An intelligent model for cost prediction in new product development projects

Abstract

Precise cost prediction of new product development (NPD) projects has been a challenge for both academia and practitioners that often requires much effort and experience. In this paper, a combination of particle swarm optimization (PSO), cross validation (CV) and support vector regression (SVR) is proposed to predict the cost of NPD projects. SVR, a novel intelligent technique for time series analysis, can overcome some shortcomings in the conventional approaches; and PSO, a new evolutionary computation technique, is utilized to set the optimal parameters of the SVR. The proposed intelligent model avoids manual selection of these parameters. The PSO solves the difficulty of setting these parameters appropriately and enhances the efficiency and capability of cost prediction. In addition, the CV is employed to train the SVR and improve the reliability of model performance. Then a real dataset of a home appliances manufacturer is provided to illustrate the proposed model and demonstrate the high performance and applicability to cost prediction of the NPD project. Finally, the effectiveness of the support vector model is compared with well-known techniques including multilayer perceptron networks (MLP), normalized radial basis function (NRBF) neural network, and pure SVR in terms of the accuracy measures. Based on the real world dataset, it is observed that the proposed model outperforms other well-known techniques.

Keywords

New product development cost prediction neural networks support vector regression cross validation particle swarm optimization

1 Introduction

New product development (NPD) has become the most important activity for numerous modern organizations. Some researchers believe that NPD is the engine of organizational success, survival, and renewal [1]. Cost prediction of NPD is of the utmost importance for the success of project managers in modern organizations. It often affects management activities such as project planning and resource allocation [2]. NPD costs tend to increase with project uncertainty, complexity and diversity in terms of tasks, resources, participants, and design characteristics [3 –5]. Therefore, accurate cost prediction should be considered during the early stages of industrial research and development.

Previous researchers have developed differentmodels for cost estimation and prediction in NPD projects, particularly in the manufacturing industry. These attempts include time series prediction models such as linear regression analysis and neural networks. Roy et al. [6] described the development of a cost-estimating methodology in an NPD environment for the cost prediction of the engineering design effort during the conceptual phases in Europe. Seo et al. [7] explored an approximate method for providing the preliminary life cycle cost based on artificial neural networks (ANNs). Cavalieri et al. [8] gave comparative results of two different parametric and ANN approaches for cost estimation in a manufacturing firm in Italy. Park and Simpson [9] proposed a production cost estimation framework to support product family design based on activity-based costing in different phases. Niazi et al. [10] reviewed the related literature on manufacturing and product cost estimation. Filomena et al. [11] utilized a target-costing model and described an experience developing early-phase cost parameters for a specific product development process effort at a mid-sized manufacturing company in Brazil. Rogozhinet al. [12] developed modified indirect cost multipliers to estimate the total cost of adding new technology in three studied cases in the automobile industry.

The review of the existing literature indicates that ANNs are the main and commonly used techniques in the NPD environment. ANN has several merits. For instance, it is able to learn a nonlinear mapping between the input and output of a process, and to approximate nonlinear functions to arbitrary accuracy. However, this technique has several inherent demerits. For instance, it needs large samples, but fast changing cost leads to the unreliability of historical data for the new prediction. Moreover, in most neural network applications, prediction accuracy is affected by difficult tasks such as the selection of network architecture, the determination of the number of hidden neurons, local minimum, and uncontrollable generalization capability [13, 14]. Thus, ANNs have some weaknesses and difficulties in estimation in the NPD environment.

Among the AI techniques, support vector machines (SVMs), which were introduced by Vapnik [15], have received much attention with remarkable results[e.g., 16–18]. In this technique, the solution of a nonlinear problem in the original lower dimensional input space can be reduced to finding a linear solution in the higher dimensional feature space in the technique [19, 20]. Currently, support vector regression (SVR) has been introduced to solve nonlinear regression prediction problems. Although the SVR can be regarded as a neural network, some features of the SVR such as its lesser number of adjustable parameters and faster learning speed, are superior to conventional neural networks. The conventional SVR in practice have some difficulties in real life situations [21]. In this regard, significant issue is the selection of parameters in the SVR. This selection as an important decision can have negative impacts on the estimation’s results [17 , 21].

