ISE 5984 · Through out this tutorial we use Electronics Inc as our example to demonstrate the use of subscripts. When is it appropriate to use subscripts in a model? Electronics

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ISE 5984

Tutorial on Subscript

By

Stephen Lisse & Thanujan Ratnarajah

05/02/2007

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Table of Contents

1. Introduction ................................................................................................................. 3

2. Creating Subscripts ..................................................................................................... 3

2.1. Single Dimensional Subscripts: .......................................................................... 3

2.2. Multi Dimensional Subscripts: ......................................................................... 10

3. Subscript Functions ................................................................................................... 14

3.1. Five main Vensim functions: ............................................................................ 14

3.2. Mapping the subscript ranges: .......................................................................... 15

4. Appendix ................................................................................................................... 16

4.1. Appendix 1 ........................................................................................................ 16

4.2. Appendix 2 ........................................................................................................ 19

5. References ................................................................................................................. 20

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1. Introduction

Subscripts are useful in complex models when certain loops need to be replicated.

Through out this tutorial we use Electronics Inc as our example to demonstrate the use of

subscripts. When is it appropriate to use subscripts in a model? Electronics Inc, an

organization specializing in producing and selling electronics items (such as DVD player,

VCR, TV and PC) hope to identify the customer adoption dynamics in different countries

(US, UK, China, Canada, and India). They could create a single customer diffusion

model and replicate it 20 times (5 countries * 4 Products) for the analysis purposes. But

this would increase the size of the model increasing CPU usage and complexity of the

model. It will hard to keep track of the model variables where there are 20 different

representations. In such cases we could se subscripts to create 20 different copies without

actually replicating it 20 times.

2. Creating Subscripts

2.1. Single Dimensional Subscripts:

For the purpose of demonstrating the use of one-dimensional subscript within the model

we use the customer base diffusion model [1]. In this particular example the modeler

hopes to understand the dynamics of a particular organization (Electronics Inc) which

produces four different products in the US market. These four products have different

market size and price.

Product Market Size Price

DVD player (P1) 100000000 50

VCR (P2) 120000000 70

PC (P3) 80000000 100

TV (P4) 250000000 150

Instead of creating four different diffusion models it is effective to use subscript within

the model to understand the dynamics.

Step 1:

Setting Subscripts: Select the ‘sub’ button located on the right hand side of the toolbar

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Figure 1

Step 2:

Select ‘new’ to add new subscripts. Select ‘edit’ to modify an existing subscript.

Figure 2

Step 3:

Create a subscript ‘Product’ which has four different elements. In this particular example

P1 denotes DVD player (Figure 3). Figure 4 explains an alternative way to define the

elements of the subscript. This particular method is useful when the subscripts have large

number of elements.

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Figure 3

Figure 4

When the elements are defined in this particular format (Figure 4), Vensim understands

that the elements are defined from P1 to P4.

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Step 4:

Apply this subscript to the cusomter.mdl. In this particular example the modeler wishes

to apply subscript to all the variables of the model.

Figure 5

After adding the subscript to the model each variable will have the name of the subscript

inside parenthesis next to them (Figure 6)

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Figure 6

Step 5:

The four different products have different market size and price. This could be defined by

adding those values specifically under Total Market Size and Price (Figure 7 and Figure

8).

If only one value is defined under Total Market size, Vensim would assume that the

market size is same for all products and would apply that value for all four elements. But

if the number values defined for these variables are greater than one but less than four or

greater than 4, Vensim would display an error message. Hence for a four element

subscript you can have one value or four separate values defined for each variable.

Anything other than that would display an error message.

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Figure 7

Figure 8

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Step 6:

Before simulating the model the modeler can decide which elements he/she would like to

see in a variable graph. You can do this by selecting the subscript control button (‘sub’-

Figure 1).

Then select which product needs to be added to the graph and adding them to the right

hand side panel (Selected elements)

Figure 9

Step 7:

Simulate the model. Figure 10 shows the dynamics of the potential adopters for the four

different products.

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Potential Adopters

400 M

300 M

200 M

100 M

0

0 1 2 3 4 5 6 7 8 9 10

Time (Year)

house

hold

s

Potential Adopters[P1] : Current




Figure 10

2.2. Multi Dimensional Subscripts:

In the previous example we concentrated on Electronics Inc’s customer diffusion models

for their four products sold in the US market. But if Electronics Inc had branches in

Canada, India, China and UK we will have to replicate the previous example model five

times to understand the dynamics in each country. But the efficient way to do this is to

use multi dimensional subscripts method. One of the dimensions we created was called

Product. We will create a second dimension called Location to capture the diffusion

models in the 5 countries where they have branches.

