Difference between revisions of "Example 1: Beer Distribution LP, Base Case"

 
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We start with an adaptation of a classic Linear Programming (LP) example:  
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[[Category: Analytica Optimizer Guide]]
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[[Category: Beer Distribution model]]
  
==Example 1: Beer Distribution LP, Base Case==
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<breadcrumbs> Analytica Optimizer Guide > Optimizing with Arrays > {{PAGENAME}}</breadcrumbs>
 +
<br />
 +
We start with an adaptation of a classic Linear Programming (LP) example.
  
===Model Description===
+
==Model Description==
  
A large brewing company operates five breweries nationwide. Each brewery distributes product among four regions. Routes include all combinations of Breweries and Market regions. The challenge is to find the shipping pattern that minimizes distribution cost while a) meeting demand in each region, and b) observing production limits at each brewery. We assume that distribution cost per case of beer is proportional to the distance between the brewery and the region.
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A large brewing company operates five breweries nationwide. Each brewery distributes product among four regions. Routes include all combinations of Breweries and Market regions. The challenge is to find the shipping pattern that minimizes distribution cost while  
 +
<ol type="a">
 +
<li> meeting demand in each region, and  
 +
<li> observing production limits at each brewery.</ol><br />We assume that distribution cost per case of beer is proportional to the distance between the brewery and the region.
  
===Setting up the Model===
+
==Setting up the Model==
  
To explore and follow this example in Analytica, run the '''Beer Distribution LP1.ana''' model in the Optimizer Examples folder.
+
To explore and follow this example in Analytica, run the <code>Beer Distribution LP1.ana</code> model in the Optimizer Examples folder.
  
 
'''Brewery and Market (indexes)'''
 
'''Brewery and Market (indexes)'''
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There are five '''Breweries''' and four '''Market''' regions:
 
There are five '''Breweries''' and four '''Market''' regions:
  
<code>Index Brewery :=  
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<pre style="background:white; border:white; margin-left: 1em;">
        ['Fairfield','Fort Collins','Jacksonville','Merrimack','St Louis']  
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Index Brewery := ['Fairfield', 'Fort Collins', 'Jacksonville', 'Merrimack', 'St Louis']  
Index Market := ['North','South','East','West']</code>
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Index Market := ['North', 'South', 'East', 'West']
 +
</pre>
 +
'''Distance (input array)'''
  
==== '''Distance (input array)''' ====
 
 
'''Distance''' between breweries and markets is represented as a table dimensioned by '''Brewery''' and '''Market'''. Units are thousands of miles:
 
'''Distance''' between breweries and markets is represented as a table dimensioned by '''Brewery''' and '''Market'''. Units are thousands of miles:
  
<code>Variable Distance :=  
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<pre style="background:white; border:white; margin-left: 1em;">
       Table(Brewery,Market)(1.8,2.2,3.4,0.1,0.3,1.2,2.4,1,2.9,1.8,0.6,3.3, 1.2,2.1,0.2,3.5,0.5,1.1,0.8,2.5)</code>
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Variable Distance :=  
 +
       Table(Brewery, Market)(1.8, 2.2, 3.4, 0.1, 0.3,1.2, 2.4, 1, 2.9,1.8, 0.6, 3.3, 1.2, 2.1, 0.2, 3.5, 0.5, 1.1, 0.8, 2.5)
 +
</pre>
  
[[File:4-3.png|400px]]
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:[[File:4-3.png|400px]]
  
 
'''Freight Price (input scalar value)'''
 
'''Freight Price (input scalar value)'''
  
 
'''Freight Price''' is a simple scalar value of $5 per 1,000 miles per case of beer.  
 
'''Freight Price''' is a simple scalar value of $5 per 1,000 miles per case of beer.  
<code>Variable Freight_price := 5</code>
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:<code>Variable Freight_price := 5</code>
 +
 
 
'''Production Limits (input array)'''
 
'''Production Limits (input array)'''
  
 
'''Production Limits''' indicate maximum capacity for each brewery in cases of beer:  
 
'''Production Limits''' indicate maximum capacity for each brewery in cases of beer:  
<code>Variable Production_limits := Table(Brewery)(600K,250K,450K,300K,240K)</code>
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:<code>Variable Production_limits := Table(Brewery)(600K, 250K, 450K, 300K, 240K)</code>
[[File:4-4.png|400px]]
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 +
:[[File:4-4.png|400px]]
  
