A Three-Level Meta-heuristic for the Vertex p-centre Problem

Transcription

1 A Three-Level Meta-heuristic for the Vertex p-centre Problem A. Al-khedhairi 1, S. Salhi 2y 1 Department of Statistics and OR, King Saud University, Saudi Arabia. 2 Centre for Heuristic Optimisation, Kent Business School, University of Kent, UK. January 13, 2007 Abstract We present an e cient three-level metaheuristic based on variable neighbourhood search and perturbation schemes for the vertex p-centre problem. Reduction tests, that take into account the characteristics of the problem, are embedded into the search to speed up the process without a signi cant loss in solution quality. Encouraging results are obtained which are then used to derive tight lower bounds. This useful information is then integrated into a well known exact method to speed up the process even further. These ideas are tested on the OR-Lib data set with excellent results. Key words: p-centre; location; Variable Neighbourhood; Multi-level heuristic, Exact method. Introduction In the p-centre problem (also known as the minimax problem) the aim is to locate p facilities and assign demand nodes (i.e., clients or customers) to these p open facilities so that the maximum distance between a demand node and the facility to which it is assigned An earlier version of this paper was given at OR48, Bath, UK. y Corresponding author; s.salhi@kent.ac.uk 1

2 is minimised. This well known location problem was rst introduced by Hakimi (1964). It has several applications including the location of emergency facilities such as ambulance stations and rehouses. There are several possible variations of the basic model. For instance, if facility locations are restricted to the nodes of the network, the problem is referred to as the vertex center problem. Centre problems which allow facilities to be located anywhere on the network are known as absolute center problems. Both versions can be either weighted or unweighted where the weight represents the population at, or the importance of, the demand nodes. Let I = set of demand nodes, I = f1; ::::; Ng: J = set of candidate facility sites, J = f1; :::::; Mg: d ij = distance between demand node i 2 I and candidate site j 2 J, p = number of facilities to be located, 8 < 1 if a facility is located at candidate site j 2 J: Z j = : 0 otherwise. 8 < 1 if demand node i 2 I is assigned to an open facility at candidate site j 2 J. Y ij = : 0 otherwise. D = maximum distance (or time) between a demand node and the nearest facility (D is also referred to as the covering distance or time). The vertex p-centre problem can be formulated as a binary linear programming problem as follows: M inimise D (1) subject to: X Y ij = 1 8i 2 I (2) j2j Y ij Z j 8i 2 I; j 2 J (3) 2

3 X Z j = p (4) j2j D X j2j d ij Y ij 8i 2 I (5) Z j, Y ij 2 f0; 1g 8i 2 I; j 2 J (6) The objective function (1) minimises the maximum distance between each demand node and its closest open facility. Constraint (2) ensures that each demand node is assigned to exactly one facility, while constraints (3) restrict demand nodes to be assigned to open facilities. Constraint (4) stipulates that exactly p facilities are to be located. Constraints (5) de ne the maximum distance between any demand node i and the nearest facility at node j. Finally, constraints (6) refer to integrality constraints. For xed values of p, the vertex p-centre problem can be solved in polynomial time. This can be done by evaluating each of the O(N p ) possible combinations of p facility sites in polynomial time (Daskin, 1995) though it may take a considerable amount of CPU time. For variable values of p, the p-centre problem is NP-hard (Kariv and Hakimi, 1979). One of the techniques to solve the vertex p-centre problem optimally is to use an auxiliary problem such as the Set Covering Problem (SCP). The objective of the SCP is to nd the minimum number of facilities and their locations so that each demand point has to be served by a facility within a speci ed maximum response time (or distance) which we refer to as radius. The solution to this problem can easily be found by solving its linear programming relaxation with occasional branch and bound applications. However, for large problems the size of the relaxed version of the SCP can be reduced using successive row and column reductions (see Daskin, 1995). This idea is to nd the smallest radius such that the optimal solution of the considered auxiliary problem yields a feasible solution to the p-centre problem. Initially, Minieka (1970) suggested a rudimentary algorithm that relies on solving a nite sequence of Set Covering Problems. The idea is to choose a threshold distance (covering distance) as radius and to check whether all demand points (customers) are covered within this radius using no more than p facilities. Based on Minieka s idea, Daskin (1995) developed an algorithm to solve this problem optimally using 3