US Market (L1)

Product Market Size Price (US $)

DVD player (P1) 100000000 50

VCR (P2) 120000000 70

PC (P3) 80000000 100

TV (P4) 250000000 150

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Canada Market (L2)


DVD player (P1) 80000000 75

VCR (P2) 90000000 80

PC (P3) 80000000 110

TV (P4) 50000000 170

UK Market (L3)


DVD player (P1) 90000000 100

VCR (P2) 110000000 140

PC (P3) 70000000 120

TV (P4) 220000000 300

India Market (L4)


DVD player (P1) 200000000 25

VCR (P2) 180000000 35

PC (P3) 90000000 50

TV (P4) 300000000 75

China Market (L5)


DVD player (P1) 400000000 25

VCR (P2) 190000000 30

PC (P3) 100000000 30

TV (P4) 400000000 20

Step 1: Please follow the same procedure explained in section 2.1 in step 1




Here pick ‘Location’ from the scroll down menu under Subscripts 2. Similarly if you

have more than two subscripts you can chose them individually.

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Figure 11

After applying the two subscripts to the entire model the variables would appear in the

following manner:

Figure 12

Step 5:

Now we have to define the market size and the price of each product for each branch

locations.

There are multiple ways to define price and market size:

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Method 1: This particular method could only be used for one or two dimensional

subscripts method.

Figure 13

Price[Product,Location] =

50,75,100,25,25;70,80,140,35,30;100,110,120,50,30;150,170,300,75.20;

The semicolons separate the prices for each product. The commas separate the price at

each location for a given product.

Method 2: This could be used for any number of dimensions.

Price[Product,Location] = TABBED ARRAY(

50 75 100 25 25

70 80 140 35 30

100 110 120 50 30

150 170 300 75 20)

Method 3: You can use excel files to import data from outside sources:

Price[Product,Location] = GET XLS DATA(

'data.xls','Test Tab','A1')

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Vensim will open the excel file named ‘data.xls’ and read the data from Test Tab.

Location data will be read across starting from A1 (A,B,C,D,E). Product will be read

down from G8 (1,2,3,4).



3. Subscript Functions

3.1. Five main Vensim functions:

SUM: Gives the sum of the subscript values

PROD: Gives the product of the subscript values

VMIN: Gives the minimum value of the subscript values

VMAX: Gives the maximum value of the subscript values

ELMCOUNT: Gives the total number of elements within the subscripts

Sum:

The variable ‘Revenue’ in the model (customer model_multiple.mdl) is defined as:

Revenue[Product,Location] = Purchase

Rate[Product,Location]*Price[Product,Location]

This would produce the revenue for each product at each location individually. In order

to calculate the total revenue of this organization (at all locations) we create a variable

‘Total Revenue’, such that:

Total Revenue = SUM(Revenue[Product!,Location!])

Vensim will total the revenue created by each product (indicated by ‘!’) at all locations.

There are is no subscript notation in the naming convention. It is because the ‘Total

Revenue’ variable creates a single value and not a matrix.

To know how much each branch (location) of the Electronic INC makes through the

entire product as revenue we created a variable ‘Revenue by Location’:

Revenue by Location[Location] = SUM(Revenue[Product!,Location])

Prod:

If a variable is defined as:

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Probability = Success rate [type 1] * Success rate [type 2]* Success rate [type 3]…….

Success rate [type n]

This could be defined by:

Probability = PROD(Success rate [type 1!])

VMIN:

Min Revenue= VMIN(Revenue by Location[Location!])

This function will return the minimum variable presented in the ‘Revenue by Location’

matrix.

VMAX:

MAX Revenue= VMAX(Revenue by Location[Location!])

This function will return the minimum variable presented in the ‘Revenue by Location’

matrix.

ELMCOUNT:

Number of products = ELMCOUNT(Product)

This function will count the total number of elements within the subscript ‘Product’. This

function can only take a single subscript dimension.

3.2. Mapping the subscript ranges:

It is sometimes necessary to map one subscript range to another. For example assume

Electronics Inc specializes in producing and selling certain products in certain locations.

Electronics Inc produces and sells only DVD players in China

Electronics Inc produces and sells only TV in US

Electronics Inc produces and sells only PC in UK

Price at Location [Location] = Price of Product [Product]

This equation will generate an error in Vensim since the subscript range in the right hand

side and the left hand side is different. We can eliminate the error by mapping the

subscript ranges.

