# System dynamics modeling with fuzzy logic application to mitigate the bullwhip effect in supply chains

Mehdi Poornikoo (Department of Maritime Operations, University of South-Eastern Norway, Vestfold, Norway)

ISSN: 1746-5664

Article publication date: 2 May 2019

Issue publication date: 18 September 2019

## Abstract

### Purpose

A plethora of studies focused on the cause and solutions for the bullwhip effect, and consequently many have successfully experimented to dampen the effect. However, the feasibility of such studies and the actual contribution for supply chain performance are yet up for debate. This paper aims to fill this gap by providing a holistic system-based perspective and proposes a fuzzy logic decision-making implementation for a single-product, three-echelon and multi-period supply chain system to mitigate such effect.

### Design/methodology/approach

This study uses system dynamics (SD) as the central modeling method for which Vensim® is used as a tool for hybrid simulation. Further, the authors used MATLAB for undertaking fuzzy logic modeling and constructing a fuzzy inference system that is later on incorporated into SD model for interaction with the main supply chain structure.

### Findings

This research illustrated the usefulness of fuzzy estimations based on experts’ linguistically and logically defined parameters instead of relying merely on the traditional demand forecasting based on time series. Despite the increased complexity of the calculations and structure of the fuzzy model, the bullwhip effect has been considerably decreased resulting in an improved supply chain performance.

### Practical implications

This dynamic modeling approach is not only useful in supply chain management but also the model developed for this study can be integrated into a corporate financial planning model. Further, this model enables optimization for an automated system in a company, where decision-makers can adjust the fuzzy variables according to various situations and inventory policies.

### Originality/value

This study presents a systemic approach to deal with uncertainty and vagueness in dynamic models, which might be a major cause in generating the bullwhip effect. For this purpose, the combination between fuzzy set theory and system dynamics is a significant step forward.

## Citation

Poornikoo, M. and Qureshi, M.A. (2019), "System dynamics modeling with fuzzy logic application to mitigate the bullwhip effect in supply chains", Journal of Modelling in Management, Vol. 14 No. 3, pp. 610-627. https://doi.org/10.1108/JM2-04-2018-0045

## Publisher

:

Emerald Publishing Limited

## 1. Introduction

In the contemporary competitive world, effective supply chain is a key to success of a business. In the past few years, successful industries have shifted from mass production to make-to-stock and customization. Thus, their approaches have moved from product-oriented to market-driven strategies (Datta et al., 2007). In such environment, competitive advantage is considered as a function of fully-unified supply chain systems (Bhamra et al., 2011). Besides, due to globalization, the complexity of supply chain systems is on the rise, and therefore, exposed to disturbances more than ever (Christopher and Peck, 2004).

Numerous studies have focused on the bullwhip effect for the past few decades, attempting to identify the cause and negative impacts of such phenomenon on the different level of supply chain systems. The bullwhip effect, also known as demand amplification, whip-lash or whip-saw (Lee et al., 1997), implies that the order variability amplifies along a supply chain. According to Lee (1997), bullwhip effect occurs when the variance of orders received by the manufacturer and supplier is much greater than that of customer’s demand, i.e. from downstream to upstream demand amplification. The four major causes of bullwhip effect are traced to price fluctuations, order batching, rationing and demand forecast updating. Walker (2005) argues that technological, process and relationship core competencies are essential factors for an organization to compete in a market. However, the bullwhip effect may inhibit the performance of these core competencies causing unintended costs, waste resources and consequently losing market share.

Several studies have attempted modeling and exploring the bullwhip effect to pinpoint the possible causes or develop strategies that would reduce the effect. Presenting the empirical evidence of bullwhip effect, Forrester (1961) highlighted the difficulties in information feedback loop between organizations as a dominant cause of the bullwhip effect. While Sterman (1989) used his “Beer Distribution Game” to prove the bullwhip effect and attributes it to the “misperception of feedback”. However, we observe three main gaps in the literature. The first one lies in the dynamics of a supply chain and the need to investigate the bullwhip effect phenomenon analytically to gain more insights on the role of conventional forecasting in the bullwhip effect. The second gap concerns the application of fuzzy set theory in the supply chain dynamics and means to deal with uncertainty/vagueness in dynamic models that might be a major cause in generating the bullwhip effect. The third gap relates to the issue of combining multiple fuzzy parameters in a system dynamics modeling framework, as there is a lack of consensus in incorporating fuzzy set theory in system dynamics domain.

