The aim of our paper is to present the new results of research work on optmizaton and simulaton for some logistc problems in the company. The System Dynamics (SD) method and the Vensim simulaton language are applied in order to solve specifc managerial problems described by Forrester in the model of supply chain. The historical model of Customer-Producer-Employment System by Forrester (Forrester, 1961) has not been examined with the sensitvity analysis, from the “automatc” testng perspectve. Optmizaton experiments have not been conducted, either. It is surprising, since the model is old and widely known. The opportunites oﬀered by the Vensim language allow us to perform such analysis. The visualizaton called “confdence bounds“ is used, to show the behaviour of chosen variables over a period of tme. The Monte-Carlo method is applied for sampling a set of numbers from within bounded domains (distributon for each searching parameters is specifed). The authors of this paper conducted numerous experiments in this scope. This paper presents their results and oﬀers some conclusions formulated at the end.
Introducton and literature review
The problem of sensitvity analysis and optmizaton performed on complex, nonlinear, dynamical and multlevels systems is very interestng from the methodological point of view, especially in the area of System Dynamics (SD). System Dynamics (Coyle, 1977, 1994, 1996, 1998, 1999; Forrester, 1961, 1969, 1971, 1972, 1975; Kasperska, 1995, 2002, 2003, 2005; Radosiński, 2001; Sterman, 2002; Wąsik, 1997, 1983) was developed in the late 1950’s and early 1960’s at the Massachusets Insttute of Technology’s Sloan School of Management by Jay W. Forrester. The approach can be applied to dynamics problems arising in complex social, managerial, economic or ecological systems. The main purpose of System Dynamics is to try to discover the “structure” that conditons the observed behaviour of the system over tme. System Dynamics tries to pose “dynamic” hypotheses that endogenously describe the observed behaviour of system.
In the area of System Dynamics method, there have not been much theory or practce related to combining simulaton and optmizaton. Although the frst trials were sufciently long ago (Keloharju, 1977, 1980, 1983; Winch, 1976), the fact is that incorporaton or embedding simulaton to optmizaton (and vice versa) has not been as popular as it should be in our view. Probably one of the main reasons was the lack of eﬀectve tools. Popular sofware packages originally used in SD modelling and simulaton, did not oﬀer possibilites of automatc optmizaton (for example: languages DYNAMO, DYSMAP (Kasperska and Mateja-Losa, Słota, 2006). Only such packages as COSMIC and COSMOS and Vensim (Ventana, 2007) make it possible to connect simulaton and optmizaton. Hence some papers published in the feld of SD (e.g. Coyle, 1996, 1998), though the work on this subject is stll scarce. The authors of this paper have some experience with the so-called embedding simulaton in optmizaton and vice versa, having conducted numerous experiments on DYNBALANCE family of models (Kasperska, 2005, 2009; Kasperska and Mateja-Losa, 2005, 2006; Kasperska, Mateja-Losa and Słota, 2000, 2001, 2003, 2006; Kasperska and Słota 2003, 2005, 2006).
The SD models usually contain several parameters. It is interestng to examine the eﬀect of their variaton on simulaton output. We select some parameters and assign maximum and minimum values along with a random distributon over which to vary them to see their impact on the model behaviour.
Vensim has a method of setng up such sensitvity simulaton. Monte Carlo multvariate sensitvity works by sampling a set of numbers from within bounded domains. To perform one multvariate test, the distributon for each specifed parameter is sampled, and the resultng values are used in a simulaton. When the number of simulatons is set, for example, at 200, this process will be repeated 200 times.
In order to perform sensitvity simulaton, the user needs to defne what kind of probability distributon values for each parameter will be drawn. The simplest distributon is the Random Uniform Distributon, in which any number between the minimum and maximum values is equally likely to occur. The Random Uniform Distributon is suitable for most sensitvity testng and is selected by default. Another commonly-used distributon is the Normal Distributon (or Bell Curve) in which the value near the mean is more likely to occur than the values far away from the mean. The results of sensitvity testng can be displayed in diﬀerent formats. Time graphs display behaviour of a variable over a period of tme. The variables spread values which combine to form individual simulaton traces.
The object of the experiments is the model named “Customer – Producton –Employment”, described in the literature by Forrester (1961) and Łukaszewicz (1975). The authors of this artcle used the descripton of the model, abbreviatons for parameters and variables afer Łukaszewicz. Our intenton is not to present the model, which is well-known, but to draw the reader’s atenton to the sensitvity and optmizaton experiments. In our paper we suggest the process of “automatc” sensitvity analysis and optmizaton by Vensim.
Analysis and study
Presentaton of the object of the experiments and the assumptons of the simulaton
Figure 1 presents the structure of the model “Customer – Producton – Employment” in Vensim conventon.
Table 1 presents the assumptons of the simulaton of the above model.
|Name of levels||Inital value of level||Name of parameters||Inital value of parameter|
|Order Filling, Inventory Reordering|
|Cash, Proft and Dividends|
|Source: based on Forrester (1961) and Łukaszewicz (1975).|
Results of the experiments on sensitvity analysis and optmizaton for some logistcs problems in the company
There are numerous logistcs problems in the “Customer – Producton – Employment” model. We would like to draw the reader’s atenton to some of them.
