System Dynamics Simulation to Determine Safety Stock for a Single-Stage Inventory System Imen Belhajali
Wafik Hachicha
Unit of Logistic, Industrial and Quality Management (LOGIQ), Higher Institute of Industrial Management Sfax, Tunisia.
[email protected]
Unit of Mechanic, Modeling and Production (U2MP) Engineering School of Sfax, Sfax, Tunisia
[email protected]
Abstract—In this paper, a single-stage inventory system, which analytical method results exist in previous literature is considered. This case is one of an adaptive base-stock policy for a single-item inventory system, where the demand process is nostationary. The demand process is modeled as an integrated moving average process for which an exponential-weighted moving average provides the optimal forecast. The case study objective is about how characterize the required inventory and how identify the safety stock requirements. For this purpose, a system dynamics approach is developed through a flow diagram model and simulated using Vensim® software. The results of the comparative study show that system dynamics approach is very useful. The system dynamics models can provide increased comprehension of, and improved insight into, the performance of an inventory system. The extension of system dynamics seems very interesting for the analysis of complex supply chain problems. In fact, analytical methods are perceived to be unhelpful by management or may require oversimplification. Keywords—inventory system; Safety stock; system dynamics; flow diagram; simulation; comparative study
I.
INTRODUCTION
Supply chain can be too complex to describe it as mathematical equations. For this reason, simulation is a commonly used technique for the design and the analysis of supply chain systems. However, supply chain simulation implies operating a model that suitably represents real systems. Supply chain simulation can generate supply chain control and improvement. Simulation technique can be used to understand all or part of the supply chain. Moreover, simulation can be used to propose and select scenarios to improve the supply chain (what-if analysis). Numerous techniques are proposed in supply chain simulation research literature. It is better to divide them in four simulation types [1]: spreadsheet simulation; system dynamics (SD); discrete-event simulation (DES); and business games. DES and SD are the two widely used modeling tools which underpin decision support systems [2].In the field of logistics and supply chain management simulation-based decision support systems provide solutions to a wide range of issues at both a strategic, operational and tactical level. Specific examples of the issues that these decision support systems address are supply chain design and reconfiguration, inventory planning and management, production scheduling and supplier
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selection. A commonly held belief exists that SD is mostly used to model problems at a strategic level, whereas DES is used at an operational/tactical level. However, it should be noted that the type of simulation that must be used will depend on the problem to be solved by the problem in each specific case. For instance, reference [2] suggest that DES has been used more frequently to model supply chains, with the exception of the bullwhip effect and complex inventory systems, which are mostly modeled using SD. System dynamics is growing at an impressive exponential rate. In relation to systems dynamic, we agree with [3], [4] about its effectiveness to model dynamic business systems, in this case supply chains. As regards the selected software, the reason it employs a modeling approach is that it combines systems dynamics concepts and discrete events simulation to represent supply chain events and uncertainty in detail, and to subsequently analyze their performance based on their structure and the existing causal relationships among their components. Local simulation is used to model the proposed case study, which is dealt with inventory system management. Inventory has two paradoxical and important sides in supply chain management and especially in manufacturing control. In fact, inventory can be considered as a savior or as a survival tool when a plant cannot produce at the desired objective rate or in moments where suppliers are not trustworthy. However, inventory can be viewed as a waste of money, a standing still investment that yields nothing mostly, when each process performances goes slickly. Inventory management is essential to generate a set of scales between these two sides. Inventory control has been a focus of the research community with different objectives. Some of these objectives are: to analyze the performance of the system based on the service levels, average inventory costs, number of stock out occasions, and the bullwhip effect [5]. The available literature addresses these problems from different perspectives and uses different tools to model, analyze and solve the systems both analytically and numerically. The analytical solutions usually depend on the mathematical basis of the fundamental and improved inventory control systems. The case study used in this paper is an adaptive base-stock policy for a single-item inventory system, where the demand
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process is no-stationary. The demand process is modeled as an integrated moving average process for which an exponentialweighted moving average provides the optimal forecast. The case study objective is about how characterize the required inventory and how identify the safety stock requirements. To this end, a system dynamics approach is developed through a flow diagram model and simulated using the Vensim simulation software. The rest of the paper is organized as follows. Section 2 describes the case study of the inventory system. Section 3 presents a brief overview of the literature on system dynamics. Then Sections 4 present the developed flow diagram model. Section 5 presents the comparative results between previous analytical approach and the proposed model. Finally, results are discussed and general guidelines are provided for future work. II.
