Not exactly sure if this belongs here, but I am doing an assignment which is based on Linear programming and well this is a Linear & Abstract Algebra forum so here goes:

A company produces chairs, tables and bookshelves. Monthly capacity for production is 3000 chairs, 1000 tables and 580 bookshelves.

The company employs 150 workers working in two 8 hour shifts 5 days a week. The average assembly time for each product is as follows:

chair - 20 minutes
table - 40 minutes
bookshelf - 15 minutes

The size of the workforce fluctuates each month, for month 1 20 workers want to take their annual leave, for month 2 25 workers want to take their annual leave and for month 3 45 workers want to take their annual leave.

The sales forecast for the three products for the next 3 months is as follows:

Month 1:

  • 2800 chairs
  • 500 tables
  • 320 bookshelves

Month 2:

  • 2300 chairs
  • 800 tables
  • 300 bookshelves

Month 3:

  • 3350 chairs
  • 1400 tables
  • 600 bookshelves


Should the company approve the annual leaves? if not, how many workers in each months could be granted annual leave while still providing sufficient production to satisfy expected demand?

MY SOLUTION:

I have to create a formulae and it's constraints based on the above question. so far what i have done is:

convert the hours to minutes and find out how much total hours is available for a single month between all the workers:
8 (hours) * 60 (minutes) * 5 (days per week) * 4 (weeks )* 150 (workers) =1 440 000 hours available per month.

I am allowed to have excess stock per month which can be transfered for sale in the consecutive month(s).

I constructed the following equation:

MIN = 2800*c+500*t+320*b;

c =20;
t = 40;
b = 15;

3000*c + 1000*t + 580*b <= 1440000;

my optimal solution (using LINGO) is 80800.00 which is how many minutes it took to create 200 chairs, 500 tables and 320 bookshelves.

Dividing 1 440 000 by 80800 we get 18 (rounded down). So that means 18 workers is all that is required to finish the order for the first month.

I know I messed up somewhere in the equation because that just doesn't seem as the correct answer, naturally if I can find the equation for 1 month I will be able to find the solution for the other 2 months aswell, so given my current "solution" is there any way to improve it?