 
'''Delivery Targets (input array)'''
 
'''Delivery Targets (input array)'''
  
 
'''Delivery Targets''' indicate the minimum quota that each Market region must receive:  
 
'''Delivery Targets''' indicate the minimum quota that each Market region must receive:  
<code>Variable Delivery_target := Table(Market)(240K,280K,700K,500K)</code>
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:<code>Variable Delivery_target := Table(Market)(240K, 280K, 700K, 500K)</code>
[[File:4-5.png|400px]]
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 +
:[[File:4-5.png|400px]]
  
 
'''Distribution Cost (intermediate array)'''
 
'''Distribution Cost (intermediate array)'''
  
 
'''Distribution Cost per Case''' is Freight Price multiplied by Distance:  
 
'''Distribution Cost per Case''' is Freight Price multiplied by Distance:  
<code>Variable Dist_per_case := Freight_price * Distance</code>
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:<code>Variable Dist_per_case := Freight_price*Distance</code>
 +
 
 
'''Shipment Quantities (input decision array)'''
 
'''Shipment Quantities (input decision array)'''
  
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'''Set Brewery and Market as intrinsic indexes'''
 
'''Set Brewery and Market as intrinsic indexes'''
  
Finally and perhaps most importantly, we consider the index status for this array. The optimized solution should be in the same form as this table. It should also be integrated such that every array element has simultaneous influence over all other elements. ''Brewery'' and '''Market''' are both intrinsic indexes in this case.
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Finally and perhaps most importantly, we consider the index status for this array. The optimized solution should be in the same form as this table. It should also be integrated such that every array element has simultaneous influence over all other elements. '''Brewery''' and '''Market''' are both ''intrinsic'' indexes in this case.
  
<code>Decision Shipment_Quantity := Array([Brewery, Market],1)
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<pre style="background:white; border:white; margin-left: 1em;">
Shipment_Quantity attribute Lower Bound := 0
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Decision Shipment_Quantity := Array([Brewery, Market], 1)
Shipment_Quantity attribute Intrinsic Indexes := [Brewery, Market</code>]
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Shipment_Quantity attribute Lower Bound := 0
 +
Shipment_Quantity attribute Intrinsic Indexes := [Brewery, Market]
 +
</pre>
  
[[File:4-6.png|400px]]
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:[[File:4-6.png|400px]]
  
 
'''Total Distribution (Objective)'''
 
'''Total Distribution (Objective)'''
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Our goal is to minimize the total distribution cost. For each route, distribution cost is the '''Distribution Cost per Case''' multiplied by the '''Shipment Quantity''' for each route. The final objective is the sum for all routes:
 
Our goal is to minimize the total distribution cost. For each route, distribution cost is the '''Distribution Cost per Case''' multiplied by the '''Shipment Quantity''' for each route. The final objective is the sum for all routes:
  
<code>Objective Total_dist_cost :=  
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<pre style="background:white; border:white; margin-left: 1em;">
       Sum(Shipment_Quantity * Dist_per_case, Brewery, Market)</code>
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Objective Total_dist_cost :=  
 +
       Sum(Shipment_Quantity*Dist_per_case, Brewery, Market)
 +
</pre>
  
 
Summing over '''Brewery''' and '''Market''' makes the objective a scalar value. This is a necessary condition for the optimization.
 
Summing over '''Brewery''' and '''Market''' makes the objective a scalar value. This is a necessary condition for the optimization.
  
<Tip="Tip">Tip Every minimizing or maximizing optimization is based on a scalar objective value. In other words, intrinsic dimensions are not allowed in the Objective. If the Objective is not scalar, Analytica assumes the dimensions are extrinsic and performs independent optimizations for each scalar element in the Objective array. (For some optimization problems, the challenge is simply to find a solution that satisfies all constraints. This type of optimization has no objective at all.)</Tip>
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<Tip Title="Tip">Tip Every minimizing or maximizing optimization is based on a scalar objective value. In other words, intrinsic dimensions are not allowed in the Objective. If the Objective is not scalar, Analytica assumes the dimensions are extrinsic and performs independent optimizations for each scalar element in the Objective array. (For some optimization problems, the challenge is simply to find a solution that satisfies all constraints. This type of optimization has no objective at all.)</Tip>
  
 
'''Supply Constraint'''
 
'''Supply Constraint'''
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Is '''Brewery''' an intrinsic index for the '''Supply Constraint''' array? The answer is YES because we need to enforce the respective Supply Constraints on all breweries simultaneously. Make sure to include '''Brewery''' in the Intrinsic Indexes list for the '''Supply Constraint''' node.
 