During the definition of subscripts (Step 3) we need to define it in the following manner:

Product: DVD, TV,PC -> Location

Location: China, US, Canada

‘->’ notation maps the Product subscript to Location subscript.

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4. Appendix

4.1. Appendix 1

1. Introduction to Subscripts

From Reference a., it is known that subscripting enables construction of very advanced

arrayed Vensim Models. Subscripting is used when a piece of model structure will need

to be repeated over and over again and a subscript allows one variable and equation to

represent a number of different distinct concepts.

a. What are subscripts

Also, as discussed in Reference b., subscripts allow a single variable to represent more

than one thing, and subscripts are only available in Vensim Professional and DSS. All

variable types except Groups, Units, String Constants, Subscript Ranges, Subscript

Elements, and Time Bases can have subscripts. Subscripts are enclosed in square brackets

[ ] directly following the variable name.

b. What are the uses of subscripts

Referring again to Reference b., a variable can have up to eight subscripts separated by

commas. Vensim will return error messages if you do not use subscript values for a

variable consistently throughout the model. Subscript elements are also sometimes

referred to as subscript constants.

One can repeat structure by creating and debugging one structure, then copying and

replicating

that structure as many times as needed. However, this may lead to complex diagrams and

hard-wiring of constant values and number of structures. Subscripting is a better way to

repeat structure. A subscript can be created and added to the one original structure,

creating as many structures as there are subscript elements. So numbers of structures and

numerical values for all structures can easily be changed. Diagrams will also be neater.

Subscripts can also allow construction of a single stock that represents many different

segments of the same physical process. This same structure can be built using subscripts

to represent each age category. By using subscripts, you can easily change the number or

characteristics of age categories by changing subscript names and values. Thus you do

not have to change the physical structure by removing or adding stocks.

Additionally, subscripts have several special functions designed to manipulate equations.

The

SUM function sums all values of a subscript marked with an exclamation mark (!). Other

functions include products, maximums, and minimums.

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Further, it should be noted that subscripts should not appear in sketches. Sketches

represent structure, and subscripts are a convenient way of replicating structure. A sketch

maintains a simpler and less cluttered view of a model by not distinguishing subscripted

and unsubscripted variables.

c. Examples of subscript usage related to modeling

Some examples of subscript use from Reference b. are discussed in the following

paragraphs.

A retail outlet such as a shoe retailer might have stores in three locations. A

model can be built for one store with structure for staff, inventory, sales, etc.

Once that model simulates properly, a subscript is created called Location, with

the subscript elements Boston, New York, and San Francisco. The subscript

Location is added to the one store structure, constants are modified to reflect the

value for each location, and now the one model contains three structures.

Each subscripted structure can be individually customized, with different

constants, multiple equations, even multiple subscripts.

Also, in a company containing a workforce, the subscript [worktype] might represent the

types of workers in the company, and the subscript [location] might represent the

physical location of the office or plant. So the equation subscript hires[worktype,

location] = desired hires[worktype, location] / time to hire worker [worktype] allows one

equation to represent the hiring of workers for all types of work and for all location

(using only two subscripts). Note that the time to hire a worker in the above equation is

considered to be different depending on worktype, but not location (therefore the

subscript is only for worktype).

Additionally, an example of the SUM function is shown below:

total workers[worktype] = SUM ( Workers[worktype, location!])

The above equation describes the summing of workers of each particular worktype (e.g.,

office staff, plant operators) over all locations. That is, total office staff summed over all

locations, total plant operators summed over all locations. The SUM function works on

the subscript location because location is marked with the exclamation mark.

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2. Creating Subscripts Within A Model

a. How to create subscripts in a simple model

The following discussion is taken from the Edit Manual of Reference b.

The Set Subscripts command lets you set subscripts on the selected variables. This is a

convenient way to add subscripts to the model variables without having to modify

individual equations. It is, of course, still possible to modify individual equations and this

is often necessary when you have multiple equations for a variable or want to use a vector

function.

When you choose the Set Subscripts command the Subscript Modification dialog will

appear:

The title bar indicates that name of the variable being changed or, if you have selected

more than one variable, the number of variables being changed.

Subscript 1 through Subscript 8 allow you to set the subscripts. Normally you will only

use one or to of these but it is possible to use all 8 so they are included for completeness.

