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Book will be sent in robust, secure packaging to ensure it reaches you securely. Book Description Springer, New Book. Shipped from UK. Established seller since Seller Inventory GB These are treated as parameters and the simulation is run for various combinations of values for these parameters. At the end of a run statistics are gathered on various measures of performance and these are then analyzed using standard techniques.
The decision-maker then selects the combination of values for the decision variables that yields the most desirable performance. Simulation models are extremely powerful and have one highly desirable feature: they can be used to model very complex systems without the need to make too many simplifying assumptions and without the need to sacrifice detail. On the other hand, one has to be very careful with simulation models because it is also easy to misuse simulation.
First, before using the model it must be properly validated. While validation is necessary with any model, it is especially important with simulation.
Second, the analyst must be familiar with how to use a simulation model correctly, including things such as replication, run length, warmup etc; a detailed explanation of these concepts is beyond the scope of this chapter but the interested reader should refer to a good text on simulation. Third, the analyst must be familiar with various statistical techniques in order to analyze simulation output in a meaningful fashion. Fourth, constructing a complex simulation model on a computer can often be a challenging and relatively time consuming task, although simulation software has developed to the point where this is becoming easier by the day.
The reason these issues are emphasized here is that a modern simulation model can be very flashy and attractive, but its real value lies in its ability to yield insights into very complex problems. However, in order to obtain such insights a considerable level of technical skill is required. A final point to keep in mind with simulation is that it does not provide one with an indication of the optimal strategy. In some sense it is a trial and error process since one experiments with various strategies that seem to make sense and looks at the objective results that the simulation model provides in order to evaluate the merits of each strategy.
If the number of decision variables is very large, then one must necessarily limit oneself to some subset of these to analyze, and it is possible that the final strategy selected may not be the optimal one. Mathematical Models : This is the final category of models, and the one that traditionally has been most commonly identified with O.
In this type of model one captures the characteristics of a system or process through a set of mathematical relationships. Mathematical models can be deterministic or probabilistic. In the former type, all parameters used to describe the model are assumed to be known or estimated with a high degree of certainty. With probabilistic models, the exact values for some of the parameters may be unknown but it is assumed that they are capable of being characterized in some systematic fashion e. However, CPM is based on a deterministic mathematical model that assumes that the duration of each project activity is a known constant, while PERT is based on a probabilistic model that assumes that each activity duration is random but follows some specific probability distribution typically, the Beta distribution.
Very broadly speaking, deterministic models tend to be somewhat easier to analyze than probabilistic ones; however, this is not universally true. Most mathematical models tend to be characterized by three main elements: decision variables, constraints and objective function s. Decision variables are used to model specific actions that are under the control of the decision-maker.
An analysis of the model will seek specific values for these variables that are desirable from one or more perspectives. Very often — especially in large models — it is also common to define additional "convenience" variables for the purpose of simplifying the model or for making it clearer. Strictly speaking, such variables are not under the control of the decision-maker, but they are also referred to as decision variables.
Constraints are used to set limits on the range of values that each decision variable can take on, and each constraint is typically a translation of some specific restriction e.
Clearly, constraints dictate the values that can be feasibly assigned to the decision variables, i. The third and final component of a mathematical model is the objective function. This is a mathematical statement of some measure of performance such as cost, profit, time, revenue, utilization, etc. It is usually desired either to maximize or to minimize the value of the objective function, depending on what it represents.
Very often, one may simultaneously have more than one objective function to optimize e. In such cases there are two options. First, one could focus on a single objective and relegate the others to a secondary status by moving them to the set of constraints and specifying some minimum or maximum desirable value for them.
This tends to be the simpler option and the one most commonly adopted. The other option is to use a technique designed specifically for multiple objectives such as goal programming. In using a mathematical model the idea is to first capture all the crucial aspects of the system using the three elements just described, and to then optimize the objective function by choosing from among all values for the decision variables that do not violate any of the constraints specified the specific values that also yield the most desirable maximum or minimum value for the objective function.
This process is often called mathematical programming. Although many mathematical models tend to follow this form, it is certainly not a requirement; for example, a model may be constructed to simply define relationships between several variables and the decision-maker may use these to study how one or more variables are affected by changes in the values of others.
Decision trees, Markov chains and many queuing models could fall into this category.
Before concluding this section on model formulation, we return to our hypothetical example and translate the statements made in the problem definition stage into a mathematical model by using the information collected in the data collection phase. To do this we define two decision variables G and W to represent respectively the number of gizmos and widgets to be made and sold next month.
There is a constraint corresponding to each of the three limited resources, which should ensure that the production of G gizmos and W widgets does not use up more of the corresponding resource than is available for use. Thus for resource 1, this would be translated into the following mathematical statement 0.
Additionally, we must also ensure that each G and W value considered is a nonnegative integer, since any other value is meaningless in terms of our definition of G and W. The completely mathematical model is:. At the lowest level one might be able to use simple graphical techniques or even trial and error. However, despite the fact that the development of spreadsheets has made this much easier to do, it is usually an infeasible approach for most nontrivial problems. Most O.
First, there are simulation techniques, which obviously are used to analyze simulation models. A significant part of these are the actual computer programs that run the model and the methods used to do so correctly. However, the more interesting and challenging part involves the techniques used to analyze the large volumes of output from the programs; typically, these encompass a number of statistical techniques. The interested reader should refer to a good book on simulation to see how these two parts fit together.
The second category comprises techniques of mathematical analysis used to address a model that does not necessarily have a clear objective function or constraints but is nevertheless a mathematical representation of the system in question.
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Examples include common statistical techniques such as regression analysis, statistical inference and analysis of variance, as well as others such as queuing, Markov chains and decision analysis. The third category consists of optimum-seeking techniques, which are typically used to solve the mathematical programs described in the previous section in order to find the optimum i.
Specific techniques include linear, nonlinear, dynamic, integer, goal and stochastic programming, as well as various network-based methods. A detailed exposition of these is beyond the scope of this chapter, but there are a number of excellent texts in mathematical programming that describe many of these methods and the interested reader should refer to one of these. The final category of techniques is often referred to as heuristics.
The distinguishing feature of a heuristic technique is that it is one that does not guarantee that the best solution will be found, but at the same time is not as complex as an optimum-seeking technique. Although heuristics could be simple, common-sense, rule-of-thumb type techniques, they are typically methods that exploit specific problem features to obtain good results. A relatively recent development in this area are so-called meta-heuristics such as genetic algorithms, tabu search, evolutionary programming and simulated annealing which are general purpose methods that can be applied to a number of different problems.
These methods in particular are increasing in popularity because of their relative simplicity and the fact that increases in computing power have greatly increased their effectiveness. In applying a specific technique something that is important to keep in mind from a practitioner's perspective is that it is often sufficient to obtain a good solution even if it is not guaranteed to be the best solution.
If neither resource-availability nor time were an issue, one would of course look for the optimum solution. However, this is rarely the case in practice, and timeliness is of the essence in many instances. In this context, it is often more important to quickly obtain a solution that is satisfactory as opposed to expending a lot of effort to determine the optimum one, especially when the marginal gain from doing so is small. The economist Herbert Simon uses the term "satisficing" to describe this concept - one searches for the optimum but stops along the way when an acceptably good solution has been found.