A Simulated Annealing Approach for the Minmax Regret Path Problem

A Simulated Annealing Approach for the Minmax Regret Path Problem
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  A Simulated Annealing Approach for the Minmax Regret PathProblem Francisco Pérez , César A. Astudillo , Matthew Bardeen , Alfredo Candia-Véjar, {castudillo, mbardeen, acandia}@utalca.clUniversidad de TalcaKm 1. Camino a los Niches Curicó, ChileAugust 5, 2012 Abstract We propose a novel neighbor generation method for a Simulated Annealing (SA) algorithmused to solve the Minmax Regret Path problem with Interval Data, a difficult problem in Com-binatorial Optimization. The problem is defined on a graph where there exists uncertainty in theedge lengths; it is assumed that the uncertainty is deterministic and only the upper and lowerbounds are known. A worst-case criterion is assumed for optimization purposes. The goal isto find a path s - t , which minimizes the maximum regret. The literature includes algorithmsthat solve the classic version of the shortest paths problem efficiently. However, the variant thatwe study in this manuscript is known to be NP-Hard. We propose a SA algorithm to tacklethe aforementioned problem, and we show that our implementation is able to find good solu-tions in reasonable times for large size instances. Furthermore, a known exact algorithm thatutilizes a Mixed Integer Programming (MIP) formulation was implemented by using a com-mercial solver (CPLEX 1 ). We show that this MIP formulation is able to solve instances up to1000 nodes within a reasonable amount of time. 1 Introduction In this work we study a variant of the well known Shortest Path (SP) problem. For the classicalversion of this problem, efficient algorithms have been known since 1959 (Dijkstra, 1959). Givena digraph G = ( V,A ) ( V  is the set of nodes and A is the set of arcs) with non-negative arc costsassociated to each arc and two nodes s and t in V  , SP consists of finding a s - t path of minimum totalcost. Dijkstra designed a polynomial time algorithm and from this, a number of other approacheshave been proposed. Ahuja et al. present the different algorithmic alternatives to solve the problem(Ahuja et al., 1993).Our interest is focused on the variant of shortest path problems where there exists uncertaintyin the objective function parameters. In this SP problem, for each arc we have a closed intervaldefiningthepossibilitiesforthearclength. A scenario isavectorwhereeachnumberrepresentsoneelement of an arc length interval. The uncertainty model used here is the minmax regret  approach(MMR), sometimes named robust deviation ; in this model the problem is to find a feasible solutionbeing α -optimal for any possible scenario with α as small as possible. One of the properties of theminmax regret model is that it is not as pessimistic as the ( absolute ) minmax model. This model 1 Although popularly referred to simply as CPLEX, its formal name is IBM ILOG CPLEX Optimization Studio.For additional information, the interested reader may consult the following URL: 1  in combinatorial optimization has largely been studied only recently: see the books by Kouvelisand Yu (Kouvelis and G., 1997), and Kasperski (Kasperski, 2008), as well as the recent reviews byAissi et al. (Aissi et al., 2009) and Candia-Véjar et al. (Candia-Véjar et al., 2011). The later alsomentions some interesting applications of the MMR model in the real world.It is known that minmax regret combinatorial optimization problems with interval data (MM-RCO) are usually NP-hard, even in the case when the classic problem is easy to solve; this isthe case of the minimum spanning tree problem, shortest path problem, assignment problems andothers; see Kasperski (2008) for details.ExactalgorithmsforMinmaxRegretPathshavebeenproposedby(Karasanetal.,2001;Kasper-ski, 2008; Montemanni and Gambardella, 2004, 2005). All these papers show that exact solutionsfor MMRP can be obtained by different methods and take into account several types of graphs anddegrees of uncertainty. However, the size of the graphs tested in these papers was limited to amaximum of 2000 nodes.In this context, our main contributions in this paper are the analysis of the performance of the CPLEX solver for a MIP formulation of MMRP, the analysis of the performance of knownheuristics for the problem and finally the analysis of the performance of a proposed SA approach forthe problem. For experiments we consider two classes of networks, random and a class of networksused in telecommunications and both containing different problem sizes. Instances containing from500 to 20000 nodes with different degrees of uncertainty were considered.In the next section, we present the formal definition of the problem and notation associated andin Section 3 a mathematical programming formulation for MMRP is presented. We also discuss themidpoint scenario and the upper limit scenario heuristics for MMRP in more detail and then presentthe general algorithm for SA. In Section 4 we formally define the neighborhood used for our SAapproach. Details of our experiments and their results are analyzed in Section 5. Finally in Section6, our conclusions and suggestions for future work are presented. 