Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 537675 7 pageshttpdxdoiorg1011552013537675

Research ArticleAn Inventory Controlled Supply Chain ModelBased on Improved BP Neural Network

Wei He

Research Center of Cluster and Enterprise Development School of Business AdministrationJiangxi University of Finance amp Economics Nanchang 330013 China

Correspondence should be addressed to Wei He 04hrirene163com

Received 22 June 2013 Revised 8 September 2013 Accepted 10 October 2013

Academic Editor Zhigang Jiang

Copyright copy 2013 Wei He This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Inventory control is a key factor for reducing supply chain cost and increasing customer satisfaction However prediction ofinventory level is a challenging task for managers As one of the widely used techniques for inventory control standard BP neuralnetwork has such problems as low convergence rate and poor prediction accuracy Aiming at these problems a new fast convergentBP neural network model for predicting inventory level is developed in this paper By adding an error offset this paper deducesthe new chain propagation rule and the new weight formula This paper also applies the improved BP neural network model topredict the inventory level of an automotive parts company The results show that the improved algorithm not only significantlyexceeds the standard algorithm but also outperforms some other improved BP algorithms both on convergence rate and predictionaccuracy

1 Introduction

Inventory control is one of the key topics for supply chainmanagement Usually inventory takes the form of raw mate-rial work in process (WIP) products semifinished productsor finished products Inventory cost is the main cost forsupply chain management A drop of just several percentagepoints of inventory cost can greatly increase the profits ofthe whole supply chain In addition sound inventory levelcan prevent shortage of material maintain the continuityof the production process and quickly satisfy customersrsquodemand Thereby exploring the optimal inventory level isvery necessary and valuable for supply chain management

To date the following inventory control problems need tobe addressed [1 2]

(1) There are highly nonlinear models which are hard toprocess

(2) There are qualitative indicators which are hard to dealwith

(3) The unchangeable indicators of inventory control lackself-adaptation

(4) Information of inventory control models is alwaysindirect and the collection of information is time-consuming and of low efficiency

(5) Inventory control models always ignore the influenceof uncertain factors such as lead time transportationconditions and change of demand

Considering the above problems traditional inventorycontrol theory is hard to meet the requirement posed bythe new environment Thanks to the uncertain feature ofinventory control and the strengths of neural network inmodel prediction this paper chooses to use BP neuralnetwork to establish inventory model and predict inventorylevel

BP neural network is a kind of nonlinear feed forwardnetwork which has good nonlinear mapping ability Theorieshave proved that BP network can approach any nonlinearmapping relationship given enough input and hidden layerswhile there is no necessity to establish a mathematical modelFurthermore by learning and training BP network can storeinformation systematically in weight matrix119882 In doing soit indicates that BP network canmemorize the characteristicsof inventory information and at the same time can adapt to

2 Discrete Dynamics in Nature and Society

the changes of inventory environment In view of the featuresof BP neural network it has great advantages in classificationand prediction

However it is acknowledged that BP neural networkalso has such problems as slow convergence and easilyconverging to local minimumwhen forecasting Consideringthe shortcomings of standard BP algorithm this paperproposes a new fast convergent BP neural network modelfor predicting inventory level By adding an error offsetthis paper deduces the new chain propagation rule and theupdated weight formula The application of the improved BPneural network model to predict the inventory level of anautomotive parts company shows that the improved algo-rithm significantly outperforms the standard algorithm andsome other improved BP algorithms both on convergencerate and prediction accuracy

This paper proceeds as follows Section 2 has a widereview of related literature Based on the standard BP neuralnetwork Section 3 introduces an improved BP neural net-work Section 4 applies the improved BP algorithm to predictthe inventory level of an automotive parts company Section 5draws some conclusions according to the results

2 Literature Review

Recently more and more scholars have applied neural net-work technique to inventory control Bansal et al used aneural network-based data mining technique to solve theproblemof inventory of a largemedical distribution company[3] Based on the neural network model described by them aprototype was conceived with data from a large decentralizedorganization The prototype was successful in reducing thetotal level of inventory by 50 in the organization whilemaintaining the same level of probability that a particularcustomerrsquos demand would be satisfied Shanmugasundaramet al [4] discussed the use of neural network-based datamining and knowledge discovery techniques to optimizeinventory levels in a large medical distribution company [4]They identified the strategic data mining techniques usedto address the problem of estimating the future sales ofmedical products using past sales data and used recurrentneural networks to predict future sales Reyes-Aldasoro etal adopted neural network technique to create a hybridframework that could be utilized for analysis modelingand forecasting purposes [5] The framework combinedtwo existing approaches and introduced a new associatedcost parameter that served as a surrogate for customersatisfaction Hong et al developed an online neural networkcontroller that optimized a three-stage supply chainWith theinventory data feedback from an RFID system the neuralnetwork controller minimized the total cost of the supplychain rapidly while satisfying a target order fulfillment ratio[6] Some of these studies further proved that the neuralnetwork technique exceeded the traditional statistical tech-nique in forecasting inventory level [7] In fact comparingwith traditional prediction methods neural network has its

own unique advantages in processing prediction problemssuch as high fault tolerance fast prediction speed avoidanceof description of complex relation between characteristicfactors and object strong adaptation and good uncertaininformation processing ability [8ndash10]

Although there are various neural network models BPneural network is the most widely used model because ofits simple structure and strong ability to learn In fact ithas been widely used in inventory control Zhang et alused the reinforcement learning technique and the BP neuralnetwork to propose a new adaptive inventory control methodfor supply chain management [11] Wang proposed a neuralnetwork-based classification approach to inventory risk levelof spare parts [12] The BP algorithm for training a neuralnetwork is used to decide the weights to connections in themodel Mansur and Kuncoro cooperated to use the marketbasket analysis (MBA) and artificial neural network (ANN)Back propagation to predict inventory level [13] In additionANNBack propagation is used to predict product inventoriesrequirementsneeds for each product Huang et al appliedback-propagation network (BPN) to evaluate the criticalityclass (I II III and IV) of spare parts [14] They foundthat the proposed BPN could successfully decrease inventoryholding costs by modifying the unreasonable target servicelevel setting which was decided by the criticality class

The BP neural network we mentioned above is referringto the standard BP neural network The standard BP neuralnetwork is based on the Widrow-Hoff rule and uses thegradient descent method to transfer the mapping of a set ofinputs to its correct output into nonlinear optimal problemsHowever the standard BP algorithm has inherent disad-vantages such as slow convergence problem of convergingto local minimum complication of system and randomnetwork structure selection [15 16]

Aiming at the weaknesses of standard BP neural net-work scholars have made further studies and proposeddifferent improved BP neural network models [17ndash25] Theimprovements of BP neural network mainly incorporate twoperspectives the direct improvements on BP neural networkand the improvements based on the proposals of other newtheories Usually the first perspective includes adding themomentum factor [17] varying the learning rate dynamically[18] and introducing resilient back propagation (RPROP)[19] The second perspective usually includes the intro-duction of simulated annealing genetic algorithm [24] andintroduction ofmultiple extendedKalman algorithm [25] Allof these improved BP algorithms can reduce the training timeto some degree and increase the prediction accuracy Somehave even applied the improved BP algorithms to forecastinginventory level and inventory control [26 27]

By adding an error offset to the error function thispaper puts forward a direct improvement on standard BPneural network Based on a dataset of an automotive partscompany it proves that the improved BP algorithm not onlyexceeds the standard BP algorithm both on convergence rateand prediction accuracy but also outperforms some otherimproved BP neural networks

Discrete Dynamics in Nature and Society 3

3 Improvement of BP Neural Network

31 Standard BP Neural Network Back-propagation algo-rithm or BP algorithm one of the most widely used algo-rithms in artificial neural network is a kind of supervisedlearning algorithm Its main purpose is to adjust weightmatrix according to the squared error between the actualoutput and target output The squared error is expressed asfollows

119864 =1

2sum

119901

(119889119901minus 119910119901)2 (1)

Here 119901 denotes the 119901th training sample 119889119901 denotes thetarget output of the 119901th training sample and 119910119901 denotes theactual output of the 119901th training sample The weights to eachneuron are revised according to the following delta rule

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901) + Δ119908

119898

119894119895 (2)

