# The Simulation of Implications of Sensor Technology on the New Product Development to Solve Lot-Sizing Problems with Fuzzy Approach

Authors: Ford Lumban Gaol; Tokuro Matsuo
Journal: Journal of Sensors
 (2020)

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## Abstract

Implications of sensor technology on new product introduction requires critical decisions on divergent marketing and operational aspects. In this paper, we employed the Bass model for a new product that used the sensor technology diffusion from the perspective of the Wagner-Whitin lot-sizing model which discusses this matter related to suitability of sensor technology on the market and operational dimensions. The objective of this study is to propose an uncertain fuzzy approach to specify the new product that applied the sensor technology introduction’s optimum time, price, production scheduling, and rate simultaneously. This model will be applied to analyze the impact of sensor technology diffusion on the market acceptance and some operational parameters, for instance, total customer population, price elasticities, startup and maintenance costs, the unit’s variable cost and research and development costs on optimized benefit, and the product’s optimized lifespan. In this experiment, we have inspiration from the version of the Sale et al. (2017) model which considers uncertainty by using fuzzy triangular numbers and performing the alpha cut method. Moreover, in this research, variable cost is considered as a Cobb-Douglas function. Initially, in Research Methodology, mathematical modeling is applied, wherein utilizing simulation and the experimental design method, data were generated. Eventually, using LINGO optimization software, the problem was solved, and for further examinations considering the existence of various parameters, design of experiment (DoE) by Design-Expert 8 software and response surface methodology was adopted to analyze and optimize the problem’s parameters. The results indicated that the objective function’s climax occurs when the higher limit of alpha being 0.7 is assumed; hence, the optimum state of demand belongs to this amount of alpha.


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## Body

## 1. Introduction

Sensor technology diffusion on new product decisions is considered a significant element to enhance the competitive advantages. The ordering and timing diversity influences organizational performances. Base on Klastorin and Tsai, risk-taking and innovativeness are considered as first movers’ qualifications. Acting primarily in new sensor technology diffusion, products’ moves illustrate several advantages; for instance, brand image, scale effects, experience, product quality’s asymmetric information, and advertising influences (Klastorin and Tsai, 2004; [1]).Sensor technology diffusion has a lot influence on the lot sizing. The sensor technology will influence the multistage inventory systems, incentives, or productivity issues that become a critical point on the lot-sizing problems. The issue on sensor technology also has advantages compared to other products due to production environment that is composed of several production centers for the automation as well as the data gathering.The unique and specific aspect of the sensors are compared to other products such as consumer products.Fast product production is assumed to have more financial independence as a result of gaining earlier cash flow, attaining legitimacy and external visibility, achieving greater market share, and boosting the possibility of survival (Schoonhoven et al., 1990; [2]). However, on the other hand, other researches depict disadvantages of moving first in new products; more extreme market risk and uncertainty and missing the opportunity of imitating first movers’ decisions are deliberated as some negative points (Klastorin and Tsai, 2004; [3]). More researches depict that being a first mover is not always a correct decision. This strategy’s achievement relies on the company’s knowledge of market requirements, size and rivals, and weakness and strength points. An accurate analysis of the firm’s internal and external situation is needed for optimal intentions (Bayus et al., 1997; [4]).There are many studies that show that two main categories have a great impression on the timing of the organizations to ship their products: organizational conditions and environmental circumstances. Below, we provide a survey for significant influencing elements: technological innovation, entrepreneurial team, organizational structure, financial resources, external power and influence, and environmental conditions.Generally, considering the results of the cost and benefit comparison in quickening the product introduction time indicates whether to be a market pioneer or a late market entrant (Bayus et al., 1997). “The process by which an innovation is communicated through certain channels over time among the members of a social system.” Regarding the diffusion’s definition, four key elements are employed:(1)
The innovation: “an innovation is an idea, practice, or object that is perceived as new by an individual or other unit of adoption.”(2)
Communication channels: communication is defined as mutual understanding attainment by means of creating and sharing information. Diffusion is known as a specific type of communication in which the information deals with innovation. A communication channel is the means of relation by which the messages are transmitted from the sender(s) to the receiver(s).(3)
Time: time as an essential element in studies is often ignored in nondiffusional researches, though it is designated in diffusion-related concepts.(4)
A social system: individuals or a group of units sharing a common problem seeking to so attain a mutual goal is defined as a social system. It is worth noting that diffusion requires a social system due to its structure [5].Moreover, Rogers has stated nine cardinal research traditions related to diffusion: anthropology, early sociology, rural sociology, education, medical sociology, communication, marketing, geography, and general sociology. Although numerous descriptions are given through divergent paradigms, the mentioned definition is the most practical and comprehensive. The history of diffusion in innovation goes back to the late 1940s. Rogers and Bass had a great influence on the matter’s progress in the 1960s and introduced effective parameters on this subject. Diffusion of innovation is categorized into two main segments: communication and administration. Based on Rogers and Yildirmaz et al., the sociology, social psychology, and technology are the three key discerned aspects from the administrative perspective of diffusion in innovation which are scrutinized in the following [5, 6].


### 1.1. Diffusion of Innovations in Sociology

The social sight of the topic includes innovation diffusion theory and perceived characteristics of innovating scopes. The innovation diffusion theory (henceforth, IDT) contains Bass, Moore, and Roger’s paradigms; Bass had a mathematical perspective illustrating a model characterizing the diffusion of the innovation procedure. Moore extended the researches on the basis of Bass’s probes into the necessity of technological innovation in firms. Furthermore, Rogers had an extreme influence on the matter. Progress on Roger’s studies with emphasis on reordering the variables led to the Perceived Characteristics of Innovating (PCI) by Moore and Benbasat.


### 1.2. Diffusion of Innovations in Social Psychology

Theory of the Reasoned Action and Theory of Planned Behavior are the two theories scrutinizing human behaviors’ comprehension in the social aspect of diffusion of innovation. To Ajzen and Fishbein, men are intellectual creatures making decisions according to an event’s data. In addition, they are able to handle the decision making progress. Hence, their behaviors are affected by a behavior’s attempt. This is known as the Theory of the Reasoned Action (henceforward, TRA). On the other hand, Bianchi and Figueiredo accentuate the condition in which an individual does not have a thorough control on his conduct, instead of the state in which human behaviors are predicted when decisions are made under full control in TRA. The novel theory is named the Theory of Planned Behavior (hereafter, TPB).


