1. DSS & Warehousing Systems
Chapter 9
Efrem Mallach
Prepared by
Luvai Motiwalla
Irwin/McGraw-Hill
Copyright © 2000 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Mathematical Models and Optimization
Introduction
Queuing models
Markov process models
Simulation, queuing theory and Markov processes compared
Optimization

3. Introduction
DSS uses mathematical models in several ways.Mathematical methods are also used for optimization,where optimization figures directly in DSS.
The two mathematical approaches that are most often used to study dynamic systems are queuing theory and Markov process analysis
Pages 346 to 347
Optimization is the act or the process of choosing the best of several alternatives.
Optimization is used in the sixth category of DSS, optimization systems. It also figure in some DSS in the seventh category, suggestion systems.
The queuing theory and the markov process analysis are based on the concept of defining the states in which the system can be, the possible transitions among those states, and the rates at which the transitions takes place.

4. Queuing Models-cont’d
Queuing models obtain the statistics of the system behavior directly, without following individual events.
Queuing theory concepts
Pages 348 to 350
Further more queuing models describe system behavior by formulas that can be evaluated for any desired set of system parameters, rather than as numbers that are calculated for one set of parameters and must be recalculated from scratch for any other set. These characteristics make queuing models a useful adjunct to simulation, especially in DSS to plan complex systems.
Queuing theory concepts:
The possible states of the system are determined by studying the system, the variables that define its state and their possible combinations.
The rate at which each state changes to each other state – that is, the rates of each possible state transition – are determined as functions of transition arrival rates into the system and service times of system process.these are in turn , determined by studying the system and its environment. These rates are called state transition rates.
If the system is in a stable situation the average transition rate into each state must necessarily equal the average transition rate out of that state.
Solving these equations yields the state probabilities themselves. From these we can determine other statistics of interest, such as average customer waiting times or average queue lengths.

5. Queuing Models Arrival and departure time distributions
Erlang distributions, Hyperexponential distributions
Queuing theory on a computer
Pages 352 to The assumption of random inter arrival and service times corresponds to an exponential distribution of these times. This distribution has a high probability of short times. It allows for long times with decreasing likelihood. An important characteristic of the exponential distribution is that knowing the elapsed time since the last event does not help predict the next one.
Erlang distributions: Using sub processes reduces the variance time and can approximate a normal service time distribution. Using several processes in a model in this way does not imply that the system itself consists of a series of processes. It is just a mathematical trick to allow us to model some types of reality more closely than an exponential distribution would. Such multistage distributions are called Erlang distributions.
Hyperexponential distributions: We can also create distributions with greater variance than the exponential distribution. This is done by replacing a real process with two parallel exponential processes in the model, one of which has a much smaller mean time than the other, and sending transactions at random to one or the other. Hyperexponential distributions are good for modeling some data communication connection times or the execution times of user commands to a computer.
Queuing theory on a computer: Because queuing theory is basically a mathematical technique that does not depend on a computer to formulate or solve its equations, the first requisite for using it in a DSS is sufficient mathematical brainpower to carry out those steps. The equations once they have been developed, are simple to put in a computer. If the overall DSS lends itself to the spreadsheet paradigm, the formula capability of any spreadsheet program is fully sufficient for queuing theory calculations.

6. Markov Process Models
A Markov process is a system that progresses from one state to another over time, where the likelihood of its being in a given state at any time step depends only on its previous state and not on its prior history.
Computer calculations for Markov process
Pages 355 to 357
Computer calculations for Markov process: Given an initial state , we can determine the probabilities of the next state from the transition probabilities shown. A matrix notation is convenient for this purpose. This matrix is called - The state transition matrix.
Because of the apparent simplicity, programming a Markov model on a computer doesn’t seem a daunting task .It just one matrix calculation after another. Two points are important before you embark on programming,
Realistic Markov transitions are quite sparse.
because of round of errors in computing, the sum of the state probability vector elements will drift away from from 1 after many iterations.

7. Simulation, Queuing Theory , and Markov Processes Compared
All three of the approaches described for modeling dynamic systems – simulation, queuing theory, and Markov processes have advantages and disadvantages in decision support applications.
Pages 357 to 358
Simulation provides a dynamic model of a dynamic system, whereas the other two approaches provide static models of a dynamic system after it has settled down to its steady state behavior. Simulation tracks what goes inside a system and forces us to infer its characteristics by summarizing a large number of observations. The others tell us these characteristics but do not give us the same visibility into its innards. Queuing theory provides the most general solution to a given problem and often the fastest as well. Simulation model results are the least general and usually take the longest time to obtain – because of both programming time and computer execution time. Markov process analyses are intermediate on both counts. Therefore at first glance one should prefer queuing theory to a Markov process analysis and the Markov process approach to simulation.

8. Optimization-cont’d
Optimization: choosing one of several numerically compared alternatives to maximize or minimize the objective function, it is an important part of many decision support systems.
Approaches to optimization
Pages 351 to 371
Approaches to optimization include complete enumeration, random search, calculus, linear programming, hill climbing,and a variety of other numerical methods

9. Optimization-cont’d
Complete enumeration: means trying all possible choices and picking the one that produces the best predicted results.
Complete enumeration: is feasible only when the options or the decision parameters are discrete and few in number. Complete enumeration can also be used as part of a larger optimization process.

10. Optimization-cont’d
Random search: here the analyst does not try to evaluate the objective function for all possible combinations of controllable variables, Rather the analyst chooses, several, perhaps a few hundred, random combinations of the controllable variables.
Random search: the random search technique cannot guarantee convergence. Its results cannot be proven optimal in a mathematical sense but are usually quite acceptable for business decision making.

11. Optimization-cont’d
The calculus approach: Calculus can optimize continuous, differentiable functions of the decision variables.
Calculus approach: Its usefulness is limited to situations in which the objective function meets these requirements and we can solve the often – complex equations that are involved.

12. Optimization-cont’d
Linear programming: can optimize systems in which the objective function depends linearly on how resources are allocated to alternative uses and where this allocation is subject to constraints, such as the available quantity of one or more resources.
Linear programming: linear programming problems are generally solved by computer. Many linear programming packages are available. Most of them use the simplex method to solve LP problems by following the edges of the feasible region.

13. Optimization-cont’d
Numerical methods: A wide variety of numerical optimization methods exists.
Numerical methods: A wide variety of numerical optimization methods exists. The analyst may be forced to use one of them when the methods usually used do not apply or do not work. One numerical method is hill climbing and the second numerical method is M.J.Box method.

14. Optimization-cont’d
Hill climbing: This is the numerical equivalent of calculus
Hill climbing: It uses numerical approximations to the derivatives and moves around the solution space, climbing the hill, of the objective function, until its approximations to the derivatives are zero within an acceptable tolerance. The process must be repeated with different starting points to make sure the overall, or global, maximum is found.

15. Optimization
M.J.Box method: it starts out as a random search does, by choosing several random values of the decision variables and evaluating the objective function at them.
M.J.Box method: rather than continuing to choose more random points, however this method uses the known values as a group to pick a new point to try. It tends to range widely over the solution space and then zero in on a maximum. As with hill climbing, it is necessary to repeat this process more than once to avoid settling on a local maximum.