty of task j in S ′ is therefore α j e ′ j = α j e j − α j p u , which represents a reduction of α j p u compared to its earli-ness penalty in S . Summing up over ¯ A ( u ) ∩ E , we obtain a reduction of the earlinesspenalties of p u α (cid:0) ¯ A ( u ) ∩ E (cid:1) , where x ( I ) = P i ∈ I x i for any x ∈ R n and any I ⊆ [1 .. n ],Similarly, the right-shifting of tasks in ¯ B ( u ) ∩ T induces an increase of the tardinesspenalties of p u β (cid:0) ¯ B ( u ) ∩ T (cid:1) , since for each j ∈ ¯ B ( u ) ∩ T we have β j t ′ j = β j ( t j + p u ).Moreover, since e u = p (cid:0) A ( u ) ∩ E (cid:1) , removing u from the early side induces areduction of its earliness penalty of α u p (cid:0) A ( u ) ∩ E (cid:1) . Similarly, introducing u on thetardy side induces an increase of its tardiness penalty of β u t ′ u = β u (cid:16) p (cid:0) B ( u ) ∩ T (cid:1) + p u (cid:17) since t u = S and S ′ , is given by the following expres-sion. ∆ u ( E , T ) = − α u p (cid:0) A ( u ) ∩ E (cid:1) + β u (cid:16) p (cid:0) B ( u ) ∩ T (cid:1) + p u (cid:17) + p u (cid:16) β (cid:0) ¯ B ( u ) ∩ T (cid:1) − α (cid:0) ¯ A ( u ) ∩ E (cid:1)(cid:17) S and S ′ are representative of ( E , T ) and ( E \{ u } , T ∪{ u } ) respectively, ∆ u ( E , T )is also the variation of f induced by the insertion of u in T . Property 4.(i) follows.The insert operation of the tardy task u on the early side applied on schedule S ′ provides schedule S . Note that this statement requires that S is u -canonical. Thisobservation allows to establish that the penalty variation induced by inserting u onthe early side in S ′ is simply − ∆ u ( E , T ), and results in Property 4.(ii). Property 4
Let ( E , T ) be apartition.(i) Forany u ∈ E , f ( E \{ u } , T ∪{ u } ) = f ( E , T ) + ∆ u ( E , T ).(ii) Forany v ∈ T , f ( E ∪{ v } , T \{ v } ) = f ( E , T ) − ∆ v ( E , T ).Let us introduce, for a given task u ∈ J , the two following constraints. u ∈ E ⇒ ∆ u ( E , T ) > I u ) u ∈ T ⇒ ∆ u ( E , T ) I ′ u )Thanks to Property 4, ( I u ) (resp. ( I ′ u )) discards every partition where u is early(resp. tardy), but which would have a lower penalty if u were tardy (resp. early).Note that the constraint ∆ u ( E , T ) > ∆ u ( E , T )
0) might be not satisfied byan optimal solution where u is tardy (resp. early).Moreover, only considering constraints ( I u ) and ( I ′ u ) for a given u is not su ffi cientto discard all insert-dominated partitions. Indeed, these constraints do not takeinto account the whole insert-neighborhood of the solutions as each one is onlycompared to its neighbor obtained by an insert operation on task u . It is thus neededto consider constraints ( I u ) and ( I ′ u ) for every u ∈ J to translate the dominance of theinsert local optimal solutions. In Section 3, we will explain how these constraintscan be used in a linear formulation. • Penalty variation induced by a swap operation
This section follows the same organization as the previous one. The objective is toobtain constraints translating the dominance of the swap locally optimal solutions.Given ( u , v ) ∈ J such that u , v , let ( E , T ) be a partition such that u ∈ E and v ∈ T , and S be a u and v -canonical representative of ( E , T ). We denote by S ′ the schedule obtained from S by swapping u and v so that S ′ is also both u and v -canonical, that is by scheduling u after all the early tasks having the same α -ratio α u / p u and v before all the tardy tasks having the same β -ratio β v / p v . S ′ is a repre-sentative of (cid:0) E \{ u }∪{ v } , T \{ v }∪{ u } (cid:1) . Let ( e , t ) (resp. ( e ′ , t ′ )) denote the earliness10nd tardiness vector of tasks in S (resp. S ′ ).If α v p v < α u p u , all the early tasks of ¯ A ( v ) are scheduled before u in S , since theV-shaped property holds, but the early tasks of A ( v ) \{ u } can be scheduled beforeor after u . This case is illustrated in Figure 4. Conversely, if α v p v > α u p u , all the earlytasks of A ( v ) are scheduled after u in S , but those of ¯ A ( v ) \ { u } can be scheduledbefore or after u . This case is illustrated in Figure 5. A ( v ) ∩ ¯ A ( u ) ∩ E S ¯ A ( u ) ∩ E A ( u ) ∩ EA ( v ) ∩ E ¯ A ( v ) ∩ E d | u S ′ ¯ A ( u ) ∩ E ∪{ v } A ( u ) ∩ EA ( v ) \{ u }∩ E ¯ A ( v ) ∩ E d | v Figure 4: Early side variation induced byswapping u and v when α v p v < α u p u A ( u ) ∩ ¯ A ( v ) ∩ E S ¯ A ( u ) ∩ E A ( u ) ∩ EA ( v ) ∩ E ¯ A ( v ) ∩ E d | u S ′ ¯ A ( u ) ∩ E A ( u ) ∩ E ∪{ v } A ( v ) ∩ E ¯ A ( v ) \{ u }∩ E d | v Figure 5: Early side variation induced byswapping u and v when α v p v > α u p u B ( v ) ∩ ¯ B ( u ) ∩ T S B ( v ) ∩ T ¯ B ( v ) ∩ TB ( u ) ∩ T ¯ B ( u ) ∩ T ¯ B ( v ) ∩ Td | v S ′ B ( v ) ∩ T ∪{ u } ¯ B ( v ) ∩ TB ( u ) ∩ T ¯ B ( u ) ∩ T \{ v } d | u Figure 6: Tardy side variation induced byswapping u and v when β v p v β u p u B ( u ) ∩ ¯ B ( v ) ∩ T S B ( v ) ∩ T ¯ B ( v ) ∩ TB ( u ) ∩ T ¯ B ( u ) ∩ Td | v S ′ B ( v ) ∩ T ¯ B ( v ) ∩ T ∪{ u } B ( u ) ∩ T \{ v } ¯ B ( u ) ∩ Td | u Figure 7: Tardy side variation induced byswapping u and v when β v p v > β u p u Similarly, we can distinguish two cases depending on the relative order of the β -ratios of u and v , as illustrated in Figures 6 and 7.As shown in Figures 4 and 5, the earliness of u in S is e u = p (cid:0) A ( u ) ∩ E (cid:1) , whilethe tardiness of u in S ′ is e ′ u = p (cid:0) B ( u ) \{ v }∩ T (cid:1) + p u ( Cf.
