### Mathematics Awareness Week 1996  ## Mathematics and Decision Making Poster ### Poster Description

The 1996 Mathematics Awareness Week poster shows a mathematical model of a high speed production floor with an automated material handling system which utilizes an overhead conveyor system to move raw material and finished goods. The model, provided by F&H Simulations, determines for the user where bottlenecks will occur, and what will happen to the throughput rate as system variables such as conveyor speeds, AGV (automated guidance vehicle) speeds and sorting rules are changed. The model takes into account the stochastic nature of shop floor production and stock replenishment.

All discrete event simulation models work on the same set of principles. Mathematical variables can be used to quantify the characteristics of the system. For example, the cycle time for a part on a machine might be represented by the variable CTIME. If the variable PRODUCED is used to represent the total number of parts produced in a 1000 hour period, then PRODUCED=1000/CTIME so long as the CTIME is a constant value for every part.

Computer simulation models handle the dynamic and random nature of a real life system by breaking the process down into discrete occurring events. The computer evaluates one event at a time, updating the system at each event. The computer model then carries out the process according to user-defined rules with the updated variables until the next event in time when the process is repeated all over again.

Deterministic mathematical formulas have been derived for simple queuing systems. The Pollaczek-Kyntchin equation determines the average waiting time for any element within this type of system (more specifically, an M/G/1 queuing system - one which has an exponentially distributed arrival rate, at a normally distributed service time, and only one server):

AverageWaitingTime = (1/2)*(1+CV^2)*(U/(1-U))*Pavg
where:
CV = standard deviation / mean
U = server utilization (busy time / total time)
Pavg = average processing time
Note that with this equation, the waiting time increases exponentially as the utilization approaches 100 percent.

 Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.