FarmFresh Foods manufactures a snack mix called TrailTime by

FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below. The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The Wend should contain no more than 1100 calories and at most 30 grams of fat. Dried fruit must be a least 30% of the volume of the mixture, and nuts must be no more than 25% of the weight of the mixture. Let X = amount (ounces) of dried fruit to put into TriafTïme. Let y = amount (ounces) of nut mix to put into TrialTime. Let z = amount (ounces) of cereal mix to put into TrialTime. Develop a model that meets these restrictions to minimizes the cost of the blend. Note: Today Canvas has some trouble showing LE and GE in the answer area. Therefore, in this problem set, we use \"greater than or equal to\" to represent \"GE\" and use \"less than or equal to\" to represent \"LE\" in the answer area. What is the objective function? max0.4.x + 0.5y + 0.2 z max 0.25x + 0.5y + z min 0.4x + 0.5y + 0.2z min 0.25x + 0.5y + z None of the above What is the constraint for calories? 150.t + 400y + 50z GE 1100 150x + 400y + 50z LE 1100 x + y + z GE 1100 x + y + z LE 1100 none of the above.

Solution

the cost of dried fruit per ounce is = $ .4
the cost of nut mix per ounce is = $ .5
the cost of cereal mix per ounce is = $ .2

we are asked to maximize the cost
let the cost function be defined as Z

=> Z(Maximize) = .4x + .5y + .2z        ----------this is the profit function

the constraints are
the trial time blend contains no more than 1100 calories

1 ounce of dried fruit has = 150 calories
1 ounce of nut mix has = 400 calories
1 ounce of cereal mix has = 50 calories

=> 150x + 400y + 50z <= 1100         ---------constraint for calories

the trial time blend needs to have atmost 30gm of fat
nut mix has 15 gm of fat / ounce
cereal mix has .2 gm of fat per ounce

=> 15y + 3z <= 30                ----------constraint for fat

total voplume of the mixture
1 ounce of dried fruit is = .3 cup by volume
1 ounce of nut mix is = .5 cup by volume
1 ounce of cereal mix is = .2 cup by volume

=> total volume of the mixture would be = (.3x + .5y + .2z)
dried fruits must be atleast 30% of the total volume of the mixture

=> .3x >= 30/100*(.3x + .5y + .2z)      ---------constraint for volume

nuts must be no more than 25% of the weight of the mixture
=> there are x , y and z ounces of the three
so the total weight of the mixture would be = (x+y+z)

=> y <= 25/100*(x+y+z) -----------constraint for the weight