### 1.3. Diffusion of Innovations in Information Technology

From the information technology aspect of diffusion of innovation, two main theories are propounded considering the technology acceptance procedure: the Technology Acceptance Model (henceforward, TAM) presented by Davis and the Unified Theory of Acceptance and Use of Technology introduced by Venkatesh, Morris, and Davis. The TAM forecasts the influence of technology derivation in a corporation, particularly productivity alterations. Through the years, this model was upgraded, and precise external factors were appended, and the Extension of the Technology Acceptance Model (TAM2) was represented by Venkatesh and Davis. Furthermore, the Technology Acceptance Model 3 (TAM3) was illustrated emphasizing on enhancing efficiency through employing new technologies by supervisors.The Unified Theory of Acceptance and Use of Technology (UTAUT) is illustrated by Venkatesh and utilizes the institutional environment as a means to perceive elements, for instance, facilitating conditions, function expectation, endeavor expectation and social influence. Afterwards, the Extended Unified Theory of Acceptance and Use of Technology (UTAUT2) was demonstrated by Venkatesh, Thong, and Xu, containing the elements from UTAUT and auxiliary parameters to fill the gap of orientation of the customer’s point of view. The hierarchy in Figure1 illustrates the embodiment of the discussed matter.Figure 1
Categorizing diffusion of innovation [7].This paper is intended to employ the Bass model for the sensor technology diffusion, and the Wagner-Whitin lot-sizing model discusses this matter related to acceptance of the product and operational dimensions. The objective of this study is to propose an uncertain fuzzy approach to specify the new product introduction’s optimum time, price, production scheduling, and rate simultaneously. This model will be applied to analyze the impact of marketing and operational parameters, for instance, total customer population, price elasticities, startup and maintenance costs, the unit’s variable cost and research and development costs on optimized benefit, and the product’s optimized lifespan. The proposed approach is the evolved version of the Sale et al. [8] model with regard to their research limitations considering a deterministic demand function and constant variable costs. To solve these limitations, some changes have been taken in the previous research preliminaries.The remainder of this research is organized as follows. First, lot sizing, sensor technology implementation, and related literature have been presented. Afterwards, considering previous researches, the novelty and contribution of this research are presented. Subsequently, the Sale et al. [8] model has been scheduled, and the changes to solve their research limitations have been illustrated (Sale, 2017). Next, by applying the design of experiment and response surface methodology, the uncertain model has been analyzed and optimal values of decision variables and objective function were emanated. Eventually, the results were designed and future researches proposed.


## 1.1. Diffusion of Innovations in Sociology

The social sight of the topic includes innovation diffusion theory and perceived characteristics of innovating scopes. The innovation diffusion theory (henceforth, IDT) contains Bass, Moore, and Roger’s paradigms; Bass had a mathematical perspective illustrating a model characterizing the diffusion of the innovation procedure. Moore extended the researches on the basis of Bass’s probes into the necessity of technological innovation in firms. Furthermore, Rogers had an extreme influence on the matter. Progress on Roger’s studies with emphasis on reordering the variables led to the Perceived Characteristics of Innovating (PCI) by Moore and Benbasat.


## 1.2. Diffusion of Innovations in Social Psychology

Theory of the Reasoned Action and Theory of Planned Behavior are the two theories scrutinizing human behaviors’ comprehension in the social aspect of diffusion of innovation. To Ajzen and Fishbein, men are intellectual creatures making decisions according to an event’s data. In addition, they are able to handle the decision making progress. Hence, their behaviors are affected by a behavior’s attempt. This is known as the Theory of the Reasoned Action (henceforward, TRA). On the other hand, Bianchi and Figueiredo accentuate the condition in which an individual does not have a thorough control on his conduct, instead of the state in which human behaviors are predicted when decisions are made under full control in TRA. The novel theory is named the Theory of Planned Behavior (hereafter, TPB).


## 1.3. Diffusion of Innovations in Information Technology

From the information technology aspect of diffusion of innovation, two main theories are propounded considering the technology acceptance procedure: the Technology Acceptance Model (henceforward, TAM) presented by Davis and the Unified Theory of Acceptance and Use of Technology introduced by Venkatesh, Morris, and Davis. The TAM forecasts the influence of technology derivation in a corporation, particularly productivity alterations. Through the years, this model was upgraded, and precise external factors were appended, and the Extension of the Technology Acceptance Model (TAM2) was represented by Venkatesh and Davis. Furthermore, the Technology Acceptance Model 3 (TAM3) was illustrated emphasizing on enhancing efficiency through employing new technologies by supervisors.The Unified Theory of Acceptance and Use of Technology (UTAUT) is illustrated by Venkatesh and utilizes the institutional environment as a means to perceive elements, for instance, facilitating conditions, function expectation, endeavor expectation and social influence. Afterwards, the Extended Unified Theory of Acceptance and Use of Technology (UTAUT2) was demonstrated by Venkatesh, Thong, and Xu, containing the elements from UTAUT and auxiliary parameters to fill the gap of orientation of the customer’s point of view. The hierarchy in Figure1 illustrates the embodiment of the discussed matter.Figure 1
Categorizing diffusion of innovation [7].This paper is intended to employ the Bass model for the sensor technology diffusion, and the Wagner-Whitin lot-sizing model discusses this matter related to acceptance of the product and operational dimensions. The objective of this study is to propose an uncertain fuzzy approach to specify the new product introduction’s optimum time, price, production scheduling, and rate simultaneously. This model will be applied to analyze the impact of marketing and operational parameters, for instance, total customer population, price elasticities, startup and maintenance costs, the unit’s variable cost and research and development costs on optimized benefit, and the product’s optimized lifespan. The proposed approach is the evolved version of the Sale et al. [8] model with regard to their research limitations considering a deterministic demand function and constant variable costs. To solve these limitations, some changes have been taken in the previous research preliminaries.The remainder of this research is organized as follows. First, lot sizing, sensor technology implementation, and related literature have been presented. Afterwards, considering previous researches, the novelty and contribution of this research are presented. Subsequently, the Sale et al. [8] model has been scheduled, and the changes to solve their research limitations have been illustrated (Sale, 2017). Next, by applying the design of experiment and response surface methodology, the uncertain model has been analyzed and optimal values of decision variables and objective function were emanated. Eventually, the results were designed and future researches proposed.