INVENTORY SYSTEM DESCRIPTION
In this paper a single-item single-stage inventory system described by reference [6] is considered. The lead-time of the replenishment policy which is denoted L in this paper, is known and remains fixe. The demand process is assumed as no-stationary stochastic process. A demand which is not satisfied by inventory is backordered. The adaptive base-stock control policy is assumed as the inventory control policy. In the next subsections, the demand process, the forecast model, and the inventory control policy are described in detail. Then the analytical results founded in literature are presented. A. Demand Process The demand process is assumed follow to an autoregressive process which is described in (1).
⎧ D1 = μ + ε1 ⎨D = D + (1 − α)ε + ε ⎩ t t −1 t −1 t
(1)
for t = 2, 3,..., where Dt denotes the demand in period t and {εt} is a white noise, which is characterized as a time series of identically and independently normally distributed (i.i.d.) random variables 2 with mean E[ε t ] = 0 and variance Var[ε t ] = σ . The parameters α and µ are known. By varying α, it is clear that we can model a range of demand processes. For instance, when α = 0, the demand follows a stationary i.i.d. process with mean µ, and variance σ2. However, for 0 < α ≤ 1, the demand process is a no-stationary process, in which larger values of a result in a less stable or more transitory process. Finally, when α = 1, the demand process is known as a random march. The demand in a period is the demand in the previous period plus a noise term, as indicating in (1). The parameter α is viewed as a measure of the inertia in the process. B. Forecast Model A first-order exponential-weighted moving average provides the minimum mean square forecast for this demand process. So, it is assumed that an exponential-weighted moving
average forecast model with parameter α and initial forecast µ is applied. Ft is denoted as the forecast in the current period as mentioning in (3).
⎧F1 = μ ⎨F = αD + (1 − α)F t t ⎩ t +1
( 2)
for t = 1, 2,..., Using equation (2) and (1), we can determine easily by induction that the forecast error is:
D t − Ft = ε t
(3)
In fact, the exponential-weighted moving average is founded as the best unbiased forecast because the forecast error is limited to a random white noise term for time period t. Therefore, it seems that there is no better forecast model for this demand process. C. Inventory Control Policy Qt is assumed the order which is placed in period t, however it will be for delivery after L periods. We consider that in each period t, we first see Dt, determine this period’s order (Qt), receive the order from L periods ago (QL-t), and then fill demand from inventory. Xt denotes the on-hand inventory (or backorders) at the finish of period t. It is described in (4) as the basic inventory equation is:
X t = X t −1 − D t − Q L − t
(4)
We consider that we can set an initial inventory level X0, and that Qt= µ for t ≤ 0 and we operate with a base-stock policy by adjusting the base stock as the demand forecast changes. The following rule is provided to do this:
Q t = D t + L(Ft +1 − Ft )
(5)
This is the myopic policy [4] for an L-period lead time, considering stationary cost parameters. That is, it minimizes the expected one-period cost a lead time into the future. Equation (5) is composed by two components to the order quantity. The first component replenishes the demand or the immediate period (Dt), using a base-stock inventory policy. The second component adjusts the base-stock level to provide somewhere to stay the change in the forecast, which changes the mean lead time demand. D. Inventory Control Policy Reference [6] finds that Xt is normally distributed with:
Std[X t ] = σ L 1 + α(L − 1) +
E[X t ] = X 0
(6)
α 2 (L − 1)(2L − 1) 6
(7)
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where t ≥ L and Std[ ] denotes the standard deviation. Therefore, the inventory in each period is normally distributed with parameters indicated in (6) and (7). The mean of this distribution is the initial inventory level, which is considered in this paper as the one control variable. In other worlds, X0 is considered as the safety stock for this inventory system. It should be noted that the standard deviation depends on the demand process parameters and the lead-time as mentioned in (8). This equation express Std[Xt] using the standard deviation of demand and the replenishment lead time L. The safety stock, indirectly the initial inventory level, should be chosen to assure some specified level of service. In general, the initial inventory level is fixed at a multiple of the standard deviation of the inventory random variable (which equals the standard deviation of demand over the lead-time): X 0 = zσ L 1 + α(L − 1) +
α 2 (L − 1)(2L − 1) 6
(8)
In this way, one assures that the probability of not stocking out in each period is (z), where ( ) is the cumulative distribution function for the well-known standard normal random variable. Thus, we can set X0 to achieve any risk service. It should be noted that we could establish conditions on the demand process that assure that Qt is nonnegative. For instance, (5) can be rewrites as (9): Q t = D t + Lα ε t
(9)
In addition, reference [6] has guaranteed that the order quantity is nonnegative if the random march is assumed bounded. e.g., kσ ≤ ε t ≤ kσ . When some of these additional assumptions are not appropriate, then the optimal solution is more complex and would require other advanced resolution techniques. In this paper, k is set equal to 3. For showing how the safety stock depends on the lead-time and on the parameter α, we use equation (8). It should be noted that, when α = 0, we get the common result that the safety stock for a stationary, i.i.d. demand process increase with the square root of L. This result (Eq. 10) is well-known in inventory management literature [7].