Here 119898 denotes the 119898th layer neural network and 119908119894119895denotes the weight on the connection from the 119894th neuron inthe (119898minus 1)th layer to the 119895th neuron in the119898th layer Δ119908119898

119894119895is

expressed as follows

Δ119908119898

119894119895= minus120578

120597119864

120597119908119894119895

(3)

Here 120578 denotes the learning rate By analyzing the aboveformula we know that the key of BP algorithm is thecalculation of 120597119864120597119908119894119895

Suppose that 119868119895 denotes the input of 119895th neuron 119874119895denotes the output of 119895th neuron and 119874119894 denotes the outputof 119894th neuron Then 119868119895 = sum119894 119908119894119895119874119894 119874119895 = 119891(119868119895)

When the 119895th neuron is the output node we have

120597119864

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 (4)

If the119895th neuron is not the output node it must be thehidden node and we have

120597119864

120597119908119894119895

= minus1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895 (5)

From the above analysis we can know that the standardBP algorithm updates the weights of its output layer andhidden layer just according to the above formula Regardedas a part of the weights the update of bias is quite similar tothat of weights so we will not give further details about itsdeduction

32 Improved BP Neural Network To improve the conver-gence rate of standard BP algorithm we propose a newalgorithm which can achieve the goal by adding an erroroffset

The essence of BP algorithm is the forward propagationof data and backward propagation of errors The weightvalue is revised according to the errors in back propagationHowever the convergence rate of standard BP algorithm is

slow and often cannot satisfy the requirements when appliedTherefore we propose a new method adding an error offsetin back propagation to greatly improve the convergencerate The latter experiment illustrates that its effect is quiteoutstanding Here we redefine the squared error as follows

119864119874 =1

2sum

119875

(119889119901minus 119910119901)2+1

2sum

119901

(119891minus1(119889119901) minus 119891minus1(119910119901))2

(6)

and (12)sum119901(119891minus1(119889119901) minus 119891minus1(119910119901))2 is the error offset and

what follows next is our deduction of 120597119864119900120597119908119894119895 from therevised squared error Consider

120597119864119900

120597119908119894119895

=120597

120597119908119894119895

(1

2(119889119895 minus 119910119895)

2

+1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

(7)

For the right-hand side of (7) the first half part is the samewith that of standardBP algorithmWhatwe need to calculateis the second half part If 119895 is the output node then 119910119895 = 119874119895119891minus1(119910119895) = 119868119895 Consider

120597

120597119908119894119895

(1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

= minus (119891minus1(119889119895) minus 119868119895)

120597

120597119908119894119895

(119891minus1(119889119895) minus 119868119895)

= minus (119891minus1(119889119895) minus 119868119895)

120597119868119895

120597119908119894119895

= minus (119891minus1(119889119895) minus 119868119895)119874119894

120597119864119900

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 minus (119891

minus1(119889119895) minus 119868119895)119874119894

(8)

The new weight formula is

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901)

+ 120578 [(119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894

+ (119891minus1(119889119895) minus 119868119895)119874119894]

(9)

If 119895th neuron is not the output node then it must be thehidden node To avoid confusion we suppose that 119896th is theoutput layer and we have

120597

120597119908119894119895

(1

2(119891minus1(119889119896) minus 119891

minus1(119910119896))2

)

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119908119894119895

(119891minus1(119889119896) minus 119891

minus1(119910119896))

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119868119895

119891minus1(119910119896) lowast

120597119868119895

120597119908119894119895

4 Discrete Dynamics in Nature and Society

= minus (119891minus1(119889119901

119895) minus 119868119896)

120597119868119896

120597119868119895

lowast 119874119894

= minus (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

(10)

The new weight formula is

119908119898

119894119895(119901 + 1)

= 119908119898

119894119895(119901)

+ 120578[[

[

1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895

+ (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

]]

]

(11)

4 Model Construction

This paper uses the dataset of an automotive parts companyto train the improved BP neural network As we knownowadays automobiles are comprised of lots of parts Theseparts are produced on the demand of automobile manu-facturers and then are sent to assembly factories to form acomplete product In this way the whole production processof an automobile exists in the form of a supply chain Torealize the highest overall efficiency it needs cooperation ofall the suppliers manufacturers wholesalers and retailersInventory control is an important aspect which reflects suchkind of cooperation In the following part this paper will usethe improved BP neural network to forecast the inventorylevel of bearingsmdashone of the components for an automobile

41 Factors Influencing Inventory Control and Selection ofSample Usually accurate inventory level is the preconditionfor good inventorymanagement For inventorymanagementinventory controlling cost and customersrsquo service levels aswell as inventory controlling quality are the main factorsto estimate the inventory level Therefore in the design ofinventory control system we mainly use these factors topredict They are described as follows [2]

(1) Various Costs They are one of the main indicators toevaluate inventory control strategy The costs mainly includeall the expenses in product purchase and production as wellas sales For enterprises analyzing inventory controlling costcan effectively reduce the overall cost of enterprises Howeverinventory controlling costs include many aspects and theseaspects can influence each other Therefore dividing inven-tory controlling cost in details and analyzing the accumulateddata of business systems to find the main factors will behelpful for enterprises to make corresponding decisions andcontrol all kinds of costs The costs mainly include orderingcost storage cost transportation cost and shortage cost

(2) Demand Level The purpose of inventory control is to bestsatisfy the demandsTherefore demand is another importantfactor influencing inventory control However demand may

be certain but also may be stochastic or seasonal Demandlevel is positively proportional to inventory level

(3) Supply Level It refers to supply level of finished productsof producers It is positively proportional to inventory level

(4) Quantity of Substitutes It refers to the types of otherparts which can substitute for the parts used It is negativelyproportional to inventory level

(5) Lead Time It refers to the period of time from sendingthe order to being ready for production It includes the timefor ordering waiting time preparatory time for suppliers todeliver goods time on transportation time for check andacceptance for warehouse entry and time for preparation foruse It is positively proportional to inventory level

(6) Customer Service Level It refers to the possibility for enter-prises to satisfy customersrsquo needs after customers proposethe ordering requirements It is negatively proportional toinventory level The higher the customer service level goesthe lower the inventory level will be In this case we use 2(very good) 1 (general) and 0 (poor) to represent the extentof the customer service level

This paper chooses the historical data of factors whichinfluence the safety inventory level and inventory data ofbearing of an automotive parts production company in oneof the middle provinces of China from March 2012 to March2013 as a sample to train the improved BP neural networkWemainly choose 100 groups of the data to train the networkand then check its prediction ability The number of trainingsamples cannot be too small otherwise the network cannotlearn enough which may result in low prediction abilityHowever too large samples will lead to redundancy At thistime the network will be overfitted Therefore this paperchooses 100 groups of data as input to train and predictand chooses inventory level as output to establish the BPneural network model In this case because the systemis nonlinear the initial value plays very important role inachieving local minimumTherefore the input sample needsto be normalized and the purpose is to make the big inputvalues also fall in the range with large gradients of activationfunction

Before network training we normalized the training dataaccording to 119863119900 = (Max119863 minus 119863119894)(Max119863 minusMin119863) and madethem within [0 1] (see Table 1)

42 Network Variables Any continuous function can berealized by a three-layer artificial neural network Thereforethis paper adopts the three-layer BPneural network structureWhen all information is input into the network the informa-tion starts by being transmitted from input layer to hiddenlayer With the work of activation function the informationis then transmitted to output layer There are 9 input factorsand the output is inventory level The selection of variables ofthe network is as follows

(1) Input LayerThe input layer includes 9 factors storage cost(X1) ordering cost (X2) shortage cost (X3) transportation

Discrete Dynamics in Nature and Society 5

Table 1 Normalized data of stock-influencing factor

Data Storagecost

Orderingcost

Shortagecost

Transportationcost

Demandlevel

Supplylevel

Quantity ofsubstitutes

Waitingtime

Servicelevel

Actualinventory

level1 10 088 094 1 065 08 025 000 0 0332 07 100 100 0 100 10 100 025 0 0383 05 040 031 0 022 00 025 058 0 0504 00 008 013 0 043 00 000 050 1 0135 05 040 038 1 065 04 000 054 0 0256 07 060 031 1 074 04 000 087 1 0387 04 032 013 1 030 02 000 037 1 0508 03 020 000 0 000 00 000 079 1 0009 03 000 013 0 013 00 025 100 1 10010 05 078 063 1 043 04 025 008 1 013