Figures 6 and 7). Note that v is removed from B ( u ) since v is not tardy in S ′ , and therefore cannot contribute tothe tardiness of u . Similarly, we have t v = p (cid:0) B ( v ) ∩ T (cid:1) + p v and e ′ v = p (cid:0) A ( v ) \{ u }∩ E (cid:1) since u is not early in S ′ . 11oreover, the tasks of A ( u ) ∩ E are identically scheduled in S and S ′ only if α v p v < α u p u . In this case, tasks of ¯ A ( u ) ∩ E are not consecutive in S ′ since v separatesthem into two blocks: tasks of ¯ A ( v ) ∩ E which have been left-shifted by p v − p u timeunits, and tasks of A ( v ) ∩ ¯ A ( u ) ∩ E which have been right-shifted by p u time units( Cf.
Figure 4).In the opposite case, i.e. if α v p v > α u p u , tasks of A ( u ) ∩ E are not consecutive in S ′ .Indeed, v separates them into two blocks: tasks of A ( v ) ∩ E which are identicallyscheduled in S and S ′ ; and tasks of A ( u ) ∩ ¯ A ( v ) ∩ E which are left-shifted by p v timeunits. Moreover, in that case tasks of A ( v ) ∩ E are left-shifted by p v − p u time units( Cf.
Figure 5).Note that, in the two previous paragraphs, as in Figures 4 to 7, we assumed that p v − p u >
0. In the contrary case, i.e. if p v − p u <
0, tasks are not left-shifted by p v − p u time units but rather right-shifted by p u − p v time units.From these observations, we can express the earliness penalty variation of allthe tasks in E \ { u } by expressing, in each case, the penalty variation induced bya block shifting as described for the insert operation. The same method can beapplied for the tasks of T \ { v } . The reader can refer to Figures 6 and 7 for illus-tration. Finally, the penalty variation between S and S ′ is given by the followingexpression. ∆ u , v ( E , T ) = − α u p (cid:0) A ( u ) ∩ E (cid:1) + β u (cid:16) p (cid:0) B ( u ) \{ v }∩ T (cid:1) + p u (cid:17) − β v (cid:16) p (cid:0) B ( v ) ∩ T (cid:1) + p v (cid:17) + α v p (cid:0) A ( v ) \{ u }∩ E (cid:1) + ( − p u + p v ) α (cid:0) ¯ A ( v ) ∩ E (cid:1) − p u α (cid:0) A ( v ) ∩ ¯ A ( u ) ∩ E (cid:1) if α v p v < α u p u ( − p u + p v ) α (cid:0) ¯ A ( u ) ∩ E (cid:1) + p v α (cid:0) A ( u ) ∩ ¯ A ( v ) ∩ E (cid:1) otherwise + ( p u − p v ) β (cid:0) ¯ B ( v ) ∩ T (cid:1) + p u β (cid:0) B ( v ) ∩ ¯ B ( u ) ∩ T (cid:1) if β v p v β u p u ( p u − p v ) β (cid:0) ¯ B ( u ) ∩ T (cid:1) − p v β (cid:0) B ( u ) ∩ ¯ B ( v ) ∩ T (cid:1) otherwise Property 5
Let ( E , T ) be apartition.Forany ( u , v ) ∈ E × T , f (cid:0) E \{ u }∪{ v } , T \{ v }∪{ u } (cid:1) = f ( E , T ) + ∆ u , v ( E , T ).For a given pair of tasks ( u , v ) ∈ J such that u , v , let us introduce the followingconstraint.( u , v ) ∈ E × T ⇒ ∆ u , v ( E , T ) > S u , v )12hanks to Property 5, ( S u , v ) discards every partition where u is early, v is tardy andwhich would have a lower penalty in the contrary case. As for the insert operation,constraint ∆ u , v ( E , T ) > u isnot early or v is not tardy. Moreover, it is needed to consider constraints ( S u , v ) forevery ( u , v ) ∈ J such that u , v to translate the dominance of swap locally optimalsolutions. In Section 3, we will explain how these constraints can be used in alinear formulation.
