A manufacturing firm produces four different products. One of the necessary raw materials is in short supply, and only R pounds are available. The selling price of product i is $S _i , per pound. Furthermore, each pound of product i uses a _i , pounds of the critical raw material. The variable cost, excluding the raw material cost, of producing x _i , pounds of product i is k _i ,xi ² , , where k _i , >O is known. Develop a mathematical model for the problem.