2 Notation and Problem Definition Let G = ( V,A ) be a digraph, where V  corresponds to the set of vertices and A is conformed by a setof arcs. With each arc ( i,j ) in A we associate a non-negative cost interval [ c − ij ,c + ij ] , c − ij ≤ c + ij , i.e.,there is uncertainty regarding the true cost of the arc ( i,j ) , and whose value is a real number thatfalls somewhere in the above-mentioned interval. Additionally, we make no particular assumptionsregarding the probability distribution of the unknown costs.The Cartesian product of the uncertainty intervals  c − ij ,c + ij  , ( i,j ) A , is denoted as S  and anyelement s of  S  is called a scenario ; S  is the vector of all possible realizations of the costs of arcs. c sij , ( i,j ) A denote the costs corresponding to scenario s .Let P be the set of all s - t paths in G . For each XP  and sS  , let F  ( s,X  ) =  ( i,j ) X  c sij , (1)be the cost of the s - t path X  in the scenario s .The classical s - t shortest path problem for a fixed scenario sS  is Problem OPT.PATH(s) . Minimize F  ( s,X  ) : XP  .Let F  ∗ ( s ) be the optimum objective value for problem OPT.PATH(s). For any XP  and sS  , the value R ( s,X  ) = F  ( s,X  ) − F  ∗ ( s ) is called the regret  for X  underscenario s . For any XP  , the value Z  ( X  ) = max sS  R ( s,X  ) , (2)2  is called the maximum (or worst-case ) regret  for X  and an optimal scenario s ∗ producing sucha worst-case regret is called worst-case scenario for X  . The minmax regret version of Problem OPT.PATH(s) is Problem MMRP . Minimize { Z  ( X  ) : XP  } .Let Z  ∗ denote the optimum objective value for Problem MMRP .For any XP  , the scenario induced by X  , s ( X  ) , for each ( i,j ) A is defined by c s ( X  ) ij =  c + ij , ( i,j ) X c − ij , otherwise.(3)Let Y  ( s ) denote an optimal solution to Problem OPT.PATH(s). Lemma 1 (Karasan et al. (Karasan et al., 2001)). s ( X  ) is a worst-case scenario for X  .According to Lemma 1, for any XP  , the worst-case regret Z  ( X  ) = F  ( s ( X  ) ,X  ) − F  ∗ ( s ( X  ))= F  ( s ( X  ) ,X  ) − F  ( s ( X  ) ,Y  ( s ( X  )) , (4)can be computed by solving just one classic SP problem according to the definition of  Y  ( S  ) givenabove. 3 Algorithms for MMRP In this section we present both a MIP formulation for MMRP, which will be used to obtain an exactsolution by using a solver CPLEX, and our SA approach for finding an approximate solution forthe problem. Two simple and known heuristics based on the definition of specific scenarios are alsopresented. 3.1 A MIP Formulation for MMRP Consider a digraph G = ( V,A ) with two distinguished nodes s and t . According with the pastsection each arc ( i,j ) ∈ A hasinterval weight  c − ij ,c + ij  and also hasabinary variable x ij associatedexpressing if the arc ( i,j ) is part of the constructed path. We use Kasperski’s MIP formulation of MMRP (Kasperski, 2008), given as follows: min  ( i,j ) ∈ A c + ij x ij − λ s + λ t (5) λ i ≤ λ  j + c + ij x ij + c − ij (1 − x ij ) , ( i,j ) ∈ A,λ ∈ R (6)  { i :(  j,i ) ∈ A } x  ji −  { k :( k,j ) ∈ A } x kj =  1 , j = s 0 , j ∈ V  - { s,t }− 1 , j = t (7) x ij ∈ { 0 , 1 } , for ( i,j ) ∈ A (8)The solver CPLEX is then used to solve the above MIP.3  3.2 Basic Heuristics for MMRP Two basic heuristics for MMRP are known; in fact these heuristics are applicable to any MM-RCO problem. These heuristics are based on the idea of specifying a particular scenario and thensolving a classic problem using this scenario. The output of these heuristics are feasible solutionsfor the MMRCO problem (Candia-Véjar et al., 2011; Conde and Candia, 2007; Kasperski, 2008;Montemanni et al., 2007; Pereira and Averbakh, 2011a,b).First we mention the midpoint scenario, s M  defined as s M  = [( c + e + c − e ) / 2] , e ∈ A . Theheuristic based on the midpoint scenario is described in Algorithm HM  . Algorithm HM( G , c ) Input: Network  G , and interval costsfunction c Output: A feasible solution Y for MMRP. 1: for all e ∈ A do 2: c s M  e = ( c + e + c − e ) / 2 3: end for 4: Y  ← OPT  ( s M  ) 5: return Y,Z  ( Y  ) Algorithm HU( G , c ) Input: Network  G , and interval costsfunction c Output: A feasible solution Y for MMRP. 1: for all e ∈ A do 2: c s U  e = c + e 3: end for 4: Y  ← OPT  ( s U  ) 5: return Y,Z  ( Y  ) We refer to the heuristic based on the midpoint scenario as HM. The other heuristic based onthe upper limit scenario will be denoted by HU and is described in Algorithm HU  .The heuristics HU and HM have been designed for rapidly