Is '''Brewery''' an intrinsic index for the '''Supply Constraint''' array? The answer is YES because we need to enforce the respective Supply Constraints on all breweries simultaneously. Make sure to include '''Brewery''' in the Intrinsic Indexes list for the '''Supply Constraint''' node.
  
<nowiki>Constraint Supply_constraint :=
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<pre style="background:white; border:white; margin-left: 1em;">
     Sum(Shipment_Quantity, Market)<=Production_Limits
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Constraint Supply_constraint :=
 +
     Sum(Shipment_Quantity, Market) <= Production_Limits
  
Supply_constraint attribute Intrinsic Indexes := [Brewery]</nowiki>
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Supply_constraint attribute Intrinsic Indexes := [Brewery]
 +
</pre>
  
 
Is '''Market''' an intrinsic index for the '''Supply Constraint'''? The answer is NO because '''Market''' is not a dimension of the array at all. The [[Sum]]() function eliminates it, leaving the array with only one dimension.
 
Is '''Market''' an intrinsic index for the '''Supply Constraint'''? The answer is NO because '''Market''' is not a dimension of the array at all. The [[Sum]]() function eliminates it, leaving the array with only one dimension.
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'''Market''' is an intrinsic index for the Demand Constraint because the solution should meet quotas for all markets. Make sure to include '''Market''' in the Intrinsic Indexes list for the '''Demand Constraint''' node.
 
'''Market''' is an intrinsic index for the Demand Constraint because the solution should meet quotas for all markets. Make sure to include '''Market''' in the Intrinsic Indexes list for the '''Demand Constraint''' node.
  
<nowiki>Constraint Demand_constraint :=
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<pre style="background:white; border:white; margin-left: 1em;">
 +
Constraint Demand_constraint :=
 
     Sum(Shipment_Quantity, Brewery) >= Delivery_Target
 
     Sum(Shipment_Quantity, Brewery) >= Delivery_Target
Demand_constraint attribute Intrinsic Indexes := [Market]</nowiki>
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Demand_constraint attribute Intrinsic Indexes := [Market]
 +
</pre>
  
 
The [[Sum]]() function eliminates '''Brewery''' from the array.
 
The [[Sum]]() function eliminates '''Brewery''' from the array.
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'''Optimization Node'''
 
'''Optimization Node'''
  
Remembering required attributes of the [[DefineOptimization]]() function as described in Chapter 1, we need to identify Decisions, Constraints, and the Objective to be minimized/maximized.
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Remembering required attributes of the [[DefineOptimization]]() function as described in the [[Quick Start]], we need to identify Decisions, Constraints, and the Objective to be minimized/maximized.
<nowiki>Variable Optimization := DefineOptimization
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<pre style="background:white; border:white; margin-left: 1em;">
    (Decision:Shipment_Quantity,
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Variable Optimization := DefineOptimization
     Constraints:Supply_Constraint, Demand_Constraint,
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    (Decision: Shipment_Quantity,
     Minimize: Total_Dist_Cost)</nowiki>
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     Constraints: Supply_Constraint, Demand_Constraint,
 +
     Minimize: Total_Dist_Cost)
 +
</pre>
 
'''Solution Node'''
 
'''Solution Node'''
  
 
We obtain the solution using the [[OptSolution]]() function. The first parameter identifies the optimization node. The second parameter (optional) identifies a specific Decision for which the solution should be represented.
 
We obtain the solution using the [[OptSolution]]() function. The first parameter identifies the optimization node. The second parameter (optional) identifies a specific Decision for which the solution should be represented.
  
<nowiki>Decision Optimized_Solution :=
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<pre style="background:white; border:white; margin-left: 1em;">
     OptSolution(Optimization, Shipment_Quantity)</nowiki>
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Decision Optimized_Solution :=
 +
     OptSolution(Optimization, Shipment_Quantity)
 +
</pre>
 
'''Optimized Objective'''
 
'''Optimized Objective'''
  
 
The [[OptObjective]]() function calculates the minimized or maximized quantity. In this case, it represents the total distribution cost.
 