Std[X t ] = σ L
(10)
When α = 1, the demand process is a random walk and we find: Std[X t ] = σ
L(L + 1)(2L + 1) 6
(11)
For this purpose, the safety stock requirement, as a function of the replenishment lead-time, behaves much differently than in the stationary demand case. So, when the demand process follows a random walk, we need increasing amounts of safety stock as the lead-time grows.
III.
SYSTEM DYNAMICS OVERVIEW
Forrester [8-9] proposes a methodology for the simulation of dynamic models: industrial dynamics; which is the origin of system dynamics [3]. System dynamics is a continuous simulation methodology that uses concepts from engineering feedback control theory to model and analyze dynamic socioeconomic systems. The mathematical description is realized with the help of ordinary differential equations. The main objective of system dynamics is to understand the structural causes that bring about the behavior of a system [3]. Some examples of inflows in the supply chain are production and sales. Some examples of outflows in the supply chain are stocks, fill rate and work in process. Systems dynamics assumes that control is carried out by varying the ratio of the variables (for instance, production and sales) which changes flows (and therefore stocks). It is also based on the feedback principle, i.e., a manager compares an objective value for a metrics with the real value and takes corrective actions, if required. In other words, the basic systems dynamics objective is to understand the structural causes that trigger system performance. This is a long-term approach. Suitable variables selection is most important (system elements analysis) because it is based on the analysis of internal logic and the system’s structural relationships. The structure consists of multiple interacting feedback loops that depict the policies and continuous processes underlying discrete events [10]. Forrester’s System Dynamics [8-9] has been used in many fields, such as management, finance, food industries, production systems, supply chain, etc. This is a rigorous method for a qualitative description of exploring supply chains as far as their processes, information, strategies and organizational limits are concerned. It facilitates modeling and the qualitative simulation analysis to design and control the supply chain structure. It also facilitates experiments with supply chains. It does not require detailed information or exact data on relationships. It focuses on the dynamic performance of the combination of feedback loops. Some recent simulation works relating to the supply chain are based on systems dynamics can be found in [11], [12], [13], etc. The main systems dynamics tools are the causal loop diagram and the flow diagram. Causal loop diagram includes the key system elements and the relationships among them, based on cause having an influence on effects. The causal loop diagram is very important to explain the final model to the user and serves as a basis to construct the flow diagram. Causal loop system is used when a complex system is studied for the first time i.e. its key system elements are not well known. The flow diagram is a translation of the causal loop diagram into a terminology that helps write equations in the computer. As mentioned by [11], Forrester suggests four main concepts, which are as follows: •
Levels describe accumulations within the system and are drawn as tanks. Levels represent the present values of the variable they contain that have resulted from the accumulated difference between inflows and outflows.
•
Flows, which transport the content of one level to another.
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•
Decision functions, which control the rates of flows between levels (drawn as valves).
•
Information channels, which connect the levels to the decision functions.
These concepts are associated with graphical representations, which allow diagrams to describe the studied system (Fig. 1 and Fig. 2). Fig. 1 depicts a very simple structure of a reservoir or level, with an inflow and an outflow. To specify the dynamic behavior, a system of equations is defined. It consists of two types of equations, which correspond to levels and decision functions (rates). Equations control the changing interactions of a set of variables, as time advances. The continuous advance of time is broken into small intervals of equal length dt. For example the equations describing the state of the level in Fig. 1 is ⎧Le(t) = Le(t − dt) + (DF1 − DF2) ∗ dt ⎨ ⎩ Le(0) = 0 Decision Function (DF1)
Level
(12)
Vensim. The Vensim simulation software has been used as the basis for this paper to study inventory system management. Regardless of the choice of software, a four-step methodology is adopted: A. Characterize The System Elements The first step consists of identifies the level, inflow, outflow, and auxiliary variables as mentioned in Fig. 1. Fig. 2. B. Write The Model Equations The relations among the elements are specified principally through equations (1, 2, 4 and 5). To do this, arithmetic formulae, software functions are used. In addition, it is necessary to assign a value to the elements, which can be a known one or an approximate one. For instance, Fig. 3 presents an overview of the first five equations of twenty two equations which are used. For example, the first one indicates that alpha parameter is attributed equal to 0.5.