The use of fuzzy sets theory in supply chain is not a radically new approach. Although most of the work is about optimizing the inventory level or replacing uncertain demand with fuzzy numbers. In the recent study, Kristianto et al. (2012) used fuzzy control to support a vendor-managed inventory (VMI). While the proposed adaptive fuzzy VMI control model appeared to reduce the bullwhip effect by eliminating the Houlihan effect and Burbidge effect, the scope of examined supply chain is limited to basic concept of VMI and lacks a holistic perspective on supply chain and the dynamics between agents. Moreover, in supply chain the implementation of fuzzy sets for decision-making process, system dynamics is relatively a novel approach. Thus, the objective of this study is set to develop a customized simulation model using system dynamics method and fuzzy rule-based inference system to evaluate the bullwhip effect in a single-product, multi-stage supply chain system. The proposed model allows users to quantify the bullwhip effect and to modify different parameters to observe their impact. We first constructed the fuzzy rule-based system in MATLAB® and then implemented into Vensim® simulation package to substitute the decision-making process based on incoming demand and inventory in hand at each echelon.

### 1.1 Supply chain bullwhip effect

Supply chain is a network of enterprises connected through upstream and downstream links and are involved in different tasks and processes that delivers values in the form of products and services to the final consumers (Christopher, 1992). The upstream and downstream relationships happen via material and information feedback flows (Towill, 1997). The bullwhip effect refers to a phenomenon in which orders received by suppliers amplifies much higher than that of the retailer. Forrester (1958) investigated a supply chain and notices how a small change in consumer’s demand leads to larger fluctuations as it travels through distribution, production for replenishment process. At each level in the supply chain, the aberration becomes greater as the orders move upstream. This is due to poor supply chain management, which is also known as Forrester effect. Forrester (1961) examined this effect in his book Industrial dynamics and concludes the bullwhip effect is a result of non-zero lead time, and imprecise forecasting by different supply chain partners when facing demand variability. However, Lee et al. (1997) identified the following four common causes of bullwhip effect are attributed to demand signal processing and non-zero lead time, order lot sizing, price fluctuations and shortage gaming. While Taylor (2000) argues that machine dependability, process capability and supply inconsistency could also be the possible reasons for the bullwhip effect, which is commonly observed as extreme level of inventory, unsatisfactory forecasts, scarce or excessive capacities, substandard consumer service, uncertain production planning and control due to accumulated backlogs and overdue shipments (Ingalls et al., 2005).

To counter bullwhip effect, enterprises frequently increase their safety stock inventories in an attempt to level production rate. However, holding a high level of expensive inventory against demand amplification would not be the most efficient way. Further, stocking up a high level of inventory adds more to the misperception of any real demand variations. According to Johnson (1998), information sharing, channel placement for swapping decision rights, decreasing order lead time and eradicating forecast updates can be used to mitigate the bullwhip effect. Wikner et al. (1991), proposed series of actions to ease up the bullwhip effect including enhancement of decision rules at each level of the supply chain, time delays reduction, eliminating part of distribution echelons, developing rules among different echelons and improvement in information sharing throughout the supply chain.

### 1.2 Fuzzy logic applications in system dynamics modeling

Modeling with soft variables is considered one of the reasons for unreliable results in SD simulations. That is why researchers attempt to improve the models’ accuracy by using artificial intelligence (Wang, 2001). Fuzzy logic, introduced by Zadeh (1965) as an extension of classical set theory based on fuzzy sets or membership functions, gives us the opportunity to express ambiguity, vague and subjective relationships with mathematical formulations. Traditionally, fuzzy logic approach is used for language processing and imprecise knowledge in expert systems, process control and pattern recognition (Karavezyris et al., 2002). Tessem and Davidsen (1994) highlighted the need for application of qualitative approach for analysis of complex dynamic systems and argue that the understanding behavior patterns and identification of dominating structure underlying these patterns are often neglected because of excessive attention to insignificant numerical details. In most cases, the problem is stated qualitatively in an outline of the key characteristics of a system’s behavior. Substantial inventory oscillations, uncontrolled inflation, policy resistance are few examples. The issue becomes worse when dealing with uncertain phenomenon such as forecasts. Nonlinear forecasting is typically more vaguely stated than problems and goals. More importantly, dealing with uncertainty associated with future behavior requires explicit representation in any given condition. Fuzzy set theory allows us to formulate vague expressions symbolically.