Table 2 Comparison of training convergence rate among standard algorithm other improved algorithms and improved algorithm of thispaper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Maximum iteration times 9897 6245 4268 4432Minimum iteration times 1456 841 985 756Average iteration times 54234 23158 20139 19687

cost (X4) demand level (X5) supply level (X6) quantity ofsubstitutes (X7) waiting time (X8) and service level (X9)

(2) Hidden Layer Usually when there are one or two hiddenlayers it has the best convergent attributes If there is nohidden layer or there are too many hidden layers theconvergent effect is not so good Theories have proved thatnetworks which have deviations and at least one S-typehidden layer and one linear output layer can approach anynonlinear function That is a three-layer BP network with ahidden layer can approach any nonlinear function

According to empirical formula ℎ = log2119868 ℎ is the

number of nodes of hidden layer and 119868 is the number of nodesof input layer We suppose ℎ = 4

(3) Output Layer The number of nodes of output layer isthe number of system objects We choose one node as theinventory level of March 2013 to be measured

(4) Selection of Initial Value and Threshold Value Becauseboth of them are two random groups of value we choose arandom value between [minus1 1]

(5) Selection of Expected Error and Number of Iterations Wechoose 10000 as the number of iterations and the expectederror is 01

5 Training Process and Experimental Result

This paper uses the neural network tool package of MATLAB76 to program the model for safety inventory level basedon BP neural network In the BP neural network model

established in this paper there are 9 inputs and the number ofneurons is relatively large We preliminarily set the trainingvariables as follows times of training are 10000 trainingtarget is 001 and learning rate is 01 The code and trainingresult is as follows

net trainParam Epochs = 10000

net trainParam goal = 01

LP lr = 01

net-train(net P T)

after 1000 trainings the training is finished

After network finishes training the network getstestedWe use the data ofMarch 2013 to testThe codeof prediction is as follows

119875 test = [05 078 063 1 043 04 025 008 1]

Out = sim (net 119875 test)

By comparing Figures 1 and 2 we can clearly see that theconvergence rate of the improved algorithm is significantlyfaster than that of standard algorithmWe select the data fromFebruary 1 2013 to February 20 2013 to test The result is asfollows

From Table 2 we can know that the improved BPalgorithm is significantly better than that of standard BPalgorithm on convergence rate In addition we also compareour improved BP algorithm with some other improved BPalgorithms The result shows that our BP algorithm also out-performs the other two improved BP algorithms mentionedin the literature review on convergence rate

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

the changes of inventory environment In view of the featuresof BP neural network it has great advantages in classificationand prediction

However it is acknowledged that BP neural networkalso has such problems as slow convergence and easilyconverging to local minimumwhen forecasting Consideringthe shortcomings of standard BP algorithm this paperproposes a new fast convergent BP neural network modelfor predicting inventory level By adding an error offsetthis paper deduces the new chain propagation rule and theupdated weight formula The application of the improved BPneural network model to predict the inventory level of anautomotive parts company shows that the improved algo-rithm significantly outperforms the standard algorithm andsome other improved BP algorithms both on convergencerate and prediction accuracy

This paper proceeds as follows Section 2 has a widereview of related literature Based on the standard BP neuralnetwork Section 3 introduces an improved BP neural net-work Section 4 applies the improved BP algorithm to predictthe inventory level of an automotive parts company Section 5draws some conclusions according to the results

2 Literature Review

Recently more and more scholars have applied neural net-work technique to inventory control Bansal et al used aneural network-based data mining technique to solve theproblemof inventory of a largemedical distribution company[3] Based on the neural network model described by them aprototype was conceived with data from a large decentralizedorganization The prototype was successful in reducing thetotal level of inventory by 50 in the organization whilemaintaining the same level of probability that a particularcustomerrsquos demand would be satisfied Shanmugasundaramet al [4] discussed the use of neural network-based datamining and knowledge discovery techniques to optimizeinventory levels in a large medical distribution company [4]They identified the strategic data mining techniques usedto address the problem of estimating the future sales ofmedical products using past sales data and used recurrentneural networks to predict future sales Reyes-Aldasoro etal adopted neural network technique to create a hybridframework that could be utilized for analysis modelingand forecasting purposes [5] The framework combinedtwo existing approaches and introduced a new associatedcost parameter that served as a surrogate for customersatisfaction Hong et al developed an online neural networkcontroller that optimized a three-stage supply chainWith theinventory data feedback from an RFID system the neuralnetwork controller minimized the total cost of the supplychain rapidly while satisfying a target order fulfillment ratio[6] Some of these studies further proved that the neuralnetwork technique exceeded the traditional statistical tech-nique in forecasting inventory level [7] In fact comparingwith traditional prediction methods neural network has its

own unique advantages in processing prediction problemssuch as high fault tolerance fast prediction speed avoidanceof description of complex relation between characteristicfactors and object strong adaptation and good uncertaininformation processing ability [8ndash10]

Although there are various neural network models BPneural network is the most widely used model because ofits simple structure and strong ability to learn In fact ithas been widely used in inventory control Zhang et alused the reinforcement learning technique and the BP neuralnetwork to propose a new adaptive inventory control methodfor supply chain management [11] Wang proposed a neuralnetwork-based classification approach to inventory risk levelof spare parts [12] The BP algorithm for training a neuralnetwork is used to decide the weights to connections in themodel Mansur and Kuncoro cooperated to use the marketbasket analysis (MBA) and artificial neural network (ANN)Back propagation to predict inventory level [13] In additionANNBack propagation is used to predict product inventoriesrequirementsneeds for each product Huang et al appliedback-propagation network (BPN) to evaluate the criticalityclass (I II III and IV) of spare parts [14] They foundthat the proposed BPN could successfully decrease inventoryholding costs by modifying the unreasonable target servicelevel setting which was decided by the criticality class

The BP neural network we mentioned above is referringto the standard BP neural network The standard BP neuralnetwork is based on the Widrow-Hoff rule and uses thegradient descent method to transfer the mapping of a set ofinputs to its correct output into nonlinear optimal problemsHowever the standard BP algorithm has inherent disad-vantages such as slow convergence problem of convergingto local minimum complication of system and randomnetwork structure selection [15 16]

Aiming at the weaknesses of standard BP neural net-work scholars have made further studies and proposeddifferent improved BP neural network models [17ndash25] Theimprovements of BP neural network mainly incorporate twoperspectives the direct improvements on BP neural networkand the improvements based on the proposals of other newtheories Usually the first perspective includes adding themomentum factor [17] varying the learning rate dynamically[18] and introducing resilient back propagation (RPROP)[19] The second perspective usually includes the intro-duction of simulated annealing genetic algorithm [24] andintroduction ofmultiple extendedKalman algorithm [25] Allof these improved BP algorithms can reduce the training timeto some degree and increase the prediction accuracy Somehave even applied the improved BP algorithms to forecastinginventory level and inventory control [26 27]

By adding an error offset to the error function thispaper puts forward a direct improvement on standard BPneural network Based on a dataset of an automotive partscompany it proves that the improved BP algorithm not onlyexceeds the standard BP algorithm both on convergence rateand prediction accuracy but also outperforms some otherimproved BP neural networks

Discrete Dynamics in Nature and Society 3

3 Improvement of BP Neural Network

31 Standard BP Neural Network Back-propagation algo-rithm or BP algorithm one of the most widely used algo-rithms in artificial neural network is a kind of supervisedlearning algorithm Its main purpose is to adjust weightmatrix according to the squared error between the actualoutput and target output The squared error is expressed asfollows

119864 =1

2sum

119901

(119889119901minus 119910119901)2 (1)

Here 119901 denotes the 119901th training sample 119889119901 denotes thetarget output of the 119901th training sample and 119910119901 denotes theactual output of the 119901th training sample The weights to eachneuron are revised according to the following delta rule

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901) + Δ119908

119898

119894119895 (2)

Here 119898 denotes the 119898th layer neural network and 119908119894119895denotes the weight on the connection from the 119894th neuron inthe (119898minus 1)th layer to the 119895th neuron in the119898th layer Δ119908119898

119894119895is

expressed as follows

Δ119908119898

119894119895= minus120578

120597119864

120597119908119894119895

(3)

Here 120578 denotes the learning rate By analyzing the aboveformula we know that the key of BP algorithm is thecalculation of 120597119864120597119908119894119895

