There are four grains G1 to G4 that contain nutrients, starch, fiber, protein, a
ID: 164975 • Letter: T
Question
There are four grains G1 to G4 that contain nutrients, starch, fiber, protein, and gluten. The composition of these grains, and their prices are given below. It is required to develop a minimum cost mixture of these grains for making a new type of multigrain flour subject to the following constraints: For taste considerations, the percent of grain 2 in the mix cannot exceed 20, and the percent of grain 3 in the mix has to be at least 30, and the percent of grain 1 in the mix has to be between 10 to 25. The percent protein content in the flour must be at least 18, the percent gluten content has to be between 8 to 13, and the percent fiber content at most 50.Explanation / Answer
We will now formulate the problem of finding the composition of the least costly blend of the grains to make the flour, using the proportions of the various grains in the blend as decision variables. Let:
pi = proportion of grain i in the blend, i = 1 to 4.
p = (p1, p2, p3, p4)T is the vector of decision variables in this problem. The linear blending assumptions are quite reasonable in this problem.
From them, we derive the percent protein content in the blend corresponding to p to be 20p1 +15p2+5p3+30p4, hence the constraint on protein content in the flour is 20p1 + 15p2 + 5p3 + 30p4 18.
Arguing the same way, we get the following LP model for this problem.
Minimize 70p1 + 40p2 + 60p3 + 80p4
Subject to 0 p2 0.2
p3 0.3
0.10 p1 0.25
20p1 + 15p2 + 5p3 + 30p4 18
8 10p1 + 20p3 + 5p4 13
40p1 + 65p2 + 35p3 + 40p4 50
p1 + p2 + p3 + p4 = 1
p4 0.
The last equality constraint expresses the condition that the sum of the proportions of the various constituents in a mixture must be equal to 1.
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