The first authors to integrate these two approaches were Pankaj and Sushil (1994), who suggested a method for qualitative analysis of causal loops using fuzzy logic to integrate the perceptions of the modelers. Their reasoning for proposing such integration was the idea that humans’ mental models are best when expressed in natural language and to construct such mental models, fuzzy logic would give us the best tool. As then, many studies have been trying to bridge fuzzy logic and system dynamics. Most of relevant studies in fuzzy logic and system dynamics integration attempted to use fuzzy variables when data is unavailable or specific variables demonstrate uncertainties. Levary (1990) proposed applying fuzzy sets concept to deal with the imprecision and vagueness in system dynamics modeling. The author then exemplifies a case where fuzzy arithmetic operations can be implemented in the level, rate and auxiliary equations and proposes using conditional statements that include fuzzy variables or fuzzy algorithms instead of conventional relationships in dynamic modeling.

Fuzzy logic and SD have been used by Ortega et al. (2000) to deal with uncertainties and ambiguities in epidemic problems such as vagueness in risk factors, contact patterns, infected conditions and hazards. They used Mamdani’s Max-Min inference method for multiple-input multiple-output (MIMO) model and used the center of area (COA) defuzzification method for calculating the crisp output. Chang et al. (2006) illustrated the fuzzy arithmetic applications in system dynamics modeling and evaluated the results for a customer–producer–employment model. Fuzzy logic was used in their model for “order quantity receiving rate” and “labor productivity” variables with triangular membership functions. However, these fuzzy variables were not interacting with each other in the model and the combination of fuzzy variables was not reflected in their research.

There exist some applications of fuzzy logic in system dynamics. For example, Ghazanfari et al. (2006) proposed causal diagrams with fuzzy relations, Xu and Li (2011) proposed a conceptual model using system dynamics and fuzzy optimization for initial, flow and level variable. Moreover, Carvalho and Tome (2000) used fuzzy cognitive maps and qualitative relation in system dynamics models; and finally, Herrera et al. (2014) developed an approach in dealing with fuzzy logic and system dynamics modeling integration.

The approach of integration system dynamics and fuzzy inference systems (FIS) for the analysis of supply chain models is a recent method that permits a better qualitative understanding of model (Guzmán and Andrade, 2009). In the most relevant research, Campuzano et al. (2010) demonstrated the application of possibility theory and fuzzy numbers for demand and orders estimation in a supply chain system dynamics model. They confirmed that using fuzzy approach would be beneficial when demand is uncertain due to incompleteness and unattainability of historical data in a dynamic environment.

Having reviewed majority of the relevant literature on fuzzy theory use in system dynamics modeling, we argue that system dynamics modeling strives for better depiction of uncertain parameters where in some cases are even cause of the problems themselves. Traditionally, lookups i.e. table functions in system dynamics modeling are used to represent soft variables, which are chiefly reliant on modeler’s subjective opinions. Many applications of fuzzy logic theory in system dynamics, thus, far have centered on modeling uncertain parameters with fuzzy numbers, but they lack holistic approach that uses FIS as a decision-making tool in supply chain dynamics. Furthermore, no evidence in previous studies was noticed where the author(s) executes fuzzy logic demonstration in a mainstream system dynamics software. Therefore, most researches have restricted the magnitude of dynamic models to map out the mathematical formulation and as a result, scope of their research does not fully capture the system dynamics principal.

## 2. Method and the model

### 2.1 System dynamics

The systems approach enables the analysis of complex, dynamic feedback systems by understanding the dynamic behavior of its elements and their interactions over time (Wolf, 2008). Feedbacks in this context means that one component of the system might influence another. For a holistic system analysis, it is crucial to take into account these feedback loops (Forrester, 1961). System dynamics has been greatly used in the study of supply chain systems with nonlinear behavior. However, the application of quantitative approaches has been mostly limited to linear supply chain systems. Therefore, experimental simulation method is mainly used in the literature for analyzing supply chain dynamics (Forrester, 1958; Sterman, 1989; Shukla et al., 2009; Poles, 2013) or to develop linearized approximation models and use the exact method for nonlinear systems (Towill, 1982; John et al., 1994; Disney and Towill, 2005; Gaalman and Disney, 2009; Zhou et al., 2010).

