Aheuristicmethodfortheinventoryroutingproblemwithtimewindows

合集下载

绿色供应链设计助推碳排量减少翻译英文

Green supply chain network design to reduce carbon emissionsSamir Elhedhli ⇑,Ryan MerrickDepartment of Management Sciences,University of Waterloo,200University Avenue,Waterloo,Ontario,Canada N2L 3G1a r t i c l e i n f o Keywords:Green supply chain design LogisticsTransportation CO 2emissions Carbon footprinta b s t r a c tWe consider a supply chain network design problem that takes CO 2emissions into account.Emission costs are considered alongside ﬁxed and variable location and production costs.The relationship between CO 2emissions and vehicle weight is modeled using a concave function leading to a concave minimization problem.As the direct solution of the resulting model is not possible,Lagrangian relaxation is used to decompose the problem into a capacitated facility location problem with single sourcing and a concave knapsack problem that can be solved easily.A Lagrangian heuristic based on the solution of the subproblem is proposed.When evaluated on a number of problems with varying capacity and cost char-acteristics,the proposed algorithm achieves solutions within 1%of the optimal.The test results indicate that considering emission costs can change the optimal conﬁguration of the supply chain,conﬁrming that emission costs should be considered when designing supply chains in jurisdictions with carbon costs.Ó2012Elsevier Ltd.All rights reserved.1.IntroductionWith the globalization of supply chains,the distance between nodes in the distribution network has grown considerably.Longer travel distances lead to increased vehicle emissions on the transportation routes,resulting in an inﬂated carbon foot-print.Hence,there is a need to effectively and efﬁciently design eco-friendly supply chains,to both improve environmental conditions and the bottom line of the work design is a logical place to start when looking to green a supply chain design.Wu and Dunn (1995)cite transportation as the largest source of environmental hazards in the logistics system.This claim is supported by the fact that transportation via combustion engine vehicles accounted for 27%of the Canadian greenhouse gas (GHG)inventory in 2007(Environment Canada,2009).And while heavy duty diesel vehicles,such as diesel tractors commonly used in logistics,account for only 4.2%of vehicles on the road,they accounted for 29.2%of Canadian GHG emissions from transportation in 2007.Thus,reducing the number of vehicle kilometers travelled through the strategic placement of nodes could play a signiﬁcant role in reducing the carbon footprint of the nation.Supply chain design models have traditionally focused on minimizing ﬁxed and operating costs without taking carbon emissions into account.Recent studies,however,started to take emissions into account.This includes Cruz and Matsypura (2009),Nagurney et al.(2007),Benjaafar et al.(2010),Merrick and Bookbinder (2010),and Ando and Taniguchi (2006).This paper develops a green supply chain design model that incorporates the cost of carbon emissions into the objective function.The goal of the model is to simultaneously minimize logistics costs and the environmental cost of CO 2emissions by strategically locating warehouses within the distribution network.A three echelon,supply chain design model is proposed that uses published experimental data to derive nonlinear concave expressions relating vehicle weight to CO 2emissions.The resulting concave mixed integer programming model is tackled using Lagrangian relaxation to decompose it by echelon and by warehouse site.The nonlinearity in one of the subproblems is eliminated by exploiting its special structure.This 1361-9209/$-see front matter Ó2012Elsevier Ltd.All rights reserved./10.1016/j.trd.2012.02.002⇑Corresponding author.E-mail address:elhedhli@uwaterloo.ca (S.Elhedhli).S.Elhedhli,R.Merrick/Transportation Research Part D17(2012)370–379371decomposition results in subproblems that require less computational effort than the initial problem.By keeping most of the features of the original problem in the subproblems,a strong Lagrangian bound is achieved.A primal heuristic is proposed to generate a feasible solution in each iteration using information from the subproblems.The quality of the heuristic is mea-sured against the Lagrangian bound.Test results indicate that the proposed method is effective inﬁnding good solutions.The remainder of the paper is organized as follows.In the next section we look at the emission data,followed by the prob-lem formulation in Section3.We then delve into the Lagrangian relaxation procedure and proposed heuristic in Sections4 and5,respectively.Finally,we test the algorithm and heuristic in Section6,and conclude in Section7.2.Emissions dataFew comprehensive data sets exist that show the relationship between vehicle weights and exhaust emissions.While the exact emission levels will depend on the engine type,terrain driven and the driver tendencies,the general relationship be-tween vehicle weight and emissions will not change(i.e.linear,concave or convex relationship).This section reviews the available emissions data and draws conclusions about the relationship between emissions and the vehicle operating weight.The most comprehensive data set of vehicular GHG emissions for is that contained in the Mobile6computer program (Environmental Protection Agency,2006).Mobile6contains an extensive database of carbon dioxide(CO2)emissions for hea-vy heavy-duty diesel vehicles obtained from full scale experiments.The database contains emissions factors for various vehi-cle weights,ranging from class2trucks up to class8b.Speed correction factors,outlined by the California Air Resources Board(Zhou,2006)for use with the Mobile6program,can also be applied to relate CO2emission levels with vehicle weight and speed of travel.Fig.2.1displays the relationship between vehicle weight and CO2emissions for various speeds of travel. The units for CO2emissions are grams(g)per vehicle kilometer travelled(VKT)and the vehicle weight is in pounds(Note that‘‘vehicle weight’’represents the empty weight plus the cargo).The speed at which a vehicle travels at is a function of the travel route,whereas as the vehicle weight is dependent on the demand requirement at lower echelons.Consequently,the function used to compute the total pollution emissions can be chosen based on the mean travel speed over the route(e.g.an average highway speed of100kph).With multiple modes of transportation(trucks),the emission cost function is the lower envelope of the individual cost functions,which is concave even for linear individual emission cost functions.As a function of total shipment,modeling the emission costs will depend on the transportation strategy ing full truckloads and a single mode,the cost func-tion could very well be modeled usingﬁxed and linear cost functions.Whereas,in the general case where multiple modes and less-than-full truckloads are allowed,the long term cost is better modeled using a concave function.3.Problem formulationLet us deﬁne the indices i=1,...,m,j=1,...,n and k=1,...,p corresponding to plant locations,potential distribution centers(DCs)and customers,respectively.A distribution center at location j has a maximum capacity V j and aﬁxed cost g j.Each customer has a demand of d k.The variable cost of handling and shipping a production unit from a plant at location i to distribution center j is c ij.Similarly,h jk denotes the average handling and shipping cost to move a production unit from distribution center j to customer k.We introduce one continuousﬂow variable and two binary location variables:x ij is theunits shipped from plant i to warehouse j ;y jk takes a value of one if customer k is assigned to distribution center j and zero otherwise;and z j takes a value of one if distribution center j is opened and zero otherwise.The capacity of the plants is as-sumed to be unlimited.The resulting MIP is:½FLM :min P m i ¼1P n j ¼1f ðx ij ÞþPn j ¼1P p k ¼1f ðd k y jk ÞþP m i ¼1P n j ¼1c ij x ij þP n j ¼1P p k ¼1h jk d k y jk þP n j ¼1g j z js :t :Pn j ¼1y jk ¼18kð1ÞP m i ¼1x ij ¼P p k ¼1d k y jk 8j ð2ÞPm i ¼1x ij 6V j z j 8j ð3ÞP p k ¼1d k y jk 6V j z j8jð4Þy jk ;z j 2f 0;1g ;x ij P 08i ;j ;kð5ÞThe ﬁrst two terms of the objective function minimize the pollution cost to the environment,where f (x )is the emissions costfunction.The rest of the terms are the ﬁxed cost of opening DCs and the handling and transportation cost to move goods between nodes.1Constraints (1)guarantee that each customer is assigned to exactly one distribution center.Constraints (2)balance the ﬂow of goods into and out of the warehouse,thus linking the decisions between echelons in the network.Con-straints (3)and (4)force capacity restrictions on the distribution centers and ensure that only open facilities are utilized.Note that constraints (1)and (2)ensure that total customer demand is satisﬁgrangian relaxationGiven the difﬁculty in solving [FLM]directly,we use Lagrangian relaxation to exploit the echelon structure of the prob-lem.It is important to select the constraints for relaxation,as relaxing more constraints may deteriorate the quality of the bound and heuristics.We relax constraints (2)using Lagrangian multipliers,l j ,since they link the echelons of the supply chain.This leads to the following subproblem:½LR-FLM :minPm i ¼1P n j ¼1f ðx ij ÞþP n j ¼1P p k ¼1f ðd k y jk ÞþP m i ¼1P n j ¼1ðc ij Àl j Þx ij þP n j ¼1P p k ¼1ðh jk d k þd k l j Þy jk þPn j ¼1g j z js :t :ð1Þ;ð3Þ;ð4Þand ð5Þwhich is separable by echelon.Furthermore,it decomposes to two subproblems:½SP1 :minPn j ¼1P p k ¼1f ðd k y jk ÞþP n j ¼1P p k ¼1ðh jk d k þd k l j Þy jk þP n j ¼1g j z js :t :ð1Þand ð4Þy jk ;z j 2f 0;1g ;8j ;kwhich determines the assignment of customers to distribution centers.As y jk is binary and due to (1),the objective can bewritten as min P n j ¼1P p k ¼1ðf ðd k Þþh jk d k þd k l j Þy jk þP nj ¼1g j z j ,making [SP1]a capacitated facility location problem with single sourcing.The second subproblem:½SP2 :minPm i ¼1P n j ¼1f ðx ij ÞþP m i ¼1P n j ¼1ðc ij Àl j Þx ijs :t :ð3Þx ij P 08i ;jcan be decomposed by potential warehouse site,resulting in n subproblems [SP2j ].½SP2j :min Pm i ¼1f ðx ij ÞþPm i ¼1ðc ij Àl j Þx ijs :t :P m i ¼1x ij 6V j z jx ij P 08i1The model does not take congestion and or breakdowns into account.Congestion and the need to keep reserve capacity for emergencies affect theenvironmental costing of the supply chain.2The model assumes that total demand has to be satisﬁed.If only partial demand is to be satisﬁed then the model has to be posed as a proﬁt maximization rather than a cost minimization model.372S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379As z j =0is trivial,we focus on the case when z j =1,which makes [SP2j ]a concave knapsack problem.An important property of concave functions is that a global solution is achieved at some extreme point of the feasible domain (Pardalos and Rosen,1986).Therefore,[SP2j ]has an optimal solution at an extreme point of x ij P 0;P mi ¼1x ij 6V j ÈÉ.This implies that at optimality at most one x ij will take the value of V j and the remaining x ij will be equal to 0.This allows us to reformulate [SP2j ]as:½SP2j :minPm i ¼1ðf ðV j Þþc ij Àl j Þx ij s :t :Pm i ¼1x ij 6V jx ij P 08iwhich is now a linear knapsack problem.The advantage of the relaxation is that [SP1]retains several important characteristics of the initial problem,such as the assignment of all customers to a single warehouse and the condition that the demand of all customers is satisﬁed.In addi-tion,[SP2]reduces to n subproblems,which can be solved with little computational effort relative to the original problem.The drawback of this relaxation is that [SP1]is a capacitated facility location problem with single sourcing that is a bit dif-ﬁcult to solve.However,[SP1]is still easier to solve than [FLM]and by retaining the critical characteristics of [FLM]in [SP1],the Lagrangian bound can be achieved in a relatively small number of iterations and reduce the overall solution time while still obtaining a high quality bound.Similarly,by exploiting the solution of [SP1]in a heuristic,high quality feasible solutions will be achieved.The Lagrangian relaxation starts by initializing the Lagrangian multipliers,and solving the subproblems.The solutions to the subproblems yield a lower bound:LB ¼½v ½SP 1 þPn j ¼1v ½SP 2jThe best Lagrangian lower bound,LB ⁄,is:LB Ã¼max l½v ½SP 1 þP n j ¼1v ½SP 2jthat can be found by solving:max lminh 2I x Pn j ¼1P p k ¼1ðf ðd k Þþh jk d k þd k l j Þy h jk þPn j ¼1g j z h j þPn j ¼1minh j 2I yj Pm i ¼1ðf ðV j Þþc ij Àl j Þx hjij ()ð6Þwhere I x is the index set of feasible integer points of the set:y h jk ;z h j :P n j ¼1y h jk ¼1;P p k ¼1d k y h jk 6V j z h j ;y h jk 2f 0;1g ;z h j 2f 0;1g ;8j ;k()and I yj is the index set of extreme points of the set:x hj ij :P m i ¼1x h jij 6V j ;x h jij P 0;8i&'We can then reformulate (6)as the Lagrangian master problem:½LMP :max h 0þPn j ¼1h js :t :h 0ÀPn j ¼1P p k ¼1d k y h jkl j 6P n j ¼1P p k ¼1ðf ðd k Þþh jk d k Þy h jk þP n j ¼1g j z h jh 2I xh j þPm i ¼1x h j ij l j6P m i ¼1f x h j ij þP m i ¼1c ij x hjij h j 2I yj ;8j[LMP]can be solved as a linear programming problem.I x &I x and I yj &I yj deﬁne a relaxation of [LMP].An initial set ofLagrangian multipliers,l ,is used to solve [SP1]and [SP2]and generate n +1cuts of the form:h 0ÀP n j ¼1Pp k ¼1d k ya jkl j 6Pn j ¼1P p k ¼1ðf ðd k Þþh jk d k Þy a jk þP n j ¼1g j za jh j þPm i ¼1x bj ij l j6P m i ¼1f x bjij þP mi ¼1c ij x b j ij 8jThe index sets x and yj are updated at each iteration as x [f ag and yj [f b j g ,respectively.S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379373The solution to [LMP]produces an upper bound,UB ,to the full master problem and a new set of Lagrangian multipliers.The new set of Lagrangian multipliers is input to [SP1]and [SP2]to generate a new solution to the subproblems and an addi-tional set of cuts to the [LMP].The procedure of iterating through subproblems and master problem solutions is terminated when the best lower bound is equal to the upper bound,at which point the Lagrangian bound is achieved.Note that the heu-ristic procedure designed to generate a feasible solution is outlined and discussed in the next section.5.A primal heuristic for generating feasible solutionsWhile the Lagrangian algorithm provides the Lagrangian bound,it does not reveal the combination of product ﬂows,cus-tomer assignments and open facilities that will produce this result.Hence,heuristics are commonly used in conjunction with Lagrangian relaxation algorithms to generate feasible solutions.To generate feasible solutions,we devise a primal heuristic based on the solution of the subproblems.Subproblem [SP1]generates the assignments of customers to distribution centers and determines if a distribution center is open or closed.Using y h jk and z hj from [SP1],the units demanded by the retailers at each distribution center can be determined.With the de-mand at each distribution center being deterministic,the original problem could be reduced to a simple continuous ﬂow transportation problem,[TP],which will always have a feasible solution.