A model of inventory management using neural networks.

Ciupan, Emilia ; Morar, Liviu ; Ciupan, Cornel 等

Abstract: The paper presents a model of inventory management. The ordering size is determined taking in account the sales history, duration of deliveries, packing restriction, trend of consumption. The consumption is studied in a larger interval divided into smaller intervals. Considering the consumption in each subinterval will determinate the average balanced consumption and the trend. The ordering size and the command point will be determined in according to the average balanced consumption, trend and transport restrictions.

Key words: management, inventory, consumption, neural network.

1. INTRODUCTION

This study proposes to create a model of inventory management specific for commercial societies which activity domain is selling.

In this case, a good management of inventory could be decisive. The distribution companies have to work with inventories, the inventory rupture being unwanted.

To determine the optimal batch and the starting-up moment of the order launching are the most crucial steps in an inventory management system.

Many works (Abrudan & Candea, 2002), (Hermanson et al., 1992) present the theoretical basis of inventory management. In the paper (Ciupan, 2005) was realized an analytical model for the inventory management. A distribution company which activity domain is IT equipments used this model.

The analytical model offers good results for companies that have continuous and complete inventory reports. We conceived a non-analytical model based on neural networks in the case when there is not a complete inventory report. The novelty consists in the development of a soft and activity study.

In the future research, we will made case studies in order to find applications for the model.

2. ANALYTICAL MODEL DESCRIPTION

This model assumes the following data as known:

- the inventory level at any given moment (St); - the history of sales; - issued and unreceived orders (Cd); - duration of deliveries (d); - the inventory rupture is prohibited; - issuing costs, maintenance costs are not considered.

We have to determine the stock command point (s) and the batch (Q). We use the history of sales on a significant time span ([T.sub.s]) to determine the average balanced consumption ([C.sub.smed]) and the trend of sales ([theta]). To make a simple balance of the consumption of [T.sub.s] we divide it in n equal intervals (days, weeks, months etc.): [t.sub.1], [t.sub.2], ... , [t.sub.n].

The average balanced consumption is determined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where:

[c.sub.i] - the consumption in the interval [t.sub.i] [p.sub.i] - consumption weight in interval [t.sub.i]

We consider the time span t1 as the first chronological times span of [T.sub.s] and [t.sub.n] the most recent one.

For trend determination, we consider k intervals: [t.sub.n], [t.sub.n-1], ... , [t.sub.n-k+1]

The trend [theta] is determined the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where 1 < p < k. The command point s is determined:

s = [mu] x [theta] x [C.sub.smed] x d (3)

The size of demand Q' is determined based on the average balanced consumption [C.sub.smed], the trend [theta], depending on the inventory level St and unreceived orders [C.sub.d]:

Q' = [mu] x [theta] x [C.sub.smed] x (d + 1) - ([S.sub.t] + [C.sub.d]) (4)

where [mu] is a further adjustment coefficient.

The adjustment coefficient may express a correction of an unspecific consumption due to an exceptional situation (past or future). A restriction about the determination of the batch is bound to the minimal batch accepted by the supplier.

We adjust the size of demand Q' in according to the packing restriction [Q.sub.min]:

Q = round (Q' / [Q.sub.min]) x [Q.sub.min] (5)

3. NEURAL NETWORK MODEL

On the other hand, we can try another approach of this activity. We consider a non-analytical approach, a model using neural networks that we will study using the designed soft. This model consists of a three-layer perceptron (fig. 1).

The layers are the following:

1. the input layer consisting of 5 neurons corresponding to the input variables [C.sub.smed], [theta], [S.sub.t], [C.sub.d], d;

2. the hidden layer consisting of variable number of neurons (we test the network using 4 and 9 neurons);

3. the output layer consisting of two neurons corresponding to the output variables s and Q.

[FIGURE 1 OMITTED]

In a first stage, we will initialize the neural network mentioned above with random values of weights for hidden and output layers neurons. In the second stage, we will train the network using a back propagation algorithm and a set of training data (Dumitrescu & Hariton, 1996; Zilouchian & Jamshidi, 2001).

The set of training data consists of a set of input values and the corresponding desired outputs. We train the network until we reach a minimum error or a fixed number of epochs.

We developed simulation software that uses neural networks for the study of the model. In addition, we can use this software for other more general applications. The soft is useful to create, teach and simulate a three-layer network. The data are introduced through strings of values. The programme allows the selection of different activation functions for the networks layers. This example uses the sigmoid activation function for the hidden layer and the purelin function for the output layer. The interface is user friendly, easy to use and the parameters of the network may be set by selection. The programme also has buttons necessary for an easy handling.