The [[OptObjective]]() function calculates the minimized or maximized quantity. In this case, it represents the total distribution cost.
<code>Variable Optimized_Objective := OptObjective(Optimization)</code>
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:<code>Variable Optimized_Objective := OptObjective(Optimization)</code>
 +
 
 
'''Status'''
 
'''Status'''
  
 
Status text will confirm that the optimizer has found a solution
 
Status text will confirm that the optimizer has found a solution
<code>Variable Status := OptStatusText(Optimization)</code>
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:<code>Variable Status := OptStatusText(Optimization)</code>
[[File:4-7.png|400px]]
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 +
:[[File:4-7.png|400px]]
  
===Verifying the Optimization Setup===
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==Verifying the Optimization Setup==
 
'''Evaluate the Optimization node'''
 
'''Evaluate the Optimization node'''
  
It is always a good idea to evaluate the Optimization node to confirm that the optimization is interpreting your problem as intended. The [[DefineOptimization]]() function evaluates as a special symbol, or array of symbols, indicating the type of optimization problem presented. In this case, «LP» confirms that this is a Linear Program as expected. Furthermore, the result shows only a single «LP» symbol. This confirms that the solution is the result of a single optimization run.  
+
It is always a good idea to evaluate the Optimization node to confirm that the optimization is interpreting your problem as intended. The [[DefineOptimization]]() function evaluates as a special symbol, or array of symbols, indicating the type of optimization problem presented. In this case, '''«LP»''' confirms that this is a Linear Program as expected. Furthermore, the result shows only a single '''«LP»''' symbol. This confirms that the solution is the result of a single optimization run.  
  
 
'''Evaluate Status'''
 
'''Evaluate Status'''
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The [[OptStatusText]]() function confirms that the optimizer has found a feasible solution satisfying all constraints.
 
The [[OptStatusText]]() function confirms that the optimizer has found a feasible solution satisfying all constraints.
  
===Checking the Result===
+
==Checking the Result==
 
'''Optimized Solution as an Array'''
 
'''Optimized Solution as an Array'''
  
 
Evaluate the Optimized Solution array to check the final result.
 
Evaluate the Optimized Solution array to check the final result.
  
[[File:4-9.png|400px]]
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:[[File:4-9.png|400px]]
  
 
In this example, we used the optional second parameter of the [[OptSolution]]() function to reference the '''Shipment Quantities''' array. This formats the output to match the same dimensions as the Decision input array. It is a two-dimensional table indexed by '''Market''' and '''Brewery.'''
 
In this example, we used the optional second parameter of the [[OptSolution]]() function to reference the '''Shipment Quantities''' array. This formats the output to match the same dimensions as the Decision input array. It is a two-dimensional table indexed by '''Market''' and '''Brewery.'''
Line 157: Line 184:
  
 
If you omit the second parameter of [[OptSolution]](), Analytica creates a local index called <tt>.DecisionVector</tt> and presents the result as a one-dimensional array. You can experiment with this output format by removing the second parameter from the [[OptSolution]]() function.  
 
If you omit the second parameter of [[OptSolution]](), Analytica creates a local index called <tt>.DecisionVector</tt> and presents the result as a one-dimensional array. You can experiment with this output format by removing the second parameter from the [[OptSolution]]() function.  
<code>Variable Optimized_solution := OptSolution(Optimization)</code>
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:<code>Variable Optimized_solution := OptSolution(Optimization)</code>
[[File:4-10.png|400px]][[File:4-10.png|400px]]
+
 
 +
:[[File:4-10.png|400px]][[File:4-10.png|400px]]
 +
 
 +
==Summary: Basic Beer Distribution LP==
 +
 
 +
The basic beer distribution example demonstrates a simple Linear Program with a two-dimensional decision array. There are two one-dimensional Constraint arrays, each over a different index. The model has only two indexes overall: '''Brewery''' and '''Market'''. Both indexes are intrinsic to the optimization. The example does not include any extrinsic indexes.
 +
 