This paper employs a new evolutionary algorithm, PSO, instead of using the common algorithms, i.e., grid algorithm and genetic algorithm due to their shortcomings and difficulties. PSO, proposed by Kennedy and Eberhart [22], was inspired by social behavior in nature such as bird flocking and fish schooling [23]. This is a population-based search algorithm that is initialized with random solutions. Unlike the grid and genetic algorithms, PSO is easy to implement, powerful, and computationally efficient. In addition, there are few parameters to adjust. PSO has been applied successfully to solve multi-dimensional optimization problem particularly function optimization [23].

The purpose of this paper is to improve the performance of nonlinear prediction in the NPD environment by using AI hybrid techniques, namely PSO, cross validation (CV) and SVR. To achieve this purpose, PSO, CV and SVR are integrated into a prediction model for the NPD project cost data. The selection of the parameters in the SVR technique is optimized by using a PSO algorithm concurrently. In addition, the CV technique is applied to the training dataset in the SVR not only to avoid over fitting but also to improve the reliability of the model performance. This proposed support vector model is validated by using a real dataset gathered from a case study in a manufacturing industry for predictingthe cost of NPD projects. In addition, comparisons are made among the proposed model and some other well-known techniques to highlight its applicability and suitability.

The remainder of this paper is structured as follows. In the next Section, the basic principles and formulations of SVR, CV, and PSO are reviewed. In Section 3, the steps of the proposed artificial intelligent (AI) model are presented for predicting the cost of an NPD project. Next, the datasets and data preparation for the empirical study are described in Section 4. In Section 5, the prediction result is illustrated and the comparison among three well-known techniques is summarized. Finally, conclusions and future research are provided in the last Section.

2 A brief overview of the techniques employed

Cost estimation and prediction is a major objective in many areas of industrial and production research, particularly NPD projects. To achieve this objective, a suitable mathematical model for cost data should be found and then future values in the time series should be predicted based on the established patterns, factors, and other related series [24, 25]. Prediction of cost time series data produced by using linear techniques cannot often be conducted accurately and precisely. Generally, real applications are not amenable to linear prediction techniques [26]. To overcome the shortcoming of the commonly used techniques, this paper focuses on the cost prediction of these projects by introducing a hybrid intelligent model based on three powerful techniques as described below.

2.1 Support vector regression

SVR is a new training technique based on SVM; however, it requires only the solution of a set of linear equations instead of the long and computationally hard quadratic programming problem involved in the standard SVM. In fact, the SVR works with a least squares cost function [27]. In the following, a brief introduction to SVR is provided.

Consider a given training set ${X_{i}, y_{i}}_{i = 1}^{N}$ , where X _i ∈ R ^P represents a p-dimensional input vector and y _i ∈ R is a scalar measured output, which represents the y _i model output. The objective is to construct a function y = f (x) which represents the dependence of the output y _i on the input x _i [28]. We define the form of this function as $y = W^{T} (X) + b,$ (1) where w is the weight vector and b is the bias term. This paper constructs this function to predict the cost of NPD projects. The regression model (1) can be constructed by using a nonlinear mapping function φ (.). By mapping the original input data onto a high–dimensional space, the nonlinear separable problem becomes linearly separable in the space. The function φ (.) : R ^p → R ^h is a nonlinear function, which maps the data into a higher-possibly infinite-dimensional feature space as illustrated in Fig. 1 [29].

The optimization problem and the equality constraints are defined by the following equations: $\begin{matrix} min J (w, e) = \frac{1}{2} w^{T} w + C \frac{1}{2} \sum_{i = 1}^{N} e_{i}^{2} \\ s . t . \\ y_{i} = w^{T} (x_{i}) + b + e_{i}, i = 1, 2, \dots, N, \end{matrix}$ (2) where e _i is the random error and C ∈ R ⁺ is a regularization parameter in optimizing the trade-off between minimizing the training errors and minimizing the model’s complexity. Now, the objective is to find the optimal parameters that minimize the prediction error of the regression model (2). The optimal model will be chosen by minimizing the cost function (3) where the errors e _i are minimized [27]. This formulation corresponds to the regression in the feature space and, since the dimension of the feature space is high (possibly infinite), this problem is difficult to solve [28]. Hence, the following Lagrange function is constructed to solve the optimization problem.