Suppose that 119868119895 denotes the input of 119895th neuron 119874119895denotes the output of 119895th neuron and 119874119894 denotes the outputof 119894th neuron Then 119868119895 = sum119894 119908119894119895119874119894 119874119895 = 119891(119868119895)

When the 119895th neuron is the output node we have

120597119864

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 (4)

If the119895th neuron is not the output node it must be thehidden node and we have

120597119864

120597119908119894119895

= minus1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895 (5)

From the above analysis we can know that the standardBP algorithm updates the weights of its output layer andhidden layer just according to the above formula Regardedas a part of the weights the update of bias is quite similar tothat of weights so we will not give further details about itsdeduction

32 Improved BP Neural Network To improve the conver-gence rate of standard BP algorithm we propose a newalgorithm which can achieve the goal by adding an erroroffset

The essence of BP algorithm is the forward propagationof data and backward propagation of errors The weightvalue is revised according to the errors in back propagationHowever the convergence rate of standard BP algorithm is

slow and often cannot satisfy the requirements when appliedTherefore we propose a new method adding an error offsetin back propagation to greatly improve the convergencerate The latter experiment illustrates that its effect is quiteoutstanding Here we redefine the squared error as follows

119864119874 =1

2sum

119875

(119889119901minus 119910119901)2+1

2sum

119901

(119891minus1(119889119901) minus 119891minus1(119910119901))2

(6)

and (12)sum119901(119891minus1(119889119901) minus 119891minus1(119910119901))2 is the error offset and

what follows next is our deduction of 120597119864119900120597119908119894119895 from therevised squared error Consider

120597119864119900

120597119908119894119895

=120597

120597119908119894119895

(1

2(119889119895 minus 119910119895)

2

+1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

(7)

For the right-hand side of (7) the first half part is the samewith that of standardBP algorithmWhatwe need to calculateis the second half part If 119895 is the output node then 119910119895 = 119874119895119891minus1(119910119895) = 119868119895 Consider

120597

120597119908119894119895

(1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

= minus (119891minus1(119889119895) minus 119868119895)

120597

120597119908119894119895

(119891minus1(119889119895) minus 119868119895)

= minus (119891minus1(119889119895) minus 119868119895)

120597119868119895

120597119908119894119895

= minus (119891minus1(119889119895) minus 119868119895)119874119894

120597119864119900

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 minus (119891

minus1(119889119895) minus 119868119895)119874119894

(8)

The new weight formula is

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901)

+ 120578 [(119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894

+ (119891minus1(119889119895) minus 119868119895)119874119894]

(9)

If 119895th neuron is not the output node then it must be thehidden node To avoid confusion we suppose that 119896th is theoutput layer and we have

120597

120597119908119894119895

(1

2(119891minus1(119889119896) minus 119891

minus1(119910119896))2

)

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119908119894119895

(119891minus1(119889119896) minus 119891

minus1(119910119896))

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119868119895

119891minus1(119910119896) lowast

120597119868119895

120597119908119894119895

4 Discrete Dynamics in Nature and Society

= minus (119891minus1(119889119901

119895) minus 119868119896)

120597119868119896

120597119868119895

lowast 119874119894

= minus (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

(10)

The new weight formula is

119908119898

119894119895(119901 + 1)

= 119908119898

119894119895(119901)

+ 120578[[

[

1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895

+ (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

]]

]

(11)

4 Model Construction

This paper uses the dataset of an automotive parts companyto train the improved BP neural network As we knownowadays automobiles are comprised of lots of parts Theseparts are produced on the demand of automobile manu-facturers and then are sent to assembly factories to form acomplete product In this way the whole production processof an automobile exists in the form of a supply chain Torealize the highest overall efficiency it needs cooperation ofall the suppliers manufacturers wholesalers and retailersInventory control is an important aspect which reflects suchkind of cooperation In the following part this paper will usethe improved BP neural network to forecast the inventorylevel of bearingsmdashone of the components for an automobile

41 Factors Influencing Inventory Control and Selection ofSample Usually accurate inventory level is the preconditionfor good inventorymanagement For inventorymanagementinventory controlling cost and customersrsquo service levels aswell as inventory controlling quality are the main factorsto estimate the inventory level Therefore in the design ofinventory control system we mainly use these factors topredict They are described as follows [2]

(1) Various Costs They are one of the main indicators toevaluate inventory control strategy The costs mainly includeall the expenses in product purchase and production as wellas sales For enterprises analyzing inventory controlling costcan effectively reduce the overall cost of enterprises Howeverinventory controlling costs include many aspects and theseaspects can influence each other Therefore dividing inven-tory controlling cost in details and analyzing the accumulateddata of business systems to find the main factors will behelpful for enterprises to make corresponding decisions andcontrol all kinds of costs The costs mainly include orderingcost storage cost transportation cost and shortage cost

(2) Demand Level The purpose of inventory control is to bestsatisfy the demandsTherefore demand is another importantfactor influencing inventory control However demand may

be certain but also may be stochastic or seasonal Demandlevel is positively proportional to inventory level

(3) Supply Level It refers to supply level of finished productsof producers It is positively proportional to inventory level

(4) Quantity of Substitutes It refers to the types of otherparts which can substitute for the parts used It is negativelyproportional to inventory level

(5) Lead Time It refers to the period of time from sendingthe order to being ready for production It includes the timefor ordering waiting time preparatory time for suppliers todeliver goods time on transportation time for check andacceptance for warehouse entry and time for preparation foruse It is positively proportional to inventory level

(6) Customer Service Level It refers to the possibility for enter-prises to satisfy customersrsquo needs after customers proposethe ordering requirements It is negatively proportional toinventory level The higher the customer service level goesthe lower the inventory level will be In this case we use 2(very good) 1 (general) and 0 (poor) to represent the extentof the customer service level

This paper chooses the historical data of factors whichinfluence the safety inventory level and inventory data ofbearing of an automotive parts production company in oneof the middle provinces of China from March 2012 to March2013 as a sample to train the improved BP neural networkWemainly choose 100 groups of the data to train the networkand then check its prediction ability The number of trainingsamples cannot be too small otherwise the network cannotlearn enough which may result in low prediction abilityHowever too large samples will lead to redundancy At thistime the network will be overfitted Therefore this paperchooses 100 groups of data as input to train and predictand chooses inventory level as output to establish the BPneural network model In this case because the systemis nonlinear the initial value plays very important role inachieving local minimumTherefore the input sample needsto be normalized and the purpose is to make the big inputvalues also fall in the range with large gradients of activationfunction

Before network training we normalized the training dataaccording to 119863119900 = (Max119863 minus 119863119894)(Max119863 minusMin119863) and madethem within [0 1] (see Table 1)

42 Network Variables Any continuous function can berealized by a three-layer artificial neural network Thereforethis paper adopts the three-layer BPneural network structureWhen all information is input into the network the informa-tion starts by being transmitted from input layer to hiddenlayer With the work of activation function the informationis then transmitted to output layer There are 9 input factorsand the output is inventory level The selection of variables ofthe network is as follows

(1) Input LayerThe input layer includes 9 factors storage cost(X1) ordering cost (X2) shortage cost (X3) transportation

Discrete Dynamics in Nature and Society 5

Table 1 Normalized data of stock-influencing factor

Data Storagecost

Orderingcost

Shortagecost

Transportationcost

Demandlevel

Supplylevel

Quantity ofsubstitutes

Waitingtime

Servicelevel

Actualinventory

level1 10 088 094 1 065 08 025 000 0 0332 07 100 100 0 100 10 100 025 0 0383 05 040 031 0 022 00 025 058 0 0504 00 008 013 0 043 00 000 050 1 0135 05 040 038 1 065 04 000 054 0 0256 07 060 031 1 074 04 000 087 1 0387 04 032 013 1 030 02 000 037 1 0508 03 020 000 0 000 00 000 079 1 0009 03 000 013 0 013 00 025 100 1 10010 05 078 063 1 043 04 025 008 1 013

Table 2 Comparison of training convergence rate among standard algorithm other improved algorithms and improved algorithm of thispaper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Maximum iteration times 9897 6245 4268 4432Minimum iteration times 1456 841 985 756Average iteration times 54234 23158 20139 19687

cost (X4) demand level (X5) supply level (X6) quantity ofsubstitutes (X7) waiting time (X8) and service level (X9)

