The Southfork Feed Company makes a feed mix from four ingredients—oats, corn, so

Question

The Southfork Feed Company makes a feed mix from four ingredients—oats, corn, soybeans, and a vitamin supplement. The company has 300 pounds of oats, 400 pounds of corn, 200 pounds of soybeans, and 100 pounds of vitamin supplement available for the mix.

The company has the following requirements for the mix:

        At least 30% of the mix must be soybeans.

        At least 20% of the mix must be the vitamin supplement.

        The ratio of corn to oats cannot exceed 2 to 1.

        The amount of oats cannot exceed the amount of soybeans.

        The mix must be at least 500 pounds.

A pound of oats costs $0.50; a pound of corn, $1.20; a pound of soybeans, $0.60; and a pound of vitamin supplement, $2.00. The feed company wants to know the number of pounds of each ingredient to put in the mix in order to minimize cost.

a.         Formulate a linear programming model for this problem.

IF POSSIBLE PLEASE TRY AND USE THE Xij "I" "J" BECAUSE THATS WHAT WE WERE LEARNING IN CHAPTER 4 BUT WHEN I SOLVED IT I COULDNT FIGURE OUT HOW TO MAKE THE DESICION VARIABLES Xij. I COULD ONLY DO IT BY MAKING X1 oats and X2 Corn and X3 Soybean and X4 Vitamin suplements. I do not need the final answer to the question I just need the MODEL. THANK YOU SO MUCH!

Explanation / Answer

Formulation: Let the mix1 have x1 pounds of Oats, x2 pounds of Corn, x3 pounds of Soybeans and x4 pounds of Vitamins. Objective Function Minimize C = 0.5x1 + 1.2x2 + 0.6x3 + 2x4; Subject to Constraints x3 0.3(x1 + x2 + x3 + x4), that is, 0.3x1 + 0.3x2 - 0.7x3 + 0.3x4 0 [Soybeans Constraint] x4 0.2(x1 + x2 + x3 + x4), that is, 0.2x1 + 0.2x2 + 0.2x3 - 0.8x4 0 [Vitamins Constraint] x2/x1 2, that is x2 2x1 or x2 - 2x1 0 [Corn to Oats Ratio Constraint] x1 x3, that is, x1 - x3 0 [Oats and Soybean Constraint] x1 + x2 + x3 + x4 500 [Total amount Constraint] x1 300 [Oats availability Constraint] x2 400 [Corn availability Constraint] x3 200 [Soybeans availability Constraint] x4 100 [Vitamins availability Constraint] x1, x2, x3, x4 0 [Non-negativity Constraints]

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