A meta-heuristic algorithm for heterogeneous fleet vehicle routing problems

Discrete Optimization
A meta-heuristic algorithm for heterogeneous fleet vehicle routing problems with two-dimensional loading constraints
Stephen C.H.Leung a ,Zhenzhen Zhang b ,Defu Zhang b ,⇑,Xian Hua b ,Ming K.Lim c
a
Department of Management Sciences,City University of Hong Kong,Hong Kong b
Department of Computer Science,Xiamen University,Xiamen 361005,China c
Derby Business School,University of Derby,Derby,UK
a r t i c l e i n f o Article history:
Received 18October 2011Accepted 16September 2012Available online 3October 2012Keywords:Routing Packing
Simulated annealing Heterogeneous fleet
a b s t r a c t
The two-dimensional loading heterogeneous fleet vehicle routing problem (2L-HFVRP)is a variant of the classical vehicle routing problem in which customers are served by a heterogeneous fleet of vehicles.These vehicles have different capacities,fixed and variable operating costs,length and width in dimension,and two-dimensional loading constraints.The objective of this problem is to minimize transportation cost of designed routes,according to which vehicles are used,to satisfy the customer demand.In this study,we proposed a simulated annealing with heuristic local search (SA_HLS)to solve the problem and the search was then extended with a collection of packing heuristics to solve the loading constraints in 2L-HFVRP.To speed up the search process,a data structure was used to record the information related to loading feasi-bility.The effectiveness of SA_HLS was tested on benchmark instances derived from the two-dimensional loading vehicle routing problem (2L-CVRP).In addition,the performance of SA_HLS was also compared with three other 2L-CVRP models and four HFVRP methods found in the literature.
Ó2012Elsevier B.V.All rights reserved.
1.Introduction
The vehicle routing problem (VRP)was firstly addressed by Dantzig and Ramser (1959),proposing the most cost-effective way to distribute items between customers and depots by a fleet of vehicles.Taking into account of the attribute of the fleet,the tra-ditional VRP has evolved to different variants.Amongst them in-clude CVRP (homogenous VRP)that only considers a constraint of vehicles having the same limited capacity (Rochat and Taillard,1995),HVRP (heterogeneous VRP)that serves customers with dif-ferent types of vehicles (Golden et al.,1984;Gendreau et al.,1999;Lima et al.,2004;Prins,2009;Brandao,2011),VRPTW (VRP with time windows)that requires the service of each customer to start within the time window subject to time windows constraints (Kolen et al.,1987);and SDVRP (split deliver VRP)that allows more than one vehicle serving a customer (Chen et al.,2007).Readers are to refer to Crainic and Laporte (1998)and Toth and Vigo (2002)for a detailed description of VRP and its variants.To solve the VPR variants above effectively,a number of metaheuristics have been applied,such as simulated annealing (Osman,1993),Tabu search (Brandao,2011;Gendreau et al.,1999),genetic algorithms (Lima et al.,2004),variable neighborhood search (Imran et al.,2009),
and ant colony optimization (Rochat and Taillard,1995;Li et al.,2009).
In the real world,logistics managers have to deal with routing and packing problems simultaneously.This results in another domain of VRP to be investigated.In the literature,there are a number of frameworks proposed to address these two problems simultaneously.Iori et al.(2007)addressed the VRP with two-dimensional packing constraints (2L-CVRP)with an algorithm based on branch-and-cut technique.Gendreau et al.(2008)proposed a Tabu search heuristic algorithm to solve large instances with up to 255customers and more than 700items in the 2L-CVRP.Zachariadis et al.(2009)developed a new meta-methodology guided Tabu search (GTS)which can obtain better results.In this work,a collection of packing heuristics was proposed to check the loading feasibility.Fuellerer et al.(2009)presented a new ant colony optimization algorithm deriving from saving-based ant colony opti-mization method and demonstrated its performance to successfully solve the 2L-CVRP.More recently,Leung et al.(2011)developed a new efficient method that consists of a series of algorithms for two-dimensional packing problems.The method has proven its capability to improve the results of most instances used by Zachariadis et al.(2009).Duhamel et al.(2011)proposed a GRASP ÂELS algorithm for 2L-CVRP,whereby the loading constraints were transformed into resource constrained project scheduling problem (RCPSP)constraints before a packing problem can be solved.However,only basic CVRP and Unrestricted version
0377-2217/$-see front matter Ó2012Elsevier B.V.All rights reserved./10.1016/j.ejor.2012.09.023
Corresponding author.Tel.:+865922582013;fax:+865922580258.