(2) Hidden Layer Usually when there are one or two hiddenlayers it has the best convergent attributes If there is nohidden layer or there are too many hidden layers theconvergent effect is not so good Theories have proved thatnetworks which have deviations and at least one S-typehidden layer and one linear output layer can approach anynonlinear function That is a three-layer BP network with ahidden layer can approach any nonlinear function

According to empirical formula ℎ = log2119868 ℎ is the

number of nodes of hidden layer and 119868 is the number of nodesof input layer We suppose ℎ = 4

(3) Output Layer The number of nodes of output layer isthe number of system objects We choose one node as theinventory level of March 2013 to be measured

(4) Selection of Initial Value and Threshold Value Becauseboth of them are two random groups of value we choose arandom value between [minus1 1]

(5) Selection of Expected Error and Number of Iterations Wechoose 10000 as the number of iterations and the expectederror is 01

5 Training Process and Experimental Result

This paper uses the neural network tool package of MATLAB76 to program the model for safety inventory level basedon BP neural network In the BP neural network model

established in this paper there are 9 inputs and the number ofneurons is relatively large We preliminarily set the trainingvariables as follows times of training are 10000 trainingtarget is 001 and learning rate is 01 The code and trainingresult is as follows

net trainParam Epochs = 10000

net trainParam goal = 01

LP lr = 01

net-train(net P T)

after 1000 trainings the training is finished

After network finishes training the network getstestedWe use the data ofMarch 2013 to testThe codeof prediction is as follows

119875 test = [05 078 063 1 043 04 025 008 1]

Out = sim (net 119875 test)

By comparing Figures 1 and 2 we can clearly see that theconvergence rate of the improved algorithm is significantlyfaster than that of standard algorithmWe select the data fromFebruary 1 2013 to February 20 2013 to test The result is asfollows

From Table 2 we can know that the improved BPalgorithm is significantly better than that of standard BPalgorithm on convergence rate In addition we also compareour improved BP algorithm with some other improved BPalgorithms The result shows that our BP algorithm also out-performs the other two improved BP algorithms mentionedin the literature review on convergence rate

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

3 Improvement of BP Neural Network

31 Standard BP Neural Network Back-propagation algo-rithm or BP algorithm one of the most widely used algo-rithms in artificial neural network is a kind of supervisedlearning algorithm Its main purpose is to adjust weightmatrix according to the squared error between the actualoutput and target output The squared error is expressed asfollows

119864 =1

2sum

119901

(119889119901minus 119910119901)2 (1)

Here 119901 denotes the 119901th training sample 119889119901 denotes thetarget output of the 119901th training sample and 119910119901 denotes theactual output of the 119901th training sample The weights to eachneuron are revised according to the following delta rule

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901) + Δ119908

119898

119894119895 (2)

Here 119898 denotes the 119898th layer neural network and 119908119894119895denotes the weight on the connection from the 119894th neuron inthe (119898minus 1)th layer to the 119895th neuron in the119898th layer Δ119908119898

119894119895is

expressed as follows

Δ119908119898

119894119895= minus120578

120597119864

120597119908119894119895

(3)

Here 120578 denotes the learning rate By analyzing the aboveformula we know that the key of BP algorithm is thecalculation of 120597119864120597119908119894119895

Suppose that 119868119895 denotes the input of 119895th neuron 119874119895denotes the output of 119895th neuron and 119874119894 denotes the outputof 119894th neuron Then 119868119895 = sum119894 119908119894119895119874119894 119874119895 = 119891(119868119895)

When the 119895th neuron is the output node we have

120597119864

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 (4)

If the119895th neuron is not the output node it must be thehidden node and we have

120597119864

120597119908119894119895

= minus1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895 (5)

From the above analysis we can know that the standardBP algorithm updates the weights of its output layer andhidden layer just according to the above formula Regardedas a part of the weights the update of bias is quite similar tothat of weights so we will not give further details about itsdeduction

32 Improved BP Neural Network To improve the conver-gence rate of standard BP algorithm we propose a newalgorithm which can achieve the goal by adding an erroroffset

The essence of BP algorithm is the forward propagationof data and backward propagation of errors The weightvalue is revised according to the errors in back propagationHowever the convergence rate of standard BP algorithm is

slow and often cannot satisfy the requirements when appliedTherefore we propose a new method adding an error offsetin back propagation to greatly improve the convergencerate The latter experiment illustrates that its effect is quiteoutstanding Here we redefine the squared error as follows

119864119874 =1

2sum

119875

(119889119901minus 119910119901)2+1

2sum

119901

(119891minus1(119889119901) minus 119891minus1(119910119901))2

(6)

and (12)sum119901(119891minus1(119889119901) minus 119891minus1(119910119901))2 is the error offset and

what follows next is our deduction of 120597119864119900120597119908119894119895 from therevised squared error Consider

120597119864119900

120597119908119894119895

=120597

120597119908119894119895

(1

2(119889119895 minus 119910119895)

2

+1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

(7)

For the right-hand side of (7) the first half part is the samewith that of standardBP algorithmWhatwe need to calculateis the second half part If 119895 is the output node then 119910119895 = 119874119895119891minus1(119910119895) = 119868119895 Consider

120597

120597119908119894119895

(1

2(119891minus1(119889119895) minus 119891

minus1(119910119895))2

)

= minus (119891minus1(119889119895) minus 119868119895)

120597

120597119908119894119895

(119891minus1(119889119895) minus 119868119895)

= minus (119891minus1(119889119895) minus 119868119895)

120597119868119895

120597119908119894119895

= minus (119891minus1(119889119895) minus 119868119895)119874119894

120597119864119900

120597119908119894119895

= minus (119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894 minus (119891

minus1(119889119895) minus 119868119895)119874119894

(8)

The new weight formula is

119908119898

119894119895(119901 + 1) = 119908

119898

119894119895(119901)

+ 120578 [(119889 minus 119910) lowast 1198911015840(119868119895) lowast 119874119894

+ (119891minus1(119889119895) minus 119868119895)119874119894]

(9)

If 119895th neuron is not the output node then it must be thehidden node To avoid confusion we suppose that 119896th is theoutput layer and we have

120597

120597119908119894119895

(1

2(119891minus1(119889119896) minus 119891

minus1(119910119896))2

)

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119908119894119895

(119891minus1(119889119896) minus 119891

minus1(119910119896))

= minus (119891minus1(119889119896) minus 119891

minus1(119910119896))

120597

120597119868119895

119891minus1(119910119896) lowast

120597119868119895

120597119908119894119895

4 Discrete Dynamics in Nature and Society

= minus (119891minus1(119889119901

119895) minus 119868119896)

120597119868119896

120597119868119895

lowast 119874119894

= minus (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

(10)

The new weight formula is

119908119898

119894119895(119901 + 1)

= 119908119898

119894119895(119901)

+ 120578[[

[

1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895

+ (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

]]

]

(11)

4 Model Construction

This paper uses the dataset of an automotive parts companyto train the improved BP neural network As we knownowadays automobiles are comprised of lots of parts Theseparts are produced on the demand of automobile manu-facturers and then are sent to assembly factories to form acomplete product In this way the whole production processof an automobile exists in the form of a supply chain Torealize the highest overall efficiency it needs cooperation ofall the suppliers manufacturers wholesalers and retailersInventory control is an important aspect which reflects suchkind of cooperation In the following part this paper will usethe improved BP neural network to forecast the inventorylevel of bearingsmdashone of the components for an automobile

41 Factors Influencing Inventory Control and Selection ofSample Usually accurate inventory level is the preconditionfor good inventorymanagement For inventorymanagementinventory controlling cost and customersrsquo service levels aswell as inventory controlling quality are the main factorsto estimate the inventory level Therefore in the design ofinventory control system we mainly use these factors topredict They are described as follows [2]

(1) Various Costs They are one of the main indicators toevaluate inventory control strategy The costs mainly includeall the expenses in product purchase and production as wellas sales For enterprises analyzing inventory controlling costcan effectively reduce the overall cost of enterprises Howeverinventory controlling costs include many aspects and theseaspects can influence each other Therefore dividing inven-tory controlling cost in details and analyzing the accumulateddata of business systems to find the main factors will behelpful for enterprises to make corresponding decisions andcontrol all kinds of costs The costs mainly include orderingcost storage cost transportation cost and shortage cost