4 the bisection method that systematically reduces the gap between the upper and lower bounds of the optimal solution and hence nds the optimal covering distance. Also, Daskin (2000) and Elloumi, Labbe and Pochet (2001) proposed two e cient exact algorithms for the vertex p-centre problem. Both algorithms solve successive subproblems and rely on carrying out an iterative search over coverage distances rather than sets of facility locations. Daskin has formulated a maximum set covering subproblem and solved it by Lagrangian relaxation, whereas Elloumi et al. used Minieka s subproblems and solved it by a combination of a greedy heuristic and the IP formulation of the subproblem. Ilhan and Pinar (2001) proposed an interesting exact solution method for the vertex p-centre problem. Their method is composed of two phases. The rst phase, called the LP-Phase, computes a lower bound to the optimal solution of the problem by solving a series of feasibility problems based on an LP formulation. This phase, which is quick, guarantees that the number of open facilities has to be at least say p, hence eliminating any redundant ILP problem with a value less than p. The second phase, referred to as the IP-Phase, uses also feasibility problems to check whether or not it is possible to serve all customers with no more than p facilities within a given radius. Very recently, Al-Khedhairi and Salhi (2005) solve the vertex p-centre problem by introducing modi cations to two exact algorithms namely the Ilhan and Pinar s (2001) and the one by Daskin (1995). For the Ilhan and Pinar s (2001) algorithm, to reduce the number of IP iterations, two enhancements are introduced in the IP phase (the second part) of that algorithm. This phase is found to be the most time consuming part. It was shown that this enhanced algorithm performs better than the original algorithm if the number of IP iterations needed to get the optimal solution is more than 4 iterations. For the Daskin s (1995) algorithm, an enhancement is used to tighten both the initial lower and upper bounds. Moreover, the Golden Section binary search, proved to be more e cient than the commonly used bisection method. The combination of these two modi cations produced a reasonable reduction in the number of IP iterations. The aim of this paper is two-folds- (i) to design an e cient three-level meta-heuristic that combines variable neighbourhood and perturbation schemes to produce tight upper bound, and (ii) to derive a corresponding tight lower bound. These two bounds when used together speed up the process of nding the optimal solution. 4

5 The Three-Level Heuristic A three-level heuristic which uses a variable neighbourhood strategy in the rst two levels and a perturbation mechanism, for diversi cation purposes, at the third level is developed. The heuristic follows the structure of multi-level heuristics as outlined by Salhi and Sari (1997) for the case of multi-depot vehicle routing problems. See Salhi (2006) for an overview of heuristic search in general. In this section we rst present the main steps of the heuristic as given in Figure 1. This is followed by some notations and de nitions as well as explanations of the main steps whenever deemed necessary. The three-level heuristic algorithm Step 1: Generate an initial feasible solution. Step 2 (Level 1): Use neighbourhood 1 (N 1 ) which consists of two smaller neighbourhoods namely N 1 1 and N 1 2 which are performed sucessively until the best solution can no longer be improved. Step 3 (Level 2): Use neighbourhood 2 (N 2 ) until no better solution can be found. If there is an improved solution go back to step 2, else go to step 4. Step 4 (Level 3): Introduce a perturbation mechanism based on the add/drop procedures. If a new solution is found go back to step 2, else stop. Figure 1: The three-level heuristic Multi-level heuristics are similar in principle to variable neighbourhood descent (VND) except that here each level represents not necessarily a neighbourhood but also possible re nement procedures. This process is repeated until none of the neighborhoods or the procedures yield improving solutions. The strengths of this scheme happen to be the same as the ones of Variable Neighbourhood Search (VNS), originally developed by Brimberg and Mladenoivc (1996), and Mladenović and Hansen (1997) for the continuous location problem (the Weber problem) and the p-median problem respectively. VNS is however simpler to use as it is based on using neighbourhoods only and also attempts to escape local optimality by using systematically a larger neighbourhood whenever the search appears to 5