E-mail address:dfzhang@ (D.Zhang).
of2L-CVRP were solved with their algorithm.Some researchers have extended their heuristics to three-dimensional problems. Gendreau et al.(2006)proposed a multi-layer Tabu search algorithm that iteratively invokes an inner Tabu search procedure to search the optimal solutions of a three-dimensional loading sub-problem.Tarantilis et al.(2009)used a guided Tabu search (GTS)approach with a combination of six packing heuristics to solve 3L-CVRP.In their work,a manual unloading problem was also tested.Furthermore,Fuellerer et al.(2010)also proposed their methods to deal with three-dimensional loading constraints.In addition,Iori and Martello(2010)provided a review in regard to vehicle routing problems with two and three-dimensional loading constraints.
Since most enterprises own a heterogeneousfleet of vehicles or hire different types of vehicles to serve their customers,it is there-fore crucial to study VRP with afleet of heterogeneous vehicles.The heterogeneousfleet VRP(HFVRP)addresses the VRP with a hetero-geneousfleet of vehicles which have various capacities,fixed costs and variable costs(Choi and Tcha,2007;Imran et al.,2009).In the literature,three versions of HFVRP have been studied.Golden et al. (1984)considered the variable costs to be uniformly spread across all vehicle types and the availability of each type of vehicle to be unlimited.Gendreau et al.(1999)considered the different variable costs for different types of vehicle.The third HFVRP was introduced by Taillard(1999)and Tarantilis et al.(2004),in which the number of available vehicles of each type is limited.Recently,Penna et al. (2011)introduced an Iterated Local Search,combined with a Vari-able Neighborhood Decent procedure and a random neighborhood ordering(ILS-RVND),to solve all variants of HFVRP.
In this paper,we combined the HFVRP with two-dimensional loading constraints,called the heterogeneousfleet vehicle routing problems with two-dimensional loading constraints(2L-HFVRP). However,to the best of our knowledge,no work has been con-ducted to address such VRP although it is a practical problem in real-world transportation and logistics industries.In2L-HFVRP, there are different types of vehicles with different capacity,fixed cost,variable cost,length and width in vehicle dimension,and
two-dimensional loading constraints.The demand of a customer is defined by a set of rectangular items with given width,length and weight.All the items belonging to one customer must be as-signed to the same route.The objective is to describe the minimum transportation costs with a function of the distance travelled,fixed and variable costs associated with the vehicles.
This paper presents a simulated annealing(SA)algorithm for 2L-HFVRP.In the literature SA has been proven to be an effective method to solve combinatorial optimization problems and it has been successfully applied to2L-CVRP(Leung et al.,2010).In this paper,a heuristic local search is used to further improve the solu-tion of SA.In addition,six promising packing algorithms,whereby five were developed by Zachariadis et al.(2009)and one by Leung et al.(2010),are also used to solve the loading constraints in 2L-HFVRP.These algorithms are extensively tested on benchmark instances derived from the2L-CVRP test problems with vehicles of different capacity,fixed and variable costs,length,and width. The comparison with several effective methods of the2L-CVRP and pure HFVRP is also given.
2.Problem description
The2L-HFVRP is defined on an undirected connected graph G=(V,E),where V={0,1,...,n}is a vertex set corresponding to the depot(vertex0)and the customers(vertices1,2,...,n)and E={e ij:i,j2V}is an edge set.For each e ij2E,a distance d ij (d ii=0)is associated.Afleet of P different types of vehicles is lo-cated at the depot,and the number of vehicles of each type is unlimited.Capacity Q t,fixed cost F t,variable cost V t,length L t and
width W t are associated to each type of vehicle t(t=1,2,...,P). The loading surface of vehicle of type t is A t=L tÃW t.On the basis that a vehicle with larger capacity usually has higher cost and greater fuel consumption,we assume that Q16Q26ÁÁÁ6Q P,F1-6F
2
6ÁÁÁ6F
P
and V16V26ÁÁÁ6V P.The traveling cost of each
edge e ij2E by a vehicle type t is C
ij t
¼V tÃd ij.The transportation cost of a route for vehicle type t is C R¼F tþ
P i<j R j
i¼1
V tÃd RðiÞ;Rðiþ1Þ, where R is the route whose start point and end point are the depot. Each customer i(i=1,2,...,n)demands a set of m i rectangular items,denoted as IT i,and the total weight of IT i equals to D i.Each item I ir2IT i(r=1,2,...,m i)has a specific length l ir and width w ir.We also denote a i¼
P m i
r¼1
w irÃl ir as the total area of the items of customer i.In2L-HFVRP,a feasible loading must satisfy the fol-lowing constraints:
(i)All items of a given customer must be loaded on the same
vehicle and split deliveries are not allowed.