3. Neighborhood based dominance in linear programming
The dominance properties described in Section 2 can be used in a linear pro-gramming framework. In this section, we provide linear inequalities translatingconstraints ( I u ), ( I ′ u ) and ( S u , v ) for all tasks u and v . We first recall the linear compact MIP formulation for the UCDDP given in Falq et al.(2021). In this formulation denoted by F , a partition is encoded by a vector δ of n binary variables. Given δ ∈ { , } J , the partition encoded by δ is (cid:0) { j ∈ J | δ j = } , { j ∈ J | δ j = } (cid:1) . In other words, for each j ∈ J , δ j indicates whether j is early or not.Moreover, F also uses X , a vector of binary variables indexed by the set J < = { ( i , j ) ∈ J | i < j } , in order to linearize products of type δ i δ j as proposed by Fortet(1959). Lemma 6 (
Fortet (1959) ) Let ( δ, X ) ∈ R J × R J < . If δ ∈ { , } J and ( δ, X ) satisfies the following inequalities: ∀ ( i , j ) ∈ J < , X i , j > δ i − δ j (1) ∀ ( i , j ) ∈ J < , X i , j > δ j − δ i (2) ∀ ( i , j ) ∈ J < , X i , j δ i + δ j (3) ∀ ( i , j ) ∈ J < , X i , j − ( δ i + δ j ) (4) then ∀ ( i , j ) ∈ J < , X i , j = ( δ i , δ j − δ i )(1 − δ j ) = − ( δ i + δ j ) − X i , j δ i δ j = δ i + δ j − X i , j P = n ( δ, X ) ∈ R J × R J < (cid:12)(cid:12)(cid:12) (1 − o and the set of its in-teger points int δ ( P ) = P ∩{ , } J ×{ , } J < . From Lemma 6, if a partition ( E , T ) isencoded by δ , there exists a unique X such that ( δ, X ) ∈ int δ ( P ). We will say that( δ, X ) encodes ( E , T ). 13o obtain an expression of the penalty of a partition from its encoding ( δ, X ),two orders on J are introduced. Let ρ and σ be two functions from [1 .. n ] to J , suchthat α ρ ( k ) p ρ ( k ) ! k ∈ [1 .. n ] and β σ ( k ) p σ ( k ) ! k ∈ [1 .. n ] are non-increasingA d -block is said ρ - σ -shaped if the early (resp. tardy) tasks are processed in de-creasing order of ρ − (resp. increasing order of σ − ). Each equivalence class ofV-shaped d -blocks admits a unique ρ - σ -shaped representative. This representativeis used to provide the following expression of the penalty of the partition encodedby ( δ, X ) ∈ { , } J ×{ , } J < . g ( δ, X ) = X j ∈ J α j ρ − ( j ) − X k = p ρ ( k ) δ j + δ ρ ( k ) − X j ,ρ ( k ) + β j σ − ( j ) − X k = p σ ( k ) − ( δ j + δ σ ( k ) ) − X j ,σ ( k ) + p j (1 − δ j ) Finally, the compact formulation for the UCDDP provided in Falq et al. (2021) isthe following. ( F ) : min ( δ, X ) ∈ int δ ( P ) g ( δ, X )Formulation F is a direct linear translation of F . Indeed, there is a one to onecorrespondence between their solution sets, i.e. between int δ ( P ) and ~ P ( J ), and g ( δ, X ) = f ( E , T ) for any ( δ, X ) encoding ( E , T ). ( I u ) , ( I ′ u ) and ( S u , v ) • Linear inequalities translating constraints ( I u ) and ( I ′ u ) for any u ∈ J Let u ∈ J . If δ ∈ { , } J encodes a partition ( E , T ), the penalty variation ∆ u ( E , T ) canbe expressed linearly from δ as follows. ∆ u ( δ ) = − α u X i ∈ A ( u ) p i δ i + β u (cid:16) X i ∈ B ( u ) p i (1 − δ i ) + p u (cid:17) + p u (cid:16) X i ∈ ¯ B ( u ) β i (1 − δ i ) − X i ∈ ¯ A ( u ) α i δ i (cid:17) Moreover, if δ ∈ { , } J encodes ( E , T ), we can translate constraint ( I u ) as follows.( E , T ) satisfies ( I u ) ⇔ (cid:0) u ∈ E and ∆ u ( E , T ) > (cid:1) or u ∈ T ⇔ (cid:0) δ u = ∆ u ( E , T ) > (cid:1) or δ u = − ∆ u ( δ ) for any δ ∈ { , } J . M u = α u p (cid:0) A ( u ) (cid:1) − β u p u + p u α (cid:0) ¯ A ( u ) (cid:1) ∆ u ( δ ) > − M u for any δ ∈ { , } J , the following inequality is satisfiedby every δ ∈ { , } J such that (1 − δ u ) = ∆ u ( δ ) > − M u (1 − δ u ) (5)Conversely, for every δ ∈ { , } J such that (1 − δ u ) =
