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Further, we have shown how a classical algorithm such as Clarke and Wright’s, enhanced with local search, can be the best choice in the context of the Inventory and Transportation Problem, both in terms of the quality of the solutions obtained and the computational time necessary to achieve them. The power of other alg... |
It could be argued that both TS and ACO require a finer tuning of their parameters in order to give an adequate performance than what was achieved here. This, however, could be interpreted as a disadvantage of their application to a variety of problem configurations and in a commercial context. |
Special attention should be given to the case of instance B35, since it is the only one for which ACO yields better results than the remaining VRP solvers (with the exception of single objective TS) and, oddly, this happens in the multiobjective case. The geographical layout of this instance is shown in Figure |
; the high eccentricity of the distribution can be seen (the depot is on one side of the shops) compared to the small number of shops. Instance A80 also has a similar value of eccentricity, but for a higher number of shops. Perhaps here lies the explanation of why ACO works better in the former but not in the latter; c... |
It must be noted that this result is subject to the specificities of the problem data, i.e. the fact that in this case the inventory cost greatly outweighs the cost of the transport. As future work we must consider a case in which the products moved are cheaper and hence the inventory cost is more in a par with the tra... |
Acknowledgements This work was part of project NoHNES - Non Hierarchical Network Evolutionary System , and has been supported by the Spanish Ministry of Science and Innovation, ref. TIN2007-68083-C02. |
Appendix I: NSGA-II NSGA-II (Deb et al.,, 2002 2000 is an non-elitist multiobjective evolutionary algorithm (MOEA) which was developed in order to overcome the problems of previous MOEAs, such as the high computational complexity of sorting non dominated solutions. These algorithms have two common features: assigning f... |
NSGA-II works as follows: Initially, a random parent population [MATH] is created, with size N. The population is sorted based on nondominance. Each solution is assigned a fitness (or rank) equal to its nondomination level (with 1 being the best level). Initially, the usual binary tournament selection, recombination, a... |
[MATH] and sorted according to nondomination in a number of fronts [MATH] . Next, a new population [MATH] of size N is created by selecting individuals from [MATH] in order of nondominance (i.e. ordered individuals from [MATH] are chosen first, then ordered individuals from [MATH] and so on until the number of individu... |
The advantages of NSGA-II with respect to previous MOEAs are the fast sorting of nondominated individuals and the preservation of diversity. |
Nondominated sorting. First, for each solution [MATH] we calculate two entities: domination count, [MATH] , i.e. the number of solutions which dominate [MATH] , and a list of solutions that [MATH] |
dominates, [MATH] . All solutions in the first nondominated front will have their [MATH] as zero. Now, for each solution with [MATH] , we visit each member [MATH] in its [MATH] and reduce its domination count by one. In doing so, if for any member the domination count becomes zero, we place it in a separate list [MATH]... |
Procedure fastNonDominatedSort() input: population [MATH] output: listOfFronts [MATH] For each [MATH] [MATH] [MATH] For each [MATH] |
if [MATH] dominates [MATH] Adds [MATH] to [MATH] else if [MATH] dominates [MATH] Increments [MATH] if [MATH] [MATH] Adds [MATH] to [MATH] |
[MATH] while [MATH] [MATH] for each [MATH] for each [MATH] Decrements [MATH] if [MATH] [MATH] Adds [MATH] to [MATH] [MATH] Increments [MATH] |
return [MATH] end; Algorithm 2 Nondominated sorting function for NSGAII. Diversity preservation. To get an estimate of the density of solutions surrounding a particular solution in the population, we calculate the average distance of two points on either side of this point along each of the objectives. This distance is... |
crowding distance , and it is calculated as shown in Algorithm . Moreover, a crowded-comparison operator [MATH] , is used in order to guide the selection process at the various stages of the algorithm towards a uniformly spread-out Pareto-optimal front. |