4. SIMULATION

We used statistical data from the company that implemented the analytical model for the teaching of the network. The inputs matrix from the training set is presented in table 1. A column of the matrix corresponds to an input vector. For this case we consider the duration of deliveries d=1.

The analytical model generates a set of data for the validation of the model. The analytical desired outputs ADR are shown in table 2. Two models were developed with the aid of the software: Net1, based on a four neurons in the intermediate layer and Net2, with nine neurons in the intermediate layer. The training errors, after 100 epochs are: 4.97 for Net1 and 3.86 for Net2. The desired outputs obtained with the nonanalytical models are shown in columns afferent to Net1 and Net2 in table 2. The errors of the created models were determined in comparison with the analytical model, the results being shown in table 2.

5. CONCLUSION

The created models give acceptable results for the products for which the input variables are relatively uniform distributed in the training interval. Good results are obtained when the values from the training data set are not dispersed in a large value interval. Very good results are obtained when the cardinal of the training data set is large enough. The number of nods in the intermediate layer does necessarily not assure good results. In the present example, the 4-branch point network in the hidden layer offers approximately the same results as the 9-point branch network.

The teaching instruction of the network depends very much on the statistical weight randomly generated in the initialization phase of the network.

A grouping and a development of models for each class of products is necessary in order to manage a wide range.

6. REFERENCES

Abrudan, I. & Candea, D. (2002). Engineering and management of manufacturing systems, Ed. Dacia, ISBN 973-35-1588-4, Cluj-Napoca

Ciupan, E. (2005). MteM, Procedings of 7Th International Conference Modern Technologies in Manufacturing, Gyenge, Cs. Ed. Mures, pp. 145-148, ISBN 973-9087-83-3, Cluj-Napoca, October, 2005

Dumitrescu, D & Hariton, C. (1996). Neural networks-theory and applications, Ed. Teora, ISBN 973-601-461-4, Bucuresti

Hermanson, R.; Edwards, J. & Maher, M. (1992). Accounting principles. IRWIN INC. BOSTON, ISBN 0-256-08916-7

Zilouchian, A & Jamshidi, M. (2001). Intelligent Control Systems using Soft Computing Methodologies, CRC Press LLC, ISBN 0-8493-1875-0

Table 1. Input data of training

 [theta]
No [C.sub.smed] [%] [S.sub.t] [C.sub.d] s Q

1 125 98 95 78 123 72
2 119 93 94 65 110 62
3 110 88 92 51 97 51
4 100 84 91 38 84 39
5 90 82 91 27 74 29
6 81 81 90 20 66 21
7 75 83 90 17 62 17
8 72 87 90 18 62 17
9 72 94 91 24 68 21
10 75 103 91 35 77 29
11 81 111 92 50 91 39
12 90 117 94 68 106 50
13 100 120 95 84 120 61
14 110 120 97 97 132 70
15 119 118 98 103 139 77
16 125 114 100 102 142 82
17 128 109 102 95 140 82
18 128 104 103 84 133 79
19 125 98 105 68 123 72
20 119 93 106 52 110 62

Table 2. Results data

ADR Net1 Net2

s Q [s.sub.1] [Q.sub.1] [s.sub.2] [Q.sub.2]

128 76 130 73 130 73
104 57 106 54 103 56
79 34 79 32 78 35
63 19 63 19 64 19
64 18 63 19 66 20
84 34 83 35 83 33
113 56 111 58 117 56
136 74 133 75 136 78
141 82 140 81 138 80
137 81 137 79 136 78
117 67 120 65 118 68
90 45 92 43 92 48
69 25 70 25 70 25
STDEV

ADR Error Net1 Error Net2

 [s.sub.1]-s [Q.sub.1]-Q [s.sub.2]-s [Q.sub.2]-Q
s Q [%] [%] [%] [%]

128 76 1,5 -3,6 1,5 -3,5
104 57 2,3 -4,9 -0,6 -1,2
79 34 0,4 -6,6 -0,9 2,6
63 19 -0,6 0 0,9 0
64 18 -2,2 5,3 2,5 11,1
84 34 -0,8 2,9 -0,8 -2,9
113 56 -1,9 3,4 3,4 0
136 74 -2,3 1,3 -0,1 5,4
141 82 -1 -1,2 -2,4 -2,4
137 81 0,2 -2,5 -0,5 -3,7
117 67 2,8 -3,1 1,2 1,5
90 45 1,6 -4,7 1,7 6,7
69 25 0,9 0 0,9 0
STDEV 1,699 3,597 1,589 4,412