## 2. Lot Sizing

Facing difficulties in replenishing sensor technology product stocks at the proper time and price and in the appropriate quality and quantity by manufacturing corporations’ managers led to endeavors to solve the lot-sizing problems (henceforth, LSP) in the 1990s [9]. The LSP specifies the minimum cost of a production of sensor technology product schedule regarding the bilateral relation between the inventory holding expenditures and the established expenses meeting the requirements (Melega et al., 2018). The aforementioned method was presented for an individual element primarily by Wagner and Whitin in 1958 and developed to multiplex components considering capacity constraints in the same year. The subject under discussion can be categorized by various specifications, for instance, demand (steady or fluctuating), time series (discrete or continuous), deadline (own or disown), and quantity of production levels (single or multilevel) (Melega et al., 2018).The concerns to the mentioned matter deliberate from divergent of sensor technology products industries’ issues. For instance, with regard to sensor technology in the paper industry, for producing gross reels, managers face the lot-sizing problem in order to determine the scale of the lots. Afterwards, the cutting stock problem is formed along with cutting the reels into tinier pieces which might satisfy the client’s requirements. Based on the results of LSP, using the sensor technology on the smaller reels might turn into paper sheets by being cut once more (Melega et al., 2018).A manifest dilemma for administrators related to the ongoing matter is the conflict between the decrement of unit prices in high quantity demands by discounts and the increase in the inventory holding expenditures. Hence, the manufacturers of sensor technology products ought to set suitable discounts, not only to encourage customers for purchasing greater orders but also to maximize the benefit. Furthermore, the clients should make the convenient decision concerning the benefits and costs deriving from large amount orders. Despite numerous modern and classic methods to achieve competitive advantages in markets, sensor technology products on the expenditure mitigation are recognized as a chief factor in boosting a firm’s market share; subsequently, shrinkage of ordering costs is considered as a priority for managers. Employing lot-sizing models besides quantity discounts succor to make the foremost purchasing decisions [9].Through time, more complex and specialized models are introduced; for instance, [10] formulated a make-to-stock (hereafter, MTS) lot-sizing problem in which one class of each product is produced; therefore, one set-up is allowed, and this case is known as lot sizing with 1 set-up (LS1S). In this model, m illustrates the number of considered items and n denotes the number of time periods which is divided into two subperiods in which N1 is allocated to production planning and contains extra production capacity and N2 for inventory management aspects. Hence, the demand for item i∈M is partitioned into two parts: dite for time period N1 and considered to be known and ditp for time period N2 which is forecasted.Furthermore, since previous articles mostly shed light on Dynamic Lot-Sizing (hereafter, DLS) models for sensor technology products, Wu et al. [11] studied DLS models for new products via pricing. In this situation, a firm’s new product is considered to be introduced to consumers by word of mouth and mass media. It is assumed that customers buy the maximum quantity of one product in the whole planning time; hence, the demand at time period t equals to the new gained buyers in the mentioned time horizon, dt=nt. Additionally, this model has the potential to include multi-item and repeated purchases, calculated, respectively, as dt=wnt and dt=Nt=∑i=1tnt in which w indicates the quantity of each customer’s purchase [11].We already conducted rigorously with the used tools, and the objective and the application field are revealed in Table1.Table 1
Research background.
AuthorYearTool/methodEmblemsvag [12]2001Simulation-based modeling, activity-based life cycle assessment (LCA)Klastorin & Tsai (2004)2004Mathematical modeling, Nash equilibrium, game theoryPflieger et al. [13]2005Simulation based modeling, life cycle assessment (LCA) modelingSrivastava & Lee [14]2005Conceptual modeling, multiple linear regression modelsXie & Simon [15]2006Simulation-based modeling, computer simulation with ARENA softwareSettanni & Emblemsvåg [16]2010Mathematical modeling, simulation-based modeling, input-output analysis (IOA), Monte Carlo methodsBauer & Auer-Srnka [17]2012Conceptual modelingVon Westarp & Schinas [18]2016Mathematical modeling, fuzzy linear programming (FLP) approachMa & Fildes [19]2017Mathematical modeling, simulation-based modeling, nonlinear integer programming modelBianchi & Figueiredo [7]2017Review articleO’Neill & Sanni [20]2018Mathematical modeling, conceptual modeling, numerical analysis, generalized economic-order-quantity (EOQ) modelZhang [21]2018Conceptual modeling, diffusion of innovation modelMelega et al. (2018)2018Review article, mathematical modelingSombultawee & Boon Itt (2018)2018Review articleHirunyawipada & Xiong [22]2018Conceptual modelingOu & Feng [23]2019Mathematical modelingFeng & Chan [24]2019Mathematical modelingCarreiro & Oliveira [25]2019Review articleAmati et al. (2019)2019Review article, conceptual modelingCurrent research2019Design of experiment, alpha cut, response surface methodologyIt is demonstrated that the fields of application in the vast majority of the cases are related to managerial purposes. Moreover, preponderance studies employed mathematical modeling and other utilized tools were conceptual modeling, simulation-based modeling and review article with the rates of 54.54%, 31.82%, 22.73%, and 9.09%, respectively. It is worth mentioning that recent researches have studied sensor technology products that optimize the life cycle and compute the optimal profit separately, yet this article scrutinizes the combination of the two aforementioned topics. The augmented features of this study are as below. Note that this research is an evolved version of the Sale et al. [8] research with regard to their research limitations considering deterministic demand function and constant variable costs. To solve these limitations, some changes have been taken in the previous research preliminaries, including
(1)
considering divergent periods’ demand as triangular fuzzy number (TFN)(2)
applying the alpha cut method in order to calculate demand’s upper and lower limits(3)
analyzing various values of alpha, demand’s upper and lower limits, and objective function(4)
employing Cobb-Douglas for the model’s variable costs(5)
exclusive analysis on the utilized Cobb-Douglas model for variable costs