(ii)All items must have afixed orientation and must be loaded with their sides parallel to the sides of the loading surface. (iii)Each vehicle must start andfinish at the depot.
(iv)Each customer can only be served once.
(v)The capacity,length and width of the vehicle cannot be exceeded.
(vi)No two items can overlap in the same route.
The objective of2L-HFVRP is to assign customer i(i=1,2,...,n) to one of the routes,so that the total transportation cost is mini-mized and all the routes fulfill the constraints.In this paper,we
200S.C.H.Leung et al./European Journal of Operational Research225(2013)199–210
consider two versions of2L-HFVRP which is the same as2L-CVRP: the Unrestricted only deals with feasible loading of the items unto the vehicles,and the Sequential considers both loading and unload-ing constraints(e.g.when visiting a customer,his/her items can be unloaded without the need to move items that belong to other cus-tomers in the same route).Fig.1gives an example of the two versions.
3.The optimization heuristics for two-dimensional loading problems
For a given route,it is necessary to determine whether all the items required by the customers can be feasibly loaded onto the vehicle.In this paper,we willfirst investigate if the total weight of items demanded by the customers exceeds the capacity of the vehicle.Otherwise,six packing heuristics are used to solve the two-dimensional loading problem.As mentioned earlier,the load-ing position of an inserted item must be feasible,i.e.it must not lead to any overlaps(for both Unrestricted and Sequential prob-lems),or sequence constraint violations(for Sequential only).The firstfive heuristics Heur i(i=1,2,...,5)are based on the work by Zachariadis et al.(2009).Each heuristic loads an item in the most suitable position selected from the feasible ones according to the individual criterion as follows:
Heur1:Bottom-leftfill(W-axis).
The selected position is the one with the minimum
W-axis coordinate,breaking ties by minimum L-
axis coordinate.
Heur2:Bottom-leftfill(L-axis).
The selected position is the one with the minimum
L-axis coordinate,breaking ties by minimum W-
axis coordinate.
Heur3:Max touching perimeter heuristic.
The selected position is the one with the maximum
sum of the common edges between the inserted
item,the loaded items in the vehicle,and the load-
ing surface of the vehicle.
Heur4:Max touching perimeter no walls heuristic.
The selected position is the one with the maximum
sum of the common edges between the inserted
item and the loaded items in the vehicle.
Heur5:Min area heuristic.
The selected position is the one with the minimum
rectangular surface.The rectangular surface corre-
sponding to the position at the circle point is shown
on the left in Fig.2.
More details of thesefive heuristics can be found in
Zachariadis et al.(2009).In order to handle a more
complex system Heur6was also used,which was
proposed in Leung et al.(2010).
Heur6:Maxfitness value heuristic.
This heuristic gives a priority to a loading point if it
can decrease the number of corner positions when
we place an item on the point.As a result,every
time an item is loaded,we will select the best load-
ing point for the item,which would increase the
probability to obtain a better loading position.
These six heuristics are called in sequence,which means if Heur1fails to produce a feasible loading solution,the more com-plex Heur i(i=2,...,6)will be called one at a time tofind the solu-tion.If a feasible solution is found,the loading process stops and the solution is stored.
During the loading process,feasible loading positions are recorded.Atfirst,only the front left corner(0,0)is available.When an item is successfully inserted,four new positions are added onto the list,and the occupied and duplicated positions are removed.As shown in Fig.2,item D is inserted in the position shown by a circle and four new positions are created.