System dynamics simulation is useful when coping with situations where feedback loops play a critical role in understanding the system’s behavior (Akkermans and Dellaert, 2005). System dynamics, developed by Forrester (1961), includes constructing the relationships between variables using causal loops, translating these relations into differential equations, exposing the system to a disturbance and then analyzing the output responses to recognize the cause and effect relationships. When formulating system dynamics simulation models, four major elements should be taken into account; levels (stocks), flows, information channels and decision functions (Forrester, 1961). Levels represent the accumulations within the system and also the existing value of the parameters. Level’s value over time depends on the inflow and the outflow rates. For instance, inventories in production control systems are level variables and the production rate and delivery rate determine the value of the inventory (level) at any given point in time. Flow rates are instantaneous flows, which run between levels. To control the rates between different levels, decision functions are used in forms of differential or algebraic equations. Finally, information channels transfer the information about the stock levels for the decision functions.

In this study, Vensim® package from Ventana Systems is used as the central modeling tool to demonstrate supply chain dynamics. Vensim® is a powerful tool for hybrid (discrete and continuous) simulation. Moreover, MATLAB® software from MathWorks has been used for undertaking fuzzy logic modeling and constructing FIS that is later on incorporated into Vensim® software for the ease of use and interaction with the main supply chain structure.

### 2.2 The case study

The case company, Iran Khodro Spare Parts and After-Sale Service Co. (ISACO), is a large-sized commercial and service company located on the west side of Tehran, Iran. In recent years, the company has experienced market disturbance due to emergence of Asian competitors and occasionally third-party products in the market. Primary observations of ISACO inventory sheets shows that the company exhibits symptoms of the bullwhip effect in their main line of products. The company struggled to estimate the market demand for their products and subsequently had difficulties with operational planning.

### 2.3 The model assumptions

The model is divided into three main divisions represented in a separate echelon. Although these divisions are part of ISACO, they have full autonomy in inventory policy and replenishments. The retailer and distributor echelons are in an aggregated form which represents total sum of all retailers and distributors. Following are the main assumptions of the proposed supply chain model:

• The demand from the customers and supply of the materials are considered exogenous variables to the model.

• There are no major constraints on capacity, labor force and quality control (defective products). Available inventory defines the order fulfillment rate, and therefore, it is the only constraint in this study. The reasoning behind this assumption is due to the reasonably short time period for this research (157 weeks) it would be fair to assume that manufacturing capacity and labor availability do not change over the course of this study.

• Even though our case company manufactures multiple products, multiple products are aggregated into a single item so as reduce the model size and for feasibility of this study.

### 2.4. The model

Figure 1 presents the stock and flow diagram (SFD) of the developed supply chain model that includes three echelons: retailer, distributor and manufacturer. The model structure provides the organizational subunits and decision milestones intertwined in different feedback loops. From the right-hand side of the model, retailer echelon receives orders from consumers. Orders then trigger the order fulfillment process. Ordering process and forecasting influence the retailer’s product flow decisions with on-hand inventory considerations. The needed items from the retailer are received by the distributor as incoming orders. Most of the rules and logics considered in retailer’s echelon are also applied to the distributor and manufacturer. For example, order fulfillment process, which is influenced by inventory and backlog, is only different in the parameters in each echelon. Further, procurement process is needed in the distributor’s echelon to adjust the in-transit flow of products. Multiple feedback mechanisms between on-hand and backlogged products, forecasted demand and lead times regulate the in-transit product flow. The left rectangular represents the manufacturer module, which includes order fulfillment, production and procurement processes. The manufacturer’s supplier is considered exogenous variable and out of the model boundary. The simulation model consists of information and material flows. Financial flow is only considered when dealing with material supply performance.

#### 2.4.1 Model validation.

Model validation is crucial to building confidence in the practicality and effectiveness of the model for its envisioned purpose. To examine the validity of the developed model, we used some of the validation guideline presented by Sterman (2000) including:

##### 2.4.1.1 Structure assessment tests.

Structure assessment tests investigates whether the constructed model is consistent with relevant knowledge of the system. In this paper, partial model tests have been performed for the rationality of individual rules as a major indicator for structure assessment tests. Therefore, the manufacturing echelon is individually tested by eliminating the links from/to distributor and supplier to examine the reliability of manufacturing level in isolation. With a constant input of 10,000 units needed from manufacturer echelon and 0 incoming orders from assembly line, the work in process and the shipment rate are expected to stabilize at 10,000 units, which demonstrated in the following Figure 2.

