In this paper, we discuss a variant of the Adaptive Neuro-Fuzzy Inference System (ANFIS) to predict two performance attributes, i.e. total cost to serve and return on working capital, following the Supply Chain Operations Reference (SCOR) model on the basis of the state of the art, This variant is based on fractional Tikhonov regularization, a kind of penalized least squares that allows tackling ill-posed problems. Additionally, it does not use backpropagation, grid partitioning or clustering. The numerical experiments revealed the good performance of the approach, encouraging further developments.
The SCOR model itself is not able to adapt proactively to any changes in the system. It can be empowered by adopting artificial intelligence (AI) techniques that have the ability to learn the cause and efect relationships from historical performance data. In [ 9, 10, 11], the performance metrics proposed by the SCOR model are combined with AI techniques in order to have predictive evaluation systems.
In [9], a fuzzy inference system (FIS) was used to represent the cause-and-efect relationships among the SCOR metrics. In [10], a neural network (NN) based model was proposed with the same purpose. Anyway, both systems present a drawback. The fuzzy inference systems require collecting the opinion of experts to tune hundreds of inference rules, while the NN does not seem to be appropriate to support decision making processes under uncertainty. In order to address such issues, in [11], a new supply chain performance evaluation system based on ANFIS was adopted to predict the performance figures of SCOR level-1 metrics based on the values of level-2 metrics. Compared against the above-mentioned methods, ANFIS, which takes into account uncertainty thanks to fuzzy sets, allowed to achieve a greater accuracy of prediction, by using historical data. As discussed in [11], there are in literature some quantitative models for supply chain performance evaluation based on the SCOR metrics and multicriteria decision-making approaches, such as TOPSIS and AHP. Anyway, as their output is a value based on a weighted linear combination of inputs, they are not suitable to takle the relationships between the SCOR level-1 and level-2 metrics; this seems to be an ability of models based on AI techniques [9, 10, 11].
In this paper, we propose a variant of ANFIS to achieve a better accuracy with lower computational cost when evaluating performance attributes of supply chains, according to the SCOR model. This variant is based on a least square approach with fractional Tikhonov regularization, with no backpropagation, no grid partitioning or clustering. This kind of regularization was introduced in [12] to takle discrete ill-posed problems. It is well known that both grid partitioning and scatter partitioning by clustering may have a significant computational cost [ 13]. The aim of the proposed approach is the simplification of the rule base, since the number of rules equals the number of terms, which is fixed as small as possible. The numerical results are promising, encouraging further developments.
The supply chain performance can be regarded as the outcome of supply chain management, taking into account the logistical drivers (facilities, inventory, transportation) and cross functional drivers (information, sourcing and pricing).
The SCOR model has been used for supply chain performance evaluation. It was proposed by the Supply Chain Council to link business processes, best practices, performance metrics, people, and technology into a unified structure [ 14]. It has been widely applied in industry, representing a common reference model. In spite of this, its use in the academic literature is rather limited. SCOR introduces five attributes [17] for performance evaluation which are: • Reliability, that is the capacity to do tasks according to plan; this also implies the predictability of a process’s output; • Responsiveness, the rate at which certain tasks are completed; practically, the time it takes to deliver items to a client; • Agility, the capacity to react to external factors and market changes in order to obtain or retain competitiveness; • Assets, the capacity to make eficient use of resources; • Costs, the costs of running supply chain operations, including the costs of labor, materials, management, and transportation.
In order to measure the success of the implementation of these strategies, the SCOR model uses some level/strategic metrics for each attribute, which are arranged in three hierarchical levels for diagnostic purposes. Level-2 metrics serve as diagnostics for level-1 metrics, implying that the performances of the level-2 metrics are informative for implementing improvements for level-1 metrics. Similarly, level-3 metrics are meant as diagnostics for level-2 metrics.[14]. Some performance attributes and their level-1 and level-2 metrics are – Level-1 metrics: order completion time; – Level-2 metrics: time for sourcing, making, delivery and delivery retail; – Level-1 metrics: return on working capital and fixed assets, cash to cash cycle time; the level-2 metrics on the return are listed in Table 2; • Responsiveness • Assets • Costs
– Level-1 metrics: total cost to serve; the items in Table 1 are level-2 metrics.
In [11], seven ANFIS schemes were used to model the causal relationships defined by SCOR, to estimate the values of level-1 metrics on the basis of level-2 metrics. This model aims to support a predictive diagnosis to identify which level-1 metric(s) underperform and, consequently to take action. Hence, the inputs are level-2 metrics, whereas the output variables represent level-1 metrics. The authors used synthetic data by following [10].
In this work, we focus only on the total cost to serve and return on working capital, by means of two ANFIS models instead of the three ones adopted in [11]. Our ANFIS models adopt fractional Tikhonov regularization, as detailed in the next section. As mentioned before, we have two models: model 1, whose output is the total cost to serve, with eight input variables, and model 2, whose output is the return on working capital, with five input variables (see section 4.1). It is worth noticing that one of the input variables of model 2 is the output of model 1. In [11], an intermediate model was adopted with three inputs and its output used with the total cost to serve as an input to the model to predict the return on working capital. Since our variant more computationally eficient than the standard ANFIS, we could avoid an intermediate model, by keeping a good accuracy. In fact, it is important an accurate prediction of the total cost to serve in order not to afect significantly the prediction of the return.
ANFIS was introduced by Jang [15] to represent the first-order Sugeno (or TSK) fuzzy inference system through a network architecture.
In general, ANFIS implements rules of the form
If 1 is 1 and ... and is then = 0 + 11 + ... + where , = 1, 2, ..., , are fuzzy sets representing linguistic attributes of the input in the -th rule ( = 1, 2, ..., ), and 1 are the unknown consequent parameters.