The items are loaded one at a time according to a given sequence.Here,two orders(Ord1,Ord2)are generated.In a given route,each customer has a unique visit order.Ord1is produced by sorting all items by reverse customer visit order,and breaking ties by decreasing area.In Ord2,all items are simply sorted by decreasing area.Both orders will be evaluated by the six heuristics to search for feasible loading solutions.The pseudo-code for the packing heuristics is given in Table1.
4.The simulated annealing meta-heuristics for2L-HFVRP
Simulated annealing(SA)is a point-based stochastic optimiza-tion method,which explores iteratively from an initial solution to a better result(Cerny,1985;Kirkpatrick et al.,1983).The search mechanism of SA has a very good convergence,and it has been widely applied in solving various NP-hard problems.Each iteration
Table1
The pseudo-code for the packing heuristics.
Is_Feasible(Route r)
if total weight of all items exceeds the capacity then
return false
end if
sort the items to generate two orderings Ord1,Ord2
for each ordering Ord i of the two orderings do
if Heur1k Heur2k Heur3k Heur4k Heur5k Heur6then
return true
end if
end for
return false
Table2
The pseudo-code of SA_HLS for the2L-HFVRP.
SA_HLS_2L-HFVRP(customer demands,vehicle information)
Generate initial Order through sorting the customers by decreasing total
weight
Assign_Vehicle(Order)to construct the initial solution
T k=T0,Iter=0//Iter is the number of iteration
while stopping criteria not met do
for i=1to Len do
if Iter<10then
generate a new Order based on the old one
Assign_Vehicle(new Order)
if the new solution is packing-feasible and better than the current
one then
accept the new solution as current solution
end if
accept the new Order based on the acceptance rule of SA
end if
stochastically select NS from{NS1,NS2,NS3},then get a feasible solution
if new solution is better than the current one then
accept the new solution as the current solution
else
accept the new solution through the acceptance probability function
end if
Local_Search(),and get a new feasible solution
if new solution is better than the best one then
replace the current solution with this new one
end if
update the best solution when the solution is better than it
end for
T k=0.9ÃT k,Iter=Iter+1
end while
return the best solution
S.C.H.Leung et al./European Journal of Operational Research225(2013)199–210201
in SA generates a candidate solution using a neighborhood func-tion.This is a vital step to develop an efficient SA.However,in many cases,the neighborhood function alone is inadequate when seeking for a global optimum solution.In addition to the proposed SA,we also use heuristic local search algorithms to improve the
solutions.Therefore,our algorithm is denoted as SA_HLS.Some mechanisms are adopted to adjust the search trajectory.
One important characteristic of SA is that it can accept a worse solution on a probabilistic manner,aiming to search for a better re-sult.With the initial temperature T 0,the temperature cooling sche-dule is T k =0.9ÃT k À1.For a specific temperature T k ,a sequence of moves are carried out,which is a Markov chain whose length is de-noted as Len .In every iteration,after applying the neighborhood function,if the new solution is better than the current solution (i.e.the cost is lower),then it is accepted.However,if the cost is higher,the new solution may be accepted subject to the accep-tance probability function p (T k ,S new ,S cur ),which depends on the difference between the corresponding cost values and the global parameter T k :
p ðT k ;S new ;S cur Þ¼exp
cos t ðS cur ÞÀcos t ðS new Þ
k
ð1Þ
where S cur and S new represent the current solution and the new solu-tion respectively.Table 2provides a framework for the proposed SA_HLS methodology.4.1.Initial solution
Good initial solutions are often a key to the overall efficiency of the metaheuristic.We construct the initial solution focusing on the demand of the customers,so that the use of different
types
Table 3
The pseudo-code for assigning customers into vehicles.