Click on the dropdown box to select a Subscript Range. When you open this dialog with

a single variable selected the subscripts showing will be those for the variable. If you

open it with more than one variable selected and each selected variable has the same

subscripts (in the same order) these will be shown. If the subscripts are not the same for

the existing variables the dialog will appear as it does above, with --none showing for

each subscript.

Apply to selected variables by lets you specify how you want to modify the equations for

the selected variables.

Setting causes the selected variables to take on the new set of subscripts. The subscripts

for these variables are changed to be the ones specified in the order specified. If a

variable already has any of these subscripts the usage will not be changed. For example if

Pop was subscripted by age and there was an equation with SUM(Pop[age!]), then

changing the subscripts to [country,age] would change the expression to

SUM(Pop[country,age!]). Similarly if you have multiple equations with Subscript

Constants these will be retained. If a variable has additional subscripts not in the list these

will be removed.

Adding at end causes the selected subscripts to be added at the end of the existing

subscripts for a variable. Thus existing subscripts will not be removed. If the variable

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already has one of the chosen subscripts it will be moved, if necessary so that it appears

at the end. For example adding age to a variable subscripted [age,sex,country] will

change the subscripts to [sex,country,age]. Just as for setting any existing occurrences of

the subscript will be maintained in their current form. You can add one or more

subscripts with this option.

Adding at beginning is the same as adding at end except the chosen subscripts appear

first, instead of last.

Removing will remove the chosen subscripts from the selected variables. If the variables

do not have any of the subscripts nothing will happen. Include subranges in subscript

lists, when checked, allows you to add subranges to variables. This can sometimes be

handy if some variables will only be computed and used on a Subrange.

Replace subranges with the specified subscript allows you to override the default

behavior for setting and adding subscripts. When this is checked if Vensim finds a

subrange for one of the chosen subscripts it will replace it with the chosen subscript,

rather than leaving it alone. Note that Vensim will still leave Subscript Elements alone. If

you want to change these first remove them, then add the subscript range.

Add a new subscript to the model allows you to add subscripts. This can also be done

from the Subscript Control or from the Equation Editor and this button is just a

convenient alternative to those techniques.

4.2. Appendix 2

Subscript Errors

Using an equation with a Subscript Range is like using a "for" or "do" loop. Although

you do not need to specify this loop explicitly, the loop concept determines what you can

and cannot do. For example, consider the following equation:

births[country,income] =frac_birth_rat[country,income] *population[country,income] ~~|

This equation is really saying for each country {USA,CANADA,MEXICO} for each

income {LOW,MIDDLE,HIGH}

births[country,income]= frac_birth_rat[country,income] *population[country,income] ~~|

end for each income end for each country

The equations for country and class determine the meaning of each country and each

income as indicated by the comments. The Subscript Ranges on the left-hand side of an

equation determine the implicit "for each" components. Thus the equation

births[country,HIGH] = frac_birth_rat[country,income] *population[country,income] ~~|

would be wrong. The left-hand side implies only a single "for each" loop on country, but

the righthand side requires one for country and one for income. In this case, you will get

a message reporting that the Subscript Range income appears on the right but not on the

left of the equation.

Family mismatches are another common problem with subscripts. All variables must

have the same number of subscripts from the same family. For example, the equation that

contains both population[country] and population[country,income] will cause a problem

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since population must be broken down either by country, or by country and income. It

cannot be broken down by country sometimes and by country and income at other times.

Even when the number of subscripts match, there can be problems if the families do not

also match. Suppose that population[country,income] appears in one place and

population[country,region] in another. It makes no sense for population to be broken

down in two different ways, both by income and by region.

Tracing Subscript Errors:

Subscript errors can be hard to trace because the equation that Vensim flags as having the

error might be correct, and another equation in error. Generally the best strategy for

finding subscript errors is to select the variable name that is causing problems as the

Workbench Variable and then click on the Uses Tree tool to show where it is used. By

looking at the equation for the variable, and then at each equation in which it is used, you

should be able to determine the source of the problem.

HINT: When working with the Sketch Editor, select a variable into the Workbench and

click on the Equation Edit tool contained in the Analysis Toolbar to directly open the

equation for a variable. When working with the Text Editor, turn on Entrain before you

select variables into the Workbench.

5. References Customer.mdl [1]: Created by Dr.Hazhir Rahmandad

Appendix 1: Vensim Reference manual

Appendix 2: Vensim Reference manual

Documents

ISE 5984 · Through out this tutorial we use Electronics Inc as our example to demonstrate the use of subscripts. When is it appropriate to use subscripts in a model? Electronics