½TP :minP n j ¼1P p k ¼1ðf ðd k Þþh jk d k Þy h jk þP m i ¼1P n j ¼1f ðx ij ÞþP m i ¼1P n j ¼1c ij x ij þPn j ¼1g j z h js :t :P m i ¼1x ij ¼P n k ¼1d k y h jk 8jx ij P 08i ;jThe ﬁrst and fourth terms in [TP]are simply constants,thus leaving only two terms in the objective function.Again,be-cause [TP]is concave there is an extreme point that is optimal,implying that each warehouse will be single-sourced by oneplant and that the goods will be transported on a single truck,as opposed to being spread over multiple vehicles.Therefore,the optimal ﬂow of units from a plant to warehouse will be equal to the quantity demanded by the warehouse or zero.We can then formulate [TP]as an assignment problem:½TP2 :min P m i ¼1f P p k ¼1d k y h jk þc ij P p k ¼1d k y hjkw ij þCs :t :P m i ¼1P p k ¼1d k y hjk w ij ¼P p k ¼1d k y h jk 8jw ij 2f 0;1g 8i ;jwhere w ij takes a value of one if warehouse j is supplied by plant i and zero otherwise.In numerical testing of the algorithm,the heuristic was activated at each iteration to ﬁnd a feasible solution.6.Numerical testingThe solution algorithm is implemented in Matlab 7and uses Cplex 11to solve the subproblems,the heuristic and the master problems.The test problems were generated similar to the capacitated facility location instances suggested by Cor-nuejols et al.(1991).The procedure calls for problems to be generated randomly while keeping the parameters realistic.The coordinates of the plants,distribution centers and customers were generated uniformly over [10,200].From the coordinates,the Euclidean distance between each set of nodes is computed.The transportation and handling costs between nodes are then set using the following relationship:c ij ¼b 1Â½10Âd ij h jk ¼b 2Â½10Âd jkwhere b is a scaling parameter to exploit different scenarios in numerical testing,and i ,j ,k are nodes.The demand of each customer,d k ,is generated uniformly on U [10,50].The capacities of the distribution centers,V j ,are set to:V j ¼j Â½U ½10;160where j is used to scale the ratio of warehouse capacity to demand.In essence,j dictates the rigidity or tightness of the problem and has a large impact on the time required to solve the problem.The capacities of the distribution centers were scaled so as to satisfy:j ¼P nj ¼1V jP pk ¼1d k!¼3;5;10The ﬁxed costs of the DCs were designed to reﬂect economies of scale.The ﬁxed cost to open a distribution center,g j ,is:374S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379g j ¼a Â½U ½0;90 þU ½100;110 ÂﬃﬃﬃﬃﬃV j qAgain,a is a scaling parameters used to test different scenarios in the numerical analysis.Just as the problem formulation was extended to include emissions costs,the test problems must also be extended.In order to compute the emission costs,the distance travelled,vehicle weight and emission rate must be known.The distance travelled can easily be determined from the randomly generated coordinates of the sites.The vehicle weight is determined by the number of units loaded on the truck (x ij or d k y jk ).To compute the weight of the vehicle,an empty vehicle weight of 15,000lb was assumed and the weight of a single production unit was assumed to be 75lb.The payload was calculated as the number of units on the truck multiplied by the weight of a single unit,which resulted in loads between 0and 45,000lb.The sum of the empty tractor-trailer weight and the payload results in a loaded vehicle weight range of 15,000–60,000lb.It is assumed that single vehicle trips would be made between nodes,thus the vehicle weights are reasonable and the emissions curve for a single truck is used.However,the emissions curve could be substituted with a best ﬁt concave line that would represent a number of vehicle trips,if so desired.Finally,the emission rate,e ,is determined using the US EPA lab data,shown in Fig.2.1(Environmental Protection Agency,2006).Using these parameters,the emission cost of the net-work,f ,is determined using the following equation:Table 6.1Comparison based on different capacity utilizations.Problem Heur.Time (%,%,%,%,s)i .j .kDCLR FCR_DC VCR ECR Iters.Quality SP1SP2Heur.MP Total Tight capacities (j =3)5.10.200.9170.4040.4890.10740.10291.7 5.3 1.7 1.3 2.55.10.400.8860.3710.5220.10740.09298.60.90.20.214.05.10.600.8580.3430.5370.12040.09998.80.80.20.222.08.15.250.9470.4860.4230.09140.05394.1 3.6 1.50.81898.15.500.9520.3800.5090.11140.02599.60.20.10.13588.15.750.9800.3390.5390.12250.00899.60.20.10.127610.20.500.9780.4540.4440.10240.02499.60.30.1$098610.20.750.9440.3980.4980.10540.06099.90.1$0$076510.20.1000.9720.4090.4850.10640.01099.90.1$0$0154310.20.1250.9810.3630.5190.11740.01399.90.0$0$0132810.20.1500.9800.3650.5270.10850.01999.80.1$0$02194Min 0.860.340.420.0940.0191.700.00$0$0Mean 0.950.390.500.11 4.180.0598.32 1.050.350.25Max0.980.490.540.1250.1099.90 5.30 1.70 1.30Moderate capacities (j =5)5.10.200.8490.4020.4890.10940.06385.09.6 3.1 2.4 1.005.10.400.8180.3310.5520.11740.07797.5 1.50.50.47.55.10.600.9170.2800.5950.12440.04197.9 1.20.50.317.08.15.250.9170.4280.4610.11150.03992.7 4.7 1.6 1.01478.15.500.8870.4020.4730.12540.03694.3 3.0 1.3 1.41808.15.750.9390.3520.5310.11640.03099.30.40.20.159210.20.500.9700.3710.5060.12340.00995.6 2.60.90.958310.20.750.9370.3440.5320.12440.02399.9$0$0$088310.20.1000.9350.3430.5310.12740.01499.60.30.10.1106110.20.1250.8970.3100.5630.12740.04199.80.1$0$0124210.20.1500.9160.2980.5780.12540.04198.80.80.30.21548Min 0.820.280.460.1140.0185.00$0$0$0Mean 0.910.350.530.12 4.090.0496.40 2.200.770.62Max 0.970.430.600.1350.0899.909.60 3.10 2.4Excess capacities (j =10)5.10.200.6230.4830.3990.11840.20499.90.10.0$096.15.10.400.6550.2590.5800.16140.03994.8 2.9 1.40.9 2.45.10.600.6400.2730.6110.11630.08395.5 2.90.90.7 1.48.15.250.9040.3560.5330.11140.04099.10.50.20.116.88.15.500.7720.3260.5230.15140.11892.7 5.0 1.3 1.0 1.88.15.750.8260.3150.5580.12760.04995.7 2.90.70.6 4.510.20.500.9020.4500.4340.11640.05099.80.1$0$010610.20.750.9970.3070.5640.12940.00196.8 1.80.70.7 6.610.20.1000.6460.3270.5380.13540.03499.70.20.1$059.210.20.1250.9270.2980.5770.12550.00899.90.1$0$019910.20.1500.6930.2970.5620.14140.03598.6 1.00.20.2173Min 0.620.260.400.1130.0092.700.10$00.0Mean 0.780.340.530.13 4.180.0697.50 1.590.500.38Max1.000.480.610.1660.2099.905.001.401.0S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379375f ðx ij Þ¼X Â0:2Âe ðx ij ÞÂd ij f ðd k y jk Þ¼X Â0:2Âe ðd k y jk ÞÂd jkX is used as a scaling parameter to test various network scenarios.The constants on the right-hand sides of the above equa-tions are used for unit conversions and to associate a dollar value to the emission quantity.For all test cases,a travel speed of 100kph was used to compute emission levels,which was assumed to be representative of highway transportation.The solution algorithm underwent rigorous testing to measure its effectiveness.Several statistics are collected during the solution procedure.Foremost,the load ratio of the open distribution centers was calculated.The DC load ratio,DCLR,relates the total capacity of all open DCs to the total units demanded by the customers,and is computed as:DCLR ¼P nj ¼1ðV j Áz j ÞP pk ¼1ðd k ÞThe cost breakdown of the best feasible network resulting from the solution algorithm is also evaluated.Three primary cost groups were considered:the ﬁxed costs to open the distribution centers (FCR_DC),the variable logistics costs (VCR)and the emissions costs (ECR).These statistics were computed as a percentage of the total system expense,denoted as Z ,using the following formulas:Table 6.2Comparison based on different dominant cost scenarios.Problem Heur.Time (%,%,%,%,s)i .j .kDCLR FCR_DC VCR ECR Iters.Quality SP1SP2Heur.MP Total Dominant ﬁxed costs 5.10.200.9600.8240.1490.02840.01091.8 5.5 1.5 1.3 1.005.10.400.9830.7440.2130.04240.003$100$0$0$02955.10.600.9970.7160.2370.04740.00099.10.50.30.112.58.15.250.9530.8460.1230.03050.00583.59.2 3.5 3.8 1.28.15.500.9440.8110.1500.03950.006100.00.00.00.032638.15.750.9940.8080.1550.03750.00198.2 1.00.40.434.010.20.500.9990.8280.1420.03050.000$100$0$0$0241010.20.750.9720.7910.1780.03250.004$100$0$0$0147810.20.1000.9830.7680.1900.04250.00299.50.30.10.149.310.20.1250.9990.7450.2160.03950.00099.90.10.00.024810.20.1500.9910.7150.2350.05050.00299.90.1$0$0194Min 0.940.720.120.0340.0083.50$0$0$0Mean 0.980.780.180.04 4.730.0097.45 1.520.530.52Max1.000.850.240.0550.01100.009.20 3.50 3.8Dominant variable costs 5.10.200.5840.1850.7700.04550.13073.314.08.2 4.50.435.10.400.6030.1270.8410.03240.05197.7 1.20.70.4 4.15.10.600.5150.1520.8180.03040.08996.5 1.7 1.10.6 2.98.15.250.7800.2160.7480.03650.05391.7 4.9 2.1 1.3 1.88.15.500.8760.1660.7990.03540.05093.6 3.6 1.8 1.0 2.08.15.750.4690.2020.7620.03540.07996.2 2.2 1.10.6 3.410.20.500.6990.2290.7330.03840.05794.7 3.1 1.40.8 3.010.20.750.7630.1960.7720.03250.02099.80.10.1$087.610.20.1000.6550.1480.8150.03740.04099.80.1$0$010710.20.1250.7500.1440.8210.03550.03499.80.1$0$012010.20.1500.7420.1360.8280.03640.02999.80.10.10.2213Min 0.470.130.730.0340.0273.300.10$0$0Mean 0.680.170.790.04 4.360.0694.81 2.83 1.510.85Max 0.880.230.840.0550.1399.8014.008.20 4.5Dominant emissions cost 5.10.200.7010.3530.3150.3325 1.01494.6 2.8 1.7 1.0 2.15.10.400.5570.3290.3360.33540.77787.5 6.3 4.0 2.10.775.10.600.6330.2300.3890.38140.50199.60.20.10.125.28.15.250.8750.3150.3140.37140.327$1000.008158.15.500.8560.2530.3840.36350.42497.6 1.40.60.4 6.28.15.750.6780.3140.3490.33740.58699.60.20.10.134.110.20.500.8140.3640.2800.35540.39397.3 1.60.70.4 5.410.20.750.7060.3060.3510.34240.56499.90.00.00.018510.20.1000.7090.2890.3620.34940.46999.50.30.10.131.310.20.1250.7880.3150.3540.33150.41199.90.10.00.020710.20.1500.6900.2870.3730.34040.63699.40.30.10.126.6Min 0.560.230.280.3340.3387.500.000.000.0Mean 0.730.310.350.35 4.270.5597.72 1.200.670.39Max 0.880.360.390.3851.011006.304.002.1376S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379FCR DC ¼P nj ¼1ðg j Áz j ÞZVCR ¼P m i ¼1P nj ¼1c ij x ij þP n j ¼1P p k ¼1h jk d k y jkECR ¼P m i ¼1P n j ¼1f ðx ij ÞþP n j ¼1P pk ¼1f ðd k y jk ÞZThe quality of the heuristic is measured by comparing the cost of the feasible solution vs.the Lagrangian bound,LR,as follows:Heuristic Quality ¼100Âheuristic solution ÀLRData on the evaluation times required to solve each section of the Lagrangian algorithm were also collected,as the solu-tion times can be used to give insight as to the relative difﬁculty of the particular parison for different capacity utilizationsWe tested the solution algorithm using a variety of cases.The ﬁrst test case considered is the base scenario,which serves as the baseline for comparison.The base case is constructed with b 1=b 2=1,a =100,X =1.The DC capacity ratio is varied from tight capacities (j =3),moderate capacities (j =5),to excess capacities (j =10).The results are shown in Table 6.1.The test statistics present several insights about the problem formulation and solution algorithm.The data shows that the rigidity of the problem (dictated by j )has a large impact on the DCLR,both in terms of the average and range of the ratio.Table 6.1shows that as the tightness of the problem is decreased (or as j is increased),the DCLR also decreases.Furthermore,the results show that the range of the DCLRs increases as j increases.Thus,it is evident that the tightness of the problem has an adverse effect on the load ratio of the distribution center.The cost breakdowns for the base scenario test cases are also presented in Table 6.1.In contrast to the DCLR,the distri-bution of costs is fairly stable across the varying DC capacity levels.Hence,the value of j has little impact on the cost dis-tribution of the network.Computational times are also shown in Table 6.1.Intuitively,the computation increases as more decisions variables are added to the problem.Additionally,the average solution time increases as the tightness of the problem increases.The data shows that the majority of the solution time is spent solving [SP1],accounting for roughly 96to 98%of the total time.The table shows that the primal heuristic produces very good feasible solutions that are less than.2%from the optimum.Contributing to the strength of the heuristic was the fact that the information we use to construct the heuristic solution is taken from [SP1].Furthermore,[SP1]retains many attributes of the original problem and is already a very strong formula-tion,which is evident by the large amount of time spent solving [SP1]parison for different dominant cost scenariosTo test the behavior of the algorithm and the characteristics of the optimal network design,we vary the cost structure to make one of the cost components dominant.Three cases are considered:dominant ﬁxed costs,dominant variable costs,and dominant emissions costs.All three are compared for the moderate capacity case (j =5).The dominant ﬁxed cost scenario enlarges the scaling parameter on the ﬁxed costs to establish the distribution centers.This case is constructed with b 1=b 2=1,a =1000,X =1.The results are displayed in Table 6.2.S.Elhedhli,R.Merrick /Transportation Research Part D 17(2012)370–379377。