$\begin{matrix} L (w, b, e; α) = J (w, e) \\ - \sum_{i = 1}^{N} α_{i} {w^{T} (x_{i}) + b + e_{i} - y_{i}}, \end{matrix}$ (3) where α = [α ₁, α ₂, …, α _n] are the Lagrange multipliers. The solution of relation (3) can be obtained by taking partial derivatives with respect to w, b, e _i and α _i, i.e., $\frac{\partial L}{\partial w} = 0 \to w = \sum_{i = 1}^{N} α_{i} (x_{i}),$ (4) $\frac{\partial L}{\partial b} = 0 \to b = \sum_{i = 1}^{N} α_{i} = 0,$ (5) $\frac{\partial L}{\partial e_{i}} = 0 \to α_{i} = {Ce}_{i}, i = 1, 2, \dots, N,$ (6)

$\begin{matrix} \frac{\partial L}{\partial α_{i}} & = & 0 \to w^{T} (x_{i}) + b + e_{i} - y_{i} = 0, \\ i & = & 1, 2, \dots, N . \end{matrix}$ (7)

The above equations in matrix form can be expressed as $[\begin{matrix} 0 & {\vec{1}}^{T} \\ \vec{1} & Ω + C^{- 1} I \end{matrix}] \times [\begin{matrix} b \\ α \end{matrix}] = [\begin{matrix} 0 \\ y \end{matrix}],$ (8) where $\begin{matrix} y^{T} = [y_{1}, y_{2}, \dots, y_{N}], \\ {\vec{1}}^{T} = [1, 1, \dots, 1], \\ α^{T} = [α_{1}, α_{2}, \dots, α_{N}], \\ Ω_{ij} = K (x_{i}), x_{j} = φ^{T} (x_{i}) φ (x_{j}), j = 1, 2, \dots, N, \end{matrix}$ and where (K (x _i, x _j)) is the kernel function and Ω is the kernel matrix. Finally, the estimated values of b and α _i, i.e., $\hat{b}$ and ${\hat{α}}_{i}$ , can be obtained by solving the linear system (8), and the resulting SVR model can be expressed as $y = f (x) = \sum_{i = 1}^{N} {\hat{α}}_{i} K (x, x_{i}) + \hat{b},$ (9) where K (x_i, x_j) is a kernel function. In this paper, a nonlinear radial basis function (RBF) kernel is considered: $K (x, x_{i}) = exp (- \frac{1}{2 σ^{2}} {∥ x - x_{i} ∥}^{2}),$ (10) where σ is the parameter of the RBF Kernel. In comparison with some other feasible kernel functions, the RBF is a more compact supported kernel and able to shorten the computational burden of training process. It also improves the generalization performance of SVR [29]. To achieve a high level of performance with SVR models, the regularization parameter C and the value of σ from the kernel function have to be set carefully.

2.2 Cross validation technique

The CV technique was introduced for estimating generalization error based on resampling [30, 31]. In general, the resulting estimates of generalization error are utilized for selecting among various models, such as different network architectures. In k-fold CV as the common type of CV technique, the data are randomly divided into k mutually exclusive subsets of approximately equal size. Among k–1 subsets, the last subset is regarded as the validation. This procedure is repeated k times, so that each subset is used once for testing. If k equals the data sample size, this is called the leave-one-out CV. In practice, the choice of the number of folds (k) depends on the size of the data. For a small data, it may be better to set a larger value for k because this leaves more examples in the training set. k = 10 is a common choice for k-fold in many real world applications [26]. The main advantage of the k-fold CV technique is to provide the best compromise between computational cost and reliable parameter estimation. It is also been applied in numerous fields of time series forecasting [26, 32].

