# How To Solve Linear Programming Word Problems A bag of food A costs and contains 40 units of proteins, 20 units of minerals and 10 units of vitamins.

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How many units of each type of toys should be stocked in order to maximize his monthly total profit profit?

Let x be the total number of toys A and y the number of toys B; x and y cannot be negative, hence x ≥ 0 and y ≥ 0 The store owner estimates that no more than 2000 toys will be sold every month x y ≤ 2000 One unit of toys A yields a profit of $2 while a unit of toys B yields a profit of$3, hence the total profit P is given by P = 2 x 3 y The store owner pays $8 and$14 for each one unit of toy A and B respectively and he does not plan to invest more than \$20,000 in inventory of these toys 8 x 14 y ≤ 20,000 What do we have to solve?

Cost C(x,y) = 10 x 12 y $\begin \ x \ge 0 \\ \ y \ge 0 \\ \ 40x 30y \ge 150 \\ \ 20x 20y \ge 90 \\ \ 10x 30y \ge 60 \\ \end$ .

Vertices: A at intersection of $$10x 30y = 60$$ and $$y = 0$$ (x-axis) coordinates of A: (6 , 0) B at intersection of $$20x 20y = 90$$ and $$10x 30y = 60$$ coordinates of B: (15/4 , 3/4) C at intersection of $$40x 30y = 150$$ and $$20x 20y = 90$$ coordinates of C : (3/2 , 3) D at at intersection of $$40x 30y = 150$$ and $$x = 0$$ (y-axis) coordinates of D: (0 , 5) Evaluate the cost c(x,y) = 10 x 12 y at each one of the vertices A(x,y), B(x,y), C(x,y) and D(x,y).

How many bags of food A and B should the consumed by the animals each day in order to meet the minimum daily requirements of 150 units of proteins, 90 units of minerals and 60 units of vitamins at a minimum cost?

Let x be the number of bags of food A and y the number of bags of food B.

Hence the store owner has to have 1333 toys of type A and 667 toys of type B in order to maximize his profit. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish.

It takes 4 hours to produce the parts of one unit of T2, 2.5 hour to assemble and 1.5 hours to polish.

Vertices: A at intersection of $$x y = 20000$$ and $$y = 0$$ , coordinates of A: (20000 , 0) B at intersection of $$x y = 17000$$ and $$y=0$$ , coordinates of B: (17000 , 0) C at intersection of $$x y = 17000$$ and $$x = 2y$$ , coordinates of C : (11333 , 5667) D at at intersection of $$x = 2y$$ and $$x y = 20000$$ , coordinates of D: (13333 , 6667) Evaluate the return R(x,y) = 1000 - 0.03 x - 0.01 y at each one of the vertices A(x,y), B(x,y), C(x,y) and D(x,y).

At A(20000 , 0) : R(20000 , 0) = 1000 - 0.03 (20000) - 0.01 (0) = 400 At B(17000 , 0) : R(17000 , 0) = 1000 - 0.03 (17000) - 0.01 (0) = 490 At C(11333 , 5667) : R(11333 , 5667) = 1000 - 0.03 (11333) - 0.01 (5667) = 603 At D(13333 , 6667) : R(13333 , 6667) = 1000 - 0.03 (13333) - 0.01 (6667) = 533 The return R is maximum at the vertex At C(11333 , 5667) where x = 11333 and y = 5667 and z = 20,000 - (x y) = 3000 For maximum return, John has to invest 333 in fund F1, 67 in fund F2 and 00 in fund F3.

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