Heuristics for Thelen’s Prime Implicant Method 1

S C H E D A E I N F O R M A T I C A E1The research is partially supported by Polish State Committee for Scientiﬁc Research (KBN)grant No.4T11C00624.126the other hand,new variants of minimization methods requiring all the prime implicants are still being developed[8,10,11].And there are a lot of other applications of a method of prime implicants generation.For example,calcu-lation of the complement of a Boolean function(in DNF),or transformation of a Boolean equation from CNF to DNF.And vice versa–as far as due to the Morgan’s laws transformation from DNF to CNF can be performed by transformation from CNF to DNF.One more application is detecting dead-locks and traps in a Petri net,which can be performed by solving logical equations[13,14].Generally,solutions of a logical equation can be easily obtained from prime implicants of its left part,if the right part is1.Also there are tasks,which can be solved by calculating the shortest prime implicant or prime implicants satisfying certain conditions.In[5]several of such logical design tasks are discussed.Covering problems,both unate and binate covering,can be easily represented as logical expressions in CNF and are usually solved by one of two approaches:BDD-based[1]or branch and bound,for which the shortest prime implicant would correspond to an optimal solution[2].The same is true for some graph problems,such as decyclisation of graphs[4].Task of detecting deadlocks in FSM networks can be reduced to task of generating a subset of prime implicants.The approach discussed in this paper can be applied(directly or with some modiﬁcations) to the whole range of mentioned problems.For generation of prime implicants several algorithms are known.The method of Nelson[9],probably historicallyﬁrst such method for CNF,is based on straightforward multiplying the disjunctions and deleting the prod-ucts that subsume other products.Such transformation is very time-and memory-consuming.More eﬃcient methods are known:an algorithm based on a search tree,proposed by B.Thelen[12],and a recursive method de-scribed in[8].Comparison of those two methods is beyond the scope of our paper;the paper is dedicated to heuristics allowing to accelerate Thelen’s method.Execution time of this algorithm depends remarkably on the order of clauses and literals in the expression.Hence we may suppose that some reordering of the expression will increase eﬃciency of the algorithm.As far as the search tree in Thelen’s method is reduced by means of certain rules (described below),it is diﬃcult to evaluate a priori eﬀects of diﬀerent variants of reordering.So it is reasonable to use a heuristic approach and to verify the heuristics statistically.Some of such heuristics are described in[5,6].The article describes some new heuristics,their analysis and comparison with known heuristics.Experiments are performed by using the randomly generated samples;the optimal combination of the heuristics is formulated on the basis of experimental results.127 2.Thelen’s algorithmThelen’s prime implicant algorithm is based on the method of Nelson[9],who has shown,that all the prime implicants of a Boolean function ina conjunctive form can be obtained by its transformation into a disjunctive form.Nelson’s transformation is performed by straightforward multiplying the disjunctions and deleting the products that subsume other products. Such transformation is very time-and memory-consuming,because all the intermediate products should be kept in memory,and their number grows exponentially.Thelen’s algorithm transforms CNF into DNF in a much more eﬃcient way.It requires linear memory for transformation and additional memory for calculated prime implicants.The subsuming products are not kept in memory.A search tree is built,such that every level of it corresponds to a clause of the CNF,and the outgoing arcs of a node correspond to the literals of the disjunction.Conjunction of all the literals corresponding to the arcs at the path from the root of the tree to a node is associated with the node.Leaf nodes of the tree are the elementary conjunctions being the prime implicants of the expression or the implicants subsuming the prime implicants calculated before.A sample tree is shown in Fig.1.The tree is searched in DFS order,and several pruning rules are used to minimize it.The rules are listed below.R1An arc is pruned,if its predecessor node-conjunction contains the com-plement of the arc-literal.R2An arc is pruned,if another non-expanded arc on a higher level still exists which has the same arc-literal.R3A disjunction is discarded,if it contains a literal which appears also in the predecessor node-conjunction.The rules above are based on the following laws of Boolean algebra:a∧a=aa∨a∧b=aa∧a=1(3)a∨1=1(4)a∧1=a(5)128Rules R1and R3follow immediately from(2)and(3).Rule R2provides that the implicants associated with the leaf nodes,if they are not prime, subsume the implicants calculated before.That means that theﬁrst calcu-lated implicant is always simple.An arc at level k with arc-literal x,such that there is a non-expanded arc with the same arc-literal at level l higher than k,is pruned by it.An implicant obtained by expanding the mentioned arc would be at least one literal shorter than the implicant which would be obtained without applying rule R2.As far as the path two times comes through literal x(at the levels k and l),according to(1),(2)the longest of those two implicants subsumes the shortest one.Hence theﬁrst calculated implicant cannot subsume the implicants calculated later,but it can be sub-sumed by them.So,applying rule R2allows to check whether an implicant is simple immediately after its calculation.It is enough to compare it with all the implicants calculated before.Due to this property the algorithm is less memory-consuming,because only prime implicants are kept.Fig.1.An example of the tree for Boolean formula:(a∨129 R4An arc j is pruned,if another already expanded arc k with the same arc-literal exists on a higher level v and if rule R2was not applied in the subtree of arc k with respect to arc p on level v which leads to arc j.But using this rule complicates the algorithm remarkably,because addi-tional information on applying rule R2has to be kept.Additional reduction reduces probability of appearing the non-prime implicants at the leaf nodes. But there is no guarantee that such implicants will not appear,and still it is necessary to perform checking,the same as in the case of tree built with using only3pruning rules.The next expression is an example for which non-prime implicants still appear even if all4rules are used:(x∨y)y(y∨z)z(x∨z) (Fig.3).3.Heuristics for Thelen’s methodOne of the possibilities of reducing the search tree is sorting the disjunc-tions by their size in ascending order.Heuristic1(Sort by Length[5]).Choose disjunction D j with the small-est number of literals.Eﬀect of this heuristics can be illustrated with a complete search tree (without arc pruning).Its size(number of nodes)can be calculated accordingto the formula:|V|=1+ni=1ij=1L j(6)where L j is the number of literals in clause number j.Let a formula consist of5clauses,each having a diﬀerent number of literals,from2to6.If they are sorted from maximal to minimal length,the complete search tree will contain1237nodes;if sorted from minimal to maximal the tree will contain only873nodes.In the second case it is30%smaller.So sorting of clauses inﬂuences the tree size remarkably.Of course for the reduced search trees relation may diﬀer.Now let us turn to the pruning rules.Note that every rule can be imple-mented only if the disjunction under consideration contains the same vari-ables as the disjunctions corresponding to the predecessor nodes.That means that if the next disjunction considers the variables which appear in the pre-vious disjunctions,there are possibilities of reduction at that level;and there is no possibility of reduction for the new variables.So we may suppose that130sorting of the clauses according to the variables also may lead to the tree reduction.Here the similar eﬀect is used as in the case of sorting by length: disjunctions containing many repeating variables allow to reduce the tree re-markably,and if such reduction can be performed not far from the root,the tree will be growing slower.So the following heuristics reorder the disjunc-tions in such a way that minimal number of new variables appears at every next level of the search tree.Heuristic2a(Sort by Literals).Choose disjunction D j with the smallest number of literals that do not appear in the disjunctions chosen before.Heuristic2b(Sort by Variables).Choose disjunction D j with the small-est number of variables that do not appear in the disjunctions chosen before.The only diﬀerence between these two variants is that heuristic2a com-pares clauses according to literals and heuristic2b according to variables. This means that131Fig.2.An example of the tree,in which eﬀects of heuristic4and rule R4are the sameappearing non-prime implicants at the leaf nodes.As far as due to rule R2an implicant can subsume only the implicants calculated before,if the implicants calculated later are in most cases shorter than those calculated earlier,then chance of subsuming is small.The next heuristic is a reversion of heuristic3.Heuristic4(Reordering Literals).Choose literal v i with the minimum frequency in the non-expanded part of the expression.In many cases(but not always)eﬀects of rule R4and heuristic4are very similar.Rule R4prunes an arc,if at a higher level there is a non-expanded arc with the same arc-literal(let it be a).That means that at the level k literal a is not the last literal in the clause.Let literalb there is a path to the node under consideration at level l.If literal a would be the last in the clause, instead of R4the rule R2would be applicable with the same eﬀect(Fig.2).We may also state that if literal a appears at level k and also at a lower level l(that means in clauses D k oraz D l(k>l)),then if b does not appear in the clauses with numbers greater than k,after applying heuristic4in the clause D k literal b will appear before a and R2will be applicable instead of R4.But if b appears in the clauses with numbers greater than k,such an eﬀect will not always occur.Here is example of heuristic4:(a∨c∨d)After applying the heuristic:(c∨d)132Such ordering of literals causes that the arc leading to non-prime impli-canta∨b∨c)(a∨c∨d)(b,appearing in clause2,appear in the next clauses with the same frequency,and without applying rule R4the algorithm will generate a non-prime implicantP H1H4R4 N N T%N T%N 14463361296752217.80702 1.739620x2677559504122.6420259633.50426855.0 121502459942.1028319.9020x2224863166127.431970228.2070228.2 70317125143547.5073824.4020x21134667149.98433324.7033324.7 3637748157536.2065415.0020x23265066024.95688133.2088133.2 1174064064181122.30113933.3025x31512752603050.84895636512.40637412.4 14490265401770 3.204380.87225x32632342282636.162264170 6.604170 6.6 91288288410960.1097014.230 Avg:133Fig.3.An example of the tree,in which there are diﬀerences between heuristic4 and rule R4implicants.But rule R4is more diﬃcult for implementation and increases a necessary memory amount.So it seems that applying heuristic4is more reasonable because allows obtaining a similar eﬀect with less eﬀort.Results of computer experiments are summarized in Tab.1.For the tests the randomly generated Boolean expressions were used.In theﬁrst column number of variables and the number of clauses of an expression are given(e.g. 20x18).T denotes the tree size(number of nodes);P denotes the number of prime implicants;N denotes the number of non-prime implicants,being the leaves of the search tree.A column‘%’shows for every heuristic the percentage of the tree size in respect of the size in the case when no heuristic is used.The experiments show that it is best to sort disjunctions according to heuristic2a,and literals in the disjunctions according to heuristic4.1344.Conclusion and further workThe presented heuristics,according to the experimental results,allow to generate all the prime implicants of a logical expression represented in the conjunctive normal form more quickly,than it can be done by using Thelen’s method with the heuristics known before.Besides of that,the presented heuristics reduce remarkably the number of the leaf nodes in the search tree corresponding to non-prime implicants.A prospective direction of future work is evaluation of eﬃciency of the proposed heuristics for solving problems mentioned in Introduction,for which Thelen’s algorithm can be applied.That may require taking into account ad-ditional optimization parameters and modiﬁcation of heuristics.One more direction is comparison between Thelen’s approach and the BDD-based ap-proach to solving problems such as covering problems.5.References[1]Brayton R.K.et al.;VIS:A System for Veriﬁcation and Synthesis,in:The Pro-ceedings of the Conf.on Computer-Aided Veriﬁcation,August1996,Springer Verlag,1102,pp.332–334.[2]Coudert O.,Madre J.K.;New Ideas for Solving Covering Problems,DesignAutomation Conference,1995,pp.641–646.[3]Coudert O.,Madre J.K.,Fraisse H.;A New Viewpoint on Two-Level LogicMinimization,Design Automation Conference,1993,pp.625–630.[4]Karatkevich A.;On Algorithms for Decyclisation of Oriented Graphs,in:Pro-ceedings of the International Workshop DESDes’01,Zielona G´o ra,Poland, 2001,pp.35–40.[5]Mathony H.J.;Universal logic design algorithm and its application the synthesisof two-level switching circuits,IEE Proceedings,136,3,1989,pp.171–177. [6]Mathony H.J.;Algorithmic Design of Two-Level and Multi-Level Switching Cir-cuits,(in German),PhD thesis,ITIV,Univ.of Karlsruhe,1988.[7]McGeer P.C.et al.;Espresso-Signature:A New Exact Minimizer for LogicFunctions,Design Automation Conference,1993,pp.618–624.[8]De Micheli D.;Synthesis and Optimization of Digital Circuits,Stanford Univ.,McGraw-Hill,Inc.,1994.135 [9]Nelson R.;Simplest Normal Truth Functions,Journal of Symbolic Logic,20,2,1955,pp.105–108.[10]Rudell R.,Sangiovanni-Vincentelli A.;Multiple-valued Minimization for PLAOptimization,IEEE Transactions on CAD/ICAS,Sept.1987,CAD-6,5,1987, pp.727–750.[11]Rytsar B.,Minziuk V.;The Set-theoretical Modiﬁcation of Boolean FunctionsMinimax Covering Method,in:Proceedings of the International Conference TCSET’2004,Lviv–Slavsko,Ukraine,2004,pp.46–48.[12]Thelen B.;Investigations of algorithms for computer-aided logic design of digitalcircuits,(in German),PhD thesis,ITIV,Univ.of Karlsruhe,1981.[13]W¸e grzyn A.,W¸e grzyn M.;Symbolic Veriﬁcation of Concurrent Logic Con-trollers by Means Petri Nets,in:Proceedings of the Third International Con-ference CAD DD’99,Minsk,Belarus,1999,pp.45–50.[14]W¸e grzyn A.,Karatkevich A.,Bieganowski J.;Detection of deadlocks and trapsin Petri nets by means of Thelen’s prime implicant method,AMCS,14,1,2004, pp.113–121.Received March8,2004。

费休氏容量法测量冻干人用狂犬病疫苗中水分含量的不确定度分析

费休氏容量法测量冻干人用狂犬病疫苗中水分含量的不确定度分析邱文娜饶玲王永智方朝东李磊江苏省泰州医药高新技术产业园区疫苗工程中心，江苏泰州 225300[摘要] 目的对费休氏容量法测量冻干人用狂犬病疫苗中水分含量的不确定度进行分析，为评价该方法的准确性和可靠性提供科学依据。

方法建立费休氏容量法测量冻干人用狂犬病疫苗中水分含量的数学模型，找出影响不确定度的因素，并对各个不确定度分量进行评估。

结果计算出各变量的不确定度，取包含因子K=2，置信概率为95%，费休氏容量法测量冻干人用狂犬病疫苗中水分含量的扩展不确定度为0.082%。

结论卡尔费休容量法检测冻干人用狂犬疫苗中水分含量时，不确定度主要由测定时卡尔费休试液的消耗体积和滴定度F值产生，该评价方法适用于冻干狂犬疫苗水分测定的不确定度评定与报告。

[关键词] 卡尔费休容量法；冻干人用狂犬病疫苗；水分；不确定度评定[中图分类号] R392 [文献标识码] A [文章编号] 2095-0616（2018）09-37-04 Uncertainty analysis of moisture content in freeze-dried rabies vaccine for human use by Karl fischer methodQIU Wenna RAO Ling WANG Yongzhi FANG Chaodong LI LeiVaccine Engineering Center of Taizhou Medical High-tech Industrial Park,Taizhou 225300,China[Abstract] Objective To analyze the uncertainty of measuring moisture content in freeze-dried rabies vaccine for human use by the Karl fischer method,and to provide a scientific basis for evaluating the accuracy and reliability of this method. Methods A mathematical model for measuring moisture content in freeze-dried rabies vaccine for human use was established by Karl fischer method,and the factors affecting the uncertainty were found and the components of the uncertainty were evaluated. Results The uncertainty of each variable was calculated and the inclusion factor K=2 was taked.The confidence probability was 95%.The expanded uncertainty of Karl fischer method for measuring moisture content in freeze-dried rabies vaccine for human use was 0.082%. Conclusion When Karl fischer method is used to detect the moisture content in freeze-dried rabies vaccine for human use,the uncertainty is mainly generated by the consumption volume and titer F value of Karl fischer test solution at the time of determination.This evaluation method is applicable to the evaluation and report of the uncertainty in the determination of moisture content in freeze-dried rabies vaccine.[Key words] Karl fischer method;Freeze-dried rabies vaccine for human use;Moisture content;Uncertainty assessment狂犬病是由狂犬病毒所致的人畜共患急性传染病，流行性广，病死率100%，有效预防该病的措施是接种狂犬病疫苗[1]。

