start impl. more business metrics

Business/Marketing/Metrics.php

@@ -14,6 +14,10 @@ declare(strict_types=1);
 
 namespace phpOMS\Business\Marketing;
 
+use phpOMS\Math\Matrix\IdentityMatrix;
+use phpOMS\Math\Matrix\Matrix;
+use phpOMS\Math\Matrix\Vector;
+
 /**
  * Marketing Metrics
  *
@@ -53,4 +57,187 @@ final class Metrics
     {
         return ($ce - $cn) / $cs;
     }
+
+    /**
+     * Calculate the customer profits
+     *
+     * @param int   $customers      Amount of customers acquired
+     * @param float $acquistionCost Acquisition cost per customer
+     * @param float $revenue        Revenues per period per customer
+     * @param float $cogs           COGS per period per customer
+     * @param float $marketingCosts Ongoing marketing costs per period per customer
+     * @param float $discountRate   Discount rate
+     * @param float $retentionRate  Retention rate (how many customers remain)
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function getBerrysCustomerProfits(
+        int $customers,
+        float $acquistionCost,
+        float $revenue,
+        float $cogs,
+        float $marketingCosts,
+        float $discountRate,
+        float $retentionRate
+    ) : float {
+        return $customers * ($revenue - $cogs) * ((1 + $discountRate) / (1 + $discountRate - $retentionRate))
+            - $customers * $marketingCosts * ((1 + $discountRate) / (1 + $discountRate - $retentionRate))
+            - $customers * $acquistionCost;
+    }
+
+    /**
+     * Life time value of a customer
+     *
+     * @param \Closure $customerProfit Profit of a customer in year t
+     * @param float    $discountRate   Discount rate
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function lifeTimeValue(\Closure $customerProfit, float $discountRate) : float
+    {
+        $ltv = 0.0;
+        for ($i = 1; $i < 1000000; ++$i) {
+            $ltv += $customerProfit($i) / \pow(1 + $discountRate, $i - 1);
+        }
+
+        return $ltv;
+    }
+
+    /**
+     * Life time value of a customer
+     *
+     * @param \Closure $customerProfit Profit of a customer in year t
+     * @param float    $discountRate   Discount rate
+     *
+     * @return float
+     *
+     * @since 1.0.0
+     */
+    public static function simpleRetentionLifeTimeValue(\Closure $customerProfit, float $discountRate, float $retentionRate) : float
+    {
+        $ltv = 0.0;
+        for ($i = 1; $i < 1000000; ++$i) {
+            $ltv += $customerProfit($i) * \pow($retentionRate, $i - 1) / \pow(1 + $discountRate, $i - 1);
+        }
+
+        return $ltv;
+    }
+
+    /**
+     * Calculate the profitability of customers based on their purchase behaviour
+     *
+     * The basis for the calculation is the migration model using a markov chain
+     *
+     * @param float $discountRate        Discount rate
+     * @param array $purchaseProbability Purchase probabilities for different periods
+     * @param array $payoffs             Payoff vector (first element = payoff - cost, other elements = -cost, last element = 0)
+     *
+     * @return Matrix A vector which shows in row i the return of the customer if he didn't buy i - 1 times before (=recency of the customer = how many periods has it been since he bought the last time)
+     *
+     * @since 1.0.0
+     */
+    public static function calclateMailingSuccess(float $discountRate, array $purchaseProbability, array $payoffs) : Matrix
+    {
+        $count = \count($purchaseProbability);
+        $profit = new Matrix($count, $count);
+        $G      = Vector::fromArray($payoffs);
+
+        $P = [
+            $G,
+            self::createCustomerPurchaseProbabilityMatrix($purchaseProbability),
+        ];
+
+        for ($i = 0; $i < $count; ++$i) {
+            if (!isset($P[$i])) {
+                $P[$i] = $P[$i - 1]->mult($P[$i - 1]);
+            }
+
+            $profit->add($P[$i]->mult($G)->mult(1 / \pow(1 + $discountRate, $i)));
+        }
+
+        return $profit;
+    }
+
+    /**
+     * Calculate V of the migration model
+     *
+     * @param float $discountRate        Discount rate
+     * @param array $purchaseProbability Purchase probabilities for different periods
+     * @param array $payoffs             Payoff vector (first element = payoff - cost, other elements = -cost, last element = 0)
+     *
+     * @return Matrix [0][0] returns the LTV
+     *
+     * @since 1.0.0
+     */
+    public static function migrationModel(float $discountRate, array $purchaseProbability, array $payoffs) : Matrix
+    {
+        $P = self::createCustomerPurchaseProbabilityMatrix($purchaseProbability);
+        $I = new IdentityMatrix(\count($purchaseProbability));
+
+        return $I->sub(
+                $P->mult(1 / (1 + $discountRate))
+            )->inverse()
+            ->mult(Vector::fromArray($payoffs));
+    }
+
+    /**
+     * Calculate the purchase probability of the different purchase states.
+     *
+     * A customer can either buy in a certain period or not.
+     * Depending on the result he either moves on to the next state (not buying) or returns to the first state (buying).
+     *
+     * @param int   $period              Period to evaluate
+     * @param array $purchaseProbability Purchase probabilities
+     *
+     * @return Matrix [
+     *      [0][0] = probability of buying in period t if customer bought in t = 1
+     *      ...
+     * ]
+     */
+    public static function migrationModelPurchaseProbability(int $period, array $purchaseProbability) : Matrix
+    {
+        $matrix = self::createCustomerPurchaseProbabilityMatrix($purchaseProbability);
+        for ($i = 0; $i < $period; ++$i) {
+            $matrix = $matrix->mult($matrix);
+        }
+
+        return $matrix;
+    }
+
+    /**
+     * Create a matrix which contains the probabilities a customer will buy in period t
+     *
+     * @param array $purchaseProbability Purchase probabilities for the different periods
+     *
+     * @latex \begin{bmatrix}
+     *      p_1 & 1 - p_1 & 0 \\
+     *      p_2 & 0 & 1 - p_2 \\
+     *      p_3 & 0 & 1 - p_3 \\
+     * \end{bmatrix}
+     *
+     * @return Matrix [
+     *      p1, 1-p1, 0,
+     *      p2, 0,    1-p2,
+     *      p3, 0,    1-p3,
+     * ] where pi = Probability that customer buys in period i
+     *
+     * @since 1.0.0
+     */
+    private static function createCustomerPurchaseProbabilityMatrix(array $purchaseProbability) : Matrix
+    {
+        $matrix = [];
+
+        $count = \count($purchaseProbability);
+        for ($i = 0; $i < $count; ++$i) {
+            $matrix[$i] = [];
+            $matrix[$i][0] = $purchaseProbability[$i];
+            $matrix[$i][$i + 1] = 1 - $purchaseProbability[$i];
+        }
+
+        return Matrix::fromArray($matrix);
+    }
 }