Assign _Vehicle (Order )iused [1,2,...,P ]={0}
for each customer i in Order do while true do
select the vehicle k which is not tabu for i and has minimum (freeD k ÀD i )ÃF k
insert the customer i at the last position of route for vehicle k if !Is_Feasible (route )then
iused [P ]=iused [P ]+1;//add a new vehicle with largest capacity Tabu this vehicle k for customer i else
if freeD k <MinD then //vehicle k cannot service any customer
iused [t ]=iused [t ]+1//assuming the type of vehicle k is t ,add one
new vehicle
end if
accept the new route,and break the loop//start to assign successive customer
end if end while end for
return generated solution
Table 4
The characteristics of items of Classes 2–5instances.Class
m i
Vertical Homogeneous Horizontal Length
Width Length Width Length Width 2[1,2][0.4L ,0.9L ][0.1W ,0.2W ][0.2L ,0.5L ][0.2W ,0.5W ][0.1L ,0.2L ][0.4W ,0.9W ]3[1,3][0.3L ,0.8L ][0.1W ,0.2W ][0.2L ,0.4L ][0.2W ,0.4W ][0.1L ,0.2L ][0.3W ,0.8W ]4[1,4][0.2L ,0.7L ][0.1W ,0.2W ][0.1L ,0.4L ][0.1W ,0.4W ][0.1L ,0.2L ][0.2W ,0.7W ]5
[1,5]
[0.1L ,0.6L ]
[0.1W ,0.2W ]
[0.1L ,0.3L ]
[0.1W ,0.3W ]
[0.1L ,0.2L ]
[0.1W ,0.6W ]
202S.C.H.Leung et al./European Journal of Operational Research 225(2013)199–210
of vehicles can be maximized.Firstly,all of the n customers are sorted on decreasing value of D i(i=1,2,...,n),where D i is the total demand of customer i(i=1,2,...,n)and the sequence is re-corded as Order.Subsequently we assign the customers one at a time from the Order list to a vehicle.The decision of which vehi-cle is assigned to a given customer is based on the least value of (freeD kÀD i)ÃF k,where freeD k is the unused capacity and F k is the fixed cost of the current vehicle k(procedure Assign_Vehicle()). Because the number of each type of vehicle is unlimited,the pro-cedure alwaysfinds a feasible solution.Table3provides a pseu-do-code for the proposed Assign_Vehicle()algorithm.iused is an array presenting the number of used vehicles of different types.MinD is the minimal demand in all the customers.When assigning one customer i to vehicle k,the feasibility is examined to ensure the loading for the modified route is feasible.Other-wise,the assignment of customer i to vehicle k is forbidden and the procedure tries to assign the customer i to another vehicle.
As shown in Table2,this procedure is used in SA.In each loop, a partial segment of Order is reversed to get a new Order.Then, we reassign the customers using this method.If a new solution is better than the current one,it becomes the new current solu-tion in order to adjust the search trajectory.This is assumed that previous solution does not have a good characteristic that can be improved easily.In order to obtain a better solution,the new Order is adopted based on the SA acceptance rule.After several steps of improvement by SA,the solution constructed is usually not comparable to the current one.So this method is only applied during thefirst ten iterations.
Table5
Dataset for different types of vehicle.
Inst A B C D
Q A L A W A F A V A Q B L C W C F C V C Q C L C W C F C V C Q D L D W D F D V D
120101010 1.025151520 1.140252530 1.260402040 1.3 220101010 1.025151520 1.140252530 1.355402040 1.5 320101010 1.030151520 1.160402040 1.2
420101010 1.040202020 1.160402030 1.2
51000101010 1.025******** 1.14000252530 1.36000402050 1.5 62000101010 1.025******** 1.14000402030 1.3
7200101010 1.0500151520 1.120002525120 6.0450040202508 8200101010 1.0500151520 1.120002525120 5.04500402025010 920101010 1.025151520 1.148402030 1.3
10200101010 1.0500151520 1.1200025251208.04500402025010 11200101010 1.0500151520 1.1200025251208.04500402025010 1220101010 1.025151520 1.140402030 1.3
132000101010 1.05000151550 2.030,000402020010
14500101010 1.01500151550 2.130002020400 3.2500040208005 155******** 1.01500151550 2.130002020400 3.2500040208005 1620101010 1.040202020 1.160402030 1.2
1720101010 1.025151520 1.140252530 1.360403040 1.4 182******** 1.0500202030 2.020********* 5.0
1920101010 1.040201020 1.160201530 1.2150402090 5.0 202000101010 1.04000201020 1.110,000301560 4.030,00040201508 2120101010 1.040201020 1.160201530 1.220040201208 2220101010 1.040201020 1.160201530 1.220040201208 2320101010 1.040201020 1.160201530 1.220040201208 2420101010 1.040201020 1.160201530 1.2100402060 3.2 2520101010 1.040201020 1.160201530 1.220040201208 2620101010 1.040201020 1.160201530 1.220040201208 2720101010 1.040201020 1.160201530 1.2100402060 3.2 2820101010 1.040201020 1.160201530 1.220040201208 29200101010 1.0500201030 2.0200040201208.0
3020101010 1.040201020 1.160201530 1.220040201208 3120101010 1.040201020 1.160201530 1.220040201208 3220101010 1.040201020 1.160201530 1.220040201208 3320101010 1.040201020 1.160201530 1.220040201208 3420101010 1.040201020 1.160201530 1.220040201208 35200101010 1.0400201020 1.51000402060 4.0
36100101010 1.020******* 1.1300302030 1.2400402040 1.3
Table6
Calibration experiment result for T0and Len.