Heuristic methods for vehicle routing problem with time windows

Heuristic methods for vehicle routing problem with time windowsK.C.Tan a,*,L.H.Lee b ,Q.L.Zhu a ,K.Ou aaDepartment of Electrical and Computer Engineering,National University of Singapore,10Kent Ridge Crescent,Singapore 119260bDepartment of Industrial and Systems Engineering,National University of Singapore,10Kent Ridge Crescent,Singapore 119260Received 7September 2000;accepted 20December 2000AbstractThis paper documents our investigation into various heuristic methods to solve the vehicle routing problem with time windows (VRPTW)to near optimal solutions.The objective of the VRPTW is to serve a number of customers within prede®ned time windows at minimum cost (in terms of distance travelled),without violating the capacity and total trip time constraints for each binatorial optimisation problems of this kind are non-polynomial-hard (NP-hard)and are best solved by heuristics.The heuristics we are exploring here are mainly third-generation arti®cial intelligent (AI)algorithms,namely simulated annealing (SA),Tabu search (TS)and genetic algorithm (GA).Based on the original SA theory proposed by Kirkpatrick and the work by Thangiah,we update the cooling scheme and develop a fast and ef®cient SA heuristic.One of the variants of Glover's TS,strict Tabu,is evaluated and ®rst used for VRPTW,with the help of both recency and frequency measures.Our GA implementation,unlike Thangiah's genetic sectoring heuristic,uses intuitive integer string representation and incorporates several new crossover operations and other advanced techniques such as hybrid hill-climbing and adaptive mutation scheme.We applied each of the heuristics developed to Solomon's 56VRPTW 100-customer instances,and yielded 18solutions better than or equivalent to the best solution ever published for these problems.This paper is also among the ®rst to document the implementation of all the three advanced AI methods for VRPTW,together with their comprehensive results.q 2001Elsevier Science Ltd.All rights reserved.Keywords :Vehicle routing problem;Time windows;Combinatorial optimisation;Heuristics;Simulated annealing;Tabu search;Genetic algorithm1.IntroductionLogistics may be de®ned as `the provision of goods and services from a supply point to various demand points'[2].A complete logistic system involves transporting raw materials from a number of suppliers or vendors,delivering them to the factory plant for manufacturing or processing,movement of the products to various warehouses or depots and eventually distribution to customers.Both the supply and distribution procedures require effective transportation management.Good transportation management can practi-cally save a private company a considerable portion of its total distribution cost.Potential cost savings constitute:lowered trucking cost due to more optimal routes and shorter distances,reduced in-house space and related costs,less penalty incurred due to untimely delivery.One of the most signi®cant measures of transportation manage-ment is effective vehicle routing.Optimising of routes for vehicles given various constraints is the origin of vehicle routing problems (VRPs).Fig.1describes a typical VRP.The solution includes tworoutes:Depot !7!8!9!11!12!Depot ;Depot !2!3!1!4!5!6!10!Depot :Sometimes the depot is denoted as 0.The vehicle routing problem with time windows (VRPTW)is a well-known non-polynomial-hard (NP-hard)problem,which is an extension of normal VRPs,encountered very frequently in making decisions about the distribution of goods and services.The problem involves a ¯eet of vehicles set off from a depot to serve a number of customers,at different geographic locations,with various demands and within speci®c time windows before returning to the depot.The objective of the problem is to ®nd routes for the vehicles to serve all the customers at a minimal cost (in terms of travel distance,etc.)without violating the capacity and travel time constraints of the vehicles and the time window constraints set by the cus-tomers.To date,there is no consistent optimising algorithm that solves the problem exactly using mathematical programming.Instead,many heuristic methods have been designed to solve VRPTW to near optima.In Marshall Fisher's survey [4],he categorised vehicle routing methods into three generations.The ®rst generation was simple heuristics developed in the 1960s and 1970s,which were mainly based on local search or sweep.Arti®cial Intelligence in Engineering 15(2001)281±2950954-1810/01/$-see front matter q 2001Elsevier Science Ltd.All rights reserved.PII:S0954-1810(01)00005-X/locate/aieng*Corresponding author.Since these earlier studies were not well documented,it is hard to compare the results they obtained 30years ago with the more recent solutions.The second genera-tion,mathematical programming based heuristics,were near-optimisation algorithms that are very different from normal heuristics.These include the generalised assign-ment problems and set partitioning to approximate the VRP.Their results are usually superior to that of simple heuristics [4,20].In fact for linear objective functions,some of these techniques are able to stretch to the optima.The third generation,or the one that is currently undergoing heavy research is exact optimisation algorithms and arti®cial intelligence methods.Among these,the most successful optimisation algorithms are K-tree,Lagrangian relaxation,etc.,while the top AI repre-sentatives in VRPTW are simulated annealing (SA),Tabu search (TS)and genetic algorithms (GAs).These algorithms are discussed brie¯y as follows:Kolen et al.[10]presented the method of branch and bound ,which is among the ®rst optimisation algorithms for VRPTW.The method calculates lower bounds using dynamic programming and state space relaxation.Branching decisions are taken on route-customer alloca-tions.The method has successfully solved the problem involving 15customers.Fisher [3]introduces an opti-misation algorithm in which lower bounds are obtained from a relaxation based on a generalisation of spanning trees called K -trees .Capacity constraints are handled by introducing a constraint requiring that some set S ,S ,C ;of the set of customers must be served by at least k (S )vehicles.This constraint is Lagrangian relaxed and the resulting problem is still a K -tree problem with modi®ed arc costs.Time window constraints are treated similarly.A constraint,requiring that not all arcs in a time violating path can be used,is generated and Lagrangian relaxed.The method has solved some of the 100-customer Solomon benchmark problems [18].One of the effective approaches at present is the shortest path composition .The fundamental observation is,the only constraint which `links'the vehicles together is that each customer in the network must be visited only once.The problem that consists of the rest of the constraints is an elementary shortest path problem with time windows and capacity constraints (ESPPTWCC)for each vehicle.Although this problem is strictly NP-hard,there are a few ef®cient dynamic programming algorithms for the slightly relaxed programs.Two decompositions have been investi-gated computationally,namely Dantzig ±Wolfe decomposi-tion and variable splitting .Desrochers et al.[25]implemented Dantzig±Wolfe decomposition,and solved up to some of the 100-customer Solomon benchmark problems.Researchers at Technical University of Denmark [9],on the other hand,suggested using variable splitting to solve the VRPTW with similar performance.Thangiah et al.[21]developed a l -interchange local search descent (LSD )method that uses a systematic insertion and swapping of customers between routes,de®ned as l -interchange operators.Due to computation burden,only 1-interchange and 2-interchange are commonly used,which allows up to one or two custo-mers to be inserted or swapped at one time.Although it is a fast algorithm,the performance is poor without the help from other heuristics.SA,®rst proposed by Kirk-patrick [8],searches the solution space by simulating the annealing process in metallurgy.The algorithm jumps to distant location in the search space initially.The step of the jumps is reduced as time goes on or as the temperature `cools'.Eventually,the process will turn into a LSD method.Osman [14]has applied SA to solve the VRP by moving one customer from one route to another or exchanging two customers from two routes.TS is a memory-based search strategy that chooses the best solution contained in N (S )that does not violate certain restrictions that prevent ually,these restrictions are stored as queues in a structure called a Tabu list .Typical restrictions prevent making a move that has been done within the last t iterations,and a solution that has been encountered in the last t iterations is usually forbidden as well.TS stops after a ®xed number of iterations.Gerdreau et al.applied TS using a neighbourhood that can be constructed by moving a single customer from one route to another.Osman and Talliard [14]used a neighbourhood that consists of all solutions obtained from inserting a customer and swapping two customers.Holland developed the GA [7]method that codes the VRPTW solutions in forms of bit strings or chromosomes.The method starts with a population of random chromo-somes.Fitter chromosomes are then selected to undergo a crossover and mutation process,as to produce children which are different from the parents but inherit certain genetic traits from the parents.This process is continued until a ®xed number of generations has been reached orK.C.Tan et al./Arti®cial Intelligence in Engineering 15(2001)281±295282Fig.1.A vehicle routing problem:a single depot VRP with 12customers.Each route starts from depot,visiting customers and ends at depot.the evolution has converged.Thangiah[22]devised a genetic sectoring heuristic with special genetic representation that keeps the polar angle offset in the genes.The algorithm follows a cluster-®rst,route-second philosophy and solved 100-customer Solomon problems to near optima.Prinetto et al.[16]proposed a hybrid GA for the travelling salesman problem(TSP)in which2-opt and Or-opt were incorporated with the GA.Blanton and Wainwright[1]presented two new crossover operators,merge cross#1and merge cross #2,which are superior to traditional crossover operators. Shaw[17]presented large neighbourhood search(LNS), a method in constraint programming,to solve VRPTW. Relatedness plays a very important part in the selection of customer to remove and re-insert into the con®guration using a constraint-based tree search.Shaw applied limited discrepancy search during the tree search to re-insert visits. The results were competitive to those obtained using opera-tions research meta-heuristics.In this paper,we further investigate and develop various advanced AI techniques including SA,TS and GA to effectively solve the VRPTW to near optimal solutions.Based on the original SA theory proposed by Kirkpatrick[8]and the work by Thangiah[21],we update the cooling scheme and develop a fast and ef®-cient SA heuristic.One of the variants of Glover's TS, strict Tabu,is evaluated and®rst used for VRPTW, with the help of both recency and frequency measures. Our GA implementation,unlike Thangiah's genetic sectoring heuristic[21],uses an intuitive integer string representation and incorporates several new crossover operations and other advanced techniques such as hybrid hill-climbing and adaptive mutation scheme. We have tested our heuristics with all56Solomon's VRPTW instances and obtained complete results for these problem sets.There are totally four heuristics tested on the instances:2-interchange method,SA, Tabu and GA.Their average performances are compared with the best-known solutions in the litera-ture.From the result analysis,our TS and GA are already close to the best ways of solving VRPTW. Totally,we found18solutions better than or equivalent to the best-known results.The discussion of results is given in Section8.In this paper,we give a mathema-tical model of VRPTW,followed by the design and implementation of the heuristics.The computational results are presented and discussed in the®nal part of the paper.2.Problem formulationThis section describes the notation and features that are common through this paper.The VRPTW constraints consist of a set of identical vehicles,a central depot node, a set of customer nodes and a network connecting the depot and customers.There are N11customers and K vehicles.The depot node is denoted as customer0.Each arc in the network represents a connection between two nodes and also indicates the direction it travels.Each route starts from the depot,visits customer nodes and then returns to the depot.The number of routes in the network is equal to the number of vehicles used.One vehicle is dedicated to one route.A cost c ij and a travel time t ij are associated with each arc of the network.In Solomon's56VRPTW100-customer instances,all distances are represented by Euclidean distance,and the speed of all vehicles is assumed to be unity.That is,it takes one unit of time to travel one unit of distance. This assumption makes the problem simpler,because numerically the travel cost c ij,the travel time t ij and the Euclidean distance between the customer nodes equal each other.Each customer in the network can be visited only once by one of the vehicles.Every vehicle has the same capacity q k and each customer has a varying demand m i.q k must be greater or equal to the summa-tion of all demands on the route travelled by vehicle k, which means that no vehicles can be overloaded.The time window constraint is denoted by a prede®ned time interval,given an earliest arrival time and latest arrival time.The vehicles must arrive at the customers not later than the latest arrival time,if vehicles arrive earlier than the earliest arrival time,waiting occurs.Each customer also imposes a service time to the route, taking consideration of the loading/unloading time of goods.In Solomon's instances,the service time is assumed to be unique regardless of the load quantity needed to be handled.Vehicles are also supposed to complete their indi-vidual routes within a total route time,which is essentially the time window of the depot.There are three types of principal decision variables in VRPTW.The principal decision variable x ijk i;j[ {0;1;2;¼;N};k[{1;2;¼;K};i±j is1if vehicle k travels from customer i to customer j,and0otherwise. The decision variable t i denotes the time when a vehicle arrives at the customer,and w i denotes the waiting time at node i.The objective is to design a network that satis®es all constraints,at the same time minimising the total travel cost.The model is mathematically formulated below:Principal decision variables:t i arrival time at node iw i wait time at node ix ijk[{0;1};0if there is no arc from node i to node j,and 1otherwise.i±j;i;j[{0;1;2;¼;N}: Parameters:K total number of vehiclesN total number of customersy i any arbitrary real numberd ij Euclidean distance between node i and node jK.C.Tan et al./Arti®cial Intelligence in Engineering15(2001)281±295283c ij cost incurred on arc from node i to jt ij travel time between node i and jm i demand at node iq k capacity of vehicle ke i earliest arrival time at node il i latest arrival time at node if i service time at node ir k maximum route time allowed for vehicle kMinimiseX Ni 0X Nj 0;j±iX Kk 1c ij x ijk 1subject to:X K k 1X Nj 1x ijk#K for i 0 2X N j 1x ijkX Nj 1x jik#1for i 0andk[{1;¼;K}3X K k 1X Nj 0;j±ix ijk 1for i[{1;¼;N} 4X K k 1X Ni 0;i±jx ijk 1for j[{1;¼;N} 5X N i 1m iX Nj 0;j±ix ijk#q k for k[{1;¼;K} 6X N i 0X Nj 0;j±ix ijk t ij1f i1w i #r k for k[{1;¼;K} 7t0 w0 f0 0 8X K k 1X Ni 0;i±jx ijk t i1t ij1f i1w i #t j for j[{1;¼;N}(9)e i# t i1w i #l i for i[{1;¼;N} 10 Formula(1)is the objective function of the problem. Constraint(2)speci®es there are maximum K routes going out of the depot.Eq.(3)makes sure every route starts and ends at the central depot.Eqs.(4)and(5)de®ne that every customer node can be visited only once by one vehicle. Eq.(6)is the capacity constraint.Eq.(7)is the maximum travel time constraint.Constraints(8)±(10)de®ne the time windows.These formulas completely specify the feasible solutions for VRPTW.3.An initial solutionMost heuristic search strategies involve®nding an initial feasible solution and then improving on that solution using local or global optimisation techniques.Here,we make use of the push forward insertion heuristic(PFIH),®rst intro-duced by Solomon[18]in1987as a method to create an initial route con®guration.PFIH is an ef®cient method to insert customers into new routes.The procedure is easy and straightforward.The method tries to insert the customer between all the edges in the current route.It selects the edge that has the lowest addi-tional insertion cost.The feasibility check tests all the constraints including time windows and load capacity. Only feasible insertions will be accepted.When the current route is full,PFIH will start a new route and repeat the procedure until all the customers are ually, PFIH gives a reasonably good feasible solution in terms of the number of vehicles used.This initial number of vehicles provides an upper bound for the number of routes in the solution.PFIH serves the role of constructing route con®guration for VRPTW.It is an ef®cient method to obtain feasible solutions.The detail information can be obtained from Solomon's paper[18].4.Local search with l-interchangeThe effectiveness of any iterative local search method is determined by the ef®ciency of the generation mechanism and the way the neighbourhood is searched.A l-inter-change generation mechanism was introduced by Osman and Christo®des[13]for the capacitated clustering problem. It is based on customer interchange between sets of vehicle routes and has been successfully implemented with a special data structure to other problems by Osman[14],Thangiah [20],etc.The local search procedure is conducted by interchan-ging customer nodes between routes.For a chosen pair of routes,the searching order for the customers to be interchanged needs to be de®ned,either systematically or randomly.In this paper,we only consider the cases l 2;which means that maximum two customer nodes may be interchanged between routes.Based on the number of l,there are totally eight interchange opera-tors are de®ned:(0,1),(1,0),(1,1),(0,2),(2,0),(2,1), (1,2),(2,2).The operator(1,2)on a route pair(R p, R q)indicates a shift of two customers from R q to R p and a shift of one customer from R p to R q.The other operators are de®ned similarly.For a given operator,the customers are considered sequentially along the routes. In both the shift and interchange process,only improved solutions are accepted if the move results in the reduction of the total cost.K.C.Tan et al./Arti®cial Intelligence in Engineering15(2001)281±295 284There are two strategies to select between candidate solutions:1.The®rst-best(FB)strategy will select the®rst solution in N l(S),the neighbourhood of the current solution,that results in a decrease in cost.2.The global-best(GB)strategy will search all solutions in N l(S),where N l(S)means the neighbourhood of current solution under l-interchange operation.GB will select the one,which will result in the maximum decrease in cost.In the following we describe the l-interchange LSD method.LSD starts from an initial feasible solution obtained by the PFIH.The PFIH solution is further improved using the l-interchange mechanism for a given number of itera-tions.The procedure of the l-interchange LSD is shown below.Algorithm1.Local search descent methodLSD-1:Obtain a feasible solution S for the VRPTW using the PFIH.LSD-2:Select a solution S0[N l S :LSD-3:If{C S0 ,C S };thenaccept S0and go to LSD-2,else go to LSD-4.LSD-4:If{neighbourhood of N l(S)has been completely searched:there are no movesthat will result in a lower cost}then go to LSD-5else go to LSD-2.LSD-5:Stop with the LSD solution.The LSD result is dependent on the initial feasible solu-tion.GB usually achieves better results than FB because it keeps track of all the improving moves but incurs more expensive computation time.On the other hand,LSD±FB is a blind search that accepts the FB result.In this paper,we implemented2-interchange GB.5.Simulated annealingSA is a stochastic relaxation technique that®nds its origin in statistical mechanics[11].The SA methodology is analogous to the annealing processing of solids.In order to avoid the meta-stable states produced by quenching, metals are often cooled very slowly,which allows them time to order themselves into stable,structurally strong, low energy con®gurations.This process is called annealing. This analogy can be used in combinatorial optimisation with the states of the solids corresponding to the feasible solu-tion,the energy at each state to the improvement in objec-tive function and the minimum energy being the optimal solution[8].SA involves a process in which the temperature is gradually reduced during the simulation.Often,the system is®rst heated and then cooled.Thus,the system is given the opportunity to surmount energetic barriers in a search for conformations with energies lower than the local-minimum energy found by energy minimisation. Unlike l-interchange,SA is a global optimisation heuristic based on probability,therefore,is able to overcome local optima.At each step of the simulation algorithm,a new state of the system is constructed from the current state by giving a random displacement to a randomly selected particle.If the energy associated with this new state was lower than the energy of the current state,the displace-ment was accepted,that is,the new state becomes the current state.If the new state had an energy higher by d joules,the probability of changing the current state to the new state isexp2d11where k is the Boltzmann constant and T the absolute temperature at present.This basic step,a metropolis step, can be repeated inde®nitely.The procedure is called a metropolis loop.It can be shown that this method of gener-ating current states led to a distribution of states in which the probability of a given state with energy e i to be the current state isexp 2e i=kTXjexp 2e j=kT12This probability function is known as Boltzmann density.One of its characteristics is that for very high temperatures,each state has almost equal chances of being the current state.At low temperatures,only states with low energies have a high probability of being the current state.These probabilities are derived for a never ending executing of the metropolis loop.The advantages of this scheme is:²SA can deal with arbitrary systems and cost functions;²SA statistically guarantees®nding an optimal solution;²SA is relatively easy to code,even for complex problems;²SA generally gives a`good'solution.However this original version of SA has some drawbacks:²Repeatedly annealing with a1/log k schedule is very slow,especially if the cost function is expensive to compute.²For problems where the energy landscape is smooth,or there are few local minima,SA is an overkillÐsimpler, faster methods(e.g.local descent)will work better.But usually one does not know what the energy landscape is.²Normal heuristic methods,which are problem-speci®c or take advantage of extra information about the system, will often be better than general methods.But SA is often comparable to heuristics.²The method cannot tell whether it has found an optimalK.C.Tan et al./Arti®cial Intelligence in Engineering15(2001)281±295285solution.Some other method(e.g.branch and bound)is required to do this.In our modi®ed version of SA,the algorithm starts with a relatively good solution resulting from PFIH.Initial temperature is set at T s 100;and is slowly decreased byT kT k2111tT k21p 13where T k is the current temperature at iteration k and t a small time constant.The square root of T k is introduced in the denominator to speed the cooling process.Here,we use a simple monotonically decreasing function to replace the 1/(log k)scheme.Our scheme gives fairly good results in much less time.The algorithm attempts solutions in the neighbourhood of the current solution randomly or system-atically and calculates the probability of moving to those solutions according toP accepting a move exp2D T k14 This is a modi®ed version of Eq.(11),where D C S0 2 C S ;C(S)is the cost of the current solution and C(S0) the cost of the new solution.If D,0;the move is always warranted.One can see that as temperature cools down,the probability of accepting a non-cost-saving move gets expo-nentially smaller.When the temperature has gone to the ®nal temperature T f 0:001or there are no more feasible moves in the neighbourhood,we reset the temperature toT r maxT r2;T b15where T r is the reset temperature,and was originally set to T s,and T b the temperature at which the best current solution was found.Final temperature is not set at zero because as temperature decreases to in®nitesimally close to zero,there is virtually zero probability of accepting a non-improving move.Thus,a®nal temperature not equal but close to zero is more realistic.To search a local neighbourhood,the2-interchange approach was adopted.Every time a GB solution is found,a2-inter-change(GB)procedure is executed to search for possible better solutions around it.The procedure terminates after a number of resets.Below is the detailed procedure of one of the SA implementations,which adopts a partial2-inter-change(FB)to search the neighbourhood.T s starting temperature of the SA method 100T f®nal temperature of the SA method 0.001T b temperature at which the current best solution was foundT r reset temperature of the SA method,originally equal to T sT k temperature of the current solution S current solutionS b current best solutionR number of resets to be donet the time constant in the range of(0,1). Algorithm2.Simulated annealingStep SA-1:Obtain a feasible solution for the VRPTW using the PFIH.Step SA-2:Improve S using the2-interchange LSD with GB strategy.Step SA-3:Set cooling parameters:T s T b T r T k 100;t 0:5:Step SA-4:Generate systematically an S0[N2 S by(2, 0)and(1,0)operations,and compute D C S0 2C S ; where N2(S)is the neighbourhood of current solution under2-interchange operation,C(S)and C(S0)means the cost of current solution and the newly generated solution,respectively.Step SA-5:If{ D#0 or(D.0and exp 2D=T k $u ; where u is a random number between[0,1]}thenset S S0.if{C S ,C S b }thenimprove S using2-interchange LSD(GB).update S b S and T b T k:Step SA-6:Set k k11:Update the temperature using Eq.(13).If{N2(S)is searched without any accepted move}then reset T r max T r=2;T b ;and set T k T r:Step SA-7:if{R resets have been made since the last S b was found}thengo to Step SA-8.else go to Step SA-4.Step SA-8:Terminate SA and print results.In general,our SA implementation is a simple and fast algorithm that solves many VRPTWs to near optima.Due to the GB approach in local neighbourhood search,the algo-rithm is able to result in stable local optimal solutions almost at all times.This is especially true if the global optimum in a problem is located very distant to the corre-sponding PFIH initial solutions.In that case SA may not have enough energy to traverse that far,given the limited number of temperature resets.6.Tabu searchTS is a memory-based search strategy,originally proposed by Glover[6],to guide the local search method to continue its search beyond a local optimum.The algo-rithm keeps a list of moves or solutions that have been made or visited in the past.This list,known as a Tabu list,is a queue of®xed or variable size.The purpose of the Tabu list is to record a number of most recent moves and prohibit anyK.C.Tan et al./Arti®cial Intelligence in Engineering15(2001)281±295 286。

