More spoilage

The question of minimizing spoilage still interests me. Today I want to consider the consequences of relaxing one of the constraints on the problem. In particular I do not assume that the two different types of fruit cost the same per unit. And I want to minimize the total amount of wasted money (i.e. the amount of spoiled fruit times the unit cost.)

Definitions:

1. There are two types of fruit A and B. The unit costs are c_A and c_B respectively.
2. We can consume z units of fruit per day.
3. The goal is to minimize the cost of the total spoiled fruit.
4. The percent of fruit that spoils at the end of each day is m_A d and m_B d respectively for A and B, where m_A, m_B are constants and d is the number of days since buying the fruit (i.e. d = 0 for the first day.)

Calculations:
First, note that by minimizing the cost of spoiled fruit at the end of each day the total cost of spoiled fruit is minimized (I should prove this and I might in a future post. I think this comes from the fact that we’re solving a convex optimization problem?).

Given x and y units of fruit for A and B respectively.
We can consume z_A and z_B units of A and B respectively, where z_A + z_B = z.

The amount of spoiled fruit at the end of the day is
m_Ad(x-z_A) + m_Bd(y-z_B), the cost of the spoiled fruit is
m_Ad(x-z_A)c_A+m_Bd(y-z_B)c_B.

Using z_B = z - z_A, the cost of the spoiled fruit is m_Ad(x-z_A)c_A+m_Bd(y-z+z_A)c_B. Expanding we get

m_Ac_Adx-m_Ac_Adz_A+m_Bc_Bdy-m_Bc_Bdz+m_Bc_Bdz_A
which reduces to = z_A(m_Bc_B - m_Ac_A)d + m_Ac_Adx + m_Bc_Bdy - m_Bc_Bdz.

Taking the derivative with respect to z_A we obtain (m_Bc_B - m_Ac_A)d.
If m_Bc_B - m_Ac_A is greater or equal to zero, then z_A = 0, z_B = z minimizes the cost of spoiled fruit; otherwise z_A = z, z_B = 0 minimizes the cost of spoiled fruit.

From the above it’s clear that introducing the factor of fruit cost does not fundamentally alter the solution. But it does introduce some interesting factors, that warrant further consideration.

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