International Journal of Scientific & Engineering Research, Volume 3, Issue 5, May-2012 1

ISSN 2229-5518

Modified s-S Inventory Model Using Artificial

Neural Network

Samuel S. Udoh, Okure U. Obot, Uduak D. George, Etebong B. Isong

Abstract— The use of Artifical Neural Network (ANN) in modifying and improving the output of exis ting scientific and economic models has been pragmatic in recent time. In this study, we modify the traditional s-S stochastic inventory model by adding the ANN predicted customers demand at lead time to the re-order quantity. The ANN was designed and implemented using Visual Basic 6.0 programming tools and Microsoft Access as the dat a- base. Data collected from petrol mega station were standardized to fit the [0,1] ANN sigmoid transfer function domain. Backpropagation algorithm was used in training the Network. At every re-order point the expected period (days) of arrival of goods was inputted into the system and the quantity of goods to be demanded (Dp ) by the customers before the arrival of new stock was predicted and added to the s-S re-order quantity. The correlation coefficient of 0.97 and 0.58 were obtained for the modified s-S and traditional s-S inventory models respectively. The modified s-S model was found to be 97% efficient in preventing stock out of petrol in a distribution depot.

Index Terms— Artificial neural network, Backpropagation algorithm, Sigmoid transfer function, Stochastic inventory model.

—————————— ——————————

nventories could be viewed as goods or materials, kept for sales, usage or processing. Virtually every segment of the economy relies on some form of inventory for their opera- tions. The ubiquitous nature of inventory has motivated this study with a view of harnessing methods and technology for effective monitoring of inventory levels to ensure continuous

availability of goods with stochastic demand.

Many researchers have worked on inventory models,

as in [1], [3], [2], [5], [7]. Gallego [3] opined that in continuous

review (*Q, R*) policy, the inventory manager continuously monitors the inventory position and places an order of size *Q *whenever the inventory position falls to or below the reorder point *R*. Tajbakhsh [7] observed that the continuous review *(Q,R) *policy is appropriate when inventory levels are re- viewed continuously. Feng *et al *[2] opined that in the case of periodic review, a slight alteration of the continous review policy is required, two levels, *s < S, are *defined and the start- ing inventory at the beginning of a period is set to *u *such that *If u ≤ s*, an order of S *– u *is placed otherwise no order is made.

Artificial Neural Network (ANN) as presented in [6],[8], [9] has been developed as generalizations of mathematical

A popular supervised training algorithm is the back propagation. It uses the historical data to adjust the network’s weight and thresholds so as to minimize the error in its predic- tions on the training set. If the network is properly trained, it has the ability to learn and model the unknown function that relates the input variables to the output variables, and can subsequently be used to make predictions where the output is not known.

Back Propagation Multilayer Perceptron Neural Network (BPMLNN) with supervised training technique proposed in [8] is applied in this research. The Proposed network compris- es three layers. The size of the input, hidden and output layers are 6,3,1 respectively.

PMS_Daily

Demand

1

PMS _Price

x2

P.MS_Availability

models of biological nervous systems. They have the advanta- geous capabilities of learning from training data, recalling memorized information, and generalizing to the unseen pat- terns.

. x3

Environmental

Factor

. 4

Seasonality

x5

o Demand “N”days Ahead

————————————————

Dr. Okure U. Obot is the Head of the Department of Computer Science,

University of Uyo, Akwa Ibom State, Nigeria. Email: abatakure@yahoo.com

Uduak D. George is lecturer in the Department of Computer Science, University of Uyo, Akwa Ibom State, Nigeria. E-mail: udwack@yahoo.com

Alternative

Product

x6

x

Etebong B. Isong is a staff of MIS Department, Akwa Ibom State Water

Company Limited, Uyo, Nigeria. E-mail: etebongisong@yahoo.co.uk

Input Layer (*X*i) Hidden Layer (*h*j) Output Layer (*O*k)

Fig. 1: Artificial neural network schema for petrol demand

Prediction.

IJSER © 2012

The research paper published by IJSER journal is about Modified s-S Inventory Model Using Artificial Neural Network 2

ISSN 2229-5518

The size of the input layer corresponds to the number of data attributes that could influence the aggregate demand of Petrol, “N”days into the future as shown in fig.1. The ANN architec- ture of fig. 1, was trained with a supervised multilayer back propagation training process proposed in [8]. The hidden layer nodes and output layer nodes were computed by equa- tions 1 and 2 respectively.