a heuristic algorithm for portfolio

434
M. Grazia Speranza
model has been applied to problems with asymmetric distributions of the rate of return (see, for instance, Zenios and Kang [6]). In Speranza [7] it was shown that taking as risk function a linear combination of the mean semi-absolute deviations, i.e. mean deviations below and above the portfolio rate of return, a model equivalent to the M A D model is obtained, whenever the sum of the coefficients of the linear combination is positive. Then in turn this model is equivalent to Markowitz model, if the rates of return are normally distributed. Moreover, it was shown that, through a suitable selection of the coefficients of the combination, 1 and 0 for the deviations below and above the average respectively, it is possible to substantially reduce the number of the constr

最优化方法有关牛顿法的矩阵的秩为一的题目

英文回答：The Newton-Raphson method is an iterative optimization algorithm utilized for locating the local minimum or maximumof a given function. Within the realm of optimization, the Newton-Raphson method iteratively updates the current solution by leveraging the second derivative information of the objective function. This approach enables the method to converge towards the optimal solution at an accelerated pacepared to first-order optimization algorithms, such as the gradient descent method. Nonetheless, the Newton-Raphson method necessitates the solution of a system of linear equations involving the Hessian matrix, which denotes the second derivative of the objective function. Of particular note, when the Hessian matrix possesses a rank of one, it introduces a special case for the Newton-Raphson method.牛顿—拉弗森方法是一种迭代优化算法，用于定位特定函数的局部最小或最大值。

salesman

1
Introduction and problem description
A variation of the classic symmetric traveling salesman problem (TSP) is studied in this paper. The variation is motivated by the observation that estimating travel times exactly is often a diﬃcult task, since they depend on many factors that are diﬃcult to predict. Uncertainty about data should be consequently taken into account. We treat the case where the only information available is represented by a set of equally possible values for each travel time. We optimize the resulting problem according to the robust deviation criterion (see Kouvelis and Yu [6]). A robust tour is, intuitively, a tour which minimizes the maximum deviation from the optimal tour over all realizations of edge costs. The robust TSP with interval data is deﬁned on an undirected graph G = {V, E }, where V is a set of vertices, with vertex 0 associated with the depot, and vertices 1, . . . , |V | representing the cities to be visited, and E is the set of edges of the graph. An interval [lij , uij ], with 0 ≤ lij ≤ uij , is associated with each edge {i, j } ∈ E , and represents the possible travel times. The objective of the optimization is to ﬁnd a Hamiltonian cycle (tour ) with the minimum cost, according to the cost function associated with the notion of robust deviation. In order to formally describe the robust TSP, we need the following deﬁnitions. A scenario R is a realization of the edge costs, i.e. a cost cR ij ∈ [lij , uij ] is chosen for each edge of the graph. The robust deviation of a tour t in scenario R is the diﬀerence between the cost of t in scenario R and the cost of a shortest tour in R. A tour t is said to be a robust tour if it has the smallest (among all possible tours) maximum (among all possible scenarios) robust deviation.

Greedy and Heuristic

Hyper-heuristics provides a high-level strategy (HLS) to obtain new heuristics by manipulating or managing a group of low-level heuristics (LLH). These new heuristic algorithms are applied to solve all kinds of NP hard problems. As the research of hyper-heuristics is still in its infancy, there is no consistent classification method for all kinds of existing hyper-heuristics in the world. According to the mechanism of high-level strategies, the existing hyper-heuristics can be roughly divided into four categories: random selection based, greedy based, meta heuristic and learning based.
Results about NP-hardness in theoretical computer science make heuristics the only viable option for a variety of complex optimization problems that need to be routinely solved in real-world applications.

高效液相色谱法测定酱油中三氯蔗糖含量不确定度的评定

崔婷婷，祁伟，何文江，等. 高效液相色谱法测定酱油中三氯蔗糖含量不确定度的评定[J]. 食品工业科技，2024，45（7）：270−275.doi: 10.13386/j.issn1002-0306.2023050252CUI Tingting, QI Wei, HE Wenjiang, et al. Evaluation of Uncertainty in Determination of Sucralose in Soy Sauce by High Performance Liquid Chromatography[J]. Science and Technology of Food Industry, 2024, 45(7): 270−275. (in Chinese with English abstract). doi:10.13386/j.issn1002-0306.2023050252· 分析检测 ·高效液相色谱法测定酱油中三氯蔗糖含量不确定度的评定崔婷婷1，祁　伟1, *，何文江1，张家琪2，黄青春1,*（1.中国地质调查局呼和浩特自然资源综合调查中心，内蒙古呼和浩特 010010；2.内蒙古自治区特种设备检验研究院乌兰察布分院，内蒙古乌兰察布 012000）摘　要：目的：建立一种高效液相色谱法测定酱油中三氯蔗糖的不确定度的评定方法。

方法：按照食品安全国家标准GB 22255-2014进行检测，依据国家计量技术规范JJF 1059.1-2012分析酱油中三氯蔗糖的含量，建立不确定度评定数学模型，分析测量过程中的不确定度因素，通过样品重复测定、标准溶液纯度、标准曲线、玻璃器具、称量过程、回收率、检测仪器对检测结果不确定度进行评定，并且进行计算。

结果：按置信区间为95%，酱油中三氯蔗糖含量为（0.048±0.007）g/kg ，k=2。

结论：建立了高效液相色谱法测定酱油中三氯蔗糖的不确定度的评定方法，最终确定酱油中三氯蔗糖的含量测定结果的不确定度来源主要有：三氯蔗糖标准溶液建立的标准曲线的拟合、回收率，明确检测酱油中三氯蔗糖含量过程中带来的各种不确定度评定因素的占比。

Heuristic

Heuristic （computer science）Heuristic optimizationIn computer science, artificial intelligence, and mathematical optimization, a heuristic is a technique designed for solving a problem more quickly when classic methods are too slow, or for finding an approximate solution when classic methods fail to find any exact solution. This is achieved by trading optimality, completeness, accuracy, or precision for speed. In a way, it can be considered a shortcut.Contents[hide]1 Definition and motivation2 Trade-off3 Examples3.1 Simpler problem3.2 Traveling salesman problem3.3 Search3.4 Newell and Simon: Heuristic Search Hypothesis3.5 Virus scanning3.6 Russell and Norvig4 Pitfalls5 See also6 ReferencesDefinition and motivation[edit]The objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. This solution may not be the best of all the actual solutions to this problem, or it may simply approximate the exact solution. But it is still valuable because finding it does not require a prohibitively long time.Heuristics may produce results by themselves, or they may be used in conjunction with optimization algorithms to improve their efficiency (e.g., they may be used to generate good seed values).Results about NP-hardness in theoretical computer science make heuristics the only viable option for a variety of complex optimization problems that need to be routinely solved in real-world applications.Trade-off[edit]The trade-off criteria for deciding whether to use a heuristic for solving a given problem include the following: Optimality: When several solutions exist for a given problem, does the heuristic guarantee that the best solution will be found? Is it actually necessary to find the best solution?Completeness: When several solutions exist for a given problem, can the heuristic find them all? Do we actually need all solutions? Many heuristics are only meant to find one solution.Accuracy and precision: Can the heuristic provide a confidence interval for the purported solution? Is the error bar on the solution unreasonably large?Execution time: Is this the best known heuristic for solving this type of problem? Some heuristics converge faster than others. Some heuristics are only marginally quicker than classic methods.In some cases, it may be difficult to decide whether the solution found by the heuristic is good enough, because the theory underlying that heuristic is not very elaborate.Examples[edit]Simpler problem[edit]One way of achieving the computational performance gain expected of a heuristic consists in solving a simpler problemwhose solution is also a solution to the initial problem. Such a heuristic is unable to find all the solutions to the initial problem, but it may find one much faster because the simple problem is easy to solve.Traveling salesman problem[edit]An example of approximation is described by Jon Bentley for solving the traveling salesman problem (TSP) so as to select the order to draw using a pen plotter. TSP is known to be NP-Complete so an optimal solution for even moderate size problem is intractable. Instead, the greedy algorithm can be used to give a good but not optimal solution (it is an approximation to the optimal answer) in a reasonably short amount of time. The greedy algorithm heuristic says to pick whatever is currently the best next step regardless of whether that precludes good steps later. It is a heuristic in that practice says it is a good enough solution, theory says there are better solutions (and even can tell how much better in some cases).[1]Search[edit]Another example of heuristic making an algorithm faster occurs in certain search problems. Initially, the heuristic tries every possibility at each step, like the full-space search algorithm. But it can stop the search at any time if the current possibility is already worse than the best solution already found. In such search problems, a heuristic can be used to try good choices first so that bad paths can be eliminated early (see alpha-beta pruning).Newell and Simon: Heuristic Search Hypothesis[edit]In their Turing Award acceptance speech, Allen Newell and Herbert A. Simon discuss the Heuristic Search Hypothesis: a physical symbol system will repeatedly generate and modify known symbol structures until the created structure matches the solution structure. Each successive iteration depends upon the step before it, thus the heuristic search learns what avenues to pursue and which ones to disregard by measuring how close the current iteration is to the solution. Therefore, some possibilities will never be generated as they are measured to be less likely to complete the solution.A heuristic method can accomplish its task by using search trees. However, instead of generating all possible solution branches, a heuristic selects branches more likely to produce outcomes than other branches. It is selective at each decision point, picking branches that are more likely to produce solutions.[2]Virus scanning[edit]Many virus scanners use heuristic rules for detecting viruses and other forms of malware. Heuristic scanning looks for code and/or behavioral patterns indicative of a class or family of viruses, with different sets of rules for different viruses. If a file or executing process is observed to contain matching code patterns and/or to be performing that set of activities, then the scanner infers that the file is infected. The most advanced part of behavior-based heuristic scanning is that it can work against highly randomized polymorphic viruses, which simpler string scanning-only approaches cannot reliably detect. Heuristic scanning has the potential to detect many future viruses without requiring the virus to be detected somewhere, submitted to the virus scanner developer, analyzed, and a detection update for the scanner provided to the scanner's users.。