(2) Demand Level The purpose of inventory control is to bestsatisfy the demandsTherefore demand is another importantfactor influencing inventory control However demand may

be certain but also may be stochastic or seasonal Demandlevel is positively proportional to inventory level

(3) Supply Level It refers to supply level of finished productsof producers It is positively proportional to inventory level

(4) Quantity of Substitutes It refers to the types of otherparts which can substitute for the parts used It is negativelyproportional to inventory level

(5) Lead Time It refers to the period of time from sendingthe order to being ready for production It includes the timefor ordering waiting time preparatory time for suppliers todeliver goods time on transportation time for check andacceptance for warehouse entry and time for preparation foruse It is positively proportional to inventory level

(6) Customer Service Level It refers to the possibility for enter-prises to satisfy customersrsquo needs after customers proposethe ordering requirements It is negatively proportional toinventory level The higher the customer service level goesthe lower the inventory level will be In this case we use 2(very good) 1 (general) and 0 (poor) to represent the extentof the customer service level

This paper chooses the historical data of factors whichinfluence the safety inventory level and inventory data ofbearing of an automotive parts production company in oneof the middle provinces of China from March 2012 to March2013 as a sample to train the improved BP neural networkWemainly choose 100 groups of the data to train the networkand then check its prediction ability The number of trainingsamples cannot be too small otherwise the network cannotlearn enough which may result in low prediction abilityHowever too large samples will lead to redundancy At thistime the network will be overfitted Therefore this paperchooses 100 groups of data as input to train and predictand chooses inventory level as output to establish the BPneural network model In this case because the systemis nonlinear the initial value plays very important role inachieving local minimumTherefore the input sample needsto be normalized and the purpose is to make the big inputvalues also fall in the range with large gradients of activationfunction

Before network training we normalized the training dataaccording to 119863119900 = (Max119863 minus 119863119894)(Max119863 minusMin119863) and madethem within [0 1] (see Table 1)

42 Network Variables Any continuous function can berealized by a three-layer artificial neural network Thereforethis paper adopts the three-layer BPneural network structureWhen all information is input into the network the informa-tion starts by being transmitted from input layer to hiddenlayer With the work of activation function the informationis then transmitted to output layer There are 9 input factorsand the output is inventory level The selection of variables ofthe network is as follows

(1) Input LayerThe input layer includes 9 factors storage cost(X1) ordering cost (X2) shortage cost (X3) transportation

Discrete Dynamics in Nature and Society 5

Table 1 Normalized data of stock-influencing factor

Data Storagecost

Orderingcost

Shortagecost

Transportationcost

Demandlevel

Supplylevel

Quantity ofsubstitutes

Waitingtime

Servicelevel

Actualinventory

level1 10 088 094 1 065 08 025 000 0 0332 07 100 100 0 100 10 100 025 0 0383 05 040 031 0 022 00 025 058 0 0504 00 008 013 0 043 00 000 050 1 0135 05 040 038 1 065 04 000 054 0 0256 07 060 031 1 074 04 000 087 1 0387 04 032 013 1 030 02 000 037 1 0508 03 020 000 0 000 00 000 079 1 0009 03 000 013 0 013 00 025 100 1 10010 05 078 063 1 043 04 025 008 1 013

Table 2 Comparison of training convergence rate among standard algorithm other improved algorithms and improved algorithm of thispaper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Maximum iteration times 9897 6245 4268 4432Minimum iteration times 1456 841 985 756Average iteration times 54234 23158 20139 19687

cost (X4) demand level (X5) supply level (X6) quantity ofsubstitutes (X7) waiting time (X8) and service level (X9)

(2) Hidden Layer Usually when there are one or two hiddenlayers it has the best convergent attributes If there is nohidden layer or there are too many hidden layers theconvergent effect is not so good Theories have proved thatnetworks which have deviations and at least one S-typehidden layer and one linear output layer can approach anynonlinear function That is a three-layer BP network with ahidden layer can approach any nonlinear function

According to empirical formula ℎ = log2119868 ℎ is the

number of nodes of hidden layer and 119868 is the number of nodesof input layer We suppose ℎ = 4

(3) Output Layer The number of nodes of output layer isthe number of system objects We choose one node as theinventory level of March 2013 to be measured

(4) Selection of Initial Value and Threshold Value Becauseboth of them are two random groups of value we choose arandom value between [minus1 1]

(5) Selection of Expected Error and Number of Iterations Wechoose 10000 as the number of iterations and the expectederror is 01

5 Training Process and Experimental Result

This paper uses the neural network tool package of MATLAB76 to program the model for safety inventory level basedon BP neural network In the BP neural network model

established in this paper there are 9 inputs and the number ofneurons is relatively large We preliminarily set the trainingvariables as follows times of training are 10000 trainingtarget is 001 and learning rate is 01 The code and trainingresult is as follows

net trainParam Epochs = 10000

net trainParam goal = 01

LP lr = 01

net-train(net P T)

after 1000 trainings the training is finished

After network finishes training the network getstestedWe use the data ofMarch 2013 to testThe codeof prediction is as follows

119875 test = [05 078 063 1 043 04 025 008 1]

Out = sim (net 119875 test)

By comparing Figures 1 and 2 we can clearly see that theconvergence rate of the improved algorithm is significantlyfaster than that of standard algorithmWe select the data fromFebruary 1 2013 to February 20 2013 to test The result is asfollows

From Table 2 we can know that the improved BPalgorithm is significantly better than that of standard BPalgorithm on convergence rate In addition we also compareour improved BP algorithm with some other improved BPalgorithms The result shows that our BP algorithm also out-performs the other two improved BP algorithms mentionedin the literature review on convergence rate

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

= minus (119891minus1(119889119901

119895) minus 119868119896)

120597119868119896

120597119868119895

lowast 119874119894

= minus (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

(10)

The new weight formula is

119908119898

119894119895(119901 + 1)

= 119908119898

119894119895(119901)

+ 120578[[

[

1198911015840(119868119895)sum

119898

(119889119896minus 119910119896) 1198911015840(119868119898) 119908119898119895

+ (119891minus1(119889119901

119895) minus 119868119896)sum

119898

1198911015840(119868119895)119908119898119895 lowast 119874119894

]]

]

(11)

4 Model Construction

This paper uses the dataset of an automotive parts companyto train the improved BP neural network As we knownowadays automobiles are comprised of lots of parts Theseparts are produced on the demand of automobile manu-facturers and then are sent to assembly factories to form acomplete product In this way the whole production processof an automobile exists in the form of a supply chain Torealize the highest overall efficiency it needs cooperation ofall the suppliers manufacturers wholesalers and retailersInventory control is an important aspect which reflects suchkind of cooperation In the following part this paper will usethe improved BP neural network to forecast the inventorylevel of bearingsmdashone of the components for an automobile

41 Factors Influencing Inventory Control and Selection ofSample Usually accurate inventory level is the preconditionfor good inventorymanagement For inventorymanagementinventory controlling cost and customersrsquo service levels aswell as inventory controlling quality are the main factorsto estimate the inventory level Therefore in the design ofinventory control system we mainly use these factors topredict They are described as follows [2]

(1) Various Costs They are one of the main indicators toevaluate inventory control strategy The costs mainly includeall the expenses in product purchase and production as wellas sales For enterprises analyzing inventory controlling costcan effectively reduce the overall cost of enterprises Howeverinventory controlling costs include many aspects and theseaspects can influence each other Therefore dividing inven-tory controlling cost in details and analyzing the accumulateddata of business systems to find the main factors will behelpful for enterprises to make corresponding decisions andcontrol all kinds of costs The costs mainly include orderingcost storage cost transportation cost and shortage cost

(2) Demand Level The purpose of inventory control is to bestsatisfy the demandsTherefore demand is another importantfactor influencing inventory control However demand may

be certain but also may be stochastic or seasonal Demandlevel is positively proportional to inventory level

(3) Supply Level It refers to supply level of finished productsof producers It is positively proportional to inventory level

(4) Quantity of Substitutes It refers to the types of otherparts which can substitute for the parts used It is negativelyproportional to inventory level