 +
==See Also==
 +
* [[DefineOptimization]]
 +
* [[OptObjective]]
 +
* [[OptSolution]]
 +
* [[Table]]
 +
* [[Sum]]
 +
* [[Index..Do]]
  
===Summary: Basic Beer Distribution LP===
 
  
The basic beer distribution example demonstrates a simple Linear Program with a twodimensional decision array. There are two one-dimensional Constraint arrays, each over a different index. The model has only two indexes overall: '''Brewery''' and '''Market'''. Both indexes are intrinsic to the optimization. The example does not include any extrinsic indexes.
+
<footer> Arrays in Optimization Models and Array Abstraction / {{PAGENAME}} / Example 2: Beer Distribution with Added Scenario</footer>

Latest revision as of 17:44, 24 May 2016



We start with an adaptation of a classic Linear Programming (LP) example.

Model Description

A large brewing company operates five breweries nationwide. Each brewery distributes product among four regions. Routes include all combinations of Breweries and Market regions. The challenge is to find the shipping pattern that minimizes distribution cost while

  1. meeting demand in each region, and
  2. observing production limits at each brewery.


We assume that distribution cost per case of beer is proportional to the distance between the brewery and the region.

Setting up the Model

To explore and follow this example in Analytica, run the Beer Distribution LP1.ana model in the Optimizer Examples folder.

Brewery and Market (indexes)

There are five Breweries and four Market regions:

Index Brewery := ['Fairfield', 'Fort Collins', 'Jacksonville', 'Merrimack', 'St Louis'] 
Index Market := ['North', 'South', 'East', 'West']

Distance (input array)

Distance between breweries and markets is represented as a table dimensioned by Brewery and Market. Units are thousands of miles:

Variable Distance := 
      Table(Brewery, Market)(1.8, 2.2, 3.4, 0.1, 0.3,1.2, 2.4, 1, 2.9,1.8, 0.6, 3.3, 1.2, 2.1, 0.2, 3.5, 0.5, 1.1, 0.8, 2.5)
4-3.png

Freight Price (input scalar value)

Freight Price is a simple scalar value of $5 per 1,000 miles per case of beer.

Variable Freight_price := 5

Production Limits (input array)

Production Limits indicate maximum capacity for each brewery in cases of beer:

Variable Production_limits := Table(Brewery)(600K, 250K, 450K, 300K, 240K)
4-4.png

Delivery Targets (input array)

Delivery Targets indicate the minimum quota that each Market region must receive:

Variable Delivery_target := Table(Market)(240K, 280K, 700K, 500K)
4-5.png

Distribution Cost (intermediate array)

Distribution Cost per Case is Freight Price multiplied by Distance:

Variable Dist_per_case := Freight_price*Distance

Shipment Quantities (input decision array)

The input decision array represents Shipment Quantities from each brewery to each region. It is a two-dimensional array indexed by Brewery and Market. The non-optimized values are not used anywhere else in the model so we can simply insert 1 as a dummy value across the array.(Initial guesses do not apply since this is a Linear Program.)

Units are cases of beer.

Staying Positive

We set the Lower Bound attribute to zero to disallow negative shipping values, along with the undesirable implication of turning perfectly good beer back into barley and hops.

Set Brewery and Market as intrinsic indexes

Finally and perhaps most importantly, we consider the index status for this array. The optimized solution should be in the same form as this table. It should also be integrated such that every array element has simultaneous influence over all other elements. Brewery and Market are both intrinsic indexes in this case.

Decision Shipment_Quantity := Array([Brewery, Market], 1)
Shipment_Quantity attribute Lower Bound := 0
Shipment_Quantity attribute Intrinsic Indexes := [Brewery, Market]
4-6.png

Total Distribution (Objective)

Our goal is to minimize the total distribution cost. For each route, distribution cost is the Distribution Cost per Case multiplied by the Shipment Quantity for each route. The final objective is the sum for all routes:

Objective Total_dist_cost := 
      Sum(Shipment_Quantity*Dist_per_case, Brewery, Market)

Summing over Brewery and Market makes the objective a scalar value. This is a necessary condition for the optimization.

Tip
Tip Every minimizing or maximizing optimization is based on a scalar objective value. In other words, intrinsic dimensions are not allowed in the Objective. If the Objective is not scalar, Analytica assumes the dimensions are extrinsic and performs independent optimizations for each scalar element in the Objective array. (For some optimization problems, the challenge is simply to find a solution that satisfies all constraints. This type of optimization has no objective at all.)