5152535
S Sec tot S Sec tot S Sec tot S Sec tot
T0
Unrest5266.65327.945159.32288.995094.74402.305111.83500.08 Seq5427.43461.795274.77510.395198.10580.955238.46680.85
3000500070009000
Len
Unrest5144.37383.375094.74402.305078.97779.145083.18968.25 Seq5328.87456.855198.10580.955206.401025.195191.071242.93
S.C.H.Leung et al./European Journal of Operational Research225(2013)199–210203
4.2.Neighborhood functions
In our work,three types of move are used to step from the current solution to the subsequent solutions.They are noted as NS i(i=1,2,3).In each loop,one of them is selected ran-domly with equal probability.To explore a larger search space, a dummy empty vehicle is added for each type of vehicle.NS1is a type of customer relocation(Or-opt)(Waters,1987),which reassigns a customer from one route to another position on the same or different route(Fig.3).It is worth noting that relo-cation between two different routes can reduce the number of vehicles required.
Waters(1987)introduced a‘‘swap’’type of route exchange which is represented by NS2(Fig.4).It is only applied to vehicles of the same type as swapping loads of heterogeneous vehicles could lead to an unfeasible route from a loading perspective. Therefore,customers’positions can only be exchanged in the cur-rent solution if they belong to vehicles of the same type.
NS3is a variant of route interchange(2-opt)(Croes,1958;Lin, 1965).As for NS2,NS3only considers vehicles of the same type (Fig.5).If the selected customers are in the same route as depicted in Fig.5a,the positions of other customers between them (and including themselves)will be reversed.If they belong to dif-ferent routes as illustrated in Fig.5b,in each route from the se-lected customer to the last customer will be grouped as a block. Between the routes the blocks will be swapped.4.3.Heuristic local search mechanism
In order to improve the quality of the solution,we also apply a heuristic local search mechanism,which consists of three methods,to the proposed SA algorithm.It is worth noting that we only apply the mechanism to the best solution with a prob-ability of5%and this is aimed to obtain more efficient solutions within a shorter period of time.The local search methods adopt thefirst improvement criterion using the neighborhood func-tions mentioned in the previous section.Because this neighbor-hood is not operated on two randomly selected customers,we define this mechanism as heuristic local search.We denote the local search methods as LS i(i=1,2,3)according to the neighborhoods NS i(i=1,2,3).These three methods are ran-domly executed.
Let us consider an instance with n customers and k vehicles.In LS1,the relocation move of one customer involves the reassign-ment of(n+k)positions.Hence,the complexity of examining NS1 neighborhood of a solution is O(n⁄(n+k)).For LS2,in the worst case whereby all customers are assigned to one type of vehicle, n2pairs of customers can be exchanged and therefore the complex-ity of NS2is O(n2).For LS3,as for LS2,the number of interchange points is(n+k),so the cardinality of pairs for interchange in NS3 is(n+k)2.As a result,examining NS3neighborhood requires O ((n+k)2)computational effort.In practice,the worst case hardly happened because the customers are usually spread out across dif-ferent types of vehicle.
Table8
Average computational results of Classes2–5for Sequential2L-HFVRP.