A total of 161 data set for petrol data set were collected from Pipelines and Product Marketing Company (PPMC) Depot, Port Harcourt. After the computation of the desired output (60 days-ahead-demand), using equation 7, 59 data set were dis- carded for lack of insufficient data to accumulate next 60 days desired demand. 102 data set were therefore available for the research. The 102 available data set were divided into three

hj = f

ok = f

m n

W ji xi j

j 1 i 1

m

wkj h j k

j 1

(1)

(2)

parts: training, validation and testing data set in the ratio of

8:1:1 respectively. The Data were stored in MS-Access 2007, and was operated upon by ANN computer program designed using Visual Basic 6.0, running on Windows Vista operating System on a pentium IV computer system.

Where: hj , Ok are hidden and output nodes respectively

W ji ,

Wkj

are the weights connecting hidden and input layer,

3 **RESULTS AND DISCUSSION**

Output and hidden layers respectively.

xi is the random va-

The proposed inventory model with effect of varying demand of product at lead time is organised such that the

riable of the input layer, j and k are the bias value at the

hidden and output layer node. Errors at the hidden layer and output layers are computed by equations 3 and 4 respec- tively

probability of stockout at lead time is greater than zero. This

could be achieved by increasing the volume of reorder point by volume of safety stock for the next order if the volume at previous reorder point could not sufficiently serve the cus-

e j h j (1

h j )( wkj ek )

(3)

tomers at lead time. Where : Lead time is the time taken for

ek = (dk - ok )( ok)(1- ok) (4)

If the error is greater than a certain predefined value, (0.001) in this case, it is propagated back into the network by means of adjustment of the weights connecting the output to the hidden layers and those connecting the hidden layers to the input lay-

the products to arrive at the depot and safety stock is invento-

ry held to protect against stockout.

In s-S inventory model, two levels, *s < S, are *de-

fined and the starting inventory at the beginning of a period is

set to *u*. The order quantity is given by equation 8

S u if u s

ers. Output and hidden layer weights are adjusted by equa- tions 5 and 6 respectively

Order Quantity =

0,

otherwise

(8)

w ji (n 1)

w ji (n)

e j xi +

wji

(5)

Where: *S *is the maximum storage capacity; *s *is the predefined

inventory level *u *is the starting inventory at the beginning of a

wkj (n 1)

wkj (n)

ek h j +

wkj (6)

period. We modify the *s-S *inventory model as follows:

(*S*

Order Quantity =

u) Dp

if u

s

(9)

Where: *n *is step in the training cycle,

wkj (n)

is the weights

0, *otherwise*

between the output and hidden node at cycle (n ), w ji (n) is the weights between the hidden and input node at cycle (n ),

Where: D p

follows:

is the predicted demand at lead time obtained as

dk and ok are the desired and computed output respectively, ej

and *e*k are hidden and output layer error components , is the momentum parameter, is the Learning rate parameter

From the Artificial Neural Network (ANN) model, the de-

sired demand of petrol *d*k , N days into the future is given as:

N

, wji

is the cumulative change in weights of the hidden

d k = xi

i 1

(10)

and input node,

wkj

is the cumulative change in

where xi is the daily demand of petrol.

weights of the output and the hidden node. The desired out- put *d*k is computed by equation 7

N

d k = xi

i 1

(7)

IJSER © 2012

The research paper published by IJSER journal is about Modified s-S Inventory Model Using Artificial Neural Network 3

ISSN 2229-5518

When the Sum of Squared Error (SSE) from the ANN model

tends to zero, the desired output

d k can be predicted by the

computed output Ok

m

given by:

Ok = f

wkj h j k

j 1

(11)

where *f *is the sigmoid 1tr1ansfer function expressed as:

f ( x)

1

1 *e *x

(12)

We set *y *to be the number of days before the arrival of new stock from the *s-S *inventory model, and thus compute

Fig 3 : Order Quantity Form.