长大列车通过弹性曲线轨道仿真求解方法研究

第53卷第1期 2018年2月西南交通大学学报JOURNAL OF SOUTHW EST JIAOTONG UNIVERSITYVol. 53 No. 1Feb. 2018文章编号：0258-2724(2018)01~0031-07 D O I：10. 3969/j. issn. 0258-2724. 2018.01.004长大列车通过弹性曲线轨道仿真求解方法研究王开云2,翟婉明2(1.石家庄铁道大学机械工程学院，河北石家庄050043; 2.西南交通大学牵引动力国家重点实验室，四川成都 610031)摘要：为解决长大列车与连续长弹性轨道的同步仿真问题，以列车通过曲线轨道为例，采用重载列车-轨道耦合动力学模型，分析了压钩力作用下轨道结构与30 t轴重列车的动态特性，提出了长大重载列车与轨道动态相互作用仿真时模型的简化求解方法.该方法将庞大的列车/轨道耦合振动系统以有限数目的三维车辆模型代替，并考虑其轨下基础结构弹性，从而极大缩减系统运动自由度.研究结果表明:列车可简化为单质点车辆模型和三维车辆模型混合的短编组列车，当模型中只包含一个三维车辆模型,且其前、后车辆均以单质点模拟时,计算结果偏低;列车承受2 200 kN压钩力并通过400 m半径曲线线路时，货车最大轮轨横向力和垂向力较多节三维货车编组模型的计算结果分别估了 24%和4%，钢轨横向、垂向位移则被低估了 20%和8% ;端部车辆采用单质点模型、中部采用三维车辆模型的车辆数至少为3时,才能较为准确地反映中间目标车辆处轮轨作用力和其下部轨道结构的动态特性.关键词：重载列车;弹性曲线;压钩力;车辆-轨道耦合动力学;仿真求解中图分类号：U270;U211 文献标志码：ASimulation Solution Method for Long andHeavy-Haul Trains Negotiating Elastic Curved TracksLIU Pengfei1,2，WANG Kaiyun2，ZHAl Wanming2(1. School of Mechanical Engineering，Shijiazhuang Tiedao University，Shijiazhuang 050043，China；2. State Key Laboratory of Traction Power，Southwest Jiaotong University，Chengdu 610031，China)Abstract：To solve the synchronisation sim ulation problem for long trains and continuous elastic track s，a representative case employing a train curving negotiation is used. A heavy-haul train-track coupled dynamic model is used to analyse the dynam ic perform ance of the track structure with a 30 t axle-load train, when a coupler com pressive force is applied to it. As im plified principle is presented to sim ulate the dynamic interaction between the long heavy-haul train and the track. The general concept behind this m ethod is replacing the large train-track coupled dynam ic system by a finite num ber of three dim ensional vehicle m odels，considering the foundation structure elasticity under the corresponding models. T h u s，the motion freedom of the system can be significantly reduced. The results indicate that a long train can be treated as a shorter mixed train，composed of a single-m ass vehicle model and a three-dim ensional wagon model. As only one wagon is sim ulated by the detailed dynam icsm odel，with the front and rear connected wagons considered as a single-m ass m odel，conservative收稿日期：2017.28基金项目：国家自然科学基金资助项目（1605315 ,51478399);河北省高等学校科学技术研究项目（BJ2016047)作者简介：刘鹏飞（986-)，男，博士，讲师，研究方向为列车与轨道动态相互作用，E-mail:lpfswj+tu@引文格式：刘鹏飞，王开云，翟婉明.长大列车通过弹性曲线轨道仿真求解方法研究[J].西南交通大学学报，2018,53(1)：31-37.LIU Pengfei，WANG Kaiyun，ZHAI Wanming. Simulation solution method for a long and heavy-haul train negotiating an elastic curved track [J]. Journal of Southwest Jiaotong University, 2018, 53(1)：31-37.32西南交通大学学报第53卷results are o b ta in e d.A s the tra in negotiates a400 m rad iu s c u rv e,sim ultaneously bearing a2 200 kN cou pler fo rc e,the la te ra l and v e rtic a l w h e e l-ra il forces in the outer w heel o f the wagon are underestim ated b y about 24% and 4%re s p e c tiv e ly,re la tive to the results ca lcula ted using a tra in m odel composed o f several th re e-d im e n sio n a l wagons.C o rre sp o n d in g ly,the la te ra l and v e rtica l displacem ent o f outer ra il are seq u e n tia lly underestim ated b y n e arly20% and8%.W hen a single-m ass m odel is used at the ends o f the t r a in,and the nu m be r o f the a d d itio n a l wagons sim ulated by the th re edim en sion al dynam ic m odel is 3 ,both the w h e e l-ra il dynam ic in te ra ctio n o f the m id d le target wagon and the tra c k dynam ic b e h a v io u r,can be refle cte d a ccu ra te ly.Key words：h e avy-h aul t r a in;elastic c u rv e;cou pler com pressive fo rc e;v e h ic le-tra c k coupledd yn a m ic s;sim u la tio n solution重载铁路运营经验表明，轴重的增加和编组加长对重载列车运行安全性和轨道结构的正常服役性能带来了严峻考验，现场测得的重载列车压钩力甚至超过2 000 k N[1]，严重时还因压钩力的横向传递引发机车扩轨掉道事故[2]，文献[3]曾通过对加拿大铁路的现场调查，统计了 1999—2006年间重载列车脱轨事故的诱发因素，其中，轨道结构服役状态和列车操纵产生的车钩力是其中的重要因素.文献[4]指出车钩压力会在车钩摆动时影响轮轨横向作用力，引起钢轨侧翻或轨距扩大，进而引发列车脱轨掉道，列车运行过程中应限制钩横摆并加强轨道结构强度.尽管车钩力对轮轨动作用力影响明显，但由于长大列车的庞大自由度，动力学建模和求解成为难点.文献[5 ]为了解决长大重载列车的三维动力学同步仿真问题，采用了计算机多核并行计雛术进行求解，以减小计擁时提高计算效率.文献[6]建立了重载列车动力学模型，各节车均能详细考虑其悬挂特性、部件振动及钩缓系统作用力，为了提高效率，采用了循环变量法.文献[7-9 ]针对压钩力作用下重载机车的运行安全性进行了深人研究，其中连挂的机车以三维动力学模型模拟，货车采用单质点代替，计算结果较好反映了钩缓作用对轮轨动力作用的影响.文献[10-11]采用车辆-轨道耦合动力学模型，考虑了基础结构的弹性，通过在单节车辆模型上施加车钩横向力，分析了机车通过曲线轨道时的轮轨接触问题和轨距扩大问题.文献[2，12-14 ]系统研究了长大编组重载列车与轨道的动力相互作用原理，建立了重载列车-轨道三维耦合动力学仿真分析模型，采用了单质点、三维车辆模型的混合模式进行求解.综合来看，重载列车运行安全性相关研究是与其轴重大、编组长及基础结构可弹性变形的特点紧密关联的.但在列车的基础上再考虑连续长弹性轨道后，仿真更加困难，再从轨道结构动态特性的角度看，重载列车相关的简化及求解方法能否较好反映列车荷载作用下的轨距扩大、轨排横移等问题尚需深入讨论.相关现场试验和理论研究表明[1_2’4’7]，重载列车受拉时，拉钩力有迫使车钩趋于对中位置的倾向，车钩的横向失稳现象得到一定抑制，而压钩力则容易触发车钩横向失稳和剧烈的车钩横向力转移，属于极端恶劣的运行工况.为此，本文以重载铁路较为恶劣的小半径曲线线路运行条件为例，并以压钩力为列车外部作用载荷，分析了30 t轴重列车的轮轨动力作用，特别是从轨道结构受力和变形方面讨论了列车模型在轨道结构动态特性分析中的适用性，对大轴重列车的模型简化及求解方法给予进一步阐述.1重载列车曲线通过仿真分析方法列车通过小半径曲线时，由于线路走向的变化车体中心线与前、后车钩会形成夹角作用在车体上的压钩力会因车钩摆角的存在派生相应的车钩横向力^、心，对车体造成外挤效果，如图1 所示.图中^为曲线半径.轮对除了发挥曲线导向作用外，还需承担较大的附加作用力以平衡车钩横向力，因此轮轨动力作用将更加剧烈.图1压钩力作用下列车通过曲线状态Fig. 1 Train curving states under couplercompressive force理论上，对于上述问题，为了进行精确计算，长大编组重载列车及其轨下结构可同时细致考虑，但万吨以上的重载列车往往包含数百节车辆，轨道绵第1期刘鹏飞，等：长大列车通过弹性曲线轨道仿真求解方法研究33延数公里，由于系统自由度数目太过庞大且涉及诸多非线性因素，采用雖积分方法求解系统振动极为耗时，降低了仿真计算的时效性.文献[1，11]的研究结果表明，重载列车和轨道动态相互作用系统可简化为图2所示模型，承受车钩力较大的车辆及其邻近车辆采用三维车辆动力学模型，并考虑轨下基础结构振动，其余车辆用单质点模型模拟即可.但采用单质点车辆模型无法反映列车通过曲线时车辆姿态变化对车钩摆角的影响，而车钩摆角的大小和方向将酿影响压钩力向轮轨界面的传递效果.此外，从轨道结构受力情况看，采用多少节三维车辆模型才能准确反映轨道的动态特性也是必须回答的问题.图2重载列车-轨道耦合动力学模型Fig. 2 Dynamic model of heavy-haul train-track system为了研究压钩力作用下车辆三维动力学模型和单质点模型的选取问题，本文设置了图3所示的 4、B、C、D、£5种列车编组简化形式，对应的三维车辆模型数〜分别为1、1、3、5、7.图3中，s,、F Vl 分别为第i根轨枕距边界的距离和此处钢轨支反力.尾部车辆以速度r匀速运行，头部车辆上施加压钩力F即可实现车钩力在车间的传递.方案4 为对比方案，车辆以匀速运行而不承受车钩力力P 代表了常规的车辆惰行运行计算工况.方案S ~ £中，前、后端部车辆采用单质点模拟，仅考虑其纵向自由度，内部车辆以精确的三维动力学模型模拟，重点关注图3中间虚线框内的師车辆轮轨作用力及其下部轨道结构的动态性能.图3重载列车简化分析方案Fig. 3 Simplified analysis scheme of heavy-haul train对于轨道结构，实际是“无限长”的，在仿真中通常可将其用有限长轨道模型来模拟.文献[11]指出，为消除轨道约束边界的影响，考虑的轨道长度往往要超过车辆本身的长度，对于轨道定点激振分析，激振传麟响最为强烈的区域是激振点前后各三跨轨枕的范围.在本例中，不同编组列车均采用了 146 m轨道长度，对于编组最长的方案也能够保证列车最外端车轮距边界距离％内的轨枕数大于40根，完全能够满足消除边界条件的影响.由于轨道承受的是列车荷载，钢轨上作用有多个轮轨力，则钢轨的垂向、横向和扭转振动微分方程可依次改写为r i=14NcZ W')，k=1~ ，(1)J=1<(0 +£^(y)〜()=-Z F L/k(si) +4NcZ Q M s)，k=1〜n l，⑵J=1r7 ,i2ns <(0+六(子）九⑴=-名风i@k('i) + 4N cZ MW]©k(*…,.)，k = 1~N T，(3)j=1式⑴~⑶中：vk、<5^k和<?T k分另1J为钢轨垂向、横向和扭转振型正则坐标；、£r、A分别为钢轨长度、弹性模量、单位长质量;N s、N。

METHOD FOR MANAGING THE ROUTING OF A COMMUNICATION

专利名称：METHOD FOR MANAGING THE ROUTING OF A COMMUNICATION SENT TO A FIRSTCOMMUNICATION TERMINAL, METHODFOR ROUTING SAID COMMUNICATION ANDCORRESPONDING DEVICES发明人：AUFFRET, Jean Marc,LEMORDANT,Philippe,DE SNOECK, Xavier申请号：EP2020/063846申请日：20200518公开号：WO2020/234246A1公开日：20201126专利内容由知识产权出版社提供专利附图：摘要：The invention relates to a method for managing the routing of a communication between a caller communication terminal and a first called communication terminal to at least one second terminal, method for routing said communication and corresponding devices. Some communication services allow a telephone communication sent to one communication terminal to be transferred to another communication terminal. Such services are called call forwarding services. A user who wishes to take advantage of call forwarding must manually set up the forwarding of communications sent to his fixed station to a chosen communication terminal. The setup operations are performed directly on the fixed terminal and are tedious. There is therefore a risk of making a significant entry error. In addition, when the user wishes to stop the call forwarding, he must once again manually change the settings on his fixed station. In the solution according to the invention, call forwarding is triggered in compliance with triggering conditions. Contextual information collected by the user's communication terminal is used to determine compliance with the triggering conditions. Therefore, such a solution offers the user adaptability, flexibility and ease of use.申请人：ORANGE地址：78 rue Olivier de Serres 75015 PARIS FR 国籍：FR更多信息请下载全文后查看。

Systems and methods for inventorying un-provisione

专利名称：Systems and methods for inventorying un-provisioned systems in a softwareprovisioning environment发明人：Michael Paul DeHaan,Adrian KarstanLikins,Seth Kelby Vidal申请号：US12391588申请日：20090224公开号：US08402123B2公开日：20130319专利内容由知识产权出版社提供专利附图：摘要：A provisioning server can utilize an inventory tool on new target machines inorder to collect specification data from the target machines, prior to provisioning software. The inventory tool can be configured to operate on the new target machines without software, such as an operating system, being installed on the new target machines. The inventory tool can be configured to communicate with the hardware of the new target machines and collect data representing the specifications of the new target machines. The inventory tool can be configured to operate and execute on any new target machine regardless of the type and configuration of the new target machine.申请人：Michael Paul DeHaan,Adrian Karstan Likins,Seth Kelby Vidal地址：Morrisville NC US,Raleigh NC US,Raleigh NC US国籍：US,US,US代理机构：Lowenstein Sandler LLP更多信息请下载全文后查看。

SYSTEMS AND METHODS FOR MANAGING FRAUDULENT OPERAT

专利名称：SYSTEMS AND METHODS FOR MANAGING FRAUDULENT OPERATIONS IN A PLURALITYOF COMPUTING DEVICES发明人：Reza Farivar,Mark Watson,Anh Truong,Galen Rafferty,Vincent Pham,JeremyGoodsitt,Austin Walters申请号：US16778583申请日：20200131公开号：US20210241277A1公开日：20210805专利内容由知识产权出版社提供专利附图：摘要：In some embodiments, a method includes receiving operation data about operations performed by computing devices managed by an entity. The operation data is stored in respective data entries of a log data storage on a server managed by an authorizing entity. A set of agents are identified from the entity associated with fraudulent operations in entries of the log data storage having positive fraud indications.A number of instances for each identified agent in the set associated with fraudulent operations are determined. A score is assigned to each agent in the set based on the number of instances that each agent was associated with fraudulent operations. An alert to an administering computing device associated with the entity is generated when the assigned score of at least one agent is greater than a predefined threshold.申请人：Capital One Services, LLC地址：McLean VA US国籍：US更多信息请下载全文后查看。

METHOD AND SYSTEM FOR DETERMINING THE AMOUNT OF LI

专利名称：METHOD AND SYSTEM FOR DETERMINING THE AMOUNT OF LIQUID IN A VESSEL FORINVENTORY TAKING发明人：AASLAID, Andrus,LAUBRE, Lauri申请号：IB2011/055107申请日：20111115公开号：WO2012/066482A1公开日：20120524专利内容由知识产权出版社提供专利附图：摘要：Method and system for determining the amount of liquid in a vessel for inventory taking. The amount of liquid is determined from the total volume of the vesseland the volume of the empty space in the vessel. An reference vessel is hermetically connected with the vessel. The volume of the empty space is determined from the difference in air pressure before and after air is forced into the vessel. The system comprises a device for measuring a volume of a vessel, the device comprising a reference vessel with known pressure, a connector for connecting the reference vessel with the vessel and a pressure sensor for measuring pressure in connected vessels. The system comprises a measuring head, comprising a reader for reading the RFID tag on the vessel: The system has a communication module for forwarding the measurement data to the cash register and warehouse management software.申请人：LM DEVELOPMENTS OÜ,AASLAID, Andrus,LAUBRE, Lauri地址：Kentmanni 15-6 EE-10116 Tallinn EE,Adamsoni 5-9 EE-10137 TallinnEE,Kentmanni Street 15-6 EE-10116 Tallinn EE国籍：EE,EE,EE代理人：KOPPEL, Mart Enn更多信息请下载全文后查看。

内河水质检测船的初步设计

国外的水质监测系统已经发展多年，美国，英国,日本等已经有广泛的应用。早在1959年，美国已经开始进行水质自动监测，到20世纪70年代，水质自动监测系统已经开始成熟。在1970年左右，美国等国就能对地表水进行自动监测。其中日本的FRP制公害监测艇附件有容积为300升的油中和剂贮藏器进行监制，当海面有废油出现可及时处理[4].这条船船长13公尺，船宽1。7公尺，吃水0。64公尺，排水量1。1吨，速度每小时19海里。美国研究的用于水下监测的水下自主航行器，代号“Starbug”的自主式水下航行器，是一种智能化的集成系统.它可以自主航行，在军事等方面发挥重要的作用。英国研制的无人监测船为双体船,设备先进，可以用GPS进行定位，并进行水质监测。它的动力系统有两种，一是用电池作为动力，而是燃烧柴油转化为动力。由此可见，国外的水质监测发展已逐步成熟，并且对水质监测船的研究也较为深入.
（1）选取较大的船宽，有利于提高稳性。
（2）B/T的增大,有利于Cb的减小，不但使阻力的有所减小，而且有利于减轻结构重量和降低造价。
（3)船宽B的大小与船舱内布置有关,选取船宽时要考虑舱内所需的设备布置等情况。
（4)选取船宽时还要考虑到河道，船坞，闸门的限制情况。
2
吃水T的确定要考虑以下因素：
(1)较小的吃水缩小了航道和港口的水深限制；
(3)The ship is designed according to the requirements to the Class C area passenger ship from steel construction rules drafted by China classification society in 2014 and the other relevant provisions of the inland river ships.