obtaining feasible solutions. Theseheuristics find a solution by solving the corresponding classic problem only twice. The first is thecomputation of the solution Y  in the specific scenario, s M  for HM or s U  for HU, and the second isthe computation of  Z  ( Y  ) (see steps 4 and 5 in Algorithm HM  and Algorithm HU  ). These heuristicshave been used in an integrated form by the sequential computing of the solutions given by HM andHU and using the best. In the evaluation of heuristics for MMR problems, some experiments haveshown that if these heuristics are considered as an initial solution for a heuristic, improved solutionsare not easy to achieve, please refer to Montemanni et al. (Montemanni et al., 2007), Pereira andAverbakh (Pereira and Averbakh, 2011a,b) and Candia-Véjar et al. (Candia-Véjar et al., 2011) fora more detailed explanation. 3.3 Simulated Annealing for MMRP Simulated Annealing (SA) is a very traditional metaheuristic, see Dréo et al. (2006) for details. Ageneric version of SA is specified in Kirkpatrick et al. (1983).We shall now describe the main concepts and parameters used within the context of the MMRPproblem, Search Space A subgraph S  of the srcinal graph G is defined such that this subgraph contains a s - t path. In S  a classical s - t shortest path subproblem is solved, where the arc costs are chosentaking the upper limit arc costs. Then, the optimum solution of these problem is evaluated foracceptation. Initial Solution The initial solution Y  0 is obtained applying the heuristic HU to the srcinal net-work  S  0 The regret Z  ( Y  0 ) is then computed.4  Initial Temperature Different values for the initial temperature were tested and t 0 = 1 was usedfor all experiments. Cooling Programming A geometric descent of the temperature was according to parameter beta.Several experiments were performed and after to consider the trade-off between the regret of the solution and time needed to compute it, β  was fixed as 0 . 94 for all experiments. Internal Loop This loop is defined by a parameter L and depended on the size of the instancestested. After initial experiments, L was fixed at 25 for instances from 500 nodes to 10000nodes. For instances with 20000 nodes the L was fixed at 50 . Neighborhood Search Moves Let S  i be the subgraph of G considered at the iteration i and let x i be the solution given by the search space at the iteration i . Then we generate a new subgraph S  i +1 of  G from S  i changing the status of some components of the vector characterizing S  i .The number of components is managed by a parameter λ and a feasible solution is obtainingsearching S  i +1 (according with the definition of Search Space) which must be tested foracceptation. Acceptation Criterion A standard probabilistic function is used to determine if a neighboring so-lution is accepted or not. Termination Criterion A fixed value of temperature (final temperature t f  ) is used as terminationcriterion with t f  = 0 . 01 .Our definition of Neighborhood Search Moves is new, but takes inspiration from that describedby Nikulin. In his paper (Nikulin, 2007), he applied this to the interval data minmax regret spanningtree problem. 4 Neighborhood Structure for the MMRP problem Given the importance of the neighborhood structure in our proposed method, we have dedicated thissection to explain it in detail. We start by defining the Local Search (LS) mechanism. Subsequentlywe detail the concepts of neighborhood structure and Search Space. After that, we explicitly de-scribe an architectural model for obtaining new candidate solution by restricting the srcinal searchspace. 4.1 Local Search (LS) Local Search (LS), described in Algorithm local-search , is a search method for a CO problem P  with feasible space S  . The method starts from an initial solution and iteratively improves it byreplacing the current solution with a new candidate, which is only marginally different. During thisinitialization phase, the method selects an initial solution Y  from the search space S  . This selectionmay be at random or taking advantage of some a priori knowledge about the problem.An essential step in the algorithm is the acceptance criterion, i.e., a neighbor is identified as thenew solution if its cost is strictly less in comparison to the current solution. This cost is a functionassumed to be known and is dependent on the particular problem. The algorithm terminates whenno improvements are possible, which happens when all the neighbors have a higher (or equal) costwhen compared to the current solution. At this juncture, the method outputs the current solution asthe best candidate. Observe that, at all iteration steps, the current solution is the best solution foundso far. LS is a sub-optimal mechanism, and it is not unusual that the output will be far from the5
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