Real Product Mix Problem

[soundguy2]

Hello. This is my first post on this forum; hopefully, I've come to the right place and someone can help me.

I have a product mix problem that involves 8 products from 4 manufacturers (2 per manufacturer). The products must be sold in certain proportions and the total number of units sold must equal a total (t) units. I’ve labeled the quantity of each product using letters a to h. I’ve used k, m, n, and t as constants.

The problem has 8 variables, but only 5 equations. From the reading I’ve done, I understand this is called an ‘under determined system’, and therefore means there’s no single solution. I suspect the solution would be an equation, but I don’t know what the equation would look like, or how I would arrive at it.
If you have an idea on how to solve this type of problem and can explain it to me, or refer me to a resource that explains how to solve, I would very much appreciate it. Please see below for the equations that describe the desired product mix.

(a + b + c + d) / (e + f + g + h) = 1
(a + c + e + g) / (b + d + f + h) = k
(a + b) / (c + d) = m
(e + f) / (g + h) = n
(a + b + c + d + e + f + g + h) = t

Also, if there’s a program that can solve this type of problem, I would love to know what it’s called. Thank you for any help you can provide.

 

[Ishuda]

Hello. This is my first post on this forum; hopefully, I've come to the right place and someone can help me.

I have a product mix problem that involves 8 products from 4 manufacturers (2 per manufacturer). The products must be sold in certain proportions and the total number of units sold must equal a total (t) units. I’ve labeled the quantity of each product using letters a to h. I’ve used k, m, n, and t as constants.

The problem has 8 variables, but only 5 equations. From the reading I’ve done, I understand this is called an ‘under determined system’, and therefore means there’s no single solution. I suspect the solution would be an equation, but I don’t know what the equation would look like, or how I would arrive at it.
If you have an idea on how to solve this type of problem and can explain it to me, or refer me to a resource that explains how to solve, I would very much appreciate it. Please see below for the equations that describe the desired product mix.

(a + b + c + d) / (e + f + g + h) = 1
(a + c + e + g) / (b + d + f + h) = k
(a + b) / (c + d) = m
(e + f) / (g + h) = n
(a + b + c + d + e + f + g + h) = t

Also, if there’s a program that can solve this type of problem, I would love to know what it’s called. Thank you for any help you can provide.

First, to get to a standard form, clear the denominators to get a different 5 equations.

[FONT=courier new]a +   b +   c +   d - e -   f -   g -   h = 0
a - k b +   c - k d + e - k f +   g - k h = 0
a +   b - m c - m d                       = 0
                      e +   f - n g - n h = 0 
a +   b +   c +   d + e +   f +   g +   h = t[/FONT]

To solve these type of equations, you can treat three of your variables as knowns and solve the usual way. See
https://en.wikipedia.org/wiki/System_of_linear_equations
for example. A simple Gauss-Jordon elimination should work.

For example, suppose you had

[FONT=courier new](1) a + b +   c  = 0
(2) a - b - m c  = t
[/FONT]

where m and t are known. We could move c over to the other side and treat it as known to get

[FONT=courier new](1) a + b = -  c
(2) a - b =  m c + t
[/FONT]

This is now a two equations in two unknowns system. Solving we get
a = [(m-1) c + t ] / 2
b = -[(m+1) c + t ] / 2
So you can choose c which will then determine a and b.

EDIT: There are ways to solve these equations in general involving least-squares solutions, Lagrange multipliers, and pseudo matrix inverses. See, for example,
http://people.csail.mit.edu/bkph/articles/Pseudo_Inverse.pdf