Supply Constraint

The Supply Constraint ensures that shipment quantities from a single brewery to all available markets do not exceed the production limit for the brewery.

The Production Limit array is already dimensioned by Brewery. Since this array is an input to Supply Constraint, we expect Brewery to be a dimension of Supply Constraint as well, even though we don’t explicitly mention the index in the defining expression. This is the basic principle of array abstraction in Analytica.

The Supply Constraint array actually defines five constraints; one for each Brewery.

Set Brewery as an intrinsic index

Is Brewery an intrinsic index for the Supply Constraint array? The answer is YES because we need to enforce the respective Supply Constraints on all breweries simultaneously. Make sure to include Brewery in the Intrinsic Indexes list for the Supply Constraint node.

Constraint Supply_constraint :=
     Sum(Shipment_Quantity, Market) <= Production_Limits

Supply_constraint attribute Intrinsic Indexes := [Brewery]

Is Market an intrinsic index for the Supply Constraint? The answer is NO because Market is not a dimension of the array at all. The Sum() function eliminates it, leaving the array with only one dimension.

Demand Constraint

The Demand Constraint ensures that each market region receives a total quantity from all breweries greater than or equal to the market demand. The input array Delivery Targets is dimensioned by Market, and therefore the Demand Constraint will also have this dimension.

The Demand Constraint array defines four constraints; one for each Market.

Set Market as an intrinsic index

Market is an intrinsic index for the Demand Constraint because the solution should meet quotas for all markets. Make sure to include Market in the Intrinsic Indexes list for the Demand Constraint node.

Constraint Demand_constraint :=
     Sum(Shipment_Quantity, Brewery) >= Delivery_Target
Demand_constraint attribute Intrinsic Indexes := [Market]

The Sum() function eliminates Brewery from the array.

Optimization Node

Remembering required attributes of the DefineOptimization() function as described in the Quick Start, we need to identify Decisions, Constraints, and the Objective to be minimized/maximized.

Variable Optimization := DefineOptimization
    (Decision: Shipment_Quantity,
    Constraints: Supply_Constraint, Demand_Constraint,
    Minimize: Total_Dist_Cost)

Solution Node

We obtain the solution using the OptSolution() function. The first parameter identifies the optimization node. The second parameter (optional) identifies a specific Decision for which the solution should be represented.

Decision Optimized_Solution :=
     OptSolution(Optimization, Shipment_Quantity)

Optimized Objective

The OptObjective() function calculates the minimized or maximized quantity. In this case, it represents the total distribution cost.

Variable Optimized_Objective := OptObjective(Optimization)

Status

Status text will confirm that the optimizer has found a solution

Variable Status := OptStatusText(Optimization)
4-7.png

Verifying the Optimization Setup

Evaluate the Optimization node

It is always a good idea to evaluate the Optimization node to confirm that the optimization is interpreting your problem as intended. The DefineOptimization() function evaluates as a special symbol, or array of symbols, indicating the type of optimization problem presented. In this case, «LP» confirms that this is a Linear Program as expected. Furthermore, the result shows only a single «LP» symbol. This confirms that the solution is the result of a single optimization run.

Evaluate Status

The OptStatusText() function confirms that the optimizer has found a feasible solution satisfying all constraints.

Checking the Result

Optimized Solution as an Array

Evaluate the Optimized Solution array to check the final result.

4-9.png

In this example, we used the optional second parameter of the OptSolution() function to reference the Shipment Quantities array. This formats the output to match the same dimensions as the Decision input array. It is a two-dimensional table indexed by Market and Brewery.

Optimized Solution as a List

If you omit the second parameter of OptSolution(), Analytica creates a local index called .DecisionVector and presents the result as a one-dimensional array. You can experiment with this output format by removing the second parameter from the OptSolution() function.

Variable Optimized_solution := OptSolution(Optimization)
4-10.png4-10.png

Summary: Basic Beer Distribution LP

The basic beer distribution example demonstrates a simple Linear Program with a two-dimensional decision array. There are two one-dimensional Constraint arrays, each over a different index. The model has only two indexes overall: Brewery and Market. Both indexes are intrinsic to the optimization. The example does not include any extrinsic indexes.

See Also


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