Introduction

Units are common

Measurement plays a central role in observations of the real world. Most observed quantities are measured in some unit (e.g. dollar, hour, meter, etc.), and the magnitude of the unit influences the mental picture that you may have of an object (e.g. ounce, kilogram, ton, etc.). When you combine such objects in a numerical relationship, the corresponding units must be commensurable. Without such consistency, the mathematical relationships become meaningless.

Why units in models

There are several good reasons to track units throughout a model. The explicit mentioning of units can enhance the readability of a model, which is especially helpful when others read and/or maintain your model. Units provide the AIMMS compiler with additional checking power to find errors in model formulations. Finally, through the use of units you can let AIMMS perform the job of unit conversion and scaling.

Standard units

The model editor in AIMMS will give you access to a large number of quantities and units, and in particular to those of the International System of Units (referred to as SI from the French “Systeme Internationale”). The SI system is an improved metric system adopted by the Eleventh General Conference of Weights and Measures in 1960. The entire SI system of measurement is constructed from the atomic base units associated with the following nine basic quantities.

Table 64 Basic SI quantities and their base units

Quantity

Atomic Base Unit

Text

length

m

meter

mass

kg

kilogram

time

s

second

temperature

K

kelvin

amount of mass

mol

mole

electric current

A

ampere

luminous intensity

cd

candela

angle

solid angle

sr

Derived quantities and units

All quantities which are not one of the nine basic SI quantities are called derived quantities. Each such quantity has a derived base unit which can be expressed in terms of the atomic base units of the basic SI quantities. Optionally, a compound unit symbol can be associated with such a derived base unit, like the symbol N for the unit kg*m/s^2. The following table illustrates some of the more well-known derived quantities and their corresponding derived base units. Note that five of them have an associated compound unit symbol. Many other derived quantities are available in AIMMS.

Table 65 Selected derived SI quantities and their base units

Quantity

Derived Base Unit

Text

area

m

square meter

volume

m

cubic meter

force

N = kg*m/s

newton

pressure

Pa = kg/m*s

pascal

energy

J = kg*m/s

joule

power

W = kg*m/s

watt

charge

C = A*s

coulomb

density

kg/m

kilogram per cubic meter

velocity

m/s

meter per second

angular velocity

rad/s

Related units

Aside from the base unit that must be associated with every quantity, it is also possible to specify a number of related units. Related units are those units that can be expressed in terms of their base unit by means of a linear relationship. A typical example is the unit km which is related to the base unit m by means of the linear relationship $$x$$ km = 1000*x m. Similarly, the unit degC (degree Celsius) is related to the base unit K through the formula $$x$$ degC = (x + 273.15) K.

Standard unit prefix notation

Frequently, related units are a multiple of their own base unit, which is reflected through a prefix notation that indicates the level of scaling. this table shows the standard SI prefix symbols and their corresponding scaling factor. Familiar examples are kton, MHz, kJ, etc. Note that any prefix can be applied to any base unit except the kilogram. The kilogram takes prefixes as if the base unit were the gram.

Table 66 Prefixes of the International System

Factor

Name

Symbol

Factor

Name

Symbol

$$10^1$$

deca

da

$$10^{-1}$$

deci

d

$$10^2$$

hecto

h

$$10^{-2}$$

centi

c

$$10^3$$

kilo

k

$$10^{-3}$$

milli

m

$$10^6$$

mega

M

$$10^{-6}$$

micro

mu

$$10^9$$

giga

G

$$10^{-9}$$

nano

n

$$10^{12}$$

tera

T

$$10^{-12}$$

pico

p

$$10^{15}$$

peta

P

$$10^{-15}$$

femto

f

$$10^{18}$$

exa

E

$$10^{-18}$$

atto

a

$$10^{21}$$

zetta

Z

$$10^{-21}$$

zepto

z

$$10^{24}$$

yotta

Y

$$10^{-24}$$

yocto

y

Flexible specification

To give you maximum freedom to choose your own quantities, units and naming conventions, AIMMS is not exclusively committed to any particular standard. However, you are encouraged to use the standard SI units and prefix symbols to make your model as readable and maintainable as possible.

Summary of terminology

Thus far you have encountered basic quantities (this table) and derived quantities (this table). Each quantity has a base unit. The base unit of a basic quantity is defined through a unit symbol, referred to as an atomic unit. All other base units are derived base units. Such units are defined through an expression in terms of other base units, which can eventually be translated into an expression of atomic base units. You have the option to associate a unit symbol with any derived base unit, which is referred to as a compound unit symbol. Whenever you have associated a unit symbol with the base unit of either a basic or derived quantity, you are also allowed to specify one or more related unit symbols by specifying the corresponding linear relationship.