Question

Consider the production scheduling problem of the perfume Polly named after a famous celebrity. The manufacturer of the perfume must plan production for the first four months of the year and anticipates a demand of 4000, 5000, 6000, and 4500 gallons in January, February, March, and April, respectively. At the beginning of the year the company has an inventory of 2000 gallons. The company is planning on issuing a new and improved perfume called Pollygone in May, so that all Polly produced must be sold by the end of April. Assume that the production cost for January and February is $5 per gallon and this will rise to $5.5 per gallon in March and April. The company can hold any amount produced in a certain month over to the next month at an inventory cost of $1 per unit. Formulate a linear optimization model that will minimize the costs incurred in meeting the demand for Polly in the period January through April. Assume for simplicity that any amount produced in a given month may be used to fulfill demand for that month.

Answer 1

Since inventory cost is $1/gallon and is greater than the cost increase between February and March, it does not pay to over-produce in any month.
Consequently there are no other options than to simply produce the monthly demand.

So to summarize,

month production cost
January 2000 10,000 [ reduced by the 2000 current inventory]
February 5000 25,000
March 6000 33,000
April 4500 24,750

Total 17,500 $92,750