##### 2.4.1.2 Extreme condition tests.

The idea of extreme condition test is to measure the robustness of the model in different scenarios. The simulation model under study is exposed to two extreme inputs for manufacturer echelon for analysis. First, extreme condition test is performed on “Incoming Orders from Assembly line” to evaluate how the model responds to zero incoming orders and merely serving distributors. Hence;

(IncomingOrdersfromAssemblyLine=0)

The result shows “zero” inventory for assembly line, meaning the model reacts sensibly.

In the next test, extreme condition was imposed on both “Incoming Orders from Assembly line” and “Incoming Orders from Distributor” simultaneously. The reasoning behind this test was to observe manufacturer response when there is absolutely no demand. As it is depicted, the model performs logically under extreme conditions, where inventory for agents stays zero throughout simulation (Figure 3).

##### 2.4.1.3 Integration error tests.

Integration error tests refer to modeling formulations and selecting an appropriate method of integration and time unit that the outcome of simulation model should not be sensitive to the choice of time step or integration method. The case study model is tested for different time steps (DTs) of 0.5, 0.25, 0.125 and 0.0625 and the results exhibit insensitivity (Figure 4).

Furthermore, sensitivity analysis, behavior reproduction tests, parameter assessment, dimensional consistency and boundary adequacy tests have also been conducted for model validation.

#### 2.4.2 Model calibration.

Using the estimated parameters from the historical data from the case study, the validated model is calibrated. The simulated output is then compared against the historical data from the case study about different variables of interest to check the goodness of the fit. For example, Figure 5 depicts a decent graphical fit of the simulated output and to the actual data of the manufacturer’s inventory level.

To measure the goodness of fit, Kolmogorov-Smirnov test (K-S test), is conducted, which is beneficial for nonparametric test for equality of continuous one-dimensional probability distribution to compare a sample with a reference probability distribution. K-S test quantifies the distance between empirical distribution of the sample and cumulative distribution of the reference. The results of the K-S test for goodness of fit are as follow:

Two-sample Kolmogorov–Smirnov test (K-S) test:

Two-sample Kolmogorov–Smirnov test:

• data: model_calibration$Inventory real and model_calibration$`Historical datà;

• D = 0.12739, p-value = 0.1564; and

• alternative hypothesis: two-sided.

Additionally, Table I demonstrates descriptive statistics for the simulation runs and historical data for the inventory level.

## 3. Fuzzy logic model structure

The developed model uses the judgment of the decision-makers for replenishment policies, which is considered as a soft variable. There are two major phases in modeling a soft variable with fuzzy logic in this model. The first phase is to find a relation between input and output, which can be presented as a FIS in MATLAB® and define fuzzy rules. The second phase is to transform the FIS into the form of a mathematical representation, which can be implemented in the constructed simulation model in Vensim®.

### 3.1 Phase 1: creating a fuzzy inference system in MATLAB®

Two inputs have been identified in this regard, which are Inventory level and Incoming Demand (which after considering Forecasting and Desired Inventory coverage translates into Desired Inventory). These two inputs have been used for each echelon while the output is the decision on Units Needed or Received Orders from another echelon. The next step is to define membership functions for Inventory level, Incoming Demand, and the Units Needed. Expert’s opinions have been used in this regard for defining the fuzzy variables and their membership functions. A total number of five membership functions have been outlined for each input and output including Extremely Low, Low, Average, High and Extremely High. Triangular and trapezoidal types of membership functions are used for defining fuzzy variables. Once again, the membership functions and the degrees of membership are based on spare parts supply chain expert’s opinions and hence, might be different from one business to another. Next step is to define the fuzzy rules in a logical way. In addition, 26 logical rules have been set via IF THEN commands consisting five membership functions for Inventory level and five membership functions for Incoming Demand (Desired Inventory). Finally, defuzzification process where the outcome of the FIS is generated in crisp values. The surface illustration of FIS with input/output value representations are demonstrated in Figures 3 and 6.