## 3. Research Methodology

### 3.1. Assumptions

(1)
The succeeding generations of sensor technology products are introduced in the same product family in a low innovated industry whilst product success is based on mode(2)
The planning horizon of sensor technology products is adequately extended, and the product life cycle is short in order to be able to plan for several product generations(3)
The manufacturer of sensor technology products employs the solo-product roll; previous product inventory is depleted by the time the new product is introduced(4)
The modified Bass model considering the price of sensor technology products is applied for each product’s demand(5)
Variable costs and demand of sensor technology products are considered dependent and uncertain, respectively(6)
Demand backlogs of sensor technology products are not allowed, and unsatisfied demand is considered as lost sales(7)
The manufacturer of sensor technology products operates in a monopolistic market, and the diminutive competitors are not capable of affecting the market


### 3.2. Modeling

The basic model of sensor technology products is presented by Sale et al. [8] with changes in demand and cost function (Sale, 2017). The demand curve in the proposed model of sensor technology products is based on the Bass model. Bass model consists of two behavioral forces: innovation and imitation. Part of the demand occurring independently from cumulative demand is related to the innovation coefficient (p) and the other part altering with the cumulative demand is linked with the imitation coefficient (q). Assuming m to be the market size, a to be the ratio of q to p, and b to be the summation of q and p, the cumulative demand for time period t is evaluate with equation (1); hence, the demand for the period t is illustrated by equation (2).
(1)Ft=1−e−bt1+ae−bt,(2)Dβ~t=m∗Ft−Ft−1.The indexβ determines that the aforementioned equation does not consider the price effects. On the one hand, manufacturer takes advantage of the earned revenue of selling its goods; on the other hand, it confronts variable unit cost, fixed set-up cost, fixed production cost, inventory cost, and new sensor technology product introduction cost. The corporation aims to maximize the acquired profit resulting from deducting the expenses from the earnings; thus, the objective function is obtained as
(3)max∑t=1Tpt−vt∗D~t−S∗σt−H∗Invt−IT.Inventory at the end of periodt is computed as follows:
(4)Invt=Invt−1+Xt−D~tfort=1toT.Remark that the decision variableXt illustrates the production level in period t and Inv0 is assumed to be zero. In order to make production of sensor technology products responsible for the demand, the following production constraint is written as
(5)∑t=1iXt≥∑t=1iD~t.The abovementioned equation indicates that the cumulative production until the desired period covers its cumulative demand. Set-ups ought to take place in periods in which production level is positive. This constraint is demonstrated as(6)m∗σt−Xt≥0fort=1toT.The net profit function is modelled as equation (7) on condition that the stock level at the first period is equal to zero.
(7)P1−V∗D~t.The optimized value ofP1 is equal to the root of derivation of equation (7) with respect to price; hence, the value for p1 is as
(8)p1=vt+ηη−1.Note thatη is assumed to be considerably bigger than 1. Equation (8) is only valid when the period of sensor technology product stock is empty. Inventory holding of sensor technology product costs result in a divergent equation for indicating the optimized price value as
(9)pt=vt+HNt+ηη−1.N determines the periods which the sensor technology products are held in the inventory before sale.


## 3.1. Assumptions

(1)
The succeeding generations of sensor technology products are introduced in the same product family in a low innovated industry whilst product success is based on mode(2)
The planning horizon of sensor technology products is adequately extended, and the product life cycle is short in order to be able to plan for several product generations(3)
The manufacturer of sensor technology products employs the solo-product roll; previous product inventory is depleted by the time the new product is introduced(4)
The modified Bass model considering the price of sensor technology products is applied for each product’s demand(5)
Variable costs and demand of sensor technology products are considered dependent and uncertain, respectively(6)
Demand backlogs of sensor technology products are not allowed, and unsatisfied demand is considered as lost sales(7)
The manufacturer of sensor technology products operates in a monopolistic market, and the diminutive competitors are not capable of affecting the market


## 3.2. Modeling

The basic model of sensor technology products is presented by Sale et al. [8] with changes in demand and cost function (Sale, 2017). The demand curve in the proposed model of sensor technology products is based on the Bass model. Bass model consists of two behavioral forces: innovation and imitation. Part of the demand occurring independently from cumulative demand is related to the innovation coefficient (p) and the other part altering with the cumulative demand is linked with the imitation coefficient (q). Assuming m to be the market size, a to be the ratio of q to p, and b to be the summation of q and p, the cumulative demand for time period t is evaluate with equation (1); hence, the demand for the period t is illustrated by equation (2).
(1)Ft=1−e−bt1+ae−bt,(2)Dβ~t=m∗Ft−Ft−1.The indexβ determines that the aforementioned equation does not consider the price effects. On the one hand, manufacturer takes advantage of the earned revenue of selling its goods; on the other hand, it confronts variable unit cost, fixed set-up cost, fixed production cost, inventory cost, and new sensor technology product introduction cost. The corporation aims to maximize the acquired profit resulting from deducting the expenses from the earnings; thus, the objective function is obtained as
(3)max∑t=1Tpt−vt∗D~t−S∗σt−H∗Invt−IT.Inventory at the end of periodt is computed as follows:
(4)Invt=Invt−1+Xt−D~tfort=1toT.Remark that the decision variableXt illustrates the production level in period t and Inv0 is assumed to be zero. In order to make production of sensor technology products responsible for the demand, the following production constraint is written as
(5)∑t=1iXt≥∑t=1iD~t.The abovementioned equation indicates that the cumulative production until the desired period covers its cumulative demand. Set-ups ought to take place in periods in which production level is positive. This constraint is demonstrated as(6)m∗σt−Xt≥0fort=1toT.The net profit function is modelled as equation (7) on condition that the stock level at the first period is equal to zero.
(7)P1−V∗D~t.The optimized value ofP1 is equal to the root of derivation of equation (7) with respect to price; hence, the value for p1 is as
(8)p1=vt+ηη−1.Note thatη is assumed to be considerably bigger than 1. Equation (8) is only valid when the period of sensor technology product stock is empty. Inventory holding of sensor technology product costs result in a divergent equation for indicating the optimized price value as
(9)pt=vt+HNt+ηη−1.N determines the periods which the sensor technology products are held in the inventory before sale.