2.3 Particle swarm optimization

PSO is a population-based meta-heuristic and evolutionary computation technique. This technique simulates social behavior among individuals to achieve precise objectives in an n-dimensional search space. PSO is an evolutionary algorithm that performs searches by using a population (called swarm) of individuals (called particles) by updating generations. To discover the optimal solution, each particle i maintains a record of the position of its previous best experience in a vector called pbest. The nbest is another best value that is tracked by the particle swarm optimizer. So far, this is the best value obtained by any particle in that particle’s neighborhood. When a particle takes the entire population as its topological neighbors, the best experience is a global best, called gbest (i.e., the best experience of all other members) [33, 34]. All particles can share information about the search space. The pbest is called the cognition part, and the gbest the social part [33]. The main features of PSO are [22 , 34]:

The technique only utilizes the performance index to illustrate the search in the problem space. Therefore, it is more appropriate for dealing with non-differentiable objective functions. This property relieves the PSO of certain assumptions on objective function, which are often needed in traditional optimization techniques.

The technique is a population-based search algorithm. This feature ensures that PSO is less likely to become trapped in a local minimum.

The technique applies probabilistic transition rules rather than deterministic rules. Hence, the PSO is a kind of stochastic optimization algorithm that is able to search a complicated and uncertain area.

The technique has the flexibility to control the balance between global and local exploration of the search space. This unique feature of PSO can overcome the premature problem and improve the search capability.

As mentioned above, to find the optimal solution, each particle moves in the direction of its pbest and gbest. For each particle i and dimension j, the velocity and position of particles can be updated by the following formulas: $\begin{matrix} v_{ij}^{t + 1} = w \cdot v_{ij}^{t} + c_{1} \cdot {rand}_{1} \cdot ({pbest}_{ij}^{t} - p_{ij}^{t}) \\ + c_{2} \cdot {rand}_{2} \cdot ({gbest}_{ij}^{t} - x_{ij}^{t}) \end{matrix}$ $x_{ij}^{t + 1} = x_{ij}^{t} + β \cdot v_{ij}^{t + 1}$

In the above equation, t is the evolutionary generation. v_ij is the velocity of particle i on dimension j, whose value is limited to the range [–v_max, v_max], x _ij is the position of particle i on dimension j, whose value is limited to the range [–x _max, x _max]. The inertia weight w is applied to balance the global exploration and local exploitation. The rand ₁ and rand ₂ are random functions in the range [0, 1], β is a constraint factor used to control the velocity weight, whose value is usually set to 1. The positive constants c ₁ and c ₂ are the personal and social learning factors, respectively. Their values are usually set to 2. According to whether a designated value of the fitness or a maximum generation is reached, the termination criterion is determined for iterations. Detailed descriptions of PSO can be found in [23].

3 Proposed support vector model

The proposed model includes three different techniques, namely PSO, CV and SVR. In this model, SVR acts as a supervised learning tool to handle input–output mapping and is focused on cost data characteristics in NPD projects, and PSO works to optimize the SVR parameters. In fact, the generalization capability and predictive accuracy of the SVR are determined by the problem parameters, including free parameters (i.e., C and σ). To select these parameters, most researchers follow the trial-and-error procedure. However, this procedure requires some luck and often is time-consuming [32]. This paper solves the above-mentioned shortcomings. Moreover, the k-fold CV is utilized to train the SVR in order to reach a more realistic evaluation of the accuracy by dividing the total dataset into multiple training and test sets, and to provide the reliable results. The proposed intelligent model is a computationally efficient combination and is helpful in the cost prediction of NPD projects. An explanation of the major steps involved in the model is provided as below:

Step 1. Training data: The model applies sequential data as training data. In this step, the sequential data reflects the identified attributes, and the training data are normalized into the same range (0, 1), which helps avoid numerical difficulties. The function utilized to normalize data is illustrated in relation (13): $x_{sca} = \frac{x_{i} - x_{min}}{x_{max} - x_{min}},$

Therefore, after effecting this transformation, all the variables become dimensionless.

Step 2. Training the SVR: In this step, SVR is deployed to handle input–output mapping. The RBF kernel is applied as reasonable choice [35]. By using the k-fold CV technique on the training dataset, the SVR training is performed to obtain the prediction model.