### 3.2 Phase 2: modeling the fuzzy inference system in system dynamics model

An extended process of transforming FIS is adopted from Usenik and Turnsek (2013), which is portrayed in Figure 4. The process begins with receiving the crisp input variable and fuzzification based on their membership functions, then applying antecedent conclusions for different scenarios. Every rule contributes one conclusion for each linguistic variable included in the consequent. To combine all of the conclusions for certain verbal values into one conclusion, the disjunction of the α values at which the verbal value has been cut is used. This process is called aggregation. As for defuzzification, there are many defuzzification methods, which can give different results. In this study, Height method is used due to simplicity and ease of use in system dynamics modeling, which finally returns the crisp output value. Variables ELD to EHD are abbreviations for “Extremely Low Demand” and “Extremely High Demand”. Similar contractions apply for ELI to EHI as “Extremely Low Inventory” and “Extremely High Inventory” (Figure 7).

The formulation used in fuzzification process attempts to mimic the fuzzy membership functions. For instance, to reproduce the ELD membership function, the following formulation is used;

ELD=IFTHENELSE(10,000<=IncomingDemand0:AND:IncomingDemand0<25,000,1,IFTHENELSE(25000<=IncomingDemand0:AND:IncomingDemand0<50,000,(50,000-IncomingDemand0)/(50,000-25,000),0))

Similarly, ELI is defined as:

ELI=IFTHENELSE(InventoryLevel<25,000,1,IFTHENELSE(25,000<=InventoryLevel:AND:InventoryLevel<35,000,(35,000-InventoryLevel)/(35,000-25,000),0))

R1 to R25 represent the fuzzy rules, which use MIN function for each fuzzy pair. MAX function is used for aggregation of all the conclusions for certain values into one conclusion, as for example ELU (Extremely Low Unit):

ELU=MAX(MAX(R5,R10),MAX(R13,MAX(R14,MAX(R15,R20))))

Finally, defuzzification based on Height method returns the crisp value for Units Needed as in the case of U1,

U1=IFTHENELSE(0<ELU,ELU*C1,0)

and so,

UnitsNeeded=(U1+U2+U3+U4+U5)/BB

The fuzzy structure has been incorporated with the rest of the supply chain model and gets initiated via a switch variable to observe the effect of fuzzy re-ordering policy on the bullwhip effect and compare it with the business as usual.

## 4. Simulation results

Following reporting guidelines for simulation-based research in social sciences (2012), model parameters for primary simulation run are listed in Table II according to minimum model reporting requirement (MMRR). The simulation model runs for 157 weeks using Euler integration type and time steps of 0.125. Exogenous variables used in this model are Incoming Demand and Incoming Orders from Assembly line. All other variables (excluding constants) are endogenous.

The analysis and results indicate the existence of the bullwhip effect within the studied supply chain. Taking actual data into consideration, the results for major stocks and flows reaffirm the demand amplification throughout the ISACO supply chain. Figure 8 presents the simulation results in the absence of fuzzy logic.

The reported bullwhip effect can be attributed to the demand signal processing and non-zero lead time, which previously called Forrester effect. Distortion in demand information has widely spread out throughout the supply chain, which is used for decision-making. In addition to that, forecasting has been the major tool for scheduling and inventory management in this model, which is typically based on historical data from immediate customers. However, the demand sent by the retailer to distributor indicates the amount of inventory replenishment from the retailer for future demand plus the desired safety stock. Therefore, the fluctuations in distributor’s demand become greater than the retailer’s demand. Subsequently, demand amplification grows over the entire supply chain. Furthermore, lengthy lead time worsen the situation due to the fact that the longer the lead time, the higher the safety stock needed for replenishment and the greater the variations. These are the major causes of the described bullwhip effect in the system, in line with the study by Lee et al. (1997) for the origin of the bullwhip effect.

### 4.1 Bullwhip effect with fuzzy logic decision-making

The implemented fuzzy logic was tested to observe the impact of fuzzy decision-making policy on the overall supply chain system and particularly the bullwhip effect. Firstly, inputs for the fuzzy system at each echelon are the Inventory level and the Desired Inventory, which perform as forecasted Incoming Demand plus the Desired Inventory Coverage. Therefore, the fuzzy policy does not replace any equations in the model and only operates as an alternative to the process of replenishment policy. The results of the fuzzy structure for major Inventory stock level variables are illustrated in Figure 9 that clearly demonstrate that the fuzzy logic decision-making policy reduced the fluctuations in the bullwhip effect.