6 reach a local optimum. In VNS the search also reverts back to the smallest neighbourhood ( rst level) for further exploration. VNS and its variants have been applied to several NP hard problems with a reasonable amount of success (Hansen and Mladenović, 2003). Notations/De nitions p j : facility number j, (j = 1; :::; p). D j : maximum service distance for facility p j, (j = 1; :::; p). D : the standard service distance, D = MaxfD j ; j = 1; :::; pg. x i : demand point i (or potential facility site i), i = f1; ::::; Ng: d ij = d(x i ; p j ) : The shortest distance between customer x i and facility p j :(i = 1; :::; N); (j = 1; :::; p): dist(i; t) : The shortest distance between customers x i and x t. Circle[j] : Contains all customers served by facility p j (i.e., all customers within a speci c distance from this facility). Let Circle[j] = fx i : d ij D j ; i = 1; :::; Ng j = 1; :::; p. Group of Circle[j] : Contains all circles represented by their facilities and their assigned customers which are within a prede ned distance from facility p j. Let Group of Circle[j] = f Circle[l] : d(p j ; p l ) D max ; l = 1; :::; pg. The choice of D max is critical. If it is too small it may become redundant but if it is too large more circles will be included. These extra circles will be too far from Circle[j] and will not contribute in improving the objective function but will consume a large amount of CPU time to nish the process. In this study, after some preliminary experiments, we set the radius of the Group of Circle[j]; D max to be 2D. Initial Solution (step 1) We use a multi-start heuristic to nd 100 di erent initial solutions, each of which consists of p integer numbers generated randomly between 1 and N. Each demand point (or node) is allocated to its nearest facility and the coverage distance is then computed. The best solution among these 100 initial solutions is retained and chosen to be our initial solution for further investigation. 6

7 Neighbourhood Structure N 1 (step 2: level 1) This neighbourhood structure consists of two neighbourhood substructures, namely N 1 1 and N 1 2. The substructuren 1 1 Starting with the initial feasible solution, for each Circle[j] the facility p j is checked for possible removal from its place to all of its customers sites (i.e., customers covered by this facility) which lie within D min (i.e., d ij D min ; x i 2 Circle[j]). If such a move produces a positive improvement (i.e., will reduce the value of D), then this new location for p j is accepted and all customers belonging to Circle[j] are then reallocated to their nearest facility among those belonging to group of Circle[j] and not to all the open facilities. If such a move is negative the next site is checked and so on until all possible sites in Circle[j] are investigated. The same process is repeated until all circles are examined. To be more e cient only those a ected circles are re-assessed (i.e., those which have provided improvement are re-examined in our search). In this neighbourhood structure N 1 1, the choice of the value D min is important. In this study after some preliminary testing we set D min = D=2. The choice of moving p j to one of the customers which are far from facility p j by more than D min will be dealt with in a di erent way using neighbourhood structure N 1 2 which is given next. This is applied only when there is no change in all circles based on N 1 1. The substructuren 1 2 This neighbourhood structure is an enlarged version of N 1 1. For each Circle[j] we rst obtain its group of circles (i.e., Group of Circle[j]) then we move, temporarily, the facility p j to each site from its customers sites given that the distance between the customer s site and the facility is greater than D min = D=2 (i.e., d(x i ; p j ) > D min ; x i 2 Circle[j]). If there is improvement all customers in group of Circle[j] will be re-allocated not as in N 1 1 where only those customers from Circle[j] are examined. If there is no improvement after examining all circles we explore neighbourhood N 2 which is given next. 7