Fitness definition: According to [32] the fitness of the training dataset can be obtained easily, but is prone to over-fitting. To handle this problem the k-fold CV technique is utilized. In this technique, after randomly dividing the training dataset into k subsets, the regression function is built with a given set of parameters (C _i, σ _i) by using k–1 subsets as the training set. The last subset is considered the validation. The above procedure is repeated k times. Consequently, the fitness function is defined as the MAPE_CV of the k-fold CV technique on the training dataset: $Fitness = min f = {MAPE}_{cv},$ $MAP E_{CV} = \frac{1}{l} \sum_{j = 1}^{l} |\frac{y_{j} - {\hat{y}}_{j}}{y_{j}}| \times 100$

where y _j is the actual value; ${\hat{y}}_{j}$ is the validation value and l is the number of subsets. The solution with a smaller MAPE_CV of the training dataset has a smaller fitness value.

In addition, the SVR utilizes free parameters which are randomly generated and employed by a PSO. By using a PSO, the intelligent model generates a population of particles and then evaluates it according to the SVR. This evolutionary algorithm is simple and straightforward for the SVR.

Step 3. Determining parameters by the PSO: In this step, the PSO is utilized to search for the SVR parameters concurrently. The sub-steps of the PSO are described below:

Step 3-1. Initialization: Initialize a defined population of particle pairs (C _i, σ _i) with random positions (x _Ci, xσ _i) and velocities (v _Ci, vσ _i), where each particle contains n variables.

Step 3-2. Fitness evaluation: In this step, the fitness value for each particle pair is evaluated by using relations (14) and (15). Set the best position (x _Ci, xσ _i) of each particle pair and its objective value equal to its initial position and objective value respectively. Set the global best position (x _Ci, xσ _i) and its objective value equal to the best initial particle pair’s position and its objective value.

Step 3-3. Update the global and personal best values: The global and personal best according to the fitness evaluation results are updated. If the current value is better, then update its objective value with the current position and objective value.

Step 3-4. Calculate the velocity of position change: Each particle flies toward a new position by obtaining the velocity of position change. The velocity of each particle is obtained by relation (11).

Step 3-5. Update position value: Each particle moves to its next position according to relation (12).

Step 3-6. Stopping criteria: The same procedures from Step 3-2 to Step 3-5 are repeated until stopping criteria (e.g., prediction accuracy and maximum iterations predefined) are satisfied. It means that near optimal parameters are obtained.

Figure 2 depicts the framework of the proposed intelligent model. This model is applied to seek a better combination of the parameters in the SVR so that a smaller MAPE_CV is obtained during prediction iterations. The main merit of the proposed model is to achieve not only higher solution accuracy with minimum incremental computational burden but also a faster speed of convergence with reliable results.

4 Application for predicting the cost of NPD projects

Prediction of cost data in NPD projects is a time series problem. Our objective is to present the prediction intelligent model with generalization capability by using the training sample provided by historical data in the NPD environment. The SVR mapping function can be written as $u_{t} = f (u_{t - 1}, u_{t - 2}, \dots, u_{t - d}),$ where u _t is the output value at time t, (u _t-1, u _t-2, …, u _t-d) is the input vector, and d is the dimension of the input vector.

This paper employed the cost data of an NPD project in a home appliances manufacturer in Iran. The studied company manufactures a wide range of home appliance products in its factory near Tehran. In addition, this company markets its products in various other cities. Every year, the company identifies market requirements and comes up a list of potential NPD projects. In this section, we do not supply more NPD project details since the subject company has reserved the information as confidential.

Executing the appropriate NPD project is one of the main factors that determine a company’s success. On the other hand, cost estimation and prediction is crucial in the early phases of these projects. Therefore, it is imperative to have a robust approach to predict the cost of NPD projects. The company considers a conceptual and operational model for moving a NPD project from idea to launch. The factors influencing cost prediction of the NPD project are given as follows:(1) NPD project complexity; (2) Technological requirements; (3) NPD project information; (4) NPD project team requirements; (5) NPD project duration; and (6) Market requirements.