企业存货管理方法研究范本

目录中英文摘要第一章绪论-------------------------------------------------------------------1 1．1研究的背景及意义------------------------------------------------------11 .2文献综述--------------------------------------------------------------11．3论文的创新与不足------------------------------------------------------2 第二章存货管理的理论概述-----------------------------------------------------3 2．1存货管理的基本概述----------------------------------------------------3 2．2存货管理的方法--------------------------------------------------------3 第三章大型零售连锁企业存货管理的相关情况分析---------------------------------53. 1大型零售连锁的基本概述-----------------------------------------------53．2存货管理对大型连锁超市的重要性---------------------------------------7 3．2.1选择发出商品计价方法对大型连锁超市的影响-------------------------73.2. 2大型连锁超市存货盘点方法的选择及其重要性-------------------------8 第四章案例分析--------------------------------------------------------------9 4．1沃尔玛超市基本情况简述-----------------------------------------------9 4．2沃尔超市存货分析-----------------------------------------------------94.2. 1沃尔玛超市财务报表分析-------------------------------------------94．2.2沃尔玛超市存货周转率分析----------------------------------------13 4．2.3沃尔玛超市内部存货管理方法--------------------------------------14 4．3不同超市存货管理方法的比较-------------------------------------------154.3.1北京联华综合超市、家乐福超市的简介-------------------------------154.3.2不同超市2011年财务报表分析--------------------------------------164．4沃尔玛超市存货管理存在的问题-----------------------------------------17 第五章大型零售连锁企业存货管理方法中存在的不足和建议------------------------195 .1大型零售连锁企业存货管理方法中存在的不足---------------------------195．2大型零售连锁企业内部存货管理的建议----------------------------------20 5．3给投资者的建议------------------------------------------------------21 第六章-----------------------------------------------------------------------22 参考文献---------------------------------------------------------------------23 附录-------------------------------------------------------------------------24 致谢-------------------------------------------------------------------------26摘要存货是零售类企业最重要的资产。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