8 Neighbourhood Structure N 2 (step 3: level 2) In this neighbourhood, two facilities are simultaneously moved from their original locations to two of their customers locations, and their contribution is then evaluated to whether or not a move is worthwhile (i.e., there is improvement). In addition, in this neighbourhood structure we restrict the two centres to be re-located not to be too far from each other so to have an e ect on the customer reallocation process, otherwise N 2 reduces to the application of N 1. As the objective in this particular covering location problem is to minimise the maximum distance, our focus is on the largest circle (i.e., the Circle[j] which has the biggest radius D j ) and its group of Circles[j] as de ned earlier in the notation subsection. In other words, starting with the largest circle and trying to decrease its radius will de nitely improve the objective function (except in case of tie). This will be possible if one of the facilities to check for swapping lies in the largest circle and the second facility lies in the group of circles to this largest circle. Note that the circles far from the largest one (i.e., outside the group of circles of the largest circle) and which have smaller radiuses do not have a direct e ect on the objective function, and hence do not need to be considered in the neighbourhood structure N 2. This reduction scheme is useful as the CPU time for such operations can be better utilised in other ways without a signi cant loss in solution quality. The procedure performed in neighbourhood N 2 is summarised as follows: 1- For each Circle[k], nd Group of Circle[k]; k = 1; : : : ; p 2- Start from the current largest circle, say Circle[j], and record its facility p j and its maximum service distance D j For each x i 2 Circle[j]), temporarily move p j to x i Choose a facility p 0 j 2 Group of Circle[j] 2.3- For each x 0 i 2 Circle[j 0 ] subject to dist(i; i 0 ) > D j, temporarily move facility p j 0 to x i 0. Examine this swap move (i.e., the swapping of p j with x i and p j 0 with x i 0 ). 8

9 2.4- If there is a positive improvement then permanently move p j and p j 0 and x i 0 respectively. to the locations of their customers sites x i reallocate all customers in all a ected circles (this will be clari ed in the reduction test section), and update all group of circles based on the new con guration and go back to step (1) Otherwise go back to step (2.3) to evaluate another x i 0. If there is no more x i 0 go back to step (2.2) to start from another j 0. When all p j 0 have been checked, go to step (2.1) to start from another x i. If all possible x i for p j have been checked go to step (2) to select the next largest circle. This scheme is performed until all circles have been evaluated and there is no more further improvement. A Perturbation Mechanism (step 4: level 3) The idea of introducing perturbation procedures is to avoid getting trapped in local optima. One way is to accept infeasible solutions. Here we allow the number of facilities of the solution to go over or under the required number of facilities (p) by a certain amount (say q), then we start dropping or adding facilities accordingly to maintain the required number of facilities. The concept of adding and dropping one or more facilities is commonly used in the heuristic literature for solving discrete combinatorial optimisation problems, see Salhi (1997). The description of our add and drop heuristics is given as follows. The Add Routing There are several ways for adding facility(s) to a current con guration. In our study and after some experiments, we decided to use the following procedure to add one facility at a time: Because our goal is to reduce the objective function which represents the value of the service distance (D j ) of the largest circle(s), so, reducing the maximum service distance of the largest circle(s), will lead to minimising the objective function value except in the presence of a tie where we allow the procedure to re-iterate. Therefore, we rst nd the largest circle (say Circle[j]) and then split it into two smaller circles. This is 9

10 performed by nding the farthest customer, in Circle[j], from the facility p j and opening a facility at its site. Let pp[j] be this new facility and let p[j 0 ] be all facilities of the Group of Circle[j]. We apply the reallocation process to a restricted problem which consists of the subset of customers belonging to Circle [j] and those in Group of Circle[j], and the subset of facilities namely p j, pp j and all the p j 0. Thus the number of facilities is now increased by one. The objective function is then evaluated by nding the maximum service distance for each facility from the p + 1 facilities. If D j is not decreased because of the presence of tie the add procedure is reapplied again. The Drop Routine There are several procedures on how to drop a facility from a current con guration. After performing some experiments we implement a simple idea of removing a facility based on the smallest circle. The main steps of this routine are given in Figure 2. - Order all the p + 1 facilities in descending order according to their maximum service distance (D j ). - Starting from the last one in the list (i.e., the facility which has the smallest D j ) and for each customer covered by this facility check its distance to its second nearest facility and record the new coverage distance for each re-allocated customer. - If all customers can be covered by (or allocated to) the other facilities without increasing the original objective function (i.e., the objective function value when the number of facilities equal to p) then record the new value of the objective function and put this facility in the list of potential facilities that can be dropped. - Pick the next facility in the list (the one above that facility we just checked it) and repeat the same process for all facilities in the ordered list. - From the above list remove the facility which yields the highest reduction in the objective function and set its corresponding objective function value as the current new value. Figure 2: The dropping scheme Note that before selecting the facility from the ordered list and to test whether it is suitable for a facility to be dropped or not, we rst make sure that its maximum service distance, D j, is less than the original objective function value. If one of its customers happens to require a larger coverage than the existing solution then there is no need for checking other customers as this facility can not be dropped. 10