0, inequality (5) is satisfied ifand only if ∆ u ( δ ) > δ ∈ { , } J satisfies (5) if and only if the partition en-coded by δ satisfies ( I u ). Property 7.(i) follows.Similarly, to translate constraint ( I ′ u ), we introduce the following constant whichis an upper bound of − ∆ u ( δ ) for any δ ∈ { , } J . M ′ u = β u p (cid:0) B ( u ) (cid:1) + β u p u + p u β (cid:0) ¯ B ( u ) (cid:1) Since we have ∀ δ ∈ { , } J , − ∆ u ( δ ) > − M ′ u , the following inequality is satisfied byevery δ ∈ { , } J such that δ u = − ∆ u ( δ ) > − M ′ u δ u (6)Conversely, if δ u = δ satisfies (6) if and only if − ∆ u ( δ ) > δ ∈ { , } J satisfies (6) if and only if if and only if thepartition encoded by δ satisfies ( I u ). Property 7.(ii) follows. Property 7
Let δ ∈ { , } J and let ( E , T ) be thepartition encoded by δ . Forany u ∈ J ,(i) δ satisfies inequality (5) if andonly if ( E , T ) satisfies constraint ( I u ).(ii) δ satisfies inequality (6) ifandonly if ( E , T ) satisfies constraint ( I ′ u ). • Linear inequalities translating constraint ( S u , v ) for any ( u , v ) ∈ J Let ( u , v ) ∈ J such that u , v . If δ ∈ { , } J encodes a partition ( E , T ), the penaltyvariation ∆ u , v ( E , T ) can be expressed linearly from δ as follows.15 u , v ( δ ) = − α u X i ∈ A ( u ) p i δ i + β u X i ∈ B ( u ) \{ v } p i (1 − δ i ) + β u p u − β v X i ∈ B ( v ) p i (1 − δ i ) − β v p v + α v X i ∈ A ( v ) \{ u } p i δ i + ( − p u + p v ) P i ∈ ¯ A ( v ) α i δ i − p u P i ∈ A ( v ) ∩ ¯ A ( u ) α i δ i if α v p v < α u p u ( − p u + p v ) P i ∈ ¯ A ( u ) α i δ i + p v P i ∈ A ( u ) ∩ ¯ A ( v ) α i δ i otherwise + ( p u − p v ) P i ∈ ¯ B ( v ) β i (1 − δ i ) + p u P i ∈ B ( v ) ∩ ¯ B ( u ) β i (1 − δ i ) if β v p v β u p u ( p u − p v ) P i ∈ ¯ B ( u ) β i (1 − δ i ) − p v P i ∈ B ( u ) ∩ ¯ B ( v ) β i (1 − δ i ) otherwiseMoreover, if δ ∈ { , } J encodes ( E , T ), we can translate constraint ( S u , v ) as follows.( E , T ) satisfies ( S u , v ) ⇔ ( u , v ) < E × T or ∆ u , v ( E , T ) > ⇔ (cid:0) u < E or v < T (cid:1) or (cid:0) u ∈ E and v ∈ T and ∆ u , v ( E , T ) > (cid:1) ⇔ (cid:16) (1 − δ u ) = δ v = (cid:17) or (cid:16)(cid:0) (1 − δ u ) = δ v = (cid:1) and ∆ u , v ( δ ) > (cid:17) ⇔ (cid:0) δ v + (1 − δ u ) (cid:1) ∈ { , } or (cid:16)(cid:0) δ v + (1 − δ u ) (cid:1) = ∆ u , v ( δ ) > (cid:17) To unify these cases into one inequality, we introduce the following constant e M u , v which is an upper bound on − ∆ u , v ( δ ) for any δ ∈ { , } J . e M u , v = α u p (cid:0) A ( u ) (cid:1) − β u p u + β v p (cid:0) B ( v ) (cid:1) + β v p v + [ p u − p v ] + α (cid:0) ¯ A ( v ) (cid:1) + p u α (cid:0) A ( v ) ∩ ¯ A ( u ) (cid:1) if α v p v < α u p u [ p u − p v ] + α (cid:0) ¯ A ( u ) (cid:1) if α v p v > α u p u + [ p v − p u ] + β (cid:0) ¯ B ( v ) (cid:1) if β v p v β u p u [ p v − p u ] + β (cid:0) ¯ B ( u ) (cid:1) + p v β (cid:0) B ( u ) ∩ ¯ B ( v ) (cid:1) if β v p v > β u p u Since we have ∆ u , v ( δ ) > − e M u , v for any δ ∈ { , } J , the inequality ∆ u , v ( δ ) > − e M u , v (cid:0) δ v + (1 − δ u ) (cid:1) is satisfied for every δ ∈ { , } J such that (cid:0) δ v + (1 − δ u ) (cid:1) =
1. In particular,this inequality is satisfied for every δ ∈ { , } J such that ( δ u , δ v ) = (0 ,
0) or (1 , ∈ { , } J such that (cid:0) δ v + (1 − δ u ) (cid:1) = i.e. such that ( δ u , δ v ) = (0 , e M u , v
0, since − e M u , v (cid:11) − e M u , v in this case. To provide aninequality satisfied by every δ ∈ { , } J such that ( δ u , δ v ) , (1 ,
0) we introduce thefollowing constant. M u , v = e M u , v if e M u , v > e M u , v / − e M u , v > − M u , v and − e M u , v > − M u , v . Therefore, the followinginequality is satisfied by every δ ∈ { , } J such that (cid:0) δ v + (1 − δ u ) (cid:1) ∈ { , } , i.e. suchthat ( δ u , δ v ) , (1 , ∆ u , v ( δ ) > − M u , v (cid:0) δ v + (1 − δ u ) (cid:1) (7)Conversely, for every (cid:0) δ v + (1 − δ u ) (cid:1) = δ ∈ { , } J such that i.e. such that ( δ u , δ v ) = (1 , ∆ u , v ( δ ) >
0. Finally, δ ∈ { , } J satisfies (7) if and only if the partition encoded by δ satisfies ( S u , v ). Property 8follows. Property 8
