Question

Linear Programming

A candy company makes three types of candy, solid-center, fruit-filled, and cream-filled, and packages these candies in three different assortments. A box of assortment I contains 4 solid-center, 4 fruit-filled, and 12 cream-filled candies, and sells for $17.95. A box of assortment II contains 12 solid-center, 4 fruit-filled, and 4 cream-filled candies, and sells for $18.45. A box of assortment III contains 8 solid-center, 8 fruit-filled, and 8 cream-filled candies, and sells for $20.85. The manufacturing costs per piece of candy are $0.01 for solid-center, $0.02 for fruit-filled, and $0.03 for cream-filled. The company can manufacture 4,800 solid-center, 4,000 fruit-filled, and 5,600 cream-filled candies weekly.

How many boxes of each type should the company produce each week in order to maximize their profits? What is the maximum profit?

*Will thumbs up for correct answer, thank you*

Answer 1

x1 = number of assortment I

x2 = number of assortment II

x3 = number of assortment III

=> profit = 17.95 x1 + 18.45 x2 + 20.85 x3 - manuf_cost

manuf_cost = (4x1+12x2+8x3)*0.02 + (4x1+4x2+8x3)*0.03 + (12x1+4x2+8x3)*0.05

= 0.8 x1 + 0.56 x2 + 0.8 x3

=> profit = 17.15 x1 + 17.89 x2 + 20.05 x3

inequalities (constraints) :

4 x1 + 12 x2 + 8 x3 <= 4800

4 x1 + 4 x2 + 8 x3 <= 4000

12 x1 + 4 x2 + 8 x3 <= 5600

4[x1]+12[x2]+8[x3] </=4800 or [x1]+3[x2]+2[x3] </=1200

4[x1]+4[x2]+8[x3] </=4000 or [x1]+[x2]+2[x3] </= 1000

12[x1]+4[x2]+8[x3] </=5600 or 3[x1]+[x2]+2[x3] </= 1400

[x1] >/=0,[x2] >/=0,[x3] >/=0

Simplex Method:

Step 1: rewrite the constraints by introducing slack variables

[x1]+3[x2]+2[x3]+[x4] </=1200

[x1]+[x2]+2[x3] +[x5] </= 1000

3[x1]+[x2]+2[x3] +[x6] </= 1400

[x1] >/=0,[x2] >/=0,[x3] >/=0, [x4] >/=0,[x5] >/=0,[x6] >/=0

The Simplex tableau's are :

0.00000 1.00000 2.00000 3.00000 0.00000

-1.00000 4.00000 12.00000 8.00000 4800.00000

-2.00000 4.00000 4.00000 8.00000 4000.00000

-3.00000 12.00000 4.00000 8.00000 5600.00000

0.00000 -18.40000 -21.14000 -22.20000 0.00000

Pivoting around element(3,1) yields :

0.00000 -3.00000 2.00000 3.00000 0.00000

-1.00000 -0.33333 10.66667 5.33333 2933.33333

-2.00000 -0.33333 2.66667 5.33333 2133.33333

1.00000 0.08333 0.33333 0.66667 466.66667

0.00000 1.53333 -15.00667 -9.93333 8586.66667

Pivoting around element(1,2) yields :

0.00000 -3.00000 -1.00000 3.00000 0.00000

2.00000 -0.03125 0.09375 0.50000 275.00000

-2.00000 -0.25000 -0.25000 4.00000 1400.00000

1.00000 0.09375 -0.03125 0.50000 375.00000

0.00000 1.06437 1.40688 -2.43000 12713.50000

Pivoting around element(2,3) yields :

0.00000 -3.00000 -1.00000 -2.00000 0.00000

2.00000 0.00000 0.12500 -0.12500 100.00000

3.00000 -0.06250 -0.06250 0.25000 350.00000

1.00000 0.12500 0.00000 -0.12500 200.00000

0.00000 0.91250 1.25500 0.60750 13564.00000

So the company should produce

200 boxes of type I

100 boxes of type II

350 boxes of type III

The profit is 13564 dollar