11 The idea behind this dropping scheme is that the facility which has a small maximum service distance is more likely, though this may not be true in certain situations, to have a small number of customers than those facilities having a large maximum service distance besides there is more chance to cover these customers with a smaller distance. This scheme has also the bene t in reducing the computational time needed to perform this drop routine. Reduction Tests The following fathoming rules are incorporated into the proposed three-level heuristic to speed up the search whenever possible. The aim is to detect those redundant moves that have no or very little e ect on the quality of the solution. Such a reduction in the computation time can then, if necessary, be used for other useful operations that may improve the solution even further. described below. We have introduced ve reduction tests which are (Test 1) If there is an improvement during the implementation of neighbourhood structure N 2 we only re-allocate those customers from the circles which are in both groups of circles involved in the search. For example, if the largest circle is Circle[j] and the other circle involved in the search which belongs to Group of Circle[j] is Circle[j 0 ], and an improvement has been achieved then we re-allocate all customers in Group of Circle[j] and Group of Circle[j 0 ] to their closest facility among those facilities belonging to their group of circles. This allocation mechanism has two bene ts. Firstly, a large saving in the CPU time is obtained to nd the best solution for each instance especially for larger values of p, and secondly, there is no, or negligible, loss in the quality to the solution generated. (Test 2) In neighbourhood N 2 we observed the following. For each Circle[j], we record all symmetrical pairs of customers sites checked for improving the objective function, so there is no need to re-check this pair again. (Test 3) In neighbourhood N 2, whenever there is an improvement we record all those a ected circles (Circle[j], j = 1; :::; p) (i.e., circles which lost or gained one or more customer). Such information is important as when the search returns from N 2 to N 1 (N

12 and N 1 2 ) as we need to concentrate the search, when using N 1 1 and N 1 2, on those a ected circles only. In other words, applying this procedure to una ected circles will not improve the objective function and hence it is useful to avoid performing such unnecessary operations. (Test 4) As the process of perturbation is performed, a restriction is applied to prevent moving back to previously visited solutions. However, each time we apply the Add routine we record the locations of the customers which have been used as a location for the new added facilities. So, when we go back from the Drop routine to the Add routine we rst check whether the chosen customer s site has already been chosen before as a location for a previously added facility or not. If this is the case then the facility which was located at this customer s site is considered to be the chosen facility to be dropped and consequently, another customer s site is then selected as the new facility to be added. This situation arises when there is more than one customer in the largest circle that yields the maximum service distance for this circle (existence of a tie). (Test 5) In the Drop procedure we construct a list of potential facilities for dropping by putting all the p + 1 facilities in descending order according to their maximum service distance. Starting from the last one in the list we check the possibility of dropping the facility if its maximum service distance is less than the objective function. We select the one that yields the most improvement. However, after performing some preliminary experiments, it was observed that in most cases, the best facility to be dropped usually occurs in the bottom half of the ordered list. Therefore, in this study we concentrate our search on the bottom half of such a list only when selecting the best one to remove. The reasoning behind this choice is that if a facility is further up in the ordered list, its maximum service distance (D j ) is likely to be higher. In addition, the facility p j in this case may serve (or cover) a larger number of customers which may increase the risk of generating a larger value of D j. Computational Results To assess the performance of our proposed heuristic, the exact method presented in Daskin (1995) is used. For consistency, both the exact method and the proposed heuristic algo- 12