Let δ ∈ { , } J and let ( E , T ) be the partition encoded by δ . For any ( u , v ) ∈ J such that u , v , δ satisfies (7) ifandonly if ( E , T ) satisfies constraint ( S u , v ).In the sequel, insert inequalities (5) and (6) and swap inequalities (7) will becalled dominance inequalities . In this section, we show how to benefit from the fact that each dominance in-equality is based on an operation. When a vector δ ∈ { , } J encoding a partition( E , T ) does not satisfy a dominance inequality, applying the corresponding oper-ation to ( E , T ) provides a partition with a strictly lower penalty. The followingproperty, resulting from properties 4, 5, 7 and 8, formally states this result. Property 9
Let δ ∈ { , } J and let ( E , T ) be thepartition encoded by δ . Forany u ∈ J ,(i) δ doesnotsatisfy (5) for u ifandonlyif u ∈ E and f (cid:0) E \{ u } , T ∪{ u } (cid:1) < f ( E , T ),(ii) δ doesnotsatisfy (6) for u ifandonlyif u ∈ T and f (cid:0) E ∪{ u } , T \{ u } (cid:1) < f ( E , T ).(iii) Moreover, forany ( u , v ) ∈ J such that u , v , δ does notsatisfy (7) for ( u , v ) if andonly if ( u , v ) ∈ E × T and f (cid:0) E \{ u }∪{ v } , T \{ v }∪{ u } (cid:1) < f ( E , T ). Proof :
Let us fix u ∈ J . From Property 7, δ does not satisfy inequality (5) if and only if( E , T ) does not satisfy constraints ( I u ), which is equivalent to u ∈ E and ∆ u ( E , T ) < sing Property 4, it is equivalent to u ∈ E and f (cid:0) E \{ u }∪{ v } , T \{ v }∪{ u } (cid:1) − f ( E , T ) < (cid:3) Property 9 will be used in Section 4 to propose a local search procedure. Thefollowing corollary, which directly derives from the negation of statements (i), (ii)and (iii), ensures that the solution provided by this local search procedure is aninsert and swap local optimum.
Corollary 10
Let δ ∈ { , } J and let ( E , T ) be thepartition encoded by δ .(i) ( E , T ) is an insert local optimum if and only if δ satisfies inequalities (5)and (6) forall u ∈ J .(ii) ( E , T ) is a swap local optimum if andonly if δ satisfies inequality (7) forall ( u , v ) ∈ J such that u , v .The following section presents experimental results to assess the practical rel-evance of the dominance inequalities.
4. Numerical results
All experiments are carried out using a single thread with Intel(R) Xeon(R)X5677, @ 3.47GHz, and 144Gb RAM. Linear programs (LP) and MIP are solvedwith Cplex 12.6.3.0.The numerical experiments are performed on the instance benchmark proposedby Biskup & Feldmann (2001a), available online on OR-Library (Biskup & Feldmann,2001b). For each number of tasks n ∈ { , , , , } , ten triples ( p , α, β ) of (cid:0) N ∗ n (cid:1) are given. For each one, we assume that d = p ( J ), so that the due date isunrestrictive. For the sake of comparison, we additionally construct instances with n ∈ { , } (resp. n ∈ { , , } ) by only considering the first n tasks of theprevious 100-task (resp. 200-task) instances. Unless otherwise specified, the gap,time and number of nodes presented in the following tables are average values overthe ten instances for a given n and the time limit is set to 3600 seconds.To measure the improvement induced by the insert or swap inequalities, wecompare the four following formulations.18 : the formulation defined in Section 3, only with inequalities (1-4) F i : the formulation obtained from F by adding (5) and (6) for all u ∈ JF s : the formulation obtained from F by adding (7) for all ( u , v ) ∈ J s.t. u , vF i + s : the formulation obtained from F s by adding (5) and (6) for all u ∈ J For a given formulation F , we distinguish two settings: a setting with all avail-able features, that is using Cplex default, denoted by F d , and a setting with lessCplex features, denoted by F l . Two types of features are disabled in this setting:the cut generation, which produces reinforcement inequalities and adds them to theformulation, and the primal heuristic procedures. The cut generation is disabled inorder to measure the impact of the dominance inequalities on the linear relaxationvalue of the formulation F , rather than their impact on the linear relaxation valueof a strengthened formulation. The primal heuristic procedures have been disabledto focus on the lower bound since we have other methods to quickly obtain goodfeasible solutions ( Cf.