To manage the new product process to improve effectiveness and efficiency, it breaks the new product project into identifiable and discrete phases (e.g., five or six phases). Each phase is designed to collect information needed to move the project forward to the next phase and/or decision point. Typically, these phases include key activities such as [36]: (1) Initial screening; (2) Preliminary marketing assessment; (3) Preliminary technical assessment; (4) Detailed market study; (5) Business/financial analysis; (6) Product development; (7) In-house product testing; (8) Customer tests of product; (9) Test market/trial sell; (10) Trial production; (11) Pre-commercialization business analysis; (12) Production start-up; and (13) Market launch.

4.1 Dataset description

In this section, in order to test the effectiveness of the proposed intelligent model, we applied it to the cost data of the NPD project. The past cost data of the project as input parameters, whether in time series format or not, have been included as attributes in the input parameters. That is to introduce the concept of time series into the proposed model. Hence, including estimation time series information in the input parameters will help predict the actual cost of the project. The experimental data should be divided into two subsets: the training dataset and the test dataset. Furthermore, the dataset should be sufficient to provide suitable training and test sets.

Cost prediction of the NPD project is the process of developing an approximation of the monetary resources required to complete the project activities [36, 37]. In fact, the prediction is based on the detailed analysis of the project activities by considering the new product features. In general, the predicted cost of the new product is calculated in an analytical way as the sum of each main activity and tasks that is constituted by the value of the resources employed in each step of the manufacturing process (e.g., raw materials, labor and equipment). For this purpose, this NPD project is updated week by week, and the cost of each key activity is calculated from its resources including labor, materials, and equipment usage during the week. In addition, 13 key activities are divided into 131 tasks in this project, and their costs are expressed in dollars. The cumulative cost data is employed to predict the time series and to facilitate the cash flow analysis.

The basic idea for the data collection is based on the concept of the project’ S-curve in an NPD environment. Since the NPD project is subject to numerous uncertain factors concerning project conditions and resources, the reasonableness of costs and S-curve should be examined. The S-curve graphically illustrates the cumulative cost progress of the NPD project over the project duration. The top manager can utilize the S-curve for the NPD project planning and control, as it provides the basis for predicting the cash flow for making financial arrangements before production and sets a target for overall progress evaluation during production.

The actual progress in terms of costs achieved so far is considered after the NPD project starts for producing an early S-curve prediction. Consequently, the data of the S-curve is updated during test and trial production phases. Table 1 illustrates the real cost dataset which is divided into a training and a test dataset in shares of 70% and 30% . Accordingly, of the 128 rows of dataset available for the project, 90 rows are selected to be the training data and 38 rows for testing the proposed model.

To obtain the project S-curve, in the training period, a rolling-based prediction procedure is conducted (see Table 1), which divides the training data into two subsets, namely fed-in (3 cost data) and fed-out (1 cost data), respectively. First, the primary 3 cost data of the fed-in subset are fed into the proposed model and the structural risk minimization principle is utilized to minimize the training error. Then, the one-step ahead prediction cost, namely the 4th predicting cost, is obtained. Second, the next 4 cost data, including 3 of the fed-in subset data (from 2nd to 3rd) pulsing the 4th data in the fed-out subset, are similarly fed into the model and the structural risk minimization principle is also employed to minimize the training error. Then, the one-step ahead predicting cost, namely the 5th predicting cost, is performed. This prediction procedure is repeated until the 131st predicting cost is obtained. Meanwhile, the training error in the training period is also provided.

4.2 Model validation and comparison results

In this section, the performance of the proposed intelligent model is validated on the real world cost prediction problem. For this purpose, a PC is utilized with the following features: Intel Pentium IV 3.0 GHz CPU, 3 MB RAM, and Windows Vista operating system.

Before constructing the training cost dataset, the primary cost data should be preprocessed. The training data is processed smoothly to eliminate singular values. The experimental data including the training cost data and test cost data are normalized, which can improve the generalization ability of AI prediction techniques. The training cost datasets are utilized to construct training sample sets according to the dimension of the input vector. As mentioned above, the dimension of the input vector is set to 3.