13 rithm are coded in C and tested on Sun Enterprise Workstation 450 running Solaris 2.6. Xpress-MP Optimisation software (Modeller Release 13.01, Optimiser Release 13.02) is used to solve the Set Covering problem optimally. The proposed heuristic algorithm is then tested on instances taken from OR-Lib (Beasley, 1990 and Reinelt, 1991) for p-median problems. The results are summarised in Table 1 where le no. represents the le number of p-median data in OR-Lib, N refers to the number of vertices in the network (or number of customers), and P is the optimal number of facilities needed to locate. CPU1 and CPU2 denote the CPU time (in seconds) elapsed, by the exact method to nd this optimal solution, and the proposed heuristic algorithm respectively. Opt Solution represents the optimal solution found by the exact method, Best Solution refers to the best solution found by the proposed algorithm, and dev (in %) is the gap between the optimal solution and our best solution. The deviation is calculated as: F opt dev = F best 100 F opt where F best is the best improved service distance (D) found by the proposed heuristic algorithm, and F opt refers to the optimal solution found by Daskin s exact method (Daskin, 1995). Table 1 shows that two instances have been solved optimally (N = 100; p = 5) and (N = 300; p = 100). 7 of the best solution use one unit extra only, and 12 other best solutions necessitate 2 extra units only when compared to the solution obtained by the exact method. The worst cases are instances number 4, 6 and 11 (Pmed4, Pmed6 and Pmed11) where the best solution found by the heuristic is 79, 89 and 64 whereas the exact solution is 74, 84 and 59, respectively (i.e., 5 units more than the optimal solution) with a deviation of 6.76%, 5.59% and 8.47% from the optimal solution, respectively. The average of the deviations over all instances is 8.19% and the worst deviation is 18.18% produced from instance number 25. In instances number 30 and 33, for example, the best solution found by our heuristic is one unit greater than the exact solution obtained by the exact method but the computing 13

14 time needed to nd this best solution is approximately half (i.e., the time decreased from and seconds to and seconds, respectively). Moreover, the instances which have their best solution two units greater than the optimal ones consume a tiny fraction of the computing time when compared to the exact method. To be more accurate large instances such as instances number 31, 35, 38, 39 and 40 required 8.81, 15.69, 0.14, and seconds whereas the exact solution needed 131.1, 204.6, 395.7, and seconds, respectively. Figure?? illustrates a comparison between the best solution found by our three-level heuristic and the optimal solution obtained by the exact method proposed in Daskin (1995). 140 The Solution Found by Exact Method and our Three level Heuristic Instance No. EXACT 3 level heuristic Figure 3: Comparison of the solution found by our three-level heuristic and exact method for the OR-Lib instances. The E ect of Each Module of The Heuristic In this section we will show how the computing time is shared between the di erent modules namely neighbourhood substructures N 1 1, N 1 2, the neighbourhood N 2, and 14

15 File Exact Method Heuristic Method No. N P Opt. Solution CPU1 (s) Best Solution dev. (%) CPU2 (s) Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Average Table 1: Comparison of OR-Lib instances for the proposed three-level heuristc and the exact algorithm of Daskin. 15

16 C P U T im e (s) the Add and the Drop routines. We rst provide an illustrative example and then produce some basic statistics to support our ndings. An Illustrative Example As an example we chose the largest instance which is number 40 (N = 900; p = 90). As shown in Figure??, neighbourhood two (N 2 ) consumes about 52 % of the total CPU time (52.44 seconds out of a total of seconds). The rst part of neighbourhood one (N 1 1 ) uses 9.5%. The Add and the Drop heuristics require approximately 7.7% and 14.6% of the total CPU time, respectively. The least time consuming task is the second part of neighbourhood one (i.e., N 1 2 ). C o m p u ta t io n e ff or t f o r e ac h m o d u le o f(pt hm eehd40) e ur is tic N1 1 N1 2 N2 A d d D ro p H e u r is t ic m o Figure 4: Illustration of computing e ort needed for each module in the heuristic for instance number 40 (N = 900; p = 90). According to this graph, to enhance the e ciency of the heuristic one way is to concentrate the e ort in speeding up the search within neighbourhood two, N 2, which appears to take up nearly half of the total time. A similar trend of results is found for most tested instances, see Al-Khedhairi (2004) for further details. 16

17 C o m p u t i n g T i m e ( s) D R O P A D D N2 N1 2 N F a c ili t y n u m be (p) r Figure 5: Sharing computing time between the di erent ve modules of our heuristic for instance number 27 (N = 600; p = 10; 20; :::; 100). A Basic Statistical Analysis To provide more reliable conclusions we carried out additional experiments using two scenarios. In the rst one we x the number of nodes (N) and vary the value of p. This will demonstrate whether increasing the value of p has the same e ect or not. In the second scenario we x p (moderate value) and investigate changes in N. Scenario 1 We x the number of nodes to be slightly large (say N = 600) and vary the value of p in step of 10 starting from 10 to 100. The results of these 10 instances are given in Figure??. It can be shown that the search in neighbourhood two, N 2, is consuming the most computing time. For instance when p = 10 and 20 the search in N 2 used around 60% and 92% of the total CPU times (6.6 and seconds out of a total of and seconds), respectively. The same trend happens for p = 70; 80; 90 and 100 where approximately 90%, 85%, 75% and 80% of 17