Problem number 1 – Too long tme of delivery from Producer to Customer. To conduct this experiment we selected the parameters: “CUD” (Time to Adjust Quoted Delivery at Factory), “CZPM” (Delay in Clerical Processing at Factory), “CWG” (Delay in Shipping at Factory), and observed the confdence bounds for variable “CD” (Delay in Quoted Delivery at Factory). The results are presented in Figure 2.
Our aim was to shorten the “CD” tme”. So we planned the optmizaton experiment consistng in minimizaton and used the optmizaton setup by Vensim, which is shown on windows on Figures 3 and 4.
The results of the experiment are presented in Table 2.
The second problem is: Too large ﬂuctuatons of labour level in the company.
To conduct this experiment we selected the parameter “CLO” (Time for Backlog Adjustment at Factory), and observed the confdence bounds for variables: “KD” (Producton Workers at Factory), “CZ” (Total Labour Change at Factory). The results are shown in Figures 5 and 6.
Our aim was to shorten the “CZ”. As above we planned the optmizaton experiment, consistng in minimizing the value of that variable. The results of the experiment are presented in Table 2.
The third problem is: Too large ﬂuctuaton of “KS” level (Cash Balance at Factory).
To conduct this experiment we selected the parameter “CWG1” (FinishedGoods Price at Factory), “SKW” (Standard Inventory Cost per Item at Factory), “WSP” (Wage Rate at Factory), “PRD” (Productvity of Labour at Factory). The observed confdence bounds for net proft are demonstrated in Figure 7.
Our aim was to maximize “PF” (Net Proft Rate at Factory). We conducted the optmizaton by Vensim (see: windows in fgures 8 and 9). The results are presented in Table 2.
|Problem (no.)||Type of optmizaton (MIN, MAX)||Scope of parameters||Optmized values of parameters||Objectve functon Inital value Final value|
|1.||MIN||CUD є (3,5)||CUD=3||CD=4.89912|
|CZPM є (0.5,1)||CZPM=0.545774|
|CWG є (0.5,1)||CWG=0.595165|
|3.||MAX||CWG1 є (90,100)||CWG1=100||PF=1.286e+006|
|SKW є (45,50)||SKW=45|
|WSP є (70,80)||WSP=80|
|PRD є (2,2.66)||PRD=2|
The possibilites of such experimentatons are practcally unlimited, however, the scope of our paper does not allow to extend the analysis.
Discussion of results
The comparison of the inital and fnal values of objectve functons in Table 2 allows us to ascertain that the choice of parameters and their scope was quite good, since the value of objectve functon has improved in all considered cases. The interestng fact is that the choice had “local” meaning (in the sense that parameters were from the sectors that were similar to objectve functons).
Łukaszewicz (1975) ascertains that in large scale models there are few sensitve parameters and a lot of insensitve parameters. It is vital to detect those sensitve parameters and thus improve the behavior of the system. Naturally the ‘trial-and-error’ process is tme and labor-consuming. This sensitve analysis by Vensim allows us to estmate the sensitveness of selected variable for the choice of given parameters in an easier way and prepare the basis for optmizaton process.
It should be stressed that the obtained results are in line with specifc assumpton about the characteristcs of the model entrance. Łukaszewicz in his paper (Łukaszewicz, 1975) recommended that the analysis of the model behavior should be under a wide spectrum of entrances, not only the classic “step” functons (like in our case) but the sinusoidal or linear ones (for example: trapezium). In books of Forrester (1961) and Łukaszewicz (1975) there are many examples of the analysis using the “trial and error” method. As a result of such analysis, the model behavior has been improved. In our paper we suggest the process of “automatc” sensitvity analysis and optmizaton by Vensim. Obviously, this simulaton language was not known by Forrester or Łukaszewicz, and their results of simulaton were tme and labor-consuming, and impressive in their tmes.
Firstly, we would like to draw a number of theoretcal conclusions:
- as Jackson (2006) said: “creatve holism is necessary in the modern world. Managers are facing ever increasing complexity, change and diversity, and the solutons they have at their disposal to cope with these issue are inadequate”. Thus we can say that the possibilites presented by SD are adequate for solving logistcs problems in the frm.
- simulaton – optmizaton experiments, on System Dynamics models allows to fnd sensitvity parameters and consequently conduct the search for optmal soluton for mult-criteria problems (objectve functons are modelled like inner elements of the model, with feedback in its structure),
- searching for optmal solutons can take into consideraton diﬀerent preferences of decision makers (diﬀerent form of objectve functon, with possibilites of weightng parameters for their factors).
- Secondly, we would like to oﬀer some practcal conclusions:
- logistcs problems in the frm can be investgated using sensitvity analysis and optmizaton by Vensim,
- the Vensim language should become popular in the environment of System Dynamics modellers, because it is an eﬀectve tool for such experiments as: simulaton – optmizaton: its sensitvity and optmizaton setups allow almost automatc search for confdence bounds or optmal value of objectve functons,
- doing simulaton with complex, large scale models, requires seeking many versions of structures, many parameters (especially sensitve parameters), including random elements. All of this is oﬀered by Vensim. Moreover, a new version of this language (see: Vensim 2013) allows its users to create interactve games. This consttutes a new directon of future investgatons for the authors.
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