Decision Function (DF2)
Inflow
Outflow
Figure 1. Flow diagram basic symbols
Where Le denote Level, DF denote Decision Function and Le(0) denote initial level. Level in Fig. 1 at time t depends on its own value at time t - dt and the value going in from decision function 1 minus the value going out to decision function 2. It is necessary to give the initial value of it to solve this equation. There are as many equations as variables. To determine the variables’ behavior, the differential equations system is integrated. Another example is presented in Fig. 2. Where X(0) means the initial state of the stock value. The stock level X(t) accumulates by integrating the flow rate (dX). Auxiliary variables (Y) control or convert other entities (g(X(t))). X
dX Y
Figure 3. Overview of the first five equations
Figure 2. Auxiliary variable integration in the flow diagram
X(t) = ∫ dX.dt + X(0) dX = f(Y(t))
(13)
Y(t) = g(X(t))
IV.
FLOW DIAGRAM MODEL AND SIMULAUION
There are different software packages to system dynamics simulation available in literature. The most widely used are (in alphabetical order): Dynamo, Ithink, Powersim, Stella and
C. Build The Flow Diagram Model Create a preliminary version of the model. This is the first model that works, although it can be improved progressively. After stabilize the model, it should be sure that the model functioning with all stable variables. Fig. 4 presents the final flow diagram model, which is corresponding to the studied single-stage inventory system. It composed of five level variables, five auxiliary variables, and seven constant inputs.
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Figure 4. Flow diagram of the inventory system under study
V.
RESULTS AND VALIDATION
The objective of the simulation is to find the analytical results provided by [6] and specifically to verify equation (7). To explore this comparison, Standard Deviation of Xt is given using undependably two approaches: Analytically using equation (7), and using the proposed system dynamics model by estimating the standard deviation of Inventory (Xt) as a simulation output. Table I presents the results for σ = 1 and various choices of α and L. TABLE I.
ANALYTICAL AND SIMULATION RESULTS
α=0
α = 0.5
α=1
L
Ref. [6]
This paper
Error This Ref. [6] (%) paper
Error (%)
Ref. [6]
This Error paper (%)
2
1.41
1.45
2.8%
1.80
1.84
2.2%
2.24
2.28
1.8%
4
2.00
2.02
1.0%
3.67
3.69
0.5%
5.48
5.5
0.4%
6
2.45
2.54
3.7%
5.89
5.98
1.5%
9.54
9.63
0.9%
8
2.83
2.94
3.9%
8.43
8.54
1.3%
14.28 14.39 0.8%
10
3.16
3.11
1.6%
11.24
11.19
0.4%
19.62 19.57 0.3%
12
3.46
3.39
2.0%
14.30
14.23
0.5%
25.50 25.43 0.3%
Figure 5. The output plots of simulation
D. Simulate The Final Model The objective of the simulation is to find the analytical results provided by previous research literature, especially by which are founded by reference [6]. After model building, Fig. 5 presents the evolution of the Inventory (Xt), the receiving (QtL), and the Demand (Dt) for σ = 1, α = 0.5, and L= 6 periods. This figure is obtained easily using Vensim Software.
In addition, Table I show the error between simulation results and the analytical results. There is no error that exceeds 4%. It can be say that the proposed approach is effective. To explore this relationship further, we plot in Fig. 6 the standard deviation of Xt as a function of the lead-time for σ = 1, and various choices of α. From Fig. 6, there are two key observations for the assumed demand process .First, when demand is no-stationary, we observe that we require
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dramatically more safety stock, in comparison with the case of stationary demand (α=0). Second, we observe that the relationship between lead-time and safety stock becomes convex for no-stationary demand. 30 Alpha = 0
Std. Dev.
25
system can be easy studied with changes in very small length of the review period, (3) Multi-stage supply chain can be considered as many stages in series. The studied inventory system can be applied by each stage. In this case, various problems such as bullwhip effect analysis and transshipment policies option can be more investigated.
Alpha = 0.5
20
REFERENCES
Alpha = 1
[1]
15 10
[2]
5 0 1
3
5
7
9
11
13
Lead-time
Figure 6. Simulated Standard deviation of Xt as a Function of Lead-Time for σ=1
Consequently, by using system dynamics, this paper confirms various previous analytical results. Extending the proposed approach in supply chain management direction is our interesting research perspective. VI.
[3] [4]
[5] [6]
[7]
CONCLUSION
In this paper a model for a single-item inventory system with a deterministic lead-time but subject to a stochastic, nostationary demand process is presented. A system dynamics approach is developed through a flow diagram model and simulated using Vensim software. The results of the comparative study show that system dynamics approach is very useful. The system dynamics simulation models can provide increased comprehension of, and improved insight into, the performance of an inventory system. In fact, once the model is built and validated, it is ready to be improved by several simple extensions, but their interests are extremely useful. For instance, (1) Stochastic lead Time can be simply implemented, (2) Asymptotic behavior, as a continuous simulation, of the
[8] [9] [10] [11]
[12]
[13]
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