Inst SA SA_HLS%Gap S Sec h Sec tot S Sec h Sec tot 1678.497.7323.43603.15 5.7331.0411.10 2753.70 6.4321.53705.03 6.0931.28 6.46 3866.5620.8941.05771.8110.3536.6110.93 4796.2822.9340.76704.878.7135.1911.48 5944.8734.6849.07802.569.3527.6215.06 6901.4425.3345.42834.769.9243.907.40 76634.1357.8694.405770.83 1.9531.4013.01 87064.9227.5274.115633.04 4.7737.4220.27 91181.8648.3956.421047.6015.7468.8411.36 108695.22114.15194.217730.73 5.8751.7811.09 119789.89171.07225.858491.9410.0758.8813.26 121707.9915.1123.731681.6132.90158.45 1.54 1335464.18183.90287.9326761.40 6.7165.3424.54 1412027.5546.68112.0511120.3011.2955.627.54 1512871.23118.99190.7211916.3024.53148.507.42 161437.7411.1545.811291.2425.73113.0210.19 172037.0124.8735.341775.4943.41198.8312.84 188364.54344.74600.495790.6230.50138.0330.77 196186.88369.25781.224303.2651.44233.9830.45 209586.34941.001534.626215.4898.14217.7335.16 2114457.73811.101613.928494.36124.15443.8541.25 2215677.581090.281798.928867.10110.40353.4043.44 2315533.281078.921615.038544.30130.44386.8244.99 246756.02595.331092.094714.08104.98287.0230.22 2522864.752937.973155.7411602.30186.28605.5349.26 2620622.13875.112351.0612380.30153.21392.2639.97 279652.102343.762607.825882.75240.04502.4039.05 2841547.502484.484178.8123585.60393.72737.3543.23 2942142.583444.275609.8122938.80486.551045.6945.57 3036243.783609.867783.7316489.70536.781033.7954.50 3149143.853493.4611927.0922033.001110.801531.2855.17 3249142.654755.4512121.4820982.701040.061487.7457.30 3350660.653911.4612173.2721906.601087.931326.9856.76 3425388.93780.5514154.5615005.001680.671887.4140.90 3512902.48160.3217513.849313.541722.961877.5627.82 366279.65256.3716302.744567.292035.082052.9927.27 Avg.27.46
Table7
Result comparison of SA_HLS and SA on Class1.
Inst SA SA_HLS%Gap
S Sec h Sec tot S Sec h Sec tot
1665.64 6.41162.40596.0729.7833.5910.45
2732.8539.30159.30679.18 5.0532.017.32
3813.87162.92182.44745.5112.3954.248.40
4745.50142.72184.50694.33 6.1518.36 6.86
5916.40214.52226.97761.19 5.6226.9916.94
6814.8520.02192.53809.56 4.7725.650.65
73387.06159.89215.363211.53 3.6729.76 5.18
83359.77212.59215.033184.45 3.2124.30 5.22
91144.65120.11219.141029.957.3940.2810.02
105400.74108.70274.945149.51 6.9725.88 4.65
115465.11217.80279.005119.40 6.9726.47 6.33
121699.5917.67261.111658.5635.8092.80 2.41
1319390.0056.72275.6714655.40 1.9332.1824.42
1410447.1039.97305.7710019.0013.5173.21 4.10
1510546.9042.11304.9210151.70 5.4955.29 3.75
161391.84108.34297.561292.5817.9476.927.13
171963.5853.33411.941770.8338.88225.599.82
184055.02343.92363.923140.5513.5435.3522.55
191980.40205.31445.781553.1132.56107.8121.58
203244.47277.20719.061956.9756.0071.5739.68
215330.48428.97703.952567.1875.00195.6651.84
225934.32160.30699.752605.9076.39174.7256.09
235811.71404.05711.092643.8493.99239.2954.51
244257.31289.19696.342555.4163.98156.4139.98
255960.19248.30908.832972.59129.04253.0950.13
265515.4843.45804.554049.6488.09180.2326.58
275093.65426.08905.033561.58159.20230.4930.08
287944.7382.841103.066858.35125.60161.0513.67
2915643.10822.161222.339695.00139.73142.6338.02
3011320.50992.751500.935663.33242.15259.8349.97
3119297.801754.361946.538054.90325.51483.4458.26
3217767.80732.971965.678408.61379.53410.8652.68
3318325.90696.911962.118555.58368.50486.2553.31
345713.851406.471998.505536.63323.50425.80 3.10
354875.690.981793.804444.59324.64401.588.84
364961.221723.82322.953669.89555.13605.3126.03
Avg.23.07
204S.C.H.Leung et al./European Journal of Operational Research225(2013)199–210。