(5) Lead Time It refers to the period of time from sendingthe order to being ready for production It includes the timefor ordering waiting time preparatory time for suppliers todeliver goods time on transportation time for check andacceptance for warehouse entry and time for preparation foruse It is positively proportional to inventory level

(6) Customer Service Level It refers to the possibility for enter-prises to satisfy customersrsquo needs after customers proposethe ordering requirements It is negatively proportional toinventory level The higher the customer service level goesthe lower the inventory level will be In this case we use 2(very good) 1 (general) and 0 (poor) to represent the extentof the customer service level

This paper chooses the historical data of factors whichinfluence the safety inventory level and inventory data ofbearing of an automotive parts production company in oneof the middle provinces of China from March 2012 to March2013 as a sample to train the improved BP neural networkWemainly choose 100 groups of the data to train the networkand then check its prediction ability The number of trainingsamples cannot be too small otherwise the network cannotlearn enough which may result in low prediction abilityHowever too large samples will lead to redundancy At thistime the network will be overfitted Therefore this paperchooses 100 groups of data as input to train and predictand chooses inventory level as output to establish the BPneural network model In this case because the systemis nonlinear the initial value plays very important role inachieving local minimumTherefore the input sample needsto be normalized and the purpose is to make the big inputvalues also fall in the range with large gradients of activationfunction

Before network training we normalized the training dataaccording to 119863119900 = (Max119863 minus 119863119894)(Max119863 minusMin119863) and madethem within [0 1] (see Table 1)

42 Network Variables Any continuous function can berealized by a three-layer artificial neural network Thereforethis paper adopts the three-layer BPneural network structureWhen all information is input into the network the informa-tion starts by being transmitted from input layer to hiddenlayer With the work of activation function the informationis then transmitted to output layer There are 9 input factorsand the output is inventory level The selection of variables ofthe network is as follows

(1) Input LayerThe input layer includes 9 factors storage cost(X1) ordering cost (X2) shortage cost (X3) transportation

Discrete Dynamics in Nature and Society 5

Table 1 Normalized data of stock-influencing factor

Data Storagecost

Orderingcost

Shortagecost

Transportationcost

Demandlevel

Supplylevel

Quantity ofsubstitutes

Waitingtime

Servicelevel

Actualinventory

level1 10 088 094 1 065 08 025 000 0 0332 07 100 100 0 100 10 100 025 0 0383 05 040 031 0 022 00 025 058 0 0504 00 008 013 0 043 00 000 050 1 0135 05 040 038 1 065 04 000 054 0 0256 07 060 031 1 074 04 000 087 1 0387 04 032 013 1 030 02 000 037 1 0508 03 020 000 0 000 00 000 079 1 0009 03 000 013 0 013 00 025 100 1 10010 05 078 063 1 043 04 025 008 1 013

Table 2 Comparison of training convergence rate among standard algorithm other improved algorithms and improved algorithm of thispaper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Maximum iteration times 9897 6245 4268 4432Minimum iteration times 1456 841 985 756Average iteration times 54234 23158 20139 19687

cost (X4) demand level (X5) supply level (X6) quantity ofsubstitutes (X7) waiting time (X8) and service level (X9)

(2) Hidden Layer Usually when there are one or two hiddenlayers it has the best convergent attributes If there is nohidden layer or there are too many hidden layers theconvergent effect is not so good Theories have proved thatnetworks which have deviations and at least one S-typehidden layer and one linear output layer can approach anynonlinear function That is a three-layer BP network with ahidden layer can approach any nonlinear function

According to empirical formula ℎ = log2119868 ℎ is the

number of nodes of hidden layer and 119868 is the number of nodesof input layer We suppose ℎ = 4

(3) Output Layer The number of nodes of output layer isthe number of system objects We choose one node as theinventory level of March 2013 to be measured

(4) Selection of Initial Value and Threshold Value Becauseboth of them are two random groups of value we choose arandom value between [minus1 1]

(5) Selection of Expected Error and Number of Iterations Wechoose 10000 as the number of iterations and the expectederror is 01

5 Training Process and Experimental Result

This paper uses the neural network tool package of MATLAB76 to program the model for safety inventory level basedon BP neural network In the BP neural network model

established in this paper there are 9 inputs and the number ofneurons is relatively large We preliminarily set the trainingvariables as follows times of training are 10000 trainingtarget is 001 and learning rate is 01 The code and trainingresult is as follows

net trainParam Epochs = 10000

net trainParam goal = 01

LP lr = 01

net-train(net P T)

after 1000 trainings the training is finished

After network finishes training the network getstestedWe use the data ofMarch 2013 to testThe codeof prediction is as follows

119875 test = [05 078 063 1 043 04 025 008 1]

Out = sim (net 119875 test)

By comparing Figures 1 and 2 we can clearly see that theconvergence rate of the improved algorithm is significantlyfaster than that of standard algorithmWe select the data fromFebruary 1 2013 to February 20 2013 to test The result is asfollows

From Table 2 we can know that the improved BPalgorithm is significantly better than that of standard BPalgorithm on convergence rate In addition we also compareour improved BP algorithm with some other improved BPalgorithms The result shows that our BP algorithm also out-performs the other two improved BP algorithms mentionedin the literature review on convergence rate

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Table 1 Normalized data of stock-influencing factor

Data Storagecost

Orderingcost

Shortagecost

Transportationcost

Demandlevel

Supplylevel

Quantity ofsubstitutes

Waitingtime

Servicelevel

Actualinventory

level1 10 088 094 1 065 08 025 000 0 0332 07 100 100 0 100 10 100 025 0 0383 05 040 031 0 022 00 025 058 0 0504 00 008 013 0 043 00 000 050 1 0135 05 040 038 1 065 04 000 054 0 0256 07 060 031 1 074 04 000 087 1 0387 04 032 013 1 030 02 000 037 1 0508 03 020 000 0 000 00 000 079 1 0009 03 000 013 0 013 00 025 100 1 10010 05 078 063 1 043 04 025 008 1 013

Table 2 Comparison of training convergence rate among standard algorithm other improved algorithms and improved algorithm of thispaper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Maximum iteration times 9897 6245 4268 4432Minimum iteration times 1456 841 985 756Average iteration times 54234 23158 20139 19687

cost (X4) demand level (X5) supply level (X6) quantity ofsubstitutes (X7) waiting time (X8) and service level (X9)

(2) Hidden Layer Usually when there are one or two hiddenlayers it has the best convergent attributes If there is nohidden layer or there are too many hidden layers theconvergent effect is not so good Theories have proved thatnetworks which have deviations and at least one S-typehidden layer and one linear output layer can approach anynonlinear function That is a three-layer BP network with ahidden layer can approach any nonlinear function

According to empirical formula ℎ = log2119868 ℎ is the

number of nodes of hidden layer and 119868 is the number of nodesof input layer We suppose ℎ = 4

(3) Output Layer The number of nodes of output layer isthe number of system objects We choose one node as theinventory level of March 2013 to be measured

(4) Selection of Initial Value and Threshold Value Becauseboth of them are two random groups of value we choose arandom value between [minus1 1]

(5) Selection of Expected Error and Number of Iterations Wechoose 10000 as the number of iterations and the expectederror is 01

5 Training Process and Experimental Result

This paper uses the neural network tool package of MATLAB76 to program the model for safety inventory level basedon BP neural network In the BP neural network model

established in this paper there are 9 inputs and the number ofneurons is relatively large We preliminarily set the trainingvariables as follows times of training are 10000 trainingtarget is 001 and learning rate is 01 The code and trainingresult is as follows

net trainParam Epochs = 10000

net trainParam goal = 01

LP lr = 01

net-train(net P T)

after 1000 trainings the training is finished

After network finishes training the network getstestedWe use the data ofMarch 2013 to testThe codeof prediction is as follows

119875 test = [05 078 063 1 043 04 025 008 1]

Out = sim (net 119875 test)

By comparing Figures 1 and 2 we can clearly see that theconvergence rate of the improved algorithm is significantlyfaster than that of standard algorithmWe select the data fromFebruary 1 2013 to February 20 2013 to test The result is asfollows

From Table 2 we can know that the improved BPalgorithm is significantly better than that of standard BPalgorithm on convergence rate In addition we also compareour improved BP algorithm with some other improved BPalgorithms The result shows that our BP algorithm also out-performs the other two improved BP algorithms mentionedin the literature review on convergence rate

