AMPL

1
AMPL

• A Mathematical
• Programming Language
• Presentation 3

2
Outline

• Set operations
• Set indexing
• Advanced syntax
• Example

3
Set Operations
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Set Operations

• Let A, B, and U be sets
• AMPL allows simple set operations
• Union U A union B
• Intersection U A inter B
• Cross Product U A , B
• These operations become useful or necessary in
  many problems

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Advantage of Set Operations

• Suppose we have 3 sets of products,
• CANDY
• TOYS
• GAMES
• In our model we may need to declare the following
  parameters for each
• Price
• Supply
• Demand

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Naïve Setup

• Param Price_C CANDY
• Param Price_T TOYS
• Param Price_G GAMES
• Param Supply_C CANDY
• Param Supply_T TOYS
• Param Supply_G GAMES
• Param Demand_C CANDY
• Param Demand_T TOYS
• Param Demand_G GAMES

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Using Set Operation Union

• Param Price CANDY union TOYS union GAMES
• Param Supply CANDY union TOYS union GAMES
• Param Demand CANDY union TOYS union GAMES

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Even Better

• Set PRODUCTS
• CANDY union TOYS union GAMES
• Param Price PRODUCTS
• Param Supply PRODUCTS
• Param Demand PRODUCTS

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Compound Sets

• Suppose you have a set called PRODUCTS of
  products to sell at a store
• Consider that instead of one store you have three
  stores in different parts of the country
• Each store has a different level of each paramater

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One Solution

• Instead of using the set PRODUCTS, make three
  different sets
• PRODUCTS_1
• PRODUCTS_2
• PRODUCTS_3
• Let each of the three numbers represent one of
  your store locations

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One Solution (cont)

• Next, define each parameter for each set of
  products
• Param Price PRODUCTS_1
• Param Supply PRODUCTS_1
• Param Demand PRODUCTS_1
• Param Price PRODUCTS_2
• Param Supply PRODUCTS_2

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Easier Solution

• For a better solution use compound sets
• Param Price PRODUCTS, STORES
• Param Supply PRODUCTS, STORES
• Param Demand PRODUCTS, STORES

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Structure of PRODUCTS, STORES

• Suppose
• PRODUCTS oreos jenga lazertag
• STORES 1 2 3
• Then PRODUCTS, STORES is the set of all
  combinations of products with stores
• (oreos,1) (oreos,2) (oreos,3)
• (jenga,1) (jenga,2) (jenga,3)
• (lazertag,1) (lazertag,2) (lazertag,3)

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Specifying Data

• In your .dat file, your declaration of Demand
  could look like this
• param Demand 1 2 3
• oreos 10 20 30
• jenga 30 33 42
• lazertag 40 30 22

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Set Indexing
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Indexing of Sets

• Indexing of a one dimensional set
• sum i in PRODUCTS costimakei
• Indexing is similar in compound sets
• One can say
• sum (i,j) in PRODUCTS, STORES
  costimakei,j
• or
• sum i in PRODUCTS, j in STORES
  costimakei,j

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When do we need indexing?

• We may declare a parameter with or without giving
  index values
• param Demand PRODUCTS, STORES
• or
• param Demand i in PRODUCTS, j in STORES

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With PRODUCTS and STORES

• The sales at each store will not exceed the
  demand for any given product at that store.
• Could be expressed as the following constraint
• subject to DEMAND (i,j) in PRODUCTS, STORES
• Demand i,j Sell i,j

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With PRODUCTS and STORES

• Suppose we have another parameter
• Param Capacity STORES
• Let the capacity of a store be the total number
  of items it can sell all together

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With PRODUCTS and STORES

• The total sales at any given store can not
  exceed the sales capacity of that store.
• Could be expressed as follows
• Subject to CAPACITY j in STORES
• Sum i in PRODUCTS Sell i, j

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Advanced Syntax

• Tips and Shortcuts

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Transposition
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How to Transpose

• Recall an earlier example
• Param Demand 1 2 3
• oreos 10 20 30
• jenga 30 33 42
• lazertag 40 30 22

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How to Transpose

• Data in Transposed form
• Param Demand (tr) oreos jenga lazertag
• 1 10 30 40
• 2 20 33 30
• 3 30 42 22

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Why Transpose

• Not Necessary, but can help with data management
• What if we had 30 stores?
• Param Demand (tr) oreos jenga lazertag
• 1 10 30 40
• 2 20 33 30
• 3 30 42 22

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Omitted Data
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Omitted Data Entry

• Example Consider the following data
• Param Cost Supply Demand
• oreos 10 20 30
• jenga 30 33 42
• lazertag 40 30 22

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Omitted Data Entry

• Suppose in addition to the data specified in the
  previous table, you have an additional parameter
  such as
• Param Calories FOOD
• This parameter would apply to oreos, but not to
  jenga or lazertag.

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Example

• Param Cost Supply Demand
• oreos 10 20 30
• jenga 30 33 42
• lazertag 40 30 22
• Param Calories
• oreos 100

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Example

• Param Cost Supply Demand Calories
• oreos 10 20 30 100
• jenga 30 33 42 .
• lazertag 40 30 22 .
• We can use . to represent omitted data

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Example Problem

• Auto Production

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Auto Production

• You own an auto company and are setting up
  operation in a new country
• You want to decide which facilities to build and
  which cars to produce
• You want to maximize your profit after one year

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Cars and Buildings

• You can make three types of cars
• Sedan
• Hybrid
• Sport
• There are two different types of car production
  plants
• Standard Car Plant
• High Tech Sport Plant

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Cars and Buildings

• There are two types of engine plants
• Standard Engine Plant
• Hybrid Engine Plant
• An advanced tech center is required for some
  technologies

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Cost, Selling Price, Market
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Building Cost
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Final Details

• Due to labor limitations you can produce no more
  than 5000 cars in the first year
• Your goal is to maximize profit