In the selection process, given two solutions with different nondomination ranks the one with the lower (better) rank will be preferred. Otherwise, if both solutions belong to the same front, then one located in a less crowded region will be preferred. |
Procedure crowdingDistance() input: population pop[popSize] for [MATH] [MATH] for each objective [MATH] Sort [MATH] using [MATH] |
[MATH] for [MATH] [MATH] end; Algorithm 3 Crowding distance assignment for NSGA-II Algorithm. For a complete description of NSGA-II, see (Deb et al.,, 2002 2000 |
Appendix II: Algorithms for solving the VRP Here we describe the algorithms employed as VRPsolvers: C&W’s algorithm, Local Search, ACO and TS, plus other additional functions. |
Algorithm CWLS input: shops[nShops], depot output: routes [Build [MATH] initial routes with one shop only] for i=1… [MATH] [MATH] |
[Calculate savings] Calculate [MATH] for each pair shops[i], shops[j] [MATH] [Best union ] repeat [MATH] Let [MATH] be the route containing shops[i] |
Let [MATH] be the route containing shops[j] if shops[i*] is the last shop in [MATH] and shops[j*] is the first shop in [MATH] and the combination is feasible |
then combine [MATH] and [MATH] delete [MATH] until there are no more savings to consider; return routes end; Algorithm 4 Clarke & Wright’s algorithm (C&W). |
Algorithm LocalSearch input: initialSolution [Initialisation] best [MATH] initialSolution costBest = costTemp [Improving each route] |
for each r in routes(initialSolution) for k=1..Max(sizeOf(r),numberOfNeighbors) select s1, s2 random shops in r tempSolution [MATH] InterchangeShops(initialSolution,s1, s2) |
if (calculateCost(tempSolution) [MATH] costBest) best [MATH] tempSolution costBest = calculateCost(best) [Improving pairs of routes] |
for r1 =1..routes(initialSolution) for r2=r1+1..routes(initialSolution) select s1 random shop in r1 select s2 random shop in r2 solTemp [MATH] InterchangeShops(initialSolution,s1, s2) |
if (calculateCost(tempSolution) [MATH] costBest) best [MATH] tempSolution costBest = calculateCost(best) return best end algorithm; |
Algorithm 5 Local Search (LS) algorithm. Algorithm ACO input: shops[nShops], numberOfIterations output: routes [Initialise values] |
routes [MATH] validRandomSolution globalCost [MATH] calculateCost( routes [Main loop] for it=1..numberOfIterations [Looking for a solution for each ant] |
for each ant i Place ant i at depot i.shopsNotVisited [MATH] shops i.solution [MATH] i.solution [MATH] reset cost, time and demand |
reset load while there are ants i for which i.shopsNotVisited [MATH] do for each ant i for which i.shopsNotVisited [MATH] nextShopToVisit [MATH] TransitionFunction() |
Update cost if nextShopToVisit == depot then reset time and load else Update time, demand and load Update i.shopsNotVisited, i.solution |
Place ant i at nextShopToVisit [Interchanges] for each ant i do [MATH] -interchanges in i.solution [Update pheromone matrix with local solution] |
Find worstPathInIteration, bestPathInIteration Reinforce bestPathInIteration in pheromone matrix Reset worstPathInIteration in pheromone matrix |
[Update global solution] if globalCost [MATH] cost(bestPathInIteration) routes [MATH] bestPathInIteration globalCost [MATH] cost(bestPathInIteration) |
[Update configuration] Update algorithm parameters: [MATH] , p return routes end algorithm; Algorithm 6 ACO - Main loop. Algorithm Transition(shopsNotVisitedYet:List(shop),currentShop:shop, |
remainingTime:float,remainingLoad:float): nextShop [MATH] if [MATH] [MATH] else [MATH] [MATH] if [MATH] AND [MATH] nextShop = j else |
nextShop = Depot return nextShop end algorithm; Algorithm 7 ACO - Transition function. Algorithm TS [Initialisation] currentSolution [MATH] initialSolution |
currentSolutionCost [MATH] calculateCost(currentSolution) [Main loop] while (iterations [MATH] MAX-ITERATIONS) bestNeighbour [MATH] getBestNeighbour (currentSolution, tabuList) |
iterations [MATH] else iterations [MATH] iterations + 1 tabuList [MATH] updateTenure return bestSolution end algorithm; Algorithm 8 |
Tabu Search (TS) algorithm - Main loop. Algorithm bestNeighbour [Initialisation] moved [MATH] false moves [MATH] getAllMoves theBestNeighbour [MATH] currentSolution |
theBestNeighbourCost [MATH] neighbourCost [MATH] [Main loop] for i=1:moves.length move [MATH] neighbour [MATH] currentSolution neighbour [MATH] move.operateOn(neighbour) |
neighbourCost [MATH] calculateCost(neighbour) isTabu [MATH] isTabu(move) [Aspiration criteria] if (neighbourCost [MATH] bestSolutionCost) |