Section 4.3).This results in eight formulation settings: F l , F il , F sl , F i + sl , F d , F id , F sd and F i + sd .For each one, inequalities (1-4), as well as inequalities (5-7) when included, areadded initially.Let us recall that only the δ variables need to be integer in F . Indeed, fromLemma 6, if δ ∈ { , } J , inequalities (1-4) ensures that X ∈ { , } J < . It is also the casefor F i , F s and F i + s . Therefore, unless otherwise specified, δ variables are set asbinary variables, while X variables are set as continuous variables. Consequently,the branching decisions only involve δ . Table 1 provides the results obtained by solving MIP to optimality, using theeight formulation settings. Each line corresponds to the ten instances of a givensize n . More precisely, Table 1 entries are the following. nd : the average number of nodes, except the root node, in the search tree, overthe instances solved to optimalityFor a given formulation setting, we choose to stop the run at a line of the table ifless than 5 over ten instances are solved to optimality. For the subsequent lines, wereport a "-" in the table.Using formulation setting F l , the ten n -task instances are solved to optimalitywithin the time limit for n up to 50. In contrast, using F il , it is the case for n up to60, using F sl for n up to 120 and using F i + sl for n up to 150. Within approximately19 minutes, F l solves 50-task instances, F il F sl F i + sl n =
50 the number of nodes goesfrom more than 53 000 for F l to only 31 for F i + sl . With this latter formulationsetting, the number of nodes is low, it is at most 200, even for large size instances.However, the time limit is reached for some 180- and 200-task instances, since thesize of the linear program solved at each node is large.In light of the four first columns of Table 1, we can conclude that, with lessCplex features, adding insert and swap inequalities significantly reduces the num-ber of nodes and hence the computation time. More precisely, adding only swapinequalities is better than adding only insert inequalities, but adding both of themprovides the best performance.The four last columns show the same improvement in terms of computationtime and number of nodes when Cplex default features are used.Let us now focus on the 4th and 5th columns to compare the impact of the dom-inance inequalities and the impact of Cplex default features. For small instances, i.e. n ∈ { , } , Cplex default features allow to solve the problem at the root node( Cf. F d columns). However, from n =
50, the number of nodes grows fast, so thatno 60-task instance can be solved within 3600 seconds. Conversely, we alreadynoticed that adding swap and insert inequalities limits the number of nodes (
Cf.F i + sl columns), so that the ten 150-task instances are solved within 3600 seconds.Finally, adding the swap and insert inequalities provides better results than addingCplex default features.Up to size 60, F i + sd solves all instances at the root node, and is faster than F i + sl .For larger instances, except 200-task instances, F i + sl and F i + sd solves the problemin similar computation times, even if F i + sl explores a smaller number of nodes:for example, it is two times smaller for n = n = F i + sd solves 4 over10 instances, while F i + sl only solves 1 over 10 instances. To conclude, F i + sl and F i + sd o ff er comparable performances, so that, for both settings, the formulationsproviding the best results are the ones with insertion and swap inequalities. To further investigate the impact of dominance inequalities, we focus in thissection on the root node of the search tree for di ff erent formulation settings. Moreprecisely, we compare the di ff erent lower bounds obtained at the root node.20n the Cplex framework, setting the node limit to 0 allows to only solve the rootnode of a MIP: the branch-and-bound algorithm is stopped before the first branch-ing. If Cplex default features are activated, the preprocessing is applied and thecuts are added before the algorithm stops. For a given formulation setting F , thecorresponding run with the node limit set to 0 is denoted by F - rn . This results ineight runs: F l - rn , F il - rn , F sl - rn , F i + sl - rn , F d - rn , F id - rn , F sd - rn , F i + sd - rn .Note that, in the Cplex framework, solving F - rn is di ff erent from solving the linearrelaxation of F , denoted by F - lp . Indeed, F - lp is obtained by setting δ variablesas continuous variables, which desactivates most of the Cplex features. In particu-lar, the reinforcement cuts cannot be added since they are not valid for the relaxedformulation. Similarly, the inference procedure on the binary variables cannot beapplied. We run the four linear relaxations F - lp , F i - lp , F s - lp and F i + s - lp . Sur-prisingly, the obtained values are the same for these four relaxations. In otherwords, adding insert and swap inequalities does not improve the linear relaxationvalue. Therefore, we only present in Table 2 the results for F - lp .To measure the quality of the nine di ff erent lower bounds obtained, we com-pute, when it is possible, the optimality gap, i.e. ( OPT − LB ) / OPT where
OPT denotes the optimal value and LB the lower bound. When the optimal value isnot known, we compute a gap using the best upper bound that we get U B , i.e. ( U B − LB ) / U B . Such gaps are indicated with a "*" in Table 2. For each of the nineruns, the entries of Table 2 are the following.L-gap : the average optimality gap of the lower bound obtained at the root nodetime : the average running time in seconds over the ten instancesThe obtained lower bound is exactly the same using F - lp , F l - rn or F sl - rn . Wededuce that with less Cplex features and without insert inequalities, setting the δ variables as binary or continuous variables, provides the same lower bound. More-over, this lower bound is quite weak, since the average optimality gap is largerthan 40% even for the 10-task instances. The computation times using F - lp and F l - rn are similar: 2 seconds for the 100-task instances and about 12 minutes forthe 500-task instances. The computation time required for F sl - rn is larger: almost20 seconds for the 100-task instances and 47 minutes for the 500-task instances.The lower bound obtained when only considering the insert inequalities isslightly better when the δ variables are set as binary variables for n ∈ { , } .Indeed, the average optimality gap is 33% instead of 41% when n =
10, and 66%instead of 68% when n =
20 (
Cf. F - lp and F il - rn columns). The computation timeusing F il - rn is comparable to the computation time using F - lp and F l - rn .21he lower bound obtained when considering both insert and swap inequalities,is significantly better when the δ variables are set as binary variables. Indeed, theaverage optimality gap is smaller than 39% for any value of n , and it is equal to0 for n =
10 (
Cf. F i + sl - rn column). The computation time for F i + sl - rn is betweenthose for F il - rn and F sl - rn : 14 seconds for the 100-task instances and about 30minutes for the 500-task instances.The lower bound provided using F d - rn , is better than the one obtained using F l - rn ,that is with less Cplex features. Indeed, the average optimality gap is 7% insteadof 41% when n =
10, and 46% instead of 94% when n = F i + sl , whose optimality gap is 0 for n = n = F d - rn increases fastwith the increase of n so that the root node cannot be solved within one hour forsizes larger than 120.Combining Cplex features with insert inequalities gives almost the same results( Cf. F id - rn column). Conversely, combining Cplex features with swap inequalitiesgives better results ( Cf. F sd - rn column). In particular, the average computationtime is reduced so that instances up to size 200 can be solved. Moreover, the op-timality gap is less than 22% for all solved instances. Finally, using F i + sd - rn giveseven better results, the average optimality gap does not exceed 15%, even for 200-task instances, which are solved in 418 seconds, instead of 1200 using F sd - rn .In a nutshell, combining insert and swap inequalities is the best to obtain alower bound at the root node. Not using Cplex features allows its fast computation( Cf. F i + sl - rn column). Conversely, using them allows to obtain a better lowerbound at the expense of the computation time ( Cf. F i + sd - rn column). In this section, we propose two upper bounds on the optimal value. The firstone is derived from the fractional solution obtained at the root node by a simplerounding procedure. The second one is obtained by applying in addition a localsearch procedure.We derive an integer solution ( δ, X ) by rounding a fractional solution ( e δ, ˜ X ), as fol-lows. 22 j ∈ J , δ j = e δ j < or (cid:0)e δ j = and α j < β j (cid:1) ∀ ( i , j ) ∈ J < , X i , j = ( δ i , δ j δ, X ) satisfies inequalities (1-4) ( Cf.