The ANFIS network structure consists of five layers. The first layer represents the fuzzification stage, returning the membership degrees of each input value into the fuzzy sets . In the second layer those membership degrees are aggregated by means of a product-type t-norm, giving some weights, representing the firing strength of a rule. In the third layer, such weights are normalized, by dividing each one by the sum of all the weights. In the fourth layer, the output 0 is obtained by summing all the partial outputs from the fourth layer. normalized weights are applied to the consequents , by giving partial outputs. The final
The membership function (MF) for can be any suitable parameterized membership function, e.g. the generalized bell-shaped function or the Gaussian function. In our model, we adopted the latter because it has a parameter less than the ones in the generalized bell-shaped MF. The Gaussian MF is () = exp − ︃( ︂( − ︂) 2)︃ where and are the premise parameters.
ANFIS uses a hybrid learning algorithm based both on backpropagation and least-squares (LS) method to determine the unknown parameters. Hence, by using the training data, one obtains a matrix equation such as H = o, where collects the unknown parameters and o the target values. The least-squares (LS) method is formulated as with the solution
min ‖M − o‖2
* = Mo, tune the premise parameters. fractional Tikhonov regularization.
where M = (M M)− 1M is the pseudoinverse of M. Backpropagation is then used to
In the variant herein proposed, the learning algorithm is based only on least squares with
The fractional Tikhonov method formalizes the following minimization problem
min ‖M − o‖2 + ‖ ‖2,
where ‖ ‖ = ( P )0.5 and P is a symmetric positive semi-definite matrix defined as [ 12]
(
When = 1, the method reduces to the standard 2-norm based Tikhonov regularization [12], that is a kind of penalized LS approach.
The solution is [12]:
P = (M M) −2 1 .
* = ∑︁ =1
+1 + (u o)v, where u and v are the vectors (columns of the matrices U and V) coming from the singular value decomposition (SVD) of the matrix M, that is M = USV , being S the diagonal matrix of singular values in decreasing arrangement. The formulas above have been deduced in [12] in the general context of penalized LS approaches, but this kind of regularization was never applied before to ANFIS (to the best authors’ knowledge).
The aim of using such learning algorithm is to avoid backpropagation and grid partitioning. The latter implies that the rules are generated by means of all possible combinations of membership functions of all inputs. This is cumbersome, since the number of fuzzy rules increases exponentially with the number of input variables.
In this section, we describe the data sets, the experiments and the results. We used synthetic data, whose ranges are fixed as suggested in [ 11], according to previous studies. By following [11], data was normalized and the performance values of the level-2 metrics were randomly generated considering the above-mentioned ranges, as detailed in the next subsection. As in [11], the data sets were split into two parts, with 70% of the samples used for the training and 30% for validation. Since, data was generated randomly in [11], we cannot compare our results against the published ones, but we replicated the experiments by using our randomly generated data sets. Hence, we compared the results by ANFIS with fractional regularization (ANFIS-F) against the ones by the standard ANFIS (as used in [11]). The standard ANFIS and the one with fractional regularization use Gaussian MFs. For the sake of completeness, we considered also the more eficient and popular ANFIS with Fuzzy C-Means clustering [ 16]. ANFIS-F was implemented in Scilab. All the experiments were run on a PC with Intel i7 9th generation processor and 16GB RAM. The adopted error measure is the Root Mean Squared Error (RMSE).
As mentioned before, we considered as performance attributes to be predicted the total cost to serve and the return on working capital. The total cost to serve is given by the sum of the direct and indirect costs to deliver products and services to customers, i.e. the sum of planning cost, sourcing cost, material landed cost, production cost, order management cost, fulfilment cost, and returns cost. Its value may vary in the range [2,000,000, 3,530,000] USD. The return on working capital represents the magnitude of investment relative to a company’s working capital position versus the revenue generated from a supply chain. Its value is assumed to vary in the range [-15, 100] %. The input values for the model 1, whose output is the total cost to serve, are listed in Table 1, with the universe of discourse (UoD) and measurement unit (MU). The input values for the model 2, whose output is the return on working capital, along with UoD and MU, are listed in Table 2. As mentioned in section 2, the total cost to serve is an input to predict the return. 4.1.1. Data Generation and Normalization The input and output variable ranges (UoD) for model 1 and model 2 are given in Table 1 and Table 2, respectively. The values were uniformly random generated. The minimum and maximum range was set for each input and output variable. There were 1000 random data points generated for each variable in the model 1, and 500 points for each variable in model 2. The values of each variable were validated to make sure they fall in the given range each time the data points were generated.
The data was normalized using the min-max normalization (in the range [0, 1]). The general formula for min-max normalization is given as: ′ =
− () () − () where ′ is the normalized value and is the original value.
Inspired by a recent work adopting ANFIS to predict the performance attributes for SC management, following the SCOR model, we introduced a more computationally eficient variant of ANFIS for the same problem. This variant is based on an LS approach with Tikhonov fractional regularization. This allowed to get a good accuracy avoiding an intermediate model, adopted in the reference work, and the lowest computational efort even when compared to the popular ANFIS equipped with FCM. In a separate work, we have tested numerically and statistically the good performance of the approach over a number of publicly available datasets. As future work, we plan to develop a multi-input multi-output neuro-fuzzy inference system to include all the SCOR attributes and to apply it to a real-world case study.
(a) (a)
Stefania Tomasiello and Muhammad Uzair acknowledge support from the European Social Fund
through the IT Academy Programme.
International Conference On Cyber Management and Engineering, CyMaEn 2021, 9497309,
2021
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