Every January Santa tidies his workshop. Each toy in the workshop may be either stored for use next year, or taken apart and rebuilt next year, or thrown away and replaced next year. Storing a toy costs £1 and uses 8 units of storage space. Taking a toy apart and rebuilding it next year costs £3 and uses 2 units of storage space. Throwing a toy away and replacing it next year costs £5 and uses 0 units of storage space. Santa has 500 units of storage space available and wants to minimize the cost.
i) Identify the relevant variable for this problem. Write an objective function and constraints for the problem in terms of these variables. What other condition must these variables satisfy?
ii) Given that Santa has a total of 100 toys in his workshop, show how the problem can be written as a linear program involving two variables. Hence determine the solution to Santa's original problem and give the minimum cost.