## 4. Numerical Example and Sensitivity Analysis

First of all, the utilized parameters of sensor technology products for experimental design and their surfaces are described; subsequently, the statistical results of modeling with the upper and lower limits and the alpha cut method are determined; finally, each parameter’s influence on the objective function is illustrated by sensitivity analysis. Table2 indicates the utilized parameters of sensor technology products for experiments and their numerical surfaces considering the Sale (2017) research.Table 2
Parameter levels.
ParameterParameter surfacesa2, 8, 14, 20b0.01, 0.04, 0.07, 0.1I4M, 14M, 22M, 30MS3K, 12K, 21K, 30Km6M, 14M, 22M, 30Mη2, 4, 6, 8H0.1, 0.4, 0.7, 1Employing Design-Expert software and response surface methodology has shed light on the parameters’ impact on the proposed model. Regarding the demand to be fuzzy, considering a 10% tolerance, the upper and lower limits are as follows [26–28]:
(10)Aα=0.9+0.1αDt′1.1−0.1αDt.The following reveals divergent parameters’ effects on the model considering the mentioned limits using the experimental design approach (Algorithm1). The coded basic model in LINGO is as follows.Algorithm 1
DATA:a = 14;b = 0.04;m =30000000;! The total size of the potential market;Elas = 2;! Elasticity;H = 0.7;! Holding costs;S = 21000;! Setup cost;I = 18000000;! New product introduction costs;Alpha = 0.7;Lambda =100;! Coef in cap-doglass function for variable cost;Gamma = 1; ! Elasticity parameter in cap-doglass function for variable cost;pp = 5;ENDDATA.SETS:Time/1...pp/:D, P, Inv, X, F, Delta, V;ENDSETS.CALC:@for (Time (t): F(t) = (1 - @exp (-b∗t))/(1 + (a∗ @exp (-b∗t))));ENDCALC.Max = (@sum (Time (t): ((P(t)- V(t))∗D(t)) - (S∗ Delta(t)) - (H∗ Inv (t))) - I)/period;@for (Time (t)| t #GT#1: Inv(t) = Inv(t-1) + X(t)- D(t));@for (Time (t) | t #EQ#1: Inv(t) = 0);@for (Time (j: @)sum(Time (t) | t #LE# j: X(t) - D(t)) > = 0);@for (Time (t): m∗ Delta(t) - X(t) > = 0);!@for (Time (t) | t #GT#1: D(t) = (0.9 + (0.1 ∗ alpha)) ∗m∗ (F(t) - F(t-1)) ∗ @pow((P(t)/P(1, -))elas));@for (Time (t) | t #GT#1: D(t) = (1.1 - (0.1 ∗ alpha)) ∗m∗ (F(t) - F(t-1)) ∗ @pow((P(t)/P(1, -))elas));@for (Time (t) | t #EQ#1: P(t) = (V(t) ∗ elas)/(elas -1));@for (Time (t) | t #GT#1: P(t) = (((V(t) + (H∗t)) ∗ elas)/(elas -1)));@for (Time (t): V(t) = lambda ∗ @pow (X(t),-gamma));@for (Time (t: @)bin(Delta(t)));@for (Time (t: @)Gin(X(t)));Period < = pp;Period > = 1;@Gin (period);Statistical results are presented in four sections, each indicating the value of alpha used for fuzzy demand function. For alpha 0.5, 0.7, 0.9, and 1, results were demonstrated separately and finally compared.


### 4.1. Section (A): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.5

The results are issued in Figures2 and 3. As depicted, a linear model is suggested by the software and contains fewer errors than other models.Figure 2
Surface modeling with lower demand limit andα=0.5.Figure 3
Scrutinizing models and suggesting a linear model.Additionally, ANOVA analysis determines that the linear model is valid, and each parameter’s influence on the model is pictured in Figure4.Figure 4
ANOVA analysis with lower demand limit andα=0.5.Figure5 presents objective function as a function of parametersSandS.Figure 5
Objective function diagram as a function ofA and S and H and S parameters with lower demand limit and α=0.5.Furthermore, Figure6 pictures the objective function based on parameterm.Figure 6
Objective function diagram as a function of parameterm with lower demand limit and α=0.5.As depicted, escalation in market size of sensor technology products has a positive correlation with the objective function. Moreover, in each problem parameter’s coefficient in regression function, response surface prediction of the mentioned issue is explicit which is indicated in Table3.Table 3
The coefficients of the parameters in the surface estimation function with a lower demand level andα=0.5.
ParameterParameter coefficient in surface estimation functiona-9.52b-466.62I-0.32S-454.55m+1.97η-30.8H+3.70Design-Expert software is employed in order to measure each parameter’s optimized value which calculates the optimized parameter values based on the targeted object by experimental design and response surface methodology. Aiming to hit the zenith of the objective function model, the obtained optimized parameter values are as noted in Table4.Table 4
Optimized value of parameters with a lower demand level andα=0.5.
ParameterOptimized parameter valuea2.0b0.04I4MS3Km30Mη2.0H0.1Table5 indicates number of periods, production amount, inventory level, price of each period, and objective function’s value.Table 5
Variable level estimation and their effect on objective function with a lower demand level andα=0.5.
Objective functionNumber of periodsX1X2X3X4X5Inv1Inv2Inv3Inv4Inv5P1P2P3P4P5-775,72623028589015121.720.12,628,2044840319022881864036027538415.58.70.87.84,520,12941323558118354539032195281212.917.61.6105,343,9005323789435112163233601962312324101713.812.913.61.639,017,8303431425676160165491.816.610.2139,707,80052704416515936221723034333992936.21.415.99.315-529,9112436742014330.69.6-2,223,99724670729026522.22.1-798,2673517560072222010798.15.68.5-2,985,145329411159590601291316.713.14.564,563,47031755153556000293126.812.12.6-672,8293128946566055021932927.220.78.9-672,82934144588932770130730.419.39.3-592,8773563535010730432000.812.33.133,379,3803233916881394038121017.44.90-5,022,771237824667038324.513.925,935,700341475435282008429410.51.78.5-550,993354026078508003631326.130.218.539,074,6503396637734230286106172.71.133,379,3804574231945563103901321122668.716.64.76.68,211,3175630587615561894617203782001893825.815.39.214.214.112,599,11049338535451572002971481900.82.517.516.1-1,430,40024509499601541430.7-5,004,8012350123000515.818-1,540,5363401124263840016320014.622.118.927,210,22035809553056502744017.116.50.1-2,422,504259283712034710.612.46,464,52342365130313894030068376262187.714.88.192,752,95055887528631321458294801804154034124.33.610.92.315.4-805,99734296326027840294161.222.5012,195,740514242074139436745399027638729133714.3171.88.310.588,075,85031142189814140363762.412.217.113,544,9105285040021605779358074437827517.33.415.315.17.9-620,046247655894024316.714.8-804,199239271585019518.828171,139,600348002116522703332538.21.8915,305,520541442860809353361860272691471310.9513.34.31.42,628,204446361846530713540120227368712.44.70.4-3,598,43832587826245901217023.216.128.17,264,81841742485933788201519619910.912.514.41.8-4,467,21536054614025001262665.26.73.5-5,005,997254961995019629.6971,200359961630256002103349.44.18-5,017,99731742110552890922034.1118.58,232,5334114955081566505103364071448.81.90.34.5-3,498,03223851279039311.224.9In Figure7, the relation between objective function and number of periods is revealed.Figure 7
Relation between objective function and number of periods with a lower demand level andα=0.5.Considering Table5 and Figure 7, it is perceived that a negative objective function value leads to less period numbers.