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Table 3 Comparison of error among standard algorithm other improved algorithms and improved algorithm of this paper

Parameter depiction Standard BPalgorithm

Improved BPalgorithm [26]

Improved BPalgorithm [27]

Improved BP algorithm ofthis paper

Prediction set error 0002687 0000938 0000921 0000780

10minus1

10minus2

10minus3

0 200 400 600 800 1000 1200 1400 1600

1791 epochsStop training

Performance is 000999644 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 1 Training convergence effect of improved algorithm

10minus1

100

10minus2

10minus3

0 1000 2000 3000 4000 5000

5721 epochsStop training

Performance is 000999002 goal is 001

Trai

ning

-blu

eG

oal-b

lack

Figure 2 Training convergence effect of standard algorithm

As prediction accuracy is concerned from Figure 3 wecan know that our improved BP algorithm exceeds signifi-cantly the standard BP algorithm

Suppose 119864 = (12)sum119901(119889119901minus 119910119901)2 is the prediction set

error From Table 3 we can clearly see that our improvedBP algorithm not only exceeds the standard BP algorithmbut also outperforms the other two improved BP algorithmsmentioned in the literature review on prediction effect

01011012013014015016017018

Valu

e

DateActuallyForecast of improved algorithmForecast of traditional algorithm

2-1

2-3

2-5

2-7

2-9

2-11

2-13

2-15

2-17

2-19

Figure 3 Prediction effect of improved algorithm

6 Conclusions

We conclude the following with the practical importance ofour findings First this paper proposes a new fast convergentBP algorithm and deduces new chain propagation rules ofneural network by introducing an error offset Secondly thispaper applies it to the prediction of inventory level of anautomotive parts company and achieves good effect From theexperimental results we can see that using neural networkto predict inventory is effective The improved BP algorithmnot only significantly exceeds the standard algorithm both onconvergence time and prediction effect but also outperformssome other improved BP algorithms on these two mainindicators In this sense this paper provides a valuablereference for inventory control of supply chain However thispaper also has limitations There are still some problems thatneed to be solved such as how to decide the number of nodesof hidden layer and the optimization of whole structure ofnetwork Apart from that the introduction of the error offsetis based on experiencesThe theoretical explanation for it stillneeds to be further discussed All these problems wait to befurther explored in future research

Acknowledgments

This work is supported by the NSFC (71361013 and 71163014)and The Education Department of Jiangxi Province Scienceand Technology Research Projects (11728)

References

[1] P W Balsmeier and W J Voisin ldquoSupply chain management atime-based strategyrdquo Industrial Management vol 38 no 5 pp24ndash27 1996

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

[2] S Minner ldquoMultiple-supplier inventory models in supply chainmanagement a reviewrdquo International Journal of ProductionEconomics vol 81-82 pp 265ndash279 2003

[3] K Bansal S Vadhavkar and A Gupta ldquoBrief applicationdescription A neural networks based forecasting techniquesfor inventory control applicationsrdquoDataMining and KnowledgeDiscovery vol 2 no 1 pp 97ndash102 1998

[4] J Shanmugasundaram M V N Prasad S Vadhavkar and AGupta ldquoUse of recurrent neural networks for strategic datamining of sales informationrdquo MIT Sloan 4347-02 Eller CollegeWorking Paper 1029-05 2002

[5] C C Reyes-Aldasoro A R Ganguly G Lemus and AGupta ldquoA hybrid model based on dynamic programmingneural networks and surrogate value for inventory optimisationapplicationsrdquo Journal of the Operational Research Society vol50 no 1 pp 85ndash94 1999

[6] S RHong S TKim andCOKim ldquoNeural network controllerwith on-line inventory feedback data in RFID-enabled supplychainrdquo International Journal of Production Research vol 48 no9 pp 2613ndash2632 2010

[7] F Y Partovi and M Anandarajan ldquoClassifying inventory usingan artificial neural network approachrdquo Computers and Indus-trial Engineering vol 41 no 4 pp 389ndash404 2002

[8] J Li Y Li J Xu and J Zhang ldquoParallel training algorithm ofBP neural networksrdquo in Proceedings of the 3rd World Congresson Intelligent Control and Automation vol 2 pp 872ndash876 July2000

[9] D E Rumelhart G E Hinton and R J Williams ldquoLearn-ing internal representations by error propagationrdquo in ParallelDistributed Processing Explorations in the Microstructure ofCognition D E Rumelhart and J L McClelland Eds vol 1chapter 8 MIT Press Cambridge Mass USA 1986

[10] N Ampazis and S J Perantonis ldquoTwo highly efficient second-order algorithms for training feedforward networksrdquo IEEETransactions on Neural Networks vol 13 no 5 pp 1064ndash10742002

[11] K Zhang J Xu and J Zhang ldquoA new adaptive inventory controlmethod for supply chains with non-stationary demandrdquo inProceedings of the 25th Control and Decision Conference (CCDCrsquo13) pp 1034ndash1038 Guiyang China May 2013

[12] W P Wang ldquoA neural network model on the forecasting ofinventory risk management of spare partsrdquo in Proceedings ofthe International Conference on Information Technology andManagement Science (ICITMS rsquo12) pp 295ndash302 Springer 2012

[13] A Mansur and T Kuncoro ldquoProduct inventory predictionsat small medium enterprise using market basket analysisapproach-neural networksrdquo Procedia Economics and Financevol 4 pp 312ndash320 2012

[14] Y Huang D X Sun G P Xing and H Chang ldquoCriticalityevaluation for spare parts based on BP neural networkrdquo in Pro-ceedings of the International Conference on Artificial Intelligenceand Computational Intelligence (AICI rsquo10) vol 1 pp 204ndash206October 2010

[15] Z Zheng ldquoReview on development of BP neural networkrdquoShanxi Electronic Technology no 2 pp 90ndash92 2008

[16] H Yu W Q Wu and L Cao ldquoImproved BP algorithm and itsapplicationrdquoComputer Knowledge and Technology vol 19 no 5pp 5256ndash5258 2009

[17] D E Rumelhart G E Hinton and R J Williams ldquoLearningrepresentations by back-propagating errorsrdquo Nature vol 323no 6088 pp 533ndash536 1986

[18] T P Vogl J K Mangis A K Rigler W T Zink and D LAlkon ldquoAccelerating the convergence of the back-propagationmethodrdquo Biological Cybernetics vol 59 no 4-5 pp 257ndash2631988

[19] M Riedmiller and H Braun ldquoDirect adaptive method forfaster backpropagation learning the RPROP Algorithmrdquo inProceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo93) vol 1 pp 586ndash591 San Francisco CalifUSA April 1993

[20] C Charalambous ldquoConjugate gradient algorithm for efficienttraining of artificial neural networksrdquo IEE Proceedings G vol139 no 3 pp 301ndash310 1992

[21] M F Moslashller ldquoA scaled conjugate gradient algorithm for fastsupervised learningrdquoNeural Networks vol 6 no 4 pp 525ndash5331993

[22] F D Foresee and M T Hagan ldquoGauss-Newton approximationto Bayesian learningrdquo in Proceedings of the IEEE InternationalConference on Neural Networks pp 1930ndash1935 June 1997

[23] R Battiti ldquoFirst and second order methods for learningbetween steepest descent and newtonrsquos methodrdquo Neural Com-putation vol 4 no 2 pp 141ndash166 1992

[24] Y Gao ldquoStudy on optimization algorithm of BP neural net-workrdquo Computer Knowledge and Technology vol 29 no 5 pp8248ndash8249 2009

[25] S Shah and F Palmieri ldquoMEKA-A fast local algorithm fortraining feed forward neural networksrdquo in Proceedings of theInternational Joint Conference on Neural Networks pp 41ndash46June 1990

[26] X P Wang Y Shi J B Ruan and H Y Shang ldquoStudy onthe inventory forecasting in supply chains based on roughset theory and improved BP neural networkrdquo in Advances inIntelligent Decision Technologies Smart Innovation Systems andTechnologies vol 4 pp 215ndash225 Springer Berlin Germany2010

[27] H Lican Z Yuhong X Xin and F Fan ldquoPrediction of invest-ment on inventory clearance based on improved BP neuralnetworkrdquo in Proceedings of the 1st International Conference onNetworking and Distributed Computing (ICNDC rsquo10) pp 73ndash75Hangzhou China October 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of