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 BLOG >> Recent

Revenue Model Sensitivity [Business Models
Posted on July 9, 2013 @ 07:36:00 AM by Paul Meagher

Yesterday I posted a completed revenue model for a season of lobster fishing. The major outputs of that model, total catch and total revenue, are actually very accurate with respect to this season's catch and revenue totals for lobster fishing. I'll talk a bit about the issue of model fit in this blog, but I mostly want to focus on what we might do with our revenue model now that we have 1) identified the major revenue factors, 2) specified the appropriate probability distributions for them, and 3) performed the appropriate math to generate revenue per unit time (e.g., per catch) and total revenue amounts. What we now need to do with our revenue model is try to understand its behavior better. When we execute the lobster fishing revenue model once it will generate one set of values for total catch and total revenue for a season. When we execute it again, it will generate a different set of values. This might leave you scratching your head as to what to make of this variability. Well, one thing we can do is rerun the model many times and use the computed season totals to generate some statistics that summarize what the mean season catch and season revenue is most likely to be. We can also compute how much variability there is in the season catch and season revenue totals by computing the standard deviation of these totals.

My latest script for revenue modelling is called lobster_fishing_sensitivity.php because its primary purpose is to do a sensitivity analysis of our lobster fishing revenue model. There is a section of the script below that contains various parameter settings. To do a sensitivity analysis all you do is start playing around with these parameter settings and see how the model output changes. This "playing around" can offer "model driven insight" into your business - how might my revenues be affected if this or that parameter value is changed (e.g., price of lobster, decay value, initial catch size, etc...). Playing around with your revenue model can help you to anticipate the future revenues of your business better.

The lobster_fishing_sensitivity.php script is well commented so I'll just present it as is and then show what the out put of the script looks like when I point my browser at it.

<?php

/**
* @script: lobster_revenue_sensitivity.php
* @author: Paul Meagher
* @modified: July 09/2013
* @about: Script to perform sensitivity analysis of a lobster fishing revenue model.
**/

// The probability distributions library sits 
// in the PDL folder at the same level as the 
// folder the script is run from.
//
// @see https://github.com/mrdealflow/PDL

require_once '../PDL/NormalDistribution.php';
require_once 
'../PDL/CategoricalDistribution.php';

// ArrayMath object I developed has some useful stats functions in it.
// @see http://www.phpmath.com/build02/IT1/docs/package.php?id=2#source

require_once '../UTIL/ArrayMath.php';

/*** PARAMETER SETTINGS FOR REVENUE MODEL - START ***/

// Set the number of catches per season.

$num_catches 40;

// Rate at which catch sizes decay (and corresponding standard deviation).

$k  = -0.03087;

// Initial catch size we will sample from (1000 lbs).

$N0 1000

// Initial catch size standard deviation we will sample from (250 lbs).

$S0 250;

// Specify categorical distribution for lobster prices (price and its probability).

$price_distribution = array('3.25'=>0.4'3.50'=>0.3'3.75'=>0.2'4.00'=>0.1); 

// Set the number of seasons you want the simulation to run for.

$num_seasons 100;

/*** PARAMETER SETTINGS FOR REVENUE MODEL - END ***/

// Model prices using a categorical distribution with price points and their probabilities.

$lobster_price_distribution  = new CategoricalDistribution($price_distribution);

for (
$s=0$s $num_seasons$s++) {
  
  
// Reset our array variables 
  
$lobster_catch   = array();
  
$lobster_price   = array();
  
$lobster_revenue = array();    

  for (
$c=0$c<$num_catches$c++) {
    
    
// Reduce the mean catch size after the first catch
    // according to exponential decay function.
    
$N $N0 exp($k $c);

    
// Reduce the standard deviation of catch size after the 
    // first catch according to exponential decay function.
    
$S $S0 exp($k $c);  
    
    
// Specify new catch size distribution for this trip.
    
$lobster_catch_distribution  = new NormalDistribution($N$S);
    
    
// Call random number generator RNG to sample values from 
    // catch size and catch price distribution.
    
$lobster_catch[$c]   = round($lobster_catch_distribution->RNG());
    
$lobster_price[$c]   = sprintf("%01.2f",$lobster_price_distribution->RNG());
    
$lobster_revenue[$c] = sprintf("%01.2f",$lobster_catch[$c] * $lobster_price[$c]);  

  }

  
// Store the season catch and revenue totals in arrays
  
$season_catch[$s]   = array_sum($lobster_catch);
  
$season_revenue[$s] = array_sum($lobster_revenue);
  
}

// ArrayMath object has some useful stats functions in its
// @see http://www.phpmath.com/build02/IT1/docs/package.php?id=2#source

$stats = new ArrayMath;

$season_catch_mean    sprintf("%01.2f"$stats->mean($season_catch));
$season_catch_stdev   sprintf("%01.2f"$stats->stdev($season_catch));

$season_revenue_mean  sprintf("%01.2f"$stats->mean($season_revenue));
$season_revenue_stdev sprintf("%01.2f"$stats->stdev($season_revenue));

echo 
"Season Catch Mean: $season_catch_mean lbs";
echo 
"<br>";
echo 
"Season Catch Standard Deviation: $season_catch_stdev lbs";
echo 
"<br>";
echo 
"Season Revenue Mean: $$season_revenue_mean";
echo 
"<br>";
echo 
"Season Revenue Standard Deviation: $$season_revenue_stdev";

?>

The output of the script looks like this:

Season Catch Mean: 23426.71 lbs
Season Catch Standard Deviation: 957.77 lbs
Season Revenue Mean: $81991.88
Season Revenue Standard Deviation: $3373.00

If we re-execute the lobster_revenue_sensitivity.php script many times, we will observe that the Season Revenue Mean value will change a bit on each execution. It will not change by that much, however, owing to some central limit stuff that is going on here. So what we are getting from this sensitivity script is a cleaner estimate of what predictions our revenue model is making. When we "fit" our revenue model to the data, it is these sensitivity stats that we should be evaluating our revenue model against and not the output of a single run of our revenue model. These sensitivity stats do in fact match up with the observed data for this season fairly well. This is a plausible revenue model for a season of lobster fishing and could be used to frame a revenue prediction (via parameter settings) for the next lobster fishing season .

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