## 5. Conclusion

Overall, from the literature review in system dynamics, the bullwhip effect and fuzzy logic theory, three specific gaps were discovered. Firstly, the need to clearly model qualitative variables in a dynamic modeling context particularly when the causes of problem are traced to ambiguity and imprecision. Secondly, the need to explicitly use fuzzy logic framework in combination with system dynamics modeling, and finally, the necessity to address the bullwhip effect from a holistic point of view that captures the underlying structure of the effect and the behavioral patterns.

Even though the theory around bullwhip effect has been well-developed in the literature and the body of knowledge is quite solid for both origin of the issue and the impacts on other levels in supply chain; the gap between theory and practice is still considerably wide. Current literature is mainly focused on the bullwhip effect existence and analysis methods are based on typical principles that owe their success more to the expert’s capability than to a systematic and organized approach to resolve the issue. On the other hand, the ongoing research on implementation of fuzzy logic approach in system dynamics framework is not straightforward, particularly when multiple soft variables need to be integrated.

Therefore, this paper attempted to develop a system dynamics simulation model for a case of supply chain system and used fuzzy logic decision-rules to mitigate the existing bullwhip effect. The model structure was customized to the case study in hand to exemplify the bullwhip effect in a single-product, three-echelon, multi-period supply chain system. Major contributions of this study can be summarized as:

• A holistic approach in designing and formulating a case study of spare part supply chain system, which includes retailer, distributor and manufacturer to grasp the underlying structure of the bullwhip effect.

• Developing and implementing FIS in a system dynamics environment to enhance the decision-making policy.

• Improvement in mitigating the bullwhip effect across supply chain with fuzzy rules by rationalizing demand signal processing through inventory in-hand and incoming demand considerations.

This research illustrated the usefulness and importance of fuzzy estimations based on experts’ linguistically and logically defined parameters instead of relying merely on the traditional demand forecasting based on time series. Despite the increased complexity of the calculations and structure of the fuzzy model, the bullwhip effect has been considerably decreased. One major practical contribution of this study would be the benefits for the industry and the case study ISACO by reducing the costs associated with the inventory fluctuations and dealing with demand uncertainties. The model enables optimization for an automated system integrated in a company where decision-makers can adjust the fuzzy variables according to various situations and inventory policies.

## Figures

#### Figure 1.

Overview of the three-echelon supply chain model

#### Figure 2.

Structure assessment tests results for manufacturer echelon

#### Figure 3.

Extreme condition tests

#### Figure 4.

Integration error tests for different DTs

#### Figure 5.

Graphical fit with the actual data

#### Figure 6.

Surface illustration of fuzzy inference system

#### Figure 7.

Fuzzy inference system implementation in system dynamics model

#### Figure 8.

Simulation results without fuzzy logic presence

#### Figure 9.

Simulation results using fuzzy logic policy

## Table I.

Descriptive statistics of model calibration

Variable Count Minimum Maximum Mean Median SD (Norm)
Selected variables for time (week) Base run
Historical inventory data for agents 157 20.42 146.2 74.16 79.33 28.81 0.3885
M inventory for agents 157 0 116.3 66.78 71.65 29.52 0.4421

## Table II.

Level variables for 3-echelon supply chain with initial values

Supply chain echelon Stock variables Formulation Initial values
Distributor echelon Backlog for orders D order rate-D fulfillment rate D Backlog for orders initial value
Forecast D change in incoming orders Received orders from clients
Inventory level D arrival rate-D shipment rate D desired inventory
Units in transit Incoming units from manufacturer-D arrival rate D desired incoming units
Manufacturer echelon Agents forecast M change in forecast Incoming orders from distributors (ISACO agents)
Backlog assembly line orders M assembly line order rate-M assembly line order fulfillment rate 0
Backlog orders from agents M order rate from agents(D)-M order fulfillment rate M backlog agent orders initial value
Inventory for agents M production rate for agents orders-M agents shipment rate M desired inventory
Inventory for assembly line M production rate for assembly line orders-M assembly line shipment rate M desired assembly line inventory
Material inventory M material delivery rate-M material usage rate M desired material inventory
Material forecast M perceived change in material forecast M material usage rate
Retailer echelon Work in process inventory M production start rate-M production rate for agents orders-M production rate for assembly line orders M desired WIP
Backlog for orders R order rate-R fulfillment rate R backlog for orders initial value
Forecast R change incoming demand 0
Inventory level Incoming units from distributor-R shipment rate R desired inventory

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## Corresponding author

Muhammad Azeem Qureshi can be contacted at: muhaqu@oslomet.no