In addition, consumer appreciation and marketability requirement suggest that
any food produced must contain at least 30% meat by weight. Formulate a linear
program whose solution provides a recipe for a batch of dog food and a batch of cat
food that maximizes the total weight of overstock used in the process. Solution. We must determine how much of each ingredient is used in each type of
food. Let
xi,- = the grams of ingredient 1' used in food 3', where i = 1, 2, 3, 4, 5 corresponds to chicken, beef, wheat, rice and water, respectively;
and j = 1,2 corresponds to cat food and dog food, respectively. We can consider
1353' 6 1R. Because there are many constraints on various relative quantities, we will
also use the following dependent variables. W;- = the total weight of food 3'. Wm = the total protein weight in food 3'. ij = the total fat weight in food j.
C, = the total calories in food 3'. All together there are 18 decision variables.
The objective is to maximize the use of overstock ingredients (by weight). That
is, we seek to maximize the objective function z = $11 + .1321 + $31 + $41 + $12 + $22 + $32 + 3342. We begin enumerating the problem constraints by deﬁning our dependent variables
in terms of our initial decision variables. W1 = $11 + $21 +3331 + $41 + 1'51- W2 = $12 + $22 + 3332 + $42 + 1‘52-
WP] = 0.27:1:11 + 0.26321 + 0.14:1:31 + 0.0273641.
Wfl = 0.1451311 + 0.159321 + 0.0243231 + 0.0025x41.
ng = 0.273312 -|— 0.263922 + 0.143332 + 0.0275642.