A vegetable oil processor wishes to determine how much shortening, salad oil, and

margarine to produce to optimize the use its current oil stock supply. At the present

time, he has $248000kg$ of soybean oil, $114000kg$ of cottonseed oil, and $2000kg$ of milk$-$

base substances. The milk$-$base substances are required only in the production of

margarine. There are certain processing losses associated with each product: $10\%$ for

shortening, $5\%$ for salad oil, and no loss for margarine. The producer's back orders

require him to produce at least $85000kg$ of shortening, $40000kg$ of salad oil, and $1100$

$kg$ of margarine. In addition, sales forecasts indicate a strong demand for all products in

the near future. The profit per kilogram and the base stock required per kilogram of each

product are given in Table$\mathrm{.}1.$ Formulate the problem to maximize profit over the next

production$-$scheduling period.

Table. $1$ Data for the vegetable oil processing problem