Lemma 6). It is thus a so-lution of F , and g ( δ, X ) is an upper bound of the optimal value. However, it isnot necessarily a solution for F i , F s and F i + s formulations, since ( δ, X ) does notnecessarily satisfy the insert and swap inequalities.In order to transform such a solution into a solution satisfying the dominanceinequalities, we can iteratively apply the operation associated to each violated dom-inance inequality, until all of them are satisfied. Algorithm 1 presents a way toimplement this procedure that we call Insert_swap_improvement . From Prop-erty 9, if an insert (resp. a swap) inequality is not satisfied, applying the appropriateinsert (resp. swap) operation provides a strictly better solution. Therefore, each so-lution is considered at most once in this procedure. Since the number of solutionsis finte, the
Insert_swap_improvement procedure finishes.The returned solution is an insert and swap local optimum, since it satisfies alldominance inequalities (
Cf.
Corrolary 10).
Insert_swap_improvement input : δ ∈ { , } J output : δ ′ encoding an insert and swap local optimum δ ′ ← δ ; is_locally_opt ← falsewhile (not is_locally_opt)is_locally_opt ← truefor u ∈ J if δ ′ u = and ∆ u ( δ ′ ) < // δ ′ does not satisfy (5) δ ′ u ←
0; is_locally_opt ← falseif δ ′ u = and ∆ u ( δ ′ ) > // δ ′ does not satisfy (6) δ ′ u ←
1; is_locally_opt ← falsefor v ∈ J \{ u } if δ ′ u = , δ ′ v = and ∆ u , v ( δ ′ ) < // δ ′ does not satisfy (7) δ ′ u ← δ ′ v ←
1; is_locally_opt ← falsereturn δ ′ Algorithm 1: the improvement procedure by insert and swap operations23ote that this algorithm can be seen as a local search procedure for the neighbor-hood associated to the insert and swap operations. Moreover, this procedure can beapplied to any integer solution. Particularly, by sake of comparison we apply it tothe solutions obtained by the heuristic "Heur II" provided by Biskup & Feldmann(2001a).We finally compare the upper bounds given by the six following heuristic solutions. BF : the solution obtained by the Biskup and Feldmann heuristic BF + : the solution obtained by applying Insert_swap_improvement to BFR F - lp R + : the solution obtained by applying Insert_swap_improvement to R R F i + sd - rn R + : the solution obtained by applying Insert_swap_improvement to R i.e. ( U B − OPT ) / OPT where
OPT denotes the optimal value and
U B the upper bound.The Biskup and Feldmann heuristic provides a solution in less than 1 second. Ap-plying rounding and
Insert_swap_improvement to a fractional solution providesa solution in less than 1 second for instances up to size 200. Therefore, the timeneeded to obtain R R + (resp. R R + ) is essentially the computationtime required to solve F - lp (resp. F i + sd - rn ) given in Table 2.As shown in Table 3, BF is a good upper bound. Indeed, its optimality gapis smaller than 0.35% for instance sizes larger than 50. However, this bound isimproved by Insert_swap_improvement : the optimality gap of BF + is smallerthan 0.02% for all the instances. With an optimality gap larger than 170%, R R + , with an optimality gap smaller than 0.01%, isvery good, and even slightly better than BF + . With an optimality gap smaller than17%, R R
1, and R + is exactly the same as R + .Finally, BF + , R + and R + are very good upper bounds. However it is worthnoticing that even if the computation time to obtain BF + is about 1 second, thebound is obtained without any guarantee, since no lower bound is provided. Con-versely, the computation time to obtain R + is larger: 25 seconds for n =
100 andabout 7 minutes for n = R + is then guaranteedto be at 14% of the optimal value for n = n =
200 (
Cf.