isTabu [MATH] false if (neighbourCost [MATH] theBestNeighbourCost AND NOT isTabu) theBestNeighbour [MATH] neighbour theBestNeighbourCost [MATH] |
neighbourCost bestNeighbourMove [MATH] move moved [MATH] false end for [Update tabu list] if moved == true tabuList.addMove(bestNeighbourMove) return theBestNeighbour end algorithm; Algorithm 9 Tabu Search - Best neighbour algorithm. |
# Source: arxiv 0909.4950 # Title: Implementing Gröbner bases for operads # Sections: all # Downloaded: 2026-03-03T02:20:33.190917+00:00 |
Implementing Gröbner bases for Operads Abstract. We present an implementation of the algorithm for computing Gröbner bases for operads due to the first author and A. Khoroshkin. We discuss the actual algorithms, the choices made for the implementation platform and the data representation, and strengths and weaknesses o... |
1. Introduction 1.1. Summary of results In an upcoming paper , the first author and Anton Khoroshkin define the concept of a Gröbner basis for finitely presented operads. In that paper, they prove the diamond lemma, and demonstrate that for an operad, having a quadratic Gröbner basis is equivalent to the existence of a... |
, an operad with a PBW basis is Koszul. Hence, an implementation of the Gröbner bases algorithm yields, in addition to a framework for exploration of operads by means of explicit calculation, a computer-aided tool for proving Koszulness. |
In this paper, we present an implementation of the Gröbner basis algorithm in the Haskell programming language . Being designed with categorical terms, Haskell provides a powerful framework for algorithms like that. What we end up with is a computer sofware package which allows to compute the Gröbner basis for a finite... |
One of the main goals of this paper is to help mathematicians who want to get familiar with this software package and use it for their needs, including changing some algorithms or adding more functionality. Consequently, this is more of an invitation to experiment with this software than a report on what it is possible... |
the computation easily yields the correct result, while for many other cases, like the pre-Lie operad for a “wrong” ordering, computations with arity [MATH] and further take enormously long. |
The actual implementation is distributed through the HackageDB repository for Haskell software projects at — software distributed through this repository are available through the automated installation tool cabal-install |
The current documentation files are kept online at 1.2. Outline of the paper The paper is organized as follows. In Section , we recall relevant background information related to operads and Gröbner bases, on one hand, and to types and functions in Haskell, on the other hand. In Section , we discuss the way we chose to ... |
1.3. Acknowledgements We wish to express our deep gratitude to Eric Hoffbeck and Henrik Strohmayer for both significant assistance in the construction of the software code, and analysis of the techniques we are using. Some of the hairier points of Haskell evaluation has been rendered clear by the helpful assistance of ... |
The first author was supported by an IRCSET research fellowship. The second author was supported by the Office of Naval Research, through grant N00014-08-1-0931. |
We are grateful to Jean–Louis Loday and Bruno Vallette who organized the “Operads 2009” meeting in CIRM Luminy, where the work on this project was started. The second author wishes to thank Dublin Institute for Advanced Studies which hosted him as a visitor during the last stage of working on this paper. |
2. Overview For exhaustive information on symmetric operads, we refer the reader to monographs and . Here, we mainly concentrate on shuffle operads, and their relationship with symmetric operads, and definitions in the symmetric case are chosen in the way that best suits this approach. |
2.1. Operads We denote by [MATH] the category of nonempty finite ordered sets (with order-preserving bijections as morphisms), and by [MATH] — the category of nonempty finite sets (with bijections as morphisms). Also, we denote by [MATH] the category of vector spaces (with linear operators as morphisms; unlike the firs... |
Definition 1 (1) (nonsymmetric) collection is a contravariant functor from the category [MATH] to the category [MATH] (2) symmetric collection (or an [MATH] -module ) is a contravariant functor from the category [MATH] to the category [MATH] |