### 4.2. Section (B): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.7

To avoid repeated results, three other alpha sections were presented briefly. It is indicated that for this amount of alpha, linear and quadratic models are suggested. Figure8 depicts the objective function as a function of divergent parameters.Figure 8
Objective function diagram as a function ofS and I and a and I parameters with lower demand limit and α=0.7.According to the analyses, the quadratic model proposed a more accurate prediction of the objective function surface. At this level of alpha, optimal values of parameters are also obtained using the Design-Expert software to maximize the objective function as presented in Table6.Table 6
Optimized value of parameters with a lower demand level andα=0.7.
ParametersOptimized parameter valueA20.0B0.04I11MS12Km30Mη8.0H0.4The relation between objective function and the number of periods is demonstrated in Figure9.Figure 9
Relation between objective function and number of periods with a lower demand level andα=0.7.


### 4.3. Section (C): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.9

Not unlike the previous amounts of alpha, linear and quadratic models of sensor technology products are recommended for the response surface by the software. Variance analysis results validate the quadric model of sensor technology products. The objective function’s graph as a function of the below mentioned parameters is pictured in Figure10.Figure 10
Objective function diagram as a function ofa and b and m and H parameters with lower demand limit and α=0.9.Intending to reach an apogee in the objective function, Table7 demonstrates the optimized value of the parameters.Table 7
Optimized value of parameters with a lower demand level andα=0.9.
ParameterOptimized parameter valueA2.0B0.04I4MS30Km22Mη2.0H1.0The relation between the number of periods and the first period’s price is indicated in Figure11. As it is apprehended, in a lower number of periods, the price in the first timespan does not exceed more than the limit, owing to the fact that the relation between the number of periods and the objective function indicates that the model’s condition leads to a lower price, resulting in a higher ratio of costs to the income; thus, a negative objective function ensues.Figure 11
Relation between the first period’s price and the number of periods with a lower demand level andα=0.9.


### 4.4. Section (D): Statistical Results of Surface Modeling with Lower Demand Limit andα=1

Given that alpha equals one, similar to the two former amounts for alpha, the linear and quadratic models for the surface are advised as in Figure12.Figure 12
Objective function diagram as a function ofS and price elasticity coefficient, S and I, and a and b with lower demand limit and α=1.Furthermore, for the stated value of alpha, the optimized amounts of parameters are also calculated as shown in Table8.Table 8
Optimized value of parameters with a lower demand level andα=1.
ParameterOptimized parameter valueA2.0B0.04I11MS30Km22Mη8.0H1.0Table9 briefly describes the result of comparing the upper and lower limits of demand with different amounts for alpha.Table 9
Comparison of the upper and lower limits of demand according to the values of alpha.
α=0.5α=0.7α=0.9α=1Lower limitHigher limitLower limitHigher limitLower limitHigher limitRecommended level for surfaceLinearLinearQuadraticLinearQuadraticQuadraticQuadraticp value of the model0.00720.00870.04930.02190.00160.02990.0008Highest amount of the objective function1.71E+082.15E+081.43E+082.33E+081.32E+081.98E+082.04E+08Objective function’s average1.96E+072.09E+071.79E+072.05E+071.32E+071.95E+072.33E+07As observed in Table9, the objective function’s climax occurs when a higher limit of alpha being 0.7 is assumed; hence, the optimum state of demand belongs to this amount of alpha. The objective function reaches the culmination while the number of periods are four with a higher demand level and α=0.7.In order to discuss the impact of the Cobb-Douglas variable cost function’s parameters on the model of sensor technology products, regarding the obtained values of the parameters from the results of the experimental design for the model in the previous section, the analysis of two parameters of the function will be performed. Table10 indicates the divergent surfaces of the two parameters for a variable cost from the Sale et al. [8] research.Table 10
Parameter surfaces in the Cobb-Douglas function.
ParameterParameter surfacesλ150, 130, 110, 90γ2, 1.5, 1.25, 1Experimental design results for Cobb-Douglas function’s parameters illustrated that the quadratic model of sensor technology products best fits the response surface. Moreover, ANOVA analysis proves the validity of the model. For the Cobb-Douglas function as a function ofλandγ, onlyλandγare demonstrated in Figure 13.Figure 13
Cobb-Douglas function’s diagram as a function ofλ and γ, as a function of λ, and as a function of γ.The software for the two aforementioned parameters predicts that as the amount ofλ diminishes, this leads to reduction in variable costs resulting in the objective function’s upsurge. Table 11 issues parameter coefficients in function surface fitting.Table 11
Optimal values of Cobb-Douglas function parameters.
ParameterParameter coefficient in surface estimation functionλ-51.02γ-1033.62λ×γ-4.15λ2+0.21γ2+331.95Moreover, considering the purpose of maximizing the objective function (profit) of sensor technology products, the software attained optimized values of Cobb-Douglas parameters as 90 and 1 forλ and γ, respectively.