18 C o m puting Tim e (s ) D R O P A D D N2 N1 2 N N u m b e r o f n od (N) es Figure 6: Sharing computing time between the ve modules of the heuristic for p = 20 and di erent number of nodes (N = 100; 200; :::; 900). the total CPU time was needed for each instance, respectively. Therefore, it seems that increasing p leads to increasing the computing time needed for exploring neighbourhood two. We believe that this increase of computing time depends in the ratio N p. If it is too small or too large the CPU time (needed for exploring neighbourhood two) tends to go down but it becomes high at approximately N p ' 30 to 40. Moreover, the graph shows that the computing time (in seconds) for neighbourhood N 1 1, Add and Drop methods is increased when the value of p is increased but still relatively negligible in percent (%) when compared to N 2. Scenario 2 Here we x the value of p to be 20 and use di erent values for N starting with N = 100 to 900 with an increment of 100. The results of these 9 instances are shown in Figure??. According to this graph, even for xed values of p and varying N, neighbourhood two, N 2, is still taking up the most 18

19 computing time. For example, for N = 100; 200 and 300 it used 0.12, 1.7 and seconds out of total CPU times of 0.18, 2.8 and 14.4 seconds (i.e., approximately 66%, 41% and 82%), respectively. In addition, for N = 700; 800 and 900 the computing time required by neighbourhood N 2 was 103.9, and seconds, respectively. Consequently, we could say that the increase in computing time needed by N 2 is not highly depended on the increase of the value of N but mainly on the value of p. Also, from the graph, we can note that the computing time required by neighbourhoods N 1 1 and N 1 2, and the Add and Drop routines is increased when the value of N is increased but still negligible in percent (%) when compared to the time spend in N 2. The Integration of Heuristic Search and Exact Methods In this section we demonstrate the usefulness of considering the strengths of both heuristic search (in our case the proposed three-level heuristic) and exact methods (in our case Daskin s algorithm, 1995) when solving the vertex p-centre problem. We put forward the following schemes for this purpose. Tightening the initial lower (L) and upper (U ) bounds The original algorithm proposed in Daskin (1995) assumes that the initial lower and upper bounds (L and U, respectively) is zero and the maximum distance value in the distance matrix, respectively. Other enhancements on these bounds are also given in Al-Khedhairi and Salhi (2005). Estimation of the lower bound (i) Set the initial upper bound (U) to the best solution found by the proposed heuristic, and the initial lower bound (L) to be L = U (1 ) and L 0 = L where is the average deviations determined from a sample of data sets. For illustration in our case we used the average value found in Table 1 which is 0:0819. It is worth noting that this value of L may not be necessary a true lower bound as it might produce a feasible 19

20 solution for the SCP. An appropriate adjustment will be introduced to deal with this issue accordingly. (ii) Also, because the value of L may not be one of the entries in the distance matrix (i.e., there may be no customer at this distance value), hence we need to adjust this value to the nearest entry in the distance matrix. We set the adjusted value of L to the coverage distance (D) and solve the SCP for this distance value. We will refer to the adjusted value of L as L a. The adjustment is given as follows: Step 1: If the SCP is feasible Set U = L a and L = L 0 (1 k) where k is the number of iterations performed (i.e., number of times the SCP is solved, initially set k = 1), and the standard deviation taken from a sample ( in our case this value is 0:0409 and it is computed from Table 1). Adjust L to the nearest entry in the matrix, set the adjusted value (L a ) to be the coverage distance (D) and solve the SCP using the new adjusted value, L a. Set k = k + 1 and repeat step 1. Step 2: Else (the SCP is infeasible) set L = L a (the last L a value) and continue to implement the original exact method of Daskin s based on the bisection method as its binary search and using the new values of L and U as its lower and upper bounds. In our examples, the presented scheme is found to be useful due to the consistency of our 3-level heuristic given its low value of its standard deviation ( = 0:04, see Table 1). In other words, in this particular application, it would not matter so much if the gap between the results obtained by the exact method and that provided by a given heuristic is large or not, as long as it is consistent. For instance even if a heuristic produces solutions that are of average quality but with a relatively small value for the standard deviation (say most of the solutions, 90% or more, happen to lie within ), such a heuristic can be useful. This is because if a given heuristic has such a property it will direct the search to the area which contains the optimal solution using the lower bound through and. Of course if the is also small, that is an extra bonus as the range of the solutions will be even tighter. 20