A heuristic method for the inventory routing problem with time windowsShu-Chu Liu ⇑,Wei-Ting LeeDepartment of Management Information Systems,National Pingtung University of Science and Technology,Pingtung 912,Taiwana r t i c l e i n f o Keywords:Vehicle routing problem with time windows (VRPTW)Inventory routing problem with time windows (IRPTW)Heuristic method NP-hardVariable neighborhood tabu search (VNTS)a b s t r a c tThis paper is to resolve the VRPTW and the inventory control decision problem simultaneously since both the vehicle routing decision with time windows and the inventory control decision affect each other and must be considered together.A mathematical model of inventory routing problem with time windows (IRPTW)is proposed.Since ﬁnding the optimal solution(s)for IRPTW is a NP-hard problem,this paper proposes a two-phase heuristic method.The ﬁrst phase is to ﬁnd the initial solution.The second phase is to improve the solution adopting the variable neighborhood tabu search (VNTS)selecting better neigh-borhood solutions,to obtain the optimal solution.Moreover,the proposed method was compared with three other heuristic methods.The experimental results indicate that the proposed method is better than the three other methods in terms of average supply chain cost (transportation cost,time window viola-tion penalty cost and inventory cost).Ó2011Elsevier Ltd.All rights reserved.1.IntroductionThe vehicle routing problem with time windows (VRPTW),which involves assigning a ﬂeet of limited capacity vehicles to serve a set of customers without violating the capacity and time constraints,has been increasingly discussed in practice recently (Calvete,Gale,Oliveros,&Sanchez-Valverde,2007;Kang,Lee,Lee,&Lee,2008;Li,Tien,&Leung,2010).The VRPTW can be di-vided into two categories (Solomon,1987):(1)soft-time windows and (2)hard-time windows.Since the time constraint of the vehi-cle routing problem with soft-time windows can ﬂexibly reﬂect the real-world situation such as transportation problem (i.e.rush hour,car accident,shortage of drivers and vehicles),customers’demand constraints,etc.,it is adopted more often than that of the vehicle routing problem with hard-time windows (Cheng &Wang,2009;Hashimoto,Yagiura,&Ibaraki,2008;Nagata &Braysy,2009).Hence,it is adopted and discussed later in this paper.In the past,the resolution of VRPTW is based on the minimal cost criterion (including transportation cost and time window vio-lation penalty cost)without considering the inventory cost (Figliozzi,2010;Hashimoto et al.,2008).However,when only one factor is accessed and minimized,the costs of other factors are increased (Moin &Salhi,2007).For example,if the vehicle rout-ing decision problem with time windows for suppliers is consid-ered and the inventory control decision problem for retailers is neglected,the vehicle routing decision can be effectively madeand however the inventory control decision cannot,causing sup-pliers not be able to reduce the supply chain cost (transportation cost,time window violation penalty cost and inventory cost)effec-tively.In contrast,if only the inventory control decision problem for retailers is considered and the vehicle routing decision problem with time windows for suppliers is ignored,the transportation and time window violation penalty cost would increase since the vehi-cle routing decision cannot be effectively made.Hence,the supply chain cost would increase.So the vehicle routing decision problem with time windows for suppliers and the inventory control deci-sion problem for retailers need to be considered simultaneously in order to ﬁnd the minimal supply chain cost.The inventory rout-ing problem with time windows (IRPTW)considering inventory and routing decisions simultaneously has gained attentions since the vendor-managed inventory (VMI)strategy was adopted in practice (The applications of IRPTW include household care indus-try (P&G),computer equipment industry (Acer),beverage industry (Hey Song Corporation,the most famous beverage production company in Taiwan),etc.Actually,a lot of applications worldwide in different supply chains including production companies and retailers have already been reported.)(Andersson,Hoff,Christian-sen,Hasle,&Lokketangen,2010;Moin &Salhi,2007).However,the IRPTW is rarely discussed in literature.Hence,the purpose of this paper is to resolve the inventory routing problem with time windows (IRPTW)so that the supply chain cost can be decreased.Since IRPTW is a NP-hard problem (VRPTW is an NP-hard prob-lem (Savelsbergh,1985).IRPTW is more complex than VRPTW.Hence,IRPTW is also an NP-hard problem),a heuristic method will be adopted to resolve this problem.However,there is no heuristic method proposed for the IRPTW until now.Since the problem0957-4174/$-see front matter Ó2011Elsevier Ltd.All rights reserved.Corresponding author.Tel.:+8860877032027911;fax:+886087740306.E-mail address:sliu@.tw (S.-C.Liu).structure of IRPTW is similar to that of VRPTW.Hence,heuristic methods used in VRPTW are reviewed in the following.Taillard, Badeau,Gendreau,Guertin,and Potvin(1997)adopted a two-phase method to resolve the VRPTW.The random insertion method is used to obtain the initial solution.Then the tabu search method,adopting cross-exchange and or-opt neighborhood search strategies,is used to improve the initial solution.Park(2001) adopted a hybrid genetic algorithm to resolve the VRPTW.Tan, Lee,Zhu,and Ou(2001)investigated three methods:tabu search (TS),simulated annealing(SA)and genetic algorithm(GA)to re-solve the VRPTW.The results show TS is better than SA and GA. Ioannou,Kritikos,and Prastacos(2003)adopted the nearest-neighbor method to resolve the VRPTW.Ichoua,Gendreau,and Potvin(2003)adopted a parallel tabu search method to resolve the VRPTW.The insertion method is used to obtain the initial solu-tion.The cross-exchange neighborhood search strategy is used to improve the initial u,Sim,and Teo(2003)adopted a two-phase method to resolve the VRPTW.Theﬁrst phase is toﬁnd the initial solution.The second phase is to improve the initial solution using the tabu search method adopting the k-opt neighborhood search strategy.Olli(2003)adopted a variable neighborhood search method to resolve the VRPTW.Chen,Hsueh, and Chang(2006)adopted the insertion method to resolve the real-time VRPTW.Hashimoto et al.(2008)proposed a heuristic method adopting time dependent concept to resolve the VRPTW. 2-opt⁄,cross-exchange and or-opt neighborhood search strategies are adopted in this method.Kang et al.(2008)adopted the tabu search method to resolve the VRPTW.The neighborhood search strategy is based on the route-perturb and route-improvement method.Fu,Eglese,and Li(2008)adopted the tabu search method to resolve the VRPTW.Figliozzi(2010)adopted an iterative route construction and improvement algorithm to resolve the VRPTW. Nagata and Braysy(2009)adopted a heuristic,insertion and guided local search strategies,to resolve the VRPTW.Li et al.(2010) adopted the tabu search to resolve the VRPTW with stochastic tra-vel and service times.Cheng and Wang(2009)adopted a decompo-sition technique and a genetic algorithm to resolve the VRPTW.In this paper,a mathematical model for the IRPTW is proposed. Sinceﬁnding the optimal solution for this model is an NP problem, a two-phase heuristic method is proposed for the IRPTW.Theﬁrst phase is toﬁnd a better initial solution based on the construction approach.The second phase is to improve the initial solution based on the variable neighborhood tabu search(VNTS),tabu search adopting different neighborhood search strategies,toﬁnd the opti-mal solution for the IRPTW effectively and efﬁciently.The reason for choosing the hybrid heuristic approach is to overcome the drawbacks of variable neighborhood search(VNS)and tabu search (TS),two well-known methods to resolve the IRPTW.In addition, the hybrid method can take advantage of VNS and TS to improve search effectiveness and efﬁciency.2.Model formulation for the inventory routing problem with time windows2.1.Assumptions and notations2.1.1.AssumptionsAccording to the literature survey,there are no other researches available for the IRPTW.Hence,the used assumptions in this paper are selected from two related research areas:the vehicle routing problem with time windows(Cheng&Wang,2009;Hashimoto et al.,2008)and the inventory control problem(Andersson et al., 2010;Moin&Salhi,2007;Tersine,1994).The details are asOnly one supplier is consideredEach retailer is served by exactly one vehicleEach route is served by one vehicleThe order quantity for each route is less than or equal to the vehicle capacityThe arriving time for each retailer or the supplier is less than or equal to the working time per dayEach route begins and ends at the same supplierThe soft-time windows are consideredA single type of products is consideredA homogenousﬂeet of vehicles is consideredThe continuous review EOQ model is usedThe order cycle time for each route is an integerNumber of retailers is knownRetailers’demands are stochasticVehicle capacity is knownVehicle dispatching cost/each time is knownTraveling cost/unit distance is knownOrdering cost/each order is knownHolding cost/unit product/unit time is knownShortage cost/each time is knownService time for each retailer is knownVehicle speed is knownProbability density function for retailer’s demand of each route during lead time is known2.1.2.NotationsKnown variablesd i daily demand for retailer iN number of retailersT planning periodb vehicle capacityc Vehicle dispatching cost/each timecm traveling cost/unit distanceA ordering cost/each orderH+holding cost/unit product/planning periodhs shortage cost/each timeST g service time of gspeed speed of vehicleP e waiting penalty cost/unit timeP l lateness penalty cost/unit timee i earliest arrival time of the time window for retailer i l i latest arrival time of the time window for retailer i WT working time per dayAuxiliary variablesi index of retailers(16i6N)g index of retailers(16g6N)or supplier(g=N+1)h index of retailers(16h6N)or supplier(h=N+1)k index of vehicles or routes(16k6K)R k order-up-to level for the replenishment of route kB(R k)expected shortage number for route k during an order cycle timef L(x)probability density function for retailer’s demand,x,ofeach route during lead time Lt i arrival time for retailer iP i(t i)time window violation penalty cost for retailer iV k retailer set for route k(16k6K)Dis kgh distance from g to h for route k13224S.-C.Liu,W.-T.Lee/Expert Systems with Applications38(2011)13223–13231Q k order quantity for route k during an order cycle time CT kUL k average demand for route k during lead time D k total demand for route k in a planning period T X 1neighborhood solutionC (X 1)category which the neighborhood solution X 1belongs toDecision variables CT korder cycle time for route k K number of vehicles (or routes)x kgh1,if g immediately precedes h on route k ;0,otherwise2.2.Model formulationBefore the model for the IRPTW is formulated,the relevant transportation cost,time window violation penalty cost and inven-tory cost are discussed ﬁrst.1.Transportation cost:The transportation cost includes the vehi-cle dispatching cost,c ,plus the traveling cost among retailers multiplying the average times of transportation in a planning period.The detailed computation of the total transportationcost is as follows:P K k ¼1P N þ1g ¼1P N þ1h ¼1ðc þcm ÂDis kgh ÞÂD kQ k .2.Time window violation penalty cost:The time window viola-tion penalty cost for each retailer i is P i (t i ).Hence,the detailed computation of the total penalty cost is as follows:P K k ¼1P i 2V k P i ðt i ÞÂD kk .3.Inventory cost:The inventory cost includes ordering cost,hold-ing cost and shortage cost.The detailed computation for thesecosts is as follows:(1)the total ordering cost ¼P K k ¼1A ÂD kk ,(2)the total holding cost ¼P Kk ¼1H þÂðQ kþR k ÀUL k Þ,(3)the total shortage cost ¼P K k ¼1hs ÂB ðR k ÞÂD kQ k.As for the computa-tion of B (R k ),please refer to Eq.(A1)in Appendix A .After the related costs are discussed,the model for the IRPTW is as follows:Min X K k ¼1X N þ1g ¼1X N þ1h ¼1ðc þcm ÂDis kgh ÞÂD k Q k Âx kgh þX i 2V k P i ðt i ÞÂD kQ k"þA ÂD k k þH þÂQ k þR k ÀUL k Þþhs ÂB ðR k ÂD kks.t.Q k 6b ;k ¼1;...;K ;ð1ÞX K k ¼1X N þ1h ¼1x kgh ¼1;g ¼1;...;N þ1;ð2ÞX N þ1g ¼1x kgh ÀX N þ1g ¼1x khg ¼0;k ¼1;...;K ;h ¼1;...;N þ1;ð3ÞX g 2V k X h 2V k X K k ¼1x kgh P 1;8ðV k ;k ;ð4ÞP i ðt i Þ¼P e ðe i Àt i Þ;if t i <e i 0;if e i 6t i 6l i P l ðt i Àl i Þ;if l i <t i8><>:9>=>;;i ¼1;...;N ;ð5Þt h 2V k ¼ðt g 2V k þST g þDis kgh =speed k ÞÂx kgh ;k ¼1;...;K ;g ¼1;...;N þ1;h ¼1;...;N þ1;ð6Þt h 2V k 6WT ;k ¼1;...;K ;ð7ÞQ k k ¼D k ¼Xi 2Vkd i ;k ¼1;...;K ;ð8ÞCT k 2positive integer ;k ¼1;...;K ;ð9Þx kgh 2f 0;1g ;k ¼1;...;K ;g ¼1;...;N þ1;h ¼1;...;N þ1:ð10ÞThe goal of the objective function is to make the supply chain cost minimum.The constraint (1)indicates that the order quantity for route k must be less than or equal to vehicle capacity.The con-straint (2)indicates there is only one vehicle in each route.The con-straint (3)indicates the vehicle k reaches the retailer and leaves from there.The constraint (4)indicates every route is started from the supplier and ended at the same supplier.The constraint (5)indi-cates the time window violation penalty cost occurs when the arri-val time t i for retailer i is earlier than e i or later than l i .The constraint (6)indicates the arrival time for h is equal to the arrival time for g ,plus the service time for g ,and plus the traveling time from g to h for route k .The constraint (7)indicates the arrival time for h in route k must be less than or equal to the working time per day.The constraint (8)indicates the order quantity divided by the order cycle time is equal to the daily demand for route k .The con-straint (9)indicates the order cycle time must be a positive integer.The constraint (10)indicates the routing decision variable is in the term of 0or 1.3.The proposed method for the inventory routing problem with time windowsTo resolve the IRPTW,two major decisions,vehicle routing and order cycle time for each route,need to be made.After both vehicle routing and order cycle time are determined,the order quantity can be obtained based on the procedure in Appendix A and the supply chain cost can be calculated based on the objective function in Section 2.2.Since the IRPTW is a NP-hard problem,this paper proposes a two-phase heuristic method.The ﬁrst phase is to ﬁnd the initial solution.The second phase is to improve the initial solu-tion based on the vehicle routing improvement procedure and the inventory control improvement procedure randomly adopting the VNTS selecting better neighborhood solutions,tabu search with insertion and exchange neighborhood search strategies.The neigh-borhood search strategies for the vehicle routing improvement procedure are explained as follows:(1)the insertion strategy is to select m retailers from different routes (m is randomly selected from U [1,d N =4e ].N is the number of retailers),delete the selected retailers from the original routes,and insert them into the other routes or the same route based on the minimal cost criterion,(2)the exchange strategy is to select w retailers (w is randomly se-lected from U [1;d N =4e ].N is the number of retailers),exchange two retailers,one from the selected group and the other from the unselected retailers,based on the minimal cost criterion and exe-cute the procedure until all the w selected retailers are exchanged with the unselected retailers.The neighborhood search strategies for the inventory control improvement procedure are explained as follows:(1)the insertion strategy is to select m routes (m is ran-domly selected from U [1;d K =4e ].K is the number of routes.),ran-domly select an integer z k (can be negetive or positive)from U [À2,2]for each selected route k and add z k to the order cycle time S.-C.Liu,W.-T.Lee /Expert Systems with Applications 38(2011)13223–1323113225w routes (w is randomly selected from U [1;d K =4e ].K is the number of routes.),exchange the order cycle times of two routes,one from the selected routes and the other from the unselected routes,based on the minimal cost criterion and execute the procedure until the order cycle times of the w selected routes are exchanged with those of unselected routes.For obtaining better solutions,the neighborhood solution X 1is classiﬁed into eight possible catego-ries:(1)inventory cost decreases,transportation cost decreases,time window violation penalty cost decreases (C (X 1)=1),(2)inventory cost decreases,transportation cost decreases,time win-dow violation penalty cost increases (C (X 1)=2),(3)inventory cost decreases,transportation cost increases,time window violation penalty cost decreases (C (X 1)=3),(4)inventory cost increases,transportation cost decreases,time window violation penalty cost decreases (C (X 1)=4),(5)inventory cost decreases,transportation cost increases,time window violation penalty cost increases (C (X 1)=5),(6)inventory cost increases,transportation cost de-creases,time window violation penalty cost increases (C (X 1)=6),(7)inventory cost increases,transportation cost increases,time window violation penalty cost decreases (C (X 1)=7),(8)inventory cost increases,transportation cost increases,time window viola-tion penalty cost increases (C (X 1)=8).The search order is based on the related information of current optimal solution.If the inven-tory cost of current optimal solution is greater than the transporta-tion cost of current optimal solution and the transportation cost of current optimal solution is greater than the time window violation penalty cost of current optimal solution,the search order is the same as that mentioned above.If the transportation cost of current optimal solution is greater than the inventory cost of current opti-mal solution and the inventory cost of current optimal solutionis13226S.-C.Liu,W.-T.Lee /Expert Systems with Applications 38(2011)13223–13231greater than the time window violation penalty cost of current optimal solution,the search order is as follows:(1)transportation cost decreases,inventory cost decreases,time window violation penalty cost decreases(C(X1)=1),(2)transportation cost de-creases,inventory cost decreases,time window violation penalty cost increases(C(X1)=2),(3)transportation cost decreases,inven-tory cost increases,time window violation penalty cost decreases (C(X1)=3),(4)transportation cost increases,inventory cost de-creases,time window violation penalty cost decreases(C(X1)=4),(5)transportation cost decreases,inventory cost increases,time window violation penalty cost increases(C(X1)=5),(6)transporta-tion cost increases,inventory cost decreases,time window viola-tion penalty cost increases(C(X1)=6),(7)transportation cost increases,inventory cost increases,time window violation penalty cost decreases(C(X1)=7),(8)transportation cost increases,inven-tory cost increases,time window violation penalty cost increases (C(X1)=8)(The VNTS searches for the neighborhood solution in theﬁrst categoryﬁrst.If it cannot be found,the VNTS searchesS.-C.Liu,W.-T.Lee/Expert Systems with Applications38(2011)13223–1323113227for the neighborhood solution in the second category.If it still can-not be found,the VNTS searches for the neighborhood solution in the third category.In the same manner,a better neighborhood solution will be found at last.).3.1.The proposed method for the IRPTWThe proposed two-phase method in this paper is to obtain the initial solutionﬁrst(phase1).Then the TS method adopting differ-ent neighborhood search strategies is applied to improve the initial solution(phase2).The detailed procedure is as follows:Phase1:Obtain the initial solution(Fig.1).Step1:(1)Set k=1.(2)Put all retailers into F.Step2:(1)Select one retailer from F randomly.(2)Put this retai-ler into V k.(3)Delete the retailer from F.Step3:Select the candidate retailer from F with the minimal angle C s(Each retailer is located with an angle C s fromthe horizontal axis originating from the supplier to theright.).Step4:Is the vehicle capacity constraint or working time con-straint violated(Please refer to the constraints(1)and(7)in the proposed model in Section2.2.)?If yes,(1)set k=k+1,(2)put the candidate retailer into V k,(3)delete the candidate retailer from F,(4)go to Step5;otherwise,(1)put the candidate retailer into V k,(2)deletethe candidate retailer from F,(3)go to Step5.Step5:Is F empty?If yes,(1)set K=k,(2)set k=1,(2)go to Step6;otherwise,go to Step3.Step6:Calculate the order cycle time for route k.(1)Calculate Q k for route k(Please refer to Appendix A for thedetails).(2)Calculate the order cycle time CT k¼Q kPi2V k d i .Step7:Translate the order cycle time CT k into an integer.(1)If CT k is an integer,go to Step8;otherwise,go to(2).(2)Calculate the cost TSC(V k,b CT k c)and TSC(V k,b CT k c+1)forroute k(According to the solution(V k,b CT k c),the orderquantity Q k for route k can be obtained based on the pro-cedure in Appendix A and the cost TSC(V k,b CT k c)for routek can be calculated based on the objective function inSection2.2.).(3)If TSCðV k;b CT k cÞ>TSCðV k;b CT k cþ1Þ;CT k¼b CT k cþ1;otherwise,CT k¼b CT k c.Step8:Is k=K?If yes,(1)set the solution{(V t,CT t)for t=1,...,K} as the initial solution X0and the temporary best solutionX⁄,(2)stops;otherwise,(1)set k=k+1,(2)go to Step6. Phase2:Improve the initial solution(Fig.2).Step9:Initialize max_step(maximal recursive number of itera-tion),step,max_category_step(maximal recursive num-ber of iteration for each category),category_step,cur_category(current category),tabu_size(size of thetabu list).Step10:Generate a candidate move(from the original solution X0 to the neighborhood solution X1)using one of the follow-ing neighborhood search strategies:(1)vehicle routingimprovement procedure:1.insertion,2.exchange,(2)inventory control improvement procedure:1.insertion,2.exchange.Step11:Is C(X1)6cur_category?If yes,(1)set cur_category=1,(2)set category_step=0,(3)go to Step13;otherwise,goto Step12.Step12:Is category_step6max_category_step?If yes,(1)set cat-egory_step=category_step+1,(2)generate a candidatemove using the last selected neighborhood search strat-egy,(3)go to Step11;otherwise,(1)set cur_category=cur_category+1,(2)set category_step=0,(3)generatea candidate move using the last selected neighborhoodsearch strategy,(4)go to Step11.Step13:Is the vehicle capacity constraint or working time con-straint violated(Check whether the vehicle capacity con-straint or working time constraint is violated.Please referto the constraints(1)and(7)in the proposed model inSection 2.2.)?If yes,go to Step10;otherwise,go toStep14.Step14:Is the move in the tabu list?If yes,go to Step10;other-wise,(1)set X0=X1,TSC(X0)=TS C(X1),(2)update thetabu list.Step15:Is TS C(X1)<TSC(X⁄)?If yes,(1)set X⁄=X1, TSC(X⁄)=TS C(X1),(2)generate a candidate move usingthe last selected neighborhood search strategy,(4)go toStep11;otherwise,go to Step16.Step16:Step16:Is step=max_step?If yes,the best solution is obtained and stops;otherwise,(1)set step=step+1(2)go to Step10.3.2.The parameter setting for the proposed methodThere are three parameters used in the proposed method: max_step,max_category_step and tabu_size.The values of max_-step,max_category_step and tabu_size are experimentally deter-mined based on the minimal cost criterion.max_step is tried from10to200(10,50,100,200)when the size of retailer number is small(retailer number N=4,8)and from1000to3000 (1000,2000,3000)when the size of retailer number is large(retai-ler number N=100).max_category_step is tried from5to30 (5,10,20,30).tabu_size is tried from4to13(4,7,10,13).putational results and comparisonsComputational experiments were conducted to examine the computation effectiveness and efﬁciency of the proposed heuristic method(H1),based on the minimal cost criterion(transportation cost,time window violation penalty cost and inventory cost),by comparing it with three other heuristic methods:H2(the heuristic method based on the minimal transportation cost and time win-dow violation penalty cost criterion and proposed by Hashimoto et al.(2008),please refer to Appendix B),H3(the heuristic method based on the minimal transportation cost and time window viola-tion penalty cost criterion and proposed by Cheng and Wang (2009),please refer to Appendix C),and H4(The heuristic method is the same as H1.However,the neighborhood solution is gener-ated randomly.Steps11and12in H1are ignored.The other steps are the same as those in H1.).All heuristic methods were coded in C++and run on a Core(TM)2Duo3G CPU with2GB RAM.Table1The parameters and their values in small-sized problems.Parameters ValuesDemands during lead time for each retailer U[0,10] Vehicle capacity1000Vehicle dispatching cost/each time25Traveling cost/unit distance1Ordering cost/each time15Holding cost/unit product planning period1Shortage cost/each time2Waiting penalty cost/unit time(min)for each retailer0.01Lateness penalty cost/unit time(min)for each retailer0.02Planning period100Speed of vehicle(unit distance/per h)50Working time per day(h)8Service time for each retailer(min)1013228S.-C.Liu,W.-T.Lee/Expert Systems with Applications38(2011)13223–13231For evaluating the proposed heuristic H1,the test problems are divided into two categories:small and larger size.For small-sized problems with up to8retailers,the solutions for H1,H2,H3and H4are compared to the optimal solution yielded by enumeration search.A set of78tests classiﬁed in two different problem sizes (4retailers(theﬁrst4retailers selected from the100-retailer data set in Solomon(1987)),8retailers(theﬁrst8retailers selected from the100-retailer data set in Solomon(1987)))was designed to evaluate the performance of the proposed heuristic solutions versus the optimal solutions.Each problem size instance contained 39tests selected from Solomon(1987),classiﬁed into four types: R1(12tests),R2(11tests),RC1(8tests),RC2(8tests).According to the literature(Soloman,1987;Herer&Levy,1997),the detailed settings for each test problem are as follows(Table1).As for the parameter setting of the proposed method H1,the values are determined based on the minimal cost criterion after experiments as follows(Please refer to Section3.2for the details.): max_step=200,max_category_step=10,tabu_size=7.Table2 shows the average solution quality and average CPU times for H1,H2,H3,H4and optimal solutions.It can be seen that the solu-tions of H1are the same as those optimal solutions and much bet-ter than those of H2,H3and H4in terms of average cost in different small-sized problem types.When the number of retailers increases,the average costs for the optimal solution,H1,H2,H3 and H4increase because the cost increases to meet the demands of increased retailers.The average CPU times are less than or equal to0.02seconds for H1,H2,H3and H4.However,the maximal aver-age CPU time for obtaining optimal solutions is around13.5s.The larger the problem size,the larger the computational time for obtaining optimal solutions.The heuristic methods are more efﬁ-cient than the optimal procedure.For larger-sized problems,the optimal solutions cannot be ob-tained in a reasonable time and there is no tight lower bound for this problem.The performance of the proposed heuristic method H1is evaluated against the solutions of H2,H3and H4.A set of 39tests(for100retailers)classiﬁed into four problem types,R1, R2,RC1and RC2,in Solomon(1987)was designed to evaluate the performance of the heuristic solutions.The detailed settings are the same as those in small-sized problems.The values of parameters for the proposed method H1are determined based on the minimal cost criterion after experiments as follows(Please refer to Section3.2for the details.):max_step=3000,max_cate-gory_step=30,tabu_size=10.Table3shows the solution quality and CPU times for H1,H2,H3,and H4for larger-sized problems. It is found that H1is much better than H2,H3and H4in terms of average cost in all problem instances.Because H1adopts the VNTS selecting better neighborhood solutions based on the mini-mal cost criterion,it can acquire the lowest cost effectively.As for the computational time,all the CPU times of H1,H2,H3and H4are less than11seconds and the four methods are efﬁcient.Table2Results for small-sized problems.Retailer number InstancetypeOptimal solution H1H2H3H4AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)4R1617.390.01617.390.014073.480.014073.480.01617.390.01 4R2617.390.01617.390.014044.970.014044.970.01617.390.01 4RC1848.650.01848.650.013760.340.013760.340.01848.650.01 4RC2848.650.01848.650.013839.650.013759.650.01848.650.01 8R11014.9113.481014.910.025894.380.025950.980.021017.490.02 8R21001.3713.461001.370.026023.140.025947.140.021004.780.02 8RC11221.4112.271221.410.024214.300.024214.300.021223.910.02 8RC21218.8112.311218.810.024306.540.024191.540.021221.570.02Table3Results for larger-sized problem.Retailer number InstancetypeH1H2H3H4AveragecostAverage CPU time(s)AveragecostAverage CPU time(s)AveragecostAverage CPU time(s)AveragecostAverage CPU time(s)100R111561.339.1642816.25 3.2634818.20 6.9112021.527.34 100R211309.279.2236746.45 3.0729487.52 6.8011851.777.53 100RC114297.2510.7552778.85 4.4441924.367.9615078.978.73 100RC214185.4510.4648081.35 4.2738432.638.2414912.028.63Table4The differences among methods under different ratios for cost structure.Instance type Ratio for coststructureH1H2H3H4AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)AveragecostAverage CPUtime(s)R1Low11561.339.1642816.25 3.2634818.20 6.9112021.527.34R1High14279.838.3443874.88 4.0237330.097.0617649.737.01R2Low11309.279.2236746.45 3.0729487.52 6.8011851.777.53R2High14115.077.5836867.96 4.0831351.187.0717361.497.13RC1Low14297.2510.7552778.85 4.4441924.367.9615078.978.73RC1High17199.568.8753741.36 4.4542278.208.1920356.637.52RC2Low14185.4510.4648081.35 4.2738432.638.2414912.028.63S.-C.Liu,W.-T.Lee/Expert Systems with Applications38(2011)13223–1323113229。