The k-fold CV technique is used in the cost dataset of the NPD project. In this paper, the value of k is set to 10. Hence, the training data is divided into 10 subsets, with each subset of the data sharing the same proportion of each subset of data. Nine subsets are employed in the training process, while the last one is used in the testing process [26]. It is repeated 10 times in order that each subset takes a turn as the test data. The MAPE_CV in this classification is calculated by averaging the individual error estimation for each run of testing. The advantages of CV are that all test subsets are independent and this leads to the robust prediction results.

The cost data of the NPD project in a time series are fed into the proposed model to predict cost data in the next test period. While improvement in the training errors occurs, two parameters of the SVR (i.e., C and σ) are optimized by the PSO concurrently. The adjusted parameters possessing the smallest test MAPE_CV value are selected as the most appropriate parameters in the case studied. Then, the test dataset is utilized to investigate the accuracy of the prediction error.

To set the parameters of the PSO, the parameters in the proposed model for the real dataset are experimentally set. The population size is 15 and the number of iterations is fixed as 100. The searching ranges for C and σ are as follows: C∈ [0, 10⁶] and σ∈ [0.001, 3000]. v _max for C and σ particles is clamped to be 10% of their search space in the NPD environment. In addition, preliminary experiments let us set the personal and social learning factors (c ₁, c ₂) = (2, 2) that achieve better accuracy. The accuracy weight w is adjusted to 0.998. The setting of these parameters of the PSO is summarized in Table 2. The optimal values of SVR parameters obtained by the proposed model are C = 20.2570 × 10⁴ and σ = 60.95. Figure 3 depicts the curve for the optimal fitness in the training stage arising from the number of generations and exhibits no significant improvements after 30 iterations.

In Fig. 4, the cost dataset and output of the proposed model and other well-known techniques are depicted. As we expected, the final prediction result of the proposed model has the minimal accuracy measures, which indicates its higher performance and applicability to cost prediction in the NPD project. The computational results also illustrate that the model outperforms the other foregoing techniques applied to this study as given in Table 3. Consequently, top managers can utilize the proposed support vector model to predict the cost data of NPD projects during the project life cycle because of its strong global search capability, ease of use, and economic benefits in the NPD environment.

5 Conclusions and future research

Project cost prediction is an important topic of wide interest in the NPD environment since it can havesignificant effects on the project’s success. Support vector machine (SVM)-based models have been utilized for the problem of cost data estimation. In recent years, support vector regression (SVR), a novel neural network technique based on SVM, has been compared with conventional techniques such as ANNs and nonlinear regression. The SVR technique has demonstrated great potential and high performance results. However, to build reliable and robust intelligent prediction models, the parameters of the SVR should be carefully set. In the related literature, limited attempts have dealt with the combination of particle swarm optimization (PSO) and SVR, although there is an excellent prediction capability for useful applications in the manufacturing industry. This paper has focused on the improvement of the SVR model by using the technique of evolutionary computation and k-fold cross validation (CV). Therefore, a hybrid AI modeling approach has been proposed for predicting the cost data of NPD projects based on time series. The main advantage of the proposed model is that, unlike conventional neural networks, it is computationally efficient because the SVR training only requires the solution of a set of linear equations. A k-fold CV is utilized to train the SVR leading to reliable and stable results. Moreover, the selection of the SVR parametersis optimized by applying the PSO concurrently. Thus, the computation time of optimization is drastically reduced. Finally, this hybrid technique can avoid over-fitting or under-fitting in the SVM-based model. A case study of a home appliance manufacturer in Iran has been presented in this study. The factors influencing cost prediction of the NPD project have been considered, and the detailed analysis of the project activities in terms of costs by the S-curve has been employed as the real dataset to demonstrate the utilization and performance of the model. Further, the proposed model has been compared with three well-known techniques for cost prediction in the NPD project. The numerical results obtained for a nonlinear time series of the real cost data have demonstrated the superior performance of the proposed support vector model against some other well-known techniques, in terms of lower prediction error and better generalization capabilities. For further research, deciding the number of input patterns in the SVR technique structurally, and optimizing other kernel functions can be suggested. In addition, to verify and extend the proposed model, results provided by public datasets and real-world applications may be tested in the future.

Footnotes

Acknowledgments

The authors would like to thank anonymous referees for their useful comments on this paper.

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