## 4.1. Section (A): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.5

The results are issued in Figures2 and 3. As depicted, a linear model is suggested by the software and contains fewer errors than other models.Figure 2
Surface modeling with lower demand limit andα=0.5.Figure 3
Scrutinizing models and suggesting a linear model.Additionally, ANOVA analysis determines that the linear model is valid, and each parameter’s influence on the model is pictured in Figure4.Figure 4
ANOVA analysis with lower demand limit andα=0.5.Figure5 presents objective function as a function of parametersSandS.Figure 5
Objective function diagram as a function ofA and S and H and S parameters with lower demand limit and α=0.5.Furthermore, Figure6 pictures the objective function based on parameterm.Figure 6
Objective function diagram as a function of parameterm with lower demand limit and α=0.5.As depicted, escalation in market size of sensor technology products has a positive correlation with the objective function. Moreover, in each problem parameter’s coefficient in regression function, response surface prediction of the mentioned issue is explicit which is indicated in Table3.Table 3
The coefficients of the parameters in the surface estimation function with a lower demand level andα=0.5.
ParameterParameter coefficient in surface estimation functiona-9.52b-466.62I-0.32S-454.55m+1.97η-30.8H+3.70Design-Expert software is employed in order to measure each parameter’s optimized value which calculates the optimized parameter values based on the targeted object by experimental design and response surface methodology. Aiming to hit the zenith of the objective function model, the obtained optimized parameter values are as noted in Table4.Table 4
Optimized value of parameters with a lower demand level andα=0.5.
ParameterOptimized parameter valuea2.0b0.04I4MS3Km30Mη2.0H0.1Table5 indicates number of periods, production amount, inventory level, price of each period, and objective function’s value.Table 5
Variable level estimation and their effect on objective function with a lower demand level andα=0.5.
Objective functionNumber of periodsX1X2X3X4X5Inv1Inv2Inv3Inv4Inv5P1P2P3P4P5-775,72623028589015121.720.12,628,2044840319022881864036027538415.58.70.87.84,520,12941323558118354539032195281212.917.61.6105,343,9005323789435112163233601962312324101713.812.913.61.639,017,8303431425676160165491.816.610.2139,707,80052704416515936221723034333992936.21.415.99.315-529,9112436742014330.69.6-2,223,99724670729026522.22.1-798,2673517560072222010798.15.68.5-2,985,145329411159590601291316.713.14.564,563,47031755153556000293126.812.12.6-672,8293128946566055021932927.220.78.9-672,82934144588932770130730.419.39.3-592,8773563535010730432000.812.33.133,379,3803233916881394038121017.44.90-5,022,771237824667038324.513.925,935,700341475435282008429410.51.78.5-550,993354026078508003631326.130.218.539,074,6503396637734230286106172.71.133,379,3804574231945563103901321122668.716.64.76.68,211,3175630587615561894617203782001893825.815.39.214.214.112,599,11049338535451572002971481900.82.517.516.1-1,430,40024509499601541430.7-5,004,8012350123000515.818-1,540,5363401124263840016320014.622.118.927,210,22035809553056502744017.116.50.1-2,422,504259283712034710.612.46,464,52342365130313894030068376262187.714.88.192,752,95055887528631321458294801804154034124.33.610.92.315.4-805,99734296326027840294161.222.5012,195,740514242074139436745399027638729133714.3171.88.310.588,075,85031142189814140363762.412.217.113,544,9105285040021605779358074437827517.33.415.315.17.9-620,046247655894024316.714.8-804,199239271585019518.828171,139,600348002116522703332538.21.8915,305,520541442860809353361860272691471310.9513.34.31.42,628,204446361846530713540120227368712.44.70.4-3,598,43832587826245901217023.216.128.17,264,81841742485933788201519619910.912.514.41.8-4,467,21536054614025001262665.26.73.5-5,005,997254961995019629.6971,200359961630256002103349.44.18-5,017,99731742110552890922034.1118.58,232,5334114955081566505103364071448.81.90.34.5-3,498,03223851279039311.224.9In Figure7, the relation between objective function and number of periods is revealed.Figure 7
Relation between objective function and number of periods with a lower demand level andα=0.5.Considering Table5 and Figure 7, it is perceived that a negative objective function value leads to less period numbers.


## 4.2. Section (B): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.7

To avoid repeated results, three other alpha sections were presented briefly. It is indicated that for this amount of alpha, linear and quadratic models are suggested. Figure8 depicts the objective function as a function of divergent parameters.Figure 8
Objective function diagram as a function ofS and I and a and I parameters with lower demand limit and α=0.7.According to the analyses, the quadratic model proposed a more accurate prediction of the objective function surface. At this level of alpha, optimal values of parameters are also obtained using the Design-Expert software to maximize the objective function as presented in Table6.Table 6
Optimized value of parameters with a lower demand level andα=0.7.
ParametersOptimized parameter valueA20.0B0.04I11MS12Km30Mη8.0H0.4The relation between objective function and the number of periods is demonstrated in Figure9.Figure 9
Relation between objective function and number of periods with a lower demand level andα=0.7.