21 Exact Method Heuristic/Exact File Optimal Best Total No. Solution # iterations CPU (s) Solution CPUH (s) CPUE(s) # iterations CPU (s) Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Pmed Average Table 2: Comparison results of using the scheme of tightening the initial lower and upper bounds, and the exact algorithm of Daskin to nd the optimal solution for the 40 OR-Lib instances. 21

22 Computational Results Table 2 reports a comparison results of using this integrating scheme. le no in column 1, Optimal Solution in column 2 and Best Solution in column 5 are as de ned in Table 1. # iterations in columns 3 and 8 refer to the number of iterations required by the exact method of Daskin (1995) and the proposed scheme suggested here respectively. CPU in column 4 reports the CPU time (in seconds) elapsed by the exact method to nd the optimal solution. CPU H and CPU E in columns 6 and 7 refer to the CPU time (in seconds) elapsed by the proposed heuristic algorithm and the scheme presented in this section, respectively. Total CPU in column 9 reports the total CPU time (i.e., CPU H and CPU E ). From Table 2 we can see that the number of iterations performed by the suggested enhancement scheme is reduced signi cantly for all instances when compared to those iterations carried out by the original implementation of the exact method. Consequently, the computational time required by this scheme is also reduced for all instances. For example, the number of iterations needed to solve 25 instances, out of the 40 instances, is reduced by at least 4 iterations which represents approximately a reduction of 50%. For example, instances number 1, 10, 16, 26, 32, 35, 38, and 39 have a reduction of 61.5 %, 52.5%, 43.6%, 51.4%, 55.4%, 49%, 64%, and 50.7% in the computational time, respectively. In instance number 15, the number of iterations required to solve this particular instance is 2 iterations only instead of 8 iterations as initially needed by the exact method. In summary, the reduction in the total number of iterations and the total computational time for all instances when compared with the exact method, decreased by 55% and 40.3%, respectively. Figure?? shows a comparison between the exact method proposed by Daskin (1995) and the integration scheme in terms of computational times required to solve each instance from the 40 OR-Lib instances, using the values of CPU, CPU H, CPU E, and Total CPU in Table 2, respectively. This gure illustrates the e ciency (i.e., the saving gained in the computational time) of using our proposed three-level heuristic alongside the enhancement scheme presented in this section to speed up the process of nding the optimal solution of the vertex p-centre problem against the exact method of Daskin (1995). 22

23 L o g (C P U Ti m e s ( e cs.)) I n s ta n c e. N o E xa c t H e u r is t ich e u r is /E t icxa c t T o t a l H e ur ist /E xa ic c t Figure 7: comparison between the exact method proposed by Daskin (1995) and enhancement scheme in terms of computational times calculated as log(cpu Time). Conclusion In this paper, a three-level heuristic that integrates Variable Neighbourhood Search metaheuristic and a perturbation mechanism is put forward to solve the vertex p-centre problem. Two neighbourhood structures are used and infeasible solutions were allowed but controlled through a perturbation mechanism using the add/drop procedures. Five reduction tests, that take into account the characteristics of the problem, are constructed to speed up the search engine. The use of such information appears to be crucial in the e ciency of the heuristic as a negligible loss of solution quality is observed. The e ect of integrating our heuristic method with a well known exact algorithm is also investigated. The results (upper bounds) obtained from the heuristic are then used to derive tighter lower which are then both used in the binary search of the exact method of Daskin (1995). This integration of heuristic search and exact method is promising, as encouraging results were obtained when tested on those data sets given in the literature. We believe that such a combination is worthwhile extending to other combinatorial problems where heuristic and exact methods complement each other rather than compete with each other. 23

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