L-gapof F i + sd - rn in Table 2). R + is a compromise between BF + and R + . Indeed, for24nstances up to size 200, R + is provided in less than 20 seconds together with alower bound, but the guarantee obtained from this lower bound is quite weak (97%for n = Cf. F l - rn in Table 2). Insert and swap operations can be used in di ff erent ways. Table 4 presents thebest way to use them depending on the expected solution quality.- To obtain an upper bound: apply rounding and Insert_swap_improvement to the fractional solution given by F - lp . ( Cf. R + column in Table 4).- To obtain an upper bound with a better guarantee than the one obtained with R + : apply rounding and Insert_swap_improvement to the fractional so-lution given by F i + sd - rn . ( Cf. R + column in Table 4).- To obtain a 5%-approached solution: use F i + sd , setting the gap limit to 5%.( Cf. F i + sd -5% column in Table 4).- To obtain an exact solution: use F i + sd . ( Cf. F i + sd column in Table 4).Table 4 sums up the performance of the four above mentioned use cases. To mea-sure the performances on the 200-task instances, no time limit is fixed. The entriesof Table 4 are the following.L-gap : the average optimality gap of the provided lower boundU-gap : the average optimality gap of the provided upper boundtime : the average running time in seconds nd : the average number of nodes except the root nodeNew experiments are conducted for the results reported in F i + sd -5% and F i + sd columns when n = ff er an overview.Table 4 shows that the number of nodes is lowered by 37.0% while the compu-tation time is only lowered by 10.8% on average for n = F i + sd reaches the time limit, only less than 100 nodesare explored. The limit for solving F i + sl is thus the size or the di ffi culty of the LPssolved at each node, rather than the number of nodes.Trying to address this issue, we implemented a separation algorithm for theinsert and swap inequalities using a callback function. The time needed to solve50-task instances using this separation algorithm and Cplex features was 1513 sec-onds with 925 nodes in average. We observe that 98% of the computation time is25sed by the UserCut Callback to add 71 inequalities in average. This is not sur-prising since the separation algorithm consists in simply evaluating the terms ofinequality (5) and (6) for the n possible tasks u , and the terms of inequality (7) forthe n possible couples ( u , v ), which results in an O ( n ) procedure.Providing a faster separation algorithm could reduce the computation time, butthe branching scheme, and then the number of nodes, would be the same. Sincethis number of nodes is quite large compared to the performance of F i + sd (whichsolves all 50-task instances at the root node), we conclude that adding dominanceinequalities through a separation procedure reduces their impact.Indeed, when initially added, the dominance inequalities allow to the Cplex pre-solve phase to fix some variables to 0 or 1. The number of LPs variables is thenreduced and the value obtained at each node is improved. When the δ variables areset as continuous variables, this presolve is not executed. It is then consistent withthe observation that adding dominance inequalities in this latter case has no impact( Cf.
Section 4.2).
5. Conclusion
In this work, we propose a new way to use neighborhood-based dominanceproperties, which results in a new kind of reinforcement inequalities. In contrastwith the commonly used reinforcement inequalities, which cut fractional points,these inequalities cut non locally optimal solutions. In particular, for the compactformulation F , we provide linear inequalities cutting all the solutions which arenot insert and swap locally optimal.From a practical point of view, we show that adding insert and swap inequali-ties greatly improves performances of F . Indeed, instead of 50-task instances, wecan now solve up to 150-task instances to optimality within one hour.Insert and swap inequalities can also be used to provide a heuristic solutionwhich is slightly better than the one proposed by Biskup & Feldmann (2001a). Forinstances up to size 200, this heuristic solution is obtained in less than 20 seconds.A lower bound providing a 15% gap can also be obtained in less than 420 seconds.We observe that insert and swap inequalities do not improve the linear relax-ation value of the compact formulation F . However, used in conjunction withCplex features, they allow to improve the lower bound obtained at the root node.Two issues follow. Firstly, for a version of F reinforced by cuts or by branchingdecisions, do dominance properties improve the linear relaxation value? Secondly,which procedure implemented in the Cplex features take advantage of the insert26nd swap inequalities? Addressing these issues requires an appropriate experimen-tal framework.Moreover, this work could be extended to other problems where the solutionscan be encoded by partitions (any kind of partition, not necessarily ordered bi-partitions). For instance, inequalities similar to insert and swap inequalities couldbe used in a generalization of UCDDP to a parallel machine framework. Indeed,if the tasks share a common due date,the dominant schedules can be encoded byordered 2 m -partitions, where m is the number of machines. This is true even ifthe common due date, the processing times and the unit earliness and tardinesspenalties depend on the machine. Beyond the scheduling field, such inequalitiescould also be used in the maximum cut problem (Karp, 1972) or in a maximum k -cut problem (Frieze & Jerrum, 1997).For other combinatorial problems where solutions do not have a partition struc-ture; some neighborhood-based dominance inequalities could also be designed us-ing appropriate operations. References
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Table 2: E ff ect of adding insert and swap inequalities on lower bounds F BF + R R + R R + n U-gap U-gap U-gap U-gap U-gap U-gap10 2.04% 0.00% 170% 0.00% 0.00% 0.00%20 0.95% 0.00% 196% 0.00% 1.33% 0.00%50 0.35% 0.02% 203% 0.00% 13.83% 0.00%60 0.26% 0.01% 170% 0.01% 16.80% 0.01%80 0.22% 0.01% 172% 0.00% 16.36% 0.00%100 0.18% 0.00% 174% 0.00% 15.72% 0.00%120 0.10% 0.00% 170% 0.00% 15.77% 0.00%150 0.10% 0.00% 171% 0.00% 15.27% 0.00%180 0.10% 0.00% 171% 0.00% 16.09% 0.00%200 0.10% 0.01% 171% 0.01% 16.28% 0.00%
Table 3: Comparison of di ff erent heuristics providing an upper bound to obtain: an upper bound a lower bound a 5%-approximation an exact solutionuse: R + R + F i + sd -5% F i + sd n L-gap U-gap time L-gap U-gap time time nd time nd50 86% 0.00% < Table 4: Di ff erent ways of using insert and swap inequalitieserent ways of using insert and swap inequalities