## 4.3. Section (C): Statistical Results of Surface Modeling with Lower Demand Limit andα=0.9

Not unlike the previous amounts of alpha, linear and quadratic models of sensor technology products are recommended for the response surface by the software. Variance analysis results validate the quadric model of sensor technology products. The objective function’s graph as a function of the below mentioned parameters is pictured in Figure10.Figure 10
Objective function diagram as a function ofa and b and m and H parameters with lower demand limit and α=0.9.Intending to reach an apogee in the objective function, Table7 demonstrates the optimized value of the parameters.Table 7
Optimized value of parameters with a lower demand level andα=0.9.
ParameterOptimized parameter valueA2.0B0.04I4MS30Km22Mη2.0H1.0The relation between the number of periods and the first period’s price is indicated in Figure11. As it is apprehended, in a lower number of periods, the price in the first timespan does not exceed more than the limit, owing to the fact that the relation between the number of periods and the objective function indicates that the model’s condition leads to a lower price, resulting in a higher ratio of costs to the income; thus, a negative objective function ensues.Figure 11
Relation between the first period’s price and the number of periods with a lower demand level andα=0.9.


## 4.4. Section (D): Statistical Results of Surface Modeling with Lower Demand Limit andα=1

Given that alpha equals one, similar to the two former amounts for alpha, the linear and quadratic models for the surface are advised as in Figure12.Figure 12
Objective function diagram as a function ofS and price elasticity coefficient, S and I, and a and b with lower demand limit and α=1.Furthermore, for the stated value of alpha, the optimized amounts of parameters are also calculated as shown in Table8.Table 8
Optimized value of parameters with a lower demand level andα=1.
ParameterOptimized parameter valueA2.0B0.04I11MS30Km22Mη8.0H1.0Table9 briefly describes the result of comparing the upper and lower limits of demand with different amounts for alpha.Table 9
Comparison of the upper and lower limits of demand according to the values of alpha.
α=0.5α=0.7α=0.9α=1Lower limitHigher limitLower limitHigher limitLower limitHigher limitRecommended level for surfaceLinearLinearQuadraticLinearQuadraticQuadraticQuadraticp value of the model0.00720.00870.04930.02190.00160.02990.0008Highest amount of the objective function1.71E+082.15E+081.43E+082.33E+081.32E+081.98E+082.04E+08Objective function’s average1.96E+072.09E+071.79E+072.05E+071.32E+071.95E+072.33E+07As observed in Table9, the objective function’s climax occurs when a higher limit of alpha being 0.7 is assumed; hence, the optimum state of demand belongs to this amount of alpha. The objective function reaches the culmination while the number of periods are four with a higher demand level and α=0.7.In order to discuss the impact of the Cobb-Douglas variable cost function’s parameters on the model of sensor technology products, regarding the obtained values of the parameters from the results of the experimental design for the model in the previous section, the analysis of two parameters of the function will be performed. Table10 indicates the divergent surfaces of the two parameters for a variable cost from the Sale et al. [8] research.Table 10
Parameter surfaces in the Cobb-Douglas function.
ParameterParameter surfacesλ150, 130, 110, 90γ2, 1.5, 1.25, 1Experimental design results for Cobb-Douglas function’s parameters illustrated that the quadratic model of sensor technology products best fits the response surface. Moreover, ANOVA analysis proves the validity of the model. For the Cobb-Douglas function as a function ofλandγ, onlyλandγare demonstrated in Figure 13.Figure 13
Cobb-Douglas function’s diagram as a function ofλ and γ, as a function of λ, and as a function of γ.The software for the two aforementioned parameters predicts that as the amount ofλ diminishes, this leads to reduction in variable costs resulting in the objective function’s upsurge. Table 11 issues parameter coefficients in function surface fitting.Table 11
Optimal values of Cobb-Douglas function parameters.
ParameterParameter coefficient in surface estimation functionλ-51.02γ-1033.62λ×γ-4.15λ2+0.21γ2+331.95Moreover, considering the purpose of maximizing the objective function (profit) of sensor technology products, the software attained optimized values of Cobb-Douglas parameters as 90 and 1 forλ and γ, respectively.


## 5. Conclusion

Acceptance of sensor technology products and operational decision making is an integral part of introducing new sensor technology products to the market. In order to succeed in presenting a new product, the basic step is to be acquainted with concepts related to diffusion of innovation, lot sizing, optimal product lifetime, etc. This paper evolved and analyzed a basic related model to determine the new sensor technology product introduction’s optimum time, price, production scheduling, and rate simultaneously.In this study, the abovementioned tenors are foremost elucidated. Hence, after determining the model’s sensor technology product assumptions, the effecting parameters, variables, objective function(s), and constraints on the model are explained. Afterwards, thereinafter mentioned mathematical models for the optimized level of the lifespan and profit of a product are presented. Thereafter, experimental design and simulation methods were used to generate data. To solve the problem, Lingo software optimized the model with Design-Expert 8 software to present experimental designs of the parameters. Subsequently, the issued model’s validity is measured employing the alpha cut method (for uncertain demand) giving 0.5, 0.7, 0.9, and 1 values to alpha and solving the method with lower and higher demand limits. Optimized levels of the parameters are calculated, and eventually, the validity of the model is proved by the ANOVA analysis.It is eventuated that the optimal amount of the profit occurs whenα=0.7 with a higher demand limit. Additionally, for considering the cost as a function, the two Cobb-Douglas function’s parameters, λ and γ, were calculated to be 90 and 1, respectively. To sum up, it is deduced that more profit and higher pace in new product introduction have a positive correlation with faster diffusion of innovation, lower price elasticity, higher market potential, and lower product introduction cost, verifying previous results.Assumptions of preceding studies have created limitations leading to discrepancy between the paper results and reality, for instance, certain demand, constant variable cost, and fixed model parameters. With a view to cover a portion of the weaknesses caused by the assumptions, this paper has considered demand to be fuzzy and employed the Cobb-Douglas function for variable cost to use the model in real cases. Considering cost experience, demand shortage constraint, and the competitors’ impact on demand will lead the researchers to a more realistic model in the future. In addition, providing a dynamic model varying its parameters as time goes by develops the model. Moreover, considering new numbers and approaches such as hesitant fuzzy and intuitionistic fuzzy would increase the uncertainty of the model.



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*Source: 3503895-2020-08-13.xml*