From Fig. 3, the order qunatity of product is computed using

the predicted demand

D y(Ok )

p N

D p as follows:

the modified s-S model of equation 9. The user inputs the

expected delivery period and clicks the Ok button for

automatic computation of the order quantity .

(13)**Table 1 **: Comparison between the traditional *s-S *and

Where: y is expected delivery period

Modified *s-S *Inventory Reorder Quantities.

Ok is the computed demand, *N *days into the future ob-

tained from the ANN model and applied in fig.2.

Fig. 2 : Petrol Inventory Form

IJSER © 2012

The research paper published by IJSER journal is about Modified s-S Inventory Model Using Artificial Neural Network 4

ISSN 2229-5518

Re-Order Qty

(litres)

Expected delivery period (days)

Fig. 4 : Graph of Reorder Quantities of *s-S *and *ms-S *models From the graph of fig.4, both traditional *s-S *and modified *s-S *(*ms-S*) models suggest the same re-order quantity when the expected days of delivery is less than or equals to one (1) day. As the expected delivery day increases, *ms-S *model scales proportionally by increasing the re-order quantity whereas the s-S model maintains almost a constant re-order quantity. The ability of *ms-S *to predict the demand of products at lead time and scale re-order quantity proportionally to expected deli- very period could prevent product stock-out at the depot. Computing the correlation coefficient between the expected period of delivery and the *ms-S *model gives 0.97 whereas *s-S *model gives 0.58, the results show that ms-S model is more efficient in preventing product stockout at the depot.

Traditional s-S inventory model has been modified in this work using Artificial Neural Network (ANN). The modified s-S (*ms-S*) model has been found to be 97% efficient in scaling the re-order quantity of petrol with specified delivery period at the depot compared to the 58% efficiency rendered by the traditional *s-S *model. We recommend the combination of neural network and fuzzy logic computing tools for modifica- tion of the *s-S *inventory model in future research.

The authors wish to thank Professor O. C. Akinyokun and Dr. O. Olabode of the Federal University of Technology Akure, Nigeria for their numerous academic contributions. Also worthy of acknowledgment is Engr. Moses Akang of Nigerian National Petroleum Corporation for his support in data collection.

**[1] **M. Dror and M. Ball “Inventory/routing: Reduction from an Annual to a Short Period Problem,” *Naval Research Logistics Quarterly*, 34(6):891-905, 1987.

[2] Q. Feng, G. Gallego, S. Sethi, H. Yan, H. Zhang “Periodic Re- view Inventory Model with Three Consecutive Delivery Modes and Forecast Updates,” *Journal of Optimization Theory and Appli- cations*, 124 ( 1): 137-155, 2005.

[3] G. Gallego, “ A minmax distribution free procedure for the (*Q*, *R*) Inventory model,” Columbia University, New York, USA . Elsevier: Operations Research Letters, *11(1) *: 55–60, 1992 .

[4] J. J. Hopefield, “ Neural Networks and Physical Systems with emergent Collective Computational Properties,” *Proceedings of the National Academy of Sciences of the USA*, 79:2554-2588, 1982.

[5] R. R. Lindeke, “Inventory Control Models: Uncertainty of

Demand, Production and Operations Management,” *IEEE *3265,

2007.

[6] O. U. Obot., O.C Akinyokun and S.S.Udoh, “Application of Neuro– Fuzzy Expert System for Diagnosis of Hypertension,” *Journal of Ni- geria Computer Society (NCS), *15(2):131-147, 2008.

[7] M. Tajbakhsh ,” On the distribution free continuous-review inventory model with a service level constraint,” Department of Industrial Engi- neering, Dalhousie University, Elsevier: Computers & Industrial Engineer- ing ,59(4): 1022–1024, 2010.

[8] S.S. Udoh and O. Olabode, “ Artificial Neural Network for the Predic- tion of the Demand of Petrol using Periodic Review, s-S, model in a Distribution Network,” *Journal of Applied Mathematical and Computa- tional Sciences (AMCOS)*, 1(2): 151-164, 2010.

[9] S. S. Udoh, “ Prediction of the Demand of Petrol at the Depot,” *An M. Tech Research, Department of Computer Science, Federal University of Technology, Akure. *Pp 100-120, 2009. (Master of Technology Thesis)

IJSER © 2012