For either type of collections, we can consider the category whose objects are collections of this type (and morphisms are morphisms of the corresponding functors). The natural forgetful functor [MATH] [MATH] leads to a forgetful functor from the category of symmetric collections to the category of nonsymmetric ones, [... |
We use the convention [MATH] in this paper. Definition 2 Let [MATH] and [MATH] be two nonsymmetric collections. Define their shuffle composition |
[MATH] by the formula [EQUATION] where the sum is taken over all shuffling surjections [MATH] , that is surjections for which [MATH] whenever [MATH] |
Let [MATH] and [MATH] be two symmetric collections. Define their (symmetric) composition [MATH] by the formula [EQUATION] where the sum is taken over all surjections [MATH] |
Each of these compositions gives a structure of a monoidal category on the category of the corresponding collections. The same definitions can be given if we replace [MATH] by another symmetric monoidal category. For our purposes, an important replacement for [MATH] will be the category of finite sets (with arbitrary m... |
Definition 3 (1) shuffle operad is a monoid in the category of nonsymmetric collections with the monoidal structure given by the shuffle composition. |
(2) symmetric operad is a monoid in the category of symmetric collections with the monoidal structure given by the (symmetric) composition. |
Definition 4 shuffle permutation of the type [MATH] is a permutation in the symmetric group [MATH] which preserves the order of the first [MATH] elements, the second [MATH] elements,…, the last [MATH] elements, and satisfies |
[EQUATION] Proposition 1 The number of shuffle permutations of the type [MATH] is equal to [EQUATION] When implementing shuffle permutations, one can use the following simple idea: In a shuffle permutation, the number whose image is [MATH] should clearly be the maximal one in its block. Moreover, if this block is of si... |
Definition 5 (1) Let [MATH] be a shuffle operad, [MATH] [MATH] , …, [MATH] . Assume that [MATH] is a shuffle permutation of the type [MATH] . Denote by [MATH] [MATH] , the [MATH] block of [MATH] |
(on which [MATH] is monotonous). Then we define [EQUATION] where [MATH] is the image of [MATH] under the isomorphism between [MATH] and [MATH] , and |
[MATH] is the monoid product map. (2) Let [MATH] be a symmetric operad, [MATH] [MATH] , …, [MATH] . Let [MATH] be an arbitrary permutation. Denote by [MATH] [MATH] , the [MATH] block of [MATH] (of size [MATH] ). Then we define |
[EQUATION] where [MATH] is the image of [MATH] under the isomorphism between [MATH] and [MATH] , and [MATH] is the monoid product map. |
It turns out that the forgetful functor is a monoidal functor between the category of symmetric operads and the category of shuffle operads. Consequently, it turns out that to study various questions of linear algebra for symmetric operads, it is sufficient to forget the full symmetric structure because the shuffle str... |
for more details. 2.2. Trees Assume that we are given a collection of disjoint finite sets [MATH] . A (rooted) tree is a non-empty directed graph [MATH] of topological genus 0 for which each vertex has at least one incoming edge and exactly one outgoing edge. We allow for some edges of a tree to be bounded by a vertex ... |
For a tree with labelled leaves, its canonical planar representative is defined as follows. In general, an embedding of a (rooted) tree in the plane is determined by an ordering of inputs for each vertex (the leftmost one being the smallest, the rightmost — the largest). To compare two inputs of a vertex [MATH] , we fi... |
is denoted by [MATH] Compositions of trees are defined as follows. Given a tree [MATH] with [MATH] leaves, trees [MATH] , …, [MATH] with [MATH] , …, [MATH] leaves respectively, we define the composition [MATH] as the tree obtained by grafting the tree [MATH] to the first leaf of [MATH] , the tree [MATH] with leaf label... |
[MATH] ; to compute it, we first compute the nonsymmetric composition of our trees and then apply [MATH] to the leaf labels of the resulting tree. |
Remark 1 Let us emphasize two practicalities. First of all, the data type for trees chosen for an implementation should be easily adjustable for performing compositions. Second, it is important here that we apply the permutation, not its inverse; usually, action of permutations on functions and mappings uses inverses, ... |
Proposition 2 (1) The collection of trees [MATH] is closed under shuffle compositions. (2) Every tree in [MATH] can be obtained from corollas by iterated shuffle compositions. |
The collection [MATH] is the free shuffle operad [MATH] in the category of finite sets; its linear span is the free shuffle operad [MATH] in the category of vector spaces. |
Here and below by a divisor of a nonsymmetric tree monomial [MATH] we mean a nonsymmetric tree monomial [MATH] whose underlying tree is embedded into the underlying tree of [MATH] in such a way that the labellings of vertices are the same. |
For a tree monomial [MATH] with the underlying nonsymmetric monomial [MATH] and a divisor [MATH] of [MATH] , let us define a tree monomial [MATH] that corresponds to [MATH] . Its vertices are already decorated, so we just need to take care of the leaf labelling. For each leaf [MATH] of [MATH] , let us consider the smal... |
Definition 6 For two tree monomials [MATH] , we say that [MATH] is divisible by [MATH] , if there exists a nonsymmetric divisor of |
[MATH] for which the corresponding tree monomial [MATH] is equal to [MATH] Remark 2 Checking divisibility is important for the Gröbner bases algorithm. Thus, one has to put effort in finding an efficient implementation. Our approach is recursive; it is very much motivated by the choice of the platform. It would be very... |
and Boyer–Moore for string divisibility. Proposition 3 A tree monomial [MATH] is divisible by [MATH] if and only if [MATH] can be obtained from [MATH] by iterated shuffle compositions with corollas. |
Assume that [MATH] is divisible by [MATH] . Take some sequence of shuffle compositions with corollas that produces [MATH] from [MATH] . This sequence can be applied to any tree monomial with the same number of arguments as [MATH] ; abusing the notation a little bit, we denote that operation on tree monomials by [MATH] ... |
Remark 3 Being able to compute the operations [MATH] in the most efficient way is crucial for the Gröbner bases algorithm. This means that much thought should be put in the data representation philosophy; keeping the divisibility information in a logical way helps to be efficient. We chose the approach where a divisor ... |
Definition 7 A tree monomial [MATH] is called a common multiple of two tree monomials [MATH] and [MATH] , if it is divisible by both [MATH] and [MATH] Tree monomials [MATH] and [MATH] are said to have a small common multiple , if they have a common multiple for which the number of vertices of the underlying tree is les... |
Remark 4 Computation of small common multiples is one of the most frequently used operations in the Gröbner basis algorithm. Our approach (described in detail below) shares certain similarities with the algorithm that lists all shuffle permutations of the given type, even though is a little bit more sophisticated. |
2.3. Gröbner bases In this section, we assume that we are working with shuffle operads over [MATH] , in particular, we assume that the set of relations [MATH] consists of linear combinations of tree monomials. For the case of symmetric operads, pre-processing of the input data is required: first, the symmetric group ac... |
In this section, we work with the free operad [MATH] , where the nonsymmetric collection [MATH] is endowed with a basis, a collection of sets [MATH] |
An ordering of tree monomials of [MATH] is said to be admissible , if it is compatible with the operadic structure, that is, replacing the operations in any shuffle composition with larger operations of the same arities increases the result of the composition. In Section 4.3 , we shall describe some admissible ordering... |
Definition 8 For an element [MATH] of the free operad [MATH] , the tree monomial [MATH] is said to be its leading term , if it is the maximal monomial that occurs in the expansion of [MATH] with a nonzero coefficient (notation: [MATH] ). This nonzero coefficient (the leading coefficient of [MATH] ) is denoted by [MATH] |
Remark 5 Whereas in the previous section we only worked with trees, from now on we use linear combinations of trees. Thus, it is important to implement working with tree polynomials, that is, linear combinations of tree monomials. The main requirement is that obtaining the leading term and the leading coefficient, bein... |
Definition 9 Assume that [MATH] and [MATH] are two elements of [MATH] for which the leading term of [MATH] is divisible by the leading term of [MATH] . The element |
[EQUATION] is called the reduction of [MATH] modulo [MATH] Definition 10 Assume that [MATH] and [MATH] are two elements of [MATH] whose leading terms have a small common multiple [MATH] . We have |
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